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[ "All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses", "All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses", "All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses", "All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses" ]
[ "Jiaojian Shi \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Weiwei Sun \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Hendrik Utzat \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n\nDepartment of Materials Science and Engineering\nStanford Univer-sity\n94305 StanfordCaliforniaUnited States\n", "Ardavan Farahvash \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Frank Y Gao \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Zhuquan Zhang \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Ulugbek Barotov \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Adam P Willard \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Keith A Nelson \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Moungi G Bawendi \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Jiaojian Shi \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Weiwei Sun \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Hendrik Utzat \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n\nDepartment of Materials Science and Engineering\nStanford Univer-sity\n94305 StanfordCaliforniaUnited States\n", "Ardavan Farahvash \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Frank Y Gao \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Zhuquan Zhang \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Ulugbek Barotov \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Adam P Willard \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Keith A Nelson \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n", "Moungi G Bawendi \nDepartment of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States\n" ]
[ "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Materials Science and Engineering\nStanford Univer-sity\n94305 StanfordCaliforniaUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Materials Science and Engineering\nStanford Univer-sity\n94305 StanfordCaliforniaUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States", "Department of Chemistry\nMassachusetts Institute of Technology\n02139Cambridge, MassachusettsUnited States" ]
[]
1 arXiv:2105.01190v1 [cond-mat.mes-hall] 3 May 2021 Photoluminescence (PL) intermittency is a ubiquitous phenomenon detrimentally reducing the temporal emission intensity stability of single colloidal quantum dots (CQDs) and the emission quantum yield of their ensembles. Despite efforts for blinking reduction via chemical engineering of the QD architecture and its environment, blinking still poses barriers to the application of QDs, particularly in single-particle tracking in biology or in single-photon sources. Here, we demonstrate the first deterministic all-optical suppression of quantum dot blinking using a compound technique of visible and mid-infrared (MIR) excitation. We show that moderate-field ultrafast MIR pulses (5.5 µm, 150 fs) can switch the emission from a charged, low quantum yield 'grey' trion state to the 'bright' exciton state in CdSe/CdS core-shell quantum dots resulting in a significant reduction of the QD intensity flicker. Quantum-tunneling simulations suggest that the MIR fields remove the excess charge from trions with reduced emission quantum yield to restore higher brightness exciton emission. Our approach can be integrated with existing single-particle tracking or super-resolution microscopy techniques without any modification to the sample and translates to other emitters presenting charging-induced PL intermittencies, such as single-photon emissive defects in diamond and two-dimensional materials.2 Colloidal quantum dots (CQDs) have now made significant commercial inroads in diverse optoelectronic applications as well as biological imaging due to their unique electronic structures, size-tunable emission, high photoluminescence (PL) quantum yield, high photo-stability, and facile chemical synthesis. 1 However, despite two decades of research, stochastic PL intermittency, also known as 'blinking', still reduces the temporal emission stability, particularly under high excitation-flux conditions in single-emitter experiment, posing a significant barrier to the wider adoption of QDs in single-emitter applications. In single-photon sources, blinking reduces the ability of QDs to produce single-photons on demand and often coincides with detrimental spectral jumping 2-6 , two processes that need to be eliminated to qualify QDs as building blocks of singlephoton sources for quantum cryptography and computing.7,8In real-time single-particle tracking in biological systems 9, 10 , the timescale of blinking coincides with the particle diffusion reducing the tracking ability.Despite the lack of a unified blinking theory 3 , it is now understood that blinking in QDs occurs primarily via two pathways i) surface trap-mediated non-radiative recombination and ii) charging-induced Auger recombination. 11 Synthetic efforts have been made towards intrinsically non-blinking quantum dots targeting both of these mechanism primarily through passivating surface trap-states and reducing the Auger recombination rate of charged states through smoothing of the potential barrier between coreand shell-layers of heteroepitaxial QDs. 12-17 Extrinsic blinking control has been com-3 paratively less explored. Adding anti-blinking agents suppresses blinking in QDs by passivating the QD surface.[18][19][20]Other approaches such as electrostatic gating have also been reported to tune the Fermi level of the QD and block the relaxation pathways through the surface, thereby suppressing non-radiative decay during OFF periods.11,21These methods, however, require the QD to be either deposited on a special substrate or immersed in a non-native environment, adding significant complexity to single-QD devices and precluding blinking control in biological environments. To date, no noninvasive active suppression of blinking has yet been demonstrated.Here, we demonstrate the first deterministic all-optical method for active reduction of QD blinking with ultrafast MIR pulses. We build on advances in ultrafast electricfield pulse technologies 22, 23 and show that off-resonant laser pulses at mid-infrared (MIR) frequencies provide sufficient strengths to overcome potential barriers in QDs while avoiding dielectric breakdown that often occurs upon applications of static electric fields. By applying MIR pulses concurrently with optical excitation of single QDs, we demonstrate that MIR pulses with an appropriate field strength can remove the excess electron of charging-induced blinking OFF states in single QDs. Therefore, MIR pulses have the potential to transiently discharge QDs in the blinking OFF states at an ultrafast speed without perturbing the equilibrium emissivity or introducing inter-band excitations.4The experimental setup is shown inFig.1a. Our experiments investigate if the MIR pulse can switch OFF states to ON states by discharging the trion state (as shown inFig. 1b) through the measurement of blinking statistics, emission lifetimes, and measuring emission spectra and emission intensities. A 1-kHz MIR pulse train at 5.5 µm is used in our experiments(Fig. 1c), which is far below the multiphoton absorption regime and away from phonon absorption frequencies of CdSe/CdS QDs. 24 We tuned the MIR exposure by modulating a mechanical shutter and quantify the exposure by the number of pulses in a burst of MIR. Under 405 nm continuous-wave (CW) laser excitation, emission from single core-shell CdSe/CdS QDs with an 8-monolayer (ML) shell thickness, deposited on a glass coverslip, was confirmed by the antibunching dip (τ = 0) from a second-order photon intensity correlation measurement, as shown in Fig. 1d. We show the effect of concurrent MIR and optical excitation in Fig. 2. We record the PL intensity traces of single QDs under 405 nm CW laser excitation. Under no MIR field, we show a representative single QD PL blinking trace (Fig. 2a) and its histogram of PL intensity distributions (Fig. 2b). The histogram in Fig. 2b shows a bimodal PL intensity distribution, corresponding to blinking ON and OFF states. With MIR excitation at a suitable field strength (here F/F max = 0.6, F max ∼ 10 MV/cm), the blinking behavior from the same QD changes as shown in Fig.2c-d, reflected by the significant decrease in the time that the dot spends in the OFF state in the blinking trace 5 as well as by the shift in the intensity histogram from bimodal to a largely unimodal ON state.The effective conversion from OFF to ON states is also reflected by an enhanced PL intensity in ensemble QDs. By imaging and selecting isolated single QDs within the field of view, we can study the dynamics and responses of statistically-averaged PL intensity (counts summed over all the dots) when MIR excitations are turned on and off.Here, MIR excitations are periodically exposed on the sample for 200 ms (a burst of 200 MIR pulses) every 5 seconds, resulting in PL intensity spikes every 5 s, as shown in Fig. 2e. No degradation in either equilibrium PL counts without MIR or enhanced PL counts with MIR is observed, suggesting the reversible nature of the PL enhancement.The enhancement of PL intensity only appears when a suitable MIR field strength is applied. With stronger MIR pulses (F/F max > 0.7) , we observe a suppression of the overall PL intensity. InFig. 2f, we show a drop in the PL intensity under an MIR exposure of 10 ms with a burst of 10 MIR pulses every two seconds at F/F max = 0.9, as shown inFig. 2f.Figure 2gillustrates the normalized PL intensity change as a function of CWlaser optical power. The MIR-induced PL intensity change is summed over multiple (∼ 100) isolated dots and normalized to their equilibrium PL intensity. The PL percentage change increases as the optical power increases and saturates at ∼ 30 W/cm 2 . The 6 extrapolated zero PL change at zero optical power indicates that the MIR itself does not produce any luminescence and only enhances PL when the dots are excited optically.The positive correlation between the PL percentage change and optical power suggests that the PL enhancement is most likely due to removal of accumulative events such as excess photo-ionized charges induced blinking.11,25 The threshold behavior in the MIR field dependence of the PL percentage change(Fig. 2h) further suggests that the MIR field is driving ionization processes and, as a result, removes excess charges inside the QDs. A crossover behavior is observed at field strengths around F/F max = 0.7 with an optical power of 20 W/cm 2 and a burst of 80 MIR pulses every 1 s, which is consistent with the observation of PL quenching under excessively high MIR fields shown inFig.2f.Figure 2idisplays the MIR exposure dependence of the PL percentage change with a fixed optical power at 20 W/cm 2 and a field strength at F/F max = 0.6. The PL percentage change increases with increasing MIR pulse exposure. The PL percentage change saturates at a burst of around 100 MIR pulses every 1 s at the field strength of F/F max = 0.6. Further increasing MIR exposure results in PL intensity degradation, which can be explained by excess charge generation with excessive pulses.PL lifetime changes further validate that MIR fields remove excess charges inside the QDs. When the MIR is off, a single QD randomly switches between ON and OFF states. The strong correlation between the PL intensity and lifetime indicates that blinking is caused by charging and discharging as expected for core-shell CdSe/CdS QDs 11, 26 7 as is seen inFig. 3athrough a fluorescence lifetime-intensity distribution (FLID) plot.With MIR excitation, the blinking OFF states disappear, and only the ON states are present with a long lifetime and high PL intensity, as shown inFig. 3b. Another representation is provided in the Supplementary Information by interrogating all the photon arrival times relative to the excitation trigger. As shown inFig. S8a, without MIR fields, the PL lifetime of the single QD is composed of fast and slow exponential decays components, which we assign to trion and exciton decay, respectively. With MIR excitations, the fast trion decay is completely suppressed, and only the slow exciton relaxation is left, implying that MIR fields efficiently remove the excess charge. Figure 3c and d show that blinking statistics are strongly modified by MIR excitation. By setting a threshold in a blinking trace separating ON and OFF states and counting their probability densities, a power-law distribution of ON-and OFF-times is obtained inFig. 3dand c, respectively. As a manifestation of self-similarity behavior that is seen in the blinking of many types of fluorophores 3 , the power-law relationship is highly robust to external perturbations. 3, 27, 28 A previous study by Hasham et al. 29 reported that a sub-bandgap CW laser can change the QD blinking power-law statistics by depleting excited states, resulting in more blinky QDs. In our experiments, we find that the MIR field depletes the OFF states and alters the OFF-time power-law statistics.Since the MIR irradiates the sample with an 80 ms burst of 80 pulses every 1 s, converting OFF states to ON states, OFF events longer than 1 s are strongly suppressed by 8 over an order of magnitude, as shown inFig. 3c. Probability cut-off times also emerge at the harmonics of the one-second excitation period. Besides the suppression of long OFF events, the MIR field also creates two short OFF events by splitting longer OFF events. As a result, we observe spikes in the probability density to the left side of the cut-off times at 1 s and at its harmonics. For ON-state statistics, more ON events longer than 1 s emerge.A change in the PL spectrum averaged over multiple single dots under the MIR fields is observed.Figure 3eshows that the MIR field induces a spectral blueshift. We subtracted the MIR-off PL spectrum from the MIR-on spectrum for better visualization, as shown inFig. 3f. The spectral blueshift manifests as the first derivative feature in the differential plot, and a second derivative component, i.e., spectral narrowing, can be deduced as well. We further obtain the change in the PL spectrum as a function of MIR field strengths inFig. 3g, which exhibits a crossover behavior. At the moderate-field regime (F/F max = 0.2 − 0.6), the MIR field blueshifts the PL spectrum, restoring excitonic emission. 30-32 At the high field regime (F/F max > 0.7), the spectrum redshifts, indicating MIR fields not only remove those excess electrons in the trion state but also introduce additional charges to the QD by ionizing the exciton state.We further examine the MIR responses on QDs with different shell thicknesses, including 10 and 14 ML QDs. We first characterize ensemble equilibrium PL. We ob-9 served a suppressed blinking behavior in giant-shell (14 ML) QDs consistent with the literature. 12 Biexciton quantum yield (BXQY) measurements were done for 8, 10, and 14 ML QDs. The BXQY can be obtained from single-dot photon-correlation measurements but this can suffer from large heterogeneity 33-35 due to the high sensitivity of competing Auger non-radiative pathways to defects and local environments. 36 Instead a solution-phase sample-averaged BXQY measurement was done to enable us to extract ensemble-averaged single-dot BXQY. 37Figure 4bshows that the BXQY for 8 ML QDs is 5.4 ± 1.0%. For 10 ML, the BXQY is 10.7 ± 1.2% and it is 21.3 ± 1.0% for 14 ML(Fig. S1). The higher BXQY in thicker-shell QDs suggests that the non-radiative pathways are of lesser importance in thicker shell QDs. In the ensemble PL lifetime measurement of QDs solution as shown inFig. 4a, 14 ML QDs exhibit an extraordinarily long emission tail over tens of microseconds, while 8 ML QDs show a much shorter exponential decay. At longer decay times, the emission from 14 ML QDs deviates from an exponential and exhibits a power-law decay behavior (the fit is shown inFig. S7), suggesting a power-law trap-detrap assisted delayed emission. [38][39][40][41] In 14 ML QDs, the delayed emission contributes a significant portion (around 30%) of the total emission.These results suggest that trion-mediated blinking is less important in 14 ML than in 8 ML QDs, and consequently, we expect the MIR response in 14 ML QDs to be different than that in 8 ML QDs.We now discuss the effect of shell-thickness on the MIR response.Figure 4c10 shows the dependence of MIR-induced PL intensity changes of 10 and 14 ML QDs under differing experimental conditions. 10 ML QDs exhibit qualitatively similar behavior with 8 ML QDs. As for the optical power dependence, the response of 10 ML QDs peaks at about three times lower optical power than 8 ML QDs, which is due to the difference in absorption cross-section for different shell-thicknesses (the absorption cross-section calibration is included inFig. S4). The trion and exciton ionization thresholds in 10 ML QDs are slightly higher than in 8 ML QDs, likely because the tunneling barrier across the shell is wider in 10 ML QDs. 14 ML QDs, as mentioned above, exhibit quite distinct MIR responses than thinner shell QDs. At all field strengths, for 14 ML QDs, the MIR blueshifts the PL spectrum(Fig. 4d) and quenches PL intensity(Fig. 4c), contrary to the PL enhancement along with blueshift at moderate field strength in 8 ML QDs. As a result, we observe a reverse effect of PL dynamics change in 14 ML QDs than in 8 ML QDs. The blinking OFF-time distribution is almost unaffected in 14 ML QDs, while the ON-time distribution is altered at 1 second and at its harmonics, contrary to the nearly unaffected ON-time distribution and altered OFF-time in 8 ML QDs. Additional data can be found in theSupplementary Information (Fig. S5 and S6).To explain the blinking suppression in 8 and 10 ML QDs, we constructed a onedimensional model of the QD confined excess electron and use numerical simulations to calculate the response to the applied time-dependent MIR field. The core-shell structure of the QD is described by a stepwise potential with a central low energy well, repre-11 senting the core, and a higher energy plateau, representing the shell, as illustrated in Fig. 5a. Trap states are modeled as an additional stepwise potential well residing at the periphery of the shell. Tunneling probabilities from the core to the trap regions are calculated over the course of an ultrafast MIR pulse by numerically integrating the timedependent Schrödinger equation under the dipole approximation (see Methods). Using a physically motivated selection of parameters, the model reproduces the observed field strength threshold behavior in thin shell QDs, as shown inFig. 5b. The lower threshold can be understood through the interplay between the length of the pulse (τ pulse ) and the spatial overlap between the core-localized and trap-localized electron eigenstates.The theoretical treatment at the high-field regime is more elusive as it requires explicit differentiation between trion and exciton states. In principle, such treatment requires a many-body model which explicitly accounts for interactions between electrons and holes 42 , however here we adopt a simpler approach by adding the exciton binding energy (estimated to be around 0.2 eV 42, 43 ) to the energy difference between the core and the shell. The larger energy barrier reduces the spatial overlap between core and trap eigenstates, shifting the field threshold for tunneling to the trap to higher values. By subtracting trap tunneling probabilities with and without the additional exciton binding energy, we obtain a curve that qualitatively reproduces the experimental field dependence over the entire range of field strengths for both the 8 ML(Fig. 5b)and 10 ML QDs(Fig. S13).12In 8 ML QDs, we conclude that MIR fields remove excess charges in QDs and consequently change PL intensity and spectrum, as visualized inFig. 5c, supported by both the experimental observations and simulations. In 8 ML QDs, blinking is caused by trion-mediated Auger recombination, corroborated by our observation of a strong correlation between PL intensity and lifetime. 11 With moderate-field MIR pulses, the trion is ionized and the excess charge is removed, restoring an exciton state. Consequently, PL intensity increases, and spectrum blueshifts. We attribute this blueshift to both the restoration of exciton emission, which always has higher energy than the trion emission, and the removal of quantum-confined Stark effect (QCSE). The excess charge in the OFF state can create an internal electric field and induce a QCSE, which distorts the electron and hole wavefunctions and consequently redshifts and broadens the emission spectrum. 44 As a result, MIR fields shift the emission to higher energy and narrow the spectrum. At a high-field regime, MIR fields not only relocate the excessive charges but also ionize the restored excitons, activating additional charge generation and subsequent trion formation that leads to Auger non-radiative processes. The charge left inside the QD again creates a local electric field and stark shifts the exciton emission, leading to a decreased PL intensity and redshifted spectra.In 14 ML QDs, we attribute the PL change under MIR fields to the removal of trapped excitons, as depicted inFig. 5d. The long tail power-law decay lifetime shown inFig. 4asuggests that a significant portion (30%) of exciton emission comes from 13 trap-detrap assisted delayed emission. In thick-shell QDs, electrons display transitory carrier separation, which has been reported to be more likely than in thin shells 38, 45 due to a larger number of traps per QD and more substantial delocalization of the electron wavefunction into the shell. 38 Thus, the electron can be transiently separated from the hole wavefunction and then slowly returns to the recombination center due to the giant shell, followed by a radiative recombination. Those charge-separated electrons are localized outside the core and tend to be readily removed by MIR fields. With MIR excitations, the trapped electrons are depleted and can no longer return to the recombination center, supported by a faster PL lifetime(Fig. S8c), therefore removing the delayed emission and decreasing the overall PL intensity. The delayed emission is usually of lower energy, 38 the removal of which results in a blueshifted spectrum, consistent with our observations inFig. 4d.In conclusion, we demonstrate that ultrafast MIR electric-field pulses can effectively remove excess charges responsible for the trion-mediated Auger recombination in blinking OFF states, thereby suppress the PL blinking and achieve near-unity quantum yield even at very high excitation flux. At high field strengths, MIR fields can further ionize excitons and cause additional charging. Experimental and simulation results support that MIR fields can manipulate carriers in QDs. Our all-optical approach enables almost non-blinking thin-shell QDs in a native environment, which is highly desirable for real-time single-molecule tracking of biological processes. 9 For exam-14 ple, fast clathrin-independent endocytosis 46, 47 involves transmembrane delivery and has been in need of a non-blinking fluorescence tag. The excitation wavelength can be readily extended to other spectral ranges due to the generality of the field-driven ionization mechanism, opening in vivo applications as one can choose a wavelength with minimal environmental absorption. The ultrafast nature of the pulse further renders cumulative heating effects negligible. Our experimental results should also motivate a hithertounexplored class of all-optical blinking control experiments with off-resonant field excitations in single emitters beyond CQDs, including nitrogen/silicon-vacancy color centers in diamond 48 and defects in two-dimensional transition metal dichalcogenides 49-52 .The potential realization of single quantum emitters free from interruptions can pave the way to potential quantum computing and quantum cryptography applications.15MethodsSingle QD sample preparationTo synthesize CdSe/CdS quantum dots, an established synthetic protocol from the literature was used. 53 To synthesize CdSe core quantum dots, 60 mg CdO, 280 mg octadecylphosphonic acid and 3 g trioctylphosphine oxide were combined in a 50 mL round bottom flask. The resulting reaction mixture was put under vacuum and heated to 150• C to remove volatile substances. After 1 hour, the mixture was heated further to 320• C under nitrogen flow to form a clear colorless solution, and 1.0 mL trioctylphosphine was added dropwise. The temperature was increased to 380 • C and the heating mantle was removed. 0.5 mL Se/trioctylphosphine (60 mg Se in 0.5 mL trioctylphosphine) was injected rapidly and the reaction mixture was cooled down to room temperature with air.The crude reaction mixture was washed with acetone and redispersed in toluene. This synthetic protocol resulted in highly monodisperse core-only quantum dots with first exciton absorption at 487 nm. To overcoat CdSe core quantum dots with CdS shells, continuous injection synthesis was used.16In a 100 mL round bottom flask, 100 nmol CdSe core QDs in toluene, 3 mL octadecene (ODE), 3 mL oleylamine and 3 mL oleic acid were mixed. The resulting mixture was degassed for 20 minutes at room temperature and for 40 minutes at 100 • C to remove volatile substances. Afterwards, the mixture was put under nitrogen flow, and the temperature was increased. When the temperature reached 200 • C, 0.08 M cadmium oleate-ODE and octanethiol-ODE (1.2 equivalents) 16 were added at 2.5 mL/hour to form 7 monolayers of CdS shell. After the completion of shell precursor injection, the reaction mixture was annealed at 310 • C for 15 minutes. Quantum dots were washed with acetone and redispersed in hexane three times.Dense colloidal QDs were diluted by a factor around 10 6 , and then were drop cast or spin coated on a cover glass for single-dot measurements.Ultrafast mid-infrared pulse generation MIR pulses were generated in a 0.5 mm thick GaSe crystal by difference-frequency mixing of the signal and idler outputs of a high-energy optical parametric amplifier (OPA).The OPA was pumped with ∼ 35 fs pulses from a commercial Ti:Sapphire regenerative amplifier (800 nm central wavelength 800 nm, 12 mJ) and has an output power at signal and idler at 2.5 mJ and 2 mJ. The signal and idler beam sizes were shrinked by a reflective telescope by a factor of 1.5 to reach optimal difference-frequency generation (DFG) efficiency. The generated MIR pulses from the GaSe crystal were collected and expanded by a pair of 3-inch diameter parabolic mirrors before being tightly focused to the sample by a third parabola with a 3-inch diameter and 2-inch effective focal length.The MIR field strengths were controlled by a pair of wire-grid polarizers (Thorlabs WP25H-K).17Photoluminescence measurementsFor PL imaging, a continuous-wave laser beam (center wavelength 405 nm, Toptica iBeam smart) was used to illuminate the sample with an up to 150 W/cm 2 excitation fluence. The PL image was collected by a 100× objective with a numerical aperture (NA) of 0.7 and imaged on an electron-multiplying charge-coupled device (EMCCD, from Andor iXon Ultra). We used a home-built confocal epifluorescence microscope to select individual QDs for single dot PL lifetime and spectral measurements. For PL spectrum characterization, successive measurements of the spectra were recorded in a spectrometer (Andor Shamrock spectrometer and Andor Newton 920). For PL lifetime measurements, a 420 ps resolution (instrumental response function) was achieved by exciting the sample with a pulsed laser source (center wavelength 405 nm, repetition rate 5 MHz, 100 ps pulse duration, Picoquant) at around 2 µJ/cm 2 and detecting the emission with an avalanche photodiode (ID Quantique, id100-vis). The lifetime was obtained by tagging the arrival time of each PL photon relative to the trigger photon using a timeto-digital converter (ID Quantique, id800-TDC). The duration of MIR irradiation was controlled by a mechanical shutter. All the above measurements were performed in an ambient environment.Solution biexciton quantum yield measurementsSolution biexciton quantum yield measurements were completed by utilizing methods 18 described by Beyler et al. 37 Samples were diluted in hexane by a factor of around 10 4 .One drop of 2 mM Cadmium oleate solution in hexane was added to reduce particle aggregation, producing an average occupation in the focal volume between 6 and 8. The solution was wicked into a rectangular capillary (VitroCom, 0.100 × 0.200 mm i.d.) and sealed with capillary tube sealant to prevent solvent evaporation. Samples were excited with a pulsed laser source (532 nm, Picoquant) at a repetition rate of 1 MHz via a home-built confocal epifluorescence setup. Excitation was focused into the sample, and emission was collected using the same infinity-corrected water-immersion objective (Nikon, Plan Apo VC 60× WI, NA 1.2). A 535 nm long-pass filter (Chroma Technology Corp) was used to separate excitation and emission. Emission was further filtered spatially by a 1:1 telescoping 50 µm pinhole (Thorlabs) and spectrally by a 532 nm notch filter (Chroma Technology Corp) before being sent to three 50:50 non-polarizing beamsplitters (Thorlabs) as described by Shulenberger et al., 54 creating four equivalent intensity beams. Each beam was focused onto a single-photon counting detector (PerkinElmer, SPCM-AQR13) by a 10 cm achromatic lens (Thorlabs). Photon arrival times were recorded by a HydraHarp 400 (Picoquant) and analyzed using home-built software published at https://github.com/nanocluster/photons.Quantum tunneling calculationElectron dynamics were calculated via a one-body, one-dimensional model with step 19 functions between the core, shell, and trap. The potential energy for free-electrons in the conduction band was given by,where r c , r s , and r t are the outer edges of the core, shell, and trap respectively. These parameters are provided in Supplementary Information. For both the 8 ML and 10 ML QDs, the core-shell conduction band energy shift of 0.32 eV was taken from several sources. 42, 55-58 The core-trap offset of 0.07 eV = 3k B T was informed by the observation that electrons can stochastically tunnel between the core and trap at room temperature,suggesting that the energy difference should be on the order of k B T . 40, 59 Finally, a vacuum ionization energy of 4.4 eV was applied for distances beyond r t . 60 In order to simulate electron tunneling in the exciton state, the core-shell energy shift was increased to 0.52 eV to roughly account for the exciton binding energy.42,43The lowest 70 eigenstates of this stationary potential were calculated using the matrix numerov method. 61 The ground state and first 3 dipole-allowed states are shown inSupplementary Information (Fig. S10). Time evolution under the MIR field was done by integrating the time-dependent Schrödinger equation using the eigenstates of the 20 stationary potential and a semiclassical representation for the laser field in the Coulomb gauge with the dipole approximation,Here c k is the coefficient of the kth eigenstate, ω kl is the frequency difference between the k and l states, and ω is the frequency of the field: 1818 cm −1 = 0.2254 eV. A sin 2 envelope was used to approximate the pulse waveform,where τ pulse = 150 fs is the approximate length of the pulse. The simulations were run for a total of 300 fs, where the field was only turned on for the first 150 fs. All observables of interests were extracted by averaging over the second 150 fs.
10.1038/s41565-021-01016-w
[ "https://export.arxiv.org/pdf/2105.01190v1.pdf" ]
233,714,906
2105.01190
6998bb894376f391a96bdd5a3b7d4abeb91e6bd9
All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses Jiaojian Shi Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Weiwei Sun Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Hendrik Utzat Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Department of Materials Science and Engineering Stanford Univer-sity 94305 StanfordCaliforniaUnited States Ardavan Farahvash Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Frank Y Gao Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Zhuquan Zhang Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Ulugbek Barotov Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Adam P Willard Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Keith A Nelson Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States Moungi G Bawendi Department of Chemistry Massachusetts Institute of Technology 02139Cambridge, MassachusettsUnited States All-optical fluorescence blinking control in quan- tum dots with ultrafast mid-infrared pulses † These authors contributed equally to this work * Data availability. The data that support the findings of this study are available from the corresponding author upon request. 21 References 1 arXiv:2105.01190v1 [cond-mat.mes-hall] 3 May 2021 Photoluminescence (PL) intermittency is a ubiquitous phenomenon detrimentally reducing the temporal emission intensity stability of single colloidal quantum dots (CQDs) and the emission quantum yield of their ensembles. Despite efforts for blinking reduction via chemical engineering of the QD architecture and its environment, blinking still poses barriers to the application of QDs, particularly in single-particle tracking in biology or in single-photon sources. Here, we demonstrate the first deterministic all-optical suppression of quantum dot blinking using a compound technique of visible and mid-infrared (MIR) excitation. We show that moderate-field ultrafast MIR pulses (5.5 µm, 150 fs) can switch the emission from a charged, low quantum yield 'grey' trion state to the 'bright' exciton state in CdSe/CdS core-shell quantum dots resulting in a significant reduction of the QD intensity flicker. Quantum-tunneling simulations suggest that the MIR fields remove the excess charge from trions with reduced emission quantum yield to restore higher brightness exciton emission. Our approach can be integrated with existing single-particle tracking or super-resolution microscopy techniques without any modification to the sample and translates to other emitters presenting charging-induced PL intermittencies, such as single-photon emissive defects in diamond and two-dimensional materials.2 Colloidal quantum dots (CQDs) have now made significant commercial inroads in diverse optoelectronic applications as well as biological imaging due to their unique electronic structures, size-tunable emission, high photoluminescence (PL) quantum yield, high photo-stability, and facile chemical synthesis. 1 However, despite two decades of research, stochastic PL intermittency, also known as 'blinking', still reduces the temporal emission stability, particularly under high excitation-flux conditions in single-emitter experiment, posing a significant barrier to the wider adoption of QDs in single-emitter applications. In single-photon sources, blinking reduces the ability of QDs to produce single-photons on demand and often coincides with detrimental spectral jumping 2-6 , two processes that need to be eliminated to qualify QDs as building blocks of singlephoton sources for quantum cryptography and computing.7,8In real-time single-particle tracking in biological systems 9, 10 , the timescale of blinking coincides with the particle diffusion reducing the tracking ability.Despite the lack of a unified blinking theory 3 , it is now understood that blinking in QDs occurs primarily via two pathways i) surface trap-mediated non-radiative recombination and ii) charging-induced Auger recombination. 11 Synthetic efforts have been made towards intrinsically non-blinking quantum dots targeting both of these mechanism primarily through passivating surface trap-states and reducing the Auger recombination rate of charged states through smoothing of the potential barrier between coreand shell-layers of heteroepitaxial QDs. 12-17 Extrinsic blinking control has been com-3 paratively less explored. Adding anti-blinking agents suppresses blinking in QDs by passivating the QD surface.[18][19][20]Other approaches such as electrostatic gating have also been reported to tune the Fermi level of the QD and block the relaxation pathways through the surface, thereby suppressing non-radiative decay during OFF periods.11,21These methods, however, require the QD to be either deposited on a special substrate or immersed in a non-native environment, adding significant complexity to single-QD devices and precluding blinking control in biological environments. To date, no noninvasive active suppression of blinking has yet been demonstrated.Here, we demonstrate the first deterministic all-optical method for active reduction of QD blinking with ultrafast MIR pulses. We build on advances in ultrafast electricfield pulse technologies 22, 23 and show that off-resonant laser pulses at mid-infrared (MIR) frequencies provide sufficient strengths to overcome potential barriers in QDs while avoiding dielectric breakdown that often occurs upon applications of static electric fields. By applying MIR pulses concurrently with optical excitation of single QDs, we demonstrate that MIR pulses with an appropriate field strength can remove the excess electron of charging-induced blinking OFF states in single QDs. Therefore, MIR pulses have the potential to transiently discharge QDs in the blinking OFF states at an ultrafast speed without perturbing the equilibrium emissivity or introducing inter-band excitations.4The experimental setup is shown inFig.1a. Our experiments investigate if the MIR pulse can switch OFF states to ON states by discharging the trion state (as shown inFig. 1b) through the measurement of blinking statistics, emission lifetimes, and measuring emission spectra and emission intensities. A 1-kHz MIR pulse train at 5.5 µm is used in our experiments(Fig. 1c), which is far below the multiphoton absorption regime and away from phonon absorption frequencies of CdSe/CdS QDs. 24 We tuned the MIR exposure by modulating a mechanical shutter and quantify the exposure by the number of pulses in a burst of MIR. Under 405 nm continuous-wave (CW) laser excitation, emission from single core-shell CdSe/CdS QDs with an 8-monolayer (ML) shell thickness, deposited on a glass coverslip, was confirmed by the antibunching dip (τ = 0) from a second-order photon intensity correlation measurement, as shown in Fig. 1d. We show the effect of concurrent MIR and optical excitation in Fig. 2. We record the PL intensity traces of single QDs under 405 nm CW laser excitation. Under no MIR field, we show a representative single QD PL blinking trace (Fig. 2a) and its histogram of PL intensity distributions (Fig. 2b). The histogram in Fig. 2b shows a bimodal PL intensity distribution, corresponding to blinking ON and OFF states. With MIR excitation at a suitable field strength (here F/F max = 0.6, F max ∼ 10 MV/cm), the blinking behavior from the same QD changes as shown in Fig.2c-d, reflected by the significant decrease in the time that the dot spends in the OFF state in the blinking trace 5 as well as by the shift in the intensity histogram from bimodal to a largely unimodal ON state.The effective conversion from OFF to ON states is also reflected by an enhanced PL intensity in ensemble QDs. By imaging and selecting isolated single QDs within the field of view, we can study the dynamics and responses of statistically-averaged PL intensity (counts summed over all the dots) when MIR excitations are turned on and off.Here, MIR excitations are periodically exposed on the sample for 200 ms (a burst of 200 MIR pulses) every 5 seconds, resulting in PL intensity spikes every 5 s, as shown in Fig. 2e. No degradation in either equilibrium PL counts without MIR or enhanced PL counts with MIR is observed, suggesting the reversible nature of the PL enhancement.The enhancement of PL intensity only appears when a suitable MIR field strength is applied. With stronger MIR pulses (F/F max > 0.7) , we observe a suppression of the overall PL intensity. InFig. 2f, we show a drop in the PL intensity under an MIR exposure of 10 ms with a burst of 10 MIR pulses every two seconds at F/F max = 0.9, as shown inFig. 2f.Figure 2gillustrates the normalized PL intensity change as a function of CWlaser optical power. The MIR-induced PL intensity change is summed over multiple (∼ 100) isolated dots and normalized to their equilibrium PL intensity. The PL percentage change increases as the optical power increases and saturates at ∼ 30 W/cm 2 . The 6 extrapolated zero PL change at zero optical power indicates that the MIR itself does not produce any luminescence and only enhances PL when the dots are excited optically.The positive correlation between the PL percentage change and optical power suggests that the PL enhancement is most likely due to removal of accumulative events such as excess photo-ionized charges induced blinking.11,25 The threshold behavior in the MIR field dependence of the PL percentage change(Fig. 2h) further suggests that the MIR field is driving ionization processes and, as a result, removes excess charges inside the QDs. A crossover behavior is observed at field strengths around F/F max = 0.7 with an optical power of 20 W/cm 2 and a burst of 80 MIR pulses every 1 s, which is consistent with the observation of PL quenching under excessively high MIR fields shown inFig.2f.Figure 2idisplays the MIR exposure dependence of the PL percentage change with a fixed optical power at 20 W/cm 2 and a field strength at F/F max = 0.6. The PL percentage change increases with increasing MIR pulse exposure. The PL percentage change saturates at a burst of around 100 MIR pulses every 1 s at the field strength of F/F max = 0.6. Further increasing MIR exposure results in PL intensity degradation, which can be explained by excess charge generation with excessive pulses.PL lifetime changes further validate that MIR fields remove excess charges inside the QDs. When the MIR is off, a single QD randomly switches between ON and OFF states. The strong correlation between the PL intensity and lifetime indicates that blinking is caused by charging and discharging as expected for core-shell CdSe/CdS QDs 11, 26 7 as is seen inFig. 3athrough a fluorescence lifetime-intensity distribution (FLID) plot.With MIR excitation, the blinking OFF states disappear, and only the ON states are present with a long lifetime and high PL intensity, as shown inFig. 3b. Another representation is provided in the Supplementary Information by interrogating all the photon arrival times relative to the excitation trigger. As shown inFig. S8a, without MIR fields, the PL lifetime of the single QD is composed of fast and slow exponential decays components, which we assign to trion and exciton decay, respectively. With MIR excitations, the fast trion decay is completely suppressed, and only the slow exciton relaxation is left, implying that MIR fields efficiently remove the excess charge. Figure 3c and d show that blinking statistics are strongly modified by MIR excitation. By setting a threshold in a blinking trace separating ON and OFF states and counting their probability densities, a power-law distribution of ON-and OFF-times is obtained inFig. 3dand c, respectively. As a manifestation of self-similarity behavior that is seen in the blinking of many types of fluorophores 3 , the power-law relationship is highly robust to external perturbations. 3, 27, 28 A previous study by Hasham et al. 29 reported that a sub-bandgap CW laser can change the QD blinking power-law statistics by depleting excited states, resulting in more blinky QDs. In our experiments, we find that the MIR field depletes the OFF states and alters the OFF-time power-law statistics.Since the MIR irradiates the sample with an 80 ms burst of 80 pulses every 1 s, converting OFF states to ON states, OFF events longer than 1 s are strongly suppressed by 8 over an order of magnitude, as shown inFig. 3c. Probability cut-off times also emerge at the harmonics of the one-second excitation period. Besides the suppression of long OFF events, the MIR field also creates two short OFF events by splitting longer OFF events. As a result, we observe spikes in the probability density to the left side of the cut-off times at 1 s and at its harmonics. For ON-state statistics, more ON events longer than 1 s emerge.A change in the PL spectrum averaged over multiple single dots under the MIR fields is observed.Figure 3eshows that the MIR field induces a spectral blueshift. We subtracted the MIR-off PL spectrum from the MIR-on spectrum for better visualization, as shown inFig. 3f. The spectral blueshift manifests as the first derivative feature in the differential plot, and a second derivative component, i.e., spectral narrowing, can be deduced as well. We further obtain the change in the PL spectrum as a function of MIR field strengths inFig. 3g, which exhibits a crossover behavior. At the moderate-field regime (F/F max = 0.2 − 0.6), the MIR field blueshifts the PL spectrum, restoring excitonic emission. 30-32 At the high field regime (F/F max > 0.7), the spectrum redshifts, indicating MIR fields not only remove those excess electrons in the trion state but also introduce additional charges to the QD by ionizing the exciton state.We further examine the MIR responses on QDs with different shell thicknesses, including 10 and 14 ML QDs. We first characterize ensemble equilibrium PL. We ob-9 served a suppressed blinking behavior in giant-shell (14 ML) QDs consistent with the literature. 12 Biexciton quantum yield (BXQY) measurements were done for 8, 10, and 14 ML QDs. The BXQY can be obtained from single-dot photon-correlation measurements but this can suffer from large heterogeneity 33-35 due to the high sensitivity of competing Auger non-radiative pathways to defects and local environments. 36 Instead a solution-phase sample-averaged BXQY measurement was done to enable us to extract ensemble-averaged single-dot BXQY. 37Figure 4bshows that the BXQY for 8 ML QDs is 5.4 ± 1.0%. For 10 ML, the BXQY is 10.7 ± 1.2% and it is 21.3 ± 1.0% for 14 ML(Fig. S1). The higher BXQY in thicker-shell QDs suggests that the non-radiative pathways are of lesser importance in thicker shell QDs. In the ensemble PL lifetime measurement of QDs solution as shown inFig. 4a, 14 ML QDs exhibit an extraordinarily long emission tail over tens of microseconds, while 8 ML QDs show a much shorter exponential decay. At longer decay times, the emission from 14 ML QDs deviates from an exponential and exhibits a power-law decay behavior (the fit is shown inFig. S7), suggesting a power-law trap-detrap assisted delayed emission. [38][39][40][41] In 14 ML QDs, the delayed emission contributes a significant portion (around 30%) of the total emission.These results suggest that trion-mediated blinking is less important in 14 ML than in 8 ML QDs, and consequently, we expect the MIR response in 14 ML QDs to be different than that in 8 ML QDs.We now discuss the effect of shell-thickness on the MIR response.Figure 4c10 shows the dependence of MIR-induced PL intensity changes of 10 and 14 ML QDs under differing experimental conditions. 10 ML QDs exhibit qualitatively similar behavior with 8 ML QDs. As for the optical power dependence, the response of 10 ML QDs peaks at about three times lower optical power than 8 ML QDs, which is due to the difference in absorption cross-section for different shell-thicknesses (the absorption cross-section calibration is included inFig. S4). The trion and exciton ionization thresholds in 10 ML QDs are slightly higher than in 8 ML QDs, likely because the tunneling barrier across the shell is wider in 10 ML QDs. 14 ML QDs, as mentioned above, exhibit quite distinct MIR responses than thinner shell QDs. At all field strengths, for 14 ML QDs, the MIR blueshifts the PL spectrum(Fig. 4d) and quenches PL intensity(Fig. 4c), contrary to the PL enhancement along with blueshift at moderate field strength in 8 ML QDs. As a result, we observe a reverse effect of PL dynamics change in 14 ML QDs than in 8 ML QDs. The blinking OFF-time distribution is almost unaffected in 14 ML QDs, while the ON-time distribution is altered at 1 second and at its harmonics, contrary to the nearly unaffected ON-time distribution and altered OFF-time in 8 ML QDs. Additional data can be found in theSupplementary Information (Fig. S5 and S6).To explain the blinking suppression in 8 and 10 ML QDs, we constructed a onedimensional model of the QD confined excess electron and use numerical simulations to calculate the response to the applied time-dependent MIR field. The core-shell structure of the QD is described by a stepwise potential with a central low energy well, repre-11 senting the core, and a higher energy plateau, representing the shell, as illustrated in Fig. 5a. Trap states are modeled as an additional stepwise potential well residing at the periphery of the shell. Tunneling probabilities from the core to the trap regions are calculated over the course of an ultrafast MIR pulse by numerically integrating the timedependent Schrödinger equation under the dipole approximation (see Methods). Using a physically motivated selection of parameters, the model reproduces the observed field strength threshold behavior in thin shell QDs, as shown inFig. 5b. The lower threshold can be understood through the interplay between the length of the pulse (τ pulse ) and the spatial overlap between the core-localized and trap-localized electron eigenstates.The theoretical treatment at the high-field regime is more elusive as it requires explicit differentiation between trion and exciton states. In principle, such treatment requires a many-body model which explicitly accounts for interactions between electrons and holes 42 , however here we adopt a simpler approach by adding the exciton binding energy (estimated to be around 0.2 eV 42, 43 ) to the energy difference between the core and the shell. The larger energy barrier reduces the spatial overlap between core and trap eigenstates, shifting the field threshold for tunneling to the trap to higher values. By subtracting trap tunneling probabilities with and without the additional exciton binding energy, we obtain a curve that qualitatively reproduces the experimental field dependence over the entire range of field strengths for both the 8 ML(Fig. 5b)and 10 ML QDs(Fig. S13).12In 8 ML QDs, we conclude that MIR fields remove excess charges in QDs and consequently change PL intensity and spectrum, as visualized inFig. 5c, supported by both the experimental observations and simulations. In 8 ML QDs, blinking is caused by trion-mediated Auger recombination, corroborated by our observation of a strong correlation between PL intensity and lifetime. 11 With moderate-field MIR pulses, the trion is ionized and the excess charge is removed, restoring an exciton state. Consequently, PL intensity increases, and spectrum blueshifts. We attribute this blueshift to both the restoration of exciton emission, which always has higher energy than the trion emission, and the removal of quantum-confined Stark effect (QCSE). The excess charge in the OFF state can create an internal electric field and induce a QCSE, which distorts the electron and hole wavefunctions and consequently redshifts and broadens the emission spectrum. 44 As a result, MIR fields shift the emission to higher energy and narrow the spectrum. At a high-field regime, MIR fields not only relocate the excessive charges but also ionize the restored excitons, activating additional charge generation and subsequent trion formation that leads to Auger non-radiative processes. The charge left inside the QD again creates a local electric field and stark shifts the exciton emission, leading to a decreased PL intensity and redshifted spectra.In 14 ML QDs, we attribute the PL change under MIR fields to the removal of trapped excitons, as depicted inFig. 5d. The long tail power-law decay lifetime shown inFig. 4asuggests that a significant portion (30%) of exciton emission comes from 13 trap-detrap assisted delayed emission. In thick-shell QDs, electrons display transitory carrier separation, which has been reported to be more likely than in thin shells 38, 45 due to a larger number of traps per QD and more substantial delocalization of the electron wavefunction into the shell. 38 Thus, the electron can be transiently separated from the hole wavefunction and then slowly returns to the recombination center due to the giant shell, followed by a radiative recombination. Those charge-separated electrons are localized outside the core and tend to be readily removed by MIR fields. With MIR excitations, the trapped electrons are depleted and can no longer return to the recombination center, supported by a faster PL lifetime(Fig. S8c), therefore removing the delayed emission and decreasing the overall PL intensity. The delayed emission is usually of lower energy, 38 the removal of which results in a blueshifted spectrum, consistent with our observations inFig. 4d.In conclusion, we demonstrate that ultrafast MIR electric-field pulses can effectively remove excess charges responsible for the trion-mediated Auger recombination in blinking OFF states, thereby suppress the PL blinking and achieve near-unity quantum yield even at very high excitation flux. At high field strengths, MIR fields can further ionize excitons and cause additional charging. Experimental and simulation results support that MIR fields can manipulate carriers in QDs. Our all-optical approach enables almost non-blinking thin-shell QDs in a native environment, which is highly desirable for real-time single-molecule tracking of biological processes. 9 For exam-14 ple, fast clathrin-independent endocytosis 46, 47 involves transmembrane delivery and has been in need of a non-blinking fluorescence tag. The excitation wavelength can be readily extended to other spectral ranges due to the generality of the field-driven ionization mechanism, opening in vivo applications as one can choose a wavelength with minimal environmental absorption. The ultrafast nature of the pulse further renders cumulative heating effects negligible. Our experimental results should also motivate a hithertounexplored class of all-optical blinking control experiments with off-resonant field excitations in single emitters beyond CQDs, including nitrogen/silicon-vacancy color centers in diamond 48 and defects in two-dimensional transition metal dichalcogenides 49-52 .The potential realization of single quantum emitters free from interruptions can pave the way to potential quantum computing and quantum cryptography applications.15MethodsSingle QD sample preparationTo synthesize CdSe/CdS quantum dots, an established synthetic protocol from the literature was used. 53 To synthesize CdSe core quantum dots, 60 mg CdO, 280 mg octadecylphosphonic acid and 3 g trioctylphosphine oxide were combined in a 50 mL round bottom flask. The resulting reaction mixture was put under vacuum and heated to 150• C to remove volatile substances. After 1 hour, the mixture was heated further to 320• C under nitrogen flow to form a clear colorless solution, and 1.0 mL trioctylphosphine was added dropwise. The temperature was increased to 380 • C and the heating mantle was removed. 0.5 mL Se/trioctylphosphine (60 mg Se in 0.5 mL trioctylphosphine) was injected rapidly and the reaction mixture was cooled down to room temperature with air.The crude reaction mixture was washed with acetone and redispersed in toluene. This synthetic protocol resulted in highly monodisperse core-only quantum dots with first exciton absorption at 487 nm. To overcoat CdSe core quantum dots with CdS shells, continuous injection synthesis was used.16In a 100 mL round bottom flask, 100 nmol CdSe core QDs in toluene, 3 mL octadecene (ODE), 3 mL oleylamine and 3 mL oleic acid were mixed. The resulting mixture was degassed for 20 minutes at room temperature and for 40 minutes at 100 • C to remove volatile substances. Afterwards, the mixture was put under nitrogen flow, and the temperature was increased. When the temperature reached 200 • C, 0.08 M cadmium oleate-ODE and octanethiol-ODE (1.2 equivalents) 16 were added at 2.5 mL/hour to form 7 monolayers of CdS shell. After the completion of shell precursor injection, the reaction mixture was annealed at 310 • C for 15 minutes. Quantum dots were washed with acetone and redispersed in hexane three times.Dense colloidal QDs were diluted by a factor around 10 6 , and then were drop cast or spin coated on a cover glass for single-dot measurements.Ultrafast mid-infrared pulse generation MIR pulses were generated in a 0.5 mm thick GaSe crystal by difference-frequency mixing of the signal and idler outputs of a high-energy optical parametric amplifier (OPA).The OPA was pumped with ∼ 35 fs pulses from a commercial Ti:Sapphire regenerative amplifier (800 nm central wavelength 800 nm, 12 mJ) and has an output power at signal and idler at 2.5 mJ and 2 mJ. The signal and idler beam sizes were shrinked by a reflective telescope by a factor of 1.5 to reach optimal difference-frequency generation (DFG) efficiency. The generated MIR pulses from the GaSe crystal were collected and expanded by a pair of 3-inch diameter parabolic mirrors before being tightly focused to the sample by a third parabola with a 3-inch diameter and 2-inch effective focal length.The MIR field strengths were controlled by a pair of wire-grid polarizers (Thorlabs WP25H-K).17Photoluminescence measurementsFor PL imaging, a continuous-wave laser beam (center wavelength 405 nm, Toptica iBeam smart) was used to illuminate the sample with an up to 150 W/cm 2 excitation fluence. The PL image was collected by a 100× objective with a numerical aperture (NA) of 0.7 and imaged on an electron-multiplying charge-coupled device (EMCCD, from Andor iXon Ultra). We used a home-built confocal epifluorescence microscope to select individual QDs for single dot PL lifetime and spectral measurements. For PL spectrum characterization, successive measurements of the spectra were recorded in a spectrometer (Andor Shamrock spectrometer and Andor Newton 920). For PL lifetime measurements, a 420 ps resolution (instrumental response function) was achieved by exciting the sample with a pulsed laser source (center wavelength 405 nm, repetition rate 5 MHz, 100 ps pulse duration, Picoquant) at around 2 µJ/cm 2 and detecting the emission with an avalanche photodiode (ID Quantique, id100-vis). The lifetime was obtained by tagging the arrival time of each PL photon relative to the trigger photon using a timeto-digital converter (ID Quantique, id800-TDC). The duration of MIR irradiation was controlled by a mechanical shutter. All the above measurements were performed in an ambient environment.Solution biexciton quantum yield measurementsSolution biexciton quantum yield measurements were completed by utilizing methods 18 described by Beyler et al. 37 Samples were diluted in hexane by a factor of around 10 4 .One drop of 2 mM Cadmium oleate solution in hexane was added to reduce particle aggregation, producing an average occupation in the focal volume between 6 and 8. The solution was wicked into a rectangular capillary (VitroCom, 0.100 × 0.200 mm i.d.) and sealed with capillary tube sealant to prevent solvent evaporation. Samples were excited with a pulsed laser source (532 nm, Picoquant) at a repetition rate of 1 MHz via a home-built confocal epifluorescence setup. Excitation was focused into the sample, and emission was collected using the same infinity-corrected water-immersion objective (Nikon, Plan Apo VC 60× WI, NA 1.2). A 535 nm long-pass filter (Chroma Technology Corp) was used to separate excitation and emission. Emission was further filtered spatially by a 1:1 telescoping 50 µm pinhole (Thorlabs) and spectrally by a 532 nm notch filter (Chroma Technology Corp) before being sent to three 50:50 non-polarizing beamsplitters (Thorlabs) as described by Shulenberger et al., 54 creating four equivalent intensity beams. Each beam was focused onto a single-photon counting detector (PerkinElmer, SPCM-AQR13) by a 10 cm achromatic lens (Thorlabs). Photon arrival times were recorded by a HydraHarp 400 (Picoquant) and analyzed using home-built software published at https://github.com/nanocluster/photons.Quantum tunneling calculationElectron dynamics were calculated via a one-body, one-dimensional model with step 19 functions between the core, shell, and trap. The potential energy for free-electrons in the conduction band was given by,where r c , r s , and r t are the outer edges of the core, shell, and trap respectively. These parameters are provided in Supplementary Information. For both the 8 ML and 10 ML QDs, the core-shell conduction band energy shift of 0.32 eV was taken from several sources. 42, 55-58 The core-trap offset of 0.07 eV = 3k B T was informed by the observation that electrons can stochastically tunnel between the core and trap at room temperature,suggesting that the energy difference should be on the order of k B T . 40, 59 Finally, a vacuum ionization energy of 4.4 eV was applied for distances beyond r t . 60 In order to simulate electron tunneling in the exciton state, the core-shell energy shift was increased to 0.52 eV to roughly account for the exciton binding energy.42,43The lowest 70 eigenstates of this stationary potential were calculated using the matrix numerov method. 61 The ground state and first 3 dipole-allowed states are shown inSupplementary Information (Fig. S10). Time evolution under the MIR field was done by integrating the time-dependent Schrödinger equation using the eigenstates of the 20 stationary potential and a semiclassical representation for the laser field in the Coulomb gauge with the dipole approximation,Here c k is the coefficient of the kth eigenstate, ω kl is the frequency difference between the k and l states, and ω is the frequency of the field: 1818 cm −1 = 0.2254 eV. A sin 2 envelope was used to approximate the pulse waveform,where τ pulse = 150 fs is the approximate length of the pulse. The simulations were run for a total of 300 fs, where the field was only turned on for the first 150 fs. All observables of interests were extracted by averaging over the second 150 fs. Photoluminescence (PL) intermittency is a ubiquitous phenomenon detrimentally reducing the temporal emission intensity stability of single colloidal quantum dots (CQDs) and the emission quantum yield of their ensembles. Despite efforts for blinking reduction via chemical engineering of the QD architecture and its environment, blinking still poses barriers to the application of QDs, particularly in single-particle tracking in biology or in single-photon sources. Here, we demonstrate the first deterministic all-optical suppression of quantum dot blinking using a compound technique of visible and mid-infrared (MIR) excitation. We show that moderate-field ultrafast MIR pulses (5.5 µm, 150 fs) can switch the emission from a charged, low quantum yield 'grey' trion state to the 'bright' exciton state in CdSe/CdS core-shell quantum dots resulting in a significant reduction of the QD intensity flicker. Quantum-tunneling simulations suggest that the MIR fields remove the excess charge from trions with reduced emission quantum yield to restore higher brightness exciton emission. Our approach can be integrated with existing single-particle tracking or super-resolution microscopy techniques without any modification to the sample and translates to other emitters presenting charging-induced PL intermittencies, such as single-photon emissive defects in diamond and two-dimensional materials. Colloidal quantum dots (CQDs) have now made significant commercial inroads in diverse optoelectronic applications as well as biological imaging due to their unique electronic structures, size-tunable emission, high photoluminescence (PL) quantum yield, high photo-stability, and facile chemical synthesis. 1 However, despite two decades of research, stochastic PL intermittency, also known as 'blinking', still reduces the temporal emission stability, particularly under high excitation-flux conditions in single-emitter experiment, posing a significant barrier to the wider adoption of QDs in single-emitter applications. In single-photon sources, blinking reduces the ability of QDs to produce single-photons on demand and often coincides with detrimental spectral jumping 2-6 , two processes that need to be eliminated to qualify QDs as building blocks of singlephoton sources for quantum cryptography and computing. 7,8 In real-time single-particle tracking in biological systems 9,10 , the timescale of blinking coincides with the particle diffusion reducing the tracking ability. Despite the lack of a unified blinking theory 3 , it is now understood that blinking in QDs occurs primarily via two pathways i) surface trap-mediated non-radiative recombination and ii) charging-induced Auger recombination. 11 Synthetic efforts have been made towards intrinsically non-blinking quantum dots targeting both of these mechanism primarily through passivating surface trap-states and reducing the Auger recombination rate of charged states through smoothing of the potential barrier between coreand shell-layers of heteroepitaxial QDs. [12][13][14][15][16][17] Extrinsic blinking control has been com-paratively less explored. Adding anti-blinking agents suppresses blinking in QDs by passivating the QD surface. [18][19][20] Other approaches such as electrostatic gating have also been reported to tune the Fermi level of the QD and block the relaxation pathways through the surface, thereby suppressing non-radiative decay during OFF periods. 11,21 These methods, however, require the QD to be either deposited on a special substrate or immersed in a non-native environment, adding significant complexity to single-QD devices and precluding blinking control in biological environments. To date, no noninvasive active suppression of blinking has yet been demonstrated. Here, we demonstrate the first deterministic all-optical method for active reduction of QD blinking with ultrafast MIR pulses. We build on advances in ultrafast electricfield pulse technologies 22,23 and show that off-resonant laser pulses at mid-infrared (MIR) frequencies provide sufficient strengths to overcome potential barriers in QDs while avoiding dielectric breakdown that often occurs upon applications of static electric fields. By applying MIR pulses concurrently with optical excitation of single QDs, we demonstrate that MIR pulses with an appropriate field strength can remove the excess electron of charging-induced blinking OFF states in single QDs. Therefore, MIR pulses have the potential to transiently discharge QDs in the blinking OFF states at an ultrafast speed without perturbing the equilibrium emissivity or introducing inter-band excitations. The experimental setup is shown in Fig.1a. Our experiments investigate if the MIR pulse can switch OFF states to ON states by discharging the trion state (as shown in Fig. 1b) through the measurement of blinking statistics, emission lifetimes, and measuring emission spectra and emission intensities. A 1-kHz MIR pulse train at 5.5 µm is used in our experiments (Fig. 1c), which is far below the multiphoton absorption regime and away from phonon absorption frequencies of CdSe/CdS QDs. 24 We tuned the MIR exposure by modulating a mechanical shutter and quantify the exposure by the number of pulses in a burst of MIR. Under 405 nm continuous-wave (CW) laser excitation, emission from single core-shell CdSe/CdS QDs with an 8-monolayer (ML) shell thickness, deposited on a glass coverslip, was confirmed by the antibunching dip (τ = 0) from a second-order photon intensity correlation measurement, as shown in Fig. 1d. We show the effect of concurrent MIR and optical excitation in Fig. 2 The effective conversion from OFF to ON states is also reflected by an enhanced PL intensity in ensemble QDs. By imaging and selecting isolated single QDs within the field of view, we can study the dynamics and responses of statistically-averaged PL intensity (counts summed over all the dots) when MIR excitations are turned on and off. Here, MIR excitations are periodically exposed on the sample for 200 ms (a burst of 200 MIR pulses) every 5 seconds, resulting in PL intensity spikes every 5 s, as shown in The positive correlation between the PL percentage change and optical power suggests that the PL enhancement is most likely due to removal of accumulative events such as excess photo-ionized charges induced blinking. 11,25 The threshold behavior in the MIR field dependence of the PL percentage change (Fig. 2h) further suggests that the MIR field is driving ionization processes and, as a result, removes excess charges inside the QDs. A crossover behavior is observed at field strengths around F/F max = 0.7 with an optical power of 20 W/cm 2 and a burst of 80 MIR pulses every 1 s, which is consistent with the observation of PL quenching under excessively high MIR fields shown in Fig. 2f. Figure 2i displays the MIR exposure dependence of the PL percentage change with a fixed optical power at 20 W/cm 2 and a field strength at F/F max = 0.6. The PL percentage change increases with increasing MIR pulse exposure. The PL percentage change saturates at a burst of around 100 MIR pulses every 1 s at the field strength of F/F max = 0.6. Further increasing MIR exposure results in PL intensity degradation, which can be explained by excess charge generation with excessive pulses. PL lifetime changes further validate that MIR fields remove excess charges inside the QDs. When the MIR is off, a single QD randomly switches between ON and OFF states. The strong correlation between the PL intensity and lifetime indicates that blinking is caused by charging and discharging as expected for core-shell CdSe/CdS QDs 11,26 as is seen in Fig. 3a through a fluorescence lifetime-intensity distribution (FLID) plot. With MIR excitation, the blinking OFF states disappear, and only the ON states are present with a long lifetime and high PL intensity, as shown in Fig. 3b. Another representation is provided in the Supplementary Information by interrogating all the photon arrival times relative to the excitation trigger. As shown in Fig. S8a, without MIR fields, the PL lifetime of the single QD is composed of fast and slow exponential decays components, which we assign to trion and exciton decay, respectively. With MIR excitations, the fast trion decay is completely suppressed, and only the slow exciton relaxation is left, implying that MIR fields efficiently remove the excess charge. Fig. 3d and c, respectively. As a manifestation of self-similarity behavior that is seen in the blinking of many types of fluorophores 3 , the power-law relationship is highly robust to external perturbations. 3, 27, 28 A previous study by Hasham et al. 29 reported that a sub-bandgap CW laser can change the QD blinking power-law statistics by depleting excited states, resulting in more blinky QDs. In our experiments, we find that the MIR field depletes the OFF states and alters the OFF-time power-law statistics. Since the MIR irradiates the sample with an 80 ms burst of 80 pulses every 1 s, converting OFF states to ON states, OFF events longer than 1 s are strongly suppressed by over an order of magnitude, as shown in Fig. 3c. Probability cut-off times also emerge at the harmonics of the one-second excitation period. Besides the suppression of long OFF events, the MIR field also creates two short OFF events by splitting longer OFF events. As a result, we observe spikes in the probability density to the left side of the cut-off times at 1 s and at its harmonics. For ON-state statistics, more ON events longer than 1 s emerge. A change in the PL spectrum averaged over multiple single dots under the MIR fields is observed. Figure 3e shows that the MIR field induces a spectral blueshift. We subtracted the MIR-off PL spectrum from the MIR-on spectrum for better visualization, as shown in Fig. 3f. The spectral blueshift manifests as the first derivative feature in the differential plot, and a second derivative component, i.e., spectral narrowing, can be deduced as well. We further obtain the change in the PL spectrum as a function of MIR field strengths in Fig. 3g, which exhibits a crossover behavior. At the moderate-field regime (F/F max = 0.2 − 0.6), the MIR field blueshifts the PL spectrum, restoring excitonic emission. 30-32 At the high field regime (F/F max > 0.7), the spectrum redshifts, indicating MIR fields not only remove those excess electrons in the trion state but also introduce additional charges to the QD by ionizing the exciton state. We further examine the MIR responses on QDs with different shell thicknesses, including 10 and 14 ML QDs. We first characterize ensemble equilibrium PL. We ob-served a suppressed blinking behavior in giant-shell (14 ML) QDs consistent with the literature. 12 Biexciton quantum yield (BXQY) measurements were done for 8, 10, and 14 ML QDs. The BXQY can be obtained from single-dot photon-correlation measurements but this can suffer from large heterogeneity 33-35 due to the high sensitivity of competing Auger non-radiative pathways to defects and local environments. 36 Instead a solution-phase sample-averaged BXQY measurement was done to enable us to extract ensemble-averaged single-dot BXQY. 37 Figure 4b shows that the BXQY for 8 ML QDs These results suggest that trion-mediated blinking is less important in 14 ML than in 8 ML QDs, and consequently, we expect the MIR response in 14 ML QDs to be different than that in 8 ML QDs. We now discuss the effect of shell-thickness on the MIR response. Figure 4c shows the dependence of MIR-induced PL intensity changes of 10 and 14 ML QDs under differing experimental conditions. 10 ML QDs exhibit qualitatively similar behavior with 8 ML QDs. As for the optical power dependence, the response of 10 ML QDs peaks at about three times lower optical power than 8 ML QDs, which is due to the difference in absorption cross-section for different shell-thicknesses (the absorption cross-section calibration is included in Fig. S4). The trion and exciton ionization thresholds in 10 ML QDs are slightly higher than in 8 ML QDs, likely because the tunneling barrier across the shell is wider in 10 ML QDs. 14 ML QDs, as mentioned above, exhibit quite distinct MIR responses than thinner shell QDs. At all field strengths, for 14 ML QDs, the MIR blueshifts the PL spectrum (Fig. 4d) and quenches PL intensity ( To explain the blinking suppression in 8 and 10 ML QDs, we constructed a onedimensional model of the QD confined excess electron and use numerical simulations to calculate the response to the applied time-dependent MIR field. The core-shell structure of the QD is described by a stepwise potential with a central low energy well, repre-senting the core, and a higher energy plateau, representing the shell, as illustrated in The theoretical treatment at the high-field regime is more elusive as it requires explicit differentiation between trion and exciton states. In principle, such treatment requires a many-body model which explicitly accounts for interactions between electrons and holes 42 , however here we adopt a simpler approach by adding the exciton binding energy (estimated to be around 0.2 eV 42,43 ) to the energy difference between the core and the shell. The larger energy barrier reduces the spatial overlap between core and trap eigenstates, shifting the field threshold for tunneling to the trap to higher values. By subtracting trap tunneling probabilities with and without the additional exciton binding energy, we obtain a curve that qualitatively reproduces the experimental field dependence over the entire range of field strengths for both the 8 ML (Fig. 5b) and 10 ML QDs (Fig. S13). In 8 ML QDs, we conclude that MIR fields remove excess charges in QDs and consequently change PL intensity and spectrum, as visualized in Fig. 5c, supported by both the experimental observations and simulations. In 8 ML QDs, blinking is caused by trion-mediated Auger recombination, corroborated by our observation of a strong correlation between PL intensity and lifetime. 11 With moderate-field MIR pulses, the trion is ionized and the excess charge is removed, restoring an exciton state. Consequently, PL intensity increases, and spectrum blueshifts. We attribute this blueshift to both the restoration of exciton emission, which always has higher energy than the trion emission, and the removal of quantum-confined Stark effect (QCSE). The excess charge in the OFF state can create an internal electric field and induce a QCSE, which distorts the electron and hole wavefunctions and consequently redshifts and broadens the emission spectrum. 44 As a result, MIR fields shift the emission to higher energy and narrow the spectrum. At a high-field regime, MIR fields not only relocate the excessive charges but also ionize the restored excitons, activating additional charge generation and subsequent trion formation that leads to Auger non-radiative processes. The charge left inside the QD again creates a local electric field and stark shifts the exciton emission, leading to a decreased PL intensity and redshifted spectra. In 14 ML QDs, we attribute the PL change under MIR fields to the removal of trapped excitons, as depicted in Fig. 5d. The long tail power-law decay lifetime shown in Fig. 4a suggests that a significant portion (30%) of exciton emission comes from trap-detrap assisted delayed emission. In thick-shell QDs, electrons display transitory carrier separation, which has been reported to be more likely than in thin shells 38, 45 due to a larger number of traps per QD and more substantial delocalization of the electron wavefunction into the shell. 38 Thus, the electron can be transiently separated from the hole wavefunction and then slowly returns to the recombination center due to the giant shell, followed by a radiative recombination. Those charge-separated electrons are localized outside the core and tend to be readily removed by MIR fields. With MIR excitations, the trapped electrons are depleted and can no longer return to the recombination center, supported by a faster PL lifetime (Fig. S8c), therefore removing the delayed emission and decreasing the overall PL intensity. The delayed emission is usually of lower energy, 38 the removal of which results in a blueshifted spectrum, consistent with our observations in Fig. 4d. In conclusion, we demonstrate that ultrafast MIR electric-field pulses can effectively remove excess charges responsible for the trion-mediated Auger recombination in blinking OFF states, thereby suppress the PL blinking and achieve near-unity quantum yield even at very high excitation flux. At high field strengths, MIR fields can further ionize excitons and cause additional charging. Experimental and simulation results support that MIR fields can manipulate carriers in QDs. Our all-optical approach enables almost non-blinking thin-shell QDs in a native environment, which is highly desirable for real-time single-molecule tracking of biological processes. 9 For exam-ple, fast clathrin-independent endocytosis 46, 47 involves transmembrane delivery and has been in need of a non-blinking fluorescence tag. The excitation wavelength can be readily extended to other spectral ranges due to the generality of the field-driven ionization mechanism, opening in vivo applications as one can choose a wavelength with minimal environmental absorption. The ultrafast nature of the pulse further renders cumulative heating effects negligible. Our experimental results should also motivate a hithertounexplored class of all-optical blinking control experiments with off-resonant field excitations in single emitters beyond CQDs, including nitrogen/silicon-vacancy color centers in diamond 48 and defects in two-dimensional transition metal dichalcogenides [49][50][51][52] . The potential realization of single quantum emitters free from interruptions can pave the way to potential quantum computing and quantum cryptography applications. Methods Single QD sample preparation To synthesize CdSe/CdS quantum dots, an established synthetic protocol from the literature was used. 53 To synthesize CdSe core quantum dots, 60 mg CdO, 280 mg octadecylphosphonic acid and 3 g trioctylphosphine oxide were combined in a 50 mL round bottom flask. The resulting reaction mixture was put under vacuum and heated to 150 • C to remove volatile substances. After 1 hour, the mixture was heated further to 320 • C under nitrogen flow to form a clear colorless solution, and 1.0 mL trioctylphosphine was added dropwise. The temperature was increased to 380 • C and the heating mantle was removed. 0.5 mL Se/trioctylphosphine (60 mg Se in 0.5 mL trioctylphosphine) was injected rapidly and the reaction mixture was cooled down to room temperature with air. The crude reaction mixture was washed with acetone and redispersed in toluene. This synthetic protocol resulted in highly monodisperse core-only quantum dots with first exciton absorption at 487 nm. To overcoat CdSe core quantum dots with CdS shells, continuous injection synthesis was used. 16 In a 100 mL round bottom flask, 100 nmol CdSe core QDs in toluene, 3 mL octadecene (ODE), 3 mL oleylamine and 3 mL oleic acid were mixed. The resulting mixture was degassed for 20 minutes at room temperature and for 40 minutes at 100 • C to remove volatile substances. Afterwards, the mixture was put under nitrogen flow, and the temperature was increased. When the temperature reached 200 • C, 0.08 M cadmium oleate-ODE and octanethiol-ODE (1.2 equivalents) were added at 2.5 mL/hour to form 7 monolayers of CdS shell. After the completion of shell precursor injection, the reaction mixture was annealed at 310 • C for 15 minutes. Quantum dots were washed with acetone and redispersed in hexane three times. Dense colloidal QDs were diluted by a factor around 10 6 , and then were drop cast or spin coated on a cover glass for single-dot measurements. Ultrafast mid-infrared pulse generation MIR pulses were generated in a 0.5 mm thick GaSe crystal by difference-frequency mixing of the signal and idler outputs of a high-energy optical parametric amplifier (OPA). The OPA was pumped with ∼ 35 fs pulses from a commercial Ti:Sapphire regenerative amplifier (800 nm central wavelength 800 nm, 12 mJ) and has an output power at signal and idler at 2.5 mJ and 2 mJ. The signal and idler beam sizes were shrinked by a reflective telescope by a factor of 1.5 to reach optimal difference-frequency generation (DFG) efficiency. The generated MIR pulses from the GaSe crystal were collected and expanded by a pair of 3-inch diameter parabolic mirrors before being tightly focused to the sample by a third parabola with a 3-inch diameter and 2-inch effective focal length. The MIR field strengths were controlled by a pair of wire-grid polarizers (Thorlabs WP25H-K). Photoluminescence measurements For PL imaging, a continuous-wave laser beam (center wavelength 405 nm, Toptica iBeam smart) was used to illuminate the sample with an up to 150 W/cm 2 excitation fluence. The PL image was collected by a 100× objective with a numerical aperture (NA) of 0.7 and imaged on an electron-multiplying charge-coupled device (EMCCD, from Andor iXon Ultra). We used a home-built confocal epifluorescence microscope to select individual QDs for single dot PL lifetime and spectral measurements. For PL spectrum characterization, successive measurements of the spectra were recorded in a spectrometer (Andor Shamrock spectrometer and Andor Newton 920). For PL lifetime measurements, a 420 ps resolution (instrumental response function) was achieved by exciting the sample with a pulsed laser source (center wavelength 405 nm, repetition rate 5 MHz, 100 ps pulse duration, Picoquant) at around 2 µJ/cm 2 and detecting the emission with an avalanche photodiode (ID Quantique, id100-vis). The lifetime was obtained by tagging the arrival time of each PL photon relative to the trigger photon using a timeto-digital converter (ID Quantique, id800-TDC). The duration of MIR irradiation was controlled by a mechanical shutter. All the above measurements were performed in an ambient environment. Solution biexciton quantum yield measurements Solution biexciton quantum yield measurements were completed by utilizing methods described by Beyler et al. 37 Samples were diluted in hexane by a factor of around 10 4 . One drop of 2 mM Cadmium oleate solution in hexane was added to reduce particle aggregation, producing an average occupation in the focal volume between 6 and 8. The Quantum tunneling calculation Electron dynamics were calculated via a one-body, one-dimensional model with step functions between the core, shell, and trap. The potential energy for free-electrons in the conduction band was given by, V (r) =                              0 0 ≤ |r| ≤ r c 0.32eV r c ≤ |r| ≤ r s 0.07eV r s ≤ |r| ≤ r t 4.4eV |r| ≥ r t ,(1) where r c , r s , and r t are the outer edges of the core, shell, and trap respectively. These suggesting that the energy difference should be on the order of k B T . 40, 59 Finally, a vacuum ionization energy of 4.4 eV was applied for distances beyond r t . 60 In order to simulate electron tunneling in the exciton state, the core-shell energy shift was increased to 0.52 eV to roughly account for the exciton binding energy. 42,43 The lowest 70 eigenstates of this stationary potential were calculated using the matrix numerov method. 61 The ground state and first 3 dipole-allowed states are shown in Supplementary Information (Fig. S10). Time evolution under the MIR field was done by integrating the time-dependent Schrödinger equation using the eigenstates of the stationary potential and a semiclassical representation for the laser field in the Coulomb gauge with the dipole approximation, c k (t) = − i E k c k (t) − 1 l F (t) ω kl ω ψ k |μ|ψ l c l (t).(2) Here c k is the coefficient of the kth eigenstate, ω kl is the frequency difference between the k and l states, and ω is the frequency of the field: 1818 cm −1 = 0.2254 eV. A sin 2 envelope was used to approximate the pulse waveform, F (t) = |F | sin 2 πt τ pulse sin(ωt),(3) where τ pulse = 150 fs is the approximate length of the pulse. The simulations were run for a total of 300 fs, where the field was only turned on for the first 150 fs. All observables of interests were extracted by averaging over the second 150 fs. Data availability. The data that support the findings of this study are available from the corresponding author upon request. This extra charge leads to the formation of another trion and subsequent non-radiative Auger decay. d, In 14 ML QDs, electrons can be trapped in the shell denoted as "trap", which can later thermally return to the recombination center and recombine radiatively, albeit slowly due to the giant shell. Under MIR irradiation, the trapped electron is pulled outside the dots (denoted as "trap*") and cannot detrap back to the recombination center to form an exciton, leading to a quenched PL and leaving a charged QD behind. Fig. 2e . 2eNo degradation in either equilibrium PL counts without MIR or enhanced PL counts with MIR is observed, suggesting the reversible nature of the PL enhancement.The enhancement of PL intensity only appears when a suitable MIR field strength is applied. With stronger MIR pulses (F/F max > 0.7) , we observe a suppression of the overall PL intensity. InFig. 2f, we show a drop in the PL intensity under an MIR exposure of 10 ms with a burst of 10 MIR pulses every two seconds at F/F max = 0.9, as shown inFig. 2f. Figure 2g 2gillustrates the normalized PL intensity change as a function of CWlaser optical power. The MIR-induced PL intensity change is summed over multiple (∼ 100) isolated dots and normalized to their equilibrium PL intensity. The PL percentage change increases as the optical power increases and saturates at ∼ 30 W/cm 2 . The extrapolated zero PL change at zero optical power indicates that the MIR itself does not produce any luminescence and only enhances PL when the dots are excited optically. Figure 3c and d show that blinking statistics are strongly modified by MIR excitation. By setting a threshold in a blinking trace separating ON and OFF states and counting their probability densities, a power-law distribution of ON-and OFF-times is obtained in is 5.4 ± 1.0%. For 10 ML, the BXQY is 10.7 ± 1.2% and it is 21.3 ± 1.0% for 14 ML(Fig. S1). The higher BXQY in thicker-shell QDs suggests that the non-radiative pathways are of lesser importance in thicker shell QDs. In the ensemble PL lifetime measurement of QDs solution as shown inFig. 4a, 14 ML QDs exhibit an extraordinarily long emission tail over tens of microseconds, while 8 ML QDs show a much shorter exponential decay. At longer decay times, the emission from 14 ML QDs deviates from an exponential and exhibits a power-law decay behavior (the fit is shown inFig. S7), suggesting a power-law trap-detrap assisted delayed emission. 38-41 In 14 ML QDs, the delayed emission contributes a significant portion (around 30%) of the total emission. Fig. 4c), contrary to the PL enhancement along with blueshift at moderate field strength in8 ML QDs. As a result, we observe a reverse effect of PL dynamics change in 14 ML QDs than in 8 ML QDs. The blinking OFF-time distribution is almost unaffected in 14 ML QDs, while the ON-time distribution is altered at 1 second and at its harmonics, contrary to the nearly unaffected ON-time distribution and altered OFF-time in 8 MLQDs. Additional data can be found in theSupplementary Information (Fig. S5 and S6). Fig. 5a . 5aTrap states are modeled as an additional stepwise potential well residing at the periphery of the shell. Tunneling probabilities from the core to the trap regions are calculated over the course of an ultrafast MIR pulse by numerically integrating the timedependent Schrödinger equation under the dipole approximation (see Methods). Using a physically motivated selection of parameters, the model reproduces the observed field strength threshold behavior in thin shell QDs, as shown inFig. 5b. The lower threshold can be understood through the interplay between the length of the pulse (τ pulse ) and the spatial overlap between the core-localized and trap-localized electron eigenstates. solution was wicked into a rectangular capillary (VitroCom, 0.100 × 0.200 mm i.d.) and sealed with capillary tube sealant to prevent solvent evaporation. Samples were excited with a pulsed laser source (532 nm, Picoquant) at a repetition rate of 1 MHz via a home-built confocal epifluorescence setup. Excitation was focused into the sample, and emission was collected using the same infinity-corrected water-immersion objective (Nikon, Plan Apo VC 60× WI, NA 1.2). A 535 nm long-pass filter (Chroma Technology Corp) was used to separate excitation and emission. Emission was further filtered spatially by a 1:1 telescoping 50 µm pinhole (Thorlabs) and spectrally by a 532 nm notch filter (Chroma Technology Corp) before being sent to three 50:50 non-polarizing beamsplitters (Thorlabs) as described by Shulenberger et al., 54 creating four equivalent intensity beams. Each beam was focused onto a single-photon counting detector (PerkinElmer, SPCM-AQR13) by a 10 cm achromatic lens (Thorlabs). Photon arrival times were recorded by a HydraHarp 400 (Picoquant) and analyzed using home-built software published at https://github.com/nanocluster/photons. parameters are provided in Supplementary Information. For both the 8 ML and 10 ML QDs, the core-shell conduction band energy shift of 0.32 eV was taken from several sources.42,[55][56][57][58] The core-trap offset of 0.07 eV = 3k B T was informed by the observation that electrons can stochastically tunnel between the core and trap at room temperature, 23 .Fig. 1 ( 0 2310Manzoni, C., Först, M., Ehrke, H. & Cavalleri, A. Single-shot detection and direct control of carrier phase drift of mid infrared pulses. Opt. Lett. 35, 757-759 (2010). 24. Nor Aliya Hamizi & Mohd Rafie Johan. Optical and FTIR studies of CdSe quantum dots. In 2010 3rd International Nanoelectronics Conference (INEC), 887-887 (2010). 25. Cherniavskaya, O., Chen, L., Islam, M. A. & Brus, L. Photoionization of individual CdSe/CdS core/shell nanocrystals on silicon with 2-nm oxide depends on surface band bending. Nano Letters 3, 497-501 (2003). 26. Javaux, C. et al. Thermal activation of non-radiative Auger recombination in charged colloidal nanocrystals. Nature Nanotechnology 8, 206-212 (2013). 27. Bharadwaj, P. & Novotny, L. Robustness of quantum dot power-law blinking. Nano Letters 11, 2137-2141 (2011). 28. Kuno, M., Fromm, D. P., Hamann, H. F., Gallagher, A. & Nesbitt, D. J. Nonexponential "blinking" kinetics of single CdSe quantum dots: A universal power law behavior. The Journal of Chemical Physics 112, 3117-3120 (2000). 29. Hasham, M. & Wilson, M. W. B. Sub-bandgap optical modulation of quantum dot blinking statistics. The Journal of Physical Chemistry Letters 11, 6404-6412 (2020). 30. Empedocles, S. A. & Bawendi, M. G. Influence of spectral diffusion on the line shapes of single CdSe nanocrystallite quantum dots. The Journal of Physical Chemistry B 103, 1826-1830 (1999). 31. Patton, B., Langbein, W. & Woggon, U. Trion, biexciton, and exciton dynamics in single selfassembled CdSe quantum dots. Phys. Rev. B 68, 125316 (2003). 32. Bracker, A. S. et al. Binding energies of positive and negative trions: From quantum wells to quantum dots. Phys. Rev. B 72, 035332 (2005). 33. Zhao, J., Chen, O., Strasfeld, D. B. & Bawendi, M. G. Biexciton quantum yield heterogeneities in single CdSe (CdS) core (shell) nanocrystals and its correlation to exciton blinking. Nano Letters 12, 4477-4483 (2012). 34. Park, Y.-S. et al. Near-unity quantum yields of biexciton emission from CdSe/CdS nanocrystals measured using single-particle spectroscopy. Phys. Rev. Lett. 106, 187401 (2011). 35. Park, Y.-S., Bae, W. K., Padilha, L. A., Pietryga, J. M. & Klimov, V. I. Effect of the core/shell interface on Auger recombination evaluated by single-quantum-dot spectroscopy. Nano Letters 14, 396-402 (2014). 36. Nair, G., Zhao, J. & Bawendi, M. G. Biexciton quantum yield of single semiconductor nanocrystals from photon statistics. Nano Letters 11, 1136-1140 (2011). 37. Beyler, A. P. et al. Sample-averaged biexciton quantum yield measured by solution-phase photon correlation. Nano Letters 14, 6792-6798 (2014). 38. Rabouw, F. T. et al. Delayed exciton emission and its relation to blinking in CdSe quantum dots. Nano Letters 15, 7718-7725 (2015). 39. Hinterding, S. O. M., Vonk, S. J. W., van Harten, E. J. & Rabouw, F. T. Dynamics of intermittent delayed emission in single CdSe/CdS quantum dots. The Journal of Physical Chemistry Letters 11, 4755-4761 (2020). 40. Jones, M., Lo, S. S. & Scholes, G. D. Quantitative modeling of the role of surface traps in CdSe/CdS/ZnS nanocrystal photoluminescence decay dynamics. Proceedings of the National Academy of Sciences 106, 3011-3016 (2009). 41. Sher, P. H. et al. Power law carrier dynamics in semiconductor nanocrystals at nanosecond timescales. Applied Physics Letters 92, 101111 (2008). Experimental scheme and single dot verification. a, Schematic illustration of a single CdSe/CdS quantum dot (QD) with mid-infrared pulse excitation and various PL probes, including PL intensity, PL spectrum, and (with pulsed rather than CW photoexcitation) time-resolved (TR) PL lifetime measurements. BPF: bandpass filter. b, In the conventional charging model for PL blinking, ON and OFF periods correspond to a neutral nanocrystal (exciton) and a charged nanocrystal (trion), respectively. During the OFF periods, ultrafast MIR fields can effectively remove the excess charge in trion and convert it to an exciton. c, The spectrum of MIR pulses centers at ∼ 5.5 µm with a bandwidth ∼ 0.5 µm. d, The second-order PL intensity correlation function measured for a single QD indicates that g 2 Fig. 2 Fig. 3 23MIR blinking control in single QDs. a, Representative PL blinking trace of a single QD without MIR excitations. b, The histogram indicates the distribution of intensities observed in the trace without MIR excitations. It shows bimodal distribution with ON and OFF events. c, Representative PL blinking trace of a single QD (the same QD of a) with MIR excitations. d, The histogram indicates the distribution of intensities observed in the trace with MIR excitations. It only has ON time fractions. e-i, MIR control of PL intensities averaged over multiple (∼100) isolated QDs. e, PL counts averaged over all dots as a function of time. MIR pulses with a field strength of F/F max = 0.4 are turned on for 200 ms (a burst of 200 pulses) every 5 s, with a zoom-in view of PL enhancement in a burst of MIR shown at right. f, Averaged PL counts as a function of time with a field strength of F/F max = 0.9 and MIR exposure of 10 ms (a burst of 10 pulses) every 2 s. g, Optical power dependence of the normalized MIR-induced PL intensity change with a burst of 80 MIR pulses every 1 s at F/F max = 0.6. h, Field dependence of MIRinduced PL intensity percentage change at a 20 W/cm 2 optical power and MIR exposure (a burst of 80 MIR pulses) every 1 s. i, MIR exposure (number of pulses burst every 1 s) dependence of MIR-induced PL intensity percentage change at an optical power of 20 W/cm 2 and a MIR field of F/F max = 0MIR fields alter PL lifetime, blinking statistics, and spectrum of single QDs. a-b, Fig. 4 Fig. 5 45MIR responses of QDs with different shell thicknesses. a, PL lifetime of ensemble QDs with 8 (red), 10 (pink), and 14 (blue) monolayer (ML) shell thickness. 14 ML QD exhibits power-law delayed emission with a long tail in decay times. b, Solution biexciton quantum yield (BXQY) data of 8 ML QDs. BXQY for 8 ML QD is 5.4 ± 1.0%. BXQY for 10 ML QDs is 10.7 ± 1.2% and 21.3 ± 1.0% for 14 ML QD (data in Fig. S1). c, Optical power and MIR field strength dependences in 10 and 14 ML QDs. d, Field dependence of the MIR-induced spectral shift in 14 ML QDs. MIR quenches PL in 14 ML QDs and introduces a spectral blueshift. Simulations and illustrations of the MIR-driven blinking control model. a, Schematic depiction of the field-driven quantum tunneling: an external electric field perturbs the energy levels of trion states in the core, shell, and the whole QD, thereby reducing the potential barriers and assisting tunneling away from the recombination center. The exciton ionization and tunneling are modeled similarly by considering the exciton binding energy. b, Quantum tunneling simulation of the field dependence of MIR-induced PL enhancement. (The experimental data is reproduced fromFig. 2h). c, For 8 and 10 ML QDs, blinking OFF states in equilibrium originate from trion emission with low quantum yield. With moderate-field MIR excitations, the excess charge is removed to the trap states on the shell surface or outside environment (denoted as "trap*"), thereby restoring excitonic emission. At a high-field regime, MIR pulses not only remove the excess charge but also ionize the exciton, leaving an additional charge inside the dot. . We record the PL intensity traces of single QDs under 405 nm CW laser excitation. Under no MIR field, we show a representative single QD PL blinking trace (Fig. 2a) and its histogram of PL intensity distributions (Fig. 2b). The histogram in Fig. 2b shows a bimodal PL intensity distribution, corresponding to blinking ON and OFF states. With MIR excitation at a suitable field strength (here F/F max = 0.6, F max ∼ 10 MV/cm), the blinking behavior from the same QD changes as shown in Fig.2c-d, reflected by the significant decrease in the time that the dot spends in the OFF state in the blinking trace as well as by the shift in the intensity histogram from bimodal to a largely unimodal ON state. Fluorescence lifetime-intensity distribution of a single QD with and without MIR excitations.Without MIR excitations, the single dot blinks between ON and OFF states, and the PL intensity strongly correlates with the lifetime. With MIR excitations, only the ON state with a long lifetime is present. c, OFF-time blinking statistics for isolated single QDs at equilibrium (blue) and during MIR excitations every one second and with a burst of 80 pulses in each period (red). 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[ "https://github.com/nanocluster/photons.Quantum", "https://github.com/nanocluster/photons." ]
[ "FAMILIES OF YOUNG FUNCTIONS AND LIMITS OF ORLICZ NORMS", "FAMILIES OF YOUNG FUNCTIONS AND LIMITS OF ORLICZ NORMS" ]
[ "Sullivan Macdonald ", "Scott Rodney " ]
[]
[]
Given a σ-finite measure space (X, µ), a Young function Φ, and a one-parameter family of Young functions {Ψ q }, we find necessary and sufficient conditions for the associated Orlicz norms of any function f ∈ L Φ (X, µ) to satisfyThe constant C is independent of f and depends only on the family {Ψ q }. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.
10.4153/s0008439523000449
[ "https://export.arxiv.org/pdf/2209.01149v3.pdf" ]
252,070,953
2209.01149
d2a57f3e1f6471ed805f929a2e20443e3cade9f8
FAMILIES OF YOUNG FUNCTIONS AND LIMITS OF ORLICZ NORMS 19 May 2023 Sullivan Macdonald Scott Rodney FAMILIES OF YOUNG FUNCTIONS AND LIMITS OF ORLICZ NORMS 19 May 2023 Given a σ-finite measure space (X, µ), a Young function Φ, and a one-parameter family of Young functions {Ψ q }, we find necessary and sufficient conditions for the associated Orlicz norms of any function f ∈ L Φ (X, µ) to satisfyThe constant C is independent of f and depends only on the family {Ψ q }. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail. Introduction It is a well-known result in classical analysis (see e.g. [5] and [6]) that if (X, µ) is a measure space and if f ∈ L r (X, µ) for some r ≥ 1, then (1.1) lim p→∞ f L p (X,µ) = f L ∞ (X,µ) . The authors of [3] investigated a similar limiting property of Orlicz norms associated with a one-parameter family of Young functions {Ψ q } defined by setting Ψ q (t) = t p log(e − 1 + t) q , where p ≥ 1 is fixed and q can be any positive real number. They showed that if f belongs to the Orlicz space L Ψq 0 (X, µ) for some q 0 > 0 then (1.2) lim q→∞ f L Ψq (X,µ) = f L ∞ (X,µ) . Here, f L Ψq (X,µ) denotes the Luxembourg norm of f with respect to Ψ q and it is given by f L Ψq (X,µ) = inf λ > 0 : X Ψ q |f | λ dµ ≤ 1 . Modifying the proof of [3, Thm. 1], we show that (1.2) holds for any family of Young functions {Ψ q } that satisfies Condition 1.1, which we call δ-admissibility. Like the authors of [3], our efforts were motivated by an application in partial differential equations, where we sought to employ a Moser iterative scheme in Orlicz spaces to study regularity of weak solutions to Poisson's equation. Ultimately those results were achieved using alternative techniques in [2]. Nevertheless, our main result Theorem 1.2 may be useful in the study of related problems. Furthermore, it illustrates a surprising relationship between Orlicz spaces defined with respect to (X, µ), and the space L ∞ (X, µ) of essentially bounded functions on X. Throughout this work we assume that (X, µ) is a positive measure space with µ(X) > 0. To state our main result concisely, we begin by defining δ-admissibility. Condition 1.1. Given δ > 0, a one-parameter family of Young functions {Ψ q } is said to be δ-admissible if lim q→∞ Ψ q (t) = ∞ for t > δ, and for 0 < t < δ one of the following holds: (i) If µ(X) = ∞ then lim q→∞ Ψ q (t) = 0, (ii) If µ(X) < ∞ then lim q→∞ Ψ q (t) < µ(X) −1 . Perhaps the simplest δ-admissible family is the 1-admissible collection obtained by taking Ψ q (t) = t q for q ≥ 1, and more examples of δ-admissible families are discussed in Section 3. Now we state our main result concerning these families of Young functions. Theorem 1.2. Let (X, µ) be a σ-finite measure space and let {Ψ q } be a one-parameter family of Young functions. Let Φ be another Young function such that for any k > 0, the composition (1.3) t Ψ −1 q (Φ(t) ) is non-decreasing on the interval [0, k] whenever q is sufficiently large. Then the identity (1.4) lim q→∞ f L Ψq (X,µ) = 1 δ f L ∞ (X,µ) holds for every f ∈ L Φ (X, µ) if and only if {Ψ q } is a δ-admissible family for some 0 < δ < ∞. Remark 1.3. (1) We note that σ-finiteness of (X, µ) is only required for the forward implication. Further, in many cases (1.3) is non-decreasing on all of [0, ∞) for large q. For example, if Ψ q (t) = t q and Φ(t) = t r for r ≥ 1, then (1.3) is non-decreasing on [0, ∞) for q ≥ r. (2) The distinction between cases (i) and (ii) in Condition 1.1 is needed to prove necessity of δ-admissibility for identity (1.4), but it is not needed for sufficiency. Indeed, if µ(X) < ∞ and 0 < t < µ(X) −1 then one cannot select sets of large enough µmeasure with which to compute the limit of Ψ −1 q (t) as q → ∞ using (1.4). We also note that Condition 1.1 is weaker than the closely related sufficient condition where lim q→∞ Ψ q (t) = 0 in (i) and (ii). Usually it is difficult to find the inverse of a given Young function in closed form. To make (1.3) easier to verify for given examples of Φ and Ψ q , notice that (1.3) is non-decreasing on [0, k] exactly when the following function is non-decreasing on [0, Φ(k)]: Φ −1 (t) Ψ −1 q (t) . So, to check that the conditions of Theorem 1.2 hold for a Young function Φ and a given family {Ψ q }, it suffices to compute the inverse of Ψ q for each q, and to know either Φ or Φ −1 . The theorem stated above implies the main result of [3], and we include a streamlined proof of this special case in Section 5. Furthermore, we observe that if Φ(t) = t then (1.3) is non-decreasing whenever Ψ q is a Young function, and from Theorem 1.2 we obtain the following result that involves no growth condition. Corollary 1.4. If (X, µ) is a σ-finite measure space and {Ψ q } is δ-admissible for 0 < δ < ∞, then (1.4) holds for every f ∈ L 1 (X, µ). The remainder of this paper is organized as follows. In Section 2 we establish preliminary results for Young functions and Orlicz spaces, and in Section 3 we discuss several examples of δ-admissible families to show how Theorem 1.2 can be applied. Section 4 is then devoted to the proof of our main result, and Section 5 examines the special case of log-bumps. We conclude with Section 6, where we demonstrate that identity (1.4) can fail if the family {Ψ q } is not δ-admissible for any finite δ > 0. Preliminaries This section contains a brief introduction to Young functions and Orlicz spaces. Our discussion is largely expository, and for a complete treatment the reader is referred to [4]. To begin, a non-negative function ψ : [0, ∞) → [0, ∞) is said to be a density if it is right continuous, non-decreasing, ψ(t) = 0 exactly when t = 0, and ψ(t) → ∞ as t → ∞. Given a density ψ, the associated function Ψ : [0, ∞) → [0, ∞) defined by Ψ(t) = t 0 ψ(s)ds is called a Young function. For our purposes, the important functional properties of Ψ are that it is continuous, strictly increasing, and convex on (0, ∞). Moreover, it is clear that Ψ(0) = 0 and that Ψ(t) → ∞ as t → ∞. Since the function Ψ(t) = t has constant density it is not a Young function according to the definition above, however for our purposes it can often be treated as one. Given a Young function Ψ and a measure space (X, µ), the Orlicz space L Ψ (X, µ) is defined as the collection of µ-measurable functions f : X → R for which the Luxembourg norm f L Ψ (X,µ) = inf λ > 0 : X Ψ |f | λ dµ ≤ 1 is finite. Equipped with this norm L Ψ (X, µ) is a Banach space; see [6]. The Orlicz classes generalize the classical Lebesgue spaces, and it is easy to verify that · L p (X,µ) = · L Ψ (X,µ) when Ψ(t) = t p for p ≥ 1. Orlicz spaces can also provide a finer scale of norms then L p (X, µ) in the following sense: if µ(X) < ∞ and Ψ q (t) = t p (1 + log(1 + t)) q for p ≥ 1 and q > 0, then for any ε > 0 we have L p+ε (X, µ) L Ψq (X, µ) L p (X, µ). These inclusions can be verified using Hölder's inequality, and their strictness follows from the examples constructed in [1]. In the sections that follow we employ several properties of the Luxembourg norm which we now establish. The first is a version of Chebyshev's inequality on the Orlicz scale. Henceforth we use the notation χ S to denote the indicator function of a set S ⊆ X. Theorem 2.1 (Chebyshev's Inequality). For any α ≥ 0, a µ-measurable function f : X → R, and a Young function Ψ, the following inequality holds: (2.1) αΨ −1 (µ({x ∈ X : |f (x)| ≥ α}) −1 ) −1 ≤ f L Ψ (X,µ) . Proof. First we establish a simpler form of (2.1) in the norm of L 1 (X, µ) using a standard argument. Fix α > 0 and define f α = αχ {|f |≥α} so that f α ≤ |f | holds pointwise and µ({|f | ≥ α}) = µ({x ∈ X : |f (x)| ≥ α}) = 1 α X f α (x)dµ ≤ 1 α X |f (x)|dµ = 1 α f L 1 (X,µ) . Next we replace α with β = α/ f Ψ . Using that Ψ is strictly increasing, it follows from the inequality above that µ({|f | ≥ β f Ψ }) = µ Ψ |f | f L Ψ (X,µ) ≥ Ψ(β) ≤ 1 Ψ(β) X Ψ |f | f L Ψ (X,µ) dµ. It is a well-known property of the Luxembourg norm, established in many standard references (e.g. [4]), that X Ψ(|f |/ f L Ψ (X,µ) )dµ ≤ 1. It follows that µ({|f | ≥ β f L Ψ (X,µ) }) ≤ Ψ(β) −1 , and since Ψ −1 is increasing this implies that Ψ −1 (µ({x ∈ X : |f | ≥ α}) −1 ) −1 ≤ β −1 . Writing β in terms of α and f gives (2.1). Equipped with this result, we can compute the Orlicz norms of indicator functions exactly. Corollary 2.2. Let S be a µ-measurable subset of X. Then χ S L Ψ (X,µ) = Ψ −1 (µ(S) −1 ) −1 . Proof. The estimate Ψ −1 (µ(S) −1 ) −1 ≤ χ S Ψ follows at once from Chebyshev's inequality. For the reverse inequality, observe that X Ψ χ S Ψ −1 (µ(S) −1 ) −1 dµ = S Ψ 1 Ψ −1 (µ(S) −1 ) −1 dµ = S 1 µ(S) dµ = 1. By the definition of the Luxembourg norm, this implies that χ S Ψ ≤ Ψ −1 (µ(S) −1 ) −1 . In this paper we work with limits of Orlicz norms that are defined by a one-parameter family of Young functions. Subject to appropriate growth conditions, these families have useful pointwise properties which we will exploit in the sections to follow. Our main condition on these families is the following generalization of Condition 1.1. Condition 2.3. Given α ≥ 0 and β ≥ α, a family of Young functions {Ψ q } is said to be (α, β)-admissible if lim q→∞ Ψ q (t) = ∞ for t > β, and for 0 < t < α one of the following holds: (i) If µ(X) = ∞ then lim q→∞ Ψ q (t) = 0, (ii) If µ(X) < ∞ then lim q→∞ Ψ q (t) < µ(X) −1 . Proposition 2.4. Let {Ψ q } be an (α, β)-admissible family. If µ(X) = ∞ and t > 0, or if µ(X) < ∞ and t ≥ µ(X) −1 , then (2.2) α ≤ lim inf q→∞ Ψ −1 q (t) ≤ lim sup q→∞ Ψ −1 q (t) ≤ β. Proof. If α = 0 then the first inequality in (2.2) holds trivially, so we assume that α > 0. Fix t > 0 if µ(X) = ∞ and t ≥ µ(X) −1 if µ(X) < ∞, and assume toward a contradiction that there exists η > 0 such that lim inf q→∞ Ψ −1 q (t) ≤ α − η. Given any 0 < ε < η then, there exists an increasing sequence {q j } such that q j → ∞ and j → ∞ and Ψ −1 q j (t) < α − ε for each j. Since each Ψ q is strictly increasing for all q > 0, we find that t < Ψ q j (α − ε) for each j. Taking the limit as j → ∞, we see from Condition 2.3 that 0 < t ≤ 0 if µ(X) = ∞, and µ(X) −1 ≤ t < µ(X) −1 if µ(X) < ∞. In any case this is a contradiction, meaning that α − η < lim inf q→∞ Ψ −1 q (t). Since η > 0 was arbitrary, we get the first inequality in (2.2). The estimates for the limit supremum in (2.2) follow in an identical fashion. If α = β = δ then Condition 2.3 is the same as δ-admissibility, and Proposition 2.4 gives lim q→∞ Ψ −1 q (t) = δ for each t > 0 when µ(X) = ∞, and for each t ≥ µ(X) −1 when 0 < µ(X) < ∞. In fact, the limit identity above is equivalent to δ-admissibility. Proposition 2.5. A family of Young functions {Ψ q } is δ-admissible if and only if lim q→∞ Ψ −1 q (t) = δ (2.3) holds for all t > 0 if µ(X) = ∞, and for all t ≥ µ(X) −1 if 0 < µ(X) < ∞. Proof. Proposition 2.4 gives the forward implication, leaving us to prove that if (2.3) holds for t in the appropriate range then {Ψ q } is δ-admissible. Regardless of whether µ(X) is finite or infinite, if Ψ q (t) ≤ M for some M > 0 and large q, then t ≤ Ψ −1 q (M) and (2.3) gives t ≤ lim q→∞ Ψ −1 q (M) = δ. Since M was arbitrary, it follows in the contrapositive that if t > δ, then lim q→∞ Ψ q (t) = ∞. Assume now that 0 < µ(X) < ∞ and suppose that (2.3) holds for all t ≥ µ(X) −1 > 0. If lim q→∞ Ψ q (t) ≥ µ(X) −1 , then given 0 < ε < 1 we have Ψ q (t) > µ(X) −1 (1 − ε) whenever q is sufficiently large. By convexity of Ψ q it follows that µ(X) −1 ≤ Ψ q (t) 1 − ε ≤ Ψ q t 1 − ε ,and so t ≥ (1 − ε)Ψ −1 q (µ(X) −1 ) whenever q is sufficiently large. Using (2.3) we take the limit to find t ≥ (1 − ε) lim q→∞ Ψ −1 q (µ(X) −1 ) = (1 − ε)δ. Since ε > 0 was arbitrary, we conclude that t ≥ δ. Thus, lim q→∞ Ψ q (t) < µ(X) −1 when t < δ. On the other hand, if µ(X) = ∞ and (2.3) is satisfied for t > 0, and if Ψ q (t) ≥ ε for some ε > 0 and for all large q, then t ≥ lim q→∞ Ψ −1 q (ε) = δ. Thus, if t < δ then lim q→∞ Ψ q (t) ≤ ε, and since ε > 0 was arbitrary we have lim q→∞ Ψ q (t) = 0. Examples There are many examples of δ-admissible families of Young functions, and moreover they are often easy to construct. In this section we showcase some families to illustrate the utility of our main result, Theorem 1.2. Example 3.1. If Ψ q (t) = t q and Φ(t) = t r for some r ≥ 1, then (1.3) is non-decreasing for t > 0 whenever q ≥ r. Moreover the family {Ψ q } is 1-admissible, and an application of Theorem 1.2 gives the well-known identity (1.1). Example 3.2. If Φ(t) = t r for r ≥ 1 and Ψ q (t) = t p log(e − 1 + t) q for fixed p ≥ 1, then Φ −1 (Ψ q (t)) t = t p r log(e − 1 + t) q r t fails the growth condition of Theorem 1.2 when r > p, and satisfies it when r ≤ p, regardless of the value of q > 0. In the latter case, Theorem 1.2 recovers identity (1. 2) for f ∈ L r (X, µ). Example 3.3. Given N ∈ N and p ≥ 1, consider the family of N th -order iterated log-bumps Ψ q (t) = t p log · · · log N times (c + t) q , where c is chosen independent of q so that Ψ q (1) = 1. This family is 1-admissible, and a straightforward adaptation of the argument in Section 5 shows that (1.3) is non-decreasing on any interval of the form [0, k] for f ∈ L Ψq 0 (X, µ) whenever q > q 0 > 0 is sufficiently large. Thus, Theorem 1.2 applies to the Orlicz norms characterized by the N th order iterated log-bumps above, giving lim q→∞ f L Ψq (X,µ) = f L ∞ (X,µ) whenever f ∈ L Ψq 0 (X, µ) for some q 0 > 0. We emphasize that in this example, the convergence of the f L Ψq (X,µ) norms to f L ∞ (X,µ) is independent of p. Thus for identity (1.4) to hold when {Ψ q } is a family of iterated log-bumps, it is not necessary to assume that f ∈ L p+ε (X, µ) for any ε > 0. Example 3.4. For any fixed Young function Φ, one can obtain a 1-admissible family using the structure of N th -order iterated log-bumps by defining Ψ q (t) = Φ(t) log · · · log N times (c + t) q , where c is chosen so that Ψ q (1) = Φ(1) for all q. Indeed, the iterated log-bumps of the form Ψ q (t) = t N j=1 log · · · log j times (c j + t) p log · · · log N times (c N + t) q (3.1) are of this type for p ≥ 1, provided that the constants c 1 , . . . , c N are chosen so that the value of Ψ q (δ) is independent of q for some δ > 0. Once again Theorem 1.2 applies to this family, allowing us to reproduce the limit in [1, Theorem 6.1]. This result is proved in [1] by means of a modification of the techniques of [3], which rely on the properties of iterated logarithms. As in the last example, a similar calculation to that employed in Section 5 shows that if Φ = Ψ q 0 for Ψ q as in (3.1), then (1.3) is non-decreasing on any bounded interval of the form [0, k] for k > 0 whenever q > q 0 is sufficiently large. Once again, we conclude that the L Ψq (X, µ) norms of f converge to f L ∞ (X,µ) , provided f ∈ L Ψq 0 (X, µ) for some q 0 > 0. There are many more families for which δ-admissibility can be established, and the interested reader is encouraged to construct their own examples. Proof of Theorem 1.2 Every Orlicz norm used in this section is defined with respect to a fixed measure space (X, µ), so we will always write · L Ψ (X,µ) = · Ψ and · L ∞ (X,µ) = · ∞ . Fix δ > 0 and suppose that {Ψ q } is a δ-admissible family of Young functions. Identity (1.4) is trivial if f ≡ 0, and we will treat the case of unbounded f separately at the end. Thus, we begin by assuming that 0 < f ∞ < ∞, and we note that it is enough to prove (1.4) when f Φ = 1. Since (1.3) is non-decreasing on [0, f ∞ ] by hypothesis for q sufficiently large, we have (4.1) f Ψq = Ψ −1 q (Φ(|f |)) |f | Ψ −1 q (Φ(|f |)) Ψq ≤ Ψ −1 q (Φ(|f |)) Ψq f ∞ Ψ −1 q (Φ( f ∞ )) . Additionally, we see that Ψ −1 q (Φ(|f |)) Ψq ≤ 1 since by definition of the Luxembourg norm, X Ψ q Ψ −1 q (Φ(|f (x)|)) dµ = X Φ(|f |)dµ ≤ 1. Moreover, in the case 0 < µ(X) < ∞ we have that Φ( f ∞ ) ≥ µ(X) −1 X Φ(|f |)dµ = µ(X) −1 . Equality holds above since f Φ = 1 and since f is bounded by assumption (see e.g. [7, Eq. (3.13)]). Using these estimates, we find from Proposition 2.5 that lim sup q→∞ f Ψq ≤ f ∞ lim q→∞ Ψ −1 q (Φ( f ∞ )) −1 = f ∞ δ . Next suppose that 0 < ε < f ∞ and let S = {x ∈ Ω : |f (x)| ≥ f ∞ − ε}. From the definition of the essential supremum and Chebyshev's inequality it follows at once that 0 < µ(S) ≤ Φ(( f ∞ −ε) −1 ), meaning that µ(S) is finite and nonzero. Moreover, Chebyshev's inequality with α = f ∞ − ε also shows that ( f ∞ − ε)Ψ −1 q (µ(S) −1 ) −1 ≤ f Ψq . From Proposition 2.5 it follows that Ψ −1 q (µ(S) −1 ) −1 → δ −1 as q → ∞, since µ(S) −1 ≥ µ(X) −1 when 0 < µ(X) < ∞. As a result we find that f ∞ − ε δ ≤ lim inf q→∞ f Ψq . Since ε > 0 was arbitrary, this gives δ −1 f ∞ ≤ lim inf q→∞ f Ψq , proving that (1.4) holds. In the case where f ∞ = ∞, choose N > 1 and set f N = min{|f |, N} so that f N ∞ = N. Applying our work above, we see that lim inf q→∞ f Ψq ≥ lim inf q→∞ f N Ψq ≥ δ −1 f N ∞ = δ −1 N. Since N may be chosen arbitrarily large, we find lim inf q→∞ f Ψq = ∞ as required. Now we show that if (1.4) holds for all f ∈ L Φ (X, µ), then the family {Ψ q } is δ-admissible. Specifically, we utilize the characterization of δ-admissible families given by Proposition 2.5 to recognize that it is enough to show that lim q→∞ Ψ −1 q (t) = δ for every t > 0 when µ(X) = ∞, and for t ≥ µ(X) −1 when µ(X) is finite and positive. Suppose first that µ(X) = ∞. Given t > 0, we use that (X, µ) is σ-finite to select sets S 1 , S 2 ⊂ X of sufficiently large measure so that with t j = µ(S j ) −1 we have 0 < t 2 < t 1 < t. Using (1.4) with f j = χ S j ∈ L Φ (X, µ) we find from Corollary 2.2 that lim q→∞ Ψ −1 q (t j ) = lim q→∞ f j −1 L Ψq (X,µ) = δ f j −1 L ∞ (X,µ) = δ. Since we may choose λ ∈ (0, 1) so that λt 2 + (1 − λ)t = t 1 , the concavity of Ψ −1 q gives λΨ −1 q (t 2 ) + (1 − λ)Ψ −1 q (t) ≤ Ψ −1 q (t 1 ) . Letting q → ∞ we find after taking a limit supremum and rearranging that lim sup q→∞ Ψ −1 q (t) ≤ δ. More, since Ψ −1 q is increasing, δ ≤ lim inf q→∞ Ψ −1 q (t). Thus, lim q→∞ Ψ −1 q (t) = δ for every t > 0. In the case that 0 < µ(X) < ∞, if t > µ(X) −1 we may proceed exactly as above, see Remark 1.3. If t = µ(X) −1 , the required estimate follows at once by applying (1.4) to f = χ X ∈ L Φ (X, µ). In any case we have established that lim q→∞ Ψ −1 q (t) = δ for t ≥ µ(X) −1 when 0 < µ(X) < ∞, and for t > 0 when µ(X) = ∞. It follows from Proposition 2.5 that {Ψ q } is δ-admissible. (4.2) 1 β f ∞ ≤ lim inf q→∞ f Ψq ≤ lim sup q→∞ f Ψq ≤ 1 α f ∞ . It may be the case that the limit of the norms f Ψq does not exist for a family {Ψ q } which is (α, β)-admissible, as we show with the following example. Let Ψ q (t) =      1 2 t q 0 ≤ t ≤ 1 2 , 1 2 (t q + (2t − 1) 2+sin q ) 1 2 < t < 1, 1 2 (t q + (2t − 1) 3 ) t ≥ 1, so that {Ψ q } is ( 1 2 , 1)-admissible. If f = χ S for S with 2 < µ(S) < ∞ then f ∞ = 1 and 1 ≤ lim inf q→∞ f Ψq ≤ lim sup q→∞ f Ψq ≤ 2. On the other hand, we can show that lim q→∞ Ψ −1 q (t) does not exist for t ∈ (0, 1 2 ), meaning that (4.3) lim q→∞ f Ψq = lim q→∞ Ψ −1 q (µ(S) −1 ) −1 does not exist. To see this, assume toward a contradiction that there is a t ∈ (0, 1 2 ) such that lim q→∞ Ψ −1 q (t) = d. First we show that d ∈ ( 1 2 , 1). Given ε > 0 we have for all large q that d−ε < Ψ −1 q (t) < d+ε and Ψ q (d − ε) < t < Ψ q (d + ε). If d < 1 2 then we can choose ε small enough that d + ε ≤ 1 2 , meaning that Ψ q (d + ε) → 0 as q → ∞. Since t < Ψ q (d + ε) for all large q this gives a contradiction for large q. Likewise if d > 1 then we can choose ε such that d − ε ≥ 1, meaning that Ψ q (d − ε) → ∞ as q → ∞. Since Ψ q (d − ε) < t < 1 2 this gives another contradiction, and it follows that 1 2 ≤ d ≤ 1. If d = 1 2 then we have 1 2 < d+ε < 1 when ε is small and Ψ q (d+ε) = 1 2 (( 1 2 +ε) q +(2ε) 2+sin q ). Choosing ε < 1 2 so small that (2ε) 2+sin q ≤ 2ε ≤ t and then taking q so large that ( 1 2 + ε) q ≤ t, we get Ψ q (d + ε) ≤ t, a contradiction. Similarly, if d = 1 then 1 2 < d − ε < 1 for small ε and Ψ q (d − ε) = (1 − ε) q + (1 − 2ε) 2+sin q 2 ≥ (1 − 2ε) 2+sin q 2 ≥ (1 − 2ε) 3 2 . Since t < 1 2 , we can choose ε sufficiently small that (1−2ε) 3 2 ≥ t, another contradiction. It follows that 1 2 < d < 1 and 1 2 < d − ε < d + ε < 1 for small ε. Consequently Ψ q (d−ε) = (d − ε) q + (2(d − ε) − 1) 2+sin q 2 and Ψ q (d+ε) = (d + ε) q + (2(d + ε) − 1) 2+sin q 2 . Taking q = π 2 + kπ for odd k ∈ Z gives t > Ψ q (d − ε) = 1 2 ((d − ε) q + 2(d − ε) − 1), while using even k gives t < Ψ q (d + ε) = 1 2 ((d + ε) q + (2(d + ε) − 1) 3 ). For ε small and q large we show that this is impossible. By convexity of the map t → t q for q ≥ 1 we have (d + ε) q ≤ 1 2 (d − ε) q + 1 2 (d + 3ε) q < (d − ε) q + (d + 3ε) q , and moreover, a straightforward calculation shows that (2(d + ε) − 1) 3 = 4d(2d − 1)(d − 1) + 4ε(2 + 6d 2 − 6d + 6dε + 2ε 2 − 3ε) + (2d − 2ε − 1). Note that 4d(2d − 1)(d − 1) < 0 when 1 2 < d < 1. From the preceding estimates we get that (d+ε) q +(2(d+ε)−1) 3 < (d−ε) q +(d+3ε) q +4d(2d−1)(d−1)+4ε(2+2ε 2 +3ε)+(2d−2ε−1). Taking ε small and q large, we can ensure that (d+3ε) q +4d(2d−1)(d−1)+4ε(2+2ε 2 +3ε) ≤ 0, and doing this gives the following contradiction: 2t < (d + ε) q + (2(d + ε) − 1) 3 ≤ (d − ε) q + 2(d − ε) − 1) < 2t. Thus, the limit in (4.3) may not exist when the δ-admissibility condition fails. Log-Bump Orlicz Norms Here we show that Theorem 1.2 implies [3, Theorem 1], which states that (1.4) holds with δ = 1 for a specific family of log-bump Young functions. Given p ≥ 1, the log-bumps are of the form Ψ q (t) = t p log(e − 1 + t) q for q > 0, and the collection of all these bumps is a 1-admissible family. Thus, [3, Theorem 1] follows from Theorem 1.2 once we demonstrate that for k > 0 the function (1.3) is non-decreasing on [0, k] when Φ(t) = Ψ q 0 (t) for some q 0 > 0 and when q is large. To do this, we first assume that q > q 0 and for t > 0 we define F (t) = t Ψ −1 q (Ψ q 0 (t)) , so that F satisfies the equation Ψ q 0 (t) = Ψ q (tF (t) −1 ). Recalling the form of Ψ q we see that F (t) p log(e − 1 + t) q 0 = log e − 1 + t F (t) q (5.1) for t > 0. It follows from the definition above that F is continuous on (0, ∞), and to extend F continuously to zero we observe that log(e − 1) q 0 lim t→0 + F (t) p = log e − 1 + lim t→0 + Ψ −1 q (Ψ q 0 (t)) q = log(e − 1) q . Defining F (0) = lim t→0 + F (t) = log(e − 1) q−q 0 p thus ensures that F is continuous on [0, ∞). Lemma 5.1. If q > q 0 then F (t) > F (0) for every t > 0. Proof. Observe that if q > q 0 then Ψ q (t) ≥ Ψ q 0 (t) when t ≥ 1, while Ψ q (t) < Ψ q 0 (t) when 0 < t < 1. In the case t ≥ 1, we use that Ψ −1 q is strictly increasing to see that t ≥ Ψ −1 q (Ψ q 0 (t)), which implies that F (t) ≥ 1 > F (0). Likewise, 0 < t < 1 gives F (t) < 1 and by (5.1) we find log(e − 1 + t) q < log e − 1 + t F (t) q = F (t) p log(e − 1 + t) q 0 . Rearranging, we see that F (t) > log(e − 1 + t) Since F (0) < c, it is easy to see that T c (0) > 0. Furthermore, a straightforward computation shows that T ′′ c (t) < 0 if and only if (5.2) q 0 c p q log (e − 1 + t) q 0 q < q log (e − 1 + t) + q − q 0 . For large q, this is achieved uniformly in c ∈ [0, M]. To see why, choose q ≥ q 0 so that M p ≤ q q 0 q log(e − 1) q−q 0 . Then c p q ≤ q q 0 log(e − 1) 1− q 0 q ≤ q q 0 log(e − 1 + 1) 1− q 0 q for each t ≥ 0, and this shows that (5.2) holds for each t ≥ 0, and we see T c (t) is strictly concave on (0, ∞). For q large as above, we find that T c (0) > 0 and T c is a continuous and strictly concave function on (0, ∞). Thus, T c (t) has a unique fixed point in [0, ∞) and so it is t 1 . This gives F injective on I since F (t 2 ) = c = F (t 1 ) shows that t 1 and t 2 are fixed points of T c (t). Since F is a continuous, injective function on I, the Intermediate Value Theorem shows that F is strictly monotone on I. Lastly, since Lemma 5.1 shows that F (0) < F (t) for t ∈ I, we conclude that F is strictly increasing on I when q is sufficiently large. Thus, with the hypotheses of Theorem 1.2 verified, we have reproved [3, Theorem 1]. Necessity of Admissibility Conditions Finally, we show that Conditions 1.1 and 2.3 are necessary for the norm limit to be related to the essential supremum of a function. This means that our admissibility conditions cannot be weakened in Theorem 1.2 or Remark 4.1. Theorem 6.1. Let {Ψ q } be a family of Young functions, let Φ be a Young function for which (1.3) is non-decreasing, and assume that there exists f ∈ L Φ (X, µ) ∩ L ∞ (X, µ) such that (6.1) 0 < lim inf q→∞ f Ψq ≤ lim sup q→∞ f Ψq < ∞. Then {Ψ q } is (α, β)-admissible for some β > 0 and α ≥ 0. Proof. First assume to the contrary that Ψ q (t) → ∞ as q → ∞ for each fixed t > 0. Arguing as in the proof of Proposition 2.4, we conclude that Ψ −1 q (t) → 0 as q → ∞ for each t > 0. Since the limit infimum in (6.1) is non-zero we have 0 < f ∞ , and as in the proof of Theorem 1.2 we can choose ε < f ∞ to see that ( f ∞ − ε) lim inf q→∞ Ψ −1 q (µ({x ∈ Ω : |f (x)| ≥ f ∞ − ε}) −1 ) −1 ≤ lim inf q→∞ f Ψq . By definition of the essential supremum, µ({x ∈ Ω : |f (x)| ≥ f ∞ − ε}) > 0, meaning that the limit on the left-hand side above diverges and f Ψq → ∞ as q → ∞, contradicting the right-hand limit of (6.1). Thus, there is a t 0 > 0 for which lim sup q→∞ Ψ q (t 0 ) < ∞, and therefore lim sup q→∞ Ψ q (t) < ∞ holds for every 0 ≤ t ≤ t 0 since each Young function is strictly increasing. This means that (α, β)-admissibility holds for some α ≥ 0 and β > 0. Similarly, suppose that Ψ q (t) → 0 as t → ∞ for each fixed t > 0, so that Ψ −1 q (t) → ∞ by the argument of Proposition 2.4. Arguing as in Section 4 we have lim sup q→∞ f Ψq ≤ f ∞ lim sup q→∞ Ψ −1 q (Φ( f ∞ )) −1 = 0, and once again this contradicts (6.1). Since each Ψ q is strictly increasing we conclude that lim inf q→∞ Ψ q (t) > 0 for large t > 0. Thus there exists α ≤ β < ∞ satisfying Condition 2.3. In the case of δ-admissibility, where α = β = δ, the argument above shows that Theorem 1.2 fails when δ is not both positive and finite. To illustrate, if f = χ [0,1] and Ψ q (t) = t p log(e + t) q then f L Ψq (R,dx) = Ψ −1 q (1) −1 , but {Ψ q } is not δ-admissible for any δ > 0. A straightforward calculation shows that lim q→∞ f L Ψq (R,dx) ≥ lim inf q→∞ f L Ψq (R,dx) = lim inf q→∞ Ψ −1 q (1) −1 = ∞ > 1 = f L ∞ (R,dx) . Thus, if Condition 1.1 fails then the induced Orlicz norms may not converge to f L ∞ (X,µ) . Date: May 2023. S. Rodney was supported by the NSERC Discovery Grant Program. S.F. MacDonald was supported by the NSERC USRA Program and the Dept. of Mathematics & Statistics at McMaster University. Remark 4. 1 . 1If we use the more general Condition 2.3 in place of Condition 1.1 in Theorem 1.2, simple modifications of the proof above show that one has the estimates k > 0, set I = [0, k] and let M = sup I F . We show that F is injective on I when q is large enough. To this end, fix t 1 ∈ I, set c = F (t 1 ), and note that F (t 1 ) = F (0) if and only if t 1 = 0. On the other hand, if t 1 > 0 then c ∈ (F (0), M] by Lemma 5.1. More, from (5.1) we see that t 1 is a fixed point of the map T c : [0, ∞) → R defined by T c (t) = c exp c Corollary 6 . 2 . 62Let {Ψ q } be a family of Young functions such that for every t > If f satisfies the remaining hypotheses of Theorem 1.2 then regardless of the value of f ∞ , lim q→∞ f L Ψq (X,µ) = ∞ (resp. lim q→∞ f L Ψq (X,µ) = 0). On The Limit of Orlicz Norms, Cape Breton University Undergraduate Thesis. A Mailhot, A. Mailhot, On The Limit of Orlicz Norms, Cape Breton University Undergraduate Thesis, 2022. See http://faculty.cbu.ca/srodney/pdf/theses/thesisam.pdf Bounded Weak Solutions to Elliptic PDE with Data in Orlicz Spaces. D Cruz-Uribe, S Rodney, J. Diff. Eq. 297D. Cruz-Uribe and S. Rodney, Bounded Weak Solutions to Elliptic PDE with Data in Orlicz Spaces, J. Diff. Eq. (297), (2021), p. 409-432 A Note on the Limit of Orlicz Norms. D Cruz-Uribe, S Rodney, Real Anal. Exchange (48D. Cruz-Uribe and S. Rodney, A Note on the Limit of Orlicz Norms, Real Anal. Exchange (48), (2023), p. 77-82 M N Rao, Z D Ren, Theory of Orlicz Spaces. Marcel-Dekker NYCM.N. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel-Dekker NYC, (1991) H L Royden, Real Analysis. MacMillan NYCthird editionH.L. Royden, Real Analysis, third edition, MacMillan NYC, (1988) W Rudin, Real and Complex Analysis. McGraw Hillthird editionW. Rudin, Real and Complex Analysis, third edition, McGraw Hill, (1987) Orlicz-Sobolev spaces and Imbedding Theorems. N S Trudinger, T K Donaldson, J. Func. Anal. 8N.S. Trudinger and T.K. Donaldson, Orlicz-Sobolev spaces and Imbedding Theorems, J. Func. Anal. (8), (1971), p. 52-75
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[ "F Munyaneza \nPhysics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa\n", "D Tsiklauri \nPhysics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa\n", "R D Viollier \nPhysics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa\n" ]
[ "Physics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa", "Physics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa", "Physics Department\nUniversity of Cape Town\n7701RondeboschSouth Africa" ]
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It has been recently shown(Munyaneza, Tsiklauri and Viollier 1998)that the analysis of the orbits of the fast moving stars close to Sgr A * (Eckart and Genzel 1997) provides a valuable tool to probe the gravitational potential near the Galactic center. As an example, we present here the results on a calculation of possible orbits of the star S0-1 in both, the black hole and degenerate neutrino ball scenarios of the central mass, based on the recent measurements of stellar proper motions at the Galactic center byGhez et al. 1998. Taking into account the error bars of their analysis, it is shown that within a few years time, the orbit of S0-1 may indeed reveal the nature of the supermassive compact dark object at the Galactic center.
10.1086/308026
[ "https://arxiv.org/pdf/astro-ph/9903242v1.pdf" ]
1,645,418
astro-ph/9903242
0b0c25f4eb1dfdc0de7f3dd6fed737b5bfa80110
16 Mar 1999 F Munyaneza Physics Department University of Cape Town 7701RondeboschSouth Africa D Tsiklauri Physics Department University of Cape Town 7701RondeboschSouth Africa R D Viollier Physics Department University of Cape Town 7701RondeboschSouth Africa 16 Mar 1999arXiv:astro-ph/9903242v1 Dynamics of the star S0-1 and the nature of the compact dark object at the Galactic centerSubject headings: black hole physics -celestial mechanics, stellar dynamics -dark matter - elementary particles -Galaxy: center It has been recently shown(Munyaneza, Tsiklauri and Viollier 1998)that the analysis of the orbits of the fast moving stars close to Sgr A * (Eckart and Genzel 1997) provides a valuable tool to probe the gravitational potential near the Galactic center. As an example, we present here the results on a calculation of possible orbits of the star S0-1 in both, the black hole and degenerate neutrino ball scenarios of the central mass, based on the recent measurements of stellar proper motions at the Galactic center byGhez et al. 1998. Taking into account the error bars of their analysis, it is shown that within a few years time, the orbit of S0-1 may indeed reveal the nature of the supermassive compact dark object at the Galactic center. Introduction The determination of the mass distribution near the center of our Galaxy and the question, whether it harbours a supermassive black hole (BH) or not, have been long-standing issues (Oort 1977;Genzel and Townes 1987;Genzel et al. 1994 andHo 1998 for a recent review). Various techniques have been used to find the mass of this supermassive compact dark object which is usually identified with the radio source Sagittarius A* (Sgr A * ) at or near the Galactic center. The most detailed information to date comes from the statistical analysis of the dynamics of stars moving in the gravitational field of the central mass distribution (Sellgren et al. 1987; Rieke and Rieke 1988;McGinn et al. 1989;Sellgren et al. 1990;Lindqvist et al. 1992;Haller et al. 1996;Eckart and Genzel 1996;Genzel et al. 1996;Eckart and Genzel 1997;Genzel et al. 1997;Ghez et al. 1998). Genzel et al. 1997 have established that the central dark object has a mass of (2.61 ± 0.76) × 10 6 M ⊙ , concentrated within a radius of 0.016 pc and located very close to Sgr A * . In the most recent observations, Ghez at al. 1998 confirm a mass of (2.6 ± 0.2) × 10 6 M ⊙ , enclosed within a radius of 0.015 pc. In the latter observations, the accuracy of the velocity measurements in the central arcsec 2 has been improved considerably, and thus the error bar on the central mass has been reduced by about a factor of 4. In both data sets, the presence of a supermassive compact dark object is revealed by the fact that several stars are moving within a projected distance of less than 0.01 pc from the central radio source Sgr A * at projected velocities in excess of 1000 km/s. For completeness, we mention here that the mass distribution at the Galactic center could also be studied through the motion of gas clouds and streamers (Lacy et al. 1980;Genzel & Townes 1987;Lacy et al. 1991). However, gas flows may be easily perturbed by non-gravitational forces such as shocks, radiation pressure, winds, magnetic fields,etc., and hence this probe is considered to be less reliable for determining the mass of the compact dark object at the Galactic center. The non-thermal spectrum of Sgr A * (Serabyn et al. 1997), that has been shown to originate from a very compact source (Rogers et al. 1994;Genzel et al. 1997;Ghez et al. 1998), and the low proper motion of Sgr A * (Backer 1996) have led many (e.g. Lynden-Bell and Rees 1971) to suggest that Sgr A * may be a supermassive BH of mass ∼ 2.6 × 10 6 M ⊙ . Supermassive BHs have also been inferred for several other galaxies such as M87 Harms et al. 1994;Macchetto et al. 1997) and NGC 4258 Myoshi et al. 1995). Taking this suggestion seriously, one is immediately faced with fundamental issues such as the prevalence of supermassive BHs in the nuclei of normal galaxies and the nature of the accretion mechanism that makes Sgr A * so much fainter than typical active galactic nuclei (Melia 1994;Narayan et al. 1995). However, as the best current observations probe the gravitational potential at radii 4 × 10 4 larger than the Schwarzschild radius of a BH of mass 2.6 × 10 6 M ⊙ (Ghez et al. 1998), it is perhaps prudent not to focus too much on the BH scenario, without having explored alternative scenarios for the supermassive compact dark object. One alternative to the BH scenario is a very compact stellar cluster (Haller et al. 1996, Sanders, 1992. However, based on the evaporation and collision time stability criteria, it is doubtful that such clusters could have survived up to the present time (see Moffat 1997 for an alternative point of view). Indeed, in the case of our Galaxy and NGC 4258, Maoz (1995Maoz ( ,1998 has found that even the lower limits to the half-mass densities of such compact clusters (1 × 10 12 M ⊙ pc −3 for NGC 4258 and 6 × 10 11 M ⊙ pc −3 for our Galaxy) are too large that they could be due to stable clusters of stellar or substellar remnants. The estimated maximal lifetimes for such dense clusters are about 10 8 years for our Galaxy and a few 10 8 years for the NGC 4258, i.e. much shorter than the age of the Universe. This seems to rule out the existence of dense clusters at the centers of the above mentioned galaxies, unless we are prepared to believe that we happen to live in a privileged epoch of the lifetime of the Universe. Note, however, that for other galaxies, such as M31, M32,M87,NGC 3115,NGC 3377,NGC 4261,NGC 4342,NGC 4486B and NGC 4594,maximal lifetimes of dense stellar clusters are in excess of 10 11 years. Moreover, it should be acknowledged that the uncertainties in the understanding of the core collapse process of such dense clusters still leave some room for speculation about a possible interpretation of the supermassive compact dark objects at the centers of galaxies (including both, our Galaxy and NGC 4258) in terms of e.g. core-collapsed clusters (Maoz 1998). But, apart from a cluster of very low mass BH's that is free of stability problems, the most attractive alternative to a dense stellar cluster is a cluster of elementary particles. In fact, in the recent past, an alternative model for the supermassive compact dark objects in galactic centers has been developed (Viollier et al. 1992(Viollier et al. , 1993Viollier 1994;Tsiklauri and Viollier 1996, 1998a,b, 1999Bilić et al. 1999). The cornerstone of this model is that the dark matter at the center of galaxies is made of nonbaryonic matter in the form of massive neutrinos that interact gravitationally forming supermassive neutrino balls in which the degeneracy pressure of the neutrinos balances their self-gravity. Such neutrino balls could have been formed in the early Universe during a first-order gravitational phase transition (Bilić and Viollier 1997,1999a. In fact, it has been recently shown that the dark matter concentration observed through stellar motion at the Galactic center Genzel et al. 1996) is consistent with a supermassive object of 2.5 × 10 6 solar masses made of self-gravitating, degenerate heavy neutrino matter (Tsiklauri & Viollier 1998a). Moreover, it has been shown that an acceptable fit to the infrared and radio spectrum above 20 GHz , which is presumably emitted by the compact dark object, can be reproduced in the framework of standard accretion disk theory (Tsiklauri & Viollier 1999;, in terms of a baryonic disk immersed in the shallow potential of the degenerate neutrino ball of 2.5 × 10 6 solar masses. The purpose of this paper is to compare the predictions of these two models for the supermassive compact dark object at the center our Galaxy, i.e. (i) the black hole scenario and (ii) the degenerate neutrino ball scenario as an example of an extended object. Both these models are not in contradiction with the technologically challenging proper motions observations and their statistical interpretation Ghez et al. 1998) . It is therefore desirable to have an additional independent dynamical test, in order to distinguish between these two possible scenarios describing the compact dark object at the center of our Galaxy. In the recent past, mainly statistical arguments involving many stars have been used to determine the gravitational potential at the Galactic center. However, in this paper, we would like to demonstrate that it is also possible to draw definite conclusions from the motion of individual stars, in particular, in the immediate vicinity of the Galactic center, where statistical arguments cannot be easily applied due to the low density of stars. To this end, we have recently calculated the orbits (Munyaneza , Tsiklauri and Viollier 1998) of the fastest moving infrared source S1 using the Genzel et al. 1997 data for a supermassive BH or a neutrino ball mass of 2.61 × 10 6 solar masses. We have shown that tracking the orbits of S1 offers a good opportunity to distinguish in a few years time between the two scenarios for the supermassive compact dark object. Here we perform a full analysis of the orbits of the same star S0-1 based on the most recent Ghez et al. 1998 data, including all the error bars of the measurements. A distance to the Galactic center of 8 kpc has been assumed throughout this paper. This paper is organized as follows: In section 2, we present the equations that describe degenerate neutrino balls and we establish some constraints on the neutrino mass based on Ghez et al. 1998 data. In section 3, we study the dynamics of S0-1 and conclude with the discussion in section 4. The compact dark object as a neutrino ball Dark matter at the Galactic center can be described by the gravitational potential Φ(r) of the neutrinos and antineutrinos that satisfies Poisson's equation ∆Φ = 4πGρ ν ,(1) where G is Newton's gravitational constant and ρ ν is the mass density of the neutrinos and antineutrinos. Neutrino matter will interact gravitationally to form supermassive neutrino balls in which self-gravity of the neutrinos is being balanced by their degeneracy pressure P ν (r) according to the equation of hydrostatic equilibrium dP ν dr = −ρ ν dΦ dr .(2) In order to solve equation (1) , one needs a relation between the pressure P ν and the density ρ ν . To this end we choose the polytropic equation of state of degenerate neutrino matter, i.e. P ν = Kρ 5/3 ν ,(3) where the polytropic constant K is given by (Viollier, 1994) K = 6 g ν 2/3 π 4/3h2 5m 8/3 ν .(4) Here, m ν denotes the neutrino mass, g ν is the spin degeneracy factor of the neutrinos and antineutrinos, i.e. g ν = 2 for Majorana and g ν = 4 for Dirac neutrinos and antineutrinos. We now introduce the dimensionless potential and radial variable, v and x , by Φ(r) = GM ⊙ a ν v ′ (x 0 ) − v(x) x ,(5)r = a ν x,(6) where x 0 is the dimensionless radius of the neutrino ball, and the scale factor a ν which plays here the role of a length unit is given by a ν = 2.1376 lyr × 17.2 keV m ν c 2 8/3 g −2/3 ν .(7) Assuming spherical symmetry, we finally arrive at the non-linear Lané-Emden equation d 2 v dx 2 = − v 3/2 x 1/2 ,(8)v(x) and its derivative v ′ (x) as M (r) = r 0 4πρ ν r 2 dr = −M ⊙ (v ′ (x)x − v(x)) .(9) In order to describe the compact dark object at the Galactic center as a neutrino ball and constrain its physical parameters appropriately, it is worthwhile to use the most recent observational data by Ghez et al. 1998, who established that the mass enclosed within 0.015 pc at the Galactic center is (2.6 ± 0.2) × 10 6 solar masses. Following the analysis of Tsiklauri & Viollier 1998a, we choose the minimal neutrino mass m ν to reproduce the observed matter distribution, as can be seen from Fig. 1, where we have added the Ghez et al. 1998 andGenzel et al. 1997 data points with error bars. In Fig. 1 we include only the neutrino ball contribution to the enclosed mass, as the stellar cluster contribution is negligible by orders of magnitude at these radii. For a M = 2.4 × 10 6 M ⊙ neutrino ball, the constraints on the neutrino mass are m ν ≥ 17.50 keV/c 2 for g ν = 2 and m ν ≥ 14.72 keV/c 2 for g ν = 4, and the radius of the neutrino ball is R ≤ 1.50 × 10 −2 pc. Using the value of M = 2.6 × 10 6 M ⊙ , the bounds on the neutrino mass are m ν ≥ 15.92 keV/c 2 for g ν = 2 or m ν ≥ 13.39 keV/c 2 for g ν = 4 and the radius of the neutrino ball turns out to be R ≤ 1.88 × 10 −2 pc . Finally, for a M = 2.8 × 10 6 M ⊙ neutrino ball, the range of neutrino mass is m ν ≥ 15.31 keV/c 2 for g ν = 2 and m ν ≥ 12.87 keV/c 2 for g ν = 4 and the corresponding neutrino ball radius R ≤ 2.04 × 10 −2 pc. Dynamics of S0-1 We investigate the motion of S0-1 that is the star closest to the Galactic center, and at the same time, also the fastest of the 15 stars in the central arcsec 2 around Sgr A * . We study the motion of S0-1 in the gravitational potential near Sgr A * , assuming that the central object is either a BH of mass M or a spatially extended object represented by a neutrino ball of mass M , that consists of self-gravitating degenerate heavy neutrino matter. The BH or neutrino ball mass M will be taken to be 2.4, 2.6 and 2.8 ×10 6 solar masses which corresponds to the range allowed by the Ghez et al. 1998 data. We use Newtonian dynamics, as the problem is essentially nonrelativistic, because the mass of the neutrino ball is much less than the Oppenheimer-Volkoff limit corresponding to this particular neutrino mass (Bilić, Munyaneza & Viollier 1999). Consequently, we can write Newton's equations of motion as x = − GM (r) (x 2 + y 2 + z 2 ) 3/2 x,(10)y = − GM (r) (x 2 + y 2 + z 2 ) 3/2 y,(11)z = − GM (r) (x 2 + y 2 + z 2 ) 3/2 z,(12) where x, y, z denote the components of the radius vector of the star S0-1 and r = x 2 + y 2 + z 2 , Sgr A * being the origin of the coordinate system. We thus assume that the center of the neutrino ball and the BH is at the position of Sgr A * . The dot denotes of course the derivative with respect to time. In the case of a BH, M (r) = M is independent of r, while in the neutrino ball scenario, M (r) is given by Eq. (9) In Fig. 2 we plot two typical orbits of S0-1 corresponding to a BH and neutrino ball mass of M = 2.6 × 10 6 M ⊙ . The input values for v x and v y are 470 km/s and -1330 km/s, respectively. The z-coordinate of the star S0-1 is assumed to be zero and the velocity component in the line-of-sight of the star S0-1, v z , has also been set equal to zero in this graph. The filled square labels denote the time in years from 1990 till 2015. In the case of a BH, the orbit of S0-1 is an ellipse, with Sgr A * being located in one focus (denoted by the star in the figure). The period of S0-1 is 12.7 years and the minimal and maximal distances from Sgr A * are 1.49 and 7.18 light days, respectively. In the case of a neutrino ball, the orbit will be bound but not closed, with minimal and maximal distances from Sgr A * of 3.98 and 42.07 light days, respectively. It can be seen from Fig. 2 that, in the case of a neutrino ball, S0-1 is deflected much less than for a BH, as the gravitational force at a given distance from Sgr A * is determined by the mass enclosed within this distance. Using Eq. (9) we can estimate the mass enclosed within a radius corresponding to the projected distance of S0-1 from Sgr A * (4.41 × 10 −3 pc ) to be ∼ 1.8 × 10 5 M ⊙ . Thus, in the case of a neutrino ball, the force acting on S0-1 is about 14 times less than in the case of a BH. This graph can serve to establish, whether Sgr A * is a BH or an extended object, due to the fact that the positions of S0-1 will differ as time goes on in the two scenarios. However, this conclusion is perhaps too optimistic, as we have not yet considered (i) the uncertainties in v x and v y , (ii) the error bars in the total mass of the BH or neutrino ball, (iii) the complete lack of information on z and v z . As a next step, we investigate the dependence of the orbits on the uncertainties in the velocity components. The results of this calculation are presented in Fig. 3 where we have set z = v z = 0. In the case of a BH, the orbits of S0-1 are ellipses , while the other 5 thick lines are bound orbits of S0-1 for the neutrino ball scenario. The spread of the orbits induced by the error bars in v x and v y is small compared to that of the recent analysis based on the Genzel et al. 1997 data (Munyaneza, Tsiklauri & Viollier 1998). The time labels, represented by filled squares on the orbits, are placed in intervals of 5 years: starting from 1995.4 up to 2005 in the case of a BH, and up to 2015 in the case of a neutrino ball. The periods of S0-1 for different orbits vary between 10 and 17 years for the BH scenario. We thus see that the error bars in v x and v y do not alter the predictions of Fig. 1 in substance. We now would like to study, how the orbits are changed if we let the mass of the neutrino ball or the BH vary within the estimated error bars (Ghez et al. 1998). In Fig. 4 and 5, we present the results of our calculations, for both scenarios, with central masses of M = 2.4 × 10 6 M ⊙ and M = 2.8 × 10 6 M ⊙ , respectively. The neutrino masses consistent with the Ghez et al. (1998) data are are m ν ≥ 17.50 keV/c 2 for a M = 2.4 × 10 6 M ⊙ neutrino ball and m ν ≥ 15.31 keV/c 2 for a M = 2.8 × 10 6 M ⊙ neutrino ball. The filled squares represent the time labels spaced by 5 year intervals as in Fig. 3. In the BH scenario, the periods of S0-1 with M = 2.4 × 10 6 M ⊙ vary between 11 years and 20 years, while in the case of a M = 2.8 × 10 6 M ⊙ , the periods vary between 9.5 and 15 years. Comparing the orbits of S0-1 in Fig. 4 and 5 with those in Fig. 3, we conclude that the errors bars in the total mass of the BH or neutrino ball make no qualitative difference for the motion of S0-1. In both scenarios of the supermassive compact dark object, all the orbits considered for three different values of the BH or neutrino ball mass are bound for z = v z = 0, as can be seen from Fig. 6 and 7, where the escape and circular velocities are plotted as functions of the distance from Sgr A * . In these graphs, we have also included the Ghez et al. (1998) data with error bars, for the 15 stars in the central arsec 2 , assuming that the velocity component and distance from Sgr A * in the line-of-sight are both zero, i.e. v z = 0 and z = 0. Thus, the data points are lower bounds on the true circular or escape velocity and radius, and the real values lie in the quarter-plane to the right-and-up of the measured data point. For instance, the innermost data point describing the star S0-1 is in both scenarios, consistent with a bound orbit if |z| and |v z | are not too large, as can be seen from the escape velocity in Fig. 6. However, S0-1 cannot be interpreted as a virialized star in the neutrino ball scenario, as is evident from the plot of the circular velocity in Fig 7; it thus would have to be an intruder star. If the projected velocity of a star at a given projected distance from Sgr A * is larger than the escape velocity at the same distance (assuming z = 0), the neutrino ball scenario is virtually ruled out, since the kinetic energy of the star would have to be very large at infinity. We now turn to the investigation of the dependence of the orbits on the z coordinate and z component of the velocity of the star S0-1. The two quantities, z and v z , are the major source of uncertainty in determining the exact orbit of the star S0-1. However, this shortcoming will not substantially affect the predictive power of our model, as we will see below. In Fig. 8 we show the results of a calculation of the dependence of the orbit on z for a M = 2.6 × 10 6 M ⊙ neutrino ball or BH. The input values for v x and v y are fixed at 470 km/s and -1330 km/s, respectively, and v z is assumed to be zero. The z-coordinate is varied from zero up to the radius of the neutrino ball, i.e. the distance from Sgr A * beyond which there is obviously no difference between the BH and the neutrino ball scenarios. In this case, the radius of the neutrino ball 1.88 × 10 −2 pc or 0.485 ′′ . The top panel represents the orbits in the case of a BH, for different values of z, while the lower panel describes the dependence of the orbit on z in the neutrino ball scenario. We conclude from this plot that, increasing |z| has the effect of shifting the orbits towards the lower right corner of the graph. This is, obviously, due to the fact that increasing |z| means going further away from the scattering center, thus yielding less deflection of the orbit. Moreover, in the neutrino ball scenario, the dependence on z is relatively insignificant, as long as |z| is smaller than the radius of the neutrino ball. This is in accordance with the fact that for small distances from the center, the potential of a neutrino ball can be approximated by a harmonic oscillator-type potential, where the Newtonian equations of motion decouple in Cartesian coordinates. The dependence of the orbits of S0-1 on v z has a similar effect as in the previous graph.Here, we have fixed z to zero and v z has been varied as an input parameter. Increasing |v z | yields a greater velocity of the star and, obviously, a fast moving star will be deflected less than a star with smaller |v z |. The results of this calculation are summarized in Fig. 9. Conclusion and discussion We have demonstrated that the orbits of S0-1 differ substantially for the BH and neutrino ball scenarios of the Galactic center, especially with the new Ghez et al. (1998) data. We have shown that using these data, the error bars in velocities of S0-1 and mass of the central object do not change the pattern of the orbits of S0-1. In the case of a BH, the orbit of S0-1 is much more curved than in the neutrino ball scenario, as long as |z| is smaller than the radius of the neutrino ball. Increasing |z| and |v z | shifts the orbits to the lower right corner of the graph, and this gives us a key to establish the allowed regions of S0-1 depending on whether it is a BH or a neutrino ball, irrespective of the values of the parameters z and v z . In Fig. 10 three orbits are plotted: the upper-leftmost orbit of S0-1 corresponding to the neutrino ball scenario (actually, line 9 of Fig. 4) and two orbits in a BH scenario with the smallest minimal and maximal distances from Sgr A * (ellipses 2 and 4 from Fig.5). This figure serves as a test to distinguish the supermassive BH scenario from the neutrino ball model of the Galactic center. It is clear that, as the observations proceed within the next year, one might be able to tell the difference between the two models of the supermassive compact dark object at the center of our Galaxy. If the star is found in the region F inside the ellipses, this will rule out both the BH and the neutrino ball scenario of Sgr A * , as seen in Fig. 10. We can estimate the minimal distance of approach to Sgr A * to be 0.909 light days. If the orbit of S0-1 ends up in the upper-left zone of the thick line, this will clearly rule out the neutrino ball scenario for the chosen lower limit of the neutrino mass. However, if S0-1 is found in the lower right corner of the same line (i.e. below the thick line), then the supermassive object can be interpreted as either a neutrino ball or a BH with a large z or v z parameter. One can of course repeat this analysis for several stars in the central arcsec 2 and use some statistical arguments: should there be no stars in the black hole zone, and many stars found in the zone for black holes and neutrino balls, the black hole interpretation of the supermassive compact dark object at the Galactic center would become less attractive, as some of the stars should be moving close to the plane perpendicular to the line-of-sight, i.e. they should have small |v z | and |z|. The neutrino masses used for the neutrino ball are lower limits. We note that increasing the neutrino mass will make the radius smaller (the neutrino ball radius scales as ∝ m −8/3 ν ) and, when it reaches the mass corresponding to the Oppenheimer-Volkoff limit, there will be little difference between the two scenarios. Acknowledgements One of us (F. Munyaneza) gratefully acknowledges funding from the Deutscher Akademischer Austauschdienst and the University of Cape Town. This work is supported by the Foundation for Fundamental Research (FFR). Ghez et al. 1998 data, the bounds on the neutrino mass are m ν ≥ 17.50 keV/c 2 for g ν = 2 or m ν ≥ 14.72 keV/c 2 for g ν = 4 and M = 2.4 × 10 6 M ⊙ . For M = 2.6 × 10 6 M ⊙ the bounds on the neutrino mass are m ν ≥ 15.92 keV/c 2 for g ν = 2 and m ν ≥ 13.39 keV/c 2 for g ν = 4. Finally, a neutrino mass range of m ν ≥ 15.31 keV/c 2 for g ν = 2 or m ν ≥ 12.87 keV/c 2 for g ν = 4 is consistent with a supermassive object of M = 2.8 × 10 6 M ⊙ . The Ghez et al. 1998 andGenzel et al. 1997 data points with error bars are also shown in this graph. Fig. 3. The periods of S0-1 vary between 9.5 years and 14.7 years in the BH scenario. Fig. 6: The escape velocity as a function of the distance from Sgr A * for BH and neutrino ball scenarios. The value of the mass of the central object is varied as indicated on the graph. The data points with error bars of 15 stars in the central arcsec 2 are taken from Ghez et al. 1998 assuming that the projected velocity and distance from Sgr A * are equal to the true velocity and distance, respectively, i.e. z = 0 and v z = 0. This graph shows that S0-1 is bound in both scenarios for different values of the mass of the central object. Fig. 7: The circular velocity as a function of the distance from Sgr A * for BH and neutrino ball scenarios. The mass of the central object is varied as indicated on the graph. The data points with error bars of 15 stars in the central arcsec 2 are taken from Ghez et al. 1998 assuming that the projected velocity and distance from Sgr A * are equal to the true velocity and distance, respectively, i.e. z = 0 and v z = 0. This graph shows that the orbits of S0-1 are almost circular in the case of the BH scenario (see text for the discussion concerning the innermost data point). Fig. 8: Projected orbits of the star S0-1 in the case of a supermassive BH (top panel) and in the case of a neutrino ball (lower panel) with M = 2.6 × 10 6 M ⊙ . In this graph we explore how the orbits are affected by the uncertainty in the z-parameter. The labels for the orbits are given in the graph. Note, that for z = 0.4849 ′′ , which corresponds to the radius of the neutrino ball for the assumed distance to the Galactic center, the orbits for a BH and neutrino ball are identical, as it should be. In this graph v x = 470 km/s, v y = −1330 km/s and v z = 0. If the star S0-1 will be found inside the ellipses (region F ), this will rule out both the BH and the neutrino ball models . If the star S0-1 will eventually be found in the upper-left zone of the graph, i.e. up and left of the thick orbit, this will rule out the neutrino ball interpretation for the chosen neutrino mass. Finally, if S0-1 will be found to the right and below the thick line, then the supermassive central object should be interpreted either as a BH with large z or as a neutrino ball. with polytropic index 3/2. The boundary conditions are chosen in such a way that v vanishes at the boundary x 0 of the neutrino ball. The mass M B of a (pointlike) baryonic star at the center of the neutrino ball is fixed by v(0) = M B /M ⊙ . The case M B = 0 corresponds to a pure neutrino ball without a pointlike source at the center. The mass enclosed within a radius r in a pure neutrino ball can be written in terms of and it reaches M (R) = M at the radius of the neutrino ball R. The initial positions and velocities for this system of equations are taken to be those of S0-1 in 1995.4, when the coordinates of S0-1 were RA = −0.107 ′′ and DEC = 0.039 ′′ . The x and y components of the projected velocity are v x = 470 ± 130 km/s and v y = −1330 ± 140 km/s(Ghez et al. 1998), respectively. Here x is opposite to the RA direction and y is in the DEC direction. Figure masses. Based on the Fig. 2 : 2Projected orbits of the star S0-1 for BH and neutrino ball scenarios with M = 2.6 × 10 6 M ⊙ and v z = z = 0. The velocity components of S0-1 are taken to be v x = 470 km/s and v y = −1330 km/s. The filled squares denote the time labels. The period of S0-1 in a the BH scenario is 12.7 years and the minimal and maximal distances from Sgr A * are 1.49 and 7.18 light days. The orbit of S0-1 in the neutrino ball scenario is bound with minimal and maximal distances from Sgr A * of 3.98 and 42.07 light-days, respectively. Fig. 3 : 3Projected orbits of the star S0-1 in the case of a BH or a neutrino ball of M = 2.6 × 10 6 M ⊙ taking into account the error bars in the velocity components. The labels for the different orbits are: 1: v x = 470 km/s and v y = −1330 km/s (median values), 2: v x = 340 km/s and v y = −1190 km/s, 3: v x = 340 km/s and v y = −1470 km/s, 4: v x = 600 km/s and v y = −1190 km/s, 5: v x = 600 km/s and v y = −1470 km/s. The periods of S0-1 for different orbits in the BH scenario vary between 10 and 17 years. The thick lines 6 to 10 correspond to the orbits in the neutrino ball scenario with the following description: 6: v x = 470 km/s and v y = −1330 km/s (median values), 7: v x = 340 km/s and v y = −1190 km/s, 8: v x = 340 km/s and v y = −1470 km/s, 9: v x = 600 km/s and v y = −1190 km/s, 10: v x = 600 km/s and v y = −1470 km/s. All the orbits in both scenarios are bound. The time labels (filled squares) on the orbits are placed in intervals of 5 years, up to the year 2005 in the case of a BH and up to 2015 in the case of a neutrino ball. Fig. 4 : 4Projected orbits of the star S0-1 in the case of a BH or neutrino ball with M = 2.4 × 10 6 M ⊙ . In this graph we explore how the orbits are affected by the uncertainty in the mass of the BH or neutrino ball. The orbits are calculated for z = v z = 0 . The description of the orbits are the same asFig. 3. The periods of S0-1 in the BH scenario vary between 11 and 20 years and all the orbits are bound in both scenarios. Fig. 5 : 5Projected orbits of the star S0-1 for M = 2.8 × 10 6 M ⊙ . All the orbits are bound and calculated for different values of the velocity components as in Fig. 9 : 9Projected orbits of the star S0-1 in the case of a supermassive BH (top panel) and in the case of a neutrino ball (lower panel) of M = 2.6 × 10 6 M ⊙ . In this graph we explore how the orbits are affected by the uncertainty in v z . The labels for the orbits are given in the graph. Here, v x = 470 km/s, v y = −1330 km/s and z = 0. Fig. 10 : 10Prediction regions for the supermassive central object. This graph combines line 9 from Fig. 4 and lines 2 and 4 from Fig. 5. 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Phys., 32, 51 This preprint was prepared with the AAS L A T E X macros v4.0. neutrino ball black hole. 4This preprint was prepared with the AAS L A T E X macros v4.0. neutrino ball black hole 2010 2010 2000 ? 2000 2015 2005 1990 1998 1998 2015 2005 1995:4
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[ "Symmetry breaking in small rotating cloud of trapped ultracold Bose atoms", "Symmetry breaking in small rotating cloud of trapped ultracold Bose atoms" ]
[ "D Dagnino \nDept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain\n", "N Barberán \nDept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain\n", "M Lewenstein \nICREA and ICFO-Institut de Ciències Fotòniques\nAv. del Canal Olímpico s/n08860Castelldefells, BarcelonaSpain\n\nInstitute for Theoretical Physics\nUniversity of Hannover\nAppelstrasse 230167HannoverGermany\n", "K Osterloh \nInstitute for Theoretical Physics\nUniversity of Hannover\nAppelstrasse 230167HannoverGermany\n", "A Riera \nDept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain\n" ]
[ "Dept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain", "Dept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain", "ICREA and ICFO-Institut de Ciències Fotòniques\nAv. del Canal Olímpico s/n08860Castelldefells, BarcelonaSpain", "Institute for Theoretical Physics\nUniversity of Hannover\nAppelstrasse 230167HannoverGermany", "Institute for Theoretical Physics\nUniversity of Hannover\nAppelstrasse 230167HannoverGermany", "Dept. ECM\nFacultat de Física\nU. de Barcelona\nE-08028BarcelonaSpain" ]
[]
We study the signatures of rotational and phase symmetry breaking in small rotating clouds of trapped ultracold Bose atoms by looking at rigorously defined condensate wave function. Rotational symmetry breaking occurs in narrow frequency windows, where the ground state of the system has degenerated with respect to the total angular momentum, and it leads to a complex wave function that exhibits vortices clearly seen as holes in the density, as well as characteristic vorticity. Phase symmetry (or gauge symmetry) breaking, on the other hand, is clearly manifested in the interference of two independent rotating clouds.
10.1103/physreva.76.013625
[ "https://export.arxiv.org/pdf/cond-mat/0610512v1.pdf" ]
4,851,685
cond-mat/0610512
529e02a8f2de3cc850436b718f0ddb3fe18a9661
Symmetry breaking in small rotating cloud of trapped ultracold Bose atoms 18 Oct 2006 D Dagnino Dept. ECM Facultat de Física U. de Barcelona E-08028BarcelonaSpain N Barberán Dept. ECM Facultat de Física U. de Barcelona E-08028BarcelonaSpain M Lewenstein ICREA and ICFO-Institut de Ciències Fotòniques Av. del Canal Olímpico s/n08860Castelldefells, BarcelonaSpain Institute for Theoretical Physics University of Hannover Appelstrasse 230167HannoverGermany K Osterloh Institute for Theoretical Physics University of Hannover Appelstrasse 230167HannoverGermany A Riera Dept. ECM Facultat de Física U. de Barcelona E-08028BarcelonaSpain Symmetry breaking in small rotating cloud of trapped ultracold Bose atoms 18 Oct 2006numbers: 7343-f0530Jp0375Hh We study the signatures of rotational and phase symmetry breaking in small rotating clouds of trapped ultracold Bose atoms by looking at rigorously defined condensate wave function. Rotational symmetry breaking occurs in narrow frequency windows, where the ground state of the system has degenerated with respect to the total angular momentum, and it leads to a complex wave function that exhibits vortices clearly seen as holes in the density, as well as characteristic vorticity. Phase symmetry (or gauge symmetry) breaking, on the other hand, is clearly manifested in the interference of two independent rotating clouds. We study the signatures of rotational and phase symmetry breaking in small rotating clouds of trapped ultracold Bose atoms by looking at rigorously defined condensate wave function. Rotational symmetry breaking occurs in narrow frequency windows, where the ground state of the system has degenerated with respect to the total angular momentum, and it leads to a complex wave function that exhibits vortices clearly seen as holes in the density, as well as characteristic vorticity. Phase symmetry (or gauge symmetry) breaking, on the other hand, is clearly manifested in the interference of two independent rotating clouds. Symmetry breaking in finite systems has been a subject of intensive debate in physics, in general (cf. the Ref. [1]), and in physics of ultracold gases in particular over the years. For Bose-Einstein condensates (BEC) two symmetries play a particular role: U (1) phase symmetry and SU (2) (or SO (3)) rotational symmetry. In the large N limit, one breaks these symmetries by hand, as proposed originally by Bogoliubov [2]. Thus, the accurate way to deal with macroscopic Bose Einstein condensates (BEC's) is by the use of a classical field, also called an order parameter, or the wave function (WF) of the condensate. This function is a single particle (SP) wave function, which is the solution of the Gross Pitaevskii (GP) equation within the mean field approximation, that characterizes the system in a proper way [3]. It has an arbitrary, but fixed phase, and for rotating systems with more than one vortex it exhibits arbitrarily places, but fixed vortex array. For dilute ultracold Bose gases (i.e. when n|a| 3 << 1 [4] where n is the density and a is the s-wave scattering length) mean field, or Bogoliubov approach is capable to reproduce very well the main properties, despite the fact for finite, fixed N and total angular momentum L, which are both constants of the motions, mean field theory cannot be exact. This observation has stimulated a lot of discussion about the nature of the phase of BEC [5,6], and particle-conserving Bogoliubov approach [7]. The modern point of view (for a recent discussion see [8]) implies that two BEC's with fixed N each one, will produce a well defined interference pattern of fringes as a result of the measurement in only one shot (comparable with the calculated n-correlation function) in contrast with the density, which would be obtained as a mean image of random interference patterns from several shots. The position of fringes in the given measurement are determined by subsequent localization of atoms arriving at detectors; the first atom is completely random, second is correlated, third even more correlated etc. [6]. Thus the information about the pattern is obtained from the many-body wave function by looking at pair, triple, ... correlations. The breaking of rotational symmetry should occur in large rotating clouds in the similar way, and a pure L-state would show, in a timeof-flight experiment, a definite interference pattern accurately represented by n-correlation functions, different from a circular symmetric profile of the single particle density. It would be a test of the meaning assigned to the measurement. Unfortunately, for large N-systems, the total angular momentum of the stationary states is not well defined and there is no qualitative difference between density and n-correlation function, usually showing in both cases vortex arrays. For small rotating clouds the situation is, however, different, as we have shown in Ref. [9]. Typically, the GS's are pure-L states for most of the values of Ω. Only, in the very narrow window of frequencies, where the ground states is degenerated with respect to L, vortex arrays can be obtained, arbitrary small symmetry breaking deformation of the trap potential leads to the appearance of symmetry breaking vortex arrays both in density and pair-correlations. Namely, in the regime of pure L-GS small systems would provide a suitable test for the meaning of the measurement distinguishing between the density or the pair-correlation output. In this Letter we study the effects of symmetry breaking in small rotating clouds of trapped ultracold Bose atoms in more depth, by introducing the rigorous definition of the condensate wave function, defined as an eigenvector of the one body density matrix operator (OBDM), corresponding to the largest eigenvalue. Such definition of the order parameter has been introduced in classical papers on off-diagonal long range order [10]. It has, however, rarely been used since its application requires the knowledge of the full many-body wave function, or at least of the exact OBDM. Since for quantum gases exact analytic solutions are either not known (2D and 3D), or very difficult to handle (1D), so far this definition has been only applied to the case of model system with harmonic forces. Here we apply for the first time to the rotating gas, using exact numerically calculated OBDM for few atom systems. We identify in this way possible states with vortices, and obtain phase characteristics of the wave function (reflecting quantized circulation of vortices), and provide unambiguous definition of the degree of condensation. With such calculated order parameter we then reproduce the density and interference patterns for two condensed clouds, and shed new light on the discussion of the origins of symmetry breaking in finite mesoscopic systems. We consider a two-dimensional system of few Bose atoms trapped in a parabolic rotating trap around the z-axis. The rotating frequency Ω is strong enough to consider the Lowest Landau Level regime with atoms interacting via contact forces. Our main goal is the description of the stationary states for different values of Ω, analyzed from the rotating frame of reference, unless otherwise stated. Our analysis is performed using the exact diagonalization formalism, valid for arbitrary interactions and densities. However, in contrast to the mean field approach, this method deals with multi-particle WF's and loses the intuitive picture provided by the mean field order parameter. Our goal is to obtain in a precise way a complex scalar field that models efficiently the system, and allows to reproduce the important features, such as the vortex states. In the regime of relatively low rotation frequency, where the degree of condensation is high and some vortices appear distributed in an ordered arrays, this scalar field plays the role of a genuine order parameter. On the other hand, it looses its capability to represent the system as Ω approaches the melting point, where the prediction [11] is that the vortex lattice disappears and the systems turns, for large systems, into a Laughlin liquid. The way to know if there is a "macro-occupied" SP wave function in the ground state |GS is to look at the eigenvalues and eigenvectors of the OBDM [4,10]. That is to say, one must solve the eigenvalue equation d r ′ n (1) ( r, r ′ )ψ * l ( r ′ ) = n l ψ * l ( r),(1) where n (1) ( r, r ′ ) = GS |Ψ † ( r)Ψ( r ′ )|GS ,(2) withΨ being the field operator. If there exist a relevant eigenvalue n 1 ≫ n l for l = 2, 3, . . ., then √ n 1 ψ 1 ( r)e iφ1(3) plays the role of the order parameter of the system, where φ 1 is an arbitrary constant phase. The WF may be expanded in the form ψ 1 ( r) = m l=0 β 1l ϕ l ( r), using the complete set of Fock-Darwin [12] WF's given by ϕ l ( r) = e ilθ r l e −r 2 /2 / √ πl!, where l labels the single particle angular momentum, and m is equal to the largest total angular momentum L involved in the expansion of the GS; length unit is here λ = h/(mω ⊥ ), and ω ⊥ denotes the trap frequency. The same SP basis is used in our numerical simulations to represent both the field operator and the multiparticle GS wave function. An alternative, and perhaps even more appropriate SP basis is determined by the functions ψ l ( r). One can define a set of canonical creation and annihilation operators for them:â † l = d r ′ ψ * l ( r ′ )Ψ( r ′ ),(4) andâ l being the hermitian conjugate ofâ † l . The Hilbert space attain then a tensor structure with respect to the modesâ l , and the new Fock (occupation number) many body basis |n 1 ⊗ |n 2 ⊗ . . .. The macro-occupied mode contains on average n 1 atoms, but this number fluctuates, and most presumably normally, i.e. the fluctuations of n 1 are of order √ n 1 ; to this aim one has to calculate GS|(â † 1â 1 ) 2 |GS . This implies that atom number fluctuations between the macro-occupied mode (condensate) and the rest of the modes (that could be regarded as phonon modes, quasi-particles) will tend to reduce the fluctuations of the phase. A natural consequence of this observation is to expect that a very fine approximation of the GS is given by the coherent state |α 1 , such that a 1 |α 1 = √ n 1 ψ 1 e iφ1 |α 1 . If n l for l = 2, 3, . . . are very small we may neglect them, and approximate the many body wave function by |α 1 ⊗ |0 2 ⊗ |0 3 ⊗ . . .. In a more precise description, we should rather approximate the GS by |α 1 ⊗ |α 2 ⊗ |α 3 ⊗ . . ., whereâ l |α l = √ n l ψ l e iφ l |α l where the phases φ l are arbitrary; one should, however, choose them to be random in order to reproduce (on average) the same OBDM as the one obtained by exact numerical diagonalization. This representation implies that the next simplifiying step would be the representation of the GS by a clasical field entering the GP equation, and containing all the involved coherent states |α k , k = 1, ...m + 1 as, Ψ( r) = m+1 k=1 √ n k ψ k e iφ k with random phases. Calculation of quantum mechanical averages would then in principle require averaging over random phases, which makes this approach technically difficult. As long as the exact GS is a state with well defined angular momentum, (a pure L-state) not degenerated with other lowest energy states in different L-subspaces, it is easy to demonstrate that the FD functions are the eigenstates of Eq.(1) and the eigenvalues n l are the occupations usually used in literature. However, at certain values of Ω where degeneracy takes place and vortex states without circular symmetry (except the case of only one centered vortex) are possible [9], the eigenfunctions of Eq.(1) are linear combinations of the FD functions and the macro-occupied function ψ 1 that represents the vortex state has expected SP angular momentum given bȳ hl = j | β 1j | 2h l j where l j are integers. A convenient definition of the degree of condensation which senses the loss of macro-occupation is given by c = n 1 −ñ N (5) where N is the total number of atoms andñ is the mean occupation calculated without the first value n 1 . In what follows, we show some results that confirm the convenience to represent the whole system by ψ 1 at certain values of Ω. As a general result, for vortex states, n 1 is always larger than the occupation of the most important FD state within the exact GS. In addition, ψ 1 provides a non-ambiguous way to characterize vortices, not only showing dimples in the density profile, but also indicating the position of each one by the change on multiples of 2π of the phase S( r) in ψ 1 ( r) =| ψ 1 ( r) | e iS( r) when moving around each vortex. In Fig.1 for N = 6 atoms, we show for three different values of Ω where degeneracy takes place, the comparison between the contour plots of the density of the exact GS and the density of ψ 1 , as well as the map of the phase S( r) of ψ 1 . In the first case (a) the GS contains two vortices that appear in a clearer way in the order parameter, as it excludes the non condensed part that smears the structure of the GS. The same picture is shown in (b) where four vortices become visible. In the second case, the map of the phase not only localizes vortices with one unit of quantized circulation, but also indicates that incipient vortices are growing at the edge of the system. In the last case (c), a six-fold symmetry is obtained not attached in this case to vortices, but to a mixed structure of dimples and bumps, a precursor of the Wigner type structure observed for few atoms in the Laughlin state at an angular momentum of L = 30 [9]. The degree of condensation as defined in Eq.(5) decreases as 0.343, 0.192 and 0.015 from (a) to (c). The order in vortices and disorder in atoms evolves to order in atoms. As Ω approaches the frequency of the trap, the occupations tend to equalize and in the Laughlin state, where n l are the FD occupations (since it is a pure L-state), and the degree of condensation tends to zero. Some excited states with large L can also be analyzed. For N = 3 and L = 9 we obtain a large vortex state with three units of circulation. Such state has been predicted in previous theoretical studies as a possible giant vortex GS (with all vorticity confined to the center of the condensate), in the presence of a small quartic potential added to the parabolic trap. In such a case stationary states for Ω > ω ⊥ are possible [13]. In our calculations the giant vortex appears as an excited state, anticipating this possibility. So far there is no experimental evidence of giant vortex structures in bosonic systems [14], they have been reported in superconductive disks [15]. Finally we show the interference pattern produced by the overlap of two initially independent condensates represented by ψ 1 functions. This study is motivated by an increasing amount of recent work revealing the possibility of obtaining very detailed experimental information on the interference pattern produced not only during the overlap of two, or more independent condensates [16], but also within a unique condensate [17]. The idea underlying our assumption is the following: we represent the two independent condensates which we call a and b by their macroscopic occupied function ψ a and ψ b respectively. By this we mean that the condensates are in two unknown coherent states |α a and |α b from which we know their order parameter except for their constant phases φ a and φ b (see Eq. (3)). At time t = 0 s the condensates are separated by a distance d and the traps are switched off. The time evolution of the system is obtained (once the transformation to the laboratory frame of reference is performed, multiplying the functions by exp(−iΩtL z )) in three steps: First, the Fourier transform of the total order parameter (the sum of the two contributions) is performed. Then, the time evolution of the Fourier components by multiplying them by exponentials of the type exp(ihk 2 t/2m) is realised; this step is done under the assumption that during the time-of-flight the interactions are irrelevant. Finally, in the third step, inverse Fourier transformation is performed. The results are shown in Fig.2 where three different times are considered. Fortunately, the uncertain about the phase relation φ = φ a − φ b is not important in the case considered, as only two terms are involved and a change on the relative phase would only produce a global shift of the interference pattern. We conclude that, we have demonstrated that the use of the eigenfunctions of the OBDM operator provides a useful and precise tool to analyze the exact GS obtained from exact diagonalization and specially the vortex states. These eigenfunctions localize and quantize the vortices and reproduce the time evolution of the interference pattern of two overlapping condensates. We want to point out that our results imply an alternative interpretation about a subject that has attracted much attention recently related with the interference pattern formation. One possibility suggested by Mullin and collaborators [8] is that the experimental measurement projects the initial condensates in Fock states into phase states, the atom distribution between the two components become uncertain and the pattern formation is possible. The other possibility discussed by Cederbaum et al. [18], is that the interference pattern appears if one includes interaction during the time-of-flight even for states that initially are Fock states. In our case, the real initial states are Fock states and no interaction is included during the time-of-flight. However, we assume that the degree of condensation of the initial states is large enough to be properly represented by an order parameter (condensate wave function). Fluctuations of the number of condensed atoms reduce the phase fluctuations and determine the order parameter phase. In effect, exact ground state manifest themselves as phase states even for small number of particles, and in this way the interference patter is produced. Note, however, that in our picture the process of determination of phase is itself random, and various phases φ k are expected to show up from shot to shot. PACS numbers: 73.43.-f,05.30.Jp, 03.75.Hh FIG. 1 : 1For N = 6 the first two pictures on each row show the density contour plot of the GS (ρ(x, y)) and the ψ1 function (ρ1(x, y)) respectively. The third picture shows tha map of the phase S( r) (see text). (a) shows a two vortex structure at Ω = 0.941 (where degeneracy between L = 10 and 12 takes place). (b) shows a four vortex structure, Ω = 0.979 (degeneracy between L = 20, 22 and 24). (c) shows a six-fold structure, Ω = 0.983 (degeneracy betwee L = 24, 26, 28 and 30). In all cases ω ⊥ = g = 1 in units of λ and u =hω ⊥ . FIG. 2 : 2Time evolution of the interference pattern during the overlap of two released condensates initially separated by a distance d = 15λ. Initially each condensate contains N = 6 atoms and their GS are characterized by L = 6 at Ω = 0.019 and by a mixture of L = 6, 8 and 10 at Ω = 0.0847 respectively (all quantities are in units of λ and u). We thank J.M. Pons and J. Taron for fruitful discussions. 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Ja- vanainen, and S.M. Yoo, Phys. Rev. Lett. 76, 161 (1996); . Y Castin, J Dalibard, Phys. Rev. A55. 4330Y. Castin, J. Dalibard, Phys. Rev. A55, 4330 (1996). . C W Gardiner, Phys. Rev. 561414C.W. Gardiner, Phys. Rev. 56, 1414 (1997); . M D Girardeau, Phys. Rev. 58775M.D. Gi- rardeau, Phys. Rev. 58, 775 (1998); . Y , Castin , R Dum, Phys. Rev. Lett. 775315Y, Castin, and R. Dum, Phys. Rev. Lett. 77, 5315 (1996). . W J Mullin, R Krotkov, F Laloë, cond-mat/0604371W.J. Mullin, R. Krotkov, and F. Laloë, cond-mat/0604371. . N Barberan, M Lewenstein, K Osterloh, D Dagnino, Phys. Rev. A. 7363623N. Barberan, M. Lewenstein, K. Osterloh, and D. Dagnino, Phys. Rev. A, 73, 063623 (2006). . V L Ginzburg, L D Landau, Zh. Eksp. Theor. Fiz. 201064V.L. Ginzburg and L.D. Landau, Zh. Eksp. Theor. Fiz. 20, 1064 (1950); . O Penrose, Phil. Mag. 421373O. Penrose, Phil. Mag. 42, 1373 (1951); . O Penrose, L Onsager, Phys. Rev. 104576O. Penrose and L. Onsager; Phys. Rev. 104, 576 (1956); . C N Yang, Rev. Mod. Phys. 34694C.N. Yang, Rev. Mod. Phys. 34, 694 (1962). . N R Cooper, N K Wilkin, J M F Gunn, Phys. Rev. Lett. 87120405N.R. Cooper, N.K. Wilkin, and J.M.F. Gunn, Phys. Rev. Lett. 87, 120405 (2001). L Jacak, P Hawrylak, A Wójs, Quantum Dots. BerlinSpringer-VerlagL. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer-Verlag, Berlin 1998). . I Danaila, Phys. Rev. A. 7213605I. Danaila, Phys. Rev. A, 72, 013605 (2005); . S Bargi, G M Kavoulakis, S M Reimann, Phys. Rev. A. 7333613S. Bargi, G.M. Kavoulakis, and S.M. Reimann, Phys. Rev. A, 73, 033613 (2006); . A L Fetter, B Jackson, Stringari, Phys. Rev. A. 7113605A.L. Fetter, B. Jackson, and . Stringari, Phys. Rev. A, 71, 013605 (2005); . H Fu, E Zaremba, Phys. Rev. A. 7313614H. Fu, and E. Zaremba, Phys. Rev. A, 73, 013614 (2006). . V Bretin, S Stock, Y Seurin, J Dalibard, Phys. Rev. Lett. 9250403V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004); . S Stock, V Bretin, F Chevy, J Dalibard, Europhys. Lett. 65594S. Stock, V. Bretin, F. Chevy, and J. Dalibard, Europhys. Lett. 65, 594 (2004). . A Kanda, B J Baelus, F M Peeters, K Kadowaki, Y Ootuka, Phys. Rev. Lett. 93257002A. Kanda, B.J. Baelus, F.M. Peeters, K. Kadowaki, and Y. Ootuka, Phys. Rev. Lett. 93, 257002 (2004). . M R Andrews, C G Townsend, H.-J Miesner, D S Durfee, D M Kunr, W Ketterle, Science. 275637M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Dur- fee, D.M. Kunr, and W. Ketterle, Science 275, 637 (1997); . O Vainio, C J Vale, M J Davis, N R Heckenberg, H Rubinsztein-Dunlop, Phys. Rev. A. 7363613O. Vainio, C.J. Vale, M.J. Davis, N.R. Heck- enberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 73, 063613 (2006); . V Gritsev, E Altman, E Demler, A Polkovnikov, ; A Polkovnikov, E Altman, E Demler, cond-mat/060275PNAS. 1036125V. Gritsev, E. Altman, E. Demler, and A. Polkovnikov, cond-mat/060275; A. Polkovnikov, E. Altman, and E. Demler, PNAS 103, 6125 (2006). . S Ritter, A Öttl, T Donner, T Bourdel, M Köhl, T Esslinger, cond-mat/0607102S. Ritter, A.Öttl, T. Donner, T. Bourdel, M. Köhl, and T. Esslinger, cond-mat/0607102. . L S Cederbaum, A I Streltsov, Y B Band, O E Alon, cond-mat/0607556L.S. Cederbaum, A.I. Streltsov, Y.B. Band, and O.E. Alon, cond-mat/0607556.
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[ "arXiv:astro-ph/0509847v1 28 Sep 2005 Torsion effects on vortex filaments and Hasimoto soliton transformation in magnetars", "arXiv:astro-ph/0509847v1 28 Sep 2005 Torsion effects on vortex filaments and Hasimoto soliton transformation in magnetars" ]
[ "L C Garcia De Andrade \nDepartamento de Física\nTeórica -IF -UERJ -Rua São Francisco Xavier 524\nRio de JaneiroRJ\n\nIntroduction\n\n" ]
[ "Departamento de Física\nTeórica -IF -UERJ -Rua São Francisco Xavier 524\nRio de JaneiroRJ", "Introduction\n" ]
[]
The role played by torsion of magnetic vortex line curves or filaments, in the equilibrium state of magnetars is investigated. When the magnetars equilibrium equations are written in terms Frenet-Serret frame it is shown that in regions of the magnetic star where the Frenet torsion is constant it induces an oscillation in the vortex filaments. By solving the magnetar equilibrium equation we shown the this behaviour also appears in the magnetic field. The first derivative of the gravitational potential with respect to the arc lenght of the vortex filament is shown to coincide with the Hasimoto soliton transformation of the Schroedinger equation for the constant torsion.
null
[ "https://export.arxiv.org/pdf/astro-ph/0509847v1.pdf" ]
119,480,960
astro-ph/0509847
ce6af345d5681a7bfa5984a6eaa3abfa1d4ac033
arXiv:astro-ph/0509847v1 28 Sep 2005 Torsion effects on vortex filaments and Hasimoto soliton transformation in magnetars L C Garcia De Andrade Departamento de Física Teórica -IF -UERJ -Rua São Francisco Xavier 524 Rio de JaneiroRJ Introduction arXiv:astro-ph/0509847v1 28 Sep 2005 Torsion effects on vortex filaments and Hasimoto soliton transformation in magnetars The role played by torsion of magnetic vortex line curves or filaments, in the equilibrium state of magnetars is investigated. When the magnetars equilibrium equations are written in terms Frenet-Serret frame it is shown that in regions of the magnetic star where the Frenet torsion is constant it induces an oscillation in the vortex filaments. By solving the magnetar equilibrium equation we shown the this behaviour also appears in the magnetic field. The first derivative of the gravitational potential with respect to the arc lenght of the vortex filament is shown to coincide with the Hasimoto soliton transformation of the Schroedinger equation for the constant torsion. Introduction The role of torsion effects in twisted vortex filaments have been recently investigated in detail by Ricca [1]. In his work he demonstrated that torsion influences cconsiderably the motion of helical vortex filaments in an imcompressible perfect fluid, where the binormal component is responsible for the displacement of the vortex filament in the fluid. Ricca's prescription follows the Moore-Saffman [2] to helices of any pitch. More recently [3] Ricca has discussed the problem of inflexional desequilibrium of inflexional magneticflux tubes, mechanism which has important implications energy balance in solar coronal loops [4] and astrophysical flows. In this work we solve the equilibrium equations of a magnetic stars ,or in brief magnetars, by addopting the method of vortex filaments and writing the magnetar equations in the Serret-Frenet frame. In this approach we are able to investigate the vortex filaments inside the magnetars by making use of the constant torsion approximation. Vortex filaments are known to exists in the interior of the Sun which is a classical example of magnetars [5]. Therefore by considering regions of magnetars where torsion is constant, we are able to show that the magnetic field oscillates in the helical form. This region can give rise to turbulent flows. The paper is organised as follows: in section II we present the model of the magnetars. In section III we present the solution of the equation based on the vortex magnetic lines where the currents and the magnetic field appear in terms of the torsion of the vortex filaments of the star. Section IV presents the discussions. Magnetars in Equilibrium The idea of the magnetars can be better understood if one considers the magnetostatic equations as ∇. B = 0 (1) ∇× H − 1 c J = 0 (2) ∇π = 1 c J× B(3) where we have considered the absence of external forces and the constitutive equations and the total pressure are given respectively by B = µ H (4) π = ρ 2 ∂ ∂ρ e (5) P = π + 1 2µ B 2(6) In the next section we shall examine the magnetostatic system above to the Serret-Frenet frame to investigate vortex magnetic filaments in magnetars. Magnetars and vortex filaments: the effects of torsion Let us now consider the magnetostatic equations in the Serret-Frenet (SF) frame ( t, N, b) where the magnetic field B can be written in terms of the SF frame along the unit vector t as B = B t. Here R is the radius of curvature and the curvature κ(s) = 1 R is constant where s is the measure of lenght along the vortex filament. This reasoning leads to ( B.∇) B = B ∂ ∂s (B t)(7) where the SF formulas are ∂ ∂s t = κ(s) N (8) ∂ ∂s N = −κ(s) t + τ (s) b (9) ∂ ∂s b = −τ (s) N(10) where τ represents the torsion of the filament. By considering the expression ∇π = 0 we obtain the following magnetostatic equations in the SF frame ∇P = t ∂ ∂s ( 1 2µ B 2 ) + N ( B 2 Rµ ) (11) ∇π = [ B 2 Rµ − ∂ ∂N ( 1 2µ B 2 )] N − ∂ ∂b ( 1 2µ B 2 ) b(12) The MHD astrophysical application we consider here is a magnetic star in equilibrium under the action of it is own gravitational field. Therefore here we consider the external force of gravity f = −∇φ (13) and the magnetostatic equations become −∇φ = 1 ρ ∇π − 1 ρc J× B (14) ∇p = 1 c J× B(15) where p := π + ρφ (16) ∂ ∂s b = −τ (s) N(17) where τ represents the torsion of the filament. By substitution of equations (11) and (12) into the magnetostatic equilibrium equation we obtain J N B ρc = ∂ ∂b ( 1 2µ B 2 ) − ∂ ∂b φ (18) J b B ρc = − 1 ρ [ B 2 Rµ − ∂ ∂N ( 1 2µ B 2 )] − ∂ ∂N φ(19) where we have considered the gravitational gradient ∇φ written in the SF frame as ∇φ = ( t ∂ ∂s + N ∂ ∂N + b ∂ ∂b )φ(20) which substitution in the magnetostatic gave rise to ∇φ = −[α b + β N](21) where α and β are given respectively by α = 1 ρc [J N B − ∂ ∂b ( 1 2µ B 2 )] (22) β = 1 ρc [J b B + B 2 ρR − B 2 Rµ − ∂ ∂N ( 1 2µ B 2 )](23) The introduction of the torsion of the filament can be done by derivation of this equation with respect to s ( ∂ ∂s α) b + ( ∂ ∂s b)α + ( ∂ ∂s β) N + ( ∂ ∂s N)β = −∇ ∂ ∂s φ (24) By making use of the SF equations we obtain [ ∂ ∂s α + τ β] b + [ ∂ ∂s β − τ α] N − κβ t = − t ∂ 2 ∂s 2 φ(25) Thus we are left with two scalar equations for α and β and φ in terms of the torsion τ ∂ ∂s α + τ β = 0 (26) ∂ ∂s β − τ α = 0 (27) ∂ 2 ∂s 2 φ − κβ = 0(28) where to simplify matters we consider that the gravitational potential of the magnetars depend only upon the variable s. By considering regions inside the Sun for example where the torsion of vortex filaments are approximately constant we are able to reduce these two scalar equations to harmonic like equations ∂ 2 ∂s 2 β + τ 2 β = 0 (29) ∂ 2 ∂s 2 α + τ 2 α = 0(30) A particular solution for these equations would be α = β = e i τ 2 s(31) which written back in terms of the magnetic field implies the following rela- tions ∂ ∂b ( 1 2µ B 2 ) = ∂ ∂N ( 1 2µ B 2 ) = 0(32) and when R goes to infinity although still inside the star. Conditions (31) implies that the magnetic field depends only on the parameter s. This particular solution presents a degeneracy in the current flows such that J b = J N = ρe i τ 2 s and the magnetic field becomes B(s) = ce i τ 2 s which presents clearly the contribution of torsion on the vortex filaments inside the magnetic star such as the sun. With these solutions for B and Js now the gravitational potential φ may be easily obtained by solving the equation (28) or equivalentely ∂ 2 ∂s 2 φ − κ c e iτ s = 0(33) which yields the Hasimoto [6,7] soliton transformation factor ψ = κe (i τ ds) in the Schroedinger nonlinear equation for the constant torsion of the vortex filaments, on its first quadrature ∂ ∂s φ = −i κ cτ e i( τ ds) (34) whose solution is φ(s) = − κ cτ 2 e (iτ s)(35) This shows that in our particular solution torsion of the vortex filaments makes a major effect on the internal structure of the magnetic star. Conclusions Several phenomena in solar physics such as hot spots has made use of the twisted vortex filament structure. In this paper we generalize the investigation of the effects of torsion to vortex filaments to general types of magnetic stars which may include for example pulsars besides the sun itself. A more general solution of the dynamical equations of the vortex filaments obtained here maybe important for the understanding of internal structure of magnetars. Evolution and Inflexional stability of twisted magnetic flux tubes. R Ricca ; R. Ricca, ; R Ricca, Fluid Dynamics Research. 36241Phys. Rev. AR. Ricca, Inflexional disequilibrium of magnetic flux-tubes ,Fluid Dynam- ics Research 36 (2005) 319. R. Ricca, Evolution and Inflexional stability of twisted magnetic flux tubes. Solar Physics 172 (1997) 241. R. Ricca, Phys. Rev. A (1999). The motion of a vortex filament with axial flow. D W Moore, P G Saffman, Phil. Transactions of R. Soc. London. 272403D.W. Moore and P.G. Saffman, The motion of a vortex filament with axial flow, Phil. Transactions of R. Soc. London 272 (1972)403. The effect of torsion on the motion of a helical vortex filament. R Ricca, Journal of Fluid Mechanics. 237241R. Ricca, The effect of torsion on the motion of a helical vortex filament, Journal of Fluid Mechanics, 237 (1994) 241. J Brat, Plasma Loops in the Solar Corona. Cambridge University PressJ. Brat et al, Plasma Loops in the Solar Corona (1991) Cambridge Uni- versity Press. A C Eringen, G A Maugin, Electrodynamics of continua II-Fluids and Complex Media. Springer-VerlagA. C. Eringen and G. A. Maugin, Electrodynamics of continua II-Fluids and Complex Media (1990) Springer-Verlag. A soliton on a vortex filament. H Hasimoto, J. Fluid Mechanics. 51477H. Hasimoto, A soliton on a vortex filament, J. Fluid Mechanics 51 (1972) 477. . W K Schief, Physics of Plasmas. 2677. W.K. Schief10465J. Plasma PhysicsW.K. Schief, Physics of Plasmas 10,7 (2003) 2677. W.K. Schief,J. Plasma Physics (2003) 65,6,465.
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[ "Least Mean Square/Fourth Algorithm with Application to Sparse Channel Estimation", "Least Mean Square/Fourth Algorithm with Application to Sparse Channel Estimation", "Keywords ; -least mean square fourth (LMS/F), adaptive sparse channel estimation (ASCE), zero-zttracting LMS/F (ZA-LMS/F), re-weighted zero-attracting LMS/F (RZA-LMS/F)." ]
[ "Guan Gui \nDepartment of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan\n", "Abolfazl Mehbodniya \nDepartment of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan\n", "Fumiyuki Adachi [email protected] \nDepartment of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan\n" ]
[ "Department of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan", "Department of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan", "Department of Communication Engineering Graduate School of Engineering\nTohoku University Sendai\nJapan" ]
[]
Broadband signal transmission over frequencyselective fading channel often requires accurate channel state information at receiver. One of the most attracting adaptive channel estimation methods is least mean square (LMS) algorithm. However, LMS-based method is often degraded by random scaling of input training signal. To improve the estimation performance, in this paper we apply the standard least mean square/fourth (LMS/F) algorithm to adaptive channel estimation (ACE). Since the broadband channel is often described by sparse channel model, such sparsity could be exploited as prior information. First, we propose an adaptive sparse channel estimation (ASCE) method using zero-attracting LMS/F (ZA-LMS/F) algorithm. To exploit the sparsity effectively, an improved channel estimation method is also proposed, using reweighted zero-attracting LMS/F (RZA-LMS/F) algorithm. We explain the reason why sparse LMS/F algorithms using -norm sparse constraint function can improve the estimation performance by virtual of geometrical interpretation. In addition, for different channel sparsity, we propose a Monte Carlo method to select a regularization parameter for RA-LMS/F and RZA-LMS/F to achieve approximate optimal estimation performance. Finally, simulation results show that the proposed ASCE methods achieve better estimation performance than the conventional one.
null
[ "https://export.arxiv.org/pdf/1304.3911v1.pdf" ]
6,871,714
1304.3911
71ebb476fa1e96a875ac13a4d9594297e99fb3eb
Least Mean Square/Fourth Algorithm with Application to Sparse Channel Estimation Guan Gui Department of Communication Engineering Graduate School of Engineering Tohoku University Sendai Japan Abolfazl Mehbodniya Department of Communication Engineering Graduate School of Engineering Tohoku University Sendai Japan Fumiyuki Adachi [email protected] Department of Communication Engineering Graduate School of Engineering Tohoku University Sendai Japan Least Mean Square/Fourth Algorithm with Application to Sparse Channel Estimation Keywords ; -least mean square fourth (LMS/F), adaptive sparse channel estimation (ASCE), zero-zttracting LMS/F (ZA-LMS/F), re-weighted zero-attracting LMS/F (RZA-LMS/F). Broadband signal transmission over frequencyselective fading channel often requires accurate channel state information at receiver. One of the most attracting adaptive channel estimation methods is least mean square (LMS) algorithm. However, LMS-based method is often degraded by random scaling of input training signal. To improve the estimation performance, in this paper we apply the standard least mean square/fourth (LMS/F) algorithm to adaptive channel estimation (ACE). Since the broadband channel is often described by sparse channel model, such sparsity could be exploited as prior information. First, we propose an adaptive sparse channel estimation (ASCE) method using zero-attracting LMS/F (ZA-LMS/F) algorithm. To exploit the sparsity effectively, an improved channel estimation method is also proposed, using reweighted zero-attracting LMS/F (RZA-LMS/F) algorithm. We explain the reason why sparse LMS/F algorithms using -norm sparse constraint function can improve the estimation performance by virtual of geometrical interpretation. In addition, for different channel sparsity, we propose a Monte Carlo method to select a regularization parameter for RA-LMS/F and RZA-LMS/F to achieve approximate optimal estimation performance. Finally, simulation results show that the proposed ASCE methods achieve better estimation performance than the conventional one. INTRODUCTION Broadband signal transmission is becoming one of the mainstream techniques in the next generation communication systems. Due to the fact that frequency-selective channel fading is unavoidable, accurate channel state information (CSI) is necessary at the receiver for coherent detection [1]. One of effective approaches is adopting adaptive channel estimation (ACE). A typical framework of ACE is shown in Fig. 1. It is well known that ACE using least mean fourth (LMF) algorithm outperforms the least mean square (LMS) algorithm in achieving a better balance between convergence and steadystate performances. Unfortunately, standard LMF algorithm is unstable due to the fact that its stability depends on the following three factors: input signal power, noise power and weight initialization [2]. To fully benefit from the obvious merits of LMS and LMF, it is logical to combine the two algorithms for ACE purposes. The combined LMS/F algorithm was first proposed by Lim and Harris [20], as a method to improve the performance of the LMS adaptive filter without sacrificing the simplicity and stability properties of LMS. Recently, many channel measurement experiments have verified that broadband channels often exhibit sparse structure. A typical example of sparse system is shown in Fig. 2, where the length of FIR is while number of dominant coefficients is . In other words, sparse channel is consisted of a very few channel coefficients and most of them are zeros. Unfortunately, ACE using LMS/F algorithm always neglects the inherent sparse structure information and it may degrade the estimation performance. In this paper, we propose sparse LMS/F algorithms with application to ASCE. Inspired by least absolute shrinkage and selection operator (LASSO) algorithm [3], to exploit channel sparsity, -norm sparse constraint function is utilized in ASCE. Similar to sparse LMS algorithms, two sparse LMS/F algorithms are termed as zeroattracting LMS/F (ZA-LMS/F) and reweighted zero-attracting LMS/F (RZA-LMS/F), respectively. The main contribution of this paper is proposing the sparse LMS/F algorithms with application to ASCE. Sparse penalized cost functions are constructed for implementing the sparse LMS/F algorithms. Two experiments are demonstrated to confirm the effectiveness of our propose methods. In the first experiment, the average mean square deviation (MSE) performance of sparse LMS/F algorithms is evaluated according to different number of nonzero coefficients. In the second experiment, the MSE performance of propose algorithms is evaluated in different reweighted factors. The remainder of this paper is organized as follows. A system model is described and standard LMS/F algorithm is introduced in Section II. In section III, sparse ASCE using ZA-LMS/F algorithm is introduced and improved ACSE using RZA-LMS/F algorithm is highlighted. Computer simulations are presented in Section IV in order to evaluate and compare performances of the proposed ASCE methods. Finally, we conclude the paper in Section V. II. STANDARD LMS/F ALGORITHM Consider a baseband frequency-selective fading wireless communication system where FIR sparse channel vector [ ] is -length and it is supported only by nonzero channel taps. Assume that an input training signal is used to probe the unknown sparse channel. At the receiver side, observed signal is given by where [ ] denotes the vector of training signal , and is the additive white Gaussian noise (AWGN) assumed to be independent with . The objective of ASCE is to adaptively estimate the unknown sparse channel estimator using the training signal and the observed signal . According to [8], we can apply standard LMS/F algorithm to adaptive channel estimation, with the cost function where is a positive threshold parameter which controls the computational complexity and stability of LMS/F algorithm. With respect to Eq. (2), the corresponding updating equation of LMS/F algorithm is given by when , LMS/F algorithm in Eq. (3) behaves like the LMF with a step size of ; and when , it reduces to the standard LMS algorithm with a step size of . According to the analysis, it is necessary to choose a proper parameter to balance between instability and estimation performance of LMS/F algorithm. Assume the -th received error as , threshold parameter controls the variable step-size as shown in Fig. 3. If we fix the , then smaller parameter achieves smaller step-size which ensures LMS/F more stable and better estimation but at the cost of higher computational complexity, and vice versa. III. SPARSE LMS/F ALGORITHMS A. ASCE using ZA-LMS/F algorithm Recall that the adaptive channel estimation method uses standard LMS/F algorithm in Eq. (2), however, the proposed method does not take advantage of the channel sparsity. This is due to the original cost function in (2) which does not utilize the sparse constraint or penalty function. To exploit the sparsity, we introduce -norm sparse constraint to the cost function in (2) where denotes a regularization parameter which balances the error term, i.e., , and sparsity of . To better understand the difference between (2) and (4), geometrical interpretation is shown in Fig. 4. Cost function in (2) cannot find sparse solution (convex point) in solution plane. Unlike (2), cost function in (4) can find sparse in solution plane due to its sparse constraint. Hence, the update equation of ZA-LMS/F algorithm is given by ( ) where and denotes the sign function which is generated from ( ) ‖ ‖ { where [ ] and . It is well known that ZA-LMS/F uses -norm constraint to approximate the optimal sparse channel estimation [9]. B. Improved ASCE method using RZA-LMS/F algorithm The ZA-LMS/F cannot distinguish between zero taps and non-zero taps since all the taps are forced to zero uniformly as show in Fig. 5. Unfortunately, ZA-LMS/F based approach will degrade the estimation performance. Motivated by reweighted -minimization sparse recovery algorithm [10] in CS [11], [12], we proposed an improved ASCE method using RZA-LMS/F algorithm. The cost function of this method is constructed by ∑ | | where is a regularization parameter which trades off the estimation error and channel sparsity. The corresponding update equation is ( ) | | where is a parameter which depends on step-size , regularization parameter and threshold . In the second term of (11), smaller than ⁄ channel coefficients | | are replaced by zeros in high probability. C. Regularization parameter selection for sparse LMS/F algorithms It is well known that regularization parameter is very important for LASSO based sparse channel estimation [13]. In [14], a parameter selection methods was proposed for LASSO based partial sparse channel estimation. To the best of our knowledge, however, there is no paper report on regularization parameter selection method for ASCE. Here, we propose an approximate optimal selection method by Monte Carlo simulation which adopts 1000 runs for achieving average performance. Parameters for computer simulation are given in Tab Fig. 5 and Fig. 6, respectively. In Fig. 5, it is easy to find that MSE performance is near optimal when regularization parameters are selected as and for and , respectively. Likewise, in Fig. 6, choosing approximate optimal regularization parameters and for RZA-LMS/F can achieve near optimal estimation performance when and , respectively. Hence, these parameters will be utilized for performance comparison with sparse LMS algorithms. IV. COMPUTER SIMULATIONS In this section, the proposed ASCE methods using (R)ZA-LMS/F algorithm is evaluated. To obtain the average performance, 1000 independent Monte-Carlo runs are adopted. The length of channel vector is set as and its number of dominant taps is set to and , respectively. Each dominant channel tap follows random Gaussian distribution as and their positions are randomly allocated within the length of which is subject to || || . The received signal-to-noise ratio (SNR) is defined as ⁄ , where is the unit transmission power. Here, we set the SNR as in computer simulation. All of the step sizes and regularization parameters are listed in Tab. II. Iterative times In the first experiment, average MSE performance of proposed methods is evaluated for and . To confirm the effectiveness of the proposed methods, we compare them with sparse LMS algorithms, i.e., ZA-LMS and RZA-LMS [15]. For a fair comparison of our proposed methods with sparse LMS methods, we utilize the same step-size, i.e., . In addition, to achieve approximate optimal sparse estimation performance, regularization parameters for two sparse LMS algorithms are adopted from the paper [16], i.e., and for ; and for 4. Average MSE performance comparison curves are depicted in Fig. 7 and Fig. 8, respectively. Obviously, LMS/F-type methods achieves better estimation performance than LMS-type ones in [15]. According to the two figures, sparse LMS/F algorithms, i.e., ZA-LMS/F and RZA-LMS/F, achieve better estimation performance than LMS/F due to the fact that sparse LMS/F algorithms utilize -norm sparse constraint function. Average MSE  ZA =1e-3  ZA =8e-4  ZA =6e-4  ZA =4e-4  ZA =2e-4  ZA =1e-4  ZA =8e-5  ZA =6e-5  ZA =4e-5  ZA =2e In the second experiment, as shown in Fig. 9, estimation performance curves of RZA-LMS/F, utilizing different reweighted factors are depicted for and . If we set the numerical values for parameters similar to Tab. II, when , RZA-LMS/F using or can achieve approximate optimal estimation performance. Fig. 9 shows that RZA-LMS/F algorithm depends on reweighted factor. Hence, the proper selection of the reweighted factor is also important when applying the RZA-LMS/F algorithm in adaptive sparse channel estimation. V. CONCLSION In this paper, a novel LMS/F algorithm was applied in ASCE. Based on the CS theory, we first proposed a novel ASCE method using ZA-LMS/F algorithm. Inspired by reweighted -norm algorithm in CS, an improved ASCE method using RZA-LMS/F algorithm was proposed. By Monte Carlo simulation, we proposed a simple method for choosing the approximate optimal regularization parameter for sparse LMS/F algorithm, i.e., ZA-LMS/F and RZA-LMS/F. Simulation results showed that the proposed ASCE methods using ZA-LMS/F and RZA-LMS/F algorithms achieve better performance than any sparse LMS methods. [16] G. Gui Fig. 1 . 1ASCE for broadband communication systems. and obtain a new cost function according to ‖ ‖ Fig. 2. A typical example of sparse multipath channel. Fig. 3 . 3Threshold parameter ( controls the variable step-size of LMS/F algorithm. Fig. 4 . 4Sparse channel estimation with -norm sparse constraint. . I. The estimation performance is evaluated by average mean square error (MSE) which is defined by ‖ ‖ where denotes the expectation operator, and are the actual channel vector and its -th iterative adaptive channel estimator, respectively. TAB. I. SIMULATION PARAMETERS. weighted factor for RZA-LMS/F Utilizing different regularization parameters, performance curves of ZA-LMS/F and RZA-LMS/F are depicted in Fig. 5 . 5ZA-LMS/F based sparse channel estimation performance depends on regularization parameter .Fig.6. RZA-LMS/F based sparse channel estimation performance depends on regularization parameter . Fig. 8 . 8Performance Fig. 9 . 9RZA-LMS/F based sparse channel estimation performance depends on re-weighted factor . , A. Mehbodniya, and F. Adachi, "Regularization selection methods for LMS-Type sparse multipath channel estimation," in submitted for The 19th Asia-Pacific Conference on Communications (APCC 2013), 2013.0 500 1000 1500 2000 2500 3000 3500 4000 10 -4 10 -3 10 -2 10 -1 10 0 Iterative times Average MSE =1 =2 =5 =10 =15 =20 =25 =30 =35 =40 =45 =50 K=2 K=4 SNR=10dB  f =0.04  RZA =0.04 =0.8 ACKNOWLEDGMENT This work was supported in part by the Japan Society for the Promotion of Science (JSPS) postdoctoral fellowship and the National Natural Science Foundation of China under Grant 61261048. Introduction of Frequency-Domain Signal Processing to Broadband Single-Carrier Transmissions in a Wireless Channel. F Adachi, H Tomeba, K Takeda, S Members, IEICE Transactions on Communicationns. 9F. Adachi, H. Tomeba, K. Takeda, and S. Members, "Introduction of Frequency-Domain Signal Processing to Broadband Single-Carrier Transmissions in a Wireless Channel," IEICE Transactions on Communicationns, vol. 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E Eweda, IEEE Transactions on Signal Processing. 603E. Eweda, "Global stabilization of the least mean fourth algorithm," IEEE Transactions on Signal Processing, vol. 60, no. 3, pp. 1473-1477, Mar. 2012. Sparse least mean forth filter with zeroattracting. G Gui, F Adachi, Submitted for IEICE Electronics ExpressG. Gui and F. Adachi, "Sparse least mean forth filter with zero- attracting," Submitted for IEICE Electronics Express, pp. 1-6, 2013. Enhancing Sparsity by Reweighted L1 Minimization. E J Candès, M B Wakin, S P Boyd, Journal of Fourier Analysis and Applications. 145-6E. J. Candès, M. B. Wakin, and S. P. Boyd, "Enhancing Sparsity by Reweighted L1 Minimization," Journal of Fourier Analysis and Applications, vol. 14, no. 5-6, pp. 877-905, Oct. 2008. Combined LMS/F algorithm. S Lim, Electroincs Letters. 336S. Lim, "Combined LMS/F algorithm," Electroincs Letters, vol. 33, no. 6, pp. 467-468, 1997. Fast Solution of L1-Norm Minimization Problems When the Solution May Be Sparse. D L Donoho, Y Tsaig, IEEE Transactions on Information Theory. 5411D. L. Donoho and Y. Tsaig, "Fast Solution of L1-Norm Minimization Problems When the Solution May Be Sparse," IEEE Transactions on Information Theory, vol. 54, no. 11, pp. 4789-4812, 2008. Enhancing Sparsity by Reweighted l1 Minimization. E J Candes, M B Wakin, S P Boyd, Journal of Fourier Analysis Applications. 145-6E. J. Candes, M. B. Wakin, and S. P. Boyd, "Enhancing Sparsity by Reweighted l1 Minimization," Journal of Fourier Analysis Applications, vol. 14, no. 5-6, pp. 877-905, 2008. Compressed Sensing. D L Donoho, IEEE Transactions on Information Theory. 524D. L. Donoho, "Compressed Sensing," IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289-1306, 2006. Robust Uncertainty Principles : Exact Signal Reconstruction From Highly Incomplete Frequency Information. E J Candes, J Romberg, T Tao, IEEE Transctions on Information Theory. 522E. J. Candes, J. Romberg, and T. Tao, "Robust Uncertainty Principles : Exact Signal Reconstruction From Highly Incomplete Frequency Information," IEEE Transctions on Information Theory, vol. 52, no. 2, pp. 489-509, 2006. Regression shrinkage and selection via the lasso. R Tibshirani, Journal of the Royal Statistical Society, Series B. 581R. Tibshirani, "Regression shrinkage and selection via the lasso," Journal of the Royal Statistical Society, Series B, vol. 58, no. 1, pp. 267-288, 1996. Partial Sparse Multipath Channel Estimation using l1-regularized LS Algorithm. G Gui, Q Wan, A M Huang, C G Jiang, IEEE TENCON2008. G. Gui, Q. Wan, A. M. Huang, and C. G. Jiang, "Partial Sparse Multi- path Channel Estimation using l1-regularized LS Algorithm," in IEEE TENCON2008, 2008, pp. 2-5. Sparse LMS for System Identification. Y Chen, Y Gu, A O Hero, Iii , IEEE International Conference on Acoustics, Speech and Signal Processing. Y. Chen, Y. Gu, and A. O. Hero III, "Sparse LMS for System Identification," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2009, no. 3, pp. 3125-3128.
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[ "Nullity Invariance for Pivot and the Interlace Polynomial", "Nullity Invariance for Pivot and the Interlace Polynomial" ]
[ "Robert Brijder \nLeiden Institute of Advanced Computer Science\nLeiden University\nThe Netherlands\n", "Hendrik Jan Hoogeboom \nLeiden Institute of Advanced Computer Science\nLeiden University\nThe Netherlands\n" ]
[ "Leiden Institute of Advanced Computer Science\nLeiden University\nThe Netherlands", "Leiden Institute of Advanced Computer Science\nLeiden University\nThe Netherlands" ]
[]
We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.
10.1016/j.laa.2011.01.024
[ "https://arxiv.org/pdf/0912.0878v3.pdf" ]
15,502,648
0912.0878
165f3515eb52a024978d047e7499bbc4f9eddfa1
Nullity Invariance for Pivot and the Interlace Polynomial 4 Aug 2010 Robert Brijder Leiden Institute of Advanced Computer Science Leiden University The Netherlands Hendrik Jan Hoogeboom Leiden Institute of Advanced Computer Science Leiden University The Netherlands Nullity Invariance for Pivot and the Interlace Polynomial 4 Aug 2010 We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof. Introduction Principal pivot transform (PPT, or simply pivot) is a matrix transformation operation capable of partially (component-wise) inverting a given matrix. PPT is originally motivated by the well-known linear complementarity problem [20], and is applied in many other settings such as mathematical programming and numerical analysis, see [19] for an overview. A natural restriction of pivot is to graphs (with possibly loops), i.e., symmetric matrices over F 2 . For graphs, each pivot operation can be decomposed into a sequence of elementary pivots. There are two types of elementary pivot operations, (frequently) called local complementation and edge complementation. These two graph operations are also (in fact, originally) defined for simple graphs. The operations are similar for graphs and simple graphs, however, for simple graphs, applicability is less restrictive. Local and edge complementation for simple graphs, introduced in [16] and [5] respectively, were originally motivated by the study of Euler circuits in 4-regular graphs and by the study of circle graphs (also called overlap graphs) as they model natural transformations of the underlying circle segments. Many other applications domains for these operations have since appeared, e.g., quantum computing [21], the formal theory of gene assembly in ciliates [11] (a research area within computational biology), and the study of interlace polynomials, initiated in [1]. In many contexts where local and edge complementation have been used, PPT has originally appeared in disguise (we briefly discuss some examples in the paper). In this paper we show that the pivot operator on matrices A (over some field) and the symmetric difference operator on sets Y have an equivalent effect w.r.t. the nullity value of the principal submatrices A[Y ] of A. We subsequently show that this nullity invariant can be formulated in terms of (a sequence of) ⋆ corresponding author: [email protected] set systems. Furthermore we discuss its consequences for pivot on graphs and in particular apply it to the interlace polynomial. It was shown in [3] that the interlace polynomial, which is defined for graphs, fulfills a characteristic recursive relation. We generalize the notion of interlace polynomial and its recursive relation to square matrices in general. In this way, we simplify the proof of the (original) recursive relation for interlace polynomials of graphs. Also, in Section 3, we recall a motivation of pivot applied to overlap graphs, and relate it to the nullity invariant. Notation and Terminology A set system (over V ) is a tuple M = (V, D) with V a finite set, called the domain of M , and D ⊆ 2 V a family of subsets of V . To simplify notation we often write X ∈ M to denote X ∈ D. Moreover, we often simply write V to denote the domain of the set system under consideration. We denote by ⊕ the logical exclusive-or (i.e., addition in F 2 ), and we carry this operator over to sets: for sets A, B ⊆ V , A ⊕ B is the set defined by x ∈ A ⊕ B iff (x ∈ A) ⊕ (x ∈ B) for x ∈ V . For sets, the ⊕ operator is called symmetric difference. We consider matrices and vectors indexed by a finite set V . For a vector v indexed by V , we denote the element of v corresponding to i ∈ V by v[i]. Also, we denote the nullity (dimension of the null space) and the determinant of a matrix A by n(A) and det(A) respectively. For X ⊆ V , the principal submatrix of A w.r.t. X is denoted by A[X]. We consider undirected graphs without parallel edges, however we do allow loops. Hence a graph G = (V, E) can be considered a symmetric V × V -matrix A = (a u,v ) over F 2 (the field having two elements): for u ∈ V , {u} ∈ E (i.e., u has a loop in G) iff a u,u = 1, and for u, v ∈ V with u = v, {u, v} ∈ E iff a u,v = 1. We denote the set of edges of G by E(G). We often make no distinction between G and its matrix representation A. Thus, e.g., we write n(G) = n(A), and, for X ⊆ V , G[X] = A[X] , which consequently is the subgraph of G induced by X. Note that as G is represented by a matrix A over F 2 , n(G) is computed over F 2 . Also, for Y ⊆ V , we define G \ Y = G[V \ Y ]. In case Y = {v} is a singleton, to simplify notation, we also write G \ Y = G \ v. Similar as for set systems, we often write V to denote the vertex set of the graph under consideration. Background: Nullity and Counting Closed Walks In this section we briefly and informally discuss an application of principal pivot transform where nullity plays an important role. In [9] a first connection between counting cycles and the nullity of a suitable matrix was established. It is shown in that paper that the number of cycles obtained as the result of applying disjoint transpositions to a cyclic permutation is described by the nullity of a corresponding "interlace matrix". It has been recognized in [18] that the result of [9] has an interpretation in terms of 2-in, 2-out digraphs (i.e., directed graphs with 2 incoming and 2 We discuss now the link with 2-in, 2-out digraphs (only in this section we consider digraphs). Let G be the 2-in, 2-out digraph of Figure 2 with V = {1, 2, 3, 4, 5, 6} as the set of vertices. Although our example graph does not have parallel edges, there is no objection to consider such "2-in, 2-out multidigraphs". Notice that the double occurrence string s = 146543625123 considered earlier corresponds to an Euler circuit C of G. We now consider partitions P of the edges of G into closed walks (i.e., cycles where repeated vertices are allowed). Note that there are 2 |V | such partitions: if in a walk passing through vertex v we go from incoming edge e of v to outgoing edge e ′ of v, then necessarily we also walk in P from the other incoming edge of v to the other outgoing edge of v. Hence for each vertex there are two "routes". Let P now be the the partition of the edges of G into 3 closed walks as indicated by Figure 3 using three types of line thicknesses. Then P follows the same route as the Euler circuit (corresponding The pivot operation, which is recalled in the next section, has the property that it can map O s1 into O s2 for any two double occurrence strings s 1 and s 2 that correspond to Euler circuits of a 2-in, 2-out digraph G, see, e.g., the survey section of [6]. Pivot In this section we recall principal pivot transform (pivot for short) for square matrices over an arbitrary field in general, see also [19]. Let A be a V ×V -matrix (over an arbitrary field), and let X ⊆ V be such that the corresponding principal submatrix A[X] is nonsingular, i.e., det A[X] = 0. The pivot of A on X, denoted by A * X, is defined as follows. If P = A[X] and A = P Q R S , then A * X = P −1 −P −1 Q RP −1 S − RP −1 Q . (1) Matrix S − RP −1 Q is called the Schur complement of P in A. The pivot can be considered a partial inverse, as A and A * X are related by the following equality, where the vectors x 1 and y 1 correspond to the elements of X. This equality is characteristic as it is sufficient to define the pivot operation, see [19,Theorem 3.1]. A x 1 x 2 = y 1 y 2 iff A * X y 1 x 2 = x 1 y 2(2) Note that if det A = 0, then A * V = A −1 . Also note by Equation (2) that the pivot operation is an involution (operation of order 2), and more generally, if (A * X) * Y is defined, then it is equal to A * (X ⊕ Y ). Nullity Invariant It is well known that any Schur complement in a matrix A has the same nullity as A itself, see, e.g., [22, Section 6.0.1]. See moreover [22, Chapter 0] for a detailed historical account of the Schur complement. We can rephrase the nullity property of the Schur complement in terms of pivot as follows. Proposition 1 (Nullity of Schur complement). Let A be a V × V -matrix, and let X ⊆ V such that A[X] is nonsingular. Then n(A * X[V \X]) = n(A[V ]). The following result is known from [20] (see also [10, Theorem 4.1.2]). Proposition 2. Let A be a V × V -matrix, and let X ⊆ V be such that A[X] is nonsingular. Then, for Y ⊆ V , det(A * X)[Y ] = det A[X ⊕ Y ]/ det A[X]. As a consequence of Proposition 2 we have the following result. Corollary 3. Let A be a V × V -matrix, and let X ⊆ V be such that A[X] is nonsingular. Then, for Y ⊆ V , (A * X)[Y ] is nonsingular iff A[X ⊕ Y ] is nonsingular. We will now combine and generalize Proposition 1 and Corollary 3 to obtain Theorem 5 below. We denote by A♯X the matrix obtained from A by replacing every row v T x of A belonging to x ∈ V \ X by i T x where i x is the vector having value 1 at element x and 0 elsewhere. Proof. By rearranging the elements of V , A is of the following form P Q R S where A[X] = P . Now A♯X is P Q 0 I where I is the identity matrix of suitable size. We have n(P ) = n(A♯X). ⊓ ⊔ We are now ready to prove the following result, which we refer to as the nullity invariant. Theorem 5. Let A be a V × V -matrix, and let X ⊆ V be such that A[X] is nonsingular. Then, for Y ⊆ V , n((A * X)[Y ]) = n(A[X ⊕ Y ]). Proof. We follow the same line of reasoning as the proof of Parsons [17] of Proposition 2 (see also [10,Theorem 4.1.1]). Let Ax = y. Then ((A♯X)x)[i] = y[i] if i ∈ X, x[i] otherwise. As, by Equation (2), ((A * X)(A♯X)x)[i] = x[i] if i ∈ X, y[i] otherwise, we have, by considering each of the four cases depending on whether or not i in X and i in Y separately, By Theorem 5, we see that the pivot operator * X on matrices and the symmetric difference operator ⊕X on sets have an equivalent effect on the nullity values of principal submatrices. (((A * X)♯Y )(A♯X)x)[i] = y[i] if i ∈ X ⊕ Y, x[i] otherwise. Note that Theorem 5 generalizes Corollary 3 as a matrix is nonsingular iff the nullity of that matrix is 0 (the empty matrix is nonsingular by convention). One can immediately see that Theorem 5 generalizes Proposition 1. Also note that by replacing Y by V \ Y in Theorem 5, we also have, equiva- lently, n((A * X)[X ⊕ Y ]) = n(A[Y ]). The "Nullity Theorem" [13, Theorem 2], restricted to square principal sub- matrices, states that if A is an invertible V × V -matrix, then, for Y ⊆ V , n(A −1 [Y ]) = n(A[V \Y ]) . Note that this is implied by Theorem 5 as A * V = A −1 . It is easy to verify from the definition of pivot that A * X is skew-symmetric whenever A is. In particular, if G is a graph (i.e., a symmetric matrix over F 2 ), then G * X is also a graph. For graphs, all matrix computations, including the determinant, will be over F 2 . Set Systems Let A be a V × V -matrix. Let M A = (V, D) be the set system with X ∈ D iff A[X] is nonsingular. Set system M A turns out to fulfill a specific exchange axiom if A is (skew-)symmetric, making it in this case a delta-matroid [4] (we will not recall its definition here as we do not use this notion explicitly). Let M = (V, D) be a set system. We define for X ⊆ V , the pivot (often called twist ) of M on X, denoted M * X, by (V, D * X) where D * X = {Y ⊕X | Y ∈ M }. By Corollary 3 it is easy to verify, see [14], that the operations of pivot on set systems and matrices match, i.e., M A * X = M A * X if the right-hand side is defined (i.e., if X ∈ M A ). Theorem 5 allows now for a generalization of this result from the set system M A of nullity 0 to a "sequence of set systems" P A for each possible nullity i. We formalize this as follows. For a finite set V , we call a sequence P = (P 0 , P 1 , . . . , P n ) with n = |V | and P i ⊆ V for all i ∈ {0, . . . , n} a partition sequence (over V ) if the nonempty P i 's form a partition of 2 V . Regarding P as a vector indexed by {0, . . . , n}, we denote P i by P [i]. Moreover, we define for partition sequence P and X ⊆ V , the pivot of P on X, denoted by P * X, to be the partition sequence (P 0 * X, P 1 * X, . . . , P n * X). Also, we call the vector (|P 0 |, |P 1 |, . . . , |P n |) of dimension n + 1, denoted by P , the norm of P . Clearly, P = P * X , i.e., the norm of P is invariant under pivot. For a V × V -matrix A we denote by P A the partition sequence over V where X ∈ P A [i] iff n(A[X]) = i. As nullity 0 corresponds to a non-zero determinant (this holds also for ∅ as det A[∅] = 1 by convention), we have M A = (V, P A [0]). We now have the following consequence of Theorem 5. Note that X ∈ P A [0] iff A * X is defined. Theorem 8. Let A be a V × V -matrix, and X ∈ P A [0]. Then P A * X = P A * X. Proof. By Theorem 5 we have for all i ∈ {0, . . . , n}, Y ∈ P A * X [i] iff n((A * X)[Y ]) = i iff n(A[X ⊕ Y ]) = i iff X ⊕ Y ∈ P A [i] iff Y ∈ P A [i] * X. ⊓ ⊔ Since the norm of a partition sequence is invariant under pivot, we have by Theorem 8, P A = P A * X . Therefore, for each i ∈ {0, . . . , n}, the number of principal submatrices of A of nullity i is equal to the number of principal submatrices of A * X of nullity i. For X ⊆ V , it it is easy to see that P A[X] is obtained from P A by removing all Y ∈ P A [i] containing at least one element outside X: P A[X] [i] = {Z ⊆ X | Z ∈ P A [i]} for all i ∈ {0, . . . , |X|}. Example 9. For matrix A from Example 6, we have P A = (P 0 , P 1 , P 2 , P 3 ) with P 0 = 2 V \ {{b, c}}, P 1 = {{b, c}}, and P 2 = P 3 = ∅. ⊓ ⊔ Example 10. For graph G from Example 7, depicted on the left-hand side of Figure 4, we have P G = (P 0 , P 1 , P 2 , P 3 , P 4 ) with By Theorem 8 we have for G * X with X = {1, 2, 3}, depicted on the right-hand side of Figure 4, P G * X = (P ′ 0 , P ′ 1 , P ′ 2 , P ′ 3 , P ′ 4 ) where P ′ 0 = {∅, {2}, {4}, {1, 2}, {1, 3}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}}, P ′ 1 = {{1}, {3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3, 4}}, P ′ 2 = {{2, 3, 4}}, P ′ 3 = P ′ 4 = ∅. We have P G = P G * X = (8, 7, 1, 0, 0). ⊓ ⊔ Example 11. In the context of Section 3, where matrix A an overlap graph O s for some double occurrence string s, we have that P Os [i] is the number of partitions of the edges of the 2-in, 2-out digraph D corresponding to s into closed walks of D, such that the number of closed walks is precisely i + 1. The value P Os [0] is therefore the number of Euler circuits in D. ⊓ ⊔ Elementary Pivots on Graphs From now on we consider pivot on graphs (i.e., symmetric V × V -matrices over F 2 ), and thus on all matrix computations will be over F 2 . Hence for graph G, M G = (V, D G ) is the set system with X ∈ D G iff det(G[X]) = 1. Also, G can be (re)constructed given M G . Indeed, {u} is a loop in G iff {u} ∈ D G , and {u, v} is an edge in G iff ({u, v} ∈ D G ) ⊕ (({u} ∈ D G ) ∧ ({v} ∈ D G )), see [7, Property 3.1]. Therefore, the function M (·) assigning to each graph G the set system M G is an injective function from the family of graphs to the family of set systems. It this way the family of graphs may be regarded as a subclass of the family of set systems. Note that M G * X is defined for all X ⊆ V , while pivot on graphs G * X is defined only if X ∈ M G (or equivalently, ∅ ∈ M G * X). In this section we recall from [14] that the pivot operation on graphs can be defined as compositions of two graph operations: local complementation and edge complementation. The pivots G * X where X is a minimal element of M G \{∅} w.r.t. inclusion are called elementary. It is noted in [14] that an elementary pivot X on graphs corresponds to either a loop, X = {u} ∈ E(G), or to an edge, X = {u, v} ∈ E(G), where both vertices u and v are non-loops. Thus for Y ∈ M G , if G[Y ] has elementary pivot X 1 , then Y \X 1 = Y ⊕X 1 ∈ M G * X1 . In this way, each Y ∈ M G can be partitioned Y = X 1 ∪ · · · ∪ X n such that G * Y = G * (X 1 ⊕ · · · ⊕ X n ) = (· · · (G * X 1 ) · · · * X n ) is a composition of elementary pivots. Consequently, a direct definition of the elementary pivots on graphs G is sufficient to define the (general) pivot operation on graphs. The elementary pivot G * {u} on a loop {u} is called local complementation. It is the graph obtained from G by complementing the edges in the neighbourhood N G (u) = {v ∈ V | {u, v} ∈ E(G), u = v} of u in G: for each v, w ∈ N G (u), {v, w} ∈ E(G) iff {v, w} ∈ E(G * {u}), and {v} ∈ E(G) iff {v} ∈ E(G * {u}) (the case v = w).V 1 = N ′ G (u) \ N ′ G (v), V 2 = N ′ G (v) \ N ′ G (u), V 3 = N ′ G (u) ∩ N ′ G (v) . Note that u, v ∈ V 3 . The graph G * {u, v} is constructed by "toggling" all edges between different V i and V j : Figure 5. The other edges remain unchanged. Note that, as a result of this operation, the neighbours of u and v are interchanged. for {x, y} with x ∈ V i , y ∈ V j and i = j: {x, y} ∈ E(G) iff {x, y} / ∈ E(G[{u, v}]), see Example 12. Figure 6 depicts an orbit of graphs under pivot. The figure also shows the applicable elementary pivots (i.e., local and edge complementation) of the graphs within the orbit. ⊓ ⊔ Interestingly, in many contexts, principal pivot transform originally appeared in disguise. For example, PPT was recognized in [15] as the operation underlying the recursive definition of the interlace polynomial, introduced in [1]. We will consider the interlace polynomial in the next section. Also, e.g., the graph model defined in [12] within the formal theory of (intramolecular) gene assembly in ciliates turns out to be exactly the elementary pivots, as noted in [8]. Furthermore, the proof of the result from [9], connecting nullity to the number of cycles in permutations, as mentioned in Section 3, implicitly uses the Schur complement (which is an essential part of PPT). The Interlace Polynomial The interlace polynomial is a graph polynomial introduced in [1,2]. We follow the terminology of [3]. The single-variable interlace polynomial (simply called interlace polynomial in [2]) for a graph G (with possibly loops) is defined by q(G) = S⊆V (y − 1) n(G[S]) . It is is shown in [3] that the interlace polynomial fulfills an interesting recursive relation, cf. Proposition 15 below, which involves local and edge complementation. As we consider here its generalization, principal pivot transform, it makes sense now to define the interlace polynomial for V × V -matrices (over some arbitrary field) in general. Therefore, we define the interlace polynomial for V × V -matrix A as As q(A) (and q ′ (A)) deals with nullity values for (square) matrices in general, one can argue that the nullity polynomial is a more appropriate name for these polynomials. We see that the coefficient a i of term a i y i of q ′ (A) is equal to P A [i] (the element of P A corresponding to i) for all i ∈ {0, . . . , n}. Therefore, we have for matrices A and A ′ , q(A) = q(A ′ ) iff q ′ (A) = q ′ (A ′ ) iff P A = P A ′ . Example 13. Let O s be the overlap graph for some double occurrence string s, and let a i be the coefficient a i of term a i y i of q ′ (O s ). We have, see Example 11, that a i is equal to the number of partitions of the edges of the 2-in, 2-out digraph D corresponding to s into closed walks of D, such that the number of closed walks is precisely i + 1. More specifically, a 0 is the number of Euler circuits in D. The interlace polynomial is originally motivated by the computation of these coefficients a i of 2-in, 2-out digraphs, see [2]. ⊓ ⊔ It is shown in [2] that the interlace polynomial is invariant under edge complementation. By Theorem 8 we see directly that this holds for pivot in general: P A * X = P A and equivalently q(A * X) = q(A). Furthermore, we show that q(A) fulfills the following recursive relation. Theorem 14. Let A be a V × V -matrix (over some field), let X ⊆ V with A[X] nonsingular, and let u ∈ X. We have q(A) = q(A \ u) + q(A * X \ u). Proof. Let P A = (P 0 , P 1 , . . . , P n ). Since X is nonempty and A[X] is nonsingular, P n = ∅. Let R = (P 0 , P 1 , . . . , P n−1 ). Let Z ∈ P i for i ∈ {0, 1, . . . , n − 1}. We have Z ⊆ V does not appear in P A\u iff u ∈ Z iff u ∈ Z ⊕ X iff Z ⊕ X does appear in P A * X\u . Hence R = P A\u + P A * X\u (point-wise addition of the two vectors), and the statement holds. ⊓ ⊔ The recursive relation for the single-variable interlace polynomial in [3] is now easily obtained from Theorem 14 by restricting to the case of elementary pivots on graphs. 1 Proposition 15 ( [3]). Let G be a graph. Then q(G) fulfills the following conditions. Fig. 2 . 2A 2-in, 2-out digraph.outgoing edges for each vertex), linking it to the interlace polynomial[2]. We discuss now this interpretation in terms of 2-in, 2-out digraphs and subsequently show the connection to the pivot operation.LetV = {1, 2, 3, 4, 5, 6} be an alphabet and let s = 146543625123 be a double occurrence string (i.e., each letter of the string occurs precisely twice) over V . The overlap graph O s corresponding to s has V as the set of vertices and an edge {u, v} precisely when u and v overlap: the vertices u and v appear either in order u, v, u, v or in order v, u, v, u in s. The overlap graph O s is given in Figure 1. One may verify that the nullity of O s is n(O s ) = 0. Consider now the subgraph O s [X] of O s induced by X = {3, 4, 5, 6}. Then it can be verified that n(O s [X]) = 2. Fig. 3 . 3Partition of the edges of a 2-in, 2-out digraph into three closed walks. to) s in vertices {1, 2}, while in the other vertices X = {3, 4, 5, 6} it follows the other route. We say that P is induced by X in s. Theorem 1 in [9] now states (applying it to the context of 2-in, 2-out digraphs) that the number of closed walks of a partition P of edges induced by X in s is n(O s [X]) + 1. In our case we have indeed |P | = 3 and n(O s [X]) = 2. For example, the partition of edges induced by {1, 3} in s corresponds to a single closed walk which may be described by the double occurrence string s ′ = 123625146543. It then holds that O s ′ is obtained from O s by pivot on {1, 3}, denoted by O s ′ = O s * {1, 3}. We notice that the partition induced by {1, 3} ⊕ {3, 4, 5, 6} = {1, 4, 5, 6} in s ′ is equal to the partition P induced by {3, 4, 5, 6} in s depicted in Figure 3. Hence n(O s * Y [Y ⊕ X]) = n(O s [X]) for X = {3, 4, 5, 6} and Y = {1, 3}. In Theorem 5 below we prove this property for arbitrary X and Y and for arbitrary square matrices (over some field) instead of restricting to overlap graphs O s . Lemma 4 . 4Let A be a V × V -matrix and X ⊆ V . Then n(A♯X) = n(A[X]). Thus we have ((A * X)♯Y )(A♯X) = A♯(X ⊕ Y ). By Lemma 4, n(A♯X) = n(A[X]) = 0, and therefore A♯X is invertible. Therefore n((A * X)♯Y ) = n(A♯(X⊕ Y )), and the result follows by Lemma 4.⊓ ⊔ Example 6 .QFig. 4 . 64Let V = {a, b, c} and let A be the V × V where the columns and rows are indexed by a, b, c respectively. We see that therefore n(A[{b, c}]) = 1. Moreover, for X = {a, b}, Graphs G and G * X of Example 7. columns of A[X] are independent and thus det A[X] = 0. We have therefore that A * X is defined, and it is given below. 5, we have n(A[{b, c}]) = n(A * X[X ⊕ {b, c}]) = n(A * X[{a, c}]). Therefore n(A * X[{a, c}]) = 1. This can easily be verified given A * X[{a, c}] Example 7 . 7Let G be the graph given on the left-hand side ofFigure 4. Let X = {1, 2, 3}. Then the X × X-matrix belonging to G[X] rows represent vertices 1, 2, 3, respectively. We see that the columns of G[X] are independent (over F 2 ) and therefore det G[X] = 1. Consequently G * X is defined and the graph is given on the right-hand side ofFigure 4. Let now Y = {1, 4}. We see that G[Y ] is a discrete graph (i.e., the graph has no edges). Therefore n(G[Y ]) = 2. Now by Theorem 5, we have n(G[Y ]) = n(G * X[X ⊕ Y ]) = n(G * X[{2, 3, 4}]). One may verify that removing vertex 1 from G * X indeed obtains a graph of nullity 2.⊓ ⊔ P 0 = 0{∅, {2}, {3}, {1, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 3, 4}}, P 1 = {{1}, {4}, {1, 2}, {2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}}, P 2 = {{1, 4}}, P 3 = P 4 = ∅. Fig. 5 .Fig. 6 . 56The other edges are left unchanged. The elementary pivot G * {u, v} on an edge {u, v} between distinct non-loop vertices u and v is called edge complementation. For a vertex x consider its closed Pivoting on an edge {u, v} in a graph with both u and v non loops. Connection {x, y} is toggled iff x ∈ Vi and y ∈ Vj with i = j. Note u and v are connected to all vertices in V3, these edges are omitted in the diagram. The operation does not affect edges adjacent to vertices outside the sets V1, V2, V3, nor does it change any of the loops. The orbit of a graph under pivot. Only the elementary pivots are shown. neighbourhood N ′ G (x) = N G (x) ∪ {x}. The edge {u, v} partitions the vertices of G connected to u or v into three sets − 1) n(A[S]) . We may (without loss of information) change variables y := y−1 in the definition of the interlace polynomial to obtain q ′ (A) = S⊆V y n(A[S]) . DiscussionWe have shown that the pivot operator * X on matrices A and the symmetric difference operator ⊕X on sets Y have an equivalent effect w.r.t. the nullity value of the principal submatrices A[Y ] of A. This nullity invariant may be described in terms of partition sequences P A , where the sets Y ⊆ V are arranged according to the nullity value of A[Y ]. We notice that interlace polynomial of a graph G corresponds to the norm P G of the partition sequence of G (where G is considered as a matrix). Hence we (may) naturally consider interlace polynomials for square matrices in general, and obtain a recursive relation for these generalized interlace polynomials. In this way, we simplify the proof of the (original) recursive relation for interlace polynomials of graphs.1 We use here the fact observed in[15] that the operations in the recursive relations of[3] are exactly the elementary pivots of Section 7, assuming that the neighbours of u and v are interchanged after applying the "pivot" operation of[3] on edge {u, v}. AcknowledgementsR. Brijder is supported by the Netherlands Organization for Scientific Research (NWO), project "Annotated graph mining". Conditions (1) and (2) follow from Theorem 14 where A is a graph G, and X is an elementary pivot (i.e., X = {u} is a loop in G or X = {u, v} is an edge in G where both u and v do not have a loop, see Section 7). Finally, if G is a discrete graph with n vertices, then, for. Y ⊆ V , Y ∈ P |y | Consequently, |P i | = n i . Thus, q ′ (G) = (y + 1) n and therefore q(G) = y nProof. Conditions (1) and (2) follow from Theorem 14 where A is a graph G, and X is an elementary pivot (i.e., X = {u} is a loop in G or X = {u, v} is an edge in G where both u and v do not have a loop, see Section 7). Finally, if G is a discrete graph with n vertices, then, for all Y ⊆ V , Y ∈ P |Y | . Consequently, |P i | = n i . Thus, q ′ (G) = (y + 1) n and therefore q(G) = y n . The interlace polynomial: a new graph polynomial. R Arratia, B Bollobás, G B Sorkin, SODA '00: Proceedings of the Eleventh Annual ACM-SIAM Symposium On Discrete Algorithms. Philadelphia, PA, USAR. Arratia, B. Bollobás, and G.B. Sorkin. The interlace polynomial: a new graph polynomial. In SODA '00: Proceedings of the Eleventh Annual ACM-SIAM Sympo- sium On Discrete Algorithms, pages 237-245, Philadelphia, PA, USA, 2000. Society for Industrial and Applied Mathematics. The interlace polynomial of a graph. R Arratia, B Bollobás, G B Sorkin, Journal of Combinatorial Theory, Series B. 922R. Arratia, B. Bollobás, and G.B. Sorkin. The interlace polynomial of a graph. Journal of Combinatorial Theory, Series B, 92(2):199-233, 2004. A two-variable interlace polynomial. R Arratia, B Bollobás, G B Sorkin, Combinatorica. 244R. Arratia, B. Bollobás, and G.B. Sorkin. A two-variable interlace polynomial. Combinatorica, 24(4):567-584, 2004. Representability of ∆-matroids. A Bouchet, Proc. 6th Hungarian Colloquium of Combinatorics, Colloquia Mathematica Societatis János Bolyai. 6th Hungarian Colloquium of Combinatorics, Colloquia Mathematica Societatis János BolyaiNorth-Holland52A. Bouchet. Representability of ∆-matroids. In Proc. 6th Hungarian Colloquium of Combinatorics, Colloquia Mathematica Societatis János Bolyai, volume 52, pages 167-182. North-Holland, 1987. Graphic presentations of isotropic systems. A Bouchet, Journal of Combinatorial Theory, Series B. 451A. Bouchet. Graphic presentations of isotropic systems. Journal of Combinatorial Theory, Series B, 45(1):58-76, 1988. Multimatroids III. Tightness and fundamental graphs. A Bouchet, European Journal of Combinatorics. 225A. Bouchet. Multimatroids III. Tightness and fundamental graphs. European Journal of Combinatorics, 22(5):657-677, 2001. Representability of ∆-matroids over GF(2). A Bouchet, A Duchamp, Linear Algebra and its Applications. 146A. Bouchet and A. Duchamp. Representability of ∆-matroids over GF(2). Linear Algebra and its Applications, 146:67-78, 1991. Pivots, determinants, and perfect matchings of graphs. R Brijder, T Harju, H J Hoogeboom, arXiv:0811.3500SubmittedR. Brijder, T. Harju, and H.J. Hoogeboom. Pivots, determinants, and perfect matchings of graphs. Submitted, [arXiv:0811.3500], 2008. Cycle decomposition by disjoint transpositions. M Cohn, A Lempel, Journal of Combinatorial Theory, Series A. 131M. Cohn and A. Lempel. Cycle decomposition by disjoint transpositions. Journal of Combinatorial Theory, Series A, 13(1):83-89, 1972. The Linear Complementarity Problem. R W Cottle, J.-S Pang, R E Stone, Academic PressSan DiegoR.W. Cottle, J.-S. Pang, and R.E. Stone. The Linear Complementarity Problem. Academic Press, San Diego, 1992. Computation in Living Cells -Gene Assembly in Ciliates. A Ehrenfeucht, T Harju, I Petre, D M Prescott, G Rozenberg, Springer VerlagA. Ehrenfeucht, T. Harju, I. Petre, D.M. Prescott, and G. Rozenberg. Computation in Living Cells -Gene Assembly in Ciliates. Springer Verlag, 2004. String and graph reduction systems for gene assembly in ciliates. A Ehrenfeucht, I Petre, D M Prescott, G Rozenberg, Mathematical Structures in Computer Science. 12A. Ehrenfeucht, I. Petre, D.M. Prescott, and G. Rozenberg. String and graph re- duction systems for gene assembly in ciliates. Mathematical Structures in Computer Science, 12:113-134, 2002. Completing a matrix when certain entries of its inverse are specified. M Fiedler, T L Markham, Linear Algebra and its Applications. 74M. Fiedler and T.L. Markham. Completing a matrix when certain entries of its inverse are specified. Linear Algebra and its Applications, 74:225-237, 1986. A generalization of Tutte's characterization of totally unimodular matrices. J F Geelen, Journal of Combinatorial Theory, Series B. 70J.F. Geelen. A generalization of Tutte's characterization of totally unimodular matrices. Journal of Combinatorial Theory, Series B, 70:101-117, 1997. Graph polynomials from principal pivoting. R Glantz, M Pelillo, Discrete Mathematics. 30624R. Glantz and M. Pelillo. Graph polynomials from principal pivoting. Discrete Mathematics, 306(24):3253-3266, 2006. Eulerian lines in finite 4-valent graphs and their transformations. A Kotzig, Theory of graphs, Proceedings of the Colloquium. Tihany, Hungary; New YorkAcademic PressA. Kotzig. Eulerian lines in finite 4-valent graphs and their transformations. In Theory of graphs, Proceedings of the Colloquium, Tihany, Hungary, 1966, pages 219-230. Academic Press, New York, 1968. Applications of principal pivoting. T D Parsons, Proceedings of the Princeton Symposium on Mathematical Programming. H.W. Kuhnthe Princeton Symposium on Mathematical ProgrammingPrinceton University PressT.D. Parsons. Applications of principal pivoting. In H.W. Kuhn, editor, Proceed- ings of the Princeton Symposium on Mathematical Programming, pages 567-581. Princeton University Press, 1970. Binary nullity, Euler circuits and interlace polynomials. L Traldi, arXiv:0903.4405European Journal of Combinatorics. To appear inL. Traldi. Binary nullity, Euler circuits and interlace polynomials. To appear in European Journal of Combinatorics, [arXiv:0903.4405], 2009. Principal pivot transforms: properties and applications. M J Tsatsomeros, Linear Algebra and its Applications. 3071-3M.J. Tsatsomeros. Principal pivot transforms: properties and applications. 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[ "Exploring sQGP and Small Systems", "Exploring sQGP and Small Systems" ]
[ "Debasish Das \nSaha Institute of Nuclear Physics\nHBNI\n1/AF700064Bidhannagar, KolkataIndia\n" ]
[ "Saha Institute of Nuclear Physics\nHBNI\n1/AF700064Bidhannagar, KolkataIndia" ]
[]
A strongly interacting Quark-Gluon Plasma (sQGP) is created in the high energy heavy ion collisions at RHIC and LHC. Our present understanding of sQGP as a very good liquid with astonishingly low viscosity is reviewed. With the arrival of the interesting results from LHC in high-energy p+p and p+A, a new endeavour to characterize the transition from these small systems to heavy ions (A+A) is now in place, since, even the small systems showed prominent similarities to heavy ions in the rising multiplicity domains. An outlook of future possibilities for better measurements is also made at the end of this brief review.2 thermalized plasma of deconfined quarks, anti-quarks, gluons and leptons, which was the primordial QGP[14]. The universe may have left the QGP phase after a few microseconds with the available quarks and gluons combining towards the formation of the mesons and baryons[15]. In our laboratories, we can probe the QGP[16,17,18,19,20,21,22]which is a deconfined system of quarks and gluons, by colliding heavy nuclei at relativistic energies. Such collisions, create QGP which can be characterized by colored partons as the dynamic degrees of freedom [23]. Smashing heavy ions, typically Au or Pb ions, at relativistic energies in the present accelerator facilities, such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), can create the QGP. The dynamics of the early universe in terms of the "Big Bang" can be studied experimentally by relativistic nucleus-nucleus collisions at RHIC and LHC in terms of "little bangs" in the laboratory [14, 15, 23]. The main epochs [23] for such a "little bang" collision are : (1) the two nuclei which are lorentz contracted and now disk-like approach each other and collide with a very small traversal time (≪ 1 fm/c).(2) The interactions start developing when the two nuclei hit each other and after such an impact the "hard" processes[11,24,25,27]i.e those which comprise of relatively large transferred momenta Q ≫ 1 GeV between the quarks, anti-quarks or gluons (partons) inside the nucleons of the two nuclei produce secondary partons with large transverse momenta p T [26]. During these times the matter is out of equilibrium and hence will need some time to equilibrate[15].(3)The "soft" collisions or those with small momentum exchange Q < 1 GeV cause copius production of particles after sometime and thermalize the QGP after about 1 fm/c[28]. The QGP now expands hydrodynamically and then cools down approximately adiabatically[13,15]. (4) The QGP then converts to a gas of hadrons and the hadrons continue to interact quasi-elastically, further accelerating the expansion and cooling the fireball until thermal freeze-out ( after ≈ 5-10 fm/c) into thousands of hadrons. The unstable hadrons decay and the stable decay products fly out to the large scale detectors surrounding the interaction region.During the hadronization process the chemical composition of the hadron gas is fixed and remains basically constant afterwards[15,23,28].By studying the behavior of the matter created in "little bangs" we can explore the
10.1142/s0217751x21300143
[ "https://export.arxiv.org/pdf/2211.05330v1.pdf" ]
239,711,898
2211.05330
285c745a86f953df5fc67c1a97e420f750a12104
Exploring sQGP and Small Systems 10 Nov 2022 Debasish Das Saha Institute of Nuclear Physics HBNI 1/AF700064Bidhannagar, KolkataIndia Exploring sQGP and Small Systems 10 Nov 20221 A strongly interacting Quark-Gluon Plasma (sQGP) is created in the high energy heavy ion collisions at RHIC and LHC. Our present understanding of sQGP as a very good liquid with astonishingly low viscosity is reviewed. With the arrival of the interesting results from LHC in high-energy p+p and p+A, a new endeavour to characterize the transition from these small systems to heavy ions (A+A) is now in place, since, even the small systems showed prominent similarities to heavy ions in the rising multiplicity domains. An outlook of future possibilities for better measurements is also made at the end of this brief review.2 thermalized plasma of deconfined quarks, anti-quarks, gluons and leptons, which was the primordial QGP[14]. The universe may have left the QGP phase after a few microseconds with the available quarks and gluons combining towards the formation of the mesons and baryons[15]. In our laboratories, we can probe the QGP[16,17,18,19,20,21,22]which is a deconfined system of quarks and gluons, by colliding heavy nuclei at relativistic energies. Such collisions, create QGP which can be characterized by colored partons as the dynamic degrees of freedom [23]. Smashing heavy ions, typically Au or Pb ions, at relativistic energies in the present accelerator facilities, such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), can create the QGP. The dynamics of the early universe in terms of the "Big Bang" can be studied experimentally by relativistic nucleus-nucleus collisions at RHIC and LHC in terms of "little bangs" in the laboratory [14, 15, 23]. The main epochs [23] for such a "little bang" collision are : (1) the two nuclei which are lorentz contracted and now disk-like approach each other and collide with a very small traversal time (≪ 1 fm/c).(2) The interactions start developing when the two nuclei hit each other and after such an impact the "hard" processes[11,24,25,27]i.e those which comprise of relatively large transferred momenta Q ≫ 1 GeV between the quarks, anti-quarks or gluons (partons) inside the nucleons of the two nuclei produce secondary partons with large transverse momenta p T [26]. During these times the matter is out of equilibrium and hence will need some time to equilibrate[15].(3)The "soft" collisions or those with small momentum exchange Q < 1 GeV cause copius production of particles after sometime and thermalize the QGP after about 1 fm/c[28]. The QGP now expands hydrodynamically and then cools down approximately adiabatically[13,15]. (4) The QGP then converts to a gas of hadrons and the hadrons continue to interact quasi-elastically, further accelerating the expansion and cooling the fireball until thermal freeze-out ( after ≈ 5-10 fm/c) into thousands of hadrons. The unstable hadrons decay and the stable decay products fly out to the large scale detectors surrounding the interaction region.During the hadronization process the chemical composition of the hadron gas is fixed and remains basically constant afterwards[15,23,28].By studying the behavior of the matter created in "little bangs" we can explore the Introduction We all know that the normal nuclear matter is made up of protons and neutrons, which in turn are made up of the quarks [1] and gluons [2]. The quarks and gluons are confined inside the colorless particles called hadrons and free colored particles do not occur. As explained by Quantum Chromo-Dynamics (QCD), the strong interaction is the governing interaction in the subatomic world [3,4]. One of the important experimental observations that QCD needs to decipher, is the confinement of the quarks and gluons [5]. The confinement property is yet not fully understood, even though qualitatively we know about the hadron properties (mesons are bound states of a quark and anti-quark and baryons are bound states of 3 quarks) from the quark model [6]. The refinements of the quark model of hadrons and the development of QCD, naturally led to expectations that matter at very high densities [7,8,9,10] may exist in a state of quasi-free quarks and gluons, the Quark-Gluon Plasma (QGP) [11,12,13]. The very early universe was different than the present times. It was too hot and dense to allow the quarks and gluons to form hadrons and was apparently filled with a phase structure of the strongly interacting matter [14,15,23,28]. The QGP reveals emerging collective behavior [28,29,30,31] that originates from the many-body interactions in QCD. The heavy-ion experiments have explored the close to perfect fluidity aspects of QGP [32,33,34,35], precisely, with varied experimental observables [36]. The new state of strongly interacting matter created in these collisions, have low shear viscosity(η) to entropy density(s) ratio, η/s, which is close to a nearly perfect fluid [15,36,37,38,39,40,41]. The paper is organised to start with a brief introduction of QGP and in Section 2 we have a brief survey of the different avenues of the formation and promulgation of the strongly coupled QGP. The term "strongly coupled QGP (sQGP)" was coined [32,42] as we have realised that QGP formed in relativistic heavy ion collisions is not a weakly coupled gas but on the other hand is more a strongly coupled liquid [33,42,43]. The realization that QGP created at RHIC is not a weakly coupled gas but a strongly coupled liquid has aroused a significant development in this research field. In Section 3 the varied probes for this dense matter formed in our laboratories and their inferences towards the understanding of the small systems like p+p and p+A collisions are discussed. Without the critial understanding of such small systems we cannot characterize the A+A collisions. Finally, we summarise by looking into the future scope for such studies that lie ahread. sQGP The results from the relativistic heavy ion collision experiments have changed the theoretical understanding of the properties of the QCD matter. Also significant know-how has evolved regarding the deconfined QCD matter created in the central interaction volume at such high energies. Previously QGP was felt to be a weakly interacting system of quarks and gluons which might be described by perturbative QCD (pQCD). However contrary to the expectations, the experimental results from RHIC [16,17,18,19], have shown that a hot, strongly interacting, nearly perfect and almost opaque relativistic liquid, also termed as the strongly coupled QGP was created in central Au+Au collisions at the top RHIC energy regime [15,32,33,34]. The comparative studies of the experimental data [16,17,18,19], and especially the elliptic flow (v 2 ) [35], in terms of the hydrodynamic models showed the nearly perfect fluid behavior of QGP. Such inferences indicate that its properties correspond to nonperturbative, strongly interacting matter. RHIC results showed that the resulting plasma could be well described by a hydrodynamic picture of a nearly ideal liquid, which show very limited internal friction or in other words very small shear viscosity (η). The created medium in such relativistic collisions, can connect to the pressure gradients by flowing apparently unobstructed [36,44,45]. Shear viscosity, η, is a characterizing parameter for fluids [44,46,47,48] and can be defined in terms of the friction force F per unit area A produced by a shear flow with transverse flow gradient ∇ y v x , F A = η ∇ y v x .(1) Small shear viscosity is a benchmark for a good fluid. Shear viscosity for a weakly coupled gas can be estimated as η = 1 3 npλ ,(2) where n is the density, p is the average momentum of the gas molecules, and λ is the mean free path. The mean free path can be expressed as λ = 1/(nσ) where σ is a preferable transport cross-section. For relativistic fluids it is more natural to normalize η to the entropy density s rather than the particle density n. It has been observed that good fluids are characterized by η/s ∼h/k B and this value is consistent with simple theoretical propositions. For all fluids, the proposed lower bound based on the results from string theory [49], is, η s ≥h 4πk B .(3) A "perfect fluid" saturates around this value by dissipating the smallest possible amount of energy. A perfect fluid thus follows the laws of fluid dynamics in the largest possible domain [47,48,49]. The experimental results from RHIC indicate that the matter produced in nuclear reactions has a small ratio of η/s [37,38]. The discovery of such a close perfect fluid nature established relativistic fluid dynamics as the new frame-work for deciphering the bulk evolution of the system [36,44]. The observations illustrate that QGP near T c is a strongly coupled one with the properties of a liquid with very low viscosity rather than that of a dilute gas [50,51]. Analysis infers [41] that the averaged specific viscosity of the QGP produced in LHC collisions is quite similar to that for the dense matter created in RHIC energy domain. So, the domain in which matter produced at RHIC/LHC is, T c < T < 2T c , was renamed into a strongly coupled QGP or "sQGP" in short [35,52,53,54,55,56,57]. On the other hand the low value for η/s could also result from an anomalous viscosity η A , originating from turbulent color magnetic and electric fields dynamically produced in the expanding quark-gluon plasma [37,51]. That is, 1/η = 1/η A + 1/η C ,(4) where η A subjugates over the collisional viscosity η C . Such arguments do not rule out a more complex structure of the gluonic component of the matter produced in the relativistic collisions [58]. At LHC energies the inital energy density(at τ 0 = 1 fm/c) is about 15 GeV f m −3 [59]. It is approximately a factor of three higher than the Au+Au collisions at the highest energy regime at RHIC. Some researchers expected that the QGP produced at the LHC would turn back to the previous picture, where quarks and gluons were more weakly coupled at higher temperature. Then the mean free path of particles in the medium and the viscosity will be significant. As a result the experimental signature will emerge as smaller flow components(v n ). But the ALICE elliptic flow v 2 results [60] have clearly shown, the opposite. The dependence of v 2 on transverse momentum is comparable with the RHIC measurements and ALICE has also established that radial flow grows with energy. Understanding sQGP was a challenge which we have researched from RHIC data. However the LHC program has added a lot to our understanding, and the paramount issues in the field now include a critical search to study the evolution between p+p, p+A collisions which are known as "small systems" and heavy ion A+A collisions, with an goal to understand "the smallest drops" of the sQGP showing collective/hydrodynamics 6 behavior [56]. Some of these assumptions are getting tested and understood carefully both in RHIC [61] and LHC [62,63] experiments. At LHC since the collision energies increase, one expects a QGP which is hotter. Such Small Systems Study of QGP requires reference measurements which is provided by the small system (p+p and p+A) collisions [28,68]. QGP is not expected to be formed in small systems as the transverse size of the overlap region is comparable to that of a single proton [20,69,70,71]. Particle production in A+A and p+A, as compared to p+p collisions, expressed as R AA , is termed as the nuclear modification factor. It has long been formulated to understand particle production mechanisms [66]. The R AA of heavy-flavor is expected to be less suppressed and elliptic flow v 2 of heavy-flavor is felt to be smaller in comparison with the light hadrons. The experimental results from ALICE, however, show that the suppression of heavy-flavor hadrons (D-meson) at high transverse momentum (p T ) and its elliptic flow v 2 are comparable to those of the light hadrons [72,73], which needs to be understood [74]. Hence looking into the p+A collisions is required [75], where medium absence provides necessary conditions, to isolate the nuclear effects from the initial hardscattering processes which we often describe as CNM [76,77,78,79]. collisions [76,77,78,79]. The nuclear modification factor of charged particles from CMS experiment [81] in p+Pb collisions, in contrast to the Pb+Pb system at top LHC energies of √ s NN =5.02 TeV, demonstrate no suppression in the 2-10 GeV/c p T region. However we visualize a weak momentum dependence for p T > 10 GeV/c in the p+Pb system, since we observe a moderate excess above unity at high p T for charged particles. Also for heavyflavor(D-meson), the nuclear modification factor, measured by ALICE experiment [82] in p+Pb collisions at same energies, show no suppression within the uncertainties in the measured p T range of 1-24 GeV/c. The strong suppression of the D-meson yields for p T > suppression in the 2<p T <10 GeV/c region, and again a rising trend around 10 GeV/c to the highest p T . The p+Pb and Pb+Pb nuclear modification factors presented in these papers [81,82,83], covering the light and heavy quarks respectively, provide stringent constraints on cold and hot nuclear matter effects. They also clearly establish why the CNM effects are of crucial importance for accurate interpretation of the measurements in heavy ion collisions and in turn advocate the necessity of studying the small system collisions. But at LHC energies do we see any new features in p+p collisions? At LHC energies the particle multiplicity is high and even reach values, which are of the same order as those found in heavy ion collisions at lower energies, and as a matter of fact, they are well above the ones observed at RHIC for peripheral Cu+Cu collisions at √ s NN = 200 GeV [84]. When LHC started with the p+p collisions, the high-multiplicity environment revealed a "ridge" which was measured by CMS [85] while studying the long-range azimuthal correlations for 2.0 < |∆η| < 4.8. The first observation of a long-range ridge-like structure at the near-side (∆φ ≈ 0) was observed for 7 TeV p+p collisions. For the high multiplicity domain of N ≈ 90 or higher, this notable feature is clearly observed for large rapidity differences |∆η| > 2. Also in the high-multiplicity p+Pb collisions at √ s NN =5.02 TeV, the azimuthal correlations for 2.0 < |∆η| < 4.0 showed a qualitatively similar long-range structure at the nearside ∆φ ≈ 0. Thus the long-range, near-side angular correlations in particle production emerged in p+p and subsequently in p+Pb collisions [86], which was further followed by an away-side structure, located at ∆φ ≈ π and exceeding the away-side jet contribution, in p+Pb collisions [87,88]. In a typical p+p collision, a ridge correlation is not expected because the system is too dilute to produce a fluid-like state. This paved the way to encourage the researchers to look for a detailed investigation of the existence of collective phenomena in p+p collisions which was known since long in heavy ion collisions [89]. The strong evidence for the collective nature of the long-range correlations was observed with the charged particles (light quarks) [63] by CMS experiment at √ s = 13 TeV. Also the elliptic flow (v 2 ) coefficients for heavy-flavor decay muons was measured by ATLAS in p+p collisions at same energy [90]. Since heavy quark yields in heavy ion collisions are expected to be modified relative to minimum bias p+p collisions [66], the obvious question arises if their production rates in high-multiplicity p+p collisions at LHC energies show any effect like J/Ψ suppression [24,25]. A stronger than linear rise of the relative production of J/Ψ as a function of multiplicity was observed for p T -integrated yields and this increase is stronger for high-pT J/Ψ mesons which we see for p+p collisions at √ s = 13 TeV [91]. An esclation of the relative J/Ψ and Υ yields [92,93,94] with the relative charged-particle multiplicity was observed in p+Pb collisions at √ s NN =8. 16 TeV [95]. The results in p+A are very similar to the results from p+p collisions [93,94,96]. The rise of the J/Ψ normalized yields are comparable to the increase observed for D-mesons [97,98] which indicate that a common mechanism may be at its origin. A plethora of new, unexpected phenomena have been observed so far in small system (p+p and p+A) collisions, which, produce remarkable similarities to heavy ion phenomenology. Summary and Outlook The more-central Au+Au collisions at RHIC, on the basis of elliptic-flow systematics, have been characterized in terms of a sQGP with small viscosity a "perfect liquid". Crucial input to our comprehension of the sQGP were inferred from the measurements of "collective flow", which in other words is the correlated emission of particles in azimuthal angle around the axis of the colliding beams. Conventionally, we have diagnosed the effects of the sQGP on the final-state particle production and correlations in A+A collisions, by using the relative to baseline measurements of p+p and p+A collisions, and thus assuming that in the smaller, and therefore shorter-lived systems, no QGP effects can happen. At LHC we found new things and even the small systems showed flow features in the rising multiplicity domains. With the increasing multiplicity, the p+p and p+A collisions enter the stage where the macroscopic description (thermodynamics and hydrodynamics) becomes applicable. While hydrodynamic models, when applied to p+A data, can explain many of the observed features, there are serious questions regarding their applicability [99]. Thus, a very detailed description of a broad range of signatures, in an even broader range of systems, will be required to finally demonstrate a full understanding of these new discoveries. Data which will be collected in Run-3 at the LHC, will be a significant addition for such studies. Better picture will be also available with the results from p+Pb collisions. Also exceptionally high-multiplicity p+p collisions are expected in Run-3 and 4 at LHC [100]. The LHC delivered nearly 30 f b −1 by the end of 2012 and propose to reach 300 f b −1 in its first 13-15 years of operation. The second long shutdown (LS-2) before Run-3 will consolidate the luminosity and reliability as well as the upgrading of the LHC injectors. After LS-3, the machine will be in the High Luminosity configuration. The High Luminosity LHC(HL-LHC) is an important and extremely challenging, upgrade [101]. The large p+p collision data sets expected to be collected at the HL-LHC will provide a compelling setting for these investigations [100,102]. Such higher multiplicities will help us to bridge the gap between the p+p and heavy ion collisions, with better detector upgrades in LHC experiments [103]. favourable high energy of LHC is more evident in the area of parton energy loss analogous to the opaque nature of the sQGP where the kinematic domain exceeds that of RHIC.The significant impact of this increase of the collision energy is the huge excess of the rates of hard probes, such as jets, electro-weak particles and heavy-flavors, including the full family of quarkonia ( cc and bb bound states)[59,64]. With a larger in-elastic crosssection, the production of bb pairs will increase more in LHC energies. The abundance ofbb pairs enable the possibility for bottom quark and anti-bottom quark pairs to recombine, following bottomonium state breakup, or combination after the pair forms from the open bottom states. The available high rates allow detail studies of the dense medium using the interactions of these probes with the medium constituents [64, 65, 66].The elastic re-scattering of the heavy quarks in the sQGP is an important element for the understanding of heavy-flavor and single-electron/muon observables in heavy ion reactions at collider energies[26,67]. The produced heavy-flavor interacts with the dense medium by exchanging energy and momentum. The ratio of the measured number of heavy-flavors in heavy ion (A+A) collisions to the expected number in the absence of nuclear or partonic matter i.e p+p collisions, is the definition of nuclear modification factor(R AA ) which is suppressed at high transverse momentum[66]. The elementary degrees of freedom and basic forces at the shortest distances are understood via small systems[15]. So a clear understanding of the small systems emerge as a necessity. The small collision systems like p+p and p+A collisions at LHC energies thus needs detail study to understand the initial and final state effects in Cold Nuclear Matter(CNM), which can provide baseline for the interpretation of heavy ion (A+A) results[28,68]. Broadly the CNM effects emcompass : (i) initial-state nuclear effects on the parton densities (i.e shadowing); (ii) coherent energy loss comprising of initial-state parton energy loss and final-state energy loss; and (iii) the final-state absorption by nucleons, which is expected to be negligible at LHC energies. The CNM effects like the change of the Parton Distribution Functions (PDFs) within the nucleons contained within the nuclei, as compared to the unbound nucleons can modify the interaction and production crosssections[80]. 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[]
[ "An Artificial Neural Network Approach For Ranking Quenching Parameters In Central Galaxies", "An Artificial Neural Network Approach For Ranking Quenching Parameters In Central Galaxies" ]
[ "Hossen Teimoorinia \nDepartment of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada\n", "Asa F L Bluck \nDepartment of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada\n\nInstitute of Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 27CH-8093ZurichSwitzerland\n", "Sara L Ellison \nDepartment of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada\n" ]
[ "Department of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada", "Department of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada", "Institute of Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 27CH-8093ZurichSwitzerland", "Department of Physics & Astronomy\nUniversity of Victoria\n3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada" ]
[ "Mon. Not. R. Astron. Soc" ]
We present a novel technique for ranking the relative importance of galaxy properties in the process of quenching star formation. Specifically, we develop an artificial neural network (ANN) approach for pattern recognition and apply it to a population of over 400,000 central galaxies taken from the Sloan Digital Sky Survey Data Release 7. We utilise a variety of physical galaxy properties for training the pattern recognition algorithm to recognise star forming and passive systems, for a 'training set' of ∼100,000 galaxies. We then apply the ANN model to a 'verification set' of ∼100,000 different galaxies, randomly chosen from the remaining sample. The success rate of each parameter singly, and in conjunction with other parameters, is taken as an indication of how important the parameters are to the process(es) of central galaxy quenching. We find that central velocity dispersion, bulge mass and B/T are excellent predictors of the passive state of the system, indicating that properties related to the central mass of the galaxy are most closely linked to the cessation of star formation. Larger scale galaxy properties (total or disk stellar masses), or those linked to environment (halo masses or δ 5 ) perform significantly less well. Our results are plausibly explained by AGN feedback driving the quenching of central galaxies, although we discuss other possibilities as well.
10.1093/mnras/stw036
[ "https://arxiv.org/pdf/1601.01258v1.pdf" ]
118,544,893
1601.01258
11ac30640af5d6664d9df9d6cec9112d15026905
An Artificial Neural Network Approach For Ranking Quenching Parameters In Central Galaxies 7 January 2016 Hossen Teimoorinia Department of Physics & Astronomy University of Victoria 3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada Asa F L Bluck Department of Physics & Astronomy University of Victoria 3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada Institute of Astronomy Department of Physics ETH Zurich Wolfgang-Pauli-Strasse 27CH-8093ZurichSwitzerland Sara L Ellison Department of Physics & Astronomy University of Victoria 3800 Finnerty RoadV8P 1A1VictoriaBritish ColumbiaCanada An Artificial Neural Network Approach For Ranking Quenching Parameters In Central Galaxies Mon. Not. R. Astron. Soc 00000007 January 2016Printed 7 January 2016(MN L a T E X style file v2.2)Galaxies: formation, evolution, star formation, environments, morphologiesblack holesAGNastronomical techniques We present a novel technique for ranking the relative importance of galaxy properties in the process of quenching star formation. Specifically, we develop an artificial neural network (ANN) approach for pattern recognition and apply it to a population of over 400,000 central galaxies taken from the Sloan Digital Sky Survey Data Release 7. We utilise a variety of physical galaxy properties for training the pattern recognition algorithm to recognise star forming and passive systems, for a 'training set' of ∼100,000 galaxies. We then apply the ANN model to a 'verification set' of ∼100,000 different galaxies, randomly chosen from the remaining sample. The success rate of each parameter singly, and in conjunction with other parameters, is taken as an indication of how important the parameters are to the process(es) of central galaxy quenching. We find that central velocity dispersion, bulge mass and B/T are excellent predictors of the passive state of the system, indicating that properties related to the central mass of the galaxy are most closely linked to the cessation of star formation. Larger scale galaxy properties (total or disk stellar masses), or those linked to environment (halo masses or δ 5 ) perform significantly less well. Our results are plausibly explained by AGN feedback driving the quenching of central galaxies, although we discuss other possibilities as well. INTRODUCTION Explaining why galaxies stop forming stars is a challenging problem in modern astrophysics. The fact that galaxies are observed to come in two broad 'types' in the local Universe is evidenced by the bimodality of several fundamental galaxy properties, including star formation rate (SFR), integrated galaxy colour, and morphology (e.g. Strateva et al. 2001, Brinchmann et al. 2004, Driver et al. 2006, Baldry et al. 2006, Wuyts et al. 2011, Peng et al. 2010, Wake et al. 2012. A compelling picture of how and why galaxies form into these distinct classes is emerging from the theoretical perspective of hierarchical assembly of dark matter haloes, and galaxy formation and feedback within these structures (e.g. Cole et al. 2000, De Lucia et al. 2006, De Lucia & Blaizot 2007, Somerville et al. 2008, Bower et al. 2008, Guo et al. 2011, Henriques et al. 2014, Vogelsberger et al. 2014a,b, Schaye et al. 2015. However, many of the details, including exactly what set of processes cause the quenching of galaxies, is still debated (e.g. Bell et al. 2012, Carollo et al. 2013, Woo et al. 2013, Bluck et al. 2014, Dekel et al. 2014, Knobel et al. 2014, Tacchella et al. 2015, Peng et al. 2015. The fraction of passive (non-star forming galaxies) in a given population has been found to depend strongly on both the stellar mass of the galaxy and the local density in which it resides (Baldry et al. 2006, Peng et al. 2010. The natural division of galaxies by whether or not they are the most massive 'central' galaxy or less massive 'satellite' galaxies in a given dark matter halo, has yielded further insight on this issue, with Peng et al. (2012) finding that central galaxies have a passive fraction mostly correlated with their stellar mass and satellites being more affected by local density. In addition to mass and local density, the structure or morphology of a galaxy also has a strong impact on the passive fraction (e.g. Driver et al. 2006, Bluck et al. 2014). More recent work has found that the central density or mass of the galactic bulge can provide a particularly tight constraint on the passive fraction (e.g. Cheung et al. 2012, Fang et al. 2013, Bluck et al. 2014, Lang et al. 2014, Omand et al. 2014. However, there is also evidence that the mass of the group or cluster dark matter halo, calculated via indirect abundance matching techniques, is a tighter constraint on the passive fraction of centrals than stellar mass (Woo et al. 2013), but not bulge mass or centralised velocity dispersion (Bluck et al. 2014. There are several viable quenching mechanisms suggested theoretically. Galaxy merging offers an initially tempting explanation because it can, in principle, explain the bimodality in SFR (or colour) and morphology (or structure) simultaneously. Galaxies with recent (major) mergers will have their disk components disrupted and diminished and their bulges enhanced (e.g. Toomre & Toomre 1972, Barnes & Hernquist 1992, Cole et al. 2000, although if the merger is gas rich disks may reform (e.g. Burkert & Naab 2004, Hopkins et al. 2013. Additionally, the merging galaxy will initially also have elevated star formation (as seen observationally in, e.g., Ellison et al. 2008, Scudder et al. 2012, Hung et al. 2013) and hence presumably gas consumption (although the observational evidence for this link is mixed, e.g. Ellison et al. 2015 and references therein), potentially leading to a significant lowering of SFR due to a lack of further fuel for star formation (as seen in recent simulations, e.g. Moreno et al. 2015). However, if the galaxy remains connected to the Universe, gas replenishment will inevitably occur from cooling of the hot gas halo, cold gas streams, and minor gas rich mergers. Therefore, merging by itself cannot account for truly (or permanently) passive systems, additional processes will be needed. This is true generally for any quenching mechanism which 'strips' gas from a galaxy but does not prevent further gas inflow, i.e. 'strangling' the galaxy (see Peng et al. 2015 for a discussion). For centrals, it is clear that a source of heat and/or mechanical disruption will be necessary to prevent cooling or accretion of gas onto a galaxy in order for it to cease forming stars. This can be achieved in numerous ways, e.g. through energetic feedback from active galactic nuclei (AGN) (e.g. McNamara et al. 2000, Nulsen et al. 2005, Hopkins et al. 2006a,b, Croton et al. 2006, Bower et al. 2008, Dunn et al. 2010, Hopkins et al. 2010, Fabian 2012, supernovae and stellar winds (e.g. Dalla & Schaye 2008, Guo et al. 2012, Vogelsberger et al. 2014a, Schaye et al. 2015, or by stabilizing virial shocks in haloes above some critical dark matter mass (e.g. Dekel & Birnboim 2006, Woo et al. 2013, Dekel et al. 2014. One other alternative is that the gas is in fact present and continues to be replenished, but somehow cannot be generated into new stellar populations, possibly due to stabilizing torques applied across giant molecular clouds from centrally concentrated mass sources (e.g. Martig et al. 2009). This latter option, however, does not appear to have strong observational support since passive galaxies are most frequently found to lack cold gas reservoirs, which must be explained by other feedback mechanisms that can by themselves account for the lack of ongoing star formation in massive galaxies (e.g. Catinella et al. 2010, Saintonge et al. 2011, Genzel et al. 2015. Due to the relative motion of satellite galaxies through the dark matter potential of the group or cluster, and across the hot gas halo, there are several additional routes available for the quenching of satellite galaxies compared to centrals. Processes such as galaxy -galaxy and host halo tidal interactions, ram pressure stripping, removal of the hot gas halo and subsequent stifling of gas supply from cooling, and pre-processing in groups prior to cluster infall can all result in the quenching of satellite galaxies (e.g. Balogh et al. 2004, Cortese et al. 2006, Moran et al. 2007, van den Bosch et al. 2007, Tasca et al. 2009, Peng et al. 2012, Hirschmann et al. 2013, Wetzel et al. 2013). These environmental processes work in concert with the mass-correlating central galaxy quenching mechanisms outlined above. Thus, the quenching of satellite galaxies is likely to be a much more complex process than that of centrals. In this first work on applying ANN techniques to galaxy quenching we focus on the simpler central galaxy population, with a publication on satellite galaxy quenching to follow (Bluck et al., in prep.). We can potentially identify the dominant central galaxy quenching mechanism, from the contenders outlined above, by investigating which galaxy properties are most closely correlated with the passive fraction. For example, the total energy available for feedback on a galaxy released via an AGN will be roughly proportional to the mass of the central supermassive black hole (Soltan 1982, Silk & Rees 1998, Fabian 1999, Bluck et al. 2011 and hence to the central velocity dispersion and bulge mass (Magorrian et al. 1998, Ferrarese & Merritt 2000, Gebhardt et al. 2000, Haring & Rix 2004, Hopkins et al. 2007, McConnell & Ma 2013. Alternatively, the total energy released from supernovae over the lifetime of a galaxy will be roughly proportional to the total stellar mass of the galaxy, as integrated star formation rate (e.g. Croton et al. 2006, Guo et al. 2011. Further, the energy available from virial shocks in dark matter haloes will be proportional to the the gravitational potential, i.e. the dark matter halo mass, and hence also to the total stellar mass of the group or cluster (Dekel & Birnboim 2006, Woo et al. 2013. Several attempts have been made to identify which galaxy properties are most closely linked to quenching for central and satellite galaxies (e.g. Peng et al. 2010, Woo et al. 2014, Bluck et al. 2014. However, these studies are typically only able to consider one or two variables at a time, motivating the need for a more inclusive and sophisticated analysis methodology. Artificial neural networks (ANNs) are a powerful tool for analysing large and complex datasets and exposing patterns in non-linear physical systems in industry, engineering and the biological sciences (see Wichchukit & O'Mahony 2010 and references therein). Their application to astrophysics has so far been somewhat limited, although there are some noticeable exceptions and successes (e.g. Andreon et al. 2001, Ball et al. 2004, Teimoorinia et al. 2012, Teimoorinia & Ellison 2014. In this work we apply ANN pattern recognition techniques to the multi-variate ranking of parameters that distinguish star forming from passive galaxies. Our aim is to use these rankings to provide observational evidence for or against the dominant quenching mechanisms of central galaxies. The paper is structured as follows: Section 2 describes our data and sample selection. Section 3 outlines the details of our ANN method and analysis methodology as applied to the SDSS. Section 4 presents our results for centrals, including single and multivariables. We discuss what drives central galaxy quenching in light of our results in Section 5, and conclude by giving a summary of our contribution in Section 6. We present an investigation of the potential for sample biases and systematics to affect the results in the Appendix. Throughout we assume a ΛCDM cosmology with {Ω M , Ω Λ , H 0 } = {0.3, 0.7, 70 km s −1 Mpc −1 }, and adopt AB magnitude units. DATA Overview Our data source is the Sloan Digital Sky Survey Data Release 7 (SDSS DR7, Abazajian et al. 2009) spectroscopic sample. We form a sub-sample of 414915 central galaxies with stellar masses in the range 9 < log(M * /M ) < 12 at z spec < 0.2. Full details of this sample, and on the stellar masses, morphologies and structures, star formation rates, and environments of these galaxies are given in Bluck et al. (2014) Section 2, and references therein. What follows in this sub-section is a brief overview of the most important details. The star formation rates for our sample are derived in Brinchmann et al. (2004), with adaptions made in Salim et al. (2007). These are based on spectroscopic emission lines for star forming galaxies with strong emission lines which are not identified as AGN, and via an empirical relationship between the strength of the 4000 Å break (D n 4000) and the specific star formation rate of a galaxy (sSFR = SFR/M * ) for non-star forming (weak or non-emission line) galaxies and AGN. AGN are determined by the Kauffmann et al. (2003) line cut applied to the Baldwin, Phillips & Terlevich (BPT, 1981) emission line diagram, at a S/N > 1. For the strong emission line galaxies which are not AGN, the SFRs are based on Hα, Hβ, [OIII] and [NII] line strengths. For both methods for deducing SFRs a fibre correction is applied, based on the colour and magnitude of light not contained within the spectroscopic fibre. Rosario et al. (2015) have demonstrated that the D n 4000 SFRs can be quite inaccurate; however, in this work we only aim to separate star forming from passive systems. Thus, the high error associated with SFRs in passive systems does not significantly impact our ability to identify them as passive. This is a more complex issue for AGN, where many galaxies could be star forming and still have their SFRs determined from the D n 4000 method. To combat this, we test the effect of removing AGN from our sample in §A2. We find that this does not alter any of our results or conclusions, and hence that our rankings are stable to possible inaccuracies in the SFRs. Stellar masses for our sample, and for the component disks and spheroids, are computed in Mendel et al. (2014) via fitting the observed ugriz magnitudes to model spectral energy distributions (SEDs). For the components, a dual Sérsic (n s = 4 bulge, n s = 1 disk) model is applied in each of the Sloan wave-bands, and combined to form a stellar mass for the bulge and disk components via SED fitting. Details on the bulge-to-total light fitting can be found in Simard et al. (2011), which is based on a GIM2D decomposition (Simard et al. 2002), and details on the mass determination is provided in Mendel et al. (2014). From this, we define the galaxy structure to be: B/T = M bulge M * = M bulge M bulge + M disk(1) where M * is the total stellar mass of the galaxy, taken here as the sum of the component bulge and disk masses. Note that since this is a mass ratio, it is not affected by ongoing star formation and hence provides an independent measure of galaxy structure. This would not be the case with a B/T parameter by light based on a single optical wave-band or a classic Sérsic index (also based on a given wave-band). Bulge effective radius is also taken from the public catalogues released in Simard et al. (2011). Halo masses are estimated from an abundance matching technique applied to the total stellar mass of the group or cluster in which each central galaxy resides. These are taken from the SDSS group catalogues of Yang et al. (2007Yang et al. ( , 2008Yang et al. ( , 2009. At M halo > 10 12 M over 90 % of galaxies are correctly assigned to groups in model data from the Millennium Simulation . Within these groups, the most massive galaxy is defined as the central and all other group members are defined as satellites of that central. This is the same sample of estimated halo masses used in other recent quenching papers (e.g. Woo et al. 2013, Bluck et al. 2014. Velocity dispersions in our sample are derived from the widths of absorption lines, made public in Bernardi et al. (2003), with up-dates to the method added in Bernardi et al. (2007). We discard all velocity dispersions which are derived from line widths with a S/N < 3.5. We also remove all cases where σ err > 50 km s −1 (only a few percent of the sample). Further, for some analyses, we restrict the sample to σ > 70 km s −1 , due to the instrumental resolution of the SDSS, although this has very little impact on our final results (see Section A4). This leaves us with ∼ 80 % of our original sample which pass these data quality cuts. For our main analyses we include the low velocity dispersions in our sample to avoid biasing our input data such that only bulge dominated galaxies are included at low stellar masses. In principle this can lead to a lower predictive power of velocity dispersion, since measurements with higher uncertainty are used, but we test for this explicitly in the Appendices and find that our results and conclusions are unaffected. We then apply an aperture correction, so that all velocity dispersions are computed at the same effective aperture. Specifically, we use the formula in Jorgensen et al. (1995), defining the central velocity dispersion as: CVD ≡ σ e/8 = R e /8 R ap −0.04 σ ap (2) where σ ap is the measured velocity dispersion in the aperture. R e is the bulge (or elliptical) effective radius and R ap is the radius of the aperture in the same units. This aperture correction typically only affects the velocity dispersion by ∼ 10%. We use three qualitatively different metrics of environment in this work: 1) group halo masses, 2) central -satellite divisions (both of which are described above) and 3) local densities. For the local densities, we utilise the normalised surface galaxy density evaluated at the n th nearest neighbour, based on values computed in Baldry et al. (2006). The local densities are calculated as: δ n = Σ n Σ n (z ± δ z )(3) where Σ n = n πr 2 p,n (4) r p,n is the projected distance (in physical units) to the n th nearest galaxy neighbour. Σ n (z ± δ z ) is the mean value of the local density parameter at the redshift range in question. This effectively normalises the density parameter accounting for the flux limit of the SDSS. Thus, a galaxy residing in a perfectly average density of space (at a given redshift) would have log(δ n ) = 0, with galaxies residing in under-densities having negative values and galaxies residing in over-densities having positive values of this parameter. In this work we set n = 5; however, none of our results are strongly affected by this choice, with identical rankings achieved for n = 3 & 10. Defining 'Passive' In order to train the ANN codes to identify passive and star forming systems, we must first have a clear definition of what constitutes a passive (or star forming) galaxy. In this work we follow the prescription for defining passive in Bluck et al. (2014) Section 3. We start by selecting only star forming emission line galaxies, which are not identified as AGN in the BPT emission line diagnostic diagram. Specifically, we select out only those galaxies which are designated as star forming by the Kauffmann et al. (2003) line cut on the BPT diagram, and additionally have a S/N > 3 in all of the relevant emission lines (Hα, Hβ, [OIII] and [NII]). The SFR -M * relationship is uni-modal for this sub-sample (see Fig. 5 in Bluck et al. 2014). We then calculate the distance any given galaxy resides at from this 'star forming main sequence'. Quantitatively, we calculate: ∆SFR = log SFR(M * , z) median(SFR SF (M * ± δM * , z ± δz))(5) where SFR S F is the the star formation rate of the star forming subsample matched at the redshift and stellar mass of each galaxy. The matching thresholds are set to 0.005 for redshift and 0.1 dex for stellar mass and are then increased (in increments of 0.005 and 0.1 dex, respectively) if necessary until a minimum of five star forming 'control' galaxies are found for each galaxy, or else the hard limits of 0.02 and 0.3 dex are reached. In most cases there are > 200 controls available per galaxy and less that one percent of galaxies are excluded from the sample due to lack of controls. The distribution of ∆SFR is highly bimodal (as with the more familiar colour bimodality, e.g. Strateva et al. 2001), and it has a clear minimum at ∆SFR = -1, i.e. at a SFR a factor of ten below the star forming main sequence (see Fig. 1). This provides a natural constraint to separate passive from star forming galaxies. The minimum of this distribution does not vary as a function of mass, morphology, or local density, hence it is a very stable and universally applicable definition for passive (see Fig. 7 in Bluck et al. 2014). Thus, we define passive and star forming galaxies to be: PA: ∆SFR -1 SF: ∆SFR > -1 In some of the analyses that follow in this paper we consider the possibility of a third classification, that of the 'green valley'. The ∆SFR limits for this configuration are given by: PA: ∆SFR -1.2 GV: -1.2 < ∆SFR < -0.6 SF: ∆SFR -0.6 None of our conclusions depend critically on whether we adopt two or three star forming classifications for our sample (see Section A3). It is important to stress at the outset that our approach implicitly assumes that there are only two (or three including the green valley) star formation states a galaxy can be in. This is a reasonable simplification given the extent of the bimodality of ∆SFR; however, our approach in this paper will not be sensitive to subtle trends in sSFR or green valley migration as is evidenced in some other works (e.g. Schawinski et al. 2014. ANN Input Parameters There is a wide variety of possible galaxy properties we could include in our ANN analysis of star forming and passive systems, however there are a few constraints that must be met. First, it is crucial to avoid using galaxy parameters which are trivially related to the SFR or colour of a galaxy. This rules out using magnitudes, colours, luminosities, as well as SFR variants (e.g. sSFR, ∆SFR) as input parameters. Also structural parameters based on single magnitudes will be highly biased by ongoing star formation in the optical, hence, we must avoid using B/T or n s parameters, if they are based on luminosities as opposed to stellar masses. We choose eight different galaxy parameters, all of which are not trivially linked to star formation, but are connected to various proposed theoretical mechanisms for quenching central galaxies. They represent a wide range in scale and hence may help to resolve which of the leading theories for galaxy quenching are most likely to be correct, and to what degree they can be impacting the evolution of central galaxies. The physical parameters of the central Local Density Parameter 0.5 -3 Mpc * Approximate 1 σ range from centre of galaxy. For photometric quantities half-light radii are used. galaxies used in this work are shown in Table 1. Note that there are parameters connected to the galaxy environments (M halo , δ 5 ), the outer regions of galaxies (M disk ), the whole galaxy (M * , B/T) and the inner regions of galaxies (CVD, M bulge , Re). This should provide a valuable test as to the scale and range of the quenching process for centrals. THE METHOD ANN In many situations linear models are not sufficient to capture complex phenomena, and thus non-linear models such as artificial neural networks (ANNs) are necessary. ANNs are among the most powerful tools in pattern recognition problems. They consist of simple mathematical units which are connected to each other in different layers and in different, often highly complicated, ways. In a multi-layer network, each layer adds its own level of nonlinearity. So, naturally, a single layer network cannot produce the non-linearity that can be seen through multiple layers. A two-layer network is strong enough to handle a multi-parameter problem such as our classification problem in this paper and is frequently applied in similar works (e.g. Ellison et al. 2016). The specific configurations are chosen based on the nature of the problem under study, and in this way ANNs can learn to detect regularities, correlations and patterns in certain sets of data. Current applications of ANNs in astronomy include star-galaxy discrimination and galaxy classification (e.g. Cortiglioni et al. 2001;Andreon et al. 2001;Ball et al. 2004;Teimoorinia 2012;Teimoorinia & Ellison 2014), but their power in data analysis has been largely untapped. Generally, input parameters (e.g. parameters in Table 1) are connected to the first layer of a network with some mathematical units (nodes) which are called neurons. The first layer can be connected to a second layer (with some new neurons, in different and complicated ways) and, at the end, the second layer are connected to the output layer. In a binary classification, the output layer contains only two nodes. Through iteration between inputs and outputs, the parameters of the mathematical nodes (weights and biases) can be fixed to optimise solving the classification problem. In this way we will have a trained network. In fact, the aim in training steps is to minimise the difference between the predicted and observed values by a performance function such as, e.g., a mean square error function. A trained network should then be validated (during the training steps or after training) by an independent data set to test performance of the trained network and also to avoid over-fitting problems. Overfitting is then evident by the result for a training set being good but for a validation set being unacceptable. An ANN model is generally 'learned' from a set of training data where, in a supervised learning mode, the training data is labelled with the 'correct' answers. Since the aim of finding a model is to provide useful predictions in future situations, questions about choosing a model are important, especially when we do not know much about the underlying nature of the process being studied. ANNs offer a powerful solution to this problem by allowing the analysis algorithm to form its own model 'organically' from iterations with the training set. In many cases, we may wish to learn a mapping from D-dimensional inputs to scalar, or G-dimensional, outputs. In other words, both the inputs and outputs may be multidimensional. In these complicated situations few techniques in the machine learning area are as effective as ANN minimisation analyses. These approaches are highly effective for many complex problems, such as finding a patterns in large datasets and in classifying non-linear multi-dimensional data between predetermined sets or classes (as in this work). In a classification problem the general goal of the ANN is for the algorithm to 'learn' a decision boundary or a threshold. Here, we have two (or three including the green valley, see Section A3) predetermined classes for the star forming states of SDSS galaxies: passive (PA) and star forming (SF). Generally, once a model is found by the network we perform a classification by comparing the posterior class probabilities, i.e. P(SF|x) and P(PA|x), in which x is our multidimensional input data. Thus, from Bayes theorem, we have: P(PA|x) = P(x|PA)P(PA) P(x) = P(x|PA)P(PA) P(x|PA)P(PA) + P(x|SF)P(SF)(6) The above equation can be written as: P(PA|x) = f(g(x)) = 1 1 + e −g(x)(7) In which f is a sigmoid (or activation) function and g is given by: g(x) = ln P(x|PA)P(PA) P(x|SF)P(SF)(8) In an our ANN approach we use a two-layered network, specifically modelling the data as: P(PA|x) = f j=1 w (2) j f i=1 w (1) i,j x i + b (1) i + b (2) j(9) in which w and b are weights and biases of the network in different layers that are fixed by the training steps. The suffix (1) indicates the first layer and (2) indicates the second layer. In this way we construct a model of the class probability given the measurement (as in Bishop 1995). Our results are stable to issues of over-fitting because we use a neural network model with typically ten neurons applied to a training set of 100,000 galaxies as input data. Moreover, we have many unused galaxies from training with which we can verify the fit on an independent 'validation set' of ∼ 100,000 different galaxies. The results from this study are always identical for both of these sets. We also apply an early stopping technique in which the training set is itself split into two sub-sets (70% training and 30% validation) to test the performance of the two sets at an early stage of development. If they show different behaviours we can identify issues and retrain accordingly. Finally, we repeat the training several times and exclude the worst cases (where a global minimum solution is not found) from our final analysis. In this manner we always concentrate on the 'best' possible results from our network model, taking All the passive galaxies (∆SFR < -1) are labeled by a value of 1 for the purposes of our ANN minimisation. We assign to all star forming galaxies (∆SFR -1) a value 0. The output of the ANN procedure will thus be a probability (between 0 and 1) for how likely each galaxy is to be passive or star forming, given the input data. Bottom panel: the same as the top panel but showing the green valley galaxies as a separate class, which are excluded from some analyses. Figure 2. Output ANN probabilities for galaxies being star forming (X = 0) or passive (X = 1) for two categories: originally determined passive galaxies (the red line) and star forming galaxies (the blue dashed line). This shows the perfect case of equation 10 (for α = 0), i.e. where we give the ANN codes all the relevant information to assign the passive state of each system. Unsurprisingly, this yields a 100% accurate classification. The original classification of the data is shown in Figure 1. the average over these as our performance indicator. Multiple application of our ANN procedure on the same problem also ensures that our results are converged, and hence have settled in a global minimum solution. In the next sub-section we give an example of our ANN approach applied to a simplified dataset to illustrate our analysis techniques. ANN Performance Test and Example In Figure 1 we show the distribution of ∆SFR for our sample of central galaxies. A cut at the minimum of this distribution (∆SFR = -1) cleanly separates the galaxies into two different groups (see Section 2.2, and Bluck et al. 2014 for full details). This is an example of a binary classification. In this kind of problem, a classifier can classify input data into two desired classes. The input values can be different physical properties (e.g. the physical galaxy parameters in Table 1) with different combinations, i.e. single or multiple variables. However, the target data are always just two different labels: SF | PA. For statistical purposes, we designate these possibilities by two real numbers, 0 and 1. In this case, one can associate an output value of zero to star forming galaxies and an output of value of 1 to passive galaxies. In practice, the output of the network will be the estimated probability that the input pattern (from the data) belongs to one of the two categories. To test our method (and illustrate our analysis technique) we use the definition of 'passive', from ∆SFR, as an input data to the ANN. But we distort it in a coherent manner to ascertain the effect of noise (or randomness) on the pattern recognition. Specifically, we define the transform: ∆SFR −→ ∆SFR + αR(10) In which R is a random number between -1 and 1. In other words, with increasing α we add more random 'noise' to our input data 'signal' (∆SFR). In a binary classification, the output of a classifier will be two different probability distributions (i.e. how likely each galaxy is to belong in each category). A trivial example is when α = 0, i.e. when we give the ANN training code all of the information it needs to decide unambiguously whether or not each galaxy belongs in the passive or star forming sample. In this case a classifier should be able to classify the data perfectly. Thus, no overlap (or misclassifications) of the two distributions is expected. We show the result of this test in Figure 2. As expected 100% of the data is correctly classified into the two categories: star forming (blue line) or passive (red line). We increase the value of α to see how the output of the network depends on increased noise, or randomness in the input data. We train the networks on a 'training set' of 50,000 passive and 50,000 star forming galaxies, randomly chosen from our parent sample. We then apply the newly formed model to an independent 'validation set' of 50,000 different passive and 50,000 different star forming galaxies. We find that our network rankings are converged, i.e. training or verifying on larger samples or running the codes for longer leads to no significant changes in the results or rankings. The output of our trained network on the independent validation set for different values of α is shown in Figure 3. As can be seen, with increasing α the ability of the ANN to distinguish between the two categories becomes diminished. In fact when we increase α by a factor of ten the two distributions become almost indistinguishable. These histograms can be used as a useful comparison to the real data analysis later on, see Section 4. In each case we can assign a performance to our classification, which we describe in detail in the next sub-section. Receiver Operating Characteristic (ROC) A Receiver Operating Characteristic or ROC plot is a statistical tool used to measure the performance of a binary classifier (e.g., Fawcett 2006). To demonstrate how we determine the performance Figure 3. Output ANN probability distributions for four example cases of our randomness parameter, α. In each plot the X-axis shows the probability that each galaxy is passive based on the best fit minimisation procedure from the ANN (where 0 = SF, 1 = PA). The red lines are for originally classified passive galaxies, with the blue lines being for originally classified star forming galaxies. The ideal case (for α = 0) is shown in Figure 2. The Yaxis shows the normalised number of galaxies in each probability binning, summing to one. As can be seen, the distributions become less separated as we increase α from top to bottom, indicating less success in predicting the passive fraction by the ANN methods as we increase the noise or randomness of the input data. These distributions can be used as a comparison to the equivalent plots for the science parameters in Figure 8. . Output ANN probability distribution for α = 0.5 case (where 0 = SF, 1 = PA) for originally classified star forming (blue) and passive (red) galaxies. The vertical grey dashed line at X = 0.7 shows a randomly selected threshold. For this threshold, the red shaded area to the right of the line gives the True Passive Rate (TPR = 0.783), and the blue shaded area to the right of the line gives the False Passive Rate (FPR = 0.069). Note that in general FPR + TPR 1, since the sum of the red area and the sum of the blue area (from X = 0 -1) is unity, not the sum of the blue and the red areas across any given threshold. of our ANN classifier we re-plot the distribution related to the value of α = 0.5 in an area format in Figure 4. The red and blue areas show the probability distributions for passive (PA) and star forming (SF) galaxies, respectively. Here, we focus on the passive galaxies which are originally labeled with value 1, although an equivalent formulation of this statistic based on the star forming sample (originally labelled as 0) is also possible. These will give equivalent results because of the binary nature of our experimental setup, i.e. P(PA) = 1 -P(SF). On the right hand side of any selected threshold (decision boundary) we will have two relevant percentage values. For example, on the right hand side of the vertical dashed line in Figure 4 (at a threshold at X = 0.7), the fraction of galaxies that are correctly classified as passive is 0.783. We call this the True Passive Rate (hereafter TPR), thus, we have TPR = 0.783. However, there are also some star forming galaxies in this region of the probability distribution which are misclassified as passive galaxies. We call this fraction the False Passive Rate (FPR). For our example threshold at X = 0.7, FPR = 0.069. Thus, for any selected threshold we will have two values: TPR and FPR. The ROC graph is obtained by plotting TPR vs. FPR for all possible thresholds (0 -1). Figure 5 shows this for α = 0.5. This curve can be used to quantify the performance of our classification (as in Bradley 1997). Higher areas under the ROC curve (hereafter AUC) indicate a better performance of the network in determining the correct star forming or passive state of galaxies. We plot ROC curves related to different values of α in Figure 6. The black dashed line is for the perfect classification (where α = 0), which yields an AUC = 1. A sample of completely random numbers (α → ∞) will generate the (diagonal) grey dashed line, with AUC = 0.5. All other values of α will yield an AUC performance between these extremes. So, from random to perfect classification the value of AUC changes from 0.5 to 1, respectively. In the engineering literature (e.g. Hosmer & Lemeshow 2000) the AUC values correspond to success 'labels', see Table 2. We obtain all AUC values associated with the different α values and plot these in Figure 7. As can be seen, the area under the curve varies from a perfect classification (at α = 0) with a value of 1 to an almost random result of 0.55 (at α=10). Thus, the AUC statistic strongly correlates with the true 'signal' in the data, in this case ∆SFR. Higher randomness or noise leads to lower AUC values. Our analysis techniques are now ready for exploitation on real data. In order to determine which galaxy properties modulate the quenching of star formation, we consider each variable from Table 1 in turn (and combinations thereof) as input to the ANN, and quantify how well they discriminate the passive and star forming populations. As described above, successful discrimination is characterized by a large AUC for that variable (or set of variables). The AUC results can then be ordered to give a quantitative ranking of the parameters' relative importance in determining whether or not a galaxy is star forming. In the following section we describe our results for central galaxies. RESULTS In this section we describe our results for central galaxies, following the method outlined in Section 3. ANN pattern recognition training is performed on 50,000 SF and 50,000 PA galaxies for each configuration of science variables considered. Once the ANN model is constructed, it is tested on a new verification set of 50,000 SF and 50,000 PA galaxies. The output probability distributions, Figure 6. Receiver Operating Characteristic (ROC) curves obtained from the ANN output probability distributions for varying values of the randomness parameter, α, shown in Figure 3. Specifically, we plot the True Passive Rate (TPR) vs. the False Passive Rate (FPR), see Section 3.3. For α = 0 the performance is perfect with AUC = 1, the AUC then decreases systematically with increasing α, up to a theoretical limit of AUC = 0.5 as α → ∞. ROC curves and AUC parameters are determined for each case. From this, a ranking of how important different galaxy properties, and sets of two and three properties, are for determining the passive state of galaxies is constructed. Single Parameters Here we use the single parameters drawn from Table 1 as input data to the ANN pattern recognition algorithm. Initially, to show the maximum potential of our data and the ANN classifier, we perform a run in which all of the parameters are used simultaneously as input data. The distribution of the output probabilities for the two original classes for this case is shown at the top of Figure 8. Two simple monotonic distributions are seen, one peaked at zero (for star formers, shown in blue) and one peaked near unity (for passive galaxies, shown in red). For example, if we choose a probability threshold at X = 0.5 we see that there are some misclassifications. Since we do not have a perfect classification, any single (or multiple) run should be compared to this run, which we hereafter label as 'ALL'. However, the success rate of the ALL run is formally 'outstanding' (see Table 2, and Hosmer & Lameshow 2000), classifying > 90% of cases in the validation set correctly. The rest of the panels in Figure 8 show the distributions of the ANN probabilities for each galaxy being passive for originally classified passive (red) and star forming (blue) galaxies, for each of the parameters in Table 1 treated singly. In general, the histograms in Fig. 8 for single runs can be compared to the test-data histograms in Fig. 3 to build some intuition for how the physical parameters perform compared to different levels of known degradation of information on the passive state. Central velocity dispersion and bulge mass perform qualitatively well, with simple monotonic distributions for each class, as with the ALL variables run. This behaviour is not seen for B/T, however, where there are many uncertain cases around probability X=0.5, although strong 'correct' peaks at the extremes of the distribution are also present (we consider whether this could be a result of ambiguous 'green valley' galaxies in Section A3). Particularly poorly separated distributions are seen for disk mass, bulge effective radius and δ 5 . For the former two parameters, the poor separation of star-forming and passive distributions is due to having a discrete value of zero for disk mass related to pure bulge (elliptical) galaxies. It is useful for the ANN code to know that there is no disk (this usually indicates a passive galaxy); however, knowing that it contains a disk does not determine the passive state with any kind of accuracy. Similarly, a very small bulge radii almost always indicates a star forming galaxy, but higher bulge radii can lead to a variety of masses, due to the underlying structure of the bulge. We test what impact spurious bulges or disks may have on our rankings in Appendix A7. When we use the density parameter, δ 5 , as the input data the output is very similar to the case where α = 10 in the previous section. The two distributions are not distinguishable indicating that this parameter acts like a random number and has no connection to passivity. We show the ROC plot (defined in Section 3.3) associated with each of the single parameters, as well as the ALL parameter run, in Figure 9. The black solid line is related to ALL, which has the best performance and the largest AUC. We estimate the associated AUC for each of the single variables and show them in Figure 10. To obtain the errors we perform many ANN runs for each single parameter and obtain the mean and standard deviation from the best well trained networks (top 10 results out of 15 total runs, each selecting a random 50,000 PA and 50,000 SF galaxies for training and a different random 100,000 galaxies for validation), ensuring an optimal solution has been found. The parameters on the X-axis of Figure 10 are ordered by their AUC values, i.e. showing most to least constraining variable. See Table 3 for the rankings and AUC values for centrals. The physical galaxy properties are ordered in Table 3 by their AUC values, and hence by how predictive they are of whether a galaxy will be forming stars or not. The ordering is largely similar to Table 1, which is sorted by the scale at which each property is measured. Thus, there is a broad (but not perfect) trend from inner to outer regions in terms of quenching predictivity. CVD, M bulge and B/T are all ranked as "excellent" by our performance metric, with CVD being the single best performing property. This result is in agreement with previous papers (e.g. Cheung Table 1 are used simultaneously as input data. The distributions of the eight single runs (single input data) are shown below. The parameters are organised from most predictive (top left) to least predictive (bottom right). These distributions can be compared to the trial case (for varying randomness, α), shown in Figure 3. Table 2 and associated text for definition. The errors are quoted as the standard deviation across the best 10 (out of 15) ANN runs, ensuring convergence. Table 1, plotting True Passive Rate (TPR) vs. False Passive Rate (FPR), see Section 3.3 for details. The best performance (largest area under the curve, AUC) is achieved for all variables used together, 'ALL', shown as a black solid line. The next best (and best single variable) is CVD followed by M bulge , i.e. it is parameters related to the inner-most regions of galaxies which perform best. An example random result is shown as the dashed black line, which performs only slightly worse than the local density (δ 5 ) parameter or bulge effective radius. fraction. Parameters associated with the galaxy's outer region or environmental metrics perform significantly less well. Such parameters include total stellar mass and halo mass which have frequently been used in the literature to parameterise the quenching of centrals (e.g. Peng et al. 2010Peng et al. , 2012Woo et al. 2013Woo et al. , 2015. Interestingly, the size of the bulge is the worst performing parameter, possibly suggesting that it is the mass and/or density of the inner region not its scale that affects star formation quenching. It is also interesting to note that bulge size is a particularly poor correlator to dynamical measurements of central black hole mass (e.g. Hopkins et al. 2007 Fig. 10 and Table 3 provide compelling evidence for the process that quenches central galaxies originating in the inner regions of galaxies. Figure 9) with respect to the single galaxy parameters input data, given in Table 1. The parameters on the X-axis are sorted in terms of their AUC values, from highest (most predictive of the passive state of galaxies) to the lowest (least predictive of the passive state of galaxies). The errors are given as the standard deviation across the best ten runs, with the data points taken as the mean of the set. The points are colour coded by their success labels, as indicated on the plot (see Table 2). Clearly, parameters related to the inner regions of galaxies perform systematically better than parameters related to the whole galaxy, the outer regions, or environments of galaxies. Implications of the Single Runs It is interesting that halo mass performs significantly better at predicting the passive state of central galaxies than local density, even though they are both ostensibly environmental parameters. Ellison, Patton & Hickox (2015) find that halo mass is strongly correlated with the presence of radio loud AGN, whereas local density is not, which may offer us an explanation through the AGN driven quenching paradigm. Additionally, there are well known strong correlations between internal galaxy properties (e.g. stellar mass, B/T ratio, M BH ) and halo mass which are much weaker for local density (e.g. Moster et al. 2010). Further, Woo et al. (2013) argue that local density is a less useful parameter for measuring environment than halo mass or cluster-centric distance because it can exist in two distinct modes: inter-halo and trans-halo, and thus its relevance to a galaxy's star formation is unclear. In any case, halo mass is certainly not the most constraining single variable, performing significantly worse than properties related to the central regions of galaxies. Thus, it is possible that its relative success over local density (and stellar mass) is a result of 'reflected glory' in that it is not a direct link to quenching but rather a result of its close correlation with inner galaxy properties. Our ANN rankings are broadly in agreement with the internal rankings of parameters made in the literature to date. However, this is the first attempt to rank the importance of all of these variables in a fully quantitative and objective manner. Specifically, we find that stellar mass has a much higher AUC than local density for centrals, in qualitative agreement with Peng et al. (2012). We also find that halo mass (derived indirectly from abundance matching) has a higher AUC than stellar mass, in agreement with Woo et al. (2013Woo et al. ( , 2015. Furthermore, we find that bulge mass is superior to all of the above in determining the passive fraction, as argued for in Bluck et al. (2014) and Lang et al. (2014). Bulge mass is also, slightly, superior to B/T structure in constraining the passive state of galaxies, as first pointed out in Bluck et al. (2014). However, bulge mass is not the best single variable found here in the ANN minimisation procedure: centralised velocity dispersion yields significantly higher AUC values (and hence tighter correlations to the star forming state of galaxies) than bulge mass. This was also argued for previously through an analysis of the passive fraction -(estimated) black hole mass relation in Bluck et al. (2015), and is consistent with the importance of central density or velocity dispersion found in several other works (e.g. Cheung et al. 2012, Wake et al. 2012, Fang et al. 2013. Figure 10 may require a reformulation of the classic 'massquenching' of Peng et al. (2010Peng et al. ( , 2012 and even the proposed updates to 'bulge-mass-quenching' of Bluck et al. (2014) or 'halomass-quenching' of Woo et al. (2013). We suggest that 'innerregion-quenching', or most probably 'black-hole-quenching' (i.e. AGN feedback) might be more appropriate given our results; we discuss this further in Section 5. Clearly environmental properties, including those from the halo, are not the most constraining single variables for regulating quenching of central galaxies, nor is stellar mass or galaxy morphology (B/T ), all of which have been previously claimed to be the dominant correlators to the passive fraction (e.g. Baldry et al. 2006, Peng et al. 2010, Woo et al. 2013). However, our result does agree with a complementary analysis, based on the area of the passive fraction relationships, presented in Bluck et al. (2015). Furthermore, the finding by Bell et al. (2008Bell et al. ( , 2012) that essentially all truly passive systems have a high Sérsic index bulge (see also Wuyts et al. 2011) is in qualitative agreement with our finding that a high central velocity dispersion and hence central density is the best predictor that a galaxy will be quenched (out of our chosen list of physical galaxy parameters). We have not included Sérsic index in our main parameter set, since it is not strictly a physical quantity, but we investigate it separately in Section 5. Finally in this sub-section, it remains interesting that the mass of the galactic disk is so un-correlated with the passive state of the system (ranked 7/8), given that in most galaxies undergoing 'normal' star formation it is the disk which is the site of gas being converted into stars. Thus, it seems that, even though disks are the sites of star formation, they are certainly not the regions from which quenching takes effect. This fact must present a serious challenge to models of galaxy quenching utilising feedback from stellar winds or supernovae in central galaxies. Out of the list of physically motivated and plausible quenching scenarios considered in this work (see Section 1), AGN feedback suggests itself as a particularly attractive explanation since it is expected to originate in the central-most regions of galaxies, and hence it is a natural (and obvious) fit to our observed ranking of single galaxy parameters. In most models which apply AGN feedback, the energy available to quench central galaxies is directly proportional to the black hole mass (e.g. Croton et al. 2006, Henriques et al. 2014, Vogelsberger et al. 2014b, Schaye et al. 2015 and this is known empirically to be tightly correlated with central velocity dispersion and bulge mass (e.g. Ferrarese & Merritt 2000, Haring and Rix 2004, Hopkins et al. 2007, McConnell et al. 2011, McConnel & Ma 2014. However, other explanations may still exist (e.g. Carollo et al. 2013) and we examine some possibilities for these in the discussion (Section 5), alongside the, perhaps more obvious, contender of AGN-feedback. We consider whether systematics from our initial sample selection can lead to a significant change in the ordering of these variables in the Appendices (see Sections A1 -A8). Generally, we find that the exact AUC values can change for the single parameters, up or down, as a result of sample selection (e.g. removing Figure 11. f AUC -parameter plot for multiple runs. The grey line is the same as in Figure 10 which shows the results for single variables. The red line shows the result for CVD + each of the rest of the variables in turn, and the blue line shows CVD + M disk + each of the other variables in turn. Note that CVD is the single best variable and M disk is the best secondary variable in conjunction with CVD. No tertiary variable gives significant improvement over CVD and M disk , although Re does perform formally the best. The black cross represents the AUC performance for all variables used simultaneously, shown for comparison. Note that the lines intersect where there are duplications of variables (i.e. for CVD and M disk ), as they should. AGN, excluding green valley galaxies, restricting the sample to lower redshifts or higher velocity dispersions) but that our rankings are almost entirely unaffected and hence are highly stable to sample variation. In the next sub-section we consider multiple parameters acting in concert as predictors of the star forming state of central galaxies. Multiple Parameters Galaxy formation and evolution is a highly complex and non-linear problem, hence there is a limited amount of information and ultimately insight that can be gleaned from assessing how a single variable affects another single variable (e.g. the predictivity of the parameters of Table 1 in determining ∆SFR). To improve on this picture one must seek to understand how galaxy properties interact together to constrain other variables, or sets of variables. Much pioneering work has already been attempted in the direction of multivariable analysis of galaxy quenching. For example, Baldry et al. (2006) and Peng et al. (2010Peng et al. ( , 2012 find that the passive fraction of galaxies is a function of two variables, M * and δ N , and that these are in principle separable. Further work has found that galaxy morphology (e.g. B/T) has a strong influence at fixed M * and δ N (Bluck et al. 2014, Lang et al. 2014) and that halo mass and central density can both affect the passive fraction of galaxies at fixed values of the other parameter . However, to date, no systematic ranking of two variable approaches for parameterizing the passive fraction exists, and certainly no higher (e.g. three variable) analyses exist. ANN techniques are ideal for problems of this type. In this sub-section we perform a systematic analysis of the predictive power of all unique sets of two and three variables drawn from Table 1. But before we discuss our results for these 55 ANN runs, we start by considering a simplified case. Our goal here is to ascertain what the second, and the third, most important variable from Table 1 is for predicting the quenching of central galaxies. We start by always giving the ANN code our estimate for the cen- tral velocity dispersion (which is found to be the best single case, see Section 4.1). We then run an ANN minimisation for each pair of variables {CVD, X}, where X represents each of the remaining variables in Table 1. We show the results on an AUC plot as a red line in Figure 11. We find that adding any of the other variables leads to some incremental improvement in the predictive power over CVD alone, but this improvement is much smaller than for the other way around, i.e. adding CVD to any of the other parameters (compare the difference between the red and grey lines in Figure 11). Disk mass and B/T are the most successful secondary parameters, with similarly high AUC results, which amount to more or less the same thing given the strong relationship between CVD and M bulge . This is a surprising result because disk mass was found to be one of the worst parameters for a single variable and yet in conjunction with central velocity dispersion it performs better than any other 2-variable set containing CVD (this is a similar result to what is found with bulge mass in Bluck et al. 2014). There is no contradiction here, however, it just reflects that having complementary information about both the central and outer regions of galaxies is useful. Nonetheless, if one must choose only one region, the inner region is much more important for constraining central galaxy quenching than the outer. To explore this further, we consider the directionality of this trend: increasing disk mass at fixed central velocity dispersion actually decreases the probability of a galaxy being passive. Thus, it is likely that the inner region (i.e. CVD) gives us information about the quenching power (most probably the AGN, given the strong correlations between CVD and M BH ) and the outer region gives us information on what remains to be quenched (e.g. gas mass or gas fraction, both of which correlate with disk mass). We continue by giving the ANN codes {CVD, M disk } in conjunction with each of the other remaining variables (i.e. 1st + 2nd best + each of the rest). This is shown as a blue line in Figure 11. Here most of the added variables offer some small improvement again, but with no clear sign of any single variable giving the highest improvement. It is particularly interesting to note that halo mass and local density (environmental parameters) lead to no significant improvement over {CVD, M disk }. Thus, even as a tertiary parameter environment is not significantly constraining of the passive state of central galaxies. This fact suggests that the quenching of galaxies is not strongly related to their dark matter haloes, once the intercorrelations with, e.g., black hole mass, central density and B/T are accounted for. However, it is not necessarily true that the best combination of two variables will contain the best single variable, nor is it necessary that the best combination of three variables will contain the best single or secondary variables. It is to the full list of unique possibilities we turn to next. In Figure 12 we show the AUC results for all unique combinations of 2-variable (top) and 3-variable (bottom) sets of parameters drawn from Table 1, i.e. we remove sets of variables which are equivalent (for example, {B/T , M * } is identical to {M bulge , M disk }). The interested reader is referred to Fig. A10 in the appendix for the full rankings of all possible combinations of variables, which we warn contains repetitious content. The top four pairs of variables (top panel, Figure 12) all contain central velocity dispersion, with parameters related to the disk or galaxy morphology being the best additional combinations. This result is qualitatively similar to what we found in Figure 11. The worst pairs frequently contain the local density parameter (δ 5 ), often in conjunction with an outer region or whole galaxy parameter (e.g. M disk ). These tend to perform significantly worse than variables which include information on the inner region of galaxies. Halo mass does perform quite well in combination with galaxy morphology, although it is significantly less predictive than some sets containing CVD. We note that the pair of variables {M * , δ 5 } ranks very poorly as 20/23 couplings of variables from Table 1, even though this has previously been considered the main dual-input for parameterizing galaxy quenching (Baldry et al. 2006, Peng et al. 2010. That said, it is important to emphasize that the use of galaxy density to constrain quenching is mostly applied to satellites in these prior works and here we focus solely on central galaxies. Also the set {M * , B/T }, which was considered as a possible optimal ranking in Bluck et al. (2014), performs only near the middle of the possible sets determined here (8/23). We do not have a central density parameter in our set of variables, however, it is likely closely coupled to CVD (as indeed is suggested in Woo et al. 2015). If this is so the combined variables of the halo and the CVD can be compared to the result of Woo et al. (2015) for halo plus central density. This combination does not perform particularly well, with a rank of 11/23. Our brief comparison to the literature should serve as a caution to anyone planning to model the quenching of galaxies via conventional techniques, these are clearly not optimal. If a two parameter fitting technique is required, the best choice, out of the variables we consider, is {CVD, B/T}. The lower panel of Figure 12 shows our results from 32 ANN runs for all unique combinations of three variable sets of the parameters in Table 1. To our knowledge, this is the first attempt to construct a systematic ranking of three-variable parameterizations of galaxy quenching. All of the top five sets contain central velocity dispersion. Thus, parameters related to the centre of galaxies are essential for predicting quenching even in sets of two and three parameters. Environmental metrics (δ 5 , M halo ) are rare in the top ten, whereas amongst the lowest ranked sets these are much more common; the very worst sets often contain two environmental metrics, further highlighting their lack of predictivity for central galaxy quenching. The best three-variable parameterization from our data is {CVD, B/T , Re}, although it performs comparably well with all of the top five or so combinations, again containing no evironmental metric. The results from these mixed runs point in a similar direction to the single variable run: whatever quenches central galaxies is mostly connected with the inner-most regions of galaxies probed in our dataset. These are the parameters which are most tightly correlated with supermassive black hole mass, and hence AGN feedback energy (see Section 5). Parameters related to the halo mass (or local galaxy density) are significantly less predictive in constraining the passive state of central galaxies, which must present a serious challenge to models of central galaxy quenching arising from the halo, or the environment generally (e.g. Dekel & Birnboim 2006, Woo et al. 2013, Dekel et al. 2014. DISCUSSION -WHAT DRIVES CENTRAL GALAXY QUENCHING? From a theoretical perspective, there are numerous physical processes associated with galaxy formation and evolution that can lead to a gradual or more sudden impact on star formation, in some cases leading to total cessation or quenching. Broadly speaking, all of these scenarios can be described as varying types and degrees of 'baryonic feedback'. There are two essential questions here: 1) why do galaxies stop forming stars (especially given that there is plenty of gas remaining in the Universe for them to convert)? and 2) why do so few baryons end up residing in galaxies, i.e. at the local gravitational minima (current estimates of ∼ 10%, Shull et al. 2012)? These two questions are highly likely to be related, with a common (set of) explanation(s). Our aim in this paper has been to identify the key parameter(s) associated with different quenching scenarios and assess how effective they are at predicting whether galaxies will be passive or star forming. From this we can give evidence for or against different models. In Section 4.1 we find that properties related to the central regions of central galaxies are most predictive of the passive state of the system, with properties related to the entirety of the galaxy or the outer regions and environments being significantly less constraining (see Figure 10 and Table 3). This immediately suggests that the source of the energy needed for quenching central galaxies might originate (or be closely coupled with) the centre of these galaxies. This is exactly as expected for the AGN-feedback driven quenching scenarios (e.g. Croton et al. 2006, Bower et al. 2008, Vogelsberger et al. 2014a,b, Schaye et al. 2015. On the other hand, with virial shock heating driven quenching we would anticipate halo mass to be the most significant parameter (e.g. Dekel & Birnboim 2006, Woo et al. 2013; with supernova feedback driven quenching we would expect stellar mass to be key (e.g. Dalla & Schaye 2009, Guo et al. 2011; and with environmental quenching we would expect a more significant dependence on both local density and halo mass (e.g. , Tasca et al. 2009, Wetzel et al. 2013, Hirschmann et al. 2013. Our conclusion that quenching originates in the centre of galaxies is somewhat different to that reached by several papers in the field (e.g. Baldry et al. 2006, Peng et al. 2010, Woo et al. 2013) although we do find accord with the conclusions of several other more recent papers (e.g. Wake et al. 2012, Bell et al. 2012, Bluck et al. 2014, Lang et al. 2014, Omand et al. 2014, Tacchella et al. 2015. In earlier work, Bell et al. (2008) presented some of the first evidence for the central bulge component being the most significant indicator of quenching by noting that a high Sérsic index bulge is ubiquitous in passive systems. The reason for most of the tension between our results and some of the literature is that we consider a more complete list of parameters than these earlier works. The internal rankings seen in the literature are recovered precisely by our analyses, we just extend this prior work by including more parameters. We are also the first to make a systematic ranking of the predictivity of pairs and triplets of variables (see Section 4.2). Here we find that parameters related to the central regions of central galaxies are still crucial to include in the most successful sets, indicating that the importance of the central region is not an artifact of multiple (other) processes acting in concert. Although AGN feedback driven quenching of central galaxies is a natural explanation of our results, it is not necessarily the only good explanation. In the remainder of this discussion we will focus on plausible alternative explanations (our conclusions are only as good as our input assumptions). A key assumption we have made in this investigation is that the quenching of galaxies is binary in nature, i.e. galaxies are either star forming or they are quenched. We do consider the possibility of intermediate (green valley) cases in the appendices (see Section A3), although even here the implicit assumption is that these are rare or non-representative cases, most probably transitory in nature. Broadly speaking this same assumption is inherent in any approach which uses passive fractions (as with much of the literature on the subject, e.g. Baldry et al. 2008, Peng et al. 2010, Woo et al. 2013, Bluck et al. 2014. However, there is mounting evi-dence that the specific star formation rates (sSFR) of galaxies might change as a function of halo mass, without significantly affecting the passive fraction and that different galaxies can migrate through the green valley at different rates depending on their morphologies (Schawinski et al. 2014). These types of subtle effects would not be noticeable in our current ANN analysis, although it would be possible and interesting to additionally train a network for predicting sSFR values (and green valley transition times) in addition to the binary quenched : star forming designation. This notwithstanding, we expect these non-binary extensions to be only minor perturbations on our general trends since galaxies do separate out convincingly into two clearly separable sub-sets in terms of their star formation rates and colours, suggesting that successful binary classification is the most important step in understanding quenching. One possibility for further consideration is that the success of a given variable (or combination of variables) at predicting whether a central galaxy will be star forming or passive is primarily a function of how accurately measured that variable is. Thus, in this scenario, well measured parameters would perform better. This is certainly true if all physical galaxy parameters are fundamentally equally predictive of quenching. However, we note that this is unlikely to be the main driver of our trends here. To illustrate this, consider bulge and disk mass. These two sub-components of galaxies are measured with more or less equal precision in the bulge disk decompositions of Mendel et al. (2014) and Simard et al. (2011). However, bulge mass is significantly more predictive of quenching than disk mass (see Fig. 10 and Table 3). One exception to this is perhaps halo mass which is inferred indirectly. It is certainly possible that improved measurements of the masses of central galaxy haloes in Sloan might improve the overall ranking of halo mass. That said, our conclusion that AGN feedback is the most probable explanation of our trends rests on the tight relationship between CVD and M BH . If we were to estimate M BH from CVD it is unlikely we would measure this with any greater precision than M halo . If this is true then it is still most likely that black hole quenching dominates over halo mass quenching for low redshift central galaxies. Nonetheless, it would certainly be interesting to revisit these analyses with dynamically measured halo and black hole masses, when sufficient numbers of each become available. Another interesting potential explanation for the apparent dominance of central velocity dispersion to quenching is that it is not the current set of galaxy properties which matter for quenching but the set of parameters at the time (or before) quenching takes effect (e.g. Carollo et al. 2013). This is unarguably true; however, estimating the parameters a galaxy had at an earlier epoch is fraught with difficulty (e.g. Torrey et al. 2015). In this first work on applying ANN techniques to the problem of galaxy quenching we choose to focus on directly measurable physical galaxy parameters. That said, by following a few lines of empirical reasoning we may conclude that a galaxy of a given stellar mass which quenched earlier than another similar mass galaxy would be denser (and hence have a higher central velocity dispersion). This follows directly from the assertion that we are looking at a same mass galaxy and a simple application of the size -mass relation as a function of redshift (Carollo et al. 2013). Arguments of this type provide an important equivocation to our interpretation: correlation does not imply (nor necessitate) causation. Thus, there are any number of possible explanations for the observed trends found in this work, of which the current example is just one possibility. It is therefore necessary to ask the follow-up question: given the observed rankings of galaxy parameters in quenching, what is the most likely phys-ical explanation for this? To aid in answering this question detailed comparisons to semi-analytic models (e.g. Henriques et al. 2014, Somerville et al. 2015 and cosmological hydrodynamical simulations (e.g. Vogelsberger et al. 2014a, Schaye et al. 2015) must be made. Bluck et al. (in prep.) will begin this process for central galaxies. It is, of course, also conceivable that some new physical parameter or set of parameters will do much better than central velocity dispersion, and may ultimately reveal a link between quenching and some other physical process than considered here. At this point it is important to reiterate that in this work we have focused exclusively on physical galaxy parameters, e.g. masses, velocities, densities and sizes. In this manner we have disregarded other parameters of potential interest, such as Sérsic indices. Wuyts et al. (2011) showed that the two peaks of the star formation rate -stellar mass plot are divided cleanly by Sérsic index, with quenched galaxies having higher values of n than star forming galaxies (a result previously considered in Bell et al. 2008). In fact, running our ANN method for n we find that it performs slightly better even than CVD (with AUC = 0.891 ± 0.003), confirming these prior results. However, we exclude n from our main analysis in this paper for two reasons: 1) n is a parameter in a fitting model, not a physical galaxy parameter, and hence it does not fit within the remit of this paper to examine which physical parameters are most tightly correlated with central galaxy quenching; and 2) n is measured in a single optical wave-band and thus can be significantly affected by ongoing star formation (or absence of star formation) in its measurement. The second point is very important to highlight, since the excellent performance of n in predicting quenching could be no more significant than attesting that star formation typically happens in disk structures. Bright blue new stars in a disk lower n and the absence of these stars in a galaxy in general yields a higher value of n. Thus, for these reasons we find the Sérsic parameter, n, to be less interesting to focus on than the other (physical) parameters in our study. Nevertheless, we mention here its excellent performance in AUC, should this be of use or interest to further research. Finally, it is interesting that the set of physical galaxy properties listed in Table 1 is not sufficient, even acting together as inputs for a sophisticated pattern recognition algorithm, to correctly determine the star forming state of all central galaxies (with ∼ 8% misclassified). There are a number of possible explanations for this effect, including, of course, inaccuracies in the measured parameters and observational errors. However, it seems likely that this set of variables is simply not an exhaustive list of all galaxy properties relevant to central galaxy quenching. A similar conclusion is made for a slightly different set of data in Knobel et al. (2015), where they conclude that galactic conformity (the tendency for passive satellites to orbit passive centrals) is evidence for 'hidden variables' in galaxy formation. Whilst this may well be true, it is also possible that there is an irreducibly probabilistic nature to whether a given galaxy will be passive or not, based in part on the chaotic evolutionary history of individual galaxies. In any case, this motivates the need to explore more variables in future statistical studies of the relationship between galactic star formation, quenching, and galaxy properties. However, global parameters may never be sufficient to be perfectly predictive of quenching, thus it may be necessary to consider more complex sets of sub-galactic variables. CONCLUSIONS In this paper we present a novel technique for assessing which galaxy properties impact the quenching of central galaxies. We train an artificial neural network (ANN) non-linear model to recognise star forming and passive galaxies (for a training and verification set each containing 100,000 galaxies). The network is provided with each of the physical galaxy parameters shown in Table 1 as input data, singly and in groups of two and three. A higher success rate of predicting whether galaxies will be star forming or passive from a given variable, or set of variables, is taken to imply a greater causal link between that parameter (or set) and the quenching mechanism(s). We quantify the performance of the network for each parameter and group of parameters by computing the area under the ROC curve (see Section 3.3), with higher AUC values signalling greater predictive power. We summarise our main contributions here: • For single variables, we find the highest AUC values, and hence predictive power, for central velocity dispersion, followed by bulge mass and B/T. All of these parameters formally rank as 'excellent' predictors of passivity in galaxies. • Parameters related to larger scale galaxy properties (e.g. M * , M disk ) or environment (M halo , δ 5 ) perform significantly less well. • The general trend in predictivity from central internal parameters to outer or external parameters provides evidence for the quenching of central galaxies originating in the mass concentration of inner regions, and being largely unrelated to their extended structures or environments (see Figure 10 and Table 3). • We suggest that the predictive success of inner-region galaxy parameters reflects the source of the quenching energy, most probably originating from black hole accretion and AGN feedback. However, we do consider other possibilities to this explanation in the discussion (Section 5). • Bulge effective radius is the worst performing parameter amongst those tested. This is not inconsistent with AGN-driven quenching, since bulge size is not strongly correlated with black hole mass, whereas bulge mass and central velocity dispersion are. • For dual and triple variable sets, inner-galaxy properties are very common amongst the best configurations, with environmental properties being rarely seen. This indicates that the importance of the inner-region parameters over outer region or environmental parameters does not diminish with the more inclusive multi-variate analysis. • Although we exclude the Sérsic index parameter, n, from our main analysis since it is not a physical galaxy property per se, we note that it performs particularly well at predicting whether galaxies will be star forming or not. This could, however, just be an artefact of this parameter tracing the light from star formation directly. We perform many tests and investigations of the effects of sample variation and potential biases and systematics on our results in the Appendix (Sections A1 -A8). We find that our rankings are very stable to issues of this type (including exclusion of green valley galaxies or AGN, volume weighting or restricting to a volume limited sample, and additional axis ratio, mass, redshift or data quality cuts). ACKNOWLEDGMENTS We thank J. Trevor Mendel, David R. Patton, Jillian Scudder and Luc Simard for helpful discussions on this work. We are partic-ularly grateful to Luc Simard and Trevor Mendel for much assis- APPENDIX A: SAMPLE VARIATION AND POSSIBLE SYSTEMATICS To demonstrate the robustness and stability of our rankings of the single galaxy parameters to sample variation, we perform different ANN runs for different carefully selected sub-samples, similar to what is shown in Figure 10. A1 Lower Redshift Cut For the main sample we use a redshift cut of z spec < 0.2 (as in Bluck et al. 2014Bluck et al. , 2015. Here we consider restricting the sample to z spec < 0.1 where we will have more reliable bulge + disk decompositions (due to higher surface brightness features at a given mass) and a higher S/N of emission lines (used for SFR and AGN determination) and the spectral continuum aiding absorption line measurements (used in calculating velocity dispersions and estimating M BH ). This sub-sample also has a higher mass/ colour completeness than the higher redshift sample (but see Section A7 for a more thorough treatment of completeness). We re-run the ANN codes for ALL and each of the single runs, and go through the methodology exactly as in Section 4.1. We show the AUC performance indicator for this more restrictive sample in Figure A1 (shown as a blue line), and overplot the previous result from Figure 10 (shown as a grey line). In general, the performance of the ANN is improved by the lower redshift cut, Figure A1. AUC -single parameter plot. The grey dashed line is the same as Figure 10 which comprises all redshifts up to our original limit of z spec < 0.2. The blue solid line is for a restrictive sub-sample of galaxies with z spec < 0.1. Figure A2. AUC -single parameter plot. The grey dashed line is the same as Figure 10 which comprises all galaxies including AGNs. The blue solid line is for the sample in which AGN galaxies are excluded. We define AGN for this analysis in §A2. indicated by higher AUC values for ALL and most single cases. This is as we might expect from increasing the S/N of our average data. However, importantly, the ranking of single variables (i.e. their ordering in terms of AUC and thus how effective they are at constraining the passive state of galaxies) is left completely unchanged. This implies that our results are robust to changes in the surface brightness of galaxy components and to the S/N of emission and absorption lines, lending more confidence to our rankings. A2 Excluding AGN We use indirect means for determining the SFRs for AGN based on the empirical relationship between the strength of the 4000 Å break and the sSFR of the galaxy (see Section 2). This is necessary because AGN contribute flux to the emission lines used to determine SFRs. However, the errors in the SFRs of AGN can be significant (Rosario et al. 2015), potentially leading to misclassifications of star forming or passive systems in our training sample. Here we consider the effect of removing all AGN from our sample. We de- Figure A3. AUC -single parameter plot. The grey dashed line is the same as Figure 10 which comprises all galaxies including green valley galaxies. The blue solid line is for the sample in which green valley galaxies are excluded, from both training and verification. fine AGN to be any galaxy which lies above the Kauffmann et al. (2003) line on the BPT diagram, at a S/N > 1. We then redo our ANN analysis for single variables and ALL. We plot the AUC result for the non-AGN sample in Figure A2 (blue line) and overplot the result for the original sample (grey line). A significant improvement in performance is seen (AUC values are generally higher). However, we find no difference in the ordering by AUC of these variables. So, whilst removing AGN from our sample (hence restricting to more reliable SFRs) improves the ANN performance, it does not affect the results of Section 4.1 in any way. A3 Excluding the Green Valley One possible source of serious systematic error in our ANN analysis can come from our initial assumption that galaxies can be decomposed cleanly into just two (binary) states in terms of their star formation, i.e. passive or star forming. This ignores the possibility that some galaxies belong in neither of these categories. In particular, galaxies lying in the 'valley' between the two peaks of ∆SFR in Figure 1 are hard to place in either of these two categories. Here we follow many authors (e.g. Strateva et al. 2001, Driver et al. 2006, Schawinski et al. 2014 in considering a third case, that of the 'Green Valley'. The definition for this class in terms of ∆SFR is given in Section 2.3. Figure A3 shows the result in terms of AUC for the sample with these green valley galaxies excluded (blue line), for comparison we overplot the original result for all galaxies (grey line). As with restricting the redshift range and excluding AGN, a significant improvement in the ANN performance is seen. This is as we might expect, since we are deliberately 'cleaning' the sample of ambiguities. However, this restriction does not lead to any difference in the ordering by AUC of the single variables, and hence does not have any impact on the ranking of how important these variables are for quenching. A4 Restricting the Velocity Dispersions Velocity dispersions with values less than 70 km/s are intrinsically less reliable than those with higher values, due to the resolution of Figure A4. AUC -single parameter plot. The grey dashed line is the same as Figure 10 which includes all velocity dispersions. The red line shows the results for a sample where velocity dispersions with σ < 70 km/s are excluded. The light cyan line shows the result for galaxies with σ < 70 km/s having their input values changed to 0 km/s. the SDSS spectra. However, placing a cut in velocity dispersion (in addition to the cut in stellar mass) would lead to a highly biased sample, where only bulge dominated galaxies (presumably more likely to be passive) are detectable at low stellar masses. So, for the initial sample we included all velocity dispersions, provided they pass our basic data quality checks (presented in Section 2). This could potentially leave us with a bias, and we investigate this possibility here. First we restrict our sample to σ > 70 km/s, and redo our ANN analysis for all single variables. The result of this procedure in terms of AUC is shown in Figure A4 as a red line, overplotted in grey is the original result for all σ. It is interesting to note that restricting the sample by velocity dispersion actually lowers the performance of the ANN, even for velocity dispersion itself! This is because a powerful piece of information is lost in this case. It seems that the presence of a compact (pressure supported) bulge is essential for a central galaxy to be quenched, and thus the opposite (where there is no bulge and hence low velocity dispersion) leads to a near certain classification of star forming. Removing the low σ cases removes the ability of the ANN code to correctly assign these cases. Therefore, we suggest that it is better to leave them in even though this could lead to a higher uncertainty of the ranking of σ. Nonetheless, the only change to the ranking caused by excluding the low velocity dispersions is the ordering of B/T and M bulge (both of which are independent of the spectral resolution of the SDSS since they are determined from the photometry alone), everything else remains unchanged. Since we are not sure of the exact values of σ < 70 km/s due to the instrumental resolution of the SDSS, and that we posit that it is just that these values are low that is useful for the ANN code, we try setting all low velocity dispersions to zero, i.e.: σ c −→ 0 (if σ < 70km/s) || σ c (if σ 70km/s)(A1) We show the result for this sample in Figure A4 as a light cyan line. Note that it is almost identically coincident with the original sample (shown in grey). This demonstrates that no information is being derived by the ANN codes from the actual values of σ < 70 km/s velocity dispersions, only that they are low. Thus, we conclude that including these low values in the sample is not biasing our results, Figure A5. AUC -single parameter plot. The grey dashed line is the same as Figure 10 which comprises all galaxies regardless of disk axis ratio. The blue solid line is for the sample in which late-type galaxies (with B/T < 0.5) are restricted to being face-on (b/a > 0.9). and moreover is actually essential to get the most optimal (and reliable) performance (given that the grey and cyan lines lie above the red line). A5 Restricting LTGs to Face-On For velocity dispersions there is an ambiguity as to the source of the kinetic energy when measured via aperture spectroscopy, i.e. contributions to σ can be made by a pressure supported bulge and/ or from disk rotation into the plane of the sky. Given that the SDSS fibre is generally centred on the middle of the galaxy light profile, for cases where the bulge dominates (and/or for very low redshifts) this effect will be small. However, where the bulge is not the dominant component of the stellar mass budget of the galaxy, significant kinematic contamination from the rotating disk can affect the measurement of σ, if the disk is inclined relative to Earth. Thus, the success of σ and M BH (which is based in part on σ) in determining the passive state of galaxies in Section 4.1 could potentially be partially attributed to measuring the disk rotation in galaxies, i.e. not actually (solely) associated with the central region. We consider this possibility in this sub-section. Here we construct a new sub-sample requiring all late-type galaxies (LTGs, defined as B/T < 0.5) to be 'face-on' with b/a > 0.9 (inc disk < 25 o ). We take these values from photometric bulge-disk decompositions performed in Simard et al. (2011). This removes ∼ 90% of LTGs but leaves the bulge dominated early-type galaxies (ETGs) unchanged. Since our purpose in this paper is to probe galaxy quenching we must carefully correct for this new bias before continuing. For this sub-sample we weight each galaxy in the ANN code by the inverse of the probability of its inclusion (which is a function of its structure, B/T), specifically we calculate (as in Bluck et al. 2015): w i = 1 1 − f rem (B/T )(A2) where f rem (B/T ) is the fraction of galaxies removed from our sample due to the b/a cut of LTGs, which varies as a function of galaxy morphology for LTGs and is of value unity for ETGs (because they are not removed). This corrects for any bias in the passive : star forming ratio of the sample, but leaves us with only face-on disks, for which σ is solely a probe of the bulge kinematics. We show the AUC plot for this sub-sample in Figure A5 as the blue line, with the original result being shown in grey for comparison. The two lines are close to being identical, and the ordering of the variables is largely the same as well. The performance of CVD does slightly better, however, indicating that it is indeed the bulge kinematics (not disk contribution) which yields the predictive power of this variable in assigning the passive state of galaxies. Some of the other single variables perform slightly less well for this sub-sample, which is most probably explained by this dataset being statistically less rich due to the removed LTGs. A6 Higher Stellar Mass Cut Our primary data sample is restricted in stellar mass to M * > 10 9 M , which is due to the relative scarcity of galaxies with lower stellar masses in the SDSS volume. For centrals, this cut in mass is probably low enough to include almost all passive galaxies (see the 1/V max weighted passive fraction -stellar mass relation presented in Fig. 8 of Bluck et al. 2014). However, it is possible that the performance of the galaxy parameters presented in Table 1 vary as a function of the stellar mass range considered. Obviously, at the extremes this will be uninteresting because all galaxies will be either passive or star forming, but at intermediate masses there may be some additional insights to be found. In this sub-section we consider the effect on the ANN ranking of single parameters of a higher mass cut of M * > 10 10 M . Our new result is shown as a blue line in Figure A6, with the original result (for M * > 10 9 M ) shown in grey for comparison. There are a few subtle differences between the mass cuts, such as disk mass performing better than stellar mass and B/T performing better than bulge mass in this sample. However, the general trend is the same, with galaxy parameters related to the inner regions of galaxies performing the best, and parameters related to the outer regions of galaxies or the local environment performing significantly worse. These changes do not in any way affect our conclusions, but it is interesting to note that the results from an ANN analysis of this type can in principle be affected by the range in masses of the input parameters. Figure A7. AUC -single parameter plot. The grey dashed line is the same as our fiducial result, shown in Figure 10, which takes all bulge -disk parameters at face value. The blue solid line is for a sample with low B/T galaxies re-categorised to pure disks and high B/T galaxies re-categorised to pure spheroids. The results for these two samples are identical within the errors, and thus the rankings remain unchanged. A7 Pure Disks and Spheroids Our morphological and structural parameters come from bulgedisk decompositions performed in Simard et al. (2011) andMendel et al. (2014). We define the structure of a galaxy to be the continuous variable B/T, which is the bulge-to-total stellar mass ratio, which is equal to one minus the disk-to-total stellar mass ratio (i.e. B/T = 1 -D/T). In this subsection we consider whether the ranking by AUC of galaxy properties is affected by the possibility that some pure disk or pure spheroid galaxies are misclassified as composite systems. The average error on an individual B/T value is ∼ ± 0.1 (see appendices in Bluck et al. 2014 for their determination from fitting of model galaxies). Thus, we allow all galaxies with B/T < 0.1 to be set to pure disks and all galaxies with B/T > 0.9 to be set to pure spheroids, which is permitted within their errors. Specifically, we define the following two mappings: We compare the AUC results for this new sample to the original runs in Figure A7. All of the parameters, including M bulge , M disk and B/T perform identically within the errors to the original run, and hence there is no change to the ranking by AUC from possible misclassifications of pure disks or spheroids. A8 Volume Limits and Weighting Due to the flux limit of the SDSS, galaxies of different masses and colours are visible in the survey to different maximum redshifts, which can lead to a bias on the ANN input sample. The usual way to deal with these effects is via volume weighting of statistics such as the passive fraction (as in, e.g., Peng et al. 2010, Woo et al. 2013, Bluck et al. 2014. The dependence of the maximum redshift, z max , each galaxy can be detected at in the SDSS on both stellar mass and (g-r) colour is presented in Figure 9 of Mendel et al. (2014). From this a maximum detection volume, V max , can be computed for each galaxy. Weighting any given statistic by Figure A8. AUC -single parameter plot. The blue line is for a mass cut of M * > 10 9.5 M . This sample is not restricted in redshift and extends out to z spec < 0.2. The blue dashed line is for volume limited sub-sample (z spec < 0.04). Figure A9. AUC -single parameter plot. The grey line shows the original (un-weighted) sample with green valley and AGN galaxies removed (see Sections A2 and A3). The blue line shows the same sample (no GV or AGN) but now weighted by 1/V max in both the training and verification sets (see Section A7). The two lines are almost identical, with only very minor differences. 1/V max corrects for the flux limit bias. The alternative to weighting is, the more familiar approach of, constructing a volume limited sample, i.e. restricting the survey to a volume at which completeness is achieved for a given stellar mass (and technically colour) limit. In this sub-section we consider both of these approaches to test whether our flux limited input sample leads to any bias in the rankings of variables. In Figure A8 we consider a slightly higher mass cut to the fiducial sample considered throughout the rest of this paper, of M * > 10 9.5 M , shown in red. This sample is not restricted in redshift and extends out to z spec < 0.2. The blue line in Figure A8 shows the AUC results for single variables for a volume limited sample where we are complete at the stellar mass limit, and at the average colour (for that mass) of the red sequence. The redshift cut for this sample is z spec < 0.04. Generally, we find that restricting to a volume limited sample does not change our results significantly. The general trend of inner galaxy properties being more predictive of quenching than outer galaxy or environmental parameters still holds true for all samples. There are, however, a few small changes. The most prominent of these is that bulge effective radius performs significantly better than local density in the volume limited case but significantly worse in the flux limited case. With this one exception, the ordering of all of the rest of the parameters is identical between the flux limited and volume limited sample, thus our ranking of galaxy parameters in quenching is highly stable to issues of completion in the input sample. Our restriction to a volume limited sample is imperfect, however, because 1) we have to assume a colour limit (here taken as the mean of the red sequence at the lower mass cut) and 2) this process necessarily reduces our sample size significantly, which impairs the power of the ANN technique. Volume weighting is a viable alternative, although there are also some issues with this approach to consider. Since the ANN procedure concentrates on finding patterns in the data, and is carefully tuned to avoid over-fitting, introducing a weight (often very large ∼ 100 -1000 in some cases) can result in amplifying outliers to the status of significant sub-patterns. Thus, before weighting we must be careful to use the 'cleanest' data set available, with the fewest 'bad' data points. Given the results of Sections A2 and A3, where we find that excluding the green valley and AGN from our sample improves the ANN performance, we also remove these galaxies from our sample before volume weighting here. In Figure A9 we show the result of our ANN minimisation procedure for un-weighted galaxies with green valley and AGN cases removed (grey line), and the same sample weighted by 1/V max (blue line). Here weighting indicates the number of times each galaxy is included in the parent sample, and hence is closely related to the probability of inclusion in the ANN training and verification sets. The two samples agree almost identically, giving the same trend in AUC performance from inner-galaxy properties to environmental properties, seen throughout the appendix and Section 4.1. The biggest difference is a noticeably worse performance of CVD relative to bulge mass and ratio. This can be explained by the fact that in the volume weighted sample greater emphasis is placed on lower values of CVD, which are intrinsically less reliable (see Section 2 and A4). In the volume limited case (above) we still see CVD performing best, and this is likely because by restricting to lower redshifts we can accurately constrain CVD to lower values. However, the directionality of the trend from inner to outer regions is left unchanged by weighting, hence we conclude that our method is not significantly affected by the initial sample setup. The primary invariance of our method to volume effects is likely a result of us selecting the same number of PA and SF galaxies for both training and verification. This reduces the effect of colour (or SFR) on our sample selection, and hence also reduces the impact of stellar mass detection thresholds, due to the strong correlation between M * and SFR or (g-r) colour. In any case, the impact of volume weighting, or restricting to a volume limited sample, is very minor on our rankings and results. Figure A10. AUC -parameter plot for all multiple runs for centrals. The top plot shows all possible combinations of two parameters as input data, and the bottom plot shows all possible combinations of three parameters as input data. They are both ordered from most to least predictive at determining the passive state of galaxies. Figure 1 . 1Distribution of ∆SFR values, defined in equation 5. Top panel: Figure 4 4Figure 4. Output ANN probability distribution for α = 0.5 case (where 0 = SF, 1 = PA) for originally classified star forming (blue) and passive (red) galaxies. The vertical grey dashed line at X = 0.7 shows a randomly selected threshold. For this threshold, the red shaded area to the right of the line gives the True Passive Rate (TPR = 0.783), and the blue shaded area to the right of the line gives the False Passive Rate (FPR = 0.069). Note that in general FPR + TPR 1, since the sum of the red area and the sum of the blue area (from X = 0 -1) is unity, not the sum of the blue and the red areas across any given threshold. Figure 5 . 5A Receiver Operating Characteristic (ROC) plot obtained from the ANN output probability distribution ofFigure 4, for α = 0.5. Specifically, we plot the True Passive Rate (TPR) vs. the False Passive Rate (FPR), see Section 3.3. We change the threshold from 1 to 0, systematically obtaining different values for TPR and FPR. As an example, the point [0.20 0.85] is obtained from threshold X = 0.5. The thresholds are indicated by the colour of the ROC curve line, labelled by the colour bar. The dashed grey line indicates the result for a random variable, with area under the ROC curve, AUC = 0.5. Figure 7 . 7Area Under the ROC Curve (AUC) vs. the randomness parameter α. The different values of AUC associated with different values of α are computed fromFigure 6. The area under the curve from a perfect classification (α = 0) changes from AUC = 1 to a completely random input data (α → ∞) with AUC = 0.5. et al. 2012, Fang et al. 2013, Bluck et al. 2014, Lang et al. 2014, Omand et al. 2014, Woo et al. 2015) that properties associated with central mass, or mass density, are the most important for determining the passive Figure 8 . 8Distributions of output probabilities (0 = SF, 1 = PA) from the ANN minimisation procedure for galaxies which are originally classified as star forming (blue lines) and passive (red lines). The top plot shows the distributions related to the ANN run where all of the parameters in Figure 9 . 9Receiver Operating Characteristic (ROC) curves for each of the distributions shown inFigure 8, for the galaxy parameters in Figure 10 . 10Area Under the Curve (AUC) -single parameter plot. This plot illustrates the area under each ROC curve (see Figure 12 . 12AUC -parameter plot for all unique set of multiple runs for central galaxies. The top panel shows all possible unique combinations of two parameters as input data, and the bottom plot shows all possible unique combinations of three parameters as input data. They are both ordered from most to least predictive at determining the passive state of galaxies. SeeFig. A10for all variables, regardless of uniqueness (i.e. containing various duplicates). Figure A6 . A6AUC -single parameter plot. The grey dashed line is the same as Figure 10 which comprises galaxies with M * > 10 9 M . The blue solid line is for a restrictive sub-sample with M * > 10 10 M . If(B/T 0.1) −→ (B/T = 0)&(M bulge = 0)&(M disk = M * )(A3)andIf(B/T 0.9) −→ (B/T = 1)&(M bulge = M * )&(M disk = 0) Table 1 . 1The physical parameters of the central galaxies used in this work.# Symbol Description Scale * 1 CVD Central Velocity Dispersion ∼ 1 kpc 2 M Bulge Bulge Stellar Mass 0.5 -4 kpc 3 Re Bulge Effective Radius 0.5 -4 Kpc 4 B/T Bulge-to-Total Stellar Mass Ratio 0.5 -8 kpc 5 M * Total Stellar Mass 2 -8 kpc 6 M Disk Disk Stellar Mass 4 -10 kpc 7 M Halo Group Halo Mass 0.1 -1 Mpc 8 δ 5 Table 2 . 2An interpretation of the AUC parameter in engineering (byHosmer & Lameshow 2000) AUC Range Description 1.0 Perfect Discrimination 0.9 -1.0 Outstanding Discrimination 0.8 -0.9 Excellent Discrimination 0.7 -0.8 Acceptable Discrimination 0.5 -0.7 Unacceptable Discrimination 0.5 No Discrimination (Random) Table 3 . 3ANN AUC ranking of single parameters for central galaxies.Rank Property AUC Success Label * ALL 0.9074 ± 0.0106 Outstanding 1 CVD 0.8559 ± 0.0039 Excellent 2 M bulge 0.8335± 0.0060 Excellent 3 B/T 0.8267 ± 0.0028 Excellent 4 M halo 0.7983 ± 0.0045 Acceptable 5 M * 0.7819 ± 0.0025 Acceptable 6 M Disk 0.7124 ±0.0016 Acceptable 7 δ 5 0.5894 ± 0.0015 Unacceptable 8 Re 0.5599± 0.0013 Unacceptable * see ), whereas central velocity dispersion and bulge mass are tightly correlated to black hole mass (e.g. Ferrarese & Merritt 2000, McConnell & Ma 2014). 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[]
[ "Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation", "Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation", "Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation", "Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation" ]
[ "Qijian Zhang \nCity University of Hong Kong\n\n", "Junhui Hou [email protected] \nCity University of Hong Kong\n\n", "Qijian Zhang \nCity University of Hong Kong\n\n", "Junhui Hou [email protected] \nCity University of Hong Kong\n\n" ]
[ "City University of Hong Kong\n", "City University of Hong Kong\n", "City University of Hong Kong\n", "City University of Hong Kong\n" ]
[]
The past few years have witnessed the prevalence of selfsupervised representation learning within the language and 2D vision communities. However, such advancements have not been fully migrated to the 3D point cloud learning community. Different from existing pre-training paradigms for 3D point clouds falling into the scope of generative modeling or contrastive learning, this paper proposes a translative pre-training framework, namely PointVST, driven by a novel self-supervised pretext task of cross-modal translation from 3D point clouds to their corresponding diverse forms of 2D rendered images. More specifically, we start by deducing view-conditioned point-wise embeddings via the insertion of a viewpoint indicator and then adaptively aggregate a view-specific global codeword fed into the subsequent 2D convolutional translation heads for image generation. Extensive experiments on various downstream tasks of 3D shape analysis and scene understanding demonstrate that PointVST shows consistent and prominent performance superiority over current state-of-the-art methods. Our code will be made publicly available.
10.48550/arxiv.2212.14197
[ "https://export.arxiv.org/pdf/2212.14197v2.pdf" ]
255,340,561
2212.14197
19134e7e7f50407051e0c32219edbf1a44373d43
Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation Qijian Zhang City University of Hong Kong Junhui Hou [email protected] City University of Hong Kong Self-Supervised Pre-training for 3D Point Clouds via View-Specific Point-to-Image Translation The past few years have witnessed the prevalence of selfsupervised representation learning within the language and 2D vision communities. However, such advancements have not been fully migrated to the 3D point cloud learning community. Different from existing pre-training paradigms for 3D point clouds falling into the scope of generative modeling or contrastive learning, this paper proposes a translative pre-training framework, namely PointVST, driven by a novel self-supervised pretext task of cross-modal translation from 3D point clouds to their corresponding diverse forms of 2D rendered images. More specifically, we start by deducing view-conditioned point-wise embeddings via the insertion of a viewpoint indicator and then adaptively aggregate a view-specific global codeword fed into the subsequent 2D convolutional translation heads for image generation. Extensive experiments on various downstream tasks of 3D shape analysis and scene understanding demonstrate that PointVST shows consistent and prominent performance superiority over current state-of-the-art methods. Our code will be made publicly available. Introduction As one of the most common and straightforward 3D data representation modalities that can faithfully record raw geometric information, point clouds have been playing a critical role in a wide range of real-world applications, such as immersive telepresence, robotics, and autonomous driving. In recent years, empowered by the continuous progress of deep set architecture design [36,37,26,53,30,45,57,70], deep learning-based frameworks that directly work on 3D point clouds have been richly investigated and applied for various downstream task scenarios of low-level geometric processing [65,52,3] and high-level semantic understanding [46,43,18]. Essentially, these approaches rely on large-scale annotated point cloud repositories [11,4,32] to achieve competitive performances in a supervised learning manner. However, manually annotating massive amounts of irregular and unstructured 3D data is known to be laborious and cumbersome, which can restrict the broad applicability of 3D point cloud representations. Fortunately, the thriving development and rapid popularization of 3D scanning technologies (e.g., LiDAR, Kinect, stereo cameras) facilitate convenient construction of largescale unlabeled 3D point cloud repositories, which strongly motivate a recent line of works [41,35,42,51,58,20,69,2] exploring self-supervised pre-training approaches for point cloud representation learning without manual annotations. Functionally, an effective pre-training process enhances the robustness and generalization ability of backbone encoders by guiding the learning of generic and transferable features, and thus reduces the amount of annotated data required for task-specific supervised learning. Still, despite the remarkable success of self-supervised pre-training within the language and 2D vision communities [8,21], its potential has not been fully realized and explored in the fast-growing area of 3D point cloud learning. Basically, the core of self-supervised learning lies in the design of appropriate pretext tasks and the specific implementation of its learning process. Therefore, depending on the difference of pretext tasks [29], existing self-supervised pre-training frameworks customized for point clouds can be broadly categorized into generative modeling [41,35,42,51] and contrastive learning [40,58,69,9,20,2]. Architecturally, the former generative paradigms are typically imple-mented as self-reconstructive pipelines, i.e., the input point cloud is encoded in the latent space to produce feature embeddings, from which the original 3D geometry information is inferred/restored. Still, these methods show the relatively limited capability of driving the backbone encoder to learn discriminative representations. As revealed in [41,67,10], the reasons could be attributed to the possibly-problematic point set similarity metrics such as Chamfer distance (CD) and earth mover's distance (EMD) that are hard to optimize during generation, and the less powerful point-tailored modeling components built upon shared multi-layer perceptrons (MLPs). The latter contrastive paradigms aim to learn feature space similarity measurements between positive and negative samples created by imposing different transformations on the given point cloud instances. These approaches show better performances and gain increasingly growing attention. Still, one may have to carefully design the network components and cautiously implement the actual parameter optimization and updating process, in order to circumvent the typical issue of model collapse [12,22]. Our Insights. Instead of following existing generative or contrastive paradigms, this paper poses a new perspective in designing self-supervised pre-training frameworks for point cloud data by innovatively introducing a translative pretext task, namely PointVST. As illustrated in Figure 1, our basic idea is to perform cross-modal translation from a given 3D point cloud to its diverse forms of 2D rendered images, such as depth, contour, and silhouette maps, under a view-conditioned processing pipeline. Fundamentally, PointVST can also be considered a generative framework in the broad sense since its ultimate goal is still to achieve "generation" at the final output end. However, such a conversion of the generation objective from the original irregular 3D geometry space to the regular 2D image domain makes a big difference to the learning effects, while naturally circumventing the aforementioned limitations of generative paradigms. For one thing, we are able to get rid of the point set similarity metrics (CD and EMD), which are proven to be weak owing to unknown correspondences between ground-truth and generated point clouds [41,67,10] and hence result in limited feature representation capability. Naturally, we can directly impose stronger supervisions by minimizing pixel-wise errors between ground-truth images and the translated results, which can be much easier to optimize. For another, we can resort to mature and powerful 2D CNNs, rather than relatively less expressive point-tailored network modules, to implement the reconstruction process. Overall, as a different way of performing 3D geometric information recovery, our proposed cross-modal translative paradigm can subtly circumvent several aspects of challenging problems induced by straightforward point cloud reconstruction. Besides, the view-specific translation mechanism facilitates capturing complete geometric structures. We conduct comprehensive experiments on various point cloud learning tasks, including 3D shape classification, part segmentation, normal estimation, and scene semantic segmentation. Under diverse evaluation protocols, our method consistently shows prominent performance superiority over current state-of-the-art approaches, indicating the great potential of the new translative pre-training paradigm. Related Works Point Cloud Representation Learning Unlike conventional 2D representation learning dealing with structured visual signals (images, videos) defined on dense and regular grids, 3D point clouds are characterized by irregularity and unorderedness. Such unstructured nature poses significant difficulties in learning discriminative geometric feature representations. Pioneered by Qi et al. [36,37], there have emerged numerous deep set architectures [26,53,45,55,30,60,59,57] that directly operate on raw point sets without any pre-processing. To overcome the dependence on large-scale annotated point cloud data, there have been a plethora of works focusing on performing unsupervised representation learning on unlabeled point clouds built upon various paradigms of selfreconstructive learning pipelines, such as auto-encoders (AEs) [25,63,6,16,71,28,15], generative adversarial networks (GANs) [54,48,1,14], and others [62,44]. However, the actual feature learning efficacy of these unsupervised approaches is still far from satisfactory. Self-Supervised Point Cloud Pretraining More recently, self-supervised learning has been adapted for point cloud pre-training pipelines, demonstrating dominating performances on various shape analysis and scene understanding tasks. Generally, previous works can be categorized into generative modeling and contrastive learning paradigms. Generative Paradigms. Based on the intuition that effective feature abstractions should contain sufficient information to reconstruct the original geometric structures, a family of generative pre-training pipelines has been proposed by designing various appropriate pretext tasks. Jigsaw [41] recovers point clouds whose parts have been randomly rearranged. OrienEst [35] tends to predict 3D rotation angles, motivated by the preceding practice for image learning [24]. GLBR [38] performs bidirectional reasoning between the global feature and different scales of local representations. cTree [42] hierarchically organizes input point cloud data in a cover-tree and predicts the decomposition patterns. OcCo [51] constructs a simple but highly effective approach based on partial point cloud completion. Contrastive Paradigms. The core of contrastive learning lies in the feature similarity measurement between positive Figure 2. Overall workflow of the proposed PointVST for self-supervised point cloud pre-training. Given an input 3D point cloud, we can deduce from the target backbone encoder BP (·) a set of point-wise features E and a global codeword g. Within the pretext task, given a randomly specified camera position, we produce a viewpoint indicator v and fuse it with backbone features to obtain view-conditioned point-wise embeddings Ev. After that, we customize AVS-Pool to aggregate from Ev a view-specific global codeword gv, which encodes partial 3D geometric patterns exactly corresponding to the viewpoint condition. Finally, we feed gv into the subsequent translation heads built upon standard 2D CNNs to generate different forms of rendered images. and negative samples. Info3D [40] takes inspiration from [33,49] to maximize the mutual information between a complete 3D object and local chunks as well as its geometrically transformed version. PointContrast [58] aims to learn invariant point-level feature mappings between transformed views of given point cloud scenes. DepthContrast [69] deals with single-view depth scans without the requirement of 3D registration and point-wise correspondences by performing instance discrimination after global feature pooling. Self-Contrast [9] proposes to contrast patches within complete point clouds for mining non-local self-similarity relationships. As an extension of BYOL [12], STRL [20] learns discriminative point cloud features via interactions of the online and target networks. CrossPoint [2] adopts a multimodal learning framework composed of a 3D point cloud branch and a textured 2D image branch, whose pre-training objective is jointly driven by the intra-modal invariance between different point cloud augmentations and cross-modal contrastive constraints between rendered image features and point cloud prototype features. More recently, inspired by the success of masked autoencoders (MAE) [17] in the 2D vision community, some researchers are devoted to adapting MAE-like pre-training pipelines [66,34,27,68], which achieve impressive performances. These approaches are particularly customized for pre-training transformer-style point cloud backbones [13,70], and thus cannot be applied to other common types of deep set architectures. By contrast, our method and the above-discussed works pursue wider applicability to common types of point cloud backbone architectures. Proposed Method Overview. Given a specific backbone point cloud feature encoder B P (·), we aim at designing an appropriate selfsupervised pretext task T (·), which can effectively facilitate learning expressive and transferable 3D geometric representations from massive unlabeled point cloud data. Formally, denote by P ∈ R N ×3 an input 3D point cloud that consists of N spatial points {x n ∈ R 3 } N n=1 . Without loss of generality, we can correspondingly obtain at the output end of B P (·) a set of point-wise features E ∈ R N ×De embedded in the high-dimensional latent space, as well as a vectorized codeword g ∈ R Dg serving as the global shape signature. Technically, as illustrated in Figure 2, the overall workflow of our PointVST pre-training framework comprises three major stages stacked in an end-to-end manner. In the very beginning, given a randomly specified camera position, we produce via positional encoding a vectorized viewpoint indicator denoted as v ∈ R Di , which is further fused with raw backbone features (E and g) to generate a set of view-conditioned point-wise embeddings denoted as E v ∈ R N ×Dv . After that, we customize an adaptive viewspecific pooling (AVS-Pool) mechanism to aggregate from E v a view-specific global codeword represented by g v . Intuitively, different from g that encodes the whole geometric structure of P conveyed by complete points, g v is supposed to depict partial 3D structural information in the projection space that exactly corresponds to its viewpoint position. In the end, g v passes through the subsequent translation heads that are built upon 2D CNNs to generate different forms of rendered images. After finishing pre-training, the backbone encoder B P (·) can be directly applied to obtain generic and discriminative point cloud representations, or integrated into various taskspecific learning pipelines for downstream fine-tuning. View-Conditioned Point-Wise Fusion By default, the input point cloud model has already been centralized and normalized into a unit sphere, such that we can configure a fixed observation distance (from the camera position to the object centroid). Thus, under the geographic coordinate system, we can describe any camera position by two parameters, i.e., latitude angle φ lat ∈ [−90 • , 90 • ], and longitude angle λ lon ∈ [0, 360 • ). Following previous practice [31] involving view-conditioned reasoning, instead of directly using raw latitude and longitude angles (φ lat , λ lon ) as the condition signal, we perform positional encoding via several learnable transformation layers to enhance flexibility. More concretely, we separately lift φ lat and λ lon into the latent space and then concatenate the resulting two highdimensional vectors to deduce the viewpoint indicator: View-Conditioned Point-Wise Embeddings (Channel Broadcasting) View-Specific v = F φ (φ lat ) ⊕ F λ (λ lon ),(1) where F φ and F λ denote non-linear transformations implemented as two consecutive fully-connected (FC) layers, and ⊕ represents feature channel concatenation. After that, we fuse raw backbone features and the viewpoint indicator to produce a set of view-specific point-wise embeddings E v , which can be formulated as: E v = M f (M e (E) ⊕ F g (g) ⊕ v),(2) where M e and M f are both implemented as shared MLPs, and F g denotes a single FC layer. Note that all the embedding values in E v tend to be non-negative, since we choose ReLU as the activation function. Adaptive View-Specific Pooling In the preceding stage, we insert the viewpoint cues in a fine-grained manner to obtain view-conditioned point-wise embeddings E v . In fact, when observing the target shape at the specified viewpoint, only a subset of visible points in P can convey effective information in the projection space of the 2D viewing plane. Therefore, to facilitate view-specific learning, we expect that features of invisible points could be adaptively omitted from the subsequent pooling operation applied on E v . To achieve this goal, we investigate adaptive view-specific pooling (AVS-Pool) that aggregates E v into a view-specific global codeword g v , as shown in Figure 3. Specifically, we start by explicitly predicting a positive scoring vector S v ∈ R N , serving as a point-wise visibility mask, by feeding E v into shared MLPs with Sigmoid as the activation function at the output end, i.e., S v = σ(M s (E v )),(3) where M s denotes three consecutive layers of shared MLPs whose final layer is implemented as a linear transformation, and σ(·) denotes the Sigmoid function. Thus, each value of S v is within the range of (0, 1) and designed to indicate the visibility status of the corresponding point. To guide the prediction of S v during the pre-training process, we accordingly introduce a visibility constraint: C v = S v − S v 1 ,(4) where S v ∈ {0, 1} denotes the ground-truth binary visibility mask (1 for visible, 0 for invisible) acquired by applying point set visibility checking [23] on input points P. By multiplying the predicted S v with E v , we achieve the effects of "masking" for suppressing the feature responses of invisible points, and then apply channel max-pooling on the masked point-wise embeddings. After that, we concatenate the pooled vector with the viewpoint indicator, and further deploy FC layers to generate the view-specific global codeword g v , which can be formulated as g v = F s (MP(S v * E v ) ⊕ v),(5) where * denotes element-wise multiplication with default channel-wise broadcasting, MP(·) represents channel-wise max-pooling, and F s denotes two consecutive FC layers. Translation Heads and Loss Functions Inheriting conventional image auto-encoders, we start by reshaping g v from the original vectorized representation to a C r -dimensional 2D feature map of dimensions H r × W r . After that, we deploy 2D CNNs as translation heads, composed of several stages of deconvolutional layers accompanied by progressive spatial up-scaling, to produce a higherresolution feature map. Finally, we feed this 2D feature map into three parallel convolutional blocks to reconstruct three forms of 2D rendered images, which are chosen as 1) depth map I d ; 2) silhouette mask I s ; and 3) contour map I c . The rendering process of these three types of images is not interfered by extrinsic factors (e.g., texture, material, illumination), and thus can purely depict geometric properties of the original 3D shape. Practically, we start by rendering the ground-truth depth map I d from the specified camera pose. After that, we can deduce the ground-truth silhouette mask I s and the ground-truth contour map I c by applying binary thresholding and Canny edge detection on I d , respectively. For supervision, we compute binary cross-entropy losses (L s and L c ) for the silhouette mask I s and the contour map I c . For the regression of depth values, we compute the L 1 loss (L d ) for the depth map I d . Thus, we can formulate the overall pre-training objective as L overall = ω v C v + ω d L d + ω s L s + ω c L c ,(6) where all weights ω v , ω d , ω s , and ω c are set to 1 empirically. Experiments We comprehensively validate the effectiveness and superiority of our proposed PointVST. In the following, we start by introducing necessary experimental setups and evaluation principles in Sec. 4.1. We present and compare downstream task performances from Sec. 4.2 to Sec. 4.5. Subsequently, we conduct extensive ablation studies in Sec. 4.6. Finally, we re-emphasize our core contribution in Sec. 4.7, while further pointing out potential limitations of the current technical implementations and future extensions. Pre-training Setups and Evaluation Schemes As a large-scale 3D shape repository consisting of more than 50,000 object models covering 55 semantic categories, ShapeNet [5] has been widely adopted as the source dataset for pre-training in many previous works (e.g., [41,35,20,2,66,34]), in which point clouds can be acquired by sampling from the original mesh models. We followed such common practice to perform pre-training on ShapeNet and evaluated on diverse downstream tasks, including object classification (Sec. 4.2), part segmentation (Sec. 4.3), normal estimation (Sec. 4.4), and scene semantic segmentation (Sec. 4.5). For works (e.g., [51]) that originally adopted inconsistent pretraining data (e.g., the smaller ModelNet [56] dataset), we used their official code to finish the pre-training process on the same data prepared from ShapeNet. Inheriting from previous closely-related approaches [51,2], two representative deep set architectures, PointNet [36] and DGCNN [53] (abbreviated as [P] and [D]), are adopted as the target point cloud backbones. Throughout our experiments, we involved multiple evaluation protocols, including linear (SVM) probing, fine-tuning, and semi-supervised learning with different amounts of partial training data. In the following (from Sec. 4.2 to Sec. 4.5), we mainly focus on comparing the proposed PointVST with influential unsupervised learning and self-supervised pre-training approaches. We especially made extensive comparisons with OcCo [51] and CrossPoint [2], the two state-of-the-art and most relevant studies that respectively represent generative and contrastive pre-training paradigms. Besides, in Sec. 4.6 we also particularly included comparisons with the very recent MAE-like approaches [34,27,68], which are specialized for transformer-style backbones and thus not applicable to generic types of deep set architectures [36,53]. Refer to Supplementary Material for more experimental results as well as detailed technical implementations. Object Classification ModelNet40 [56] is a commonly-used 3D shape classification benchmark dataset totally consisting of 12311 object models (9843 for training and 2468 for testing) covering 40 semantic categories. In our experiments, each input point Method OAcc 3D-GAN [54] 83.3 Latent-GAN [1] 85.7 FoldingNet [63] 88.4 PointCaps [71] 88.9 MTFL [16] 89.1 SelfContrast [9] 89.6 VIP-GAN [14] 90.2 [P]-Jigsaw [41] 87.3 [P]-OrienEst [35] 88.6 [P]-STRL [20] 88.3 [P]-OcCo [51] 88 cloud is uniformly composed of 1024 points without auxiliary attributes. Given a pre-trained backbone, we froze its model parameters, and then directly applied it to each point cloud within both the training and testing sets to extract the corresponding collections of vectorized global codewords, based on which a linear SVM is trained and tested. Table 1 compares the linear classification performances of different unsupervised and self-supervised methods under the measurement of overall accuracy (OAcc), in which our PointVST outperforms all the competing methods with large margins. In particular, compared with the second best method CrossPoint that also shares cross-modal characteristics, we still demonstrate prominent performance superiority, i.e., 1.1% and 0.9% accuracy improvements for Point-Net and DGCNN backbones, respectively. Besides, we further explored the rotation robustness of different methods, which can better reveal the representation capability of the learned geometric features. Accordingly, all the competing methods are equipped with random rotation for data augmentation purposes during pre-training. Quantitative results are presented in Table 2. For the z/z setting, all the competing methods show relatively satisfactory performances, and our PointVST still takes the leading position. For the much more challenging SO3/SO3 setting, though all methods suffer from significant degradation, our PointVST shows larger performance gains. In addition to transferring features exported from frozen backbone encoders, we also attempted to use the pre-trained model parameters for backbone network initialization, after which the whole learning framework is fine-tuned in a task-specific manner. Here, we experimented with ScanOb-jectNN [47] (the OBJ-BG split), a much more challenging real-scanned object classification benchmark. Point clouds in this dataset are typically noisy, incomplete, non-uniform, and interfered by background context. As illustrated in Table 3, our PointVST achieves the best performances with both backbones, effectively demonstrating our transferability from synthetic models to real-world point cloud scans. Part Segmentation ShapeNetPart [64] is a popular large-scale 3D object segmentation dataset with 50 classes of part-level annotations coming from 16 different object-level categories. Under the official split, there are 14007 models for training and 2874 models for testing. Each input point cloud uniformly contains 2048 points without auxiliary attributes. In contrast to the preceding task scenario (shape classification) of global geometry recognition, here the point-wise semantic prediction task requires to extract much more finegrained and discriminative feature descriptions. Given a pre-trained backbone, we performed fine-tuning for the whole segmentation framework. Table 4 reports the performances of different approaches under the measurement of mean interaction-over-union (mIoU). It is observed that our PointVST significantly outperforms other methods. Meanwhile, we noticed that the performance improvements relative to their DGCNN baselines (with random initialization) brought by OcCo and CrossPoint are relatively limited (with only 0.3% and 0.4% gains, respectively), while ours can reach 2.3%. Even for the less expressive PointNet backbone, its fine-tuning result reaches 86.8%, which is highly competitive. Besides, we further explored fine-tuning using different amounts of labeled training data. As illustrated in Figure 4, our PointVST consistently shows significant performance gains over all the competing methods. Normal Estimation Previous self-supervised pre-training works focus on the evaluation of high-level semantic understanding tasks such as classification and segmentation, but ignore verifications on low-level geometry processing tasks that are also highly valuable in various applications. To fill in this gap, we further performed normal estimation by fine-tuning from pretrained backbones on ModelNet40, where each input point cloud contains 1024 points and the corresponding groundtruth normal vectors can be easily deduced from mesh faces. In practice, considering that the computed ground-truth normals usually suffer from flipped directions, we adopted the absolute cosine distance (proposed in [36]) as the optimization objective as well as evaluation metric, given by NRErr = 1 − abs( n gt · n pr n gt 2 · n pr 2 ), where n pr and n gt denote the regressed and ground-truth normal vectors, respectively. Quantitative results are presented in Table 5, where an interesting observation is that OcCo outperforms CrossPoint for both backbones, and the performance gains brought by CrossPoint become even negligible. This phenomenon may indicate that a straightforward contrastive learning strategy implemented across heterogeneous 2D image and 3D point cloud features can barely benefit low-level fine-grained geometry processing. Besides, as for fine-tuning with different amounts of labeled training data, as shown in Figure 4, CrossPoint still shows inferior performances against OcCo, while our PointVST consistently takes the leading position with obvious margins. We perform fine-tuning from pre-trained backbones using limited training data (5%, 10%, 25%, 50%, 75%). The same partial training set is used for all the competing methods for fair comparisons. Scene Segmentation In addition to object-centric tasks, we also experimented with indoor scene segmentation on S3DIS [4], a commonlyused benchmark dataset, containing 271 single rooms from 6 different areas, with over 270 million densely annotated points covering 13 semantic categories. The pre-processing procedures follow the original baselines [36,53]. Following the evaluation protocol of previous work [51], here we aim to verify the transferability of the pre-trained backbones from the source object-level domain to the target scene-level domain. As shown in Table 6, all the competing methods can bring different degrees of performance boosts relative to randomly initialized baselines, and our PointVST also shows obvious performance superiority. Ablation Study Image translation objectives. As described in Eq. 6, in the full implementation, our translation objectives involve three different forms of 2D rendered images, i.e., depth, contour, and silhouette maps. To reveal their necessity and influence, we conducted experiments under the setting of ModelNet40 linear SVM classification by respectively removing each of the three loss terms. As indicated in Table 7, all the three image supervisions turn to be necessary. The removal of L d leads to the most severe performance degradation, because depth maps convey the richest geometric information. Mesh Rendering Point Cloud Rendering 3D Points Figure 5. Visual comparisons of ground-truth images rendered from meshes and sparse point clouds. Image rendering strategies. Different ways of ground-truth image creation can influence the actual pre-training effects. Although mesh rendering is chosen in our implementation, we demonstrate that point cloud rendering also works well. Here, for the setting of ModelNet40 SVM classification, we adopted Pytorch3D [39] library's rasterizer to render images directly from point clouds in the pre-training phase. Typical visual examples are presented in Figure 5. As reported in Table 8, compared with mesh rendering (with 90.2% and 92.1% OAcc for backbones [P] and [D]), switching to point cloud rendering causes slight performance degradation, but still turns to be highly effective. Pre-training data domains. In different previous works, the source point cloud data domains selected for implementing the pre-training process can be object-level [51,66,2,34], scene-level [58,69], or both [20,27] (see Supplementary Material for a detailed summary). To explore the scalability of PointVST for pre-training with scene-level point clouds, here we adopted the popular ScanNet [7] dataset to prepare the corresponding pre-training data, where the 2D ground-truth images are directly rendered from point clouds of scene blocks/patches. We report fine-tuning (with backbone [D]) performances in Table 9, where the preceding results with ShapeNet pretraining are also pasted (in the first and third rows) for convenience. It is observed that PointVST shows satisfactory improvements under different transfer modes. And we also noticed that better results can be achieved given a smaller source-target domain gap (e.g., O ⇒ O, S ⇒ S). Effectiveness of AVS-Pool. In our technical implementation, deducing a view-specific global codeword from viewconditioned point-wise embeddings relies on the proposed AVS-Pool. To evaluate its necessity, we designed multiple variants of PointVST by replacing AVS-Pool with different pooling strategies. In addition to common average-pooling and max-pooling, we also introduced a more advanced attentive aggregation technique [61] that has been integrated in various low-level and high-level point cloud processing tasks [18,19,50]. As shown in Table 10, average-pooling suffers from the most severe performance degradation due to its lack of selectivity, max-pooling and attentive pooling show better performances since they are more suitable for view-specific aggregation. The comparison between AVS-Pool-Uns and the full implementation can validate the necessity of our proposed visibility constraint C v . Moreover, as indicated in the last row for the GT-Vis variant, although directly using ground-truth visibility seems more straightforward, it tends to be sub-optimal compared with adaptively learning the visibility mask. We reason that the adaptive prediction process itself can guide the model to better extract view-specific patterns. Transformer-oriented pre-training. As mentioned earlier, MAE-like pre-training frameworks [66,34,27,68] that are specialized for transformer-style 3D point cloud backbones also attract much attention for their impressive performances, despite the inapplicability to many other widelyused (non-transformer) deep set architectures. To fairly form apples-to-apples comparisons, we applied PointVST to pre-train a standard transformer backbone [34,27] (abbreviated as [T]), and then performed fine-tuning for object classification on ModelNet40 and part segmentation on ShapeNetPart. Note that Point-M2AE [68] is built upon a more advanced multi-scale transformer (denoted as [Tm]) backbone. Following these works, we used test voting [30] on ModelNet40, but not for ShapeNetPart. As illustrated in Table 11, our PointVST still shows leading performances when applied to the transformer-style backbones, which are known to have greater model capacity. Discussions As the very first attempt towards a new design paradigm of self-supervised point cloud pre-training driven by crossmodal translation, our ultimate goal, as well as the core contribution, is to verify the viability and explore the potential of such direction. And through extensive experiments, the proposed PointVST did consistently show prominent performance superiority, despite the concise technical implementation without sophisticated learning mechanisms. To better motivate the subsequent explorations along the proposed direction, below we discuss the limitations of our current framework and also some promising future efforts. Limitations and extensions. The current technical implementation is designed to infer geometric appearance rendered from an outer viewpoint, which can cause information loss for point clouds that suffer from severe self-occlusion or have rich spatial structures inside the underlying surface. A more flexible or adaptively-learned viewpoint specification strategy is needed to relieve this issue. Besides, it might be indirect to acquire ground-truth 2D images corresponding to 3D point clouds with extremely sparse/noisy distributions. Under such circumstances, the rendering results can be unsatisfactory, and thus we may have to switch to using realistic RGB camera images instead of synthetic groundtruth data. As a much more sophisticated problem setting, this can be left for future study. Besides, another promising direction is to explore more advanced and effective ways of deducing view-specific feature representations, in addition to the current visibility-based scheme. Conclusion This paper aims at designing self-supervised pre-training frameworks for learning highly expressive and transferable geometric feature representations from unlabeled 3D point clouds. The core contribution is a new cross-modal translative pretext task, namely PointVST, posing a paradigm shift compared with previous generative and contrastive counterparts. Through extensive experiments on diverse evaluation protocols and downstream task scenarios, PointVST shows state-of-the-art performances with consistent and prominent gains. Our highly encouraging results indicate a much more promising way of integrating 2D visual signals into the pretraining pipelines of 3D geometric signals. 2.4) Length of View-Specific Global Codeword. In all our experiments, we uniformly configured the length of the produced view-specific global codeword g v as 2048. Here, we further experimented with 1024 and 4096 to explore its influence. As compared in Table 17, the performances of our [P]-PointVST and [D]-PointVST respectively decrease by 0.2% and 0.1% when using larger image dimension of 256 × 256. This may indicate that the actual pre-training effects can be weakened when the pretext task becomes much more difficult (i.e., recovering more detailed geometric details in the image domain). When the image dimension is set to smaller 64 × 64, the performance degradation further reaches 0.4% and 0.5% for [P]-PointVST and [D]-PointVST, respectively. Figure 6, we adopted the classic t-SNE [?] technique for visualizing the distribution of the learned feature representations exported from our pre-trained backbone encoders on the test split of ModelNet10. We can observe that our PointVST can contribute satisfactory intra-class clustering and inter-class separability effects in the pre-trained backbone feature encoders. 2.6) Visualization of Feature Distribution. As shown in Pretext Task Performances Under the development protocols of self-supervised pre-training, the core focus lies in whether the target backbone encoder is effectively guided to learn transferable features and to what extent the downstream task performance can be boosted. However, to facilitate a more intuitive and comprehensive understanding, we still provided both qualitative and quantitative results of the pretext task performance within our proposed PointVST. 3.1) Prediction Results of Point-Wise Visibility Scores. As introduced in Section. 3.2 of the manuscript, in our proposed AVS-Pool, we tend to explicitly predict the positive scoring vector S v that indicates point-wise visibility status with respect to the specified viewpoint. We performed quantitative evaluations by binarizing the predicted S v with appropriate thresholds and compared it with the corresponding ground-truth mask S v . As reported in Figure 7, the overall accuracy of the predicted S v reaches over 94%, meaning that the corresponding learnable layers are effectively optimized to provide reliable per-point visibility information. Note that, as demonstrated in Table 10 of the manuscript, the task of visibility prediction itself is conducive to the learning effects of the overall pre-training pipeline. Besides, in Figure 8, we provided typical visual examples by simultaneously presenting the complete point clouds, the predicted and the ground-truth subsets of visible points (with respect to different randomly specified viewpoints). 3.2) Translation Results of 2D Rendered Images. In our implementation, we included three different forms of image translation objectives (single-channel images), i.e., depth I d , silhouette I s , and contour I c . In Figure 9, we provided typical visual examples of our translated results and their corresponding ground-truth images (i.e., I d , I s , and I c ) (with respect to different randomly specified viewpoints). Figure 9. Visual examples of our translated images. Given (a) input point clouds, we compare ground-truth ((b), (d)) (f)) and translated ((c), (e), (g)) silhouette, contour, and depth images with respect to randomly specified viewpoints. Generally, it is observed that PointVST is able to produce reasonable view-specific image translation results. Still, since recovering fine-grained geometric details in the 2D image domain from sparse 3D point clouds is highly difficult, the resulting translation results show relatively coarse patterns. For example, it can hardly capture complex object contours, and the translated depth distributions are also less accurate in complex regions. Figure 1 . 1Illustration of view-specific cross-modal translation from a 3D point cloud to diverse forms of 2D rendered images. Figure 3 . 3Illustration of AVS-Pool, which is customized for aggregating view-conditioned point-wise embeddings Ev into a vectorized view-specific global codeword gv. Figure 4 . 4(a) Semi-supervised part segmentation on ShapeNet-Part. (b) Semi-supervised normal estimation on ModelNet40. Figure 6 . 6Visualization of features exported from our pre-trained PointNet and DGCNN backbones on ModelNet10. Figure 7 . 7Overall accuracy of the predicted point-wise visibility status when pre-training the DGCNN (classification-oriented) backbone encoder. We applied different thresholding values (from 0.20 to 0.80 with the uniform interval of 0.05) for the binarization of the raw score values in Sv. Figure 8 . 8Visual examples of point-wise visibility checking, where we presented (b) ground-truth and (c) our predicted visible points with respect to randomly specified viewpoints by masking the corresponding (a) input point clouds P with S v and Sv. Table 1 . 1Linear SVM classification on ModelNet40. Table 2. Linear SVM classification for handling inputs with z/z and SO3/SO3 rotations on ModelNet40. For the z/z setting, each input is rotated around the ground-axis with a uniform interval of 15 • . For the SO3/SO3 setting, each input is repeatedly rotated 16 times by randomly sampling from the 3D rotation group SO(3)..8 [P]-CrossPoint [2] 89.1 [P]-PointVST 90.2 [D]-Jigsaw [41] 90.6 [D]-OrienEst [35] 90.8 [D]-STRL [20] 90.9 [D]-OcCo [51] 90.7 [D]-CrossPoint [2] 91.2 [D]-PointVST 92.1 Method ModelNet40 z/z SO3/SO3 [P]-OcCo [51] 83.9 50.6 [P]-CrossPoint [2] 86.5 58.4 [P]-PointVST 87.6 64.7 [D]-OcCo [51] 86.3 55.7 [D]-CrossPoint [2] 88.2 74.8 [D]-PointVST 90.0 78.3 Table 3 . 3Real-scanned object classification on ScanObjectNN (OBJ-BG) by fine-tuning from pre-trained backbones.Method OAcc Method OAcc [P]-Random 73.3 [D]-Random 82.8 [P]-OcCo [51] 79.8 [D]-OcCo [51] 84.5 [P]-CrossPoint [2] 80.2 [D]-CrossPoint [2] 86.2 [P]-PointVST 80.7 [D]-PointVST 89.3 Table 4. Part segmentation on ShapeNetPart by fine-tuning from pre-trained backbones. Method mIoU Method mIoU [P]-Random 83.7 [D]-Random 85.1 [P]-OcCo [51] 84.4 [D]-OcCo [51] 85.4 [P]-CrossPoint [2] 85.0 [D]-CrossPoint [2] 85.5 [P]-PointVST 86.8 [D]-PointVST 87.4 Table 5 . 5Normal regression on ModelNet40 by fine-tuning from pre-trained backbones.Method NRErr Method NRErr [P]-Random 0.176 [D]-Random 0.102 [P]-OcCo [51] 0.155 [D]-OcCo [51] 0.084 [P]-CrossPoint [2] 0.169 [D]-CrossPoint [2] 0.098 [P]-PointVST 0.132 [D]-PointVST 0.076 Table 6. Scene-level semantic segmentation on S3DIS under 6- fold cross-validation. Method mIoU Method mIoU [P]-Random 47.6 [D]-Random 56.1 [P]-OcCo [51] 54.5 [D]-OcCo [51] 58.3 [P]-CrossPoint [2] 54.2 [D]-CrossPoint [2] 57.9 [P]-PointVST 55.7 [D]-PointVST 60.1 Table 7 . 7Effects of pre-training PointVST with different image translation objectives under ModelNet40 linear SVM classification. Within brackets, we emphasize the performance degradation of each variant relative to the full implementation. Table 8 . 8Comparisonsof pre-training PointVST with ground- truth images rendered from meshes (M-Rendered) and points (P-Rendered) under ModelNet40 linear SVM classification. Table 9 . 9Performances of[D]-PointVST with different source (to be pre-trained) and target (to be fine-tuned) data domains. O and S denote object-level and scene-level, respectively.Source Data Target Data Transfer Mode mIoU ShapeNet S3DIS O ⇒ S 60.1 ScanNet S3DIS S ⇒ S 63.7 ShapeNet ShapeNetPart O ⇒ O 87.4 ScanNet ShapeNetPart S ⇒ O 86.5 Table 10. Effectiveness verifications of AVS-Pool under Mod- elNet40 linear SVM classification, where Avg-Pool, Max-Pool, Att-Pool, and AVS-Pool-Uns respectively denote average-pooling, max-pooling, attentive pooling [61], and AVS-Pool without ex- plicitly supervising Sv (i.e., removing Cv from L overall ). GT-Vis means replacing adaptive visibility mask prediction with ground- truth visibility. Within brackets, we emphasize the performance degradation of each variant relative to the full implementation. Table 11 . 11Comparison with MAE-like pre-training approaches specialized for transformer-style point cloud backbones.Method ModelNet40 ShapeNetPart [T]-Point-MAE [34] 93.8 86.1 [T]-MaskPoint [27] 93.8 86.0 [Tm]-Point-M2AE [68] 94.0 86.5 [T]-PointVST 94.1 87.6 Table 15 . 15Influence of position encoding (P.E.) for view representation on ModelNet40 linear SVM classification.View Representation [P]-PointVST [D]-PointVST w/ P.E. 90.2 92.1 w/o P.E. 89.8 91.9 Table 16 . 16Influence of the length of view-specific global codeword gv on ModelNet40 linear SVM classification.As reported inTable 16, reducing the length to 1024 leads to different degrees of performance degradation for both [P]-PointVST and [D]-PointVST. When increasing the length to 4096, the performance of [D]-PointVST decreases by 0.4%, while [P]-PointVST is slightly boosted by 0.1%. 2.5) Dimension of Translated 2D Images. Throughout our experiments, we uniformly configured the dimension of translated images as 128 × 128. Here, we also experimented with larger 256 × 256 and smaller 64 × 64 dimensions.Codeword Length [P]-PointVST [D]-PointVST 2048 90.2 92.1 1024 89.7 91.8 4096 90.3 91.7 Table 17 . 17Influence of different dimensions of translated images on ModelNet40 linear SVM classification.Image Dimension [P]-PointVST [D]-PointVST 128 × 128 90.2 92.1 256 × 256 90.0 92.0 64 × 64 89.8 91.6 Supplementary Materials6. Detailed Technical Configurations 1.1) View-Conditioned Point-Wise Fusion. When specifying camera positions for 2D image rendering, we uniformly set the observation distance as d = 2. As presented in Eq. (1) of the manuscript for obtaining the vectorized viewpoint indicator v ∈ R Di , both F φ and F λ comprise two consecutive FC layers whose output channels are {64, 128}. Thus, we uniformly have the dimension of the resulting viewpoint indicator D i = 256. After that, when performing point-wise fusion (Eq. (2)), the output channels of both M e and F g are 1024, meaning that the input dimension of M f is 1024 + 1024 + 256 = 2304. And we set its output channels as 1024, such that the dimension of the resulting view-conditioned point-wise embeddings To learn the visibility mask S v in a supervised manner, we adopted a classic hidden point removal algorithm[23]developed in the graphics community, which directly operates on raw point sets while maintaining simple and fast. 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For data preparation on ScanNet [7], we cropped sub-regions from the whole point cloud scene (with grid-subsampling [18]) using both kNN-based and area-based strategies, such that each cropped point cloud block/patch contains 4096 3D spatial points (without auxiliary attributes). After rendering the ground-truth depth map I d , we can straightforwardly deduce the binary ground-truth silhouette mask I s , i.e., all non-empty depth values are set as 1, and can produce the ground-truth contour map I c by applying Canny edge detection. Additionally, we also applied the classic morphological dilation operation on the binary edges to strengthen thin/broken structuresSource Data Preparation for Pre-training. For data preparation on ShapeNet [5], we used the commonly-adopted Poisson disk sampling algorithm to generate point cloud data (without auxiliary attributes) from the original polygon mesh models. The number of points is set as 2048 for this setting. During the rendering process, the resulting image resolution is configured as 128 × 128, which can effectively maintain necessary geometric details while achieving satisfactory pre-training efficiency. For data preparation on ScanNet [7], we cropped sub-regions from the whole point cloud scene (with grid-subsampling [18]) using both kNN-based and area-based strategies, such that each cropped point cloud block/patch contains 4096 3D spatial points (without auxiliary attributes). After rendering the ground-truth depth map I d , we can straightforwardly deduce the binary ground-truth silhouette mask I s , i.e., all non-empty depth values are set as 1, and can produce the ground-truth contour map I c by applying Canny edge detection. Additionally, we also applied the classic morphological dilation operation on the binary edges to strengthen thin/broken structures. Additional Results and Analyses. Additional Results and Analyses Transferability is a critical factor when evaluating the superiority of self-supervised representation learning approaches. Depending on different data characteristics, common point cloud processing tasks can be classified as object-level and scene-level. Accordingly, different previous works configured different experiment settings for the corresponding verifications. As summarized in Table 12, compared with most previous influential works on self-supervised point cloud representation learning, in this paper, we explored more comprehensive transfer modes between object-level and scene-level point cloud data domains. which can better validate the effectiveness and generalizability of our proposed PointVSTComprehensive Verifications of Domain Transfer Modes. Transferability is a critical factor when evaluating the superi- ority of self-supervised representation learning approaches. Depending on different data characteristics, common point cloud processing tasks can be classified as object-level and scene-level. Accordingly, different previous works configured different experiment settings for the corresponding verifications. As summarized in Table 12, compared with most previous influential works on self-supervised point cloud representation learning, in this paper, we explored more comprehensive transfer modes between object-level and scene-level point cloud data domains, which can better validate the effectiveness and generalizability of our proposed PointVST. Domain transfer modes involved in previous influential works on self-supervised 3D point cloud representation learning. The left and the right side of "⇒" respectively denotes the source data domain (for pre-training) and the target data domain (for fine-tuning). For simplicity, we use the notations "O" and "S" to respectively represent the object-level. Table 12. ShapeNet [5], ModelNet [56]) and scenelevel (e.g., ScanNet [7], S3DIS [4]) point cloud data domains. In the last column, we particularly denote whether the corresponding work considers the data domain gap between object-level and scene-level point clouds. Previous Works Source-to-Target Transfer Modes With Gap Across O and STable 12. Domain transfer modes involved in previous influential works on self-supervised 3D point cloud representation learning. The left and the right side of "⇒" respectively denotes the source data domain (for pre-training) and the target data domain (for fine-tuning). For simplicity, we use the notations "O" and "S" to respectively represent the object-level (e.g., ShapeNet [5], ModelNet [56]) and scene- level (e.g., ScanNet [7], S3DIS [4]) point cloud data domains. In the last column, we particularly denote whether the corresponding work considers the data domain gap between object-level and scene-level point clouds. Previous Works Source-to-Target Transfer Modes With Gap Across O and S Experiments on Outdoor Scene Semantic Segmentation. Toronto3D [73] is a large-scale LiDAR point cloud dataset for benchmarking real-scene semantic segmentation, which consists of 78.3 million densely-annotated points covering approximately 1KM of outdoor urban space. There are totally 8 different categories of semantic labels. Table 13. Outdoor scene segmentation on Toronto3D by fine-tuning from the DGCNN backbone pre-trained on ShapeNet. Method mIoUExperiments on Outdoor Scene Semantic Segmentation. Toronto3D [73] is a large-scale LiDAR point cloud dataset for benchmarking real-scene semantic segmentation, which consists of 78.3 million densely-annotated points covering ap- proximately 1KM of outdoor urban space. There are totally 8 different categories of semantic labels. Table 13. Outdoor scene segmentation on Toronto3D by fine-tuning from the DGCNN backbone pre-trained on ShapeNet. Method mIoU Comparisons of Toronto3D segmentation performances with different source domains for backbone pre-training, including two object datasets (ShapeNet and ScanObjectNN) and one scene dataset. Table 14. ScanNetTable 14. Comparisons of Toronto3D segmentation performances with different source domains for backbone pre-training, includ- ing two object datasets (ShapeNet and ScanObjectNN) and one scene dataset (ScanNet). Following the official split [73], the section L002 is used for testing and the rest three sections (L001, L003, L004) are used for training. The raw data pre-processing procedures follow. Following the official split [73], the section L002 is used for testing and the rest three sections (L001, L003, L004) are used for training. The raw data pre-processing procedures follow [19]. road marking and road) given only 3D coordinates of points. Hence, as also explored in [19], we further included RGB color attributes as additional input information (for all the competing methods) to make the corresponding evaluations more discriminative. Table 13 reports quantitative performances of different approaches, where we experimented with the DGCNN backbone pretrained on ShapeNet. Our PointVST achieves 2.6% improvement over the baseline DGCNN and outperforms both OcCo and CrossPoint. Besides, as shown in Table 14, we also explored the effects of pre-training with different domains of source datasets. We can observe that the downstream performance is still competitive even with a much smaller pre-training dataset. Particularly, as analyzed in [19], it is basically impossible to differentiate some classes. Switching to ScanNet for pre-training leads to better performances since the domain map between source and target data becomes smallerParticularly, as analyzed in [19], it is basically impossible to differentiate some classes (e.g., road marking and road) given only 3D coordinates of points. Hence, as also explored in [19], we further included RGB color attributes as additional input information (for all the competing methods) to make the corresponding evaluations more discriminative. Table 13 reports quantitative performances of different approaches, where we experimented with the DGCNN backbone pretrained on ShapeNet. Our PointVST achieves 2.6% improvement over the baseline DGCNN and outperforms both OcCo and CrossPoint. Besides, as shown in Table 14, we also explored the effects of pre-training with different domains of source datasets. We can observe that the downstream performance is still competitive even with a much smaller pre-training dataset (i.e., ScanObjectNN). Switching to ScanNet for pre-training leads to better performances since the domain map between source and target data becomes smaller. As introduced in Section 3.1 (Eq. (1)) of the manuscript, the vectorized viewpoint indicator v is obtained via simple positional encoding. Here, we removed the two non-linear transformation layers (F φ , F λ ) and directly used raw latitude and longitude angles (φ lat , λ lon ) for the subsequent processing procedures. Positional Encoding of Raw Camera Positions. As reported in Table 15, compared with the full implementation, the removal of positional encoding respectively leads toPositional Encoding of Raw Camera Positions. As introduced in Section 3.1 (Eq. (1)) of the manuscript, the vectorized viewpoint indicator v is obtained via simple positional encoding. Here, we removed the two non-linear transformation layers (F φ , F λ ) and directly used raw latitude and longitude angles (φ lat , λ lon ) for the subsequent processing procedures. As reported in Table 15, compared with the full implementation, the removal of positional encoding respectively leads to 4% and 0.2% performance degradation for our. PointVST and [D]-PointVST on ModelNet40 linear SVM classification4% and 0.2% performance degradation for our [P]-PointVST and [D]-PointVST on ModelNet40 linear SVM classification.
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[ "MULTIWAVE TOMOGRAPHY WITH REFLECTORS: LANDWEBER'S ITERATION", "MULTIWAVE TOMOGRAPHY WITH REFLECTORS: LANDWEBER'S ITERATION" ]
[ "Plamen Stefanov ", "Yang Yang " ]
[]
[]
We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space.
10.3934/ipi.2017018
[ "https://arxiv.org/pdf/1603.07045v3.pdf" ]
18,065,098
1603.07045
a092c2f298b74fa6c69e73f784bb34c10fd99f35
MULTIWAVE TOMOGRAPHY WITH REFLECTORS: LANDWEBER'S ITERATION Plamen Stefanov Yang Yang MULTIWAVE TOMOGRAPHY WITH REFLECTORS: LANDWEBER'S ITERATION We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space. Introduction In this paper, we apply the well known Landweber iteration method for the multiwave tomography problem with reflectors. We also compare it to the Averaged Time Reversal (ATR) method developed by the authors in [33] leading to an inversion of a Neumann series. We also present a convergence analysis for the Landweber method for linear problems in Hilbert spaces. Even though the Landweber iteration method is widely used and well studied, we prove some results that we have not found in the literature. In numerical inversions, we always work in finite dimensional spaces, but there are good reasons to understand the method better in the continuous model first, and then to understand how well the discretization approximates the continuous model. As we demonstrate in this work, for the inverse problem we study involving wave propagation, the discrete model based on a second order finite difference scheme is always unstable regardless of whether the continuous one is stable or not. On the other hand, stable problems in our case behave in a stable way numerically, even though they are not (when discretized). We analyze the convergence rates of the iterations, their relation to the a priori stability of the problem, of the spectrum of the operator and on the spectral measure of the object we want to recover, and present numerical evidence of those results. While there are more advanced computational inversion methods for general linear or non-linear inverse problems, including such with regularization, our goal is not to develop "the best one". Without real life data and understanding the real measurements challenges, that would be not so useful anyway. We would like to have methods that would allow us to test the mathematics of the problem: how the geometry affects the stability, for example; and partial data on the boundary. We are also interested in how the discretization and the numerical solver to model the PDE affects the problem, regardless of how it is solved after that. The multiwave tomography problem with reflectors (the term "cavity" is often used, as well) we study models the following. We send one type of waves to an object we want to image, like a part of the human body like microwave radiation (in thermoacoustic tomography), laser rays (in photoacoustic tomography) or elastic waves. Those waves have a low resolution (the laser ray would diffuse inside) but they are partly absorbed by the cells, which cause them to emit ultrasound waves of high enough frequency allowing for a high resolution. The emitted sound waves are measured on the boundary, or on a part of it, and we want to recover the source; after that, we want to recover the absorption rate. We do not study the latter problem here. This problems is well studied under the assumption that the waves travel freely away from the body, see, e.g., [13,21,22,23,38,30,31,26,32,7,8,5] and the references there. In some experimental setup, see [10], reflectors are placed around the body and the measurements are made on a part of it. The mathematical model of this is presented below in (2.1), first studied in [24,17,7,8]. In [33] and [25], see also [1], two different stable ways to solve the problem were proposed, both based on exponentially convergent Neumann series, expanding some ideas introduced first in [30]. The inversion in [1,25] works for the case of partial boundary data as well; while in this case, in [33], we know that it gives a parametrix for partial boundary data but there is no proof yet that in converges to a solution. We tested it in that case anyway, and we also do it in this paper. Recently, Acosta and Montalto [2] studied a model taking account attenuation, see also [18]. Landweber iterations for the whole space problem were done recently in [7,8] and compared to the sharp time reversal method in [30] implemented numerically in [26]. Another recent work is [5], where several different optimizations are compared, based on different regularization terms. The model and known theoretical results Let Ω ⊂ R n be a bounded open subset. Consider (2.1)        (∂ 2 t − c 2 (x)∆)u = 0 in (0, T ) × Ω, ∂ ν u| (0,T )×∂Ω = 0, u| t=0 = f, ∂ t u| t=0 = 0, where ∂ ν is the normal derivative. One could consider a Riemannian metric g and then ∆ would be the Laplace-Beltrami operator, see [33] and section 6. The measurement operator is (2.2) L : f → u| (0,T )×∂Ω . The goal is to inverse L and recover f . In the partial data problem, we are given (2.3) L Γ : f → u| (0,T )×Γ , where Γ ⊂ ∂Ω is a relatively open subset. We recall some uniqueness and stability results about this problem, see also [33]. Uniqueness follows from unique continuation [34]. The sharpest time T for uniqueness if given by (2.4) T 0 := max x∈Ω 0 dist(x, Γ), where dist(x, Γ) is the distance between a point and a set in the metric c −2 dx 2 . If T < T 0 , Lf determines f uniquely only at the points staying at distance less than T from Γ. If T > T 0 , there is a global uniqueness. A sharp stability condition follows from the Bardos, Lebeau and Rauch [6]. In view of the intended applications, we would assume that ∂Ω is strictly convex and that supp f is compactly supported in Ω but [6] covers the general case. (a) We say that the stability condition is satisfied if every broken unit speed geodesic γ(t) with γ(0) ∈Ω 0 has at least one common point with Γ for |t| < T , i.e., if γ(t) ∈ Γ for some t ∈ (−T, T ). (b) We call the point (x, ξ) ∈ T * Ω 0 \ 0 a visible singularity if the unit speed geodesic γ through (x, ξ/|ξ|) has a common point with Γ for |t| < T . We call the ones for which γ never reachesΓ for |t| ≤ T invisible ones. We denote by T 1 = T 1 (Γ, Ω) the least upper bound of all T for which the stability condition is satisfied. As mentioned in [33], the visible and invisible singularities do not cover the while T * Ω 0 \ 0 since some rays may hit ∂Γ or t = T . The complement of those two sets is of measure zero however. Visible singularities can be recovered stably, and non-visible cannot. We will not give formal definitions (see, e.g., [29]). In numerical computations, this means that visible singularities, like edges, etc., can in principle be recovered well (the would look "sharp") but the actual reconstruction may require some non-trivial efforts. Non-visible singularities cannot be recovered well and the typical way to deal with this is to recover some regularized version of them, like blurred edges. If the stability condition is satisfied, them one has an H 1 → H 1 stability estimate [33]. The proof of this follows from the fact that L is an FIO of order 0 associated with a local diffeomorphism. That diffeomorphism can be described in the following way. For every (x, ξ) ∈ T * Ω 0 \ 0, we take the geodesic γ x,ξ/|ξ| (t), where |ξ| is the norm of the covector ξ at x in the metric c −2 dx 2 . We also identify vectors and covectors by the metric. When γ x,ξ/|ξ| hits ∂Ω, we take that point and the projection of the tangent vector on ∂Ω. That correspondence is a local diffeomorphism, and here it is essential that we have guaranteed that γ x,ξ/|ξ| hits ∂Ω transversely. By the stability condition, such a contact with ∂Ω exists. After that, we reflect the geodesic by the usual law of reflection, and look for another contact, etc. We consider both positive and negative t. That would give us a multi-valued map but locally (near the starting singularity and locally near the image), each branch is a diffeomorphism. Then L is a restriction of a first order elliptic FIO (say, the same operator but defined on a larger open set containing the closure of (0, T ) × Γ), to (0, T ) × Γ. In particular, that means that it is bounded from L 2 (Ω 0 ) to L 2 ((0, T ) × Γ). We consider L 2 (Ω 0 ) as subspace of L 2 (Ω). The stability condition guarantees that at least one of the contacts with ∂Ω is interior for that set, and this is enough for building a microlocal parametrix. This, together with the uniqueness result, implies the following theorem. We refer to [29] for a more general case, and to [6] for this particular one. Theorem 2.1. Let ∂Ω be strictly convex and fix Ω 0 Ω, an open Γ ⊂ Ω and T > 0. Then if the stability condition is satisfied, i.e., if T > T 1 , then there exist 0 < µ ≤ C so that µ f L 2 (Ω 0 ) ≤ Lf L 2 ((0,T )×Γ) ≤ C f L 2 (Ω 0 ) , ∀f ∈ L 2 (Ω 0 ). In fact, in the reconstructions, we work in the topologically equivalent space L 2 (Ω 0 , c −2 dx). This is the measure appearing in the energy Ω (|∇f 1 | 2 + |f 2 |c −2 ) dx; and our numerical simulations show that the error is smaller if we use that measure in the definition of L * later. If there are invisible singularities, then the estimate above cannot hold with µ > 0 and it fails even if we replace the L 2 norm by weaker Sobolev ones. We review the Averaged Time Reversal method in section 6. One of the essential differences is that there, the energy space is (topologically equivalent to) H 1 (Ω) while in the Landweber iterations we choose to work in L 2 (Ω). We could do Landweber iterations considering L as an operator from H 1 0 (Ω) to H 1 ((0, T ) × Γ), as well or even as an operator from H 1 0 (Ω) to L 2 ((0, T ) × Γ). In fact, we tried the latter numerically. Then L is a smoothing operator of degree 1 (an elliptic FIO of order −1 associated with a local diffeomorphism), and this makes the problem always unstable in those spaces. The reconstructions we get are very blurred as one would expect. Finally, if one wants to achieve some regularizing effect with a regularizing parameter, similar to Tikhonov regularizations, one could introduce the following H 1 norm f 2 H 1 ε = (0,T )×Ω |f | 2 c −2 + ε|∇f | 2 dx, where ε > 0 is a parameter. Then we think of L as the operator L : H 1 ε (Ω) → L 2 ((0, T ) × ∂Ω), restricted to functions supported in Ω 0 . This replaces L * in the inversions by (1 − εc 2 ∆ D ) −1 L * , where ∆ D is the Dirichlet Laplacian in Ω. Another modification would be to keep L 2 (Ω, c −2 dx) but to replace L 2 ((0, T ) × ∂Ω) by a certain H −1 space with a parameter. This would put a low pass frequency filter to the right of L * rather than to the left. One such reconstruction is shown in Figure 12. The plan of this work is the following. We review the Landweber iterations in section 3 and study the convergence or the divergence, the dependence on the step γ, etc. In section 4, we compute the adjoint of the measurement operator L with full or partial data. We present numerical examples in section 5. A review of the averaged time reversal method is included in section 6. Acknowledgments. The authors thank Francois Monard and Jianliang Qian for their advice during the preparation of this work. Reference [41] was pointed out to the authors by Francois Monard. Landweber iterations As explained in the Introduction, one of the goals of this paper is to analyze further the Landweber iterations method for linear inverse problems, its relation to the stability or the instability of the problem and to the power spectrum of the function we want to recover (defined as the differential dµ f (λ) of P λ f 2 , where P λ is the spectral projection of L * L). We also study the effect of the choice of the step γ in the iterations and how they are influenced by noise or data not in the range. We do not study regularized versions of the iterations for severely ill posed linear problems, or non-linear problems. We do not study stopping criteria and their effect on the error, either. For analysis of the linear Landweber method, see, e.g., [36,37,15,9,20] and the references there. For the non-linear one, see, e.g., [16]. Some of the analysis below can be found in the literature, for examaple in [15], but we did not find all of it in the literature. 3.1. Landweber Iterations. Let L : H 1 → H 2 be a bounded operator where H 1,2 are Hilbert spaces. We want to solve the linear inverse problem of inverting L. Assume no noise first. Then we write (3.1) Lf = m in the form (3.2) (I − (I − γL * L))f = γL * m. The operator K := I − γL * L is self-adjoint with spectrum in the interval (3.3) 1 − γ L 2 , 1 − γµ 2 , where µ ≥ 0 is any of the stability constants in the stability estimate (3.4) µ f ≤ Lf , which might be zero if there is no stability or even injectivity. We choose µ to be the largest number with that property, i.e., µ 2 is the bottom of the (closed) spectrum of L * L. If L * L has a discrete spectrum, then µ would be the smallest singular value. In other words, L and µ are the sharp constants for which (3.5) 0 ≤ µ 2 ≤ L * L ≤ L 2 . Then (3.3) is the smallest closed interval containing the spectrum spec(K) of K but the spectrum itself could have gaps. By (3.3), we see that K is a strict contraction ( K < 1) if and only if (3.6) − 1 < 1 − γ L 2 , 1 − γµ 2 < 1, i.e., (3.7) 0 < γ < 2 L 2 , 0 < µ. In other words, γ has to be small enough and the problem has to be stable. Then f can be reconstructed by the following Neumann expansion (3.8) ∞ j=0 K j γL * m = ∞ j=0 (I − γL * L) j γL * m which converges uniformly and exponentially because (I − γL * L) j L * m ≤ K j L * m . This implies the following scheme. Landweber Iterations. Set f 0 =0, f k =f k−1 − γL * (Lf k−1 − m), k = 1, 2, . . . . (3.9) The scheme is used even when the data m is "noisy", i.e., when m ∈ Ran L, or when there is no uniqueness. Sometimes, the following criterion is used. Let C > 1 be a prescribed constant. The iteration is terminated when the condition Lf k − m H 2 < Cδ is violated for the first time, where δ is an a priori bound of the noise level, i.e., a constant for which m − Lf ≤ δ, where f is the unknown function we want to reconstruct and we think of m as a noisy measurement. Then we take f k for such a k as an approximation of f . If there is stability, i.e., when µ > 0, this implies f k − f ≤ Cδ/µ. It is well known that Landweber Iterations is just a gradient decent method for the functional f −→ Lf − m 2 H 2 . This makes some of the properties we describe below more geometric; for example if m ∈ Ran L, a minimizer, if exists, would be the same as a minimizer for m replaced by m projected on Ran L. Let us denote K by K γ now. To estimate the rate of convergence, we need to estimate how close K γ is to 1. By (3.3) and (3.6), (3.10) K = max 1 − γ L 2 , |γµ 2 − 1| = max |1 − γ L 2 |, 1 − γµ 2 . In Figure 1, we plot log 10 K for µ 2 = 1, L 2 = 20 for a range of γ's. This gives us the asymptotic rate of convergence in the uniform norm, since the error if we use the N -th partial sum is (I − K ) −1 K N , which log 10 is N log 10 K − log 10 (I − K ). In Figure 3, we present a numerical evidence of that behavior. Note that in the analysis below, with exact data, the factor (I − K ) −1 is actually removed, see (3.18), for example. We have (3.11) 1 − K γ = min γµ 2 , 2 − γ L 2 =: ν. Since the expressions in the parentheses are respectively an increasing and a decreasing function of γ, the maximum is achieved when they are equal, i.e., for γ equal to (3.12) γ * = 2 µ 2 + L 2 . With γ = γ * , (3.13) 1 − K γ * = 2µ 2 µ 2 + L 2 =: ν * , which is well known, see e.g., [9, pp.66-67]. Then the series is dominated by the geometric series C (1 − ν * ) j which converges exponentially, and therefore, (3.8) converges also uniformly (in the operator norm). When 0 < ν * 1 however, the speed of convergence decreases and the sensitivity to computational errors increases. For general γ ∈ (0, 2/ L 2 ), the Landweber iterations are dominated by C (1 − ν) j (and this is sharp), see (3.11). Note that ν as a function of γ is piecewise linear with a maximum at γ * , indeed but since the slope of the second factor in (3.11) is larger (and typically, much larger) by absolute value (which is L 2 ) than the slope of the first one, µ 2 , increasing γ form some very small value to γ * would improve the convergence gradually but after passing γ * , the convergence will deteriorate very fast. We observed this behavior in our tests. Note that the ratio of the two slopes is the condition number L 2 /µ 2 of L * L. If µ = 0, then K = 1. We can use the fact however that K j is applied to L * m only. Then f can be reconstructed by the following formula, see (3.8), (3.14) ∞ j=0 (I − γL * L) j γL * Lf (with Lf = m given). A zero eigenvalue of L * L would actually create a series of zeros, that still converges. Below, we actually allow L to have a non-trivial kernel Ker L = Ker L * L and study f 's modulo that kernel. The proof is a standard application of functional analysis, see [29], for example. Definition 3.1. We will call the problem stable, if either of the conditions above holds. We always assume that µ is the maximal constant (which exists) for which Proposition 3.1(b) holds, when it does. Such a condition is satisfied, if L * L is an elliptic ΨDO, see for example [29]. When the problem is stable, (3.15) L : (Ker L) ⊥ −→ Ran L is invertible with a norm of the inverse equal to 1/µ. A more detailed analysis is done by the spectral theorem below. Let S N f = N −1 j=0 (I − γL * L) j γL * Lf be the N -the partial sum in (3.14). Modifications. If f belongs to a subspace of H 1 , we can replace L by LΠ * , where Π is the orthogonal projection to that space (ΠΠ * = I) and then L * L is replaced by ΠL * LΠ * and still apply the scheme. Also, for every bounded non-negative operator P in H 2 , we can replace L * L by L * P L. If H 2 is an L 2 space with respect to some measure, a different choice of the measure would insert such a P there. In that case, P can be a multiplication by a non-negative function, which can be used to satisfy compatibility conditions in the case we study, for example. Exact data. Theorem 3.1 (Inversion with exact data). Assume (3.16) 0 < γ < 2/ L 2 . (a) Then for every f ∈ H 1 , the series (3.14) converges to the orthogonal projection f 1 of f to (Ker L) ⊥ . (b) The series (3.14) converges uniformly (i.e., in the operator norm) if and only if the problem is stable. In that case, it converges to f 1 exponentially fast in the operator norm; more precisely, with ν as in (3.11), (3.17) S N f − f 1 ≤ (1 − ν) N f 1 . Remark 3.1. As we showed above, ν is maximized for γ = γ * . Then ν = ν * , and S N f − f 1 ≤ L 2 − µ 2 L 2 + µ 2 N f 1 . Proof. Let |L| = spec(|L|) λ dP λ be the spectral decomposition of |L| := (L * L) 1/2 , where dP λ is the spectral measure supported on spec(|L|) = spec(L * L), which is contained in [0, L ]. In that representation, the series (3.14) consists of the terms (1 − γλ 2 ) j γλ 2 . Recall that if |L| is an N × N matrix, then P λ = λ j ≤λ P λ j , where λ j and P λ j are the eigenvalues of |L| and the projections to the corresponding eigenspaces, respectively. Then dP λ = λ j ≤λ P λ j δ(λ − λ j ) dλ, and for any f ∈ R N , the spectral measure dµ f of f we use below is given by dµ f = d(P λ f, f ) = λ j ≤λ P λ j f 2 δ(λ − λ j ) dλ. In particular, for every g ∈ L 2 , we have spec(|L|) g(λ) dµ f = λ j ≤λ |c j | 2 g(λ j ), where c j are the scalar projections (the Fourier coefficients) of f w.r.t. to an orthonormal basis of eigenfunctions corresponding to λ j . Using the identity (1 − q)(1 + q + · · · + q N −1 ) = 1 − q N , we get S N f = N −1 j=0 (I − γλ 2 ) j γλ 2 dP λ f = 1 − (1 − γλ 2 ) N dP λ f. Let f = f 0 + f 1 be the orthogonal decomposition of f with f 0 ∈ Ker L = Ker |L| and f 1 ⊥ Ker |L|. Clearly, S N f 0 = 0. For S N f 1 we have the same formula as above with dP λ replaced by dP 1 λ , where the latter is the restriction of dP λ to the orthogonal complement of Ker |L|. On that space, λ = 0 is not in the pure point spectrum spec pp (|L|) = spec pp (L * L) but may still be in the spectrum. We refer to [27] for the definition and the basic properties of the pure point and continuous spectra of bounded self-adjoint operators. Then (3.18) S N f − f 1 2 = spec(|L|) |1 − γλ 2 | 2N dµ f 1 , where dµ f 1 = d(f 1 , P λ f 1 ) is the spectral measure associated with f 1 . The integrand converges to g(x) = 0 for x = 0 and g(0) = 1, as N → ∞. It is bounded by the integrable function 1, see (3.6). By the Lebesgue dominated convergence theorem, the limit in (3.18), as N → ∞, is spec(|L|) g dµ f 1 and the latter is zero because µ f 1 ({0}) = 0. We will provide a more direct proof of the latter revealing better the nature of the problem. We claim that (3.19) lim δ→0+ spec(|L|)∩[0,δ) dµ f 1 = 0. Indeed, as δ 0, the integral above (which is just the measure of spec(|L|) ∩ [0, δ) with respect to µ f 1 ) is a monotonically decreasing non-negative function of δ and therefore, it has a limit which is its infimum. By the properties of Borel measures, if the limit is positive, that means that {0} has a positive measure. Therefore, 0 would be in the pure point spectrum which we eliminated by restricting L on the orthogonal complement of Ker(|L|). Therefore, the limit in (3.19) is zero, as claimed. Choose ε > 0 now. Let δ > 0 be such that the l.h.s. of (3.19) does not exceed ε. Then by (3.18), S N f − f 1 2 = spec(|L|)∩[0,δ) |1 − γλ 2 | 2N dµ f 1 + spec(|L|)∩[δ,∞) |1 − γλ 2 | 2N dµ f 1 ≤ ε + spec(|L|)∩[δ,∞) |1 − γλ 2 | 2N dµ f 1 . (3.20) For λ ≥ δ in spec(|L|), the term |1 − γλ 2 | is bounded away from 1, and therefore tends to 0 exponentially fast, as N → ∞. Choose N 1 so that the last integral in (3.20) does not exceed ε. Then S N f − f 1 2 ≤ 2ε for N 1, which completes the proof of (a). This argument shows that the rate of convergence depends heavily on the spectral density (measure) of f near λ = 0 for λ > 0 but not for λ = 0. To prove (b), notice first that the norm of S N − I on (Ker L) ⊥ is given by the maximum of (1 − γλ 2 ) N on spec(|L| (Ker L) ⊥ ), see [27, Theorem VII.1(g)]. For uniform convergence, we need that maximum to be strictly less than 1, which proves the first part of (b). When this happens, then (1 − γλ 2 ) N ≤ (1 − ν) N . 3.3. Noisy data. Let the measurement Lf be "noisy", i.e., we are given m ∈ H 2 not in the range of L and want to "solve" Lf = m. A practical solution of such problem is to find some kind of approximation of the actual f assuming that m is close in some sense to the range. We will study what happens if we solve (3.2) with a Neumann series without assuming (3.1). L * Lf = L * m (note that L * m ∈ (Ker L) ⊥ ) orthogonal to Ker L. The convergence is uniform with a rate (see (3.17) ) (1 − ν) N /µ. (b) If the problem is not stable, then there is no uniform convergence. Moreover, there is no even strong convergence. The set of m for which (3.8) is unbounded (and hence, diverges) is residual and in particular, dense, i.e., its complement is of first Baire category. Proof. Recall that every bounded operator L has a polar decomposition of the type L = U |L|, see [28,VI.4], true also for bounded operators between different Hilbert spaces, where U is a partial isometry. We have Ker U = Ker L, Ran U = Ran L, and U is an isometry on (Ker L) ⊥ . We can write (3.2) as (3.21) (I − (I − γL * L))f = γ|L|U * m. For U * , we have Ker U * = (Ran L) ⊥ and Ran U * = (Ker L) ⊥ . Set m * = U * m. We have m * ⊥ Ker L. Then (3.21) reduces to (3.22) (I − (I − γL * L))f = γ|L|m * . ForS N m := N −1 j=0 (I − γL * L) j γ|L|m * we now haveS N m = spec(|L|) N −1 j=0 (I − γλ 2 ) j γλ dP λ m * = spec(|L|) g N (λ) dP λ m * , g N (λ) := 1 − (1 − γλ 2 ) N λ . The function g N is smooth even near λ = 0. Therefore, in the spectral representation,S N = g N (λ). To prove (a), assume that the problem is stable. Then L * L is invertible on (Ker L) ⊥ by the spectral theorem. The function 1/λ is a bounded function on the spectrum, and it is the spectral representation of the inverse of |L| restricted to (Ker L) ⊥ that we will denote temporarily by |L 1 | −1 . Notice that |L 1 | −1 U * = L −1 1 , where L 1 : (Ker L) ⊥ → Ran L (the latter is closed) is the restriction of L as in (3.15). Therefore, denoting by m 1 the orthogonal projection of m to (Ker L) ⊥ , we have L −1 1 m 1 = |L 1 | −1 U * m = |L 1 | −1 m * which is the spectral representation is λ −1 m * . Therefore,S N m − L −1 1 m 1 = − spec(|L|) (1 − γλ 2 ) N λ dP λ m * . Note that f = L −1 1 m 1 is the unique solution of L * Lf = L * m orthogonal to Ker L. Then (3.23) S N m − L −1 1 m 1 2 = spec(|L|) (1 − γλ 2 ) 2N λ 2 dµ m * . The convergence then follows as in Theorem 3.1 because then the integrand is uniformly bounded on the spectrum. Assume now that the problem is unstable. Assume thatS N m is bounded (which would be true if it converges) for every m * ∈ (Ker L) ⊥ with a bound that may depend on m * . By the uniform boundedness theorem, the norms S N must be uniformly bounded, as well. Therefore, by [27, Theorem VII.1(g)] (3.24) S N = g N (λ) ≤ C, ∀N, ∀λ ∈ spec(|L|). On the other hand, (1)), as N → ∞. This contradicts (3.24) since in the non stable case, spec(|L|) contains a sequence of positive numbers converging to 0; and in particular proves lack of uniform convergence. We also proved that there cannot be strong convergence in this case. g N (N −1/2 ) = √ N (1 − e −γ + oλ N = C 2 / √ N . The proof is then completed by noticing that the set of m for which theS N m is unbounded is known to be residual since S N is unbounded. Residual sets are called sometimes generic. We give a more direct proof of the last statement, revealing the structure of those m for which the iterations diverge. We will show in particular, that the span of such m is infinitely dimensional. Notice first that spec(L * L) contains a sequence λ n 0. Let I k = (λ k , λ k+1 ]. By the standard proof that N × N is countable, where N = {1, 2, . . . } (or by its conclusion), we can label I k and λ k as I l,m and λ i,m so that the map k → (l, m) is a bijection. For each l, λ l,m → 0, as m → ∞ because the latter is a subsequence of λ k with rearranged terms. Then for every l, on the range H l of the spectral projection corresponding to ∪ k I l,k , the series (3.8) diverges for some m so that U * m ∈ H l . On the other hand, U is unitary from (Ker L) ⊥ to Ran L; therefore, U * : Ran L → (Ker L) ⊥ is unitary, as well. Since H l ⊂ (Ker L) ⊥ , U * m ∈ H l identifies m ∈ Ran L uniquely by applying the inverse of U * . This proves the existence of m = m l which makes (3.8) divergent, and those m l 's are mutually orthogonal. They are mapped to a linearly independent set under U * . Note that we actually proved something more: for every l, "almost all" elements m in H l force (3.8) to diverge: only those with finitely many non-zero projections do not. On the other hand, there is an infinitely dimensional space of m for which we have strong convergence, for example all m with a spectral measure dµ m * such that 1/λ is integrable. Remark 3.2. Since N −1/2 g(x/ √ N ) = 1 − (1 − γx 2 /N ) N x ∼ 1 − e −γx 2 x , as N → ∞, it follows that max 0≤λ≤ √ 2/γ |g N | ∼ C √ N attained at λ ∼ C 1 / √ N . Remark 3.3. It is worth noting that in the stable case (a), the Landweber solution f = L −1 1 m 1 can also be described in the following way. The noisy data is first projected to Ran L (which is closed). Denote that projection by m 1 . Then the exact solution is found as in the case of exact data: the solution of Lf = m 1 orthogonal to Ker L. This is called sometimes the Moore-Penrose inverse. If there is no stability, m is projected to the closure of Ran L, because in m * = U * m the result would be the same if we replace m by m 1 . Then we take m * = U * m 1 . Then (3.21) is equivalent to |L| 2 f = |L|m * . This is an equation in (Ker L) ⊥ , equivalent there to |L|f = m * . We have m * ∈ (Ker |L|) ⊥ and the later is the closure of Ran |L| but it is guaranteed to be in range if and only if that range is closed, i.e., if there is stability. Remark 3.4. If H 1 is finitely dimensional, then the problem is always stable. If we discretize a problem in an infinitely dimensional space, we get a finitely dimensional one. One may think that there is no convergence problem then. There are two potential problems with this argument however. First, the gap could be very small causing stability problems in numerical inversions. Second, the discretization could be a very poor approximation of the continuous problem near the bottom of the spectrum causing a very small gap (optimal µ) even though the continuous model may have a not such a small one. In our simulations, for example, we get eigenvalues of order 10 −15 while the "significant ones" are of order 1. Remark 3.5. The proof of Theorem 3.2 reveals that the rate of convergence or possible convergence with noisy data depends on the spectral measure of m * , which we defined as the orthogonal projection of U * m onto (Ker L) ⊥ . Choosing a different initial guess. We may think f 0 in (3.9) as an initial guess. We may also think of f 1 = γL * Lm as such. When K < 1, we have a sequence generated by a contraction map, and this sequence would converge regardless of the initial f 0 . We then choose f 0 not necessarily zero and set, as before, f k = f k−1 − γL * (Lf k−1 − m), k = 1, 2, . . . . Then the limit f , if exits, must satisfy f = f − γL * (Lf − m), which is equivalent to L * Lf = L * m, which is the starting point in Section 3.1, see (3.2). If f 0 is already close enough to f , we would expect the iterations to converge faster. In fact, we could gain speed if f − f 0 has a better power spectrum w.r.t. L * L, i.e., has a smaller spectral measure there. It is easy to show that f N = (1 + K + · · · + K N −1 )γL * m + K N f 0 . Therefore, f N − f 0 = (1 + K + · · · + K N −1 )γL * m + (K N − I)f 0 = (1 + K + · · · + K N −1 )(γL * m + (K − I)f 0 ) = (1 + K + · · · + K N −1 )γL * (m − Lf 0 ). Therefore, the Landweber iterations in this case correspond to the ones we studied so far for the equation L * L(f − f 0 ) = L * (m − Lf 0 ), i.e., f is replaced with f − f 0 and m is replaced by m − Lf 0 . Even though the new equation is equivalent to the old one, if f − f 0 is small in a certain sense or it has a lower spectral measure near the origin, the series may converge faster. Application to multiwave tomography with reflectors We apply the abstract theory in the previous section to multiwave tomography with reflectors. To have L as a bounded L 2 → L 2 map, we will restrict supp f to a compact subset of Ω, and assume convexity. The reason for that is to exclude possible singularities issued from supp f hitting ∂Ω tangentially. We also refer to [35] for a discussion of the mapping properties of a similar operator (the solution of the wave equation without reflections, i.e., propagating in R n ) to R × ∂Ω in case of tangential rays. First we derive the expression for (4.1) L * : L 2 ((0, T ) × ∂Ω) → L 2 (Ω 0 , c −2 dx). We know that L * must exist as a bounded operator in the spaces indicated above, and C ∞ 0 ((0, T ) × ∂Ω) is dense in L 2 ((0, T ) × ∂Ω); therefore L * restricted to the latter space defines L * uniquely. L : C ∞ 0 ((0, T ) × ∂Ω) → L 2 (Ω, c −2 dx), g → −∂ t v| t=0, x∈Ω 0 , where v is the solution of the following initial boundary value problem (4.3)        (∂ 2 t − c 2 (x)∆)v = 0 in (0, T ) × Ω, ∂ ν v| (0,T )×∂Ω = g, v| t=T = 0, ∂ t v| t=T = 0. Proof. First we assume f ∈ C ∞ 0 (Ω 0 ) in (2.1) and g ∈ C ∞ 0 ((0, T ) × ∂Ω) in (4.3) . Note that (4.3) is uniquely solvable, and one way to construct a solution is to take a smooth v 0 satisfying the boundary condition in (4.3), supported in (0, T ) ×Ω and set v = v 0 + v 1 , where v 1 = − v 0 , and v 1 has zero Cauchy data on t = T and satisfies the homogeneous Neumann boundary condition. Here, is the wave operator and the source problem for v 1 can be solved by Duhamel's principle using the well posedness of the boundary value problem for the Neumann problem with Cauchy data in H 1 (Ω) × L 2 (Ω), see e.g., [14] or [33] in this context. In [33], in the energy space, H 1 is replaced by the same space modulo constants (the kernel of the Neumann realization of −c 2 ∆) but for Cauchy data (C, 0) with C constant, the solution is trivially u = C. Let v be a solution of (4.3), and let u solve (2.1). Then (4.4) 0 = Ω T 0 1 c 2 (x) ∂ 2 t v − ∆v u dtdx = Ω 1 c 2 (x) T 0 ∂ 2 t vu dtdx − Ω T 0 ∆vu dtdx. For the first term in (4.4), integration by parts in t twice gives Ω 1 c 2 (x) T 0 ∂ 2 t vu dt dx = Ω 1 c 2 (x) ∂ t vu T 0 − T 0 ∂ t v∂ t u dt dx = − Ω 1 c 2 (x) (∂ t v| t=0 )f dx − Ω 1 c 2 (x) T 0 ∂ t v∂ t u dt dx = − Ω 1 c 2 (x) (∂ t v| t=0 )f dx − Ω 1 c 2 (x) v∂ t u T 0 − T 0 v∂ 2 t u dt dx = − Ω 1 c 2 (x) (∂ t v| t=0 )f dx + Ω 1 c 2 (x) T 0 v∂ 2 t u dt dx, where we have used that u| t=0 = f , ∂ t u| t=0 = 0 and v| t=T = ∂ t v| t=T = 0. For the second term in (4.4), applying Green's formula yields Ω T 0 (∆v)u dt dx = Ω T 0 v∆u dt dx + T 0 ∂Ω gu dS(x) dt − T 0 ∂Ω v∂ ν u dS(x) dt. Notice that the last term on the right-hand side actually vanishes as ∂ ν u = 0. Inserting these expressions into (4.4) and observing that ∂ 2 t u = c 2 (x)∆u we have 0 = − Ω 0 c −2 (x)(∂ t v| t=0 )f dx − T 0 ∂Ω gu dS(x)dt = Ω 0 (L g)f c −2 dx − T 0 ∂Ω g Lf dS(x) dt, which justifies (4.2). Since C ∞ 0 (Ω 0 ) and C ∞ 0 ((0, , T )×∂Ω) are dense in L 2 (Ω 0 , c −2 dx) and L 2 ((0, T )× ∂Ω), respectively, the closure of L is indeed the adjoint operator of L. Remark 4.1. The proof of the theorem reveals the following. Note first that if g is smooth but not C ∞ 0 (more specifically, not vanishing at t = T ), the compatibility condition for the first derivatives of v is violated at t = T and x ∈ ∂Ω since v| t=T = 0 implies ∂ ν v| t=T = 0 as well, which requires g| t=T = 0. In that case, there is no C 1 solution in the closed cylinder; and for a C 2 solution, we need to satisfy even one more compatibility condition: ∂ ν g| t=T = 0. We are interested in L * L, and we cannot hope to have g| t=T = g t | t=T = 0 for g = Lf . Therefore, when computing L * Lf , we (almost) always must compute a weak solution of (4.3) and therefore deal with g violating the compatibility conditions. The way to overcome that difficulty is to notice that we are constructing an L 2 → L 2 operator, see (4.1). Let χ ε be the characteristic function of [0, T − ε], with a fixed ε > 0. Then χ ε g → g in L 2 for any g ∈ L 2 but the rate of convergence is g-dependent (for the multiplication operator, we have χ ε → Id strongly only). Therefore, if we replace g by χ ε g and solve the resulting problem (the jump of χ ε g at t = T − ε is not a problem), we would get a good approximation for 0 < ε 1. On the other hand, in the iterations above, we apply L * to g = Lf for a sequence of g's. One could use the results in [6] to investigate whether χ ε g → g uniformly in g for g in the range of L for T > 0 satisfying the stability condition but we will not do this. Another way is just to replace L by χ ε L, and therefore, L * will also be replaced by L * χ ε . In fact, we could put any real valued L ∞ function there, as long as it has a positive lower bound in a set on (0, T ) × ∂Ω which set set satisfy the stability condition. We compare the Averaged Time Reversal method [33] with the Landweber method. We use a standard second order finite difference scheme on an N ×N grid, and we restrict the image to a smaller square by multiplying by zero in the frame staying at 3% from the border. This does not affect the SL phantom but it affects L * (and L) in the iterations; we have χL * Lχ instead of L * L in the iterations, where χ is the cutoff function. This does not eliminate completely geodesics through supp f tangent to the boundary if c is not constant but by [6], the stability is still preserved. The errors are presented in Figure 3. The thinner curves in the top part are the errors on a log 10 scale for different choices of the parameter γ. Since we use the step dt = dx/( √ 2 max(c)), dx = dy = 2/(N − 1) in the numerical implementation, this effectively rescales the t-axis by the coefficient 1.5 √ 2, and therefore, changes the L 2 norm on [0, T ] × ∂Ω. The choice of γ depends on that factor, as well. We see in Figure 3 that the optimal γ seems to be γ ≈ 0.055, based on 50 iterations. The convergence is highly sensitive to increasing γ beyond this level. On the other hand, decreasing γ reduces the accuracy but this happens much slower. Choosing the optimal γ is based on trials and errors in this case. In Figure 3, we show the errors vs. γ for 10, 30 and 50 iterations. The errors are consistent with the theoretical analysis in Section 3.2. The error curves for each fixed number of steps (10, 30 or 50) have shapes similar to that in Figure 1. Note that to compare Figure 3, and Figure 1, one has to divide the values in the former graphs by the number of the steps. The theoretical analysis of the uniform convergence rate illustrated in Figure 1, allows us to estimate µ 2 roughly. The slopes of the curves in Figure 3 up to the optimal γ, normalized to one step, give a slope in the range [1,2]. Note that it is closer to 2 for N = 10, and close to 1 for N = 50. Converting to natural log, we get a rough estimate of µ 2 in [2.3, 4.6]. Estimating L is not practical from that graph. We computed the matrix L of L in its discrete representation for N = 101, and Ω 0 being an interior 94 × 94 square. We halved N because computing L and its spectrum is very expensive for N = 201. The errors in that case are similar to what we presented for N = 101. We computed L on that grid. The eigenvalues of L * L are plotted in Figure 4, the smooth curve with the vertical axis on the left. We computed the numerical representation L * w of L * using the wave solver to solve (4.2) as well. The actual numerical inversion uses L * w L, not L * L. The spectral representation looks a bit different then but the locations of the lower part of the spectrum and the power spectrum of the SL phantom remain almost the same. Some of the eigenvalues have imaginary parts small relative to their real ones. The largest computed eigenvalue of L * L is λ 8836 ≈ 50.5, and the smallest 30 are of order 10 −15 , which can be thought of as 0 computed with a double numerical precision. We also get λ 1000 ≈ 0.036. So for all practical purposes, L * w L has a kernel of dimension at least 1, 000! This seems to contradict the fact that L is stable. The computed eigenfunctions of L * L (not L * w L) are shown in Figure 5. The eigenfunction problem is unstable but for our purposes, it is enough to know that there is a certain approximate orthonormal (enough) base so that on the first, say 1, 000 eigenfunctions, L has a norm of several orders of magnitude smaller than L(SL) / SL , where SL is the SL phantom. We verified numerically that L acting on those computed eigenfunctions have small norms, as the small eigenvalues suggest, and that the basis is orthonormal to a good precision. We also generated linear combinations of the first 1, 000 eigenfunctions with random coefficients and computed the norm of L on them. The relative norm of the SL phantom on that space is around 0.0016, which means that the total spectral measure there is the square of that, i.e., approximately 2.42 × 10 −6 . Also, using so generated phantoms, the iterations break. It turns out that the eigenfunctions corresponding to the extremely small eigenvalues have sharp jumps from minimal to maximal values from cell to cell like a chess board. The plotted 1, 000-th eigenfunction has mostly high frequency content as well, see Figure 5. The 7, 500-th one has a much lower high frequency content. This suggests that the extremely small eigenvalues are due to the poor accuracy of the wave equation solver for frequencies close to Nyquist, see also section 5.4. This can also be confirmed by using semiclassical methods. The operator L * L − z for 0 < z 1 is elliptic because z is separated from the range of the (elliptic) principal symbol. Therefore, possible eigenfunctions with eigenvalues z should be smooth and should also have low oscillations. Therefore, eigenfunction #1 eigenfunction #1,000 eigenfunction #7,500 Figure 5. The eigenfunctions of L * L corresponding to λ 1 , λ 1,000 and λ 7,500 the spectrum of the discrete L * w L contains small eigenvalues not approximating the spectrum of L * L. Why do the iterations still work, and we are getting a lower bound of the spectrum µ 2 in [2.3, 4.6] (based on one f , indeed but we tried a few other ones)? It turns out that our f (the SL phantom) does not have much spectral content in the bottom of the spectrum. In Figure 4, we plot the computed squares of the Fourier coefficients (the power spectrum) of the SL phantom with respect to the eigenfunction basis of L * w L, see the rough curve there and the vertical axis on the right. It turns out that the SL phantom has very small Fourier coefficients until, say, λ 3,000 ≈ 4.26. We can take this as an effective value for µ 2 since the analysis above says that the bottom on the spectrum should be taken on the support of the spectral measure of f , see (3.18). This correlates well with our estimate µ 2 ∈ [2.3, 4.6] based on Figure 3. In the top of the spectrum, there is a similar but a smaller gap with the highest significant eigenvalue λ 8,718 ≈ 31.84. Therefore, L 2 ≈ 31.84 on the support of the spectral measure of the SL phantom is a good estimate. That gives us, roughly speaking, γ * = 0.055, which is close to our numerical results, see Figure 3. We recall that the computed errors are based on inversion with L * w L (even though we do not compute the matrices L * w and L in the inversion) while the spectral representation is done based on L * L. It is well known that finite difference schemes for differential operators require a priori boundedness of the higher order derivatives, therefore the conclusions above should not be surprising. High discrete frequencies propagate with slower speeds and the numerical solution is a good approximation only at low frequencies, see e.g., [41,3]. Also, the discretization of the phantom matters. We render the SL phantom on a higher dimensional grid first, then resample it to the grid we work with. This way, we have a better sampling of the continuous function the SL phantom represents. This does not affect the rates of convergence much, and without that, we still have a very low spectral content for very small eigenvalues, even though it is not that low. The original and the reconstructions however do not show the typical aliasing artifacts like staircase edges, etc., and a still visibly "sharp" enough. 5.1.2. Data not in the range. In the example above, we add the data on one of the sides to another one. The difference, and hence the new data is not in the range by unique continuation. The reconstruction shows obvious artifacts as one would expect. The convergence behavior as a function of γ however, does not change in a radical way, see Figure 6. In particular, the iterations start Figure 7. In the continuous model, see Theorem 3.2, the iterations would converge at a rate depending on the spectral measure of the noise; which could be very slow. They converge to f +f as above. In the numerical case however, the problem is actually unstable, as explained above. The data now has a small but not negligible spectral part in the lower part of the spectrum due to the nature of the noise. This is responsible for a very slow divergence. Note that when the "noise" has a different spectral character, as in Figure 6, we are closer to convergence. The ATR method with this example is more sensitive to noise and gives significantly worse errors and images. 5.1.4. Conclusions in the stable case. We summarize the conclusions in this section in the following. We want to emphasize again that some of them, like the first one are well known. • The finite difference solver for the wave equation is a poor approximation of the continuous model for (discrete) frequencies close to Nyquist. We refer to section 5.4 for a further discussion. Even the sampling of the phantom could be a poor approximation (aliased). • The discrete realization L of L is unstable even if L is stable. This is not a problem with "generic" phantoms because they have a very small spectral content at the bottom of the spectrum of L * L which corresponds to asmall hight frequency content (w.r.t. the discrete Fourier transform). • The discrete realization L * w of L * used in the inversion is based on back-projection at each step and it is not the same as L * . The differences do not affect the inversions visibly but create small but measurable imaginary parts of the eigenvalues of L * w L. Most of the differences are related to modes with frequencies close to Nyquist. • Despite this, the inversion with "generic" phantoms works as predicted by the theory of the continuous model because they have small projections to the eigenfunctions of L * L with small eigenvalues (the unstable subspace). If we base the theory on the discrete model, we would have to conclude that the problem is very unstable. • In case of noise, the iterations converge (to something different than f ) if the noise does not have low spectral content (corresponding to higher frequencies in the discrete Fourier transform). When the noise has a significant low spectral content (high frequencies), the inherent instability of the discretization causes a slow divergence. 5.2. A unstable example; partial boundary data. We take a constant speed c = 1 so that we can compute easily the invisible singularities. We use data on a part of the boundary as indicated in Figure 8: the bottom and the left-hand side plus 20% of the adjacent sides. We take T = 1.8 which is smaller than the side of the box Ω, equal to 2. We have uniqueness since T 0 = 1.8 in this case. For stability however, we need T > 2. Singularities from some neighborhood of the top of the SL phantom traveling vertically and close to it, do not reach Γ for time T = 1.8. Therefore, such edges cannot be reconstructed stably. Our goal is to test the convergence of both methods and the dependence on γ in the Landweber one. The reconstructions are shown in Figure 8. The second and the third one are done with the Landweber method but in the second one, we use the fact that we know that the SL phantom is supported in Ω 0 , being a slightly smaller square. Numerically, this means that we work with χL * Lχ instead of L * L, (i.e., L is replaced by Lχ, where χ is the characteristic function of Ω 0 . In the third reconstruction, there is no such restriction. The cutoff improves the error a bit. The "ripples" artifacts are due to the sharp cutoff of the data at T = 1.8. If we introduce a gradual cutoff χ 1 (t) with χ 1 (T ) = 0, i.e., if we use χL * χ 1 Lχ instead of L * L, this removes the visible "ripples" but blurs the edge of the SL phantom in a larger neighborhood of the invisible singularities on the top. Despite the slightly larger error, the ATR reconstruction has less artifacts. The errors for various values of γ are shown in Figure 9. The convergence is slower than in the stable case after 10 or so steps, and the improvement with γ increasing below reaching its optimal value is much slower. The errors are based on a cutoff to Ω 0 , which is the better case. With the cutoff, the Landweber and the ATR errors are closer. On the other hand, the Landweber method is more flexible w.r.t. introducing such weights. The Landweber method performs better in this case in terms of the L 2 error. We analyzed the eigenvalues of L * L next, and the Fourier coefficients of the SL phantom w.r.t. its eigenvalues. The results are shown in Figure 10. This situation is reversed now, compared to the stable case. There are still a lot of "zero" (extremely small) eigenvalues with highly oscillatory eigenfunctions. The non-"zero" Fourier coefficients however start much earlier than the non-"zero" eigenvalues. Since our spectral analysis shows that we need to restrict our considerations on the support of the spectral measure only, we can take λ ≈ 800 as a rough lower bound for this support. Then we have practically zero eigenvalues up to, roughly speaking, λ 1,800 , where the SL phantom has a non-negligible spectral measure. The norm of the projection of the SL phantom to the space spanned by the first 1, 000 eigenvalues is around 40% of the total one. This means instability and it is in contrast to Figure 4. 5.2.1. Unstable example with data not in the range and with noise. We add data on one side to another one, as in the stable case. The errors did not look much different than the noise case below. Next, we add Gaussian noise with 0.1 standard deviation. The data Lf ranges in the interval [−1, 1.5]. The errors reach a minimum and start diverging slowly for γ ≤ 0.16, and diverge fast for γ larger than that, see Figure 11 on the left. This suggests that γ * ≈ 0.16, for that particular image, at least, and correlates well with Theorem 3.2 and its proof which proves divergence with generic perturbations of the data not in the range in the unstable case but indicates a slow divergence. Note that the optimal γ looks similar to that in the noise free case, see Figure 9. On the other hand, instead of a convergent series, we get a slowly divergent one. Finally, we add a filter χ between L and L * w which cuts high frequencies and also frequencies outside the characteristic cone on each side; in other words, we replace L * w by L * w χ. Since χ is a Fourier multiplier by a non-negative function, χ is a non-negative operator. The errors get smaller and the iterations do not start to diverge even after 200 iterations, see Figure 11. This is in line Figure 11. A unstable case with Gaussian noise. Left: error curves with γ ranging from 0.03 to 0.17 (the diverging curve). The critical value of γ looks close to that in the zero noise case in Figure 9. The iterations for γ ≤ 0.16 diverge slowly in contrast with the noise free case in Figure 9. Right: Filtered data, γ ≤ 0.15. The errors look like in Figure 9. with Theorem 3.2 and its proof since the filtering of the noise in the data reduced significantly the spectral content of the noisy data (w.r.t. L * L) in the low part of the spectrum, where the instability is manifested. After that, the problem behaves as that with data not in the range but with small high frequency content (w.r.t. the Fourier transform), see Figure 9, where the number of iterations is 50 vs. 200 in Figure 11 on the right. The reconstructions are shown in Figure 12. Even though the filtered reconstruction has a smaller error, it is only marginally better in recovering detail. Left: unfiltered data. Right: filtered data. 5.3. Discontinuous sound speed. We choose a sound speed with a jump across a smaller square. Such speeds model thermoacoustic tomography in brain imaging, see, e.g., [40,31,39]. Singularities in this case reflect from the internal boundary and may refract, as well. Since the speed outside that boundary is faster, this creates rays that do not refract. This makes the problem inside that rectangle potentially unstable. The reconstructions are shown in Figure 13. The ATR method works a bit better not only by providing a slightly better error but there are less visible artifacts. The Landweber reconstruction handles noise better however. 5.4. Forward data generated on a different grid. We present now examples where the data was created on a finer grid. As we mentioned above, the finite difference wave solver is inaccurate for large frequencies. This is known as numerical (grid) dispersion, see, e.g., [4,11,19,12,3]. High frequency waves propagate slower, which explains the terms dispersion. This effect gets worse with time. This creates "ripples" in the wave fronts of high frequency waves, with oscillations in the back of the front. It was experimentally found in [4,11,19] that in order to get a satisfactory performance for a second order finite difference scheme, say at distance 30-40 wavelengths ( [11]), one needs to sample in the spatial variables each wavelength at a rate 4-5 higher than the Nyquist one, which is two point per wavelength. We refer to [12,3] for further discussion and for ways to improve the performance with other solvers. Such errors would be canceled in the backprojection, which is a well known phenomenon in numerical inverse problems. For this reason, it is often suggested that the forward data should be generated by a different solver. This does not solve the dispersion problem however. The backward solver (if based on finite differences) would still be inaccurate for high frequencies, if they are present in the data. The different forwards solver may have the same problem. A natural attempt seems to be to generate data by a known analytic solution or using a much finer grid. This would guarantee the good accuracy of the data but the problem with the backward solver at high frequencies remains. The practical solution (again, if we backproject by finite differences) is to make sure that the data has mostly low frequency content relative to the grid used in the back-projection or to use a higher order scheme which still suffers form dispersion but allows higher frequencies [11]. Another solution would be to use other solvers not so much limited at higher frequencies but this is behind the scope of this work. A Landweber reconstruction of the SL phantom with data computed on a finer grid with c = 1 is shown in Figure 14. The ripple artifacts are due to high frequency waves in the backprojection propagating slower than the wave speed 1. Since they become worse when T increases, it is important to keep T low. The critical stability time here is √ 2 and we chose T = 2. In an idealized real life situation, where we have properly sampled boundary data, textitif we still want to use a finite difference scheme, we should do the numerical inversion on a grid much finer than the one determined by Nyquist rate of the signal. That would increase the cost, of course. To simulate such a situation, we should take a continuous phantom with a finite frequency band, oversample it by a factor of five, for example, keep T low, solve the forward problem and sample it on a coarser grid depending on the frequency content of the data. To do the backprojection, we should increase the size of the grid and do finite differences. But making the grid coarser after we have computed Λf and finer right after it basically means that we could stay either with the original finer one in our simulations or change it but keep the data severely oversampled. As mentioned above, a more efficient solution would be to use different solvers for back propagation. Based on the discussion above, we chose a low frequency phantom in Figure 15 below with data generated by the finer grid used in Figure 14. The reconstruction is much more accurate. A reconstruction with the variable speed used above, see section 5.1.1 yields visibly similar results with a relative L 2 error 4%. Figure 15. Same situation as in Figure 14 but the phantom contains Gaussians with low frequency content only. Left: original. Right: reconstruction with 50 iterations. The relative L 2 error is 1.8%, the L ∞ error is 3.5%. Review of the Averaged Time Reversal Method In this section we review briefly the averaged time reversal method proposed in [33]. The treatment there works for any fixed Riemannian metric g on Ω and the wave equation (∂ 2 t − c 2 (x)∆ g )u = 0 in (0, T ) × Ω where ∆ g is the Laplace-Beltrami operator in the metric g and c(x) > 0 is a smooth function. In applications g is the Euclidean metric and ∆ g is the Euclidean Laplacian. Here we sketch the method as well as the resulting algorithm only in the Euclidean case for simplicity of notions. More general and detailed exposition can be found in [33]. 6.1. Complete data. We start with the complete data case, i.e. when Γ = ∂Ω. Here and below L 2 (Ω) = L 2 (Ω, c −2 dx). We sometimes adopt the notation u(t) = u(t, ·). Define the Dirichlet space H D (Ω) as the completion of C ∞ 0 (Ω) under the Dirichlet norm The function φ is often referred to as the harmonic extension of h(T ). When h = Lf we have v(0) = ALf , which can be viewed as an approximation of the initial data f in the multiwave tomography model in R n [30]. Indeed, introduce the error operator (6.2) Kf := f − ALf. In the multiwave tomography model in R n , it is shown in [30] that if T is such that there is stability, K is a contraction on H D (Ω), that is, K H D (Ω)→H D (Ω) < 1. Moreover, K is compact. This makes the operator I − K invertible on H D (Ω) and one thus deduces from (6. 2) that f = (I − K) −1 ALf . Inserting the expansion (I − K) −1 = I + K + K 2 + . . . gives the Neumann series reconstruction algorithm in [30]. In the multiwave tomography model in Ω, the error operator K : H D (Ω) → H N (Ω) is no longer a contraction regardless of how large T is. In fact one could construct high frequency solutions propagating along a single broken geodesic to show that K H D (Ω)→H N (Ω) = 1. This breaks down the inversion of I − K. The idea suggested in [33] is to average the time reversal operator A with respect to T . The averaging process causes some partial cancellation of the microlocal singularities with opposite signs at t = 0 when T > T 1 , see Definition 2.1, making the averaged error operator a microlocal contraction (but not compact anymore). We rewrite the time reversal operator A for the purpose of averaging later. Setṽ = v − Ph(T ) with v the solution of (6.1), thenṽ solves and Ah = v(0) =ṽ(0) + Ph(T ). We consider the terminal time as a parameter now, and call it τ with τ ∈ [0, T ]. We replace T by τ in (6.3). Denote the corresponding solution byṽ τ and the corresponding time reversal operator by A(τ ). Note that (6.4) A(τ )h =ṽ τ (0) + Ph(τ ) and note thatṽ τ solves a similar problem as (6.3) but the boundary data is replaced by H(τ − t)(h(t) − h(τ )) with H the Heaviside function. Choose a weight function χ ∈ C ∞ 0 (R) which is positive and has integral equal to one over [0, T ]. We define the averaged time reversal operator A by multiplying (6.4) by χ(τ ) and integrating it over [0, T ]: Ah := Finally, define A 0 := Π 0 A where Π 0 projects the result onto H D (Ω 0 ). The projection of the second term actually vanishes since it is harmonic. The following theorem gives an explicit reconstruction of f when h = Lf . Theorem 6.1. Let (Ω, c −2 dx 2 ) be non-trapping with a strictly convex boundary as a Riemannian manifold and let Ω 0 Ω. Suppose T > T 1 (∂Ω, Ω) and denote K 0 := Id − A 0 L on H D (Ω 0 ). Then K 0 H D (Ω 0 )→H D (Ω 0 ) < 1. In particular, Id − K 0 is invertible on H D (Ω 0 ), and the inverse problem has an explicit solution of the form In other words, the modification to the time reversal (6.1) is that we impose the homogeneous Neumann boundary condition (which is satisfied by the forward solution u) on ∂Ω\Γ where the measurement is unavailable. The averaged time reversal operator is then constructed in a similar manner. We were only able to prove however that the averaging method with partial data however provides a parametrix recovering the singularities of f which do not hit the edge of [0, T ] × Γ [33]. This does not cause a significant loss of singularities as those that hit the edge of Γ form a set of measure zero in the cotangent bundle. Then we have K ≤ 1 but we do not know if K < 1. This does not guarantee convergence of the Neumann series but we use such series numerically nevertheless. Date: April 21, 2016. First author partly supported by NSF Grant DMS-1301646. Definition 2 . 1 . 21Let ∂Ω be strictly convex with respect to c −2 dx 2 . Fix Ω 0 Ω, an open Γ ⊂ Ω and T > 0. Figure 1 . 1log 10 error at each step vs. γ with exact data. This is the graph of log 10 K as a function of γ for µ 2 = 1, L 2 = 20. The optimal γ * is γ * = 2/21 ≈ 0.0952. The error increases fast on the right of γ * . with µ > 0 for f ⊥ Ker L, (b) spec(L * L) has a gap (0, µ) with some µ > 0, (c) Ran L is closed. Theorem 3 . 2 ( 32Noisy data). Assume(3.16). Then the Neumann series (3.8) converges for every m ∈ H 2 if and only if the problem is stable. Moreover, (a) If the problem is stable, L * L is invertible on (Ker L) ⊥ with a norm of the inverse 1/µ 2 . Then the limit is the unique solution of Figure 2 . 2The functions g N (λ) with γ = 1 and N = 5, 20, 40, 80. As N increases, the maximum increases as C 1 √ N and its location shifts to the left to Theorem 4. 1 . 1The adjoint operator of L defined in (2.2) is the closure of Corollary 4 . 1 . 41The adjoint L * Γ to L Γ defined in(2.3) is EL * , where E is the operator of extension as zero of functions defined on (0, T ) × Γ to (0, T ) × ∂Ω.Proof. We have L Γ = RL, where R is the restriction to (0, T ) × Γ. Since R * = E, this completes the proof.5. Numerical examples 5.1. A stable example: variable speed, whole boundary. 5.1.1. Exact data. We start with the Shepp-Logan (SL) phantom with a variable speed c = 1 + 0.3 sin(πx) + 0.2 cos(πy) in the box [−1, 1] 2 . The speed takes values between 0.5 and 1.5. We take T = 4 and observations on the whole boundary. That time is large enough to guarantee stability. Figure 3 .Figure 4 . 34Error vs. γ for the Landweber iterations in the stable example. Plotted are errors after 10 iterations (boxes), after 30 iterations (diamonds) and after 50 ones (dots). Right: Error vs. the number of the iterations. The bottom curve with the circles are the ATR errors. The other curves correspond to γ ranging from 0.1 to 0.06, as on the left, starting form the top to the bottom in the top left corner. A stable case. The smooth increasing curve represents the eigenvalues of L * Lin the discrete realization on an 101×101 grid restricted to a 94×94 grid. There are small imaginary parts along the horizontal line in the middle. L * is computed as the adjoint to the matrix representation of L. The rough curve represents the squares of the Fourier coefficients of the SL phantom. Figure 6 . 6Error plots in the stable case with data not in the range. Approximate convergence.to diverge for γ between 0.04 and 0.05. As we saw in Theorem 3.2, the iterations would converge to f +f , where f is the original f , andf is the unique solution of Lf 1 = δm, where δm is the "noise", and δm is its orthogonal projection to Ran L. The error curves are consistent with the distance from f to the iterations f n converging exponentially fast to f +f . They have minima corresponding to the closest element of f n to f .5.1.3. The stable case with noise. We add Gaussian noise with standard deviation 0.1. The data Lf ranges in the interval [−1.2, 1.7]. The noise curves are shown in Figure 7 . 7Reconstruction and error plots in the stable case with noise with 200 iterations; γ = 0.01, 0.02, 0.03, 0.04. A slow divergence. For γ = 0.05 we get a fast divergence. Figure 8 . 8Speed 1 with T = 1.8 with invisible singularities on the top. From left to right: (a) ATR with 50 steps, a cut near the border, 34% error. (b) Landweber with a cut near the borders, error 31%; (c) Landweber without a cut near the borders, error 36%. Figure 9 . 9Error vs. γ in the unstable case. Left: errors after 10 iterations (boxes), after 30 iterations (diamonds) and after 50 ones (dots). The vertical axis is on a log 10 scale and the horizontal axis represents γ. The lower curve, corresponding to 50 iterations, is flatter than in the stable case. On the right, the curve with the diamond marks is the error with the ATR method. Figure 10 . 10The unstable case. The smooth increasing curve represents the eigenvalues of L * L in the discrete realization on an 101 × 101 grid restricted to a 94 × 94 grid. L * is computed as the adjoint matrix rather than with the wave equation solver. The rough curve represents the squares of the Fourier coefficients of the SL phantom. The arrows indicate the effective lower bound of the power spectrum of the SL phantom, and the first positive eigenvalue modulo small errors, respectively. Figure 12 . 12The unstable case, reconstructions with the Landweber method with noise. Figure 13 . 13Reconstructions with a discontinuous speed, plotted on the left. The original is black and white disks on a gray background. Data on the marked part of the boundary, T = 4. Center: ATR reconstruction, error 1%. Right: Landweber reconstruction, error 1.5%. Figure 14 . 14Left: the original SL phantom. Right: The reconstructed SL phantom with c = 1 and T = 2. The phantom was originally on a 201 × 201 grid with ∆t = ∆x/ √ 2. The data was computed on a grid rescaled by a factor of 5.7 in the spatial variables and 7.41 times in the time variable; and then rescaled to the the original one on the boundary before inversion. The ripple artifacts are due to high frequency waves in the time reversal moving slower than the speed c = 1. Neumann space H N (Ω) to be the quotient space of H 1 (Ω) modulo constant functions. For a subdomain Ω 0 ⊂ Ω with smooth boundary, we identify H D (Ω 0 ) with the subspace of H D (Ω) consisting of functions supported in Ω 0 . Introduce the operator Π 0 : H D (Ω) → H D (Ω 0 ) by Π 0 f := h where h solves ∆h = ∆f in Ω 0 , h| ∂Ω 0 = 0. Π 0 is in fact the orthogonal projection of H D (Ω) onto H D (Ω 0 ), see [33, Lemma 2]. Let u be the solution of (2.1) and L the measurement operator. We construct a (non-averaged) time reversal operator A as follows. Given h ∈ H 1 ([0, T ] × ∂Ω), define Ah := v(0) where v is the T ) × Ω, v| (0,T )×∂Ω = h, v| t=T = Ph(T ), ∂ t v| t=T = 0.Here P is the Poisson operator such that φ := Ph(T ) is the solution of ∆φ = 0 in Ω, φ| ∂Ω = h(T ). T ) × Ω, v| (0,T )×∂Ω = h(t) − h(T ), v(T ) = ∂ tṽ (T ) = 0. 0 h, h := Lf.This theorem leads to the following iterative reconstruction algorithm:f 1 = A 0 h, h := Lf, f n = (Id − A 0 L)f n−1 + A 0 h, n = 2, 3, . . . . 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[ "LAGRANGIAN DESCRIPTION OF N=2 MINIMAL MODELS AS CRITICAL POINTS OF LANDAU-GINZBURG THEORIES", "LAGRANGIAN DESCRIPTION OF N=2 MINIMAL MODELS AS CRITICAL POINTS OF LANDAU-GINZBURG THEORIES" ]
[ "M T Grisaru \nPhysics Department\nDipartimento di Fisica dell'Università di Milano and INFN\nBrandeis University Waltham\n02254MAUSA\n", "D Zanon \nSezione di Milano\nVia Celoria 16I-20133MilanoItaly\n" ]
[ "Physics Department\nDipartimento di Fisica dell'Università di Milano and INFN\nBrandeis University Waltham\n02254MAUSA", "Sezione di Milano\nVia Celoria 16I-20133MilanoItaly" ]
[]
We discuss a two-dimensional lagrangian model with N = 2 supersymmetry described by a Kähler potential K(X,X) and superpotential gX k which explicitly exhibits renormalization group flows to infrared fixed points where the central charge has a value equal that of the N = 2, A k−1 minimal model. We consider the dressing of such models by N=2 supergravity: in contradistinction to bosonic or N = 1 models, no modification of the β-function takes place.
null
[ "https://arxiv.org/pdf/hep-th/9501126v1.pdf" ]
118,695,728
hep-th/9501126
35f9801626e92f4efc5281641796f7c932a37f3f
LAGRANGIAN DESCRIPTION OF N=2 MINIMAL MODELS AS CRITICAL POINTS OF LANDAU-GINZBURG THEORIES 27 Jan 1995 M T Grisaru Physics Department Dipartimento di Fisica dell'Università di Milano and INFN Brandeis University Waltham 02254MAUSA D Zanon Sezione di Milano Via Celoria 16I-20133MilanoItaly LAGRANGIAN DESCRIPTION OF N=2 MINIMAL MODELS AS CRITICAL POINTS OF LANDAU-GINZBURG THEORIES 27 Jan 1995 We discuss a two-dimensional lagrangian model with N = 2 supersymmetry described by a Kähler potential K(X,X) and superpotential gX k which explicitly exhibits renormalization group flows to infrared fixed points where the central charge has a value equal that of the N = 2, A k−1 minimal model. We consider the dressing of such models by N=2 supergravity: in contradistinction to bosonic or N = 1 models, no modification of the β-function takes place. It has been recognized for some time that N = 2 minimal models can be viewed as critical points of Landau-Ginzburg theories, and a considerable body of literature has developed around this idea 1,2,3,4,5,6,7 . It is generally believed that in a field-theoretic language such models are described, at and away from the fixed points, by N = 2 superspace actions of the form S = d 2 xd 4 θ K(X,X) + d 2 xd 2 θ W (X) + d 2 xd 2θW (X) (1) where K(X,X) is the Kähler potential, function of chiral and antichiral superfields X,X, while the superpotential W (X) is a quasi-homogeneous polynomial in the chiral superfields. These ideas have been tested in numerous ways, but no complete lagrangian models have been constructed which exhibit explicitly this behavior. Away from the fixed points, along the renormalization group trajectories, the N = 2 nonrenormalization theorem ensures that the form of the superpotential remains unchanged while the Kähler potential flows according to quantum corrections in such a way that at the fixed points the resulting action describes superconformally invariant systems. In a complete lagrangian description one would like to exhibit a suitable Kähler potential such that, for example, for a simple Landau-Ginzburg superpotential gX k the system flows to an infrared fixed point where the central charge is the one of the A k−1 N=2 minimal model. We present here such a Kähler potential. Generalizations to other minimal models are straightforward. 8 We first examine the situation at the fixed point. In the absence of the superpotential W , a generic N = 2 σ-model is classically (super)conformally invariant. In the presence of the superpotential the stress-energy tensor (the supercurrent actually) acquires a classical trace (a supertrace) and for a general K(X,X) no improvement term can be found to make the theory superconformally invariant. In fact, for a given superpotential, the condition of scale invariance fixes the Kähler potential and the improvement term up to a normalization factor. As we shall see, it is this normalization factor that determines the central charge, and it is a specific normalization factor that gets selected, when we start away from the critical point, by the renormalization group flow. (The critical point form of our lagrangian has been also described by Marshakov 5 and, in its Liouville version, by Liao and Mansfield 6 , but in these references the normalization factor could not be determined.) We work in Minkowski space with light-cone variables x | = = 1 √ 2 (x 0 + x 1 ) , ∂ | = = 1 √ 2 (∂ 0 + ∂ 1 ) x = = 1 √ 2 (x 0 − x 1 ) , ∂ = = 1 √ 2 (∂ 0 − ∂ 1 )(2) and ✷ ≡ ∂ µ ∂ µ = 2∂ | = ∂ = , ∂ = 1 x | = = 2πiδ (2) (x)(3) The superspace spinorial coordinates are θ + , θ − ,θ + ,θ − , and the corresponding covariant derivatives satisfy {D + ,D + } = i∂ | = , {D − ,D − } = i∂ =(4) with all other anticommutators vanishing. For a kinetic term d 2 xd 4 θ XX the chiral field propagator is < X(x, θ)X(0) >= − 1 2πD 2 D 2 δ (4) (θ) ln[m 2 (2x | = x = + ℓ 2 )](5) where m and ℓ are infrared and ultraviolet cutoffs respectively. We have defined D 2 ≡ D + D − andD 2 ≡D +D− . We couple the system described by Eq. (1) to N = 2 supergravity and in order to conveniently describe the above-mentioned improvement of the supercurrent we include a chiral "dilaton" term, so that the action takes the form S = d 2 xd 4 θE −1 K(e iH.∂ X, e −iH.∂X ) + d 2 xd 2 θe −2σ W (X)(6)+ d 2 xd 2θ e −2σW (X) + d 2 xd 2 θ e −2σ R Ψ(X) + d 2 xd 2θ e −2σRΨ (X) Here the vector superfield H and the chiral compensator σ are the supergravity prepotentials. At the linearized level we have the explicit expressions 8 E −1 = 1 − [D + , D + ]H = − [D − , D − ]H | = R = 4D +D− [σ + D +D+ H = + D −D− H | = ] R = 4D + D − [σ −D + D + H = −D − D − H | = ](7) The general solution of the constraints of N = 2 supergravity is given in Ref. 9. The invariance of the supergravity-coupled system under local supersymmetry transformations is expressed by the conservation law D − J | = = D + J , D − J | = =D +J(8) where the supercurrent is given by J | = ≡ δS δH = | H,σ=0 = 2[D + XD +X K XX − 2D + D + Ψ + 2D +D+Ψ ](9) and the supertrace by J ≡ δS δσ | H,σ=0 = −2[W − 2D +D−Ψ ](10) We have introduced the Kähler metric K XX = ∂ 2 K ∂X∂X(11) Superconformal invariance requires the supertrace J to vanish. For the superpotential W = gX k the equations of motion (with the notation K X ≡ ∂ X K, etc.) D +D− K X + W X = 0(12) give W = − 1 kD +D− (XK X )(13) Using this expression in Eq. (10), the condition for superconformal invariance, J = 0, J = 0, requires XK X = −2kΨ(X) ,XKX = −2kΨ(X)(14) modulo a linear superfield which gives no contributions to the action. We have assumed that Ψ,Ψ are local, and (anti)chirality and dimensionality require them to be functions of X,X respectively. The equations above can be immediately integrated and give K = α ln X lnX Ψ = − α 2k ln X ,Ψ = − α 2k lnX(15) with arbitrary constant α. Using the field redefinition X ≡ e Φ the corresponding lagrangian can be recast in Liouville form. We compute now the conformal anomaly of our model, and show that the central charge of the fixed point theory equals the central charge of N = 2, A k−1 minimal models when α is suitably chosen. It is given by the coefficient in front of the induced supergravity effective action R✷ −1R obtained by integrating out the fields X,X, and can be determined by contributions to the H = self-energy, from which the covariant expression can be reconstructed. We compute away from the fixed point, using an effective configuration-space propagator < X(x, θ)X(x ′ , θ ′ ) >= − K XX 2πD 2 D 2 δ (4) (θ −θ ′ ) ln{m 2 [2(x−x ′ ) | = (x−x ′ ) = + ℓ 2 ]} (16) where K XX is the inverse of the Kähler metric (cf. Ref. 12, Eq. (3.13); additional terms, involving derivatives of the Kähler metric in the propagator do not give relevant contributions). The couplings to H = can be read from the supercurrent in Eq. (9). The relevant vertex from the Kähler potential is 2i d 4 θ H = D + XD +X K XX(17) This leads to the one-loop contribution − 1 π 2 H =D − D −D+ D + (x − x ′ ) 2 | = H =(18) From the dilaton term coupling 4 d 4 θ(Ψ − Ψ)∂ | = H =(19) we have the tree level contribution − 16 π Ψ X K XXΨX H =D − D −D+ D + (x − x ′ ) 2 | = H =(20) Using the relation between the Kähler potential and the improvement term at the fixed point given in Eq. (15), we obtain the final result − 1 π 2 H =D − D −D+ D + (x − x ′ ) 2 | = H = 1 + 4πα k 2 ⇒ 1 4π R 1 ✷R 1 + 4πα k 2(21) From this expression we obtain the central charge of the system c = 1 + 4πα k 2(22) For α = − k 2π it equals the correct value for the N = 2, A k−1 minimal model, c = 1 − 2 k(23) We exhibit now a system which flows in the IR region to the superconformal theory defined above. It has two properties: its β-function is one-loop exact, so that we can make all-orders statements about the flows; and, although it contains arbitrary parameters, the flow to the fixed point uniquely picks out values which give the correct central charge for identification with the A k−1 minimal models. The theory is described by the superpotential gX k and the Kähler metric K XX = 1 1 + bXX + c(XX) 2(24) corresponding to the Kähler potential K = dXdXK XX = XX − b 4 (XX) 2 + b 2 − c 9 (XX) 3 + · · ·(25) Quantum corrections give rise to divergences that can be reabsorbed by renormalization of the parameters b, c, and wave-function renormalization. Actually it is convenient to rescale the field, X → a − 1 2 X, so that the Kähler metric and superpotential become (with a redefinition of the parameter c) Thus as in a standard σ-model approach one renormalizes the Kähler metric including the parameter a (this is equivalent to wave-function renormalization). According to Eq. (26), since the superpotential is not renormalized, a renormalization of the parameter a leads to a corresponding renormalization of the coupling constant g. K XX = 1 a + bXX + c(XX) 2 , ga − k 2 X k(26) At the one-loop level the divergence is proportional to the Ricci tensor, − ( 1 2π ln m 2 ℓ 2 )R XX = ( 1 2π ln m 2 ℓ 2 ) ab + 4acXX + bc(XX) 2 [a + bXX + c(XX) 2 ] 2(27) so that the Kähler metric, including one-loop corrections, becomes K XX + ∆K XX = 1 a(1 − Λb) + (b − 4Λac)XX + c(1 − Λa)(XX) 2(28) where Λ ≡ 1 2π ln m 2 ℓ 2 . The original parameters in the classical lagrangian are then expressed in terms of renormalized ones: a = Z a a R , b = Z b b R , c = Z c c R g = µZ g g R(29) where µ is the renormalization mass scale, and Z g Z − k 2 a = 1 as required by the N = 2 nonrenormalization theorem. From Eq.(28) we find Z a = 1 + b( 1 2π ln µ 2 ℓ 2 ) Z b = 1 + 4ac b ( 1 2π ln µ 2 ℓ 2 ) Z c = 1 + b( 1 2π ln µ 2 ℓ 2 ) Z g = 1 + bk 2 ( 1 2π ln µ 2 ℓ 2 )(30) Defining t = ln µ, the renormalized parameters satisfy the following renormalization group equations (in the following we drop the subscript R) da dt = − 1 π ab db dt = − 4 π ac dc dt = − 1 π cb dg dt = −(1 + b 2π k)g(31) Conformal invariance is achieved at the zeroes of the coupling β-functions. In particular we are looking for a nontrivial IR fixed point for the coupling constant g, i.e. such that b(t) → − 2π k as t → −∞. Thus we study the solutions of the system in Eq. (31) and select the relevant trajectories. The equations in (31) have two invariants, the ratio a c = ρ (32) and the combination, which we choose to make positive and parametrize suitably, b 2 − 4ac = b 2 − 4ρc 2 = (πλ) 2(33) Here ρ and λ are arbitrary constants parametrizing individual trajectories. In the b-c plane we obtain two types of trajectories, hyperbolas or ellipses, depending on the sign of ρ. Since we are interested in trajectories with two fixed points we write the elliptical solutions, with ρ < 0. (The bosonic model studied in Ref. 10, written in a different coordinate system, has ρ = 1.) b(t) = πλ tanh λt a(t) = ± πλ √ −ρ 2 (cosh λt) −1 c(t) = ∓ πλ 2 √ −ρ (cosh λt) −1 g(t) = g 0 e −t [cosh λt] − k 2(34) The wanted IR fixed point is reached by flowing along trajectories which have λ = 2 k (35) In this case the superfield a − 1 2 X acquires anomalous dimension 1/k in the corresponding IR conformal theory, while a and c flow to zero. Therefore, the effective lagrangian with Kähler potential K(X,X, a(t), b(t), c(t)) and superpotential W (X, g(t), a(t)) has the following behaviour in the infrared, t → −∞ , K(t) → − k 2π ln X lnX , W (t) → g 0 X k (36) The improvement term at the IR fixed point has Ψ = 1 4π ln X. Changing variables, X ≡ e Φ , leads to the Liouville lagrangian L = − k 2πΦ Φ + g 0 e kΦ(37) with negative kinetic term and with normalization determined by the superpotential (cf. Refs. 5,6). We emphasize that imposing conformal invariance at the one-loop level, i.e. R XX = 0, is sufficient to insure the absence of divergences at higher-loop orders since the Riemann tensor trivially vanishes as well. Moreover, while in the bosonic or in the N = 1 supersymmetric theories the dilaton term contributes to the metric β-function, in the N = 2 case no metric-dilaton mixing occurs due to the chirality of Ψ. Thus at the conformal point we obtain exact, all-order results. The case of two fields, with superpotential gX n +g ′ X k Y m can be treated in similar fashion 8 . The flows to the IR fixed point lead to a central charge which agrees with the results for the various minimal models described by a two-field Landau-Ginzburg potential. At the fixed point the model we have discussed is conformally invariant and therefore integrable. We have examined its integrability along the flow, by looking for a higher-spin conserved current. We observe that for the case k = 1 the model reduces to the supersymmetric complex sine-Gordon system studied by Napolitano and Sciuto 13 , which is indeed classically integrable. In particular, these authors have shown that a spin 3/2 conserved current exists for their model. However, for k > 1 we have been unable to construct a conserved spin 3/2 current. Although we suspect that no conserved current exists for our models, integrability along the flows remains an open question. Finally, we discuss effects due to gravitational dressing 14,15,16,17 . It has been shown that for bosonic theories one-loop β-functions in the presence of induced gravity are related to the corresponding ones computed in the absence of gravitational effects by the universal formula β G = κ + 2 κ + 1 β(38) Here κ is the level of the gravitational SL(2, R) current algebra which can be expressed in terms of the matter central charge as κ + 2 = 1 12 [c − 13 − (1 − c)(25 − c)](39) In the semiclassical limit, c → −∞, the dressing becomes β G → (1 + 6 c )β(40) The above result was obtained by treating induced gravity in light-cone gauge and making use of the corresponding Ward identities 15 , or in conformal gauge 16 where induced gravity is described by the Liouville action, making use of the fact that one must distinguish between the scale defined by the fiducial metric, and the physical scale defined by the Liouville field. It can also be checked explicitly in perturbation theory by performing a calculation for the standard bosonic σ-model coupled to induced gravity in light-cone gauge. A similar treatment is possible for N = 1 and N = 2 induced supergravity 18 . The argument is particularly simple in (super)conformal gauge. The idea 16,17 is that, in the presence of the Liouville mode, the physical scale gets modified with respect to the standard renormalization scale. In two-dimensional gravity the only dimensionful object, which provides the physical scale, is the cosmological constant term. In lightcone gauge the cosmological constant is just a c-number, so that the usual scaling is physical and the modifications of the matter β-functions arise through new divergent contributions due to the gravitational couplings. In conformal gauge instead the oneloop matter divergence does not receive gravitational corrections. However in this case the cosmological term is renormalized and thus it determines the new physical scale that enters in the computation of the dressed matter β-functions. In (super)conformal gauge the computation can be performed treating on equal footing the bosonic, N = 0, and N = 1, 2 induced (super)gravities. In all cases the presence of the Liouville field φ in the cosmological constant term determines the physical scale Λ through d 2 zd 2N θe α + φ ⇔ d 2 zd 2N θΛ −2s(41) where α + is the positive solution of − 1 2 α(α + Q) = s(42) i.e. α + = 1 2 −Q + Q 2 − 8s(43) For the various theories of interest one has, from the dimensionality of d 2 zd 2N θ, s = The (super)gravity modification of one-loop matter β-functions comes from the chain rule relating derivatives with respect to the physical scale Λ and the renormalization scale µ in the absence of gravity β G = ∂lnµ ∂lnΛ β = − 2s α + Q β(45) Using the above expressions, one finds for the ordinary gravity case, N = 0, the result in Eq. (38). For N = 1, 2 using also the expressions for the level κ of the light-cone supergravity Kač-Moody algebra N = 1 : κ + 3 2 = 1 8 c − 5 − 1 − c)(9 − c) N = 2 : κ + 1 = 1 4 (c − 1)(46) one finds N = 1 : β G = κ + 3 2 κ + 1 β N = 2 : β G = β(47) These results can be verified perturbatively in light-cone gauge. Details are presented in a separate publication 18 . We conclude that there is no supergravity dressing of β-functions in the N = 2 case. We also note that even in the presence of supergravity the nonrenormalization theorems continue to hold so that there is no correction to the superpotential. Therefore, the RG flow results discussed above are not modified by the presence of induced N = 2 supergravity. Acknowledgments D. Zanon thanks the Physics Department of Harvard University for hospitality during the period when some of this work was done. ( A related metric, with a = c, has been discussed in a bosonic σ-model context by Fateev et al 10 . The authors of Ref. 11 have speculated on the relevance of such metrics for studying N = 2 flows.). . D Kastor, E Martinec, S Shenker, Nucl. Phys. 316590D. Kastor, E. Martinec and S. Shenker, Nucl. Phys B316 (1989) 590. . C Vafa, N Warner, Phys. Lett. 21851C. Vafa and N. Warner, Phys. Lett. 218B (1989) 51; . W Lerche, C Vafa, N Warner, Nucl. Phys. 324427W. Lerche, C. Vafa and N. Warner, Nucl. Phys B324 (1989) 427. . P S Howe, P C West, Phys. Lett. 227397P.S. Howe and P.C. West, Phys. Lett. B227 (1989) 397 ; . S Cecotti, L Girardello, A Pasquinucci, Nucl. Phys. 328701S. Cecotti, L. Girardello and A. Pasquinucci, Nucl. Phys. B328 (1989) 701; . Int. J. Mod. Phys. 62427Int. J. Mod. Phys. A6 (1991) 2427. . A Marshakov, A Morozov, Phys. Lett. 23597A. Marshakov and A. Morozov, Phys. Lett. B235 (1990) 97. . H C Liao, P Mansfield, Phys. Lett. 255237H.C. Liao and P. Mansfield, Phys. Lett. 255B (1991) 237. . E Witten, Int. J. Mod. Phys. 94783E. Witten, Int. J. Mod. Phys. A9 (1994) 4783. . M T Grisaru, D Zanon, Phys. Lett. 33277M.T. Grisaru and D. Zanon, Phys. Lett B332 (1994) 77. Prepotentials for (2,2) Supergravity. M T Grisaru, M Wehlau, Int. J. Mod. Phys. A. to be publishedM.T. Grisaru and M. Wehlau, Prepotentials for (2,2) Supergravity, Int. J. Mod. Phys. A (to be published). . V A Fateev, E Onofri, Al B Zamolodchikov, Nucl. Phys. 406521V.A. Fateev, E. Onofri and Al.B. Zamolodchikov, Nucl. Phys. B406 (1993) 521. . P Fendley, K Intrilligator, Nucl. Phys. 413653P. Fendley and K. Intrilligator, Nucl. Phys. B413 (1994) 653. . M T Grisaru, A E M Van De Ven, D Zanon, Nucl. Phys. 277388M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Nucl. Phys. B277 (1986) 388. . E Napolitano, S Sciuto, Phys. Lett. 11343E. Napolitano and S. Sciuto, Phys. Lett. B113 (1982) 43. . C Schmidhuber, Nucl. Phys. 404342C. Schmidhuber, Nucl. Phys. B404 (1993) 342. . I R Klebanov, I I Kohan, A M Polyakov, Phys. Rev. Lett. 713243I.R. Klebanov, I.I. Kohan and A.M. Polyakov, Phys. Rev. Lett. 71 (1993) 3243. Bohr Institute preprint NBI-HE-93-63. J Ambjorn, K Ghoroku, J. Ambjorn and K. Ghoroku, Bohr Institute preprint NBI-HE-93-63. . Y Tanii, S Kojima, N Sakai, Phys. Lett. 32259Y. Tanii, S. Kojima and N. Sakai, Phys. Lett. B322 (1994) 59. M T Grisaru, D Zanon, Supergravity dressing of N=1 and N=2 models. to be publishedM.T. Grisaru and D. Zanon, Supergravity dressing of N=1 and N=2 models (to be published).
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[ "A meson-exchange πN model up to energies √ s ≤ 2.0 GeV", "A meson-exchange πN model up to energies √ s ≤ 2.0 GeV" ]
[ "Shin Nan Yang \nDepartment of Physics and Center for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan\n", "Guan Yeu Chen \nDepartment of Physics and Center for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan\n", "S S Kamalov \nBogoliubov Laboratory for Theoretical Physics\nJINR\n141980Dubna, Moscow RegionRussia\n" ]
[ "Department of Physics and Center for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan", "Department of Physics and Center for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan", "Bogoliubov Laboratory for Theoretical Physics\nJINR\n141980Dubna, Moscow RegionRussia" ]
[]
A meson-exchange πN model, previously constructed using three-dimensional reduction scheme of the Bethe-Salpeter equation for a model Lagrangian involving π, η, N, ∆, ρ, and σ fields, is extended to energies up to 2 GeV by including the ηN channel and all the four stars πN resonances up to the F −waves. The effects of other 2π channels are taken into account phenomenologically. The extended model gives an excellent fit to both πN phase shifts and inelasticity parameters in all channels up to the F −waves. However, a few of the extracted resonance parameters differ considerably from the PDG values.
10.1016/j.nuclphysa.2007.03.151
[ "https://arxiv.org/pdf/nucl-th/0610076v1.pdf" ]
118,698,138
nucl-th/0610076
74028cb89421d310425c4e89bfffa800fd32f11a
A meson-exchange πN model up to energies √ s ≤ 2.0 GeV arXiv:nucl-th/0610076v1 19 Oct 2006 Shin Nan Yang Department of Physics and Center for Theoretical Sciences National Taiwan University 10617TaipeiTaiwan Guan Yeu Chen Department of Physics and Center for Theoretical Sciences National Taiwan University 10617TaipeiTaiwan S S Kamalov Bogoliubov Laboratory for Theoretical Physics JINR 141980Dubna, Moscow RegionRussia A meson-exchange πN model up to energies √ s ≤ 2.0 GeV arXiv:nucl-th/0610076v1 19 Oct 20061 A meson-exchange πN model, previously constructed using three-dimensional reduction scheme of the Bethe-Salpeter equation for a model Lagrangian involving π, η, N, ∆, ρ, and σ fields, is extended to energies up to 2 GeV by including the ηN channel and all the four stars πN resonances up to the F −waves. The effects of other 2π channels are taken into account phenomenologically. The extended model gives an excellent fit to both πN phase shifts and inelasticity parameters in all channels up to the F −waves. However, a few of the extracted resonance parameters differ considerably from the PDG values. INTRODUCTION Pion-nucleon scattering is one of the main sources of information for the baryon spectrum. In addition, it also plays a fundamental role in the description of nuclear dynamics for which the πN off-shell amplitude serves as the basic input to most of the existing nuclear calculations at intermediate energies. Knowledge about the off-shell πN amplitude is also essential in interpreting the experiments performed at the intermediate-energy electron accelerators in order to unravel the internal structure of these hadrons [1,2,3]. It is hence important to have a sound theoretical description of the πN interaction. It is commonly accepted that Quantum Chromodynamics (QCD) is the fundamental theory of the strong interaction. However, due to the confinement problem, it is still practically impossible to derive the πN interaction directly from QCD. On the other hand, models based on meson-exchange (MEX) pictures have been very successful in describing the NN scattering. Over the past decade, MEX approach has also been applied by several groups [4,5,6,7,8,9] to construct models for πN scattering. In previous works we have constructed several MEX πN models within the threedimensional reduction scheme of the Bethe-Salpeter equation [2,9] and investigated their sensitivity with respect to various three-dimensional reduction schemes. The model Lagrangian included only π, N, ∆, ρ, and σ fields. It was found that all the resulting mesonexchange models can yield similarly good descriptions of πN scattering data up to 400 MeV. The model obtained with the Cooper-Jennings reduction scheme [10] was recently extended up to a c.m. energy of 2 GeV in the S 11 channel by including the ηN channel and a set of higher S 11 resonances [11]. The effects of the other ππN channels like the σN, ρN, and π∆, instead of including them directly in the coupled-channels calculation, were taken into account by introducing a phenomenological term in the resonance propagators. An excellent fit to the t-matrix in both πN and ηN channels was obtained. Here we further extend the model to other higher partial waves up to the F −waves. MESON-EXCHANGE MODEL FOR πN SCATTERING The MEX πN model we previously constructed was obtained by using a three-dimensional reduction scheme of the Bethe-Salpeter equation for a model Lagrangian involving π, N, ∆, ρ, and σ fields. Details can be found in Ref. [9]. As the energy increases, two-pion channels like σN, ηN, π∆, ρN as well as a nonresonant continuum of ππN states become increasingly important, and at the same time more and more nucleon resonances appear as intermediate states. In Ref. [11] the πN model constructed in [9] was extended for the S 11 partial wave by explicitly coupling the π, η and ππ channels and including the couplings with higher baryon resonances. For example, in the case when there is only one resonance R contributing, the Hilbert space was enlarged to include a bare S 11 resonance R which acquires a width by its coupling with the πN and ηN channels through the Lagrangian L I = ig (0) πN RR τ N · π + ig (0) ηN RR Nη + h.c.,(1) where N, R, π, and η denote the field operators for the nucleon, bare R, pion and eta meson, respectively. Then the full t-matrix can be written as a system of coupled equations, t ij (E) = v ij (E) + k v ik (E) g k (E) t kj (E) ,(2) where i and j denote the π, or η channel and E = W is the total center mass energy. In general, the potential v ij is a sum of non-resonant (v B ij ) and bare resonance (v R ij ) terms, v ij (E) = v B ij (E) + v R ij (E) .(3) The non-resonant term v B ππ for the πN elastic channel contains contributions from the s-and u-channel, pseudovector Born terms and t-channel contributions with ω, ρ, and σ exchange. The parameters in v B ππ are fixed from the analysis of the pion scattering phase shifts for the S− and P −waves at low energies (W < 1300 MeV) [9]. In channels involving η, v B iη is taken to be zero since the ηNN coupling is very small. The bare resonance contribution v R ij (E) = h (0) † iR h (0) jR E − M (0) R ,(4) where h iR and M R denote the bare vertex operator for π/η + N → R and bare mass of R, respectively, arises from the excitation and de-excitation of the resonance R. The matrix elements of the potential v R ij (E) can be symbolically expressed in the form v R ij (q, q ′ ; E) = f i (Λ i , q; E) g (0) i g (0) j f j (Λ j , q ′ ; E) E − M (0) R + i 2 Γ R 2π (E) ,(5) where q and q ′ are the pion (or eta) momenta in the initial and final states, and g (0) i/j is the resonance vertex couplings. As in [9], we associate with each external line of the particle α in a Feynman diagram a covariant form factor F α = [n α Λ 4 α /(n α Λ 4 α + (p 2 α − m 2 α ) 2 )] nα , where p α , m α , and Λ α are the four-momentum, mass, and cut-off parameter of particle α, respectively, and n c = 10. As a result, f i depends on the product of three cut-off parameters. In Eq. (5) we have included a phenomenological term Γ R 2π (E) in the resonance propagator to account for the ππN decay channel. Therefore, our "bare" resonance propagator already contains some renormalization or "dressing" effects due to the coupling with the ππN channel. With this prescription we assume that any further non-resonant coupling mechanisms with the ππN channel are small. The form of Γ R 2π (E) can be found in [11] and is characterized by two parameters, a cut-off Λ R and the 2π decay width at the resonance Γ (0)R 2π . Consequently, one isolated resonance will contain five free parameters, M (0) R , Γ (0)R 2π , Λ R , g (0) i and g (0) j . The generalization of the coupled channels model to the case of N resonances with the same quantum numbers is then given by v R ij (q, q ′ ; E) = N n=1 v Rn ij (q, q ′ ; E),(6) with free parameters for the bare masses, 2π widths, coupling constants, and cut-off parameters for each resonance. After solving the coupled channel equations, the next task is the extraction of the physical (or "dressed") masses, partial widths, and branching ratios of the resonances. It is well-known this procedure is definitely model dependent, because the background and the resonance contributions can not be separated in a unique way. In this work, we employ the procedure used in Ref. [3] where, in the case of pion-nucleon elastic scattering with only one resonance contributing, the full t-matrix is written as follows, t πN (E) = t B πN (E) + t R πN (E),(7) where t B πN (E) = v B πN + v B πN g 0 (E) t πN (E) , t R πN (E) = v R πN + v R πB g 0 (E) t πN (E).(8) The "background" t B πN includes contributions not only from the background rescattering but also from intermediate resonance excitation. This is compensated by the fact that the resonance contribution t R πN now contains only the terms that start with the bare resonance vertex. In terms of self-energy and vertex functions, one obtains the result [12] t R πN (E) =h πR (E)h (0) πR (E) E − M (0) R (E) − Σ R (E) ,(9) wherē h πR (E) = (1 + g 0 (E) t B πN (E))h (0) † πR , Σ R 1π (E) = h (0) πR g 0hπR (E).(10) h πR (E) describes the dressed vertex of R → πN [3]. Σ R 1π is the self-energy of the dressed R arising from one-pion intermediate states and Σ R (E) = Σ R 1π +Σ R 2π with Σ R 2π (E) = 2iΓ R 2π . The information about the physical mass and the total width of the resonance R are contained in the dressed resonance propagator given in Eq. (9). The complex self-energy Σ R (E) leads to a shift from the real "bare" mass to a complex and energy-dependent value. We characterize the resonance by energy-independent parameter that is obtained by solving the equation E − M (0) R − Re Σ R (E) = 0 .(11) The solution of this equation, E = M R , corresponds to the energy at which the dressed propagator in Eq. (9) is purely imaginary. The "physical" or "dressed" mass and the width of the resonance is then defined by, M R = M (0) R + Re Σ R (M R ), Γ R = −2 Im Σ R (M R ).(12) When there are N resonances contributing in the same channel, Eq. (8) can be generalized to take the form of t πN (E) = t B πN (E) + N i=1 t R i πN (E) .(13) The contribution from each resonance R i can be expressed in terms of the bare h (0) πR i and dressedh πR i vertex operators as well as the resonance self energy derived from one-pion Σ R i 1π (E) and two-pion Σ R i 2π (E) channels, that is t R i πN (E) =h πR i (E)h (0) πR i (E) E − M (0) R i − Σ R i 1π (E) − Σ R i 2π (E) ,(14) where M (0) R i is the bare mass of R i . The vertices for resonance excitation are obtained, in analogous to Eqs. (8)(9)(10), from the following two equations: h πR i (E) = (1 + g 0 (E) t B i πN (E))h (0) † πR i (15) t B i πN (E) = v B i (E) + v B i (E) g 0 (E) t B i πN (E) ,(16) where v B i (E) = v B πN + N j =i v R j πN (E). The one-pion self-energies corresponding to Eq. (10), is Σ R i 1π = h (0) πR i g 0h † πR i . We wish to emphasize that in the formulation we present above, namely in Eqs. (13)(14)(15)(16), the N resonances are treated in a completely symmetrical way and the self-energy and the dressing of any resonance receive contributions from all other resonances. RESULTS AND DISCUSSIONS In Fig. 1, we compare our results for the real and imaginary parts of the t-matrix in some selected channels in S−, P −, D− and F −waves up to 2 GeV c.m. energy with the experimental data as obtained in the SAID partial wave analysis [13]. One sees that we are able to describe the data very well. The bare and physical resonance masses, and widths extracted according to Eq. (12) are presented in Table 1. Even though our model describes the data for t-matrix well as Figure 1. The best fit of the real and imaginary parts of the πN scattering t-matrix using dynamical MEX model (solid curves). The dashed curves give the background contribution. Experimental data are the results of the partial wave analysis from Ref. [13]. seen in Fig. 1, the resonance properties we extract as given in Table 1 do show several differences when compared with PDG values [14]. The most notable ones are that (1). we require two resonances not listed in PDG: S 11 (1878) and D 13 (1946); (2). the masses and widths we obtain for the 2nd and 3rd resonances in S 31 and P 11 deviates substantially from the PDG values. The PDG values for (M R , Γ R ) for these resonances are S 31 : (1900± 50, 190±50), (2150±100, 200±100) and P 11 : (1710±30, 180±100), (2125±75, 260±100); (3). the width we obtain for F 15 (2000) is only 58 MeV which is much smaller that the PDG value of 490 ± 310 MeV. SUMMARY We have extended our previously constructed meson-exchange πN model to energies up to 2 GeV by including the ηN channel and all the four stars πN resonances up to the F −waves. The effects of other 2π channels are taken into account phenomenologically. We have treated, in any given channel, all the contributing resonances in a completely symmetrical manner such that every resonance is dressed by the presence of all other resonances. The extended model gives an excellent fit to both πN phase shifts and inelasticity parameters in all channels up to the F −waves. However, a few of the resonance parameters differ substantially from the PDG values. This πN model will be applied to evaluate the contribution of the pion cloud to the photopion reactions up to 2 GeV c.m. energy as was done in Ref. [3] so that the photoexcitation strengths of all resonances below 2 GeV can be reliably extracted. Table 1 Bare M (0) R and physical M R resonance masses and total width in MeV. 1st res. 2nd res. 3rd res. N * M (0) R . S N Yang, J. Phys. 11205S.N. Yang, J. Phys. G11 (1985) L205; . Phys. Rev. 401810Phys. Rev. C40 (1989) 1810. . C C Lee, S N Yang, T.-S H N Lee ; L131; S, Yang, Chin. J. Phys. 17485J. Phys.C.C. Lee, S.N. Yang, and T.-S.H. Lee, J. Phys. G17 (1991) L131; S.N. Yang, Chin. J. Phys. 29 (1991) 485. . S S Kamalov, S N Yang, Phys. Rev. Lett. 834494S.S. Kamalov and S.N. Yang, Phys. Rev. Lett. 83 (1999) 4494. . B C Pearce, B Jennings, Nucl. Phys. 528655B.C. Pearce and B. Jennings, Nucl. Phys. A528 (1991) 655. . F Gross, Y Surya, Phys. Rev. 47703F. Gross and Y. Surya, Phys. Rev. C47 (1993) 703. . C Schütze, J W Durso, K Holinde, J Speth, Phys. Rev. 492671C. Schütze, J.W. Durso, K. Holinde and J. Speth, Phys. Rev. C49 (1994) 2671. . A D Lahiff, I R Afnan, Phys. Rev. C. 6024608A.D. Lahiff and I.R. Afnan, Phys. Rev. C 60 (1999) 024608. . V Pascalutsa, J A Tjon, Phys. Rev. 6154003V. Pascalutsa and J.A. Tjon, Phys. Rev. C61 (2000) 054003. . C T Hung, S N Yang, T.-S H Lee, Phys. Rev. C. 6434309C.T. Hung, S.N. Yang, and T.-S.H. Lee, Phys. Rev. C 64 (2001) 034309; . J. Phys. 201531J. Phys. G20 (1994) 1531. . M Cooper, B Jennings, Nucl. Phys. 500553M. Cooper and B. Jennings, Nucl. Phys. A500 (1989) 553. . G Y Chen, S Kamalov, S N Yang, D Drechsel, L Tiator, Nucl. Phys. 723447G.Y. Chen, S. Kamalov, S.N. Yang, D. Drechsel, and L. Tiator, Nucl. Phys. A723 (2003) 447. . S S Hsiao, C T Hung, J L Tsai, S N Yang, Y B Dong, Few-Body Systems. 2555S.S. Hsiao, C.T. Hung, J.L. Tsai, S.N. Yang, and Y.B. Dong, Few-Body Systems 25 (1998) 55. . R A Arndt, W J Briscoe, I I Strakovsky, R L Workman, Phys. Rev. 6935213R.A. Arndt, W.J. Briscoe, I.I. Strakovsky and R.L. Workman, Phys. Rev. C69 (2004) 035213. . W M Yao, J. Phys. 331Particle Data GroupW.M. Yao et al., [Particle Data Group], J. Phys. G33 (2006) 1.
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[ "The Reticulation of a Universal Algebra", "The Reticulation of a Universal Algebra" ]
[ "George Georgescu \nFaculty of Mathematics and Computer Science Academiei 14\nUniversity of Bucharest\n010014BucharestRORomania\n", "Claudia Mureşan [email protected] \nFaculty of Mathematics and Computer Science Academiei 14\nUniversity of Bucharest\n010014BucharestRORomania\n" ]
[ "Faculty of Mathematics and Computer Science Academiei 14\nUniversity of Bucharest\n010014BucharestRORomania", "Faculty of Mathematics and Computer Science Academiei 14\nUniversity of Bucharest\n010014BucharestRORomania" ]
[]
The reticulation of an algebra A is a bounded distributive lattice L(A) whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of A, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra A from a semi-degenerate congruencemodular variety C in the case when the commutator of A, applied to compact congruences of A, produces compact congruences, in particular when C has principal commutators; furthermore, it turns out that weaker conditions than the fact that A belongs to a congruence-modular variety are sufficient for A to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from C is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and C, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.
10.7561/sacs.2018.1.67
[ "https://arxiv.org/pdf/1706.04270v1.pdf" ]
52,821,356
1706.04270
16631c8474c4825e761af1f4d8be0b63375705fe
The Reticulation of a Universal Algebra 13 Jun 2017 March 28, 2018 George Georgescu Faculty of Mathematics and Computer Science Academiei 14 University of Bucharest 010014BucharestRORomania Claudia Mureşan [email protected] Faculty of Mathematics and Computer Science Academiei 14 University of Bucharest 010014BucharestRORomania The Reticulation of a Universal Algebra 13 Jun 2017 March 28, 20182010 Mathematics Subject Classification: primary: 08B10; secondary: 08A3006B1006F3503G25 Keywords: (congruence-modularcongruence-distributive) varietycommutator(primecompact) congru- encereticulation The reticulation of an algebra A is a bounded distributive lattice L(A) whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of A, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra A from a semi-degenerate congruencemodular variety C in the case when the commutator of A, applied to compact congruences of A, produces compact congruences, in particular when C has principal commutators; furthermore, it turns out that weaker conditions than the fact that A belongs to a congruence-modular variety are sufficient for A to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from C is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and C, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra. Introduction The reticulation of a commutative unitary ring R is a bounded distributive lattice L(R) whose prime spectrum of ideals is homeomorphic to the prime spectrum of ideals of R. Its construction has appeared in [32], but it has been extensively studied in [52], where it has received the name reticulation. The mapping R → L(R) sets a covariant functor from the category of commutative unitary rings to that of bounded distributive lattices, through which properties can be transferred between these categories. In [7], the reticulation has been defined and studied for non-commutative unitary rings and it has been proven that such a ring has a reticulation (with the topological definition above) iff it is quasi-commutative. Over the past two decades, reticulations have been constructed for orderred algebras related to logic: MValgebras [8,9], BL-algebras [37,20,38], residuated lattices [41,42,43,44,45,46], 0-distributive lattices [49], almost distributive lattices [50], Hilbert algebras [13], hoops [16]. All these algebras posess a "prime spectrum" which is homeomorphic to the prime spectrum of filters or ideals of a bounded distributive lattice; their reticulations consist of such bounded distributive lattices, whose study involves obtaining a construction for them and using that construction to transfer properties between these classes of algebras and bounded distributive lattices. The purpose of the present paper is to set the problem of constructing a reticulation in a universal algebra framework and providing a solution to this problem in a case as general as possible, that includes the cases of the varieties above and generalizes the constructions which have been obtained in those particular cases. Apart from the novelty of using commutator theory [18,39] for the study of the reticulation, essentially, the tools needed for obtaining reticulations in this very general setting are quite similar to those which have been put to work for the classes of algebras above, and it turns out that many types of results that hold for their reticulations can be generalized to our setting. In order to obtain strong generalizations, we have worked with hypotheses as weak as possible; all our results in this paper hold for semi-degenerate congruence-modular varieties whose members have the sets of compact congruences closed with respect to the commutator, with just a few exceptions that necessitate, moreover, principal commutators. The present paper is structured as follows: Section 2 presents the notations and basic results we use in what follows; Section 3 collects a set of results from commutator theory which we use in the sequel; in Section 4, we present the standard construction of the Stone topologies on prime spectra, specifically the prime spectrum of ideals of a bounded distributive lattice and the prime spectrum of congruences of a universal algebra whose commutator fulfills certain conditions. The results in the following sections that are not cited from other papers, or mentioned as being either known or quite simple to obtain, are new and original. In Section 5, we construct the reticulation for universal algebras whose commutators fulfill certain conditions, prove that this construction has the desired topological property and obtain some related results. In Section 6, we provide some examples of reticulations, study particular cases, such as the congruencedistributive case, show that our construction generalizes constructions for the reticulation which have been obtained for particular varieties, and prove that our construction preserves finite direct products of algebras without skew congruences. In Section 7, we obtain some arithmetical properties on commutators that we need in what follows, as well as algebraic properties regarding the behaviour of surjections with respect to commutators and to certain types of congruences. In Section 8 we study the behaviour of Boolean congruences with respect to the reticulation, in the general case, but also in particular ones, such as the case of associative commutators or that of semiprime algebras. In Section 9, we define a reticulation functor; our definition is not ideal, as it only acts on surjections; extending it to all morphisms remains an open problem. In this final section, we also show that the reticulation preserves quotients, and that it is a Boolean lattice exactly in the case of hyperarchimedean algebras, which we also characterize by several other conditions on their reticulation. These characterizations serve as an example for the transfer of properties to and from the category of bounded distributive lattices which the reticulation makes possible. We intend to further pursue the study of the reticulation in this universal algebra setting and use it to transfer more properties between the variety of bounded distributive lattices and the kinds of varieties that allow a construction for the reticulation. A theme for a potentially extensive future study is characterizing those varieties with the property that the reticulations of their members cover the entire class of bounded distributive lattices. Preliminaries In this section, we recall some properties on lattices and congruences in universal algebras. For a further study of the following results on universal algebras, we refer the reader to [1], [12], [27], [34]. For those on lattices, we recommend [5], [11], [17], [26], [51]. i for all i ∈ I, then the direct product of (X i ) i∈I as a family of binary relations shall be denoted just as the one for sets, because there will be no danger of confusion when using this notation: i∈I X i = {((a i ) i∈I , (b i ) i∈I ) | (∀ i ∈ I) ((a i , b i ) ∈ X i )} ⊆ M 2 . Unless mentioned otherwise, the operations and order relation of a (bounded) lattice shall be denoted in the usual way, and the complementation of a Boolean algebra shall be denoted by ¬ . Throughout this paper, whenever there is no danger of confusion, any algebra shall be designated by its support set. All algebras shall be considerred non-empty; by trivial algebra we shall mean one-element algebra, and by non-trivial algebra we shall mean algebra with at least two distinct elements. Any direct product of algebras and any quotient algebra shall be considerred with the operations defined canonically. For brevity, we shall denote by A ∼ = B the fact that two algebras A and B of the same type are isomorphic. Let L be a bounded lattice. By Id(L) we shall denote the set of the ideals of L, that is the non-empty subsets of L which are closed with respect to the join and to lower bounds. By Filt(L) we shall denote the set of the filters of L, that is the ideals of the dual of L: the non-empty subsets of L which are closed with respect to the meet and to upper bounds. For any M ⊆ L and any a ∈ L, (M ], respectively [M ), shall denote the ideal, respectively the filter of L generated by M , and the principal ideal, ({a}] = {x ∈ L | a ≥ x}, respectively the principal filter, and F ∨ G = [F ∪ G) for all J, K ∈ Id(L) and all F, G ∈ Filt(L), and they are distributive iff L is distributive; moreover, they are complete lattices, with [{a}) = {x ∈ L | a ≤ x},i∈I J i = [ i∈I J i ) and i∈I F i = ( i∈I F i ] for any families (J i ) i∈I ⊆ Id(L) and (F i ) i∈I ⊆ Filt(L). Obviously, for any a, b ∈ L, (a] ∨ (b] = (a ∨ b], (a] ∩ (b] = (a ∧ b], [a) ∨ [b) = [a ∧ b) and [a) ∩ [b) = [a ∨ b). If L is a complete lattice, then, for any family (a i ) i∈I ⊆ L, i∈I (a i ] = ( i∈I a i ], i∈I (a i ] = ( i∈I a i ], i∈I [a i ) = [ i∈I a i ) and i∈I [a i ) = [ i∈I a i ) . By PId(L), respectively PFilt(L), we shall denote the set of the principal ideals, respectively the principal filters of L. We shall denote by Max Id (L), respectively Max Filt (L), the set of the maximal ideals, respectively the maximal filters of L, that is the maximal elements of the set of proper ideals of L, Id(L) \ {L}, respectively that of proper filters of L, Filt(L) \ {L}. By Spec Id (L) we shall denote the set of the prime ideals of L, that is the proper ideals P of L such that, for any x, y ∈ L, x ∧ y ∈ P implies x ∈ P or y ∈ P . Dually, Spec Filt (L) shall denote the set of the prime filters of L, that is the proper filters P of L such that, for any x, y ∈ L, x ∨ y ∈ P implies x ∈ P or y ∈ P . For any algebra A, Con(A) shall denote the set of the congruences of A, and Max(A) shall denote the set of the maximal congruences of A, that is the maximal elements of the set of proper congruences of A: Con(A) \ {∇ A }. Let θ ∈ Con(A), a ∈ A, M ⊆ A and X ⊆ A 2 , arbitrary. Then a/θ shall denote the congruence class of a with respect to θ, M/θ = {x/θ | x ∈ M }, p θ : A → A/θ shall be the canonical surjective morphism: p θ (a) = a/θ for all a ∈ A, X/θ = {(x/θ, y/θ) | (x, y) ∈ X} and Cg A (X) shall be the congruence of A generated by X. It is well known that (Con(A), ∨, ∩, ∆ A , ∇ A ) is a bounded lattice, orderred by set inclusion, where φ ∨ ψ = Cg A (φ ∪ ψ) for all φ, ψ ∈ Con(A); moreover, this is a complete lattice, in which a, b). The set of the principal congruences of A shall be denoted by PCon (A). K(A) shall denote the set of the finitely generated congruences of A, which coincide to the compact elements of the lattice Con (A). Clearly, i∈I φ i = Cg A ( i∈I φ i ) for any family (φ i ) i∈I ⊆ Con(A). For any a, b ∈ A, the principal congruence Cg A ({(a, b)}) shall also be denoted by Cg A (PCon(A) ⊆ K(A) and ∆ A ∈ PCon(A), because ∆ A = Cg A (x, x) for any x ∈ A. Throughout the rest of this paper, τ shall be a universal algebras signature, C shall be an equational class of τ -algebras A and B shall be algebras from C and f : A → B shall be a morphism in C. Unless mentioned otherwise, by morphism we shall mean τ -morphism. We recall that A is said to be congruence-modular, respectively congruence-distributive, iff the lattice Con(A) is modular, respectively distributive, and that C is said to be congruence-modular, respectively congruence-distributive, iff every algebra in C is congruence-modular, respectively congruence-distributive. The Commutator This section is composed of results on the commutator in arbitrary and in congruence-modular varieties, which are either previously known of very easy to derive from previously known results. For a further study of these results, see [1], [21], [34], [48]. Out of the various definitions for commutator operations on congruence lattices, we have chosen to work with the term condition commutator, from the following definition. Recall that, in algebras from congruence-modular varieties, all definitions for the commutator give the same commutator operation. For any term t over τ , we shall denote by t A the derivative operation of A associated to t. Definition 3.1. [39] Let α, β ∈ Con(A). For any µ ∈ Con(A), by C(α, β; µ) we denote the fact that the following condition holds: for all n, k ∈ N and any term t over τ of arity n + k, if (a i , b i ) ∈ α for all i ∈ 1, n and (c j , d j ) ∈ β for all j ∈ 1, k, then (t A (a 1 , . . . , a n , c 1 , . . . , c k ), t A (a 1 , . . . , a n , d 1 , . . . , d k )) ∈ µ iff (t A (b 1 , . . . , b n , c 1 , . . . , c k ), t A (b 1 , . . . , b n , d 1 , . . . , d k )) ∈ µ. We denote by [α, β] A = {µ ∈ Con(A) | C(α, β; µ)}; we call [α, β] A the commutator of α and β in A. Remark 3.2. Let α, β ∈ Con (A). Clearly, C(α, β; ∇ A ). Since Con(A) is a complete lattice, it follows that [α, β] A ∈ Con(A). Furthermore, according to [39,Lemma 4.4,(2)], for any family (µ i ) i∈I ⊆ Con(A), if C(α, β; µ i ) for all i ∈ I, then C(α, β; i∈I µ i ). Hence C(α, β; [α, β] A ), and thus [α, β] A = min { µ ∈ Con(A) | C(α, β; µ)}, which is exactly the definition of the commutator from [40]. Definition 3.3. The operation [·, ·] A : Con(A) × Con(A) → Con(A) is called the commutator of A. Theorem 3.4. [21] If C is congruence-modular, then, for each member M of C, [·, ·] M is the unique binary operation on Con(M ) such that, for all α, β ∈ Con(M ), [α, β] M = min{µ ∈ Con(M ) | µ ⊆ α ∩ β and, for any member N of C and any surjective morphism h : M → N in C, µ∨Ker(h) = h * ([h(α∨Ker(h)), h(β ∨Ker(h))] N )}. Theorem 3.5. [31] If C is congruence-distributive, then, in each member of C, the commutator coincides to the intersection of congruences. For brevity, most of the times, we shall use the remarks in this paper without referencing them, and the same goes for the lemmas and propositions that state basic results. • increasing in both arguments, that is, for all α, β, φ, ψ ∈ Con(A), if α ⊆ β and φ ⊆ ψ, then [α, φ] A ⊆ [β, ψ] A ; • smaller than its arguments, so, for any α, β ∈ Con (A), [α, β] A ⊆ α ∩ β. If C is congruence-modular, then the commutator is also: • commutative, that is [α, β] A = [β, α] A for all α, β ∈ Con(A); • distributive in both arguments with respect to arbitrary joins, that is, for any families (α i ) i∈I and (β j ) j∈J of congruences of A, [ i∈I α i , j∈J β j ] A = i∈I j∈J [α i , β j ] A . Remark 3.7. Assume that [·, ·] A is commutative. Then the distributivity of [·, ·] A in both arguments w.r.t. arbitrary joins is equivalent to its distributivity in one argument w.r.t. arbitrary joins, which in turn is equivalent to its distributivity w.r.t. the join in the case when Con(A) is finite, in particular when A is finite. Obviously, if [·, ·] A equals the intersection and it is distributive w.r.t. the join (by Proposition 3.6, the latter holds if C is congruence-modular), then A is congruence-distributive. Lemma 3.8. [21] If C is congruence-modular and S is a subalgebra of A, then, for any α, β ∈ Con (A), [α ∩ S 2 , β ∩ S 2 ] S ⊆ [α, β] A ∩ S 2 .i ∈ 1, n, α i , β i ∈ Con(M i ). Then: [ n i=1 α i , n i=1 β i ] M = n i=1 [α i , β i ] Mi . Remark 3.10. By Theorem 3.4 and Remark 2.1, if C is congruence-modular, α, β, θ ∈ Con(A) and f is surjective, then [f (α∨Ker(f )), f (β∨Ker(f ))] B = f ([α, β] A ∨Ker(f )), thus [(α∨θ)/θ, (β∨θ)/θ] B = ([α, β] A ∨θ)/θ, hence, if θ ⊆ [α, β] A , then [α/θ, β/θ] A/θ = [α, β] A /θ. Definition 3.11. [21] Let φ be a proper congruence of A. Then φ is called a prime congruence of A iff, for all α, β ∈ Con(A), [α, β] A ⊆ φ implies α ⊆ φ or β ⊆ φ. φ is called a semiprime congruence of A iff, for all α ∈ Con(A), [α, α] A ⊆ φ implies α ⊆ φ. The set of the prime congruences of A shall be denoted by Spec (A). Spec(A) is called the (prime) spectrum of A and Max(A) is called the maximal spectrum of A. Following [34], we say that C is semi-degenerate iff no non-trivial algebra in C has one-element subalgebras. For instance, the class of unitary rings and any class of bounded orderred structures is semi-degenerate. • any proper congruence of A is included in a maximal congruence of A; • any maximal congruence of A is prime. Remark 3.13. By Lemma 3.12, if A is non-trivial and C is congruence-modular and semi-degenerate, then A has maximal congruences, thus it has prime congruences. (iii) C has no skew congruences, that is, for any algebras M and N from C, Con(M × N ) = {θ × ζ | θ ∈ Con(M ), ζ ∈ Con(N )}. 2] If f is surjective, then, for any a, b ∈ A, any X ⊆ A 2 , any θ ∈ Con(A) and any α, β ∈ [Ker(f )): (i) f (θ ∨ Ker(f )) = Cg B (f (θ)); f (α ∨ β) = f (α) ∨ f (β); (ii) f (Cg A (a, b) ∨ Ker(f )) = Cg B (f (a), f (b)); f (Cg A (X) ∨ Ker(f )) = Cg B (f (X)); (iii) (Cg A (a, b) ∨ θ)/θ = Cg A/θ (a/θ, b/θ); (Cg A (X) ∨ θ)/θ = Cg A/θ (X/θ). We say that A has principal commutators iff, for all α, β ∈ PCon(A), we have [α, β] A ∈ PCon(A), that is iff PCon(A) is closed with respect to the commutator of A. Following [1], we say that C has principal commutators iff each member of C has principal commutators. We say that C has associative commutators iff, for each member M of C, the commutator of M is an associative binary operation on Con(M ). 1 , b 1 ), . . . , (a n , b n )}) | n ∈ N * , a 1 , b 1 , . . . , a n , b (A). From this, it is immediate that K(A) is closed with respect to finite joins, and, if A has principal commutators and [·, ·] A is commutative and distributive w.r.t. the join (for instance if C is congruencemodular), then K(A) is also closed with respect to the commutator of A. Remark 3.18. K(A) = {Cg A (∅)} ∪ {Cg A ({(an ∈ A} = {∆ A } ∪ { n i=1 Cg A (a i , b i ) | n ∈ N * , a 1 , b 1 , . . . , a n , b n ∈ A} = { n i=1 Cg A (a i , b i ) | n ∈ N * , a 1 , b 1 , . . . , a n , b n ∈ A}, since ∆ A ∈ PCon Remark 3.19. If C is congruence-distributive, then, as shown by Theorem 3.5: • C has principal commutators iff C has the principal intersection property (PIP); • K(M ) is closed with respect to the commutator for each member M of C iff C has the compact intersection property (CIP). As a particular case of Remark 3.18, if C is congruence-distributive and has the PIP, then C has the CIP. Example 3.20. [1], [10], [25], [31,Theorem 2.8], [33], [36] As shown by Theorem 3.5, any congruence-distributive variety has associative commutators. The variety of commutative unitary rings is semi-degenerate, congruencemodular, with principal commutators and associative commutators, and it is not congruence-distributive. Out of the semi-degenerate congruence-distributive varieties with the CIP, we mention semi-degenerate filtral varieties. Out of the semi-degenerate congruence-distributive varieties with the PIP, we mention: bounded distributive lattices, residuated lattices (a variety which includes Gödel algebras, product algebras, MTL-algebras, BL-algebras, MV-algebras) and semi-degenerate discriminator varieties (out of which we mention Boolean algebras, n-valued Post algebras, n-valued Lukasiewicz algebras, n-valued MV-algebras, n-dimensional cylindric algebras, Gödel residuated lattices). The Stone Topologies on Prime and Maximal Spectra In what follows, we present the Stone topologies on the prime and maximal spectra of ideals and filters of a bounded distributive lattice and those of congruences of an algebra with the greatest congruence compact from a congruence-modular variety; in particular, the following hold for algebras from semi-degenerate congruencemodular varieties. The results in this section are either previously known or very easy to derive from previously known results; see, for instance, [30]. Let L be a bounded distributive lattice. For any I ∈ Id(L) and any a ∈ L, we shall denote by • for any family (J i ) i∈I ⊆ Id(L), V Id,L ( i∈I J i ) = i∈I V Id,L (J i ) and D Id,L ( i∈I J i ) = i∈I D Id,L (J i ); • thus, for any a, b ∈ L, V Id,L (a ∧ b) = V Id,L (a) ∪ V Id,L (b), D Id,L (a ∧ b) = D Id,L (a) ∩ D Id,L (b), V Id,L (a ∨ b) = V Id,L (a) ∩ V Id,L (b) and D Id,L (a ∨ b) = D Id,L (a) ∪ D Id,L (b); • if L is a complete lattice, then, for any family ( Throughout the rest of this section, we shall assume that [·, ·] A is commutative and distributive w.r.t. arbitrary joins. For each θ ∈ Con(A), we shall denote by a i ) i∈I ⊆ L, V Id,L ( i∈I a i ) = i∈I V Id,L (a i ) and D Id,L ( i∈I J i ) = i∈I D Id,L (J i ); • if I ∈ Id(L),V A (θ) = Spec(A) ∩ [θ) = {φ ∈ Spec(A) | θ ⊆ φ} and by D A (θ) = Spec(A) \ V A (θ) = {ψ ∈ Spec(A) | θ ψ}. We shall also denote, for any a, b ∈ A, by V A (a, b) = V A (Cg A (a, b)) = {φ ∈ Spec(A) | (a, b) ∈ φ} and by D A (a, b) = D A (Cg A (a, b)) = {ψ ∈ Spec(A) | (a, b) / ∈ ψ}. The proof of the following result is straightforward. Proposition 4.2. [1] (Spec(A), {D A (θ) | θ ∈ Con(A)}) is a topological space, having {D A (a, b) | a, b ∈ A} as a basis and in which, for all α, β ∈ Con(A) and any family (α i ) i∈I ⊆ Con(A), the following hold: (i) D A (∆ A ) = ∅ and D A (∇ A ) = Spec(A); V A (∆ A ) = Spec(A) and V A (∇ A ) = ∅; (ii) D A ([α, β] A ) = D A (α ∩ β) = D A (α) ∩ D A (β) =; V A ([α, β] A ) = V A (α ∩ β) = V A (α) ∪ V A (β); (iii) D A ( i∈I α i ) = i∈I D A (α i ); V A ( i∈I α i ) = i∈I V A (α i ). {D A (θ) | θ ∈ Con(A)} is called the Stone topology on Spec(A). Obviously, its family of closed sets is {V A (θ) | θ ∈ Con(A)}, and {V A (a, b) | a, b ∈ A} is a basis of closed sets for this topology. The Stone topology on Spec(A) induces the Stone topology on Max(A), namely {D A (θ) ∩ Max(A) | θ ∈ Con(A)}. Remark 4.3. Let α, β ∈ Con(A). Then, clearly: • V A (α) ⊆ V A (β) iff Spec(A) \ D A (α) ⊆ Spec(A) \ D A (β) iff D A (β) ⊆ D A (α); • if α ⊆ β, then V A (β) ⊆ V A (α) and D A (α) ⊆ D A (β). Proposition 4.4. If C is congruence-modular and semi-degenerate, then, for any α ∈ Con(A): D A (α) = Spec(A) iff V A (α) = ∅ iff α = ∇ A . Proof. D A (α) = Spec(A) iff Spec(A) \ D A (α) = ∅ iff V A (α) = ∅. Since Spec(A) ⊆ Con(A) \ {∇ A }, we have V A (∇ A ) = ∅, which was also part of Proposition 4.2. If α = ∇ A , then, according to Lemma 3.12, there exists a φ ∈ Spec(A) such that α ⊆ φ, that is V A (α) = ∅. Remark 4.5. Recall that, if f is surjective, then the map α → f (α) is a lattice isomorphism from [Ker(f )) to Con(B). Now assume that C is congruence-modular. Then this map is an order isomorphism from Max(A) ∩ [Ker(f )) to Max(B). Furthermore, this map is an order isomorphism from Spec(A) ∩ [Ker(f )) to Spec(B) (see also [1], [25], [47]). Hence, if Ker(f ) ⊆ α ∈ Con(A), then V B (f (α)) = f (V A (α)) and [f (α)) ∩ Max(B) = f ([α) ∩ Max(A)). Therefore, for all θ ∈ Con(A), the map α → α/θ is a lattice isomorphism from [θ) to Con(A/θ), an order isomorphism from Max(A) ∩ [θ) to Max(A/θ) and an order isomorphism from Spec (A) ∩ [θ) to Spec(A/θ); hence, if θ ⊆ α ∈ Con(A), then V A/θ (α/θ) = {ψ/θ | ψ ∈ V A (α)} and [α/θ) ∩ Max(A/θ) = {ψ/θ | ψ ∈ [α) ∩ Max(A)}. The Construction of the Reticulation of a Universal Algebra and Related Results Throughout this section, we shall assume that [·, ·] A is commutative and distributive w.r.t. arbitrary joins, and that ∇ A ∈ K(A). For every θ ∈ Con(A), we shall denote by ρ A (θ) the radical of θ, that is the intersection of the prime congruences of A which include θ: ρ A (θ) = {φ ∈ Spec(A) | θ ⊆ φ} = φ∈VA(θ) φ. Remark 5.1. Let α, β ∈ Con(A) and φ ∈ Spec(A). Then, clearly: (i) V A (∇ A ) = ∅, and thus ρ A (∇ A ) = ∇ A ; (ii) ρ A (φ) = φ; moreover, ρ A (α) = α iff α is the intersection of a family of prime congruences of A; (iii) if α ⊆ β, then V A (α) ⊇ V A (β), hence ρ A (α) ⊆ ρ A (β); (iv) if α ⊆ φ, then ρ A (α) ⊆ φ, since φ ∈ V A (α). Following [1], for any α, β ∈ Con(A) and every n ∈ N * , we denote by [α, β] 1 A = [α, β] A and [α, β] n+1 A = [[α, β] n A , [α, β] n A ] A , and by (α, β] 1 A = [α, β] A and (α, β] n+1 A = (α, (α, β] n A ] A . Lemma 5.2. For all n ∈ N * , any α, β ∈ Con(A) and any family (α i ) i∈I ∈ Con(A): (i) α ⊆ ρ A (α); (ii) V A (α) = V A (ρ A (α)); (iii) V A ( i∈I α i ) = V A ( i∈I ρ A (α i )); (iv) V A ([α, β] n A ) = V A ([α, β] A ) = V A (α ∩ β) = V A (α) ∪ V A (β); (v) V A ([α, α] n A ) = V A ([α, α] A ) = V A (α). Proof. (i) Trivial. (ii) By (i) and Remark 4.3, V A (ρ A (α)) ⊆ V A (α). If φ ∈ V A (α), then φ ∈ V A (ρ A (α)), according to Remark 5.1, (iv), thus V A (α) ⊆ V A (ρ A (α)). Hence V A (α) = V A (ρ A (α)). (iii) By (ii) and Proposition 4.2, (iii), V A ( i∈I α i ) = i∈I V A (α i ) = i∈I V A (ρ A (α i )) = V A ( i∈I ρ A (α i )). (iv) By Proposition 4.2, (ii). V A ([α, β] A ) = V A (α ∩ β) = V A (α) ∪ V A (β) Now we prove that V A ([α, β] n A ) = V A (α) ∪ V A (β) by induction on n ∈ N * . V A ([α, β] 1 A ) = V A ([α, β] A ) = V A (α) ∪ V A (β). Now let n ∈ N * such that V A ([θ, ζ] n A ) = V A (θ) ∪ V A (ζ) for all θ, ζ ∈ Con(A). Then V A ([α, β] n+1 A ) = V A ([[α, β] n A , [α, β] n A ] A ) = V A ([α, β] n A ) ∪ V A ([α, β] n A ) = V A ([α, β] n A ) = V A (α) ∪ V A (β). (v) By (iv). Proposition 5.3. For all α, β, θ ∈ Con(A), the following hold: (i) ρ A (α) ⊆ ρ A (β) iff α ⊆ ρ A (β) iff V A (α) ⊇ V A (β); (ii) ρ A (α) = ρ A (β) iff V A (α) = V A (β); (iii) if θ ⊆ α, then ρ A/θ (α/θ) = ρ A (α)/θ; (iv) ρ A/θ (∆ A/θ ) = ρ A (θ)/θ; (v) ρ A/θ ((α ∨ θ)/θ) = ρ A (α ∨ θ)/θ. Proof. (i) Clearly, if V A (α) ⊇ V A (β), then ρ A (α) ⊆ ρ A (β). If ρ A (α) ⊆ ρ A (β), then, since α ⊆ ρ A (α), it follows that α ⊆ ρ A (β). Finally, if α ⊆ ρ A (β), then V A (α) ⊇ V A (ρ A (β)) = V A (β), by Remark 5.1, (iii), and Lemma 5.2, (ii). (ii) By (i). (iii) If θ ⊆ α, then we may write: ρ A/θ (α/θ) = ψ∈V A/θ (α/θ) ψ = φ∈VA(α) φ/θ = ( φ∈VA(α) φ)/θ = ρ A (α)/θ. (iv) By (iii), ρ A/θ (∆ A/θ ) = ρ A/θ (θ/θ) = ρ A (θ)/θ. (v) By (iii). Proposition 5.4. For any n ∈ N * , any α ∈ Con(A) and any family (α i ) i∈I ⊆ Con(A): (i) if C is congruence-modular and semi-degenerate, then: ρ A (α) = ∇ A iff α = ∇ A ; (ii) ρ A ([α, β] n A ) = ρ A ([α, β] A ) = ρ A (α ∩ β) = ρ A (α) ∩ ρ A (β); (iii) ρ A ([α, α] n A ) = ρ A ([α, α] A ) = ρ A (α); (iv) ρ A (ρ A (α)) = ρ A (α); (v) ρ A ( i∈I ρ A (α i )) = ρ A ( i∈I α i ); (vi) if C is congruence-modular and semi-degenerate, then: i∈I ρ A (α i ) = ∇ A iff i∈I α i = ∇ A . Proof. (i) By Lemma 5.2, (i), ∇ A ⊆ ρ A (∇ A ), thus ρ A (∇ A ) = ∇ A . If α = ∇ A , then there exists φ ∈ V A (α), thus ρ A (α) ⊆ φ ∇ A . (ii) By Remark 5.1, (iii), Lemma 5.2, (iv), and Proposition 4.2, (ii), ρ A ([α, β] n A ) = ρ A ([α, β] A ) = ρ A (α ∩ β) = φ∈VA(α∩β) φ = φ∈VA(α)∪VA(β) = φ∈VA(α) φ ∩ φ∈VA(β) φ = ρ A (α) ∩ ρ A (β). (iii) By (ii). (iv) By Remark 5.1, (iii), and Lemma 5.2, (ii). (v) By Remark 5.1, (iii), and Lemma 5. 2, (iii), ρ A ( i∈I ρ A (α i )) = ρ A ( i∈I α i ). (vi) By (v) and (i), i∈I ρ A (α i ) = ∇ A iff ρ A ( i∈I ρ A (α i )) = ∇ A iff ρ A ( i∈I α i ) = ∇ A iff i∈I α i = ∇ A . The radical congruences of A are the congruences α of A such that α = ρ A (α). Let us denote by RCon(A) the set of the radical congruences of A. Proof. By [1, Lemma 1.6], the radical congruences of A coincide to its semiprime congruences, that is the congruences θ of A such that, for all α ∈ Con(A), [α, α] A ⊆ θ implies α ⊆ θ. Clearly, if [·, ·] A = ∩, then every congruence of A is semiprime, and thus radical. Most of the previous results on the radicals of congruences are known, but, for the sake of completeness, we have provided short proofs for them. For any α, β ∈ Con(A), let us denote by α • ∨ β = ρ A (α ∨ β). For any family (α i ) i∈I ⊆ Con(A), we shall denote by • i∈I α i = ρ A ( i∈I α i ). Proposition 5.8. (RCon(A), • ∨, ∩, ρ A (∆ A ), ρ A (∇ A ) = ∇ A ) is a bounded lattice, orderred by set inclusion. Moreover, it is a complete lattice, in which the arbitrary join is given by the • defined above. Proof. Of course, ∩ is idempotent, commutative and associative, and, clearly, (iv) and • ∨ is commutative. Now let α, β, γ ∈ Con(A) and R = {ρ A (α), ρ A (β), ρ A (γ)} ⊆ RCon(A); we shall use Proposition 5.4, (ii),(v): α, β, γ ∈ Con(A), ρ A (α) • ∨ ρ A (α) = ρ A (ρ A (α) ∨ ρ A (α)) = ρ A (α ∨ α) = ρ A (α), so • ∨ is idempotent; ρ A (α) • ∨ (ρ A (β) • ∨ ρ A (γ)) = ρ A (α) • ∨ ρ A (ρ A (β) ∨ ρ A (γ)) = ρ A (α) • ∨ ρ A (β ∨ γ) = ρ A (ρ A (α) ∨ ρ A (ρ A (β ∨ γ))) = ρ A (ρ A (α) ∨ ρ A (β ∨ γ)) = ρ A (α ∨ (β ∨ γ)) = ρ A (α ∨ β ∨ γ) = ρ A (ρ A (α) ∨ ρ A (β) ∨ ρ A (γ)) = • θ∈R θ, thus, by the commutativity of • ∨, we also have (ρ A (α) • ∨ ρ A (β)) • ∨ ρ A (γ) = ρ A (γ) • ∨ (ρ A (α) • ∨ ρ A (β)) = • θ∈R θ, hence ρ A (α) • ∨ (ρ A (β) • ∨ ρ A (γ)) = (ρ A (α) • ∨ ρ A (β)) • ∨ ρ A (γ), so • ∨ is associative; ρ A (α) • ∨ (ρ A (α) ∩ ρ A (β)) = ρ A (α) • ∨ ρ A (α ∩ β) = ρ A (ρ A (α) ∨ ρ A (α ∩ β)) = ρ A (α ∨ (α ∩ β)) = ρ A (α) and ρ A (α) ∩ (ρ A (α) • ∨ ρ A (β)) = ρ A (α ∩ (ρ A (α) • ∨ ρ A (β))) = ρ A (α∩ρ A (ρ A (α)∨ρ A (β))) = ρ A (α∩ρ A (α∨β)) = ρ A (ρ A (α∩ρ A (α∨β))) = ρ A (ρ A (α))∩ρ A (ρ A (α∨β))) = ρ A (α) ∩ ρ A (α ∨ β)) = ρ A (α ∩ (α ∨ β)) = ρ A (α) , so the absorption laws hold. Of course, for all θ, ζ ∈ RCon(A), θ ∩ ζ = θ iff θ ⊆ ζ. Therefore (RCon(A), • ∨, ∩) is a lattice, orderred by set inclusion. From Remark 5.1, (iii) and (i), we obtain that this lattice has ρ A (∆ A ) as first element and ρ A (∇ A ) = ∇ A as last element. Now let us consider a family ( α i ) i∈I ⊆ Con(A), M = {ρ A (α i ) | i ∈ I} ⊆ RCon(A) and let us denote by θ = • i∈I ρ A (α i ) = ρ A ( i∈I ρ A (α i )) = ρ A ( i∈I α i ), by Proposition 5.4, (v). Then θ ∈ RCon(A) and ρ A (α i ) ⊆ θ for all i ∈ I. Now, if ζ ∈ RCon(A) and ρ A (α i ) ⊆ ζ for all i ∈ I, then i∈I ρ A (α i ) ⊆ ζ, so, by Remark 5.1, (iii), and Proposition 5.4, (v), ζ = ρ A (ζ) ⊇ ρ A ( i∈I ρ A (α i )) = • i∈I ρ A (α i ) = θ. Therefore θ = sup(M ) in the bounded lattice RCon (A), hence this lattice is complete. Let us define a binary relation ≡ A on Con(A) by: α ≡ A β iff ρ A (α) = ρ A (β), for any α, β ∈ Con(A). ≡ A ∩(K(A)) 2 shall also be denoted by ≡ A . Remark 5.9. Clearly, ≡ A is an equivalence on Con(A), thus also on K(A). On RCon(A), ≡ A coincides to the equality, that is to ∆ RCon(A) , because, for any α, β ∈ Con(A), ρ A (α) ≡ A ρ A (β) iff ρ A (ρ A (α)) = ρ A (ρ A (β)) iff ρ A (α) = ρ A (β). So, trivially, ≡ A is a congruence of the lattice RCon(A). On Con(A), ≡ A preserves the commutator, ∩, ∨ and • ∨, even and • over arbitrary families of congruences, in particular it is a congruence of the lattice Con (A). Indeed, if α, α ′ , β, β ′ ∈ Con(A) such that α ≡ A α ′ and β ≡ A β ′ , that is ρ A (α) = ρ A (α ′ ) and ρ A (β) = ρ A (β ′ ), then, by Proposition 5.4, (ii), ρ A ([α, β] A ) = ρ A (α ∩ β) = ρ A (α) ∩ ρ A (β) = ρ A (α ′ ) ∩ ρ A (β ′ ) = ρ A (α ′ ∩ β ′ ) = ρ A ([α ′ , β ′ ] A ), thus [α, β] A ≡ A [α ′ , β ′ ] A ≡ A α ∩ β ≡ A α ′ ∩ β ′ . Now, if (α i ) i∈I ⊆ Con(A) and (α ′ i ) i∈I ⊆ Con(A) such that, for all i ∈ I, α i ≡ A α ′ i , that is ρ A (α i ) = ρ A (α ′ i ), then, by Proposition 5.4, (iv) and (v), ρ A ( • i∈I α i ) = ρ A (ρ A ( i∈I α i )) = ρ A ( i∈I α i ) = ρ A ( i∈I ρ A (α i )) = ρ A ( i∈I ρ A (α ′ i )) = ρ A ( i∈I α ′ i ) = ρ A (ρ A ( i∈I α ′ i )) = ρ A ( • i∈I α ′ i ), hence • i∈I α i ≡ A • i∈I α ′ i ≡ A i∈I α i ≡ A i∈I α ′ i . Moreover, as shown by Proposition 5.4, (ii), (iv) and (v), just as in the calculations above, for all α, β ∈ Con(A) and all (α i ) i∈I ⊆ Con(A), [α, β] A ≡ A α ∩ β and • i∈I α i ≡ A i∈I α i . Note, also, that, for all α ∈ Con(A), α ≡ A ρ A (α), by Proposition 5.4, (iv). For all α ∈ Con(A), let us denote by α the equivalence class of α with respect to ≡ A , and let L( A) = K(A)/ ≡ A = { θ | θ ∈ K(A)}. Let λ A : Con(A) → Con(A)/ ≡ A be the canonical surjection: λ A (θ) = θ for all θ ∈ Con(A) ; we denote in the same way its restriction to K(A), with its co-domain restricted to L(A), that is the canonical surjection λ A : K(A) → L(A). Let us define the following operations on Con(A), where the second equalities follow from Remark 5.9, as does the fact that these operations are well defined: • for all α, β ∈ Con(A), α ∨ β = α ∨ β = α • ∨ β and α ∧ β = α ∩ β = [α, β] A ; • 0 = ∆ A = ρ A (∆ A ) and 1 = ∇ A = ρ A (∇ A ). Remark 5.10. By Proposition 5.4, (i), if C is congruence-modular and semi-degenerate, then, for any α ∈ Con(A), α = 1 iff α = ∇ A . Lemma 5.11. (Con(A)/ ≡ A , ∨, ∧, 0, 1) is a bounded distributive lattice and λ A : Con(A) → Con(A)/ ≡ A is a bounded lattice morphism. Moreover, Con(A)/ ≡ A is a complete lattice, in which i∈I α i = i∈I α i and i∈I α i = i∈I α i for any family (α i ) i∈I ⊆ Con(A), and the meet is completely distributive with respect to the join, thus Con(A)/ ≡ A is a frame. Proof. By Remark 5.9, ≡ A is a congruence of the bounded lattice Con(A), hence (Con(A)/ ≡ A , ∨, ∧, 0, 1) is a bounded lattice and the canonical surjection λ A : Con(A) → Con(A)/ ≡ A is a bounded lattice morphism, in particular it is order-preserving. It is straightforward, from the fact that the lattice Con(A) is complete and the surjectivity of the lattice morphism λ A , that the lattice Con(A)/ ≡ A is complete and its joins and meets of arbitrary families of elements have the form in the enunciation. By Proposition 3.6, for any families (α i ) i∈I and (β j ) j∈J of congruences of A, ( i∈I α i ) ∧ ( j∈J β j ) = (( i∈I α i ) ∩ ( j∈J β j )) = ([ i∈I α i , j∈J β j ] A ) = ( i∈I j∈J [α i , β j ] A ) = i∈I j∈J [α i , β j ] A = i∈I j∈J ( α i ∧ β j ) , that is the meet is completely distributive with respect to the join in Con(A)/ ≡ A , thus Con(A)/ ≡ A is a frame, in particular it is a bounded distributive lattice. We shall denote by ≤ the partial order of the lattice Con(A)/ ≡ A . Proposition 5.12. (RCon(A), • ∨, ∩, ρ A (∆ A ), ρ A (∇ A ) = ∇ A ) is a frame, isomorphic to Con(A)/ ≡ A . Proof. Let ϕ : Con(A)/ ≡ A → RCon(A), for all α ∈ Con(A), ϕ( α) = ρ A (α). If α, β ∈ Con(A), then the following equivalences hold: α = β iff α ≡ A β iff ρ A (α) = ρ A (β) iff ϕ( α) = ϕ( β) , hence ϕ is well defined and injective. By Remark 5.6, ϕ is surjective. By Proposition 5.4, (ii) and (v), for all α, β ∈ Con Proposition 5.4,(v), and Lemma 5.11 show that ϕ preserves arbitrary joins). Therefore ϕ is a lattice isomorphism, thus an order isomorphism, hence it preserves arbitrary joins and meets. From this and Lemma 5.11 we obtain that RCon(A) is a frame and ϕ is a frame isomorphism. (A), ϕ( α ∧ β) = ϕ( α ∩ β) = ρ A (α ∩ β) = ρ A (α) ∩ ρ A (β) and ϕ( α ∨ β) = ϕ( α ∨ β) = ρ A (α ∨ β) = ρ A (α) ∨ ρ A (β) (actually, Throughout the rest of this section, we shall assume that K(A) is closed with respect to the commutator. For any θ ∈ Con(A) and any I ∈ Id(L(A)), we shall denote by: • θ * = { α | α ∈ K(A), α ⊆ θ} = λ A (K(A) ∩ (θ]) ⊆ L(A), where (θ] = (θ] Con(A) ∈ PId(Con(A)); • I * = {α ∈ K(A) | α ∈ I} = α∈λ −1 A (I) α ∈ Con(A); note that λ −1 A (I) is non-empty, because ∆ A ∈ K(A) and ∆ A = ρ A (∆ A ) = 0 ∈ I. Lemma 5.14. For all θ ∈ Con(A): • θ * ⊆ ( θ ] Con(A)/ ≡ A ∩ L(A) and θ * ∈ Id(L(A)); • if θ ∈ K(A), then θ * = ( θ ] Con(A)/ ≡ A ∩ L(A) = ( θ ] L(A) ∈ PId(L(A)). Proof. Let θ ∈ Con (A), and, in this proof, let us denote by θ = ( θ ] Con(A)/ ≡ A and, in the case when θ ∈ K(A), by ( θ ] = ( θ ] L(A) . θ * = { α | α ∈ (θ] ∩ K(A)}. For all α ∈ (θ] ∩ K(A), we have α ∈ L(A) and α ⊆ θ, thus α ≤ θ in Con(A)/ ≡ A , hence α ∈ θ ∩ L(A), therefore θ * ⊆ θ ∩ L(A). ∆ A ∈ K(A) and ∆ A ⊆ θ, thus ∆ A ∈ θ * , so θ * is non-empty. Since K(A) is closed w.r.t. [·, ·] A , α ∨ β, [α, β] A ∈ K(A) for any α, β ∈ K(A). Let x, y ∈ θ * , which means that x = α and y = β for some α, β ∈ K(A) ∩ (θ]. Then α ∨ β ∈ K(A) ∩ (θ], thus x ∨ y = α ∨ β = α ∨ β ∈ θ * . Now let x ∈ θ * and y ∈ L(A) such that x ≥ y, so that y = x ∧ y. Then x = α for some α ∈ K(A) ∩ (θ] and y = β for some β ∈ K(A). Thus [α, β] A ∈ K(A) and [α, β] A ⊆ α ∩ β ⊆ α ⊆ θ, hence [α, β] A ∈ K(A) ∩ (θ], therefore y = x ∧ y = α ∧ β = [α, β] A ∈ θ * . Hence θ * ∈ Id(L(A)). Now assume that θ ∈ K(A), so that θ ∈ L(A). By the above, θ * ⊆ θ ∩ L(A) = ( θ ]. Let x ∈ ( θ ], so that there exists an α ∈ K(A) with α = x ≤ θ, thus [α, θ] A = α ∩ θ = α = x. But [α, θ] A ∈ K(A) ∩ (θ], so x = [α, θ] A ∈ θ * . Therefore we also have ( θ ] ⊆ θ * , hence θ * = ( θ ] ∈ PId(L(A)). By the above, we have two functions: • θ ∈ Con(A) → θ * ∈ Id(L(A)); • I ∈ Id(L(A)) → I * ∈ Con(A). Lemma 5.15. The two functions above are order-preserving. Proof. For any θ, ζ ∈ Con(A) such that θ ⊆ ζ, we have (θ] ⊆ (ζ], hence θ * ⊆ ζ * . For any I, J ∈ Id(L(A)) such that I ⊆ J, we have λ −1 A (I) ⊆ λ −1 A (J), thus I * ⊆ J * . Lemma 5.16. Let α ∈ K(A) and I ∈ Id(L(A)). Then: α ⊆ I * iff α ∈ I. Proof. "⇐:" If α ∈ I, then α ∈ λ −1 A (I), thus α ⊆ I * . "⇒:" If α ⊆ I * = {β ∈ K(A) | β ∈ I}, then, since α ∈ K(A), it follows that there exist an n ∈ N * and β 1 , . . . , β n ∈ K(A) such that β 1 , . . . , β n ∈ I and α ⊆ Proof. Let φ ∈ Spec(A). Then φ ⊆ (φ * ) * by Lemma 5.17, (i). n i=1 β i , hence α ⊆ n i=1 β i = n i=1 β i ∈ I, thus α ∈ I.Now let β ∈ K(A) such that β ∈ φ * = { α | α ∈ K(A), α ⊆ φ}, which means that β = α for some α ∈ K(A) with α ⊆ φ. Since β = α, we have ρ A (β) = ρ A (α), while α ⊆ φ gives us ρ A (α) ⊆ ρ A (φ) = φ, where the last equality follows from the fact that φ ∈ Spec (A). Hence β ⊆ ρ A (β) ⊆ φ. Therefore (φ * ) * = {γ ∈ K(A) | γ ∈ φ * } ⊆ φ. Hence φ = (φ * ) * . Lemma 5.20. For any φ ∈ Spec(A), we have φ * ∈ Spec Id (L(A)). Proof. Let φ ∈ Spec(A). Then φ * ∈ Id(L(A)) = Id (K(A)/ ≡ A ). Let α, β ∈ K(A) such that [α, β] A = α∧ β ∈ φ * = { γ | γ ∈ K(A), γ ⊆ φ}. Then there exists a γ ∈ K(A) such that γ ⊆ φ and γ = [α, β] A , thus ρ A (γ) = ρ A ([α, β] A ) and ρ A (γ) ⊆ ρ A (φ) = φ since φ ∈ Spec(A). Hence [α, β] A ⊆ ρ A ([α, β] A ) ⊆ φ, hence α ⊆ φ or β ⊆ φ since φ ∈ Spec(A) . But this means that α ∈ φ * or β ∈ φ * . Therefore φ * ∈ Spec Id (L(A)). Lemma 5.21. For any P ∈ Spec Id (L(A)), we have P * ∈ Spec(A). Proof. Let P ∈ Spec Id (L(A)). Then P * ∈ Con(A). Let α, β ∈ PCon(A) such that [α, β] A ⊆ P * . Then α, β ∈ K(A), so that [α, β] A ∈ K(A), and [α, β] A ⊆ {γ ∈ K(A) | γ ∈ P }, hence there exist an n ∈ N * and γ 1 , . . . , γ n ∈ K(A) such that γ 1 , . . . , γ n ∈ P and [α, β] A ⊆ n i=1 γ i . But then n i=1 γ i = n i=1 γ i ∈ P , hence α ∧ β = [α, β] A ∈ P , thus α ∈ P or β ∈ P since P ∈ Spec Id (L(A)). By Lemma 5.16, it follows that α ⊆ P * or β ⊆ P * . Therefore P * ∈ Spec(A). By Lemmas 5.20 and 5.21, we have these restrictions of the functions defined above: Let θ ∈ Con(A) and φ ∈ V A (θ), that is φ ∈ Spec(A) and θ ⊆ φ. Then, by Lemmas 5.21 and 5.15, φ * ∈ Spec Id (L(A)) and θ * ⊆ φ * , so φ * ∈ V Id,L(A) (θ * ), and we have u(φ) = φ * . Hence u(V A (θ)) ⊆ V Id,L(A) (θ * ). Now let P ∈ V Id,L(A) (θ * ), that is P ∈ Spec Id (L(A)) and θ * ⊆ P . Then, by Lemma 5.17, (i), and Lemmas 5.15 and 5.21, θ ⊆ (θ * ) * ⊆ P * ∈ Spec(A), thus P * ∈ V A (θ), and we have u(P * ) = u(v(P )) = P . Hence Proposition 5.24. V Id,L(A) (θ * ) ⊆ u(V A (θ)). Therefore u(V A (θ)) = V Id,L(A) (θ * ), [5], [26] If L and M are bounded distributive lattices whose prime spectra of ideals, endowed with the Stone topologies, are homeomorphic, then L and M are isomorphic. (θ) = {(a, b) ∈ A 2 | (∃ n ∈ N * ) ([Cg A (a, b), Cg A (a, b)] n A ⊆ θ)}, so ρ A (∆ A ) = {(a, b) ∈ A 2 | (∃ n ∈ N * ) ([Cg A (a, b), Cg A (a, b)] n A = ∆ A )}. Proposition 5.28. For any θ ∈ Con(A), (θ * ) * = ρ A (θ). Proof. For every β ∈ K(A) such that β ∈ θ * = { γ | γ ∈ K(A), γ ⊆ θ}, there exists an α ∈ K(A) such that α ⊆ θ and α = β, thus β ⊆ ρ A (β) = ρ A (α) ⊆ ρ A (θ). Therefore (θ * ) * = {γ ∈ K(A) | γ ∈ θ * } ⊆ ρ A (θ). Now let (a, b) ∈ ρ A (θ), so that, according to Proposition 5.27, Lemma 5.17, (i), and Lemma 5.16, for some n ∈ N * , (Cg A (a, b)), thus Cg A (a, b) = ([ Cg A (a, b), Cg A (a, b)] n A ) ∈ θ * , hence (a, b) ∈ Cg A (a, b) ⊆ (θ * ) * by Lemma 5.16. Therefore ρ A (θ) ⊆ (θ * ) * . Hence (θ * ) * = ρ A (θ). Corollary 5.30. The maps: [Cg A (a, b), Cg A (a, b)] n A ⊆ θ ⊆ (θ * ) * , hence ([Cg A (a, b), Cg A (a, b)] n A ) ∈ θ * . But ρ A ([Cg A (a, b), Cg A (a, b)] n A ) = ρ A • θ ∈ RCon(A) → θ * ∈ Id(L(A)), • I ∈ Id(L(A)) → I * ∈ RCon (A) are frame isomorphisms and inverses of each other. Proof. By Corollary 5.29, (ii), for all I ∈ Id(L(A)), we have I * ∈ RCon(A), hence the second map above is well defined. By Lemma 5.17, (ii), for all I ∈ Id(L(A)), (I * ) * = I. By Proposition 5.28, for all θ ∈ RCon(A), θ = ρ A (θ) = (θ * ) * . Hence these functions are inverses of each other, thus they are bijections. By Lemma 5.15, these maps are order-preserving, thus they are order isomorphisms, hence they preserve arbitrary joins and meets, therefore they are frame isomorphisms. Some Examples, Particular Cases and Preservation of Finite Direct Products Throughout this section, we shall assume that [·, ·] A is commutative and distributive w.r.t. arbitrary joins and ∇ A ∈ K(A). These hypotheses are sufficient for the following results we cite from other works to hold. We shall denote by HSP(A) the variety generated by A. In the following examples, we determine the prime spectra by using [1, Proposition 1.2], which says that, for each proper congruence φ of A: φ is prime iff φ is meetirreducible and semiprime. So, if we know that HSP(A) is congruence-modular, then we only have to calculate [α, α] A for every α ∈ Con(A). The complete tables of the commutators for the following algebras show that their commutators are commutative and distributive w.r.t. the join. Of course, since each of the algebras M from the following examples is finite, we have ∇ M ∈ K(M ). We have used the method in [40] to calculate the commutators, excepting those in groups, where we have used the commutators on normal subgroups; recall that the variety of groups is congruence-modular [39]. Following [1], we say that A is: Abelian iff [∇ A , ∇ A ] A = ∆ A ; solvable iff [∇ A , ∇ A ] n A = ∆ A for some n ∈ N * ; nilpotent iff (∇ A , ∇ A ] n A = ∆ A for some n ∈ N * . For any n ∈ N * , we shall denote by L n the n-element chain. By ⊕ we shall denote the ordinal sum of bounded lattices. [39], any Abelian group is an Abelian algebra, hence its reticulation is trivial. As a fact that may be interesting by its symmetry, if A is finite and its commutator equals the intersection, so that Con(A) is a finite distributive lattice, then L(Con(A)) = Con(Con(A)) = Con(L(A)). It might also be interesting to find weaker conditions on A under which L(Con(A)) ∼ = Con(L(A)). [22], [28], it follows that L(A) = K(A) ∼ = PFilt (A), which is the dual of the reticulation of a residuated lattice obtained in [41], [42], [43], where the reticulation has the prime spectrum of filters homeomorphic to the prime spectrum of filters, thus to that of congruences of A by the above, so this duality to the construction of L(A) from Section 5 was to be expected. Remark 6.5. If A is a commutative unitary ring and Id(A) is its lattice of ideals, then it is well known that Id(A) ∼ = Con (A). If, for all I ∈ Id(A), we denote by √ I the intersection of the prime filters of A which include I, then [7, Lemma, p. 1861] shows that, for any J ∈ Id(A), there exists a finitely generated ideal K of A such that √ J = √ K. From this, it immediately follows that the lattice L(A) is isomorphic to the reticulation of A constructed in [7]. Remark 6.6. Let n, k ∈ N * and assume that C is congruence-modular, S is a subalgebra of A, α, β ∈ Con(A), If A is simple, that is Con(A) = {∆ A , ∇ A } ⊆ K(A) ⊆ Con(A), so that K(A) = Con(A) = {∆ A , ∇ A },M 1 , . . . , M n are algebras from C, M = n i=1 M i and, for all i ∈ 1, n, α i , β i ∈ Con(M i ). From Lemma 3.8, it is immediate that [α ∩ S 2 , β ∩ S 2 ] k S ⊆ [α, β] k A ∩ S 2 and (α ∩ S 2 , β ∩ S 2 ] k S ⊆ (α, β] k A ∩ S 2 . Hence, if A is Abelian or solvable or nilpotent, then S is Abelian or solvable or nilpotent, respectively. From Proposition 3.9, it is immediate that [ n i=1 α i , n i=1 β i ] k M = n i=1 [α i , β i ] k Mi and ( n i=1 α i , n i=1 β i ] k M = n i=1 (α i , β i ] k Mi . From this, it is easy to prove that: M is Abelian or solvable or nilpotent iff M 1 , . . . , M n are Abelian or solvable or nilpotent, respectively. The following are the subgroups of C 8 , respectively S 3 , all of which are normal, and the proper ones are cyclic, thus Abelian: 1 , −1 , i , j , k and C 8 , respectively 1 , t , u , v , c and S 3 , so C 8 and S 3 have the following congruence lattices and commutators, which suffice to conclude that Spec(C 8 ) = Spec(S 3 ) = ∅, since we are in a congruence-modular variety, and thus L(C 8 ) ∼ = L(S 3 ) ∼ = L 1 , by Remark 6.1: ∆ C8 =≡ 1 ∇ C8 =≡ C8 r r r r r r ❅ ❅ ❅ ❅ ❅ ❅ ≡ i ≡ j ≡ k ≡ −1 ∆ S3 =≡ 1 ∇ S3 =≡ S3 r r r r r r ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ ◗ ≡ t ≡ u ≡ v ≡ c θ [θ, θ] C8 ∆ C8 ∆ C8 ≡ −1 ∆ C8 ≡ i ≡ −1 ≡ j ≡ −1 ≡ k ≡ −1 ∇ C8 ≡ −1 θ [θ, θ] S3 ∆ S3 ∆ S3 ≡ t ∆ S3 ≡ u ∆ S3 ≡ v ∆ S3 ≡ c ∆ S3 ∇ S3 ∇ S3 Notice, also, that C 8 is solvable, as we have announced, thus, according to Remark 6.6, so is any finite direct product whose factors are subgroups of C 8 , which, of course, is Abelian if all those subgroups are proper. a 0 a b c d b a 0 c b b x b c 0 a a y c b a 0 Proposition 6.12 (the reticulation preserves finite direct products without skew congruences). Let M be an algebra from C such that the direct product A × M has no skew congruences. Then: 0 z d b a 0 0 ∆ U ∇ U r r r r r r ❅ ❅ ❅ ❅ α β γ δ [·, ·] U ∆ U α β γ δ ∇ U ∆ U ∆ U ∆ U ∆ U ∆ U ∆ U ∆ U α ∆ U δ δ δ δ α β ∆ U δ δ δ δ β γ ∆ U δ δ γ δ γ δ ∆ U δ δ δ ∆ U δ ∇ U ∆ U α β γ δ ∇ U U is not[·, ·] N ∆ N ψ ψ 1 φ ξ ξ 1 χ χ 1 ∇ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ψ ∆ N ψ 1 ψ 1 ∆ N ∆ N ∆ N ψ 1 ψ 1 ψ 1 ψ 1 ∆ N ψ 1 ψ 1 ∆ N ∆ N ∆ N ψ 1 ψ 1 ψ 1 φ ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ∆ N ξ ∆ N ∆ N ∆ N ∆ N ξ 1 ξ 1 ξ 1 ξ 1 ξ 1 ξ 1 ∆ N ∆ N ∆ N ∆ N ξ 1 ξ 1 ξ 1 ξ 1 ξ 1 χ ∆ N ψ 1 ψ 1 ∆ N ξ 1 ξ 1 χ 1 χ 1 χ 1 χ 1 ∆ N ψ 1 ψ 1 ∆ N ξ 1 ξ 1 χ 1 χ 1 χ 1 ∇ N ∆ N ψ 1 ψ 1 ∆ N ξ 1 ξ 1 χ 1 χ 1 χ 1 θ ρ N (θ) ∆ N φ ψ ψ ψ 1 ψ φ φ ξ ξ ξ 1 ξ χ ∇ N χ 1 ∇ N ∇ N ∇ N 0 r r r r ❅ ❅ ❅ ❅ ξ ψ 1 L(N ) (i) for all ∅ = X ⊆ A 2 and all ∅ = Y ⊆ M 2 , Cg A×M (X ×Y ) = Cg A (X)×Cg M (Y ), and the map (α, µ) → α×µ is a lattice isomorphism from Con(A) × Con(M ) to Con(A × M ); (ii) PCon(A × M ) = {α × µ | α ∈ PCon(A), µ ∈ PCon(M )} and K(A × M ) = {α × µ | α ∈ K(A), µ ∈ K(M )}. If C is congruence-modular and ∇ M ∈ K(M ), then: • for all α ∈ Con(A) and all µ ∈ Con(M ), ρ A×M (α × µ) = ρ A (α) × ρ M (µ); • ≡ A×M =≡ A × ≡ M and L(A × M ) ∼ = L(A) × L(M ). Proof. A × M has no skew congruences, that is Con (A × M ) = {α × µ | α ∈ Con(A), µ ∈ Con(M )}. (i) By Remark 6.11, Cg A×M (X × Y ) = {θ ∈ Con(A × M ) | X × Y ⊆ θ} = {α × µ | α ∈ Con(A), µ ∈ Con(M ), X × Y ⊆ α × µ} = {α × µ | α ∈ Con(A), µ ∈ Con(M ), X ⊆ α, Y ⊆ µ} = ( {α ∈ Con(A) | X ⊆ α}) × ( {µ ∈ Con(M ) | Y ⊆ µ}) = Cg A (X) × Cg M (Y ). This also shows that the map (α, µ) → α× µ is a lattice isomorphism from Con(A)× Con(M ) to Con(A× M ), because it is clearly injective, it is surjective by the above, it preserves the intersection by Remark 6.11 and, for all α, β ∈ Con(A) and all µ, ν ∈ Con(M ), (α×µ)∨(β ×ν) = Cg A×M ((α×µ)∪(β ×ν)) ⊆ Cg A×M ((α∪β)×(µ∪ν)) = ({(a 1 , u 1 ), . . . , (a n , u n )}) | n ∈ N * , a 1 , . . . , a n ∈ A, u 1 , . . . , Cg A (α ∪ β) × Cg M (µ ∪ ν) = (α ∨ β) × (µ ∨ ν), since, clearly, (α × µ) ∪ (β × ν) ⊆ (α ∪ β) × (µ ∪ ν), but, also, (α × µ) ∨ (β × ν) ∈ Con(A × M ), thus (α × µ) ∪ (β × ν) ⊆ (α × µ) ∨ (β × ν) = γ ∨ σ for some γ ∈ Con(A) and σ ∈ Con(M ), so α × µ ⊆ γ ∨ σ and β × ν ⊆ γ ∨ σ, hence α ⊆ γ, β ⊆ γ, µ ⊆ σ and ν ⊆ σ, so (α ∨ β) ⊆ γ and (µ ∨ ν) ⊆ σ, hence (α ∨ β) × (µ ∨ ν) ⊆ γ ∨ σ = (α × µ) ∨ (β × ν) ⊆ (α ∨ β) × (µ ∨ ν), therefore (α ∨ β) × (µ ∨ ν) = (α × µ) ∨ (β × ν). (ii) By (i), for all a, b ∈ A and all u, v ∈ M , Cg A×M ((a, u), (b, v)) = Cg A (a, b) × Cg M (u, v),u n ∈ M } = { n i=1 Cg A×M (a i , u i ) | n ∈ N * , a 1 , . . . , a n ∈ A, u 1 , . . . , u n ∈ M } = { n i=1 (Cg A (a i ) × Cg M (u i )) | n ∈ N * , a 1 , . . . , a n ∈ A, u 1 , . . . , u n ∈ M } = {( n i=1 Cg A (a i )) × ( n i=1 Cg M (u i )) | n ∈ N * , a 1 , . . . , a n ∈ A, u 1 , . . . , u n ∈ M } = {Cg A (a 1 , . . . , a n ) × Cg M (u 1 , . . . , u n ) | n ∈ N * , a 1 , . . . , a n ∈ A, u 1 , . . . , u n ∈ M } = {α × µ | α ∈ K(A), µ ∈ K(M )}, since the above also hold if some of the elements a 1 , . . . , a n or u 1 , . . . , u n coincide. Now assume that C is congruence-modular and ∇ M ∈ K(M ). Then, by Remark 6.11, for any α ∈ Con(A) and any µ ∈ Con(M ), ρ A×M (α×µ) = {χ ∈ Spec(A×M ) | α×µ ⊆ χ} = {φ×∇ M | φ ∈ Spec(A), α×µ ⊆ φ×∇ M }∩ {∇ A × ψ | ψ ∈ Spec(M ), α × µ ⊆ ∇ A × ψ} = {φ × ∇ M | φ ∈ Spec(A), α ⊆ φ} ∩ {∇ A × ψ | ψ ∈ Spec(M ), µ ⊆ ψ} = ( {φ | φ ∈ Spec(A), α ⊆ φ}×∇ M )∩(∇ A × {ψ | ψ ∈ Spec(M ), µ ⊆ ψ}) = (ρ A (α)×∇ M )∩(∇ A ×ρ M (µ)) = (ρ A (α) ∩ ∇ A ) × (∇ M ∩ ρ M (µ)) = ρ A (α) × ρ M (µ). Hence, for all θ, ζ ∈ Con(A × M ), we have: θ = α × µ and ζ = β × ν for some α, β ∈ Con(A) and µ, ν ∈ Con(M ), and thus: θ ≡ A×M ζ iff ρ A×M (θ) = ρ A×M (ζ) iff ρ A×M (α× µ) = ρ A×M (β × ν) iff ρ A (α)× ρ M (µ) = ρ A (β)× ρ M (ν) iff ρ A (α) = ρ A (β) = ρ A (β) and ρ M (µ) = ρ M (ν) iff α ≡ A β and µ ≡ M ν. Now let ϕ : L(A)×L(M ) → L(A×M ), for all α ∈ K(A) and all µ ∈ K(M ), ϕ( α, µ) = α × µ. By (ii), ϕ is well defined and surjective and fulfills: ν) and, similarly, ϕ(( α, µ) ∧ ( β, ν)) = ϕ( α, µ) ∧ ϕ( β, ν). By the form of ≡ A×M above, ϕ is injective. Hence ϕ is a lattice isomorphism. Example 6.13. Let V be the variety generated by the variety of lattices and that of groups. Then, according to [15, Theorem 1, Lemma 1, Proposition 3] and [35], V is congruence-modular and any algebra M from V is of the form M = (L, ∨, ∧) × (G, ·, ⋆), where (L, ∨, ∧) is a lattice, (G, ·) is a group and x ⋆ y = x −1 · y for all x, y ∈ G, and the direct product above has no skew congruences, thus, by Proposition 6.12, Con(M ) ∼ = Con(L) × Con(G) and L(M ) ∼ = L(L) × L(G), since each congruence of the group G also preserves the operation ⋆. Thus, for instance, in we consider the lattice P from Example 6.10 and the group (S 3 , •) from Example 6.7, and we denote σ ⋆ τ = σ −1 • τ for all σ, τ ∈ S 3 , and M = (P, ∨, ∧) × (S 3 , •, ⋆), then M is a finite algebra from V which is not congruence-distributive, because Con(M ) ∼ = Con(P) × Con(S 3 ) and Con(S 3 ) is not distributive, and ϕ(( α, µ) ∨ ( β, ν)) = ϕ( α ∨ β, µ ∨ ν) = ϕ( α ∨ β, µ ∨ ν) = ((α ∨ β) × (µ ∨ ν)) ∧ = ((α × µ) ∨ (β × ν)) ∧ = (α × µ) ∨ (β × ν) = ϕ( α, µ) ∨ ϕ( β,L(M ) ∼ = L(P) × L(S 3 ) ∼ = L(P) × L 1 ∼ = L(P) ∼ = Con(P) ∼ = L 2 ⊕ L 2 2 . Throughout this section, we shall assume that [·, ·] A is commutative and distributive w.r.t. arbitrary joins and ∇ A ∈ K(A). Lemma 7.1. For all n ∈ N * and all α, β ∈ Con (A), [α, β] n+1 A = [[α, β] A , [α, β] A ] n A . Proof. Let α, β ∈ Con(A). We proceed by induction on n. By its definition, [α, β] 2 A = [[α, β] A , [α, β] A ] A . Now let n ∈ N * such that [α, β] n+1 A = [[α, β] A , [α, β] A ] n A . Then, by the induction hypothesis, [α, β] n+2 A = [[α, β] n+1 A , [α, β] n+1 A ] A = [[[α, β] A , [α, β] A ] n A , [[α, β] A , [α, β] A ] n A ] A = [[α, β] A , [α, β] A ] n+1 A . Lemma 7.2. If the commutator of A is associative, then, for any n ∈ N * and all α, β ∈ Con (A), [α, β] n+1 A = [[α, α] n A , [β, β] n A ] A . Proof. Assume that the commutator of A is associative, and let us also use its commutativity, along with Lemma 7.1. Let α, β ∈ Con(A). We apply induction on n. For n = 1, [α, β] 2 A = [[α, β] A , [α, β] A ] A = [[α, α] A , [β, β] A ] A . Now let n ∈ N * such that [α, β] n+1 A = [[α, α] n A , [β, β] n A ] A . Then [α, β] n+2 A = [[α, β] n+1 A , [α, β] n+1 A ] A = [[[α, α] n A , [β, β] n A ] A , [[α, α] n A , [β, β] n A ] A ] A = [[[α, α] n A , [α, α] n A ] A , [[β, β] n A , [β, β] n A ] A ] A = [[α, α] n+1 A , [β, β] n+1 A ] A . Lemma 7.3. For all n, k ∈ N * and all α, β, φ, ψ, α 1 , α 2 , . . . , α k ∈ Con(A): (i) if α ⊆ β and φ ⊆ ψ, then [α, φ] n A ⊆ [β, ψ] n A ; (ii) if k ≤ n, then [α, β] n A ⊆ [α, β] k A ; (iii) if k ≥ 2 and n ≥ 2, then [α, β] k·n A ⊆ [[α, β] k A , [α, β] k A ] n A ; (iv) [α ∨ β, α ∨ β] n A ⊆ α ∨ [β, β] n A ; (v) [α ∨ β, α ∨ β] n·k A ⊆ [α, α] k A ∨ [β, β] n A ; (vi) [α ∨ β, α ∨ β] n 2 A ⊆ [α, α] n A ∨ [β, β] n A ; (vii) [α 1 ∨ . . . ∨ α k , α 1 ∨ . . . ∨ α k ] n k A ⊆ [α 1 , α 1 ] n A ∨ . . . ∨ [α k , α k ] n A . Proof. (i) By Proposition 3.6, through induction on n. (ii) For all p ∈ N * , [α, β] p+1 A = [[α, β] p A , [α, β] p A ] A ⊆ [α, β] p A , hence the inclusion in the enunciation. (iii) Assume that n ≥ 2. We apply induction on k, (ii) and Lemma 7.1. For k = 2, we have: [[α, β] 2 A , [α, β] 2 A ] n A = [[[α, β] A , [α, β] A ] A , [[α, β] A , [α, β] A ] A ] n A = [α, β] n+2 A ⊇ [α, β] 2n A . Now take a k ≥ 2 that fulfills the inclusion in the enunciation for all α, β ∈ Con (A). Then [[α, β] k+1 A , [α, β] k+1 A ] n A = [[[α, β] A , [α, β] A ] k A , [[α, β] A , [α, β] A ] k A ] n A ⊇ [[α, β] A , [α, β] A ] k·n A = [α, β] k·n+1 A ⊇ [α, β] k·n+n A = [α, β] (k+1)·n A . (iv) We apply induction on n. For n = 1 and all α, β ∈ Con (A), we have [α ∨ β, α ∨ β] A = [α, α] A ∨ [α, β] A ∨ [β, α] A ∨ [β, β] A ⊆ α ∨ [β, β] A . Now let n ∈ N * such that [α ∨ β, α ∨ β] n A ⊆ α ∨ [β, β] n A for all α, β ∈ Con(A). Then, by the induction hypothesis and the case n = 1, we have, for all α, β ∈ Con(A): [α ∨ β, α ∨ β] n+1 A = [[α ∨ β, α ∨ β] n A , [α ∨ β, α ∨ β] n A ] A ⊆ [α ∨ [β, β] n A , α ∨ [β, β] n A ] A ⊆ α ∨ [[β, β] n A , [β, β] n A ] A = α ∨ [β, β] n+1 A . (v) We apply Lemma 7.3. For n = 1, [α, α] k A ∨[β, β] A ⊇ [α∨β, α∨β] k A . For k = 1, [α, α] A ∨[β, β] n A ⊇ [α∨β, α∨β] n A . For k ≥ 2 and n ≥ 2, [α, α] k A ∨ [β, β] n A ⊇ [[α, α] k A ∨ β, [α, α] k A ∨ β] n A ⊇ [[α ∨ β, α ∨ β] k A , [α ∨ β, α ∨ β] k A ] n A ⊇ [α ∨ β, α ∨ β] k·n A . (vi) Take k = n in (v). (vii) We apply induction on k. The statement is trivial for k = 1. Let k ∈ N * that fulfills the equality in the enunciation for any congruences of A, and let α 1 , . . . , α k , α k+1 ∈ Con(A). By (v), it follows that [α 1 ∨ . . . ∨ α k ∨ α k+1 , α 1 ∨. . .∨α k ∨α k+1 ] n k+1 A = [(α 1 ∨. . .∨α k )∨α k+1 , (α 1 ∨. . .∨α k )∨α k+1 ] n k ·n A ⊆ [α 1 ∨. . .∨α k , α 1 ∨. . .∨α k ] n k A ∨ = [α k+1 , α k+1 ] n A ⊆ [α 1 , α 1 ] n A ∨ . . . ∨ [α k , α k ] n A ∨ [α k+1 , α k+1 ] n A . For all θ, ζ ∈ Con(A), we shall denote by θ → ζ = {α ∈ Con(A) | [θ, α] A ⊆ ζ} and by θ ⊥ = θ → ∆ A = {α ∈ Con(A) | [θ, α] A = ∆ A }. Remark 7.4. For all θ, ζ ∈ Con(A), θ → ζ = max{α ∈ Con(A) | [θ, α] A ⊆ ζ}, because, if we denote by M = {α ∈ Con(A) | [θ, α] A ⊆ ζ}, then [θ, θ → ζ] A = [θ, α∈M α] A = α∈M [θ, α] A ⊆ ζ, hence θ → ζ ∈ M . Lemma 7.5. For all α, β, γ ∈ Con(A), [α, β] A ⊆ γ iff α ⊆ β → γ. Proof. "⇒:" β → γ = {θ ∈ Con(A) | [β, θ] A ⊆ γ}. Since [β, α] A = [α, β] A ⊆ γ, it follows that α ⊆ β → γ. "⇐:" We have α ⊆ β → γ = {θ ∈ Con(A) | [β, θ] A ⊆ γ}, hence [β, α] A = [α, β] A ⊆ [β, β → γ] A = [β, {θ ∈ Con(A) | [β, θ] A ⊆ γ}] A = {[β, θ] A | θ ∈ Con(A), [β, θ] A ⊆ γ} ⊆ γ. For the following results, recall, also, the equivalences in Proposition 3.15. Lemma 7.6. For all α, β ∈ Con(A) such that [α, ∇ A ] A = α: α → β = ∇ A iff α ⊆ β. Proof. α → β = ∇ A iff ∇ A ⊆ α → β iff α = [∇ A , α] A ⊆ β,(i) if α ∨ β = ∇ A , then [α, β] A = α ∩ β; (ii) if α ∨ β = α ∨ γ = ∇ A , then α ∨ [β, γ] A = α ∨ (β ∩ γ) = ∇ A ; (iii) if α ∨ β = ∇ A , then [α, α] n A ∨ [β, β] n A = ∇ A for all n ∈ N * . Proof. (i) Assume that α ∨ β = ∇ A . Since (α ∩ β) → [α, β] A = {θ ∈ Con(A) | [α ∩ β, θ] A ⊆ [α, β] A } and [α ∩ β, β] A ⊆ [α, β] A and [α, α ∩ β] A ⊆ [α, β] A , it follows that α ⊆ (α ∩ β) → [α, β] A and β ⊆ (α ∩ β) → [α, β] A , hence ∇ A = α ∨ β ⊆ (α ∩ β) → [α, β] A , therefore (α ∩ β) → [α, β] A = ∇ A , thus α ∩ β ⊆ [α, β] A by Lemma 7.6. Since the converse inclusion always holds, it follows that α ∩ β = [α, β] A . (ii) Assume that α ∨ β = α ∨ γ = ∇ A , so that ∇ A = [∇ A , ∇ A ] A = [α ∨ β, α ∨ γ] A = [α, α] A ∨ [β, α] A ∨ [α, γ] A ∨ [β, γ] A ⊆ α ∨ [β, γ] A ⊆ α ∨ (β ∩ γ) ⊆ ∇ A , hence α ∨ [β, γ] A = α ∨ (β ∩ γ) = ∇ A . (iii) We apply induction on n. Assume that α∨β = ∇ A , so that, by (ii), α∨[β, β] A = ∇ A , thus [α, α] A ∨[β, β] A = ∇ A , hence the implication holds in the case n = 1. Now, if n ∈ N * fulfills the implication in the enunciation for all α, β ∈ Con(A), and assume that α ∨ β = ∇ A , so that [α, α] n A ∨ [β, β] n A = ∇ A . Then, by the case n = 1, it follows that [α, α] n+1 A ∨ [β, β] n+1 A = [[α, α] n A , [α, α] n A ] A ∨ [[β, β] n A , [β, β] n A ] A = ∇ A . Lemma 7.9. If [γ, ∇ A ] A = γ for all γ ∈ Con(A) , then, for all α ∈ B(Con(A)) and all θ ∈ Con(A), [α, θ] A = α∩θ. Proof. Let θ ∈ Con(A) and α ∈ B(Con(A)), so that there exists a β ∈ Con(A) with α ∨ β = ∇ A and α ∩ β = ∆ A . Then the following hold: [α, θ] A ⊆ α ∩ θ = [∇ A , α ∩ θ] A = [α ∨ β, α ∩ θ] A = [α, α ∩ θ] A ∨ [β, α ∩ θ] A ⊆ [α, α ∩ θ] A ∨ (β ∩ α ∩ θ) ⊆ [α, θ] A ∨ ∆ A = [α, θ] A , hence [α, θ] A = α ∩ θ. We have followed the argument from [29,Lemma 4]. • if C is congruence-modular and semi-degenerate, then f (B(Con(A))∩[Ker(f ))) ⊆ f ({α∨Ker(f ) | α ∈ B(Con(A))}) ⊆ B (Con(B)). (ii) For all θ ∈ Con(A): • {α/θ | α ∈ PCon(A) ∩ [θ)} ⊆ {(α ∨ θ)/θ | α ∈ PCon(A)}) = PCon(A/θ); • {α/θ | α ∈ K(A) ∩ [θ)} ⊆ {(α ∨ θ)/θ | α ∈ K(A)}) = K(A/θ); • if C is congruence-modular and semi-degenerate, then {α/θ | α ∈ B(Con(A)) ∩ [θ)} ⊆ {(α ∨ θ)/θ | α ∈ B(Con(A))}) ⊆ B(Con(A/θ)). Proof. The first inclusion in each statement is trivial. (i) By Lemma 3.17, (ii), for the statements on principal and on compact congruences. Now let α ∈ B(Con(A)), so that α ∨ β = ∇ A and [α, β] A = ∆ A for some β ∈ Con(A), hence, by Lemma 3.17, (i), and Remark 3.10, f (α ∨ Ker(f )) ∨ f (β ∨ Ker(f )) = f (α ∨ Ker(f ) ∨ β ∨ Ker(f )) = f (∇ A ) = ∇ B and [f (α ∨ Ker(f )), f (β ∨ Ker(f )] B = f ([α, β] A ∨ Ker(f )) = f (∆ A ) = ∆ B , therefore f (α ∨ Ker(f )) ∈ B(Con(B)). (ii) By (i) for f = p θ . Proposition 7.12. (i) Assume that f is surjective. Then: if ∇ A ∈ PCon(A), then ∇ B ∈ PCon(B), while, if ∇ A ∈ K(A), then ∇ B ∈ K(B). (ii) ∇ A ∈ PCon(A) iff ∇ A/θ ∈ PCon(A/θ) for all θ ∈ Con(A). ∇ A ∈ K(A) iff ∇ A/θ ∈ K(A/θ) for all θ ∈ Con(A). Proof. (i) By Lemma 7.11, (i). (ii) By (i) for the direct implications, and the fact that A/∆ A is isomorphic to A, for the converse implications. Lemma 7.13. If C is congruence-modular, then, for all n ∈ N * and any α, β ∈ Con(A): (i) if f is surjective, then [f (α ∨ Ker(f )), f (β ∨ Ker(f ))] n B = f ([α, β] n A ∨ Ker(f )); (ii) for any θ ∈ Con(A), [(α ∨ θ)/θ, (β ∨ θ)/θ] n A/θ = ([α, β] n A ∨ θ)/θ; (iii) for any θ ∈ Con(A) and any X, Y ∈ P(A 2 ), [Cg A/θ (X/θ), Cg A/θ (Y /θ)] n A/θ = ([Cg A (X), Cg A (Y )] n A ∨ θ)/θ. Proof. (i) We proceed by induction on n. For n = 1, this holds by Remark 3.10. Now take an n ∈ N * such that [f (α ∨ Ker(f )), f (β ∨ Ker(f ))] n B = f ([α, β] n A ∨ Ker(f )). Then, by the induction hypothesis and Remark 3.10, [f (α ∨ Ker(f )), f (β ∨ Ker(f ))] n+1 B = [[f (α ∨ Ker(f )), f (β ∨ Ker(f ))] n B , [f (α ∨ Ker(f )), f (β ∨ Ker(f ))] n B ] B = [f ([α, β] n A ∨ Ker(f )), f ([α, β] n A ∨ Ker(f ))] B = f ([[α, β] n A , [α, β] n A ] n A ∨ Ker(f )) = f ([α, β] n+1 A ∨ Ker(f )). (ii) Take f = p θ in (i). (iii) Take α = Cg A (X) and β = Cg A (Y ) in (ii) and apply Lemma 3.17, (iii). Boolean Congruences versus the Reticulation Throughout this section, we shall assume that [·, ·] A is commutative and distributive w.r.t. arbitrary joins and ∇ A ∈ K(A). We call A a semiprime algebra iff ρ A (∆ A ) = ∆ A . So A is semiprime iff ∆ A ∈ RCon(A). Remark 8.1. By Proposition 5.7, if the commutator of A equals the intersection, then A is semiprime, hence, if C is congruence-distributive, then every member of C is semiprime. Proposition 8.2. A/ρ A (∆ A ) is semiprime. Proof. By Proposition 5.3, (iv), and Proposition 5.4, (iv), ρ A/ρA(∆A) (∆ A/ρA(∆A) ) = ρ A (ρ A (∆ A ))/ρ A (∆ A ) = ρ A (∆ A )/ρ A (∆ A ) = ∆ A/ρA(∆A) . Lemma 8.3. If A is semiprime, then, for all α, β ∈ Con(A): • λ A (α) = 0 iff α = ∆ A ; • [α, β] A = ∆ A iff α ∩ β = ∆ A . Proof. Let α, β ∈ Con(A). Since λ A (∆ A ) = 0 and [α, β] A ⊆ α ∩ β, the converse implications always hold. Now assume that A is semiprime. If λ A (α) = 0 = λ A (∆ A ), then α ⊆ ρ A (α) = ρ A (∆ A ) = ∆ A , thus α = ∆ A . If [α, β] A = ∆ A , then λ A (α ∩ β) = λ A ([α, β] A ) = λ A (∆ A ) = 0, hence α ∩ β = ∆ A by the above. Lemma 8.4. For any θ ∈ Con(A), the following hold: (i) ρ A (θ) = {α ∈ Con(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)} = {α ∈ K(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)} = {α ∈ PCon(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)}; (ii) for any α ∈ K(A), α ⊆ ρ A (θ) iff there exists a k ∈ N * such that [α, α] k A ⊆ θ. Proof. (i) By Proposition 5.27 and the fact that PCon( A) ⊆ K(A) ⊆ Con(A), ρ A (θ) = {Cg A (a, b) | (a, b) ∈ A 2 , (∃ k ∈ N * ) ([Cg A (a, b), Cg A (a, b)] k A ⊆ θ)} = {α ∈ PCon(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)} ⊆ {α ∈ K(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)} ⊆ {α ∈ Con(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)} ⊆ {α ∈ PCon(A) | (∃ k ∈ N * ) ([α, α] k A ⊆ θ)}, where the last inclusion holds because, for any α ∈ Con(A), if k ∈ N * is such that [α, α] k A ⊆ θ, then, for any (a, b) ∈ α, [Cg A (a, b), Cg A (a, b)] k A ⊆ [α, α] k A ⊆ θ, thus α = (a,b)∈α Cg A (a, b) ⊆ {Cg A (a, b) | (a, b) ∈ A 2 , (∃ k ∈ N * ) ([Cg A (a, b), Cg A (a, b)] k A ⊆ θ)}. Hence the equalities in the enunciation. (ii) The converse implication follows directly from (i). For the direct implication, from (i) it follows that, for any α ∈ K(A) such that α ⊆ ρ A (θ), there exist non-empty families (β j ) j∈J ⊆ K(A) and (k j ) j∈J ⊆ N * such that α ⊆ j∈J β j and [β j , β j ] kj A ⊆ θ for all j ∈ J. Since α ∈ K(A), it follows that there exist an n ∈ N * and j 1 , . . . , j n ∈ J such that α ⊆ n i=1 β ji . Let j = max{j 1 , . . . , j n } ∈ N * . Then [β ji , β ji ] k A ⊆ [β ji , β ji ] ki A ⊆ θ for each i ∈ 1, n, thus, by Lemma 7.3, (vii), [α, α] k n A ⊆ [ n i=1 β ji , n i=1 β ji ] k n A ⊆ n i=1 [β ji , β ji ] k A ⊆ θ. Proposition 8.5. (i) ρ A (∆ A ) = {α ∈ Con(A) | (∃ k ∈ N * ) ([α, α] k A = ∆ A )} = {α ∈ K(A) | (∃ k ∈ N * ) ([α, α] k A = ∆ A )} = {α ∈ PCon(A) | (∃ k ∈ N * ) ([α, α] k A = ∆ A )}; (ii) for any α ∈ K(A), α ⊆ ρ A (∆ A ) iff there exists a k ∈ N * such that [α, α] k A = ∆ A . Proof. By Lemma 8.4. Corollary 8.6. A is semiprime iff, for any α ∈ K(A) and any k ∈ N * , if [α, α] k A = ∆ A , then α = ∆ A . Throughout the rest of this section, we shall assume that K(A) is closed w.r.t. the commutator of A and [θ, ∇ A ] A = θ for all θ ∈ Con(A); see also Proposition 3.15. For any bounded lattice L, we shall denote by B(L) the set of the complemented elements of L. If L is distributive, then B(L) is the Boolean center of L. Although Con(A) is not necessarily distributive, we shall call B(Con(A)) the Boolean Center of Con (A). So B(Con(A)) is the set of the α ∈ Con(A) such that there exists a β ∈ Con(A) which fulfills α ∨ β = ∇ A and α ∩ β = ∆ A , thus also [α, β] A = ∆ A . Proof. Let α ∈ B(Con(A)), so that α ∨ β = ∇ A and α ∩ β = ∆ A for some β ∈ Con(A). Now let ∅ = (α i ) i∈I ⊆ Con(A) such that α ⊆ i∈I α i , so that β ∨ i∈I α i = ∇ A ∈ K(A), thus ∇ A = β ∨ n j=1 α ij for some n ∈ N * and some i 1 , . . . , i n ⊆ I, hence, by Proposition 7.8, (i), α = [α, Proof. λ A (B(Con(A))) = B(Con(A))/ ≡ A . Now we use Lemma 8.8. Let α ∈ B(Con(A)) ⊆ K(A), so that λ A (α) ∈ L(A) and, for some β ∈ B(Con(A)) ⊆ K(A), we have α ∨ β = ∇ A and α ∩ β = ∆ A . Then λ A (β) ∈ L(A), Throughout the rest of this section, C shall be congruence-modular and semi-degenerate. Proof. (i) By Lemma 8.11, λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean morphism. By Remark 5.10, λ A (α) = 1 iff α = ∇ A , hence this Boolean morphism is injective. (ii) Assume that A is semiprime, and let x ∈ B(Con(A)/ ≡ A ), so that x ∨ y = 1 and x ∧ y = 0 for some y ∈ B(Con(A)/ ≡ A ). Hence there exist α, β ∈ Con(A) such that x = λ A (α) and y = λ A (β), thus 1 = x ∨ y = λ A (α) ∨ λ A (β) = λ A (α ∨ β) and 0 = x ∧ y = λ A (α) ∧ λ A (β) = λ A (α ∩ β), therefore α ∨ β = ∇ A and α ∩ β = ∆ A , by Remark 5.10 and Lemma 8.3. Hence α ∈ B(Con(A)), thus x = λ A (α) ∈ λ A (B(Con(A))) = B(Con(A))/ ≡ A , therefore, by Lemma 8.11, B(Con(A)/ ≡ A ) ⊆ λ A (B(Con(A))) = B(Con(A))/ ≡ A ⊆ B(L(A)) ⊆ B(Con(A)/ ≡ A ), hence λ A (B(Con(A))) = B(Con(A))/ ≡ A = B(L(A)) = B(Con(A)/ ≡ A ). Therefore λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is surjective, so, by (i), it is a Boolean isomorphism. ∇ A ] A = [α, β ∨ n j=1 α ij ] A = [α, β] A ∨ [α, n j=1 α ij ] A = ∆ A ∨ [α, n j=1 α ij ] A = [α, n j=1 α ij ] A ⊆ n j=1 α ij , hence α ∈ K(A).1 = λ A (∇ A ) = λ A (α ∨ β) = λ A (α) ∨ λ A (β) and 0 = λ A (∆ A ) = λ A (α ∩ β) = λ A (α) ∧ λ A (β), (iii) Assume that the commutator of A is associative, and let x ∈ B(L(A)) ⊆ L(A) = λ A (K(A)), so that x∨y = 1 and x ∧ y = 0 for some y ∈ B(L(A)) and there exist α, β ∈ K(A) such that x = λ A (α) and y = λ A (β). Then λ A (α ∨ β) = λ A (α) ∨ λ A (β) = x ∨ y = 1 = λ A (∇ A ), hence α ∨ β = ∇ A by Remark 5.10. We also have Con(A))), hence B(L(A)) ⊆ λ A (B (Con(A))), thus B(L(A)) ⊆ λ A (B(Con(A))) = B(Con(A))/ ≡ A ⊆ B(L(A)) ⊆ B(Con(A)/ ≡ A ) by Lemma 8.11, therefore λ A (B(Con(A))) = B(Con(A))/ ≡ A = B(L(A)) ⊆ B(Con(A)/ ≡ A ). Therefore λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is surjective, so, by (i), it is a Boolean isomorphism. Lemma 8.13. If A is semiprime and α ∈ Con(A), then: α ∈ B(Con(A)) iff λ A (α) ∈ B(L(A)). λ A ([α, β] A ) = λ A (α) ∧ λ A (β) = x ∧ y = 0 = λ A (∆ A ),x = λ A (α) = λ A ([α, α] k A ) ∈ λ A (B( Proof. We apply Lemma 8.11, which, first of all, gives us the direct implication. For the converse, assume that λ A (α) ∈ B(L(A)) = B(Con(A)/ ≡ A ), so that there exists a β ∈ Con(A) with λ A (α ∨ β) = λ A (α) ∨ λ A (β) = 1 = λ A (∇ A ) and λ A (α ∩ β) = λ A (α) ∧ λ A (β) = 0, thus α ∨ β = ∇ A and α ∩ β = ∆ A by Remark 5.10 and Lemma 8.3. Therefore α ∈ B(Con(A)). For any Ω ⊆ Con(A), let us consider the property: (A, Ω) for all α, β ∈ Ω and all n ∈ N * , there exists a k ∈ N * such that [[α, α] Lemma 8.25. If A is hyperarchimedean, then A/θ is hyperarchimedean for all θ ∈ Con(A). Proof. Let θ ∈ Con(A). For any a, b ∈ A, there exists an n ∈ N * such that [Cg A (a, b), Cg A (a, b)] n A ∈ B(Con(A)). Then, according to Lemma 7.13, (iii), and Lemma 7.11, (ii), [Cg A/θ (a/θ, b/θ), Cg A/θ (a/θ, b/θ)] n A/θ = ([ Cg A (a, b), Cg A (a, b)] n A ∨ θ)/θ ∈ B(Con(A/θ)), therefore A/θ is hyperarchimedean. Proof. Let θ ∈ K(A), so that θ = α 1 ∨ . . . ∨ α n for some n ∈ N * and α 1 , . . . , α n ∈ PCon(A). Since A is hyperarchimedean, there exists a k ∈ N * such that, for all i ∈ 1, n, [α i , α i ] k A ∈ B(Con(A)), thus λ A (α i ) = λ A ([α i , α i ] k A ) ∈ λ A (B(Con(A))) ⊆ B(L(A)) by Lemma 8.11, so that λ A (θ) = λ A (α 1 ) ∨ . . . ∨ λ A (α n ) ∈ B(L(A)). Hence A Reticulation Functor Throughout this section, C shall be congruence-modular and semi-degenerate and such that, in each of its members, the set of the compact congruences is closed w.r.t. the commutator. Also, the morphism f : A → B shall be surjective, so that the map ϕ f : Con(A) → Con(B), ϕ f (α) = f (α ∨ Ker(f )) for all α ∈ Con(A), is well defined. Remark 9.1. By Lemma 7.11, (i), ϕ f (K(A)) = K(B). For any algebra M from C and any X ⊆ M 2 , let us denote V M (X) = V M (Cg M (X)). Then, by the proof of [1, Proposition 2.1] and Lemma 3.17, (i), for all α ∈ Con(A), {f (φ) | φ ∈ V A (α)} = f (V A (α)) = V B (f (α)) = V B (Cg B (f (α))) = V B (f (α ∨ Ker(f ))) = V B (ϕ f (α)). Con(A) | | K(A) L(A) Con(B) K(B) L(B) λ A ❄ λ B ❄ ϕ f ϕ f L(f ) ✲ ✲ ✲ We shall denote by N the set of the natural numbers and by N * = N \ {0}. For any set M , P(M ) shall be the set of the subsets of M , id M : M → M shall be the identity map, and we shall denote by ∆ M = {(x, x) | x ∈ M } and ∇ M = M 2 . For any family (M i ) i∈I of sets and any M ⊆ i∈I M i , whenever there is no danger of confusion, by a = (a i ) i∈I ∈ M we mean a i ∈ M i for all i ∈ I, such that a ∈ M . For any sets M , N and any function f : M → N , we shall denote by Ker(f ) = {(x, y) ∈ M 2 | f (x) = f (y)}, and the direct and inverse image of f in the usual way; we shall denote, simply, f = f 2 : P(M 2 ) → P(N 2 ) and f * = (f 2 ) −1 : P(N 2 ) → P(M 2 ); so, for any X ⊆ M 2 and any Y ⊆ N 2 , f (X) = {(f (a), f (b)) | (a, b) ∈ X} and f * (Y ) = {(a, b) ∈ M 2 | (f (a), f (b)) ∈ Y }, thus Ker(f ) = f * (∆ N ). Also, if X i ⊆ M 2 generated by a shall also be denoted by (a], respectively [a); whenever we need to specify the lattice L, we shall denote [M ) L , (M ] L , [a) L and (a] L instead of [M ), (M ], [a) and (a], respectively. It is well known that (Id(L), ∨, ∩, {0}, L) and (Filt(L), ∨, ∩, {1}, L) are bounded lattices, with J ∨ K = (J ∪ K] Proposition 3.6. [39, Lemma 4.6,Lemma 4.7,Theorem 8.3] The commutator is: Proposition 3.9. [48, Theorem 5.17, p. 48] Assume that C is congruence-modular, and let n ∈ N * , M 1 , . . . , M n be algebras from C, M = n i=1 M i and, for all Lemma 3 . 312. [1, Theorem 5.3] If C is congruence-modular and semi-degenerate, then: Proposition 3.14.[34] C is semi-degenerate iff, for all members M of C, ∇ M ∈ K(M ). Proposition 3.15. [21, Theorem 8.5, p. 85] If C is congruence-modular, then the following are equivalent: (i) for any algebra M from C, [∇ M , ∇ M ] M = ∇ M ; (ii) for any algebra M from C and any θ ∈ Con(M ), [θ, ∇ M ] M = θ; Lemma 3 . 316. (i) If C is congruence-modular and semi-degenerate, then C fulfills the equivalent conditions from Proposition 3.15. (ii) If C is congruence-distributive, then C fulfills the equivalent conditions from Proposition 3.15. Proof. (i) This is exactly [1, Lemma 5.2]. (ii) Clear, from Theorem 3.5. Lemma 3. 17 . [ 4 , 174Lemma 1.11], [53, Proposition 1. V Id,L (I) = Spec Id (L) ∩ [I) = {P ∈ Spec Id (L) | I ⊆ P }, D Id,L (I) = Spec Id (L) \ V Id,L (I) = {Q ∈ Spec Id (L) | I Q}, V Id,L (a) = V Id,L ((a]) = {P ∈ Spec Id (L) | a ∈ P } and D Id,L (a) = D Id,L ((a]) = Spec Id (L) \ V Id,L (a) = {Q ∈ Spec Id (L) | a / ∈ Q}. By replacing Spec Id (L) with Spec Filt (L), in the same way we can define V Filt,L (F ), D Filt,L (F ), V Filt,L (a) and D Filt,L (a) for any F ∈ Filt(L) and any a ∈ L. Remark 4 . 1 . 41The following hold, and their duals hold for filters:• for any J, K ∈ Id(L), V Id,L (J ∩ K) = V Id,L (J) ∪ V Id,L (K) and D Id,L (J ∩ K) = D Id,L (J) ∩ D Id,L (K); then: D Id,L (I) = Spec Id (L) iff V Id,L (I) = ∅ iff I = L; • D Id,L ({0}) = ∅ and V Id,L ({0}) = Spec Id (L); • if L is distributive (so that the Prime Ideal Theorem holds in L and, hence, any ideal of L equals the intersection of the prime ideals that include it) and I ∈ Id(L), then: D Id,L (I) = ∅ iff V Id,L (I) = Spec Id (L) iff I = {0}. As shown by Remark 4.1, {D Id,L (I) | I ∈ Id(L)} is a topology on Spec Id (L), called the Stone topology, having {D Id,L (a) | a ∈ L} as a basis and, obviously, {V Id,L (I) | I ∈ Id(L)} as the family of closed sets and {V Id,L (a) | a ∈ L} as a basis of closed sets. Since Max Id (L) ⊆ Spec Id (L), {D Id,L (I) ∩ Max Id (L) | I ∈ Id(L)} is a topology on Max Id (L), which is also called the Stone topology, and it has {D Id,L (a) ∩ Max Id (L) | a ∈ L} as a basis, {V Id,L (I) ∩ Max Id (L) | I ∈ Id(L)} as the family of closed sets and {V Id,L (a) ∩ Max Id (L) | a ∈ L} as a basis of closed sets. Dually, we have the Stone topologies on Spec Filt (L) and Max Filt (L). Spec Id (L), Max Id (L), Spec Filt (L) and Max Filt (L) are called the (prime) spectrum of ideals, maximal spectrum of ideals, (prime) spectrum of filters and maximal spectrum of filters of L, respectively. Remark 5 . 5 . 55By Remark 5.1, (ii), Spec(A) ⊆ RCon(A); moreover, the elements of RCon(A) are exactly the intersections of prime congruences of A. Remark 5 . 6 . 56RCon(A) = {α ∈ Con(A) | α = ρ A (α)} = {ρ A (α) | α ∈ Con(A)}.Indeed, the first of these equalities is the definition of RCon(A) and the second equality follows from Proposition 5.4,(iv). Proposition 5 . 7 . 57If the commutator of A equals the intersection, in particular if C is congruence-distributive, then RCon(A) = Con(A). Proposition 5 . 513. L(A) is a bounded sublattice of Con(A)/ ≡ A , thus it is a bounded distributive lattice. Proof. Since ∇ A ∈ K(A), we have 1 = ∇ A ∈ L(A). By Remark 3.18, ∆ A ∈ K(A), thus 0 = ∆ A ∈ L(A). If K(A) is closed with respect to the commutator, then, for each α, β ∈ K(A), we have [α, β] A ∈ K(A), thus α ∧ β = [α, β] A ∈ L(A) Again by Remark 3.18, for each α, β ∈ K(A), α ∨ β = α ∨ β ∈ L(A). Hence L(A) is a bounded sublattice of Con(A)/ ≡ A , which is distributive by Lemma 5.11, thus L(A) is a bounded distributive lattice. Lemma 5. 17 . 17(i) For any θ ∈ Con(A), θ ⊆ (θ * ) * . (ii) For any I ∈ Id(L(A)), I = (I * ) * .Proof. (i) Let θ ∈ Con(A). For any (a, b) ∈ θ, Cg A (a, b) ∈ PCon(A) ⊆ K(A) and Cg A (a, b) ⊆ θ, thus Cg A (a, b) ∈ K(A) ∩ (θ], hence Cg A (a, b) ∈ θ * , therefore Cg A (a, b) ⊆ (θ * ) * by Lemmas 5.14 and 5.16, so (a, b) ∈ (θ * ) * . Hence θ ⊆ (θ * ) * .(ii) For any x ∈ L(A), by Lemma 5.16, the following equivalences hold: x ∈ (I * ) * iff there exists an α ∈ K(A) such that α ⊆ I * and x = α iff there exists an α ∈ K(A) such that α ∈ I and x = α iff x ∈ I. Therefore (I * ) * = I.Proposition 5.18. (i) The map I ∈ Id(L(A)) → I * ∈ Con(A) is injective. (ii) The map θ ∈ Con(A) → θ * ∈ Id(L(A)) is surjective. Proof. (i) Let I, J ∈ Id(L(A)) such that I * = J * . Then (I * ) * = (J * ) * , so I = J by Lemma 5.17, (ii). (ii) Let I ∈ Id(L(A)), and denote θ = I * ∈ Con(A). Then θ * = (I * ) * = I by Lemma 5.17, (ii). Lemma 5. 19 . 19For any φ ∈ Spec(A), φ = (φ * ) * . • u : Spec(A) → Spec Id (L(A)), for all φ ∈ Spec(A), u(φ) = φ * ; • v : Spec Id (L(A)) → Spec(A), for all P ∈ Spec Id (L(A)), v(P ) = P * . Proposition 5.22. u and v are homeomorphisms, inverses of each other, between the prime spectrum of A and the prime spectrum of ideals of L(A), endowed with the Stone topologies. Proof. By Lemma 5.17, (ii), for all P ∈ Spec Id (L(A)), we have u(v(P )) = P . By Lemma 5.19, for all φ ∈ Spec(A), we have v(u(φ)) = φ. Thus u and v are bijections and they are inverses of each other. thus u is closed, hence u is open, so v is continuous. Now let I ∈ Id(L(A)). Then, according to Proposition 5.18, (ii), I = θ * for some θ ∈ Con(A). By the above, u(V A (θ)) = V Id,L(A) (θ * ) = V Id,L(A) (I), hence v(V Id,L(A) (I)) = v(u(V A (θ))) = V A (θ), therefore v is closed, hence v is open, thus u is continuous.Hence u and v are homeomorphisms. Corollary 5 . 523 (existence of the reticulation). L(A) is a reticulation for the algebra A. Corollary 5 . 525 (uniqueness of the reticulation). The reticulation of A is unique up to a lattice isomorphism.Corollary 5.26. If C is congruence-modular and semi-degenerate, then u and v induce homeomorphisms, inverses of each other, between the maximal spectrum of A and the maximal spectrum of ideals of L(A), endowed with the Stone topologies. Proof. By Lemma 3.12, Proposition 5.22 and the fact that, as Lemma 5.15 ensures us, u and v are orderpreserving, and hence they are order isomorphisms between the posets (Spec(A), ⊆) and (Spec Id (L(A)), ⊆). Proposition 5 . 27 . [ 1 , 5271Proposition 4.1] For any θ ∈ Con(A), ρ A Corollary 5 . 529. (i) For all θ ∈ Con(A), ρ A (θ) * = θ * . (ii) For all I ∈ Id(L(A)), ρ A (I * ) = I * .Proof. (i) By Lemma 5.17, (ii), and Proposition 5.28, θ * = ((θ * ) * ) * = ρ A (θ) * . (ii) By Proposition 5.28 and Lemma 5.17, (ii), we have ρ A (I * ) = ((I * ) * ) * = I * . Remark 6 . 1 . 61If Spec(A) = ∅, then ρ A (α) = ∇ A for all α ∈ Con(A), hence≡ A = ∇ K(A) , thus L(A) = K(A)/∇ K(A) ∼ = L 1 . If Spec(A) = {φ} for some φ ∈ Con(A) \ {∇ A }, then: ρ A (θ) = φ = ρ A (∆ A ) for all θ ∈ (φ], and ρ A (θ) = ∇ A = ρ A (∇ A ) for all θ ∈ Con(A) \ (φ], therefore, since ∆ A , ∇ A ∈ K(A), L(A) = K(A)/ ≡ A ∼ = L 2 . Obviously, if A is Abelian,then A is nilpotent and solvable and Spec(A) = ∅. Moreover, by [1, Proposition 1.3], if A is solvable or nilpotent, then Spec(A) = ∅. Thus, if A is solvable or nilpotent, in particular if A is Abelian, then L(A) ∼ = L 1 . For instance, according to thus L(A) = {0, 1}, so we are situated in one of the following two cases: either A is Abelian, so that L(A) ∼ = L 1 , or the commutator of A equals the intersection, so that Spec(A) = {∆ A } and thus L(A) ∼ = L 2 . Proposition 6 . 2 . 62If the commutator of A equals the intersection, in particular if C is congruence-distributive, then K(A) is a bounded sublattice of the bounded distributive lattice Con(A) and λ A : K(A) → L(A) is a lattice isomorphism, thus we may take L(A) = K(A). Proof. Assume that [·, ·] A = ∩. ∆ A ∈ PCon(A) ⊆ K(A). By Remark 3.18, K(A) is closed w.r.t. the join, and we are under the assumptions that ∇ A ∈ K(A) and K(A) is closed w.r.t. the commutator, so w.r.t. the intersection. Hence K(A) is a bounded sublattice of Con(A). By Proposition 5.7, ≡ A = ∆ K(A) , thus L(A) = K(A)/∆ K(A) ∼ = K(A) and the canonical surjection λ A : K(A) → L(A) is a lattice isomorphism. Remark 6.3. If Con(A) = K(A), in particular if A is finite, then L(A) = Con(A)/ ≡ A , so, if, furthermore, the commutator of A equals the intersection, in particular if C is congruence-distributive, then L(A) ∼ = Con(A) by Proposition 6.2, thus we may take L(A) = Con(A). Remark 6 . 4 . 64By Proposition 6.2, if A is a residuated lattice, then L(A) = K(A). If we denote by Filt(A) the set of the filters of A and by PFilt(A) the set of the principal filters of A, then, since Con(A) ∼ = Filt(A) and the finitely generated filters of A are principal filters Example 6. 7 . 7For any group (G, ·), any x ∈ G and any normal subgroup H of G, let us denote by x the subgroup of G generated by x and by ≡ H the congruence of G associated to H:≡ H = {(y, z) ∈ G 2 | yz −1 ∈ H}.As shown by the following commutators calculations, the cuaternions group,C 8 = {1, −1, i, −i, j, −j, k, −k}, is a solvable algebra which is not Abelian, while the group S 3 = {1, t, u, v, c, d} of the permutations of the set 1, 3, where 1 = id 1,3 , t = (1 2), u = (1 3), v = (2 3), c = (1 2 3) and d = c • c, has Spec(S 3 ) = ∅, without being solvable or nilpotent. Example 6 . 8 . 68This is the algebra from [2, Example 6.3] and [3, Example 4.2]: U = ({0, a, b, c, d}, +), with + defined by the following table, which has the congruence lattice represented below, where U/α = {{0, a}, {b, c, d}}, U/β = {{0, b}, {a, c, d}}, U/γ = {{0, c, d}, {a, b}} and U/δ = {{0}, {a}, {b}, {c, d}}: + 0 a b c d Abelian, nor is it solvable or nilpotent, as shown by the table of [·, ·] U above, but Spec(A) = ∅, thus L(U ) ∼ = L 1 by Remark 6.1. Example 6 . 9 . 69Let M = ({a, b, x, y, z}, +) and N = ({a, b, c, x, y}, +), with + defined by the following tables. Then Con(M ) and Con(N ) have the Hasse diagrams below, where: • M/α = {{a, b}, {x, y, z}}, M/β = {{a, b}, {x, y}, {z}}, M/γ = {{a, b}, {x, z}, {y}}, M/δ = {{a, b}, {x}, {y, z}} and M/ε = {{a, b}, {x}, {y}, {z}}; • N/χ = {{a, b, c}, {x, y}}, N/χ 1 = {{a, b, c}, {x}, {y}}, N/ξ = {{a, b}, {c}, {x, y}}, N/ξ 1 = {{a, b}, {c}, {x}, {y}}, N/ψ = {{a}, {b, c}, {x, y}}, N/ψ 1 = {{a}, {b, c}, {x}, {y}} and N/φ = {{a}, {b}, {c}, {x, y}}. Note that, despite the fact that M is congruence-modular and N is congruence-distributive, neither HSP(M ), nor HSP(N ) is congruence-modular, because S = ({a, b}, +) ∼ = (L 2 , max) ∼ = (Z 2 , ·) is a subalgebra of both M and N , and it can be easily checked that S 2 is not congruence-modular. Thus neither HSP(M ), nor HSP(N ) is semidegenerate, which is also obvious from the fact that ({a}, +) is a subalgebra of both M and N . We have: [θ, ζ] M = ε for all θ, ζ ∈ [ε) and, of course, [∆ M , θ] M = [θ, ∆ M ] M = ∆ M for all θ ∈ Con(M ), hence Spec(M ) = {∆ M } and thus L(M ) ∼ = L 2 , while [·, ·] N is given by the following table, thus Spec(N ) = {ψ, ξ}, so ρ N is defined as follows and hence L(M ) ∼ = L 2 2 : Example 6. 10 . 10Here are some finite congruence-distributive examples, thus in which the reticulations are isomorphic to the congruence lattices. Regarding the preservation properties fulfilled by the reticulation, these examples show that there is no embedding relation between the reticulation of an algebra and those of its subalgebras: if E is the following bounded lattice, then, for instance,{0, x, y, 1} = L 4 = L 2 ⊕ L 2 ⊕ L 2 , D = {0, a, x, b, 1} and P = {0, a, x, y, 1} are bounded sublattices of E. We have: L(E) ∼ = Con(E) = {∆ E , µ, ∇ E } ∼ = L 3 , where E/µ = {{0}, {a}, {x, y}, {b}, {1}}, L(L 4 ) ∼ = Con(L 4 ) = Con(L 2 ⊕ L 2 ⊕ L 2 ) ∼ = Con(L 2 ) 3 ∼ = L3 2 , L(D) ∼ = Con(D) = {∆ D , ∇ D } ∼ = L 2 and L(P) ∼ = Con(P) = {∆ P , α, β, γ, ∇ P } ∼ = L 2 ⊕ L 2 2 , where P/α = {{0, x, y}, {a, 1}}, P/β = {{0, a}, {x, y, 1}} and P/γ = {{0}, {a}, {x, y}, Remark 6.11. By [24, Lemma 3.3], in any variety, arbitrary intersections commute with arbitrary direct products of congruences. If C is congruence-modular and M is an algebra from C such that A × M has no skew congruences, then Spec(A × M ) = {φ × ∇ M | φ ∈ Spec(A)} ∪ {∇ A × ψ | ψ ∈ Spec(M )}. This follows from Proposition 3.9 in the same way as in the congruence-distributive case, treated in [24, Proposition 3.5,(ii)]. hence the expression of PCon(A × M ) in the enunciation. From this and the second statement in (i), we obtain: K(A × M ) = {Cg A×M . 7 . 7By the above, if [·, ·] A is associative, then (Con(A), ∨, ∩, [·, ·] A , →, ∆ A , ∇ A ) is a residuated lattice, and, if a C is a congruence-distributive variety, then (Con(A), ∨, ∩, [·, ·] A , →, ∆ A , ∇ A ) is, moreover, a Gödel algebra.Proposition 7.8. If [θ, ∇ A ] A = θ for all θ ∈ Con(A), for any α, β, γ ∈ Con(A): Remark 7 . 10 . 710By Lemma 7.9, if [γ, ∇ A ] A = γ for all γ ∈ Con(A), then, in B(Con(A)), the commutator of A equals the intersection, in particular the intersection in B(Con(A)) is distributive with respect to the join.Lemma 7.11. (i) If f is surjective, then: • f (PCon(A) ∩ [Ker(f ))) ⊆ f ({α ∨ Ker(f ) | α ∈ PCon(A)}) = PCon(B); • f (K(A) ∩ [Ker(f ))) ⊆ f ({α ∨ Ker(f ) | α ∈ K(A)}) = K(B); Remark 8 . 7 . 87Obviously, ∆ A , ∇ A ∈ B(Con(A)). Proposition 8 . 9 . 89If K(A) = B(Con(A)), then L(A) = B(L(A)). ∆ Proof. If K(A) = B(Con(A)), then L(A) = λ A (K(A)) = λ A (B(Con(A))) ⊆ B(L(A)) ⊆ L(A) by Lemma 8.11, thus L(A) = B(L(A)). Lemma 8.10. For any σ, θ ∈ Con(A):θ ⊥ = {α ∈ PCon(A) | [α, θ] A = ∆ A } = {α ∈ K(A) | [α, θ] A = ∆ A } = {α ∈ Con(A) | [α, θ] A = ∆ A } = max{α ∈ Con(A) | [α, θ] A = ∆ A }, thus: σ ⊆ θ ⊥ iff [σ, θ] A = ∆ A . Proof. Let M = {α ∈ Con(A) | [α, θ] A = ∆ A }. For all α ∈ M and all (a, b) ∈ α, [Cg A (a, b), θ] A ⊆ [α, θ] A = ∆ A , thus Cg A (a, b) ∈ M ∩ PCon(A). Hence θ ⊥ = A = ∆ A , hence θ ⊥ ∈ M , thus θ ⊥ = max(M ). If σ ⊆ θ ⊥ , then [σ, θ] A ⊆ [θ ⊥ , θ] A = ∆ A , thus [σ, θ] A = ∆ A ,and conversely: if [σ, θ] A = ∆ A , then σ ∈ M , thus σ ⊆ max(M ) = θ ⊥ . Lemma 8.11. λ A (B(Con(A))) = B(Con(A))/ ≡ A ⊆ B(L(A)) ⊆ B(Con(A)/ ≡ A ) and λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean morphism. hence λ A (α) ∈ B(L(A)). Therefore λ A (B(Con(A))) ⊆ B(L(A)). Since L(A) is a bounded sublattice of the bounded distributive lattice Con(A)/ ≡ A , it follows that B(L(A)) is a Boolean subalgebra of B(Con(A)/ ≡ A ). Hence λ A (B(Con(A))) = B(Con(A))/ ≡ A ⊆ B(L(A)) ⊆ B(Con(A)/ ≡ A ). λ A : Con(A) → Con(A)/ ≡ A is a (surjective) bounded lattice morphism. Hence λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is well defined and it is a bounded lattice morphism, thus it is a Boolean morphism. Proposition 8 . 812. (i) The Boolean morphism λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is injective. (ii) If the commutator of A is associative, then λ A (B(Con(A))) = B(L(A)) = B(Con(A))/ ≡ A ⊆ B(Con(A)/ ≡ A ) and λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean isomorphism. (iii) If A is semiprime, then λ A (B(Con(A))) = B(L(A)) = B(Con(A))/ ≡ A = B(Con(A)/ ≡ A ) and λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean isomorphism. k A , [β, β] k A ] A ⊆ [α, β] n A Proof. (i) The direct implication is Proposition 8.9. For the converse, let α ∈ K(A), so that λ A (α) ∈ L(A) = B(L(A)), thus α ∈ B(Con(A)) by Lemma 8.13. Hence K(A) ⊆ B(Con(A)), thus K(A) = B(Con(A)) by Lemma 8.8. (ii) By Lemma 8.23, we obtain that A is hyperarchimedean and semiprime, and B(Con(A)) = K(A) = Con(A), hence L(A) = B(L(A)) by Proposition 8.9, and thus λ A : Con(A) = B(Con(A)) → B(L(A)) = L(A) is a Boolean isomorphism, according to Proposition 8.12. Lemma 8 . 26 . 826If A is hyperarchimedean, then L(A) is a Boolean lattice. λ A (K(A)) = L(A) ⊆ B(L(A)), thus L(A) = B(L(A)), so L(A) is a Boolean lattice. thus [α, β] A ⊆ ρ A (α ∩ β) = ρ A (∆ A ), and, since K(A) is closed with respect to the commutator, we have [α, β] A ∈ K(A), thus, according to Proposition 8.5, (ii), [[α, α] k A , [β, β] k A ] A = [α, β] k+1 A = [[α, β] A , [α, β] A ] k A = ∆ A for some k ∈ N * ; we have applied Lemmas 7.1 and 7.2.But α∨β = ∇ A , thus [α, α] k A ∨[β, β] k A = ∇ A , hence [α, α] k A ∩[β, β] k A = [[α, α] k A , [β, β] k A ] A = ∆ A by Proposition 7.8,(iii)and (i). Therefore [α, α] k A ∈ B(Con(A)), thus Proposition 8.19. For any θ ∈ Con(A): We call A a hyperarchimedean algebra iff, for all α ∈ PCon(A), there exists an n ∈ N * such that [α, α] n A ∈ B(Con(A)). Con(A)) holds. Notice, from the proof of statement (iii) from Proposition 8.12, that this statement, and thus the fact that λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean isomorphism, also hold if property (A, K(A)) is fulfilled. Remark 8.14. By Lemma 7.2, if the commutator of A is associative, then (A. instead of the associativity of the commutator of A. Open problem 8.15. Under the current context, determine whether (A, K(A)) always holds; if it doesn't, then determine whether (A, K(A)) is equivalent to the associativity of the commutator of ARemark 8.14. By Lemma 7.2, if the commutator of A is associative, then (A, Con(A)) holds. Notice, from the proof of statement (iii) from Proposition 8.12, that this statement, and thus the fact that λ A | B(Con(A)) : B(Con(A)) → B(L(A)) is a Boolean isomorphism, also hold if property (A, K(A)) is fulfilled, instead of the associativity of the commutator of A. Open problem 8.15. Under the current context, determine whether (A, K(A)) always holds; if it doesn't, then determine whether (A, K(A)) is equivalent to the associativity of the commutator of A. . ∨ , [· , ·] A = ∩ , ⊥ , ∆ A , Lemma 8.16. (B(Conis a Boolean algebraLemma 8.16. (B(Con(A)), ∨, [·, ·] A = ∩, ⊥, ∆ A , ∇ A ) is a Boolean algebra. Let α, β ∈ B(Con(A)), so that there exist α, β ∈ B(Con(A)) such that α ∨ α = β ∨ β = ∇ A and α ∩ α = β ∩ β = ∆ A . Then, by Remark 7.10, the following hold: (α ∨ β) ∩ α ∩ β = (α ∩ α ∩ β) ∨ (β ∩ α ∩ β) = ∆ A ∨ ∆ A = ∆ A and, since α ∩ β ⊆ β, it follows that = ∇ A . Analogously, (α ∨ β) ∩ α ∩ β = ∆ A and α ∨ β ∨ (α ∩ β) = ∇ A. Proof. We follow, in part, the argument from [29, Lemma 4. Hence α ∨ β, α ∩ β ∈ B(Con(A)). Clearly, ∆ A , ∇ A ∈ B(Con(A)Proof. We follow, in part, the argument from [29, Lemma 4]. Let α, β ∈ B(Con(A)), so that there exist α, β ∈ B(Con(A)) such that α ∨ α = β ∨ β = ∇ A and α ∩ α = β ∩ β = ∆ A . Then, by Remark 7.10, the following hold: (α ∨ β) ∩ α ∩ β = (α ∩ α ∩ β) ∨ (β ∩ α ∩ β) = ∆ A ∨ ∆ A = ∆ A and, since α ∩ β ⊆ β, it follows that = ∇ A . Analogously, (α ∨ β) ∩ α ∩ β = ∆ A and α ∨ β ∨ (α ∩ β) = ∇ A . Hence α ∨ β, α ∩ β ∈ B(Con(A)). Clearly, ∆ A , ∇ A ∈ B(Con(A)). ∇ A ) is a bounded distributive lattice, and, by its definition, it is also complemented, thus it is a Boolean lattice. By a well-known characterization of the complement in a Boolean lattice, for any θ ∈ B(Con(A)), the complement of θ in B(Con(A)) is θ = max{α ∈ B(Con(A)) | α ∩ θ = ∆ A } = max{α ∈ B(Con(A)) | [α, θ] A = ∆ A } ⊆ max{α ∈ Con(A) | [α, θ] A = ∆ A } = θ ⊥ according to Lemma 8.10, thus ∇ A = θ ∨ θ ⊆ θ ∨ θ ⊥ , so θ ∨ θ ⊥ = ∇ A. ∨ , [ · , ·] A = ∩ , ∆ , Therefore B(Con(A)) is a bounded sublattice of Con(A). By Remark 7.10, it follows that (B(Con(A)). Again by Lemma 8.10, ∆ A = [θ, θ ⊥ ] A = θ ∩ θ ⊥ . Therefore θ ⊥ ∈ B(Con(A)) and θ ⊥ is the complement of θ in B(Con(A)Therefore B(Con(A)) is a bounded sublattice of Con(A). By Remark 7.10, it follows that (B(Con(A)), ∨, [·, ·] A = ∩, ∆ A , ∇ A ) is a bounded distributive lattice, and, by its definition, it is also complemented, thus it is a Boolean lattice. By a well-known characterization of the complement in a Boolean lattice, for any θ ∈ B(Con(A)), the complement of θ in B(Con(A)) is θ = max{α ∈ B(Con(A)) | α ∩ θ = ∆ A } = max{α ∈ B(Con(A)) | [α, θ] A = ∆ A } ⊆ max{α ∈ Con(A) | [α, θ] A = ∆ A } = θ ⊥ according to Lemma 8.10, thus ∇ A = θ ∨ θ ⊆ θ ∨ θ ⊥ , so θ ∨ θ ⊥ = ∇ A . Again by Lemma 8.10, ∆ A = [θ, θ ⊥ ] A = θ ∩ θ ⊥ . Therefore θ ⊥ ∈ B(Con(A)) and θ ⊥ is the complement of θ in B(Con(A)). For any bounded lattice L and any I ∈ Id(L), we shall denote by Ann(I) the annihilator of I in L: Ann(I) =. For any bounded lattice L and any I ∈ Id(L), we shall denote by Ann(I) the annihilator of I in L: Ann(I) = Throughout the rest of this paper, all annihilators shall be considerred in the bounded distributive lattice L(A), so they shall be ideals of the lattice L(A). {a ∈ L | (∀ x ∈ I) (a ∧ x = 0)}. It is immediate that, if L is distributive, then Ann(I) ∈ Id(L). Recall that L(A) = λ A (K(A){a ∈ L | (∀ x ∈ I) (a ∧ x = 0)}. It is immediate that, if L is distributive, then Ann(I) ∈ Id(L). Throughout the rest of this paper, all annihilators shall be considerred in the bounded distributive lattice L(A), so they shall be ideals of the lattice L(A). Recall that L(A) = λ A (K(A)). . • Ann, α * ) = {λ A (β) | β ∈ K(A), λ Aα, β] A ) = 0}• Ann(α * ) = {λ A (β) | β ∈ K(A), λ A ([α, β] A ) = 0}; • if A is semiprime, then Ann(α * ) = {λ A (β) | β ∈ K(A). α, β] A = ∆ A }• if A is semiprime, then Ann(α * ) = {λ A (β) | β ∈ K(A), [α, β] A = ∆ A }. (α)]) (x ∧ λ A (β) = 0)} = {λ A (β) | β ∈ K(A), λ A (α) ∧ λ A (β) = 0)} = {λ A (β) | β ∈ K(A), λ A. Proof. By Lemma 5.14, Ann(α * ) = Ann((λ A (α)]) = {λ A (β) | β ∈ K(A), (∀ x ∈ (λ Aα, β] A ) = 0}. By Lemma 8.3, if A is semiprime, then, for any β ∈ K(A), λ A ([α, β] A ) = 0 iff [α, β] A = ∆ A , hence the second equality in the enunciationProof. By Lemma 5.14, Ann(α * ) = Ann((λ A (α)]) = {λ A (β) | β ∈ K(A), (∀ x ∈ (λ A (α)]) (x ∧ λ A (β) = 0)} = {λ A (β) | β ∈ K(A), λ A (α) ∧ λ A (β) = 0)} = {λ A (β) | β ∈ K(A), λ A ([α, β] A ) = 0}. By Lemma 8.3, if A is semiprime, then, for any β ∈ K(A), λ A ([α, β] A ) = 0 iff [α, β] A = ∆ A , hence the second equality in the enunciation. For any α ∈ Con(A) and any I ∈ Id(L(A)), if Ann(α * ) ⊆ I, then α ⊥ ⊆ I * . If A is semiprime and α ∈ K(A). Lemma 8.18. then the converse implication holds, as wellLemma 8.18. For any α ∈ Con(A) and any I ∈ Id(L(A)), if Ann(α * ) ⊆ I, then α ⊥ ⊆ I * . If A is semiprime and α ∈ K(A), then the converse implication holds, as well. A ) = λ A (∆ A ) = 0. Now let x ∈ α * , so that x = λ A (γ) for some γ ∈ K(A) with γ ⊆ α. Then x = λ A (γ) ≤ λ A (α), hence x ∧ λ A (β) = λ A (γ) ∧ λ A (β) ≤ λ A (α) ∧ λ A (β) = 0, so x ∧ λ A (β) = 0, thus λ A (β) ∈ Ann(α * ) ⊆ I, therefore β ⊆ I * by Lemma 5.16. According to Lemma 8.10, α ⊥ = {β ∈ K(A) | [α, β] A = ∆ A } ⊆ I * . For the converse implication, assume that A is semiprime, α ∈ K(A) and α ⊥ ⊆ I * , and let x ∈ Ann(α * ), which means that x = λ A (β) for some β ∈ K(A) with. that Ann(α * ) ⊆ I and let β ∈ K(A) such that [α, β] A = ∆ A , hence λ A (α) ∧ λ A (β) = λ AFor the direct implication. α, β] A = ∆ A , according to Lemma 8.17. Hence, by Lemmas 8.10 and 5.16, β ⊆ α ⊥ ⊆ I * , thus x = λ A (β) ∈ I, therefore Ann(α ⊥ ) ⊆ IProof. For the direct implication, assume that Ann(α * ) ⊆ I and let β ∈ K(A) such that [α, β] A = ∆ A , hence λ A (α) ∧ λ A (β) = λ A ([α, β] A ) = λ A (∆ A ) = 0. Now let x ∈ α * , so that x = λ A (γ) for some γ ∈ K(A) with γ ⊆ α. Then x = λ A (γ) ≤ λ A (α), hence x ∧ λ A (β) = λ A (γ) ∧ λ A (β) ≤ λ A (α) ∧ λ A (β) = 0, so x ∧ λ A (β) = 0, thus λ A (β) ∈ Ann(α * ) ⊆ I, therefore β ⊆ I * by Lemma 5.16. According to Lemma 8.10, α ⊥ = {β ∈ K(A) | [α, β] A = ∆ A } ⊆ I * . For the converse implication, assume that A is semiprime, α ∈ K(A) and α ⊥ ⊆ I * , and let x ∈ Ann(α * ), which means that x = λ A (β) for some β ∈ K(A) with [α, β] A = ∆ A , according to Lemma 8.17. Hence, by Lemmas 8.10 and 5.16, β ⊆ α ⊥ ⊆ I * , thus x = λ A (β) ∈ I, therefore Ann(α ⊥ ) ⊆ I. Con(A)) for all α ∈ PCon(A), then A is hyperarchimedean. Thus, if PCon(A) ⊆ B(Con(A)) and A has principal commutators, then A is hyperarchimedean. A ∈ B, Remark 8.22. If [α, α. If the commutator of A equals the intersection, for instance if C is congruence-distributive, then: A is hyperarchimedean iff PCon(A) ⊆ B(Con(A)Remark 8.22. If [α, α] A ∈ B(Con(A)) for all α ∈ PCon(A), then A is hyperarchimedean. Thus, if PCon(A) ⊆ B(Con(A)) and A has principal commutators, then A is hyperarchimedean. If the commutator of A equals the intersection, for instance if C is congruence-distributive, then: A is hyperarchimedean iff PCon(A) ⊆ B(Con(A)). By Lemmas 8.16 and 8.8, the following equivalences hold: PCon(A) ⊆ B(Con(A)) iff K(A) ⊆ B(Con(A)) iff K(A) = B(Con(A)). By Lemmas 8.16 and 8.8, the following equivalences hold: PCon(A) ⊆ B(Con(A)) iff K(A) ⊆ B(Con(A)) iff K(A) = B(Con(A)). Boolean iff Con(A) = B(Con(A)), which implies that the commutator of A equals the intersection, according to Remark 7.10, and thus, since PCon(A) ⊆ Con(A) = B(Con(A)), A is hyperarchimedean. Remark 8.23. By Lemma 8.16, the lattice Con(A) is. while Remark 8.1 ensures us that A is semiprime. From Lemma 8.8, we obtain the following equivalences: Con(A) is a Boolean lattice iff B(Con(A)) = Con(A) iff B(Con(A)) = K(A) =Remark 8.23. By Lemma 8.16, the lattice Con(A) is Boolean iff Con(A) = B(Con(A)), which implies that the commutator of A equals the intersection, according to Remark 7.10, and thus, since PCon(A) ⊆ Con(A) = B(Con(A)), A is hyperarchimedean, while Remark 8.1 ensures us that A is semiprime. From Lemma 8.8, we obtain the following equivalences: Con(A) is a Boolean lattice iff B(Con(A)) = Con(A) iff B(Con(A)) = K(A) = Of course, since L(A) is a bounded distributive lattice, L(A) is a Boolean algebra iff. L(A) = B(L(A)Con(A). Of course, since L(A) is a bounded distributive lattice, L(A) is a Boolean algebra iff L(A) = B(L(A)). i) If A is semiprime, then: K(A) = B(Con(A)) iff L(A) = B(L(A)). (ii) If Con(A) is a Boolean lattice, then A is hyperarchimedean and semiprime and L(A) is isomorphic to Con(A. Proposition 8.24.. in particular L(A) is a Boolean lattice, as wellProposition 8.24. (i) If A is semiprime, then: K(A) = B(Con(A)) iff L(A) = B(L(A)). (ii) If Con(A) is a Boolean lattice, then A is hyperarchimedean and semiprime and L(A) is isomorphic to Con(A), in particular L(A) is a Boolean lattice, as well. L(A) → L(B), for all α ∈ K(A), L(f )( α) = ϕ f (α). Let us define L(f. that is L(f )(λ A (α)) = λ B (f (α ∨ Ker(f )Let us define L(f ) : L(A) → L(B), for all α ∈ K(A), L(f )( α) = ϕ f (α), that is L(f )(λ A (α)) = λ B (f (α ∨ Ker(f )) K(A) → K(B) is well defined and surjective. Let α, β ∈ K(A) such that λ A (α) = λ A (β), so that ρ A (α) = ρ A (β), thus V A (α) = V A (β), hence V B (ϕ f (α)) = f (V A (α)) = f. V AProof. By Remark 9.1, the restriction ϕ f | K(AProof. By Remark 9.1, the restriction ϕ f | K(A) : K(A) → K(B) is well defined and surjective. Let α, β ∈ K(A) such that λ A (α) = λ A (β), so that ρ A (α) = ρ A (β), thus V A (α) = V A (β), hence V B (ϕ f (α)) = f (V A (α)) = f (V A (β)) thus ρ B (ϕ f (α)) = ρ B (ϕ f (β)), so λ B (ϕ f (α)) = λ B (ϕ f (β)). = V B, that is L(f )(λ A (α= V B (ϕ f (β)), thus ρ B (ϕ f (α)) = ρ B (ϕ f (β)), so λ B (ϕ f (α)) = λ B (ϕ f (β)), that is L(f )(λ A (α)) = (B) are surjective, thus so is their composition, and, since L(f ) • λ A = λ B • ϕ f , it follows that L(f ) is surjective. By Remark 3.10, Lemma 3.17, (ii), and Proposition 5.4, (ii) and (v), for all α, β ∈ K(A), the following hold: L(f )( α ∧ β) = L(f )(λ A (α) ∧ λ A (β)) = L(f )(λ A ([α, β] A )) = λ B (ϕ f ([α, β] A )) = λ B (f ([α, β] A ∨ Ker(f ))) = λ B ([f (α ∨ Ker(f )), f (β ∨ Ker(f ))] B )) = λ B (f (α ∨ Ker(f ))) ∧ λ B (f (β ∨ Ker. and Remark 9.1. Hence L(f ) is well defined. λ B : K(B) → L(B) ϕ f | K(A) : K(A) → KL(f )(λ A (β)); we have used Proposition 5.3, (ii. = λ B (ϕ f (α)) ∧ λ B (ϕ f (βL(f )(λ A (β)); we have used Proposition 5.3, (ii), and Remark 9.1. Hence L(f ) is well defined. λ B : K(B) → L(B) ϕ f | K(A) : K(A) → K(B) are surjective, thus so is their composition, and, since L(f ) • λ A = λ B • ϕ f , it follows that L(f ) is surjective. By Remark 3.10, Lemma 3.17, (ii), and Proposition 5.4, (ii) and (v), for all α, β ∈ K(A), the following hold: L(f )( α ∧ β) = L(f )(λ A (α) ∧ λ A (β)) = L(f )(λ A ([α, β] A )) = λ B (ϕ f ([α, β] A )) = λ B (f ([α, β] A ∨ Ker(f ))) = λ B ([f (α ∨ Ker(f )), f (β ∨ Ker(f ))] B )) = λ B (f (α ∨ Ker(f ))) ∧ λ B (f (β ∨ Ker(f ))) = λ B (ϕ f (α)) ∧ λ B (ϕ f (β)) = (f )( β) and L(f )( α ∨ β) = L(f )(λ A (α) ∨ λ A (β)) = L(f )(λ A (α ∨ β)) = λ B (ϕ f (α ∨ β)) = λ B (f (α ∨ β ∨ Ker(f ))) = λ B (f (α ∨ Ker(f ) ∨ β ∨ Ker(f ))) = λ B (f (α ∨ Ker. L(f )(λ A (α)) ∧ L(f )(λ A (β)) = L(f )( α) ∧ L∨ λ B (f (β ∨L(f )(λ A (α)) ∧ L(f )(λ A (β)) = L(f )( α) ∧ L(f )( β) and L(f )( α ∨ β) = L(f )(λ A (α) ∨ λ A (β)) = L(f )(λ A (α ∨ β)) = λ B (ϕ f (α ∨ β)) = λ B (f (α ∨ β ∨ Ker(f ))) = λ B (f (α ∨ Ker(f ) ∨ β ∨ Ker(f ))) = λ B (f (α ∨ Ker(f ))) ∨ λ B (f (β ∨ α)) ∨ λ B (ϕ f (β)) = L(f )(λ A (α)) ∨ L(f )(λ A (β). = λ B (ϕ f= λ B (ϕ f (α)) ∨ λ B (ϕ f (β)) = L(f )(λ A (α)) ∨ L(f )(λ A (β)) Clearly, if C is an algebra from C and g : B → C is a surjective morphism in C, then L(g • f ) =. Remark 9.3.Remark 9.3. Clearly, if C is an algebra from C and g : B → C is a surjective morphism in C, then L(g • f ) = L(f ). the definition of L to the whole category C, with the image in D01. of courseL(g) • L(f ). the definition of L to the whole category C, with the image in D01, of course. then we may take L(f ) = ϕ f | K(A) : K(A) → K(B), with K(A) and K(B) bounded sublattices of Con(A) and Con(B), respectively. For any bounded lattice morphism h : L → M. Remark 9.5. By Proposition 6.2, if C is congruence-distributive. let us denote by Ker Id (h) = h −1 ({0}) = {x ∈ L | h(x) =Remark 9.5. By Proposition 6.2, if C is congruence-distributive, then we may take L(f ) = ϕ f | K(A) : K(A) → K(B), with K(A) and K(B) bounded sublattices of Con(A) and Con(B), respectively. For any bounded lattice morphism h : L → M , let us denote by Ker Id (h) = h −1 ({0}) = {x ∈ L | h(x) = so that L/Ker Id (h) ∼ = h(L) by the Main Isomorphism Theorem (for lattices and lattice ideals). 0} ∈ Id, 0} ∈ Id(L), so that L/Ker Id (h) ∼ = h(L) by the Main Isomorphism Theorem (for lattices and lattice ideals). For any θ ∈ Con(A), the lattices L(A/θ) and L(A)/θ * are isomorphic. Proposition 9.6 (the reticulation preserves quotientsProposition 9.6 (the reticulation preserves quotients). For any θ ∈ Con(A), the lattices L(A/θ) and L(A)/θ * are isomorphic. (A)). p θ : A → A/θ is a surjective morphism in C, so we can apply the construction above: For all α ∈ Con(A), ϕ p θ (α) = p θ (α ∨ Ker(p θ )) = (α ∨ θ)/θ, so, for all α ∈ K(A), L(p θ )( α) = (α ∨ θ)/θ ∈ L(A/θ). Thus, for any α ∈ K(A): α ∈ Ker Id (L(p θ )) iff L(p θ )( α) = ∆ A/θ iff (α ∨ θ)/θ = θ/θ, that is λ A/θ ((α ∨ θ)/θ) = λ A/θ (θ/θ), iff ρ A/θ ((α ∨ θ)/θ) = ρ A/θ (θ/θ) iff ρ A (α ∨ θ)/θ = ρ A (θ)/θ iff ρ A (α ∨ θ) = ρ A (θ) iff ρ A (α ∨ θ) ⊆ ρ A (θ) iff α ∨ θ ⊆ ρ A (θ) iff α ⊆ ρ A (θ) iff α ∈ (ρ A (θ)) * = θ * , hence Ker Id (L(p θ )) = θ *. Proof. Recall that θ * = λ A (K(A) ∩ (θ]) = { α | α ∈ K(A), α ⊆ θ} ∈ Id(Lwe have applied Proposition 5.3, (iii. Remark 5.1, (iii), Proposition 5.3, (i), and Corollary 5.29, (i). Proposition 9.2 ensures us that the lattice morphism L(p θ ) is surjective, so, from the Main Isomorphism Theorem, we obtain: L(A/θ) ∼ = L(A)/θ * . Proposition 9.7. The lattices L(A) and L(A/ρ A (∆ A )) are isomorphicProof. Recall that θ * = λ A (K(A) ∩ (θ]) = { α | α ∈ K(A), α ⊆ θ} ∈ Id(L(A)). p θ : A → A/θ is a surjective morphism in C, so we can apply the construction above: For all α ∈ Con(A), ϕ p θ (α) = p θ (α ∨ Ker(p θ )) = (α ∨ θ)/θ, so, for all α ∈ K(A), L(p θ )( α) = (α ∨ θ)/θ ∈ L(A/θ). Thus, for any α ∈ K(A): α ∈ Ker Id (L(p θ )) iff L(p θ )( α) = ∆ A/θ iff (α ∨ θ)/θ = θ/θ, that is λ A/θ ((α ∨ θ)/θ) = λ A/θ (θ/θ), iff ρ A/θ ((α ∨ θ)/θ) = ρ A/θ (θ/θ) iff ρ A (α ∨ θ)/θ = ρ A (θ)/θ iff ρ A (α ∨ θ) = ρ A (θ) iff ρ A (α ∨ θ) ⊆ ρ A (θ) iff α ∨ θ ⊆ ρ A (θ) iff α ⊆ ρ A (θ) iff α ∈ (ρ A (θ)) * = θ * , hence Ker Id (L(p θ )) = θ * ; we have applied Proposition 5.3, (iii), Remark 5.1, (iii), Proposition 5.3, (i), and Corollary 5.29, (i). Proposition 9.2 ensures us that the lattice morphism L(p θ ) is surjective, so, from the Main Isomorphism Theorem, we obtain: L(A/θ) ∼ = L(A)/θ * . Proposition 9.7. The lattices L(A) and L(A/ρ A (∆ A )) are isomorphic. 6, ρ A (∆ A ) * = ∆ * A , hence the lattice L(A/ρ A (∆ A )) is isomorphic to L(A)/ρ A (∆ A )) * = L(A)/∆ * A , which, in turn, is isomorphic to. By Corollary 5.29, (i), and Proposition 9. and thus to L(A), since the algebras A/∆ A and A are isomorphicProof. By Corollary 5.29, (i), and Proposition 9.6, ρ A (∆ A ) * = ∆ * A , hence the lattice L(A/ρ A (∆ A )) is isomorphic to L(A)/ρ A (∆ A )) * = L(A)/∆ * A , which, in turn, is isomorphic to L(A/∆ A ), and thus to L(A), since the algebras A/∆ A and A are isomorphic. 7 show that the reticulation of any algebra M from a semi-degenerate congruence-modular variety. such that K(M ) is closed with respect to the commutator of M and ∇ M ∈ K(M ). Remark 9.8. Propositions 8.2 and 9. is isomorphic to the reticulation of a semiprime algebra from the same varietyRemark 9.8. Propositions 8.2 and 9.7 show that the reticulation of any algebra M from a semi-degenerate congruence-modular variety, such that K(M ) is closed with respect to the commutator of M and ∇ M ∈ K(M ), is isomorphic to the reticulation of a semiprime algebra from the same variety. B(L(A)) and B(Con(A/ρ A (∆ A ))) are isomorphic Boolean algebras. 9Corollary 9Corollary 9.9. B(L(A)) and B(Con(A/ρ A (∆ A ))) are isomorphic Boolean algebras. A/ρ A (∆ A ) is semiprime, thus the Boolean algebra B(Con(A/ρ A (∆ A ))) is isomorphic to B(L(A/ρ A (∆ A ))). 12By Propositions 8.2, 8.. which in turn is isomorphic to B(L(A)Proof. By Propositions 8.2, 8.12 and 9.7, A/ρ A (∆ A ) is semiprime, thus the Boolean algebra B(Con(A/ρ A (∆ A ))) is isomorphic to B(L(A/ρ A (∆ A ))), which in turn is isomorphic to B(L(A)). Recall the well-known Nachbin's Theorem, which states that, given a bounded distributive lattice L, we have: L is a Boolean algebra iff Max Id (L) = Spec Id (L) iff Max Filt (L) = Spec Filt (L). Recall the well-known Nachbin's Theorem, which states that, given a bounded distributive lattice L, we have: L is a Boolean algebra iff Max Id (L) = Spec Id (L) iff Max Filt (L) = Spec Filt (L). Proposition 9.10. The following are equivalent: (i) A is hyperarchimedean. Proposition 9.10. The following are equivalent: (i) A is hyperarchimedean; A/ρ A (∆ A ) is hyperarchimedean. A/ρ A (∆ A ) is hyperarchimedean; Max(A) = Spec(A). (iii) Max(A) = Spec(A); L(A) is a Boolean lattice. L(A) is a Boolean lattice; If A is semiprime, that is ρ A (∆ A ) = ∆ A , so that A/ρ A (∆ A ) = A/∆ A is isomorphic to A, then (i) is equivalent to (ii) and (v) is equivalent to (vi). Proof. By Nachbin's Theorem, Proposition 5.22 and Corollary 5.26, (iii) is equivalent to (iv). Trivially, (vi) implies (iv), while the converse holds by Corollary 9. 9Now let us drop the condition that A is semiprime. But A/ρ A (∆ A ) is semiprime. according to Proposition 8.2, hence, by the above, (ii) is equivalent to Max(A/ρ A (∆ AProof. By Nachbin's Theorem, Proposition 5.22 and Corollary 5.26, (iii) is equivalent to (iv). Trivially, (vi) implies (iv), while the converse holds by Corollary 9.9. If A is semiprime, that is ρ A (∆ A ) = ∆ A , so that A/ρ A (∆ A ) = A/∆ A is isomorphic to A, then (i) is equiv- alent to (ii) and (v) is equivalent to (vi). Now let us drop the condition that A is semiprime. But A/ρ A (∆ A ) is semiprime, according to Proposition 8.2, hence, by the above, (ii) is equivalent to Max(A/ρ A (∆ A )) = A )) and to the fact that L(A/ρ A (∆ A )) is a Boolean lattice, which, in turn, is equivalent to (iv) by Proposition 9. 7. But, as shown by Lemma 3.12 and Remark 4.5, Max(A/ρ A (∆ A )) = Spec(A/ρ A (∆ A )) iffSpec(A/ρ A (∆ A )) and to the fact that L(A/ρ A (∆ A )) is a Boolean lattice, which, in turn, is equivalent to (iv) by Proposition 9.7. 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[ "Topol ogi calM ul ti vorti cesSol uti onsofthe Sel f -D ualM axwel l -Chern-Si m ons-Hi ggs System", "Topol ogi calM ul ti vorti cesSol uti onsofthe Sel f -D ualM axwel l -Chern-Si m ons-Hi ggs System" ]
[ "D Ongho \nPO ST EC H790-784PohangK orea\n", "C Hae \nPO ST EC H790-784PohangK orea\n" ]
[ "PO ST EC H790-784PohangK orea", "PO ST EC H790-784PohangK orea" ]
[]
A bstract W e study exi stence and vari ous behavi ors of topol ogi cal m ul tivorti ces sol uti ons ofthe rel ati vi sti c sel f-dualM axwel l -C hern-Si m ons-H i ggssystem . W e rstprove exi stence ofgeneraltopol ogi calsol uti ons by appl yi ng vari ati onalm ethods to the new l y di scovered m i ni m i zi ng functi onal . T hen,by an i terati on m ethod we prove exi stence oftopol ogi calsol uti onssati sfyi ng som eextra condi ti ons,w hi ch wecal ladm i ssi bl e sol uti ons. W e establ i sh asym ptoti c exponenti aldecay esti m ates forthese topol ogi calsol uti ons. W e al so i nvesti gate the l i m i ti ng behavi ors of the adm i ssi bl e sol uti ons as param eters i n our system goes to som e l i m i ts. Forthe A bel i an H i ggsl i m i twe obtai n strong convergence resul t,w hi l e forthe C hern-Si m onsl i m i twe onl y obtai ned thatouradm i ssi bl e sol uti ons are weakl y approxi m ati ng one ofthe C hern-Si m ons sol uti ons.
10.1006/jdeq.1996.3224
[ "https://export.arxiv.org/pdf/cond-mat/9711212v1.pdf" ]
5,606,513
cond-mat/9711212
b45acc6bffe7e6fc2ecbe73850e9e5d49e0f5dd6
Topol ogi calM ul ti vorti cesSol uti onsofthe Sel f -D ualM axwel l -Chern-Si m ons-Hi ggs System 21 Nov 1997 D Ongho PO ST EC H790-784PohangK orea C Hae PO ST EC H790-784PohangK orea Topol ogi calM ul ti vorti cesSol uti onsofthe Sel f -D ualM axwel l -Chern-Si m ons-Hi ggs System 21 Nov 1997 A bstract W e study exi stence and vari ous behavi ors of topol ogi cal m ul tivorti ces sol uti ons ofthe rel ati vi sti c sel f-dualM axwel l -C hern-Si m ons-H i ggssystem . W e rstprove exi stence ofgeneraltopol ogi calsol uti ons by appl yi ng vari ati onalm ethods to the new l y di scovered m i ni m i zi ng functi onal . T hen,by an i terati on m ethod we prove exi stence oftopol ogi calsol uti onssati sfyi ng som eextra condi ti ons,w hi ch wecal ladm i ssi bl e sol uti ons. W e establ i sh asym ptoti c exponenti aldecay esti m ates forthese topol ogi calsol uti ons. W e al so i nvesti gate the l i m i ti ng behavi ors of the adm i ssi bl e sol uti ons as param eters i n our system goes to som e l i m i ts. Forthe A bel i an H i ggsl i m i twe obtai n strong convergence resul t,w hi l e forthe C hern-Si m onsl i m i twe onl y obtai ned thatouradm i ssi bl e sol uti ons are weakl y approxi m ati ng one ofthe C hern-Si m ons sol uti ons. Introduction Si nce the pi oneeri ng works by G i nzburg and Landau on the superconducti vi ty there are m any studi es on the A bel i an H i ggs system [ 10] , [ 12] (and references therei n). In parti cul ar i n [ 10] Ja e and Taubes establ i shed the uni que exi stence ofgeneral ni te energy m ul ti -vorti ces sol uti ons for the B ogom ol ' nyiequati ons. (See al so [ 12]for m ore constructi ve exi stence prooftogether w i th expl i ci t num eri calsol uti ons. ) M ore recentl y, m oti vated l argel y by the physi cs of hi gh tem perature superconducti vi ty the sel f-dual C hern-Si m ons system (hereafter C hern-Si m ons system ) was m odel ed i n [ 2]and [ 3] . (See [ 1]for a generalsurvey. ) T he generalexi stence theorem of topol ogi cal sol uti ons for the correspondi ng B ogom ol ' nyi equati ons was establ i shed i n [ 11] by a vari ati onal m ethod, and i n [ 8] by an i terati on argum ent. For the nontopol ogi calboundary condi ti on we have onl y generalexi stence resul t for the radi alsol uti ons for vorti ces i n a si ngl e poi nt [ 9] . W e recal lthat i n the Lagrangi an ofthe C hern-Si m onssystem there i s no M axwel lterm appeari ng i n theA bel i an H i ggssystem ,w hi l etheform eri ncl udesthe C hern-Si m onsterm w hi ch i snotpresenti n the l ater. N ai ve i ncl usi on of both of the two term s i n the Lagrangi an m akes the system non sel f-dual (i . e. there i s no B ogom ol ' nyitype equati ons for the nontri vi algl obalm i ni m i zer of the energy functi onal . ) In [ 5] , however, a sel f-dual system i ncl udi ng both of the M axwel l and the C hern-Si m ons term s,so cal l ed (rel ati vi sti c) sel f-dualM axwell -C hern-Sim ons-H iggssystem wassuccessful l y m odel ed usi ng theN = 2 supersym m etry argum ent [ 4] , [ 6] . It was found that we need extra neutral scal ar el d to m ake the system sel f-dual . In thi s paper we rst prove generalexi stence theorem for topol ogi cal m ul ti -vortex sol uti ons ofthe correspondi ng B ogom ol ' nyiequati ons of thi s system by a vari ati onal m ethod. T hen, usi ng an i terati on argum ent we constructi vel y prove exi stence of a cl ass of sol uti ons enjoyi ng som e extra condi ti ons. W e cal ltopol ogi calsol uti ons sati sfyi ng these extra condi ti ons the adm issibl e topol ogicalsol utions. W e prove asym ptoti c exponenti aldecay esti m ates for the vari ous term s i n our Lagrangi an for the generaltopol ogi calsol uti ons. O ne ofthe m ost i nteresti ng facts for the adm i ssi bl e topol ogi cal sol uti ons i s that these sol uti ons are real l y "i nterpol ated" between the A bel i an H i ggs sol uti on and the C hern-Si m ons sol uti ons i n the the fol l ow i ng sense: for xed el ectri c charge,w hen the C hern-Si m ons coupl i ng constant goes to zero, our sol uti on converges to the sol uti on of the A bel i an H i ggs system . T he convergence i n thi s case i s very strong. O n the other hand,w hen both the C hern-Si m ons coupl i ng constant and the el ectri c charge goes to i n ni ty w i th som e constrai nts between the two constants,we proved thatoursol uti on i s"weakl y approxi m atel y" sati sfyi ng the B ogom ol ' nyi equati ons for C hern-Si m ons system . In the exi stence prooffor the adm i ssi bl e topol ogi calsol uti ons,al though we used i terati on m ethod i n R 2 di rectl y, we coul d start i terati on i n a bounded dom ai n w i th sui tabl e boundary condi ti on to obtai n a sol uti on i n thatdom ai n,and then enl arge thi s dom ai n to the w hol e ofR 2 as i s done i n [ 8] ,and [ 12]i n a m uch si m pl er case than ours. In thi s way i t woul d be possi bl e to obtai n an expl i ci t num eri calsol uti on. T he organi zati on ofthi s paper i s fol l ow i ng. In the secti on 1 we i ntroduce the acti on functi onalfor the sel f-dual M axwel l -C hern-Si m ons-H i ggs system ,and deduce a system ofsecond order el l i pti c parti aldi erenti alequati ons w hi ch i s a reduced versi on ofthe B ogom ol ' nyisystem . In the secti on 2 by a vari ati onalm ethod we prove a generalexi stence oftopol ogi calsol uti ons. T hen,we i ntroduce the noti on ofadm i ssi bl e topol ogi calsol uti on. In the secti on 3,4 and 5 we prove exi stence ofadm i ssi bl etopol ogi calsol uti onsby an i terati on m ethod.In the secti on 5,i n parti cul ar,we establ i sh exponenti al decay esti m atesforoursol uti ons.In thesecti on 6 weprovestrong convergence ofthe adm i ssi bl e topol ogi calsol uti ons to the A bel i an H i ggs sol uti on. T he l ast secti on consi ders the C hern-Si m ons l i m i t,and we prove that adm i ssi bl e topol ogi cal sol uti ons are weakl y consi stent to the C hern-Si m ons equati on i n thi s l i m i t. (A fter nishing this work,we found thatthere was a study on the non rel ativistic version ofour m odelby Spruck-Yang in [7].) P relim inaries T he Lagrangi an densi ty forthe (rel ati vi sti c)sel f-dualM axwel l -C hern-Si m ons-H i ggs system i n (2 + 1)-D Lagrangi an densi ty m odel ed by C . Lee etal [ 5]i s L = 1 4 F F + 4 F A + D D + 1 2 @ N @ N q 2 N 2 j j 2 1 2 (qj j 2 + N q) 2(1) w here i sa com pl ex scal ar el d,N i sa realscal ar el d,A = (A 0 ;A 1 ;A 2 ) i s a vector el d, F = @ A @ A , D = @ iqA , = 0;1;2, @ 0 = @ @t ,@ j = @ @x j ,j = 1;2,q > 0 i s the charge ofel ectron,and > 0 i s a coupl i ng constant for C hern-Si m ons term . T he acti on functi onal for thi s system i s gi ven by A = Z R 3 Ldx: ( T he stati c energy functi onalfor the above system i s E = Z R 2 j D 0 j 2 + j D j j 2 + 1 2 F 2 j0 + 1 2 F 2 12 + 1 2 (@ j N ) 2 + q 2 N 2 j j 2 + 1 2 (qj j 2 + N q) 2 dx:(3) w i th the G auss l aw constrai nt ( + 2q 2 j j 2 )A 0 = F 12 :(4) T hi si sthe Eul er-Lagrange equati on w i th vari ati on ofthe acti on taken w i th respect to A 0 . Integrati ng by parts,usi ng (4), we obtai n from the energy functi onal E = Z R 2 j (D 1 iD 2 ) j 2 + j D 0 iq N j 2 + 1 2 (F j0 @ j N ) 2 + 1 2 j F 12 (qj j 2 + N q)j 2 dx q Z R 2 F 12 dx:(5) T hi s i m pl i es the l ower bound for the energy E q Z R 2 F 12 dx ; w hi ch i s saturated by the sol uti ons ofthe equati ons(the B ogom ol ' nyi equati ons) for ( ;A ;N ) A 0 = N (6) (D 1 iD 2 ) = 0 (7) F 12 (qj j 2 + N q)= 0 (8) A 0 = ( q(1 j j 2 )+ 2 A 0 )+ 2q 2 j j 2 A 0(9) w here the upper(l ower) si gn corresponds to posi ti ve(negati ve) val ues of R R 2 F 12 dx,and (9) fol l ow s from the G auss l aw com bi ned w i th (8). If( ;A ;N ) i s a sol uti on that m akes E ni te,then ei ther 2n j arg(z z j ); n j 2 Z + ; w here z = (x;y) i s the canoni cal coordi nates i n R 2 , and each z j = (x j ;y j ) i s a zero of w i th w i ndi ng num ber n j ,w hi ch corresponds to the m ul ti pl i ci ty ofthe j th vortex. A ftersi m i l arreducti on procedure si m i l ar to [ 10] , we obtai n the equati ons(we have chosen the upper si gn. ) n j l n j z z j j 2 1 + j z z j j 2 ! ; and we set u = v + f to rem ove the si ngul ar i nhom ogeneous term i n (10). T hen (10) and (11) u = 2q 2 (e u 1) 2q A 0 + 4 m X j= 1 n j (z z j )(10)becom e v = 2q 2 (e v+ f 1) 2q A 0 + g;(12) w here g = m X j= 1 4n j (1 + j z z j j 2 ) 2 E xistence of a V ariationalSolution Sol vi ng (12) for A 0 ,and substi tuti ng thi s i nto (13),we obtai n 2 v ( 2 + 4q 2 e v+ f ) v + 4q 4 e v+ f (e v+ f 1) = 2q 2 j r (v + f)j 2 e v+ f 4q 2 ge v+ f + g 2 g(15) Ifwe form al l y set = 0 i n thi s equati on,then we have ( v 2q 2 (e v+ f 1) g)= 0 w hi ch,i fwe ask v 2 H 2 (R 2 ),recovers the A bel i an H i ggs system studi ed i n [ 10] . O n the other hand, i f we take the l i m i t ;q ! 1 w i th q 2 = = l xed num ber,then after form al l y droppi ng the l ower order term s i n o(q) and o( ),we obtai n v = 4l 2 e v+ f (e v+ f 1)+ g: T hi s i s the equati on correspondi ng to the pure C hern-Si m ons system studi ed i n [ 8] , [ 11] ,etc. Later i n thi s paper we provi de ri gorous justi cati ons for these two l i m i ti ng behavi orsofthesol uti ons.B y a di rectcal cul ati on we nd that the equati on (15) i s a vari ati onalequati on ofthe fol l ow i ng functi onal . F (v) = Z 1 2 j vj 2 ( g 2 g)v + 2q 4 (e v+ f 1) 2 + 1 2 2 j r vj 2 + 2q 2 e v+ f j r (v + f)j 2 dx(16) T heabovefuncti onali swel l -de ned i n H 2 (R 2 )si ncee f j r fj 2 2 L 1 (R 2 ). W e now prove exi stence ofsol uti on of (15) i n H 2 (R 2 ). Further regul ari ty then v 2 H 2 (R 2 ) fol l ow s from the standard regul ari ty resul ts for the nonhom ogeneous bi harm oni c equati ons. T heorem 1 T he functional(16) is coercive,and weakl y l ower sem icontinuous in H 2 (R 2 ), and thus there is a gl obal m inim izer of the functional(16) in H 2 (R 2 ). Proof: Ifa sequence fv k g converges to v weakl y i n H 2 (R 2 ),v k ! v strongl y i n L 1 (B R ) and i n H 1 (B R ) for any bal lB R = B (0;R ) R 2 by R el l i ch' s com pactness theorem . T huswe observe that to prove the l ower sem i -conti nui ty of the functi onal (16), i t i s su ci ent to prove the l ower sem i -conti nui ty of R R 2 e v+ f j r (v + f)j 2 dx. W e have l i m i nf k! 1 Z R 2 e v k + f j r (v k + f)j 2 dx l i m i nf k! 1 Z B R e v k + f j r (v k + f)j 2 dx = Z B R e v+ f j r (v + f)j 2 dx: Letti ng R ! 1 ,we obtai n the desi red weak l ower sem i -conti nui ty. O n the other hand,we note that the coerci vi ty ofF i n H 2 (R 2 ),i s a si m pl e corol l ary ofthe i nequal i ty: kvk 2 L 2 (R 2 ) C (1 + k vk 2 L 2 (R 2 ) + ke v+ f 1k 2 L 2 (R 2 ) ); (17) si nce we have j Z ( g 2 g)vdxj C + kvk 2 L 2 (R 2 ) for any > 0,and by the C al deron-Zygm und i nequal i ty we have kD 2 vk L 2 (R 2 ) C k vk L 2 (R 2 ) : Forthe proofof(17),we justrecal lthe i nequal i ty (4. 10)i n [ 11] ,w hi ch i m m edi atel y i m pl i es; kvk 2 L 2 (R 2 ) C (1 + kr vk 2 L 2 (R 2 ) + ke v+ f 1k 2 L 2 (R 2 ) ): N ow ,for any > 0 we have Z j r vj 2 dx = Z v vdx kvk 2 L 2 (R 2 ) + C k vk 2 L 2 (R 2 ) : Taki ng sm al lenough,(17) fol l ow s. T hi s com pl etes the proofofthe theorem . P roposition 1 Let (v;A 0 ) be any topol ogical sol ution of (12)-(13), and v q a be the nite energy sol ution ofthe A bel ian H iggs system . T hen the foll owing conditions are equival ent. (i) A 0 0 (ii) v f (iii) A 0 q (e v+ f 1) (iv) v v q a . Proof: (i)) (ii),(i)) (iv): W e assum e A 0 0. Letv q a be the sol uti on ofthe A bel i an H i ggs system ,i . e. v q a sati s es v q a = 2q 2 (e v q a + f 1)+ g:(18) T he exi stence and uni queness of v q a 2 H 2 (R 2 )\ C 1 (R 2 ) sati sfyi ng v q a f i s wel l -know n[ 10] . From (12) w i th A 0 0 we have v 2q 2 (e v+ f 1)+ g: T hus, (v q a v) 2q 2 (e v q a f e v+ f )= 2q 2 e + f (v q a v) by the m ean val ue theorem ,w here i s between v and v ;q . B y the m axi m um pri nci pl e we have v v q a f: (ii)) (i): W e assum e v f. From (13) we have A ( 2 + 2q 2 e v+ f )A : T hus (i ) fol l ow s from the m axi m um pri nci pl e. (i)) (iii): W e assum e A 0 0. Set G = q (1 e v+ f ). T hen, we com pute. G = q j r (v + f)j 2 e v+ f q (v + f)e v+ f q e v+ f [ 2q 2 (e v+ f 1) 2q A 0 ] = 2q 2 e v+ f G + 2q 2 A 0 T hus,we have (G + A 0 ) ( 2 + 2q 2 e v+ f )(G + A 0 )+ 2q 2 A 0 ( 2 + 2q 2 e v+ f )(G + A 0 ): Si nce G ! 0 as j zj ! 1 , we have G + A 0 by the m axi m um pri nci pl e. Iteration Schem e In thi s secti on we construct an approxi m ate m ul ti -vorti ces sol uti on sequence ofour B ogom ol ' nyiequati ons by an i terati on schem e. Later thi s approxi m ate sol uti on sequence w i l l be found to converge to an adm i ssi bl e topol ogi cal sol uti on. O ur i terati on schem e i s si m i l ar to that of [ 8] ,but i s substanti al l y extended i n form . D e nition 2 W e set v 0 = v q a , A 0 0 = 0, where v q a is the nite energy sol ution of the A bel ian H iggs system . D e ne (v i ;A i 0 ) 2 H 2 (R 2 ) \ C 1 (R 2 ),i 1 iterativel y as foll ows: Firstde ne v i from (v i 1 ;A i 1 0 ) by sol ving: ( d)v i = 2q 2 (e v i 1 + f 1) 2q A i 1 0 + g dv i 1 ;(19) and then de ne A i 0 from (v i ;A i 1 0 ) by sol ving: ( 2 2q 2 e v i + f d)A i 0 = q(1 e v i + f ) dA i 1 0 :(20) H ere,d 2q 2 is a constant that willbe xed l ater. Lem m a 1 T he schem e (19)-(20) is well -de ned, and the iteration sequence (v i ;A i 0 ) satis es the m onotonicity,i.e. v i v i 1 v 0 f A i 0 A i 1 0 A 0 0 = 0: Proof: W e proceed by an i nducti on. For i= 1 we have from (19) ( d)v 1 = 2q 2 (e v 0 + f 1)+ g dv 0 O n the other hand v 0 sati s es v 0 = 2q 2 (e v 0 + f 1)+ g: T hus we have ( d)(v 1 v 0 )= 0: From thi s we obtai n v 1 = v 0 2 H 2 (R 2 )\ C 1 (R 2 ),and obvi ousl y v 1 v 0 f: N ow from (20) for i= 1 we have ( 2 2q 2 e v 1 + f d)A 1 0 = q(1 e v 1 + f ):(21) U si ng the m ean val ue theorem ,we obtai n Z R 2 (1 e v 1 + f ) 2 dx Z R 2 (v 1 + f) 2 e + f dx Z R 2 (v 1 + f) 2 dx < 1 ; w here v 1 < < f,and we used the fact f 2 L 2 (R 2 ). W e al so have 0 e v 1 + f < 1. T hus, by the standard resul t of the l i near el l i pti c theory the equati on (21) de nes A 1 0 2 H 2 (R 2 )\ C 1 (R 2 ). Furtherm ore, si nce q(1 e v 1 + f ) 0; by the m axi m um pri nci pl e appl i ed to (21) we have A 1 0 A 0 0 = 0: T husLem m a 1 i strue fori= 1. Supposethe l em m a i strue up to i 1. C l earl y (19)-(20)de ne(v i ;A i 0 )2 H 2 \ C 1 (R 2 )from (v i 1 ;A i 1 0 ). W e onl y need to observe that Z R 2 (e v i + f 1) 2 dx = Z R 2 (v i + f) 2 e + f dx < 1 and 0 e v i + f 1 i fv i 1 2 L 2 (R 2 ) and v i 1 f. W e al so have ( d)(v i v i 1 ) = 2q 2 (e v i 1 + f e v i 2 + f ) 2q (A i 1 A i 2 ) d(v i 1 v i 2 ) d(e v i 1 + f e v i 2 + f (v i 1 v i 2 )) = d(e f+ 1)(v i 1 v i 2 ); w here v i 1 + f v i 2 + f,v i v i 1 . O n the other hand, ( d)(A i 0 A i 1 0 ) = q(e v i + f e v i 1 + f ) + ( 2 + 2q 2 e v i + f )(A i 0 A i 1 0 ) + 2q 2 (e v i + f e v i 1 + f )A i 1 0 d(A i 1 0 A i 2 0 ) ( 2 + 2q 2 e v i + f )(A i 0 A i 1 0 ); w here we used the assum pti on that our l em m a hol ds up to i 1,and v i v i 1 . T herefore,by the m axi m um pri nci pl e, A i 0 A i 1 0 Lem m a 1 i s thus proved. Lem m a 2 T he iteration sequence (v i ;A i 0 ) satis es the inequal ity A i 0 q (e v i + f 1): for each i= 0;1;2; . Proof: W e use i nducti on agai n. Lem m a 2 i s true for i= 1. W e set G i = q (1 e v i + f ). Suppose Lem m a 2 hol ds for i 1,then G i = q r (r (v i + f)e v i + f ) = q j r (v i + f)j 2 e v i + f q (v i + f)e v i + f q e v i + f (2q 2 (e v i 1 + f 1) 2q A i 1 0 + d(v i v i 1 )) 2q 2 e v i 1 + f G i + q (e v i + f e v i 1 + f q de v i + f (v i v i 1 ) 2q 2 e v i 1 + f G i q de v i + f (v i v i 1 ) by A i 0 0,i 0. T herefore ( d)(G i + A i 0 ) ( 2 + 2q 2 e v i + f )(A i 0 + G i ) dG i dA i 1 0 q de v i + f (v i v i 1 ) ( 2 + 2q 2 e v i + f )(A i 0 + G i ) d(A i 1 0 + G i + q (e v i + f e v i 1 + f ); w here we used the m ean val ue theorem i n the l ast step, and used the fact v i v i 1 . R ew ri ti ng i t,we have ( 2 2q 2 e v i + f d)(G i + A i 0 ) d(A i 1 0 + G i 1 ) 0 by the i nducti on hypothesi s. B y m axi m um pri nci pl e we have A i 0 + G i 0. T hi s com pl etes the proofofLem m a 2. M onotonicity of F (v i ) In thi s secti on we w i l lprove the fol l ow i ng: Lem m a 3 Letfv i g given asin D e nition 2 and F (v)isgiven in (16). W e have F (v i ) F (v i 1 ) F (v 0 ):(22) To prove thi s we rstl y begi n w i th: Lem m a 4 Let(v i ;A i 0 ) be as in D e nition 2,then Z R 2 j 1 e v i + f + q A i 0 j+ d 2q 2 j v i v i+ 1 j dx = 2 q 2 m X j= 1 n j for alli 0. From Lem m a 1 and 2 we have 1 e v i + f + q A i 0 ; v i v i+ 1 0: W e onl y need to prove Z R 2 1 e v i + f + q A i 0 + d 2q 2 (v i v i+ 1 ) dx = 1 2q 2 Z R 2 g dx: Fi x R > 0. Integrati ng (19) over B R = fj zj< R g,we obtai n Z B R (1 e v i + f + q A i 0 )+ d 2q 2 (v i v i+ 1 ) dx = 1 2q 2 Z B R (g v i+ 1 )dx (23) B y di vergence theorem Z B R v i = Z @B R @v i @r d : W e note that v i 2 H 1 (R 2 ). T hus Z @B R j r v i j d 2 R Z @B R j r v i j 2 d 1=2(24) by H ol der' s i nequal i ty. Let H (r)= Z @B r j r v i j 2 d ; then Z R 2 j r v i j 2 dx = Z 1 0 H (r)dr < + 1 T herefore there exi sts an i ncreasi ng sequence ofradi i ,fr k g 1 k= 1 ,such that l i m k! 1 r k = + 1 ;and H (r k )< o(r k ) r k O therw i se,there exi sts > 0 andr > 0 such that H (r)> r for r >r, but then R R 2 j r v i j 2 dx = 1 . T hus (24) i m pl i es Z @B r k j r v i j d (2 r k H (r k )) 1=2 (2 o(r k )) 1=2 : T herefore l i m k! 1 Z @B r k j r v i j d = 0: C hoose R = r k ,and l et k ! 1 i n (23),then we have l i m k! 1 Z B r k 1 e v i + f + q A i 0 + d 2q 2 (v i v i+ 1 ) dx = 1 2q 2 Z R 2 g dx: T hi s,together w i th Z R 2 gdx = m X j= 1 8 n j Z 1 0 rdr (1 + r 2 ) 2 = 4 m X j= 1 n j com pl etes the proofofthe l em m a. A sa corol l ary ofLem m a 4,wecan getthefol l ow i ng uni form bound. C orollary 1 Let(v i ;A i 0 ) be as in D e nition 1,and de ne S = sup i 1 Z R 2 e v i + f j r (v i + f)j 2 dx 1 2(25) T hen S (4 ProofofLem m a 3: From (16) we have e v i + f j r (v i + f)j 2 dx = Z R 2 d(v i 1 v i )e v i + f + 2q 2 e v i + f (1 e v i 1 + f + q A i 1 0 ) dx Z R 2 d(v i 1 v i )+ 2q 2 (1 e v i 1 + f + q A i 1 0 ) dxF (v i 1 ) F (v i ) = Z R 2 1 2 j (v i v i 1 )j 2 v i (v i v i 1 ) + ( g 2 g)(v i v i 1 )+ 2 2 j r (v i v i 1 )j 2 2 r (v i v i 1 ) r v i + II + III dx; w here II = 2q 4 (e v i + f 1) 2 (e v i 1 + f 1) 2 III = 2q 2 e v i + f j r (v i + f)j 2 e v i 1 + f j r (v i 1 + f)j 2 : W e al so set I = v i (v i v i 1 )+ ( g 2 g)(v i v i 1 ) 2 r (v i v i 1 ) r v i : T hen F (v i 1 ) F (v i ) = Z R 2 1 2 j (v i v i 1 )j 2 + 2 2 j r (v i v i 1 )j 2 + I + II + III dx:(26) W e rstl y esti m ate I. From (19) we have A i 1 0 = 1 2q 2q 2 (e v i 1 + f 1)+ g d(v i 1 v i v i ): Putti ng thi s i nto (20) after substi tuti ng iw i th i 1 i n (20),we have 2 v i (d + 2 + 2q 2 e v i 1 + f ) v i + (d 2 + 2q 2 de v i 1 + f )v i = (2q 2 e v i 1 + f d) v i 1 + 2q 2 j r (v i 1 + f)j 2 e v i 1 + f 4q 2 ge v i 1 + f + g 2 g + d( 2 + 2q 2 e v i 1 + f )v i 1 4q 4 e v i 1 + f (e v i 1 + f 1) 2q d(A i 1 0 A i 2 0 ): (27) M ul ti pl yi ng (27) by v i v i 1 ,i ntegrati ng by parts,we have Z R 2 h v i (v i v i 1 )+ 2 r (v i v i 1 ) r v i ( g 2 g)(v i v i 1 ) i dx = Z R 2 dj r (v i v i 1 )j 2 + (d 2 + 2q 2 de v i 1 + f )(v i v i 1 ) 2 + 2q d(A i 1 0 A i 2 0 )(v i v i 1 ) IV V dx; w here we set IV = 4q 4 e v i 1 + f (e v i 1 + f 1)(v i v i 1 ) V = 2q 2 e v i 1 + f (v i v i 1 ) (v i + v i 1 ) + j r (v i 1 + f)j 2 e v i 1 + f (v i v i 1 )+ 2ge v i 1 + f (v i v i 1 ) : R ecal l i ng thede ni ti on ofI and observi ng ( A i 1 0 A i 2 0 )(v i v i 1 ) 0, we get I+ IV + V dj r (v i v i 1 )j 2 + (d 2 + 2q 2 de v i 1 + f )(v i v i 1 ) 2 : (28) To cal cul ate I + II + III,we observe II IV = 4q 4 (v i v i 1 ) e v i 1 + f (e v i 1 + f 1) e + f (e + f 1) = 4q 4 (v i v i 1 )(v i 1 )e + f (2e + f 1); w hereweused m ean val uetheorem repeatedl y w i th v i v i 1 . T hus II IV 4q 4 (v i v i 1 ) 2 e v i 1 + f :(29) N ow we have III = 2q 2 [ (e v i + f e v i 1 + f )j r (v i + f)j 2 + e v i 1 + f (j r (v i + f)j 2 j r (v i 1 + f)j 2 )] = 2q 2 [ e + f (v i v i 1 )j r (v i + f)j 2 + e v i 1 + f r (v i v i 1 ) r (v i + v i 1 + 2f)] = V I + V II; V = 2q 2 e v i 1 + f (v i v i 1 )[ (v i + v i 1 + 2f) + j r (v i 1 + f)j 2 ] = V III + IX ; w here v i 1 v i by m ean val ue theorem . W e now cal cul ate III V = (V II V III) IX + V I. B y i ntegrati on by parts we obtai n Z R 2 [V II V III ] dx = 2q 2 Z R 2 h e v i 1 + f (v i v i 1 )r (v i 1 + f) r (v i + v i 1 + 2f) i dx = Z R 2 [ X ] dx; X IX = 2q 2 e v i 1 + f (v i v i 1 )r (v i + f) r (v i 1 + f) = X I: Si nce V I 0,we have V I + X I 2q 2 (v i v i 1 ) e v i + f r (v i + f) r (v i v i 1 ) + (e v i + f e v i 1 + f )r (v i + f) r (v i 1 + f) 2q 2 j v i v i 1 j e v i + f j r (v i + f)j j r (v i v i 1 )j + j e v i + f e v i 1 + f j j r (v i 1 + f)j 2 + j e v i + f e v i 1 + f j j r (v i v i 1 )j j r (v i 1 + f)j 2q 2 j v i v i 1 j (e v i + f j r (v i + f)j + e v i 1 + f j r (v i v i 1 )j j r (v i 1 + f)j ) + e + f j v i v i 1 j j r (v i 1 + f)j 2 by the m ean val ue theorem w here we used the factj e v i + f e v i 1 + f j e v i 1 + f i n the l ast step. W e use H ol der' s i nequal i ty and i nterpol ati on i nequal i ty to obtai n Z R 2 [V I + X I ] dx 2q 2 ke v i + f r (v i + f)k L 2 (R 2 ) + k e v i 1 + f r (v i 1 + f)k L 2 (R 2 ) k(v i v i 1 )k L 1 (R 2 ) kr (v i v i 1 )k L 2 (R 2 ) 2q 2 kv i v i 1 k 2 L 1 (R 2 ) Z R 2 e v i + f j r (v i + f)j 2 dx 4q 2 C Skv i v i 1 k 1 2 L 2 (R 2 ) k (v i v i 1 )k 1 2 L 2 (R 2 ) kr (v i v i 1 )k L 2 (R 2 ) 2q 2 C S 2 kv i v i 1 k L 2 (R 2 ) k (v i v i 1 )k L 2 (R 2 ) ; w here C i s an absol ute constant and we set S = sup i 1 Z R 2 e v i + f j r (v i + f)j 2 dx 1 2 as i n C orol l ary 1. A ppl yi ng Young' s i nequal i ty,we have Z R 2 [III V ] dx C q 4 (S 2 + S 4 )kv i v i 1 k 2 L 2 (R 2 ) C q 2 Skr (v i v i 1 )k 2 L 2 (R 2 ) 1 4 k (v i v i 1 )k 2 L 2 (R 2 )(30) C om bi ni ng w i th (28),(29) and (30),(26) becom es F (v i 1 ) F (v i ) 1 4 k (v i v i 1 )k 2 L 2 (R 2 ) + (d 2 C )kv i v i 1 k 2 L 2 (R 2 ) + (d C )kr (v i v i 1 )k 2 L 2 (R 2 ) ; w here C i s an absol ute constant dependi ng on q and P m j= 1 n j . Taki ng d l arge enough,and usi ng the C al deron-Zygm und i nequal i ty,we have nal l y F (v i 1 ) F (v i ) C kv i v i 1 k H 2 (R 2 ) ; w hi ch i s a stronger form of(22). T hi s com pl etes the proofofLem m a 3. C orollary 2 Letv be any adm issibl e topol ogicalsol ution of(12)-(13), and v q a be the nite energy sol ution ofthe A bel ian H iggssystem . T hen, we have F (v) F (v q a ): Proof: Just substi tute v i = v;v i 1 = v q a i n the proof of Lem m a 3, and i nstead ofLem m a 4 we use Z R 2 j 1 e v+ f + q A 0 j dx = 1 2q 2 Z R 2 g = 2 q 2 m X j= 1 n j ; w hi ch fol l ow si m m edi atel y from i ntegrati on of(12)and Proposi ti on 1. E xistence ofA dm issible Solutionsand A sym ptotic D ecay B ased on the previ ous esti m ates for the i terati on sequence fv i g, i n thi ssecti on,we prove the exi stence ofadm i ssi bl e topol ogi calsol uti ons ofour B ogom ol ' nyiequati ons (12)-(13). W e al so establ i sh asym ptoti c exponenti al decay esti m ates of these sol uti ons as j zj ! 1 . A s a corol l ary of these decay esti m ates we prove that the acti on (2) and, hence the energy functi onal(3) are ni te. Fi rstl y we prove T heorem 2 G iven z j 2 R 2 ,n j 2 Z + with j = 1; ;m ,there exists a sm ooth sol ution ( ;A ;N ) to (6)-(9) such that = 0 ateach z = z j with corresponding winding num bers n j ,and satisfying 0 1 j j 2 q N = q A 0 (31) for allq; > 0 Proof: B y (22) the m onotone decreasi ng sequence fv i g sati s es F (v i ) F (v 0 )< 1 8i= 1;2; : T hi s i m pl i es by (17) and (16) that kv i k H 2 (R 2 ) < C F (v i ) C F (v 0 ) 8i= 1;2; : (32) T hus sup i 0 kv i k H 2 (R 2 ) < 1 : O n the other hand,from (19) A i 0 = 1 2q h 2q 2 (e v i + f 1)+ g + d(v i+ 1 v i ) v i+ 1 i w hi ch bel ongs to L 2 (R 2 ) uni form l y by (32). T hus,sup i 0 kA i 0 k L 2 (R 2 ) < 1 ,and sup i 0 k A i 0 k L 2 (R 2 ) < 1 by (20). C om bi ni ng thi s w i th the C al deron-Zygm und i nequal i ty and the standard i nterpol ati on i nequal i ty,we obtai n sup i 0 (v + f + i ) w here = P m j= 1 2n j arg(z z j ). For for = 1 2 (A 1 iA 2 ) and @ z = 1 2 (@ 1 i@ 2 ) we al so de ne = i@ z ln : kA i 0 k H 2 (R 2 ) < 1 T hus there exi sts v;A 0 2 H 2 (R 2 ) and a subsequence (v i ;A i 0 ) such that v i ! v and A i 0 ! A Expl i ci t com putati on show s A 1 = 1 2 (@ 2 v + @ 2 b);(35) and W e now establ i sh asym ptoti c exponenti aldecay esti m ates for adm i ssi bl e topol ogi calsol uti ons ofour B ogom ol ' nyiequati ons. A 2 = 1 2 ( @ 1 v @ 1 b);(36) T heorem 3 Let ( ;A 0 ;N ) be any adm issibl e topol ogical sol ution of the B ogom ol 'nyiequations (6)- (9). Suppose > 0 is given,then there exists r 0 = r 0 ( )> 0 and C = C such that R em ark: W e note that was chosen(see the proof bel ow ) so that 2 2 + 2 = ,thus 0 < < 1. 0 1 j j 2 ; j N j ; j F 12 j C e q(1 ) 1 2 jzj (37) j D j ; j r A 0 j C e q(1 ProofofT heorem 3: From (10) and (11), u 2 = 2j r uj 2 + 2u u 4q 2 (e u 1)u 4q A 0 u; A 2 0 = 2j r A 0 j 2 + 2A 0 A 0 2( 2 + 2q 2 e u )A 2 0 + 2 q(1 e u )A 0 for j zj> sup j fj z j j g. Let E = u 2 + 2A 2 0 ,then we have E 4q 2 e u 1 u u 2 + 2e u A 2 0 + 4 2 A 2 0 + 4 qA 0 (1 e u u) 4q 2 (u)E + 4 2 A 2 0 8 qA 0 u(39) w here we used the i nequal i ty t e t 1,and set (t)= m i nfe t ; e t 1 t g. N ote that 8 qj A 0 uj 4( 2 + 2 q 2 )A 2 0 + 4 2 2 + 2 q 2 q 2 u 2 = 4( 2 + 2 q 2 )A 2 0 + 4 2 2 + 2 q 2 u 2 = 4 2 A 2 0 + 4 q 2 E ; si nce 2 2 + 2 = . T hus,(39) becom es E 4( (u) )q 2 E(40) Si nce u ! 0 as j zj! 1 ,gi ven > 0,we can choose r 0 so l arge that (u) 1 on j zj> r 0 . T hus,by com pari ng E w i th the functi on and j F 12 j q(1 j j 2 )+ j N j ; w hi ch fol l ow s from (8). N ext, we esti m ate the asym ptoti c decay of j D i j 2 . W e observe j D j 2 = j (@ iqA ) j 2 = 1 4 e u j (@ (u + iq ) iq(@ ? u + @ )j 2 = 1 4 e u j @ u iq@ ? uj 2 ; w here we used the notati on (@ ? ) = (@ 0 ; @ 2 ;@ 1 ). T hus j D j 2 C j r uj 2 . T herefore i t i s su ci ent to have decay esti m ate for j r uj 2 . A di rect cal cul ati on gi ves j r uj 2 = 2j r 2 uj 2 + 2r u r u 4q 2 r u r (e u 1 q A 0 ) = 4q 2 e u j r uj 2 4q r u r A 0 j r A 0 j 2 = 2j r 2 A 0 j 2 + 2r A 0 r A 0 2( 2 + 2q 2 e u )j r A 0 j 2 (2q e u 4q 2 e u A 0 )r A 0 r u for j zj> sup j fj z j j g. W e set J = j r uj 2 + 2j r A 0 j 2 ,then we have J 4q 2 e u J + 4 2 j r A 0 j 2 (8 q + 4q 2 e u j A 0 j )j r uj j r A 0 j Proof: Si nce N = A 0 2 H 1 (R 2 ) and j j 1, Z R 2 (@ N ) 2 dx < 1 ; Z R 2 N 2 j j 2 dx < 1 : From (4) and (10) we have F 12 = qj j 2 N q 2 L 2 (R 2 ). C l earl y F 0i = @ i A 0 2 L 2 (R 2 ); i= 1; 2: T herefore Z R 2 j F j 2 dx < 1 : W e now consi der the C hern-Si m ons term . Fi rstl y we have Z R 2 j F 12 A 0 j dx Z R 2 j F 12 j 2 dx 1 2 Z R 2 j A 0 j 2 dx 1 2 < 1 : T hus i t su ces to prove F 01 A 2 = @ 1 A 0 A 2 ; F 02 A 1 = @ 2 A 0 A 1 2 L 1 (R 2 ):(42) Si nce A 1 ;A 2 2 L p (R 2 ) for al lp 2 (2;1 ]from (35)-(36),and r A 0 2 L q (R 2 ), for q 2 [ 1;1 ] from (38), (42) fol l ow s i m m edi atel y by the H ol der i nequal i ty. T herefore we al so have that F A 2 L 1 (R 2 ): Fi nal l y from (38) we have j D j 2 2 L 1 (R 2 ): T hi s com pl etes the proofofthe corol l ary. A belian H iggs Lim it In thi s secti on we prove that,for q xed,the sequence ofadm i ssi bl e topol ogi calsol uti ons,(v ;q ;A ;q 0 )convergesto (v q a ;0),as goesto zero, w here v q a i s the ni te energy sol uti on of the A bel i an H i ggs system . Fi rstl y we establ i sh: Lem m a 5 Let (v ;q ;A ;q 0 ) be any adm issibl e topol ogical sol ution of (12) and (13). T hen,for each xed q 2 (0;1 ),we have sup 0< < 1 kv ;q k H 2 (R 2 ) < 1 ; sup 0< < 1 kA ;q 0 k H 2 (R 2 ) :< 1(43) T hus,by the Sobol ev em bedding we have sup 0< < 1 kv ;q k L 1 (R 2 ) < 1 ; sup 0< < 1 kA ;q 0 k L 1 (R 2 ) < 1(44) Proof: Let 2 (0;1). From C orol l ary 2 and (17) we have kv ;q k H 2 (R 2 ) (1 + 2 )C 1 + C 2 F (v ;q ) (1 + 2 )C 1 + C 2 F (v q a ) C (1 + 2 ); w here C 1 ;C 2 and C are constants i ndependent of . T hus the rst i nequal i ty of (43) fol l ow s. N ow , taki ng L 2 (R 2 ) i nner product (13) w i th A ;q 0 ,we have after i ntegrati on by part Z R 2 j r A ;q 0 j 2 + ( 2 + 2q 2 e v ;q + f )j A ;q 0 j 2 dx = Z R 2 q(1 e v ;q + f )j A ;q 0 jdx 2 2 Z R 2 j A ;q 0 j 2 dx + q 2 2 Z R 2 (1 e v ;q + f ) 2 dx 2 2 Z R 2 j A ;q 0 j 2 dx + q 2 2 Z R 2 j v ;q + fj 2 dx: T hus,by Young' s i nequal i ty and the rsti nequal i ty i n (43) we obtai n Z R 2 j r A ;q 0 j 2 dx + Z R 2 ( 2 + 4q 2 e v ;q + f )j A ;q 0 j 2 dx C ;(45) w here C i s i ndependent of . From (13) and (45) we have Z R 2 j A ;q 0 j 2 dx 2 2 q 2 Z R 2 (1 e v ;q + f ) 2 dx + 2 Z R 2 ( 2 + 2q 2 e v ;q + f ) 2 j A ;q 0 j 2 dx 2 2 q 2 Z R 2 j v ;q + fj 2 e 2( + f) dx + 2( 2 + 2q 2 ) Z R 2 ( 2 + 4q 2 e v ;q + f )j A ;q 0 j 2 dx 2q 2 Z R 2 j v ;q + fj 2 dx + 2(1 + q 2 ) Z R 2 ( 2 + 4q 2 e v ;q + f )j A ;q 0 j 2 dx C fora constantC i ndependentof ,w here 2 (v ;q + f;0),and we used the m ean val ue theorem . T hus,by the C al deron-Zygm und i nequal i ty Z R 2 j D 2 A ;q 0 j 2 dx C :(46) B y Sobol ev' s em beddi ng for a bounded dom ai n, for any bal l B R = fj zj< R g R 2 ,we have. kA ;q 0 k L 1 (B R ) C ; w hereC i si ndependentof . W e take R > m ax 1 j m fj z j j+ 1g. T hen, Z R 2 j A ;q 0 j 2 dx = Z B R j A ;q 0 j 2 dx + Z R 2 B R j A ;q 0 j 2 dx R 2 kA ;q 0 k 2 L 1 (B R ) + ke v ;q + f k L 1 (R 2 B R ) Z R 2 B R e v ;q + f j A ;q 0 j 2 dx C 1 (R )+ C 2 ke f k L 1 (R 2 B R ) Z R 2 e v ;q + f j A ;q 0 j 2 dx C (R );(47) w here C 1 (R );C 2 and C (R ) are i ndependent of . C om bi ni ng (45)-(47), we obtai n the second i nequal i ty i n (43). T hi s com pl etes the proofofLem m a 5. T heorem 4 Let v ;q ;A ;q 0 be the adm issibl e topol ogical sol utions of (12) and (13), and v q a a nite energy sol ution of the A bel ian H iggs system . Letq be xed. For allk 2 Z + we have v ;q ! v q a ; and A ;q 0 ! 0 in H k (R 2 ): as ! 0. Proof: W e have by m ean val ue theorem (v ;q v q a )= 2q 2 (e v ;q + f e v q a + f ) 2 qA ;q 0 = 2q 2 e + f (v ;q v q a ) 2 qA ;q 0(48) w here 2 (v ;q ;v q a ). M ul ti pl yi ng (48) by v ;q v q a , we have after i ntegrati on by parts Z R 2 j r (v ;q v q a )j 2 + 2q 2 e + f (v ;q v q a ) 2 dx = 2 q Z R 2 A ;q 0 (v ;q v q a )dx 2 qkA ;q 0 k L 2 (R 2 ) kv ;q v q a k L 2 (R 2 ) C w here C i s i ndependent of by Lem m a 5. Si nce k k L 1 (R 2 ) kv ;q k L 1 (R 2 ) + kv q a k L 1 (R 2 ) C i ndependentl y of < 1,we have from the above esti m ate Z R 2 j r (v ;q v q a )j 2 + e f j v ;q v q a j 2 dx ! 0 as ! 0. Let = [ m j= 1 fj z z j j< g. N ow , Z j v ;q v q a j 2 dx m 2 kv ;q v q a k 2 L 1 (R 2 ) C 2 : w here C i s i ndependent of by Lem m a 5. T hus,for any gi ven > 0, we can choose i ndependentl y of so that Z j v ;q v q a j 2 dx 2 : For such we have Z R 2 j v ;q v q a j 2 dx = Z j v ;q v q a j 2 dx + Z R 2 j v ;q v q a j 2 dx 2 + sup R 2 fe jfj g Z R 2 e f j v ;q v q a j 2 dx 2 + 2 = for su ci entl y sm al l ,i . e. Z R 2 j v ;q v q a j 2 dx ! 0 as ! 0. C om bi ni ng the above resul ts,we obtai n v ;q ! v q a i n H 1 (R 2 ) as ! 0: N ow we prove the convergence for A ;q 0 . M ul ti pl yi ng (13) by A ;q 0 and i ntegrati ng,we esti m ate Z R 2 j r A ;q 0 j 2 + ( 2 + 2q 2 e v ;q + f )j A ;q 0 j 2 dx q Z R 2 (1 e v ;q + f )j A ;q 0 jdx qkA ;q 0 k L 2 (R 2 ) k1 e v ;q + f k L 2 (R 2 ) C kv ;q + fk L 2 (R 2 ) C : w here we used Lem m a 5 i n the rst and thi rd step and use the fact 1 e t t for t 0 i n the second step. U si ng the fact j v ;q j < C uni form l y i n < 1,we obtai n from thi s Z R 2 j r A ;q 0 j 2 + e f j A ;q 0 j 2 dx ! 0 as ! 0. Si nce j A ;q 0 j< C uni form l y i n < 1,by Lem m a 5 we can deduce A ;q 0 ! 0 i n H 1 (R 2 ) si m i l arl y to the case ofv ;q . From these resul tstogetherw i th uni form boundskv ;q k;kA ;q 0 k C ,appl yi ng the standard el l i pti c regul ari ty to (48) and (13) repeatedl y,we obtai n (v ;q ; A ;q 0 )! (v q a ; 0) i n [ H k (R 2 )] 2 ; 8k 1 C hern-Sim ons Lim it In thi s secti on we study the behavi ors ofv ;q ;A ;q 0 as ;q ! 1 w i th l= q 2 = kept xed for the adm i ssi bl e topol ogi calsol uti ons. A l though we coul d notobtai n the strong convergence to a sol uti on ofthe C hern-Si m ons equati on, i nstead, we w i l l prove that the sequence fv ;q g i s "weakl y approxi m ati ng" the C hern-Si m ons equati on: v = 4l 2 e v+ f (e v+ f 1)+ g: W e denote l= q 2 = the xed num ber,and ;q = qA ;q 0 throughout thi s secti on. T heorem 5 Letf(v ;q ;A ;q 0 )g be a sequence ofadm issibl e topol ogical sol utions of(12)-(13). For any 2 C 1 0 (R 2 ) we have l i m ;q! 1 Z R 2 h v ;q 4l 2 e v ;q + f (e v ;q + f 1) g i dx = 0:(49) For proof of thi s theorem we rstl y establ i sh the fol l ow i ng l em m a w hi ch i s i nteresti ng i n i tsel f. Lem m a 6 Letf(v ;q ;A ;q 0 )g be given as in T heorem 5. For any xed p 2 [ 1;1 ) we have l i m ;q! 1 k ;q l(e v ;q + f 1)k L p (R 2 ) = 0: Proof: Fi rstl y we have from v ;q + f 0 ke v ;q + f 1k L 1 (R 2 ) 1: A l so,from 0 ;q l(e v ;q + f 1), k ;q k L 1 (R 2 ) lke v ;q + f 1k L 1 (R 2 ) l: From (12) we have Z R 2 j ;q l(e v ;q + f 1)j dx = Z R 2 h ;q l(e v ;q + f 1) i dx = 1 q Z R 2 gdx: T hus l i m ;q! 1 k ;q l(e v ;q + f 1)k L 1 (R 2 ) = 0: B y a standard i nterpol ati on i nequal i ty k ;q l(e v ;q + f 1)k L p (R 2 ) k ;q l(e v ;q + f 1)k 1 p L 1 (R 2 ) k ;q l(e v ;q + f 1)k 1 1 p L 1 (R 2 ) (2l) 1 1 p k ;q l(e v ;q + f 1)k 1 p L 1 (R 2 ) ! 0 as ;q ! 1 w i th q 2 = = l xed. ProofofT heorem 5: From (13) added by (14) q= we obtai n (v ;q + 2l q ;q )= g + 4le v ;q + f ;q : M ul ti pl yi ng 2 C 1 0 (R 2 ),and i ntegrati ng by parts,we obtai n j ;q l(e v ;q + f 1)j dx = 0: Z R 2 v ;q = 4l 2 Z R 2 h e v ;q + f (e v ;q + f 1)+ g i dx + 2l q Z R 2 ;q dx + 4l Z R 2 e v ; T hus,T heorem 5 fol l ow s. R em ark: Ifwe coul d have uni form L 1 (R 2 ) esti m ate of r v ;q ,then we coul d prove exi stence ofsubsequence fv ;q g and i ts L q loc (R 2 ) (1 q < 2)-l i m i t v such that v i s a sm ooth sol uti on ofthe C hern-Si m ons equati on. 2 ! 1 and N = A 0 ! 0 as j xj! 1 . T he form er i s cal l ed non-topol ogi cal , and the l atter i s cal l ed topol ogi cal . In thi s paper we are consi deri ng onl y topol ogi cal boundary condi ti on. A 0 = q(1 e v+ f )+ ( 2 + 2q 2 e v+ f )A 0 20 ( iv)) (ii): T hi s i s obvi ous,and i ncl uded i n the above proof.(iii)) (i): A ssum i ng (i i i ),we have from (13) A 0 2q 2 e v+ f A 0 : T hus,(i )fol l ow sagai n by the m axi m um pri nci pl e. T hi scom pl etesthe proofofthe proposi ti on. D e nition 1 W e calla topol ogicalsol ution (v;A 0 )satisfying any one of the four conditions in Proposition 1 by an adm issibl e topol ogical sol ution. and we used the m ean val ue theorem . Si nce e w i 1 and v i 1 v i 2 0 by i nducti on hypothesi s,we obtai n ( d)(v i v i 1 ) 0:A ppl yi ng m axi m um pri nci pl e agai n,we have prove our m ai n l em m a i n thi s secti on. 0 both weakl y i n H 2 (R 2 )and strongl y both i n H 1 loc (R 2 )and i n L 1 loc (R 2 ) by R el l i ch' scom pactnesstheorem . T hel i m i tsv;A 0 2 H 2 (R 2 )sati s es (12)-(13) i n the weak sense, and by repeatedl y usi ng the standard l i nearel l i pti c regul ari ty resul twe have v;A 0 2 C 1 (R 2 ):M oreover,by constructi on we have ng ourreducti on procedurefrom (12)-(13)to theB ogom ol ' nyi equati ons (6)-(9), we we nd that the el ds A ; ;N ( = 0;1;2) sati sfy the B ogom ol ' nyiequati ons (6)-(9). In parti cul ar (32) fol l ow s i m m edi atel y from (34) and (35). i n j zj r 0 ,usi ng the m axi m um pri nci pl e, we deduce j uj 2 ; j A 0 j 2 C e on j zj> r 0 ,w here C was xed to com pare E w i th on fj zj= r 0 g.(37) fol l ow s from the fact 0 1 j j 2 = 1 e u j uj ; N = A 0 ; 0 j )J + 4 2 j r A 0 j 2 8 qj r uj j r A 0 j ; (41) w here we used j r uj j r A 0 j 1=2J i n the second i nequal i ty. (41) i s the sam e form as (39),observi ng i n case of(41) we have e u (1 1 2 j A 0 j )! 1 as j zj! 1 : G i ven > 0, we appl y Young' s i nequal i ty to the term j r uj j r A 0 j si m i l arl y to the previ ous case,and get J 4q 2 (1 )J w hen j zj> r 0 for r 0 l arge enough. T he above equati on i s the sam e as (40). T hus J sati s es the esti m ate (38). T hi s com pl etes the proofofT heorem 5. N ow we com pl ete proofofour exi stence theorem by provi ng that the sol uti onsconstructed i n T heorem 2 m ake ouracti on i n (2) ni te. T hi s fol l ow s i fwe prove thatany adm i ssi bl e topol ogi calsol uti on m akes the acti on ni te. C orollary 3 Let (A ; ;N ) be the sol ution of the B ogom ol 'nyi equations (6)-(9) constructed T heorem 2,then we have A = A (A ; ;N )< 1 : q + f [ ;q l(e v ;q + f 1)] dx: ;q + f [ ;q l(e v ;q + f 1)] dx k k L 1 (R 2 ) d l i ke to deepl y thank to ProfessorC hoonkyu Lee for i ntroduci ng theprobl em si ssued i n thi spaperforthem ,and m any hel pfuldi scussi ons.T hi sresearch i ssupported parti al l y by K O SEF(K 94073, K 95070),B SR I(N 94121),G A R C -K O SEF and SN U (95-03-1038). Sel f-D ual C hern-Sim ons T heories, Spri nger Lecture N ote i n Physi cs. G Unne, M. 36G . D unne, Sel f-D ual C hern-Sim ons T heories, Spri nger Lecture N ote i n Physi cs,M 36, (1995). . J Ong, Y , P Y Pac, Phys.R ev.Lett. 642230J.H ong,Y .K i m and P.Y .Pac,Phys.R ev.Lett.64,pp.2230, (1990). . R , Jacki , E J , Phys.R ev.Lett. 642234R .Jacki w and E.J.W ei nberg,Phys.R ev.Lett.64,2234,(1990) . B H Lee, C Lee, H , Phys.R ev.D. 454588B .H .Lee,C .Lee and H .M i n.Phys.R ev.D ,45,pp.4588 (1990). Sel f-D ual M axwell C hern-Sim ons sol itons. C Lee, K Lee, H , Phys.Lett.B. 252C . Lee, K . Lee and H . M i n, Sel f-D ual M axwell C hern-Sim ons sol itons,Phys.Lett.B ,252,pp.79-83(1990). . C Lee, K Lee, E J , Phys.Lett.B. 243105C .Lee,K .Lee and E.J.W ei nberg,Phys.Lett.B 243,pp.105- (1990). Existence T heorem s for Periodic N onrel ativistic M axwell -C hern-Sim ons Sol itons. J Spruck, Y Yang, to appear i n J.D i . Eqns.J. Spruck and Y . Yang, Existence T heorem s for Periodic N on- rel ativistic M axwell -C hern-Sim ons Sol itons,to appear i n J.D i . Eqns.(1996). Topol ogical Sol utions in the Sel f-D ual C hern-Sim ons T heory: Existence and A pproxim ation. J Spruck, Y Yang, A nn.Inst. H enriPoi ncar e. 12J. Spruck and Y . Yang, Topol ogical Sol utions in the Sel f-D ual C hern-Sim ons T heory: Existence and A pproxim ation,A nn.Inst. H enriPoi ncar e,12,pp.75-97(1995) T he Existence ofN on-Topol ogicalSol itons in the Sel f-D ualC hern-Sim ons T heory. J Spruck, Y Yang, C om m .M ath.Phys. 149J.Spruck and Y .Yang,T he Existence ofN on-Topol ogicalSol itons in the Sel f-D ualC hern-Sim ons T heory,C om m .M ath.Phys.149, pp.361-376(1992). . A , C Taubes, Vorti Oston, A . Ja e and C . Taubes, Vorti ces and M onopol es, B i rkh auser, B oston,1980. T he Existence ofC hern-Sim onsVortices. R , C om m .M ath. Phys. 137R .W ang,T he Existence ofC hern-Sim onsVortices,C om m .M ath. Phys. ,137,pp.587-597(1991) A brikosov's Vortices in the C riticalC oupl ing. S , Y Yang, SIA M J.M ath.A nal. 23S.W ang and Y .Yang,A brikosov's Vortices in the C riticalC ou- pl ing,SIA M J.M ath.A nal .Vol .23,1992,pp.1125-1140.
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[ "Tile Hamiltonians for Decagonal Phases", "Tile Hamiltonians for Decagonal Phases" ]
[ "Michael Widom \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA\n", "Ibrahim Al-Lehyani \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA\n", "Marek Mihalkovic \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA\n" ]
[ "Department of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA", "Department of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA", "Department of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA" ]
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A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions between and within tiles. We study Al-Co-Ni and Al-Co-Cu decagonal quasicrystals described as decorated Hexagon-Boat-Star (HBS) tiles using ab-initio methods. A dominant term in the TH counts the number of H, B and S tiles, favoring tilings of H and B only. In our model for Al-Co-Cu, chemical ordering of Cu and Co along tile edges defines tile edge arrowing. Unlike the edge arrowing of Penrose matching rules, however, the energetics for Al-Co-Cu do not force quasiperiodicity. Energetically favored structures resemble crystalline approximants to which the actual quasicrystalline compounds transform at low temperature.
10.1016/j.jnoncrysol.2003.11.018
[ "https://export.arxiv.org/pdf/cond-mat/0212605v1.pdf" ]
44,488,530
cond-mat/0212605
e591cafdb1e408dec54cc6c42be898538732ecbc
Tile Hamiltonians for Decagonal Phases arXiv:cond-mat/0212605v1 26 Dec 2002 March 22, 2022 Michael Widom Department of Physics Carnegie Mellon University 15213PittsburghPAUSA Ibrahim Al-Lehyani Department of Physics Carnegie Mellon University 15213PittsburghPAUSA Marek Mihalkovic Department of Physics Carnegie Mellon University 15213PittsburghPAUSA Tile Hamiltonians for Decagonal Phases arXiv:cond-mat/0212605v1 26 Dec 2002 March 22, 2022 A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions between and within tiles. We study Al-Co-Ni and Al-Co-Cu decagonal quasicrystals described as decorated Hexagon-Boat-Star (HBS) tiles using ab-initio methods. A dominant term in the TH counts the number of H, B and S tiles, favoring tilings of H and B only. In our model for Al-Co-Cu, chemical ordering of Cu and Co along tile edges defines tile edge arrowing. Unlike the edge arrowing of Penrose matching rules, however, the energetics for Al-Co-Cu do not force quasiperiodicity. Energetically favored structures resemble crystalline approximants to which the actual quasicrystalline compounds transform at low temperature. Explaining thermodynamic stability is a fundamental problem in the field of quasicrystals. Competing explanations range from energetic stabilization utilizing matching rules such as those that force quasiperiodicity in the Penrose tiling [1], to entropic stabilization [2] focusing on the configurational entropy available in random tiling models [3,4]. Experimental evidence so far has not unambiguously settled the matter, and the true situation is certainly more complex than either of the two extremes just described. With the advent of plausible atomistic quasicrystal models and advances in first-principles calculation methodology we hope further theoretical progress may be made in this area. Our approach reported here is based on a "tiling Hamiltonian", in which a family of low energy atomistic structures is placed in 1:1 correspondence with a family of tilings of the plane. The energetics of the tiling Hamiltonian is defined in a manner that closely approximates the ab-initio energetics of the atomistic structures. The energetics we derive proves remeniscient of Penrose "matching rules" (which force global quasiperiodicity in minimumenergy structures) but differs in crucial aspects. Indeed, we find that our tile Hamiltonian does not favor quasiperiodicity. Quasiperiodicity may occur at high temperatures as a result of random tiling configura-tional entropy. At low temperatures energy favors transformation to crystalline phases, which is indeed often observed experimentally [5,6,7,8,9]. Penrose tiles are fat and thin rhombi (Fig 1a). Edges are assigned single-and double-arrow decorations (as shown) which must match for common edges in adjacent tiles. Perfect quasicrystals obey these rules everywhere. The double-arrow matching rule [4] causes rhombi to associate into hexagon (H), boat (B) and star (S) shapes (with relative frequency H:B:S= √ 5τ : √ 5:1), while the single-arrow rules force quasiperiodicity in the HBS tiling. It has been shown previously [10,11] that plausible atomistic structures of AlCoNi and AlCoCu can be described as HBS tilings decorated with atoms (Fig 1b). Hence, we may consider the Penrose rhombus double-arrow rules to be satisfied by definition of our basic HBS tiles. In a tiling model of quasicrystals, the actual atomic interactions in the system Hamiltonian can be replaced with effective interactions between and within tiles [12]. The resulting tile Hamiltonian is a rearrangement of contributions to the actual total energy. In a simple atomic interaction picture (pair potentials for example) the relation between the actual atomic interactions and the tile Hamiltonian is straightforward. It might be difficult to find the relations between them for more complicated atomic interactions (many body potentials, or full ab-initio energetics, for example) but it is theoretically possible. The tile Hamiltonian includes terms which depend only on the number of tiles, and includes other terms for tile interactions. The tile Hamiltonian greatly simplifies our understanding of the relationship between structure and energy, and is a reasonable way to describe the tiling ensemble. Are Penrose single-arrow matching rules enforced by energetics of real materials? For a simple model [11] of Al 70 Co 9 Ni 21 in which both edge sites are occupied by Ni atoms there is no source of symmetry-breaking at short length scales able to define an orientation of the tile edges. The energetics of structures based on HBS tiles decorated in this manner depends primarily on the numbers of H, B and S tiles. As seen in Fig. 2, certain phason flips convert an HS pair into a BB pair (or vice-versa). Pairpotential-based total energy calculations of these two structures [11,15] reveal that structure (a) containing the BB pair is lower in energy than (b) containing the HS pair by 0.2 eV. The physical origin of this energy difference lies in the number of 72 • vertices, which drops by 1 in the transition HS → BB. At a 72 • vertex transition metal pairs are close neighbors, causing a reduction in the number of energetically favorable [16] aluminum-transition metal near neighbor interactions. Hence we may express the tile Hamiltonian as H = E s N s(1) where N s is the number of star tiles present, and the coefficient E s = 0. time the magnitude (and even the sign) of this term is unknown [15]. Additional corrections relating to the number of 144 • vertices are small relative to the term shown [11,15]. Hence we focus our attention on the two-dimensional behavior defined by Hamiltonian (eq. (1)). Monte Carlo simulations show that S tiles are infrequent at T = 1000K and completely absent in the lowest energy structures, which are random HB tilings with relative frequency H:B=1:τ . A typical structure is illustrated in Fig. 3a. The situation for AlCoCu is more complicated than for AlCoNi, due to the chemical alternation of Co/Cu pairs on tile edges. Cockayne and Widom [10] suggested that tile edges could be assigned arrow direction based on their Co/Cu decorations (Fig.1b). The physical origin of Co/Cu chemical ordering rests on the status of Cu as a Noble Metal with completely filled d orbitals, unlike normal transition metals such as Co. Energetically, it turns out to be highly favorable for Co/Cu pairs to orient such that the Co atoms are further removed from 72 • vertices than Cu atoms. For consistency with Penrose matching rules, we thus define the arrow to point from Cu towards Co. When the HBS tiles are decorated consistently with the Penrose matching rules, all arrows point outwards from 72 • vertices, minimizing the energy associated with chemical ordering of Co/Cu. However other tilings (such as the random HB tiling illustrated in Fig. 3a) contain "zig-zag" structures. The middle of the three bonds in a zig-zag can never be oriented to point outwards from each of its 72 • vertices, leading to a minimum energy cost for each zigzag, E zz . Hence we define our tile Hamiltonian H = E s N s + E zz N zz(2) where N zz is the number of zig-zags present, while the coefficients E s = 0.2 eV and E zz = 0.12 eV have been derived from full ab-initio calculations [13]. Eq. (2) is actually a simplification of the full tile Hamiltonian [13,14] which captures accurately the energetics of the lowest energy structures. Although we assign the energy cost to the zig-zag shape, its origin is the frustration of the central bond orientation, and not a feature of the shape itself. We have found a few special approximants containing neither stars nor zig-zags. The simplest of these approximants (and the largest phason strain) covers space by translation of a single boat tile (see tiling B1 in Ref. [13]). The next larger of these approximants (but the smaller phason strain) covers space with "lightbulb" objects (see Fig. 4a) consisting of two boats and a hexagon. Other star-and zig-zagfree structures have been found that are basically superstructures of the lightbulb tiling. For large quasicrystal approximants of low phason strain it appears impossible to simultaneously eliminate both stars and zig-zags. Were we to start with a phason strain-free random HB tiling, containing N tiles (N/τ 2 tiles of type H and N τ tiles of type B), a series of tile flips could segregate the tiles into a zig-zag-free lightbulb tiling adjoining a pure H tiling. Counting up the tile numbers, we see that the lightbulb tiling contains N/τ 2 type B tiles and hence N/2τ 2 type H tiles. This leaves (1/τ −1/2τ 2 )N extra H tiles remaining to form a pure H tiling which contains 1 zig-zag per H tile. Accordingly, we conjecture (1/τ − 1/2τ 2 )N is the minimum number of zig-zags possible in an HB tiling of N tiles at composition H 1 B τ . The number of S tiles present in an ideal HBS tiling of N tiles total works out to (2/τ − 1/τ 2 )N , just twice the apparent minimum number of zig-zags. Indeed, we believe this may be the minimal allowed value of N s + 2N zz in zero phason strain tilings. If this were true, then the density of stars in a zero phason strain Penrose tiling is the minimum possible density of stars in any zig-zag-free tiling. The lightbulb tiling illustrated in Fig. 4a exhibits a unit cell of a 72 • rhombus with an edge length of 2(cos π 10 + cos 3π 10 )L ≈ 20Å where L = 6.4Å is the edge length of the HBS tiling for AlCoCu. Such a crystal structure appears when decagonal Al 65 Co 20 Cu 15 is annealed at low temperatures. It is seen in HREM as a rhombic lattice of ring contrasts identified as 20Å clusters. HREM images of the atomic structure associated with our lightbulb tiling (when decorated with atoms as in Fig. 2) contain nearly complete ring contrasts . Fig 4b illus-trates a simple model high-resolution structure image [18] obtained by superposing Gaussian functions at each atomic position with weight proportional to the atomic number (to do a better job of HREM modeling we should incorporate chemical and phason stacking disorder in our structure model and perform dynamical diffraction analysis of the electron microscope imaging). Dark spots correspond to atomic columns and white to empty channels. This type of image should resemble HREM images from a thin sample near the Scherzer defocus. Fig. 4 bears a qualitative resemblence to the HREM patterns of low-temperature Al 65 Co 20 Cu 15 in Ref. [6]. Thus it may be that our tiling Hamiltonian gives an indication of the structure of the low temperature phase, and explains its appearance as driven by the need to eliminate star tiles and zig-zags. The precise values of the coefficients E s and E zz in equations (1) and (2) can be questioned because they wre calculated with atoms placed at ideal sites. Their values will change if atomic relaxation is allowed, although we expect the general form of the tile Hamiltonians and the magnitudes and signs of the terms to be preserved. Small changes in chemical composition can lead to surprisingly large changes in the tile Hamiltonian by altering the atomic interactions specifically at those points where the unfavorable star or zig-zag energies originate. Such an effect could explain why the low temperature structure observed for Al 63 Co 17.5 Cu 17.5 Si 2 (a 72 • rhombus with a 51Å edge length [7]) differs from that found [6] in Al 65 Co 20 Cu 15 . In general, variation of the tile Hamiltonian parameters can lead to transitions such . as that illustrated in Fig. 3. A chemistry dependence is also found in the case of AlCoNi, where small changes in composition lead to a wide array of different structure types [5]. Transitions as composition (or temperature) is changed may be related to changes in the values of terms in a tile Hamiltonian. For example, a change from Al 70 Co 9 Ni 21 to Al 72 Co 11 Ni 17 results in CoAl pairs replacing NiNi pairs on tile edge sites at 72 • vertices [11]. Consequently the energy cost of 72 • vertices, and hence E s is reduced on average. However, the Co/Al pairs carry an edge orientation (similar to Co/Cu pairs) so we need to add a zig-zag energy into the Hamiltonian (1), resulting in a new Hamiltonian like (2). Although the true low temperature phase at this composition is not certain, at a nearby composition of Al 71 Co 14.5 Ni 14.5 the system indeed takes on one of two structures based on tilings by 72 • rhombi with 20Å edge lengths [8,9]. One structure, known as PD2, has the unit cell of the lightbulb tiling (Fig. 4). The other structure, known as PD1, pairs rhombi into "chevron" structures ( Fig. 5) in which, again, both stars and zigzags may be avoided. Both of these structures have an 8Å periodicity in the stacking direction, so an additional term related to phason stacking faults may need to be included in our tile Hamiltonians. In conclusion, we show that an ensemble of low energy quasicrystal and approximant structures may be modeled using very simple tiling Hamiltonians. The tile Hamiltonians representing Al-Co-Ni and Al-Co-Cu favor crystalline structures at low temperatures but may exhibit quasicrystals in equilibrium at high temperatures. The favored low energy crystal struc-tures resemble the transformation products actually observed in these compounds at low temperatures. Fig. 1 . 1(A) HBS tiles and their decompositions to Penrose tiles. (b) Atomic decorations for AlCoCu. In (b), only TM and symmetry breaking Al atoms are shown. For AlCoNi both edge sites are decorated with Ni atoms. Fig. 2 . 22 eV. To fully model decagonal AlCoNi (indeed any decagonal phase) we should add into the Hamiltonian (eq. (1)) terms representing phason stacking disorder. Unfortunately, at this b c Space can be tiled in many ways using HBS tiles. Both these approximants contain equal numbers of each atom type. Structures in (a) and (b) differ by the phason flip outlined in (b) with a dashed line, which converts two B tiles into and H and an S. Fig. 3 . 3Typical low temperature configurations. (a) Tile Hamiltonian (eq. (1)), or (eq. (2)) with Es > 2Ezz. Wide gray bonds identify "zig-zag" structures. (b) Tile Hamiltonian (eq. (2)) with Es < 2Ezz. Star tiles are shaded gray for emphasis. Fig. 4 . 4zig-zag-and star-free "lightbulb tiling" structure. (a) Tiling has orthorhombic cell with lattice constants 23.1Å × 31.8Å (dashed lines) containing inscribed 72 • rhombus with 20Å edge length. (b) Model HREM structure image[18] showing 20Å ring contrasts at vertices of 72 • rhombus. Fig. 5 . 5Chevron tiling of 72 • rhombi with 20Å edge lengths. The rhombic unit cell of dimensions 37.6Å × 39.7Å matches the low temperature structure PD1 of Al-Co-Ni[9] AcknowledgmentsThe authors thank C.L. Henley and S. Naidu for useful discussions. IA wishes to thank King Abdul Aziz University (Saudi Arabia) for supporting his study and MW acknowledges support by the National Science Foundation under grant DMR-0111198. We thank the Pittsburgh Supercomputer Center for computer time used for this study. K Ingersent, Quasicrystals: the state of the art. D.P. DiVincenzo and P.J. SteinhardtSingaporeWorld Scientific110K. Ingersent, in Quasicrystals: the state of the art, D.P. DiVincenzo and P.J. Steinhardt (eds.) (World Scientific, Singapore, 1991) 110. . M Widom, K J Strandburg, R H Swendsen, Phys. Rev. Lett. 58706M. Widom, K.J. Strandburg and R.H. Swendsen, Phys. Rev. Lett. 58 (1987) 706. . V Elser, Phys. Rev. Lett. 541730V. Elser, Phys. Rev. Lett. 54 (1985) 1730. C L Henley, Quasicrystals: the State of the Art. P.J. Steinhardt and D.P. DiVincenzoSingaporeC.L. Henley, in Quasicrystals: the State of the Art, edited by P.J. Steinhardt and D.P. DiVincenzo (World Scien- tific, Singapore, 1991) . S Ritsch, C Beeli, H.-U Nissen, T Godecke, M Scheffer, R Luck, Phil. Mag. Lett. 7867S. Ritsch, C. Beeli, H.-U. Nissen, T. Godecke, M. Scheffer, and R. Luck, Phil. Mag. Lett., 78. 67 (1998) . K Hiraga, Wei Sun, F J Lincoln, Japanese Journal of Applied Physics Letters. 30302K. Hiraga, Wei Sun, and F.J. Lincoln, Japanese Journal of Applied Physics Letters,30, L302 (1991) . M Fettweis, P Launois, F Denoyer, R Reich, M Lambert, Phys. Rev. B. 4915573M. Fettweis, P. Launois, F. Denoyer, R. Reich and M. Lambert, Phys. Rev. B 49, 15573 (1994) . K Hiraga, F J Lincoln, W Sun, Mat. Trans. JIM. 32308K. Hiraga, F.J. Lincoln and W. Sun, Mat. Trans. JIM 32, 308 (1991) . M Doblinger, R Wittmann, D Gerthsen, B Grushko, Mat. Sci. Eng. A. 294296131M. Doblinger, R. Wittmann, D. Gerthsen and B. Grushko, Mat. Sci. Eng. A 294-296 (2000) 131. . E Cockayne, M Widom, Phys. Rev. Lett. 81598E. Cockayne and M. Widom, Phys. Rev. Lett. 81, 598 (1998) . M Mihalkovic, I Al-Lehyani, E Cockayne, C L Henley, N Moghadam, J A Moriarty, Y Wang, M Widom, Phys. Rev. B. 65104205M. Mihalkovic, I. Al-Lehyani, E. Cockayne, C.L. Henley, N. Moghadam, J.A. Moriarty, Y. Wang and M. Widom, Phys. Rev. B 65, 104205 (2002) . M Mihalkovic, W.-J Zhu, C L Henley, M Oxborrow, Phys. Rev. B. 53M. Mihalkovic, W.-J. Zhu, C.L. Henley and M. Oxbor- row, Phys. Rev. B. 53, 9002, (1996) . I Al-Lehyani, M Widom, Phys. Rev. B. to appearI. Al-Lehyani and M. Widom, Phys. Rev. B (to appear, 2002) . M Widom, I Al-Lehyani, Y Wang, E Cockayne, Matter. Sci. Eng. 295M. Widom, I. Al-Lehyani, Y. Wang and E. Cockayne, Matter. Sci. Eng. A295, 8 (2000) . C L Henley, M Mihalkovic, M Widom, J. Alloys and Compounds. to appearC.L. Henley, M. Mihalkovic and M. Widom, J. Alloys and Compounds (to appear, 2002) . I Al-Lehyani, M Widom, Y Wang, N Moghadam, G M Stocks, J A Moriarty, Phys. Rev. 6475109I. Al-Lehyani, M. Widom, Y. Wang, N. Moghadam, G.M. Stocks and J.A. Moriarty, Phys. Rev. B64, 075109 (2001) . C L Henley, Phys. Rev. B. 34797C.L. Henley, Phys. Rev. B 34, 797 (1986) K Hiraga, Quasicrystals: The state of the art. D.P. DiVincenzo and P.J. SteinhardtWorld ScientificK. Hiraga, in Quasicrystals: The state of the art, eds. D.P. DiVincenzo and P.J. Steinhardt (World Scientific, 1991) p. 95-110.
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[ "BIMS-PU: Bi-Directional and Multi-Scale Point Cloud Upsampling", "BIMS-PU: Bi-Directional and Multi-Scale Point Cloud Upsampling" ]
[ "Yechao Bai ", "Xiaogang Wang ", "Marcelo H Ang Jr", "Daniela Rus " ]
[]
[]
The learning and aggregation of multi-scale features are essential in empowering neural networks to capture the finegrained geometric details in the point cloud upsampling task. Most existing approaches extract multi-scale features from a point cloud of a fixed resolution, hence obtain only a limited level of details. Though an existing approach aggregates a feature hierarchy of different resolutions from a cascade of upsampling sub-network, the training is complex with expensive computation. To address these issues, we construct a new point cloud upsampling pipeline called BIMS-PU that integrates the feature pyramid architecture with a bi-directional up and downsampling path. Specifically, we decompose the up/downsampling procedure into several up/downsampling sub-steps by breaking the target sampling factor into smaller factors. The multi-scale features are naturally produced in a parallel manner and aggregated using a fast feature fusion method. Supervision signal is simultaneously applied to all upsampled point clouds of different scales. Moreover, we formulate a residual block to ease the training of our model. Extensive quantitative and qualitative experiments on different datasets show that our method achieves superior results to state-of-the-art approaches. Last but not least, we demonstrate that point cloud upsampling can improve robot perception by ameliorating the 3D data quality.
10.1109/lra.2022.3183932
[ "https://arxiv.org/pdf/2206.12648v1.pdf" ]
249,904,317
2206.12648
0497894fedd1a548d5e18a8a9c4ec2a994c6f3c7
BIMS-PU: Bi-Directional and Multi-Scale Point Cloud Upsampling Yechao Bai Xiaogang Wang Marcelo H Ang Jr Daniela Rus BIMS-PU: Bi-Directional and Multi-Scale Point Cloud Upsampling 10.1109/LRA.2022.3183932BAI et al.: BIMS-PU. CITATION INFORMATION: 1Index Terms-Deep Learning for Visual Perception; Computer Vision for Automation The learning and aggregation of multi-scale features are essential in empowering neural networks to capture the finegrained geometric details in the point cloud upsampling task. Most existing approaches extract multi-scale features from a point cloud of a fixed resolution, hence obtain only a limited level of details. Though an existing approach aggregates a feature hierarchy of different resolutions from a cascade of upsampling sub-network, the training is complex with expensive computation. To address these issues, we construct a new point cloud upsampling pipeline called BIMS-PU that integrates the feature pyramid architecture with a bi-directional up and downsampling path. Specifically, we decompose the up/downsampling procedure into several up/downsampling sub-steps by breaking the target sampling factor into smaller factors. The multi-scale features are naturally produced in a parallel manner and aggregated using a fast feature fusion method. Supervision signal is simultaneously applied to all upsampled point clouds of different scales. Moreover, we formulate a residual block to ease the training of our model. Extensive quantitative and qualitative experiments on different datasets show that our method achieves superior results to state-of-the-art approaches. Last but not least, we demonstrate that point cloud upsampling can improve robot perception by ameliorating the 3D data quality. I. INTRODUCTION T HE importance of 3D data has become evident in applications like autonomous driving, robotics, medical imaging, etc. Recent studies [1], [2], [3], [4], [5] have shown that point cloud is a compact and efficient 3D representation. However, real-scanned point clouds produced by depth camera and LiDAR are often sparse, noisy, and irregular [6], [7], [8]. As point cloud upsampling can improve the quality of realscanned data by increasing the point density and uniformity, it has drawn increasing attention in computer vision and robotics community. The upsampled point cloud has to preserve the geometric detail of the underlying surface. To this end, several state-of-the-art point cloud upsampling methods [9], [10], [11], The result demonstrates that our approach has distinct advantages at challenging places like intersections, the narrow gap between two surfaces, and slender objects. We credited it to the effective multi-scale feature fusion, which enables the model to leverage local and global contextual information. [12] exploit the multi-scale features of the point cloud. PU-Net [9] progressively increases the ball queries of each point to extract multi-scale local region features and concatenates them to generate the upsampled features. Following a similar principle as PU-Net, PU-GCN [11] designs an Inception DenseGCN module, which has parallel DenseGCN branches of different receptive fields, to encode multi-scale context of point clouds. However, the level of geometric detail in the aggregated multi-scale feature is limited as the input resolution is fixed. To get fine-grained details, MPU [10] breaks the upsampling network into successive subnets to progressively upsample the point clouds. Although MPU preserves better details, its back-and-forth conversion between high-dimension feature space and 3D spatial space leads to increased computation complexity and training difficulty. In this work, we adapt the feature pyramid architecture [13], [14], [15], [16] for the point cloud upsampling task and construct a bidirectional and multi-scale upsampling module. Concretely, instead of obtaining the multi-scale feature directly from a feature extractor, our method generates multi-scale point cloud features from a bi-directional up and downsampling pathway inspired by the back projection mechanism [17] developed for image super-resolution [18], [19], [20], [21]. The back projection mechanism uses an iterative up and downsampling procedure to minimize reconstruction errors. Our method decomposes the upsampling/downsampling procedure into substeps with a smaller ratio in feature space and generates the multi-scale upsampled point cloud features in a parallel manner. The goal is to decrease the optimization difficulty via decomposing the task into multiple simpler sub-tasks [22]. Next, we leverage a fast weighted feature fusion method to aggregate the resultant multi-scale features. Subsequently, upsampled point cloud features of each scale get reconstructed into 3D point clouds. Supervision signals are applied simultaneously at each output scale. The multi-scale supervision guides the network's training to fuse multi-scale features to be more discriminative. To enable the model to learn complex mapping with additional height and width, we formulate a simple yet highly effective residual block based on the residual learning concept [23] to expand/squeeze the number of channels of a point feature during feature expansion and point reconstruction. The quantitative and qualitative results show that the bi-directional multi-scale up/downsampling pathway improves the fine-grained geometric details of the upsampled point cloud. This is because it enables the up/downsampling operators to be trained with features of different resolutions produced by multi-scale features fusion. Lastly, to verify the value of point cloud upsampling to the robotics community, we design an experiment to show that upsampling is beneficial to point cloud classification, a fundamental robot perception task. In summary, our contributions are: N × 3 N × C 1 … F up lQ l Point Reconstruction Unit l 2 l N × C 2 2 l N × 3 … Loss l F up 1Q 1 Point Reconstruction Unit 1 2N × C 2 2N × 3 Loss 1 F up LQ L Point Reconstruction Unit L 2 L N × C 2 2 L N × 3 Loss L r′ = 8 Output Q 3 • Design a bi-directional multi-scale upsampling module; • Propose training with multi-scale supervision to facilitate the multi-scale feature fusion to produce more discriminative features; • Conduct extensive quantitative and qualitative experiments on synthetic and real-world datasets to show that our method achieves superior results to state-of-the-art approaches; • Design an experiment to verify the value of point cloud upsampling to robot perception. II. RELATED WORKS Learning-based point cloud upsampling. PointNet [1] and the multi-scale variant PointNet++ [2] propose networks that directly consume point cloud for several 3D recognition tasks. Based on PointNet++, PU-Net [9] designs a pointbased network for the point cloud upsampling task. It adopts the hierarchical feature learning mechanism [2] for feature extraction and upsamples point cloud at patch-level. EC-Net [24] designs an edge-aware network and a joint loss to deliberately improve the consolidation near the edge. However, it requires expensive edge annotation for training. To generate outputs of large upsampling factor, MPU [10] progressively upsampled the point cloud to different levels of resolution with a cascade of sub-networks. Unlike previous encoder-decoder networks, PU-GAN [25] incorporates the adversarial training concept into a point upsampling network. It uses a selfattention unit to leverage the long-range context dependencies in the upsampling module. The geometric-centric network, PUGeo-Net [26], explicitly learns the first and second fundamental forms for point cloud upsampling. However, it requires the normals of points as a supervision signal, which is not directly available in a real-scanned point cloud. PU-GCN [11] focuses on improving the upsampling module and the feature extraction module for point cloud upsampling. It integrates the graphical convolutional network into the upsampling module and designs an Inception-based feature extraction module. The recently proposed Dis-PU [12] disentangle the point upsampling tasks into dense point generation and point spatial refinement. To achieve this, they design a network that consists of two cascaded sub-networks. Though both MPU [10] and our methods produce multi-resolution point clouds, our method is not a kind of progressive upsampling method. There are two distinctive differences. First, MPU [10] consists of a cascade of sub-networks, whereas our method produces multi-scale outputs in a single network. Second, the MPU progressively trains L subnets using 2L + 1 stages, whereas we apply multiscale supervision signals to each output scale simultaneously, so training can be completed in only one go. PU-GCN [11] and Dis-PU [12] extract local and global features to learn fine-grained details using graphical convolutional network and attention mechanism. However, their method has a high computation demand due to intensive use of kNN and selfattention operation. Our method uses a hierarchical network architecture that is computationally efficient to extract multiscale features to grasp the fine-grained patterns. Multi-scale feature representation and aggregation. Multiscale feature representation and aggregation are one of the main problems in visual perception tasks. Lin et al. [13] proposes a top-down architecture with lateral connections to fuse multi-scale features instead of directly using the pyramidal feature hierarchy for prediction. Taking one step further, Liu et al. [14] adds a bottom-up pathway to enhance the entire feature hierarchy. Ghiasi et al. [15] adopts a neural architecture search to discover a new feature pyramid architecture. Tan et al. [16] revises the architecture design to be more intuitive and principled. The main difference between our work and the pyramid feature architecture in 2D visual perception tasks is that the latter obtains the multi-scale feature directly from feature extractors. But our approach generates a pyramidal point cloud feature from a bi-directional up/downsampling pathway with shortcut connections. III. APPROACH A. Overview Given a sparse point cloud P of N points, our network outputs L denser point cloud {Q 1 , . . . ,Q l , . . . ,Q L } wherê Q l ∈ R r N ×3 , r = 2 l is the intermediate upsampling factor and r = 2 L is the desired upsampling factor. Q is the ground truth point cloud with rN points. The upsampled point clouds should lie on the underlying surface of the object and have a uniform distribution. Our network consists of three parts: point feature extraction, point feature expansion and point reconstruction. Point feature extraction. The feature extractor learns point feature F ∈ R N ×C1 from the input point cloud P ∈ R N ×d , C 1 > d. In this case, the feature of the input is the 3D coordinates of the point cloud, namely d = 3. We adopt the feature extractor in MPU [10] which uses dynamic graph convolution [27] to extract point features from local neighborhoods via kNN search in feature space and exploit a dense connection to facilitate information reuse. Point feature expansion. In this part, we propose a bidirectional multi-scale upsampling module to expand the point features. It takes the point feature F as input and outputs L upsampled point features {F up 1 , . . . , F up l , . . . , F up L }, where F up l ∈ R r N ×C2 , F up L ∈ R rN ×C2 and C 2 < C 1 . The bidirectional up and downsampling path preserves the finegrained geometric details by learning the up/down-sampling operators with global and local context provided by the point features of different resolutions. Point reconstruction. To reconstruct the L upsampled point features from latent space to coordinate space, we assign each F l 1 F l 2 F r 2 F r 1 F m 0 F m 1 F up 2 F up 1 N × C 1 2N × C 2 4N × C 2 4N × C 2 2N × C 2 2N × C 2 N × C 2 2N × C 2 4N × C 2 HR LR (a) PointOutput: N × C 2 N × 2C 2 Reshape MLP (c) Downsampling operator Output: 2N × C 2 Input: N × C 2 Duplicate 2D Grid MLP 2N × 2 (b) Upsampling operator 2N × C 2 B. Bi-directional Multi-scale Upsampling Module The proposed bi-directional multi-scale upsampling module consists of two sets of upsampling operators and one set of downsampling operators as shown in Fig 3(a). The number of operator in each set is L, where 2 L = r. The scaling factor of each operator is 2. We illustrate the up/downsampling operator in Fig 3(b)/(c). This upsampling operator combines the duplicated point features and a 2D grid [28], [10] that adds spatial variation to help spread out the points. Then it uses shared MLP to produce the upsampled point features. The downsampling operator reshapes the feature and then uses shared MLP to generate the downsampled point features. We leverage hierarchical network architecture which is computationally efficient to learn local/global fine grained patterns instead of using self-attention operator or graphical convolutional network. Given point feature F from feature extraction as input, upsampling module first maps the lowresolution (LR) point feature to a high-resolution (HR) point feature using the first set of upsampling operators on the left side. Then, it maps the HR point feature back to the LR point feature using the set of downsampling operators in the middle. Lastly, the reconstructed LR point feature is mapped to a HR feature point by the second set of upsampling operators on the right side. Our upsampling module uses the shortcut connections and a weighted feature fusion method to achieve a fast and efficient multi-scale feature fusion. For simplicity the desired upsampling factor r is set to 4. Concretely, we describe the two upsampled point feature outputs F up 1 , F up 2 , and an intermediate fused feature F m 1 in Fig. 3. Fig. 4: Illustration of the residual block. F up 2 = w 1 · F l 2 + w 2 · Up (F r 1 ) w 1 + w 2 + F up 1 = w 1 · F l 1 + w 2 · F m 1 + w 3 · Up (F m 0 ) w 1 + w 2 + w 3 + F m 1 = w 1 · F l 1 + w 2 · Down F l 2 w 1 + w 2 + (1) Input Feature Output Feature N × C in N × C out N × C out N × C out N × C out where Up and Down is the up and downsampling operation; w i is a learnable weight that represents the importance of each input, which is ensured to be positive by applying ReLU [29] after each w i ; is a small value to avoid numerical instability. C. Residual Block In this work, we design a lightweight residual block, as illustrated in Fig. 4, to expand/squeeze the channel of point feature during feature expansion and point reconstruction. Residual learning have proven to be effective at easing the optimization of a network during training [23]. Our intention is to use the residual block to allow the network to learn a complex mapping with increased model complexity namely the scalable bidirectional pathway and cross-scale shortcut links in our model. D. Multi-scale Supervision. Our model generates multiple intermediate upsampled point clouds in one feed-forward pass. Supervision signals are applied simultaneously at each output scale. Notably, we do not downsample the ground truth to create the multiscale supervision label to avoid potential artificial artifacts and laborious effort. The multi-scale supervision guides the network's training to fuse multi-scale features to be more discriminative. Additionally, decomposing the task into multiple simpler sub-tasks decrease the optimization difficulty. It allows our model to have a large representation capacity to learn complicated mappings, thus achieve fewer outliers and render better geometric details. We train our upsampling network with multi-scale supervision using a robust joint loss: L = L i=1 α i · L joint Q,Q i L joint = L CD + λ · L rep(2) where Q andQ i are the ground truth and multi-scale output point cloud respectively; α i , λ ∈ [0, 1] are weighting factors; L CD (·) means Chamfer Distance [3] which measures the average closest point distance between two point sets; L rep (·) means Repulsion Loss [9] which encourage the generated points to distribute more uniformly. IV. EXPERIMENTS A. Implementation Details Datasets For quantitative/qualitative comparisons between models, we employ two synthetic datasets and one realscanned dataset. (1) PU-GAN's dataset provides 120 training and 27 testing objects. (2) PU1K dataset proposed in PU-GCN [11] consists of 1,020 training and 127 testing objects. (3) The real-scanned dataset ScanObjectNN [30] contains 2,902 point cloud objects in 15 categories manually filtered and selected from SceneNN [31] and ScanNet [32]. Each object has 2,048 points. Since the ground truth is not available, we only conduct qualitative experiments on the real-world dataset with pre-trained models. To demonstrate point cloud upsampling is beneficial to 3D object classification, we employ both synthetic and real-scanned datasets in our experiment in Section IV-E. The synthetic dataset is the ModelNet40 [33] dataset, which contains point clouds of 40 common object categories sampled from 100 unique CAD models per category. The real-scanned dataset is ScanObjectNN [30]. Training details For training, we use the training data provided by PU-GAN's dataset [25] and PU1K [11] and follows the settings in PU-GAN [25] and PU-GCN [11] for model comparison. Concretely, the ground truth point clouds Q has 1,024 points; the input point clouds P of 256 points are randomly downsampled from the ground truth point cloud Q on the fly during training; the upsampling ratio r is 4. We train our model for 400 epochs using the Adam optimizer with an initial learning rate of 0.001. The batch size is 64 on PU1K and 28 on PU-GAN's dataset. We decrease the learning rate by a factor of 0.7 for every 40 epochs. Data augmentation techniques applied includes random rotation/scaling/shifting. In the point feature extraction unit the k for k-nearest neighbors search is 16. C 1 and C 2 in the feature expansion unit are 648 and 128. We use α 1 = 0.6, α 2 = 1.0 and α 1 = 0.6, α 2 = 0.8, α 3 = 1.0 in Eq. 2 for r = 4 and r = 16 respectively. We implemented our network using PyTorch and all experiments are conducted on an Nvidia RTX 2080 GPU. Testing details For testing, we adopt the commonly used patch-based strategy [9], [10], [25], [11], [12] as follows. First, use Poisson disk sampling to generate the ground-truth object point clouds Q of rN points from object mesh and then downsample it to get sparse input point clouds P of N points. Second, apply the farthest point sampling [2] to the input point clouds to get query points and extract overlapping input patches of 256 points around each query point using kNN. Next, feed all input patches to an upsampling model and combine the output overlapping point clouds of 1,024 points to get the dense object point cloud. Lastly, apply farthest point sampling to produce a uniform and dense object point cloud that contains rN points. Both PU-GCN and Dis-PU reported model comparisons results using PU-GAN's dataset. We notice two differences in test settings between them and PU-GAN's in their experiment. (1) PU-GAN [25] and Dis-PU [12] use Monte-Carlo downsampling while PU-GCN uses Poisson downsampling 1 to generate the sparse input point cloud P . (2) PU-GAN [25] and PU-GCN [11] use input point cloud of 2,048 points but Dis-PU [12] We can see that our model produces a more accurate reconstruction with fewer red points and preserves better geometric details at challenging areas. GAN [25] to use the widely used Monte-Carlo sampling. Regarding the number of test input points, we also follow PU-GAN [25]'s setting to use 2,048 points for consistency between model comparisons conducted on PU-GAN's dataset and PU1K dataset. Evaluation metrics The evaluation metrics are (i) Chamfer distance (CD); (ii) Hausdorff distance (HD) [34]; (iii) pointto-surface distance (P2F). A lower evaluation metric indicates a better performance. B. Quantitative Comparisons We conduct comparisons on two datasets PU-GAN's dataset [25] and PU1K [11]. The results are shown in Tables I and II, respectively. The recently proposed PUGeo-Net [26] is not included in the comparison as it requires the accurate normal of point for training, which is not directly available in point clouds. Comparisons on PU-GAN's dataset. Quantitative comparisons between models on PU-GAN's dataset under different upsampling ratios are presented in Table I. The results show that our method performs competitively to state-of-the-art approaches under both small and large upsampling ratios. Notably, our model has the smallest model size when r = 4 and is 37% lighter than the runner-up model Dis-PU. Though PU-GCN MPU DIS-PU Ours Input Fig. 6: Qualitative comparison of upsampled (16×) point clouds generated using different methods and its 3D mesh reconstruction. The sparse inputs of 2,048 points are from real-scanned dataset ScanObjectNN [30]. While Dis-PU [12] has an advantage in surface smoothness, our method outperforms others in terms of producing a more accurate reconstruction that has fewer surface defects and outliers. our model size grows as the upsampling ratio increases, when r = 16, it is still comparable to the size of Dis-PU and much smaller than PU-Net and MPU. The input and ground truth test data are generated following PU-GAN's setting when testing r = 4 and r = 16. We train the models on PU-GAN [25]'s dataset using their released source code and report the test performance using the best model obtained from training. For r = 4, we directly use the result of PU-Net, MPU, and PU-GAN from PU-GAN [25]'s paper. Though Dis-PU [12] and PU-GCN [11] have conducted model comparisons using PU-GAN's dataset under r = 4 and r = 16, we don't use the results in their paper because they used different test settings from PU-GAN [25], which is discussed in Section IV-A-Testing in detail. Comparisons on PU1K. Quantitative comparisons on PU1K dataset are presented in Table II. We conduct two sets of experiments. One uses test input point cloud generated using Poisson downsampling provided by PU-GCN [11]. The other one uses input point cloud generated using Monte-Carlo downsampling to produce a more realistic and non-uniform distribution. For PU-Net [9], MPU [10], and PU-GCN [11], TABLE II: Quantitative comparisons with the state-of-theart on PU1K. We conduct two sets of experiments. One uses input point cloud generated using Poisson downsampling as in the paper [11]. Another one uses input point cloud generated using Monte-Carlo downsampling which produces more realistic non-uniform distribution distribution. The units of CD, HD, and P2F are 10 −3 . The best result is highlighted in bold letters and the runner-up is highlighted with an underline. we use the pre-trained model provided by PU-GCN [11] to get their results. As the pre-trained model of PU-GAN [25] and Dis-PU [12] is not provided, we train the models on PU1K using their released codes and get the result using the best model. In both sets of experiments, our model presents competitive performance. We can see that all models generally performs better when the input point cloud is generated using Monte-Carlo downsampling as it is the downsampling method used during training (see Section IV-A-Training details). Our superior performances on the Monte-Carlo setting verified that our model is more robust to non-uniform points. We also get lower errors on CD and P2F, which shows that our model is able to reconstruct more accurate object shapes. C. Ablation Study We analyze the contribution of each component of our network on the PU-GAN's dataset in Table III, which includes multi-scale fusion, multi-scale supervision, and residual block. We remove each component from the full model one by one (from bottom to top) and measure the performances in terms of CD, HD and P2F. As shown in Table III, all components contribute to the full model since removing any component hampers the performance. D. Qualitative Comparisons We compare our model qualitatively with other methods on two synthetic datasets and one real-word dataset. The results are shown in Fig. 5 and 6. In Fig. 5, the color indicates the nearest distance of each output point to the ground truth surface. We observe three distinct advantages of our approach: 1) Generate points with lower error in the area near the sharp edge, and the edges are cleaner and sharper. 2) Produces fewer outliers in challenging areas like the joint, intersection, and the narrow gap between two surfaces. 3) Preserve better geometric details of slender objects. Specifically, the number of points generally increases along the longitudinal direction of the objects. In Fig. 6, we compare the upsampled point cloud generated using different methods and its 3D mesh reconstruction, the noisy and sparse inputs are from the real-scanned dataset ScanObjectNN [30], where we set r = 16. While Dis-PU [12] has an advantage in surface smoothness, our method outperforms others in terms of producing a more accurate reconstruction that has fewer surface defects and outliers. The qualitative results suggest that our model possesses a better understanding of the global and local context relationship and is capable of generating high-fidelity object details. E. Benefit of upsampling to point cloud classification Point cloud object classification is a fundamental task to robot perception which is crucial to downstream tasks like object detection and semantic segmentation. To demonstrate the value of point cloud upsampling to the robotic community, we design an experiment to show that upsampling is beneficial to point cloud object classification. In this experiment, We employ a synthetic dataset ModelNet40 [33] and a real-scanned dataset ScanObejctNN [30] and choose two widely used models PointNet++ 2 [2] and PointNet 3 [1] to perform point cloud shape classification on ModelNet40 and ScanObjectNN respectively. The PointNet++ is pre-trained on ModelNet40, whereas PointNet is pre-trained on ScanOb-jectNN's hardest variant PB T50 RS. First, we use farthest point sampling and random sampling to sample the testing point clouds to point clouds of 512 points as ground truth data and point clouds of 128 points as input data. Next, we upsample the input point clouds by four times using a set of point cloud upsampling models. The upsampling models are pre-trained on PU-GAN's [25] dataset used in Table I. We compare the classification accuracy of the groundtruth, randomly-downsampled, and upsampled point clouds in Table IV. The results show that applying point cloud upsampling to sparse and nonuniform point clouds to generate denser point clouds effectively improves the classification result in both synthetic and real-scanned data. Interestingly the classification performance improvement is more significant on real-scanned data. This demonstrates that the upsampling models can generate new points according to the distribution pattern of the underlying surface. Further, the quantitative classification accuracy comparison indicates the superiority of our model in reconstruction accuracy. Experiments could be designed following a similar principle to show the effect of point cloud upsampling on semantic segmentation and part segmentation. V. CONCLUSION In this work, we propose a bi-directional multi-scale upsampling approach for 3D point cloud upsampling. We decompose a bi-directional up/downsampling pathway into subup/downsampling steps of smaller scaling factors to produce a pyramidal multi-scale point feature hierarchy. The point features in the hierarchy are fused and reconstructed to point clouds of different resolutions. Supervision signals are applied to each output point cloud to ensure that the feature fusion produces discriminative features. A simple yet effective residual block is proposed to reduce the optimization difficulty. Extensive quantitative and qualitative results on synthetic and real-world datasets demonstrate that our method achieves superior results compared to state-of-the-art approaches. We demonstrate that point cloud upsampling can improve robot perception by ameliorating the 3D data quality using a simple experiment. Fig. 1 : 1was recommended for publication by Editor Cesar Cadena upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Research Foundation, Prime Minister's Office, Singapore, under its CREATE program, Singapore-MIT Alliance for Research and Technology (SMART) Future Urban Mobility (FM) IRG. (Yechao Bai and Xiaogang Wang are co-first authors.)(Corresponding author: Yechao Bai.) 1 Yechao Bai, Xiaogang Wang and Marcelo H. Ang Jr are with the Department of Mechanical Engineering, National University of Singapore, Singapore. {yechao.bai, xiaogangw}@u.nus.edu, [email protected] 2 Daniela Rus is with the Massachusetts Institute of Technology, Cambridge, MA, USA. [email protected] Comparison between our model and stateof-art methods. The color indicates the nearest distance of each output point to the ground truth surface. Fig. 2 : 2Overview of the bi-directional multi-scale point cloud upsampling network (BIMS-PU). The architecture of our model has L pathways to upsample a given sparse point cloud P to increasingly denser outputs Q 1 , . . . Q L at the same time, as depicted in the lower left of the figure. Learning a multi-scale point cloud upsampling task enables our model to capture rich geometric details of different levels of resolution. FFig. 3 : 3Illustration of the point feature upsampling unit in Fig. 2. For simplicity, the target upsampling factor is 4. upsampled feature with one 2-layer Multi-Layer Perceptron (MLP) to regress the 3D coordinates. Fig. 5 : 5uses input point cloud of 1,024 points. Because the Monte-Carlo downsampling generates a realistic and non-uniform point cloud distribution, while the Poisson downsampling produces a uniform point cloud distribution, and the former is also used during training for input point cloud generation. Hence, we follow PU-Qualitative comparisons of point cloud upsampling on synthetic dataset. The color indicates the nearest distance of each output point to the ground truth surface. ©2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. see https://www.ieee.org/publications/rights/index.html for more information. 1arXiv:2206.12648v1 [cs.CV] 25 Jun 2022 … BIMS-PU Input P r′ = 2 Output Q 1 r′ = 4 Output Q 2 P F Point Feature Upsampling Unit Point Feature Extraction Unit TABLE I : IQuantitative comparisons with the state-of-theart on PU-GAN's dataset. The units of CD, HD, and P2F are 10 −3 . The best result is highlighted in bold letters and the runner-up is highlighted with an underline.Methods 4× 16× Size CD↓ HD↓ P2F↓ Size CD↓ HD↓ P2F↓ PU-Net [9] 10.1M 0.72 8.94 6.84 24.5M 0.38 6.36 8.44 MPU [10] 23.1M 0.49 6.11 3.96 92.5M 0.19 5.58 3.52 PU-GAN [25] 9.6M 0.28 4.64 2.33 9.6M 0.23 6.09 3.31 PU-GCN [11] 9.7M 0.27 4.38 2.80 9.7M 0.18 4.72 3.15 Dis-PU [12] 13.2M 0.24 4.63 2.23 13.2M 0.16 8.14 2.43 Our 8.3M 0.28 4.28 2.05 15.6M 0.16 4.70 2.59 TABLE III : IIIAblation study. Experiments are conducted on PU-GAN's dataset. The units of CD, HD and P2F are 10 −3 . MS is the abbreviation of multi-scale. The effectiveness of our proposed components (residual block, multi-scale supervision, and multi-scale feature fusion) for point cloud upsampling is validated. Model Residual MS Supervision MS Fusion CD↓ HD↓ P2F↓ A 0.30 6.05 2.77 B 0.28 5.24 2.17 C 0.28 4.87 2.04 Full 0.28 4.28 2.05 TABLE IV : IVClassification accuracy comparison. We use random sampling and farthest point sampling on the testing point cloud in synthetic dataset ModelNet40[33] and realscanned dataset ScanObjectNN[30] to generate point clouds of 128 points as input and point clouds of 512 points as ground truth. Then we upsample the input point cloud by 4 times using a set of upsampling models. The classification accuracy comparison is conducted between input/upsampled/ground-truth point cloud. The results indicate that point cloud upsampling is beneficial to point cloud classification, and our model is superior in generating new points that reflect the underlying surface of point clouds. Overall and average class accuracy are shown in %.Model ModelNet40 [33] ScanObjectNN [30] OA. Cls Acc. OA Cls Acc. PU-GAN [25] 84.2 80.0 71.5 67.4 PU-GCN [11] 84.6 80.6 70.9 67.0 Dis-PU [12] 85.0 80.4 71.1 66.8 Ours 87.1 82.5 72.0 68.2 Input 79.4 74.1 61.8 56.3 Ground truth 92.5 88.7 74.9 70.6 https://github.com/guochengqian/PU-GCN/issues/3#issuecomment-888289259 https://github.com/yanx27/Pointnet Pointnet2 pytorch 3 https://github.com/hkust-vgd/scanobjectnn Pointnet: Deep learning on point sets for 3d classification and segmentation. 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[ "https://github.com/guochengqian/PU-GCN/issues/3#issuecomment-888289259", "https://github.com/yanx27/Pointnet", "https://github.com/hkust-vgd/scanobjectnn" ]
[ "Constraints on Models for TeV Gamma Rays from Gamma-Ray Bursts", "Constraints on Models for TeV Gamma Rays from Gamma-Ray Bursts" ]
[ "P Chris Fragile ", "Grant J Mathews ", "John Poirier ", "Tomonori Totani ", "\nUniversity of California\nLawrence Livermore National Laboratory\n94550LivermoreCA\n", "\nCenter for Astrophysics\nDepartment of Physics, Notre Dame\nTheory Division\nUniversity of Notre Dame\nNational Astronomical Observatory\n46556, 181-8588Mitaka, TokyoINJapan\n" ]
[ "University of California\nLawrence Livermore National Laboratory\n94550LivermoreCA", "Center for Astrophysics\nDepartment of Physics, Notre Dame\nTheory Division\nUniversity of Notre Dame\nNational Astronomical Observatory\n46556, 181-8588Mitaka, TokyoINJapan" ]
[]
We explore several models which might be proposed to explain recent possible detections of high-energy (TeV) gamma rays in association with low-energy gamma-ray bursts (GRBs). Likely values (and/or upper limits) for the source energies in low-and high-energy gamma rays and hadrons are deduced for the burst sources associated with possible TeV gamma-ray detections by the Project GRAND array. Possible spectra for energetic gammas are deduced for three models: 1) inverse-Compton scattering of ambient photons from relativistic electrons; 2) proton-synchrotron emission; and 3) inelastic scattering of relativistic protons from ambient photons creating highenergy neutral pions, which decay into high-energy photons. These models rely on some basic assumptions about the GRB properties, e.g. that: the low-and high-energy gamma rays are produced at the same location; the time variability of the high-energy component can be estimated from the FWHM of the highest peak in the low-energy gamma ray light curve; and the variabilityluminosity relation of Fenimore & Ramirez-Ruiz(2000)gives a reliable estimate of the redshifts of these bursts. We also explore the impact of each of these assumptions upon our models. We conclude that the energetic requirements are difficult to satisfy for any of these models unless, perhaps, either the photon beaming angle is much narrower for the high-energy component than for the low-energy GRB or the bursts occur at very low redshifts ( < ∼ 0.01). Nevertheless, we find that the energetic requirements are most easily satisfied if TeV gamma rays are produced predominantly by inverse-Compton scattering with a magnetic field strength well below equipartition or by protonsynchrotron emission with a magnetic field strength near equipartition. PACS numbers: 98.70.Rz, 98.70.Sa, 95.55.Vj
10.1016/j.astropartphys.2003.08.005
[ "https://export.arxiv.org/pdf/astro-ph/0206383v3.pdf" ]
16,001,225
astro-ph/0206383
af9d56c9c24036048cb0ed41460d9fb8223e9fc2
Constraints on Models for TeV Gamma Rays from Gamma-Ray Bursts 5 Sep 2003 P Chris Fragile Grant J Mathews John Poirier Tomonori Totani University of California Lawrence Livermore National Laboratory 94550LivermoreCA Center for Astrophysics Department of Physics, Notre Dame Theory Division University of Notre Dame National Astronomical Observatory 46556, 181-8588Mitaka, TokyoINJapan Constraints on Models for TeV Gamma Rays from Gamma-Ray Bursts 5 Sep 2003acceleration of particlescosmic raysgamma ray burstsgamma ray sources * Electronic address: fragile@llnlgov 2 We explore several models which might be proposed to explain recent possible detections of high-energy (TeV) gamma rays in association with low-energy gamma-ray bursts (GRBs). Likely values (and/or upper limits) for the source energies in low-and high-energy gamma rays and hadrons are deduced for the burst sources associated with possible TeV gamma-ray detections by the Project GRAND array. Possible spectra for energetic gammas are deduced for three models: 1) inverse-Compton scattering of ambient photons from relativistic electrons; 2) proton-synchrotron emission; and 3) inelastic scattering of relativistic protons from ambient photons creating highenergy neutral pions, which decay into high-energy photons. These models rely on some basic assumptions about the GRB properties, e.g. that: the low-and high-energy gamma rays are produced at the same location; the time variability of the high-energy component can be estimated from the FWHM of the highest peak in the low-energy gamma ray light curve; and the variabilityluminosity relation of Fenimore & Ramirez-Ruiz(2000)gives a reliable estimate of the redshifts of these bursts. We also explore the impact of each of these assumptions upon our models. We conclude that the energetic requirements are difficult to satisfy for any of these models unless, perhaps, either the photon beaming angle is much narrower for the high-energy component than for the low-energy GRB or the bursts occur at very low redshifts ( < ∼ 0.01). Nevertheless, we find that the energetic requirements are most easily satisfied if TeV gamma rays are produced predominantly by inverse-Compton scattering with a magnetic field strength well below equipartition or by protonsynchrotron emission with a magnetic field strength near equipartition. PACS numbers: 98.70.Rz, 98.70.Sa, 95.55.Vj I. INTRODUCTION Evidence has been accumulating for the arrival of ∼ GeV-TeV gamma rays in coincidence with low-energy (∼ MeV) gamma-ray bursts (GRBs). For example, EGRET detected seven GRBs which emitted high energy photons in the ∼ 100 MeV to 18 GeV range [1,2,3]. There have also been results suggestive of gamma rays beyond the GeV range [4,5], although these were not claimed as firm detections. Evidence has also been reported for TeV emission in one burst out of 54 BATSE GRBs in the field of view of the Milagrito detector [6]. In another paper Poirier et al. [7] reported suggestive evidence for sub-TeV gamma rays arriving in coincidence with GRBs which occurred near zenith above the Gamma Ray Astrophysics at Notre Dame (GRAND) air shower array. In that experiment, most of the eight bursts analyzed were associated with at least some marginal excess (∼ 1σ) of muons including the event detected by Milagrito. One burst evidenced a possible detection at the 2.7σ level. As shown in Poirier et al. [7], if this detection is real, then the output in energetic gammas is likely to dominate the energetics of the burst. Although these data are not overwhelmingly convincing, they are at least suggestive that detectable energetic TeV gamma rays might be associated with low-energy gamma-ray bursts [8]. Moreover, these new detections, if real, can not be explained by a simple extrapolation of the BATSE spectrum [7], particularly if intergalactic absorption is taken into account [9,10]. A new ∼ TeV component in the GRB spectrum seems to be required. The present work is therefore an attempt to interpret these new possible detections in the context of three models, which might be proposed for the production of TeV gammas in a GRB. These are: 1) inverse-Compton scattering of ambient photons from relativistic electrons in the burst environment; 2) proton-synchrotron emission [10,11,12,13]; and 3) inelastic scattering of relativistic protons from ambient photons creating high-energy neutral pions, which decay into high-energy photons [14,15]. We here briefly outline the underlying physics and characteristic energetic gamma-ray spectra associated with each of these possible models. We then derive limits on the parameters of these models based upon the detection limits from the Project GRAND array. Based upon this, we conclude that it is difficult for any of these models to satisfy the energetic requirements unless the photon beaming angle is very narrow for the high-energy component. Of the models considered, the most likely are inverse-Compton scattering or proton-synchrotron emission. We note, however, that these conclusions rely upon a few assumptions. For example, we have assumed that the low-and high-energy gamma rays are produced at the same location; that the time variability of the high-energy component can be estimated from the FWHM of the highest peak in the low-energy gamma ray light curve; and that the variability-luminosity relation of Fenimore & Ramirez-Ruiz (2000) gives a reliable estimate of the redshifts of these bursts. We explore the impact of each of these assumptions and find that, unless the bursts occur at very low redshifts ( < ∼ 0.01), the energetic requirements remain difficult to satisfy. II. LOW-ENERGY GRB PROPERTIES The mystery of the astrophysical origin for low-energy gamma ray bursts (GRBs) has been with us for some time. As of yet there is no consensus explanation, although there is mounting evidence for an association with supernovae [16]. A likely scenario is a burst environment involving collisions of an ultra relativistic e + − e − plasma fireball [17,18,19]. These fireballs may produce low-energy gamma rays either by "internal" collisions of multiple shocks [20,21], or by "external" collisions of a single shock with ambient interstellar material [22]. In either of these paradigms it is possible for very energetic gammas to be produced through inverse-Compton scattering of ambient photons off the relativistic electrons. Furthermore, it seems likely that baryons would be accelerated along with the pair plasma to very high energies [12,14,23]. Synchrotron emission from energetic protons [11,12,13,24] or possibly hadro-production of pions in the burst environment [15] and subsequent π 0 gamma decay could also lead to an additional spectral component of very energetic gammas. In any case, it is at least plausible that energetic gammas could arrive in coincidence with a gamma-ray burst. This was the premise of Project GRAND's search for high-energy gammas arriving in coincidence with BATSE GRB observations. It is also possible, however, that low-and high-energy gamma-ray components are generated in different regions or phases of a burst. This could lead to substantially different arrival times for each component. This was in fact the case for the 18 GeV photon observed by EGRET, which arrived ∼ 4500 s after the low-energy emission had ended. Observations of TeV gammas could provide important clues as to the baryon loading, Lorentz factor, and ambient magnetic field of the relativistic fireball. Our goal in this paper is to constrain the possible spectrum and source of energetic photons using the Project GRAND data. Hence, we restrict ourselves to considering high-energy gammas produced concurrently with low-energy gammas consistent with the search technique employed by Project GRAND. III. FITS TO OBSERVED GRB SPECTRA Table I summarizes some of the features of the BATSE and Project GRAND observations of the eight GRBs analyzed in Poirier et al. [7], where a detailed explanation of the Project GRAND results can be found. In the present paper, we will denote quantities in the frame of the observer by the superscript "ob". We will also use ǫ γ to denote the low energy GRB photons and distinguish them from the high-energy component, denoted E γ . The observed BATSE spectra are fit with a broken power law of the form [25] dφ γ (ǫ ob γ ) dǫ ob γ = a      ǫ ob,MeV γb β l −β h ǫ ob,MeV γ −β l , if ǫ ob γ < ǫ ob γb , ǫ ob,MeV γ −β h , if ǫ ob γ ≥ ǫ ob γb ,(1) where β h ≈ 2, β l ≈ 1, and ǫ ob γb ≈ 1 MeV is the break energy of the observed spectrum. Although these bursts are often better fit by using an exponential to join the two components, a broken power law is adequate for the present discussion. It maintains a simple analytic form for the equations, and as we shall see, the precise low energy form is almost irrelevant as long as a break energy exists. In what follows we will use values of a, β h , β l , and ǫ ob γb derived from optimum fits to the BATSE spectra for all events except the Milagrito event for which the BATSE fluence was too weak to obtain a reliable spectral fit. The BATSE fit parameters corresponding to equation (1) are listed in Table II. The fit parameters for these bursts were provided at our request by M. S. Briggs at the Marshall Space Flight Center. We also include the variability time scale ∆t for these bursts, which was estimated as the full width at half maximum (FWHM) of the brightest peak in each light curve. The light curves of GRBs typically show a wide range of timescale variability, so this choice may not be justified. In §VII we explore the dependence of our conclusions on a wide range of values for ∆t. In Table III, we give estimated redshifts for each burst. GRB 990123 is the only burst in this group for which an optical counterpart was detected. This burst, therefore, is known to have occurred at a redshift of z = 1.6 [26]. The redshifts for most of the remaining bursts were estimated using the variability-luminosity relation of Fenimore & Ramirez-Ruiz [27]. This method relies on the apparent correlation between the time variability of a burst, which can be measured from its light curve, and the absolute luminosity of the burst, which we wish to infer. This provides us with a straightforward method for converting GRB observables into luminosities and redshifts [27]. Following Fenimore & Ramirez-Ruiz [27], we first fit a quadratic polynomial to the background in the non-burst portions of the BATSE 64 ms four channel data (i.e. DISSC data). Let b i be the binned background counts from this polynomial fit. If g i are the observed binned counts during the actual burst event, then the net count is c i = g i − b i . We then rebin the counts by dilating the time samples by Y = (1 + z)/(1 + z b ), where z is the redshift we wish to estimate and z b is a baseline redshift. Following [27], we take z b = 2. The new, dilated net counts, C i , represent what the time history would look like at z = z b . The variability is then defined to be the average mean-square of the variations in C i relative to a smoothed time history, as V = Y −0.24 1 N (C i − < C >) 2 − B i C 2 p ,(2) where C p is the peak of the dilated net count during the burst and < C > is the count smoothed with a square-wave window with a length equal to 15% of the duration of the burst. The Y −0.24 term corrects the variability for the energy-dependence of the time scale of a GRB [27]. The B i term (dilated background counts in a sample) accounts for the Poisson noise. The sum is taken over the N samples that exceed the background by at least 5σ. The estimated variabilities are listed in Table III. Based upon the fits of Fenimore & Ramirez-Ruiz [27], we can relate this variability V to the peak isotropic luminosity L 256 averaged over 256 ms in a specified energy range, ǫ l,p to ǫ u,p (i.e. L 256 erg s −1 in the 50 to 300 keV band). This variability-luminosity relation is L 256 /(4π) = 1.9 × 10 51 V 0.86 erg s −1 .(3) This peak luminosity depends upon the redshift, the observed spectral shape, and the observed peak photon flux P 256 (also averaged over 256 ms and over the same energy range) as L 256 = 4πD 2 z P 256 < ǫ γ > ,(4) where D z is the co-moving distance and < ǫ γ > is the average photon energy in the luminosity bandpass per photon in the count bandpass. For this work we compute luminosities for an isotropic burst environment, such that Ω = 4π, where Ω is the unknown opening angle of the burst. If GRBs emit in a jet, our inferred luminosities and energies are diminished by Ω/4π. The co-moving distance D z for a flat Ω M + Ω Λ = 1 model is simply given by D z = c H 0 z 0 Ω M (1 + z ′ ) 3 + Ω Λ −1/2 dz ′ .(5) For the purposes of the present discussion, we will adopt the currently popular Ω M = 0.3, [28,29,30]. From the observed photon spectrum the average photon energy in the luminosity bandpass per photon in the count bandpass is Ω Λ = 0.7, H 0 = 72 km s −1 Mpc −1 model< ǫ γ >= ǫu,p ǫ l,p ǫ γ φ[ǫ γ /Y ]dǫ γ ǫu ǫ l φ[ǫ γ ]dǫ γ ,(6) where ǫ l and ǫ u are the limits on the BATSE energy range (≈ 20 − 1500 keV). We can now iteratively solve equations (2-6) until the estimate for z converges. This method of estimating the redshift of GRBs was found [31] to be consistent with other estimates that rely upon an apparent relation between the luminosity and the time lag between hard and soft energy peaks. The correlation between these two independent methods argues in favor of their reliability [32]. However, in §VII we explore the impact of systematically larger and smaller redshifts on our conclusions. Additionally, we can estimate the effective 4π luminosity at the source in the BATSE energy band time-averaged over the full T 90 interval, L γ , by L γ = 4πD 2 z ǫu ǫ l dǫ γ ǫ γ dφ[ǫ γ /(1 + z)] dǫ γ .(7) This is the luminosity estimate we use for the rest of the work presented in this paper. Both luminosity estimates are listed in Table III for each burst. A. GRB 970417a GRB 970417a was the one burst (of 54 in the field of view) for which the Milagrito collaboration reported evidence of TeV emission during the duration of this burst within the BATSE error circle [6]. For this reason, we have included it in our analysis. Interestingly, this was a relatively weak BATSE burst with a fluence of 3.9 × 10 −7 erg cm −2 in all four BATSE energy channels (> 20 keV). Because this is such a weak low-energy burst, it is difficult to obtain reliable fits for the BATSE spectral parameters. For this reason, we have instead used the following average values of all bright bursts from Preece et al. [33]: β l = 1.0, β h = 2.25, and ǫ ob γb = 225 keV. We can then use these average parameters and the observed flux to fix the normalization, a. The weak signal in the BATSE band also prevents us from using the variability-luminosity relation described above to determine the redshift of this burst. Instead we adopt z = 0.7 based upon the analysis of Totani [10]. IV. THE MODELS A. Inverse-Compton Spectrum One possible source for energetic gamma rays is the inverse-Compton (IC) scattering of low-energy ambient photons by relativistic electrons. Indeed, such IC photons are thought to be the source of observed high-energy photons from active galactic nuclei such as Mk-421 [34]. The inverse-Compton-scattering spectrum is generally written as dΦ γ dE γ IC ∝ E −1 γ dE e dΦ e dE e dxf e−s (x) ,(8) where dΦ e /dE e is the spectrum of electrons characteristic of a Fermi mechanism, dΦ dE = A      E −α , if E < E b , E b E −α−1 , if E ≥ E b ,(9) where A is a normalization constant and α ≈ 2 ± 0.2 [35]. The flux of the synchrotron photons f e−s is f e−s (x) =      x −(α−1)/2 , below the cooling break, x −α/2 , above the cooling break, where x ∝ E γ /E 2 e .(10) Evaluating the integrals, we arrive at the simplified expression (dΦ γ /dE γ ) IC ∝ E −δ γ ,(11) where the spectral index δ is given by δ =      (α + 1)/2 , for E γ < E cool , (α + 2)/2 , for E γ > E cool ,(12) where α ∼ 2 is the electron spectral index and E cool is the energy scale at which the electron cooling time becomes comparable to the system lifetime. The inverse-Compton cooling time in the frame of the shock is 1 t IC = 1.3 × 10 −4 γ e L ob γ,51 (1 + z) 4 Γ 6 300 ∆t 2 s −1 ,(13) where γ e is the electron Lorentz factor and Γ = 300Γ 300 is the bulk Lorentz factor, both in the shock frame; L ob γ,51 = L ob γ /10 51 erg s −1 where L ob γ = L γ /(1 + z) 2 ; and ∆t = (1 + z)r d /Γ 2 c is the variability timescale (in seconds) for the GRB, where r d is the radius at which low-energy gamma rays are emitted, both measured in the observer frame. The total fractional power radiated in inverse-Compton photons can be estimated from the ratio of the expansion time to the cooling time: f IC = Γ∆t (1 + z)t IC (14) = 3.9 × 10 −2 γ e L ob γ,51 (1 + z) 3 Γ 5 300 ∆t .(15) The cooling frequency corresponds to f IC = 1. The relation between the Compton-scattered photon energy and γ e in the observer's frame is E ob γ ∼ γ 2 e ǫ ob γ , where again ǫ ob γ is the energy of the BATSE photons in the observer's frame. For simplicity we will take ǫ ob γ = ǫ ob γb . Therefore, the cooling break energy can be written as E ob cool = 0.66 Γ 10 300 ∆t 2 ǫ ob,MeV γb (L ob γ,51 ) 2 (1 + z) 6 GeV .(16) Relativistic Klein-Nishina (KN) corrections to the Compton spectrum may be important in the GRB environment over some range of energies. For completeness we include these by introducing a parameter Γ KN = E ob γ ǫ ob γ (1 + z) 2 /Γ 2 m 2 e c 4 . When Γ KN ≫ 1, the KN effect is important. We adopt Γ KN = 1 as the the lower boundary of the KN regime. This implies a lower limit to the observed gamma energy at which the KN effects should be considered, E ob KN ≡ 24 Γ 2 300 (1 + z) 2 ǫ ob,MeV γb GeV .(17) In the KN regime the emissivity of IC radiation per electron is independent of the electron energy and the cooling time becomes proportional to the electron energy [see e.g. 36]. The emissivity is reduced by a factor of ∼ Γ −2 KN compared with the classical IC formula (see Eq. [15]), and the observed photon energy is E ob γ ∼ Γγ e m e c 2 /(1 + z). We can thus define the cooling frequency in the KN regime, by setting f KN IC = f IC Γ −2 KN = 1. This gives E KN,ob cool = 140 L ob γ,51 Γ 2 300 ∆t(ǫ ob,MeV γb ) 2 GeV,(18) where we have used E ob γ ∼ Γγ e m e c 2 /(1 + z) for the observed Compton-scattered photon energy. In the KN regime, the cooling is efficient in the lower photon energy range of E γ < E KN cool , i.e., contrary to the situation in the non-KN regime. In the KN regime, the photon index becomes δ =      α , for E γ < E KN cool , α + 1 , for E γ > E KN cool .(19) To summarize, the IC spectrum can be modeled as dΦ γ dE γ IC = A IC γ                    E ob γ −(α+1)/2 , if E γ < E cool , (E cool ) 1/2 E ob γ −(α+2)/2 , if E cool ≤ E γ < E KN , (E cool ) 1/2 (E KN ) α/2−1 E ob γ −α , if E KN ≤ E γ < E KN cool , (E cool ) 1/2 (E KN ) α/2−1 E KN cool E ob γ −(α+1) , if E γ ≥ E KN cool ,(20) where all energies are assumed to be measured in GeV. For typical parameters, all three quantities, E cool , E KN , and E KN cool are around 1-100 GeV. In any case, the spectrum beyond E γ > ∼ 100 GeV must be steep with δ ∼ 3. The resultant IC gamma-ray spectrum should cut off above E ob,max γ,IC ≈ E ob,max e = 4.2 × 10 5 Γ 5/2 300 ∆t 1/2 ξ 1/4 B (L ob γ,51 ) 1/4 (1 + z) 2 GeV ,(21) due to the cut-off in the spectrum of ultra-relativistic electrons at energies above E ob,max e . B. Spectrum of Proton-Synchrotron Gamma Rays It is generally believed that the expanding plasma contains at least some baryons. Indeed, some baryons within the jet are required to increase the burst duration and luminosity [cf. 37]. In the region where the electrons are accelerated, protons may also be accelerated up to ultra-high energies > 10 20 eV [14] producing a spectrum characteristic of a Fermi mechanism, (eq. [9]). The possibility of energetic protons producing ∼TeV gammas by synchrotron emission has been discussed in a number of papers [10,11,12,13,24]. This mechanism has the desirable characteristic that the low-energy photons produced by electron-synchrotron emission and the high-energy photons from energetic protons can be produced simultaneously in the same environment [12,13]. In this model one assumes that there is a magnetic field present in the burst environment with approximate equipartition between the magnetic energy density B 2 /8π and the total energy density, U. That is, B 2 8π = ξ B U ,(22) where ξ B is a fraction of order unity. Following Totani [13] we assume an optimally efficient proton-synchrotron environment in which U ∼ U p ∼ (m p /m e )U γ , where U γ is the photon energy density of BATSE gamma rays in the frame of the burst. This latter quantity can be related to the GRB luminosity utilizing L γ = 4πr 2 d Γ 2 cU γ . We can then rewrite Eq. (22) in terms of the variables introduced in the previous section. This gives B = 1.4 × 10 5 ξ 1/2 B (L ob γ,51 ) 1/2 (1 + z) 2 Γ 3 300 ∆t G .(23) The photon energy from proton-synchrotron emission in the observer's frame is then E ob γ,p−s = ΓΓ 2 p ehB (1 + z)m p c = (E ob p ) 2 ehB(1 + z) m 3 p Γc 5 ,(24) where Γ p = E p /m p c 2 is the relativistic gamma factor of the protons in the frame of the fireball. This leads to the desired relation between the observed proton energy and the observed gamma energy, E ob p = C × E ob,GeV γ,p−s 1/2 ,(25) where, C = 1.6 × 10 9 Γ 2 300 ∆t 1/2 ξ 1/4 B (L ob γ,51 ) 1/4 (1 + z) 3/2 GeV . The final quantity needed to derive the energetic gamma spectrum is the cooling rate due to synchrotron emission. In the frame of the shock, the synchrotron cooling rate is 1 t p−sync ≡ − 1 E p dE p dt = 4e 4 B 2 E ob p (1 + z) 9m 4 p c 7 Γ = 7.5 × 10 −2 ξ 3/4 B (L ob γ,51 ) 3/4 (1 + z) 7/2 Γ 5 300 ∆t 3/2 E ob,GeV γ 1/2 s −1 .(27) The fractional energy loss to synchrotron photons then becomes f p−sync (E ob p ) ≃ Γ∆t (1 + z)t p−sync =      E ob p E ob pb,p−s , if E ob p < E ob pb,p−s , 1 , if E ob p ≥ E ob pb,p−s ,(28) where E ob pb,p−s = 7.5 × 10 7 Γ 6 300 ∆t ξ B L ob γ,51 (1 + z) 4 GeV .(29) Finally, we note that a proton flux Φ p (E p ) will produce an energetic gamma flux of Φ γ = E p E γ f π (E p )Φ p .(30) The observed high-energy photon spectrum can then be derived from equations (9), (30), and (25), dΦ γ dE γ = dΦ p dE p × dE p dE γ × dΦ γ dΦ p ,(31) to yield dΦ γ dE γ p−s = 1 2 A p−sync p C 2−α ×      E ob,GeV γb,p−s −1/2 E ob,GeV γ −(α+1)/2 , if E ob γ < E ob γb,p−s , E ob,GeV γ −(α+2)/2 , if E ob γ ≥ E ob γb,p−s ,(32) where A p−sync p is a normalization constant to be determined from observations. The break energy is E ob γb,p−s = 2.1 × 10 −3 Γ 8 300 ∆t ξ 3/2 B (L ob γ,51 ) 3/2 (1 + z) 5 GeV . The gamma-ray spectrum should cut off above E ob,max γ,p−s ≈ 15Γ 300 /(1 + z) TeV [13] due to the cut-off in the spectrum of ultra-relativistic protons at energies above E ob,max p = 2.1 × 10 11 Γ 5/2 300 ∆t 1/2 ξ 1/4 B (L ob γ,51 ) 1/4 (1 + z) 2 GeV .(34) C. Spectrum of Photo-Pion Gamma Rays Along with undergoing synchrotron radiation, protons accelerated in the burst environment may collide with photons in the expanding fireball to produce secondary pions, which subsequently decay into high-energy gammas and neutrinos. This source of gamma rays seems unlikely due to the fact that it results from a secondary strong interaction and therefore has a small cross-section relative to electromagnetic interactions. Nevertheless an estimate of this spectrum is straightforward, so we include it here. Another alternative possibility might be pion production via proton-proton collisions [cf. 20]. However, the proton density in the frame of the shock must be small to ensure a low optical depth for gammas. Hence, p − γ collisions are favored over p − p, although as we will show, even this preferred reaction places unreasonable energetic requirements on the burst environment. Following Waxman & Bahcall [15], the energy loss rate due to pion production is 1 t π = 1 2Γ 2 p ∞ Eo dEσ π (E)ξ(E)E ∞ E/2Γp dǫ ǫ 2 dφ γ (ǫ) dǫ ,(35) where E 0 ≈ 0.15 GeV is the threshold for pion production. In the first integral, σ π is the cross section for pion production due to a collision with a photon of energy ǫ γ in the rest frame of the proton, and ξ(E) is the average fractional energy lost to the pion. The second integral is over the low energy GRB spectrum, where φ γ (ǫ γ ) is the photon flux in the frame of the proton. The evaluation of t π can be simplified [15] by integrating the pion production cross section over the broken-power-law GRB spectrum (Eq. [1]) transformed back to the frame of the expanding plasma. Approximating the integral over the pion production cross section by the contribution from the peak of the ∆-resonance [as in 15] we deduce t π for a general spectral power law index β i : 1 t π = cU γ ǫ γb σ peak ξ peak (1 + β i ) ∆E peak E peak × min 1, 2Γ p ǫ γb E peak β i −1 s −1 ,(36) where σ peak ≈ 5 × 10 −28 cm 2 is the ∆-resonance cross section, while E peak = 0.3 GeV and ∆E peak ≃ 0.2 GeV are the energy and width of the resonance, respectively. The fractional energy lost at the peak is ξ peak ≈ 0.2, and U γ is the photon energy density of BATSE gamma rays in the frame of the fireball, as before. As before we estimate the fractional power radiated as f π = Γ∆t (1 + z)t π = 4.5 × 10 −4 (1 + β i ) L ob γ,51 (1 + z) 2 ǫ ob,MeV γb Γ 4 300 ∆t      × E ob p E ob pb,π β h −1 , if E ob p < E ob pb,π , ×1 , if E ob p ≥ E ob pb,π ,(37) where ǫ ob γb ≈ 1 MeV is the break energy of the two power laws of the observed GRB spectrum. The last factor in equation (37) describes a break in the proton spectrum. In the observer frame this break energy is [15] E ob pb,π = 1.3 × 10 7 Γ 2 300 (1 + z) 2 ǫ ob,MeV γb GeV .(38) Roughly half of the energy lost by the protons goes into π + 's, which quickly decay into neutrinos and positrons (through µ + s). In this work, we have ignored the effects of these decay products on the emerging gamma-ray spectrum. For neutrinos this is reasonable since there is very little chance of them interacting further. The positrons, however, may influence the gamma-ray spectrum through positron-synchrotron radiation [24] or pair annihilation. The other half of the energy lost by the protons goes into π 0 's, which then decay into two photons. The mean pion energy is ξ peak E p . When the π 0 decays, the energy is shared equally among the photons. Hence, each gamma ray has an average energy E γ = ξ peak E p /2 .(39) Now from equation (31) dΦ γ dE γ π = f π 2 ξ peak 2−α AE −α γ = A π p D π ×      (1 + β h ) −1 E ob,GeV γb,π 1−β h E ob,GeV γ β h −α−1 , if E ob γ < E ob γb,π , (1 + β l ) −1 E ob,GeV γ −α , if E ob γ ≥ E ob γb,π ,(40) where A π p is a normalization constant and D π = 4.5 × 10 −4 L ob γ,51 (1 + z) 2−α ǫ ob,MeV γb Γ 4 300 ∆t 2 ξ peak 2−α .(41) The observed break energy E ob γb,π in the pion decay gamma spectrum is given by E ob γb,π = ξ peak 2 E ob pb,π ≈ 1.3 × 10 6 Γ 2 300 (1 + z) 2 ǫ ob,MeV γb GeV .(42) Above this energy the spectrum should obey the dΦ γ /dE γ ∼ E −2 γ of the protons and below this break energy, the exponent should be harder by one power, i.e. dΦ γ /dE γ ∼ E −1 γ . As a practical matter, photons with energy as high as the break energy will not be observed, as they will be extinguished by pair production as described below. V. PHOTON, PROTON, AND ELECTRON LUMINOSITIES AT THE SOURCE From the above it is clear that the three models considered here imply different spectral shapes for the high-energy gamma component. Figure 1 compares the initial spectra for all three mechanisms normalized to reproduce the Project GRAND observations for GRB 971110 (as explained in Section 7). This would correspond to the unrealistic limit of no self or intergalactic absorption. Nevertheless, this illustrates the fact that these mechanisms have significantly different energetic requirements. Clearly, the most favorable energetically are the inverse-Compton and proton-synchrotron models. Figure 2 further illustrates this point by reproducing the source proton and electron spectra required by the various models, again normalized to reproduce the Project GRAND observations. Also included in this figure is the source electron spectrum required to only produce the observed BATSE data for this burst, following the electron-synchrotron model. The Project GRAND result, if it represents a real detection, requires a much higher flux of electrons to produce sufficient high-energy gamma rays through inverse-Compton scattering. This is probably a troubling requirement for the inverse-Compton model. To calculate these spectra we have used the normalization of the gamma-ray spectrum from the observed muon excess to determine the normalization of the associated ultrarelativistic proton and electron spectra. This is straightforward for the proton-synchrotron and photo-pion models, since the proton normalization appears explicitly in our final expressions (equations [40] and [32]). For the inverse-Compton model, we have followed Sari & Esin [38] to estimate the electron normalization from the following approximate relation between the low-energy electron-synchrotron spectrum, assumed to be observed by BATSE, and the inverse-Compton spectrum in the energy range of Project GRAND E γ dΦ γ dE γ IC ≃ 0.5r d σ T nǫ γ dφ γ dǫ γ e−s .(43) We take the electron number density n to be 1 cm −3 . The electron-synchrotron spectrum is assumed to have a form directly comparable to equation (32). We evaluate equation (43) at E γ = E ob cool and ǫ γ = ǫ ob γb to find A e−sync e . On the other hand, if one attributes [14] the observed cosmic-ray excess above 10 20 eV to the energetic protons accelerated in GRBs, then an independent estimate of the proton normalization at the source can be obtained. Following Waxman [14] we note that the observed cosmic ray flux above 10 20 eV is ∼ 3 × 10 −21 cm −2 s −1 sr −1 [39], corresponding to an average universal number density of energetic cosmic rays of n CR ∼ 10 −30 cm −3 . If this density is due to GRBs then the number of protons with energies greater than 10 20 eV produced per GRB must be N(E p > 10 20 ) = n CR /ν γ τ CR ∼ 5 × 10 44 ,(44) where ν γ ≈ 2 × 10 −10 (h/70) 3 Mpc −3 yr −1 is the cosmological GRB rate [40,41,42], and τ CR ≈ 3×10 8 yr is the lifetime of protons with E p > 10 20 eV. With our nominal E −2 p spectral form, we deduce an absolute normalization of the proton spectrum emerging from an average GRB source of dN p /dE p = 5 × 10 55 E −2 GeV GeV −1 . This is to be compared with the number of photons with E > 1 MeV emerging from a nominal GRB. If we assume that 10 53 erg is released in gammas above 1 MeV for an energetic photon spectrum of dΦ γ /dE γ ∝ E −2 γ , then the normalized spectrum for gammas above 1 MeV would be dN γ /dE γ = 6 × 10 58 E −2 GeV GeV −1 . Thus, one expects about 1200 gammas per proton from such a burst. Since both the proton-synchrotron and photo-pion models are based upon the same underlying baryon content in the relativistic plasma, it is instructive to summarize the relative efficiency of these two mechanisms for generating energetic gamma rays in GRBs. Let us consider a typical burst with β h = 2 and a proton spectrum with α = 2. Then in the region E ob γ ∼ 1 TeV, the fractional energy loss into these two mechanisms is f π = 1.7 × 10 −7 L ob γ,51 (1 + z) 4 Γ 6 300 ∆t E ob γ TeV ,(45) The factor of 1.7 × 10 −7 suggests that proton-synchrotron emission will dominate over the photo-pion production for the parameters and gamma-ray energy considered. Only for sufficiently small ξ B ( < ∼ 10 −4 ) does the efficiency of the proton-synchrotron mechanism begin to deviate from unity as ξ 3/4 B . A direct comparison between inverse-Compton scattering and proton-synchrotron emission is not possible since these two mechanisms rely upon different underlying energy sources, i.e. relativistic electrons for inverse-Compton scattering and relativistic protons for protonsynchrotron emission. Nevertheless, we note that both of these methods can be very efficient. In the region E ob γ ∼ 1 TeV, f IC ∼ 1 and f p−sync ∼ 1 for the range of parameters considered. This suggests that inverse-Compton scattering provides a very competitive mechanism for the generation of energetic gammas. Furthermore, it makes no requirement on the preexistence of a magnetic field or baryon loading in the fireball plasma. Another important difference among all of these mechanisms is their associated cut-off energies. As noted above, there should be a cut-off for the proton-synchrotron spectrum around E ob,max γ,p−s ≈ 15 TeV, corresponding to a cut-off in the ultra-relativistic proton energies. The inverse-Compton spectrum has a much larger cut-off at around E ob,max γ,IC ≈ 400 TeV, corresponding to a cut-off in the relativistic electron energies. For pion decay, however, the spectrum may extend all the way to 10 7 TeV [15], but with a break at around 1300 TeV. These differences have a large effect on the implied total source luminosities of the bursts, as is apparent in Figure 1. VI. PAIR PRODUCTION OPTICAL DEPTH The spectra derived above must be corrected for two effects, both of which are due to pair production by energetic photons. First, within the burst environment, energetic gamma rays will interact with other photons to produce e + − e − pairs. If this process is highly efficient, ∼ TeV gamma rays may not be able to escape from the burst. Even if some photons escape, this self-absorption will affect the implied source luminosity. The second effect is due to absorption along the line-of-sight from the burst environment. Here the energetic gamma rays interact with the intergalactic infrared and microwave backgrounds. This effect can cause a dramatic shift in the spectrum of energetic gammas in the TeV range depending upon the distance to the burst. A. Internal Optical Depth from Pair Production A photon of energy E γ interacts mainly with target photons of energy ǫ γ ∼ 2m 2 e c 4 /E γ in the shock frame. We can approximate the cross section for pair-production as 3σ T /16, where σ T = 6.6 × 10 −25 cm 2 is the Thomson cross section. Then the optical depth can be approximated as τ γγ,int ∼ 3 16 σ T D 2 z r 2 d ǫ γ dφ γ [ǫ γ /(1 + z)] dǫ γ ∆t Γ(1 + z) ,(47) where ∆tΓ/(1 + z) is the width of the emitting region as measured in the shock frame. This formula is similar to that of Waxman & Bahcall [15], except that this form takes into account the spectral break in the low-energy GRB photons. This break is important since it implies that the optical depth is proportional to ∼ E γ for E ob γ < ∼ 2m 2 e c 4 Γ 2 /ǫ ob γb ≈ 50 GeV, but roughly constant for higher energies. The Waxman & Bahcall [15] result is only valid in the lower energy range. The proper energy dependence is important because the internal optical depth can be of order unity. In the case of GRB 971110, the internal optical depth at 100 GeV is ≈ 3 implying that some energetic gammas could emerge. The thin lines in Figure 1 show how the three source spectra normalized to fit GRB 971110 would be modified by internal absorption. The change in the energy dependence of the internal optical depth is apparent above about 50 GeV. For the remainder of the bursts, we were unable to place meaningful constraints on the internal optical depth due to large uncertainties in the fit parameters, particularly β h and Γ. For these bursts, we have taken the very optimistic assumption of τ γγ,int = 0. B. Intergalactic Optical Depth from Pair Production Another important constraint on the observed burst spectra comes from the absorption of photons via pair production during collisions with the intergalactic infrared background. In this work we use a calculated optical depth for intergalactic absorption based upon the standard formulation [e.g. 9]. We use a model for the luminosity evolution of background light from Totani, Yoshii, & Sato [43]. The dust emission component is calculated assuming a dust emission spectrum similar to that of the Solar neighborhood. The fraction of light absorbed by dust is adjusted to reproduce the observed far infrared background from COBE [44]. This method is summarized in Totani [10]. It is consistent with other optical-depth calculations [cf. 9,45]. The resultant e + −e − pairs may then further modify the original gamma-ray spectrum by providing a medium for some of the remaining gamma rays to undergo intergalactic inverse-Compton scattering. This process results in complicated showers of secondary electrons and gammas. These secondary gammas may be observed, but over a much longer time-scale, since much of this secondary light traverses a longer path length. The flux from these secondary gamma rays is probably below the detection threshold of current arrays such as Project GRAND. Nevertheless, this might be a noteworthy effect. Figure 3 shows final spectra in the observer's frame when both internal and intergalactic absorption are taken into account, assuming the burst is arriving from a redshift of z = 0.6 (appropriate for GRB 971110). Even though the source spectra are vastly different, the observed high-energy gamma spectra are quite similar. Hence, the implied source energy requirement may be the only way to distinguish between the models. For illustration, Figure 3 also includes the observed BATSE spectrum for this burst, fit with both the Band (Eq. [1]) and electron-synchrotron models. VII. RESULTS In this work we use the muon observations of Project GRAND to fix the normalization (or upper limit) in each of the models described above for the various bursts analyzed. That is, the spectral shape is fixed from equations (40), (32), or (20), and the number of muons expected is then computed using N µ = dA × T90 × ∞ E min dE ob γ P µ (E ob γ ) dΦ γ (E ob γ ) dE ob γ exp(−τ γγ ) ,(48) where dA is the collecting area of the Project GRAND array (the effective area at the time of GRB 971110 was approximately 6.3 × 10 5 cm 2 ), E min is the primary gamma-ray detection threshold for Project GRAND (∼ 10 GeV), and P µ ≈ 7.0 × 10 −5 (E GeV γ ) 1.17 is the probability per primary for a muon to reach detection level at Project GRAND. This probability (valid for E γ > ∼ 3 GeV) was computed by Fasso & Poirier [46] using the Monte Carlo atmospheric absorption code, FLUKA. Here τ γγ includes both the internal and intergalactic optical depth estimated for each burst as described above. For illustration, Table IV summarizes some estimates of the relative magnitudes of internal and intergalactic optical depths at two energy scales, 100 GeV and 1 TeV. In practice, the integral in equation (48) is cut off at ≈ 30 TeV since the optical depth is quite high for photons above this energy. The normalization constants for each model are then adjusted so that N µ agrees with the 2σ upper limits set by Project GRAND, except for GRB 971110 where we used the observed mean value (see Table I). Here we adopt typical values for the degree of equipartition ξ B and the relativistic Lorentz factor Γ, namely ξ B ≈ 1 and Γ ≈ 300. Below we explore the dependence of our results on a broad range of these parameters. We can then use our normalized gamma spectra to estimate the total energy emitted in high-energy gammas at the source. For the inverse-Compton model, we can also estimate the energy emitted in electrons, noting again the characteristic spectrum of a Fermi mechanism given in equation (9). Similarly we can estimate the energy emitted in protons for the protonsynchrotron and photo-pion models. Implied energies for photon emission into 4π are given in Tables V, VI, and VII for the inverse-Compton, proton-synchrotron, and photo-pion models, respectively. We also list the required energies of the source protons or electrons, as appropriate. Our results assume α = 2 for the accelerated electron and proton spectra. For GRB 971110 we estimated the statistical uncertainties of our results using Monte Carlo techniques to explore the parameter space of each of the models assuming Gaussian error distributions. As evidenced by the large uncertainties, the models are not well constrained at present. The distinction between the inverse-Compton and proton-synchrotron mechanisms has an important consequence on the magnetic field of the GRB, specifically the degree of equipartition, ξ B . In the case of inverse-Compton scattering, the ratio of the inverse-Compton luminosity L IC γ to electron-synchrotron luminosity L sync should be equal to the ratio of the IC target photon energy density U γ to the magnetic field energy density U B since both mechanisms arise from the same population of hot relativistic electrons. Thus, L IC γ L sync = U γ U B ,(49) where U B = ξ B U and U is the total rest-frame energy density of the emission region. If we adopt the generally accepted view that the ∼MeV gamma rays seen by BATSE are caused by electron-synchrotron radiation then L sync = L γ . For GRB 971110, L IC γ /L γ ∼ 10 4 , which implies ξ B U < ∼ 10 −4 U γ .(50) Since the energy density of the ∼MeV BATSE gamma rays is likely to be higher than any other radiation source available for inverse-Compton scattering, we identify this energy density as U γ . This association has been implicit throughout this paper. On the other hand, since L IC γ ∼ 10 4 L γ the energy density of the ∼TeV gamma rays must be greater than U γ by about 10 4 . We can then use 10 4 U γ as a lower limit for the total rest-frame energy density U. Combining this with Eq. (50) we get the following upper limit on the degree of equipartition ξ B < ∼ 10 −8 .(51) Note also that Eq. (49) holds only in the classical regime. In the KN regime ξ B is even smaller because inverse-Compton scattering becomes less efficient. Hence, the inverse-Compton model is at odds with models that propose GRBs as the source of ultra-high energy cosmic rays, since those models require a magnetic field near equipartition, ξ B ∼ 1 [cf. 14]. In mechanisms are generally less sensitive to changes in Γ over the range 100 < ∼ Γ < ∼ 1000 or changes in ∆t over the range 0.1 < ∼ ∆t < ∼ 10. The sharp increase in the energy content of the protons and electrons at small Γ enters primarily through the internal optical depth. Figure 5 shows that the proton-synchrotron mechanism is highly efficient over a fairly broad range of ξ B ( > ∼ 0.01). Regardless of the mechanism, if GRB 971110 is indeed a detection and our estimated redshift of z = 0.6 is valid, the implied energy in energetic (E γ > 1 GeV) gamma rays is more than 100 times higher than the energy in the low-energy gamma rays. This renders the already challenging energetic requirements on the GRB source engine to be even more difficult. It may be possible to alleviate this difficulty if, perhaps, the photon beaming angle is much narrower for the high-energy component than for the low-energy GRB. Another possibility is that the redshift is much lower than our estimated value for this burst. In Figure 7 we explore the energetic requirements for GRB 971110 over a very broad range of redshifts from 0.005 to 3. This range is consistent with most currently measured GRB redshifts. This figure illustrates quite clearly the critical role that intergalactic absorption plays in driving up the energetic requirements and highlights the need for accurately determining the true redshifts of GRBs. VIII. CONCLUSION We have analyzed the eight GRBs discussed in Poirier et al. [7], which occurred above the Project GRAND array. We have studied the implied energies in energetic (∼TeV) gamma rays (and associated electrons and protons) of such GRBs in the context of three possible mechanisms: inverse-Compton scattering, proton-synchrotron radiation, and photo-pion production. Our analysis suggest that all of these models face significant energetic requirements. Gamma-ray production by either inverse-Compton scattering or proton-synchrotron radiation is probably the most efficient process. Although it can not be claimed that TeV gammas have unambiguously been detected in association with low-energy GRBs, we have argued that there is enough mounting evidence to warrant further study. Furthermore, we have shown that if TeV gammas continue to be observed, then they present some interesting dilemmas for GRB physics. In view of their potential as a probe of the GRB source environment, we argue that further efforts to measure energetic gamma rays in association with low-energy gamma-ray bursts are warranted. · · · a · · · a 2.8 31. 940526 · · · a · · · a 0.4 ± 0.2 9.6 ± 4.0 980420 · · · a · · · a 1.3 ± 0.8 20. ± 6. 960428 · · · a · · · a 3.1 ± 2.0 30. ± 10. 980105 · · · a · · · a 1.6 ± 1.2 22. ± 8. 980301 · · · a · · · a 6.8 ± 3.4 44. ± 13. 970417a · · · a · · · a 0.5 ± 0.3 12. ± 4. a Large uncertainties in the fit parameters made constraints on the internal optical depth uninformative in these cases. For these bursts, we have made the optimistic assumption τ γγ,int = 0. Here we estimate the total energy escaping in the gamma-ray component from each burst using the inverse-Compton model. We also estimate the total energy in the electron component at the source. a We quote the isotropic energies. These entries must be multiplied by Ω/(4π) to get the true energies. Here we estimate the total energy escaping in the gamma-ray component from each burst using the proton-synchrotron model. We also estimate the total energy in the proton component at the source. a We quote the isotropic energies. These entries must be multiplied by Ω/(4π) to get the true energies. Here we estimate the total energy escaping in the gamma-ray component from each burst using the photo-pion model. We also estimate the total energy in the proton component at the source. a We quote the isotropic energies. These entries must be multiplied by Ω/(4π) to get the true energies. Nevertheless, several points are worth noting from the tables. For the most significant possible detection (GRB 971110), the energetic requirements for the IC model (E T ot,IC γ = (2.6 ± 8.6) × 10 55 erg) and the proton-synchrotron model (E T ot,p−sync γ = (3. ± 10.) × 10 55 erg) are much less than that for a photo-pion mechanism (E T ot,π γ = (4. ± 27.) × 10 62 erg), as expected. Figures 4, 5, and 6 we explore the dependence of our results upon a broad range of possible values for Γ, ξ B , and ∆t, respectively. These uncertainties were not formally included in our statistical estimates. Nevertheless, these figures support our qualitative conclusions. Specifically, the energetic requirements of all three models are in excess of that for the GRB for a broad range of parameters. Indeed most parameter variations in Figures 4 -6 only exacerbate this problem. However, the inverse-Compton and proton-synchrotron FIG. 1 : 1Illustrative gamma-ray spectra for the three models discussed in the text. Spectra have been normalized to produce the muon excess observed by Project GRAND for GRB 971110. The curves begin at the detection threshold for Project GRAND and run to the respective cut-offs of each model. The thick lines are the raw source spectra. The thin lines illustrate the effects of internal pair-production optical depth on the source spectra. The change in energy dependence of the internal optical depth is apparent above about 50 GeV. FIG. 2 : 2Illustrative electron and proton spectra in the source frame for the three models discussed in the text. Spectra have been normalized to produce the muon excess observed by Project GRAND or to fit the observed BATSE spectrum for GRB 971110. FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 : 3456Illustration of the effects of internal and intergalactic pair-production optical depth on the source spectra shown inFigure 1. This calculation assumes that the source (GRB 971110) is at a redshift of z = 0.6. The change in energy dependence of the internal optical depth is apparent above about 50 GeV. For illustration, this plot also includes the observed BATSE spectrum for this burst, fit with both the Band (Eq.[1]) and electron-synchrotron models. Plot of the energy escaping in gamma rays (accounting for τ γγ,int ) and the source energy in protons or electrons for the inverse-Compton, proton-synchrotron, and photo-pion models as a function of the Lorentz boost of the GRB fireball. These curves correspond to spectra that were normalized to fit the observed muon excess for GRB 971110. The sharp increase in the energy content of the protons and electrons at small Γ enters primarily through the internal optical depth. Plot of the energy escaping in gamma rays (accounting for τ γγ,int ) and the source energy in protons for the proton-synchrotron model as a function of the magnetic equipartition of the GRB fireball. These curves correspond to spectra that were normalized to fit the observed muon excess for GRB 971110. The offset between the photon and proton spectra for ξ B > ∼ 0.01 is due solely to absorption due to the internal optical depth, τ γγ,int . Plot of the energy escaping in gamma rays (accounting for τ γγ,int ) and the source energy in protons or electrons for the inverse-Compton, proton-synchrotron, and photo-pion models as a function of the time variability of the GRB fireball. These curves correspond to spectra that were normalized to fit the observed muon excess for GRB 971110. FIG. 7 : 7Plot of the energy escaping in gamma rays (accounting for τ γγ,int ) and the source energy in protons or electrons for the inverse-Compton, proton-synchrotron, and photo-pion models as a function of the redshift of the GRB fireball. These curves correspond to spectra that were normalized to fit the observed muon excess for GRB 971110. TABLE I : IProject GRAND's Response to Selected BATSE Bursts Angles RA, Dec, and δθ in degrees and T90 in seconds. Upper limits are 2σ confidence level. c RA, Dec, and δθ for GRB 970417a are based upon the Milagrito data.GRB Trig T90 a RA a Dec a δθ a N µ b 971110 6472 195.2 242 50 0.6 467 ± 171 990123 7343 62.5 229 42 0.4 < 75 940526 2994 48.6 132 34 1.7 < 76 980420 6694 39.9 293 27 0.6 < 133 960428 5450 172.2 304 35 1.0 < 213 980105 6560 36.8 37 52 1.4 < 107 980301 6619 36.0 148 35 1.3 < 150 970417a 6188 7.9 290 c 54 c 0.5 c 20 ± 17 a b TABLE II : IIObserved and Inferred GRB Properties in BATSE Energy Range a Spectral fits were not available for GRB 970417a. Properties are based upon assumed redshift z ≈ 0.7, observed BATSE fluence, and average GRB spectral shape.b Average parameters for all bright GRBs considered inPreece et al. (2000).a β l β h ǫ ob γb ∆t GRB (cm −2 s −1 MeV −1 ) (MeV) (s) 971110 0.0751 ± 0.0095 1.02 ± 0.04 2.33 ± 0.11 0.404 ± 0.041 1.8 990123 1.93 ± 0.14 0.60 ± 0.01 3.11 ± 0.07 1.29 ± 0.07 4.1 940526 ≤ 0.217 1.01 ± 0.02 > 3.2 > 1.28 1.2 980420 0.0815 ± 0.0105 0.18 ± 0.08 2.57 ± 0.27 0.553 ± 0.033 1.8 960428 0.0610 ± 0.0086 0.58 ± 0.09 2.49 ± 0.24 0.433 ± 0.033 0.7 980105 0.0332 ± 0.0081 0.69 ± 0.06 2.82 ± 0.19 0.286 ± 0.031 1.2 980301 0.0068 ± 0.0020 0.54 ± 0.20 3.26 ± 1.41 0.317 ± 0.033 2.3 970417a 0.0100 ± 0.0061 a 1.0 ± 0.5 b 2.25 ± 0.75 b 0.250 ± 0.150 b 1.1 TABLE III : IIIVariability-Luminosity Redshift Estimates TABLE IV : IVInternal & Intergalactic Optical DepthsInternal Optical Depth Intergalactic Optical Depth GRB E γ = 100 GeV 1 TeV E γ = 100 GeV 1 TeV 971110 2.7 ± 3.5 2.9 ± 3.6 0.4 ± 0.2 9.6 ± 4.0 990123 TABLE V : VInferred Properties of Inverse-Compton Model . ± 13.) × 10 7 25. ± 7. 75. ± 48. (2.6 ± 8.6) × 10 55 (1. ± 15.) × 10 60A IC γ E ob cool E ob KN E KN,ob cool E T ot,IC γ a E T ot,IC e a GRB (cm −2 s −1 GeV −1 ) (GeV) (GeV) (GeV) (erg) (erg) 971110 4. ± 80. (1990123 < 6.2 × 10 −2 0.04 2.8 23. < 3.5 × 10 55 < 2.6 × 10 57 940526 < 1.2 × 10 −2 14. 7.3 5.1 < 2.5 × 10 54 < 2.5 × 10 54 980420 < 4.2 × 10 −2 0.3 9.8 56. < 2.4 × 10 55 < 1.6 × 10 56 960428 < 1.7 × 10 −2 0.004 8.2 380 < 2.0 × 10 56 < 2.4 × 10 59 980105 < 9.6 × 10 −2 0.04 17. 350 < 3.9 × 10 55 < 1.4 × 10 57 980301 < 2.1 × 10 −1 0.05 6.5 73. < 3.4 × 10 56 < 1.0 × 10 58 970417a < 1.3 × 10 −2 8.6 33. 63. < 2.8 × 10 54 < 2.8 × 10 54 TABLE VI : VIInferred Properties of Proton-Synchrotron ModelA p−sync p E ob γb E T ot,p−sync γ a E T ot,p−sync p a GRB (cm −2 s −1 GeV −1 ) (GeV) (erg) (erg) 971110 1. ± 22. (1. ± 22.) × 10 2 (3. ± 10.) × 10 55 (1. ± 21.) × 10 58 990123 < 1.9 × 10 −2 6.2 × 10 −5 < 9.6 × 10 55 < 9.6 × 10 55 940526 < 1.4 × 10 −2 1.3 × 10 −2 < 5.3 × 10 54 < 5.3 × 10 54 980420 < 4.1 × 10 −2 8.9 × 10 −4 < 5.4 × 10 55 < 5.4 × 10 55 960428 < 2.0 × 10 −2 5.8 × 10 −5 < 2.7 × 10 56 < 2.7 × 10 56 980105 < 3.8 × 10 −2 4.0 × 10 −4 < 5.6 × 10 55 < 5.6 × 10 55 980301 < 9.3 × 10 −2 2.5 × 10 −4 < 7.0 × 10 56 < 7.0 × 10 56 970417a < 6.7 × 10 −2 3.0 × 10 −2 < 5.9 × 10 54 < 5.9 × 10 54 TABLE VII : VIIInferred Properties of Photo-Pion ModelA π p E ob γb E T ot,π γ a E T ot,π p a GRB (cm −2 s −1 GeV −1 ) (10 5 GeV) (erg) (erg) 971110 (1. ± 13.) × 10 11 13. ± 4. (4. ± 27.) × 10 61 (2. ± 27.) × 10 69 990123 < 1.3 × 10 20 1.5 < 1.2 × 10 74 < 1.1 × 10 78 940526 < 1.8 × 10 13 4.0 < 2.0 × 10 65 < 1.3 × 10 70 980420 < 7.8 × 10 15 5.3 < 2.5 × 10 69 < 2.1 × 10 73 960428 < 3.9 × 10 18 4.4 < 5.3 × 10 73 < 1.1 × 10 77 980105 < 2.3 × 10 17 9.4 < 1.8 × 10 71 < 7.8 × 10 74 980301 < 3.1 × 10 27 3.5 < 3.4 × 10 81 < 4.8 × 10 85 970417a < 5.8 × 10 12 18. < 3.2 × 10 64 < 1.2 × 10 69 AcknowledgmentsThe authors would like to thank M. S. Briggs for providing fits to these BATSE data. This research has made use of data obtained from the High Energy Astrophysics Science Archive . E J Schneid, Astron. Astrophys. 25513Schneid, E. J., et al. 1992, Astron. Astrophys., 255, L13 . K Hurley, Nature. 372652Hurley, K. 1994, Nature, 372, 652 J R Catelli, B L Dingus, E J Schneid, Gamma Ray Bursts. C. A. MeeganNew YorkAIPCatelli, J. R., Dingus, B. L. & Schneid, E. J. 1997, in Gamma Ray Bursts, ed. C. A. Meegan (New York: AIP) . M Amenomori, Astron. 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[ "FEWFEDWEIGHT: Few-shot Federated Learning Framework across Multiple NLP Tasks", "FEWFEDWEIGHT: Few-shot Federated Learning Framework across Multiple NLP Tasks" ]
[ "Weilong Dong \nCollege of Intelligence and Computing\nTianjin University\nTianjinChina\n", "† ‡ ", "Xinwei Wu \nCollege of Intelligence and Computing\nTianjin University\nTianjinChina\n", "† ‡ ", "Junzhuo Li \nSchool of New Media and Communication\nTianjin University\nTianjinChina\n", "‡ ", "Shuangzhi Wu \nByteDance Lark AI\nBeijingChina\n", "Chao Bian [email protected] \nByteDance Lark AI\nBeijingChina\n", "Deyi Xiong [email protected] " ]
[ "College of Intelligence and Computing\nTianjin University\nTianjinChina", "College of Intelligence and Computing\nTianjin University\nTianjinChina", "School of New Media and Communication\nTianjin University\nTianjinChina", "ByteDance Lark AI\nBeijingChina", "ByteDance Lark AI\nBeijingChina" ]
[]
Massively multi-task learning with large language models has recently made substantial progress on few-shot generalization. However, this is usually performed in a centralized learning fashion, ignoring the privacy sensitivity issue of (annotated) data used in multiple tasks. To mitigate this issue, we propose FEWFEDWEIGHT, a few-shot federated learning framework across multiple tasks, to achieve the best of both worlds: privacy preservation and cross-task generalization. FEWFED-WEIGHT trains client models in isolated devices without sharing data. It broadcasts the global model in the server to each client and produces pseudo data for clients so that knowledge from the global model can be explored to enhance fewshot learning of each client model. An energy-based algorithm is further proposed to weight pseudo samples in order to reduce the negative impact of noise from the generated pseudo data. Adaptive model weights of client models are also tuned according to their performance. We use these model weights to dynamically aggregate client models to update the global model. Experiments on 118 NLP tasks show that FEWFEDWEIGHT can significantly improve the performance of client models on 61% tasks with an average performance improvement rate of 30.5% over the baseline and substantially outperform FedAvg[McMahan et al., 2017]and other decentralized learning methods.
10.48550/arxiv.2212.08354
[ "https://export.arxiv.org/pdf/2212.08354v1.pdf" ]
254,823,272
2212.08354
3585e08f2491859679b761eae8444afe7ec62f74
FEWFEDWEIGHT: Few-shot Federated Learning Framework across Multiple NLP Tasks 16 Dec 2022 Weilong Dong College of Intelligence and Computing Tianjin University TianjinChina † ‡ Xinwei Wu College of Intelligence and Computing Tianjin University TianjinChina † ‡ Junzhuo Li School of New Media and Communication Tianjin University TianjinChina ‡ Shuangzhi Wu ByteDance Lark AI BeijingChina Chao Bian [email protected] ByteDance Lark AI BeijingChina Deyi Xiong [email protected] FEWFEDWEIGHT: Few-shot Federated Learning Framework across Multiple NLP Tasks 16 Dec 2022 Massively multi-task learning with large language models has recently made substantial progress on few-shot generalization. However, this is usually performed in a centralized learning fashion, ignoring the privacy sensitivity issue of (annotated) data used in multiple tasks. To mitigate this issue, we propose FEWFEDWEIGHT, a few-shot federated learning framework across multiple tasks, to achieve the best of both worlds: privacy preservation and cross-task generalization. FEWFED-WEIGHT trains client models in isolated devices without sharing data. It broadcasts the global model in the server to each client and produces pseudo data for clients so that knowledge from the global model can be explored to enhance fewshot learning of each client model. An energy-based algorithm is further proposed to weight pseudo samples in order to reduce the negative impact of noise from the generated pseudo data. Adaptive model weights of client models are also tuned according to their performance. We use these model weights to dynamically aggregate client models to update the global model. Experiments on 118 NLP tasks show that FEWFEDWEIGHT can significantly improve the performance of client models on 61% tasks with an average performance improvement rate of 30.5% over the baseline and substantially outperform FedAvg[McMahan et al., 2017]and other decentralized learning methods. Introduction Large language models [Brown et al., 2020;Liu et al., 2021b] (LLMs) are capable of performing few-shot learning on downstream tasks, where a small number of annotated instances are provided to teach the model a new task [Liu et al., 2021b,c;. Significant interest has recently emerged on transforming a wide array of downstream tasks into a unified form, e.g., human-readable prompts or instructions Sanh et al. [2021]; , and continuing to train/finetune LLMs on the transformed data so as to endow LLMs with few/zero-shot task generalization [Sun et al., 2021;Xu et al., 2022]. The goal of such efforts might be towards general linguistic intelligence that can reuse previously acquired linguistic knowledge and adapt to a new task quickly [Yogatama et al., 2019]. Intuitively and empirically, cross-task generalization benefits from massive and diversified downstream tasks Sanh et al. [2021]; . However, in real-world scenarios, the data of different tasks are usually distributed across isolated clients (e.g., devices, users or institutions) and privacy-sensitive. Local private data are not legally allowed to leave their owners/clients by law, e.g., EU GDPR Voigt and Von dem Bussche [2017]. Owing to privacy risk or legality, collecting sufficient multi-task data for centralized training becomes difficult. To address this issue, Federated Learning (FL) [Konečnỳ et al., 2016;McMahan et al., 2017] is proposed, where each client participates in a training process without sharing private data with other clients. In this paper, our key interest is to combine federated learning with multi-task learning in the fewshot setting, where each client owns a small amount of private data (illustrated in Figure 1), in the goal of achieving the best of both worlds: cross-task generalization and data privacy. This scenario is pervasive, for example, different hospitals own a small number of annotated privacy-sensitive examples on different tasks (e.g., medical QA, named entity recognition or relation extraction over electronic health records (EHR), EHR summarization, medical image captioning). The challenges of this few-shot federated learning across multiple tasks are at least two-fold: (1) teaching models to few-shot learn and (2) sharing cross-task knowledge across clients without sharing private data. To deal with the aforementioned research questions, we propose a weighted few-shot federated learning framework (FEWFEDWEIGHT) for massive tasks to harness LLM-based few-shot learning with federated learning. Specifically, we construct a global model at the server, which aggregates information from client models located in each client and facilitates knowledge transfer across clients and tasks. In order to few-shot learn efficiently in each client, we use the global model to produce pseudo labeled data for each client model. To reduce the negative impact from noise in the generated pseudo data, we further propose an energy-based algorithm to weight the generated examples. In order to aggregate the trained client models into the global model, we introduce a dynamic aggregation method, which estimates the weights of client models according to the performance of the corresponding client model on the pseudo examples generated by the global model and the annotated examples. The main contributions of our work are summarized as follows: • We propose a federated learning framework for multi-task learning in the few-shot and private setting, which is equipped with two components: an energy-based weighting algorithm for updating the weights of pseudo examples generated by the global model and a dynamic aggregation method based on the performance of client models. and (2) the client models trained by FEWFEDWEIGHT gain 30.5% over the original model and they can generalize well cross multiple tasks. Preliminary: Federated Learning In federated learning, each client has its own private data which are not shared during training. Let U = {u 1 , u 2 , ..., u K } be a set of clients, where K is the number of clients. Apart from the clients, there is a central server. The server initializes a global model, sends the global model to a subset of clients for updating and aggregates the updated client models into the global model. We denote the server as S. In the classic FL framework FedAvg [McMahan et al., 2017], S aggregates the client models as follows: F global = N i=1 δ i · F i ,(1) where F global is the aggregated global model, F i is the local model on client u i and δ i is the model weight for F i . In FedAvg, the model weight δ is estimated according to the amount of data on each client, which is computed as δ j = nj K j=1 nj , where n j is the number of training samples on u j . In addition to the global federated learning methods, prior works [Smith et al., 2017;Marfoq et al., 2021] have explored personalized federated learning (PFL) for multi-task learning. In PFL, all clients jointly train a global model, but each client can built its own personalized client model by interpolating the global model and its client model. During training, a personalized client model is able to exploit information of other clients from the global model without sharing their data. A common method in PFL is to add a regularization term in the loss function. Specifically, the loss of each client model is computed as L i =L i + ||F global − F i || 2 , whereL i is the original loss on u i . We choose this PFL setting in our work for tackling different tasks. Methodology In this section, we introduce our proposed FEWFEDWEIGHT for few-shot federated learning across multiple tasks. The architecture of our module is illustrated in Figure 2. We first overview FEWFED-WEIGHT ( §3.1) and then elaborate the two essential components in our framework: data augmentation with energy-based sample weighting ( §3.2) and dynamic model aggregation ( §3.3). Overview Algorithm 1 formalizes the entire flow of FEWFEDWEIGHT. We denote the training data as {(X, Y )}, X = {X 1 , ..., X K }, Y = {Y 1 , ..., Y K }, where (X i , Y i ) is the training data on client u i .F (t+1) i , L (i) ← Client(F (t) global , X i , Y i ) 5: Compute F (t+1) global based on Eq. (1) & (5) 6: Client: 7: Generate pseudo labelŝ Y i = Predict(F (t) i , X i ) 8: F (t+1) i , L (i) ← Client Model Updating(F (t) i , X i , Y i ,Ŷ i ) 9: return F (t+1) i , L (i) 10: end for At the beginning, the server S uses a pre-trained language model as the initial global model F (0) i on u i is initialized as F (0) global . In the t-th training epoch, S sends F (t) global to each client. Instead of replacing F (t) i with F (t) global , F (t) global is used to predict pseudo labelsŶ i on X i , which will be described in subsection 3.2 (Line 7 in Algorithm 1). Then an energy-based weighting algorithm is performed to evaluateŶ i 's quality and estimate weights w pseudo . With the estimated weights, the client model F (t) i is updated on the combination of (X i , Y i ) and (X i ,Ŷ i ) (Line 8 in Algorithm 1). The updated model F global , which will be elaborated in subsection 3.3 (Line 5 in Algorithm 1). Data Augmentation Pseudo Data Generation with Global Knowledge Data augmentation can ease the data scarcity issue in few-shot learning [Feng et al., 2021]. In our case, each client can synthesize pseudo data using its own client model. However, this cannot allow each client model to leverage information of other clients and the client model trained on its own data is not able to generalize across other tasks. Hence, we use the global model to synthesize data for each client. Raffel et al. [2020a] and , where different tasks are unified into the same text-to-text format to enable knowledge transfer across tasks, we define a prompt template for task conversion and unification: <task name>: <x i > [SEP] <y i >. Take question answering as an example, x i is reformulated as: <question> [SEP] <choices>, y i is the corresponding answer. Suppose the unlabeled data on u i is X i = {x (i) 1 , x (i) 2 , ..., x (i) ni } and corresponding label is Y i = {y (i) 1 , y (i) 2 , ..., y (i) ni }. Following At the t-th FL training epoch, u i receives the global model F to predict the labels of X i . We denote the predicted pseudo labels asŶ i = {ŷ (i) 1 ,ŷ (i) 2 , ...,ŷ (i) ni }. When predictingŶ i , F (t) global actually performs the task with X i being fed as input. The generated output with the highest probability will be taken asŶ i . The predicted pseudo data are then combined with the original local data to train the client model. On the training instances from the pseudo data {X i ,Ŷ i }, the client model is updated with an energy- Energy-based Pseudo Instance Weighting Although the pseudo instances synthesized by the global model contain global knowledge, they are noisy. We hence propose an energy-based tokenwise weighting method to reduce the negative impact of noise from these pseudo instances. introduce the following energy function to perform out-of-distribution detection in image classification: E(x; f ) = −T · log K i e fi(x)/T (2) where T is the temperature parameter, K is the number of image classes, x is the input image and f i (x) is the output logit in dimension i (i.e., the logit value for class i). The energy E(x; f ) is higher for unobserved samples and lower for observed ones. The essence of the energy function indicates that noisy instances have higher energy than clean instances. However, adapting Eq. (2) to our model is nontrivial as we are confronted with two challenges. First, our task is a generation task rather than classification, which means that the number of sequences (classes) grows exponentially with sequence length. In order to deal with this, we estimate token-wise energy instead of sequence-wise energy as token-wise generation can be treated as a multi-class classification problem (prediting a word from the vocabulary). Second, due to the large size of the vocabulary used in our model, energy for different tokens calculated by Eq. (2) would be almost indistinguishable from each other, as shown in Figure 3. Although the dimension of logits is large, only logits with high values have a substantial impact on the final prediction. We hence choose the top-k logits to calculate the energy as follows: E(a; f ) = −T · log i e fi(a)/T , i ∈ K(3) where a is the predicted token, f (a) denotes the logits corresponding to a, K is the set of indices of logits with the top-k values. A noisy token will have larger energy and should have a lower weight. The instance weight is hence computed by normalizing token-wise energy at the sequence level: w a pseudo = −E(a; f ) l i −E(i; f )(4) where l is the length of a predicted pseudo label. The training loss for this pseudo label is hence defined as: Lŷ = l i w i pseudo · L î y , where L î y is the NLL loss of token i. Dynamic Aggregation Aggregation weights in FL [McMahan et al., 2017; and PFL [Smith et al., 2017; are usually estimated according to the number of training instances in clients, which is not adaptive to the training performance of client models. Since different clients have different tasks and the degree of difficulty to learn is varying across tasks, it would be beneficial if aggregation weights could match the task difficulty of clients (i.e., paying more attention to harder tasks). To achieve this purpose, we estimate the aggregation weight δ i in Eq. (1) as follows: δ i = L (i) K i L (i) ,(5) where L (i) = L (i) pseudo + L (i) annotated , which is the total training loss on client u i , the sum of the loss on the weighted pseudo samples and that on the annotated samples. The weight δ i for client model F i is proportional to F i 's training loss. A large weight suggests that tasks in client u i are challenging. Similar to the essence of AdaBoost [Freund et al., 1996], the client model with a larger weight (i.e., with poorer performance) plays a more important role in model aggregation and thus the challenging tasks are paid more attention to. Experiments We examined the effectiveness of the proposed FEWFEDWEIGHT on a wide range of tasks. Setup Datasets We followed to use 118 tasks from the Huggingface Datasets [Lhoest et al., 2021], which are categorized into 4 groups: classification, QA, conditional generation and other. All tasks were reframed into the same text-to-text format. The number of clients was set to 4 by default. The numbers of tasks in the clients were set as evenly as possible: 29, 29, 29, 31, respectively. Each client had randomly selected and diversified tasks so that we could evaluate the cross-task generalization performance in the client models. Details of the selected tasks and task distribution over clients are shown in Appendix A.1. Few-shot Setting We used the few-shot sampling method of for our few-shot learning. For generation tasks (e.g., summarization, dialogue), 32 training examples were selected for each task. For classification tasks (e.g., natural language inference, sentiment analysis), 16 examples were selected for each task. Since labeled data are rare in real-world scenario, our test sets also followed the same few-shot setting as the training sets. To eliminate random factors, we chose 5 different random seeds to select few-shot samples. These selected samples collectively construct the training set and test set. Training Details We chose BART-Base [Lewis et al., 2020] (140M parameters) as our client and global model. The training hyperparameters for FEWFEDWEIGHT were set as follows: 20 as the number of training epochs, 3e-5 for learning rate with a linear warm up, 8 for the batch size. The top-7 largest logit values were selected to compute the energy and the temperature parameter for the energy function was set to 1. Baselines We compared the proposed FEWFEDWEIGHT against the following baselines: : A general framework for personalized federate learning, which achieves personalization by regularizing client models towards their average. Ditto mainly focuses on fairness and robustness in FL instead of the limited data and task heterogeneity. • Meta Weighting: We follow Ren et al. [2018] to use a meta-learning based method to weight pseudo samples in our data augmentation component (see Section 3.2) for each client. This meta weighting approach serves as a strong baseline to compare with our energy-based weighting. More details of the meta-learning based weighting are provided in Appendix A.2. Evaluation Metrics Evaluation metrics for different tasks vary widely. Averaging metrics of different tasks hence doesn't make sense. In order to evaluate the performance of FEWFEDWEIGHT in comparison to baselines and tackle the problem of incompatable metrics, we define a unified metric: average performance improvement rate (APIR). The performance improvement rate (PIR) refers to how much the performance has been improved over the base model. The PIR is computed as follows: PIR = m new − m base m base ,(6) where m new refers to the performance of the new model while m base is the performance of the base model in terms of the widely-used metric for the corresponding task. In our experiments, the base model refers to Data Silo by default. The APIR is the average of PIRs across all tasks. Each client has its own APIR and the APIR averaged over the four clients is reported as the APIR for the corresponding method. Overall Performance The overall results are shown in Table 1. We observe that FedAvg is not able to well handle fewshot multi-task learning. Although FedAvg leverages the data from all clients through federated learning, it performs just slightly better than Data Silo. The AIPR of Data Silo DA demonstrates that data augmentation is beneficial to few-shot learning. Although each client can only use its own private data in Data Silo DA, both its APIR and the number of Win tasks are better than those of FedAvg. The recently proposed PFL framework, Ditto, achieves better APIR than FedAvg and Data Silo DA. However, it is worse than Meta Weighting. As a strong baseline, Meta Weighting achieves promising performance. Due to its global data augmentation and meta-learning based pseudo sample weighting, the dilemma between data scarcity in few-shot learning and noise in data augmentation can be alleviated to some extent. However, its time complexity is very high due to the meta-learning FEWFEDWEIGHT achieves 30.5% in APIR over Data Silo. Its APIR is close to that of Centralized Training and the number of Win tasks is even larger than that of Centralized Training. In order to have a clear comparison with classic FedAvg, we illustrate their PIRs on non-tie tasks in Figure 4. Ablation Study FEWFEDWEIGHT contains three essential components: data augmentation with global knowledge, energy-based pseudo data weighting and dynamic aggregation, which collectively contribute to the advantage of FEWFEDWEIGHT. We further conducted ablation experiments to investigate the effectiveness of the three components. Results are shown in Table 2. FedAvg+Data Augmentation is similar to Data Silo DA in that both use augmented data, but the latter synthesizes pseudo data with its client model. The APIR of FedAvg + DA is 3.3% higher than that of Data Silo DA. We conjecture that this improvement could attribute to cross-task knowledge transfer brought by pseudo data synthesized by the global model. The energy-based pseudo data weighting significantly outperforms data augmentation by 15.5% in APIR while the dynamic aggregation achieves an improvement of 9.5% over data augmentation, suggesting that the two components are effective and beneficial to FEWFEDWEIGHT. The three components together achieve further improvements over both the data augmentation + pseudo data weighting and data augmentation + dynamic aggregation. Varying the Number of Clients To verify the stability of FEWFEDWEIGHT over different numbers of clients, we conducted experiments in 4 settings with different numbers of clients. In order to have a fair comparison, we fixed the number of tasks on each client as the number of clients participating in FL training is varying. Five #Clients APIR Win Lose Tie 2 23.2% 8 0 2 4 29.5% 12 5 3 8 29.8% 24 9 7 12 22.8% 37 9 14 Table 3: Results of different number of clients. The total number of tasks varies with the number of clients. Methods Generating Training Total FedAvg/Ditto 0 2 2 FEWFEDWEIGHT/DataSilo DA 13 4 17 Meta Weighting 13 8 21 tasks were assigned to each client. We evaluated FEWFEDWEIGHT on 2, 4, 8, 12 clients respectively. Results are reported in Table 3. We observe that FEWFEDWEIGHT has a remarkable stability across different numbers of clients. The APIR of 2 clients and 12 clients is slightly lower than that of the other two settings although they win in almost all tasks. We conjecture that increasing the number of clients can improve the number of win tasks as more information is incorporated. But it's hard to improve the APIR over all tasks as these tasks are distributed over more clients. Training Cost We further compared FEWFEDWEIGHT with other baselines in terms of the training time and communication time between clients and the server. The additional training time of FEWFEDWEIGHT mainly comes from the data augmentation component because it needs to predict pseudo labels for all samples in each epoch. The time for generating pseudo data is the same for all methods with data augmentation (e.g., Data Silo DA, Meta Weighting). The training/generating time of one epoch over different methods is shown in Table 4. FEWFEDWEIGHT doesn't require extra time except for generating and training on pseudo data (The training time is twice as long as FedAvg because the size of total training data is doubled) while Meta Weighting requires more training time on pseudo data (details of meta-weighting is shown in Appendix A.2). The communication cost of FEWFEDWEIGHT is almost the same as FedAvg or Ditto, as the extra communication is brought by sending the training loss L (i) to the server S. Such transmission time is almost negligible compared to the training time of client model F i . Analysis on the Impact of Dynamic Aggregation In order to look into the difference between FedAvg and FEWFEDWEIGHT, we adopted the model weights of FedAvg in the first 10 training epochs while those of FEWFEDWEIGHT are used in the last 10 training epochs. The curves of weights of the 4 client models during training are shown in Figure 5. In the first 10 epochs, FedAvg assigns the smallest model weight to client 2 because the amount of training data in client 2 is the smallest. When we use FEWFEDWEIGHT in the second 10 epochs, the model weight of client 2 becomes the largest due to its worst performance. Its weight then becomes lower, demonstrating that its performance is getting better gradually in the second 10 epochs. Additionally, the weights of the four client models tend to be stably changing as the number of training epochs increases. Related Work Few-shot Multi-task Learning in NLP Large language models [Brown et al., 2020;Zhang et al., 2021b;Perez et al., 2021] achieve substantial progress on few-shot learning, thanks to the implicit multi-task learning in the pretraining of language models Radford et al. [2019]; Sanh et al. [2021]. A variety of efforts [Narayan et al., 2018;Raffel et al., 2020b; have been devoted to mapping a wide range of NLP downstream tasks into a unified form. For example, Narayan et al. [2018] redefines the NLU tasks as generating specified answers through automatic prompts. Natural language templates (prompts) [Liu et al., 2021c; have also been explored to reframe different NLP tasks into a unified text-to-text generation task. Such efforts facilitate the standardization of few-shot multi-task learning process [Bragg et al., 2021;. Significantly different from these studies, we explore few-shot multi-task learning in a decentralized environment. Data Augmentation Data augmentation is to automatically increase the amount and diversity of training data without explicitly collecting new data [Feng et al., 2021]. It has been widely used in computer vision and natural language processing to address issues such as low-resource language processing , bias mitigation [Zhao et al., 2018], imbalanced data [Chawla et al., 2002], few-shot learning [Lee et al., 2021]. In our work, we automatically generate pseudo data to alleviate the problem of insufficient data in local client training. Federated Learning Federated Learning (a.k.a collaborative learning) [Konečnỳ et al., 2016;Chen et al., 2019] is a decentralized machine learning technique, where heterogeneous training data are stored locally on each client and are not exchangeable across clients during the training process. Textual data are crucial for many NLP tasks, which may be privacy-sensitive, especially for those collected from edge devices (e.g., mobile phones). In this aspect, recent years have witnessed growing interest in the intersection of NLP and federated learning [Garcia Bernal, 2020;Liu et al., 2021a]. FL with Few-shot or Multi-task Learning The inherent decentralization, heterogeneity and privacy-awareness in federated learning make it naturally suitable to unite FL with multi-task learning and few-shot learning Zhang et al. [2021a]. For few-shot federated learning, methods such as adversarial learning [Fan and Huang, 2021] or self-supervision [Shome and Kar, 2021] have been exploited in the federated learning setting to learn from scarce data or use unlabeled data to alleviate the lack of supervised data. And federated learning can be naturally embedded into the multi-task learning framework due to its distributed learning [Smith et al., 2017;Marfoq et al., 2021;. In this aspect, Smith et al. [2017] pioneer and frame federated multi-task learning with MOCHA, where relationships among tasks (clients) are separately modeled during the optimization of weights of tasks. Our work is different from these methods in that we focus on the scenario of both few-shot and multi-task learning. Conclusion In this paper, we have presented FEWFEDWEIGHT, enabling few-shot learning across massive NLP tasks with federated learning. In this new framework, the global model synthesizes pseudo samples for each client model, which are weighted by an energy-based algorithm. Aggregation weights of client models are estimated according to their performance during training. Experiments on 118 different tasks demonstrate the effectiveness of the proposed FEWFEDWEIGHT. Algorithm 2 Meta Weighting Input: X i , Y i ,Ŷ i , F i , T i , l 1: for t = 0, 1, 2, ..., T i − 1 do 2: Sample B ∈ (X i , Y i ) andB ∈ (X i ,Ŷ i ) 3: First Step: 4:ỹ i ← Forward(F i , x j ), x j ∈B 5: lB ← bi 1 l(ỹ j ,ŷ j ) 6: The 118 tasks were randomly dispatched to 4 clients, which is shown in Table 5. The 118 tasks can be categorized into 4 groups according to : classification, QA, conditional generation and other. The categories of tasks are shown in Table 6. F ′ i ← Optimize(F i , A.2 Meta Weighting The meta weighting algorithm is shown in Algorithm 2. At the beginning, we randomly sample a mini-batchB ∈ (X i ,Ŷ i ) and B ∈ (X i , Y i ). In the first step, backpropagation is performed onB and the client model F i is updated. We denote the original client model as F i and the updated client model as F ′ i . F ′ i will be evaluated on B to get the sample weight. In the second step, the loss function is changed. Assuming the original loss function is l B = bi 1 l(ȳ j , y j ), where b i is the batch size of B andB,ȳ j is the prediction of F ′ i and l is per-sample loss function. In this paper, we use CrossEntropy to compute l. The new loss function is computed as follows:l B = bi j=1 ǫ j · l(ȳ j , y j ), where ǫ j is the perturbation to the per-sample loss. ǫ j is initialized to 1 for all samples. When F ′ i performs backpropagation on B withl B , we get the grad to ǫ, ∇ǫ = ∂lB ∂ǫ . The sample weight is computed as:ŵ = max(−∇ǫ, 0). (8) To stabilize the training process, we normalize weights of samples from the same mini-batch as: w j =ŵ j b i 1ŵ j . In the third step, F i is updated on B andB with sample weights w. The loss is computed as: l ′B = bi j=1 w j · l(ỹ j ,ŷ j ) + l(ỹ j , y j ), whereỹ j is the prediction of F i . Figure 1 : 1Isolated multi-task data in local devices. Figure 2 : 2The diagram of FEWFEDWEIGHT. the number of training epochs T . S broadcasts F (0) global to each client and the initial client model F loss are sent to S for model aggregation. The server S computes the new global model F (t+1) Figure 3 : 3Energy distributions over top-k logits and all logits. The energy distribution over all logits (orange curve) is "crowded" and the energy values for different tokens are almost indistinguishable from each other. based weighting algorithm to minimize a loss L (i) pseudo , while on the instances from the original local data {X i , Y i }, the client model is trained as usual to optimize a loss L Figure 4 : 4PIR results of FedAvg and FEWFEDWEIGHT on different tasks. Figure 5 : 5Client model weights learned by FedAvg vs. FEWFEDWEIGHT. ỹ i ← Forward(F i , x j ), x j ∈B 14: l ′B ← bi 1 w j · l(ỹ j ,ŷ j ) + l(ỹ j , y j ) • We conduct experiments on 118 open-source tasks to evaluate the proposed FEWFED-WEIGHT framework. To the best of our knowledge, this is the first attempt to perform federated learning on massive NLP tasks. • Experiments demonstrate that (1) FEWFEDWEIGHT significantly outperforms the classic federated learning FedAVG, the personalized federated learning Ditto, centralized training, data silo data augmentation and meta weighting based data augmentationGlobal Model Client Model Global Model Model Aggregation Client Model Client Model … Client 1 … Global Model Client Model Client K Client Models and Loss Aggregation Weight = ( ) ( ) ( ) ( ) … ( ) Local Data Pseudo Data Predict Global Model Client Model Forward Client Model Forward Logits Select top-k logits Energy-based Weighting Training Loss Logits Backward Optimize Broadcasting Global Model Reweight Pseudo tokens : APIR results of different methods. Win/Lose/Tie is the number of tasks on which the corresponding method is better/worse than or as good as the base model (i.e., data silo without data augmentation).• Data Silo: Each client trains a model with its own private data. This method serves as the base model for PIR evaluation.• Data Silo DA: Data augmentation is used for the Data Silo training and the numbers of training samples after data augmentation in clients are the same as those in FEWFED-WEIGHT. • Centralized Training: We train a central model with all data put together. Note that the Centralized Training does not take data privacy into account, which is to show the oracle result at the cost of data privacy. • FedAvg [McMahan et al., 2017]: A classical FL algorithm, in which clients participate in training a global model but there are no personalized client models. • DittoMethod APIR Win Lose Tie Data Silo DA 8.2% 44 29 45 FedAvg 5.1% 40 41 37 Ditto 14.1% 45 34 39 Meta Weighting 22.6% 49 32 37 FEWFEDWEIGHT 30.5% 72 7 39 Centralized Training 36.9% 65 16 37 Table 1 : Ablation study results of FEWFEDWEIGHT. procedure shown in Appendix A.2. Additionally, it is worse than FEWFEDWEIGHT in terms of APIR.Method APIR Win Lose Tie FedAvg + Data Augmentation 11.5% 43 33 42 + Pseudo Data Weighting 27.0% 74 6 38 + Dynamic Aggregation 21.0% 64 16 38 FEWFEDWEIGHT 30.5% 72 7 39 Table 2 Table 4 : 4The time cost of different methods on one epoch of the local training of a client (i.e., time units per epoch). Flex: Unifying evaluation for few-shot nlp. 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Client Tasks Client 1 blimp-determiner-noun-agreement-with-adj-irregular-1, aeslc, glue-mrpc, math qa, quarel, kilt ay2, tweet eval-stance atheism, lama-squad, tab fact, aqua rat, tweet eval-emoji, glue-wnli, codah, spider tweet eval-offensive, wiki qa, blimp-ellipsis n bar 1, openbookqa, sms spam, acronym identification, ethos-national origin, definite pronoun resolution, hellaswag, superglue-wsc, numer sense, ade corpus v2-dosage, blimp-ellipsis n bar 2, squad-no context, e2e nlg cleaned Client 2 google wellformed query, xsum, wiqa, qa srl, tweet eval-stance abortion, ade corpus v2-effect, sick, ethos-religion, commonsense qa, jeopardy, biomrc, superglue-multirc, ethos-race, eli5-askh, glue-qqp, paws, ethos-directed vs generalized, glue-sst2, mocha, tweet eval-hate, glue-rte, hate speech offensive, blimp-anaphor number agreement, lama-conceptnet, superglue-wic, boolq, kilt hotpotqa, aslg pc12, quartz-no knowledge Client 3 tweet eval-stance climate, tweet eval-sentiment, qasc, medical questions pairs, break-QDMR-high-level, imdb, glue-mnli, ethos-gender, trec-finegrained, crows pairs, adversarialqa, onestop english, duorc, web questions, yelp review full, swag, proto qa, scitail, tweet eval-stance feminist, limit, common gen, scicite, blimp-irregular past participle adjectives, social i qa, anli, kilt zsre, cosmos qa, superglue-record, squad-with context Client 4 emotion, blimp-existential there quantifiers 1, sciq, race-middle, kilt wow, wino grande, rotten tomatoes, superglue-cb, poem sentiment, ropes, piqa, quail, climate fever, lama-google re, search qa, wiki auto, mc taco, blimp-wh questions object gap, hotpot qa, emo, kilt nq, kilt trex, quartz-with knowledge, aeslc, dbpedia 14, yahoo answers topics, app reviews, superglue-copa, blimp-anaphor gender agreement, hate speech18, gigaword Table 5: Tasks assigned to different clients. Type Tasks Classification anli, medical questions pairs, paws, glue-rte, onestop english, poem sentiment, sick, glue-sst2, scicite, rotten tomatoes, climate fever, glue-qqp, scitail, sms spam, dbpedia 14, emotion, glue-wnli, superglue-cb, emo, ethos-gender, imdb, glue-mrpc, ethos-directed vs generalized, ethos-race, tab fact, ethos-national origin, superglue-wic, superglue-wsc, glue-mnli, google wellformed query, wiki qa, hate speech18, hate speech offensive, ethos-religion, trec-finegrained, tweet eval-emoji, wiki auto, tweet eval-hate, tweet eval-offensive, tweet eval-sentiment, tweet eval-stance abortion, tweet eval-stance atheism, tweet eval-stance climate, tweet eval-stance feminist, yahoo answers topics QA lama-conceptne, adversarialqa, lama-google re, lama-squad, math qa, mc taco, aqua rat, wino grande, numer sense, biomrc, openbookqa, qasc, quail, quarel, quartz-no knowledge, search qa, ropes, quartz-with knowledge, race-middle, boolq, sciq, codah, commonsense qa, jeopardy, social i qa, squad-no context, duorc, squad-with context, superglue-copa, eli5-askh, swag, superglue-multirc, superglue-record, web questions, hellaswag, wiqa, kilt zsre, kilt nq, kilt ay2, kilt trex, cosmos qa Conditional Generation aeslc, spider, gigaword, xsum, kilt wow Other acronym identification, ade corpus v2-dosage, ade corpus v2-effect, limit, app reviews, crows pairs, mocha, aslg pc12, blimp-anaphor gender agreement, piqa, yelp review full, proto qa, common gen. Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, Kai-Wei Chang, Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies2Gender bias in coreference resolution: Evaluation and debiasing methods. Short Papers. blimp-determiner noun agreement with adj irregular 1, qa srl, blimp-ellipsis n bar 1, blimp-ellipsis n bar 2, blimp-existential there quantifiers 1, hotpot qa, e2e nlg cleaned, blimp-anaphor number agreement, blimp-irregular past participle adjectives, definite pronoun resolution, break-QDMR-high-level, blimp-wh questions object gap Table 6: The categories of 118 tasksJieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. Gender bias in coreference resolution: Evaluation and debiasing methods. In Proceedings of the 2018 Confer- ence of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pages 15-20, 2018. Client Tasks Client 1 blimp-determiner-noun-agreement-with-adj-irregular-1, aeslc, glue-mrpc, math qa, quarel, kilt ay2, tweet eval-stance atheism, lama-squad, tab fact, aqua rat, tweet eval-emoji, glue-wnli, codah, spider tweet eval-offensive, wiki qa, blimp-ellipsis n bar 1, openbookqa, sms spam, acronym identification, ethos-national origin, definite pronoun resolution, hellaswag, superglue-wsc, numer sense, ade corpus v2-dosage, blimp-ellipsis n bar 2, squad-no context, e2e nlg cleaned Client 2 google wellformed query, xsum, wiqa, qa srl, tweet eval-stance abortion, ade corpus v2-effect, sick, ethos-religion, commonsense qa, jeopardy, biomrc, superglue-multirc, ethos-race, eli5-askh, glue-qqp, paws, ethos-directed vs generalized, glue-sst2, mocha, tweet eval-hate, glue-rte, hate speech offensive, blimp-anaphor number agreement, lama-conceptnet, superglue-wic, boolq, kilt hotpotqa, aslg pc12, quartz-no knowledge Client 3 tweet eval-stance climate, tweet eval-sentiment, qasc, medical questions pairs, break-QDMR-high-level, imdb, glue-mnli, ethos-gender, trec-finegrained, crows pairs, adversarialqa, onestop english, duorc, web questions, yelp review full, swag, proto qa, scitail, tweet eval-stance feminist, limit, common gen, scicite, blimp-irregular past participle adjectives, social i qa, anli, kilt zsre, cosmos qa, superglue-record, squad-with context Client 4 emotion, blimp-existential there quantifiers 1, sciq, race-middle, kilt wow, wino grande, rotten tomatoes, superglue-cb, poem sentiment, ropes, piqa, quail, climate fever, lama-google re, search qa, wiki auto, mc taco, blimp-wh questions object gap, hotpot qa, emo, kilt nq, kilt trex, quartz-with knowledge, aeslc, dbpedia 14, yahoo answers topics, app reviews, superglue-copa, blimp-anaphor gender agreement, hate speech18, gigaword Table 5: Tasks assigned to different clients. Type Tasks Classification anli, medical questions pairs, paws, glue-rte, onestop english, poem sentiment, sick, glue-sst2, scicite, rotten tomatoes, climate fever, glue-qqp, scitail, sms spam, dbpedia 14, emotion, glue-wnli, superglue-cb, emo, ethos-gender, imdb, glue-mrpc, ethos-directed vs generalized, ethos-race, tab fact, ethos-national origin, superglue-wic, superglue-wsc, glue-mnli, google wellformed query, wiki qa, hate speech18, hate speech offensive, ethos-religion, trec-finegrained, tweet eval-emoji, wiki auto, tweet eval-hate, tweet eval-offensive, tweet eval-sentiment, tweet eval-stance abortion, tweet eval-stance atheism, tweet eval-stance climate, tweet eval-stance feminist, yahoo answers topics QA lama-conceptne, adversarialqa, lama-google re, lama-squad, math qa, mc taco, aqua rat, wino grande, numer sense, biomrc, openbookqa, qasc, quail, quarel, quartz-no knowledge, search qa, ropes, quartz-with knowledge, race-middle, boolq, sciq, codah, commonsense qa, jeopardy, social i qa, squad-no context, duorc, squad-with context, superglue-copa, eli5-askh, swag, superglue-multirc, superglue-record, web questions, hellaswag, wiqa, kilt zsre, kilt nq, kilt ay2, kilt trex, cosmos qa Conditional Generation aeslc, spider, gigaword, xsum, kilt wow Other acronym identification, ade corpus v2-dosage, ade corpus v2-effect, limit, app reviews, crows pairs, mocha, aslg pc12, blimp-anaphor gender agreement, piqa, yelp review full, proto qa, common gen, blimp-determiner noun agreement with adj irregular 1, qa srl, blimp-ellipsis n bar 1, blimp-ellipsis n bar 2, blimp-existential there quantifiers 1, hotpot qa, e2e nlg cleaned, blimp-anaphor number agreement, blimp-irregular past participle adjectives, definite pronoun resolution, break-QDMR-high-level, blimp-wh questions object gap Table 6: The categories of 118 tasks.
[]
[ "Fermionic atoms in a Three Dimensional optical lattice: Observing Fermi Surfaces, Dynamics and Interactions", "Fermionic atoms in a Three Dimensional optical lattice: Observing Fermi Surfaces, Dynamics and Interactions" ]
[ "Michael Köhl \nInstitute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland\n", "Henning Moritz \nInstitute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland\n", "Thilo Stöferle \nInstitute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland\n", "Kenneth Günter \nInstitute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland\n", "Tilman Esslinger \nInstitute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland\n" ]
[ "Institute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland", "Institute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland", "Institute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland", "Institute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland", "Institute of Quantum Electronics\nETH Zürich\nCH-8093Hönggerberg, ZürichSwitzerland" ]
[]
We have studied interacting and non-interacting quantum degenerate Fermi gases in a threedimensional optical lattice. We directly image the Fermi surface of the atoms in the lattice by turning off the optical lattice adiabatically. Due to the confining potential gradual filling of the lattice transforms the system from a normal state into a band insulator. The dynamics of the transition from a band insulator to a normal state is studied and the time scale is measured to be an order of magnitude larger than the tunneling time in the lattice. Using a Feshbach resonance we increase the interaction between atoms in two different spin states and dynamically induce a coupling between the lowest energy bands. We observe a shift of this coupling with respect to the Feshbach resonance in free space which is anticipated for strongly confined atoms.The exploration of quantum degenerate gases of fermionic atoms is driven by the ambition to get deeper insight into long-standing problems of quantum manybody physics, such as high temperature superconductivity. Very recently, the cross-over regime between a strongly interacting two-component Fermi gas and a molecular Bose-Einstein condensate has been studied in harmonic traps[1,2,3,4,5]. These experiments mark a milestone towards the understanding of superfluidity of fermionic atoms. However, the analogy to an electron gas in a solid is limited since there the electrons experience a periodic lattice potential. The lattice structure is in fact a key ingredient for most models describing quantum many-body phenomena in materials. It has been suggested that strongly interacting fermionic atoms in optical lattices could be employed for studies of high-Tcsuperconductivity [6], Mott-insulating phases [7], Bose condensation of fermionic particle-hole pairs [8], or interacting spin systems [9].Here we report on an experiment bridging the gap between current ultracold atom systems and fundamental concepts in condensed matter physics. A quantum degenerate Fermi gas of atoms is prepared in the crystal structure of a three dimensional optical lattice potential created by three crossed standing laser waves. The unique control over all relevant parameters in this system allows us to carry out experiments which are not feasible with solid-state systems.It was conceived by D. Jaksch et al. that ultracold atoms exposed to the periodic potential of an optical lattice are an almost ideal realization of a Hubbard model[10]. This model is elementary to describe the quantum physics of many electrons in a solid. It takes into account a single band of a static lattice potential and assumes the interactions to be purely local[11]. Ultracold atoms in an optical lattice give a very direct access to the underlying physics. The fundamental parameters include the tunnel coupling between adjacent lattice sites, the atom-atom interactions and the dimensionality of the system. Previous experiments with far-detuned three dimensional optical lattices[12,13,14]were always carried out with bosonic atoms, and experiments with fermions were restricted to a single standing wave[15]. In the latter situation many atoms can reside in each standing wave minimum but formation of a band insulator is prevented by the weak transverse confinement. The observed inhibition of transport[16] is due to localized states and therefore differs qualitatively from the band insulator which we create in the three dimensional optical lattice.The experiments are performed in a modified apparatus previously used to study bosonic rubidium atoms in optical lattices[14,17]. A mixture of bosonic 87 Rb and fermionic 40 K atoms is captured in a magneto-optical trap. For magnetic trapping we optically pump the potassium atoms into the |F = 9/2, m F = 9/2 and the rubidium atoms into the |F = 2, m F = 2 hyperfine ground state, with F being the total angular momentum and m F the magnetic quantum number. The mixture is evaporatively cooled using microwave radiation to selectively remove the most energetic rubidium atoms from the trap. The potassium cloud is sympathetically cooled by thermal contact with the rubidium atoms[18]. After reaching quantum degeneracy for both species with typically 6 × 10 5 potassium atoms at a temperature of T /T F = 0.32 (T F = 260 nK is the Fermitemperature of the non-interacting gas) we remove all the rubidium atoms from the trap. The potassium atoms are then transferred from the magnetic trap into a crossed beam optical dipole trap whose laser beams possess a wavelength of λ = 826 nm and are focused at the position of the Fermi gas to 1/e 2 -radii of 50 µm (x-axis) and 70 µm (y-axis). The initial trapping frequencies are ω x = 2π × 93 Hz, ω y = 2π × 154 Hz and ω z = 2π × 157 Hz. When loading the optical trap we turn off the magnetic confinement in such a way that a variable homogenous magnetic field remains present. In the optical trap we
10.1103/physrevlett.94.080403
[ "https://export.arxiv.org/pdf/cond-mat/0410389v3.pdf" ]
3,128,205
cond-mat/0410389
56326ce03b68a04cda2f48a630712981fa9222df
Fermionic atoms in a Three Dimensional optical lattice: Observing Fermi Surfaces, Dynamics and Interactions 9 Mar 2005 Michael Köhl Institute of Quantum Electronics ETH Zürich CH-8093Hönggerberg, ZürichSwitzerland Henning Moritz Institute of Quantum Electronics ETH Zürich CH-8093Hönggerberg, ZürichSwitzerland Thilo Stöferle Institute of Quantum Electronics ETH Zürich CH-8093Hönggerberg, ZürichSwitzerland Kenneth Günter Institute of Quantum Electronics ETH Zürich CH-8093Hönggerberg, ZürichSwitzerland Tilman Esslinger Institute of Quantum Electronics ETH Zürich CH-8093Hönggerberg, ZürichSwitzerland Fermionic atoms in a Three Dimensional optical lattice: Observing Fermi Surfaces, Dynamics and Interactions 9 Mar 2005(Dated: March 23, 2022)numbers: 0375Ss0530Fk3450-s7118+y We have studied interacting and non-interacting quantum degenerate Fermi gases in a threedimensional optical lattice. We directly image the Fermi surface of the atoms in the lattice by turning off the optical lattice adiabatically. Due to the confining potential gradual filling of the lattice transforms the system from a normal state into a band insulator. The dynamics of the transition from a band insulator to a normal state is studied and the time scale is measured to be an order of magnitude larger than the tunneling time in the lattice. Using a Feshbach resonance we increase the interaction between atoms in two different spin states and dynamically induce a coupling between the lowest energy bands. We observe a shift of this coupling with respect to the Feshbach resonance in free space which is anticipated for strongly confined atoms.The exploration of quantum degenerate gases of fermionic atoms is driven by the ambition to get deeper insight into long-standing problems of quantum manybody physics, such as high temperature superconductivity. Very recently, the cross-over regime between a strongly interacting two-component Fermi gas and a molecular Bose-Einstein condensate has been studied in harmonic traps[1,2,3,4,5]. These experiments mark a milestone towards the understanding of superfluidity of fermionic atoms. However, the analogy to an electron gas in a solid is limited since there the electrons experience a periodic lattice potential. The lattice structure is in fact a key ingredient for most models describing quantum many-body phenomena in materials. It has been suggested that strongly interacting fermionic atoms in optical lattices could be employed for studies of high-Tcsuperconductivity [6], Mott-insulating phases [7], Bose condensation of fermionic particle-hole pairs [8], or interacting spin systems [9].Here we report on an experiment bridging the gap between current ultracold atom systems and fundamental concepts in condensed matter physics. A quantum degenerate Fermi gas of atoms is prepared in the crystal structure of a three dimensional optical lattice potential created by three crossed standing laser waves. The unique control over all relevant parameters in this system allows us to carry out experiments which are not feasible with solid-state systems.It was conceived by D. Jaksch et al. that ultracold atoms exposed to the periodic potential of an optical lattice are an almost ideal realization of a Hubbard model[10]. This model is elementary to describe the quantum physics of many electrons in a solid. It takes into account a single band of a static lattice potential and assumes the interactions to be purely local[11]. Ultracold atoms in an optical lattice give a very direct access to the underlying physics. The fundamental parameters include the tunnel coupling between adjacent lattice sites, the atom-atom interactions and the dimensionality of the system. Previous experiments with far-detuned three dimensional optical lattices[12,13,14]were always carried out with bosonic atoms, and experiments with fermions were restricted to a single standing wave[15]. In the latter situation many atoms can reside in each standing wave minimum but formation of a band insulator is prevented by the weak transverse confinement. The observed inhibition of transport[16] is due to localized states and therefore differs qualitatively from the band insulator which we create in the three dimensional optical lattice.The experiments are performed in a modified apparatus previously used to study bosonic rubidium atoms in optical lattices[14,17]. A mixture of bosonic 87 Rb and fermionic 40 K atoms is captured in a magneto-optical trap. For magnetic trapping we optically pump the potassium atoms into the |F = 9/2, m F = 9/2 and the rubidium atoms into the |F = 2, m F = 2 hyperfine ground state, with F being the total angular momentum and m F the magnetic quantum number. The mixture is evaporatively cooled using microwave radiation to selectively remove the most energetic rubidium atoms from the trap. The potassium cloud is sympathetically cooled by thermal contact with the rubidium atoms[18]. After reaching quantum degeneracy for both species with typically 6 × 10 5 potassium atoms at a temperature of T /T F = 0.32 (T F = 260 nK is the Fermitemperature of the non-interacting gas) we remove all the rubidium atoms from the trap. The potassium atoms are then transferred from the magnetic trap into a crossed beam optical dipole trap whose laser beams possess a wavelength of λ = 826 nm and are focused at the position of the Fermi gas to 1/e 2 -radii of 50 µm (x-axis) and 70 µm (y-axis). The initial trapping frequencies are ω x = 2π × 93 Hz, ω y = 2π × 154 Hz and ω z = 2π × 157 Hz. When loading the optical trap we turn off the magnetic confinement in such a way that a variable homogenous magnetic field remains present. In the optical trap we We have studied interacting and non-interacting quantum degenerate Fermi gases in a threedimensional optical lattice. We directly image the Fermi surface of the atoms in the lattice by turning off the optical lattice adiabatically. Due to the confining potential gradual filling of the lattice transforms the system from a normal state into a band insulator. The dynamics of the transition from a band insulator to a normal state is studied and the time scale is measured to be an order of magnitude larger than the tunneling time in the lattice. Using a Feshbach resonance we increase the interaction between atoms in two different spin states and dynamically induce a coupling between the lowest energy bands. We observe a shift of this coupling with respect to the Feshbach resonance in free space which is anticipated for strongly confined atoms. The exploration of quantum degenerate gases of fermionic atoms is driven by the ambition to get deeper insight into long-standing problems of quantum manybody physics, such as high temperature superconductivity. Very recently, the cross-over regime between a strongly interacting two-component Fermi gas and a molecular Bose-Einstein condensate has been studied in harmonic traps [1,2,3,4,5]. These experiments mark a milestone towards the understanding of superfluidity of fermionic atoms. However, the analogy to an electron gas in a solid is limited since there the electrons experience a periodic lattice potential. The lattice structure is in fact a key ingredient for most models describing quantum many-body phenomena in materials. It has been suggested that strongly interacting fermionic atoms in optical lattices could be employed for studies of high-Tcsuperconductivity [6], Mott-insulating phases [7], Bose condensation of fermionic particle-hole pairs [8], or interacting spin systems [9]. Here we report on an experiment bridging the gap between current ultracold atom systems and fundamental concepts in condensed matter physics. A quantum degenerate Fermi gas of atoms is prepared in the crystal structure of a three dimensional optical lattice potential created by three crossed standing laser waves. The unique control over all relevant parameters in this system allows us to carry out experiments which are not feasible with solid-state systems. It was conceived by D. Jaksch et al. that ultracold atoms exposed to the periodic potential of an optical lattice are an almost ideal realization of a Hubbard model [10]. This model is elementary to describe the quantum physics of many electrons in a solid. It takes into account a single band of a static lattice potential and assumes the interactions to be purely local [11]. Ultracold atoms in an optical lattice give a very direct access to the underlying physics. The fundamental parameters include the tunnel coupling between adjacent lattice sites, the atom-atom interactions and the dimensionality of the system. Previous experiments with far-detuned three dimensional optical lattices [12,13,14] were always carried out with bosonic atoms, and experiments with fermions were restricted to a single standing wave [15]. In the latter situation many atoms can reside in each standing wave minimum but formation of a band insulator is prevented by the weak transverse confinement. The observed inhibition of transport [16] is due to localized states and therefore differs qualitatively from the band insulator which we create in the three dimensional optical lattice. The experiments are performed in a modified apparatus previously used to study bosonic rubidium atoms in optical lattices [14,17]. A mixture of bosonic 87 Rb and fermionic 40 K atoms is captured in a magneto-optical trap. For magnetic trapping we optically pump the potassium atoms into the |F = 9/2, m F = 9/2 and the rubidium atoms into the |F = 2, m F = 2 hyperfine ground state, with F being the total angular momentum and m F the magnetic quantum number. The mixture is evaporatively cooled using microwave radiation to selectively remove the most energetic rubidium atoms from the trap. The potassium cloud is sympathetically cooled by thermal contact with the rubidium atoms [18]. After reaching quantum degeneracy for both species with typically 6 × 10 5 potassium atoms at a temperature of T /T F = 0.32 (T F = 260 nK is the Fermitemperature of the non-interacting gas) we remove all the rubidium atoms from the trap. The potassium atoms are then transferred from the magnetic trap into a crossed beam optical dipole trap whose laser beams possess a wavelength of λ = 826 nm and are focused at the position of the Fermi gas to 1/e 2 -radii of 50 µm (x-axis) and 70 µm (y-axis). The initial trapping frequencies are ω x = 2π × 93 Hz, ω y = 2π × 154 Hz and ω z = 2π × 157 Hz. When loading the optical trap we turn off the magnetic confinement in such a way that a variable homogenous magnetic field remains present. In the optical trap we prepare a spin mixture with (50 ± 4)% in each of the |F = 9/2, m F = −9/2 and |F = 9/2, m F = −7/2 spin states using a sequence of two radio frequency pulses. By lowering the depth of the optical trap on a time scale of 2 seconds we further evaporatively cool the potassium gas. This is done at a bias magnetic field of B = 227 G, which is well above the magnetic Feshbach resonance centered at B 0 = 202.1 G [1] and the s-wave scattering length between the two fermionic spin states is a = 118a 0 (a 0 is the Bohr radius). At the end of the evaporation we reach temperatures between T /T F = 0.2 and 0.25 with 5 × 10 4 to 2 × 10 5 particles, respectively. Prior to loading the atoms into the optical lattice we tune the magnetic field to B = (210 ± 0.1) G, such that the s-wave scattering length between the two states vanishes. Then the standing wave laser field along the vertical z-axis is turned on. Subsequently, the optical dipole trap along the y-axis is turned off and a standing wave laser field along the same axis is turned on, followed by the same procedure along the x-axis. In order to keep the loading of the atoms into the lattice as adiabatic as possible the intensities of the lasers are slowly increased (decreased) using exponential ramps with time constants of 10 ms (25 ms) and durations of 20 ms (50 ms), respectively. In its final configuration the optical lattice is formed by three orthogonal standing waves with mutually orthogonal polarizations and 1/e 2 -radii of 50 µm (x-axis) and 70 µm (y-axis and z-axis), which are derived from the same lasers as for the optical dipole trap. The laser fields of the three beams have a linewidth of the order of 10 kHz and their frequencies are offset with respect to each other by between 15 and 150 MHz. The resulting optical potential depth V x,y,z is proportional to the laser intensity and is conveniently expressed in terms of the recoil energy E r =h 2 k 2 /(2m), with k = 2π/λ and m being the atomic mass. The lattice depth was calibrated by modulating the laser intensity and studying the parametric heating. The calibration error is estimated to be < 10%. The potential created by the optical lattice results in a simple cubic crystal structure and the gaussian intensity profiles of the lattice beams give rise to an additional confining potential which varies with the laser intensity. As a result, the sharp edges characterizing the T = 0 distribution function for the quasi momentum in the homogeneous case [19] are expected to be rounded off. A quantitative picture can be obtained by considering a tightbinding Hamiltonian to describe non-interacting fermions in an optical lattice with an additional harmonic confinement [20]. At T = 0 the inhomogeneous system is characterized by the total atom number N and by the characteristic length ζ over which the potential shift due to the harmonic confinement equals the tunnel coupling matrix element J. One finds ζ α = 2J/mω 2 α , with the frequencies of the external harmonic confinement given by ω α (α = x, y, z). The density distribution scaled by ζ α and the momentum distribution of the atoms in the lattice only depend on the characteristic density ρ c = N d 3 ζxζy ζz , where d is the lattice spacing [7]. For a three-dimensional lattice with 20 × 20 × 20 sites we have numerically calculated the characteristic density for the onset of a band insulator to be ρ c ≃ 60. For this value of ρ c the occupation number at the center of the trap is larger than 0.99. It has been pointed out that a fermionic band insulator in an optical lattice with confining potential constitutes a high fidelity quantum register [21]. In the experiment we probe the population within the first Brillouin zones by ramping down the optical lattice slowly enough for the atoms to stay adiabatically in the lowest band whilst quasi-momentum is approximately conserved [22]. We lower the lattice potential to zero over a timescale of 1 ms. After 1 ms we switch off the homogeneous magnetic field and allow for total of 9 ms of ballistic expansion before we take an absorption image of the expanded atom cloud. The momentum distribution obtained from these time of flight images, shown in Fig. 1, reproduces the quasi-momentum distributions of the atoms inside the lattice. With increasing characteristic density the initially circular shape of the Fermi surface develops extensions pointing towards the Bragg planes and finally transforms into a square shape completely filling the first Brillouin zone deeply in the band insulator. We have observed population of higher bands if more atoms are filled into the lattice initially. In Fig. 2 the experimental data for momentum distributions along the line with quasi-momentum q y = 0 are compared to the results of numerical simulations using the same characteristic densities. When imaging the cloud along the x-direction we find a homogeneous filling of the band in the vertical (z-) direction, probably due to the change in the harmonic confinement while loading the lattice combined with the presence of gravity. This asymmetry between the horizontal x-, y-, and the vertical z-directions vanishes when the gas approaches the band insulating regime. We have examined the level of adiabaticity of our loading scheme into the optical lattice by transferring the atoms from the band insulator back into the crossed beam dipole trap. There we find a temperature of 0.35 T F when the initial temperature prior to loading into the lattice was 0.2 T F . We have studied the dynamic response of the noninteracting Fermi gas to a change in the characteristic density from a value deep in the band insulating regime to a value below. In the latter regime the fermions are delocalized over several sites of the optical lattice and an interference pattern is observed when the atoms are abruptly released from the lattice. The width of the interference peaks is a measure of the length scale over which the atoms are delocalized in the lattice or, equivalently, their coherence length. We change the characteristic density in situ by varying the strength of the lattice laser beams. Starting from an initial characteristic density of ρ c = 16 in an optical lattice with a potential depth of 5 E r we create a band insulator with a characteristic density of ρ c = 2700 at a potential depth of 15 E r . After holding the atoms for 5 ms we reduce the potential depth back to 5 E r , using an exponential ramp with duration and time constant t r . This is followed by a rapid switch off of the lattice (see Fig. 3a). We measure the width of the central momentum peak in the time of flight images for different durations t r and obtain the time scales τ x = (2.7 ± 0.4) ms and τ y = (3.8 ± 0.3) ms in the x-and y-direction, respectively. This corresponds to approximately ten times the timescale for tunnelling given by h/2zJ at a potential depth of 5 E r , where z is the coordination number of the lattice. This non-trivial dynamics appears to be significantly slower than the time scale measured for the transition of a Mott-insulating state to a superfluid state using bosonic atoms in an optical lattice [13]. The comparatively slow dynamics of delocalization of the fermions when approaching the normal state is most likely due to Pauli blocking which prevents tunneling of atoms in regions where the lowest band is full and the atoms are well localized. We investigate the interacting regime in the lattice starting from a non-interacting gas deep in a band insulator with V x = 12 E r and V y = V z = 18 E r and corresponding trapping frequencies of ω x = 2π × 50 kHz and ω y = ω z = 2π × 62 kHz in the individual minima. A short radio-frequency pulse is applied to transfer all atoms from the |F = 9/2, m F = −7/2 into the |F = 9/2, m F = −5/2 state, with the atoms in the |F = 9/2, m F = −9/2 remaining unaffected. We ramp the magnetic field with an inverse sweep rate of 12 µs/G to different final values around the Feshbach resonance (see Fig. 4a) located at B = 224 G [23]. The sweep across the Feshbach resonance goes from the side of repulsive interactions towards the side of attractive interactions. When using this direction of the sweep there is no adiabatic conversion to molecules. After turning off the optical lattice adiabatically and switching off the magnetic field we measure the momentum distribution. To see the effect of the interactions we determine the fraction of atoms transferred into higher bands. For final magnetic field values well above the Feshbach resonance we observe a significant increase in the number of atoms in higher bands along the weak axis of the lattice, demonstrating an interaction-induced coupling between the lowest bands. Since the s-wave interaction is changed on a time scale short compared to the tunnelling time between adjacent potential minima we may regard the band insulator as an array of harmonic potential wells. It has been shown that increasing the s-wave scattering length for two particles in a harmonic oscillator shifts the energy of the two-particle state upwards until the next oscillator level is reached [24]. In our case this leads to a population of higher energy bands. The fraction of atoms transferred could be limited by the number of doubly occupied lattice sites and tunnelling in the higher bands. The number of doubly occupied sites could be measured by studying the formation of molecules in the lattice. In addition, we observe a shift of the position of the Feshbach resonance from its value in free space to larger values of the magnetic field (see Fig. 4a), which has been predicted for tightly confined atoms in an optical lattice [25]. This mechanism for a confinement induced resonance is related to the phenomenon predicted for one-dimensional quantum gases [26] which has as yet escaped experimental observation. For a quantitative description of this strongly interacting Fermi gas on a lattice a multi-band Hubbard model could be considered but these are even in the static case notoriously difficult or even impossible to solve with present methods [27]. In conclusion we have created a fermionic manyparticle quantum system on a lattice. We have demonstrated the dynamic control over the parameters of the system such as filling and interactions which is not feasible in solid state systems. For the non-interacting static regime we find good agreement between our measurements and a theoretical model. Both the dynamic measurements and the strongly interacting case pose challenges for the present theoretical understanding of manyparticle fermionic systems on optical lattices. PACS numbers: 03.75.Ss, 05.30.Fk, 34.50.-s, 71.18.+y FIG. 1 : 1Observing the Fermi surface. Time of flight images obtained after adiabatically ramping down the optical lattice. The characteristic density increases from left to right. (a) 3500 atoms per spin state and a potential depth of the optical lattice of 5 Er. Images (b)-(e) were obtained with 15000 atoms per spin state. The potential depths of the optical lattices were 5 Er (b), 6 Er (c), 8 Er (d) and 12 Er (e). The images show the optical density (OD) integrated along the vertically oriented z-axis after 9 ms of ballistic expansion. FIG. 2 : 2Analysis of the density distributions. The dots are cuts through the measured density distribution for quasi momentum qy = 0 after adiabatically ramping down the optical lattice. (a) Normal state with ρc = 14.5, (b) band insulator with ρc = 137, (c) band insulator with ρc = 2500. We have numerically calculated the momentum distribution function of fermions in the lowest band of a three-dimensional lattice with 20 × 20 × 20 sites and characteristic lengths ζx/d = 3.2, ζy/d = 2.6, ζz/d = 2.5 ((a) and (b)) and ζx/d = 1, ζy/d = 0.8, ζz/d = 0.8 (c), assuming zero temperature (solid lines). Experimental data of (c) are averaged over 5 images. Imperfect adiabaticity during the switch-off of the optical lattice may cause the rounding-off of the experimental data at the edge of the Brillouin zone in (b) and (c). The calculated momentum distribution function is scaled to match the experimental data using identical scale factors for all graphs. FIG. 3 : 3Restoring phase coherence. (a) Control sequence for the depth of the optical lattice. (b) Pseudocolor image of the momentum distribution after releasing the atoms from the initial optical lattice of 5 Er and 7 ms ballistic expansion. It reveals the central momentum peak and the matter wave interference peaks at ±2hk. The data are averaged over 5 repetitive measurements. (c) Width of the central momentum peak obtained from Gaussian fits to the atomic density distribution. The initial width is determined by the momentum spread of an atom localized in the vibrational ground state of a lattice well. The 10% difference in this size comes from slightly different magnifications of the imaging system in the two orthogonal directions. The difference in the asymptotic values of the width can most likely be attributed to the loading sequence of the lattice and to the asymmetry of the confining potentials due to the different beam waists. The error bars show the statistical error of 4 repetitive measurements. FIG. 4 : 4Interaction-induced transition between Bloch bands. (a) Transferring fermions into higher bands using a sweep across the Feshbach resonance (filled symbols). The inverse magnetic field sweep rate is 12 µs/G. The line shows a sigmoidal fit to the data. The open symbols show a repetition of the experiment with the atoms prepared in the spin states |F = 9/2, mF = −9/2 and |F = 9/2, mF = −7/2 where the scattering length is not sensitive to the magnetic field. The magnetic field is calibrated by rf spectroscopy between Zeeman levels. Due to the rapid ramp the field lags behind its asymptotic value and the horizontal error bars represent this deviation. (b) Fraction of atoms in higher bands for a final magnetic field of 233 G for different magnetic field sweep rates. The vertical error bars show the statistical error of 4 repetitive measurements. (c) Momentum distribution for a final magnetic field of B = 233 G and a 12 µs/G sweep rate. Arrows indicate the atoms in the higher bands. We would like to thank G. Blatter, C. Bruder, H. P. Büchler, S. Jonsell, A. Muramatsu, M. Rigol, C. Schori, P. Törmä and M. Troyer for insightful discussions, and SNF, SEP Information Sciences and QSIT for funding. . C A Regal, M Greiner, D S Jin, Phys. Rev. Lett. 9240403C. A. Regal, M. Greiner, D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004). . M Bartenstein, Phys. Rev. Lett. 92120401M. Bartenstein et al. Phys. Rev. Lett. 92, 120401 (2004). . M W Zwierlein, Phys. Rev. Lett. 92120403M.W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004). . J Kinast, Phys. Rev. Lett. 92150402J. Kinast et al., Phys. Rev. Lett. 92, 150402 (2004). . T Bourdel, Phys. Rev. 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[ "S parameter and parity doubling below the conformal window", "S parameter and parity doubling below the conformal window", "S parameter and parity doubling below the conformal window", "S parameter and parity doubling below the conformal window" ]
[ "David Schaich [email protected] \nfor the Lattice Strong Dynamics (LSD) Collaboration † Physics Department & Center for Computational Science\nDepartment of Physics\nBoston University\n02215BostonMA\n\nUniversity of Colorado\n80309 ‡BoulderCO\n", "David Schaich [email protected] \nfor the Lattice Strong Dynamics (LSD) Collaboration † Physics Department & Center for Computational Science\nDepartment of Physics\nBoston University\n02215BostonMA\n\nUniversity of Colorado\n80309 ‡BoulderCO\n" ]
[ "for the Lattice Strong Dynamics (LSD) Collaboration † Physics Department & Center for Computational Science\nDepartment of Physics\nBoston University\n02215BostonMA", "University of Colorado\n80309 ‡BoulderCO", "for the Lattice Strong Dynamics (LSD) Collaboration † Physics Department & Center for Computational Science\nDepartment of Physics\nBoston University\n02215BostonMA", "University of Colorado\n80309 ‡BoulderCO" ]
[ "XXIX International Symposium on Lattice Field Theory 10-16", "XXIX International Symposium on Lattice Field Theory 10-16" ]
Recently the Lattice Strong Dynamics Collaboration reported a reduction of the electroweak S parameter for SU(3) gauge theory with N f = 6 fermions in the fundamental representation, compared to scaled-up QCD. Here I provide additional details of our calculation. I discuss our use of conserved lattice currents; the relation to vector-axial parity doubling; finite-volume effects; and the sensitivity of our results to the number of fermion doublets with chiral electroweak couplings. Results presented here include additional data, and do not affect our previously-published conclusions.
10.22323/1.139.0087
[ "https://export.arxiv.org/pdf/1111.4993v2.pdf" ]
119,198,113
1111.4993
3e80f4af657ab06c65ac1b21758fd6f1c7b6ec41
S parameter and parity doubling below the conformal window July 2011 David Schaich [email protected] for the Lattice Strong Dynamics (LSD) Collaboration † Physics Department & Center for Computational Science Department of Physics Boston University 02215BostonMA University of Colorado 80309 ‡BoulderCO S parameter and parity doubling below the conformal window XXIX International Symposium on Lattice Field Theory 10-16 Squaw Valley, Lake Tahoe, CaliforniaJuly 2011* Speaker. † http://www.yale.edu/LSD ‡ Present address c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. S parameter and parity doubling below the conformal window David Schaich Recently the Lattice Strong Dynamics Collaboration reported a reduction of the electroweak S parameter for SU(3) gauge theory with N f = 6 fermions in the fundamental representation, compared to scaled-up QCD. Here I provide additional details of our calculation. I discuss our use of conserved lattice currents; the relation to vector-axial parity doubling; finite-volume effects; and the sensitivity of our results to the number of fermion doublets with chiral electroweak couplings. Results presented here include additional data, and do not affect our previously-published conclusions. Introduction The application of lattice gauge theory to strongly-interacting physics beyond QCD is at present a very active field [1]. While much of the current interest is motivated by the possibility that new strong dynamics may play a role in electroweak symmetry breaking [2,3], improving our general understanding of strong dynamics is an important theoretical goal in its own right. The standard picture of strongly-interacting SU(N) gauge theories is that as we increase the number N f of fermions in a given representation, an infrared fixed point will develop at some critical N f may possess the dynamical scale separation that characterizes "walking" theories, as well as parity doubling between vector (V ) and axial-vector (A) spectra that can reduce the electroweak S parameter to phenomenologically viable values [4]. The Lattice Strong Dynamics Collaboration approaches these questions by using QCD as a baseline. We consider SU(3) gauge theory and steadily increase the number of fundamental fermions, comparing our results against the familiar case N f = 2. We use computationally expensive domain wall fermions for better control over lattice artifacts. Our first studies focused on the N f = 6 model, which while not truly walking exhibits some of the associated phenomena: by matching IR scales between N f = 2 and N f = 6 calculations, we observed an enhancement in the N f = 6 chiral condensate [5] and a reduction of the S parameter relative to scaled-up QCD [6]. Here I provide additional details of our S parameter calculation that were not discussed in Ref. [6]. Results presented here also include additional data, and do not affect the conclusions of Ref. [6]. We can identify three main ingredients in our expression for the S parameter, S = 4πN D lim Q 2 →0 d dQ 2 Π V −A (Q 2 ) − ∆S SM . (1.1) The term ∆S SM accounts for the three Nambu-Goldstone bosons (NGBs) eaten by the W ± and Z, and is discussed in detail by Ref. [6]. In Section 2 I review our calculation of the transverse V -A polarization function Π V −A (Q 2 ), and relate it to the vector and axial spectra in Section 3. Finally, N D is the number of doublets with chiral electroweak couplings; in Section 4 I show how it affects our results for the S parameter. Currents and correlators On the lattice, the transverse V -A polarization function Π V −A (Q 2 ) is determined from Π µν V −A (Q) = δ µν − Q µ Q ν Q 2 Π V −A (Q 2 ) − Q µ Q ν Q 2 Π L V −A (Q 2 ) = Z ∑ x e iQ·(x+ µ/2) Tr V µa (x)V νb (0) − A µa (x)A νb (0) . (2.1) Here Q = 2 sin(πn/L) are lattice momenta, while Q = 2πn/L; these are spacelike Q 2 = −q 2 > 0. The current correlators mix two types of domain wall currents. V µa and A µa are non-conserved "local" currents defined on the domain walls; in terms of five-dimensional fermion fields Ψ(x, s), V µa (x) = 1 2 Ψ(x, L s − 1)γ µ (1 + γ 5 )τ a Ψ(x, L s − 1) + Ψ(x, 0)γ µ (1 − γ 5 )τ a Ψ(x, 0) A µa (x) = 1 2 Ψ(x, L s − 1)γ µ (1 + γ 5 )τ a Ψ(x, L s − 1) − Ψ(x, 0)γ µ (1 − γ 5 )τ a Ψ(x, 0) . (2.2) The conserved currents V µa and A µa are point-split, and summed over the fifth dimension: V µa (x) = L s −1 ∑ s=0 j µa (x, s) A µa (x) = L s −1 ∑ s=0 sign s − L s − 1 2 j µa (x, s), (2.3) j µa (x, s) = 1 2 Ψ(x + µ, s)(1 + γ µ )U † x,µ τ a Ψ(x, s) − Ψ(x, s)(1 − γ µ )U x,µ τ a Ψ(x + µ, s) . (2.4) The Fourier transform in Eqn. 2.1 involves (x + µ/2) because the conserved currents are point-split on the link (x, x + µ). The flavor matrices τ a are normalized to Tr τ a τ b = δ ab /2. Although the conserved and local currents must agree in the continuum limit, at finite lattice spacing only the former satisfy a Ward identity ( Q µ Π µν VV = 0, Fig. 1). Because the correlators involve both currents, Eqn. 2.1 includes the renormalization factor Z, which we compute nonperturbatively, Z = 0.85 (0.73) for N f = 2 (6). Our chiral lattice fermions ensure that Z = Z A = Z V . In principle, it would be best to work entirely with the conserved currents V µa and A µa instead of using the mixed correlators in Eqn. 2.1. In practice, evaluating conserved-conserved correlators such as V µa (x)V νb (0) requires O(L s ) inversions, increasing the computational cost of the calculation by roughly an order of magnitude. As emphasized in Ref. [7], lattice artifacts cancel in the V -A difference of the mixed correlators, allowing us to use these less expensive quantities. This is illustrated in the left panel of Fig. 2 : even though Π µν Q ν = 0 since V νa and A νa are not conserved, Π µν VV (Q 2 ) − Π µν AA (Q 2 ) Q ν ≈ 0. In the right panel, we see that this does not hold if we use only local currents in the correlators. Parity doubling and finite volume effects Because chiral perturbation theory cannot reliably be applied to our N f = 6 calculations [8], we extract the slope Π V −A (0) by fitting our data to a simple four-parameter rational function, Π V −A (Q 2 ) = a 0 + a 1 Q 2 1 + b 1 Q 2 + b 2 Q 4 . (3.1) This "Padé(1,2)" functional form has the correct asymptotic behavior Π V −A (Q 2 ) ∼ Q −2 at large Q 2 , and also resembles the single-pole dominance approximation to the V -A dispersion relation Π V −A (Q 2 ) = −F 2 P + Q 2 12π ∞ 0 ds π R V (s) − R A (s) s + Q 2 . (3.2) (F P is the pseudoscalar decay constant.) That is, with the single-pole dominance approximation R(s) = 12π 2 F 2 δ (s − M 2 ), this dispersion relation becomes Π (pole) V −A (Q 2 ) = −F 2 P + Q 2 F 2 V M 2 V + Q 2 − Q 2 F 2 A M 2 A + Q 2 ,(3.3) which reproduces the form of Eqn. 3.1 when we apply the corresponding approximation to the first Weinberg sum rule, F 2 P = F 2 V − F 2 A . Because the lattice data contain information about the entire spectrum, the fit parameters in Eqn. 3.1 do not directly correspond to the combinations of meson masses and decay constants predicted by the pole-dominance Eqn. 3.3. Uncorrelated fits of our data to Eqn. 3.1 produce stable results with χ 2 /do f 1 as we vary the Q 2 fit range. Our results for Π V −A (0) are shown as colored points in the left panel of Fig. 3. The black points in that plot are pole-dominance predictions based on Eqn. 3.3. Both the direct fit results and the pole-dominance predictions show a reduction for N f = 6 compared to N f = 2 at light pseudoscalar masses M P M V 0 , where M V 0 is the vector meson mass in the chiral limit. The pole-dominance predictions are systematically lower than the direct results, consistent with the expectation that states neglected by the single-pole dominance approximation would provide additional positive contributions. The lightest N f = 2 points in Fig. 3 are empty because they correspond to a fermion mass m so small that finite-volume effects may be significant. Finite-volume effects are a concern for the S parameter calculation because they can produce spurious parity doubling that artificially reduces Π V −A (0) . This is illustrated in the right panel of Fig. 3 for N f = 6 calculations on 16 3 ×32 volumes: Π V −A (0) → 0 as m → 0, which would naïvely suggest a negative S parameter from Eqn. 1.1. The associated distortion of the spectrum provides clear evidence that this is merely a finite-volume effect: as m decreases, the 16 3 ×32 pseudoscalar mass M P freezes around M 2 P ≈ 1.2M 2 V 0 , which is not the case for the 32 3 ×64 results also shown in the plot. Returning to the lightest N f = 2 points, the pole-dominance prediction for Π V −A (0) decreases due to spurious parity doubling from finite-volume effects. However, we do not see a similar reduction in the direct fit result. Instead, this point clearly continues the trend established at heavier masses, and the corresponding N f = 2 results for S (Fig. 4, below) reproduce the prediction obtained by scaling up QCD phenomenology, lim M 2 P →0 S = 0.32(3) [4]. This suggests that the Padé fits may be less sensitive than spectral quantities to these finite-volume effects, increasing our confidence that the reduction observed for N f = 6 is physical. S parameter results Realistic models of dynamical electroweak symmetry breaking must produce exactly three massless NGBs to be eaten by the W ± and Z. Any additional pseudo-Nambu-Goldstone bosons (PNGBs) must acquire masses from standard-model and other (e.g., extended-technicolor) interactions in order to satisfy experimental constraints. On the lattice, however, we perform calculations with N 2 f − 1 degenerate massive PNGBs. When we use Eqn. 1.1 to determine the S parameter from the Π V −A (0) results shown in Fig. 3, the ∆S SM term removes the contribution only of the three would-be NGBs. (To be more precise, the I 3 = 0 NGB does not contribute, and ∆S SM cancels the contribution of the |I 3 | = 1 pair.) The remaining N 2 f − 4 PNGBs introduce chiral-log terms ∝ log[M 2 V 0 /M 2 P ] that would diverge in the chiral limit M 2 P → 0. To guide the eye, we include in Fig. 4 simple linear fits accounting for the N D -dependent chiral-log divergence that remains for N f > 2. We fit the lightest three solid points to the form The blue N f = 6 curves allow us to estimate the fermion mass m at which we could directly observe chiral log effects. The necessary m is too small for us to explore on our present 32 3 ×64 volumes. Again, in a realistic phenomenological context, we must have only three massless NGBs, with N 2 f − 4 massive PNGBs. To estimate a definite value for the N f = 6 S parameter in this situation, we can imagine freezing the masses of all N 2 f − 4 PNGBs at some finite value (such as M 2 P = 0.38M 2 V 0 at the minimum of the N D = 1 blue curve in Fig. 4), and then taking only the three NGBs to the chiral limit M 2 P → 0. A qualitative picture of this scenario is sketched in Fig. 5. S = A + Bx + − 1 12π log (1/x) (4.1) where x ≡ M 2 P /M 2 V up to the loss of asymptotic freedom) the system is IR-conformal. Approximately-conformal systems with N f N (c) Figure 1 : 1On every configuration, Q µ Π µν VV = 0 when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each Q 2 value distinguish different ν. Figure 2 : 2On every configuration, lattice artifacts Π µν Q ν = 0 cancel in the V -A difference when one conserved current is used in each correlator (left), but not when only non-conserved local currents are used (right). The horizontal offsets around each Q 2 value distinguish different µ. Figure 3 : 3The slope of Π V −A (Q 2 ) at Q 2 = 0, plotted versus M 2 P /M 2 V 0 .Left: N f = 2 and 6 results on 32 3 ×64 volumes from direct fits to Eqn. 3.1 (colored), compared to pole-dominance predictions (black). Right: N f = 6 results on 16 3 ×32 and 32 3 ×64 volumes. Fig. 4presents our S parameter results for N f = 2 and 6, considering two possible values of N D for N f = 6. The plot on the left presents the case in which every fermion possesses chiral electroweak couplings, N D = N f /2 = 3. The minimal case in which only a single doublet has chiral couplings (N D = 1) is shown on the right. In both cases the N f = 6 results show a reduction compared to rescaling N f = 2, before diverging in the chiral limit. With N D = 1 the S parameter can be significantly closer to the experimental value S ≈ −0.15(10) for M (re f ) H ∼ 1 TeV[9]. Figure 4 : 4S parameter for N f = 2 and 6, for the maximum N D = 3 (left) and minimum N D = 1 (right). The bands correspond to fits explained in the text. 0 and counts the pairs of PNGBs with I 3 = 0, Figure 5 : 5S parameter for N f = 2 and 6 with N D = 1, imagining that we freeze the masses of allN 2 f − 4 PNGBs at M 2 P = 0.38M 2 V 0 ,as described in the text. U.S. National Science Foundation through TeraGrid resources provided by the National Institute for Computational Sciences under grant number TG-MCA08X008; 2 and Boston University's Scientific Computing Facilities. http://www.usqcd.org http://www.xsede.org The conformal window on the lattice. L , Del Debbio, arXiv:1102.4066PoS. 20104hep-latL. Del Debbio, "The conformal window on the lattice", PoS Lattice 2010 (2011) 004, arXiv:1102.4066 [hep-lat]. Strong dynamics and electroweak symmetry breaking. C T Hill, E H Simmons, 10.1016/S0370-1573(03)00140-6arXiv:hep-ph/0203079Phys. Rept. 381hep-phC. T. Hill and E. H. Simmons, "Strong dynamics and electroweak symmetry breaking", Phys. Rept. 381 (2003) 235-402, arXiv:hep-ph/0203079 [hep-ph]. EWSB Theory on the Eve of Higgs Boson Exclusion/Discovery. S Rychkov, arXiv:1109.1180PoS. 201129hep-phS. Rychkov, "EWSB Theory on the Eve of Higgs Boson Exclusion/Discovery", PoS EPS-HEP2011 (2011) 029, arXiv:1109.1180 [hep-ph]. Estimation of oblique electroweak corrections. M E Peskin, T Takeuchi, 10.1103/PhysRevD.46.381Phys. Rev. 46M. E. Peskin and T. Takeuchi, "Estimation of oblique electroweak corrections", Phys. Rev. D46 (1992) 381-409. Toward TeV Conformality. T Appelquist, LSD CollaborationA Avakian, LSD CollaborationR Babich, LSD CollaborationR C Brower, LSD CollaborationM Cheng, LSD CollaborationM A Clark, LSD CollaborationS D Cohen, LSD CollaborationG T Fleming, LSD CollaborationJ Kiskis, LSD CollaborationE T Neil, LSD CollaborationJ C Osborn, LSD CollaborationC Rebbi, LSD CollaborationD Schaich, LSD CollaborationP Vranas, LSD Collaboration10.1103/PhysRevLett.104.071601arXiv:0910.2224Phys. Rev. Lett. 10471601hep-phLSD Collaboration, T. Appelquist, A. Avakian, R. Babich, R. C. Brower, M. Cheng, M. A. Clark, S. D. Cohen, G. T. Fleming, J. Kiskis, E. T. Neil, J. C. Osborn, C. Rebbi, D. Schaich, and P. Vranas, "Toward TeV Conformality", Phys. Rev. Lett. 104 (2010) 071601, arXiv:0910.2224 [hep-ph]. Parity Doubling and the S Parameter Below the Conformal Window. T Appelquist, LSD CollaborationR Babich, LSD CollaborationR C Brower, LSD CollaborationM Cheng, LSD CollaborationM A Clark, LSD CollaborationS D Cohen, LSD CollaborationG T Fleming, LSD CollaborationJ Kiskis, LSD CollaborationM Lin, LSD CollaborationE T Neil, LSD CollaborationJ C Osborn, LSD CollaborationC Rebbi, LSD CollaborationD Schaich, LSD CollaborationP Vranas, LSD Collaboration10.1103/PhysRevLett.106.231601arXiv:1009.5967Phys. Rev. Lett. 106231601hep-phLSD Collaboration, T. Appelquist, R. Babich, R. C. Brower, M. Cheng, M. A. Clark, S. D. Cohen, G. T. Fleming, J. Kiskis, M. Lin, E. T. Neil, J. C. Osborn, C. Rebbi, D. Schaich, and P. Vranas, "Parity Doubling and the S Parameter Below the Conformal Window", Phys. Rev. Lett. 106 (2011) 231601, arXiv:1009.5967 [hep-ph]. The S Parameter in QCD from Domain Wall Fermions. P A Boyle, RBC ; UKQCD CollaborationL Debbio, RBC ; UKQCD CollaborationJ Wennekers, RBC ; UKQCD CollaborationJ M Zanotti, RBC ; UKQCD Collaboration10.1103/PhysRevD.81.014504arXiv:0909.4931Phys. Rev. 8114504hep-latRBC and UKQCD Collaboration, P. A. Boyle, L. Del Debbio, J. Wennekers, and J. M. Zanotti, "The S Parameter in QCD from Domain Wall Fermions", Phys. Rev. D81 (2010) 014504, arXiv:0909.4931 [hep-lat]. Lattice study of ChPT beyond QCD. E T Neil, LSD CollaborationA Avakian, LSD CollaborationR Babich, LSD CollaborationR C Brower, LSD CollaborationM Cheng, LSD CollaborationM A Clark, LSD CollaborationS D Cohen, LSD CollaborationG T Fleming, LSD CollaborationJ Kiskis, LSD CollaborationJ C Osborn, LSD CollaborationC Rebbi, LSD CollaborationD Schaich, LSD CollaborationP Vranas, LSD CollaborationarXiv:1002.3777PoS. 0988hep-latLSD Collaboration, E. T. Neil, A. Avakian, R. Babich, R. C. Brower, M. Cheng, M. A. Clark, S. D. Cohen, G. T. Fleming, J. Kiskis, J. C. Osborn, C. Rebbi, D. 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[]
[ "Gravity-induced resonances in a rotating trap", "Gravity-induced resonances in a rotating trap" ]
[ "Iwo Bialynicki-Birula \nCenter for Theoretical Physics\nPolish Academy of Sciences Al\nLotników 32/4602-668WarsawPoland\n", "Tomasz Sowiński \nCenter for Theoretical Physics\nPolish Academy of Sciences Al\nLotników 32/4602-668WarsawPoland\n" ]
[ "Center for Theoretical Physics\nPolish Academy of Sciences Al\nLotników 32/4602-668WarsawPoland", "Center for Theoretical Physics\nPolish Academy of Sciences Al\nLotników 32/4602-668WarsawPoland" ]
[]
It is shown that in an anisotropic harmonic trap that rotates with the properly chosen rotation rate, the force of gravity leads to a resonant behavior. Full analysis of the dynamics in an anisotropic, rotating trap in 3D is presented and several regions of stability are identified. On resonance, the oscillation amplitude of a single particle, or of the center of mass of a many-particle system (for example, BEC), grows linearly with time and all particles are expelled from the trap. The resonances can only occur when the rotation axis is tilted away from the vertical position. The positions of the resonances (there are always two of them) do not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity of rotation.
10.1103/physreva.71.043610
[ "https://export.arxiv.org/pdf/quant-ph/0412004v2.pdf" ]
30,102,109
quant-ph/0412004
966ecc2d5d17a0b6ffae390cd4b4ccad0438afaa
Gravity-induced resonances in a rotating trap arXiv:quant-ph/0412004v2 9 Feb 2005 Iwo Bialynicki-Birula Center for Theoretical Physics Polish Academy of Sciences Al Lotników 32/4602-668WarsawPoland Tomasz Sowiński Center for Theoretical Physics Polish Academy of Sciences Al Lotników 32/4602-668WarsawPoland Gravity-induced resonances in a rotating trap arXiv:quant-ph/0412004v2 9 Feb 2005numbers: 0375Kk4530+s4550-j It is shown that in an anisotropic harmonic trap that rotates with the properly chosen rotation rate, the force of gravity leads to a resonant behavior. Full analysis of the dynamics in an anisotropic, rotating trap in 3D is presented and several regions of stability are identified. On resonance, the oscillation amplitude of a single particle, or of the center of mass of a many-particle system (for example, BEC), grows linearly with time and all particles are expelled from the trap. The resonances can only occur when the rotation axis is tilted away from the vertical position. The positions of the resonances (there are always two of them) do not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity of rotation. I. INTRODUCTION The main purpose of this work is to expose the role of gravity in the dynamics of particles in a rotating trap. In addition, we present a complete analysis of the stability regions for a rotating trap in 3D. We prove that in the generic case there are three separate regions of stability with different characteristics. Gravity induced resonances are relevant only if they occur in the regions of stability, otherwise, they are swamped by the exponential behavior of trajectories. Harmonic traps are often used in optics and atomic physics (especially in the form of TOP traps [1] in the study of Bose-Einstein condensates) and yet a complete theory of these devices has not been developed. The solution to the problem of a trap rotating around one of the trap axes is effectively two-dimensional and its solution has been known for at least hundred years. In the classic textbook on analytical dynamics by Whittaker [2] we find a solution of a mathematically equivalent problem of small oscillations of "a heavy particle about its position of equilibrium at the lowest point of a surface which is rotating with constant angular velocity about a vertical axis through the point". The quantum-mechanical counterpart of the Whittaker problem has also been completely solved [3,4] and the statistical mechanics of a classical gas was studied in [5]. In this work we present a complete solution to the problem of the motion of a particle moving in a most general anisotropic rotating harmonic trap in 3D and in the presence of gravity. This is an exactly soluble problem but technical difficulties apparently served so far as a deterrent in developing a full description. A full description of the particle dynamics in a rotating anisotropic trap in three dimensions so far has not been given despite a new significance of this problem brought about by experimental and theoretical studies of Bose-Einstein condensates and the accompanying thermal clouds in rotating traps * Electronic address: [email protected] [5,6,7,8,9,10,11,12]. Explicit formulas describing the complete mode structure in the three-dimensional case are indeed quite cumbersome [16] because we deal here with third-order polynomials and on top of that they have rather complicated coefficients. However, many important features may be exhibited without straining the reader's patience. In particular, we can identify various stability regions for an arbitrary orientation of the angular velocity and we can give conditions for a resonance. The standard arrangement [5,6,7,8,9,10,11,12] is to choose a vertical axis of rotation of the trap. Slight tiltings of this axis were introduced to excite the scissors modes [13,14,15]. However, for such very small tilting angles the effects described in the present paper would not be noticeable. In the case of a vertical axis of rotation, there are no resonances. The only effect of gravity is a displacement of the equilibrium position. The situation completely changes and new phenomena will occur when the axis of rotation is tilted away from the vertical position. In this generic case, for every anisotropic three-dimensional trap there exist two (not three as one might expect) characteristic frequencies at which resonances occur. The motion in a trap that is rotating at a resonant frequency will become unbounded and all particles will be expelled from the trap. The position of the resonance does not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity. All our results are valid not only for a single particle but also for the center of mass motion in manybody (classical or quantum) theory since for all quadratic Hamiltonians the center of mass motion completely separates from the internal motion [16,17,18]. Therefore, a trap rotating at the resonant frequency will not hold the Bose-Einstein condensate. Owing to the linearity of the equations of motion for a harmonic trap, all conclusions hold both in classical and in quantum theory. A resonant behavior caused by an application of a static force may seem counterintuitive, but it is explained by the fact that in a rotating frame the force of gravity acts as a periodically changing external force. II. EQUATIONS OF MOTION The best way to analyze the behavior of particles in a uniformly rotating trap is to first perform the transformation to the rotating frame. In this frame the harmonic trap potential is frozen but the force of gravity is rotating with the angular velocity of the trap rotation. In the rotating frame the Hamiltonian has the form H = p 2 2m + r·Ω·p + m 2 r·V ·r − mr·g(t).(1) The potential matrixV is symmetric and positive definite. The eigenvalues of this matrix are the squared frequencies of the oscillations in the non-rotating trap. The angular velocity matrixΩ is related to the components of the angular velocity vector through the formula Ω ik = ǫ ijk Ω j . The vector of the gravitational acceleration g(t), as seen in the rotating frame, can be expressed in the form g(t) = g + g ⊥ cos(Ωt) − (n × g ⊥ ) sin(Ωt),(2) where n denotes the direction and Ω denotes the length of the angular velocity vector Ω. The parallel and the transverse components of the gravitational acceleration vector g = g(0) are defined as g = n(n·g) and g ⊥ = g − n(n·g), respectively. Note that the time-dependent part vanishes when the rotation axis is vertical. The equations of motion determined by the Hamiltonian (1) have the following form dr(t) dt = p(t) m −Ω·r(t),(3a)dp(t) dt = −mV ·r(t) −Ω·p(t) + mg(t).(3b) These equations describe an oscillator in a rotating frame displaced by a constant force (the longitudinal part of g) and driven by a periodic force (the transverse part of g). It is convenient to rewrite the expression (2) as a real part of a complex function g(t) = ℜ g + (g ⊥ + i(n × g ⊥ ))e iΩt .(4) In compact notation Eqs. (3a) have the form dR(t) dt =M(Ω)·R(t) + ℜ(G + G ⊥ e iΩt ),(5) where R(t) = r(t) p(t) ,M(Ω) = −Ω m −1Î −mV −Ω ,(6)G = m 0 g , G ⊥ = m 0 g ⊥ + i(n × g ⊥ ) .(7) We shall now replace the equations of motion by their complex counterpart dW(t) dt =M(Ω)·W(t) + G + G ⊥ e iΩt .(8) The physical trajectory in phase space is described by the real part of the complex vector W(t). Let us introduce a basis of six eigenvectors ofM(Ω) M(Ω)X k = iω k (Ω)X k , k = 1, . . . , 6(9) and expand W(t) and G(t) in this base as follows W(t) = 6 k=1 α k (t) e iω k (Ω)t X k ,(10)G = 6 k=1 γ k X k , G ⊥ = 6 k=1 γ k ⊥ X k .(11) Owing to a simple block structure ofM(Ω), the basis vectors X k can be determined by reducing effectively the problem to three dimensions. We use this method in the Appendix B to determine the resonant solution. The equation of motion (8) can be rewritten now as a set of equations for the coefficient functions α k (t) dα k (t) dt = γ k e −iω k (Ω)t + γ k ⊥ e i(Ω−ω k (Ω))t , k = 1, . . . , 6.(12) It is clear now that the mode amplitude α k (t) will grow linearly in time -the signature of a resonance -whenever either one of the two terms on the right hand side becomes time independent. This happens to the first term if one of the frequencies ω k (Ω) vanishes but the corresponding coefficient γ k does not vanish. This case is not interesting, since it means that we are just at the border of the lower instability region and the trap is not holding particles, as discussed in the next section. The second term becomes time independent when the angular velocity of trap rotation Ω satisfies the resonance condition Ω = ω k (Ω) and, of course, γ k ⊥ = 0. This resonance is different from a resonance in a standard periodically driven oscillator. In the present case the characteristic frequencies of the trap depend on the frequency Ω of the driving force. Therefore, the position of the resonance has to be determined selfconsistently. A full description of these gravity induced resonances requires the knowledge of the behavior of ω k (Ω)'s as functions of Ω. In particular, it is important to know whether a resonance occurs in a region where the system undergoes stable oscillations. This will be discussed in the next Section. III. REGIONS OF STABILITY The stability of motion for a harmonic oscillator is determined by the values of its characteristic frequencies ω -the roots of the characteristic polynomial. In the present case, these frequencies are determined by the characteristic equation for the matrixM(Ω) Det M (Ω) − iω = 0.(13) The characteristic polynomial is tri-quadratic Q(χ) = χ 3 + A χ 2 + B χ + C, χ = ω 2 ,(14) where the coefficients A, B i C can be expressed in a rotationally invariant form [16] A = −2Ω 2 − Tr{V }, B = Ω 4 +Ω 2 (3n·V ·n−Tr{V })+ Tr{V } 2 −Tr{V 2 } 2 , C = Ω 2 (Tr{V }−Ω 2 )n·V ·n−Ω 2 n·V 2 ·n−Det{V }. (15) Stable oscillations take place when all characteristic ω's are real. This means that all three roots of the polynomial Q(χ) must be real and positive. Without rotation, when Ω = 0, the three roots of Q(χ) are equal to the eigenvalues of the potential matrixV . We have then a simple system of three harmonic oscillators vibrating independently along the principal directions of the trap. As Ω increases, our system will, in general, go through two regions of instability: the lower region when one of the roots of Q(χ) is negative and the upper region when two roots are complex. We shall exhibit this behavior by plotting the zero contour lines of Q(χ) in the Ωχ-plane. We assume that the trap potential and the direction of rotation are fixed and we treat the characteristic polynomial Q(χ) as a function of Ω and χ only. Contour lines representing the zeroes of Q(χ) in the generic case are shown in Fig. 1. There is a region of Ω, where only one real root of Q(χ) exists. However, this region is bounded, so for sufficiently large Ω the system is always stable. It has been argued in Ref. [16] that there is always a region of instability when one of the roots of Q(χ) is negative. The corresponding modes grow exponentially with time. As seen in Fig. 1, this region of instability is bounded by the two values Ω 1,2 at which the curve crosses the vertical axis. These values are given by the zeroes of C, treated as a biquadratic expressions in Ω Ω 1,2 = b ± √ b 2 − 4ac 2a ,(16) where a = n·V ·n, b = Tr{V }n·V ·n − n·V 2 ·n, and c = Det{V }. Since a, b, and c are positive and b 2 ≥ 4ac, both values Ω 1 and Ω 2 are real. A degenerate case is possible, when Ω 1 = Ω 2 then the region of instability shrinks to zero. In order to determine, when this can happen, we may use the (explicitly non-negative) representation of the discriminant b 2 − 4ac given in Ref. [16]. Assuming for definitness that V x < V y < V z , we find that this happens in two cases: when the trap is not fully anisotropic (V x = V y or V y = V z ) or the axis of rotation lies in the xz-plane and its azimuthal angle satisfies the condition sin 2 θ = (1 − V x /V y )/(1 − V x /V z ). The second possibility has not been noticed in Ref. [16]. Graphical representation of the solutions for n x = 1, n y = n z = 0 is shown in Fig. 2. In this plot only one instability region is present, where one of the roots of (14) is negative. Owing to the stabilizing effect of the Coriolis force, for fast rotations the system becomes again stable. We can also see that one of the characteristic frequencies remains constant, it does not vary with Ω. It is so, because the rotation does not influence the motion in the direction of the rotation axis. This is the degenerate case described by Whittaker and thoroughly studied in connection with the BEC traps [3,4,5,10]. In this case there is only one instability region, where the square of the frequency is negative. When the direction of angular velocity is not parallel to one of the axes of the trap, a second kind of instability appears. In addition to the region of (purely exponential) instability, described before, when one root of Q(χ) was negative there is also, in general, an additional region of (oscillatory) instability where two roots are complex. Since the coefficients of the characteristic polynomial are real, the square of the second frequency is complex conjugate to the first one. In this case the instability has the form of expanding oscillations. In Fig. 3 we show how the second kind of instability develops when the rotation axis is tilted away a little bit from the z direction. We will show now, that the second kind of instability does not occur in the most common case, when the rotation vector is along one of the axes of the trap. In particular, it can never happen in a two dimensional trap. Without any loss of generality, we may assume that the direction of rotation is along the z axis. In this degenerate case the characteristic polynomial (14) factorizes because the motion in the z direction is not influenced by rotation, Q(χ) = (χ − V z ) ×(χ 2 − χ(2Ω 2 + V x + V y ) + Ω 4 − Ω 2 (V x + V y ) + V x V y ).(17) The factor quadratic in χ has real zeros if its discriminant ∆ is positive and this is, indeed, the case since ∆ = 8Ω 2 (V x + V y ) + (V x − V y ) 2 ≥ 0.(18) It follows, however, from the topology of the curves representing the zeros that the second kind of instability always exists if the direction of rotation does not coincide with one of the axes of the harmonic trap. IV. GRAVITY INDUCED RESONANCES There are, in general, two resonant values of Ω. They occur when χ = Ω 2 . In this case, from Eq. (14) we get a biquadratic equation for the resonant values of Ω: DΩ 4 + EΩ 2 + F = 0,(19) where D = −2( Tr{V } − n·V ·n ), E = Tr{V } 2 −Tr{V 2 } 2 + Tr{V }n·V ·n− n·V 2 ·n, falls into the range of the first region of stable oscillations where we would expect the confinement of particles in the trap. The upper value may fall in the lower region of stability (Fig. 4), in the lower region of instability (Fig. 5), or in the higher region of stability (Fig. 6). A degenerate case is also possible (cf. Fig. 7) when the two resonance frequencies coincide but this happens only under very special circumstances (see Appendix A). The existence of only two resonant frequencies (we would expect three resonant frequencies for a three dimensional oscillator) clearly shows that we are dealing here with a more complex dynamical system than a simple driven harmonic oscillator. F = −Det{V }.(20) The essential difference between the resonant and the nonresonant behavior is illustrated in Figs. 8-10. The role of gravity is best seen by comparing Fig. 8 (with gravity) and Fig. 11 (gravity switched off). The calculations were performed in the coordinate system in which the potential matrixV is diagonal and it was assumed that the force of gravity at t = 0 is directed along the z axis of the trap. All four figures were generated for the same trap and under the same initial conditions: the particle is initially placed in the center of the trap and it is given the initial velocity of 1 cm/s in the x direction. These simple initial conditions result in the excitation of all the modes of the oscillator. However, at resonance the mode growing linearly with t dominates the time evolution. This can be seen by comparing, for the same parameters of the trap and at the resonant frequency Ω/2π = 6.49421 Hz, the motion depicted in Fig. 8, that was calculated numerically, with the motion depicted in Fig. 12. The second plot was obtained from an analytic solution of the equations of motion (8), that contains only the resonant part. This analytic solution is derived in the Appendix B. Both plots are essentially the same except for very small differences that are due to the fact that the initial conditions assumed in Fig. 8 require some admixture of the nonresonant solutions while the analytic solution does not contain any nonresonant pieces. Gravity induced resonances are significant only if they occur in the region of stability. In all regions of instability, where the trajectories exhibit exponential growth, the resonances cannot be detected. V. CONCLUSIONS We have analyzed the stability of motion in an anisotropic, rotating harmonic trap in 3D. We have found that, in general, there are three regions of stability. The second region and the third region merge only when the rotation axis coincides with the one of the trap axes. We have demonstrated the presence of resonances in a rotating harmonic trap subjected to the force of gravity. The nature of these resonances is different from the standard forced harmonic oscillator since the resonant frequencies must be determined selfconsistently. The resonances occur at two rotation rates but they exist only when the rotation axis is tilted away from the vertical direction. However, the rotation axis can still be directed along one of the trap axes. The lower resonance always falls in the region of stable oscillations where as the higher resonance may fall in the lower region of stability, in the first region of instability, or in the higher region of stability. Resonant rotation rates depend solely on the properties of the trap and not on the masses of particles. The resonances cause the escape of a particle, or the center of mass for a collection of interacting particles, from the Hz, respectively. The angular velocity vector has the direction (1, 1, 1) and its length Ω/2π = 6.49421 Hz satisfies the resonance condition, as calculated in the Appendix A. The distances in this plot are measured in centimeters. The direction of the gravitational force (g = 9.81ms −2 ) is assumed to coincide at t = 0 with the z-axis of the trap. We can see that the amplitude of oscillations increases linearly with time. Fig. 8, except that the gravitational field has been turned off. The angular velocity Ω/2π = 6.49421 Hz has the resonant value. The scale in this plot is reduced by a factor of 100 as compared to Fig. 8. Clearly, there is no sign of any resonant behavior. trap. In view of recent experiments in which the axis of rotation was tilted away from the trap axis [14,15], it should be possible to confirm experimentally the existence of the gravity-induced resonances for Bose-Einstein condensates in rotating traps. In particular, the presence of resonances makes it possible to quickly expel all par- Q(Ω 2 ) ≡ −2Ω 4 (Tr{V } − n·V ·n) + Ω 2 Tr{V } 2 − Tr{V 2 } 2 + Tr{V } n·V ·n − n·V 2 ·n − Det{V } = 0. (A1) In the coordinate system aligned with the principal axes of the trap, the solutions of this equation have the form Ω 2 ± = (1 − n 2 x /2)V y V z + · · · ± ((1 − n 2 x /2)V y V z + . . . ) 2 − 2V x V y V z ((1 − n 2 x )V x + . . . ) 2(1 − n 2 x )V x + . . . ,(A2) where V x , V y , V z denote the diagonal elements ofV and the dots stand everywhere for two additional terms obtained by the cyclic substitutions x → y → z → x. These solutions are always real since the discriminant ∆ -the expression under the square root -is never negative. This is seen from the following representation of ∆ as a sum of two nonnegative terms (for definitness, we have assumed here that V x < V y < V z ) ∆ = (1 − n 2 x /2)V y V z − (1 + n 2 y /2)V x V z − (1 + n 2 z /2)V x V y 2 + 4V x n 2 z (V z − V x )(V y V z + V 2 y /2) + n 2 y (V y − V x )(V y V z + V 2 z /2) .(A3) This representation of ∆ can also be used to find the necessary condition for the two resonant values to merge into one. The second term vanishes (for a nondegener-ate trap) only if n x = ±1 and the first term then vanishes when the trap frequencies satisfy the condition (cf. Fig. 7. 1 2V x = 1 V y + 1 V z . (A4) The lower value Ω − of the resonant angular velocity never exceeds the lowest critical angular velocity at which the system becomes unstable. To prove this, we may compare the expression (A1) with the following expression that determines the critical value [16] Q(0) ≡ −Ω 4 n·V ·n + Ω 2 (Tr{V } n·V ·n − n·V 2 ·n) − Det{V } = 0. Both these expressions may be represented by inverted parabolas that cross the y-axis at the same negative value −Det{V }. Since the derivative of (A1) with respect to Ω 2 at the crossing point is larger than the derivative of (A5), the parabola (A1) is steeper at the crossing point. Therefore, it must cross the x-axis at a lower value of Ω 2 and the lower resonant value lies below the boundary of the stability region. APPENDIX B: ANALYTIC RESONANT SOLUTION A resonant solution is the one that has in the mode expansion (10) only the term oscillating with the resonant frequency Ω. The amplitude of these oscillations is growing linearly in time. In order to find an explicit form of this solution, we substitute into the equations of motion (8) the following Ansatz W r = (At + B)e iΩt + C,(B1) where A, B, and C are six-dimensional vectors. Upon substituting this form into the equations of motion and comparing the terms that have the same time dependence, we obtain the following equations for the three unknown vectorsˆM (Ω)·A = iΩA, (B2) M (Ω)·B = iΩB + A − G ⊥ , (B3) M (Ω)·C = −G .(B4) The first equation says that the vector A is directed along the resonant eigenmode. From the second equation we can determine the length of this vector. The vector B can be determined from the second equation only up to a component along A. This is so, because by changing the origin of the time scale, we may always add such a component to B. Finally, the vector C is determined from the third equation. It is now a matter of pure algebra to find these three vectors. Six-dimensional problems are quite cumbersome but we may replace them here by their three-dimensional counterparts, owing to a simple block structure of the matrixˆM (Ω). In addition, let us note that we need only the upper three components of the solution W r (t) to determine the trajectory. We shall call them a, b, and c, respectively. Thus the physical trajectory for the resonant solution will have the form r(t) = ℜ((at + b)e iΩt + c). Eliminating the lower three components, we obtain the following set of threedimensional equationsN (Ω)·a = 0, (B5a) N (Ω)·b = iΩb + a − h,(B5b)N (0)·c = −g ,(B5c) whereN (ω) = ω 2 −2iωΩ−Ω 2 −V and h = g ⊥ +i(n×g ⊥ ). N (ω) =   ω 2 + Ω 2 (1 − n 2 x ) − V x −Ω 2 n x n y − 2iωΩn z ) −Ω 2 n x n z + 2iωΩn y ) −Ω 2 n x n y + 2iωΩn z ) ω 2 + Ω 2 (1 − n 2 y ) − V y −Ω 2 n y n z − 2iωΩn x ) −Ω 2 n x n z − 2iωΩn y ) −Ω 2 n y n z + 2iωΩn x ) ω 2 + Ω 2 (1 − n 2 z ) − V z   .(B6) The best way to find solutions of Eqs. (B5) is to expand the vectors a and b in the basis of the eigenvectors of the matrixN (Ω) a = α 0 e 0 + α + e + + α − e − , b = β 0 e 0 + β + e + + β − e − . Upon substituting these formulas into Eq. (B5a) one finds that α + and α − vanish, β 0 is arbitrary, and the remaining three coefficients can be calculated from Eq. (B5b). Everywhere in this Appendix Ω stands for one of the two resonant frequencies given by the expression (A2). The characteristic polynomial of the matrixN (Ω) is λ 3 + λ 2 (V x + V y + V z − 5Ω 2 ) + 4λ(Ω 4 − 3Ω 2 (V x + V y + V z ) + V y V z + V x V z + V x V y ).(B8) One of the eigenvalues vanishes and the remaining two eigenvalues λ ± have the form λ ± = 5Ω 2 − V x − V y − V z 2 ± 9Ω 4 + 2(V x (1 + 2n 2 x ) + . . . ) + V 2 x − 2V y V z + . . . 2 ,(B9) where the dots have the same meaning as in the Appendix A. Knowing the eigenvalues, we may write down explicit expressions for the projectorsP 0 ,P + , andP + corresponding to the eigenvectors ofN (Ω), P 0 = (N (Ω) − λ + )(N (Ω) − λ − ) λ + λ − ,(B10a)P + =N (Ω)(N (Ω) − λ − ) λ + (λ + − λ − ) ,(B10b)P − =N (Ω)(N (Ω) − λ + ) λ − (λ − − λ + ) .(B10c) The eigenvectors e 0 and e ± may be obtained by acting with the projectors on any generic vector (not an eigenvector ofM (Ω)). We shall choose h = g ⊥ + i(n × g ⊥ ) as this vector because h appears already in Eq. (B5c). Thus, the (unnormalized) eigenvectors, needed in the expansions (B7a), can be written in the form e 0 =P 0 ·h,(B11a)e + =P + ·h,(B11b)e − =P − ·h.(B11c) Taking into account the orthogonality of the eigenvectors, we may find the coefficients α 0 and β ± that lead to the following explicit expression for the vectors a, b a = e † 0 ·e 0 2e † 0 ·(iΩ +Ω)·e 0 e 0 , (B12a) b = i(e † 0 ·e 0 ) (e † + ·Ω·e 0 ) (e † 0 ·(iΩ +Ω)·e 0 ) (e † + ·e + ) − 1 e + λ + + i(e † 0 · e 0 ) (e † − ·Ω·e 0 ) (e † 0 ·(iΩ +Ω)·e 0 ) (e † − ·e − ) − 1 e − λ − .(B12b) Finally, the vector c is found by solving the equationŝ M (0)·c = −n(n·g) and it has the following components c = (n·g) [n x (V y − Ω 2 )(V z − Ω 2 ), n y (V z − Ω 2 )(V x − Ω 2 ), n z (V x − Ω 2 )(V y − Ω 2 )] V x V y V z − Ω 2 (n 2 x V x (V y + V z ) + n 2 y V y (V z + V x ) + n 2 z V z (V x + V y )) + Ω 4 (n 2 x V x + n 2 y V y + n 2 z V z ) . (B13) [2] E. T. Whittaker, A Treatise on the Analytical Dynam- FIG. 1 :FIG. 2 :FIG. 3 : 123The contour lines in this figure represent the zeroes of the characteristic polynomial(14) for Vx = 1, Vy = 2, Vz = 3 and nx = 1/ √ 3, ny = 1/ √ 3, nz = 1/ √ 3 plotted as functions of the magnitude of angular velocity Ω and the square of the characteristic frequency χ = ω 2 . In addition to the lower region of instability (between dashed lines) where one of the roots of (14) is negative, there is also an upper region (between dotted lines) where there exists only one real root (determined by the continuation of the line that begins at χ = 3, not seen in this figure because it corresponds to a very large value of χ). Since there is no absolute scale of frequencies involved in the analysis of stability regions, we have chosen the lowest trap frequency as a unit. This plot is for the same trap as inFig. 1but for the rotation around the trap axis nx = 0, ny = 0, nz = 1. In this degenerate case there exists only one region of instability, when one of the roots of (14) is negative.The properties of the two solutions Ω 2 ± of Eq. (19) are described in the Appendix A. We may also analyze the resonance condition graphically by superposing the parabola χ = Ω 2 (shown as a thick line inFigs. 4-7)on the plots of characteristic frequencies. The lower value is more interesting because, as shown in the Appendix A, This plot shows how the degenerate case(Fig. 2)merges with the general case(Fig. 1). Here we have chosen the following values of the parameters: Vx = 1, Vy = 2, Vz = 3 and nx = sin(1/10), ny = 0, nz = cos(1/10). FIG. 4 : 4Vx = 1, Vy = 2, Vz = 3 nx = sin( 2π 5 ), ny = 0, nz = cos( 2π 5 ). Two horizontal lines enclose the lower region of instability. Both resonant frequencies lie in the lower region of stability. FIG. 5 : 5Vx = 1, Vy = 2, Vz = 3 nx = sin( π 4 ), ny = 0, nz = cos( π 4 ). Higher resonant frequency lies in the lower region of instability. FIG. 6 : 6Vx = 1, Vy = 2, Vz = 3 nx = sin( π 60 ), ny = 0, nz = cos( π 60 ). Resonant frequencies lie in two different regions of stability. FIG. 7 : 7In the degenerate case, when two resonant frequencies coincide, their common value is also equal to the lowest frequency of the trap. In this case the trap parameters are: Vx = 1, Vy = 10/3, Vz = 5, and nx = 1. FIG. 8 : 8The trajectory of a particle in the coordinate frame rotating with the trap. The characteristic frequencies of the trap in the x, y, and z directions are 10 Hz, 15 Hz, and 20 FIG. 9 : 9The trajectory of a particle under the same conditions as inFig. 8, except that the value of the angular velocity Ω/2π = 4 Hz is below the resonance. The scale in this plot is reduced by a factor of 10 as compared toFig. 8. FIG. 10 : 10The trajectory of a particle under the same conditions as inFig. 8, except that the value of the angular velocity Ω/2π = 9 Hz is above the resonance. The scale in this plot is reduced by a factor of 10 as compared toFig. 8. FIG. 11 : 11The trajectory of a particle under the same conditions as in FIG. 12 : 12The trajectory of a particle described by the analytic formulas given in the Appendix B. The trap parameters and the angular velocity are the same as inFig. 8. values of the angular velocity Ω are the ones that coincide with a characteristic frequency of the trap. Thus, the resonant values of Ω are obtained by solving the biquadratic equation(19) [ 1 ] 1W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). of Particles and Rigid Bodies. CambridgeCambridge University Press207Reprinted by Dover in 1944of Particles and Rigid Bodies, Cambridge University Press, Cambridge, 1904 (Reprinted by Dover in 1944), p. 207. . M Linn, M Niemeyer, A L Fetter, Phys. Rev. A. 6423602M. Linn, M. Niemeyer, and A. L. Fetter, Phys. Rev. A 64, 023602 (2001). . M Ö Oktel, Phys. Rev. A. 6923618M.Ö. Oktel, Phys. Rev. A 69, 023618 (2004). . D Guéry-Odelin, Phys. Rev. A. 6233607D. Guéry-Odelin, Phys. Rev. A 62, 033607 (2000). . A Recati, F Zambelli, S Stringari, Phys. Rev. Lett. 86377A. Recati, F. Zambelli and S. Stringari, Phys. Rev. Lett. 86, 377 (2001). . K W Madison, F Chevy, V Bretin, J Dalibard, Phys. Rev. Lett. 864443K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard , Phys. Rev. Lett. 86, 4443 (2001). . S Sinha, Y Castin, Phys. Rev. Lett. 87190402S. Sinha and Y. Castin, Phys. Rev. Lett. 87, 190402 (2001). . J J Garcia-Ripoll, V M Perez-Garcia, Phys. Rev. A. 6413602J. J. Garcia-Ripoll and V. M. Perez-Garcia, Phys. Rev. A 64, 013602 (2001). . P Rosenbusch, D S Petrov, S Sinha, F Chevy, V Bretin, Y Castin, G Shlyapnikov, J Dalibard, Phys. Rev. Lett. 88250403P. Rosenbusch, D. S. Petrov, S. Sinha, F. Chevy, V. Bretin, Y. Castin, G. Shlyapnikov, and J. Dalibard, Phys. Rev. Lett. 88, 250403 (2002). . M Cozzini, S Stringari, V Bretin, P Rosenbusch, J Dalibard, Phys. Rev. A. 6721602M. Cozzini, S. Stringari, V. Bretin, P. Rosenbusch, and J. Dalibard, Phys. Rev. A 67, 021602(R) (2003). . J R Abo-Shaeer, C Raman, W Ketterle, Phys. Rev. Lett. 8870409J. R. Abo-Shaeer, C. Raman, and W. Ketterle, Phys. Rev. Lett. 88, 070409 (2002). . D Guéry-Odelin, S Stringari, Phys. Rev. Lett. 834452D. Guéry-Odelin and S. Stringari, Phys. Rev. Lett. 83, 4452 (1999). . O M Maragò, S A Hopkins, J Arlt, E Hodby, G Hechenblaikner, C J Foot, Phys. Rev. Lett. 842056O. M. Maragò, S. A. Hopkins, J. Arlt, E. Hodby, G. Hechenblaikner, and C. J. Foot, Phys. Rev. Lett. 84, 2056 (2000). . N L Smith, W H Heathcote, J M Krueger, C J Foot, Phys. Rev. Lett. 9380406N. L. Smith, W. H. Heathcote,J. M. Krueger, and C. J. Foot, Phys. Rev. Lett. 93, 080406 (2004). . I Bialynicki-Birula, Z Bialynicka-Birula, Phys. Rev. A. 6563606I. Bialynicki-Birula and Z. Bialynicka-Birula, Phys. Rev. A 65, 063606 (2002). . W Kohn, Phys. Rev. 1731242W. Kohn, Phys. Rev. 173, 1242 (1961). . J F Dobson, Phys. Rev. Lett. 732244J. 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[ "A FREE-FLOATING PLANETARY MASS MEMBER OF THE TW HYA ASSOCIATION", "A FREE-FLOATING PLANETARY MASS MEMBER OF THE TW HYA ASSOCIATION" ]
[ "Adam C Schneider ", "James Windsor ", "Michael C Cushing ", "J Davy Kirkpatrick ", "Edward L Wright " ]
[]
[]
We present WISEA J114724.10−204021.3, a young, low-mass, high probability member of the TW Hya association. WISEA J114724.10−204021.3 was discovered based on its red AllWISE color (W1−W2 = 0.63 mag) and extremely red 2MASS J − K S color (> 2.64 mag), the latter of which is confirmed with near-infrared photometry from the VISTA Hemisphere Survey (J − K S = 2.57±0.03). Follow-up near-infrared spectroscopy shows a spectral type of L7 ± 1 as well as several spectroscopic indicators of youth. These include a peaked H-band shape and a steeper K-band slope, traits typically attributed to low surface gravity. The sky position, proper motion, and distance estimates of WISEA J114724.10−204021.3 are all consistent with membership in the ∼10 Myr old TW Hya association. Using the age of the TW Hya association and evolutionary models, we estimate the mass of WISEA J114724.10−204021.3 to be 5−13 M Jup , making it one of the youngest and lowest mass free-floating objects yet discovered in the Solar neighborhood.
10.3847/2041-8205/822/1/l1
[ "https://arxiv.org/pdf/1603.07985v1.pdf" ]
30,068,452
1603.07985
471993b8191cea47b95af5afcaa9b987f25eb73d
A FREE-FLOATING PLANETARY MASS MEMBER OF THE TW HYA ASSOCIATION Adam C Schneider James Windsor Michael C Cushing J Davy Kirkpatrick Edward L Wright A FREE-FLOATING PLANETARY MASS MEMBER OF THE TW HYA ASSOCIATION Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: stars: brown dwarfs We present WISEA J114724.10−204021.3, a young, low-mass, high probability member of the TW Hya association. WISEA J114724.10−204021.3 was discovered based on its red AllWISE color (W1−W2 = 0.63 mag) and extremely red 2MASS J − K S color (> 2.64 mag), the latter of which is confirmed with near-infrared photometry from the VISTA Hemisphere Survey (J − K S = 2.57±0.03). Follow-up near-infrared spectroscopy shows a spectral type of L7 ± 1 as well as several spectroscopic indicators of youth. These include a peaked H-band shape and a steeper K-band slope, traits typically attributed to low surface gravity. The sky position, proper motion, and distance estimates of WISEA J114724.10−204021.3 are all consistent with membership in the ∼10 Myr old TW Hya association. Using the age of the TW Hya association and evolutionary models, we estimate the mass of WISEA J114724.10−204021.3 to be 5−13 M Jup , making it one of the youngest and lowest mass free-floating objects yet discovered in the Solar neighborhood. INTRODUCTION Young, late-type L dwarfs (brown dwarfs with ages < 100 Myr and T eff 1600 K) show striking spectroscopic similarities to young, directly-imaged exoplanets (e.g., Gizis et al. 2012, Faherty et al. 2013. Unlike exoplanets, however, free-floating brown dwarfs are much simpler to observe because they lack a nearby, bright host sun. As such, young, low-mass brown dwarfs are ideal laboratories for investigating the physical conditions likely to be present in giant exoplanets, thus offering critical checks of theory. While young brown dwarfs belonging to nearby (d <100 pc) young associations are beginning to be found in greater numbers (e.g., Gagné et al. 2015), very few young, late-type L (>L5) dwarfs are currently known ( 10; see Figure 4 of Gagné et al. 2015). However, it is these young, late-type L dwarfs that are crucial as comparison objects for directly imaged young brown dwarf and planetary companions (e.g., Bowler et al. 2014, Gauza et al. 2015, Hinkley et al. 2015, Bonnefoy et al. 2015. If a substellar object can be connected to a young, nearby moving group, then its age can be firmly established, thereby providing a vital anchor point for lowmass evolutionary models. The young (10 ± 3 Myr - Bell et al. 2015) TW Hya association (TWA) is one of the nearest regions of recent star formation. Its proximity (∼50 pc) and young age make it an excellent testbed for studying early phases of stellar and substellar evolution. Here we report the discovery of WISEA J114724.10−204021.3, a new L7 member of the TW Hya association. 2. IDENTIFICATION OF WISEA J114724.10−204021.3 As noted in Schneider et al. (2014), young, latetype L dwarfs occupy a unique region of 2 Micron All-Sky Survey (2MASS; Skrutskie et al. 2006) and Widefield Infrared Survey Explorer (WISE; Wright et al. 2010) color space compared to field L dwarfs because of their extremely red near-infrared colors (see their Figure 5). WISEA J114724.10−204021.3 (hereafter WISEA 1147−2040) was found as part of a larger program focused on finding young, late-type L dwarfs based on their 2MASS and AllWISE 5 colors. Briefly, candidate young L dwarfs were chosen by requiring that they have a 2MASS J − K S color between 2.0 and 3.5 mag and an AllWISE W1 (3.6 µm) − W2 (4.5 µm) color between 0.3 and 0.9 mag. Candidates were also required to have their K S , W1, and W2 passband uncertainties be greater than zero (i.e., not upper limits). Note that we did not require objects to be well detected in the J-band, anticipating the existence of objects so red in J − K S color that they may be detected at the K S band, but not at J (as is the case for WISEA 1147−2040). Lastly, we required that the separation between the 2MASS and AllWISE source positions of candidates be greater than 1 , thereby ensuring each candidate shows appreciable proper motion between the 2MASS and AllWISE epochs ( 100 mas yr −1 , considering the ∼10 year time baseline between 2MASS and AllWISE). We then scrutinized each candidate individually by inspecting available optical (DSS and SDSS), near-infrared (2MASS), and midinfrared (AllWISE) images to ensure each candidate is a point source (i.e., not extended or blended) with noticeable proper motion. Of the ∼50 returned candidate red L dwarfs, WISEA 1147−2040 was picked out early as a strong young L dwarf candidate because of its uncommonly red 2MASS J − K S color (>2.64 mag) and therefore worthy of follow-up spectroscopic observations. The basic properties and photometry of WISEA 1147−2040 References. -(1) AllWISE; (2) 2MASS; (3) VHS; (4) This work; (5) BANYAN II (Malo et al. 2013, Gagné et al. 2014a); (6) Bell et al. (2015) a From model fitting (Section 4.2). b From the estimated bolometric luminosity (Section 5). are given in Table 1. OBSERVATIONS 3.1. IRTF/SpeX WISEA 1147−2040 was observed with the upgraded SpeX spectrograph ) on the night of 2016 Feb 12 UT at the NASA Infrared Telescope Facility (IRTF) on Mauna Kea. The observations were made in prism mode with a 0. 5 wide slit, which achieves a resolving power (λ/∆λ) of ∼150 over the range 0.8 -2.5 µm. We oriented the 15 long slit along the parallactic angle and obtained a series of 18 120s exposures at two different nod positions along the slit for a total exposure time of 2160s. The A0V star HD 101122 was observed at a similar airmass for telluric correction purposes. The spectrum was reduced using the SpeXtool reduction package (Cushing et al. 2004;Vacca et al. 2003). The extracted spectrum was flux calibrated using the VHS K S photometry. Figure 2. While none of the L-type standards provide a good match to the spectrum of WISEA 1147−2040, the J-band portion most closely resembles that of the L7 standard. We also compare the spectrum of WISEA 1147−2040 to the known, young, latetype L dwarfs 2MASS J03552337+1133437 (Reid et al. 2006, Faherty et al. 2013, WISEP J004701.06+680352.1 (Gizis et al. 2012), and WISE J174102.78−464225.5 (Schneider et al. 2014) in the right panel of Figure 2. WISEA 1147−2040 best matches the spectrum of WISE J174102.78−464225.5, an L7 type brown dwarf with likely membership in the β Pictoris or AB Doradus moving groups (Schneider et al. 2014). Note that Gagné et al. (2015) type WISE J174102.78−464225.5 as L5:−L7:γ. Based on these comparisons, we estimate a spectral type of L7 ± 1 (very red) for WISEA 1147−2040. Evidence of Youth WISEA 1147−2040 has several spectroscopic and photometric traits that provide strong evidence of its young age. Because young brown dwarfs are still contracting to their final radii, they have lower surface gravities than field age brown dwarfs with the same mass. One of the consequences thought to be due to low surface gravity is an unusually dusty atmosphere. Such excessively dusty Kellogg et al. 2015). While there are a few examples of L dwarfs with very red near-infrared colors that are not young (e.g., Kirkpatrick et al. 2010), bona fide young L dwarfs are typically found to have redder near-infrared colors than field dwarfs of the same spectral type (Cruz et al. 2009, Faherty et al. 2013. WISEA 1147−2040's red J − K S color is the first indication that it is young. The near-infrared spectrum of WISEA 1147−2040 has a distinctly peaked H-band appearance and a steeper Kband slope. For a field age brown dwarf with a normal surface gravity, the H and K band portions of their spectra are predominantly shaped by H 2 O and collisionallyinduced absorption (CIA) of H 2 . For a young brown dwarf, where the surface gravity is much lower, the effects of CIA are greatly reduced, resulting in the triangular H-band and steeper K-band shapes seen in their spectra (Rice et al. 2011). Allers & Liu (2013) defined the H-cont index to assess how peaked the H-band portion of a spectrum is. We measure an H-cont index value of 0.968 for WISEA 1147−2040, which is decidedly different than the field L dwarf population, instead aligning well with other low gravity objects (see Figure 23 of Allers & Liu 2013 and Figure 5 of Gagné et al. 2015). Canty et al. (2013) defined the H 2 (K) index as a measure of the K-band continuum shape and showed it could easily dis-tinguish young brown dwarfs from field brown dwarfs for late-M spectral types. Schneider et al. (2014) extended the H 2 (K) index into the L dwarf regime and showed that it could also be used to distinguish low gravity for L spectral types. We measure an H 2 (K) value for WISEA 1147−2040 of 1.035, again aligning well with other low surface gravity L dwarfs (see Figure 10 of Schneider et al. 2014 and Figure 14 of Gagné et al. 2015). Other low gravity indices (e.g., FeH z , VO z ) are only functional for spectral types L5 and are therefore unsuitable for WISEA 1147−2040 (Allers & Liu 2013). We can also find evidence of whether or not WISEA 1147−2040 has a low surface gravity by comparing its near-infrared spectrum to models with varying surface gravities. We compare the atmospheric models of Allard et al. (2012) using the method of Cushing et al. (2008). We find a best fitting temperature of 1500 ± 100 K and surface gravity of 4.0 ± 0.5, a surface gravity much lower than a typical field L dwarf. The best-fitting model is shown in Figure 3. The combination of an extremely red J − K S color, triangular H-band shape, steeper K-band slope, and a low surface gravity estimate from model fitting lead us to conclude that WISEA 1147−2040 has a low surface gravity and is therefore young. Kirkpatrick et al. (2008) show that low-gravity features only manifest in objects with ages less than that of the Pleiades, so we take ∼100 Myr as the upper age limit for WISEA 1147−2040. Membership in the TW Hya Association Because WISEA 1147−2040 shows youthful characteristics, we can now evaluate whether or not it belongs to one of the young nearby associations. We first use the BANYAN II (Malo et al. 2013, Gagné et al. 2014a) moving group membership evaluation tool, which takes the position and proper motion of a source and, through the use of a naive Bayesian classifier analysis, assesses membership probabilities for several nearby, young moving groups. According to BANYAN II, WISEA 1147−2040 has an 84.32% chance of belonging to TWA, under the assumption that it is young. We can first evaluate the feasibility of TWA membership for WISEA 1147−2040 by inspecting its sky position relative to other TWA members, as this particular association is confined to a particular area of the sky (compared to most other young moving groups). The top panel in Figure 4 shows the position and proper motion vector of WISEA 1147−2040 along with confirmed TWA members from Schneider et al. (2012a), Schneider et al. (2012b, and Murphy et al. (2015), as well as high probability (>50%) candidate TWA members from Gagné et al. (2015) and the recently announced L7 candidate TWA member 2MASS J11193254−1137466 from Kellogg et al. (2015). The figure shows that WISEA 1147−2040 is in close proximity to the other members of TWA, and is thus viable as a TWA candidate member. BANYAN II also provides predicted radial velocity and distance values, assuming WISEA 1147−2040 is a TWA member, of 9.61 km s −1 and 31.3 ± 3.8 pc, respectively. While a higher resolution spectrum will be required to measure the radial velocity of WISEA 1147−2040, we can compare photometric and kinematic distance estimates to those predicated by BANYAN II to see if they are in agreement. We estimate the distance to WISEA 1147−2040 in two ways. First, we estimate its distance photometrically. Note that because WISEA 1147−2040 has such red near-infrared colors, the absolute magnitude-spectral type relations for field brown dwarfs cannot be used for WISEA 1147−2040 with all available photometric bands. However, Faherty et al. (2013) show that the young, very red L5γ brown dwarf 2MASS J035523.37+113343.7 and the field L5 dwarf 2MASS J1507476−162738 have very similar absolute flux values at the K-band, while 2MASS J035523.37+113343.7 emits less flux than 2MASS J1507476−162738 at wavelengths shorter than K, and emits more flux at wavelengths redward of K. We investigated whether or not this trait applies to other young, very red L7 dwarfs with measured parallaxes, specifically PSO J318.5338−22.8603 ) and WISEP J004701.06+680352.1 (Gizis et al. 2012). Using the absolute magnitude-spectral type relations of Dupuy & Liu (2012), we calculate a K MKO photometric distance for PSO J318.5338−22.8603 and a spectral type of L7 of 25.1 ± 0.2 pc, which agrees quite well with the measured parallax distance of 24.6 ± 1.4 pc , especially when compared to the distances found using the J MKO (39.5 ± 0.7 pc) and W2 (17.9 ± 0.2 pc) band relations. Similarly, for WISEP J004701.06+680352.1 and a spectral type of L7, we calculate a K S photometric distance of 13.5 ± 0.2 pc, agreeing well with the measured distance of 12.2 pc (Gizis et al. 2015). Again, the K S photometric distance is much more accurate than the J (18.5 ± 0.6 pc) and W2 (10.3 ± 0.1 pc) band photometric distances. Therefore, we use the K S absolute magnitudespectral type relation of Dupuy & Liu (2012) to estimate a photometric distance to WISEA 1147−2040 of 31.2 ± 1.5 pc. We also estimate a kinematic distance to WISEA 1147−2040 following the "moving cluster" or "convergent point" method outlined in Mamajek (2005). This method works because the proper motions of comoving stars appear to converge to a single point on the celestial sphere from Earth's frame of reference. Mamajek (2005) provides coordinates of the convergent point for TWA of (α cp = 103.2 ± 1.5, δ cp = -30.7 ± 1.5), while Ducourant et al. (2014) find (α cp = 102.4 ± 1.4, δ cp = -27.3 ± 0.6). For this analysis, we evaluate the kinematic distance to WISEA 1147−2040 using both sets of convergent point coordinates. We use mean U V W space velocities for TWA members of (U, V, W ) = (-11.12 ± 0.90, -18.88 ± 1.56, -5.63 ± 2.78) km s −1 from Gagné et al. (2014a). For WISEA 1147−2040, we find kinematic distances of 32.6 pc and 32.0 pc for the Mamajek (2005) and Ducourant et al. (2014) convergent points, respectively. Radial velocities can also be estimated from the convergent point method, for which we find values of 9.2 and 8.7 km s −1 for the Mamajek (2005) and Ducourant et al. (2014) convergent points, respectively. These radial velocity values agree very well with those estimated from BANYAN II. Both the photometric distance from the K S magnitude and the kinematic distance estimates of WISEA 1147−2040 agree remarkably well with each other, as well as the predicted distance from BANYAN II. If we use either the photometric or kinematic distance to WISEA 1147−2040 as an additional input into BANYAN II, we find TWA membership probabilities of ∼96%. Using the BANYAN II predicted distance, we compare the galactic XY Z coordinates of WISEA 1147−2040 with bona fide and high probability candidate members of TWA in the bottom panel of Figure 4. Measured parallaxes for TWA members come from Weinberger et al. (2013) and Ducourant et al. (2014), where distances with the smallest uncertainties were chosen for objects found in both studies. We use the kinematic distances to TWA 3A, TWA 6, TWA 30A, and TWA 30B from Ducourant et al. (2014) and the kinematic distances to TWA 33 and TWA 34 from Schneider et al. (2012b). We use the quoted BANYAN distances to TWA 35 and TWA 36, as well as the high probability candidates from Table 4 of Gagné et al. (2015) from Murphy et al. (2015) and Gagné et al. (2015), respectively. We also include the L7 TWA candidate member 2MASS J11193254−1137466 from Kellogg et al. (2015), using their distance estimate of ∼25 pc. The figure shows that WISEA 1147−2040 has XY Z positions consistent with other TWA members. Based on WISEA 1147−2040's young age, sky position, and the excellent agreement between its photometric distance estimate, kinematic distance estimate, and its predicted distance from BANYAN II, we conclude that WISEA 1147−2040 is a member of the TW Hya association. DISCUSSION Based on WISEA 1147−2040's T eff and log g estimates from its spectrum, and an age of 10 ± 3 Myr for TWA (Bell et al. 2015), we estimate a mass of 6−13 M Jup and 9−11 M Jup from the COND evolutionary models (Baraffe et al. 2003) and the (f sed = 2) models, respectively. Note that Liu et al. (2013) conclude that physical properties for PSO J318.5338−22.8603 (and other young, red L dwarfs) derived from near-infrared spectra are unreliable, instead using a combination of near-infrared spectra, photometry, and a distance measurement to determine a bolometric luminosity. Mass, effective temperature, and surface gravity were then determined using the measured bolometric luminosity and moving group age. While a parallax measurement for WISEA 1147−2040 is yet unavailable, we can use the distance estimate from BANYAN II (31.3 ± 3.8 pc), combined with our flux calibrated near-infrared SpeX spectrum and WISE W1 and W2 magnitudes to estimate a preliminary bolometric luminosity. Any wavelengths not covered by the SpeX near-infrared spectrum or AllWISE photometry we fill in with the best fitting model from Figure 2. We Monte Carlo the uncertainties for the flux calibrated spectrum, AllWISE photometry, and distance estimate to determine the uncertainly of the calculated luminosity. We find log(L bol /L ) = -4.42 ± 0.11, which corresponds to T eff values of ∼1100−1200 K at an age of 10 ± 3 Myr , much lower than the T eff of WISEA 1147−2040 found via model fitting. This luminosity corresponds to a mass estimate of 5−6 M Jup . The differences between the parameters derived via fitting models to the near-infrared spectrum and those determined from the bolometric luminosity are similar to the differences seen for PSO J318.5338−22.860 in . Both the mass estimates from model fitting and from the bolometric luminosity make WISEA 1147−2040 the lowest mass free floating confirmed member of the TW Hya association and one of the lowest mass brown dwarfs in the Solar neighborhood. In the TW Hya association, only the planetary mass companion 2M1207b (Chauvin et al. 2004(Chauvin et al. , 2005) has a lower mass. As such, WISEA 1147−2040 provides an exceptional laboratory for investigating the chemistry and cloud structure in young, planetary mass objects. A higher resolution spectrum of WISEA 1147−2040 would provide both a radial velocity, helping to secure TWA membership, as well as additional gravity sensitive diagnostics (e.g., K I equivalent widths). A trigonometric parallax for WISEA 1147−2040 would also further confirm it's membership in TWA. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This publication made use of observations obtained as part of the VISTA Hemisphere Survey, ESO Progam, 179.A-2010 (PI: McMahon). 3. 2 .Figure 1 . 21VISTA/VHS WISEA 1147−2040 was observed by the Visible and Infrared Survey Telescope for Astronomy (VISTA; Emer-Three color composite VHS image centered on the position of WISEA 1147−2040 (Y = blue, J = green, K S = red). son et al. 2004) in the Y , J, and K S photometric bands (Figure 1) as part of the VISTA Hemisphere Survey (VHS; PI McMahon, Cambridge, UK). WISEA 1147−2040 was well detected in all three bands and the photometry from VHS is listed inTable 1. These observations confirm the very red J − K S color of WISEA 1147−2040 (J − K S = 2.57 ± 0.03 mag). Spectral Typing We determined a spectral type for WISEA 1147−2040 following the method outlined in the Appendix of Schneider et al. (2014). A comparison of the SpeX spectrum of WISEA 1147−2040 with several near-infrared spectral standards (Kirkpatrick et al. 2010) from the Spex Prism Spectral Library (SPL, Burgasser 2014) is shown in the left panel of Figure 2 . 2Left: The IRTF/SpeX spectrum of WISEA 1147−2040 (black) compared with the near-infrared L6 (2MASSI J1010148−040649; Reid et al. 2006), L7 (2MASSI J0103320+193536; Cruz et al. 2004), and L8 (2MASSW J1632291+190441; Burgasser 2007) standards (red). Right: The IRTF/SpeX spectrum of WISEA 1147−2040 (black) compared with 2MASS J03552337+1133437 (L5γ; Faherty et al. 2013), WISEP J004701.06+680352.1 (L7 INT-G; Gizis et al. 2015), and WISE J174102.78−464225.5 (L7 (very red); Schneider et al. 2014). Each spectrum is normalized by the mean flux from 1.27 to 1.32 µm. atmospheres then give rise to very red near-infrared colors. WISEA 1147−2040 has an extremely red J − K S color (2.57 ± 0.03 mag) determined from its VHS photometry. In fact, only three free floating L dwarfs are known to have redder J − K S colors; the 20 Myr old β Pictoris moving group member PSO J318.5338−22.8603 (J − K S = 2.837 mag; Liu et al. 2013), the extremely dusty L7 dwarf ULAS J222711−004547 (J − K S = 3.04 mag; Marocco et al. 2014), and the recently discovered TWA candidate member 2MASS J11193254−1137466 (J −K S = 2.62 mag; Figure 3 . 3The IRTF/SpeX spectrum of WISEA 1147−2040 (black) compared with the best fitting BT-Settl model (red; T eff = 1500 K, log g = 4.0). Table 1 1WISEA J114724.10−204021.3 PropertiesParameter Value Ref. Identifiers AllWISE J114724.10-204021.3 1 2MASS 11472421−2040204 2 Observed Properties α (J2000) 11:47:24.10 1 δ (J2000) −20:40:21.3 1 µα −122.1±12.0 mas yr −1 4 µ δ −74.5±11.3 mas yr −1 4 J (2MASS) >17.511 mag 2 H (2MASS) 15.764 ± 0.112 mag 2 K S (2MASS) 14.872 ± 0.106 mag 2 Y (VHS) 19.160 ± 0.067 mag 3 J (VHS) 17.445 ± 0.029 mag 3 K S (VHS) 14.872 ± 0.011 mag 3 W1 13.718 ± 0.026 mag 1 W2 13.090 ± 0.030 mag 1 W3 >12.155 mag 1 W4 >8.913 mag 1 J − K S (2MASS) >2.64 mag 2 J − K S (VHS) 2.57 ± 0.03 mag 3 J − K S (synthetic) 2.61 ± 0.03 mag 4 W 1 − W 2 0.63 ± 0.04 mag 1 Inferred Properties Spectral Type (NIR) L7 ± 1 (red) 4 Photometric Distance 31.2 ± 1.5 pc 4 Kinematic Distance 32−33 pc 4 Predicted Distance 31.3 ± 3.8 pc 4,5 T eff a 1500 ± 100 K 4 log g a 4.0 ± 0.5 4 Mass a 6−13 M Jup 4 T eff b 1100−1200 K 4 Mass b 5−6 M Jup 4 Age 10 ± 3 Myr 6 http://http://wise2.ipac.caltech.edu/docs/release/allwise/expsup/ arXiv:1603.07985v1 [astro-ph.SR] 25 Mar 2016 . 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[]
[ "A fitted finite volume method for stochastic optimal control Problems", "A fitted finite volume method for stochastic optimal control Problems" ]
[ "Christelle Dleuna Nyoumbi es:[email protected] \nInstitut de Mathématiques et de Sciences Physiques (IMSP)\nUniversité d'Abomey-Calavi\n01B.P. 613Porto-NovoBenin\n", "Antoine Tambue \nDepartment of Computer science\nElectrical engineering and Mathematical sciences\nWestern Norway University of Applied Sciences\nInndalsveien 285063Bergen\n\nCenter for Research in Computational and Applied Mechanics (CERECAM)\nDepartment of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7701RondeboschSouth Africa\n\nThe African Institute for Mathematical Sciences(AIMS)\n6-8 Melrose Road7945MuizenbergSouth Africa\n" ]
[ "Institut de Mathématiques et de Sciences Physiques (IMSP)\nUniversité d'Abomey-Calavi\n01B.P. 613Porto-NovoBenin", "Department of Computer science\nElectrical engineering and Mathematical sciences\nWestern Norway University of Applied Sciences\nInndalsveien 285063Bergen", "Center for Research in Computational and Applied Mechanics (CERECAM)\nDepartment of Mathematics and Applied Mathematics\nUniversity of Cape Town\n7701RondeboschSouth Africa", "The African Institute for Mathematical Sciences(AIMS)\n6-8 Melrose Road7945MuizenbergSouth Africa" ]
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In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems.The computational challenge is due to the nature of the HJB equation, which may be a secondorder degenerated partial differential equation coupled with optimization. In the work, we discretize the HJB equation using the fitted finite volume method and show that matrix resulting from spatial discretization is an M-matrix. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.
10.3934/math.2021186
null
211,204,928
2002.08464
f7302b78d94ca42408999f0dac7821b704ebab55
A fitted finite volume method for stochastic optimal control Problems 19 Feb 2020 Christelle Dleuna Nyoumbi es:[email protected] Institut de Mathématiques et de Sciences Physiques (IMSP) Université d'Abomey-Calavi 01B.P. 613Porto-NovoBenin Antoine Tambue Department of Computer science Electrical engineering and Mathematical sciences Western Norway University of Applied Sciences Inndalsveien 285063Bergen Center for Research in Computational and Applied Mechanics (CERECAM) Department of Mathematics and Applied Mathematics University of Cape Town 7701RondeboschSouth Africa The African Institute for Mathematical Sciences(AIMS) 6-8 Melrose Road7945MuizenbergSouth Africa A fitted finite volume method for stochastic optimal control Problems 19 Feb 2020Preprint submitted to Mathematics and Computers in Simulation February 21, 2020(Christelle Dleuna Nyoumbi), [email protected] (Antoine Tambue)Stochastic Optimal ControlHJB Equationsfinite volume methodfinite difference method * Corresponding author In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems.The computational challenge is due to the nature of the HJB equation, which may be a secondorder degenerated partial differential equation coupled with optimization. In the work, we discretize the HJB equation using the fitted finite volume method and show that matrix resulting from spatial discretization is an M-matrix. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method. Introduction We consider the numerical approximation of the following controlled Stochastic Differential Equation (SDE) defined in R n (n ≥ 1) by dx t = b(t, x t , α t )dt + σ(t, x t , α t )dω t , x(0) = x 0 (1) where b : [0, T ] × R n × A → R n (t, x t , α t )) → b(t, x t , α t )(2) is the drift term and σ : [0, T ] × R n × A → R n×d (t, x t , α t )) → σ(t, x t , α t )(3) the d-dimensional diffusion coefficients. Note that ω t are d-dimensional independent Brownian motion on (Ω, F, (F t ) t≥0 , P), α = (α t ) t≥0 is an F-adapted process, valued in A closed convex subset of R m (m ≥ 1) and satisfying some integrability conditions and/or state constraints. Precise assumptions on b and σ to ensure the existence of the unique solution x t of (1) can be found in [8]. Given a function g from R n into R and f from [0, T ] × R n × A into R, the value function is defined by v(t, x) = sup α∈A E T t f (s, x, α) ds + g(x T ) , x ∈ R n ,(4) and the resulting Hamilton Jacobi-Bellamn (HJB) equation (see [10])is given by        v t (t, x) + sup α∈A [L α v(t, x) + f (t, x, α)] = 0 on [0, T ) × R n v(T, x) = g(x), x ∈ R n(5) where L α v(t, x) = n i=1 (b(t, x, α)) i ∂v(t, x) ∂x i + n i,j=1 (a(t, x, α)) i,j ∂ 2 v(t, x) ∂x i ∂x j , and a(t, x, α) = ( 1 2 (σ(t, x, α))(σ(t, x, α)) T ) i,j . The existence and uniqueness of the viscosity solution of the HJB equation (5) is well known and can be found in [8]. Equation (5) is a initial value problem. There are two unknown functions in this equation, the value function v and the optimal control α. In most practical situations, (5) is not analytically solvable therefore numerical approximations are the only tools appropriate to provide reasonable approximations. Numerical approximation of HJB-equation of type (5) is therefore an active research area and has attracted a lot of attentions [20,18,11,10,15,14,16,13,9]. While solving numerically HJB equation, the keys challenge are the low regularity of the solution of HJB equation and the lack of appropriate numerical methods to tackle the degeneracy of the differential operator in HJB equation. Indeed adding to the standard issue that we usually have when solving degenerated PDE, we need to couple with an optimization problem at every grid point and every time step. In terms of existing numerical methods, there are two basic threads of literature concerning controlled HJB equations. A standard approach is based on Markov chain approximation. In financial terms, this approach is equivalent to an explicit finite difference method. However, these methods are well-known to suffer from time step limitations due to stability issues [6]. A more recent approach is based on numerical methods such as finite difference method which ensure convergence to the viscosity solution of the HJB equation couple with an optimization problem at each time [12] . For many stochastic optimal control problems such as Merton's control problem, the linear operator is degenerated when the spatial variables approach the region near to zero. This degeneracy has an adverse impact on the accuracy when the finite difference method is used to solve the PDE (see [7], chapter 26). This degeneracy also has an adverse impact on the accuracy of our stochastic optimal control problems since its numerical resolution implies the resolution of PDE, coupled with optimization problem. In this article, we propose a numerical scheme based on a finite volume method suitable to handle the degeneracy of the linear operator while solving numerically the HJB equation in dimension 1 and 2. The method is coupled with implicit time-stepping method for temporal discretization method and the iterative method presented in [9] for optimization problem at every time step. More precisely, this method is based on fitted finite volume technique proposed in [1] to solve the degenerated Black Sholes equations. Note that to the best of our knowledge, such method has not been used to solve the stochastic optimal control problem (5). The merit of the method is that it is absolutely stable in time because of the implicit nature of the time discretisation and the corresponding matrix after spatial discretization is a positive-definite M -matrix. Numerical simulations prove that our proposed method is more accurate that the standard method based on finite difference spatial discretization. The rest of this article is organized as follows. In section 2, we present the finite volume method with the fitting technique for dimension 1 and 2. We will also show that the system matrix of the resulting discrete equations is an M -matrix. In section 3, we will present the temporal discretization and optimization problem in dimension 1 and 2. Numerical experiments using Matlab software will be performed in section 4 to demonstrate the accuracy of the proposed numerical method. We conclude the work at section 5 by summarizing our finding. Spatial discretization As we already know, the resolution of the HJB equation (5) involves a spatial discretisation, a temporal discretisation and an optimisation problem at every grid point and each time step. The goal of this section is to provide the spatial discretization of the HJB equation (5) solving our stochastic optimal control problem (4). Details in this section can be found in [4], where such methods have been used to solve the degenerated Black Sholes equation for option pricing with constant coefficients. Spatial discretization based on fitted finite volume method in dimension 1 Consider the more generalized HJB equation (5) in dimension 1 (n = 1) which can be written in the form. ∂v(x, t) ∂t + sup α∈A ∂ ∂x a(x, t, α) x 2 ∂v(x, t) ∂x + b(x, t, α) x v(x, t) + c(x, t, α) v(x, t) = 0,(7) where a(t, x, α) > 0, α = α(x, t) and bounded. As usual, we truncate the problem in the finite interval I = [0, x max ]. Let the interval I = [0, x max ] be divided into N 1 sub-intervals I i := (x i , x i+1 ), i = 0 · · · N 1 − 1 with 0 = x 0 < x 1 < · · · · · · < x N 1 = x max . We also set x i+1/2 = x i + x i+1 2 and x i−1/2 = x i−1 + x i 2 for each i = 1 · · · N 1 − 1 . If we define x −1/2 = x 0 and x N 1 +1/2 = x max integrating both size of (7) over J i = x i−1/2 , x i+1/2 and taking α i = α(x i , t), we have x i+1/2 x i−1/2 ∂v ∂t dx + x i+1/2 x i−1/2 sup α i ∈A ∂ ∂x x a(x, t, α i ) x ∂v ∂x + b(x, t, α i ) v + c(x, t, α i ) v dx = 0(8) Applying the mid-points quadrature rule to the first and the last point terms, we obtain the above dv dt l i + sup α i ∈A x i+1/2 ρ(v) x i+1/2 − x i−1/2 ρ(v) x i−1/2 + c(x i , t, α i ) v i l i = 0,(9)for i = 1, 2, · · · N 1 − 1, where l i = x i+1/2 − x i−1/2 is the length of J i . v i denotes the nodal approxi- mation to v(τ, x i ) and ρ(v) is the flux associated with v defined by ρ(v) := a(x, t, α i ) x ∂v ∂x + b(x, t, α i ) v.(10) Clearly, we now need to derive approximation of the flux defined above at the mid-point x i+1/2 , of the interval I i for i = 2, · · · N 1 −1. This discussion is divided into two cases for i ≥ 1 and I 0 = (0, x 1 ). Case I: Approximation of ρ at x i+1/2 for i ≥ 2. The term a(x, t, α i ) x ∂v ∂x + b(x, t, α i ) v is approximated by solving the boundary value problem a(x, t, α i ) x ∂v ∂x + b(x i+1/2 , t, α i ), v, = 0, x ∈ I i (11) v(x i ) = v i (t), v(x i+1 ) = v i+1 (t).(12) Integrating (11) yields the first-order linear equations ρ i (v)(t) = a(x, t, α i ) x ∂v ∂x + b(x i+1/2 , t, α i ) v = C 1(13) where C 1 denotes an additive constant. As in [4], the solution is given by v(t) = C 1 b(x i+1/2 , t, α i ) + C 2 x − b(x i+1/2 , t, α i ) a(x i+1/2 , t, α i ) .(14) Note that in this deduction we have assumed that b(x i+1/2 , t, α i ) = 0. By setting β i (t) = b(x i+1/2 , t, α i ) a(x i+1/2 , t, α i ) , using the boundary conditions in (11) yields v i (t) = C 1 b(x i+1/2 , t, α i ) + C 2 x −β i (t) i and v i+1 (t) = C 1 b(x i+1/2 , t, α i ) + C 2 x −β i (t) i+1(15) Solving the following linear system with respect to C 1 and C 2 yields        v i (t) = C 1 b(x i+1/2 , t, α i ) + C 2 x −β i (t) i v i+1 (t) = C 1 b(x i+1/2 , t, α i ) + C 2 x −β i (t) i+1(16) yields ρ i (v)(t) = C 1 = b(x i+1/2 , t, α i ) x β i (t) i+1 v i+1 (t) − x β i (t) i v i (t) x β i (t) i+1 − x β i (t) i (17) ρ i (v)(t) provides an approximation to the ρ(v)(t) at x i+1/2 . Case II: This is the degenerated zone. The aims here is to approximate ρ at x 1/2 in the sub-interval I 0 . In this case, the following problem is considered (a(x 1/2 , t, α 1 ) x ∂v ∂x + b(x 1/2 , t, α 1 ) v) = C 2 in [0, x 1 ] (18) v(0) = v 0 (t), v(x 1 ) = v 1 (t) where C 2 is an unknown constant to be determined. Following [4], integrating (18) yields ρ 0 (v)| 1/2 (t) = a(x 1/2 , t, α i ) x 1/2 ∂v ∂x + b(x 1/2 , t, α 1 ) v = b(x 1/2 , t, α 1 ) v 0 (t) + C 2 x 1/2 .(19) Since x 1/2 = x 1 + x 0 2 with x 0 = 0, we have C 2 x 1 = (a(x 1/2 , t, α 1 ) + b(x 1/2 , t, α 1 ))(v 1 (t) − v 0 (t)). Therefore we have ρ 0 (v)| 1/2 (t) = 1 2 (a(x 1/2 , t, α 1 ) + b(x 1/2 , t, α 1 ))v 1 (t) − (a(x 1/2 , t, α 1 ) − b(x 1/2 , t, α 1 ))v 0 (t) .(20) By replacing ρ by its approximated value, (9) becomes for i = 0, 1, · · · , N 1 − 1 dv i (t) dt + sup α i ∈A 1 l i   x i+1/2 b(x i+1/2 , t, α i ) x β i (t) i+1 v i+1 (t) − x β i (t) i v i (t) x β i (t) i+1 − x β i (t) i(21)−x i−1/2 b(x i−1/2 , t, α i ) x β i−1 (t) i v i (t) − x β i−1 (t) i−1 v i−1 (t) x β i−1 (t) i − x β i−1 (t) i−1 + c i (t, α i ) v i (t) l i   = 0 By setting τ = T − t and including the boundary conditions, we have the following system of Ordinary Differential Equation (ODE) coupled with optimisation problem.        −v τ (τ ) + sup α∈A N 1 −1 [A(α, τ ) v(τ ) + G(α, τ )] = 0 v(0) given,(22) which can be rewritten as      v τ (τ ) + inf α∈A N 1 −1 [E(α, τ ) v(τ ) + F (α, τ )] = 0 v(0) given,(23) where v(τ ) = (v 1 (τ ), · · · , v N 1 −1 (τ )) and F (α, τ ) = (F 1 (α 1 , τ ), · · · , F N 1 −1 (α N 1 −1 , τ )) includes all Dirichlet boundary and final conditions, (61) is an M-matrix for any α ∈ A. A(α, τ ) = −E(α, τ ) and G(α, τ ) = −F (α, τ ) are defined as for i = 1, · · · , N 1 − 1 E i,i+1 (α i , τ ) = −x i+1/2 b i+1/2 (τ, α i ) x β i (τ ) i+1 l i (x β i (τ ) i+1 − x β i (τ ) i ) ,(24)E i,i (α i , τ ) = x i+1/2 b i+1/2 (τ, α i ) x β i (τ ) i l i (x β i (τ ) i+1 − x β i (τ ) i ) + x i−1/2 b i−1/2 (τ, α i ) x β i−1 (τ ) i l i (x β i−1 (τ ) i − x β i−1 (τ ) i−1 ) − c i (τ, α i ) ,(25)E i,i−1 (α i , τ ) = −x i−1/2 b i−1/2 (τ, α i ) x β i−1 (τ ) i−1 l i (x β i−1 (τ ) i − x β i−1 (τ ) i−1 ) ,(26)E 1,1 (α 1 , τ ) = x 1+1/2 b 1+1/2 (τ, α 1 ) x β 1 (τ ) 1 l 1 (x β 1 (τ ) 2 − x β 1 (τ ) 1 ) + 1 4 l 1 x 1 (a 1/2 (τ, α 1 ) + b 1/2(τ,α 1 ) ) − c 1 (τ, α 1 ) (27) E 1,2 (α 1 , τ ) = −x 1+1/2 b 1+1/2 (τ, α 1 ) x β 2 l 1 (x β 1 (τ ) 2 − x β 1 (τ ) 1 ) (28) G(α, τ ) =               − 1 4 l 1 x 1 (a 1/2 (τ, α 1 ) − b 1/2 (τ, α 1 )) v 0 0 . . . 0 −x N 1 −1/2 b N 1 −1/2 (τ, α N 1 −1 ) x β i (τ ) N 1 l N 1 −1 (x β N 1 −1 (τ ) N 1 − x β N 1 −1 (τ ) N 1 −1 ) v N 1               . Theorem 2.1. Assume that c i (τ, α) < 0, i = 1, · · · , N 1 − 1, let h = max 1≤i≤N 1 l i . If h is relatively small then the matrix E(α, τ ) in the system Proof. Let us show that E(α, τ ) has positive diagonal, non-positive off diagonal, and is diagonally dominant. We first note that b i+1/2 (τ, α) x β i (τ ) i+1 − x β i (τ ) i = a i+1/2 (τ, α) β i (τ ) x β i (τ ) i+1 − x β i (τ ) i > 0,(29)for i = 1, · · · , N 1 − 1, and all b i+1/2 (τ, α) = 0, b i−1/2 (τ, α) = 0, with a i+1/2 (τ, α) > 0 and a i−1/2 (τ, α) > 0. This also holds when b i+1/2 (τ, α) → 0 and b i−1/2 (τ, α) → 0, that is lim b i+1/2 (τ,α)→0 b i+1/2 (τ, α) x β i (τ ) i+1 − x β i (τ ) i = b i+1/2 (τ, α) e β i (τ ) ln(x i+1 ) − e β i (τ ) ln(x i ) = b i+1/2 (τ, α) β i (τ ) ln(x i+1 ) − β i (τ ) ln(x i ) (30) = a i+1/2 (τ, α) ln x i+1 x i −1 > 0, lim b i−1/2 (τ,α)→0 b i−1/2 (τ ) x β i−1 (τ ) i − x β i−1 (τ ) i−1 = b i−1/2 (τ, α) e β i−1 (τ ) ln(x i ) − e β i−1 (τ ) ln(x i−1 ) = b i−1/2 (τ, α) β i−1 (τ ) ln(x i ) − β i−1 (τ ) ln(x i−1 ) = a i−1/2 (τ, α) ln x i x i−1 −1 > 0 . Using the definition of E(α, τ ) given above, we see that E i,i 0, E i,i+1 0, E i,i−1 0 i = 2, · · · , N 1 − 1, |E i,i | ≥ |E i,i−1 | + |E i,i+1 | because x β i (τ ) i+1 ≈ x β i (τ ) i + x β i (τ )−1 i β i (τ ) h, x β i−1 (τ ) i−1 ≈ x β i−1 (τ ) i − x β i−1 (τ )−1 i β i−1 (τ ) h and |E i,i | − |E i,i+1 | − |E i,i−1 | = − b i+1/2 (τ ) x β i (τ ) i+1 − x β i (τ ) i >0 h β i β i x β i −1 i →0 →0 + b i−1/2 (τ, α) x β i−1 (τ ) i − x β i−1 (τ ) i−1 >0 h β i−1 β i−1 x β i−1 −1 i →0 →0 − c i (τ, α). Note that for i = 1, we have E 1,1 ≥ 0 if a 1/2 (τ, α) + b 1/2 (τ, α), are nonnegative and c 1 (τ, α) < 0. So E(α, τ ) is diagonally dominant and is therefore an M-matrix. Spatial discretization based on fitted finite volume method in dimension 2 Here we consider the following two dimensional problem ∂v(x, y, t) ∂t + sup α∈A [∇ · (k(x, y, t, α)) + c(x, y, t, α) v(x, y, t)] = 0,(31) where k(x, y, t, α) = A(x, y, t, α) · ∇v(x, y, t) + b v(x, y, t) is the flux, b = (x b 1 (x, y, t, α), y b 2 (x, y, t, α)) T and A =   a 11 a 12 a 21 a 22   , We will assume that a 21 = a 12 . We also also define the following coefficients, which will help us to build our scheme smoothly a 11 (x, y, t, α) = a(x, y, t, α) x 2 , a 22 (x, y, t, α) = a(x, y, t, α)y 2 and a 12 = a 21 = d 1 (x, y, t, α)xy. As usual the two dimensional domain is truncated to I x × I y , where I x = [0, x max ] and I y = [0, y max ] be divided into N 1 and N 2 sub-intervals: I x i := (x i , x i+1 ), I y j := (y j , y j+1 ), i = 0, · · · , N 1 − 1, j = 0, · · · , N 2 − 1 with 0 = x 0 < x 1 < · · · · · · < x N 1 = x max and 0 = y 0 < y 1 < · · · · · · < y N 2 = y max . This defines a mesh on I x × I y with all the mesh lines perpendicular to one of the axes. We also set x i+1/2 = x i + x i+1 2 , x i−1/2 = x i−1 + x i 2 , y j+1/2 = y j + y j+1 2 , y j−1/2 = y j−1 + y j 2 , for each i = 1, · · · , N 1 − 1 and each j = 1, · · · , N 2 − 1. We denote N = (N 1 − 1) × (N 2 − 1). These mid-points form a second partition of I x × I y if we define x −1/2 = x 0 , x N 1 +1/2 = x max , y −1/2 = y 0 and y N 2 +1/2 = y max . For each i = 0, 1, · · · , N 1 and j = 0, 1, · · · , N 2 , we set h x i = x i+1/2 − x i−1/2 and h y j = y j+1/2 − y j−1/2 . We now discuss the finite volume method for (31). Integrating both size of (31) over R i,j = x i−1/2 , x i+1/2 × y j−1/2 , y j+1/2 , we have x i+1/2 x i−1/2 y j+1/2 y j−1/2 ∂v ∂t dx dy + x i+1/2 x i−1/2 y j+1/2 y j−1/2 sup α∈A [∇ · (k(v(x, y, t, α))) + c(x, y, t, α) v(x, y, t)] dx dy = 0,(32) for i = 1, 2, · · · , N 1 − 1, j = 1, 2, · · · , N 2 − 1. Applying the mid-points quadrature rule to the first and the last point terms, we obtain the above dv i,j (t) dt l i,j + sup α i,j ∈A ∂R i,j ∇ · (k(v(x, y, t, α i,j )) dx dy + c i,j (t, α i,j ) v i,j (t) l i,j = 0 (33) for i = 1, 2, · · · N 1 − 1, j = 1, 2, · · · N 2 − 1 where l i,j = x i+1/2 − x i−1/2 × y j+1/2 − y j−1/2 is the length of R i,j , and v i,j (t) denotes the nodal approximation to v(x i , y j , t). We now consider the approximation of the middle term in (33). Let n denote the unit vector outward-normal to ∂R i,j . By Ostrogradski Theorem, integrating by parts and using the definition of flux k, we have R i,j ∇ · (k(v)) = ∂R i,j k(v(x, y, t, α i,j )) · n ds = (xi+1/2,yj+1/2) (xi+1/2,yj−1/2) a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v dy(a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v dy (35) ≈ a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v | (xi+1/2,yj) · h y j . To achieve this, it is clear that we now need to derive approximations of the k(v(x, y, t, α i,j )) · n defined above at the mid-point x i+1/2 , y j , of the interval I x i for i = 0, 1, · · · N 1 − 1. This discussion is divided into two cases for i ≥ 1 and i ∈ I 0 = (0, x 1 ). This is really an extension of the one dimensional fitted finite volume presented in the previous section. Case I: For i ≥ 2. Remember that a 11 (x, y, t, α) = a(x, y, t, α) x 2 , we approximate the term a 11 ∂v ∂x + x b 1 v by solv-ing the following two points boundary value problem a(x i+1/2 , y j , t, α i,j ) x i+1/2 ∂v ∂x + b 1 (x i+1/2 , y j , t, α i,j ) v = 0 (36) v(x i , y j , t) = v i,j (t), v(x i+1 , y j , t) = v i+1,j (t). Integrating (36) yields the first-order linear equations a(x i+1/2 , y j , t, α i,j ) x i+1/2 ∂v ∂x + b 1 (x i+1/2 , y j , t, α i,j ) v = C 1(37) where C 1 denotes an additive constant. Following the one dimensional fitted finite volume presented in the previous section, we have C 1 = b 1i+1/2,j (t, α i,j ) x β i,j (t) i+1 v i+1,j − x β i,j (t) i v i,j x β i,j (t) i+1 − x β i,j (t) i .(38) Therefore, a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v ≈ x i+1/2   b 1i+1/2,j (t, α i,j ) x β i,j (t) i+1 v i+1,j − x β i,j (t) i v i,j x β i,j (t) i+1 − x β i,j (t) i + d 1 y ∂v ∂y   ,(39) where β i,j (t) = b 1i+1/2,j (t, α i,j ) a i+1/2,j (t, α i,j ) and a 12 = a 21 = d 1 (x, y, t, α) x y. Finally, we use the forward difference, ∂v ∂y ≈ v i,j+1 − v i,j h y j Finally, a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v (xi+1/2,yj) · h y j ≈ x i+1/2   b 1i+1/2,j (t, α i,j ) x β i,j (t) i+1 v i+1,j − x β i,j (t) i v i,j x β i,j (t) i+1 − x β i,j (t) i + d 1i,j (t, α i,j ) y j v i,j+1 − v i,j h y j   · h y j . (40) Simillary, the second term in (34) can be approximated by a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v (xi−1/2,yj) · h y j ≈ x i−1/2   b 1i−1/2,j (t, α i,j ) x β i−1,j (t) i v i,j − x β i−1,j (t) i−1 v i−1,j x β i−1,j (t) i − x β i−1,j (t) i−1 + d 1i,j (t, α i,j ) y j v i,j+1 − v i,j h y j   · h y j . (41) Case II: For j ≥ 2. For the third term we want to approximate the integral by a constant, that is (xi+1/2,yj+1/2) (xi−1/2,yj+1/2) a 21 ∂v ∂x + a 22 ∂v ∂y + y b 2 v dx (42) ≈ a 21 ∂v ∂x + a 22 ∂v ∂y + y b 2 v | (yj+1/2,xi) · h x i . Following the first case of this section, we have a 21 ∂v ∂x + a 22 ∂v ∂x 2 + y b 2 v (xi,yj+1/2) · h x i ≈ y j+1/2   b 2i,j+1/2 (t, α i,j ) y j+1β i,j (t) v i,j+1 − y jβ i,j (t) v i,j y j+1β i,j (t) − y jβ i,j (t) + d 1i,j (t, α i,j ) x i v i+1,j − v i,j h x i   · h x i . (43) Similary, the fourth term in (34) can be approximated by a 21 ∂v ∂x + a 22 ∂v ∂y + y b 2 v (xi,yj−1/2) · h x i ≈ y j−1/2   b 2i,j−1/2 (t, α i,j ) yβ i,j−1 (t) j v i,j − yβ i,j−1 (t) j−1 v i,j−1 yβ i,j−1 (t) j − yβ i,j−1 (t) j−1 + d 1i,j (t, α i,j ) x i v i+1,j − v i,j h x i   · h x i ,(44) for j = 2, · · · , N 2 − 1, whereβ i,j (t) = b 2i,j+1/2 (t, α i,j ) a i,j+1/2 (t, α i,j ) with a 22 (x, y, t, α) =ā(x, y, t, α) y 2 . Case III: Approximation of the flux at I 0 . Note that the analysis in case I does not apply to the approximation of the flux on [0, x 1 ] because (36) is degenerated. Therefore, we reconsider the following form (a(x 1/2 , y j , t, α 1,j ) x 1/2 ∂v ∂x + b 1 (x 1/2 , y, t, α 1,j ) v) ≡ C 2 in [0, x 1 ] (45) v(x 0 , y j ) = v 0,j , v(x 1 , y j ) = v 1,j , where C 2 is an unknown constant to be determined. Integrating (45), we can deduce that a 11 ∂v ∂x + a 12 ∂v ∂y + x b 1 v (x1/2,yj) · h y j ≈ x 1/2 1 2 (a x 1/2 ,j (t, α 1,j ) + b 1x 1/2 ,j (t, α)) v 1,j − (a x 1/2 ,j (t, α 1,j ) − b 1x 1/2 ,j (t, α 1,j )) v 0,j(46)+d 11,j (t, α 1,j ) y j v 1,j+1 − v 1,j h y j · h y j . Case IV: Approximation of the flux at J 0 . Using the same procedure for the approximation of the flux at I 0 , we deduce that a 21 ∂v ∂x + a 22 ∂v ∂y + y b 2 v (xi,y1/2) · h x i ≈ y 1/2 1 2 (ā i,y 1/2 (t, α i,1 ) + b 2i,y 1/2 (t, α)) v i,1 − (ā i,y 1/2 (t, α i,1 ) − b 2i,y 1/2 (t, α i,1 )) v i,0 (47) +d 1i,1 (t, α i,1 ) x i v i+1,1 − v i,1 h x i · h x i . By replacing the flux by his value for i = 1, · · · , N 1 − 1 and j = 1, · · · , N 2 − 1, equation (33) becomes dv i,j dt (48) + sup α i,j ∈A 1 l i,j   x i+1/2   b 1i+1/2,j (t, α) x β i,j (t) i+1 v i+1,j − x β i,j (t) i v i,j x β i,j (t) i+1 − x β i,j (t) i + d 1i,j (t, α i,j ) y j v i,j+1 − v i,j h y j   · h y j −x i−1/2   b 1i−1/2,j (t, α i,j ) x β i−1,j i v i,j − x β i−1,j (t) i−1 v i−1,j x β i−1,j (t) i − x β i−1,j (t) i−1 + d 1i,j (t, α i,j ) y j v i,j+1 − v i,j h y j   · h y j +y j+1/2   b 2i,j+1/2 (t, α i,j ) y j+1β i,j (t) v i,j+1 − y jβ i,j (t) v i,j y j+1β i,j (t) − y jβ i,j (t) + d 1i,j (t, α i,j ) x i v i+1,j − v i,j h x i   · h x i −y j−1/2   b 2i,j−1/2 (t, α i,j ) y jβ i,j−1 (t) v i,j − y j−1β i,j−1 (t) v i,j−1 y jβ i,j−1 (t) − y j−1β i,j−1 (t) +d 1i,j (t, α i,j ) x i v i+1,j − v i,j h x i · h x i + c i,j (t, α i,j ) v i,j l i,j = 0 By setting τ = T − t and including the boundary conditions, we have the following system        sup α∈A N e i,j i−1,j (τ, α) v i−1,j + e i,j i,j (τ, α) v i,j + e i,j i+1,j (τ, α) v i+1,j + e i,j i,j−1 (τ, α)v i,j−1 + e i,j i,j+1 (τ, α)v i,j+1 − dv i,j dτ = 0, with v(0) given,(49) where for i = 1, · · · , N 1 − 1, j = 1, · · · , N 2 − 1 and N = ( N 1 − 1) × (N 2 − 1), we have e 1,j 0,j = − 1 4 l 1,j h y j x 1 (a x 1/2 ,j (τ, α 1,j ) − b 1x 1/2 ,j (τ, α 1,j )) v 0,j(50)e 1,j 1,j = 1 4 l 1,j h y j x 1 (a x 1/2 ,j (τ, α 1,j ) + b 1x 1/2 ,j (τ, α 1,j )) − 1 2 c 1,j (τ, α 1,j ) + d 11,j (τ, α 1,j ) x i h y j l 1,j + x 1+1/2 h y j b 11+1/2,j (τ, α 1,j ) x β 1,j (τ ) 1 l 1,j x β 1,j (τ ) 2 − x β 1,j (τ ) 1 (51) e 1,j 2,j = −d 11,j (τ, α 1,j ) x i h y j l 1,j − x 1+1/2 h y j b 11+1/2,j (τ, α 1,j ) x β 1,j (τ ) 2 l 1,j x β 1,j (τ ) 2 − x β 1,j (τ ) 1 (52) e i,1 i,0 = − 1 4 l i,1 h x i y 1 (ā i,y 1/2 (τ, α i,1 ) − b 2i,y 1/2 (τ, α i,1 )) v i,0 (53) e i,1 i,1 = 1 4 l i,1 h x i y 1 (ā i,y 1/2 (τ, α i,1 ) + b 2i,y 1/2 (τ, α i,1 ) − 1 2 c i,1 (τ, α i,1 ) + d 1i,1 (τ, α i,1 ) y j h x i l i,1 + y 1+1/2 h x i b 2i,1+1/2 (τ, α i,1 ) yβ i,1 (τ ) 1 l i,1 yβ i,1 (τ ) 2 − yβ i,1 (τ ) 1 (54) e i,1 i,2 = −d 1i,1 (τ, α i,1 ) y j h x i l i,1 − y 1+1/2 h x i b 2i,1+1/2 (τ, α i,1 ) y 2β i,1 (τ ) l i,1 y 2β i,1 (τ ) − y 1β i,1 (τ ) (55) e i,j i+1,j = −d 1i,j (τ, α i,j )) h y j l i,j − x i+1/2 h y j b 1i+1/2 (τ, α i,j ) x β i,j (τ ) i+1 l i,j x β i,j (τ ) i+1 − x β i,j (τ ) i (56) e i,j i−1,j = −x i−1/2 h y j b 1i−1/2,j (τ, α i,j )) x β i−1,j (τ ) i−1 l i,j x β i−1,j (τ ) i − x β i−1,j (τ ) i−1 (57) e i,j i,j = d 1i,j (τ, α i,j )) x i h y j l i,j + x i+1/2 h y j b 1i+1/2,j (τ, α i,j )) x β i,j (τ ) i l i,j x β i,j (τ ) i+1 − x β i,j (τ ) i(58)+ x i−1/2 h y j b 1i−1/2,j (τ, α i,j )) x β i−1,j (τ ) i l i,j x β i−1,j (τ ) i − x β i−1,j (τ ) i−1 − c i,j (τ, α i,j )) d 1i,j (τ, α i,j )) y j h x i l i,j + y j+1/2 h x i b 2i,j+1/2 (τ, α i,j )) y jβ i,j (τ ) l i,j y j+1β i,j (τ ) − y jβ i,j (τ ) + y j−1/2 h x i b 2i,j−1/2 (τ, α i,j )) y j β i,j−1 (τ ) l i,j y jβ i,j−1 (τ ) − y j−1β i,j−1 (τ ) e i,j i,j+1 = −d 1i,j (τ, α i,j )) y j h x i l i,j − y j+1/2 h x i b 2i,j+1/2 (τ, α i,j )) y j+1β i,j (τ ) l i,j y j+1β i,j (τ ) − y jβ i,j (τ ) (59) e i,j i,j−1 = −y j−1/2 h x i b 2i,j−1/2 (τ, α i,j )) y j−1β i,j−1 (τ ) l i,j y jβ i,j−1 (τ ) − y j−1β i,j−1 (τ ) .(60) As for one dimension case, (49) can be rewritten as the Ordinary Differential Equation (ODE) coupled with optimization problem      dv(τ ) dτ + inf α∈A N [E(τ, α) v(τ ) + F (τ, , α)] = 0, with v(0) given, (61) or        dv(τ ) dτ = sup α∈A N [A(τ, α) v(τ ) + G(τ, α)] with v(0) given, where A(τ, α) = −E(τ, α), v = (v 1,1 , · · · , v 1,N 2 −1 , · · · , v N 1 −1,1 , · · · , v N 1 −1,N 2 −1 ) and G(τ, α) = −F (τ, α) includes boundary condition. Theorem 2.2. Assumme that c i,j (τ, α) < 0, d 1i,j (τ, α) > 0, i = 1, · · · , N 1 −1, j = 1, · · · , N 2 −1, and let h = max Proof. The Proof follows the same lines of that of Theorem 2.2. Temporal discretization and optimization This section is devoted to the numerical time discretization method for the spatially discretized optimization problem using the fitted finite volume method. We will present it in one and two dimensional. Let us re-consider the differential equation coupled with optimization problem given in (22) or (62) by v τ (τ ) = sup α∈A N [A(τ, α)v(τ ) + G(τ, α)](63) v(0) given, For temporal discretization, we use a constant time step ∆t > 0, of course variable time steps can be used. The temporal grid points given by ∆t = τ n+1 − τ n for n = 1, 2, . . . m − 1. We denote v(τ n ) ≈ v n , A n (α) = A(τ n , α) and G n (α) = G(τ n , α). For θ ∈ 1 2 , 1 , following [9], the θ-Method approximation of (63) in time is given by v n+1 − v n = ∆t sup α∈A N θ [A n+1 (α) v n+1 + G n+1 (α)](64)+(1 − θ) [A n (α) v n + G n (α)]) . As we can notice, to find the unknown v n+1 we need also to solve an optimization. Let α n+1 ∈ arg sup α∈A N θ ∆t A n+1 (α) v n+1 + G n+1 (α) + (1 − θ) ∆t [A n (α) v n + G n (α)] .(65) Then, the unknown v n+1 is solution of the following equation [I − θ ∆t A n+1 (α n+1 )] v n+1 = [I + (1 − θ) ∆t A n (α n+1 )] v n(66)+ [θ ∆t G n+1 (α n+1 ) + (1 − θ)∆t G n (α n+1 )], Note that, for θ = 1 2 , we have the Crank Nickolson scheme and for θ = 1 we have the Implicit scheme. Unfortunately (64)-(65) are nonlinear and coupled and we need to iterate at every time step. The following iterative scheme close to the one in [9] is used. 1. Let (v n+1 ) 0 = v n , 2. Letv k = (v n+1 ) k , 3. For k = 0, 1, 2 · · · until convergence ( v k+1 −v k ≤ , given tolerance) solve α k i ∈ arg sup α∈A N θ ∆t A n+1 (α)v k + G n+1 (α) i + (1 − θ) ∆t [A n (α) v n + G n (α)] i (67) α k = (α k ) i (68) [I − θ ∆t A n+1 (α k )]v k+1 = [I + (1 − θ) ∆t A n (α k )]v n(69)+ [θ ∆t G n+1 (α k ) + (1 − θ)∆t G n (α k )], 4. Let k l being the last iteration in step 3, set v n+1 :=v k l , α n+1 := α k l . Numerical experiments The goal of this section is carried out on test problems in both 1 and 2 space dimensions to validate the numerical scheme presented in the previous section. All computations were performed in Matlab 2013 using the estimate parameters coming from [17] and [9]. We will present two problems with exact solution and one problem without exact solution modelling cash management in finance. Problem 4.1. Consider the following Merton's stochastic control problem such that α = α(t, x) is a feedback control belongs in (0, 1)        v(x, t) = max α∈ (0,1) E 1 p x p (T ) . dx t = b αt (t, x t ) dt + σ αt (t, x t ) dω t (70) where b αt (t, x t ) = x t [r + α t (µ − r)], σ αt (t, x t ) = x t α t σ, 0 < p < 1 is coefficient of risk aversion, r, µ, σ are constants, x t ∈ R and ω t Brownian motion. We assume µ > r. For this problem, the corresponding HJB equation is given by        v t (t, x) + sup α∈(0,1) [L α v(t, x)] = 0 on [0, T ) × R v(T, x) = x p p , x ∈ R (71) where L α v(t, x) = (b α (t, x)) ∂v(t, x) ∂x + (a α (t, x)) ∂ 2 v(t, x) ∂x 2 ,(72) and a α (t, x) = 1 2 (σ α (t, x)) 2 . The divergence form of the HJB (71) is given by ∂v(t, x) ∂t + sup α∈ (0,1) ∂ ∂x a(t, x, α) x 2 ∂v(t, x) ∂x + b(t, x, α) x v(t, x) + c(t, x, α) v(t, x) = 0, (73) where a(t, x, α) = 1 2 σ 2 α 2 b(t, x, α) = r + (µ − r) α − σ 2 α 2 c(t, x, α) = −(r + α (µ − r) − σ 2 α 2 ). The Domain where we compare the solution is Ω = [0, x max ], where Dirichlet Boundary conditions is used at the boundaries. Of course the value of the boundary conditions are taken to be equal to the exact solution. The exact solution given in [8] is given at every ( x i , τ n ) by v (τ n , x i ) = e p×(n×∆t−T )×ρ × (x i ) p p ,(74)ρ = r + (µ − r) 2 σ 2 (1 − p) + 1 2 (p − 1) σ 2 (µ − r) σ 2 (1 − p) 2 , 0 < p < 1(75) We use the following L 2 (Ω × [0, T ]) norm of the absolute error v m − v L 2 [Ω×[0,T ]] = m−1 n=0 N 1 −1 i=1 (τ n+1 − τ n ) × l i × (v n i − v (τ n , x i , )) 2 1/2 ,(76) where v m is the numerical approximation of v computed from our numerical scheme. The 3 D graphs of the Implicit Fitted Finite Volume ( θ = 1) with its corresponding exact solution is given at Figure 1 and We compare the fitted finite volume method and the finite difference method in Table 1 Time subdivision 200 150 100 50 Error of Implicit Fitted Finite Volume method 3.34 E-01 6.81 E-01 1.01 E-00 1.33 E-00 Error of Implicit Finite difference method 3.37 E-01 6.89 E-01 1.02 E-00 1.34 E-00 α 1 = α 1 (t, x) and α 2 = α 2 (t, y) are a feedback control in (0, 1) v(t, x, y) = max (α 1 ,α 2 )∈ (0,1)×(0,1) E 1 p x p (T ) × 1 p y p (T ) ,(77) subject to      dx t = b 1 α 1t (t, x t ) dt + σ α 2t (t, x t ) dω 1t dy t = b 2 α 2t (t, y t ) dt + σ α 2t (t, y t ) dω 2t (78) where b 1 α 1t (t, x t ) = x t [r 1 + α 1t (µ 1 − r 1 )] , b 2 α 2t (t, y t ) = y t [r 2 + α 2t (µ 2 − r 2 )] , σ α 1t (t, x t ) = x t α 1t σ, σ α 2t (t, y t ) = y t α 2t σ, 0 < p < 1 is coefficient of risk aversion, r 1 , µ 1 , r 2 , µ 2 σ are constants, x t y t ∈ R and ρ the correlation of the two Brownian motion. We assume that µ 1 > r 1 and µ 2 > r 2 . For this problem, the corresponding HJB equation is given by        v t (t, x, y) + sup (α 1 ,α 2 )∈(0,1)×(0,1) [L α 1 ,α 2 v(t, x, y)] = 0 on [0, T ) × R × R v(T, x, y) = x p p × y p p , x, y ∈ R(79) where L α v(t, x, y) = (b 1 α 1 (t, x)) ∂v(t, x, y) ∂x + (b 2 α 2 (t, y)) ∂v(t, x, y) ∂y + 1 2 (σ α 1 (t, x)) 2 ∂ 2 v(t, x, y) ∂x 2 + 1 2 (σ α 2 (t, y)) 2 ∂ 2 v(t, x, y) ∂y 2 + (σ α 1 (t, x)) (σ α 2 (t, y)) ∂ 2 v(t, x, y) ∂x 2 , and the two dimensional divergence form is given by ∂v(t, x, y) ∂t + sup α∈ (0,1)×(0,1) [∇ · (k(t, x, y, α)) + c(t, x, y, α) v(t, x, y)] = 0,(80) where k(t, x, y, α) = A(t, x, y, α) · ∇v(t, x, y) + b(t, x, y, α) · v(t, x, y) A =   a 11 a 12 a 21 a 22   , a 11 (t, x, y, α) = 1 2 σ 2 α 2 1 x 2 , a 22 (t, x, y, α) = 1 2 σ 2 α 2 2 y 2 , a 12 (t, x, y, α) = a 21 (x, y, t, α) = 1 2 σ 2 α 1 α 2 x y. By identification, a(t, x, y, α) = 1 2 σ 2 α 2 1 a(t, x, y, α) = 1 2 σ 2 α 2 2 b 1 (t, x, y, α) = r 1 + α 1 (µ 1 − r 1 ) − 1 2 σ 2 α 1 α 2 − σ 2 α 2 1 , b 2 (t, x, y, α) = r 2 + α 2 (µ 2 − r 2 ) − 1 2 σ 2 α 1 α 2 − σ 2 α 2 2 , c(t, x, y, α) = − [r 1 + (µ 1 − r 1 ) α 1 ] − [r 2 + (µ 2 − r 2 ) α 2 ] + σ 2 α 2 1 + α 2 2 + α 1 α 2 , d 1 (t, x, y, α) = 1 2 σ 2 α 1 α 2 . The two dimensional Antsaz exact solution [8] at (τ n , x i , y j ) is given by v (τ n , x i , y j ) = e p×(n×∆t−T )×ρ × (x i ) p p × (y j ) p p , ρ = sup α 1 ,α 2 ∈(0,1)×(0,1) r 1 + r 2 + (µ 1 − r 1 ) α 1 + (µ 2 − r 2 ) α 2 + 1 2 σ 2 α 2 1 (p − 1) + 1 2 σ 2 α 2 2 (p − 1) + σ 2 α 1 α 2 p , 0 < p < 1/2.N 1 −1 i=1 N 2 −1 j=1 (τ n+1 − τ n ) × hx i × hy j × (v n i,j − v (τ n , x i , x j , )) 2 1/2 ,(81) where v m is the numerical approximation of v computed from our numerical scheme. The 3 D graphs of the Implicit Fitted Finite Volume ( θ = 1 at the final time T = 1) with its corresponding exact solution is given at Figure 3 and Figure 2, with N 1 = 50, N 2 = 45, r 1 = 0.0449/2, µ 1 = 0.0657/2, r 2 = 0.044/2, µ 2 = 0.065/2, σ = 0.2537/2 and p = 0.5255/2. We compare the fitted finite volume method and the finite difference method in Table 2. Again, we can observe the accuracy of the fitted scheme comparing to the finite difference scheme. Problem 4.3. We consider a optimal Cash Management under a stochastic volatility Model problem coming from [21]. We assume that the firm invests its cash in a bank account and a stock in a portfolio of value w t at time t, and the proportion of wealth invested in the stock at time t is u t . The interest rate earned in the bank account is r 1 , the return from the stock at time t has two components, the cash dividend rate r 2 , the capital gain rate R t . The dynamic of the capital gain rate R t is assumed to be governed by the stochastic process dR t = [β 1 R t + f ] dt + σ t dW 1t ,(82) and the volatility σ t with modeled by dσ t = α σ t dt + β σ t dW 2t . Suppose that the firm has a demand rate d t for cash at time t, and that the demand rate d(t) is normally distributed with mean 0 and variance 0.2. We assume that u t ∈ [0, 1] and the wealth dynamics for this cash management problem is given by dw t = w t u t r 2 dt + w t (1 − u t ) r 1 dt + w t R t dt − d(t) w t dt.(84) The objective of the firm is to maximize the expectation of the total holdings at the terminal time T . The portfolio optimization problem is given by J(w, R, σ, T ) = max u∈[0,1] E {w T } .(85) subject to              dw t = w t u t r 2 dt + w t (1 − u t ) r 1 dt + w t R t dt − d(t) w t dt, dR t = [β 1 R t + f ] dt + σ t dW 1t dσ t = α σ t dt + β σ t dW 2t We assume that the two Brownian motions are correlated, that is dW 1t dW 2t = ρ dt. For this problem of optimal Cash Management the analytical solution is not available, so our numerical scheme will to provide approximated solution. The corresponding HJB equation for this optimal cash management problem (85) is given by 1] {(f + β 1 R) J R + w (u r 2 + (1 − u) r 1 + u R − d(t)) J w + (86) 1/2 σ 2 J RR + β 2 σ 2 J σσ + 2 ρ β σ 2 J σ R + α σ J σ = 0, with terminal condition J(·, T ) = w T . To simplify the problem, we assume that J(w, R, σ, t) = wH(R, σ, t). J t + max u∈[0, Therefore (86) [∇ · (k(R, σ, t, u)) + c(R, σ, t, u) H(R, σ, t)] = 0, . Figure 5 shows a sample of fitted finite volume solution of the wealth rate H at the point (1/2, 1/2) from t = 1 to t = 10 with N 1 = 10, N 2 = 10, R max = 1/2, σ max = 1/2. We can estimate the mean and moment of H using Monte Carlo Method by generating many samples of H. Conclusion We presented a fitted finite volume method to solve the HJB equation from stochastic optimal control problems coupled with implicit temporal discretization. The optimization problem is solved at every time step using iterative method. It was shown that the corresponding system matrix is an M-matrix, so the maximum principle is preserved for the discrete system. Numerical experiments in 1 and 2 dimensions are performed to prove the accuracy of the fitted finite volume method comparing to the standard finite difference methods. ,j . If h is reatively small then the matrix E(τ, α) = e matrix for any α ∈ A N . Figure 2 . 2For our computation, we have [0, 10] for computational domain with N = 1500, r = 0.0449, µ = 0.0657, σ = 0.2537, p = 0.5255 and T = 1. Figure 1 : 1Implicit Fitted Finite Volume. Figure 2 : 2Ansatz Analytical solution We use the following L 2 (Ω × [0, T ]), Ω = [0, x max ] × [0, y max ] norm of the absolute error v m − v L 2 [Ω×[0,T ]] Figure 3 : 3Implicit Fitted Finite Volume (θ = 1) at finite time T = 1. Figure 4 : 4Ansatz Analytical solution at finite time T = 1 f + β 1 R) H R + (u r 2 + (1 − u) r 1 + u R − d(t)) H+ (87) 1/2 σ 2 J RR + β 2 σ 2 H σσ + 2 ρ β σ 2 H σ R + (α σ) H σ = 0with terminal condition H(R, σ, T ) = 1. The HJB equation (87) is a problem with two state variables R and σ. The divergence form of the problem (87) is then given by ∂H(R, σ, t) ∂t + sup u ∈ [0,1] Figure 5 : 5A fitted finite volume sample solution of the wealth rate H at the point 1/2, 1/2 Table 1 : 1Comparison of the implicit fitted finite volume method and implicit finite difference method.From Table 1, we can observe the accuracy of the implicit fitted finite volume comparing to the implicit finite difference method. Problem 4.2. Consider the following two dimensional Merton's stochastic control model such that Error of Fitted Finite Volume method 4.08 E-02 7.84 E-02 1.14 E-01 1.47 E-01 Error of Finite difference method 4.23 E-02 7.93 E-02 1.16 E-01 1.48 E-01Time subdivision 200 150 100 50 Table 2 : 2Errors table for fitted finite volume method and finite difference method in dimension 2. AcknowledgementsThe first author was supported by the project African Center of Excellence in Mathematics and Applied Sciences (ACE-MSA) in Benin and the European Mathematical Society (EMS).Referenceswhere k(R, σ, t, u) = A(R, σ, t, u) · ∇H(R, σ, t) + b(R, σ, t, u) · H(R, σ, t)By identification,Because we have a stochastic volatility model, to solve the PDE equation, we have considered the following boundary conditions of Heston modelBecause the PDE has two second derivatives in the two spatial directions, four boundary conditions are needed. This comes from the fact that the two second order derivatives give rise to two unknown integration constants. To meet this requirement, at the boundary σ = 0 it is considered inserting σ = 0 into the PDE to complete the set of four boundary conditions:The HJB equation is solved using some parameters values in[21]given in the following tabular A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities. C.-S Huang, C.-H Hung, S Wang, Computing. 773C.-S Huang, C.-H Hung and S. Wang, A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities, Computing 77(3) (2006) 297-320. Fitted finite volume method for a generalized Black Scholes equation transformed on finite interval. R Valkov, Numerical Algorithms. 651R. Valkov, Fitted finite volume method for a generalized Black Scholes equation transformed on finite interval, Numerical Algorithms 65 (1) (2014) 195-220. N V Krylov, Controlled diffusion processes, Applications of Mathematics. New YorkSpringer-VerlagN. V. Krylov, Controlled diffusion processes, Applications of Mathematics, Springer-Verlag, New York, 1980 A Novel fitted finite volume method for the Black-Scholes equation governing option pricing. S Wang, IMA J. Numer. Anal. 24S. Wang, A Novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal 24 (2004) 699-720. The pricing of options on assets with stochastic volatilities. J Hull, A White, J.Finance. 422J. Hull, A. White, The pricing of options on assets with stochastic volatilities, J.Finance 42(2) (1987) 281-300. Numerical Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance. Peter Forsyth, George Labahn, Journal of Computational Finance. 112Peter Forsyth and George Labahn, Numerical Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance. Journal of Computational Finance, 11(2) (2007) 1-43. The Best of Wilmott 1: Incorporating the Quantitative Finance Review. P Wilmott, John Wiley & SonsP. Wilmott, The Best of Wilmott 1: Incorporating the Quantitative Finance Review. John Wiley & Sons, 2005. H Pham, Optimisation et contrôle stochastique appliquésà la finance, Mathématiques et applications. New YorkSpringer-verlagH. Pham, Optimisation et contrôle stochastique appliquésà la finance, Mathématiques et applications, Springer-verlag New York, 2000. Numerical Solution of the Hamilton-Jacobi-Bellman Equation for Stochastic Optimal Control Problems. H Peyrl, F Herzog, H P Geering, WSEAS Int. Conf. on Dynamical Systems and control. H. Peyrl, F. Herzog and H. P. Geering, Numerical Solution of the Hamilton-Jacobi-Bellman Equation for Stochastic Optimal Control Problems, WSEAS Int. Conf. on Dynamical Systems and control, Venice, Italy, November 2-4, 2005, 489-497. On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probability Theory and Related Fields. N V Krylov, 117N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probability Theory and Related Fields, 117 (2000) 1-16. The rate of convergence of finite-difference approximations for Bellman's equations with Lipschitz coefficients. N V Krylov, Applied Mathematics and Optimization. 52N.V. Krylov, The rate of convergence of finite-difference approximations for Bellman's equa- tions with Lipschitz coefficients, Applied Mathematics and Optimization, 52 (2005) 365-399. On finite-difference approximations for normalized Bellman's equations. I Gyöngy, D Šiška, Applied Mathematics and Optimization. 60297I. Gyöngy, D.Šiška, On finite-difference approximations for normalized Bellman's equations, Applied Mathematics and Optimization, 60(2009), Article number: 297. On the rate of convergence of approximations schemes for Bellman equations associated with optimal stopping time problems. E R Jakobsen, Mathematical Models and Methods in Applied Sciences. 1305E.R. Jakobsen, On the rate of convergence of approximations schemes for Bellman equations associated with optimal stopping time problems, Mathematical Models and Methods in Applied Sciences, 13 (05) (2003), 613-644. Viscosity solutions of Hamilton-Jacobi equations. M G Crandall, P Lions, Transactions of the American Mathematical Society. 2771M.G. Crandall, P.L Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1) (1983) 1-42. Some properties of viscosity solutions of Hamilton-Jacobi equations. M G Crandall, L C Evans, P L Lions, Transactions of the American Mathematical Society. 2822M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton- Jacobi equations, Transactions of the American Mathematical Society, 282(2), (1984), p. 487-502. M G Crandall, P L Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation. 43M.G. Crandall, P.L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, 43 (1984) 1-19. Merton's portfolio problem, constant fraction investment strategy and frequency of portfolio rebalancing newblock Master Thesis. J Holth, University of OsloJ. Holth, Merton's portfolio problem, constant fraction investment strategy and frequency of portfolio rebalancing newblock Master Thesis, University of Oslo, http://hdl.handle.net/10852/10798, 2011, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies. N V Krylov, Electronic Journal of Probability. 42N.V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electronic Journal of Probability, 4 (2) (1999) 1-19. User's guide to viscosity solutions of second order partial differential equations. M G Crandall, H Ishii, P L Lions, American Mathematical Society27M.G. Crandall, H. Ishii and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations, American Mathematical Society, 27 (1992) 1-67. On application of an alternating direction method to Hamilton-Jacobi-Bellman equations. C.-S Huang, S Wang, K L Teo, Journal of Computational and Applied Mathematics. 27C.-S. Huang, S. Wang and K.L.Teo, On application of an alternating direction method to Hamilton-Jacobi-Bellman equations. Journal of Computational and Applied Mathematics, 27 (2004) 153-166. On Optimal Cash Management under a Stochastic Volatility Model. N Song, W.-K Ching, T.-K Siu, C , K.-F Yiu, East Asian Journal on Applied Mathematics. 32N. Song, W.-K. Ching, T.-K. Siu and C. K.-F. Yiu, On Optimal Cash Management under a Stochastic Volatility Model. East Asian Journal on Applied Mathematics, 3 (2) (2013) 81-92 . On convergence of a Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities. C.-S Huang, C.-H Hung, S Wang, IMA J. Numer. Anal. 30C.-S Huang, C.-H.Hung and S. Wang, On convergence of a Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities, IMA J. Numer. Anal. 30 (2010) 1101-1120.
[]
[ "UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY", "UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY" ]
[ "Guillaume Lample \nFacebook AI Research\n\n\nUMR 7606\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLIP6\n", "Ludovic Denoyer [email protected] \nUMR 7606\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLIP6\n", "Aurelio Marc&apos; ", "Ranzato [email protected] \nFacebook AI Research\n\n" ]
[ "Facebook AI Research\n", "UMR 7606\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLIP6", "UMR 7606\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLIP6", "Facebook AI Research\n" ]
[]
Machine translation has recently achieved impressive performance thanks to recent advances in deep learning and the availability of large-scale parallel corpora. There have been numerous attempts to extend these successes to low-resource language pairs, yet requiring tens of thousands of parallel sentences. In this work, we take this research direction to the extreme and investigate whether it is possible to learn to translate even without any parallel data. We propose a model that takes sentences from monolingual corpora in two different languages and maps them into the same latent space. By learning to reconstruct in both languages from this shared feature space, the model effectively learns to translate without using any labeled data. We demonstrate our model on two widely used datasets and two language pairs, reporting BLEU scores up to 32.8, without using even a single parallel sentence at training time.
null
[ "https://arxiv.org/pdf/1711.00043v1.pdf" ]
3,518,190
1711.00043
73c4eb346a2bafaaedeede6c816e17a8062f88ce
UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY Guillaume Lample Facebook AI Research UMR 7606 Sorbonne Universités UPMC Univ Paris 06 CNRS LIP6 Ludovic Denoyer [email protected] UMR 7606 Sorbonne Universités UPMC Univ Paris 06 CNRS LIP6 Aurelio Marc&apos; Ranzato [email protected] Facebook AI Research UNSUPERVISED MACHINE TRANSLATION USING MONOLINGUAL CORPORA ONLY Under review as a conference paper at ICLR 2018 Machine translation has recently achieved impressive performance thanks to recent advances in deep learning and the availability of large-scale parallel corpora. There have been numerous attempts to extend these successes to low-resource language pairs, yet requiring tens of thousands of parallel sentences. In this work, we take this research direction to the extreme and investigate whether it is possible to learn to translate even without any parallel data. We propose a model that takes sentences from monolingual corpora in two different languages and maps them into the same latent space. By learning to reconstruct in both languages from this shared feature space, the model effectively learns to translate without using any labeled data. We demonstrate our model on two widely used datasets and two language pairs, reporting BLEU scores up to 32.8, without using even a single parallel sentence at training time. INTRODUCTION Thanks to recent advances in deep learning (Sutskever et al., 2014;Bahdanau et al., 2015) and the availability of large-scale parallel corpora, machine translation has now reached impressive performance on several language pairs . However, these models work very well only when provided with massive amounts of parallel data, in the order of millions of parallel sentences. Unfortunately, parallel corpora are costly to build as they require specialized expertise, and are often nonexistent for low-resource languages. Conversely, monolingual data is much easier to find, and many languages with limited parallel data still possess significant amounts of monolingual data. There have been several attempts at leveraging monolingual data to improve the quality of machine translation systems in a semi-supervised setting (Munteanu et al., 2004;Irvine, 2013;Irvine & Callison-Burch, 2015;Zheng et al., 2017). Most notably, Sennrich et al. (2015) proposed a very effective data-augmentation scheme, dubbed "back-translation", whereby an auxiliary translation system from the target language to the source language is first trained on the available parallel data, and then used to produce translations from a large monolingual corpus on the target side. The pairs composed of these translations with their corresponding ground truth targets are then used as additional training data for the original translation system. Another way to leverage monolingual data on the target side is to augment the decoder with a language model (Gulcehre et al., 2015). And finally, Cheng et al. (2016); He et al. (2016a) have proposed to add an auxiliary auto-encoding task on monolingual data, which ensures that a translated sentence can be translated back to the original one. All these works still rely on several tens of thousands parallel sentences, however. Previous work on zero-resource machine translation has also relied on labeled information, not from the language pair of interest but from other related language pairs (Firat et al., 2016;Johnson et al., 2016;Chen et al., 2017) or from other modalities (Nakayama & Nishida, 2017;Lee et al., 2017). The only exception is the work by Ravi & Knight (2011); Pourdamghani & Knight (2017), where the machine translation problem is reduced to a deciphering problem. Unfortunately, their method is limited to rather short sentences and it has only been demonstrated on a very simplistic setting comprising of the most frequent short sentences, or very closely related languages. (autoencoding): the model is trained to reconstruct a sentence from a noisy version of it. x is the target, C(x) is the noisy input,x is the reconstruction. Right (translation): the model is trained to translate a sentence in the other domain. The input is a noisy translation (in this case, from source-to-target) produced by the model itself, M , at the previous iteration (t), y = M (t) (x). The model is symmetric, and we repeat the same process in the other language. See text for more details. In this paper, we investigate whether it is possible to train a general machine translation system without any form of supervision whatsoever. The only assumption we make is that there exists a monolingual corpus on each language. This set up is interesting for a twofold reason. First, this is applicable whenever we encounter a new language pair for which we have no annotation. Second, it provides a strong lower bound performance on what any good semi-supervised approach is expected to yield. The key idea is to build a common latent space between the two languages (or domains) and to learn to translate by reconstructing in both domains according to two principles: (i) the model has to be able to reconstruct a sentence in a given language from a noisy version of it, as in standard denoising auto-encoders (Vincent et al., 2008). (ii) The model also learns to reconstruct any source sentence given a noisy translation of the same sentence in the target domain, and vice versa. For (ii), the translated sentence is obtained by using a back-translation procedure (Sennrich et al., 2015), i.e. by using the learned model to translate the source sentence to the target domain. In addition to these reconstruction objectives, we constrain the source and target sentence latent representations to have the same distribution using an adversarial regularization term, whereby the model tries to fool a discriminator which is simultaneously trained to identify the language of a given latent sentence representation (Ganin et al., 2016). This procedure is then iteratively repeated, giving rise to translation models of increasing quality. To keep our approach fully unsupervised, we initialize our algorithm by using a naïve unsupervised translation model based on a word by word translation of sentences with a bilingual lexicon derived from the same monolingual data (Conneau et al., 2017). As a result, and by only using monolingual data, we can encode sentences of both languages into the same feature space, and from there, we can also decode/translate in any of these languages; see Figure 1 for an illustration. While not being able to compete with supervised approaches using lots of parallel resources, we show in section 4 that our model is able to achieve remarkable performance. For instance, on the WMT dataset we can achieve the same translation quality of a similar machine translation system trained with full supervision on 100,000 sentence pairs. On the Multi30K-Task1 dataset we achieve a BLEU above 22 on all the language pairs, with up to 32.76 on English-French. Next, in section 2, we describe the model and the training algorithm. We then present experimental results in section 4. Finally, we further discuss related work in section 5 and summarize our findings in section 6. UNSUPERVISED NEURAL MACHINE TRANSLATION In this section, we first describe the architecture of the translation system, and then we explain how we train it. NEURAL MACHINE TRANSLATION MODEL The translation model we propose is composed of an encoder and a decoder, respectively responsible for encoding source and target sentences to a latent space, and to decode from that latent space to the source or the target domain. We use a single encoder and a single decoder for both domains (Johnson et al., 2016). The only difference when applying these modules to different languages is the choice of lookup tables. Let us denote by W S the set of words in the source domain associated with the (learned) words embeddings Z S = (z s 1 , ...., z s |W S | ), and by W T the set of words in the target domain associated with the embeddings Z T = (z t 1 , ...., z t |W T | ), Z being the set of all the embeddings. Given an input sentence of m words x = (x 1 , x 2 , ..., x m ) in a particular language , ∈ {src, tgt}, an encoder e θenc,Z (x, ) computes a sequence of m hidden states z = (z 1 , z 2 , ..., z m ) by using the corresponding word embeddings, i.e. Z S if = src and Z T if = tgt; the other parameters θ enc are instead shared between the source and target languages. For the sake of simplicity, the encoder will be denoted as e(x, ) in the following. These hidden states are vectors in R n , n being the dimension of the latent space. A decoder d θ dec ,Z (z, ) takes as input z and a language , and generates an output sequence y = (y 1 , y 2 , ..., y k ), where each word y i is in the corresponding vocabulary W . This decoder makes use of the corresponding word embeddings, and it is otherwise parameterized by a vector θ dec that does not depend on the output language. It will thus be denoted d(z, ) in the following. To generate an output word y i , the decoder iteratively takes as input the previously generated word y i−1 (y 0 being a start symbol which is language dependent), updates its internal state, and returns the word that has the highest probability of being the next one. The process is repeated until the decoder generates a stop symbol indicating the end of the sequence. In this article, we use a sequence-to-sequence model with attention (Bahdanau et al., 2015). The encoder is a bidirectional-LSTM which returns a sequence of hidden states z = (z 1 , z 2 , ..., z m ). At each step, the decoder, which is an LSTM, takes the previous hidden state, the current word and a context vector given by a weighted sum over the encoder states. OVERVIEW OF THE METHOD We consider a dataset of sentences in the source domain, denoted by D src , and another dataset in the target domain, denoted by D tgt . These datasets do not correspond to each other, in general. We train the encoder and decoder by reconstructing a sentence in a particular domain, given a noisy version of the same sentence in the same or in the other domain. At a high level, the model starts with an unsupervised naïve translation model obtained by making word-by-word translation of sentences using a parallel dictionary learned in an unsupervised way (Conneau et al., 2017). Then, at each iteration, the encoder and decoder are trained by minimizing an objective function that measures their ability to both reconstruct and translate from a noisy version of an input training sentence. This noisy input is obtained by dropping and swapping words in the case of the auto-encoding task, while it is the result of a translation with the model at the previous iteration in the case of the translation task. In order to promote alignment of the latent distribution of sentences in the source and the target domains, our approach also simultaneously learns a discriminator in an adversarial setting. The newly learned encoder/decoder are then used at the next iteration to generate new translations, until convergence of the algorithm. At test time and despite the lack of parallel data at training time, the encoder and decoder can be composed into a standard machine translation system. DENOISING AUTO-ENCODING Training an autoencoder of sentences is a trivial task, if the sequence-to-sequence model is provided with an attention mechanism like in our work 1 . Without any constraint, the auto-encoder very quickly learns to merely copy every input word one by one. Such a model would also perfectly copy sequences of random words, suggesting that the model does not learn any useful structure in the data. To address this issue, we adopt the same strategy of Denoising Auto-encoders (DAE) (Vincent et al., 2008)), and add noise to the input sentences (see Figure 1-left), similarly to Hill et al. (2016). Considering a domain = src or = tgt, and a stochastic noise model denoted by C which operates on sentences, we define the following objective function: L auto (θ enc , θ dec , Z, ) = E x∼D ,x∼d(e(C(x), ), ) [∆(x, x)](1) wherex ∼ d(e(C(x), ), ) means thatx is a reconstruction of the corrupted version of x, with x sampled from the monolingual dataset D . In this equation, ∆ is a measure of discrepancy between the two sequences, the sum of token-level cross-entropy losses in our case. Noise model C(x) is a randomly sampled noisy version of sentence x. In particular, we add two different types of noise to the input sentence. First, we drop every word in the input sentence with a probability p wd . Second, we slightly shuffle the input sentence. To do so, we apply a random permutation σ to the input sentence, verifying the condition ∀i ∈ {1, n}, |σ(i) − i| ≤ k where n is the length of the input sentence, and k is a tunable parameter. In our experiments, both the word dropout and the input shuffling strategies turned out to have a critical impact on the results, see also section 4.5. Using both strategies at the same time is what gave us the best performance. In practice, we found p wd = 0.1 and k = 3 to be good parameters. CROSS DOMAIN TRAINING The second objective of our approach is to constrain the model to be able to map an input sentence from a the source/target domain 1 to the target/source domain 2 , which is what we are ultimately interested in at test time. The principle here is to sample a sentence x ∈ D 1 , and to generate a corrupted translation of this sentence in 2 . This corrupted version is generated by applying the current translation model denoted M to x such that y = M (x). Then a corrupted version C(y) is sampled (see Figure 1-right). The objective is thus to learn the encoder and the decoder such that they can reconstruct x from C(y). The cross-domain loss can be written as: L cd (θ enc , θ dec , Z, 1 , 2 ) = E x∼D 1 ,x∼d(e(C(y), 2), 1 ) [∆(x, x)](2) where y = M (x) and ∆ is again the sum of token-level cross-entropy losses. ADVERSARIAL TRAINING Intuitively, the decoder of a neural machine translation system works well only when its input is produced by the encoder it was trained with, or at the very least, when that input comes from a distribution very close to the one induced by its encoder. Therefore, we would like our encoder to output features in the same space regardless of the actual language of the input sentence. If such condition is satisfied, our decoder may be able to decode in a certain language regardless of the language of the encoder input sentence. Note however that the decoder could still produce a bad translation while yielding a valid sentence in the target domain, as constraining the encoder to map two languages in the same feature space does not imply a strict correspondence between sentences. Fortunately, the previously introduced loss for cross-domain training in equation 2 mitigates this concern. Also, recent work on bilingual lexical induction has shown that such a constraint is very effective at the word level, suggesting that it may also work at the sentence level, as long as the two monolingual corpora exhibit strong structure in feature space. In order to add such a constraint, we train a neural network, which we will refer to as the discriminator, to classify between the encoding of source sentences and the encoding of target sentences (Ganin et al., 2016). The discriminator operates on the output of the encoder, which is a sequence of latent vectors (z 1 , ..., z m ), with z i ∈ R n , and produces a binary prediction about the language of the encoder input sentence: p D (l|z 1 , ..., z m ) ∝ The discriminator is trained to predict the language by minimizing the following cross-entropy loss: L D (θ D |θ, Z) = −E (xi, i) [log p D ( i |e(x i , i ))], where (x i , i ) corresponds to sentence and language id pairs uniformly sampled from the two monolingual datasets, θ D are the parameters of the discriminator, θ enc are the parameters of the encoder, and Z are the encoder word embeddings. The encoder is trained instead to fool the discriminator: L adv (θ enc , Z|θ D ) = −E (xi, i) [log p D ( j |e(x i , i ))](3) with j = 1 if i = 2 , and vice versa. Final Objective function The final objective function at one iteration of our learning algorithm is thus: L(θ enc , θ dec , Z) =λ auto [L auto (θ enc , θ dec , Z, src) + L auto (θ enc , θ dec , Z, tgt)]+ λ cd [L cd (θ enc , θ dec , Z, src, tgt) + L cd (θ enc , θ dec , Z, tgt, src)]+ λ adv L adv (θ enc , Z|θ D ) where λ auto , λ cd , and λ adv are hyper-parameters weighting the importance of the auto-encoding, cross-domain and adversarial loss. In parallel, the discriminator loss L D is minimized to update the discriminator. TRAINING In this section we describe the overall training algorithm and the unsupervised criterion we used to select hyper-parameters. ITERATIVE TRAINING The final learning algorithm is described in Algorithm 1 and the general architecture of the model is shown in Figure 2. As explained previously, our model relies on an iterative algorithm which starts from an initial translation model M (1) (line 3). This is used to translate the available monolingual To jump start the process, M (1) simply makes a word-by-word translation of each sentence using a parallel dictionary learned using the unsupervised method proposed by Conneau et al. (2017), which only leverages monolingual data. Algorithm 1 Unsupervised Training for Machine Translation 1: procedure TRAINING(D src , D tgt , T ) 2: Infer bilingual dictionary using monolingual data (Conneau et al., 2017) 3: M (1) ← unsupervised word-by-word translation model using the inferred dictionary 4: for t = 1, T do The intuition behind our algorithm is that as long as the initial translation model M (1) retains at least some information of the input sentence, the encoder will map such translation into a representation in feature space that also corresponds to a cleaner version of the input, since the encoder is trained to denoise. At the same time, the decoder is trained to predict noiseless outputs, conditioned on noisy features. Putting these two pieces together will produce less noisy translations, which will enable better back-translations at the next iteration, and so on so forth. UNSUPERVISED MODEL SELECTION CRITERION In order to select hyper-parameters, we wish to have a criterion correlated with the translation quality. However, we do not have access to parallel sentences to judge how well our model translates, not even at validation time. Therefore, we propose the surrogate criterion which we show correlates well with BLEU (Papineni et al., 2002), the metric we care about at test time. For all sentences x in a domain 1 , we translate these sentences to the other domain 2 , and then translate the resulting sentences back to 1 . The quality of the model is then evaluated by computing the BLEU score over the original inputs and their reconstructions via this two-step translation process. The performance is then averaged over the two directions, and the selected model is the one with the highest average score. Given an encoder e, a decoder d and two non-parallel datasets D src and D tgt , we denote M src→tgt (x) = d(e(x, src), tgt) the translation model from src to tgt, and M tgt→src the model in the opposite direction. Our model selection criterion M S(e, d, D src , D tgt ) is: Figure 3 shows a typical example of the correlation between this measure and the final translation model performance (evaluated here using a parallel dataset). M S(e, d, D src , D tgt ) = 1 2 E x∼Dsrc [BLEU (x, M src→tgt • M tgt→src (x))] + 1 2 E x∼Dtgt [BLEU (x, M tgt→src • M src→tgt (x))](5) EXPERIMENTS In this section, we first describe the datasets and the pre-processing we used, then we introduce the baselines we considered, and finally we report the extensive empirical validation proving the effectiveness of our method. We will release the code to the public once the revision process is over. DATASETS In our experiments, we consider the English-French and English-German language pairs, on three different datasets. WMT'14 English-French We use the full training set of 36 million pairs, we lower-case them and remove sentences longer than 50 words, as well as pairs with a source/target length ratio above 1.5, resulting in a parallel corpus of about 30 million sentences. Next, we build monolingual corpora by selecting the English sentences from 15 million random pairs, and selecting the French sentences from the complementary set. The former set constitutes our English monolingual dataset. The latter set is our French monolingual dataset. The lack of overlap between the two sets ensures that there is not exact correspondence between examples in the two datasets. The validation set is comprised of 3,000 English and French sentences extracted from our monolingual training corpora described above. These sentences are not the translation of each other, and they will be used by our unsupervised model selection criterion, as explained in 3.2. Finally, we report results on the full newstest2014 dataset. WMT'16 English-German We follow the same procedure as above to create monolingual training and validation corpora in English and German, which results in two monolingual training corpora of 1.8 million sentences each. We test our model on the newstest2016 dataset. Multi30k-Task1 The task 1 of the Multi30k dataset (Elliott et al., 2016) has 30,000 images, with annotations in English, French and German, that are translations of each other. We consider the English-French and English-German pairs. We disregard the images and only consider the parallel annotations, with the provided training, validation and test sets, composed of 29,000, 1,000 and 1,000 pairs of sentences respectively. For both pairs of languages and similarly to the WMT datasets above, we split the training and validation sets into monolingual corpora, resulting in 14,500 monolingual source and target sentences in the training set, and 500 sentences in the validation set. BASELINES Word-by-word translation (WBW) The first baseline is a system that performs word-by-word translations of the input sentences using the inferred bilingual dictionary (Conneau et al., 2017). This baseline provides surprisingly good results for related language pairs, like English-French, where the word order is similar, but performs rather poorly on more distant pairs like English-German, as can be seen in Table 1. Word reordering (WR) After translating word-by-word as in WBW, here we reorder words using an LSTM-based language model trained on the target side. Since we cannot exhaustively score every possible word permutation (some sentences have about 100 words), we consider all pairwise swaps of neighboring words, we select the best swap, and iterate ten times. We use this baseline only on the WMT dataset that has a large enough monolingual data to train a language model. Oracle Word Reordering (OWR) Using the reference, we produce the best possible generation using only the words given by WBW. The performance of this method is an upper-bound of what any model could do without replacing words. Table 1: BLEU score on the WMT and Multi30k-Task1 datasets using greedy decoding. Supervised Learning We finally consider exactly the same model as ours, but trained with supervision, using the standard cross-entropy loss on the original parallel sentences. UNSUPERVISED DICTIONARY LEARNING To implement our baseline and also to initialize the embeddings Z of our model, we first train word embeddings on the source and target monolingual corpora using fastText (Bojanowski et al., 2017), and then we apply the unsupervised method proposed by Conneau et al. (2017) to infer a bilingual dictionary which can be use for word-by-word translation. Since WMT yields a very large-scale monolingual dataset, we obtain very high-quality embeddings and dictionaries, with an accuracy of 84.48% and 77.29% on French-English and German-English, which is on par with what could be obtained using a state-of-the-art supervised alignment method (Conneau et al., 2017). On the Multi30k datasets instead, the monolingual training corpora are too small to train good word embeddings (more than two order of magnitude smaller than WMT). We therefore learn word vectors on Wikipedia using fastText 2 . EXPERIMENTAL DETAILS Discriminator Architecture The discriminator is a multilayer perceptron with three hidden layers of size 1024, Leaky-ReLU activation functions and an output logistic unit. Following Goodfellow (2016), we include a smoothing coefficient s = 0.1 in the discriminator predictions. Training Details The encoder and the decoder are trained using Adam (Kingma & Ba, 2014), with a learning rate of 0.0003, β 1 = 0.5, and a mini-batch size of 32. The discriminator is trained using RMSProp (Tieleman & Hinton, 2012) with a learning rate of 0.0005. We evenly alternate between one encoder-decoder and one discriminator update. We set λ auto = λ cd = λ adv = 1. Table 1 shows the BLEU scores achieved by our model and the baselines we considered. First, we observe that word-by-word translation is surprisingly effective when translating into English, obtaining a BLEU score of 16.77 and 10.09 for fr-en on respectively Multi30k-Task1 and WMT datasets. Word-reordering only slightly improves upon word-by-word translation. Our model instead, clearly outperforms these baselines, even on the WMT dataset which has more diversity of topics and sentences with much more complicated structure. After just one iteration, we obtain a BLEU score of 27.48 and 12.10 for the en-fr task. Interestingly, we do even better than oracle reordering, suggesting that our model not only reorders but also correctly substitutes some words. After a few iterations, our model obtains BLEU of 32.76 and 15.05 on Multi30k-Task1 and WMT datasts for the English to French task, which is rather remarkable. Similar observations can be made for the other language pairs we considered. Comparison with supervised approaches Here, we assess how much labeled data are worth our two large monolingual corpora. On WMT, we trained a standard supervised model on both language pairs, using various amounts of parallel data. Figure 4-right shows the resulting performance. Our unsupervised approach obtains the same performance than a supervised model trained on about 100,000 parallel sentences, which is impressive. Of course, adding more parallel examples allows the supervised approach to outperform our method, but the good performance of our unsupervised method suggests that it could be very effective for low-resources languages where no parallel data are available. Moreover, these results open the door to the development of semi-supervised translation models, which will be the focus of future investigation. Figure 4-left illustrates the quality of the learned model after each iteration of the learning process in the language pairs of Multi30k-Task1 dataset, other results being provided in Table 1. One can see that the quality of the obtained model is high just after the first iteration of the process. Subsequent iterations yield significant gains although with diminishing returns. At iteration 3, the performance gains are marginal, showing that our approach quickly converges. Table 2 shows examples of translations of three sentences on the Multi30k dataset, as we iterate. Iteration 0 corresponds to the word-by-word translation obtained with our cross-lingual dictionary, which clearly suffers from word order issues. We can observe that the quality of the translations increases at every iteration. EXPERIMENTAL RESULTS Iterative Learning Ablation Study We perform an ablation study to understand the importance of the different components of our system. To this end, we have trained multiple versions of our model with some missing components: the discriminator, the cross-domain loss, the auto-encoding loss, etc. Table 3 shows that the best performance is obtained with the simultaneous use of all the described elements. The most critical component is the unsupervised word alignment technique, either in the form of a back-translation dataset generated using word-by-word translation, or in the form of pretrained embeddings which enable to map sentences of different languages in the same latent space. On the English-French pair of Multi30k-Task1, with a back-translation dataset but without pretrained embeddings, our model obtains a BLEU score of 25.29 and 26.10, which is only a few points below the model using all components. Similarly, when the model uses pretrained embeddings but no back-translation dataset (when λ cd = 0), it obtains 25.44 and 27.14. On the other hand, a model that does not use any of these components only reaches 8.78 and 9.15 BLEU. Source un homme est debout près d' une série de jeux vidéo dans un bar . Iteration 0 a man is seated near a series of games video in a bar . Iteration 1 a man is standing near a closeup of other games in a bar . Iteration 2 a man is standing near a bunch of video video game in a bar . Iteration 3 a man is standing near a bunch of video games in a bar . Reference a man is standing by a group of video games in a bar . Source une femme aux cheveux roses habillée en noir parleà un homme . Iteration 0 a woman at hair roses dressed in black speaks to a man . Iteration 1 a woman at glasses dressed in black talking to a man . Iteration 2 a woman at pink hair dressed in black speaks to a man . Iteration 3 a woman with pink hair dressed in black is talking to a man . Reference a woman with pink hair dressed in black talks to a man . Source une photo d' une rue bondée en ville . Iteration 0 a photo a street crowded in city . Iteration 1 a picture of a street crowded in a city . Iteration 2 a picture of a crowded city street . Iteration 3 a picture of a crowded street in a city . Reference a view of a crowded city street . Table 3: Ablation study on the Multi30k-Task1 dataset. The adversarial component also significantly improves the performance of our system, with a difference of up to 5.33 BLEU in the French-English pair of Multi30k-Task1. This confirms our intuition that, to really benefit from the cross-domain loss, one has to ensure that the distribution of latent sentence representations is similar across the two languages. Without the auto-encoding loss (when λ auto = 0), the model only obtains 20.02, which is 8.05 BLEU points below the method using all components. Finally, performance is greatly degraded also when the corruption process of the input sentences is removed, as the model has much harder time learning useful regularities and merely learns to copy input data. RELATED WORK A similar work to ours is the style transfer method with non-parallel text by Shen et al. (2017). The authors consider a sequence-to-sequence model, where the latent state given to the decoder is also fed to a discriminator. The encoder is trained with the decoder to reconstruct the input, but also to fool the discriminator. The authors also found it beneficial to train two discriminators, one for the source and one for the target domain. Then, they trained the decoder so that the recurrent hidden states during the decoding process of a sentence in a particular domain are not distinguishable according to the respective discriminator. This algorithm, called Professor forcing, was initially introduced by Lamb et al. (2016) to encourage the dynamics of the decoder observed during inference to be similar to the ones observed at training time. Before that, Hu et al. (2017) trained a variational autoencoder (VAE) where the decoder input is the concatenation of an unstructured latent vector, and a structured code representing the attribute of the sentence to generate. A discriminator is trained on top of the decoder to classify the labels of generated sentences, while the decoder is trained to satisfy this discriminator. Because of the nondifferentiability of the decoding process, at each step, their decoder takes as input the probability vector predicted at the previous step. Perhaps, the most relevant prior work is by He et al. (2016b), who essentially optimizes directly for the model selection metric we propose in section 3.2. One drawback of their approach, which has not been applied to the fully unsupervised setting, is that it requires to back-propagate through the sequence of discrete predictions using reinforcement learning-based approaches which are notoriously inefficient. In this work, we instead propose to a) use a symmetric architecture, and b) freeze the translator from source to target when training the translator from target to source, and vice versa. By alternating this process we operate with a fully differentiable model and we efficiently converge. CONCLUSION We presented a new approach to neural machine translation where a translation model is learned using monolingual datasets only, without any alignment between sentences or documents. The principle of our approach is to start from a simple unsupervised word-by-word translation model, and to iteratively improve this model based on a reconstruction loss, and using a discriminator to align latent distributions of both the source and the target languages. Our experiments demonstrate that our approach is able to learn effective translation models without any supervision of any sort. Figure 1 : 1Toy illustration of the principles guiding the design of our objective function. Left p D ( |z j ), with p D : R n → [0; 1], where 0 corresponds to the source domain, and 1 to the target domain. Figure 2 : 2Illustration of the proposed architecture and training objectives. The architecture is a sequence to sequence model, with both encoder and decoder operating on two languages depending on an input language identifier that swaps lookup tables. Top (auto-encoding): the model learns to denoise sentences in each domain. Bottom (translation): like before, except that we encode from another language, using as input the translation produced by the model at the previous iteration (light blue box). The green ellipses indicate terms in the loss function. data, as needed by the cross-domain loss function of Equation 2. At each iteration, a new encoder and decoder are trained by minimizing the loss of Equation 4 -line 7 of the algorithm. Then, a new translation model M (t+1) is created by composing the resulting encoder and decoder, and the process repeats. θ discr ← arg min L D , θ enc , θ dec , Z ← arg min L 8:M (t+1) ← e (t) • d (t) Figure 3 : 3Unsupervised model selection. BLEU score of the source to target and target to source models on the Multi30k-Task1 English-French dataset as a function of the number of passes through the dataset at iteration (t) = 1 of the algorithm. BLEU correlates very well with the proposed model selection criterion, see Equation 5. Figure 4 : 4Left: BLEU as a function of the number of iterations of our algorithm on the Multi30k-Task1 datasets. Right: The curves show BLEU as a function of the amount of parallel data on WMT datasets. The unsupervised method which leverages about 10 million monolingual sentences, achieves performance (see horizontal lines) close to what we would obtain by employing 100,000 parallel sentences. Table 2 : 2Unsupervised translations. Examples of translations on the French-English pair of the Multi30k-Task1 dataset. Iteration 0 corresponds to word-by-word translation. 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[ "https://github.com/facebookresearch/fastText" ]
[ "Machine Learning with Probabilistic Law Discovery: A Concise Introduction", "Machine Learning with Probabilistic Law Discovery: A Concise Introduction" ]
[ "Alexander Demin [email protected] \nErshov Institute of Informatics Systems\nNovosibirskRussia\n", "Denis Ponomaryov \nErshov Institute of Informatics Systems\nNovosibirskRussia\n" ]
[ "Ershov Institute of Informatics Systems\nNovosibirskRussia", "Ershov Institute of Informatics Systems\nNovosibirskRussia" ]
[]
Probabilistic Law Discovery (PLD) is a logic based Machine Learning method, which implements a variant of probabilistic rule learning. In several aspects, PLD is close to Decision Tree/Random Forest methods, but it differs significantly in how relevant rules are defined. The learning procedure of PLD solves the optimization problem related to the search for rules (called probabilistic laws), which have a minimal length and relatively high probability. At inference, ensembles of these rules are used for prediction. Probabilistic laws are human-readable and PLD based models are transparent and inherently interpretable. Applications of PLD include classification/clusterization/regression tasks, as well as time series analysis/anomaly detection and adaptive (robotic) control. In this paper, we outline the main principles of PLD, highlight its benefits and limitations and provide some application guidelines.
10.26516/1997-7670.2023.43.91
[ "https://export.arxiv.org/pdf/2212.11901v1.pdf" ]
254,974,372
2212.11901
191a6d5fa7b151e18ac4be1aa3b7f6b6ffb07e29
Machine Learning with Probabilistic Law Discovery: A Concise Introduction Alexander Demin [email protected] Ershov Institute of Informatics Systems NovosibirskRussia Denis Ponomaryov Ershov Institute of Informatics Systems NovosibirskRussia Machine Learning with Probabilistic Law Discovery: A Concise Introduction Probabilistic Rule LearningKnowledge Discoveryinterpretable Machine Learning Probabilistic Law Discovery (PLD) is a logic based Machine Learning method, which implements a variant of probabilistic rule learning. In several aspects, PLD is close to Decision Tree/Random Forest methods, but it differs significantly in how relevant rules are defined. The learning procedure of PLD solves the optimization problem related to the search for rules (called probabilistic laws), which have a minimal length and relatively high probability. At inference, ensembles of these rules are used for prediction. Probabilistic laws are human-readable and PLD based models are transparent and inherently interpretable. Applications of PLD include classification/clusterization/regression tasks, as well as time series analysis/anomaly detection and adaptive (robotic) control. In this paper, we outline the main principles of PLD, highlight its benefits and limitations and provide some application guidelines. Introduction Despite the popularity of neural network based and boosting models, there is still a big interest to logic based methods of Machine Learning, which support explicit representation of learned hypotheses. The inherent interpretability of these methods is what makes them particularly useful in critical domains such as, e.g., information security, medicine, automated control, etc. Of particular interest in the field of logic based Machine Learning is Probabilistic Law Discovery (PLD) [1], which is a variant of probabilistic rule learning. It allows for balancing between the completeness of the set of the learned hypotheses and computational expenses, and in the limit it guarantees learning the complete set of hypotheses true on data. While having some similarities with Decision Tree/Random Forest methods, PLD based models uniquely combine ensembling features with the property of being inherently interpretable. The explicitness of hypotheses learned by PLD allows for building glass-box classification, clusterization, regression, or adaptive control models, which also support straightforward integration of domain knowledge. The transparency of PLD based models makes them accessible for post-hoc meta-analysis to support transfer learning, conceptual abstraction, symmetry detection, etc. Similarly to some other logic based ML methods, the disadvantages of PLD are due to the complexity of rule learning, which is related to NP-hard problems and thus, direct implementations of PLD face efficiency problems when applied to datasets with big numbers of features. Currently several implementations of PLD are known, which combat the dimensionality problem with the help of heuristics. They have been benchmarked on different ML tasks against other well-known models, e.g., decision tress, neural networks, associative rules, in domains such as medicine [14], finance [11,15,16], bioinformatics [12], adaptive control [10,7,9,8,6]. The aim of this paper is to provide a concise and accessible introduction into Probabilistic Law Discovery, which covers the base learning algorithms, optimization techniques, and application guidelines. The exposition is based on the latest implementation of PLD, which provides a reasonable balance between the completeness of the learned hypotheses and computational complexity. Principles of Probabilistic Law Discovery Probabilistic Law Discovery is based on learning probabilistic rules on data as expressions in a human-readable formal language. Conceptually close to PLD are the decision tree/random forest methods, but the main difference is in how the most informative rules are defined and how they are learned. Probabilistic Rules A (probabilistic) rule is an expression of the form P 1 (x), . . . , P n (x) → R(x) (1) where x is a variable and R, P 1 , . . . , P n , n 0, are predicates. P 1 (x), . . . , P n (x) is the premise of the rule and R(x) is the conclusion. The rule size is the number of predicates in the premise. It is a common requirement that all predicates must be "simple enough" to compute. In applications, this requirement is specified as the existence of a polynomial/linear/logarithmic algorithm (wrt the size of the dataset) to compute the predicates. For instance, if there is a predicate HasChildren then there must be a procedure to compute in at most polynomial time (wrt the input data) for any object o whether HasChildren(o) holds on data. In general, a predicate can be given as a formula of first order logic with one free variable (called the object variable). Complex predicates (such as, e.g., HasChildren(x) ∨ HasBrother(x) or ∃y Brother(x, y)) can be written in terms of base predicates (HasChildren, HasBrother). In order to use PLD, one must first define a set of base predicates and procedures to compute them and then (if needed) one can introduce complex predicates as formulas over the base predicates. The following are examples of rules: CheapItem(x) → HighDemand(x) HasChildren(x), ∃y Brother(x, y) → Daughter(x) For object-feature datasets, the base predicates can be taken as corresponding to the features. However, with complex predicates one can formulate more expressive rules. For example, the predicate HasChildren(x) ∨ HasBrother(x) gives the set of objects that have at least one of the features. The predicate ∃y Brother(x, y) employs a relationship of objects. Depending on the choice of predicates we can see more or less information in the data. For an object-feature dataset D = O, F , let p be a probability measure on the set of objects O. The probability of a rule P 1 (x), . . . , P n (x) → R(x) on D is defined as the value of p(P 1 , . . . , P n , R) divided by p(P 1 , . . . , P n ), where p(P 1 , . . . , P n , R) is the probability measure of those objects o ∈ O, for which all P 1 (o), . . . , P n (o), R(o) hold (the meaning of p(P 1 , . . . , P n ) is defined similarly). In applications of PLD, the frequency probability measure is typically used, and thus, the rule probability reflects the number of objects for which both, the premise and conclusion hold, in relation to the number of objects, for which only the premise is true. Probabilistic Law Learning Given a predicate language, the natural question is which rules best reflect the regularities hidden in the data. One can enumerate rules in a brute-force fashion and estimate their probabilities, but the number of all rules in the given language is large and for many rules the probability may be close to zero, which means they are not too informative. PLD is based on the assumption that important are those rules, which have a minimal size and relatively high probability. This corresponds to the classical trade-off between the size and informativeness of compressed data representation (the so called Minimum Description Length principle). Essentially, PLD is solving a certain optimization problem of minimizing rule length, while maximizing the probability. A rule P 1 (x), . . . , P n (x) → R(x) is said to be a probabilistic law on a given dataset if it has a non-zero probability p and the following holds: the probability of any other rule with the same conclusion and a premise given by a proper subset of predicates P 1 , . . . , P n , is strictly less than p. Thus, P 1 (x), . . . , P n (x) → R(x) is the shortest rule (by inclusion of premises) with conclusion R, which has probability p. Note that the definition leaves the possibility that there may exist a rule with a superset of predicates in the premise and with the same conclusion, for which the probability is greater than p. It is known that the problem to find short rules having a given probability is computationally difficult. There may be exponentially many shortest rules, which have probability 1 on data [17], and the problem to decide whether there exists one with at most k predicates in the premise is NP-complete [18]. This implies that it is hard to compute the set of all probabilistic laws on a given dataset. The learning procedure of PLD is implemented by a heuristic algorithm, which has the following properties: for a given dataset and a predicate language, it outputs a set of rules in this language, which are probabilistic laws on this dataset in general the algorithm does not guarantee to find all the probabilistic laws on the dataset it allows for balancing between the completeness of the obtained set of laws and computation time in the limit (having unbounded computational resources) the algorithm computes the complete set of probabilistic laws on the given data The algorithm implements a directed enumeration. For a predicate R, it outputs a set of probabilistic laws with conclusion R. In practice, however, the computations can be organized in such a way as to obtain probabilistic laws for all conclusion predicates in one pass. The algorithm enumerates the rules with conclusion R in a directed manner, starting from the rule with the empty premise (∅ → R), by refinement (i.e., by adding predicates one-by-one to the premise). Clearly, refining a rule may change its probability. The principle hyperparameter of the algorithm is the base rule enumeration depth (denoted as d). The algorithm enumerates all rules whose premise consists of at most d predicates and selects those ones which are probabilistic laws. Since the base enumeration is complete, it is guaranteed that all probabilistic laws with at most d predicates in the premise are computed. After that, the algorithm selects from the obtained probabilistic laws those ones, which have exactly d predicates in the premise, and it starts refining only these rules (by successively adding predicates one by one to the premises), while checking whether their probability increases (i.e., whether the resulting rule remains a law). This stage of the algorithm is called additional enumeration. The premise of a rule is extended with new predicates as long as its probability increases. If, after an addition of a predicate, the probability is not increased, then such a refinement of the rule is discarded. Figure 1 illustrates the situation when the algorithm computes the probabilistic law P 1 , . . . , P 4 → R. When the predicate P 5 is added, the probability drops down, but as the predicate P 6 is added, it reaches a value that dominates the probabilities of all the rules with shorter premises (by set inclusion). That is, the rule P 1 , . . . , P 6 → R is a longer probabilistic law. In this situation, the algorithm does not find the rule P 1 , . . . , P 6 → R at additional enumeration and thus, it does not guarantee to find global maxima (note the red point in Figure 1 vs the local maxima depicted by the green point). In this sense, the algorithm can be classified as a hill-climbing one. The heuristic used in the algorithm is based on the assumption that probabilistic laws are typically arranged into chains, in which each subsequent law is obtained from the previous one by refinement with a single predicate, for example: { P 1 , P 2 → R , P 1 , P 2 , P 3 → R , P 1 , P 2 , P 3 , P 4 → R }. At the base enumeration step, the algorithm tries to capture the beginning fragments of these chains, and then, in the additional enumeration, it tries to find other laws from these chains. Requirement to Input Datasets The input data must be converted into a tabular object-feature representation. Categorical features must be converted to Boolean ones (e.g., by using one-hot encoding). Numeric features must be quantized and converted to Boolean ones, for example, by using an iterative splitting into ranges of values greater/smaller than the median. Application Scenarios PLD based models are used for the following tasks: probabilistic prediction of features (the classification task) [5,14,12] identification of features/specific feature values that an object must have in order to be assigned to a particular class (abductive classification) combining features into subsets closed wrt probabilistic laws and computing subsets of objects corresponding to these closed subsets (hierarchical object and feature clusterization) [2,3,4] prediction of value intervals for numeric features (interval regression) anomaly detection and time series analysis [13,15,16] building self-learning agent systems that interact with environments (reinforcement learning) [19] control of modular systems with many degrees of freedom, in particular, adaptive robotic control [10,7,9,8,6] We comment on solutions to these tasks in Section 5. The implementation of the Probabilistic Law Discovery algorithm is based on the construction of a rule derivation graph. The nodes in this graph are the rules enumerated by the algorithm and there is an edge from a rule r to r if r is a subrule of r , i.e. they have the same conclusion and the set of predicates from the premise of r is a proper subset of predicates from the premise of r . Figure 2 illustrates an example derivation graph over some dataset and the predicate language {R, A, B, C, D, E, . . .}. In this figure, only those edges are shown, which correspond to refinements computed by an (example) run the PLD algorithm. Probabilistic laws in Figure 2 are marked with the red color and those rules, which are not refined, are marked with a cross. In particular, A → R is not a probabilistic law (i.e., it has a probability less or equal to the probability of ∅ → R). In this example illustration, all rules up to the depth d = 2 are enumerated and it turns out that there are probabilistic laws A, C → R and B, C → R of depth 2. At the additional enumeration phase, these rules are refined, thus giving the probabilistic law B, C, D → R, which is refined further. Using the graph structure to represent rules allows for optimizing the search for probabilistic laws in several ways. By using the graph, one can quickly find subrules of a given rule when checking the conditions for being a probabilistic law. Also the graph allows one to store the statistics on the calculated rules so as not to count them twice. Since the same rule can be a subrule of several other child rules, its statistics (the data to compute the probability and significance of the rule) can be reused. Let R be some target predicate (e.g., a feature to be predicted). To compute a set of probabilistic laws with conclusion R, the derivation graph is used by the PLD algorithm as follows. Initially, the root node of the graph is generated with the rule ∅ → R. Statistics for this rule, such as probability, confidence, etc., are computed. Auxiliary sets REG 0 and N odes 0 are initialized to consist of this single rule. At step 1 k d, where d is the base enumeration depth (d 1), the next graph level is built, which consists of the nodes obtained as the refinement of the rules from N odes k−1 with a single predicate. The resulting nodes is the set N odes k . For each node N from N odes k : statistics for N is calculated -N is connected by an edge to each subrule of N from the previous graph layer it is verified whether N is a probabilistic law; the set REG k is defined to consist of all probabilistic laws from N odes k , which meet a statistical criterion At step k > d (additional enumeration), the set of all single-predicate refinements of probabilistic laws from REG k−1 is computed. The resulting nodes is the set N odes k . For each node N from N odes k : it is checked whether N meets a statistical criterion subrules of N size k − 1 are searched in the previous graph layer by using F indP arents procedure as follows. If there is a subrule r, for which p(r) p(N ), then N is not a probabilistic law and the procedure stops. Otherwise, each found subrule r is connected with node N by an edge. If some subrule of node N is not found in the previous layer, then a node N sub is created for this subrule in the graph and its statistics is calculated. If p(N sub ) p(N ) then the procedure stops. Finally, F indP arents is applied recursively to each subrule node of N . The set REG k is defined to consist of the probabilistic laws N which meet a statistical criterion. The PLD algorithm stops at a step k d (and it outputs the union of REG i , for all 0 i k) if either of the following conditions holds: the set REG k is empty (i.e., there are no probabilistic laws at level k in the graph); the number k equals to the maximal rule size M axSize (a hyperparameter for setting the maximal number of predicates in premises of rules considered by the algorithm). We note that the rule ∅ → R in REG 0 provides some basic information about how likely the objects from the dataset are to have feature R (under no further conditions). It is used, for example, in the classification task, in order to avoid prediction failure in cases where the attributes of an object being classified "cover" none of the premises of probabilistic laws with conclusion R. Note also that probabilistic laws are filtered out wrt a statistical criterion. For example, it can be the case that there is a single object in a dataset with the feature CheapP roduct and this object has also the feature HighDemand. In this case the rule CheapP roduct(x) → HighDemand(x) has probability 1 on the data, but it applies only to the single object and therefore is not informative for generalization to new samples. Because of this the PLD algorithm estimates the statistical significance of probabilistic laws. In implementations of PLD a statistical significance test with a confidence interval a is used, which is a hyperparameter of the algorithm. To further restrict the number of considered rules, the algorithm implements several optimizations, which we discuss below. Optimizations The PLD algorithm faces two principal computational problems. The algorithm implements a search in a rule space of size exponential wrt the number of predicates (features) from the given predicate language. Despite the additional enumeration heuristic, the number of rules considered by the algorithm may still be too large to complete computation within reasonable resources. The second problem is related to the estimation of rule statistics. For instance, computing rule confidence (the number of objects, for which the premise of a rule is true), when implemented naively, requires a full scan of the dataset. To solve the first problem, the following additional criteria and hyperparameters are used to reduce the number of probabilistic laws considered at additional enumeration: -Probability threshold. If the probability of a law is below a given threshold, then it is dropped (and is not further refined). -Statistical significance threshold. If the statistical significance of a probabilistic law is under a specified threshold, then it is dropped. -Probability gain threshold. Those refinements of a probabilistic law r are dropped that give a probability gain (wrt the value of p(r)) less than a specified threshold. -Probability gain thresholds for each level in the derivation graph. Similar to the previous criterion, a probability gain threshold can be set separately for each level in the graph. This allows for restricting the number of laws specifically for each level, in case there is a blow-up in the number of laws at certain graph levels. To solve the second computational problem, the derivation graph is used to speed up subrule search and statistics computation. In particular, the algorithm employs index caching for dataset objects: for a rule r, references to all those objects are stored, which r is applicable to (i.e., on which the rule premise is true). Then to compute statistics for a refinement r of r only these objects are used, one does not need to scan the complete dataset. This optimization significantly speeds up computations in practice. Clearly, the downside of caching is the need to store multiple indexes, which requires additional memory. Hyperparameters and Tuning The following list summarizes the main hyperparameters of PLD: base rule enumeration depth d maximum rule size (maximal enumeration depth) M axSize probability threshold for rules confidence (statistical significance) threshold for rules probability gain threshold (global threshold) for laws probability gain thresholds for each level of the derivation graph and for each law size (level/size specific threshold) One of the shortcomings of the current PLD implementation is related to the problem that it is hard to tell in advance how much time/memory will be required for searching probabilistic laws up to a given size M axSize. It might happen that there are too many probabilistic laws at some enumeration level. When iterating over refinements of these laws at the next level one faces combinatorial explosion: it is impossible to complete the search within acceptable time or memory limit for storing the derivation graph and caching statistics. One of the important aspects here is hyperparameter tuning, for which we recommend the following approach. One can make first a test run of PLD on a given training data with all thresholds set to zero. If an explosive growth in the number of rules is observed the thresholds are increased. Either the global probability gain threshold is increased or the level/size specific ones, if explosion occurs at a specific level of the derivation graph. The procedure is repeated until the algorithm is able to iterate to the specified size M axSize within acceptable time/resources. Another way is to adjust hyperparameters wrt the quality of predictions based on the computed probabilistic laws. In situation when PLD cannot enumerate laws up to a given size M axSize we have a choice: either find shorter probabilistic laws by reducing M axSize, or implement enumeration up to the required size by dropping some shorter laws rules by adjusting other thresholds. In this case, one can proceed in the standard way: select a small test subset of the training data and choose a option which provides the best prediction on this test subset. Applications The rule learning approach implemented by PLD is employed in different ML tasks as follows. Classification For each class label, a predicate (called predictor ) is introduced into the language and the PLD algorithm is run to learn probabilistic laws with these predicates in the conclusion. The resulting laws are used for classification of data objects in the following way. For an object, a subset of laws applicable to its features is selected. A law is applicable if the set the predicates from its premise is a subset of the predicates corresponding to the object features. Then the laws with maximal probability values are selected. If there is a single law of this kind then the class label for the object is defined by the predictor from the conclusion of the law. The probability for this label is defined as the probability value of the law. If there are several such laws then the object is not assigned a label (in this case classification fails). Clusterization Object and predicate (feature) clusters are defined via probabilistic laws in the following way. For a subset of predicates F an agreement measure of F is calculated as the difference between the sum of probabilities of laws P 1 (x), . . . , P n (x) → R(x), n 0 s.t. {P 1 , . . . , P n , R} ⊆ F and the sum of probabilities of those laws, whose premise is in F, but the conclusion is not. The measure reflects the difference between the total probability of laws true on F and the total probability of laws false on F. Predicate (feature) clusters are then defined via local maxima of the agreement measure: adding or removing any single predicate from a cluster yields a lower measure value. The set of laws true on a cluster F is called a characteristic set of F. Then two objects are assigned to the same (object) cluster if their feature sets have similar agreement measures wrt a characteristic set of some feature cluster. The result of a PLD based clusterization obtained this way is a partially ordered (wrt set inclusion) hierarchy of feature and object clusters. Regression In terms of PLD, regression is solved in several ways. The first one employs quantization: the rule language is extended with predicates, which correspond to certain ranges of feature values. The PLD algorithm is applied to learn probabilistic laws with these predictor predicates in the conclusion. Then regression is reduced to classification with the predictors being the class labels. Another approach employs averaging: if for an object o we have k 1 probabilistic laws with predictors R 1 , . . . , R k applicable to o and the ranges, which correspond to predictors, support averaging, then the average value for these ranges is returned as the resulting value. The third approach employs rules with binary inequality predicates for interval regression. We provide here an illustrating example. In general, PLD can support rules with n-ary predicates and functional terms via grounding (we comment on this extension of PLD in Section 7). If the language contains the predicate < and functions P rice, Demand, a variant of PLD can be applied to learn laws of the form P rice(x) < P rice(y) → Demand(x) > Demand(y) (where x, y are object variables and P rice(x) < P rice(y) is a more convenient writing for the atom < (P rice(x), P rice(y))). Then based on the laws P rice(item 3 ) < P rice(item 1 ) → Demand(item 3 ) > Demand(item 1 ) and P rice(item 1 ) < P rice(item 2 ) → Demand(item 1 ) > Demand(item 2 ) one obtains the interval (Demand(item 2 ), Demand(item 3 )) for Demand(item 1 ), with concrete values given by Demand function. Anomaly Detection Predicates are defined to correspond to the features of interest in the input data. Those probabilistic laws learned by PLD are selected which have probability above a certain threshold (for example, these can be laws with a probability greater than 0.9). They are considered as the rules describing the normal behavior of the system and thus, violation of these rules should indicate an abnormal event. Then the task is reduced to learning a parameter corresponding to the proportion of violated rules that should indicate an anomaly. Time Series Analysis The predicate language is defined to describe past states of the given time series in terms of important features such as extreme points, technical indicators, etc. The time series data is converted into a tabular form, where each row represents predicate values for each slice of interest of the time series. Then PLD is applied to learn probabilistic laws from the resulting tabular data and the laws are used for prediction in the same way as in classification. Adaptive Control One of the PLD based approaches is close to classical Reinforcement Learning. The agent history is analyzed to infer the laws that best predict the value of the reward based on the observed state and performed (series of) action(s). The agent decides on the next action based on the rules which are applicable to the observed state (given by a set of predicates) and provide the maximal reward prediction. If no rule is applicable to the observed state, the agent makes a random action. In the second approach, PLD is used for learning laws that predict transitions between states. The premise of every such law contains a state description in term of agent's sensor predicates and an action predicate, which corresponds to one of the available actions. The conclusion consists of predicates, which describe the state obtained after making this action. Action strategies (policies) are learned from the transition rules by grouping them into chains. The agent is able to reason about plausibility of these policies even though some of them may represent unseen trajectories. This approach is more sample efficient than the first one, but it consists of several learning components which in general require more computing resources. An important aspect in both approaches is that PLD allows for building complex hierarchical control systems with support for automatic subgoal discovery. An agent is able to dynamically extend the predicate language with shortcuts (new sensor predicates) for important intermediate subgoal states that must be achieved on the way to the primary goal. If a subgoal predicate is present in the premise of a transition rule used by the agent, it switches to achieving this subgoal by using the appropriate policies for the subgoal. Limitations -Probabilistic Rule Discovery is computationally expensive. The worst-case complexity of the base enumeration phase of PLD algorithm can be estimated as N d , where N is the number of predicates in the language and d is the base enumeration depth. It is problematic to use PLD on datasets with large numbers of features (one can reduce the complexity by lowering the parameter d, but then the algorithm may miss many informative rules). -It is problematic to use PLD for datasets containing a large number of continuous features. This is due to the fact that PLD is a discrete learning model and thus, every continuous feature has to be quantized into a set of discrete ones (for example, by using one-hot coding and/or iterative splitting by medians). This, in turn, leads to problems related to the large number of discrete features and besides, with coarse splitting, some important information may be lost. In particular, because of these reasons the application of PLD for audio/video/image data is limited. -Similarly, it is hard to use PLD for regression tasks in general. It is required to either quantize data or predict the relationships of target feature values to the values of other features in terms of comparison predicates like . -In order to apply PLD, one has to explicitly define the language of predicates (features), which essentially requires feature engineering. There exists a PLD based approach that supports feature discovery by using the so-called Probabilistic Formal Concepts [2,3]. These combine base features into groups of interrelated ones, but anyway the set of base features must be explicitly given. Advantages -Probabilistic laws learned by PLD are explicit and human readable. PLD based solutions to ML tasks listed in Section 5 are inherently interpretable and thus, PLD can be used as a basis for building explainable ML models. -The form of probabilistic laws resembles rules of decision trees. However, in general, for every target (predictor) predicate, the PLD algorithm learns not just a single probabilistic law, but several ones. In this aspect, PLD is closer to ensemble methods on decision trees. However, many such methods, such as, e.g., Random Forest, rely heavily on randomness in the construction of the trees and also they employ non-trivial operators for combining predictions over an ensemble. This makes these methods difficult to interpret and requires additional mechanisms for explaining outputs. In contrast, PLD has both advantages: interpretability and the feature of making decisions based on an "ensemble" of probabilistic laws. -In contrast to black-box models, the explicitness of the rule language used by PLD allows for embedding domain knowledge into the model. For example, the knowledge can be introduced as rules with probability 1 (ground truth), or with a lower probability. When injected into the model, these rules are used for prediction together with learned probabilistic laws. -The explicitness of probabilistic laws makes it possible to perform post-hoc analysis of a trained PLD model. In fact, the learned laws can be viewed as explicit instructions (an algorithm) for decision making. These can be reused for building PLD models for other datasets (transfer learning). The explicitness of the learned algorithm makes it available for meta-analysis in order to identify, for example, deeply correlated features (as implemented in the method of Probabilistic Formal Concepts), or those features/specific feature values that an object must have in order to be assigned to a particular class (abductive classification), or detection of symmetries in the structure of laws and features, which is important for sample efficiency of self-learning systems with many degrees of freedom. 7 Directions for Improvement Support for n-ary Predicates The choice of predicates has a direct impact on the expressiveness of PLD models. In general, PLD allows for using use not only unary, but also n-ary predicates to support laws of the form P 1 (x 1 ) , . . . , P n (x n ) → R (y)(2) where x 1 , . . . , x n are (possibly non-disjoint) sets of variables and y is a subset of variables from the union of x i , for i = 1, . . . , n. Besides object variables, functional terms can be used. Rules with n-ary predicates and functional terms are more expressive, for example, they can express relationships of features for pairs of objects like: P rice(x) < P rice(y) → Demand(y) < Demand(x) Support for the extended rule language in PLD is currently implemented by grounding. For every combination of values for the variables of a n-ary predicate, a unary base predicate is introduced into the language. Clearly, this approach is computationally inefficient. More efficient ways to support n-ary predicates would significantly increase the expressiveness of PLD. For example, rule 3 with the binary comparison predicate is an example of a regularity that is hard to express with models based on neural networks. Learned Quantization As PLD does not support continuous features, they must be converted into Boolean ones, for example, by quantization. This implies additional difficulties as the quantization granularity needs to be carefully adjusted for each continuous feature so as to not discard information and avoid a blow-up in the number of features. Quantization can be built into the learning algorithm of PLD as follows. When enumerating rules by refinement, we first take those predicates for addition into the premise, which divide the range of the corresponding feature values by the median into two subranges. Then, after some of these predicates corresponding to a value range r is added to the premise, for further refinement, we consider those predicates that split r into smaller subranges, and we proceed similarly in further refinements. As a result, the range of feature values will be split up as long as the rule probability increases and its confidence is maintained. Thus, there will be no need to quantize the features in advance, the required quantization granularity will be adjusted automatically in the process of learning. Enumeration Reduction To improve the efficiency of learning, different hyperparameters are used in PLD to control the form and significance thresholds for rules (Section 4). The disadvantage of this approach is that it is hard to know in advance how much computational resources will be spent to search for probabilistic laws under the specified parameters. One has to make several test runs of the learning procedure to assess the complexity of computations, which is inefficient. An alternative way would be to select some limited number of the most promising rules at each level of enumeration according to some criterion. The maximal number of rules considered in the enumeration can be limited depending on the available computational resources and the selection criterion can be based, e.g, on the probability gain for rules after refinement, the entropy criterion (like in decision trees), etc. Conclusion In this paper, we have presented the main concepts of Probabilistic Law Discovery and discussed its advantages and limitations. Based on our analysis, we can summarize recommendations on application of PLD as follows: -PLD can be used to build interpretable ML models for tasks, which require explainable decision making it is not efficient to use PLD for tasks which do not require explainability and which involve datasets with large numbers of features or which contain many continuous features (for example, image/audio/video datasets) -PLD can be considered as an alternative to common ML methods in tasks where explainability is not required, but where datasets contains mostly Boolean or categorical features Fig. 1 . 1Probability change for rules with conclusion R and different premises Fig. 2 . 2An example rule derivation graph for base enumeration depth d = 2 The principle problem of rule learning can be attacked by improving the optimization techniques already present in PLD, as well as by adopting other techniques from Operations Research and Machine Learning. Applications of PLD to the variety of ML problems (including classification/clusterization/regression, time series/anomaly analysis, and adaptive control) evidence the high potential of models based on rule learning. We believe that elements of PLD based models can be reused to build interpetable solutions to various ML tasks on top of logic based and rule learning models like Decision Trees/Random Forests, and others. Relational Approch to Knowledge Discovery and its Applications. A V Demin, E E Vityaev, Proc. ZONT Conference. 2013. ZONT Conference. 2013Novosibirsk1in RussianDemin A.V., Vityaev E.E. Relational Approch to Knowledge Discovery and its Applications. Proc. ZONT Conference. 2013, vol. 1, Novosibirsk, pp. 122-130. (in Russian) Probabilistic Generalization of Formal Concepts. Programming and Computer Software. E E Vityaev, A V Demin, D K Ponomaryov, Pleiades Publishing38Vityaev E.E., Demin A.V., Ponomaryov D.K. Probabilistic Generalization of For- mal Concepts. Programming and Computer Software. 2012, vol. 38(5), Pleiades Publishing, pp. 219-230. Probabilistic Concepts in Formal Contexts. E E Vityaev, A V Demin, D K Ponomaryov, Lecture Notes in Computer Science. 7162Springer VerlagVityaev E.E., Demin A.V., Ponomaryov D.K. Probabilistic Concepts in Formal Contexts. Lecture Notes in Computer Science. 2012, vol. 7162, Springer Verlag, pp. 394-410. Data Clusterization with the Logic Probabilistic Approach to Knowledge Discovery. Information Technology in the Humanities. A V Demin, IAET SB RAS. 21in RussianDemin A.V. Data Clusterization with the Logic Probabilistic Approach to Knowl- edge Discovery. Information Technology in the Humanities. 2015, vol. 21, IAET SB RAS, Novosibirsk, pp. 28-33. (in Russian) A V Demin, E E Vityaev, Method, Classification Information Technology in the Humanities. 2010. Novosibirsk15in RussianDemin A.V., Vityaev E.E. A Method for "Natural" Classification Information Technology in the Humanities. 2010, vol. 15, IAET SB RAS, Novosibirsk, pp. 16-22. (in Russian) Deep Learning of Adaptive Control Systems Based on a Logicalprobabilistic Approach. A V Demin, Series Mathematics. 2021. Irkutsk38Bulletin of Irkutsk State UniversityDemin A.V. Deep Learning of Adaptive Control Systems Based on a Logical- probabilistic Approach. Bulletin of Irkutsk State University, Series Mathematics. 2021, vol. 38, Irkutsk, pp. 65-83. Adaptive Locomotion Control System for Robots with Arbitrarily Modular Design. A V Demin, Procedia Computer Science. 2020. edia Computer Science. 2020Elsevier169Demin A.V. Adaptive Locomotion Control System for Robots with Arbitrarily Modular Design. Procedia Computer Science. 2020, vol. 169, Elsevier, pp. 829- 834. Adaptive Control of Multiped Robot. A V Demin, E E Vityaev, Procedia Computer Science. 145ElsevierDemin A.V., Vityaev E.E. Adaptive Control of Multiped Robot. Procedia Com- puter Science. 2018, vol. 145, Elsevier, pp. 629-634. Adaptive Control of Modular Robots. Biologically Inspired Cognitive Architectures (BICA) for Young Scientists. A V Demin, E E Vityaev, Advances in Intelligent Systems and Computing. Springer636Demin A.V., Vityaev E.E. Adaptive Control of Modular Robots. Biologically In- spired Cognitive Architectures (BICA) for Young Scientists, Advances in Intelli- gent Systems and Computing. 2018, vol. 636, Springer, pp. 204-212. Learning in a Virtual Model of the C. elegans Nematode for Locomotion and Chemotaxis. A V Demin, E E Vityaev, Biologically Inspired Cognitive Architectures. 7ElsevierDemin A.V., Vityaev E.E. Learning in a Virtual Model of the C. elegans Nematode for Locomotion and Chemotaxis. Biologically Inspired Cognitive Architectures. 2014, vol.7, Elsevier, pp. 9-14. The Development of a Universal DISCOVERY Knowledge Mining System and its Applications. A V Demin, E E Vityaev, 7NovosibirskBulletin of Novosibirsk State University: Information Technologies. in RussianDemin A.V., Vityaev E.E. The Development of a Universal DISCOVERY Knowl- edge Mining System and its Applications. Bulletin of Novosibirsk State University: Information Technologies. 2009, vol. 7(1), Novosibirsk, pp. 73-83. (in Russian) Transcription Factor Binding Site Discovery by the Probabilistic Rules. I Khomicheva, A V Demin, E E Vityaev, Proc. 2nd Workshop on Data mining in Functional Genomics and Proteomics. The 18th European Conference on Machine Learning and the 11th European Conference on Principles and Practice of Knowledge Discovery in Databases. 2nd Workshop on Data mining in Functional Genomics and Proteomics. The 18th European Conference on Machine Learning and the 11th European Conference on Principles and Practice of Knowledge Discovery in DatabasesWarsaw, PolandKhomicheva I., Demin A.V., Vityaev E.E. Transcription Factor Binding Site Discovery by the Probabilistic Rules. Proc. 2nd Workshop on Data mining in Functional Genomics and Proteomics. The 18th European Conference on Ma- chine Learning and the 11th European Conference on Principles and Practice of Knowledge Discovery in Databases. 2007, Warsaw, Poland, pp.104-109. Regularity Discovery and Anomaly Detection in Network Traffic Bulletin of Novosibirsk State University: Information Technologies. E E Vityaev, B K Kovalerchuk, A M Fedotov, V B Barahnin, S D Belov, D S Durdin, A V Demin, 6Novosibirskin RussianVityaev E.E., Kovalerchuk B.K., Fedotov A.M., Barahnin V.B., Belov S.D., Dur- din D.S., Demin A.V. Regularity Discovery and Anomaly Detection in Network Traffic Bulletin of Novosibirsk State University: Information Technologies. 2008, vol. 6(2), Novosibirsk, pp. 57-68. (in Russian) The Development of a Universal DISCOV-ERY Knowledge Mining System and its Applications in. A V Demin, E E Vityaev, T L Poloz, Medical Diagnostics Proc. ZONT conference. Novosibirsk1in RussianDemin A.V., Vityaev E.E., Poloz T.L. The Development of a Universal DISCOV- ERY Knowledge Mining System and its Applications in Medical Diagnostics Proc. ZONT conference. 2007, vol. 1, Novosibirsk, pp. 63-70. (in Russian) A V Demin, E E Vityaev, Financial Time Series: Prediction and Detecting Dynamics Change. Proc. ZONT conference. in RussianDemin A.V., Vityaev E.E. Financial Time Series: Prediction and Detecting Dy- namics Change. Proc. ZONT conference. 2009, pp. 79-86. (in Russian) Bulletin of 7th Intl. Ershov conference Perspectives of Informatics Systems. A V Demin, E E Vityaev, A Technology for Predicting Financial Time Series. in RussianDemin A.V., Vityaev E.E. A Technology for Predicting Financial Time Series. Bulletin of 7th Intl. Ershov conference Perspectives of Informatics Systems. 2009, pp. 114-119. (in Russian) On the Intractability of Computing the Duquenne-Guigues Base. S O Kuznetsov, J. of Universal Computer Science. 108S.O. Kuznetsov. On the Intractability of Computing the Duquenne-Guigues Base. J. of Universal Computer Science, 2004, vol. 10, no. 8. Discovering All Most Specific Sentences. D Gunopulos, R Khardon, H Mannila, S Saluja, H Toivonen, R S Sharma, ACM Transactions on Database Systems. 282D. Gunopulos, R. Khardon, H. Mannila, S. Saluja, H. Toivonen, and R.S. Sharma. Discovering All Most Specific Sentences. ACM Transactions on Database Systems, 2003, vol. 28, no. 2. Interpretable Reinforcement Learning with Multilevel Subgoal Discovery. A V Demin, D K Ponomaryov, Proc. 21st International Conference on Machine Learning and Applications. 21st International Conference on Machine Learning and ApplicationsDemin A.V. and Ponomaryov D.K. Interpretable Reinforcement Learning with Multilevel Subgoal Discovery. Proc. 21st International Conference on Machine Learning and Applications. 2022.
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[ "GENERALIZED RIEMANN FUNCTIONS, THEIR WEIGHTS, AND THE COMPLETE GRAPH", "GENERALIZED RIEMANN FUNCTIONS, THEIR WEIGHTS, AND THE COMPLETE GRAPH" ]
[ "Nicolas Folinsbee ", "Joel Friedman " ]
[]
[]
By a Riemann function we mean a function f : Z n → Z such that f (d) is equals 0 for d 1 + · · · + dn sufficiently small, and equals d 1 + · · · + dn + C for a constant, C, for d 1 + · · · + dn sufficiently large. By adding 1 to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions.To each Riemann function we associate a related function W : Z n → Z via Möbius inversion that we call the weight of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function f satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for the complete graph.For the complete graph, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to another linear time algorithm to compute the Baker-Norine rank, via a formula likely related to one of Cori and Le Borgne, but seemingly simpler, namelyOur study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.
10.37236/11281
[ "https://arxiv.org/pdf/2205.13592v1.pdf" ]
249,152,287
2205.13592
d7085f6556faef8ccbe8bab5683e47ff455a8d59
GENERALIZED RIEMANN FUNCTIONS, THEIR WEIGHTS, AND THE COMPLETE GRAPH Nicolas Folinsbee Joel Friedman GENERALIZED RIEMANN FUNCTIONS, THEIR WEIGHTS, AND THE COMPLETE GRAPH By a Riemann function we mean a function f : Z n → Z such that f (d) is equals 0 for d 1 + · · · + dn sufficiently small, and equals d 1 + · · · + dn + C for a constant, C, for d 1 + · · · + dn sufficiently large. By adding 1 to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions.To each Riemann function we associate a related function W : Z n → Z via Möbius inversion that we call the weight of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function f satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for the complete graph.For the complete graph, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to another linear time algorithm to compute the Baker-Norine rank, via a formula likely related to one of Cori and Le Borgne, but seemingly simpler, namelyOur study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions. The main goal of this article is to give a combinatorial study of what we call Riemann functions and their weights. Our main motivation is to gain insight into the special case that is the Graph Riemann-Roch fomula of Baker and Norine [BN07]; the Baker-Norine formula has received a lot of recent attention [CB13,Bac17,MS14,CLM15], as has its generalization to tropical curves and other settings in recent years [Bac17, GK08, HKN13, JM13, AC13, MS13, AM10,CDPR12]. We were first interested in weights to address a question posed in [BN07] regarding whether or not their Graph Riemann-Roch formula could be understood as an Euler characteristic equation; this is partially answered in [FF]. However, weights are interesting for a number of purely combinatorial reasons: first, a Riemann-Roch formula is simpler to express in terms of the weight of the Riemann function. Second, the weights of the Riemann-Roch functions of certain graphs are very simple to write down. For example, in this article we build on the methods of Cori and Le Borgne [CB13] to give a very simple formula for the weights of the Baker-Norine rank function of a complete graph; this will allow us to prove a likely simpler variant of their algorithm to compute the values of this rank function. Furthermore, for the above reasons, as well as its connections to sheaves and Euler characteristics in [FF], we suspect that weights may be a useful way to describe many Riemann functions. This article has two types of results: foundational results on Riemann functions and Riemann-Roch type formulas, and calculations of the weights of Baker-Norine rank functions of two types of graphs. Let us briefly summarize the results, assuming some terminology that will be made precise in Section 2. 1.1. Riemann Functions and Weights. By a Riemann function we mean a function f : Z n → Z such that f (d) = f (d 1 , . . . , d n ) is initially zero, meaning f (d) = 0 for deg(d) = d 1 + · · · + d n sufficiently small, and eventually-meaning for deg(d) sufficiently large-equals deg(d) + C for a constant, C ∈ Z, which we call the offset of f . By adding 1 to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. If f : Z n → Z is any function that is initially zero, then there is a unique, initially zero W such that f (d) = d ≤d W (d ) where ≤ the usual partial order on Z n (i.e., d ≤ d means d i ≤ d i for all i = 1, . . . , n); we call W the weight of f . If f is a Riemann function, then W is also eventually zero; much of what we prove about Riemann functions also holds for generalized Riemann functions, which we define as any initially zero function f whose weight is eventually zero. Returning to a Riemann function f : Z n → Z with offset C, for any K ∈ Z n there exists a unique function f ∧ K such that for all d ∈ Z n we have (1) f (d) − f ∧ K (K − d) = deg(d) + C, and we refer to as a generalized Riemann-Roch formula; f ∧ K is also a Riemann function. Furthermore, if f ∧ K = f for some f, K, then the formula reads f (d) − f (K − d) = deg(d) + C, which is the usual type of Riemann-Roch formula, both the classical formula of Riemann-Roch, and the Baker-Norine analog. Hence, our view of Riemann-Roch formulas is more "happy-go-lucky" than is common in the literature: for each f, K there is a generalized Riemann-Roch formula (1); we study any such formula, and view the case where f ∧ K = f as a special case which we call self-duality. We are interested in weight functions, W , for a number of reasons: (1) the weights of the Baker-Norine rank (plus 1) of the graphs we study in this article turn out be be simple to describe and very sparse (i.e., mostly 0); by contrast, at least for the complete graph, the Baker-Norine function is more difficult to compute. Hence the weights may be a more efficient way to encode certain Riemann functions of interest. (2) For a Riemann function f : Z n → Z, the weight of f ∧ K turns out to equal (−1) n W * L , where L = K + 1 (where 1 = (1, . . . , 1)), and W * L is the function W * L (d) = W (L − d); hence it seems easier to check self-duality using the weight, W , rather than directly on f . (3) In [FF], we model Riemann functions by restricting f : Z n → Z to two of its variables, while holding the other n − 2 variables fixed; if f satisfies self-duality, a two-variable restriction, f : Z 2 → Z, of f will generally not be self-dual; however K ∈ Z 2 can be described as a restriction of f ∧ K (for any K ∈ Z n ). Since self-duality isn't preserved under restrictions, but generalized Riemann-Roch formulas behave well under restrictions, it seems essential to work with generalized Riemann-Roch formulas (1) in [FF] or whenever we wish to work with restrictions of Riemann functions to a subset of their variables. (4) In certain Riemann functions of interest, such as those considered by Amini and Manjunath [AM10], self-duality does not generally hold, and yet one can always work with weights and generalized Riemann-Roch formulas. (5) The formalism of weights applies to generalized Riemann functions, which is a much wider class of functions, and we believe likely to be useful in future work to model other interesting functions. In this case (1) is replaced by f (d) − f ∧ K (K − d) = h(d), where h is the unique modular function that eventually equals f (see Section 3). One might expect such formulas to hold when, for example f = f (d) is the sum of even Betti numbers of a sheaf depending on a parameter d ∈ Z n , whose Euler characteristic equals a modular function h. 1.2. The Weight of the Baker-Norine rank for Two Types of Graphs. The second type of result in this article concerns the weights of the Baker-Norine rank function (plus 1) for two types of graphs, namely graphs on two vertices and the complete graph, K n , on n vertices. Both types of weight functions are quite simple and very sparse (i.e., mostly 0). For K n we build on the ideas of Cori and Le Borgne [CB13] to compute the weight of the Baker-Norine rank. A side effect of this computation is a formula for the Baker-Norine rank: r BN,Kn (d) = −1+ i = 0, . . . , deg(d) n−2 j=1 (d j −d n−1 +i) mod n ≤ deg(d)−i , where the "mod" function above returns a value in {0, . . . , n − 1}; this looks related to a formula given by Cori and Le Borgne. We also explain that-like the Cori and Le Borgne algorithm-there is an algorithm that computes this function in time O(n). Our proof of this formula is self-contained, although uses some of the observations of Cori and Le Borge including one short and rather ingenious idea of theirs regarding the Baker-Norine function on a complete graph. 1.3. Organization of this Article. The rest of this article is organized as follows. In Section 2 we give some basic terminology, including the definition of a Riemann function and some examples, which (after subtracting 1) includes the Baker-Norine rank. In Section 3 we discuss what we mean by the weight of a Riemann function; this leads to a notation of generalized Riemann functions, which share many of the properties of Riemann functions. In Section 4 we define what we mean by a Riemann-Roch formula; we describe the equivalent condition on weights, which is simpler; these ideas generalize in a natural way to the setting of generalized Riemann functions. In Section 5 we compute the weight of the Baker-Norine rank for graphs on two vertices, joined by any number of edges. In Section 6 we compute the weight of the Baker-Norine rank for a complete graph on n vertices, and we give a formula for the Baker-Norine rank, which-like a related formula of Cori and Le Borgne-allows the rank to be computed in linear time in n. In Section 7 we prove our main theorems-stated earlier-that characterize modular functions used to define generalized Riemann functions. Basic Terminology and Riemann Functions In this section we introduce some basic terminology and define the notion of a Riemann function. Then we give some examples of Riemann functions. 2.1. Basic Notation. We use Z, N to denote the integers and positive integers; for a ∈ Z, we use Z ≤a to denote the integers less than or equal to a, and similarly for the subscript ≥ a. For n ∈ N we use [n] to denote {1, . . . , n}. We use bold face d = (d 1 , . . . , d n ) to denote elements of Z n , using plain face for the components of d; by the degree of d, denoted deg(d) or at times |d|, we mean d 1 + . . . + d n . We set Z n deg 0 = {d ∈ Z n | deg(d) = 0}, and for a ∈ Z we similarly set Z n deg a = {d ∈ Z n | deg(d) = a}, Z n deg≤a = {d ∈ Z n | deg(d) ≤ a}. We use e i ∈ Z n (with n understood) be the i-th standard basis vector (i.e., whose j-th component is 1 if j = i and 0 otherwise), and for I ⊂ [n] (with n understood) we set (2) e I = i∈I e i ; hence in case I = ∅ is the empty set, then e ∅ = 0 = (0, . . . , 0), and similarly e [n] = 1 = (1, . . . , 1). For n ∈ N, we endow Z n with the usual partial order, that is d ≤ d iff d i ≤ d i ∀i ∈ [n],Z n → Z is a Riemann function if for some C, a, b ∈ Z we have (1) f (d) = 0 if deg(d) ≤ a; and (2) f (d) = deg(d) + C if deg(d) ≥ b; we refer to C as the offset of f . In our study of Riemann functions, it will be useful to introduce the following terminology. Definition 2.2. If f, g are functions Z n → Z, we say that f equals g initially (respectively, eventually) if f (d) = g(d) for deg(d) sufficiently small (respectively, sufficiently large); similarly, we say that that f is initially zero (respectively eventually zero) if f (d) = 0 for deg(d) sufficiently small (respectively, sufficiently large). Therefore f : Z n → Z is a Riemann function iff it is initially zero and it eventually equals the function deg(d) + C, where C is the offset of f . 2.3. The Baker-Norine Rank and Riemann-Roch Formula. In this article we study examples of the Baker-Norine rank for various graphs. In this subsection we briefly review its definition and its properties; for more details, see [BN07]. We will consider graphs, G = (V, E) that are connected and may have multiple edges but no self-loops. Recall that if G = (V, E) is any graph, then its Laplacian, ∆ G equals D G − A G where D G is the diagonal degree counting matrix of G, and A G is the adjacency matrix of G. Definition 2.3 (The Baker-Norine rank function of a graph). Let G = (V, E) be a connected graph without self-loops (but possibly multiple edges) on n vertices that are ordered as v 1 , . . . , v n . Hence we view its Laplacian, ∆ G , as a map Z n → Z n . Let L = Image(∆). We say that d, d ∈ Z n are equivalent, written d ∼ d , if d − d ∈ L, and say that d is effective if d ≥ 0. Let N be the elements of Z n that are not equivalent to an effective element of Z n ; in particular deg(d) < 0 ⇒ d ∈ N . Consider (3) f (d) = ρ L 1 (d, N ) = min d ∈N d − d L 1 , where · L 1 is the usual L 1 -norm (x 1 , . . . , x n ) L 1 = |x 1 | + · · · + |x n |. We also write f = f G , to emphasize the graph G, although its definition as a function Z n → Z also depends on the ordering v 1 , . . . , v n of its vertices. The Baker-Norine rank of d, denoted r BN (d), is f (d) − 1. Since f (d) = 0 iff d ∈ N , which is the case if deg(d) < 0, it follows f is initially zero, and hence r BN (d) initially equals −1. We remark that for f (d) ≥ 0 we easily see that both: (1) f (d) equals the largest integer m ≥ 0 such that for any a ≥ 0 and of degree m we have that d − a is equivalent to an effective element of Z n , and (2) f (d) = 1 + min i∈[n] f (d − e i ). The Baker-Norine Graph Riemann-Roch formula states that for all d we have (4) r BN (d) − r BN (K − d) = deg(d) + 1 − g where (1) g = 1 + |E| − |V | (which is non-negative since G is connected), and (2) K = deg G (v 1 ) − 2, . . . , deg G (v n ) − 2 , where deg G (v) is the degree of v in G, i.e., the number of edges incident upon v in G. It follows that for all d ∈ Z n (5) f (d) − f (K − d) = deg(d) + 1 − g. It follows that for d such that deg(d) > deg(K) = i deg G (v i ) − 2 = 2|E| − 2|V | we have f (K − d) = 0; hence (6) deg(d) > 2|E| − 2|V | ⇒ f (d) = deg(d) + 1 − g, i.e., f (d) eventually equals deg(d) + 1 − g. Hence f is a Riemann function with offset C = 1 − g. The Baker-Norine formula is an analog of the classical Riemann-Roch formula for algebraic curves or Riemann surfaces; we briefly discuss this in Subsection 2.5. 2.4. Generalizations of the Baker-Norine Rank. Many variants of the Baker-Norine rank have been studied. We remark that in literature that generalizes that Baker-Norine rank, e.g., [AM10], one typically studies the function r = f − 1 where f is as in (3) for various N , and hence r is initially −1 instead of initially 0. Example 2.4. Amini and Manjunath [AM10] generalized Definition 2.3 by taking L ⊂ Z n deg 0 be any lattice of full rank in Z n deg 0 (i.e., rank n − 1); it this case the definitions of "equivalent," "effective," and of N in Definition 2.3 carry over; they show that f as in (3) is a Riemann funtion with offset is 1 − g max (L), with g max (L) as defined on page 5 there. They also give conditions on L so that a Riemann-Roch analog (5) holds; one of their conditions is that all maximal points of N have the same degree (i.e., g min = g max as in [AM10]); they give a second, more technical condition. To generalize the above examples, let us give some conditions on a subset N ⊂ Z n which ensure that f in (3) gives a Riemann function. Proposition 2.5. Let n ∈ N and N ⊂ Z n such that (1) for some m, m ∈ Z we have b = deg(d) − M − Cn ≥ 0 we have d = d − C1 − be 1 has degree M ; hence for some d ∈ N ∩ Z n M we have d − d = a where |a 1 | + · · · + |a n | ≤ C; hence |a i | ≤ C for all i. It follows that setting a to be a = d − d = d − (a + d) = C1 + be 1 − a, we have a 1 = C + a 1 + b and for i ≥ 2, a i = C + a i , and hence all a i ≥ 0. Hence the L 1 distance of d to d is at most a 1 + · · · + a n = deg Remark 2.6. Condition (2) of Proposition 2.5 on N above follows from the following stronger condition: for any N ⊂ Z n , say that d ∈ Z n is an invariant translation of N if for all d ∈ Z n , d ∈ N iff d + d ∈ N . We easily see that the set, T = T (N ) of all invariant translations is a subgroup of the additive group Z n , and that (7) implies that T ⊂ Z n deg 0 . If T is a full rank subgroup of Z n deg 0 (i.e., of rank n − 1), then condition (2) of Proposition 2.5 is automatically satisfied. f (d) = min{deg(d − d ) | d ∈ N , d ≤ d} = min{deg(d − d ) | f (d ) = 0}. Furthermore, if N is a downset, then for any i ∈ [n], any path from a d ∈ Z n to a d ∈ N translates to a path of the same length from d − e i to d − e i , which again lies in N . Hence if N is a downset, then f = f (d) as in (3) is a non-decreasing function of d. Remark 2.8. We remark that if L ⊂ Z n deg 0 is not of full rank in Example 2.4, then condition (2) of Proposition 2.5 fails to hold, and we easily see that f in (3) fails to be a Riemann function. 2.5. Examples Based on Riemann's Theorem. All the above discussion is based on the classical Riemann's theorem and Riemann-Roch theorem. However, we use these examples only for illustration, and they are not essential to our discussion of the Baker-Norine rank functions and of most of the rest of this article. Let X be an algebraic curve over an algebraically closed field k, and K be its function field; one understands either (1) K is a finite extension of k(x) where x is an indeterminate (i.e., transcendental) and X is its set of discrete valuations (e.g., [Lan82], Section 1.2), or (2) X is projective curve in the usual sense (e.g., [Har77], Section 4.1), and K is its function field. (For k = C one can also view X as a compact Riemann surface, and K as its field of meromorphic functions.) To each f ∈ K \ {0} one associates the divisor (i.e., Weil divisor) equal to (f ) = v∈X ord v (f )v [Lan82] 1 . For each divisor D one sets L(D) = {0} ∪ {f ∈ K | (f ) ≥ −D}, where we regard 0 ∈ K as having divisor (0) ≥ −D for all D; this makes L(D) ⊂ K a k-linear subspace, and we set l(D) = dim k L(D). For a divisor D, we use deg(D) to denote the sum of the Z-coefficients in D. For f ∈ K \ {0}, f has the same number of zeroes and poles, counted with multiplicity, i.e., deg((f )) = 0. It follows that l(D) = 0 when deg(D) < 0. Riemann's theorem says that for the genus g ∈ Z ≥0 of X, for any divisor D with deg(D) sufficiently large, l(D) = deg(D) + 1 − g. Hence for any points P 1 , . . . , P n ∈ X we have (9) f (d) def = l(d 1 P 1 + · · · + d n P n ) is a Riemann function. The Riemann-Roch formula states that l(D) = l(ω − D) + deg(D) + 1 − g where ω is the canonical divisor, i.e., the divisor associated to any 1-form. Example 2.9. Let K be an elliptic curve, i.e., a curve of genus g = 0, and P 1 , P 2 two points of the curve. The Riemann-Roch theorem implies that f (d) = 0 if deg(d) < 0 and f (d) = deg(d) − 1 if deg(d) > 0. Hence it remains to determine f (d) for d = (d 1 , −d 1 ) of degree 0, and f (d 1 , −d 1 ) is either 0 or 1. If P 1 − P 2 has infinite order in the group law (which, for fixed P 1 , holds for all but countably many P 2 ), then f (d 1 , −d 1 ) = 1 iff d 1 = 0; by contrast, if P 1 − P 2 has order r ∈ N, then f (d 1 , −d 1 ) = 1 iff d 1 is divisible by r. Riemann Functions from other Riemann Functions. Example 2.10. If for some k, n ∈ N, f 1 , . . . , f 2k+1 are Riemann functions, then so is f 1 − f 2 + f 3 − · · · − f 2k + f 2k+1 . One can restrict any Riemann function to a subset of its variables, the others taking fixed values, to get a Riemann function on fewer variables. In [FF] the restriction to two variables is the most important. Let us define the appropriate notation. Example 2.11. Let f : Z n → Z be any Riemann function with f (d) = deg(d) + C for deg(d) sufficiently large. Then for any distinct i, j ∈ [n] and d ∈ Z n , the function f i,j,d : Z 2 → Z given as (10) f i,j,d (a i , a j ) = f d + a i e i + a j e j is a Riemann function Z 2 → Z, and for a i + a j large we have (11) f i,j,d (a i , a j ) = a i + a j + C , where C = deg(d) + C. We call f i,j,d a two-variable restriction of f ; we may similarly restrict f to one variable or three or more variables, and any such restriction is clearly a Riemann function. [It turns out that in [FF], it is important that that C depends only on d and not on i, j.] 2.7. Typical Properties of Riemann Functions. Let us describe some typical properties of Riemann functions above. Definition 2.12. We say that a function f : Z n → Z is (1) slowly growing if for all d ∈ Z n and i ∈ [n] we have f (d) ≤ f (d + e i ) ≤ f (d) + 1, and (2) p-periodic for a p ∈ N if for all i, j ∈ [n] and all d ∈ Z n we have f (d + p e i − p e j ) = f (d). We easily see: (1) f in (9) is always slowly growing, but not generally periodic; (2) f in (3), then (3) is slowly growing whenever N is a downset (as remarked above); (3) in Example 2.4, f is p-periodic for any p such that each element of Z n deg 0 /L has order divisible by p (hence this holds for p = |Z n deg 0 /L|); (4) in Example 2.11, if f : Z n → Z is either slowly growing or p-periodic for some p, then the same holds of any restriction of f to two (or any number) of its variables. The Weight of a Riemann Function, and Generalized Riemann Functions In this section we define the weights of a Riemann function, a notion central to this article. Since a Riemann function Z 2 → Z eventually equals d 1 +d 2 +C, one may consider that one possible generalization of this notion for a function Z 3 → Z might be a function that eventually equals a polynomial of degree two in d 1 , d 2 , d 3 . In fact, most everything we say about Riemann functions hold for a much larger class of functions Z n → Z which we call generalized Riemann functions; this includes all polynomials of d 1 , . . . , d n of degree n − 1, but many more functions. Weights and Möbuis Inversion. If f : Z n → Z is initially zero, then there is a unique initially zero W ∈ Z n → Z for which (12) f (d) = d ≤d W (d ), since we can determine W (d) inductively on deg(d) set (13) W (d) = f (d) − d ≤d, d =d W (d ). Recall from (2) the notation e I for I ⊂ [n]. Proposition 3.1. Consider the operator m on functions f : Z n → Z defined via (14) (mf )(d) = I⊂[n] (−1) |I| f (d − e I ), and the operator on functions W : Z n → Z that are initially zero given by (15) (sW )(d) = d ≤d W (d ), Then if f is any initially zero function, and W is given by the equation f = sW (i.e., W is defined inductively by (13)), then W = mf . The above can be viewed as the Möbius inversion formula for the partial order ≤ on Z n . Proof. We have f (d) = 0 whenever deg(d) ≤ b for some b, and then (14) shows that (mf )(d) = 0 for deg(d) ≤ b as well. Since there is a unique initially zero W with sW = f , it suffices to show that smf = f . Since f is initially zero, for any d ∈ Z n write (smf )(d) as (smf )(d) = d ≤d I⊂[n] (−1) |I| f (d − e I ) which is a double sum of finitely many terms since f is initially zero; hence we may rearrange terms, set d = d − e I and write this double sum as d ≤d f (d ) a d , where a d = I s.t. d +e I ≤d (−1) |I| ; to compute a d , setting J = {j ∈ [n] | d j < d j }, we have I s.t. d +e I ≤d (−1) |I| = I⊂J (−1) |I| which equals 1 if J = ∅ and otherwise equals 0. It follows that a d = 1, and for d = d, we have a d = 0. Definition 3.2. Throughout this article we reserve the symbols m, s for their meanings in (12) and (14). If f, W are initially zero functions Z n → Z with f = sW , we say that f counts W and that W is the weight of f . A function h : Z n → Z is modular if f ∈ ker m (i.e., mf is the zero function). We say that f : Z n → Z is a generalized Riemann function if (1) f is initially zero, and (2) f eventually equals a modular function, i.e., for some h ∈ ker m we have f (d) = h(d) for deg(d) sufficiently large. 3.2. Weights of Riemann Functions Z 2 → Z. We will be especially interested in Riemann functions Z 2 → Z and their weights W = mf . It is useful to notice that for such functions we that that for any fixed d 1 and d 2 sufficiently large, f (d 1 , d 2 ) − f (d 1 − 1, d 2 ) = 1, and hence, for fixed d 1 , (16) ∞ d2=−∞ W (d 1 , d 2 ) = 1, and similarly, for fixed d 2 we have (17) ∞ d1=−∞ W (d 1 , d 2 ) = 1. Viewing W as a two-dimensional infinite array of numbers indexed in Z × Z, one can therefore say that W : Z 2 → Z is a Riemann weight iff all its "row sums" (16) and all its "column sums" (17) equal one. Examples and Classification of Generalized Riemann Functions. At times it is convenient to write m using the "downward shift operators," t i for i ∈ [n], where t i is the operator on functions Z n → Z given by (18) (t i f )(d) = f (d − e i ); one easily verifies that the t i commute with one another, and that m = (1 − t 1 ) . . . (1 − t n ), (where 1 is the identity operator). In particular, it follows that if f = f (d) is independent of its i-th variable, then (1 − t i )f = 0, and hence mf = 0. In particular mf = 0 if (1) f is a sum of functions, each of which is independent in some variable, and, in particular, (2) if f is a polynomial of degree at most n−1. Hence deg(d)+C is a modular function for any n ≥ 1, and hence a Riemann function is, indeed, a generalized Riemann function. We now characterize modular functions in two different ways. Theorem 3.3. A function h : Z n → Z is modular iff it can be written as a sum of functions each of which depends on only n − 1 of its n variables. We postpone its proof to Section 7. The following description of modular functions will be needed when we discuss what we call Riemann-Roch formulas. Theorem 3.4. If a ∈ Z, n ∈ N, and h is any integer-valued function defined on d ∈ Z n with a ≤ deg(d) ≤ a + n − 1, then h has a unique extension to a modular function Z n → Z. We also postpone the proof of this theorem to Section 7. According to this theorem, if h 1 , h 2 are two modular functions, then h 1 and h 2 are equal whenever they are eventually equal (i.e., h 1 (d) = h 2 (d) for deg(d) sufficiently large), then h 1 = h 2 . In particular, if f : Z n → Z is a generalized Riemann function, then the modular function h that is eventually equal to f is uniquely determined. 3.4. The Weight of the Baker-Norine Rank and Other Functions Initially Equal to −1. Since the Baker-Norine rank and many similar functions are initially equal to −1, we make the following convention. Definition 3.5. If r : Z n → Z is a function that is initially equal to −1, by the weight of r we mean the function mr, which clearly equals mf with f = 1 + r. We also note that in the above definition, for any i ∈ [n] we have (1 − t i )r = (1 − t i )f . Hence, as soon as we apply either all of m, or merely one of its factors 1 − t i , there is no difference in working with r or f . When computing the weight of Baker-Norine type functions, we often use the more suggestive r BN rather than f = 1 + r BN . Riemann-Roch Formulas and Self-Duality In this section we express Riemann-Roch formulas more simply in terms of the weight of the Riemann function. Definition 4.1. Let f : Z n → Z be a generalized Riemann function, and h the modular function eventually equal to f . For K ∈ Z n , the K-dual of f , denoted f ∧ K , refers to the function Z n → Z given by (19) f ∧ K (d) = f (K − d) − h(K − d). We equivalently write (20) f (d) − f ∧ K (K − d) = h(d) and refer to this equation as a generalized Riemann-Roch formula. In particular, if f is a Riemann function with offset C, then h(d) = deg(d) + C, and (20) means that (21) f (d) − f ∧ K (K − d) = deg(d) + C. The usual Riemann-Roch formulas-the classical one and the Baker-Norine formula-are cases where f ∧ K = f equals f for some f, K. Hence the above definition is very loose: it says that for any generalized Riemann function, f , and any K ∈ Z n , there is always a "generalized Riemann-Roch formula;" we refer to the special cases where f = f ∧ K for some K as self-duality in Definition 4.4 below. In Subsection 1.1 we explained some reasons we work with generalized Riemann-Roch formulas; briefly, these reasons are: (1) requiring self-duality would eliminate many interesting Riemann functions, such as the general ones considered by [AM10], and likely some interesting generalized Riemann functions; and (2) self-duality does not behave well under fixing some of the variables of a Riemann function and considering the resulting restriction. We now give remarks, a theorem, and examples regarding generalized Riemann-Roch formulas. Definition 4.2. If W : Z n → Z is any function and L ∈ Z n , the L-dual weight of W , denoted W * L refers to the function given by W * L (d) = W (L − d). It is immediate that (W * L ) * L = W . Theorem 4. 3. Let f : Z n → Z be a generalized Riemann function, and W = mf . Let K ∈ Z n and let L = K + 1. (1) we have (22) m f ∧ K = (−1) n W * L = (−1) n (mf ) * L . (2) f ∧ K is a generalized Riemann function, and a Riemann function if f is. (3) (f ∧ K ) ∧ K = f . (4) f ∧ K = f iff W * L = (−1) n W . Proof. Proof of (1): applying m to (19) we have (23) (m f ∧ K )(d) = I⊂[n] (−1) |I| f ∧ K (d − e I ) which, in view of (19), equals I⊂[n] (−1) |I| f (K − d + e I ) − h(K − d + e I ) .(24) Substituting J = [n] \ I, for any g : Z n → Z we can write I⊂[n] (−1) |I| g(K − d + e I ) = J⊂[n] (−1) n−|J| g(K − d + 1 − e J ) = (−1) n J⊂[n] (−1) |J| g(K − d + 1 − e J ) = (−1) n (mg)(K − d + 1) = (−1) n (mg) * L (d). Taking g = f −h, and using mf = W and mh = 0, we have (24) equals (−1) n W * L (d), and since this also equals (19) we get (22). Proof of (2): f is a generalized Riemann function iff W = m is of finite support, which is equivalent to W * L being of finite support; hence f is a generalized Riemann function iff f ∧ K is. Moreover, f is a Riemann function iff in addition (20) has h(d) = deg(d) + C; in this case (21) with d replaced with K − d is equivalent to f (K − d) − f ∧ K (d) = h(K − d) for all d, which reversing the sign gives f ∧ K (d) − f (K − d) = −h(K − d) = − deg(K − d) + C = deg(d) + C , where C = C − deg(K). Proof of (3): we may write (22) as f ∧ K = s(−1) n (mf ) * L , and hence (f ∧ K ) ∧ K = s(−1) n (mf ∧ K ) * L = s(−1) n (−1) n W * L * L = sW = f. Proof of (4): f ∧ K = f (since both functions are initially zero) iff mf ∧ K = mf , and by (22) this is equivalent to (−1) n W * L = W . Definition 4.4. We say that a generalized Riemann function f : Z n → Z is self-dual if either of the equivalent conditions holds: (1) for some K ∈ Z n , f ∧ K = f ; (2) for some L ∈ Z n , W * L = (−1) n W . Let us remark on the uniqueness of K and L in the above definition: if W * L1 = W * L2 , it follows that for all d ∈ Z n , W (d) = (W * L2 ) * L2 (d) = (W * L1 ) * L2 (d) = W * L1 (L 2 − d) = W (L 1 − L 2 + d), and therefore W is translation invariant by L 1 − L 2 ; since f = sW , and s commutes with translation, f is also translation invariant by L 1 − L 2 . Similarly, if f ∧ K1 = f ∧ K2 , then W * L1 = W * L2 where L j = K j + 1, and L 1 − L 2 = K 1 − K 2 , and hence f and W are both translation invariant by K 1 − K 2 . Hence f and W have the same set of invariant translations, T ⊂ Z n deg 0 . Hence K and L in Definition 4.4 are unique up to a translation by the set T . We remark that the condition (−1) n W * L = W seems to have more direct symmetry than the equivalent condition f ∧ K = f ; furthermore, in the examples of the W that we compute in Sections 5 and 6, the W are very sparse (i.e., mostly 0), and so verifying (−1) n W * L = W seems simpler. Of course, the classical or Graph Riemann-Roch formulas, in terms of our Definition 4.4, are assertions that self-duality holds in these cases. r BN,G (d) − r BN,G (K − d) = deg(d) + 1 − g, where g = |E| − |V | + 1 and K = i e i (deg G (v i ) − 2). Since f = r BN,G + 1 is the associated Riemann function, the left-hand-side above also equals f (d)−f ∧ K (K−d), and hence f = f ∧ K is self-dual. Example 4.6. Amini and Manjunath [AM10] give conditions for f as in (3) with N as in Example 2.4 to satisfy self-duality. The first is that all maximal points of N have the same degree (g min = g max in [AM10]); the second is more technical. However, to us these Riemann functions seem interesting to study whether or not self-duality holds. The Weight of Two Vertex Graphs and Riemann Functions of Two Variables In this section we prove the following theorem. In this subsection we make some remarks on weights that we call "perfect matchings." Definition 5.2. Let W be a function Z 2 → Z that is initially and eventually zero. We say that W is a perfect matching if there exists a permutation (i.e., a bijection) π : Z → Z such that (25) W (i, j) = 1 if j = π(i), and 0 otherwise. It follows that for π as above, π(i) + i is bounded above and below, since W is initially and eventually 0. Of course, if W is r-periodic, i.e., for all d ∈ Z 2 , W (d) = W (d + (r, −r)), then π is skew-periodic in the sense that π(i + r) = π(i) − r for all i ∈ Z. Proposition 5.3. Let f : Z 2 → Z be a slowly growing Riemann function, i.e., for i = 1, 2 and any d ∈ Z 2 we have f (d) ≤ f (d + e i ) ≤ f (d) + 1. Let W = mf be the weight of f . Then W takes only the values 0 and ±1. Furthermore, for any d ∈ Z 2 , let a = f (d) (26) W (d) = 1 ⇐⇒ f (d − e 1 ) = f (d − e 2 ) = f (d − e 1 − e 2 ) = a − 1, and(27)W (d) = −1 ⇐⇒ f (d − e 1 ) = f (d − e 2 ) = a = f (d − e 1 − e 2 ) + 1. We say that f is supermodular when W (d) ≥ 0 for all 0; in this case W is a perfect matching. Proof. For d ∈ Z 2 , let a = f (d). Then f (d − e 1 − e 2 ) is between a − 2 and a, since f is slowly growing. We proceed by a case analysis: ( (26) and (27). If W never takes the value −1, then (16) implies that for each d 1 there is a unique d 2 with W (d 1 , d 2 ) = 1, so setting π(d 1 ) = d 2 gives a map π : Z → Z; then (17) implies that π has an inverse. (2) if f (d) ≥ 1, then there is a path from d to N as in (3) of positive length through the points of Z 2 , and hence for some i = 1, 2 we have f (d − e i ) = f (d) − 1; then Proposition 5.3 implies that W (d) ≥ 0. It follows that W is a perfect matching, and hence W is given by (25) for some perfect matching π; since f is r-periodic, it suffices to determine π(i) for i = 0, 1, . . . , r − 1. Let us do so by finding some values of f . Since (0, 0) ∈ L, we have f (0, 0) = 1, and for all i ≥ 0, f (i, 0) ≥ 1. But (i, 0) − e 2 cannot be effective for i ≤ r − 1, since then for some m ∈ Z we would have (i, −1) ≥ m(r, −r), which implies both m ≤ i/r < 1 and m ≥ 1/r > 0, which is impossible. Hence for 0 ≤ i ≤ r − 1 we have f (i, 0) = 1. On the other hand, we can prove that for i ≥ 0 we have f (i, i) ≥ i + 1, using induction on i: for i = 0 we have f (0, 0) = 1, and for the inductive claim with i ≥ 1, since (i, i) is effective we have f (i, i) = 1 + max f (i − 1, i), f (i, i − 1) ≥ 1 + f (i − 1, i − 1) ≥ 1 + i by the inductive hypothesis. For 0 ≤ i ≤ r − 1, since f (i, 0) = 1 and f (i, i) ≥ i + 1, the fact that f is slowly growing implies that f (i, j) = j + 1 for 0 ≤ j ≤ i. Similarly, for such i, j with 0 ≤ i ≤ j , f (i, j) = i + 1. Using this, it follows that for i = 0, . . . , r − 1 we have W (i, i) = f (i, i) − 2f (i, i − 1) + f (i − 1, i − 1) = i − 2(i − 1) + i − 1 = 1. It follows that π(i) = i for 0 ≤ i ≤ r − 1, and the theorem follows. Notice that this computation proves the Riemann-Roch formula in this case: this computation shows that W = W * L for L = (r − 1, r − 1). Hence f = f ∧ K for K = (r − 2, r − 2), and therefore adding these last two equations, the f (K) cancels and we get 0 = 2(r − 2) + 2C, and so C = 2 − r is the offset. Hence f (d) − f (K − d) = deg(d) + Cf (d) − f (K − d) = deg(d) − r + 2. 6. The Weight of the Riemann-Roch Rank of the Complete Graph and Related Graphs The point of this subsection is to give a self-contained computation of the remarkably simple and sparse weight function of the Baker-Norine rank for the complete graph. Our proof uses many standard ideas in the graph Riemann-Roch literature [BN07, Bac17, AM10, CB13], but also one rather ingenious idea of Cori and Le Borgne [CB13]. 6.1. Proof Overview and Computer-Aided Computations. Our analysis of the weights for the complete graph and the resulting formula of the Baker-Norine function is based on seeing some remarkable patterns in computer-aided computation. Explaining this also serves as an overview for our proofs below, and motivates the notation that we introduce. Let G be a graph on n-vertices ordered v 1 , . . . , v n . To compute the Baker-Norine function, r BN of a graph (and the resulting weight, W ), we note tht r BN (d) = −1 if deg(d) < 0; it suffices to compute r BN (d) on Z n deg 0 , then on Z n deg 1 , then Z n deg 2 , etc. Since r BN and W are invariant under the image of the Laplacian, ∆ G , it suffices to determine the value of r BN on a set of representatives of Pic i (G) = Z n deg i /Image(∆ G ) for i = 0, 1, . . .. To do so, it is natural to: find a set of "convenient coordinates" for Pic 0 (G) = Z n deg 0 /Image(∆ G ), meaning a set B and a bijection ι : B → Pic 0 (G) such that the computations below are easy to do for i = 0, 1, . . ., namely: (1) for all b ∈ B, determine if ι(b)+ie n is not effective, i.e., if r BN (ι(b)+ie n ) = −1; and (2) for all other b ∈ B we compute r BN (b + ie n ) via the formula r BN (b + ie n ) = 1 + min j∈[n] r BN (b + ie n − e j ); hence we need a reasonably fast algorithm to determine the element of B that is equivalent to ι −1 (b + e n − e j ). [We are finished when i ≥ deg(L) where L = K + 1 where K is the Baker-Norine canonical divisor, and hence when i ≥ 2(|E| − |V |) + |V | = 2|E| − |V |; we may use W = (−1) n W * L to finish when i ≥ |E| + (1 − |V |)/2.] Of course, one can replace e n above by any of e 1 , . . . , e n−1 , or, more generally, any element of Z n of degree 1; our choice of e n is convenient for the representatives of B below. It turns out that there is a very convenient choice for B suggested in [CB13]: namely, we give their proof that every element of Z n is equivalent to a unique element of A given by | a 1 , . . . , a n−2 ∈ {0, . . . , n − 1}, a n−1 = 0 , i.e., some element of the form (a 1 , . . . , a n ) A = a∈ A = {0, . . . , n − 1} n−2 × {0} × Z ⊂ Z n The only problem is that the group law in Pic(K n ) is a bit tricky to write down, since if a, a ∈ A, then the element of A that is equivalent to a + a has, for all i ≤ n − 2, its i-th coordinate equal to (a i + a i ) mod n, but the n-th coordinate needs to take into account the number of i such that a i + a i ≥ n. In other words, the addition law on the first n − 2 coordinates of A is that of (Z/nZ) n−2 (and the (n − 1)-th coordinate is always 0), but addition on the n-th coordinate depends on the first n−2 coordinates; in other words, the addition law on A induced by the law on Pic gives an isomorphism between A and a semidirect product (Z/nZ) n−2 Z. Of course, since A ⊂ Z n , this type of complicated addition law cannot be helped: the order of any nonzero element of Z n is infinite, whereas the order of each element in Pic 0 is finite; hence if Pic 0 is nontrivial (or, equivalently, G is not a tree), then no set of representatives of Pic can have a simple addition law. To get a simpler addition law, we define a second set of coordinates: namely, we set B = {0, . . . , n − 1} n−2 , we define ι : B → Pic 0 via ιb = b 1 , . . . , b n−2 , 0, −b 1 − · · · − b n−2 ∈ Z n deg 0 . In order to avoid writing ι all the time, for (b, i) ∈ B × Z we set b, i = ι(b) + ie n , which equals b 1 , . . . , b n−2 , 0, i − b 1 − · · · − b n−2 ∈ Z n deg i . Hence we leave the first n − 1 coordinates as is in A, but we form b, i to have degree i. In this way b, i + b , i has degree i+i , has (n−1)-th coordinate 0, and has the first n−2 coordinates given by addition in (Z/nZ) n−2 ; hence the addition law in Pic in the second coordinates (b, i), is just addition on (Z/nZ) n−2 × Z. The theorems we give below simply reflect the patterns that we saw, namely: we first noticed that the weights W = mr BN for the complete graph were very sparse, i.e., mostly 0's, and the non-zero values of W followed a simple pattern. Then, since m = (1 − t 1 ) . . . (1 − t n ) (recall that t i is the "downward shift operator" given in (18)), we tried computing some subset of the 1 − t i applied to r BN to find a simple pattern. After a number of unsuccessful attempts, we discovered that (1 − t n−1 )r BN had a remarkably simple pattern, namely that for small n, (1 − t n−1 )r BN b, i = 1 if b 1 + · · · + b n ≤ i 0 otherwise. From this one also easily sees the pattern (1 − t n )(1 − t n−1 )r BN b, i = 1 if b 1 + · · · + b n = i 0 otherwise. The rest of this section is devoted to proving that these patterns above, which we observed for small n, indeed hold for all n. Our starting point for the proof requires some important techniques of [CB13], which are more simply stated in terms of the representatives A of Pic(K n ) = Z n /Image(∆ Kn ) used by used in [CB13]. 6.2. Maximal Decrease. The following is a standard tool used in studying the graph Riemann-Roch rank, used by Baker-Norine [BN07] and many subsequent papers. It is valid in the general setting of (3) when N is a downset. Recall from Definition 2.12 that f : Z n → Z if for all j ∈ [n] and d ∈ Z n we have f (d) ≤ f (d + e j ) ≤ f (d) + 1. If so, an easy induction argument (on deg (d − d )) shows that if d , d ∈ Z n with d ≤ d, then (28) f (d ) ≥ f (d) − deg(d − d ). Definition 6.1. Let f : Z n → Z be slowly growing. Let d , d ∈ Z n with d ≤ d. We say that f is maximally decreasing from d to d if equality holds in (28), or equivalently f (d) = f (d ) + deg(d − d ). The following is Lemma 5 of [CB13], but is used in most papers we have seen involving the Baker-Norine rank, e.g., [BN07, Bac17, AM10]. Proposition 6.2. Let f : Z n → Z be slowly growing. Then for any d , d , d ∈ Z n , f is maximally decreasing from d to d iff it is maximally decreasing from both d to d and from d to d . The proof is immediate from the fact that the two inequalities f (d) − f (d ) ≤ deg(d − d ), f (d ) − f (d ) ≤ deg(d − d ) both hold with equality iff their sum does, and their sum is f (d) − f (d ) ≤ deg(d − d ). We remark that f is slowly growing whenever it is of the form (3) where N is a downset such that Z n deg≤m ⊂ N for some m (so that f takes on finite values). We also remark that in this case d ∈ Z n , and d ∈ N is such that d − d = min d ∈N d − d , then f is maximally decreasing from d to d . A Generalization of a Fundamental Lemma of Cori and Le Borgne. Next we give an elegant and rather ingenious observation of [CB13] (half of the proof of Proposition 10 there) that is the starting point of their (and our) study the Baker-Norine rank for the complete graph; we state their observation in slightly more general terms. Lemma 6.3. Fix n ∈ N, and let K n = (V, E) be the complete graph on vertex set V = [n], i.e., E consists of exactly one edge joining any two distinct vertices. Consider the Baker-Norine rank r BN : Z n → Z on K n . If a ≥ 0 then (29) a n−1 = 0 ⇒ r BN (a) = r BN (a − e n−1 ) + 1. Of course, by symmetry (29) holds with both occurrences of n − 1 replaced by any j ∈ [n]. Proof. Since a ≥ 0, r BN (a) ≥ 0, and hence r BN is maximally decreasing from a to a − b for some b ≥ 0 with r BN (a − b) = −1. Since r BN (a − b) = −1, we must have a j − b j ≤ −1 for some j ∈ [n] ; fix any such j. Then b j ≥ a j + 1 ≥ 1; setting a = a − b j e j we have a − b ≤ a ≤ a, and hence r BN is maximally decreasing from a to a . But the vector (30) a = a − a j e j − (b j − a j )e n−1 is merely the vector a followed by an exchange of the (n−1)-th and j-th coordinates (if j = n − 1, then a = a ). Hence a , a have the same degree and same value of r BN ; hence f is also maximally decreasing from a to a . Since b j − a j ≥ 1, (30) implies a ≤ a − e n−1 ≤ a; since f is maximally decreasing from a to a , f is maximally decreasing from a to a − e n−1 as well, and hence (29) holds. Remark 6.4. If n, m ∈ N, we use K m n = (V, E) to denote the graph with V = [n] and m edges between any two vertices (so K 1 n = K n ). Then r BN,K m n (d) is again a symmetric function of its variables (d 1 , . . . , d n ) = d, and the same argument shows that for any b ∈ Z ≥0 , a ≥ b1 and a n−1 = b implies that f (d) = f (d − e n−1 ) + 1. We believe it is possible to use this observation, specifically for b = m, to give an analog of Theorem 6.17 below regarding K m n . 6.4. The First Coordinates for Pic, D'après Cori-Le Borgne. Let us recall some more standard graph Riemann-Roch terminology (see, e.g., [BN07,CB13], and then give our first set of coordinates for the Picard group of a graph. These coordinates are those found in the Algorithm at the end of Section 2.1 of [CB13]. Recall Z n deg i consists of the elements of Z n of degree i. Recall [BN07] the Picard group of a graph, G, with n vertices v 1 , . . . , v n is defined as Pic(G) = Z n /Image(∆ G ); since Image(∆ G ) consists entirely of vectors of degree 0, Pic(G) is the union over i ∈ Z of (31) Pic i (G) = Z n deg i /Image(∆ G ). It is known that for all i, | Pic i (G)| equals (1/n) det (∆ G ), where det denotes the product of the nonzero eigenvalues of ∆ G (and Kirchoff's theorem says that this is the number of unrooted spanning trees of G). For G = K n it is a standard fact that this number of trees is n n−2 , i.e., (32) | Pic i (K n )| = n n−2 . Next we pick a convenient set of representatives for each class in Z n /Image(∆ Kn ). Notation 6.5. For any n ∈ N, we let a 1 , . . . , a n−2 ∈ {0, . . . , n − 1}, a n−1 = 0} = {0, . . . , n − 1} n−2 × {0} × Z (we usually simply write A since n will be understood and fixed); in addition, for i ∈ Z, we use A deg i to denote the set (33) A = A(n) = {a ∈ Z n |A deg i def = A ∩ Z n deg i = {a ∈ A | deg(a) = i}. In the above notation, note that a ∈ A deg i ⇐⇒ a n = i − a 1 − · · · − a n−2 and hence a ∈ A deg i ⇒ a n ≥ 0 ⇐⇒ a 1 + · · · + a n−2 ≤ i (34) a ∈ A deg i ⇒ a n = 0 ⇐⇒ a 1 + · · · + a n−2 = i (35) Lemma 6.6. Fix n ∈ N, and let K n = (V, E) be the complete graph on vertex set V = [n]. Then for all d ∈ Z n there exists a unique a ∈ A = A(n) with d ∼ a (i.e., d − a ∈ Image(∆ Kn )), given by: for j ∈ [n − 2], a j = (d j − d n−1 ) mod n, i.e., a j is the element of {0, . . . , n − 1} congruent to d j − d n−1 modulo n, a n−1 = 0, and a n = deg(d) − a 1 − · · · − a n−2 . Proof. Existence is shown in "Algorithm" at the end of Section 2.1 of [CB13]: we note that the image of ∆ G contains (1, . . . , 1, 1 − n) and, for any j ∈ [n], n(e j − e n ). For any d we get an equivalent vector with (n − 1)-th coordinate 0 by subtracting multiples of (1, . . . , 1, 1 − n); then we find an equivalent vector with the first n − 2 coordinates between 0 and n−1 by subtracting multiples of n(e j −e n ) for j ∈ [n−2]. Note that the above algorithm determines a map µ : Z n → A that such that (36) ∀d ∈ Z n , d ∼ µ(d), i.e., d and µ(d) are equivalent modulo Image(K n ). To prove that each d is equivalent to a unique element of A, we need to show that if a, a ∈ A are equivalent, i.e., a − a ∈ Image(∆ Kn ), then we must have a = a . Note that if a, a are equivalent, then they have the same degree and hence both lie in A deg i for the same i. Hence it suffices to show that each element of A deg i is in a distinct class of Pic i (K n ). Let us rephrase this condition. Note that since A deg i ⊂ Z n deg i , the quotient map Z n deg i → Z n deg i /Image(∆ Kn ) = Pic i (K n ) restricts to a map ν i : A deg i → Pic i (K n ). To show that each element of A deg i is in its own class of Pic i (K n ) simply means that ν i is injective. Let us prove this. So fix an i ∈ Z. Choosing a set of representatives, P i ⊂ Z n i for Pic i ; in view of (36), µ restricted to P i gives a map of sets µ| Pi : P i → A deg i that takes each element in the domain to a vector equivalent to it; hence this gives a map of sets µ i : Pic i → A deg i such that µ i takes each p ∈ Pic i to an element that lies in p. It follows that the map ν i µ i is the identity map on Pic i . But we easily see that A deg i has size n n−2 , since if a = (a 1 , . . . , a n ) ∈ A deg i then a 1 , . . . , a n−2 ∈ {0, . . . , n − 1}, and any a 1 , . . . , a n−2 ∈ {0, . . . , n − 1} determine the values of a n−1 , a n , namely a n−1 = 0, a n = i − a 1 − · · · − a n−2 . Since ν i µ i is the identity map on Pic i , and this map factors through the set A deg i of the same size, both ν i and µ i must be bijections. Hence ν i is an injection, which proves the desired uniqueness property. Here is how we often use the above theorem. 6.5. An Intermediate Weight Calculation: (1 − t n−1 )r BN . In this section we prove that the pattern we noticed in computer-aided calculation for small values of n can be proved to hold for all n. Theorem 6.8. Fix n ∈ N, and let K n = (V, E) be the complete graph on vertex set V = [n]. Consider the Baker-Norine rank r BN : Z n → Z on K n . For any a ∈ A deg i , (37) a 1 + · · · + a n−2 ≤ i ⇐⇒ a n ≥ 0 ⇐⇒ r BN (a) = r BN (a − e n−1 ) + 1. We remark that (37) generalizes Proposition 10 of [CB13]. Proof. For all a ∈ A, a ≥ 0 iff a n ≥ 0, since all other coordinates of a are nonnegative. For a ∈ A deg i , in view of (34) when get a ≥ 0 ⇐⇒ a n ≥ 0 ⇐⇒ a 1 + · · · + a n−2 ≤ i. Hence Lemma 6.3 implies that for a ∈ A deg i , (38) a 1 + · · · + a n−2 ≤ i ⇒ r BN (a) = r BN (a − e n−1 ) + 1. We now prove the reverse implication by, roughly speaking, giving a calculation that shows that there is "no more room" for r BN (a) − r BN (a − e i ) to be 1 otherwise, given that we know the offset of 1 + r BN,Kn . Let us make this precise. For any i ∈ Z, let M i = {a ∈ A deg i | r BN (a) = r BN (a − e n−1 ) + 1} and let N i = {a ∈ A deg i | a 1 + · · · + a n−2 ≤ i} . Then (38) implies M i ≥ N i , and (37) holds provided that we can show M i = N i for all i. Since a ∈ A implies that a 1 , . . . , a n−2 ≥ 0, it follows that for i ≤ −1 we have M i = N i = 0; similarly, since a 1 , . . . , a n−2 ≤ n − 1 for a ∈ A, we have a 1 + · · · + a n−2 ≤ (n − 1)(n − 2); hence for i ≥ n(n − 2) we have a 1 + · · · + a n−2 ≤ n(n − 2) ≤ i, and hence for such i we have N i = | Pic i | = n n−2 , and hence M i = n n−2 as well. Our strategy will be to show that for sufficiently large ∈ N we have M 0 + · · · + M = N 0 + · · · + N ; if so, then the inequalities M i ≥ N i must hold with equality (i.e., there is "no room" for some N i to be strictly smaller than M i ). Let us take a large ∈ N; and consider M 0 + · · · + M : for each a ∈ A deg we have r BN (a) = − g and r BN a − e n−1 ( + 1) = −1, and hence (39) i=0 r BN (a−ie n−1 )−r BN (a−(i+1)e n−1 ) = r BN (a)−r BN a−e n−1 ( +1) = −g+1. But for all j, A j is a set of Pic j representatives; hence for fixed i, as a varies over A , and a − ie n varies over a set of Pic −i representatives; hence a∈A r BN (a − ie n−1 ) − r BN (a − (i + 1)e n−1 ) = p∈Pic −i r BN (p) − r BN (p − e n−1 ) = a ∈A −i r BN (a ) − rBN(a − e n−1 ) = M −i (since r BN (a ) − r BN (a − e n−1 ) is either 0 or 1, and M −i counts the total number equal to 1). Hence summing (39) over all a ∈ A we get (40) M + M −1 + · · · + M 0 = n n−2 ( − g + 1). Next consider N 0 + · · · + N for large: note that for all (a 1 , . . . , a n−2 ) ∈ {0, . . . , n − 1} n−2 and i ∈ Z, we have either a 1 + · · · + a n−2 ≤ i or a 1 + · · · + a n−2 ≥ i + 1 (i.e., exactly one of the two inequalities above holds), and hence either a 1 + · · · + a n−2 ≤ i or (n − 1 − a 1 ) + · · · + (n − 1 − a n−2 ) ≤ (n − 1)(n − 2) − i − 1. Since (a 1 , . . . , a n−2 ) → (n − 1 − a 1 , . . . , n − 1 − a n−2 ) is a bijection of {0, . . . , n − 1} n−2 to itself, it follows that for all i and all a 1 , . . . , a n−2 ∈ {0, . . . , n − 1}, either (a 1 , . . . , a n−2 ) ∈ {0, . . . , n−1} n−2 is counted once either in N i , or (n−1−a 1 , . . . , n− 1 − a n−2 ) is counted once in N (n−2)(n−1)−i−1 ; hence N i + N (n−2)(n−1)−i−1 = n n−2 . Hence for all i ∈ Z we have N 0 + · · · + N (n−2)(n−1)−1 = (n − 2)(n − 1)n n−2 2 , and for ≥ (n − 1)(n − 2) − 1 we have N 0 + . . . + N = (n − 2)(n − 1)n n−2 2 + n n−2 − (n − 1)(n − 2) + 1 =n n−2 (n − 1)(n − 2) 2 + − (n − 1)(n − 2) + 1 =n n−2 ( − g + 1), in view of the fact that g = 1 + |E| − |V | = 1 + n(n − 1) 2 − n = 2 + n 2 − n − 2n 2 = (n − 1)(n − 2) 2 . Hence, from (40) we have N 0 + . . . + N = n n−2 ( − g + 1) = M 0 + · · · + M for large. But since M i ≥ N i for all i, we must have N i = M i for all 0 ≤ i ≤ ; hence N i = M i for all i. 6.6. A New Rank Formula for the Complete Graph and an Algorithm. Cori and Le Borgne [CB13] (after Proposition 6, bottom of page 9 and in [CLB16],Proposition 13) describe an O(n) algorithm that computes r BN (d) for the complete graph K n . Also, they show that when d is a sorted parking configuration, meaning that 0 ≤ d i < i for i < n and d 1 ≤ d 2 ≤ · · · ≤ d n−1 (and d n is unconstrained), they show (see Theorem 12 [CLB16]) that setting q = (d n + 1)/(n − 1) , r = (d n + 1) mod (n − 1) one has r BN (d) = −1 + n i=1 max 0, q − i + 1 + d i + χ i ≤ r , where χ(P ) is 1 if P is true, and 0 if P is false. Here we give another formula for the rank, perhaps related to the above formula; by contrast, our formula holds for a ∈ A, but easily generalizes to all d ∈ Z n . The formula is a corollary to Theorem 6.8. Corollary 6.9. Let n ∈ Z, and A be as in (33). For any a ∈ A we have In particular, for any d ∈ Z n we have (42) r BN,Kn (d) = −1+ i = 0, . . . , deg(d) n−2 j=1 (d j −d n−1 +i) mod n ≤ deg(d)−i . Proof. Since a − (deg(a) + 1)e n−1 has negative degree, we have (43) deg(a) i=0 r BN (a − ie n−1 ) − r BN (a − (i + 1)e n−1 ) = r BN (a) − (−1). According to Theorem 6.8, for a fixed i, r BN (a − ie n−1 ) − r BN (a − (i + 1)e n−1 ) equals 1 or 0 according to whether or not the unique a ∈ A that is equivalent to a − ie n−1 satisfies (44) a 1 + · · · + a n−2 ≤ deg(a ). According to Lemma 6.6, since the (n − 1)-th component of a − ie n−1 is −i, a is given as ∀j ∈ [n − 2], a j = (a j + i) mod n, and (a n−1 = 0) and deg(a ) = deg(a) − i. Hence (44) holds iff n−2 j=1 (a j + i) mod n ≤ deg(a) − i. Hence, in view of (43) we have (41). To prove (42), we note that any d ∈ Z n is equivalent to a ∈ A, where a j = (d j − d n−1 ) mod n for j ≤ n − 2, and deg(a) = deg(d). Remark 6.10. In the proof above we are making use of the fact that if f : Z n → Z is any function that is initially equal to a constant, then then f (d) = (1 − t) + (1 − t n−1 )t n−1 + (1 − t n−1 )t 2 n−1 + · · · f (d) where the right-hand-side represents a finite sum, since for any fixed d, for sufficiently large m ∈ N we have (1 − t n−1 )t m n−1 f (d) = 0. One can similarly write, for any i ∈ [n], (1 − t i ) −1 = 1 + t i + t 2 i + · · · with the right-hand-side representing a finite sum when applied to an initially vanishing function f at any given value d. It follows that if f, f are initially zero, then (45) (1 − t i )f = h ⇐⇒ f = (1 + t i + t 2 i + · · · ) h. At times one of the two conditions above is easier to show that the other, at times not. For example, Theorem 6.8 above gives us a formula for f = (1 − t n−1 )r BN over a ∈ A; in Theorem 6.15 we determine h = (1 − t n )f , but it is just as easy to apply either side of (45) with i = n. On the other hand, to compute the weight of r BN in Theorem 6.17, with h as above and W = (1 − t 1 ) . . . (1 − t n−2 )h, the above formula seems easier to verity than the equivalent h = (1 + t 1 + t 2 1 + · · · ) . . . (1 + t n−2 + t 2 n−2 + · · · )W. Next we briefly give a linear time algorithm to compute r BN of the complete graph based on (41) or (42) in Corollary 6.9. First, for simplicity, take an arbitrary d ∈ Z n and note that the equivalent a ∈ A has a i = (d i − d n−1 ) mod n for i ≤ n − 2 and deg(a) = deg(d). Hence it suffices to show how to compute (41) with a ∈ A. Setting g(i) = n−2 j=1 (a j + i) mod n we have that g(i + n) = g(i) for all i, and (46) g(i) = −m i n + n−2 j=1 a j , where m i is the number of j ∈ [n − 2] such that a j + i ≥ n, i.e., with a j ≥ n − i. Next, we claim that we can compute m 0 , . . . , m n−1 in linear time: indeed, by a single pass through a 1 , . . . , a n−2 , one can count for each k = 1, . . . , n − 1 the number, m k = {j ∈ [n − 2] | a j = k} , i.e., the number of j for which a j = k; then one computes m 0 , . . . , m n−1 by setting m 0 = 0 and for k = 1, . . . , n − 1 setting m k = m n−k + m k−1 . Once we compute m 0 , . . . , m n−1 , we can compute g(0), . . . , g(n − 1) in linear time by computing j a j (once) and then applying (46) for each i = 0, . . . , n − 1. Now note that for k = {0, . . . , n − 1}, we have that for any i ∈ {0, . . . , deg(a)} with i mod n = k, we have g(i) = g(k), and hence the condition Notation 6.11. For any n ∈ N and i ∈ Z, we use (1) B = B(n) to denote the set {0, . . . , n − 1} n−2 (and usually we just write B since n will be fixed); and (2) for any b ∈ B and i ∈ Z, we use b, i to denote Definition 6.12. For fixed n ∈ Z, we refer to B = B(n) and the map B × Z → Z n in (48) as the second coordinates of Pic(K n ) representatives. n−2 j=1 (a j + i) mod n ≤ deg(a) − i is equivalent to i + g(k) ≤ deg((48) b, i = (b 1 , . . . , b n−2 , 0, i − b 1 − · · · − b n−2 ) ∈ A deg i ⊂ Z n deg i ⊂ Z n . (3) if c ∈ Z n−2 , Proposition 6.13. Let n ∈ N, and let notation be as in Notation 6.5 and 6.11. Consider the complete graph, K n , and equivalence modulo Image(∆ Kn ). Then: (1) for each b ∈ B and i ∈ Z, (b 1 , . . . , b n−2 ), i = (a 1 , . . . , a n ), where a 1 = b 1 , . . . , a n−2 = b n−2 , a n−1 = 0, and a n = i − b 1 − · · · − b n−2 . (2) For all i ∈ Z, the set B × {i} is taken via ·, · bijectively to A deg i , and hence to a set of representatives of Pic i . b, i + b , i ∼ (b + b ) mod n, i + i . Similarly for subtraction, i.e., with − everywhere replacing +. Proof. (1) is immediate from the notation. (2) follows from (1). (3) follows from (1) and Lemma 6.6. (4) follows from(3). Example 6.14. Applying the above proposition, we see that (49) e 1 ∼ e 1 , 1 , . . . , e n−2 ∼ e n−2 , 1 , e n−1 ∼ (n − 1)1, 1 , e n ∼ 0, 1 , where we use e i to denote the vector in Z n or in Z n−2 , as appropriate. Moreover, equality holds in all the above, except for e n−1 , where e n−1 ∼ (n − 1)1, 1 = n − 1, . . . , n − 1, 0, 1 − (n − 2)(n − 1) . 6.8. Computation of (1 − t n )(1 − t n−1 )r BN . Theorem 6.15. Fix n ∈ N, and let K n = (V, G) be the complete graph on vertex set V = [n], i.e., E consists of exactly one edge joining any two distinct vertices. Consider the Baker-Norine rank r BN : Z n → Z on K n . (1) If a ∈ A deg i , then (1 − t n )(1 − t n−1 )r BN,Kn (a) = 1 if a 1 + · · · + a n−2 = i, and 0 otherwise. (2) For all b ∈ B and i ∈ Z, (51) (1 − t n )(1 − t n−1 )r BN,Kn ( b, i ) = 1 if b 1 + · · · + b n−2 = i, and 0 otherwise. Proof. The left-hand-side of (50) equals (1 − t n )(1 − t n−1 )r BN,Kn (a) = (1 − t n−1 )r BN,Kn (a) − (1 − t n−1 )r BN,Kn (a − e n ). Note that if a ∈ A deg i , then a − e n = (a 1 , . . . , a n−2 , 0, i − 1 − a 1 − · · · − a n−2 ) ∈ A deg i−1 . By Theorem 6.8, (1 − t n−1 )r BN,Kn (a) is 1 or 0 according to whether or not a 1 + · · · + a n−2 ≤ i or not, and similarly with a replaced by a − e n ∈ A deg i−1 , according to whether or not a 1 + · · · + a n−2 ≤ i − 1. Hence we conclude (50). (2) (i.e., (51)) follows immediately from (1) (i.e., (50)). When going through the weight calculations in the next two sections, it may be helpful to visualize consequences of Theorem 6.8 in the case n = 4, and to consider what (51) means in terms of the b, i coordinates, namely that b 1 + b 2 = i; see Figure 1. However, (51) implies that (1 − t n )(1 − t n−1 )r BN,Kn ( b, i ) = g(b 1 + · · · + b n−2 − i), for some function g (namely the "Dirac delta function at 0," i.e., the function that is 1 at 0 and otherwise 0). We find it conceptually simpler to prove a theorem that applies (1 − t 1 ) . . . (1 − t n−2 ) to any function of b, i of the form g(b 1 + · · · + b n−2 − i). Here is the result. It will be helpful to introduce the following "tensor" notation: if J ⊂ [n − 2], then set (52) t J = j∈J t j . Proposition 6.16. Let h : Z n → Z be any function that is invariant under translation by the image of the Laplacian of the complete graph. Say that for all (b, i) ∈ B × Z, h( b, i ) = g(b 1 + · · · + b n−2 − i) for some function g, i.e., h depends only on the value of b 1 + · · · + b n−2 − i. Then (1) if j ∈ [n − 2] and b ∈ B = {0, · · · , n − 1} n−2 has b j > 0, then for all i ∈ Z we have (53) ((1 − t j )h)( b, i ) = 0; (2) let j ∈ [n − 2] and J ⊂ [n − 2] with j / ∈ J ; if b ∈ B = {0, · · · , n − 1} n−2 has b j > 0, then for all i ∈ Z we have (54) (1 − t j )t J h ( b, i ) = 0 (using the "tensor" notation (52)); (3) if b ∈ B with b = 0 (hence b j > 0 for some j ∈ [n − 2]),(55) ( 1 − t 1 ) . . . (1 − t n−2 )h ( b, i ) = 0; and (4) (in the remaining case, b = 0) (56) (1 − t 1 ) . . . (1 − t n−2 )h ( 0, i ) = n−2 k=0 (−1) k n − 2 k g(i − kn). We remark that the proof below shows that claims (1) and (2) above hold, more generally, whenever h( b, i ) = g(b 1 , . . . , b j−1 , b j − i, b j+1 , . . . , b n−2 ) for some g, i.e., h is an arbitrary function, except that its dependence on b j and i is only on b j − i and the rest of the b j with j = j. Proof. Our proof will constantly use (49). Proof of (1): if b j > 0, then b − e j ∈ B, and hence b, i − e j = b − e j , i − 1 , and hence (1 − t j )h ( b, i ) = h( b, i ) − h( b − e j , i − 1 ) = g (b 1 + · · · + b n−2 ) − i − g (b 1 + · · · + b n−2 − 1) − (i − 1) = 0. This gives (53). Proof of (2): let b = (b − e J ) mod n. Since j / ∈ J we have b j = b j > 0, and hence b − e j ∈ B. Hence t J h ( b, i ) = h b , i − |J | t j t J h ( b, i = h b − e j , i − |J | − 1 . Hence the same calculation as in the previous paragraph (with b replacing b and i − |J | replacing i) gives (54). Proof of (3): we have (1 − t 1 ) . . . (1 − t n−2 ) = J ⊂[n−2]\{j} (−1) |J | (1 − t j )t J , and so (54) implies (55). Proof of (4): for any J ⊂ [n − 2], using (49) we have 0, i − e J ∼ (n − 1)e J , i − |J| , and hence f 0, i − e J = f (n − 1)e J , i − |J| = g((n − 1)|J| − i + |J|) = g(n|J| − i). Since (1 − t 1 ) . . . (1 − t n−2 ) = J⊂[n−2] (−1) |J| t J ,Proof. Setting h( b, i ) = (1 − t n−1 )(1 − t n )r BN ( b, i ), (51) shows that h( b, i ) = g(b 1 + · · · + b n−2 − i), where g(0) = 1 and elsewhere g vanishes. Since W = (1 − t 1 ) · · · (1 − t n−2 )h, we may apply Proposition 6.16 and conclude: (1) if b ∈ B is nonzero, then (55) implies that W ( b, i ) = 0, and (2) if b = 0, then W ( 0, i ) = n−2 k=0 (−1) k n − 2 k g(nk − i). Hence W ( 0, i ) = 0 unless i is of the form nk, with 0 ≤ k ≤ n − 2, in which case W ( 0, nk ) = (−1) k n − 2 k . 6.11. Remark on Theorem 6.17. Another important consequence of Theorem 6.17 is that, by symmetry, for any d ∈ Z n , and any distinct i, j ∈ [n] we have (1 − t i )(1 − t j )W (d) ≥ 0. In [FF] this will imply that when we can model f = 1 + r BN,Kn as Euler characteristics of a family of sheaves in a sense explained there. Fundamental Domains and the Proofs of Theorems 3.3 and 3.4 In this section we prove the Theorems 3.3 and 3.4. We do so with a tool that we call a cubism of Z n . However, Theorems 3.3 has a more direct proof without using cubisms, so we first give the direct proof. In fact, the direct proof will motivate the definition of a cubism. i : Z n → Z for each i ∈ [n] such that (1) h i = h i (d) is independent of the i-th variable, d i , and(2)(59) ∀d ∈ D n coord , f (d) = n i=1 h i (d). Hence the function i h i above is an extension of f to all of Z n such that each h i is independent of its i-th variable. Before giving the formal proof, let us explain the ideas for small n. The case n = 1 is immediate. The proof for n = 2 is as follows: consider (60) g(d 1 , d 2 ) = f (d 1 , 0) + f (0, d 2 ) − f (0, 0) : since g(d 1 , 0) = f (d 1 , 0) + f (0, 0) − f (0, 0) = f (d 1 , 0) we have f (d) = g(d) whenever d 2 = 0; by symmetry, the same is true if d 1 = 0; hence g = f on all of D 2 coord . But we easily write the right-hand-side of (60) as h 1 (d 2 ) + h 2 (d 1 ), by setting, say, h 1 (d 2 ) = f (0, d 2 ) − f (0, 0) and setting h 2 (d 1 ) = f (d 1 , 0). Similarly for n = 3, and g(d 1 , d 2 , d 3 ) = f (d 1 , d 2 , 0)+f (d 1 , 0, d 3 )+f (0, d 2 , d 3 )−f (d 1 , 0, 0)−f (0, d 2 , 0)−f (0, 0, d 3 )+f (0, 0, 0). For all n ≥ 4, we simply need to introduce convenient notation. Proof. For d ∈ Z n and I ⊂ [n], introduce the notation d I = i∈I d i e i . Consider the function g : Z n → Z given by f (d I )(−1) n−1−|I| (which makes sense, since d I ∈ D n coord whenever I = [n]). We claim that g = f when restricted to d ∈ D n coord ; by symmetry it suffices to check the case d n = 0, whereupon the term f (d I ) with n / ∈ I cancels the term corresponding to I ∪ n, except for the single remaining term where I = {1, . . . , n − 1}. Hence for d n = 0, g(d) = f (d), and, by symmetry, g = f on all of D n coord . Now we see that the right-hand-side (61) is of the desired form i h i as in the statement of the lemma, by setting h i = i / ∈I, 1,...,i−1∈I f (d I )(−1) n−1−|I| ; since for each I ⊂ [n] with I = [n] there is a unique i ∈ [n] such that i / ∈ I but 1, . . . , i − 1 ∈ I (namely the lowest value of i not in I), we have i h i equals the right-hand-side (61). Theorem 7.2. Let n ∈ N and D n coord be as in (58). Then any function f : D n coord → Z has a unique extension to a modular function h : Z n → Z. Proof of Theorem 3.3. One direction is immediate; it suffices to show that any modular function, h, can be written as a sum of functions, each of which depends on only n − 1 of its variables. So consider the restriction of h to D n coord ; then this restriction determines a unique modular function, which must be h. But then Theorem 7.2 implies that h = i h i , where each h i is independent of its i-th variable. Fundamental Modular Domains. Let us restate what we proved in the previous subsection. Definition 7.3. Let D ⊂ Z n . We call D a fundamental modular domain (respectively subfundamental, superfundamental) if for every function f : D → Z there exists a unique (respectively, at least one, at most one) modular function h : Z n → Z such that f = h on D. We remark that our terminology results from the following almost immediate facts: a subset of a subfundamental modular domain is subfundamental, and a strict subset of a fundamental domain is not fundamental; similarly for supersets and superfundamental domains. In the last subsection, Theorem 3.3 was proven via Theorem 7.2, which proved that D n coord is a fundamental modular domain. Theorem 3.4 essentially states that for any n ∈ N and a ∈ Z, D = {d ∈ Z n | a ≤ deg(d) ≤ a + n − 1} is a fundamental modular domain. We can prove both ideas by the method of a cubism, that we now explain. 7.3. Cubisms: Motivation, Definition, and Implication of Domain Fundamentality. The proof of Theorem 7.2 can be viewed as follows: we ordered the elements of N n by a function rank(d) = d 1 + · · · + d n − (n − 1), (so the minimum rank of an element of Z n is 1), and proved by induction on m ≥ 1 that there is a unique extension of a function h : D n coord → Z to all points of rank at most m so that (mh)(d) = 0 for all d of rank at most m. Let us generalize this idea. Definition 7.4. For d ∈ Z n , the d-cube refers to the set Cube(d) = {d ∈ Z n | d − 1 ≤ d ≤ d}. We refer to the set of all d-cubes as the set of n-cubes. If D ⊂ Z n , we say that function r : Z n → N is a cubism of D if, setting In the last paragraph of this section we remark that in some cubisms it is more convenient to replace the partial ordering of the n-cubes induced by the function r : Z n → N above with, more generally, a well-ordering or a partial ordering such that each subset has a minimal element. Example 7.5. In Figure 2 we illustrate an example of a cubism of D, with D = D coord as above, suggested by the above proof of Theorem 7.2 and n = 2 (so the n-cubes are really squares). To show that r attains only positive integer values, we can write r as r(d) = 1 + n i=1 max(d i − 1, −d i ); since max(d i − 1, −d i ) is non-negative for any d i ∈ Z, r attains only positive values. We leave the verification of (1) and (2) in the definition of a cubism to the reader. We also remark that-unlike the above example-there is no need for r −1 ({m}) to be finite; in fact, the next example shows that it can be convenient for r −1 {m} to be infinite. We easily see that these single points are distinct as d varies over all d / ∈ D, and it follows that r is a cubism of D. Example 7.7. One can show by a cubism argument that the set D ⊂ Z 2 given by {(0, 0)} ∪ {d ∈ Z 2 | deg(d) = ±1} is fundamental, by defining r(d) to be |d 1 | if deg(d) = 1 and otherwise || deg(d)|−1|; we depict this cubism in Figure 3. It follows that any subset of D is subfundamental (e.g., removing (0, 0)), and any superset of D is superfundamental. It is intriguing-but not relevant to this article-to consider the various other fundamental modular domains of Z n . We also note that in Example 7.7, it may be simpler to first extend a function D → Z along all points of degree 0, whereupon the extension is defined on all points of degree between −1 and 1, and then further extend the function to all of Z n . In this case one can view the set of 2-cubes as a well-ordered set, where all points of degree 0 are ordered before all points of degrees not between −1 and 1. One can therefore define a more general cubism as any well-ordering of the n-cubes of Z n , or, more generally, any partial ordering such that each subset of n-cubes has a minimal element. The proofs of all theorems easily generalize to these more general notions of a cubism. D i , i = 0 D i , i = 1 D i , i = 2 D i , i = 3 of a Fundamental Lemma of Cori and Le Borgne 20 6.4. The First Coordinates for Pic, D'après Cori-Le Borgne 20 6.5. An Intermediate Weight Calculation:(1 − t n−1 )r BN 22 6.6. A New Rank Formula for the Complete Graph and an Algorithm 24 6.7. The Second Coordinates for Pic 26 6.8. Computation of (1 − t n )(1 − t n−1 ( 7 ) 7Z n deg≤m ⊂ N ⊂ Z n deg≤m , and (2) setting M to be the largest degree of an element of N , then there exists a C such that if d ∈ Z n deg M , then then some d ∈ N ∩Z n deg M has d−d 1 ≤ C. Then f as in (3) is a Riemann function with offset −M . Proof. Since d ∈ N for deg(d) ≤ m, we have that f is initially zero. By induction on deg(d), we easily show that for any d with deg(d) > M , the L 1 distance from d to Z ≤M is at least deg(d) − M . Hence (d) ≥ deg(d) − M ; let us show that equality holds for deg(d) ≥ M + Cn. Say that d ∈ Z n satisfies deg(d) ≥ M + Cn. Then setting (d) − deg(d ) = deg(d) − M, and hence f (d) ≤ deg(d) − M . Hence, (8) holds with equality whenever deg(d) ≥ M + Cn. Let us make some further remarks on examples provided by Proposition 2.5. Remark 2 . 7 . 27In typical examples N above is a downset, i.e., d ∈ N and d ≤ d implies that d ∈ N . In this case if the closest point in N to some d ∈ Z n is d ∈ N , then clearly (1) d ≤ d, and (2) with f as in(3), f (d) = deg(d − d ); we easily verify the converse, i.e., Example 4. 5 . 5The Baker-Norine[BN07] Graph Riemann-Roch theorem for a graph, G = (V, E), with V = {v 1 , . . . , v n } can be stated as Theorem 5. 1 . 1Let G be a graph on two vertices, v 1 , v 2 with r ≥ 1 edges joining v 1 and v 2 . Let r BN : Z 2 → Z be the Baker-Norine rank, let f = 1 + r BN , i.e., f is as in(3)in Definition 2.3. Then d is in the image of the Laplacian iff d is an integral multiple of (r, −r). Let W = mf be the weight of f . Then W (0, 0) = W (1, 1) = . . . = W (r − 1, r − 1) = 1; furthermore W (d) = 1 if d is equivalent to one of (i, i) with i = 0, . . . , r − 1, and otherwise W (d) = 0. 5.1. Perfect Matchings and Slowly Growing Riemann Functions. 1) if f (d − e 1 − e 2 ) = a = 2, then f (d − e 1 ) differs by at most 1 from both a and a − 2, and hence f (d − e 1 ) = a − 1; similarly f (d − e 2 ) = a − 1, and so W (d) = 0.(2) if f (d − e 1 − e 2 ) = a, then since f is non-decreasing we have f (d − e i ) = a for i = 1, 2, and hence W (d) = 0;(3) if f (d − e 1 − e 2 ) = a − 1, then since f is non-decreasing we have that for each i = 1, 2, f (d − e i ) is either a or a − 1; this gives four cases to check, which imply Proof of Theorem 5.1. The rows of the Laplacian of G are (r, −r) and (−r, r), and hence the image, L, of the Laplacian equals the integer multiples of (r, −r).First let us prove that f is supermodular by a case analysis: indeed,(1) if f (d) = 0, then f (d ) = 0 for d ≤ d and hence W (d) = 0; f (K) = C, and taking d = K we get f (K) − 1 = deg(K) + C = 2(r − 2) + C; Corollary 6 . 7 . 67Fix an n ∈ N. For each i ∈ Z, A deg i is a set of representatives of the classes Pic i (K n ) in Z n deg i . Similarly, for any d ∈ Z n , as a ranges over A deg i , a − d ranges over a set of representatives of A deg i where i = i − deg(d). ( 41 ) 41r BN,Kn (a) = −1+ i = 0, . . . , deg(a) j +i) mod n ≤ deg(a)−i . a), and hence the number of such i, for k fixed, is deg(a) − g(k) + n /n . a) − g(k) + n /n , which completes an O(n) time algorithm to compute r BN . 6.7. The Second Coordinates for Pic. To complete our computation of the weight of r BN of the complete graph, we use a new set of coordinates. As explained in Subsection 6.1, the second coordinates turn out to represent Pic as a product (47) Pic = (Z/nZ) n−2 × Z. we use c mod n to denote the component-wise application of modn, i.e., c mod n = c 1 mod n, . . . , c n−2 mod n ∈ B = {0, . . . , n − 1} n−2 . ( 3 ) 3For all i ∈ Z, each d ∈ Z n deg i is equivalent to a unique element of the form b, i with b ∈ B, namely with b = d 1 − d n−1 , . . . , d n−2 − d n−1 mod n,where mod n is the component-wise application of mod n, i.e., b i = (d i − d n−1 ) mod n ∈ {0, . . . , n − 1}.(4) For any b, b ∈ B = {0, . . . , n − 1} n−2 and any i, i ∈ Z, we have Figure 1 . 1The non-zero values of of (1 − t n−1 )(1 − t n )r BN ( b, i ) for n = 4, b = (b 1 , b 2 ) ∈ {0, 1, 2, 3} 2 , namely 1 if b 1 + b 2 = i,and 0 otherwise. 6.9. A Generalization of the Weight Calculation. To compute the weight of the Baker-Norine rank on K n , we need to apply (1 − t 1 ) . . . (1 − t n−2 ). . t 1 ) . . . (1 − t n−2 )h 0, i = Computation of W . Theorem 6.17. Fix n ∈ N, and let K n = (V, E) be the complete graph on vertex set V = [n]. Consider the Baker-Norine rank r BN : Z n → Z on K n . The weight, W = m(r BN,Kn ), is given by (57) W ( b, i ) = (−1) n−2 if b = 0 and i = n for some = 0, . . . , n − 2, and 0 otherwise. 7. 1 . 1Proof of Theorem 3.3 Without Reference to Cubisms. Lemma 7.1. Let n ∈ Z, and let D n coord ⊂ Z n given by (58) D n coord = {d |d i = 0 for at least one i ∈ [n]}. Then for any f : D n coord → Z, there exist functions h (− 1 ) 1Proof. The existence of the extension of h is guaranteed by Lemma 7.1. Let us prove uniqueness. By symmetry it suffices to show that the values of h on the setN n = {d | d i > 0 for all i ∈ [n]} are uniquely determined. But if h is modular|I|+1 h(d − e I ).Now we prove by induction on m that for all m ≥ n, if d ∈ N n and deg(d) = m, then h(d) is uniquely determined. The base case is m = n, where the only element of degree n in N n is d = 1. But for each I ⊂ [n] with I = ∅, 1 − e I ∈ D n coord ; hence (62) uniquely determines h(1). To prove the inductive claim: let d ∈ N n with deg(d) = m; for all I ⊂ [n] with I = ∅, d − e I ≥ 0 and d − e I and has degree less than m. Hence (62) determines h(d) in terms of values of h that, by induction, have already been determined. ∈ Z ≥0 (hence D 0 = D), we have (1) if m ≥ 1 and r(d) = r(d ) = m, then (64) Cube(d) ∩ Cube(d ) ∈ D m−1 ,and (2) for all m ≥ 1 and d ∈ Z n with r(d) = m we have (65) Cube(d) \ D m−1 = 1. Proposition 7. 6 . 6If D ⊂ Z n has a cubism, then D is fundamental. Proof. Fix a function f : D → Z, and set g 0 = f . Let us prove by induction on m ∈ N that there is a unique function D m → Z such that (1) (mg m )(d) = 0 for all d with r(d) ≤ m; (2) the restriction of g m to D m−1 equals g m−1 ; and (3) the value of g m on each c ∈ D m \ D m−1 is determined by the equation (mg m )(d) = 0 for a unique d ∈ D m−1 such that c ∈ Cube(d) \ D m−1 , via the equation (66) − g m (c)(−1) deg(d−c) = c ∈Cube(d)\{c} g m−1 (c )(−1) deg(d−c ) . Figure 2 . 2A cubism for D n coord with n = 2. 7. 5 . 5Other Examples of Cubisms and the Proof of Theorem 3.4. Proof of Theorem 3.4. LetD = {d | a ≤ deg(d) ≤ a + n − 1}.Define r : Z n → N asr(d) = deg(d) − a + n + 1 if deg(d) ≥ a + n, and a + n − deg(d) if deg(d) < a + n.SettingD 0 = D and, for m ∈ N, D m as in (63), we easily see that that if r(d) = m then Cube(d) \ D m−1 consists of a single point, namely d if deg(d) ≥ a + n, and otherwise the single point d − 1. Figure 3 . 3A Cubism for Example 7.7. where [n] = {1, 2, . . . , n}. 2.2. Riemann Functions. In this section we define Riemann functions and give examples that have appeared in the literature.Definition 2.1. We say that a function f : Here ordv(f ) is (1) 0 if f (v) is finite and non-zero, (2) the multiplicity of the zero at v if f (v) = 0, and (3) minus the multiplicity of the pole at v if f (v) = ∞. The base case m = 1 is argued almost exactly as the inductive claim from m − 1 to m; so we will prove the base case m = 1, leaving in m everywhere.For m = 1, we have that D m−1 = D 0 = D, and (65) implies that for each d withThis determines g m (d) via (66) with c =d, since all other c ∈ Cube(d) in the sum (67) either lie in D or have rank at most m − 1; (64) shows that for distinct d, d of rank m, the correspondingd,d are distinct, so that it is possible to set the value of g m as required on alld that are the unique element of Cube(d) \ D m−1 for some d of rank m.For the inductive step, we assume the claim holds for m − 1, and we repeat the same argument above. This shows that g m : D m → Z exist for all m with the desired properties. Now define h : Z n → Z as follows:We claim that h above is modular: indeed, for any d ∈ Z n , if m = r(d), then mg m (d) = 0 and D m contains Cube(d); since g m+1 , g m+2 , . . . are all extensions of g m , we have mh(d) = mg m (d) = 0. Now we claim that h is the unique modular function Z n → Z whose restriction to D is f : indeed, assume that h is another such modular function, and that h = h ; then the definition of h implies that there exists an m such that g m does not equal the restriction of h to D m ; consider the smallest such m. [Straying a bit, one could define a subcubism by replacing the = 1 in (65) by ≥ 1, and the same proof shows that a D with a subcubism is subfundamental; similarly for supercubism and ≤ 1.] 7.4. Second Proof of Theorem 7.2. The proof of Theorem 7.2 above can be viewed as giving a cubism (e.g.,Figure 2for n = 2). Let us formalize this.Second proof of Theorem 7.2. For each d ∈ Z n , let r(d) = |d 1 | + · · · + |d n | + {i ∈ [n] | d i ≤ 0} − n + 1; more intuitively, r(d) is just the L 1 distance of the furthest point in Cube(d) to D n coord , since if all d i ≥ 1 then the furthest point is just d, and r(d) is just d 1 + · · · + d n − n + 1, and otherwise we need minor corrections for those d i ≤ 0. Now we claim that r is a cubism. Riemann-Roch theory for weighted graphs and tropical curves. Omid Amini, Lucia Caporaso, MR 3046301Adv. Math. 240Omid Amini and Lucia Caporaso, Riemann-Roch theory for weighted graphs and trop- ical curves, Adv. Math. 240 (2013), 1-23. MR 3046301 Riemann-Roch for sub-lattices of the root lattice An. Omid Amini, Madhusudan Manjunath, Electron. J. Combin. 171Research Paper 124, 50. MR 2729373Omid Amini and Madhusudan Manjunath, Riemann-Roch for sub-lattices of the root lattice An, Electron. J. Combin. 17 (2010), no. 1, Research Paper 124, 50. MR 2729373 Riemann-Roch theory for graph orientations. 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[]
[ "SNe Ia Redshift in a Non-Adiabatic Universe", "SNe Ia Redshift in a Non-Adiabatic Universe" ]
[ "Rajendra P Gupta *correspondence:[email protected] \nMacronix Research Corporation\n9 Veery LaneK1J 8X4OttawaCanada\n\nNational Research Council\nOttawaCanada (retired\n" ]
[ "Macronix Research Corporation\n9 Veery LaneK1J 8X4OttawaCanada", "National Research Council\nOttawaCanada (retired" ]
[]
Absract: By relaxing the constraint of adiabatic universe used in most cosmological models, we have shown that the new approach provides a better fit to the supernovae Ia redshift data with a single parameter, the Hubble constant 0 , than the standard ΛCDM model with two parameters, 0 and the cosmological constant Λ related density Ω Λ . The new approach is compliant with the cosmological principle. It yields the 0 = 68.28 (±0.53) km s -1 Mpc -1 with an analytical value of the deceleration parameter 0 = −0.4. The analysis presented is for a matter only, flat universe. The cosmological constant Λ may thus be considered as a manifestation of a non-adiabatic universe that is treated as an adiabatic universe.
10.3390/universe4100104
[ "https://arxiv.org/pdf/1810.12090v1.pdf" ]
53,627,527
1810.12090
9d4508d281eb7f6ecff60c79eaad7c495a8ecf4a
SNe Ia Redshift in a Non-Adiabatic Universe Rajendra P Gupta *correspondence:[email protected] Macronix Research Corporation 9 Veery LaneK1J 8X4OttawaCanada National Research Council OttawaCanada (retired SNe Ia Redshift in a Non-Adiabatic Universe 1galaxiessupernovae, distances and redshiftscosmic microwave background radiationdistance scalecosmology theorycosmological constantHubble constantgeneral relativity PACS: 9880-k9880Es9862Py _____________________________________________________________________________________ Absract: By relaxing the constraint of adiabatic universe used in most cosmological models, we have shown that the new approach provides a better fit to the supernovae Ia redshift data with a single parameter, the Hubble constant 0 , than the standard ΛCDM model with two parameters, 0 and the cosmological constant Λ related density Ω Λ . The new approach is compliant with the cosmological principle. It yields the 0 = 68.28 (±0.53) km s -1 Mpc -1 with an analytical value of the deceleration parameter 0 = −0.4. The analysis presented is for a matter only, flat universe. The cosmological constant Λ may thus be considered as a manifestation of a non-adiabatic universe that is treated as an adiabatic universe. Introduction The redshift of the extragalactic objects, such as supernovae Ia (SNe Ia) is arguably the most important of all cosmic observations that are used for modeling the universe. Two major explanations of the redshift are the tired light effect in the steady state theory and the expansion of the universe [1]. However, since the discovery of the microwave background radiation by Penzias and Wilson in 1964 [2], the acceptable explanation for the redshift by mainstream cosmologist has steadily shifted in favour of the big-bang expansion of the universe, and today alternative approaches for explaining the redshift are not acceptable by most cosmologists. The situation has been most succinctly expressed by Vishwakarma and Narlikar in a recent paper [3] as follows: "… a recent trend in the analysis of SNeIa data departs from the standard practice of executing a quantitative assessment of a cosmological theory-the expected primary goal of the observations [4,5]. Instead of using the data to directly test the considered model, the new procedure tacitly assumes that the model gives a good fit to the data, and limits itself to estimating the confidence intervals for the parameters of the model and their internal errors. The important purpose of testing a cosmological theory is thereby vitiated." Interestingly, it is the close analysis of the cosmic microwave background that has created tension between the Hubble constant derived from the spectral data and from the microwave background data [6,7]. The status of the expanding universe and steady state theories has been recently reviewed by López-Corredoira [8] and Orlov and Raikov [9]. They concluded that based on the currently available observational data it is not possible to unambiguously identify the preferred approach to cosmology. It has been phenomenologically shown that the tired light may be due to Mach effect, which may contribute dominantly to the cosmological redshift [10]. While the paper's assumption that observed redshift may be a combination of the expansion of the universe and tired light effect appears to be sound, it incorrectly divided the distance modulus of the light emitting source between the two components rather than keeping the proper distance of the source the same and dividing the redshift. This was corrected in a subsequent paper [11] which showed that a hybrid of Einstein de Sitter cosmological model and the tired light (Mach effect) model gave an excellent fit to the SNe Ia data while at the same time providing analytically the deceleration parameter and the ratio of the contribution of the two models. Consequently, the new model was dubbed Einstein de Sitter Mach (EDSM) model. Using Poisson's work on the motion of point particles in curved spacetime [12], Fischer [13] has shown analytically that gravitational back reaction may be responsible for the tired light phenomenon and could account for some or most of the observed redshift. His finding may also be related to Mach effect. The EDSM model required a luminosity flux correction factor proportional to 1/√1 + that was left unexplained [11], being the redshift. This inspired us to look at the fundamentals of cosmological modeling and see if some of the assumptions need revisiting. Most cosmological models are based on one or more of the following assumptions: 1. Cosmological principle: The universe is homogeneous and isotropic -at large scale. 2. Adiabatic expansion: The energy does not enter or leave a volume of the universe. 3. Perfect fluid: The equation of state follows simple energy-pressure proportionately law. 4. Interaction free components: Fluid equation for each component is independent. We believe that the adiabatic expansion of the universe is the weakest among all the above assumptions. After all Einstein's incorporation of the cosmological constant in his field equations in itself comprises a breach of adiabatic assumption. More recently, Komatsu and Kimura [14,15] have suggested a nonadiabatic model. Their approach has been to modify the Friedmann and acceleration equations by adding extra terms and derive the continuity (fluid) equation from the first law of thermodynamics, assuming non-adiabatic expansion caused by the entropy and temperature on the horizon. The solution of the equations is thus based on multiple unknown parameters that need to be determined by fitting the SNe Ia data. We believe if the model is sound then we would not need any fitting parameter other than the Hubble constant. Theory The Friedmann equation, coupled with the fluid equation and the equation of state, provides the dynamics of the universe and thus the evolution of the scale factor . It does not give the redshift directly. The redshift is taken to represent the expansion, and only the expansion, of the universe, and thus scale factor is considered to be directly observable through the relation = 1/(1 + ). The relation ignores other causes that may contribute to the redshift. If the redshift is indeed contributed partially by other factors, such as by the Mach effect, then the scale factor determined by said equations will not equate to 1/(1 + ). Unless the said equations are modified to take into account other factors, they cannot be considered to represent the cosmology correctly. Since energy density is common to all the three equations, and evolution of density is governed by the fluid equation, we will try to look at it with a magnifying glass. The starting point for the fluid equation in cosmology is the first law of thermodynamics [1,16]: = + ,(1) where is the thermal energy transfer into the system, is the change in the internal energy of the system, and = is the work done on the system having pressure to increase its volume by . Normally, is set to zero on the ground that the universe is perfectly homogeneous and that there can therefore be no bulk flow of thermal energy. However, if the energy loss of a particle, such as that of a photon through tired light phenomenon, is equally shared by all the particles of the universe (or by the 'fabric' of the universe) in the spirit of the Mach effect [17] then can be non-zero while conserving the homogeneity of the universe. We will thus abandon the assumption that = 0. The first law of thermodynamics for the expanding universe then yields: ̇+̇=̇. (2) We now apply it to an expanding sphere of commoving radius and scale factor ( ). Then the sphere volume ( ) = . Since the internal energy of the sphere with energy density ( ) is ( ) = ( ) ( ), its rate of change may be written as ̇=̇+̇= (̇+ 3̇) .(4) If we assume the energy loss ̇ to be proportional to the internal energy of the sphere ̇= − = − ,(5) where is the proportionality constant, then Equation (2) may be written as ̇+ 3̇( + ) + = 0,(6) which is the new fluid equation for the expanding universe. Using the equation of state relation = , and rearranging Equation (6), we may write + 3(1 + ) + = 0.(7) Assuming to be constant in the equation of state, this can be integrated to yield ln( ) + 3(1 + ) ln( ) + + = 0. Here is the integration constant. Now = 0 corresponds to the scale factor = 1 and = 0 , giving = − ln( 0 ) − 0 . We may then write Equation (8) as ( ) = 0 −3(1+ ) ( 0 − ) .(9) Let us now examine the simplest form of the Friedmann equation (single component, flat universe) with G as the gravitational constant. It may be written [16] as (̇) 2 = ( 8 3 2 ).(10) Substituting from Equation (9), we get ̇2 = ( 8 0 3 2 ) −(1+3 ) ( 0 − ) .(11) Since 0 ≡ ( 0 ) = 1, it can be shown that it has the following solution [Appendix A, Equations (A1) to (A8)]: = / 0 = ( 1− − 2 1− − 0 2 ) 2 3+3 ,(12)≈ ( 0 ) 2 3+3 (1 + 1 4 ( 2 3+3 ) ( 0 − ) + ( 2 )).(13) This reduces to the standard expression for the scale factor in adiabatic universe ( = 0). Since the Hubble parameter is defined as ( ) =̇/ , differentiating Equation (12) with respect to and rearranging, we get [Appendix A, Equations (A9) to (A??)]: ̇= ( 3+3 ) ( 2 − 1) −1 , or(14)e 2 = 1 + ( 1 ( ) ) ( 3+3 ), or(15)2 = ln (1 + ( 1 ( ) ) ( 3+3 )), or (16) 0 = 2 3+3 ( 1 0 ) when ⇒ 0.(17) Here Equations (15) and (16) can be used to determine the age of the universe in the non-adiabatic universe provided we know . They reduce to Equation (17) in the limit of ⇒ 0. It is the standard expression in adiabatic universe for the age of the universe in terms of the Hubble constant for a single component flat universe. We see from Equation (11) that at = 0 , ̇( 0 ) = √( 8 0 3 2 ). We can therefore write the expression for the age of the universe in terms the energy density as 0, = 2 ln (1 + ( 2 ) ( 1 1+ ) √ 2 6 0 ) , and(18)0,0 = ( 1 1+ ) √ 2 6 0 when ⇒ 0 .(19) Equation (18) is the expression for the age of the universe for the single component flat non-adiabatically expanding universe and Equation (19) is the standard expression for adiabatically expanding universe obtained in the limit of zero . We need to know in order to get 0 in the non-adiabatic universe. If we like, we could resolve the Friedmann Equation (11) into adiabatic and non-adiabatic components: ̇2 = ( 8 0 3 2 ) −(1+3 ) [1 + ( 0 − ) + 1 2 2 ( 0 − ) 2 … ].(11′) Here 1 st term in the square bracket is the adiabatic term that is used in most cosmological models and the remaining terms represent the non-adiabatic correction. The non-adiabatic correction is non-existent at = 0 , i.e. = 0, and negligible when is close to 0 , i.e. ≪ 1. Thus, we can resort to adiabatic universe as the boundary condition when finding certain analytical parameters and correlations. Since we know the analytically derived value of the deceleration parameter 0 = −0.4 from the adiabatic EDSM model [11], let us first workout the expression for the same from its standard definition and see if can be expressed in terms of 0 . 0 ≡ − (̈̇2) = 0 .(20) Equation (14) may be differentiated and rearranged to obtain the expression for 0 as follows. (14), or (25) Up until now we have not used any observational data. In order to proceed further, we need to know the Hubble constant 0 . The observational data is usually provided in the form of distance modulus and the redshift . In an expansion only model, we may write the distance modulus as [1,16] ̈( ) = ( 3+3 ) [ ̇( ) ( 2 − 1) −1 − ( ) ( 2 − 1) −2 2 ( 2 )],(21)= ( 3+3 ) ( 2 − 1) −1 [̇( ) − ( 2 ) ( ) ( 2 − 1) −1 2 ] ,(22)= (̇( ) ( ) )̇( ) [1 − ( 2 ) ( ( ) ( ) ) ( 2 − 1) −1 2 ], or (23) ̈( ) ( ) 2 ( ) = 1 − ( 2 ) ( ( ) ( ) ) ( 2 − 1) −1 2 ,(24)= 1 − 3+3 2 2 from Equation= −1 + ( 3(1+ ) 2 ) 2 , or 0 = −1 + ( 3(1+ ) 2 ) 0 2 .(26) Here is the luminosity distance of the source emitting the photons at time whose redshift is being measured, and is the proper distance of the source in mega parsecs observed at time 0 . When all the redshift is allocated to the expansion of the universe, 1 + = 1/ ( ). We may then write Equation (29) ( ) = ∫ (1 + )/( ) ( 0 ) ( ) .(30) Equation (12) can now be used to determine / for substitution in Equation (30). Since we are observing redshift in the matter dominated universe, we may simplify Equation (12) by taking = 0, and rewrite it as 1 + = 1 = (1 − − 0 2 ) 2 3 (1 − − 2 ) − 2 3 , or = ( 3 ) (1 + ) (1 − − 2 ) −1 (− − 2 ).(31) We can use Equation (31) We can include Mach effect contribution to the redshift following the approach in an earlier paper [11] and recalculate the distance modulus . Using subscript M for Mach effect and X for expansion effect and equating the proper distance expressions for the two, and since 1 + = (1 + )(1 + ) and 0 = = , we may write We will now consider how various parameters compare between the adiabatic models and the non-adiabatic model developed here. If we compare Equation (5) What is the scale factor here? In a standard adiabatically expanding universe, = 0, and = 1/(1 + ). However, in the EDSM model the redshift has two components, due to the expansion of the universe and due to the Mach effect, with 1 + = (1 + )(1 + ) [11]. We should therefore replace in Equation (42) Recalling that the standard expression for the radiation energy density evolution is given by ( ) = ,0 −4 = ,0 (1 + ) 4 ,(44) we find that ( ) = ( )/(1 + ) 3 , and is a fraction of the energy density for a given without the Mach effect. Since Equation (9) is valid also for matter with = 0, we get in the adiabatic universe with = 0, ( ) = ,0 −3 .(45) Comparing it with Equation (43) we see that the ratio of the radiation density and mass density is proportional to (1 + ), the same as in the standard expansion models [1,16]. The ratio is inclusive of the factor − ( 0 − ) in the non-adiabatic universe. Results The database used in this study is for 580 SNe Ia data points with redshifts 0.015 ≤ ≤ 1.414 as compiled in the Union2 , database [18] updated to 2017. We used Matlab curve fitting tool to fit the data using non-linear least square regression. To minimize the impact of large scatter of data points, we applied the 'Robust Bisquare' method in Matlab. This tool fits data by minimizing the summed square of the residuals, and reduces the weight of outliers using bi-square weights. This scheme minimizes a weighted sum of squares, where the weight given to each data point depends on how far the point is from the fitted line. Points farther from the line get reduced weight. Robust fitting with bisquare weight uses an iteratively reweighted least square algorithm. The Goodness of Fit in Matlab is given by parameters SSE (sum of squares due to errors, i.e. summed square of residuals) that is minimized in the fitting algorithm; R-Square that indicates the proportionate amount of variation in the response variable explained by the independent variable in the model (larger the R-squared, more the variability explained by the model); and RMSE (root mean square error, i.e. standard error of the regression)-closer the value to zero, better is the data fit. Figure 1 shows the curves fitted to the data set using Equation (40) and Equation (37), and using standard ΛCDM model [11] for comparison. The expression used for CDM model is = 5log [ 0 ∫ /√Ω ,0 (1 + ) 3 + 1 − Ω ,0 ] 0 + 5 log(1 + ) + 25.(46) The first one has been labeled as EDSM-NA since it is the non-adiabatic version of the EDSM model (flat, matter only, including Mach effect) of reference [11]. The second curve is labeled as EdeS-NA as it is the non-adiabatic version of the Einstein de Sitter model (flat, matter only universe). The third one is the curve for standard ΛCDM model. There is no visible difference between the three curves when the full data fit curves are viewed in the left display of the figure, and only very slight visible difference when the zoomed-in right display at high is viewed. Corresponding goodness-of-fit numbers are presented in Table 1. The goodness-of-fit numbers differ only slightly. In Figure 3 we Discussion From Figure 1 and Table 1, it is difficult to state unarguably which model is better. Nevertheless, based on the fact that non-adiabatic models yield data fit using only one fit parameter, whereas the ΛCDM model requires two fit parameters, the preferred model would be one of the non-adiabatic models. And since the analytical value of the deceleration parameter used in this work is derived by equating the Mach proper distance and expansion proper distance [11], both the non-adiabatic models -EdeS-NA and EDSM-NA implicitly involve Mach effect. Our choice of the non-adiabatic model will thus be determined by studying which model fits other cosmological observations better. It should be mentioned that non-adiabatic modeling has been tried by some cosmologist, most recently by Komatsu and Kimura [14,15]. Their approach has been to modify the Friedmann and acceleration equations by adding extra terms and derive the continuity (fluid) equation from the first law of thermodynamics, assuming non-adiabatic expansion caused by the entropy and temperature on the horizon. The solution of the equations is thus based on multiple unknown parameters that need to be determined by fitting the SNe Ia data. Our approach here modifies only the fluid equation from the first law of thermodynamics on the assumption that the system energy gain or loss is proportional to the energy of the system -Equation (5). No adjustable parameters, other than the universal Hubble constant, are required to fit the data. The deceleration parameter that is needed in our non-adiabatic formulation is analytically obtained from the EDSM model [11]. It may therefore be concluded that the luminosity flux correction factor of reference [11] proportional to 1/√1 + is due to the non-adiabatic nature of the universe. Similarly, one could say that the cosmological constant Λ approximates the non-adiabatic nature of the universe when studied in an adiabatic approximation of the universe. The lowest value of the Hubble constant in this work is obtained with the EDSM-NA model without compromising the goodness-of-fit. It is closer to the Hubble constant obtained from the cosmic microwave background (CMB) data, such as from Plank and WMAP space crafts, than from the ΛCDM model or the EdeS-NA model. At 68.28 km s -1 Mpc -1 it is almost right at the weighted average of 68.1 km s -1 Mpc -1 reported from WMAP and Planck data points [19]. However, it may just be a coincidence. As discussed by Bonnet-Bidaud [20], the origin of CMB is not fully settled as yet. Another important thing to note is that the ratio of Mach and expansion contribution to the redshift has now changed. In reference [11] expansion contribution at the current epoch ( = 0 or = 0) was only 40%. If we expand Equation (39) in the limit of very small , we can see that = 0.6 and since = + in this limit, we find that is 62.5% of . The luminosity flux correction factor can be considered as responsible for this discrepancy. This means that 62.5% of Hubble constant 0 is due to the expansion of the universe and remaining due to Mach effect. All the expansion related cosmological It should be emphasized that the main merit of the model presented here is that it can fit the data with a single parameter, the Hubble constant 0 . There have been several models developed in the past, such as based on the modified tired light approach in plasma cosmology by Lorenzo Zaninetti [21], that can give excellent fit to the data with one additional parameter which has to be determined by fitting the data. Conclusion The cosmological model presented in this communication is based on relaxing the assumption that universe dynamics is adiabatic within the confines of the cosmological principle. The fact that a single parameter yields a better fit to the SNe Ia data using the non-adiabatic model presented here than the two parameter fit of the same data using ΛCDM model establishes the superiority of the new model. The Hubble constant obtained by the two models is almost the same, in fact the non-adiabatic Machexpansion hybrid model EDSM-NA gives a lower value, 0 = 68.28 (±0.53) km s -1 Mpc -1 , very close to 68.1 km s -1 Mpc -1 sought by cosmic microwave background data from Plank and WMAP space crafts. It may therefore be possible to dispense with the cosmological constant after all, and corresponding perpetually elusive dark energy, in the spirit of Einstein who always wanted to correct his greatest mistake! Appendix A In this Appendix, our objective is to show how to obtain Equations (12) and (14) from Equation (11). Here 1 is the integration constant that needs to be determined from the boundary condition; it should reduce to the standard expression for ( ) [16] in the non-adiabatic universe when = 0. Since the scale factor ( 0 ) ≡ 1, dividing Equation (A3) by the same by setting = 0 , we get For 0 = 0−0.4 and = 0 (i.e. matter only universe), Equation (26) yields 0 2 = 0.4 or = −1.833/ 0 . Substituting these values in Equation (15) at = 0 yields = −1.8 0 and the age of the universe no simple analytical solution for the integral in Equation (36). Substituting ≡ (1 − − 0 ) = −1.5 and = −1.8 0 from above -Equation (26) and the paragraph following it, and defining 0 ≡ / 0 , we may write the distance modulus as not possible to express analytically (or ) in terms of and write directly in terms of . Nevertheless, Equation (39') can be numerically solved for for any value of , and distance modulus calculated to include Mach effect as well as expansion effect using the expression with the standard Mach effect photon energy loss equation − = 0 then for radiation energy = − 0 and the Mach effect redshift is given by 1 + = exp( 0 / ) [10] with = ( 0 − ). And, since = 1/3 for radiation, Equation (9) may be written for radiation as Figure 1 . 1Fitted data curves for the three models inTable 1. The first one has been labeled as EDSM-NA since it is the non-adiabatic version of the EDSM model (flat, matter only, including Mach effect) in reference[11]. The second curve is labeled as EdeS-NA as it is the non-adiabatic version of the Einstein de Sitter model (flat, matter only universe). The third one is the curve for ΛCDM model. The left figure shows the complete fitted curves for the 580 points data set whereas the right figure is the zoom-in of the fit in the high region to enhance the difference among the fitted curves. Figure 2 2depicts the evolution of dimensionless parameters: a) scale factor , b) Hubble parameter / 0 , c) and deceleration parameter , against the dimensionless time 0 ( − 0 ) for the standard ΛCDM model and the non-adiabatic model developed here. While the shapes of the curves are different, the trends are similar. We notice that the ΛCDM curves are steeper in most of the plotted region. have plotted inverse of the expansion scale factor 1 = (1 + ) against the inverse of the standard scale factor 1 = (1 + ). The curve can be approximated with a power law expression = 1.1345 0.4714 except at rather low values. Also we have included a curve showing −3 (1 + ) = (1 + ) 3 (1 + ) against −4 = (1 + ) 4 to show that the radiation energy density scaling is altered drastically by the inclusion of Mach effect. This curve may be approximated with a power law expression = 1.4604 0.6035 except at small values. Figure 2 . 2Evolution of dimensionless parameters -scale factor (left figure), Hubble parameter / 0 (middle figure), and deceleration parameter (right figure)against dimensionless time 0 ( − 0 ) for the standard ΛCDM model and the Non-Adiabatic model EdeS-NA. Figure 3 . 3Evolution of inverse of scale factors and radiation density -Inverse expansion scale factor 1/ = (1 + ) against the inverse standard scale factor 1/ = (1 + ) for low values (left figure) and high values (middle figure); radiation energy density scaling (right figure) for the Non-Adiabatic model EDSM-NA . of this equation from the free online solver Wolfram Alpha (http://www.wolframalpha. Table 1 . 1Parameter and goodness-of-fit for the two models. 0 is in km s -1 Mpc -1 . SSE stands for sum of squares due to errors and RMSE for root mean square error.Parameter Parameter H 0 H 0 Low H 0 High Ω m,0 Ω m,0 Low Ω m,0 High SSE R-Square RMSE EDSM-NA 68.28 68.81 67.75 None NA NA 24.58 0.9958 0.2060 40 EdeS-NA 69.01 69.54 68.48 None NA NA 24.10 0.9959 0.2040 37 ΛCDM 69.85 70.71 69.01 0.2877 0.2489 0.3266 24.35 0.9959 0.2053 46 Equation used Goodness of Fit Model 95% Confidence 95% Confidence P J E Peebles, Principles of physical cosmology. Princeton, NJPrinceton University PressP. 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[ "DENSITY JUMPS NEAR THE VIRIAL RADIUS OF GALAXY CLUSTERS", "DENSITY JUMPS NEAR THE VIRIAL RADIUS OF GALAXY CLUSTERS" ]
[ "Anna Patej ", "Abraham Loeb " ]
[]
[]
Recent simulations have indicated that the dark matter halos of galaxy clusters should feature steep density jumps near the virial radius. Since the member galaxies are expected to follow similar collisionless dynamics as the dark matter, the galaxy density profile should show such a feature as well. We examine the potential of current datasets to test this prediction by selecting cluster members for a sample of 56 low-redshift (0.1 < z < 0.3) galaxy clusters, constructing their projected number density profiles, and fitting them with two profiles, one with a steep density jump and one without. Additionally, we investigate the presence of a jump using a non-parametric spline approach. We find that some of these clusters show strong evidence for a model with a density jump. We discuss avenues for further analysis of the density jump with future datasets.
10.3847/0004-637x/824/2/69
[ "https://arxiv.org/pdf/1509.07506v1.pdf" ]
119,025,898
1509.07506
25f29ee2a01f03f541297545077e553b8528000a
DENSITY JUMPS NEAR THE VIRIAL RADIUS OF GALAXY CLUSTERS Anna Patej Abraham Loeb DENSITY JUMPS NEAR THE VIRIAL RADIUS OF GALAXY CLUSTERS arXiv:1509.07506v1 [astro-ph.CO] 24 Sep 2015 Draft version September 28, 2015 Draft version September 28, 2015Preprint typeset using L A T E X style emulateapj v. 5/2/11 Recent simulations have indicated that the dark matter halos of galaxy clusters should feature steep density jumps near the virial radius. Since the member galaxies are expected to follow similar collisionless dynamics as the dark matter, the galaxy density profile should show such a feature as well. We examine the potential of current datasets to test this prediction by selecting cluster members for a sample of 56 low-redshift (0.1 < z < 0.3) galaxy clusters, constructing their projected number density profiles, and fitting them with two profiles, one with a steep density jump and one without. Additionally, we investigate the presence of a jump using a non-parametric spline approach. We find that some of these clusters show strong evidence for a model with a density jump. We discuss avenues for further analysis of the density jump with future datasets. INTRODUCTION Galaxy clusters contain a representative sample of the matter in the universe: the dominant constituent is dark matter, while the baryonic components include hot gas and the galaxies themselves (for a review, see Voit 2005). A longstanding analytical prediction for cosmological structure formation is the existence of a shock bounding the hot, gaseous intracluster medium and a corresponding jump in the dark matter profile, coincident with the virial radius of the galaxy cluster (e.g., Bertschinger 1985). More recently, simulations by Diemer & Kravtsov (2014) (hereafter DK14) have reinforced the expectation that the dark matter halo itself should exhibit a sharp density steepening near the virial radius (see also Adhikari, Dalal, & Chamberlain 2014). The member galaxies of a cluster are expected to trace the cluster's dark matter profile in the cluster outskirts, as they are subject to similar collisionless dynamics; accordingly, we may expect to see such a feature not just in the dark matter profile but also in the galaxy density profile. Tully (2010) examined the distributions of the galaxies in the Coma and Virgo clusters and detected sharp density cut-offs at radii of 3 and 2 Mpc, respectively, from the cluster centers, which he identified with the caustics of second turnaround. These correspond roughly to the virial radii of these clusters (e.g., Kubo et al. 2007;Karachentsev et al. 2014). The analysis of Trentham & Tully (2009) identified a similar feature in the galaxy group NGC 1023. More, Diemer, & Kravtsov (2015) mentioned that hints of such a jump may have been seen in other data sets. For instance, Tully (2015) measured the second turnaround radius, which is associated with the aforementioned density jumps, of several groups and clusters. Additionally, Rines et al. (2013) (hereafter R13) presented spectroscopic velocities for cluster members in 58 galaxy clusters, from which they reconstructed the density profiles, which appear to show a deficit of galaxies with respect to an NFW profile (Navarro, Frenk, & White 1997). However, they noted that their results at large radii from the cluster center may be affected by having few spectroscopically observed cluster members in the exterior regions. In this paper, we focus on the R13 sample of clusters and use public data from the Sloan Digital Sky Survey (SDSS) (York et al. 2000) to select cluster members photometrically. Rather than use spectroscopic velocities, we construct the radial number density profiles of member galaxies and fit them with two functional forms to examine whether we can detect a feature consistent with the predicted density jump in any of the clusters using available data sets. Where applicable, we assume the standard ΛCDM cosmology with Ω Λ = 0.73 and Ω m = 0.27, consistent with the parameters selected by DK14 to enable direct comparison. To compare results across clusters we also use the measure R ∆ , which is the radius within which the mean mass densityρ = ∆ρ b (z), where ρ b is a specified cosmological background density and ∆ is the density contrast with respect to ρ b (z). DATA SDSS Catalogs To probe the existence of the density jump predicted by DK14, we focus our attention on the sample of 56 lowredshift (0.1 < z < 0.3) galaxy clusters from R13 (of their original 58 clusters, we do not include MS0906/A750 in our samples, since these clusters are nearly coincident and hence it is difficult to make a clean selection of member galaxies for each). Since their cluster sample was selected from regions of sky covered by SDSS, we use publicly available data from the most recent Data Release 12 (Alam, et al. 2015) to select cluster members. We obtain photometric catalogs from the SDSS Sky-Server SQL server 3 . For each cluster, we query galaxies within 1.5 degrees of the cluster center (which, from R13, is the X-ray center) in both right ascension (RA) and declination. We further restrict our query to sources in the 'Galaxy' view with observed r-band magnitude between 14 and 22 and which have 'clean' photometry as determined by the SDSS pipeline. For each galaxy, we obtain the dereddened 'model' magnitudes (which we will use throughout the remainder of this work), corrected for Galactic extinction according to Schlegel, Finkbeiner, & Davis (1998) by the SDSS pipeline, and we also select photometric redshift estimates and associated redshift quality estimates from the 'Photo-z' table. Cluster Member Selection We will test two cluster member selections. The first, which we will refer to as Selection A, is based solely on photometry, primarily comprising red cluster galaxies selected from their location on a color-magnitude diagram but also including some galaxies with sufficiently secure photometric redshift estimates that are blueward of the red sequence. The second, which we call Selection B, folds in the spectroscopy of R13 to include additional cluster members and to reject some of the photometrically selected galaxies whose redshift estimates place them beyond the cluster. These selection procedures are outlined in more detail in the following sections. Red Sequence Cluster Member Selection Both of our methods of cluster member selection are based upon the target selection process of R13, which relied on the red sequence method of Gladders & Yee (2000). The red sequence refers to a linear feature in the color-magnitude diagram of galaxies in a cluster field that arises from the population of red, early-type galaxies that have been observed to comprise the majority of cluster members. By selecting galaxies within a range around this line, we can obtain a sample of cluster members. As in R13, we construct a diagram of g − r vs r and fix the slope of the red sequence line to −0.04. We then select the appropriate intercept by examining the distribution of galaxies within 5 arcminutes of the cluster center. We can also use the spectroscopic data publicly available from R13; by matching their spectroscopic catalogs to SDSS photometry, we can additionally calibrate our red sequence selection, as will be described in Section 2.2.3. Having thus identified the linear red sequence feature, we select galaxies that are within an offset of 0.1 magnitude above the red sequence line, and 0.15 magnitude below the red sequence line, as illustrated in Figure 1. This is in contrast to the choice of ±0.3 magnitude in R13, which was found to be much wider than the actual red sequence as indicated by the spectroscopy. The asymmetry of the limits above and below the red sequence is motivated by the results of Section 5.2 and Figure 16 of R13, which suggest that there are fewer cluster members above the red sequence than below it, as confirmed by the righthand panel of Figure 1, which shows the distribution of spectroscopically confirmed members. Contribution of Non-Red Sequence Galaxies Since the red sequence method selects a specific population of galaxies, it is worth examining whether the exclusion of other types of galaxies in the cluster affects our result. The analysis of Dressler (1980), which examined the galactic populations of over 50 galaxy clusters, found that blue, late-type galaxies comprise a small fraction of the populations of rich galaxy clusters, although this fraction increases as the density of galaxies decreases. This is supported by the spectroscopic data provided by R13, who find that within the virial radius of the cluster the fraction of blue galaxies is 10%, while over the entire radial range for which they obtained spectra, the fraction is 15%. Since we have photometric redshift estimates, we use these to also select cluster members that are blueward of the red sequence. We restrict our attention to galaxies that have secure photometric redshifts, as indicated by the 'photoErrorClass' and 'zErr' tags; the former we select to be equal to 1, and the latter we restrict to be less than 0.03. We then mark as additional cluster members galaxies that were not originally selected via the red sequence method and whose photometric redshift estimate is within 0.03 of the cluster redshift provided by R13. The population of galaxies with photometric redshift estimates of sufficient quality as indicated by the SDSS pipeline is too small for most clusters to use as the basis for our analysis; within the central 1.5 h −1 Mpc, there are on average only about 60 galaxies with photometric redshifts satisfying |z c − z g | < 0.03 and subject to the above quality cuts, many of which were already picked up via the red sequence method. Additionally, basing a selection on these redshifts would exclude most cluster members with r 19. Accordingly, as noted above, we must combine them with the galaxies selected via the red sequence method. This final selection, denoted Selection A, for an example cluster is shown in the lefthand panel of Figure 2. Additional Tests of the Cluster Member Selection While neither the SDSS photometric redshifts nor the spectroscopy of R13 identify enough cluster members for our analysis, we can use this additional information to refine the red sequence selection in a second selection. As before, after selecting red sequence cluster members using the procedure outlined in Section 2.2.1, we add in bluer cluster members based on SDSS photometric redshifts as described in Section 2.2.2. However, in this case, we can further refine the selection by also rejecting cluster members if the SDSS photometric redshifts are sufficiently secure and the members in question have redshifts in the range |z c − z g | < 0.03, where z c is the cluster redshift and z g is the photometric redshift estimate. We can use the spectroscopic data of R13 similarly. After matching their tables to the SDSS catalogs, we add in cluster members as identified by the spectroscopy and reject galaxies that were originally selected by the red sequence method or photometric redshifts but are identified as non-members by R13. This method of cluster member selection, which we term 'Selection B,' is summarized in the righthand panel of Figure 2. This selection is more observationally expensive than Selection A, so it is worth testing both methods to see whether the additional information makes a difference in the results. On average, the numbers of cluster members selected by these two methods differ by about 5% within 1.5 h −1 Mpc of the cluster center. We will test the effect of this selection on our analysis in Section 4. As an additional step in verifying our cluster member selection, we construct a projected radial density plot consisting of the total radial density profiles of all The red sequence cluster member selection for example cluster A1068. Left: in the color-magnitude diagram, we initially plot the location of galaxies within 5 arcminutes of the cluster center, which should be dominated by cluster members, to provide an indication of the red sequence. Middle: The galaxies selected by the red sequence method using the limits described in Section 2.2.1; we plot the galaxies thus selected that are within 2.5 h −1 Mpc of the cluster center. The labelled intercept (g − r) i is selected to be the value at r = 16. Right: The cluster members as selected via the spectroscopy of R13, which can be used to verify the red sequence selection. sources in our catalog (subject to r < 20) as well as the non-cluster density profile, computed by subtracting the counts of cluster members from the total number of galaxies in each bin. In the total profile, we expect to see a steep increase in galaxies at small radial distances that flattens out at large R. After subtracting out the contribution from these cluster members, the resulting non-cluster member profile should be roughly flat. Figure 3 shows these density plots for four example clusters, using Selection A. METHOD Approach Our fiducial analysis relies on cluster members selected as described in Section 2, subject to a magnitude cut of r < 20. We determine the number counts of galaxies in bins of 0.1 h −1 Mpc, from which we construct the projected number density profile of member galaxies, N (R), with R indicating the projected radius, using the cluster centers listed in R13. We use Poissonian error bars. We employ two methods to test for evidence of the density jump. In the first, we fit the profiles using the fitting formula provided by DK14 to incorporate their predicted density steepening: n DK (r) = n in (r) 1 + r r t β − γ β + n m b e r 5R 200 −se + 1 .(1) As suggested by DK14, we fix β = 6 and γ = 4, a choice that yields a dependence of r t on the mass accretion rate, Γ: r t = 0.62 + 1.18e −2Γ/3 × R 200(2) We further select the NFW profile with two parameters, n s and r s , as our inner density profile n in : n NFW (r) = n s r/r s (1 + r/r s ) 2 .(3) We note that DK14 used the Einasto function (Einasto 1965) instead of the NFW for the inner profile, but in the regime of interest, the distinction between the two is negligible. We will refer to the model given by Equation 1 with n in (r) = n NFW (r) as the 'Density Jump' model, or 'DJ' model in abbreviation. Since the density jump feature occurs around the virial radius, we use a fitting range of R < 2R 200 , which is beyond the range that is typically fitted well by an NFW profile (we also exclude the inner R < 0.1R vir in the fit, consistent with DK14). Accordingly, the two models that we will compare via fitting will be the DJ profile and a profile given by an NFW profile with an outer term: n(r) = n NFW (r) + n m b e r 5R 200 −se + 1 .(4) These profiles are projected numerically and fitted to our galaxy density data. The free parameters in our fit are n s , r s , b e , and s e for both profiles; the full DJ formula has one additional free parameter, Γ. We restrict the fits to the range 0 < Γ < 5 and to reasonable values of n s and r s , the latter of which is restricted by the value of the NFW concentration parameter, c = R 200c /r s . Observations have suggested that this value is lower for galaxy profiles than for dark matter (e.g., Lin, Mohr, & Stanford 2004;Hansen et al. 2005;Budzynski et al. 2012), and we set the range as 2.0 < c < 6.0 in our fits. We refer to the Appendix of DK14 for appropriate ranges for s e (0.5 − 2.0) and b e (0.1 − 4.0). We additionally fix n m , which will be discussed further in Section 3.2, and we select as our upper limit of integration R = 10R vir , same as used by DK14. We then examine the results of the fits using the Akaike Information Criterion (AIC; Akaike 1974) and the Bayesian Information Criterion (BIC; Schwarz 1978), which provide a means of comparing models fitted to data. As the use of these methods in astrophysics and cosmology has been discussed in a number of papers (e.g., Takeuchi The cluster member selection for example cluster A1068 using Selection A, which comprises galaxies selected using the red sequence method and galaxies with SDSS photometric redshifts, which add in some galaxies blueward of the red sequence. Bottom: The cluster member selection for this same cluster using Selection B, which combines the red sequence, photometric redshifts, and spectroscopic redshifts. ther details. To apply these criteria, we compute the following statistics for each fit: AIC = χ 2 + 2p + 2p(p + 1) N − p − 1 ,(5)BIC = χ 2 + p ln(N ),(6) where p is the number of parameters in the fit, N is the number of data points being fitted, and χ 2 is the standard minimized goodness-of-fit parameter. In the case of the AIC, we have also included a correction term of 2p Our second method consists of smoothing the profiles by fitting a smoothing cubic spline to our data. For a data set with measured values y i and errors σ i at a set of points r i , the smoothing spline f (r i ) is constructed to satisfy the condition (p + 1)/(N − p − 1)N i=1 y i − f (r i ) σ i 2 ≤ S,(7) where S is a constant that interpolates between smoothing and fitting: that is, when S = 0, the spline is forced to pass through every data point, so that there is no smoothing, whereas as S is increased, the curve becomes smoother at the expense of the fit (de Boor 2001). Reinsch (1967) argues that the smoothing parameter S should be chosen in the range N − √ 2N ≤ S ≤ N + √ 2N , where N is the number of data points over which we construct the spline, if the σ i are estimates of the standard deviation in y i . We accordingly choose three values within this range to compare to an NFW fit: S = N − √ 2N , N , N + √ 2N . The NFW model is expected to be a good fit to the inner parts of the profile, so we use the analytical expression for the projected NFW density (e.g., Wright & Brainerd 2000) to fit the cluster galaxy density profiles within R 200 , establishing the values of n s and r s . We then calculate the logarithmic derivative d log(N )/d log(R) of the splines for each cluster to test for the presence of the density jump feature, comparing it to the logarithmic derivative of the NFW fit. Fixed Parameters As noted in Section 1, we define measures of cluster size such that the mean mass density inside the radius R ∆ isρ = ∆ρ b (z); commonly used values are R 500 and R 200 . The other quantity that needs to be specified is the background density ρ b (z); one choice often used in observational work is the critical density ρ c (z) ≡ 3H(z) 2 /8πG, where H(z) is the Hubble constant at redshift z, which is given by H(z) 2 = H 2 0 Ω m (1 + z) 3 + Ω Λ with H 0 = 100 h km/s/Mpc. Another choice is to use the mean matter density ρ m (z) = ρ c (z)Ω m (z), where Ω m (z) = Ω m (1 + z) 3 / Ω m (1 + z) 3 + Ω Λ , which is used with ∆ = 200. As noted above, DK14 use the mean matter density ρ m (z) to define the radius R 200 = R 200m used in the fitting formula given by Equation (1). However, R13 measures R 200 = R 200c for their sample of clusters using the critical density as reference, so we need to convert their measure to that of DK14. To do so, we note that a given mean densityρ may be written in two ways: ρ = ∆ c ρ c (z) = ∆ m ρ m (z).(8) If we specify that ρ = ρ NFW , then in the outskirts of the cluster (i.e., including near R 200 ), we have: ρ(R ∆ ) ∝ 1 R 3 ∆ .(9) A1132 All Galaxies Non-cluster Galaxies Cluster Galaxies Fig. 3.-Plots of the projected densities of galaxies with r < 20 in four cluster fields: A1068, A1302, A646, and A1132. Both the total galaxy density (black points) and the cluster density (red points) are expected to rise steeply towards the cluster center. The blue points show the density after removing cluster members from the total count, with a line drawn to indicate the average value beyond 20 arcminutes for comparison. The non-cluster galaxy distribution should be roughly flat if we have adequately selected cluster members, as for A1068 and A1302. A646 and A1132, on the other hand, show some residual overdensity. Combining this relation with Equation (8) yields R 200m R 200c = 1 Ω m (z) 1/3 .(10) For z = 0.1 − 0.3, this implies that R 200m ≈ 1.4 × R 200c .(11) Accordingly, for simplicity, we will henceforth refer to R 200m as R 200 , and the values from R13 will be converted to this measure using Equation (11). Lastly, we need to establish an appropriate value for n m . We note that the original prescription of DK14 defines Equation (1) in terms of mass densities ρ rather than number densities n, and their fitting results assume a fixed value of ρ m (z) = ρ c (z)Ω m (z). The translation into a number density needs to take into account the impact of the primary selection function by which we obtain cluster members, the red sequence method. In the absence of a cluster, this method would select a population of galaxies that lies in the appropriate region of colormagnitude space; in this case, the projected density of these galaxies is expected to be roughly constant across the field of view. Recalling the outer profile term of Equation (1), n out (r) = n m b e r 5R 200 −se + 1 ,(12) we can analytically determine the contribution to the surface density of the last, constant term. The surface density is the line-of-sight integral, N (R) = 2 ∞ R n(r)r √ r 2 − R 2 dr;(13) this integral diverges for a constant n(r). However, in practice we must truncate this integral at some maximum radius R max . As noted above, DK14 use R max = 10R vir ≈ 9R 200 . In that case, N m (R) = 2n m 9R200 R r √ r 2 − R 2 dr,(14)N m (R) = 2n m R 200 81 − R R 200 2 .(15) For the scales of interest in our fits, R 2R 200 (and even a bit beyond), this value is roughly constant, N m ≈ 17.6n m R 200 .(16) The redshift dependence of N m can be determined by applying the red sequence method in test fields that are not centered on low redshift clusters. If we select a population of galaxies with this method, then we can construct the projected density in a given radial bin i as N m,i = N i π R 2 i,max − R 2 i,min ,(17)= 1 D A (z) 2   N i π θ 2 i,max − θ 2 i,min   ,(18) where N i is the number of galaxies in the ith bin and D A (z) denotes the angular diameter distance. We expect that the angular projected density (the term in brackets) over the radial range is a roughly constant value, which we denote by η. Then N m (z) = η D A (z) 2 = η(1 + z) 2 D c (z) 2 ,(19) where D c is the comoving distance, which at the small redshifts considered here, is given by D c (z) ≈ (c/H 0 )z. Upon absorbing the factor of c/H 0 into η, we have: N m (z) = η (1 + z) 2 z 2 .(20) To obtain the value of η, we apply our red sequence cuts in random test fields from SDSS (also subject to our initial magnitude cut of r < 20) and fit N with the above function. This yields η ≈ 0.08 h 2 Mpc −2 . Accordingly, combining Equation (16) with (20), we find that: n m (z, R 200 ) = 4.55 × 10 −3 R 200 (1 + z) 2 z 2 .(21) We fix this value individually for each cluster using its measured redshift and R 200 (the latter converted as discussed above) from R13. RESULTS Fitting The results of fitting the projected number density profiles with Equations (1) and (4) are summarized in Table 1 for both the fiducial method and variations, the latter of which will be discussed in the next section. However, as the DJ model has one additional parameter than the NFW+Outer model, we then compare the two fits using the AIC and BIC. We compute, in each case, ∆IC = IC NFW+O − IC DJ . Since lower values of IC are favored, this quantity will be positive if the criteria indicate evidence in favor of the DJ model. In Table 1, we thus first list in the column, '% χ 2 DJ /ndf < χ 2 NFWO /ndf,' the number of galaxy clusters for which the reduced χ 2 is lower for the DJ model and second the number of galaxy clusters whose fits pass a general quality cut, including requiring that χ 2 /ndf < 3 for the DJ model and a more generous χ 2 /ndf < 5 for the other. The next two columns indicate how many clusters have ∆IC = IC NFW+O − IC DJ > 0 for each criterion, which would suggest evidence in favor of the DJ model. The last two columns indicates how many clusters have one of the ∆IC > 5, indicating strong evidence in favor of this model. Our fiducial method uses a binning of ∆R = 0.1 h −1 Mpc and employs a magnitude cut of r < 20. We use two selection methods to identify cluster members; as discussed in more detail in Section 2, Selection A is based on SDSS photometry and Selection B refines the first with R13 spectroscopy. We see that refining the cluster member selection yields a higher number of clusters for which the DJ model is a better description than a simple NFW model with an outer term. However, the results of individual clusters are consistent between the selection methods -the seven clusters with the strongest evidence in favor of the density jump using Selection A (A655, A1033, A1246, A1437, A1689, A1914, and A2034) continue to have ∆IC > 5 using the second selection. These clusters (along with the more marginal case A1835, which has only one ∆IC > 5 using Selection A) are shown in Figure 4 for Selection A fits and Figure 5 for Selection B; however, using Selection B several other clusters also reveal strong (and for some even stronger) evidence for a density jump. In both cases, the cluster with the strongest evidence for a density jump is A1689. Furthermore, the increase in the number of clusters with IC evidence in favor of the DJ model using Selection B suggests that it may be beneficial to investigate these profiles further with a cleaner cluster member selection, with either a sample of galaxies with high-quality photometric redshift estimates or with a sufficiently dense sample of galaxies with secure spectroscopic redshifts. Additionally, a few of these clusters (particularly A1246) show some residual overdensity at roughly θ < 5 ′ of the cluster center after the cluster member selection (c.f. Figure 3), which could be indicative either of having missed some population of galaxies in our selection or that there are some other agglomerations of galaxies along the line of sight that contribute to the overdensity. High-quality, dense redshift estimates would help resolve this ambiguity. We additionally test the fits by making the radial bins twice as large (∆R = 0.2 h −1 Mpc). In this case, the reduced χ 2 values are a bit worse for Selection A, as indicated in Table 1. We find that the larger binning yields the greatest drop in the number of clusters with evidence favorable towards the DJ model using the AIC; the BIC results, however, are virtually unchanged from the earlier case. The discrepancy is likely caused by the smaller number of data points over which we fit relative to the number of parameters, which makes affects the correction term for the AIC. Smoothing Splines In addition to fitting the projected galaxy density profiles with Equations (1) and (4), we also fit a smoothing cubic spline as described in Section 3.1. This latter approach has the benefit of being model-independent; we use the spline fits to construct a smooth logarithmic derivative of the profile to test for the existence of a density jump. First, to provide context for the spline results, in Figure 6, we plot the DJ model at fixed Γ = 4 for various redshifts, which determine n m , and for an example fixed redshift z = 0.15, we also show the variation with Γ. The bottom panel of each plot shows the corresponding logarithmic derivatives. The effect of the redshift dependence is to make the density jump steeper at higher redshift due to the lower background n m . At a fixed n m , a smaller value of Γ means a shallower density jump, whose maximum amplitude slope is attained at larger radii, although the values of b e and s e , which are free parameters in our fits, also contribute to the variation in the amplitude. Figure 6 can be compared to Figure 13 of DK14. While the behavior of the density jump is qualitatively the same, we note that in our analysis, the maximum slope amplitudes are smaller than those predicted by DK14 due to the elevated background; accordingly, in our case the maximum slope amplitudes are governed not only by the mass accretion rate Γ but also the redshift of the cluster. However, the density jump location also depends on Γ. Accordingly, we model our spline fits on Figure 6. We show the cubic spline fits for the same subset of clusters selected via the information criteria in the fiducial analysis. Figure 7 shows the spline fits for Selection A, while Figure 8 is its counterpart for Selection B. The spline fits do appear to pick up a modest steepening of slope, consistent with the low redshift of these clusters. Since at low redshift we can expect to almost exclusively detect very large density jumps (Γ ≈ 5), which are likely uncommon, it is perhaps not surprising that the number of clusters for which the information criteria suggest strong evidence for the jump is fairly low. DISCUSSION We have tested whether cluster galaxy density profiles show evidence for a density jump feature near the virial radius using two methods: profile fitting and spline smoothing. We have examined the evidence in favor of the presence of a steep density jump in the galaxy density profiles of clusters, and also investigated the results via spline smoothing. There does appear to be some dependence of our results on cluster redshift. In our fiducial analysis of Section 4.1, the seven clusters that showed strong evidence (both ∆IC > 5) for the density jump using Selection A spanned the redshift range of roughly z = 0.1 − 0.2, which omits the higher redshift clusters (although an eighth cluster, A1835, with one ∆IC > 5 value, is at z = 0.25). However, employing Selection B enlarges the sample of strong evidence clusters, and extends this to about z = 0.1 − 0.25, effectively the entire range of R13. More massive clusters tend to show stronger evidence for a density jump: these same 8 clusters have virial masses in the range M vir = (2 − 11) × 10 14 h −1 M ⊙ , while a significant fraction of the clusters in the R13 sample have M vir = (0.3 − 2) × 10 14 h −1 M ⊙ (and may also be considered galaxy groups). This trend continues even when considering the additional clusters that pass the criteria using Selection B. This behavior can be expected, as higher mass clusters tend to have higher values of Γ and their profiles are likely to be better sampled than lower mass systems. We see variations based on the cluster member selection. Our primary selection uses only photometric data, but redoing the analysis with the inclusion of the R13 spectroscopy does shift the results; in particular, the refined selection yields a larger sample of clusters that show some evidence for the jump. This suggests that additional data is required to make a firm detection of the density jump. This additional data could be in the form of dense spectroscopy out to large radii of cluster member galaxies, building upon the R13 catalog, which would provide the most secure cluster member determination. Otherwise, high-quality photometric redshifts could also improve upon our estimates. Additionally, it is worth noting that in interpreting these results it is necessary to keep in mind some of the other factors that can impact the jump signature. In particular, cluster asphericity and contamination by other small groups and clusters along the line of sight can contribute to a diminishing of the jump signal. The latter of these can be addressed with redshift estimates. For the first, we note that the signature of the jump becomes more pronounced and thus easier to detect when Γ is large; however, clusters with large values of Γ may be more likely to have disturbed shapes and substructures, which when examined in projection could obscure the signature of the density jump. More generally, it would be useful to compare the results in Table 1 to simulations. Mock halo catalogs could give an indication of the conditions needed for a cluster to show a discernible jump and thus the fraction of clusters in which we can expect to find this signature, which would provide context for the fractions we find using ob-servational data. While this is a promising avenue for future analyses, such a comparison is beyond the scope of this work. SUMMARY AND CONCLUSIONS Using the cluster sample of R13 and optical data from SDSS, we have searched for the signature of a density jump feature near the virial radius (∼ R 200 ) predicted by the simulations of DK14. Our fiducial analysis selects cluster members from SDSS photometric catalogs using the red sequence method and photometric redshifts, and compares this to a selection refined by the inclusion of the spectroscopy of R13. After constructing the radial density profiles of the clusters, we fit two models, one with a density jump -Equation (1) -and one without -Equation (4) -and used the Akaike and Bayesian Information Criteria (AIC and BIC) to examine the evidence in favor of the density jump model. These criteria indicate that, using our fiducial methods, at least 10% showed strong evidence (∆IC > 5) for the model that includes a density jump. The clusters with strong evidence for the density jump tend to have a higher mass (M vir 2 × 10 14 h −1 M ⊙ ), as expected. The cluster with the strongest evidence for the jump in this sample is Abell 1689. We examined varying some of the parameters in our analysis. We find that using a larger bin size yields fewer clusters with strong evidence (if we stipulate a strong result for both criteria; the AIC appears to be most affected by the binning) for the density jump. However, if we require only one of the information criteria to yield strong evidence, then we are in excellent agreement with the fiducial analysis. We additionally tested our results using cubic smoothing splines. The spline analysis appears to indicate some steepening of slope in these clusters, but the comparison to the theoretical predictions of DK14 is complicated by background and projection effects. In the case of the former, a high background can diminish the signal, while the latter acknowledges that since clusters can have diverse shapes, the density jump is not necessarily localized at a single radius, which can lead to modifications of the signal in projection. Finally, varying the cluster member selection methods does appear to have an effect on the results; in particular, including the most secure cluster member indicators (spectroscopic redshifts) increases the number of clusters for which we have strong evidence in favor of a model with a density jump. Thus, our conclusions are limited by the availability of data for the identification of galaxy cluster members. While the red sequence method provides a means of identifying the majority of cluster members, which tend to be red, early type galaxies, it does not readily provide a means of rejecting interloping red galaxies from higher redshifts or for including the smaller population of member galaxies blueward of the red sequence. While the spectroscopic observations of R13 provide an excellent start, it would be useful to extend their samples with additional redshift estimates to improve the completeness in the cluster outskirts, mitigating observational selection effects that can mimic a density steepening. Accordingly, at present, neither the extant photometry nor spectroscopy of these clusters provides adequate numbers of confirmed member galaxies out to sufficiently large radii to significantly improve upon our methods. However, in order to better constrain the density jump feature and precisely determine its amplitude and position, future efforts will require either secure photometric redshifts or additional spectroscopy to refine the cluster galaxy density profiles. Future work using mock halo catalogs could help pinpoint the conditions under which clusters can be expected to show a discernible density jump to help guide these observational endeavors. Fig. 1.-The red sequence cluster member selection for example cluster A1068. Left: in the color-magnitude diagram, we initially plot the location of galaxies within 5 arcminutes of the cluster center, which should be dominated by cluster members, to provide an indication of the red sequence. Middle: The galaxies selected by the red sequence method using the limits described in Section 2.2.1; we plot the galaxies thus selected that are within 2.5 h −1 Mpc of the cluster center. The labelled intercept (g − r) i is selected to be the value at r = 16. Right: The cluster members as selected via the spectroscopy of R13, which can be used to verify the red sequence selection. Fig. 2 . 2Liddle 2007;Broderick et al. 2011;Tan & Biswas 2012), we simply mention the most salient qualities here and refer the reader to these works for fur--Top: Fig. 4 .Fig. 5 . 45-Plots of the projected number density profiles for the 8 clusters with the highest ∆AIC and ∆BIC values in the fiducial analysis using Selection A. Three fitted functions are shown: a base NFW, fitted interior to R 200 , an NFW+Outer model, given by Equation(4)and fitted interior to 2R 200 , and the full Density Jump model given by Equation 1 with an inner NFW profile, fitted interior to 2R 200 . -Plots of the projected number density profiles for the 8 clusters with ∆IC > 5 using both selection methods, with fits shown for the profiles constructed using Selection B. The curves are the same as inFigure 4. Fig. 6 . 6-The predicted signature of the density jump is a steepening of slope that can be analyzed using the logarithmic derivative of the DK14 model for four combinations of se and be values. The redshift dependence enters via nm(z, R 200 ) as given by Equation(21), and we show the result for three values of z in the range of the R13 clusters at fixed Γ = 4. For the z = 0.15 case we also show the variation of the profile with Γ. An NFW profile (dashed) is drawn for comparison. All other model parameters are fixed for all curves shown. Fig. 7 . 7-The top panel shows spline fits for various values of the smoothing parameter S for the clusters in Figure 4. The lower panel displays the logarithmic derivatives of the splines. An NFW fit and its logarithmic derivative (dashed line) are shown for comparison. Fig. 8 . 8-Same asFigure 7, but for Selection B profiles. which is recommended for small values of N(Burnham & Anderson 2002. The model that is preferred by these criteria is the one with the lower IC = AIC, BIC value. If we compute ∆IC = IC high − IC low , then, roughly, values of ∆IC = 1 − 5 are indicative of 'positive' evidence in favor of the model with lower IC and values of ∆IC > 5 denote 'strong' evidence (e.g.,Liddle 2007;Broderick et al. 2011). TABLE 1 Information 1Criteria Results http://skyserver.sdss.org/dr12/en/tools/search/sql.aspx We are very grateful to Andrey Kravtsov and Benedikt Diemer for their detailed and helpful comments on a draft of this paper. We would also like to thank the staff of [email protected] for advice regarding downloading data from SDSS.This work was partly supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144152. This work was also supported by NSF Grant No. AST-1312034. This work relied on tools from Numpy (van der Walt, Colbert, & Varoquaux 2011), Scipy(Oliphant 2007), and Matplotlib(Hunter 2007), as well as ROOT (http://root.cern.ch/). This paper relies on data from SDSS-III. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.The . S Adhikari, N Dalal, R T Chamberlain, JCAP. 1119Adhikari, S., Dalal, N., & Chamberlain, R. T. 2014, JCAP, 11, 019 . H Akaike, IEEE Transactions on Automatic Control. 19716Akaike H. 1974, IEEE Transactions on Automatic Control, 19, 716 . S Alam, ApJS. 21912Alam, S., et al. 2015, ApJS, 219, 12 . E Bertschinger, ApJS. 5839Bertschinger, E. 1985, ApJS, 58, 39 A Practical Guide to Splines. C De Boor, Springer-VerlagNew YorkRevised Editionde Boor, C. 2001, A Practical Guide to Splines, Revised Edition (New York: Springer-Verlag) . 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[ "H igh-P ressure A m orphous N itrogen", "H igh-P ressure A m orphous N itrogen" ]
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T he phase di agram and stabi l i ty l i m i ts of di atom i c sol i d ni trogen have been expl ored i n a w i de pressure{tem perature range by several opti cal spectroscopi c techni ques. A new l y characteri zed narrow -gap sem i conducti ng phase has been found to exi st i n a range of80{270 G Pa and 10{510 K .T he vi brati onaland opti calproperti es ofthe phase produced under these condi ti ons i ndi cate that i t i s l argel y am orphous and back transform s to a new m ol ecul ar phase. T he band gap ofthe phase i s found to decrease w i th pressure i ndi cati ng possi bl e m etal l i zati on by band overl ap above 280 G Pa.
10.1103/physrevb.64.052103
[ "https://export.arxiv.org/pdf/cond-mat/0105101v1.pdf" ]
119,343,638
cond-mat/0105101
d8613d7d3282dff76e1c0b6c9ae1a196d4147117
H igh-P ressure A m orphous N itrogen 4 May 2001 H igh-P ressure A m orphous N itrogen 4 May 2001Eugene G regoryanz,A l exander F.G oncharov,R ussel lJ.H em l ey and H o-kwang M ao G eophysicalLaboratory and C enter for H igh Pressure Research, C arnegie Institution ofW ashington, 5251 B road B ranch Road N W ,W ashington D .C .20015 U .S.A T he phase di agram and stabi l i ty l i m i ts of di atom i c sol i d ni trogen have been expl ored i n a w i de pressure{tem perature range by several opti cal spectroscopi c techni ques. A new l y characteri zed narrow -gap sem i conducti ng phase has been found to exi st i n a range of80{270 G Pa and 10{510 K .T he vi brati onaland opti calproperti es ofthe phase produced under these condi ti ons i ndi cate that i t i s l argel y am orphous and back transform s to a new m ol ecul ar phase. T he band gap ofthe phase i s found to decrease w i th pressure i ndi cati ng possi bl e m etal l i zati on by band overl ap above 280 G Pa. D espi te earl y theoreti calpredi cti ons for a transform ati on ofni trogen to a m onoatom i cstate, 1{3 rel i abl eexperi m entalevi dence becam e avai l abl e onl y qui te recentl y. 4;5 O pti cal spectroscopy, vi sual observati ons and el ectricalconducti vi ty m easurem entsshowed thatthe m ateri al transform s to a sem i conducti ng non-di atom i c phase at 150 G Pa (190 G Pa at 80 K ). 4;5 T he transi ti on to the nonm ol ecul ar state was predi cted to be hi ndered by a l arge energy barri er and accom pani ed by a l arge volum e di sconti nui ty and hysteresi s. 2;3 T he l atter was conrm ed by vi sual observati ons w hi ch i ndi cated that the hi gh-pressure phase can be preserved to 17 G Pa at l ow tem peratures. 5 C haracteri zati on of the hi gh-pressure phase (cal l ed here)sti l lrem ai nsan i m portanti ssue because ofthe l ack of structuralstudi es and system ati c spectroscopi c data atdi erent P{T condi ti ons. O pti calabsorpti on spectra 4 reveal the presence of a l ow -frequency l ogari thm i c U rbach tai l 6 and a hi gher energy regi on,w hi ch obeys the em pi ri calTaucl aw . 7 T hi stypeofabsorpti on edgei stypicalforam orphoussem i conductors(see R ef.8).A l though bei ng qui te di agnosti c,thi sobservati on sti l lrequi resconrm ati on,as a hi ghl y di sordered hi gh-pressure structure i sconsi stentw i th the nature ofthe transform ati on;i . e. a l arge vol um e change ata reconstructi ve phase transi ti on can cause l arge shear stresses because ofi nhom ogeneous nucl eati on ofthe hi gh-pressure phase (see R ef.9). M oreover,experi m entsdem onstratethattwo phasescoexi sti n a w i de pressure range (at300 K ), 4 thusm aki ng the characteri zati on ofthe phase even m ore di cul t. H ere we presentnew opti caldata overa w i de P{T range i ndi cati ng that hi gh-pressure non-m ol ecul ar phase i s a l argel y am orphous, narrow gap sem i conductor to at l east 268 G Pa. W e exam i ne the stabi l i ty l i m i ts of the di atom i c m ol ecul ar state and present evi dence for new transform ati ons,i ncl udi ng m etal l i zati on by band overl ap above 280 G Pa. Four experi m ents were perform ed at room tem perature w i th the m axi m um pressures vari ed from 180 to 268 G Pa. A bove 200 G Pa pressure was determ i ned usi ng tunabl e red l i nes ofT i : sapphi re l aser com bi ned w i th ti m e resol vi ng techni que (Fi g. 1). For l ow -tem perature m easurem ents we used a conti nuous-ow H e cryostat, w hi ch al l owed i nfrared and in situ R am an/ uorescence m easurem ents. 10 H i gh-tem perature R am an and vi si bl e transm i ssi on m easurem ents were perform ed w i th an external l y heated cel l . 11 In thi scase,i nfrared m easurem ents were done on sam pl es quenched to room tem perature. Fi g. 2 show s representati ve IR and vi si bl e transm i ssi on spectra dem onstrati ng the e ect oftem perature on the sem i conducti ng opti cal edge characteri sti c of the phase. T he spectra presented correspond to the conditi ons w hen no m ol ecul ar phase i s present i n the sam pl e asdeterm i ned from vi brati onalspectroscopy (see bel ow ). N o vari ati on ofthe shape and posi ti on ofthe band gap can be detected from transm i ssi on spectra at di erent tem peraturesand constantpressure of200 G Pa.Fi g.2b show s that the l ow -energy porti on of the spectra pl otted i n l ogari thm i c scal e (U rbach pl ot) have a constant sl ope ( ) i n a 10{200 K range. T hi s al so agrees w i th 300 K data, 4 thus show i ng that i s not tem perature dependent. Si m i l arspectra have been reported foram orphous phosphorus at zero pressure. 13 T hi s i s typi calfor sol i d am orphous sem i conductors, 8 because the random m i cro el d i s caused by stati c di sorder i n the system as opposed to crystal l i ne m ateri al s 6 w here the vi brati ons generate a tem perature-dependent dynam i caldi sorder. D eterm i nati on ofthe band gap from ourdata i sa compl i cated i ssue,because there i s no characteri sti c feature ofthe spectra w hi ch can be associ ated w i th the band gap (e. g. ,R ef.14). T hi s i s especi al l y i m portant for our m easurem ents,si nce we essenti al l y dealw i th sam pl es ofvari ousthi ckness(w hi ch i sa functi on ofanvi lgeom etry and pressure). A s the resul t,vi sualobservati ons ofthe sampl e above 230 G Pa showed that i t i s red or yel l ow i sh i n transm i ssi on and bl ack i n re ecti on,w hi ch i s consi stent w i th the sem i conducti ng state. T he col or ofthe sam pl e (com pare w i th the observati onsofdark ni trogen i n R efs. 4, 5, 15, 16) m ay be expl ai ned by i ts thi ckness (ofthe order of 1 m ) com pared to the sam pl es brought to 150 G Pa (up to 5 m ). A t the hi ghest pressure (268 G Pa), vi si bl e transm i ssi on spectra cl earl y show the presence of the fundam entalabsorpti on edge characteri sti c ofsem iconductors (Fi g. 2a). T hi s resul t i s i n agreem ent w i th di rect el ectri calm easurem ents perform ed to 240 G Pa. 5 T he hi gh-energy absorpti on edge, w hi ch can be ob-served i n thi s case, corresponds to el ectroni c transiti onsbetween extended states(unl i keU rbach absorpti on, w hi ch i s presum abl y caused by transi ti ons from l ocali zed to extended states). Extrapol ati on of the absorpti on spectra pl otted as(h ) 0:5 versush gi vesthe val ue of opti cal gap. 7 T hese val ues at di erent pressures are show n i n Fi g. 3. N ote that data from di erent experim ents agree,despi te the di erent sam pl e thi ckness and the fact that som e ofthe data are taken on pressure rel ease i n a m etastabl e pressure regi on (see bel ow ). W e observed a m onotoni cred-shi ftoftheband gap w i th pressure (see al so Fi g. 2a). T he pressure dependence ofthe band gap i s subl i near m ai nl y due to contri buti on from the poi nts obtai ned on decom pressi on. T he extrapol ati on ofthe band gap val ues gi ves m etal l i zati on at pressuressl i ghtl y above300 G Pa.Li nearextrapol ati on ofthi s curve to hi gherpressures(nottaki ng i nto accountpoi nts obtai ned upon decom pressi on)gi vesa val ue of280 G Pa. W e now present tem perature m easurem ents ofthe vibrati onalproperti es of the phase. Type II di am onds were used form i d-IR m easurem entsto avoi d i nterference w i th the characteri sti c absorpti on of the sam pl e. T he representati ve absorpti on spectra at di erent tem peratures (see Fi g. 2) cl earl y show the presence of a broad 1700 cm 1 IR band (com pare w i th R ef.4). Its presence was al so observed i n the sam pl e heated to 495 K at 117 G Pa (see bel ow ).T he posi ti on ofthe band and i tsdam pi ng (i f tted as one band) does not depend on pressure and tem perature w i thi n the error bars. T he R am an spectrum ofthe phase obtai ned on heati ng (see bel ow ) does not show any trace of the m ol ecul ar phase (see Fi g. 4b). C areful exam i nati on of the spectrum i n thi s case showed a weak broad band at 640 cm 1 i n agreem ent w i th the data ofR ef.4 and a shoulder near 1750 cm 1 (both i ndi cated w i th arrow s i n Fi g. 3b). T he l ater m ay i ndi cate the presence of the second R am an peak cl ose to the posi ti on of the observed IR band,but can al so be due to R am an i n the stressed di am ond. T he broad two-peak structure ofthe phonon spectrum ofthem ateri ali sconsi stentw i th i tsam orphous nonm ol ecul arnature. For an am orphousstate,the spectrum observed woul d cl osel y resem bl ea densi ty ofphonon states 18 w i th the m axi m a correspondi ng roughl y to the zone boundary acousti cand opti c vi brati onsofan underl yi ng structure. 19 T he onl y l atti ce dynam i cs cal cul ati ons for hypotheti cal hi gh-pressure crystal l i ne phases of nitrogen are avai l abl e for the cubi c gauche phase, 20 and cal cul ated phonon frequenci es are i n a good agreem ent w i th our m easurem ents. T he vi brati onal spectroscopy and band gap structure i ndi cate the absence of a l ongrange order. T he m ateri alcan sti l lpossess som e shortrange order,for exam pl e rel ated to pyram i dalcoordi nati on of ni trogen atom s. T he absence of the l ong-range ordercan bedueto the structural exi bi l i ty becauseeach atom form s bonds w i th onl y three other atom s out of6 nearest nei ghbors. 19 W e probed the forward and reverse transform ati on of them ol ecul arto phasei n di erentregi onsofP-T space. W e used IR transm i ssi on spectra asdi agnosti csofthedegree oftransform ati on to the nonm ol ecul ar phase. T he absence of IR bands correspondi ng to vi brons and l atti ce m odes ofthe m ol ecul ar phase 4 was used as a cri teri a. Si nce both the m ol ecul ar and nonm ol ecul ar phases are transparent i n the m i d-IR ,the am ount ofthe phase presenti ssi m pl y proportionalto theam pl i tudeofthecorrespondi ng IR peaks. T hi s i s unl i ke the si tuati on w i th R am an spectra w hi ch are attenuated by absorpti on the phase. W e exam i ned the transform ati on at 205 K and el evated pressures and found that i t starts at 155 G Pa and com pl etes at 185 G Pa. T hi s i s shi fted to hi gher pressures com pared to our 300 K data 4 and i s i n agreem ent w i th the trend reported i n R ef. 5. T he sam pl e has been cool ed dow n to 10 K at 200 G Pa and warm ed up after subsequent rel ease ofpressure at 130-150 G Pa. IR and vi si bl e transm i ssi on spectra and R am an spectra cl earl y showed the persi stence of phase w i thoutany reversetransform ati on dow n to 120 G Pa.A tthi spoi ntthe pressure dropped to 87 G Pa and the sam pl e transform ed i nstantaneousl y back to a transparent phase (cal l ed 0 here). T he m ol ecul ar nature of thi s phase i s con rm ed by i ts R am an spectrum (Fi g. 4a) al though the posi ti ons ofthevi bron l i nesdo notcorrespond to thoseobserved on pressure i ncrease (Fi g. 4c). T hi s m eans that am orphous phase back transform sto a m ol ecul arphase w hi ch di ers from the one observed on upstroke. O n further rel ease ofpressure (to 60 G Pa) we observe the R am an spectra w hi ch are si m i l ar to those of or phases i n posi ti ons and i ntensi ti es ofvi bron peaks. T he qual i ty ofthe spectra (on pressure rel ease,the sam pl e thi ns dow n l eadi ng to the consi derabl e l ossofi ntensi ty)doesnotal l ow to establ i sh the presence/absenceofthe weakervi bron m odes characteri sti c of phase and unam bi guousl y determ i ne w hether 0 phase back transform s to or phase. In the heati ng experi m ent we rst exposed the sampl e to 495 K at 117 G Pa. T he e ect of tem perature caused a gradualtransform ati on (starti ng at450 K )si mi l ar to that observed at 300 and 200 K .T he com pari son ofR am an m odes reveal ed m ore than a 10-fol d decrease ofi ntensi ty i n the R am an vi bronsand no observabl e l atti ce m odes. Q uenchi ng of the sam pl e to room tem perature di d not change the col or and the vi si bl e absorpti on spectra. Surpri si ngl y,the i nfrared spectra reveal ed the presence ofm ol ecul ar vi brons,i ndi cati ng an i ncompl ete transform ati on (about 30% ofnonm ol ecul ar phase judgi ng from the i nfrared acti vi ty). D uri ng the second heati ng the sam pl e was com pl etel y transform ed to the phase. T hen pressure was dropped to 105 G Pa at 460 K causi ng an i nstantaneous reverse transform ati on to a transparent m ol ecul ar phase. T he spectralposi ti ons of the bands and thei r num ber do not correspond to those observed at thi s pressure on com pressi on but are si m il arto those obtai ned duri ng the unl oadi ng at300 K (see above). Increasi ng pressure to 135 G Pa at 510 K drove the di rect transform ati on i nto the phase agai n. Fi g. 5 sum m ari zes our data for the phase di agram of ni trogen obtai ned i n a course ofextensi ve P{T m easure-m ents. Substanti alhysteresi s i s observed for the transform ati on from and back to the m ol ecul ar phase,so the observed curves shoul d be treated as ki neti c boundari es. Forthedi recttransform ati on,ourdata arei n good agreem entw i th theresul tsofvi sualobservati onsofR ef.5.O ur hi gh-tem perature data show thatthe hysteresi sbecom es qui te sm al lat tem peratures above 500 K .T here i s l arge hysteresi s at l ower tem perature such that the m ol ecul ar phase can be m etastabl y retai ned beyond the { 0 boundary (above approxi m atel y 100 G Pa; see al so R ef. 5). T hus, the observati on of another m ol ecul ar phase ( 0 ) i n thi s P-T condi ti ons m eans that thi s phase i s eitherki neti cal y favored ortherm odynam i cal l y stabl e w i th respect to the phase. Ifthe potenti albarri er between two crystal l i ne phases i s hi gh (m ol ecul ar di ssoci ati on i s requi red i n our case), the transi ti on m ay be preem pted by a transform ati on to m etastabl e phase,w hi ch m ay be am orphous. 21 Transm i ssi on spectra of N 2 as a functi on of temperature. Spectra are shi fted verti cal l y for the cl ari ty. T he characteri sti c peak ofthe phase i s shaded. Inset (a) show s the pressure dependence of the absorpti on spectra of N 2 at very hi gh pressures and room tem perature. G ray l i nes represent the Tauc ts to the spectra i n an appropri ate spectral range. T he determ i nati on ofthe energy gap from these m easurem ents i s obscured by addi ti onall osses caused by a presence ofa ne ruby pow der i n the cham ber. T he hi gh-energy absorpti on edge i s m ost probabl y due to stress i nduced absorpti on ofdi am ond anvi l s 12. (b) U rbach pl ots at 200 G Pa and di erenttem peratures(shi fted verti cal l y). G ray l i nes are gui des to the eye. FIG . 1 . 1T hi s denesan i ntri nsi c stabi l i ty l i m i t(e. g. ,spi nodal )forthe diatom i c m ol ecul arstate ofni trogen. In vi ew ofthe am orphouscom ponentofthe hi gherpressure phase,the transi ti on m ay be consi dered as a type of pressure-i nduced am orphi zati on. A s such, the transform ati on boundary coul d track the m etastabl e extensi on ofthe m el ti ng l i ne ofthe m ol ecul ar phase,and i fso i t shoul d have a negati ve sl ope (consi stentw i th negati ve V and posi ti ve S for a transi ti on to dense am orphous state. 21 ) A l ternati vel y,one can vi ew thi s i n term s ofan i ntri nsi c (el asti c or dynam i cal ) i nstabi l i ty of the structure of the m ol ecul ar sol i d. In thi s sense, the behavi or of the m ateri al paral l el s other am orphi zi ng system s that undergo coordi nati on changes (see R ef.23). T he authors are gratefulto Y . Fei for the hel p w i th hi gh-tem peratureexperi m ent.Speci althanksto J.B adro and M .Som ayazul u forthei rcom m entson hi gh-pressure am orphi zati on. T hi s work i s supported by N SLS,N SF and D O E. R uby uorescence spectrum at pressure 268 G Pa and 300K .R uby w as exci ted w i th 730 nm l i ne ofT i : Sapphi re l aser. FIG .2. FIGFIG . 5 . 5.3. B and gap ofthe phase as a functi on ofpressure. Sol i d ci rcl es represent i ncreasi ng pressure and open ci rcl es decreasi ng pressure. Li near and quadrati c extrapol ati on are show n i n dashed l i nes. arrow s);the tted curve show speak at1700 cm 1 . (c)R am an shi fts ofm ol ecul ar phases versus pressure. D otted l i nes represent shi fts on com pressi on. Sol i d l i nes and open ci rcl es represent shi fts on decom pressi on from the phase. Phase di agram of ni trogen i n a w i de P{T range. Fi l l ed ci rcl es, dashed-dotted l i nes and arrow s are the data from thi s w ork. O pen ci rcl es and dashed l i ne are from vi sual observati onsofR ef.5. A rrow sshow thepathsal ong w hi ch the transform ati on boundari es w ere crossed. T he l ength of the arrow represents the w i dth of the tw o-phase regi on or pressure uncertai nty (on a pressure rel ease). Phase boundari es at l ow pressures are from R efs.22, 24. T he phase boundari es for ; and loc phases are not show n. T here i s evi dence for furtherpressure-i nduced transform ati ons ofthe hi gh-pressure m ol ecul ar phase but the products are bel i eved to be cl osel y rel ated. 15;25 D otted l i ne i s extrapol ati on of the -transform ati on boundary. . A K , R Lesar, Phys.R ev.Lett. 541929A .K .M cM ahan and R .LeSar,Phys.R ev.Lett.54,1929 (1985). . R M , R J Eeds, Phys. R ev. B. 345082R . M . M arti n and R . J. N eeds, Phys. R ev. B 34, 5082 (1986). . C Ot, L H Yang, A K , Phys.R ev. B. 4614419C .M ai l hi ot,L.H .Yang and A .K .M cM ahan,Phys.R ev. B 46,14419 (1992). . A F Oncharov, E A Regoryanz, H K Ao, Z Li, R J , Phys.R ev.Lett. 851262A .F.G oncharov,E.A .G regoryanz,H .K .M ao,Z.Li u and R .J.H em l ey,Phys.R ev.Lett.85,1262 (2000). . M I Erem Ets, R J Ey, H K , E Regoryanz, to be publ i shedM . I.Erem ets,R .J.H em l ey,H .K .M ao and E.G regoryanz (to be publ i shed). . F Rbach, Phys.R ev.B. 921324F.U rbach,Phys.R ev.B 92,1324 (1953). . J Tauc, R Ri Gorovi Ciand, A Vancu, Phys.Status Sol i di. 15627J.Tauc,R .G ri gorovi ciand A .Vancu,Phys.Status Sol i di 15,627 (1966). N F Ott, E A , El ectronic Processes in N on-C rystall ine M aterial s2nd edi ti on. C l arendon PressN .F.M ott and E.A .D avi s,El ectronic Processes in N on- C rystall ine M aterial s2nd edi ti on (C l arendon Press,O xford 1979). . R Oz, J.G eophys.R es. 9210352R .Jeanl oz,J.G eophys.R es.92,10352 (1987). . A Oncharov, V Struzhki N, R Ey, H Ao, Z Li U, I N, Science and Technol ogy of H igh Pressure. A .G oncharov,V .Struzhki n,R .H em l ey,H .M ao, Z.Li u, i n Science and Technol ogy of H igh Pressure,edi ted by M . . H Anghnani, W J , M F , U ni versi ti es Press1H yderabad,Indi a,H onol ul u,H aw ai iH .M anghnani ,W .J.N el l i s and M .F.N i col(U ni versi ti es Press,H yderabad,Indi a,H onol ul u,H aw ai i ,1999),Vol .1, pp.90-95. A Tribute to Roger G .B urns. Y Fei, Ineralspectroscopy, M .W .Schafer (G eochem i calSoci ety243H oustonY .Fei ,M ineralSpectroscopy: A Tribute to Roger G .B urns, edi ted by M . D .D yar,C .M cC am m on and M .W .Schafer (G eochem i calSoci ety,H ouston,1966) p.243. Y K Vohra, i n Recent Trends in H igh Pressure Research. A . K .Si ngh (O xford & IB H Publ i shi ng C o. ,C al349Y .K .Vohra,i n Recent Trends in H igh Pressure Research, edi ted by A . K .Si ngh (O xford & IB H Publ i shi ng C o. ,C al - cutta 1991),p.349. . S , Phys.R ev.B. 5912940S.K ni ef,Phys.R ev.B 59,12940 (1999). . R Ei Chl I N, D Schi Ferl, D Arti N, C Vanderborgh, C , Phys.R ev.Lett. 551464R .R ei chl i n,D .Schi ferl ,D .M arti n,C .Vanderborgh and C . M i l l s,Phys.R ev.Lett.55,1464 (1985). . P M , H K Ao, R J , Physi ca 139&. 14016P.M .B el l ,H .K .M ao,R .J.H em l ey,Physi ca 139& 140B , 16 (1986); H em l ey (unpubl i shed resul ts). R J , R .J.H em l ey (unpubl i shed resul ts). . Y K Vohra, H , H Luo, A Uo, Phys. Lett. 571007Y . K . Vohra, H , H i a, H . Luo and A . R uo , A ppl . Phys. Lett.57,1007 (1990). M H Rodski I N Light, Sol ids Topi cs i n A ppl i ed Physi cs. M . C ardona (Spri nger-Verl ag8M . H . B rodski i n Light Scattering in Sol ids Topi cs i n A p- pl i ed Physi cs, vol . 8, edi ted by M . C ardona (Spri nger- Verl ag,N ew York 1983). A l lthree-coordi nated structures can be obtai ned by di storti on of a si m pl e cubi c phase (6-fol d coordi nated w hi ch i s unstabl e) and are rel ated to phonon i nstabi l i ti es i n di erent hi gh-sym m etry poi nts ofthe B ri ll oui n zone. T heoreti calcal cul ati ons 3 revealseveralcandi dates for the hi gh-pressures phase. 20T heoreti calcal cul ati ons 3 revealseveralcandi dates for the hi gh-pressures phase.A l lthree-coordi nated structures can be obtai ned by di storti on of a si m pl e cubi c phase (6-fol d coordi nated w hi ch i s unstabl e) and are rel ated to phonon i nstabi l i ti es i n di erent hi gh-sym m etry poi nts ofthe B ri l - l oui n zone. 20 . T W , Phys.R ev.B. 489327T . W .B arbee III,Phys.R ev.B 48,9327 (1993). . S M Forrecentrevi Ew Ssee, S K Sharm, Si, Prog. M at. Sci. 401Forrecentrevi ew ssee S.M .Sharm a and S.K .Si kka,Prog. M at. Sci . 40, 1 (1996); . P , P , Eur J , 9907P. R i chet and P. G i l l et, Eur. J. M i neral .9,907,(1997). . D A Young, C Zha, R , J Yen, M , A Zi Nn, D Schi Ferl, S K I Nkead, R , D Pi, Ck, Phys. R ev.B. 355353D . A .Young,C .Zha,R .B oehl er,J.Yen,M .N i col ,A .Zi nn, D .Schi ferl ,S.K i nkead,R .H anson and D .Pi nni ck,Phys. R ev.B 35,5353 (1987). . R J , A Jephcoat, H Ao, L Ng, M Anghnani, 33452LondonR . J. H em l ey, A .Jephcoat, H .M ao, L. M i ng, M . M angh- nani ,N ature (London)334,52 (1988); T hi s i ncl udes the l i kel i hood that m ateri al produced on com pressi on i sheterogeneous(i . e.partl y crystal l i ne);see R . J.H eml ey. J Adro, J. -L , P , Teter i n Physics M eets M ineral ogy H .A oki. R .H em l eyPress42173C am bri dge U ni vJ.B adro,J. -L.B ar- rat,and P.G i l l et,Europhys.Lett.42,643 (1998).T hi s i n- cl udes the l i kel i hood that m ateri al produced on com pres- si on i sheterogeneous(i . e.partl y crystal l i ne);see R . J.H em - l ey,J.B adro and D . M .Teter i n Physics M eets M ineral ogy H .A oki ,Y .Syom o and R .H em l ey (eds)(C am bri dge U ni v. Press,2000) p.173. . R , M Jordan, L , H Jodl, J Hem, Phys. 1086849R .B i ni ,M .Jordan,L.U l i vi ,H .Jodl ,J.C hem .Phys 108, 6849 (1998); . R B I Ni, L Vi, J Reutz, H Jodl, J Hem, Phys. 1128522R .B i ni ,L.U l i vi ,J.K reutz,H .Jodl ,J.C hem . Phys 112,8522 (2000). . H , A P Jephcoat, Phys.R ev.Lett. 83332H .O l i jnyk and A .P.Jephcoat, Phys.R ev.Lett.83, 332 (1999).
[]
[ "Comment on \"Macrospopic Equation for the Roughness of Growing Interfaces in Quenched Disorder\"", "Comment on \"Macrospopic Equation for the Roughness of Growing Interfaces in Quenched Disorder\"" ]
[ "Juan M López ", "José J Ramasco \nInstituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria\nE-39005SantanderSpain\n\nFractal Concepts in Surface Growth (Cambridge University Press\nCambridge\n", "Miguel A Rodríguez \nInstituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria\nE-39005SantanderSpain\n\nFractal Concepts in Surface Growth (Cambridge University Press\nCambridge\n", "- L Barabási ", "H E Stanley ", "\nDepartment of Mathematics\nDepartamento de Física Moderna\nImperial College\n180 Queen's GateSW7 2BZLondonUnited Kingdom\n", "\nUniversidad de Cantabria\nAvenida Los Castros s/n, Santander E-39005Spain\n" ]
[ "Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria\nE-39005SantanderSpain", "Fractal Concepts in Surface Growth (Cambridge University Press\nCambridge", "Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria\nE-39005SantanderSpain", "Fractal Concepts in Surface Growth (Cambridge University Press\nCambridge", "Department of Mathematics\nDepartamento de Física Moderna\nImperial College\n180 Queen's GateSW7 2BZLondonUnited Kingdom", "Universidad de Cantabria\nAvenida Los Castros s/n, Santander E-39005Spain" ]
[]
In a recent Letter [1] Braunstein and Buceta introduced a 'macroscopic' equation for the time evolution of the width of interfaces belonging to the directed percolation depinning (DPD) universality class[2]. From numerical simulations of the DPD model, they inferred an ansatz (Eq.(1) in Ref.[1]) for the time derivative of the interface width (called DSIW in Ref.[1]) at the depinning transition. Braunstein and Buceta found that their formula fitted the numerical data at the depinning trasition, for q c = 0.539 and β = 0.63, with the appropriate election of some arbitrary constants.Here we argue that, contrary to what it is claimed in Ref.[1], Braunstein and Buceta's formula does not describe the 'macroscopic' behaviour of the interface. The formula proposed in Ref [1] for the DSIW is an approximation to the very short times regime (when less than one layer has been completed), which is not significant for the description of the surface dynamics at large scales. We obtain analitically the short time behaviour of the DPD model, which is valid for any q and explains the apperance of an exponential term in the formula of Ref.[1] for the DSIW.Let us consider the DPD model in a system of size L and a density q of blocked cells (p = 1 − q density of free cells). We are interested in the very short times regime when the first monolayer still has not been completed, i.e. the number of growth attempts N is N ≪ L (this corresponds to times t = N/L ≪ 1). In this regime, the probability of having a column i with height h i > min(h i−1 , h i+1 )+2 is negligible and the columns are growing almost independently. The growth at this early stage can be seen as a random deposition (RD) process[3]in which every column grows in one unit with probability p/L. The short time regime of the DPD model is then like RD, which is solvable exactly, but with the additional ingredient of a density q of blocked sites.One can see that, within this approximation, the probability of having a column with height h after N growth attempts is given bywhere s = 1/L is the probability of attempting to growth a column and the usual approximation s r (1 − s) N −r N !/[(N − r)!r!] ≈ (N s) r exp(−N s)/r! has been made. From the probability (1), one can calculate the interface width W 2 = h 2 − h 2 and then the time derivative, which leading terms arewhere t = N s = N/L is the time in the units used in Ref.[1]. This formula gives the exact time evolution of dW 2 dt for any q (not only at q c = 0.539) and is valid for times t ≪ 1. For times t > 1 differences between neighbouring columns are likely to be larger than 2 resulting in horizontal correlations and the break down of (2). A comparison of Eq.(2) with numerical simulations of the DPD model is presented inFigure 1.Our calculation suggests that the exponential term in the ansatz of Ref.[1] is actually produced by the usual random deposition-like dynamics, which occurs in any growth model [3] for short times. [1] L.A. Braunstein and R.C. Buceta, Phys. Rev. Lett. 81, 630 (1998). [2] L.-H. Tang and H. Leshhorn, Phys. Rev. A 45, R8309 (1992); S.V. Buldyrev et. al., Phys. Rev. A 45, R8313 (1992). [3] A.0.0 5.0 10.0 ln (t) −3.0 −2.0 −1.0 0.0 ln (dW 2 /dt ) t −0.5 t 0.258 FIG. 1.Numerical results for the DPD model in a system of size L = 2 13 for qc = 0.539 (circles) and q = 0.3 (squares). Continuous lines correspond to Eq.(2) and fit the data for t ≪ 1. For larger times our approximation is not valid any longer and the power law t 2β−1 takes over with β = 0.623 and β = 0.3 for qc = 0.539 and q = 0.3 respectively (dotted lines).
10.1103/physrevlett.82.1337
[ "https://arxiv.org/pdf/cond-mat/9903131v1.pdf" ]
118,966,593
cond-mat/9903131
32c723cf3e40c89a028787c3512dd3b53e47ca37
Comment on "Macrospopic Equation for the Roughness of Growing Interfaces in Quenched Disorder" 8 Mar 1999 Juan M López José J Ramasco Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria E-39005SantanderSpain Fractal Concepts in Surface Growth (Cambridge University Press Cambridge Miguel A Rodríguez Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas -Univer-sidad de Cantabria E-39005SantanderSpain Fractal Concepts in Surface Growth (Cambridge University Press Cambridge - L Barabási H E Stanley Department of Mathematics Departamento de Física Moderna Imperial College 180 Queen's GateSW7 2BZLondonUnited Kingdom Universidad de Cantabria Avenida Los Castros s/n, Santander E-39005Spain Comment on "Macrospopic Equation for the Roughness of Growing Interfaces in Quenched Disorder" 8 Mar 1999 In a recent Letter [1] Braunstein and Buceta introduced a 'macroscopic' equation for the time evolution of the width of interfaces belonging to the directed percolation depinning (DPD) universality class[2]. From numerical simulations of the DPD model, they inferred an ansatz (Eq.(1) in Ref.[1]) for the time derivative of the interface width (called DSIW in Ref.[1]) at the depinning transition. Braunstein and Buceta found that their formula fitted the numerical data at the depinning trasition, for q c = 0.539 and β = 0.63, with the appropriate election of some arbitrary constants.Here we argue that, contrary to what it is claimed in Ref.[1], Braunstein and Buceta's formula does not describe the 'macroscopic' behaviour of the interface. The formula proposed in Ref [1] for the DSIW is an approximation to the very short times regime (when less than one layer has been completed), which is not significant for the description of the surface dynamics at large scales. We obtain analitically the short time behaviour of the DPD model, which is valid for any q and explains the apperance of an exponential term in the formula of Ref.[1] for the DSIW.Let us consider the DPD model in a system of size L and a density q of blocked cells (p = 1 − q density of free cells). We are interested in the very short times regime when the first monolayer still has not been completed, i.e. the number of growth attempts N is N ≪ L (this corresponds to times t = N/L ≪ 1). In this regime, the probability of having a column i with height h i > min(h i−1 , h i+1 )+2 is negligible and the columns are growing almost independently. The growth at this early stage can be seen as a random deposition (RD) process[3]in which every column grows in one unit with probability p/L. The short time regime of the DPD model is then like RD, which is solvable exactly, but with the additional ingredient of a density q of blocked sites.One can see that, within this approximation, the probability of having a column with height h after N growth attempts is given bywhere s = 1/L is the probability of attempting to growth a column and the usual approximation s r (1 − s) N −r N !/[(N − r)!r!] ≈ (N s) r exp(−N s)/r! has been made. From the probability (1), one can calculate the interface width W 2 = h 2 − h 2 and then the time derivative, which leading terms arewhere t = N s = N/L is the time in the units used in Ref.[1]. This formula gives the exact time evolution of dW 2 dt for any q (not only at q c = 0.539) and is valid for times t ≪ 1. For times t > 1 differences between neighbouring columns are likely to be larger than 2 resulting in horizontal correlations and the break down of (2). A comparison of Eq.(2) with numerical simulations of the DPD model is presented inFigure 1.Our calculation suggests that the exponential term in the ansatz of Ref.[1] is actually produced by the usual random deposition-like dynamics, which occurs in any growth model [3] for short times. [1] L.A. Braunstein and R.C. Buceta, Phys. Rev. Lett. 81, 630 (1998). [2] L.-H. Tang and H. Leshhorn, Phys. Rev. A 45, R8309 (1992); S.V. Buldyrev et. al., Phys. Rev. A 45, R8313 (1992). [3] A.0.0 5.0 10.0 ln (t) −3.0 −2.0 −1.0 0.0 ln (dW 2 /dt ) t −0.5 t 0.258 FIG. 1.Numerical results for the DPD model in a system of size L = 2 13 for qc = 0.539 (circles) and q = 0.3 (squares). Continuous lines correspond to Eq.(2) and fit the data for t ≪ 1. For larger times our approximation is not valid any longer and the power law t 2β−1 takes over with β = 0.623 and β = 0.3 for qc = 0.539 and q = 0.3 respectively (dotted lines). In a recent Letter [1] Braunstein and Buceta introduced a 'macroscopic' equation for the time evolution of the width of interfaces belonging to the directed percolation depinning (DPD) universality class [2]. From numerical simulations of the DPD model, they inferred an ansatz (Eq.(1) in Ref. [1]) for the time derivative of the interface width (called DSIW in Ref. [1]) at the depinning transition. Braunstein and Buceta found that their formula fitted the numerical data at the depinning trasition, for q c = 0.539 and β = 0.63, with the appropriate election of some arbitrary constants. Here we argue that, contrary to what it is claimed in Ref. [1], Braunstein and Buceta's formula does not describe the 'macroscopic' behaviour of the interface. The formula proposed in Ref [1] for the DSIW is an approximation to the very short times regime (when less than one layer has been completed), which is not significant for the description of the surface dynamics at large scales. We obtain analitically the short time behaviour of the DPD model, which is valid for any q and explains the apperance of an exponential term in the formula of Ref. [1] for the DSIW. Let us consider the DPD model in a system of size L and a density q of blocked cells (p = 1 − q density of free cells). We are interested in the very short times regime when the first monolayer still has not been completed, i.e. the number of growth attempts N is N ≪ L (this corresponds to times t = N/L ≪ 1). In this regime, the probability of having a column i with height h i > min(h i−1 , h i+1 )+2 is negligible and the columns are growing almost independently. The growth at this early stage can be seen as a random deposition (RD) process [3] in which every column grows in one unit with probability p/L. The short time regime of the DPD model is then like RD, which is solvable exactly, but with the additional ingredient of a density q of blocked sites. One can see that, within this approximation, the probability of having a column with height h after N growth attempts is given by P (N, h) = (N sp) h h! e −N s + qp h N r=h+1 (N s) r r! e −N s ,(1) where s = 1/L is the probability of attempting to growth a column and the usual approximation s r (1 − s) N −r N !/[(N − r)!r!] ≈ (N s) r exp(−N s)/r! has been made. From the probability (1), one can calculate the interface width W 2 = h 2 − h 2 and then the time derivative, which leading terms are dW 2 dt = pe −qt + 2p 2 e −qt e −qt − 1 q + t ,(2) where t = N s = N/L is the time in the units used in Ref. [1]. This formula gives the exact time evolution of dW 2 dt for any q (not only at q c = 0.539) and is valid for times t ≪ 1. For times t > 1 differences between neighbouring columns are likely to be larger than 2 resulting in horizontal correlations and the break down of (2). A comparison of Eq.(2) with numerical simulations of the DPD model is presented in Figure 1. Our calculation suggests that the exponential term in the ansatz of Ref. [1] is actually produced by the usual random deposition-like dynamics, which occurs in any growth model [3] for short times. /dt )t −0.5 t 0.258 FIG. 1. Numerical results for the DPD model in a system of size L = 2 13 for qc = 0.539 (circles) and q = 0.3 (squares). Continuous lines correspond to Eq.(2) and fit the data for t ≪ 1. For larger times our approximation is not valid any longer and the power law t 2β−1 takes over with β = 0.623 and β = 0.3 for qc = 0.539 and q = 0.3 respectively (dotted lines). . L A Braunstein, R C Buceta, Phys. Rev. Lett. 81630L.A. Braunstein and R.C. Buceta, Phys. Rev. Lett. 81, 630 (1998). . L.-H Tang, H Leshhorn, Phys. Rev. A. 458309L.-H. Tang and H. Leshhorn, Phys. Rev. A 45, R8309 (1992); . S V Buldyrev, Phys. Rev. A. 458313S.V. Buldyrev et. al., Phys. Rev. A 45, R8313 (1992). A.-L Barabási, H E Stanley, Fractal Concepts in Surface Growth. CambridgeCambridge University PressA.-L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, (1995).
[]
[ "A framework for high-fidelity particle tracking on massively parallel systems", "A framework for high-fidelity particle tracking on massively parallel systems" ]
[ "Patrick Kopper \nInstitute of Aircraft Propulsion Systems\nUniversity of Stuttgart\nPfaffenwaldring 670569StuttgartGermany\n", "Anna Schwarz \nInstitute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany\n", "Stephen M Copplestone \nboltzplatz -numerical plasma dynamics GmbH\nSchelmenwasenstr. 3470567StuttgartGermany\n", "Philip Ortwein \nInstitute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany\n", "Stephan Staudacher \nInstitute of Aircraft Propulsion Systems\nUniversity of Stuttgart\nPfaffenwaldring 670569StuttgartGermany\n", "Andrea Beck \nInstitute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany\n" ]
[ "Institute of Aircraft Propulsion Systems\nUniversity of Stuttgart\nPfaffenwaldring 670569StuttgartGermany", "Institute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany", "boltzplatz -numerical plasma dynamics GmbH\nSchelmenwasenstr. 3470567StuttgartGermany", "Institute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany", "Institute of Aircraft Propulsion Systems\nUniversity of Stuttgart\nPfaffenwaldring 670569StuttgartGermany", "Institute of Aerodynamics and Gas Dynamics\nUniversity of Stuttgart\nPfaffenwaldring 2170569StuttgartGermany" ]
[]
Particle-laden flows occur in a wide range of disciplines, from atmospheric flows to renewable energy to turbomachinery. They generally pose a challenging environment for the numerical prediction of particleinduced phenomena due to their often complex geometry and highly instationary flow field which covers a wide range of spatial and temporal scales. At the same time, confidence in the evolution of the particulate phase is crucial for the reliable prediction of non-linear effects such as erosion and fouling. As a result, the multiscale nature requires the time-accurate integration of the flow field and the dispersed phase, especially in the presence of transition and separation. In this work, we present the extension of the open-source highorder accurate CFD framework FLEXI towards particle-laden flows. FLEXI is a massively parallel solver for the compressible Navier-Stokes-Fourier equations which operates on (un-)structured grids including curved elements and hanging nodes. An efficient particle tracking approach in physical space based on methods from ray-tracing is employed to handle intersections with curved boundaries. We describe the models for a one-and two-way coupled dispersed phase and their numerical treatment, where particular emphasis is placed on discussing the background and motivation leading to specific implementation choices. Special care is taken to retain the excellent scaling properties of FLEXI on high performance computing infrastructures during the complete tool chain including high-order accurate post-processing. Finally, we demonstrate the applicability of the extended framework to large-scale problems.
10.1016/j.cpc.2023.108762
[ "https://export.arxiv.org/pdf/2211.05458v1.pdf" ]
253,447,140
2211.05458
003f2d28f24c00a579d92a1c18d986b8e41eb6f1
A framework for high-fidelity particle tracking on massively parallel systems Patrick Kopper Institute of Aircraft Propulsion Systems University of Stuttgart Pfaffenwaldring 670569StuttgartGermany Anna Schwarz Institute of Aerodynamics and Gas Dynamics University of Stuttgart Pfaffenwaldring 2170569StuttgartGermany Stephen M Copplestone boltzplatz -numerical plasma dynamics GmbH Schelmenwasenstr. 3470567StuttgartGermany Philip Ortwein Institute of Aerodynamics and Gas Dynamics University of Stuttgart Pfaffenwaldring 2170569StuttgartGermany Stephan Staudacher Institute of Aircraft Propulsion Systems University of Stuttgart Pfaffenwaldring 670569StuttgartGermany Andrea Beck Institute of Aerodynamics and Gas Dynamics University of Stuttgart Pfaffenwaldring 2170569StuttgartGermany A framework for high-fidelity particle tracking on massively parallel systems high-orderdiscontinuous Galerkinhigh-performance computingparticle-laden flowlarge eddy simulation Particle-laden flows occur in a wide range of disciplines, from atmospheric flows to renewable energy to turbomachinery. They generally pose a challenging environment for the numerical prediction of particleinduced phenomena due to their often complex geometry and highly instationary flow field which covers a wide range of spatial and temporal scales. At the same time, confidence in the evolution of the particulate phase is crucial for the reliable prediction of non-linear effects such as erosion and fouling. As a result, the multiscale nature requires the time-accurate integration of the flow field and the dispersed phase, especially in the presence of transition and separation. In this work, we present the extension of the open-source highorder accurate CFD framework FLEXI towards particle-laden flows. FLEXI is a massively parallel solver for the compressible Navier-Stokes-Fourier equations which operates on (un-)structured grids including curved elements and hanging nodes. An efficient particle tracking approach in physical space based on methods from ray-tracing is employed to handle intersections with curved boundaries. We describe the models for a one-and two-way coupled dispersed phase and their numerical treatment, where particular emphasis is placed on discussing the background and motivation leading to specific implementation choices. Special care is taken to retain the excellent scaling properties of FLEXI on high performance computing infrastructures during the complete tool chain including high-order accurate post-processing. Finally, we demonstrate the applicability of the extended framework to large-scale problems. Introduction Particle-laden flows with a dilute dispersed phase have long been of scientific interest due to their wide range of occurrences [1,2]. Particles from natural and artificial sources can remain suspended in the surrounding fluid for almost indefinite time and affect a multitude of disciplines such as pollutant and climate prediction [3,4,5], human pathogen transport [6,7], aeronautical applications [8,9,10] or sprays [11,12]. Following [13], three distinct classes for the coupling of the fluid and the particle phase can be distinguished, categorizing the coupling into one-, two-, and four-way coupled approaches. Both one-way and two-way coupled fluid-particle phases are sufficiently accurate to describe dilute dispersed phases. The particles are mainly driven by the large scales of the fluid flow, while the reverse influence of the particles on the flow and thus on the turbulent scales is neglected for one-way coupling, but considered in the two-way coupled regime. Following the assumption of a dilute disperse phase, contact forces acting on a particle are only considered through interactions with solid walls, i.e., inter-particle collisions can be neglected. Dropping the aforementioned assumption leads to four-way coupled phases which resolve collisions but involve a computationally expensive search for collision partners [14]. As was shown by Balachandar and Eaton [15], both the Lagrangian point-particle approach and the particle-resolved Direct Numerical Simulation (DNS) are well suited for the numerical treatment of the considered coupling regimes. In the Lagrangian point-particle ansatz, an Eulerian field solver for the continuous fluid phase is combined with a scheme for the Lagrangian movement of the dispersed particles which are modeled as point masses. The point-particle approximation restricts the application range of the Euler-Lagrange ansatz to particle sizes in the range of the Kolmogorov scale, i.e., the smallest turbulent flow scale. A particle-resolved Direct Numerical Simulation (DNS) is recommended for larger particle sizes. In this approach, the geometry and hence the flow scales around a particle are resolved, thus this ansatz is limited to small numbers of particles. Compared to particle-resolved DNS, an Euler-Lagrange (EL) approach is less computationally intensive while still providing individual particle information which is inevitably lost in Euler-Euler or dusty gas approaches. However, since the forces induced by the interaction of the particles with the surrounding fluid are not resolved by the Euler-Lagrange approach as compared to the particleresolved DNS, they have to be modeled appropriately. A detailed overview of the numerical treatment of the considered coupling regimes is given in [15,16]. Beyond the choice of the coupling regime, a common characteristic of particle-laden flows is the complexity of numerical simulations due to the distinct properties of the different phases and the wide range of encountered scales. As the accuracy of the particle movement is directly dependent on the resolved scales of the underlying flow field, approaches which aim to reduce the complexity, e.g., the physical order, of the problem such as the Reynolds-Averaged Navier-Stokes equations (RANS) have shown severe deficiencies in unsteady and turbulent flow fields [17]. Large-Eddy Simulations (LES) offer the required spatial and temporal resolution but come with significant demands towards the computing resources. Thus, for cases where compromising the solution quality in favor of reduced computational effort is deemed unacceptable, the focus is placed on the efficient solution of the particle-laden LES on highly parallel systems. An example of such a case can be found in turbomachinery applications, where the boundary layer dynamics are unsteady and transition, separation and wake interactions play a dominant role [18]. High-order methods are well suited for the accurate solution of such problems due to their inherent low dissipation error. So far, the EL ansatz has been successfully coupled to high-order finite volume [19], finite differences [20] and discontinuous Galerkin [17] schemes to enable high-order accurate particle tracking and interpolation in a compressible regime. In [19], the compressible Navier-Stokes-Fourier equations (NSE) are discretized by a 5 th -order WENO scheme which features a local time stepping and a block-based multiresolution ansatz to localize particles efficiently within the Eulerian mesh. While the framework shows near-optimal parallel performance on highly-parallel systems when restricted to (potentially multiple) continuous phases with a level-set approach [21], to the author's best knowledge no scaling results have been published for particle-laden flows. The authors in [20] placed less emphasis on efficiency and utilized a finite difference method which solves the volume-filtered NSE with arbitrary order in space and time. All frameworks have in common that they are able to capture and handle discontinuities in the solution appropriately. However, high-order finite volume and finite differences schemes pose challenges to their efficient parallelization due to their inherent non-local solution representation which results in wide communication stencils stretching across multiple elements. Furthermore, to author's best knowledge both frameworks are not publicly available as open-source. In order to remedy some of these shortcomings, high-order methods based on a local solution representation have gained significant research interest in recent years, as they offer excellent scaling properties and are flexible enough to simulate complex geometries. One such approach is the aforementioned Discontinuous Galerkin (DG) method which was shown to be well-suited for compressible flow simulations involving turbulence and aeroacoustics [22,23]. In this work, the open-source high-order accurate CFD framework FLEXI 2 is utilized to solve the compressible Navier-Stokes-Fourier equations using the discontinuous Galerkin Spectral Element Method (DGSEM) on (un-)structured grids featuring curved faces and hanging nodes. FLEXI is equipped with pre-and post-processing tools [24] and has been successfully applied to a wide variety of problems including airfoil and turbomachinery simulations [25,18] as well as data-driven shock-capturing [26,27]. In this work, we aim to present a comprehensive description of the extension of FLEXI to incorporate particle tracking on (un-)structured grids with possibly curved elements. In addition, since FLEXI is designed to operate efficiently on highly parallel systems, we show that the extended framework preserves the excellent scaling properties of FLEXI. The proposed framework features the following characteristics. Both the approximation of the dispersed and the fluid phase are high-order accurate in space and time for arbitrary orders. The particle tracking (and the fluid phase) are designed to run efficiently on highly parallel systems and complex geometries including possibly curved elements and hanging nodes. The particles can be tracked across discontinuities and shocks in the solution in combination with the shock capturing scheme of FLEXI [28,24], similar to [19,20]. In combination, these features enable an efficient, high-order accurate particle tracking in a compressible carrier phase on arbitrary core counts. In addition, the presented framework is open-source 3 , easily extendable, and can be embedded in machine-learning frameworks [29]. The focus of this work is on the modeling, challenges and applications of discrete particles embedded in a continuous flow field. In section 2, we present the underlying equations for both the continuous and discrete phase including particle-wall interactions. We follow by a description of the numerical treatment for these equations in section 3. Section 4 focuses on the actual implementation, including parallelization, load balancing, and post-processing. Extensive validation studies are presented in section 5 before demonstrating the scaling capabilities in section 6. We show two possible applications of the framework, first the ash deposition within a low pressure turbine cascade and second the particle-laden flow around a finite wallmounted cylinder in section 7. We close with a brief conclusion and outlook in section 8. Theory Continuous Phase The fluid field is governed by the compressible unsteady Navier-Stokes-Fourier equations, given in vectorial form as dq dt + ∇ · F (q, ∇q) = S,(1) where q = [ρ, ρu 1 , ρu 2 , ρu 3 , ρe] T is the vector comprising the conservative variables, S is a source term, ρ the fluid density, u i the i-th component of the velocity vector and e the total energy per unit mass. The source term S accounts for the influence of the dispersed phase on the fluid in two-or four-way coupled regimes. The physical flux F is composed of the inviscid Euler and the viscous fluxes. The equation system is closed by the equation of state of a calorically perfect gas. The dynamic viscosity µ is obtained from Sutherland's law [30], while the heat flux is given by Fourier's law. Following Stokes' hypothesis, the bulk viscosity is set to zero. Dispersed Phase The particles are treated as discrete points which move in a Lagrangian manner according to the following ordinary differential equation dx p dt = v p(2) with the particle position in physical space x p = [x p,1 , x p,2 , x p,3 ] T and the particle velocity obtained from the integration of eq. (3). The framework is (mainly) intended for simulations which can assume a one-way coupled fluid and dispersed phase, i.e., dilute flows where the volume fraction is φ < 10 −6 [13]. However, two-way coupling is available under the previous assumption that a particle is a discrete point with zero radius whose influence remains element-local. The equation of motion for an isolated particle in unsteady uniform fluid is described by the Basset-Boussinesq-Oseen equation, based on the works by Basset [31], Boussinesq [32] and Oseen [33]. This equation was deduced under the assumption of unsteady Stokes flow with a particle diameter d p in the range of the Kolmogorov scale ν of the surrounding fluid and for small particle Mach and Reynolds numbers, i.e., Re p = |u f − v p |d p ρ f µ < 1, M p = |u f − v p | c → 0, with the speed of sound c of the fluid. The first attempt towards an equation for unsteady non-uniform fluid flow was derived by Tchen [34]. Including the considerations and improvements of Corrsin and Lumley [35], Auton et al. [36] as well as Maxey and Riley [37], Gatignol [38], the resulting equation is commonly called the Maxey-Riley-Gatignol (MRG) equation. A generalized version which prescribes the particle acceleration in the Lagrangian frame of reference is given as m p dv p dt = F D + m f Du f Dt + m f 2 Du f Dt − dv p dt + F B + F L + g (m p − m f ) ,(3) where m p and v p are the mass and velocity of the particle, m f is the mass of the displaced fluid volume, u f is the (theoretical) fluid velocity neglecting disturbance caused by the particle itself and interpolated to the particle center of mass, and g denotes the gravitational acceleration. The substantial derivative of the fluid velocity was introduced for the first time by [36] and is given as Du f Dt = ∂u f ∂t + u f · ∂u f ∂x , du f dt = ∂u f ∂t + v p · ∂u f ∂x , together with the full derivative along the particle trajectory. The original MRG equation includes the Faxén correction [39] to account for the local curvature of the fluid velocity field (non-uniform flow), the influence of which is generally neglected as it is considered to be small compared to the other forces [40]. The terms on the right represent the drag force F D , undisturbed fluid stresses with contributions from viscous effects and pressure gradients, F P , the added mass F AM , the Basset force F B encompassing the history term, the combined lift force F L consisting of the Saffman and Magnus force as well as the combined term for buoyancy and gravity, respectively. The added or virtual mass considers the work required for the acceleration of the adjacent fluid due to the boundary layer surrounding a particle. The Basset history term [41] causes eq. (3) to turn into a fractional-order differential equation, thus inhibiting the use of standard numerical integration schemes [42,43]. The approximate integration of this term is prescribed in section 2.2.4. Further modifications have enabled the extension of eq. (3) to higher Reynolds numbers, a detailed overview is provided in [44], and the compressible regime [45]. As the compressible formulation is restricted to small Mach and Reynolds numbers and for reasons of simplicity, in this work, only the incompressible form of the MRG equation is considered. Similar to subgrid scale (SGS) models for the continuous phase, the influence of the unresolved scales can be approximated by an appropriate SGS model, e.g., [46,47,48]. Stokes Number. An estimate of the contribution of the right-hand side of eq. (3) on the particle trajectory can be determined with the Stokes number. The Stokes number characterizes the particle behavior in relation to the fluid time scale and is defined as St = u * τ p L * ,(4) which indicates whether a particle is mainly driven by its inertia (higher Stokes numbers) or follows the fluid streamlines (small Stokes numbers). In eq. (4), τ p = d 2 p ρ p (18µ) −1 denotes the particle relaxation time, u * the characteristic flow velocity and L * the characteristic length, see e.g. [49]. Drag Force The quasi-steady drag force is described exactly through Stokes' law for spherical particles with Re p < 1. To account for higher particle Reynolds numbers, a correction factor f D is employed, yielding the generalized drag force as F D = 3πµd p f D (u f − v p )(5) with f D → 1 for Stokes flow. The drag factor for spherical particles with higher Reynolds number is obtained from the empirical model of Schiller and Naumann [50] as f D = 1 + 0.15Re 0.687 p : Re p < 800.(6) For non-spherical particles, the drag factor is extended to the four-parameter general drag correlation proposed by Haider and Levenspiel [51] as f D = 1 + ARe B p + Re p 24 C 1 + D/Re p where the parameters A through D depend on the particle sphericity and are obtained from a least squares fit based on experimental measurements and published in [51]. Following [52], compressibility effects are non-negligible for particle Mach numbers of M p > 0.6. Thus, the drag factor of [52], where f D = f (Re p , M p ), can be employed to account for Mach number influence. Saffman Lift Force The Saffman lift force accounts for the buoyancy caused by a velocity gradient in the surrounding fluid flow and is modeled after [53,54] as F S = 6.46 4 C S d 2 p ρ f µ |ω| (ω × (u f − v p )) with the vorticity ω = ∇ × u f and the correction of [55] for higher Reynolds numbers C S = 1 − 0.3314 √ ϑe −0.1Rep + 0.3314 √ ϑ : Re p ≤ 40, 0.0524 ϑRe p : Re p > 40, where ϑ = dp|ω| 2|u f −vp| : 0.005 < ϑ < 0.4. Magnus Force The Magnus force results from the rotation of particles in motion and is modeled according to [56] as F M = π 8 C M d 3 p ρ f (Ω × (u f − v p )) |u f − v p | |Ω| with an empirical correction by [57] for higher Reynolds numbers, given as C M = 0.45 + 4 Re ω Re p − 0.45 exp(−0.05684Re 0.4 ω Re 0.7 p ) : 1 2 < Re ω Re p < 3, 10 < Re p < 140. Here, the rotational Reynolds number is Re ω = d 2 p |Ω|ρ f 4µ , the relative fluid-particle angular velocity Ω = 1 2 (∇ × u f ) − ω p and the angular particle velocity ω p = ∇ × v p . The latter results from the temporal integration of an additional ordinary differential equation derived by [58], which reduces to I dω p dt = M = ρ f d 5 p 64 C w Ω|Ω|(7) in the steady-state case, where M is the torque, I = π 60 ρ p d 5 p the moment of inertia of a spherical particle and C w = 6.45 √ Re w + 32.1 √ Re w : 20 < Re w < 1000 is a correction factor for higher Reynolds numbers proposed by [59]. Basset Force The history term in the Basset force addresses the temporal delay of the particle boundary layer due to viscous effects, i.e., the acceleration history, and is given by F B = 3 2 d 2 p √ πρ f µ t t0 d(u f − v p ) dτ (t − τ ) −1/2 dτ + (u f − v p ) | t0 √ t − t 0(8) with the original Basset kernel [41] and a correction term of [60] for particles with (u f − v p ) | t=t0 = 0. Following [61], the derivative in the first term in eq. (8) is approximated by a linear term and the integral is solved by the use of a trapezoidal-based method to handle the singularity in the upper limit. The number of preceding terms K has to be chosen appropriately to find an trade-off between integration accuracy and memory consumption. Two-Way Coupling Following the particle-source-in-cell approach proposed by [62], for the two-way coupling, the forces acting on the particles and the corresponding work appear as a source term, S = [0, S m,1 , S m,2 , S m,3 , S e ] in the momentum equations and the energy equation, respectively. The source terms for the momentum equations S m = [S m,1 , S m,2 , S m,3 ] and the energy equation S e at a point x ijk , i, j, k ∈ N >0 are given by S m = −P F D + m f Du f Dt + m f 2 Du f Dt − dv p dt + F B + F L , x ijk ,(9)S e = −P S m · v p , x ijk ,(10) with the projection operator P{·, ·}, which projects the source term onto the grid. It has to be noted that the assumption of an undisturbed fluid velocity u f is violated in a two-way coupled ansatz and recent publications have proposed several approaches to reduce the error introduced through this assumption [63]. Within this work, the influence of the source term is assumed to be restricted to the nearest degree of freedom (DoF), i.e., the projection operator for an arbitrary variable a is P{a, x ijk } = a V ijk to ensure conservation, where V ijk is the volume spanned by the nearest degree of freedom ijk. The particle-induced source term imposes an additional time step restriction, apart from the convective and viscous restrictions. For its derivation, only the drag force with f D = 1 is considered. The eigenvalue of the resulting ordinary differential equation is τ −1 p , which yields a maximum allowable time step of dt ≤ τ p . A high-order time integration scheme allows larger time steps, e.g., dt ≤ 2.75τ p for a fourth-order Runge-Kutta scheme [20]. Intersection with Solid Walls The impact of particles on a wall is approximated by means of a hard sphere model, where the particle intersections are handled in an a posteriori manner, i.e., dt stage > dt collision , with the time step dt collision required to resolve a collision in time and the actual time step dt stage . The change of momentum of a particle between two instances in time is given by m p,2 (v p,2 + 2(v p,1 · n)n) − m p,1 v p,1 = J, where (·) 2 denotes variables after the impact and (·) 1 before it. The normal vector of the boundary is designated by n. The change of momentum is J = 0 for a perfectly reflective wall, where a purely elastic deformation of the particle and the wall is assumed. A plastic deformation of the particle or the wall results in J = 0. In this case, the particle properties after impact are determined via coefficients of restitution (CoR); for a quantity x it reads e x = x 2 (x 1 ) −1 . CoRs are generally approximated by so-called rebound models based on empirical correlations, often with physical constraints and tunable parameters which depend on the particles' characteristics and the surface material. The common rebound models describe the change in the particle trajectory under the assumption that the particle mass does not change upon impact. However, the presented framework allows for a change of the particle mass if such a model is available. Further details on the implemented rebound models are given in [64,65,66,67]. The change of the angular particle velocity during a wall impact is determined according to [44] as I 2 ω 2 − I 1 ω 1 = − d p,1 2 [n × (m p,2 v p,2 − m p,1 v p,1 )] . Numerical Methods In the following, we briefly discuss the numerical treatment of the governing equations for the fluid and dispersed phase. Discontinuous Galerkin Spectral Element Method The Navier-Stokes-Fourier equations are solved via the Discontinuous Galerkin Spectral Element Method (DGSEM). For this, the computational domain Ω ⊆ R 3 is discretized by non-overlapping, (non-)conforming hexahedral elements, where the six element faces Γ k , k = 1, . . . , 6 are allowed to be curved. Curved faces are approximated in a tensor product manner by one-dimensional Lagrange polynomials l up to degree N geo as Γ k (m, n) = Ngeo i,j=0 Γ k (m i , n j )l i (m)l j (n), (m, n) ∈ [−1, 1] 2 .(11) Details on the mapping from reference to physical space are provided in [68]. To obtain an efficient discretization scheme, the governing equations are transformed into the reference coordinate system ξ = [ξ 1 , ξ 2 , ξ 3 ] T of the reference element E = [−1, 1] 3 via the mapping x = χ(ξ, t), x ∈ Ω. The discrete L 2 projection onto the test space composed of polynomials φ(ξ) up to degree N , followed by an application of Green's identity yields the weak form, given as E J ∂q h ∂t φ(ξ)dξ + ∂E (F · n) * φ(ξ)dS − E F (q h , ∇q h ) · ∇ ξ φ(ξ)dξ = 0,(12) with the Jacobian J of the mapping and the outward pointing normal vector n. In eq. (12), F denotes the contravariant flux vector and (F · n) * the numerical flux normal to the element face. The elementlocal solution q h = q h (ξ, t) is approximated by a tensor product of one-dimensional nodal Lagrange basis functions l of degree N q h (ξ, t) = N i,j,k=0q ijk (t)l i (ξ 1 )l j (ξ 2 )l k (ξ 3 ),(13) with the nodal degrees of freedomq ijk (t). Equation (12) is numerically integrated on the interpolation points by the Legendre-Gauss quadrature with (N + 1) 3 Legendre-Gauss-Lobatto points. This collocation of integration and interpolation points allows for a highly efficient scheme. The Euler fluxes at the cell boundaries are approximated via the numerical flux by Roe, see [69], with the entropy fix by Harten et al. [70]. The viscous fluxes are computed with the BR1 scheme of Bassi and Rebay [71]. The flux is split according to [72,73,74] to mitigate aliasing errors which can lead to stability issues. Moreover, a suitable shock capturing procedure [28] is required, as high-order schemes are subject to oscillations in the vicinity of discontinuities, also known as Gibbs phenomenon. For this purpose, the DG operator in these cells is replaced by a second-order accurate finite volume (FV) (subcell) scheme with (N + 1) 3 integral means, which reduces the loss of resolution caused by the inherent higher dissipation of the lower-order FV operator. The reader is referred to [28,75,22,76] for further details on DGSEM, the shock capturing procedure and applications. Following the method of lines approach, the explicit low-storage fourth-order accurate Runge-Kutta (RK) scheme by Carpenter and Kennedy [77] is employed for the integration in time. The open-source framework FLEXI 4 is used as a solver for the fluid and dispersed phase which includes the numerical methods mentioned below. Particle Push and Tracking Particles are advanced in time with the same RK scheme as the fluid phase. At each stage, the particle push is determined which includes the interpolation of the conserved variables onto the particle position, the calculation of the corresponding force on the discrete particle, and the subsequent integration of the particle trajectories in time using the updated particle acceleration. The resulting particle path is obtained through particle tracking which involves checking for element boundary intersections and subsequent application of appropriate boundary conditions. The first part of this section will focus on the particle push, while the particle tracking will be discussed in the second one. Interpolation The particle position in reference space ξ p = [ξ 1 p , ξ 2 p , ξ 3 p ] T enables the interpolation of the conserved variables onto the center of mass of the particle from which the primitive variables v(ξ p , t) = (ρ, u 1 , u 2 , u 3 , p) T can be obtained. With eq. (13), the interpolation for a DG element is defined as q h (ξ p , t) = N i,j,k=0q ijk (t)l i (ξ 1 p )l j (ξ 2 p )l k (ξ 3 p ),(14) while the second-order FV subcells reduce the particle interpolation to a (linear) interpolation of the conserved variables to the particle position. The particle position in reference space ξ p is generally determined by an iterative procedure, e.g., via Newton's method [78], to find the root of χ(ξ p ) − x p = 0. This necessitates an initial estimate ξ 0 p of ξ p , which can be chosen based on the following three approaches. The first approach uses the nearest, in the sense of the discrete L 2 norm, interpolation point, while the second employs the nearest mesh point and is available only if curved elements are considered. Lastly, the third approach is to use the mean distance x p − x bary , x i,bary − x bary , i = 1, . . . , 6 between two opposite element faces as initial estimate, where x bary is the barycenter of the element and x i,bary the barycenter of the six element faces. An adequate approach is generally chosen depending on the mesh element type, e.g., the first approach is utilized for linear meshes and the second approach for curved element meshes. Integration in Time In each Runge-Kutta stage, the conserved variables are interpolated onto the particle position using eq. (14), and the particle trajectory is obtained by the integration of eq. (2) and eq. (3) in time. Hence, the initial particle trajectory (neglecting boundary conditions) in stage n, i.e., t ∈ [t n , t n+1 ] describes the path traveled by the particles within dt stage = t n+1 − t n , given by x p (t; α) = x p (t n ) + α t |t| , α ∈ [0, |t|], t = x p (t n+1 ) − x p (t n ),(15) where t denotes the particle trajectory and α the displacement. Tracking The particle tracking in FLEXI advances particles in time through movement in physical space. While intersection handling is also performed in physical space, the code additionally offers the option to locate particles through an interpolation ansatz in reference space, cf. Ortwein et al. [79]. Tracking in Physical Space. Each element face is checked for an intersection with x p (t) = x p (t n )+tv p , starting from the element in which the particle resides at t n . Intersections are considered if 0 < dt inter ≤ dt stage , i.e., the particle reaches the face within the current time step. If the element face is an internal face, the particle is considered to have moved to the adjacent element and the algorithm is repeated from x p (t)| t=t n +dtinter . If the element face corresponds to a boundary, the corresponding boundary condition (open, reflective, periodic) is applied, and the algorithm is continued on the modified trajectory for dt remaining = dt stage − dt inter . A special case denotes the intersection with a non-conforming mortar side, where one big element face is matched to two or four small element faces [24]. Here, only the mapping from small to big faces is unique since the adjacent element can be determined directly. For the opposite direction, when encountering an intersection with a big mortar side, the code inverts the trajectory and performs the intersection search on the associated small element faces. Hence, the intersection is again unique and after a second inversion the algorithm can be continued from x p (t)| t=t n +dtinter on the small mortar face. Localization in Reference Space. This tracking approach inherently does not consider boundary interactions. In the first step, all particles whose starting positions x p (t n ) are near boundaries are again traced in physical space. Subsequently, the final element for all particles is determined by identifying the particles' host cells with the closest barycenters through a Fast Init Background Mesh (FIBGM, see section 4.1.2) and calculating the position in reference space through Newton's method. The particle is considered inside the element if x p (ξ) ∈ [−1, 1] 3 , which offers a built-in verification. The first step is omitted if x p (t n ) is far from boundary faces, so that always dt inter > dt stage . Intersection Handling In the following, the treatment of an intersection with an element face is briefly discussed. Details on the procedure and further literature are given by [79]. To efficiently compute an intersection of the particle path with an arbitrary (curved) face, each element face is described by Bézier polynomials of degree N geo , given in a tensor product manner as P(ξ, η) = Ngeo m=0 Ngeo n=0P mn B m (ξ)B n (η) with the Bernstein polynomials B of degree N geo , the Bézier control pointsP and the reference space (ξ, η) ∈ [−1, 1] 2 . In order to compute the intersections of the particle trajectory given in eq. (15) with an element face, the roots of x p (t > t n , α) = x p (t n ) + α t |t| ! = p(ξ, η)(16) have to be found, i.e., α, ξ and η. In eq. (16), p(ξ, η) denotes the equation of the element face. However, the evaluation of eq. (16) with p(ξ, η) = P(ξ, η) is time consuming and, hence, depending on whether the element faces are curved, more efficient approaches are applied, as described below. The type of the element face (planar rectangular, bilinear, planar quadrilateral, curvilinear) is determined by placing a bounding box around their physical extent [79,80,81]. If this box is empty, the element face is planar, and bilinear or curvilinear otherwise. The bounding boxes are constructed for each element face at the beginning of the simulation to reduce the computational effort during runtime. A further benefit is that the bounding box allows to efficiently check whether a particle could possibly intersect an element face or not. The intersections can then be calculated as follows. Planar Rectangular Faces. In the planar case, the particle displacement is calculated according to α = (x m · n) − (x p (t n ) · n) (t · n) with the mid point of the Bézier surface x m . The equation of the element face for a planar rectangular face is given by p(ξ, η) = aξ + bη + c, where the coefficients a to c are based on the corner Bézier points. The resulting equation system is solved analytically to obtain (ξ, η), and an intersection occurs if (ξ, η) ∈ [−1, 1] 2 . Bilinear or Planar Quadrilateral Faces. The intersection with a bilinear or quadrilateral side is computed according to [82], where the surface is described by a bilinear patch of the form p(ξ, η) = aξη + bξ + cη + d. The parameters a to d are determined via the corner Bézier points of the face, as described in [82], and an intersection is obtained if α ≤ |t|. Curved Faces. To reduce the complexity, the Bézier surface is projected onto a local coordinate system spanned by two planes which are orthogonal to each other. The first plane is defined by the use of the particle trajectory and an arbitrary vector n 1 orthogonal to it. The second plane is defined in the same way, with the additional condition that the two planes are orthogonal, i.e., n 2 = n 1 × t. An intersection with this new 2D plane is handled via Bézier clipping [83], a method from ray-tracing, or Newton's method [79]. The latter is applied if the bounding box is flat (curvilinear planar face) and solves P(ξ, η) = 0, i.e., the Jacobian of the Bézier polynomials is required. In Bézier clipping, the intersections of the particle path with a Bézier surface, i.e., p(ξ, η) = P(ξ, η), are calculated. For this, two orthonormal vectors are defined by the use of the 2D projected Bézier control points. With the help of these vectors, the convex hull of the projected Bézier points is formed. This convex hull is utilized to mark regions where intersections are most probable. The other regions are clipped away with the de Casteljau subdivision, and the procedure is repeated until the intersection point is found. Further details can be found in the respective literature [83,79]. Implementation Details FLEXI is designed to efficiently utilize arbitrary core counts on massively parallel systems. The DG method is well-suited for these applications since the inter-core exchange of information is limited to the numerical flux through the element faces along the process boundaries. Additionally, the element-local discretization of DGSEM enables latency hiding through non-blocking communication. Particles, by contrast, pose additional challenges to efficient parallelization, since both the number of particles per cell as well as the number of particles crossing the processor boundaries are varying and cannot be determined a priori. At the same time, immediate communication of particles moving to another processor would be detrimental to performance. To permit the completion of the particle tracking on the local processor as well as latency-hiding on the discrete phase, FLEXI follows the halo-region approach to extend the process-local domain. The section below follows this outline. First, details on the employed parallelization strategy for pure fluid flow are given. Subsequently, the generation of the halo region and the latency hiding of the runtime communication for both emission and tracking are presented. After that, a potential load imbalance due to unevenly distributed particles among the processes is addressed. To preserve the parallel tool chain of FLEXI from pre-to post-processing, the section concludes by presenting the extension of the post-processing to particles. It should be noted that the particle tracking is developed in cooperation with the PICLas framework 5 which focuses on solutions to non-equilibrium gas and plasma flows [84,85]. For an in-depth review on the current state of the particle parallelization approach shared by both codes and their application to non-equilibrium flows, the reader is referred to [86]. Parallelization The continuous phase in FLEXI is based on the distributed memory paradigm, i.e. each core is assigned and restricted to its individual memory address space. In order to ensure memory locality and improve cache hit rate, all elements are pre-sorted along a space-filling Hilbert curve during mesh generation by the open-source mesh generator HOPR [68]. The distribution of the mesh over the cores is performed by utilizing this curve to obtain a continuous segment for each core. Associated information, such as side connectivity and node coordinates, are stored non-uniquely along the same curve and are thus also available as contiguous segment. The space-filling curve approach extends to the on-disk storage format which allows massively parallel access to non-overlapping data regions through the HDF5 library [87]. Runtime information exchange is handled through non-blocking Message Passing Interface (MPI) communication [88]. Latency hiding is extensively used to ensure maximal time intervals available for communication without stalling the code. Since inter-processor information exchange is limited to direct neighbors which are known a priori, FLEXI performs in-memory re-ordering of side and node information. Thus, memory locality is facilitated, and the data is inherently stored in the linear buffers required for MPI communication. Halo Region While the continuous phase is calculated in reference space, particles are tracked in physical space. Subsequently, the complete geometry information along a path x p | t=t → x p | t=t+dtstage is required to complete their time integration. A halo region provides geometry information within a given distance around the local domain [79]. This region permits each core to perform particle tracking until the final particle position is achieved and avoids unnecessary communication during a time increment. The sorting along a space-filling curve allows for fast domain decomposition, but the position along the SFC provides no information about the cell location in physical space. As a consequence, an efficient search in physical space must be performed to create the halo region required for a performant particle tracking in the parallel context. Distributed approaches encounter a severe performance bottleneck on massively parallel systems as each encountered halo element requires grid information local to one processor to be communicated to a multitude of other processors. To avoid this limitation, the Lagrangian particle implementation is based on MPI-3 shared memory programming, which results in a hybrid memory code when particles are enabled. For this, the complete raw mesh information, which is composed exclusively of element information, face connectivity and node coordinates without derived metrics, is stored in a shared memory region on each compute-node. Based on this information, a communication-free two-step search algorithm is performed to identify halo elements [86]. These elements are subsequently added to the elements locally considered for particle tracking. While the metrics are restricted to the local and halo elements, the identifiers from the global mesh are kept in order to ensure consistent numbering throughout the computational domain. Detailed information on this topic can be found in [86]. Cartesian Background Mesh In addition to the halo element search, tracking a particle in physical space necessitates an efficient scheme for the identification of the element in which the particle resides. As previously stated, the element identification stored for a particular grid element allows for no correlation to its position in physical space. Thus, the task to correlate the corresponding element to any given position would involve an elaborate search over potentially the entire grid. To alleviate this problem, a Cartesian Background Mesh (BGM) is created [79]. Upon code initialization, the computational domain is overlaid with an I,J,K-identifiable Cartesian grid and the mapping from each BGM cell to all overlapping unstructured mesh elements is built. This reduces the potential mesh elements associated with each position, ideally down to a single candidate. Emission The information associated with each BGM cell is also utilized in the particle emission step. During emission, a potentially large number of initial positions in physical space must be mapped to their corresponding elements. Depending on the quality of the initial guess, this can result in a costly identification step and is thus again parallelized. A processor takes part in this step if its local mesh region has at least partial overlap with the complete emission region. However, in contrast to tracked particles, there is no guarantee that an initial position in physical space corresponds to either a local or halo element on a respective processor. Moreover, without having knowledge of the element ID associated with this particle, a processor is unable to identify the corresponding processor. To alleviate this problem, the number of associated mesh elements and the affiliated MPI ranks are stored for each BGM cell during the initialization. Then, the first step after calculating all physical particle positions is to inquire whether the local compute node has all associated mesh elements of a BGM cell available in shared memory. If this is not the case, the particle positions are collected and sent to all processors associated with the BGM cell through non-blocking MPI communication. The localization of the remaining particles is then used to hide this communication. Particles which are localized during this step and unambiguously matched to another processors, i.e. residing either on the same compute-node or in the halo region, are sent exclusively to that core. In the last step, each processor locates all received particle positions only in the processor-local mesh elements. Thus, these particles are discarded by all processors except the relevant processor. Latency Hiding Any communication between processing cores incurs latency costs, which are greatly exacerbated when performing inter-node communication as is generally the case in today's highly parallel systems. Efficient parallelization thus requires the minimization of the amount of exchanged information while at the same time maximizing the time available for the completion of the remaining part without stalling the code. DG schemes are well-suited for this task since the volume integral is a purely local operation which can be utilized to hide the latency for the communication of the cell face information across MPI borders. A detailed description of the efficient parallelization of the DG operator for the pure fluid phase (including FV subcells) is given in [24]. However, the presence of a dispersed phase adds another challenge towards efficient parallelization as the number of particles crossing an MPI boundary are generally not known a priori. This results in the necessity for a two-stage communication, as depicted in fig. 1. In the first step, only the number of exchange particles is communicated, thus enabling the receiving core to open the corresponding MPI buffers. Clearly, this step is latency-dominated rather than bandwidth-dominated. For this reason, we elected to hide this step behind the volume integral that is also utilized for the latency hiding of the fluid phase. The second step involves the transfer of the actual particle information. Since this step is more bandwidth-intense, the communication is hidden behind the surface integral, which avoids a stacking on the DG communication and maximizes the time before data starvation occurs. The result is a highly-efficient operator for particle-laden flow. Load Balance Pure DG schemes generally require a fixed effort per grid element. While the exact computing time will be machine dependent, it is sensible to presume that it remains constant across the machine. Given the additional assumption of negligible stalling through communication latency, an ideal load balance distribution can be straightforwardly achieved by evenly distributing the elements among the processors. However, the distribution of the Lagrangian particles is generally not known a priori. Additionally, the ratio of computational time required to advance one degree of freedom in time compared to a particle integration step is machine-dependent. Thus, an efficient load balancing must take both the varying particle distribution and the ratio of computing efforts into account. FLEXI follows a warm restart-based load balancing approach [89]. Runtime measurements are performed at fixed time intervals to determine the actual CPU time, which is then assigned to the individual mesh elements. If a sufficiently high imbalance is detected, load balancing is performed through in-memory redistribution of the elements along the space-filling curve. While this procedure does not seek to minimize communication as, e.g., graph-based distributions attempt, the domain decomposition step is generally faster compared to more advanced approaches [90,91], and the resulting distribution performs similarly given sufficient latency hiding. Post-Processing In order to visualize our results in a highly parallel manner, we extended the custom-built visualization tool chain described in [76] with respect to particles. This tool chain can be used in combination with the open-source software ParaView [92]. The interested reader is referred to [76] for details on the post-processing and visualization tools for the fluid phase. Our post-processing tools include a plugin for ParaView written in C/C+ and a standalone visualization tool written in Fortran. In addition to the fluid flow, both methods overlay the particle location in the domain and any historical information of any particle that passed a boundary condition or impacted upon a wall since the last write-out. In order to keep this information synchronized, the instantaneous particles' characteristics as well as historical impact data are written at the same time and to the identical HDF5 file as the nodal fluid field information. For the instantaneous particles, the particle position in physical space, the particle velocities, the species and the number of impacts with a wall are written. The old and the new particle trajectories, velocities and kinetic energies as well as the species, the impact time, the boundary it crossed and the number of reflections so far are saved for the impacting particles while we chose to visualize these particles on their impact position. Additional particle information can be saved if provided by their respective models outlined in section 2.2. Validation Before turning to actual application cases, we validate the various building blocks for particle-laden flow, from temporal integration to two-way coupling. For the validation of the fluid phase, see e.g. [75,93,25]. In the following, all forces in eq. (3), except the drag force, are neglected, the drag factor in eq. (6) is utilized and one-way coupling is assumed, unless stated otherwise. High-Order Time Integration First, the high-order time integration of the particles is validated. For this, the particle transport and momentum equations given in eq. (3) and eq. (2) were analytically integrated in time for f D = 1. A stationary flow field with a linear velocity profile was considered, i.e., ρ = p = 1 and u = [y/x p,2 | t=0 , 0, 0] T . The particles were initialized at x p = [0, 2, 0.5] T with v p = 0, and the particle momentum was m p dvp dt = F D [1, 0, 0] T +m p g with g = [0, g, 0] = [0, −9.81, 0] T . The fluid velocity at the particle position u f = [x p,2 /x p,2 | t=0 , 0, 0] T is based on the particle path in y-direction due to the linear velocity profile and normalized by x p,2 | t=0 = 2 which yields u f,1 = xp,2 xp,2|t=0 = 1 + gt 2 2xp,2|t=0 . Thus, the analytical integration results in v p,1,ex = 1 − e − t τp + g 2x p,2 | t=0 t 2 − 2tτ p + 2τ 2 p − 2τ 2 p e − t τp , x p,1,ex = t − τ p 1 − e − t τp + g 2x p,2 | t=0 t 3 − t 2 τ p + 2τ 2 p t + 2τ 3 p e − t τp − 1 , v p,2,ex =gt + v p,2 | t=0 , x p,2,ex = 1 2 gt 2 + x p,2 | t=0 . The study was preformed on a computational domain of Ω = [0, 2] 3 discretized with two elements in each direction. Three particles were investigated, each with a different Stokes number St = {0.1, 1, 10}, until t = 0.4 with dt = {0.0125 · 2 k : k ∈ N, k ∈ [0, 4]}. The particle relaxation time was calculated via eq. (4), u * = y/2 and L * = 2. Figure 3a depicts the discrete L 1 norm between the analytically, (·) ex , and numerically, (·) num , integrated particle position, defined as L 1 = |x p,ex − x p,num | and the theoretical slope. The 4 th -order accuracy in time is achieved for all Stokes numbers considered in this study under the condition that f D = 1. Particle Push Following [94], the effect of the forces, F D , F P , F AM and F B , on a particle is quantified by the use of a turbulent channel flow at Re τ = 175 and Re = 4050 laden with particles of two different Stokes numbers. The particle density is set to 2.65ρ, ρ = 1, and particle diameters of d p = 0.005δ and d p = 0.01δ were investigated, where δ = 1 is the channel half height. For each particle diameter, three species were initialized, one for each of the forces, F P , F AM and F B . The computational domain Ω = [0, 4πδ] × [0, 2δ] × [0, 4πδ/3] was discretized by x elements with N = 5. A total of 3072 particles were uniformly emitted at four planes distributed equidistantly along the streamwise direction. Since a fixed time step dt is required for the temporal integration of the Basset force [61], the particles were advanced in time by the explicit Euler scheme with dt = 10 −5 . As the authors in [94] omit the description of the numerical treatment of the Basset force, the number of previous time steps was chosen as K = 20 and K = 100 for d p = 0.005δ and d p = 0.01δ, respectively. Particles were emitted at 4T * and statistics have been accumulated over 0.1T * , with the characteristic time T * = δ uτ . As depicted in fig. 2, the results are in agreement with the literature. Cylinder in Cross-Flow Haugen and Kragset [95] have shown that the impaction efficiency of a cylinder in a particle-laden crossflow is well-suited for code validation. The impaction efficiency denotes the ratio of emitted particles which travel in the direction of the cylinder to the number of particles that actually collide with it. They identified particles with small Stokes numbers to be most sensitive since those show the strongest influence on changes in the boundary layer while performing their simulations with Reynolds numbers Re D ranging from 20 to 6600 based on the cylinder diameter D. This validation study follows their example by focusing on the Re D = 421 case. Figure 3b compares the results generated by FLEXI with the data from Haugen and Kragset as well as theoretical considerations of a cylinder at Re D = 491 by Muhr [96]. FLEXI shows excellent agreement with the literature with a slight discrepancy for high Stokes numbers. Two-Way Coupling A comparison of the Mach angle of a supersonic particle traveling through a quiescent fluid with the theoretical value is a suitable test case to validate the source term introduced by two-way coupled Euler-Lagrangian particles. Parallel Performance Following the validation of the framework, we now demonstrate the scaling capabilities of the extended FLEXI and the effectiveness of load balancing. Scaling Since FLEXI is designed as massively parallel code, scaling must be retained throughout the framework. In this work, the scaling was tested on a Cartesian box with elementary dimensions 2L×L×L, L ∈ R >0 , with a freestream flow field and approx. 175 000 particles per unit cube. The domain size was then extended with increasing processor counts to maintain an equal workload per core. To increase the practical applicability of the test case, additional code features were enabled. The flow and particles were initially convected with mean velocity u f = v p = [1, 1, 1] while synthetic turbulence was generated in the first half of the domain through a Recycling Rescaling Anisotropic Linear Forcing (RRALF) method [97]. A sponge zone was applied in the final decile to dampen fluctuations before reaching the outflow boundary. The polynomial degree was set to N = 5, resulting in a sixth-order accurate scheme. Simulations were performed on the HPE Apollo Hawk system at the High Performance Computing Center (HLRS) in Stuttgart with dual-socket AMD EPYC TM nodes (128 cores per node) and an InfiniBand HDR200 interconnect. The code was compiled with the GNU compiler version 9.2.0 with the libraries mpt 2.23, hdf5 1.10.5 and aocl 3.0. Each run was repeated 3 times to eliminate fluctuations in overall machine load. Parallel efficiency in terms of weak and strong scaling is depicted in fig. 5. The interconnect on Hawk is deployed in a 9-dimensional enhanced hypercube topology resulting in diminishing bandwidth as the node number increases. This is observable as jump in parallel efficiency with the performance leveling out again for higher node counts. Nonetheless, FLEXI demonstrates excellent scaling with parallel efficiency for the weak scaling case at > 85 % including for the 256 nodes (32 768 core) run. Higher core numbers are possible but require changing some internal counters to 8 byte-integers which was omitted for this tests. Strong scaling temporarily even exceeds the ideal speedup due to relieved memory pressure and the additional latency hiding capability introduced with the particle load. Load Balancing Load balancing is evaluated using the setup from section 5.3. Instantaneous particle positions colored by species are shown in fig. 6a. While only the non-reflected particles are necessary for the validation in section 5.3, we opted to keep reflected particles in order to further validate the boundary intersection accuracy. Figure 6b depicts the average load per processor, sampled across 100 iterations, with red areas denoting high relative computational load and blue areas indicating minimal loading. In the top half, the load before load balancing is illustrated, where most of the computational effort is focused on the axial direction upstream and downstream of the cylinder position. The load drops in the vicinity of the cylinder as the grid element size decreases towards the cylinder. The bottom half depicts the load after balancing using the same scale. While the distribution is not perfectly uniform, it has moved considerably closer to equilibrium. Applications In this section, we briefly demonstrate the applicability of FLEXI to more challenging large-scale test cases, the ash deposition in a turbine linear cascade and the particle-laden flow around a wall-mounted cylinder. In the following, all forces except the drag force are neglected and the drag factor in eq. (6) is employed, unless stated otherwise. Furthermore, a one-way coupled fluid and dispersed phase is assumed. Turbine Linear Cascade Ash deposition was identified as a major contribution to performance degradation of turbines, especially on modern specimens with hot section temperatures of or exceeding 1350 K [98]. In this application, we simulate the ash deposition on a T106C low-pressure turbine linear cascade. The setup was chosen as described by [99] with an exit Mach number of M = 0.65 and Reynolds number of Re = 80 000. The mesh is fully periodic in pitchwise and spanwise direction with the Mach number distribution shown in fig. 7a. More details on the mesh and fluid setup can be found in [17]. Ash density was estimated at ρ p = 990 kg/m 3 , and 16 particle species with Stokes numbers ranging from 0.01 to 1000.0 were injected over the entire inlet boundary with the local fluid velocity. As the deposition rate is highly variable and thus strongly dependent on the chosen model, only the initial impaction efficiency is illustrated in fig. 7c. Note that similar to fig. 3b, the impaction efficiency will approach unity as the shading region of the blade extends across the complete passage for heavy particles. Wall-Mounted Cylinder Particles suspended in the ingested air are a major source of fan and compressor erosion of jet engines. This more challenging setup investigates the particle-laden flow around a wall-mounted cylinder at Re D = 32 000 which is representative of the leading edge of a transonic compressor blade. The numerical setup is based on experiments by Kawamura [100] who evaluated wall-mounted cylinders at varying height, H, to diameter, D, ratios H/D. For this study, the domain of the H/D = 8 case was discretized using a fully hexahedral mesh with 788 112 elements and N = 7. In order to retain the high-order geometry near the cylinder surface, the mesh features full volume curving with N geo = 4 using agglomeration to generate high-order inner element mappings. The turbulent inflow boundary layer was generated using the RRALF approach. The generated inflow turbulence with an inlet Mach number of M = 0.7 is illustrated in fig. 8 using the instantaneous isosurface of the Q-criterion colored by the velocity component in downstream direction. The RRALF region covers the left side of the graph, distinguishable by a small discontinuity in the velocity magnitude due to the upstream pressure field of the cylinder. The RRALF is shown to produce a statistically stable boundary layer flow with a Reynolds number of Re θ = 3299.12 based on momentum thickness at the cylinder position. Particles were emitted at the end of the RRALF region with the particle velocity chosen as the instantaneous fluid velocity. The particle density is set to ρ p = 2500 kg/m 3 . A total of 11 particle species were simulated with Stokes numbers ranging from 0.001 to 10.0. The instantaneous particle position was sampled every 8.3 × 10 −8 s, with the cumulative particle distribution at height z ≤ H depicted in fig. 9. Lower (St = 0.1, fig. 9a) and higher (St = 10, fig. 9c) Stokes numbers result in a more pronounced particle-free region compared to the medium Stokes number case in fig. 9b which shows the fastest return to (almost) uniform distribution, albeit with a strong variation in particle momentum. Especially the high Stokes number case yields a local increase in particle density with the impact extending several times the diameter of the cylinder downstream. Here, the working high-order boundary treatment is directly evident in fig. 9c, as particles reflect off the curved surface and move upstream in the laminar flow outside the wall boundary layer until the downstream force causes a reversal of the velocity vector. The resulting bimodal distribution between reflected and non-reflected particles needs to be considered when predicting compressor erosion, particularly of the subsequent stages. Conclusion and Outlook Particle-laden flows pose significant challenges towards their efficient and accurate numerical solution, especially in a high performance setting. In the context of the compressible Navier-Stokes equations, the focus of one-or two-way coupled Euler-Lagrangian solvers is more on the time-accurate particle tracking than on the efficiency on highly parallel systems. In this work, we aimed to alleviate this deficiency and presented the extension of the open-source massively parallel solver FLEXI towards particle-laden flows. Since the Eulerian code base was previously validated, this work focused the modeling of the dispersed phase and its numerical treatment. Particular emphasis was given to the implementation of the particle tracking on parallel systems with arbitrary and possibly curved element faces. Since extensibility was an explicit goal, we chose to extensively discuss the background and motivation leading to specific implementation choices. Subsequently, each building block of our proposed framework was validated on its own, and we have verified the predictive performance of our implementation. Furthermore, we illustrated the excellent scaling properties of our framework using a canonical test case. Finally, the complete framework was applied to more challenging test cases to demonstrate its applicability to large-scale problems. In the future, we seek to extend this open-source framework to incorporate further features such as the tracking of larger particles with a level-set ansatz or the extension to four-way coupling. At the same time, our project is open towards any external contribution. Figure 1 : 1Flow chart of the discontinuous Galerkin operator for particle-laden flow. Figure 2 : 2Discrete Probability Density Function (PDF) of the forces F y,i acting on the particles in wall-normal direction. The forces are normalized by the drag force in wall-normal direction. The dashed lines highlight the results of[94]. Discrete L1 error between the analytically and numerically integrated particle position in x-direction. Impaction efficiency over Stokes number for the Re D = 491 cylinder. Figure 3 : 3Validation for the discrete phase. Figure 4 : 4For this, a computational domain of Ω = [0, 2] × [0, 1] × [0, 1] was discretized by 199 × 99 × 1 elements with N = 5. The fluid was initially at rest, i.e., ρ = 1, p = 94.464286, u = 0 and µ = 10 −5 , and the particles moved continuously into the domain at x p = [0.0, 0.5, 0.5] with v p = [23, 0, 0], resulting in a particle Mach number of M p = |u f −vp| c = 2 with the speed of sound c = κ p ρ . The Mach angle θ is given by sin(θ) = 1Mp , resulting in θ = 30 • , which is reproduced by the numerical results, as depicted infig. 4. Validation of the two-way coupling. Instantaneous density (left) and pressure profile (right) at t = 0.7. Weak scaling with 6912 DoFs/core and 1562 particles/core. Strong scaling with 5.66 · 10 7 DoFs and 1.28 · 10 7 particles. Figure 5 : 5Parallel efficiency for the structured rectangular mesh. Average load per processor, top: before load balancing, bottom: after load balancing. Figure 6 : 6Instantaneous particle positions and load distribution for the Re D = 491 cylinder. Instantaneous isosurfaces of the Q-criterion colored by Mach number with Schlieren visualization in the background. Figure 7 : 7T106C low-pressure turbine cascade results. Figure 8 : 8Instantaneous isosurface of the Q-criterion colored by downstream velocity. Figure 9 : 9Particle distribution at z ≤ H for different Stokes numbers colored by velocity magnitude |vp|. Extrapolate u → u − , u +U Lifting Flux Lifting Volume Operator Lifting Surface Integral Extrapolate ∇U → ∇u − , ∇U + Particle Push Particle Tracking Volume Operator DG Particle Comm Fluxes Surface Integral Particle Source Term ∂U dt Explicit Runge Kutta Time Integration Send u + Receive u + Send Lifting Flux Receive Lifting Flux Send ∇u + Receive ∇u + Send Flux Receive Flux Send # of Particles Receive # of Particles Send Particle Data Receive Particle Data https://github.com/flexi-framework/flexi https://github.com/flexi-framework/flexi-particle www.flexi-project.org https://github.com/piclas-framework/piclas AcknowledgementsThe research presented in this paper was funded in parts by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC 2075 -390740016 and by the DFG Rebound -420603919. We acknowledge the support by the Stuttgart Center for Simulation Science (SimTech). 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[ "https://github.com/flexi-framework/flexi", "https://github.com/flexi-framework/flexi-particle", "https://github.com/piclas-framework/piclas" ]
[ "Prepared for submission to JHEP Three-body resonances in the ϕ 4 theory", "Prepared for submission to JHEP Three-body resonances in the ϕ 4 theory" ]
[ "Marco Garofalo [email protected] \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n", "Maxim Mai \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n\nInstitute for Nuclear Studies\nDepartment of Physics\nThe George Washington University\n20052WashingtonDCUSA\n", "Fernando Romero-López \nCenter for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n", "Akaki Rusetsky \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n\nTbilisi State University\n0186TbilisiGeorgia\n", "Carsten Urbach \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n" ]
[ "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Institute for Nuclear Studies\nDepartment of Physics\nThe George Washington University\n20052WashingtonDCUSA", "Center for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA", "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Tbilisi State University\n0186TbilisiGeorgia", "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany" ]
[]
We study the properties of three-body resonances using a lattice complex scalar ϕ 4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two-and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the threeparticle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory.
10.1007/jhep02(2023)252
[ "https://export.arxiv.org/pdf/2211.05605v2.pdf" ]
253,447,330
2211.05605
bc766d594e733bd5648aac9775952aa1f5ba17c1
Prepared for submission to JHEP Three-body resonances in the ϕ 4 theory 7 Mar 2023 Marco Garofalo [email protected] Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Maxim Mai Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Institute for Nuclear Studies Department of Physics The George Washington University 20052WashingtonDCUSA Fernando Romero-López Center for Theoretical Physics Massachusetts Institute of Technology 02139CambridgeMAUSA Akaki Rusetsky Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Tbilisi State University 0186TbilisiGeorgia Carsten Urbach Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Prepared for submission to JHEP Three-body resonances in the ϕ 4 theory 7 Mar 2023 We study the properties of three-body resonances using a lattice complex scalar ϕ 4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two-and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the threeparticle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory. Introduction The hadronic three-body problem marks the current frontier of the theoretical and computational progress in hadron spectroscopy. Its understanding is crucial to various systems of high relevance, such as the Roper resonance and its large branching ratios to ππN channels [1][2][3][4][5] and other mesonic resonances, such as the ω(782) [6] decaying into π + π − π 0 , or the search for spin exotics [7] decaying only to three-body final states. Understanding such systems has been a major challenge for a long time, and only recently it came into reach due to rapid theoretical and computational advances. Specifically, enormous progress has been achieved connecting the spectrum of three-body systems in finite and infinite volume via the so-called quantization condition, for reviews see Refs. [63][64][65][66][67][68]. Several weakly interacting systems of three mesons at maximal isospin have indeed been calculated in Lattice QCD [34,37,54,[69][70][71][72][73][74][75][76] and successfully studied using threebody quantization conditions. The strategy for resonant three-body systems has been laid out [46,77,78], and its first application to the axial a 1 (1260)-resonance from Lattice QCD has been accomplished [77]. With only a few available Lattice QCD results for resonant three-body systems [4,77], it was so far not possible to uncover the appearance of such interesting effects as the avoided level crossing or test various strategies in extracting infinite-volume quantities. In this work, we attempt to close this gap by using ϕ 4 -theory, which is a viable testbed for formalism in a controlled setup. For previous work see Refs. [79][80][81]. Due to reduced computational costs compared to lattice QCD, the exploration of parameter space is much more flexible. In addition, one can control the particle content and the resonance parameters freely. On the analysis side, we utilize state-of-the-art quantization conditions, namely the Relativistic Field Theory (RFT) [15,16] and Finite-Volume Unitarity (FVU) [33,34] approach, testing and comparing those for the first time on the same set of data. We note that this also tests the performance of the Non-Relativistic Effective Field Theory (NREFT) [21,22] approach, which in its Lorentz-invariant formulation [55] is algebraically identical to the FVU quantization condition, at least within the approximations used in this work. At their core, all these approaches aim to separate the power-law volume dependence of the threeparticle scattering amplitude from the exponentially suppressed one, which necessitates singling out the classes of Feynman diagrams in which the intermediate particles can go onshell. Obviously, this goal is achieved by different means in the different formalisms, but the final result is equivalent in the sense that all configurations of three particles being on-shell are accounted for. The differences include the chosen cutoff in the spectator momentum, how exponentially suppressed terms are accounted for, and the particular choice of the parametrization for the (sub)-system dynamics. Relations between different parametrizations are, in general, non-trivial and involve integral equations, see Refs. [54,62,82]. Still, an empirical comparison of both finite-volume approaches (FVU/RFT) on the same set of lattice results has not been performed. Potentially, there might be scenarios where one or another approach may be more advantageous to describe the actual data. This defines the second goal of this work, allowing one to identify possible systematics of analysis tools for future studies. This paper is organized as follows: in section 2 we show the toy model with an explicit resonance coupling to the three-particle final states. In section 3 we recap both finite-volume quantization conditions and discuss the workflow for extraction of scattering parameters. Next, in section 4 we present our analysis of the finite-volume spectrum at different values of the couplings in the action. Furthermore, in section 5 we present the determination of the mass and width of the resonance based on the fitted scattering quantities. We conclude with the summary in section 6. Description of the Model Scalar models were already used to study resonances of two particles in [83,84], providing the necessary background for carrying out similar studies also in lattice QCD. Here we study a similar model which has a resonance coupled to the three-body state. The Euclidean model is composed of two complex scalar fields ϕ i (i = 0, 1) with non-degenerate (bare) masses m 0 < m 1 and Lagrangian L = i=0,1 1 2 ∂ µ ϕ † i ∂ µ ϕ i + 1 2 m 2 i ϕ † i ϕ i + λ i (ϕ † i ϕ i ) 2 + g 2 ϕ † 1 ϕ 3 0 + h.c. . (2.1) The terms proportional to the bare coupling g make the heavy particle (with field ϕ 1 ) unstable since it can decay into three light particles, each of them associated with field ϕ 0 . The Lagrangian has a global symmetry ϕ 0 → e iα ϕ 0 and ϕ 1 → e i3α ϕ 1 , which prevents the mixing of operators transforming in different ways under this symmetry, for instance, the operator ϕ 0 cannot mix with the operator ϕ 3 0 . Thus, the mixing of one light particle with three light particles is forbidden. This will be useful for the extraction of the spectrum. To study the problem numerically, we define the theory on a finite hypercubic lattice with periodic boundary conditions, lattice spacing a, and volume V = T · L 3 , where T denotes the Euclidean time extent and L the spatial extent of the lattice. We define the derivatives in the Lagrangian on a lattice as finite differences ∂ µ ϕ i (x) = 1 a (ϕ i (x + aµ) − ϕ i (x)) . (2.2) In the following, we set the lattice spacing a = 1 for convenience. Redefining m 2 i = 1 − 2λ i κ i − 8,λ i = 4κ 2 i λ i ,ĝ = 4g κ 3 0 κ 1 , ϕ i = √ 2κ i φ i ,(2.3) the discretized lattice action reads S = x i=0,1 − κ i φ † i (x) µ [φ i (x + µ) + φ i (x − µ)] + φ † i (x)φ i (x) +λ i (φ † i (x)φ i (x) − 1) 2 +ĝ 2 φ † 1 (x)φ 3 0 (x) + h.c. . (2.4) As a further simplification, we study the model in the limit λ i → ∞ for both couplings i = 0, 1, this is often referred to as the Ising limit (a more detailed discussion of this limit can be found in section 2.4.2 of [85]). In this limit, the only non-zero contribution to the path integral over the field comes from the configurations that satisfy φ † i (x)φ i (x) = 1. Thus, the integral over the complex variable φ i is reduced to the integral of an angle θ, representing the phase of the field φ i = e iθ i and the action simplifies to The model is most likely trivial and reduces to a free theory in the continuum limit [86]. However, with a small but finite lattice spacing and with energies below the cutoff scale, the model effectively describes an interacting continuum field theory. S = x i=0,1 − κ i φ † i (x) µ [φ i (x + µ) + φ i (x − µ)] +ĝ 2 φ † 1 (x)φ 3 0 (x) + h.c. . (2.5) κ 0 κ 1 g L 0. Simulation algorithm We generate ensembles, using the Metropolis-Hastings algorithm 1 . For each point x a new configuration is proposed φ (x) from a uniform distribution, and it is accepted with probability P = min {1, exp (−∆S)}, where ∆S is the variation of the action (2.5). When simulating even lattice sizes, the lattice can be divided, as usual, into two sectors (even/odd), where all the points within one sector can be updated in parallel. This strategy cannot be used for odd lattice sizes, where we have to divide the lattice into three sectors instead, which can be updated simultaneously. We implement the simulation algorithm using the Kokkos C++ library [88,89], which provides abstractions for both parallel execution of code and data management in order to write performance-portable applications. The list of ensembles, generated in this work, is compiled in Table 1. We used 2 · 10 7 configurations for each ensemble, generated from 200 replicas each of 10 5 thermalized configurations. We skip the first 10 4 configurations in each replica for thermalization. For the light mass M 0 , we measured the integrated autocorrelation time τ int ∼ 0.8, in units of the Monte Carlo time. We bin the configurations in blocks of 10 5 (the entire replica), which we expect to be safely larger than the τ int of any of the observables investigated here. We resample the resulting 200 configurations using the Jackknife technique to propagate uncertainties to all derived quantities. Observables We measure the mass of the light particle from an exponential fit to the appropriate twopoint correlation functions at large time distances φ † 0 (t)φ 0 (0) ≈ |A φ 0 →0 | 2 e −M 0 t + e −M 0 (T −t) , (2.6) withφ i (t) = x φ i (t, x) being a field projected onto the zero spatial momentum, M 0 the mass of one particle φ 0 and the matrix element A φ 0 →0 = φ 0 |φ † 0 |0 . Analogously, the energy of the two light particle system E 2 can be determined from φ † 0 (t) 2 φ 0 (0) 2 ≈ |A 2φ 0 →0 | 2 e −E 2 t − e −E 2 (T −t) + |A φ 0 →φ 0 | 2 e −M 0 T (2.7) where the matrix elements are A 2φ 0 →0 = 2φ 0 | φ † 0 (t) 2 |0 and A φ 0 →φ 0 = φ 0 | φ † 0 (t) 2 |φ † 0 . When the coupling is zero g = 0 the particle φ 1 is stable thus its mass M 1 can be measured as for M 0 from the exponential fit of φ † 1 (t)φ 1 (0) ≈ A 1→0 e −M 1 t + e −M 1 (T −t) . (2.8) If g > 0, then one φ 1 particle can mix with the three φ 0 particles. Hence we consider the operators φ 3 0 and φ 1 with identical quantum numbers to construct the correlator matrix C(t) = φ † 0 (t) 3 φ 0 (0) 3 φ † 0 (t) 3φ 1 (0) φ † 1 (t) φ 0 (0) 3 φ † 1 (t)φ 1 (0) . (2.9) We solve the generalized eigenvalue problem (GEVP) [90][91][92][93] C(t) v n = λ(t, t 0 ) C(t 0 )v n (2.10) for all t, keeping fixed t 0 = 3 in lattice units. From the eigenvalues λ(t), the energy levels E 3 and E 3 can be extracted with the help of an exponential fit λ(t, t 0 ) ∝ e −E 3 (t−t 0 ) + e −E 3 (T −(t−t 0 )) , (2.11) λ (t, t 0 ) ∝ e −E 3 (t−t 0 ) + e −E 3 (T −(t−t 0 )) . (2.12) We check that our results are stable by comparing the extracted energy levels to the case where we also include the operators φ 1 φ † 0 φ 0 and φ 3 0 φ † 0 φ 0 in the correlator matrix. All our ensembles have time extent T = 64, and we have checked that this allows us to neglect all the uncertainties due to finite T safely. To reduce the statistical error of the correlators, we use translational invariance and average over all possible combinations with the same source-sink separation e.g. φ † 0 (t)φ 0 (0) = 1 T T t =0 φ † 0 (t )φ 0 (t − t ) . (2.13) For the implementation of the GEVP we use [94], and the values of the energies measured in our ensembles are reported in Table 2. Quantization Conditions We now describe the two-and three-particle quantization conditions. In the two-particle case, this is the well-established Lüscher formalism [95]. In the S-wave only limit, the two-particle spectrum is given by solutions of the following equation Table 2. q * cot δ = 2 γL √ π Z P 00 1, q * L 2π .(3. Energy levels determined in this work. M 0 is the one-particle mass and E 2 is the two-particle energy. When g = 0, M 1 labels the mass of the heavy particle, and E 3 the lowest three-particle energy level. When g > 0, E 3 and E 3 correspond to two different three-particle energy levels. All energies are given in units of the lattice spacing. Here Z P 00 is the Lüscher zeta function in the moving frame, the Lorentz boost factor γ = E 2 (P )/E CM is defined in terms of center of mass energy E 2 CM = E 2 2 (P ) − P 2 with P the total momentum of the system, while the relative momentum of the two particles is given by q * 2 = E 2 CM /4 − M 2 0 . The right-hand side of Eq. (3.1) can be computed from the spectrum at a finite volume while the left-hand side is related to the scattering amplitude in the infinite volume. All the data quoted in this paper are in the P = 0 frame, γ = 1, and we only consider the trivial irreducible representation A 1 of the octahedral group. As mentioned above, we utilize both the RFT and the FVU approach in the three-particle sector. The RFT and FVU approaches have been shown to be formally equivalent, and a non-trivial integral-equation-type relation between the three-particle interaction parameters has been established [54,62,82]. All approaches use scheme-dependent quantities to parametrize three-body effects: K df,3 for RFT and C for FVU. The schemes differ, for example, due to different implementations of the spectator-momentum cutoff or different forms of the one-particle exchange terms. In the following, we recap both approaches in the case of three identical scalars and no two-to-three processes. Relativistic Field Theory approach The RFT approach [15,16] is derived by classifying all power-law finite-volume effects emerging from all Feynman diagrams to all orders in perturbation theory in a generic relativistic field theory. In the case of three identical scalars with mass M 0 and no two-tothree transitions, the quantization condition reads det [F 3 (E 3 , P , L) + 1/K df,3 (E * 3 )] = 0 , (3.2) where E * 3 = E 2 3 − P 2 is the center-of-mass (c.m.) three-particle energy, and F 3 and K df,3 are matrices in a space labeled by the finite-volume momentum, p = 2π L n, n ∈ Z 3 , of one of the particles (denoted the "spectator") and the angular momentum of the other two in their two-particle c.m. frame. The above determinant acts in this space. In the simplest case, with only S-wave interactions, taking the zero total momentum P = 0 and assuming K df,3 to be only a function of the overall energy of the system E 3 , the quantization condition can be reduced to F iso 3 (E 3 , L) = −1/K iso df,3 (E 3 ) . (3.3) This is usually referred to as the isotropic approximation, which neglects higher partial waves in the three-particle system and two-particle subsystems, and is expected to be valid close to the threshold. The right-hand side of this equation is the three-particle K-matrix, and it parametrizes three-particle short-range interactions. Note that K iso df,3 is a schemedependent unphysical object. The connection to the physical amplitude will be discussed below in section 5. The left-hand side of Eq. (3.3) contains finite-volume information and the two-particle scattering phase shift. The relevant expressions to compute F iso 3 are: F iso 3 (E 3 , L) = 1 L 3 kp F s 3 −F s 1 (K s 2 ) −1 +F s +G sF s kp , K s 2 kp = δ kp 32πω k E * 2,k q * 2,k cot δ + |q * 2,k | (1 − H(k)) , F s kp = δ kp 1 L 3 UV a −PV UV a H(k) 4ω k 4ω a ω k+a (E 3 − ω k − ω a − ω k+a ) , G s kp = H(k)H(p) L 3 2ω k 2ω p (P − p − k) 2 − M 2 0 . (3.4) Here, the vectors p and k label the finite-volume momenta of the spectator particle, and K s 2 , F s and G s are matrices in a space with p, k indices. The on-shell energies for particles with momentum x are denoted by ω x = x 2 + M 2 0 , while the c.m. energy of the interacting pair and relative c.m. momentum are given by E * 2 2,k = (E 3 − ω k ) 2 − k 2 = E 2 3 + M 2 0 − 2E 3 ω k , q * 2 2,k = E * 2 2,k /4 − M 2 0 . Moreover, k and p are the four momenta of the spectator particles and P = (E 3 , 0) in the overall c.m. frame. In F s , the integral is defined a ≡ d 3 a/(2π) 3 , while the sum over a runs over all finite-volume momenta. The principal value (PV) prescription is defined as in Ref. [15]. The superscript "UV" in the sum and integral indicate that an ultraviolet cutoff is required to separately evaluate the sum and integral. A method for evaluating numerically the sum minus integral can be found in appendix B of Ref. [31]. The expressions in Eqs. (3.4) contain a smooth cutoff function H(k) as defined in Eqs. (28) and (29) of Ref. [15], which we display here for completeness H(k) = J E * 2 2,k − (1 + α)M 2 0 (3 − α)M 2 0 , J(z) =        0 , z ≤ 0 ; exp − 1 z exp − 1 1−z , 0 < z < 1 ; 1 , 1 ≤ z . (3.5) In this work, we keep α = −1, which ensures that all matrices appearing in the quantization condition are finite. In particular, the cutoff function restricts k < k max , where k max is defined by E * 2 2,k kmax = 0. We use the implementation of the quantization condition provided in Ref. [61] and the associated repository [96]. Finite-Volume Unitarity approach The FVU approach is based on the unitarity relations for the three-to-three body scattering amplitude. Hereby, the bookkeeping of various configurations of three particles, going onshell, is simplified by utilizing the so-called isobar-spectator language [97]. The isobar can be thought of as an intermediate auxiliary field which, in particular, can also describe a system of two repulsively interacting particles [34,71,74]. The notion of an isobar is closely related to the particle pair in the RFT approach -namely, a full propagator of an isobar coincides with the two-particle Green function in a particular partial-wave channel. Using this amplitude and employing constraints on intermediate momenta due to the (periodic) boundary conditions on the lattice, this approach yields the FVU three-body quantization condition. In the c.m. frame, it is algebraically identical to the Lorentz-invariant NREFT quantization condition as mentioned before, and hence, if needed, one could use the same procedure to transform the equation to the moving frames. In this paper, however, we work explicitly in the c.m. frame and the need for such a transformation does not arise. Alternatively, one might consider the "relativization" of the one-particle exchange term similar to the last line in Eq. (3.4) and setting a cutoff low enough to ensure that no spurious energy levels emerge in the spectrum. For the present case with only S-wave interactions the FVU three-body quantization condition reduces to det B + C + E L · K −1 − Σ F V kp = 0 . (3.6) Explicitly, the above matrices in the space of spectator momenta are defined as [B(E 3 )] kp = B(k, p; E 3 ) = −1 2ω k+p (E 3 − ω k − ω p − ω k+p ) , K −1 (E 3 ) kp = δ 3 kpK −1 2 E * 2 2,p , (3.7) Σ F V (E 3 ) kp = δ 3 kp Σ F V (E * 2 2,p , M 0 L, p) , [E L ] kp = δ 3 kp (M 0 L) 3 2ω p . Here, we used the same nomenclature as introduced in the previous section. Ignoring the exponentially suppressed e −M 0 L terms, the only volume-dependent terms are given by the kinematical function E L , the one-particle exchange diagram with propagator B, and the two-body self-energy term Σ F V evaluated in the finite volume. The form of the latter two is fixed by ensuring two-and three-body unitarity in the infinite volume as discussed before. In particular, the two-body self-energy term reads Σ F V (E * 2 2,p , M 0 L, p) = J p (E * 2 2,p ) (M 0 L) 3 s E * 2 2,p (4ω 2 s ) 1 2ω s 1 E * 2 2,p − 4ω 2 s (3.8) and Lorentz boost with the three-momentum p, s = s + p s · p p 2 J p (E * 2 2,p ) − 1 + J p (E * 2 2,p ) 2 , J p (E * 2 2,p ) = E * 2 2,p E * 2 2,p + p 2 ,(3.9) see Ref. [37]. The matrix C in the isotropic approximation is a matrix in the spectator momenta, where all entries are identical and depend only on the total energy E 3 . It is a volume-independent term that, together withK −1 2 , encodes the three-and two-body dynamics, respectively. Hence, they cannot be fixed from principles of the S-matrix theory alone, but only from the fits to the actual finite-volume spectra. We note that fixing C requires the calculation of the three-body spectrum, whereasK −1 2 can be fixed either from the two-body spectrum alone or from a combined fit of the two-and three-body energy levels. Specifically, using the standard Lüscher approach (3.1) we express the two-body term asK −1 2 (E * 2 2,p ) = −q * ,2 2,p cot δ 16π E * 2 2,p + Re Σ IV (E * 2 2,p ) ,(3.10) through the two-body S-wave phase-shift δ with q * ,2 2,p defined as in the previous section. The relative c.m. momentum is defined as in section 3.1, and the infinite-volume self-energy reads Σ IV (E * 2 2,p ) = d 3 s (2π) 3 1 2ω s E * 2 2,p (4ω 2 s ) 1 E * 2 2,p − 4ω 2 s . (3.11) We note that other parametrizations of the two-body termK −1 2 can be chosen as well. Finally, we note that all available three-body quantization conditions, Eqs. (3.6) and (3.2), are infinite-dimensional in the spectator momentum space, which calls for a truncation of the momentum space. Various approaches to this issue have been discussed in the literature, such as the inclusion of the form-factors [16,33,37], hard cutoff, or over-subtractions [74,77,98]. These schemes all come with various (dis)advantages. Here we work with a hard cutoff |p| < Λ with Λ = √ 8π/L, which is sufficient to access the kinematical region of interest, see section 4. Parametrization of the two-and three-body forces Finding the solutions of the quantization conditions in Eqs. (3.1), (3.3), and (3.6) allows one to predict the energy eigenvalues, given the knowledge of volume-independent quantities. This also means that we can extract these volume-independent quantities from finite-volume spectra. In particular, we are interested in constraining q * cot δ for the two-body sector, and K iso df,3 (RFT) and C (FVU) in the three-body sector. In practice, we need to parametrize the energy dependence of the quantities, describing the interactions, with a small set of parameters. For the two-particle interactions, it is customary to use the effective range expansion: q * cot δ = 1 a + O(q * 2 ) , (3.12) where a is the scattering length. In the three-body sector, we will use parametrizations that include an explicit pole to accommodate a resonance. In particular, C in FVU and K iso df,3 in RFT will be parametrized as C = c 0 E 2 3 − m 2 R + c 1 , K iso df,3 = c 0 E 2 3 − m 2 R + c 1 ,(3.13) where c i and c i (i = 0, 1) are numerical constants to be determined from the data. When the final-state rescattering is weak, one can relate the sign of c 0 (c 0 ) to the residua of the two-point correlation function. The latter has a definite sign as discussed in Ref. [39], which implies that c 0 (c 0 ) should be negative. Previous studies [31,77] indicate that these parametrizations can describe resonances. Moreover, it should be pointed out that the parametrization given in Eq. (3.13) is already rather general. For example, on physical grounds, one may exclude a double pole or a cut in the variable E 2 3 . The latter can be directly ruled out from unitarity considerations in the low-energy region. Regarding the former, it is expected that the weak repulsive finalstate interactions will lead to a small splitting of the double pole, resulting in two nearby poles in the scattering amplitude -an implausible scenario in the model studied in this work. We, therefore, refrain from considering these rather exotic scenarios and concentrate on the simple parametrization of Eq. (3.13). The only freedom left in this expression is adding polynomial terms in E 2 3 to the background or adding more poles. It is also worth mentioning that, under certain circumstances, even a three-body force without poles can lead to a dynamical generation of resonances (see Ref. [77]). In this case, the background would be described by a higher-order polynomial which mimics the Taylor expansion of the pole term at low-energy. Unlike Ref. [77], our model produces weakly repulsive two-particle interactions, and so, a dynamical generation of poles is not expected. In addition, fits to a higher-order polynomials are very unstable. For these reasons, we opt to simply use the model in Eq. (3.13), which as will be seen, will provide a good description of the data. Since the two-particle subsystem is not resonant, finding solutions of Eq. (3.1) is straightforward. In contrast, the presence of a resonance in the three-particle spectrum could make the problem numerically unstable. To ameliorate this problem, we multiply the quantization conditions by the denominator of the three-body force. Thus, the quantization condition becomes (E 3 3 − m 2 R ) det B + C + E L · K −1 − Σ F V = 0 , (3.14) (E 3 3 − m 2 R ) (1/F iso 3 (E 3 , L) + K iso df,3 (E 3 )) = 0 . (3.15) Given the parametrizations in Eqs. (3.13), the solution of the above quantization condition will give the predicted energy levels. These modified quantization conditions do not have a pole at E = m R for FVU or m R for RFT. However, in the case of c 0 or c 0 equal to zero, they will both have roots, describing a three-particle system with constant three-body force, and one stable particle with no finite-volume effects and constant mass m R or m R . The final step involves extracting the parameters by performing a fit to the energy levels. This is the so-called "spectrum method" see, e.g. Refs. [75,99] . In our case, we simultaneously fit the two-and three-particle spectra, finding the values of the parameters p n such that the correlated χ 2 -function becomes minimal: χ 2 = i,j E i (p n ) − E data i C −1 ij E j (p n ) − E data j . (3.16) HereC is the covariance matrix of the lattice energy levels E data i in the two-and threeparticle spectrum. Moreover, E i (p n ) are the predicted energy levels obtained by solving the quantization conditions with the given parameters. Analysis of finite-volume spectra In this section, we present our numerical results. To summarize, we observe that FVU and RFT lead to qualitatively identical data descriptions, i.e., the best fit with both formalisms gives similar χ 2 (see Table 3). After the numerical demonstration of the equivalence of the two formalisms, we present an investigation of our model in the limit of zero coupling g and a check that the scenario without a pole in the three-particle amplitude is not compatible with our data (section 4.2). Numerical comparison of FVU and RFT We fit our models to the data, measured for our ensembles with g = 4.43, 8.87 and 17.81 (Table 2), as described in section 3.3. Our best-fit results are reported in Table 3 Table 3. Summary of the FVU and RFT fits to the two-(E 2 ) and three-body (E 3 ) levels, including (cross)correlations. For each bare coupling g, the results represent three and four-parameter fits, respectively. resulting spectrum prediction is plotted in Fig. 1. In that figure, the three panels correspond to the three non-zero values of the coupling g. In all three panels, we plot ∆E/M 0 as a function of LM 0 . Here, ∆E represents the energy shift in the two-(black squares) and three-particle (blue circles) systems, respectively (∆E = E j − jM 0 with j = 2, 3 for the two-respectively three-particle systems). The bands represent our best fits to the data with the RFT (red stripes) and the FVU (shaded blue) parametrizations. In the two-body sector, we obtain compatible results within the RFT and FVU approach for the scattering length a. Note, however, that in the three-body sector, the parameters are not directly comparable due to the scheme dependence discussed above. Nevertheless, we find that the parametrization used in RFT and the one in FVU Eq. (3.13) can fit the data with good χ 2 and they both give consistent predictions of the energy levels. We also performed a fit with and without the parameters c 1 or c 1 , which correspond to the background term in the three-body force. The inclusion of this extra parameter in the fit gives a small reduction of the χ 2 in the cases of g = 8.87 and g = 17.81, while it is essential to fit the data at g = 4.43. We observe that c 0 and c 0 are non-zero within errors and their mean values increase with the bare parameter g. This can also be appreciated in the spectrum: the avoided level crossing, which is characteristic of a resonance, becomes wider with increasing values of g (see Fig. 1). The values for m R and m R reported in Table 2 are close to each other even if they are different between errors. The similarity may be due to the pole in the amplitude being very close to the real axis (see section 5.3), thus m R is not so far from the physical parameter M R . Testing the resonance hypothesis In this section, we study the manifestation of a resonance in the finite-volume spectrum with its signature as an avoided level crossing. First, we consider the case of vanishing coupling g, i.e., when the particle φ 1 becomes stable and decoupled from φ 0 . This setting can be seen as a benchmark, since it corresponds to the well-studied ϕ 4 theory, and all scattering Here, the φ 1 -particle is stable (green triangles on the plot). The green solid band represents the fit result of all the φ 1 energy levels to a constant, while the red striped bands are the fit to the two-and three φ 0 -particle energy levels with the RFT quantization condition Eq. (4.1). Right: In this panel we show results for g = 8.87. We compare two fit models, where the first assumes the that φ 1 -particle is a resonance and the second assumes that it is stable. The red-striped bands represent the fit (3.13) with c 1 set to zero, reported in the sixth row of Table 3, while the gray crosshatch band is the result of the fit with K iso df,3 constant and a constant energy level equal M 1 . In both fits the two-particle sector is fitted with the Lüscher quantization condition with parametrization given in Eq. (3.1). quantities are expected to be described by very simple parametrizations. Our choice of κ 0 and κ 1 is such that the energy levels of the particle φ 1 and three-particle φ 0 cross around LM 0 ∼ 5.6 (Fig. 2). Note that this crossing does not imply φ 1 → 3φ 0 transitions, which are excluded based on the symmetries of the theory at g = 0. The energy level corresponding to one heavy particle φ 1 can be simply extracted from the exponential fit at large time separation to the two-point correlation function Eq. 2.8. We fit the value of M 1 at each volume as a constant since we only expect exponentially suppressed finite-volume effects. The two and three φ 0 -particles are fitted with the RFT formalism with K iso df,3 = c and q * cot δ = 1/a. The best-fit values are χ 2 dof = 1.8 (4.1) c M 2 0 = 1351(490) M 1 /M 0 = 3.03431(32) a M 0 = −0.1514(18) . (4.2) This means that in the limit g → 0, the scenario of φ 1 as a stable particle decoupled from φ 0 is supported by the data, and has a reasonably good fit quality, as expected. 2 Now, we turn on the interaction between the φ 0 and φ 1 fields and repeat the above test. In particular, we want to check if the interpretation of φ 1 as a stable particle would also be supported by data in this case at non-zero g-values. We do so by fitting either {c, M 1 } as before, or the form given in Eq. (3.13). The result of both fits can be found in Fig. 2, where the non-resonant fit is represented as the gray crosshatch band. As can be seen, the non-resonant fit fails to describe the data close to the avoided level crossing. The best-fit result is (22) . χ 2 dof = 3.1 , cM 2 0 = −161 ± (880) , m R /M 0 = 3.02142(16) , aM 0 = −0.1553 Note that the χ 2 dof of this fit is much worse than the one of the benchmark fit displayed in Eq. (4.1), which rules out this model for the scattering quatities. On the other hand, a fit with a pole in the K iso df,3 matrix (3.13) with two parameters (i.e. with c 1 = 0) gives a χ 2 dof = 1.6 as reported in the sixth line of Table 3 and displayed in Fig. 2 as a solid red band.When including the pole, the χ 2 dof is of the same order of the benchmark fit at g = 0. Infinite-volume scattering After having determined the two-and three-body parameters, the goal is to extract physical resonance parameters, namely, the resonance pole position. However, a technical complication in the three-particle finite-volume formalism(s) is that the three-body parameters, K iso df,3 or C, are scheme-dependent and therefore unphysical. In order to remove the scheme dependence, a set of integral equations leading to the physical scattering amplitude needs to be solved. To do so, we use the state-of-the-art tools that have been developed separately for each method, Refs. [100][101][102] for the FVU approach, and Refs. [58,73] for the RFT. Pole position in the FVU approach In the FVU approach, the infinite-volume scattering amplitude is extracted as follows. First, the two-body scattering amplitude is simply proportional to 1/(Σ IV −K −1 2 ), whereas the three-body analog is more complex. In particular, the connected isobar-spectator scattering amplitude projected to the S-wave reads T 00 (k, p; E 3 ) = B 00 (k, p; E 3 ) + C(E 3 ) − ∞ 0 d 2 4π 2 ω (B 00 (k, ; E 3 ) + C(E 3 )) τ (E * 2, )T 00 ( , p; E 3 ) ,(5.1) where = | | and 1/τ (E * 2, ) = −q * 2, cot δ/(16πE * 2, ) − i Im Σ IV (E * 2 2, ), see Eq. (3.11). Here, the S-wave projected one-particle exchange is calculated from Eq. The connection between T 00 and the three-body scattering amplitude M 3 can be found in Ref. [97], here we ony need that the poles of T 00 are the same of M 3 . The complexity in solving the one-dimensional integral equation (5.1) lies in the fact that the interaction kernel (one-particle exchange term B 00 ) develops non-trivial cuts. Here, we use the method of the integration contour deformation [102], see also Refs. [100,101] for recent applications. One begins with choosing a complex spectator momentum contour (SMC), along which the integration in is performed. The choice is made to ensure one does not hit the singularities of the kernel B, Eq. (3.7), i.e., zeroes of ω +k (E 3 − ω +k − ω − ω k ). In practice, this is an iterative process, since the momenta and k should be also located on this contour. These in turn determine the values of ω +k and E 3 − ω +k − ω − ω k . The latter should not become zero when both and k are taken somewhere on the contour (otherwise, one would have to choose another contour). The blue and black points in the left panel of Fig. 3 demonstrate this explicitly. They are generated as follows. The real part of the quantity E 3 is fixed somewhere near the expected location of the resonance (We draw the figure for E 3 /M 0 = 3.017−0.001i, but we have convinced ourselves that the picture remains the same in the relevant interval of the values of E 3 ∈ C). Then, the different values of momenta k are chosen on the contour and equations ω +k = 0 and E 3 − ω +k − ω − ω k = 0 are solved for , which defines two regions, denoted by blue and black dots in the figure. It is seen that the blue and the black areas do not cross the red contour ∈ SMC and thus the denominator never vanishes. Hence, the singularity of the kernel B is indeed avoided. In the next step, one picks the self-energy momentum contour (SEC), along which the integration over the momentum s is performed in Σ IV , see Eq. (3.11). This should be done in order to ensure that the integrand in this integral never becomes singular. The right panel of Fig. 3 shows the quantities E * 2 2, and 4ω 2 s where and s run along the SMC and SEC, respectively, and the same value for E 3 is chosen. It is seen that these two quantities never coincide and, hence, the integrand always stays regular. The choice of the contours is, in principle, a matter of taste. In the present work, we have adopted the following choice: SMC : {t − i 0.6(1 − e −t/0.3 )(1 − e (t−Λ)/0.3 )| t ∈ (0, Λ)} SEC : {t − i 1.675 arctan (0.6t) | t ∈ (0, ∞)} ,(5.3) which does not hit any singularities for the range of energies considered in this work. In Fig. 3 we have chosen the hard cutoff at Λ = 1.2M 0 , see section 3.2. For the selfenergy integration in Eq. (3.11) the cutoff can be safely removed, since the over-subtracted integrand falls sufficiently quickly at large integration momenta. Transforming now the (regular) integrals along these contours into finite sums, the integral equation (5.1) can be simply solved as a matrix equation T 00 (E 3 ) = 1 1 + (B 00 (E 3 ) + C(E 3 )) · W · τ (E * 2, ) · (B 00 (E 3 ) + C(E 3 )) ,(5.4) where the bold symbols denote matrices over spectator momentum ∈ SMC. The integration weights µ on the chosen contour are encoded in the matrix W pq = δ pq p 2 /(4π 2 ω p )µ p . Finally, the resonance poles can be found as roots of the equation det[1 + (B 00 (E 3 ) + C(E 3 )) · W · τ (E * 2, )] = 0 . (5.5) Owing to the fact that the integration contour lies in the lower half of the complex energy plane, the quantity T 00 (E 3 ) is automatically evaluated on the second Riemann sheet for Im E 3 < 0. We refer the reader to the Refs. [100,101] for more details. Pole position in the RFT approach In the case of RFT, the divergence-free scattering amplitude is given by: M df,3 (k i ; p i ) = M 3 (k i ; p i ) − S D (u,u) (k, p) ,(5.6) where M 3 is the full scattering amplitude that depends on the four-momenta of incoming and outgoing particles, D (u,u) is a subtraction term that cancels physical divergences present in three-particle scattering, S is a symmetrization operator that sums over the three choices of spectator momentum for both initial and final state, and k and p are the spectator momenta. Since we focus on S-wave interactions, we omit partial-wave indices in the interacting pair. A resonance appears as a pole in M 3 in the complex plane, which is inherited by M df,3 . Note that the pole position in M 3 is the same as in the quantity T 00 from 5.4. Explicitly, in the isotropic approximation, M df,3 is given by: M df,3 (E * 3 ) = S L(k) 1 1/K iso df,3 + F ∞ 3 R(p) . (5.7) The quantities L(k), R(p) and F ∞ 3 will be defined below. Since the numerator is not divergent, it suffices to find complex roots of 1/K iso df,3 + F ∞ 3 = 0. Moreover, in the isotropic limit, all involved quantities are only functions of the energy. All necessary equations to evaluate these quantities are given here: D (u,u) s (p, k) = −M s 2 (E * 2,p )G s (p, k, )M s 2 (E * 2,k ) − M s 2 (E * 2,p ) kmax 0 k 2 dk (2π) 2 ω k G s (p, k , )D (u,u) s (k , k) , G s (p, k, ) = − H(p)H(k) 4pk log 2pk − (E 3 − ω k − ω p ) 2 + p 2 + k 2 + m 2 − i −2pk − (E 3 − ω k − ω p ) 2 + p 2 + k 2 + m 2 − i . (5.8) In the formulae above, k and p are the magnitudes of the three-momenta while is a positive parameter necessary to define G s . Note that we have been working with a finite , in order to avoid the singularities on the real axis, and the physical solution can be obtained by taking the → 0 limit of the subsequent solutions. The cutoff function H is defined in (3.5) while M s 2 (k) is the physical s-wave two-particle scattering amplitude 1 M s 2 (k) = q * 2,k cot δ 16πE * 2,k + ρ(k) , ρ(k) = 1 16πE * 2,k −iq * 2,k E * 2 2,k ≥ 4m 2 ; |q * 2,k | E * 2 2,k < 4m 2 . (5.9) Finally the quantities L(k), R(p) and F ∞ 3 R(k) = L(k) = 1 3 − 2ω k M s 2 (k) ρ(k) − kmax 0 k 2 dk (2π) 2 ω k D (u,u) s (k, k ) ρ(k ) , (5.10) F ∞ 3 = k 2 dk (2π) 2 ρ(k)L(k) , ρ(k) = H(k)ρ(k) 2ω k . (5.11) To solve numerically the expressions in Eq. (5.8) on the real axis, we follow the procedure outlined in Refs. [58,73]. We namely replace kmax 0 dk , with a discrete sum k ∆k containing N terms. Then, the first expression in Eq. with the N × N matrices G p,k ( ) = G s (p, k, ) , M p,k = δ pk M s 2 (E * 2,p ) , (5.13) P p,k = δ pk k 2 ∆k (2π) 2 ω k (5.14) and D (u,u) s (p, k) = lim N →∞ lim →0 D(N, ) . (5.15) Once this function is available, everything else is straightforward to evaluate. The final step is to perform an analytic continuation into the complex plane. However, since the interaction is weak we expect the resonance to be very close to the real axis, and we can simply extrapolate from the real axis to the complex plane. This avoids issues with the analytic continuation of the cutoff function in Eq. (5.8). More specifically, we fit the real and imaginary part of F ∞ 3 to a simple polynomial in energy to build an interpolating function, see Fig. 4. An example of this is shown in Fig. 4. Finally, we use that function to find the zeros of the denominator of Eq. (5.7). Results for the mass and the width of the resonance Once the three-body forces are determined as in section 4.1, we need to solve an integral equation in both approaches FVU and RFT to extract the physical information. We solve the RFT integral equation using = 10 −7 and N = 2000, while in FVU we discretize the contour with 200 points, in both cases we did not observe any residual discretization effects. The final step is to find positions in the complex plane, such that the three-to-three amplitude has a pole: M 3 = R −1 E 3 − M R + iΓ/2 + R 0 ,(5.16) for a range of energy |E 3 −M R | < Γ, the above is known as Breit-Wigner parametrization. In the FVU, the pole positions are extracted directly on the second Riemann sheet, calculating then the mass and the width of the resonance via M R − iΓ/2 = E * 3 . In Fig. 5 we observe that FVU and RFT give compatible predictions of the pole position within errors and, thus, physical parameters M R and Γ as reported in Table 4. As expected, the width increases with increasing values of the coupling g. The decay width of one particle into three identical particles can be computed as Γ = 1 2M R 3! dQ 1→3 |M 1→3 | 2 ,(5.17) where the factor 1/3! is a symmetry factor taking into account that the final particles are identical while dQ 1→3 is the integral over the three-particle phase space which for total momentum P = 0 reads dQ 1→3 = (2π) 4 δ 4 (p 1 + p 2 + p 3 − P ) 3 i=0 dp i (2π) 3 2ω p i ,(5.18) and M 1→3 is the scattering amplitude with one initial φ 1 and three final φ 0 . As a matter of fact, the phase space factor is responsible for the small size of the width of the found resonance. To exemplify this and also to compare the obtained widths with the tree level expectation (∼ g 2 ), we plot the ratio ΓM R / dQ 1→3 as a function of g 2 in Fig. 6. We observe Table 4. that the results for the combination ΓM R / dQ 3 have a slope of O(10 −1 ). Furthermore, for lower values of g, the relation seems linear but lower than the tree-level prediction O(1) 3 . Finally we observe deviation from the linearity for the highest point in g. Conclusion We have determined the properties of resonances with the three-particle decay modes in the complex ϕ 4 theory. This has been achieved after several steps: (i) generating field configurations and computing the finite-volume energy levels, (ii) analyzing the spectrum with (different) finite-volume formalisms, and (iii) solving the integral equations to compute the pole position of the three-particle amplitude in the complex energy plane. The model of choice contains two complex scalars with masses M 1 > 3M 0 , and an explicit term in the Lagrangian, allowing a one-to-three decay. By solving the Generalized Eigenvalue problem, we have determined the energy levels of two and three particles. Given the affordable computational cost of this theory, we have carried out the simulations at several lattice volumes and parameters in the action. More details about the theory can be found in section 2, and a summary of the energy levels is provided in Table 2. Regarding the analysis of the spectra, we have used two versions of the three-particle finitevolume formalisms: the RFT and FVU. Indeed, this is the first time that the same dataset has been analyzed using the two formalisms. By fitting the energy levels using the quantization conditions, we have obtained the two-and three-body scattering parameters. Our findings support the statement that comparable descriptions of the finite-volume spectrum can be achieved with either formalism, i.e., with similar χ 2 in the fits. Figure 1 shows the lattice spectra and the different fits with the two approaches. We have found that in order to describe the energy levels and the observed avoided level crossing, an explicit pole in the three-body forces, K df,3 for RFT, and C for FVU, is needed. The scattering parameters in the three-body sector, obtained with the two formalisms are neither directly comparable nor physical, as they come with a particular scheme-and cutoff dependence. For this reason, we have evaluated physical observables, such as the mass and the width of the resonance. The computation of this quantity involves solving integral equations and performing an analytic continuation into the complex energy plane. In this way, we find completely consistent numerical results for these observables, see Fig. 5 for the main result of this work. We can indeed conclude that the physical observables, computed in this work, have a small systematic dependence on the underlying choice of parametrization for the three-body interactions. We have therefore demonstrated the practical equivalence of the different available threebody methods in this controlled setup. Future work will involve applying the same steps to the QCD resonances. Some additional complications will then be needed to be addressed (e.g. nonidentical particles, multichannel scattering, spin. etc.), but the workflow presented in this work will generally remain. = M R − iΓ/2 computed in FVU and RFT using the parametrizations of the three-body force of Eq. (3.13) and the best fits from Table 3. The error reported is only statistical. Figure 1 . 1The energy shift ∆E/M 0 = E j /M 0 − j, j = 2, 3 of the two-and three-particle systems as a function of LM 0 for three values of g. These data points are compared with the best fit results (with 4 parameters) of RFT (red stripes) and FVU (shaded blue) approaches to the energy levels, respectively. Figure 2 . 2Interacting energy level shifts of two-(black squares) and three-particle (blue circles) systems, as functions of LM 0 . Left: The case of g = 0: BFigure 3 . 300 (k, p; E 3 ) = 1 4π dΩpdΩkY * 00 (k)B(k, p; E 3 )Y 00 (p) . On the left panel the integration contour ∈ SMC is shown. It is chosen to avoid the regions in the complex plane Γ s = { |ω +k = 0, ∀ k∈SMC } and Γ x = { |E 3 − ω − ω k − ω +k = 0, ∀ k∈SMC } for a representative three-body energy E 3 /M 0 = 3.017 − 0.001i. In the right panel, the quantities E * 2 2, and 4ω 2 s are displayed, where and s run along the SMC and SEC, respectively, for the same value of E 3 . N, ) = M · G( ) · M − M · G( ) · P · D(N, ) (5.12) Figure 4 . 4Real and imaginary part of F ∞ 3 for g = 17.81, in the small interval E 3 /M 0 ∈ [3.02075, 3.02123] a line is sufficient to interpolate the data. Figure 5 . 5Comparison of the pole positions between the FVU and RFT three-particle formalisms. The pole position is related to the mass and width of the resonance reported in Figure 6 . 6Values of Γ multiplied by M R normalized with the three particle phase space dQ 1→3 computed in RFT (red squares) and FVU (blue circles). Table 1. Ensembles used in this work. The time extent is always kept fixed to T = 64.148522 0.134228 0 20-24 0.147957 0.131234 4.43 20-25 0.147710 0.131062 8.87 21-26 0.147145 0.131062 17.81 21-27 Table 4 . 4Values of the mass and width of the resonance E pole3 For this model, more advanced algorithms are available, see e.g. Ref.[87]. However, given the large values of the bare mass in our ensembles(Table 1), we do not expect a significant speed-up compared to the Metropolis-Hastings algorithm. Our implementation is available at https://github.com/HISKP-LQCD/ Z2-phi4/tree/complex-ising It should be stressed once more that we did not include a pole term in the parametrization of the kernel, since the results at g = 0 indicate that the residue of this term must vanish, as g → 0. Furthermore, three particles with weak repulsive interactions are not expected to produce a shallow bound state and hence the energy level, which has almost no dependence on L, can be safely interpreted as a one-heavy-particle state. 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[ "https://github.com/HISKP-LQCD/", "https://github.com/HISKP-LQCD/hadron,", "https://github.com/ferolo2/QC3_release." ]
[ "Optimal Power Allocation for Integrated Visible Light Positioning and Communication System with a Single LED-Lamp", "Optimal Power Allocation for Integrated Visible Light Positioning and Communication System with a Single LED-Lamp" ]
[ "Shuai Ma ", "Ruixin Yang ", "Bing Li ", "Yongyan Chen ", "Hang Li ", "Youlong Wu ", "Majid Safari ", "Shiyin Li ", "Naofal Al-Dhahir " ]
[]
[]
In this paper, we investigate an integrated visible light positioning and communication (VLPC) system with a single LED-lamp. First, by leveraging the fact that the VLC channel model is a function of the receiver's location, we propose a system model that estimates the channel state information (CSI) based on the positioning information without transmitting pilot sequences. Second, we derive the Cramer-Rao lower bound (CRLB) on the positioning error variance and a lower bound on the achievable rate with on-off keying modulation. Third, based on the derived performance metrics, we optimize the power allocation to minimize the CRLB, while satisfying the rate outage probability constraint. To tackle this non-convex optimization problem, we apply the worst-case distribution of the Conditional Value-at-Risk (CVaR) and the block coordinate descent (BCD) methods to obtain the feasible solutions. Finally, the effects of critical system parameters, such as outage probability, rate threshold, total power threshold, are revealed by numerical results.
10.1109/tcomm.2022.3204659
[ "https://export.arxiv.org/pdf/2208.14268v1.pdf" ]
251,928,939
2208.14268
f051e2fb2ed37fa20a4ddd5495d0ba8d294613fc
Optimal Power Allocation for Integrated Visible Light Positioning and Communication System with a Single LED-Lamp Aug 2022 Shuai Ma Ruixin Yang Bing Li Yongyan Chen Hang Li Youlong Wu Majid Safari Shiyin Li Naofal Al-Dhahir Optimal Power Allocation for Integrated Visible Light Positioning and Communication System with a Single LED-Lamp Aug 20221Index Terms-Visible light communicationVisible light posi- tioningPower allocationCramer-Rao lower bound In this paper, we investigate an integrated visible light positioning and communication (VLPC) system with a single LED-lamp. First, by leveraging the fact that the VLC channel model is a function of the receiver's location, we propose a system model that estimates the channel state information (CSI) based on the positioning information without transmitting pilot sequences. Second, we derive the Cramer-Rao lower bound (CRLB) on the positioning error variance and a lower bound on the achievable rate with on-off keying modulation. Third, based on the derived performance metrics, we optimize the power allocation to minimize the CRLB, while satisfying the rate outage probability constraint. To tackle this non-convex optimization problem, we apply the worst-case distribution of the Conditional Value-at-Risk (CVaR) and the block coordinate descent (BCD) methods to obtain the feasible solutions. Finally, the effects of critical system parameters, such as outage probability, rate threshold, total power threshold, are revealed by numerical results. Shuai Ma, Ruixin Yang, Bing Li, Yongyan Chen, Hang Li, Youlong Wu, Majid Safari, Shiyin Li, and Naofal Al-Dhahir Abstract-In this paper, we investigate an integrated visible light positioning and communication (VLPC) system with a single LED-lamp. First, by leveraging the fact that the VLC channel model is a function of the receiver's location, we propose a system model that estimates the channel state information (CSI) based on the positioning information without transmitting pilot sequences. Second, we derive the Cramer-Rao lower bound (CRLB) on the positioning error variance and a lower bound on the achievable rate with on-off keying modulation. Third, based on the derived performance metrics, we optimize the power allocation to minimize the CRLB, while satisfying the rate outage probability constraint. To tackle this non-convex optimization problem, we apply the worst-case distribution of the Conditional Value-at-Risk (CVaR) and the block coordinate descent (BCD) methods to obtain the feasible solutions. Finally, the effects of critical system parameters, such as outage probability, rate threshold, total power threshold, are revealed by numerical results. Index Terms-Visible light communication, Visible light positioning, Power allocation, Cramer-Rao lower bound. I. INTRODUCTION With the explosively increasing number of Internet of Things (IoT) devices in beyond fifth generation (B5G) networks, the crisis of radio frequency (RF) spectrum shortage becomes increasingly challenging, which makes it more difficult for the RF wireless systems to meet the high speed data transmission and high accuracy positioning demands simultaneously [1]. It is worth noting that more than 50% of voice traffic and 70% of wireless data traffic occur in indoor environments [2]. Since the indoor activity requires both illumination and network access, visible light communication (VLC) [3] and visible light positioning (VLP) [4], which apply the ubiquitous light emitting diodes (LEDs) as access points (APs) and anchor nodes, are promising technologies for indoor IoT applications. Comparing with the conventional radio frequency (RF) wireless technologies, the distinct advantages of VLC and VLP are multifold [4] [5], including no electromagnetic interference, high energy efficiency, high security, and low-cost. VLC utilizes the simple intensity modulation and direct detection (IM/DD) mechanism for information transmission, and has attracted significant research interests as a breakthrough technology for B5G networks [6]. Extensive studies S. Ma have been reported to improve VLC networks performance. For example, by adopting the alternating direction method of multipliers (ADMM), a distributed coordinated interference management scheme is proposed in [7] for VLC networks. To balance energy and bandwidth efficiency, both power allocation and rate splitting are optimized in [8] for DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM). In [9], in order to jointly optimize the post-equalizer, the precoder and the DC offset, a gradient projection-based procedure is presented to minimize the sum mean squared error (MSE) of the received symbols. Furthermore, VLC has been commercialized in industry. Some startup companies and existing industry giants, such as pureLiFi, Philips, and Oledcomm, are providing VLC commercial solutions in home and business buildings. Owning to its short wavelength and low multipath interference, VLP can achieve high indoor positioning accuracy, which can facilitate various applications, such as indoor navigation, location aware services, logistic management, and assets tracking, to name few. By exploiting different visible light characteristics, existing VLP schemes can apply time of arrival (TOA) [10] [11], time difference of arrival (TDOA) [12], angle of arrival (AOA) [13] and received signal strength (RSS) [14] techniques for positioning. Among the above VLP schemes, the RSS based scheme is widely adopted due to its simplicity and ubiquity, where the distance between the lamp base station (BS) and the device is calculated based on the channel model. For example, by using the weighted k-nearest-neighbor (K-NN), a multi-LEDs positioning system is designed in [15] based on sparse fingerprints. In [16], an artificial neural network (ANN)-based position estimator is proposed for 3D RSS-based VLP systems. Most of the existing works only focus on VLC or VLP individually. In practical indoor applications, an integrated system with both the communication and positioning functions is highly desirable. So far, only few works considered the integration of VLC and VLP. Specifically, VLC systems based on orthogonal frequency division multiplexing access (OFDMA) [17], [18] were proposed to estimate the receiver position. An integrated visible light positioning and communication (VLPC) system was designed in [19] by using filter bank multicarrier-based subcarrier multiplexing (FBMC-SCM). Such FBMC-SCM-assisted VLP with its high signal processing complexity was proposed to reduce the out-of-band interference (OOBI). Towards an OFDMA VLPC network, the authors in [20] jointly optimized the AP selection, bandwidth allocation, adaptive modulation, and power allocation to maximize the data rate while satisfying positioning accuracy constraints. In [21], a modified experience replay actorcritic (MERAC) reinforcement learning (RL) approach was presented to maximize the sum rate under the users' minimum data rates and positioning accuracy requirements. In [22], the authors proposed a coordinated resource allocation approach to maximize the sum rate while satisfying the minimum data rates and positioning accuracy requirements of devices. In the above considered VLPC systems, it is required that at least two lamp signals are captured at the receiver simultaneously for effective positioning. Unfortunately, such multilamp setup may not fit in many practical scenarios, such as in a tunnel, corridor, and staircase, where the lamps are sparsely installed. In these scenarios, the multi-lamp based method will not be as efficient as in a large and flat room. In terms of system design, most of the existing VLPC literatures [4], [23] mainly focus on optimizing the resource allocation in different frequency bands to guarantee quality of service (QoS) of communication and positioning. However, some fundamental issues have not been well investigated. Particularly, does the positioning benefit or compromise the communication? How are the two performances related? Given the limited power consumption, how to balance the two performances while taking the positioning error into account? In this paper, we aim to address the above mentioned fundamental issues, as well as to provide a robust beamforming and power allocation scheme. The main contributions of this paper are summarized as follows: • We establish a VLPC system model with a single LEDlamp and a mobile user with multiple photoelectric detectors (PD). By leveraging the fact that the VLC channel model is a function of the receiver's location, the lamp estimate the channel state information (CSI) based on the positioning results, instead of transmitting pilot sequences for CSI estimation, which can significantly reduce the system overhead. • We derive the Cramer-Rao lower bound (CRLB) on the positioning error variance, which is used as the VLP performance metric. In addition, we derive the achievable rate expression for on-off keying (OOK) modulation, and its closed-form lower bound. Furthermore, by exploiting CRLB and achievable rate expressions, we reveal the inner relationship between VLP and VLC for the first time, i.e., derive the distribution of the CSI error of VLC based on the positioning error of VLP, and obtain a rate outage probability of VLC. • Based on the derived model and metrics, we further investigate a joint positioning and communication power allocation and beamforming problem to minimize the CRLB subject to rate outage constraints and power constraints. The outage probability constraint makes the optimization problem non-convex. Then, we apply the Conditional Value-at-Risk (CVaR) and the block coordinate descent (BCD) method techniques to convert the original problem into two convex VLP and VLC sub-problems. Finally, we develop a BCD algorithm for robust VLPC design, in which the positioning and communication power allocation are iteratively optimized until convergence. The rest of this paper is organized as follows. We present the VLPC system model in Section II. The key performance metrics for the VLPC system are derived in Section III. In Section IV, we investigate the chance constrained robust design. Extensive simulation results are presented in Section V. Section VI concludes the paper. Moreover, Table I and II present the means of the key notations and the main acronyms of this paper, respectively. Notations: Boldfaced lowercase and uppercase letters represent vectors and matrices, respectively. M {1, ..., M }. The transpose and trace of a matrix are denoted as (·) T and Tr (·), respectively. · 2 denotes 2-norm. N denotes the Gaussian distribution. 0 denotes a column vector where all elements are 0. R n represents the space of n-dimensional real matrices. S n represents the space of n-dimensional real symmetric matrices. The wireless channel of the VLPC system has two types of links, i.e. the line-of-sight (LOS) link and the non line-ofsight (NLOS) link. Generally, the influence of the LOS link is much stronger than that of the NLOS link [24]. In order to facilitate the theoretical analysis, the design of VLPC system is based only on the LOS link, but both the LOS link and NLOS link are considered in simulation verification. According to the Lambert radiation model [25], the LOS path gain between the LED and the ith PD within field-of-view (FoV) can be expressed as h i = (m + 1) A PD 2πd 2 i cos m (φ i ) cos (ϕ i ) gT f .(1) Here, m is the order of Lambertian emission and m = − ln 2 ln(cos θ 1/2) , where θ 1/2 is the semi-angle at half power. Other parameters are defined as follows: A PD denotes the PD area; d i is the distance between the LED and the ith PD; φ i and ϕ i are the radiance and incidence angles, respectively; g denotes the gain of the optical concentrator, and is given by g = n 2 r sin 2 (ψFoV) , where n r denotes the refractive index, and ψ FoV represents the FoV of receiver; and T f denotes the gain of the optical filter. Without loss of generality, assume that the PDs are pointing straight upward [26]. Based on the geometric relationship, the LOS path gain (1) parameters can be specified as d i = l − u i 2 , (2a) cos (φ i ) = (u i − l) T n LED l − u i 2 = z l − z i l − u i 2 ,(2b)cos (ϕ i ) = (l − u i ) T n i l − u i 2 = z l − z i l − u i 2 ,(2c) where n LED = [0, 0, −1] T and n i = [0, 0, 1] T are unit direction vectors of the LED and the ith PD, respectively. After substituting the above equations into (1), the LOS path gain can be expressed as h i = α(z l − z i ) m+1 l − u i m+3 2 ,(3) where α = (m+1)APDgT f 2π . As shown in Fig. 2, the operational frame consists of three subframes: positioning subframe (downlink), feedback subframe (uplink) and data transmission subframe (downlink). The corresponding durations are T p , T u and T c , respectively. More specific, during the positioning subframe, the LED lamp transmits the positioning symbols to the MU, which estimates the PDs' locations based on the RSS, and the corresponding positioning information will be used for channel estimation in the next sub-frame. Then, during the feedback subframe, the MU sends feedback signals of the PDs' locations to the lamp which estimates the CSI between the LED and the PDs. Finally, during the data transmission subframe, the lamp transmits data symbols to the MU according to the estimated CSI. This model can be extented into the multi-user system though proper multiple access methods. Due to the positioning theory, the positioning signal and subframe can be shared directly by all users without the multiple access. However, the multiple acess method, such as OFDMA, time division multiple address (TDMA), is necessary for the multi-user uplink feedback and downlink data transmission. Thus, The problem and solution also should refer to the classical theory of the multi-user networks. A. Positioning Signal Model and Measurements In the following, we will specify the signal model in order to analyze the operation in each subframe. Let s p (t) denote the positioning symbol generated at the lamp at time t, and |s p (t)| ≤ A, E {s p (t)} = 0, E s 2 p (t) = ε, where A > 0 is the peak amplitude. For t ∈ [0, T p ], the transmitted positioning signal x p (t) of the LED is given as x p (t) = P p s p (t) + I DC ,(4) where P p indicates the allocated transmission power to the positioning symbol, and I DC > 0 denotes the direct current (DC) bias. To guarantee that the transmitted signal is non-negative, the power P p should satisfy P p A ≤ I DC .(5) Given the human eye safety requirement, the LED optical power is limited, i.e. P p A + I DC ≤ P max o , where P max o denotes the maximum optical power. Thus, the power level P p should also satisfy P p ≤ P max o − I DC A .(6) Besides, due to practical circuit limitations, the electrical power of the transmitted signal is constrained as E x 2 p (t) ≤ P max e , i.e., P p ε + I 2 DC ≤ P max e ,(7) where P max e denotes the maximum LED electrical power. Based on (5), (6) and (7), the constraint of the power P p can be written as 0 ≤ P p ≤ min I 2 DC A 2 , (P max o − I DC ) 2 A 2 , P max e − I 2 DC ε . (8) Then, the received positioning signal at the ith PD can be expressed as y p,i (t) = h i x p (t) + n p,i (t) ,(9) where n p,i denotes the received additive white Gaussian noise (AWGN), which includes shot noise and thermal noise [27], and n p,i ∼ N 0, σ 2 p . Theoretically, the electrical power of the received positioning signal is given by P r,i = E y 2 p,i (t) = P p ε + I 2 DC h 2 i + σ 2 p,i ,(10) where σ 2 p,i denotes noise power. Combining (3) and (10), we have the following M equations            (z l −zu−vz,1) m+1 l−u−v1 m+3 2 = 1 α Pr,1−σ 2 p,1 Ppε+I 2 DC 1 2 , . . . (z l −zu−vz,M ) m+1 l−u−vM m+3 2 = 1 α Pr,M −σ 2 p,M Ppε+I 2 DC 1 2 .(11) To transform the Equations in (11) into a concise form, we define the auxiliary variable η i (u) = (z l − z u − v z,i ) m+1 l − u − v i m+3 2 − 1 α P r,i − σ 2 p,i P p ε + I 2 DC 1 2 .(12) Thus, Equation (11) can be equivalently reformulated as follows η i (u) = 0, i ∈ M.(13) Here Equation (13) can be solved by using off-the-shelf optimization solvers, such as FSOLVE in MATLAB [28]. In general, the positioning error is inevitable. Letû and e p denote the estimated MU location and the corresponding positioning error, where e p = [e x , e y , e z ] T . Their relationship can be written as e p = u −û.(14) Generally, the positioning error e p can be assumed to follow the Gaussian distribution [29]- [31], and then CRLB can be achieved by the maximum-likelihood (ML) estimator [32], [33]. We use f ep (e p ) to denote the probability density distribution of e p , which follows a Gaussian distribution with mean 0 and covariance matrix E p , i.e., e p ∼ N (0, E p ). B. MU Feedback and Channel Estimation When t ∈ [T p , T p + T u ], the estimated location of the MÛ u will be sent to the lamp, which also serves as an anchor node. Based onû, the lamp can estimate the CSI between the LED and the PDs. Specifically,ĥ = ĥ 1 , ...,ĥ M T ∈ R M×1 denotes the estimated CSI vector; ∆h = [∆h 1 , ..., ∆h M ] T ∈ R M×1 denotes the CSI estimation error vector. Let h i andĥ i denote the perfect and estimated CSI between the LED and the ith PD, and ∆h i denote the estimated CSI error, i.e., h i =ĥ i + ∆h i . According to (3), the estimated CSÎ h i is a function of the estimated UE's locationû given bŷ h i = α(z l −ẑ u − v z,i ) m+1 l −û − v i m+3 2 .(15) Based on (3) and (15), the estimated CSI error ∆h i is given as ∆h i = α(z l −ẑ u − v z,i − e z ) m+1 l −û − v i − e p m+3 2 − α(z l −ẑ u − v z,i ) m+1 l −û − v i m+3 2 .(16) C. Data Transmission Let s OOK (t) denote the data symbol transmitted from the LED, and s OOK (t) takes value 0 or A with equal probability, i.e., Pr {s OOK (t) = 0} = 1 2 , and Pr {s OOK (t) = A} = 1 2 , where A is the peak amplitude of the symbol. Due to s OOK ≥ 0, I DC can be 0 in the data transmission. For t ∈ [T p + T u , T p + T u + T c ], the LED transmitted data signal x c (t) can be expressed as x c (t) = P c s OOK (t) ,(17) where P c indicates the allocated communication power of the LED. Similarly, the communication power P c should also meet the eye safety constraint, i.e. √ P c A ≤ P max o , where P max o denotes the maximum optical power. Thus, the communication power P c should satisfy P c ≤ P max o A .(18) Under practical circuit limitations, the electrical power of the transmitted signal is constrained as E x 2 c (t) ≤ P max e , i.e., P c E s 2 OOK (t) = P c A 2 2 ≤ P max e ,(19) where P max e denotes the maximum LED electrical power. Based on (18) and (19), the power P c should satisfy 0 ≤ P c ≤ min (P max o ) 2 A 2 , 2P max e A 2 .(20) At the receiver, let v = [v 1 , ..., v M ] T ∈ R M×1 denote the receive beamforming vector of the MU, and v = 1. Therefore, the received data signal at MU can be expressed as y c (t) = v T ĥ + ∆h x c (t) + z c ,(21) where z c ∆ = v T n c , and n c ∈ R M×1 denotes the receiver Gaussian noise vector, i.e., n c ∼ N 0, σ 2 c I . III. PERFORMANCE METRICS A. Cramer-Rao Lower Bound The CRLB represents a lower bound on the variance of the positioning estimation error. Hence, we adopt CRLB as the performance metric for the positioning accuracy in this paper. Specifically, we consider three-dimensional MU location estimation. Considering the received signal model in (9), the likelihood function of y p,i (t) can be written as f (y p,i (t) ; u) = 1 √ 2πσ p e − (y p,i (t)−h i xp(t)) 2 2σ 2 p .(22) Therefore, the log-likelihood function of the received signal Λ (u) = ln M i=1 f (y p,i (t) ; u) = ln κ − 1 2σ 2 p M i=1 Tp 0 (y p,i (t) − h i x p (t)) 2 dt,(23) where κ is a constant that does not depend on the unknown parameters. Recalling the definition given in (14), and denoting by E p the covariance matrix of the positioning error e p . Then, according to the definition of the CRLB on the variance of any unbiased estimator [35], a lower limit on the variance of the ith element in an unbiased estimate vectorû is given by i [E p ] ii ≥ i J −1 p ii ,(24) where [·] ii denotes the diagonal element of a matrix, and J p denotes the Fisher Information matrix (FIM), which is defined as [J p ] ij = −E ∂ 2 Λ (u) ∂u i ∂u j ,(25) for i ∈ {1, 2, 3}, j ∈ {1, 2, 3}. Likewise, [·] ij denotes the element on the ith row and jth column of a matrix. We show in Appendix A that the FIM for (23) is given by J p = T p P p ε + I 2 DC σ 2 p Q,(26) where Q =         M i=1 ∂hi ∂xu ∂hi ∂xu M i=1 ∂hi ∂xu ∂hi ∂yu M i=1 ∂hi ∂xu ∂hi ∂zu M i=1 ∂hi ∂xu ∂hi ∂yu M i=1 ∂hi ∂yu ∂hi ∂yu M i=1 ∂hi ∂yu ∂hi ∂zu M i=1 ∂hi ∂xu ∂hi ∂zu M i=1 ∂hi ∂yu ∂hi ∂zu M i=1 ∂hi ∂zu ∂hi ∂zu         ,(27a)∂h i ∂x u = −α (m + 3) (z l − z u − v z,i ) m+1 (x u + v x,i − x l ) l − u − v i m+5 ,(27b)∂h i ∂y u = −α (m + 3) (z l − z u − v z,i ) m+1 (y u + v y,i − y l ) l − u − v i m+5 , (27c) ∂h i ∂z u = − (m + 1) α(z l − z u − v z,i ) m l − u − v i m+3 + (m + 3) α(z l − z u − v z,i ) m+2 l − u − v i m+5 .(27d) Moreover, let B (Hz) denote the bandwidth of VLC link. Combine the bandwidth of VLC link, the variance of positioning error e p is lower bounded by Tr (E p ) ≥ Tr J −1 p = Bσ 2 p Tr Q −1 T p (P p ε + I 2 DC ) .(28) In the following, we will use (28) as the positioning performance metric which is to be minimized. B. Achievable Rate for OOK When considering OOK modulation, the input signal no longer follows the Gaussian distribution, and thus the Shannon capacity formula based on the Gaussian assumption cannot be directly applied. In Appendix B, we show that the mutual information is given by I (x c ; y c ) = − 1 2 E zc        log 2     e − z 2 c 2σ 2 c + e − ( v T h √ Pc A+zc ) 2 2σ 2 c 2            − 1 2 E zc        log 2     e − z 2 c 2σ 2 c + e − ( −v T h √ Pc A+zc ) 2 2σ 2 c 2            − 1 2 ln 2 .(29) Due to the expectation operation, the expression (29) is not analytically tractable, and can only be calculated numerically at the expense of high computational complexity. To strike a balance between complexity and analytical tractability, we derive a closed-form lower bound on the mutual information (29). Let R L c (∆h) denote a lower bound on the achievable rate. Using Jensen's Inequality and combining h =ĥ + ∆h, we show in Appendix C that R L c (∆h) with the bandwidth B is given by R L c (∆h) = 3B − B ln 2 − 2Blog 2   1 + e − ( v T (ĥ+∆h)) 2 Pc A 2 4Bσ 2 c   .(30) The CSI error ∆h affects the achievable rate, and ∆h is a function of the positioning error e p as shown in Equation (16), which also depends on the positioning signal power P p . Therefore, both the positioning signal power P p and communication signal power P c affect the achievable rate, and their allocations need to be carefully optimized. IV. ROBUST POWER ALLOCATION FOR VLPC DESIGN In this section, we investigate the positioning error variance minimization problem via a robust power allocation design by considering the rate outage probability constraint. Different from existing works, this paper considers the application of the VLPC system in the 3D case, and establishes the connection between the rate outage probability and the positioning error for the first time. A. Problem Formulation We derive the rate outage probability by investigating the relationship between the positioning error and CSI error. Based on (16), the CSI error ∆h is a function of the positioning error e p . Thus, let ∆h i ∆ = g i (e p ) denote the CSI error function of the ith PD, i.e., ∆h i = g i (e p ) = α(z l −ẑ u − v z,i − e z ) m+1 l −û − v i − e p m+3 2 − α(z l −ẑ u − v z,i ) m+1 l −û − v i m+3 2 ,(31) where i ∈ M. Then, we can write e p = g −1 i (∆h i ), and the probability density function f hi (∆h i ) is given by f hi (∆h i ) = f ep g −1 i (∆h i ) ∂g −1 i (∆h i ) ∂∆h i .(32) Unfortunately, an explicit expression of the function e p = g −1 i (∆h i ) is difficult to derive. Nonetheless, we can numerically calculate both the mean and covariance matrix of the CSI error vector ∆h. Specifically, let µ = E {∆h} ∈ R M×1 denote the mean vector of the estimated CSI error ∆h, which is given as E {∆h i } = g i (e p ) f ep (e p ) de p .(33)Furthermore, let D = E (∆h − µ) (∆h − µ) T ∈ R M×M denote the covariance matrix of the estimated CSI error vector ∆h. Then, the element on the ith row and jth column of D is given by [D] ij = E {(∆h i − E {∆h i }) (∆h j − E {∆h j })} ,(34) where i ∈ M, and j ∈ M. The exact distribution of CSI errors ∆h is unknown except for its first and second-order moments. Then, we may define a set P of distributions for ∆h as follows P = {P : E P {∆h} = µ, Var P {∆h} = D} ,(35) where P denotes an arbitrary distribution with the mean µ and covariance matrix D. The set P in (35) determines the CSI error variation, and the rate outage probability. Now, we can formulate the positioning error variance minimization through robust power allocation problem as follows min Pp,Pc,v Tr J −1 p (36a) s.t. sup ∆h∼P, P∈P Pr R L c (∆h) ≤r ≤ P out ,(36b)P p + P c ≤ P T , (36c) 0 ≤ P p ≤ P max p , (36d) 0 ≤ P c ≤ P max c ,(36e)v 2 = 1,(36f) wherer denotes the minimum rate requirement, P out denotes the maximum tolerable outage probability, P T denotes the total power, P max p ∆ = min I 2 DC A 2 , (P max o −IDC) 2 A 2 , P max e −I 2 DC ε , and P max c ∆ = min (P max o ) 2 A 2 , 2P max e A 2 . B. Proposed Robust VLPC Method The main challenge of problem (36) lies in the chance constraint (36b), which does not have a closed-form expression. Hence, we will reformulate constraint (36b). Combined with the lower bound on the achievable rate in (30), the inequality R L c (∆h) ≤r can be equivalently rewritten as v T ĥ + ∆h 2 ≤ δ P c ,(37) where δ ∆ = − 4Bσ 2 c A 2 ln 2 3 2 − 1 2 ln 2 −r 2B − 1 . By using the following equivalence relationship V = vv T ⇔ V 0, rank (V) = 1,(38) and neglecting the non-convex rank constraint rank (V) = 1, constraints (37) and (36f) can be respectively relaxed as ∆h T V∆h + 2ĥ T V∆h +ĥ T Vĥ ≤ δ P c ,(39a)Tr (V) = 1, V 0,(39b) In words, we exploit the semidefinite relaxation (SDR) technique to relax (37) to a semidefinite program (SDP). Then, the outage constraint (36b) can be recast as Pr ∆h T V∆h + 2ĥ T V∆h +ĥ T Vĥ − δ P c ≤ 0 ≤ P out .(40) An effective approach to proceed is to transform (40) into a distributionally robust chance constraint. Then, we can find the worst-case distribution among all the possible distributions from the ambiguity set, i.e., inf P∈P Pr P ∆h T (−V)∆h + 2∆h T (−V)ĥ +ĥ T (−V)ĥ + δ P c ≤ 0 ≥ 1 − P out ,(41) where inf P∈P Pr P {·} denotes the distribution that can achieve the minimum value of the probability. To further deal with the intractability of (41), we introduce a CVaR-based method [36], which is known as a good convex approximation of the worst-case chance constraint. Lemma 1 (CVaR-Based Method): For a constraint function L that is concave or quadratic in ξ, the distributionally robust chance constraint is equivalent to the worst-case constraint, given by [37] inf P∈P (42) where the expression P − CVaR ρ {L (ξ)} denotes the CVaR of function L (ξ) at threshold ρ under distribution P, which is defined as Pr P {L (ξ) ≤ 0} ≥ 1 − ρ ⇔ sup P∈P {P − CVaR ρ {L (ξ)}} ≤ 0,P − CVaR ρ {L (ξ)} = inf β∈R β + 1 ρ E P (L (ξ) − β) + .(43) Here, R is the set of real numbers, (z) + = max {0, z}, and β ∈ R is an auxiliary variable introduced by CVaR. The worstcase CVaR on the right hand side of (42) can be converted into a group of SDPs, which will be shown in the following lemma. Lemma 2: Let L (ξ) = ξ T Qξ + q T ξ + q 0 denote a quadratic function of ξ, ∀ξ ∈ R n . The worst-case CVaR can be computed as [37] sup P∈P {P − CVaR ρ {L (ξ)}} = min β,M β + 1 ρ Tr (ΩM) (44a) s.t.M 0, M ∈ S n+1 , (44b) M − Q 1 2 q 1 2 q T q 0 − β 0,(44c) where M is an auxiliary matrix variable, and Ω is a matrix defined as Ω = Σ + µµ T µ µ T 1 ,(45) where µ ∈ R n and Σ ∈ S n are the mean vector and covariance matrix of random vector ξ, respectively. Let define the continuous quadratic function L (∆h) = ∆h T (−V)∆h + 2∆h T (−V)ĥ +ĥ T (−V)ĥ + δ Pc . By Lemma 2, the worst-case chance constraint in (41) can be computed by the optimization problem as similarly as the problem (44). Then, according to the Lemma 1, the problem can be equivalent to the following CVaR constraints: β + 1 P out Tr (ΩM) ≤ 0, (46a) M − −V −V Tĥ −ĥ T V −ĥ T Vĥ + δ Pc − β 0, (46b) M 0, M ∈ S 4 ,(46c) where M and β are two auxiliary variables, and Ω = D + µµ T µ µ T 1 . Therefore, the original distributionally chance-constrained problem (36) can be reformulated as follows Note that, problem (47) is still non-convex given that the optimization variables P p and V are coupled together in constraint (46a). However, the problem (47) can be decomposed into two convex subproblems with two decoupling variables blocks: {P c , V, M, β} and P p , respectively. It means that when one of blocks is fixed, the problem becomes convex in the remaining block of variables, which is called the multiconvex problem [38]. To solve this kind of multi-convex problem, we propose an efficient BCD algorithm [39] for robust VLPC design with variables coupling, which can guarantee to globally converge to the stationary point [40], [41]. Then, at every iteration, the two convex subproblems, i.e., VLP subproblems and VLC subproblems, are alternatively optimized with respect to one block variable while the remaining blocks are held fixed. More specifically, for the kth iteration, the VLP and VLC subproblems are optimized follows. 1) VLP subproblem: For fixing variables P (k−1) c , the positioning power P which can be solved using the interior point methods, such as CVX [42]. 2) VLC subproblem: With given positioning power P In summary, the overall BCD algorithm for robust VLPC design is listed in Algorithm 1. The solution of the BCD Algorithm 1 is a stationary point of the joint optimization problem (36) [43], [44]. Note that, due to the SDR, the rank of V (k) may not be 1. For rank V (k) = 1, the optimal beamformer v can be calculated by eigenvalue decomposition. When rank V (k) > 1, we can calculate a high-quality feasible solution v of problem (49) based on the Gaussian randomization procedure [45]. Meanwhile, the two SDP problem can be efficiently solved with a worst case complexity O max {m, n} 4 n 0.5 log δ −1 , where n is the problem size n, m denotes the number of constraints m, and δ represents the accuracy of SDP [45]. And the proposed BCD algorithm has a sub-linear convergence rate, O 1 k , where k is the index of iteration [46]. Algorithm 1 Block Coordinate Descent Algorithm for Robust VLPC Design Input: Initializer, P out , P k ← k + 1; V. SIMULATION RESULTS In this section, we present simulation results to evaluate the effectiveness of the proposed VLPC system design. Consider a VLPC system in a room with size 5 × 5 × 3m 3 , where one corner of the room is the origin (0, 0, 0) of the Cartesian coordinate system (X, Y, Z). Assume that the LED location is (2.5, 2.5, 3) and the MU is equipped with M = 3 PDs. Moreover, as shown in Fig. 3, we verify the performance of the proposed optimization method for four different horizontal locations of the MU, i.e., U 1 (1, 1, z u ), U 2 (1.5, 1.5, z u ), U 3 (2, 2, z u ) and U 4 (2.5, 2.5, z u ), where the PDs are arranged according to an equilateral triangle with side length L. The other simulation parameters are summarized in Table III. A. Positioning Performance First of all, it should be noted that the positioning error in this section is the root mean square error (RMSE), which is the error between the average estimated value obtained from multiple measurements and the true value. The positioning symbol with normalized power is considered, i.e., ε = 1. The received SNR is defined as the received SNR at the 1th PD, i.e, SNR = 10lg Fig. 4 (a) illustrates the positioning error at the four test points versus received SNR, where z u = 1m. We may observe that the positioning error decreases rapidly at first and then slowly, and finally converge to a constant as SNR increases. This is because as SNR increases, the influence of the noise decreases. For high SNR, the localization performance is negligibly affected by the noise, but still affected by the NLOS link. In addition, it can be seen from the Fig. 4 (a) that the positioning error at test points U 1 , U 2 , U 3 and U 4 gradually decreases at the same SNR. This is because U 4 is the closest to the lamp, while U 1 is the farthest. Therefore, when the total transmit power is constant, the SNR at U 4 , U 3 , U 2 and U 1 decreases. In Fig. 4 (b), we plot the positioning error at test point U 1 versus SNR, for different MU heights. The shapes of the curves are similar to Fig. 4 (a), and the reasons are also similar. Fig. 5 shows the positioning error versus side length L at high and low SNR. As we can see, with the increase of relative distance L, the positioning error first decreases rapidly and then slowly when the SNR=5dB, while the position error first decreases slowly and then remains unchanged when the SNR=15dB. This is because the larger the relative distance L is, the greater the difference in signal intensity received by each PD will be. Thus, the solution of the nonlinear equations in (13) will be more accurate, especially at low SNR, and the impact of the signal strength difference on accuracy is more obvious. At the same time, when the relative distance L reaches a certain value, additional increases will not help. In addition, when L is constant, the positioning error at U 1 , U 2 and U 3 still decreases sequentially, and the gap between the positioning errors at U 2 and U 3 becomes smaller with increasing L. Pph 2 1 Bσ 2 p . B. Communication Performance To evaluate the communication performance, we first introduce the non-robust VLPC design scheme, which ignores the CSI uncertainty ∆h, and the estimated CSI h is viewed as the perfect CSI h. Moreover, in our simulations, we choose the following basic parameters: length of positioning subframe T p = 0.12 sec side length L = 0.1 m and the test point U 3 with z u = 1.5 m. Fig. 6 (a) depicts the CDF of achievable rate of the non-robust VLPC design, the robust VLPC design with the maximum tolerated outage probabilities P out = 1% and 5%, and the equal power allocation design (P p = P c ), where the total power is P T = 10W . It can be seen that the outage probability of the non-robust VLPC design is 50%, which significantly exceeds the maximum tolerated outage probability requirement. On the other hand, the outage probability of the proposed robust VLPC design is lower than 5%, which meets the outage probability requirement. Fig. 6 (b) depicts the CDF of CRLB with the same parameters as Fig. 6 (a). As can be seen from Fig. 6 (b), the positioning error of the robust VLPC design with P out = 1% is lager than that of the design with P out = 5%. Combined with Fig. 6 (a), the robust VLPC design allocates more power to communication than non-robust design under the premise of minimizing positioning accuracy in order to meet the minimum rate requirements of the system. Therefore, when the total power is limited, the positioning power decreases correspondingly, resulting in the increase of positioning error. Compared with the equal power allocation scheme, the robust VLPC scheme allocates less power to communication under the condition of satisfying the rate requirement, resulting in less positioning error. Thus, Fig. 6 demonstrates the effectiveness of our proposed robust VLPC design. Fig. 7 (a) shows the positioning power P p and the communication power P c of the robust VLPC design versus the outage probability P out withr = 10Mbit/sec and P T = 10W . From Fig. 7 (a), with increasing outage probability P out , the positioning power P p increases, while the communication power P c decreases. This is because as the outage probability P out decreases, the probability that the communication rate is the thresholdr decreases, and the robust design becomes more conservative. Moreover, under the same simulation conditions as Fig. 7 (a), Fig. 7 (b) depicts the CRLB and average communication rateR c of the robust VLPC design versus the outage probability P out . We observe that, as the outage probability P out increases, the CRLB decreases, and the average communication rateR c also decreases. This is because for a given total power, the communication power P c and the positioning power P p are both related to the communication rate, and there exists a tradeoff between them. Fig. 8 show the influence of the total power P T on the robust VLPC design. Fig. 8 (a) shows the optimized power allocation versus P T with P out = 5% andr = 5Mbit/sec. From Fig. 8 (a), we can observe that as the total power P T increases, both the positioning power P p and the communication power P c increase because more power can be allocated for both positioning and communication power to meet the positioning performance requirements and rate constraints. In addition, Fig. 8 (b) shows the CRLB and average rateR c versus P T with the same parameters as Fig. 8 (a). From Fig. 8 (b), we can observe that as the total power P T increases, the CRLB decreases and the average communication rateR c increases. This is intuitive since higher total available power improves both the positioning and communication. Fig. 9 show the influence of the rate thresholdsr on the robust VLPC design, where P T = 8W and P out = 5%. Fig. 9 (a) shows the optimized power allocation versus different rate thresholdsr. We can see that the allocated positioning power P p decreases while P c increases as the rate thresholdr increases. This is because the robust VLPC system needs more communication power to meet the rate thresholdr. Moreover, Fig. 9 (b) shows the CRLB and average rateR c versusr. It can be seen that as the rate thresholdr increases, the CRLB increases because the positioning power P p decreases asr increases. In addition, the average rate increases because the communication power P c increases asr increases. VI. CONCLUSION In this paper, we reveal the intrinsic relationship between VLP and VLC based on the relationship between CSI and location, i.e., the positioning information can be used to estimate the CSI. Then, both the CRLB for VLP and the achievable rate of VLC are derived. Furthermore, a robust power allocation scheme is proposed under practical optical constraints, and QoS requirements. To tackle the rate outage constraints, the worst-case distribution of the CVaR is conservatively approximated to a more tractable form. Then, we propose a BCD Algorithm for robust VLPC design, in which the VLP and VLC sub-problems are iteratively optimized. Finally, our simulation results demonstrate the effectiveness of the proposed VLPC scheme for both localization and communications. APPENDIX A DERIVATION OF EQUATION (26) The Fisher Information matrix (FIM) is given by Then, based on the first partial derivatives of the likelihood function in (23), we have J p =      −E ∂ 2 Λ(u) ∂x 2 u −E ∂ 2 Λ(u) ∂xu∂yu −E ∂ 2 Λ(u) ∂xu∂zu −E ∂ 2 Λ(u) ∂yu∂xu −E ∂ 2 Λ(u) ∂y 2 u −E ∂ 2 Λ(u) ∂yu∂zu −E ∂ 2 Λ(u) ∂zu∂xu −E ∂ 2 Λ(u) ∂zu∂yu −E ∂ 2 Λ(u) ∂z 2 u      .(50)∂Λ (u) ∂x u = − 1 σ 2 p M i=1 Tp 0 h i x 2 p (t) − y p,i (t) x p (t) ∂h i ∂x u dt,(51a)∂Λ (u) ∂y u = − 1 σ 2 p M i=1 Tp 0 h i x 2 p (t) − y p,i (t) x p (t) ∂h i ∂y u dt,(51b)∂Λ (u) ∂z u = − 1 σ 2 M i=1 Tp 0 h i x 2 p (t) − y p,i (t) x p (t) ∂h i ∂z u dt.(51c) Furthermore, according to the second partial derivatives of the likelihood function, we obtain as ∂h i ∂x u = −α (m + 3) (z l − z u − v z,i ) m+1 (x u + v x,i − x l ) l − u − v i m+5 ,(54a)∂h i ∂y u = −α (m + 3) (z l − z u − v z,i ) m+1 (y u + v y,i − y l ) l − u − v i m+5 , (54b) ∂h i ∂z u = − (m + 1) α(z l − z u − v z,i ) m l − u − v i m+3 + (m + 3) α(z l − z u − v z,i ) m+2 l − u − v i m+5 . (54c) APPENDIX B DERIVATION OF EQUATION (29) For brevity, we drop the time index t throughout this appendix. Let s 1 and s 2 denote values A and 0, respectively. According to (21), the PDF of y c can be written as Since log 2 (x) is a concave function with respect to x, according to (56e), a lower bound on the mutual information is derived as , R. Yang, B. Li, Y. Chen and S. Li are with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China. (e-mail: [email protected]). Fig. 1 : 1VLPC system, as shown in Fig. 1, where the lamp is equipped with a single LED that points straight downward, and a mobile user (MU) has a receiver with M PDs (M ≥ 3) 1 . Let l = [x l , y l , z l ] T , u = [x u , y u , z u ] T and u i = [x i , y i , z i ] T denote the locations of the LED, the MU and the ith PD, respectively, where i ∈ M. Moreover, let v i = [v x,i , v y,i , v z,i ] T denote the offset of the ith PD to the MU, i.e., u i = u + v i . System model illustration. Fig. 2 : 2The frame structure of the considered VLPC system. t. (36c), (36d), (36e), (39b), (46a), (46b), (46c). t. (36e), (39b), (46a), (46b), (46c). V (k) , M (k) , β (k) by solving VLC subproblem (49) with fixed P V (k) . Fig. 3 : 3Locations of MU and LED. Fig. 4 : 4(a) Positioning error versus SNR when the test point is chosen at different locations, where L = 0.1m; (b) Positioning error versus SNR when the transceiver height difference ∆z = z l − zu is different, where L = 0.1m. Fig. 5 : 5Positioning error versus L(cm), where ∆z = 2m. Fig. 6 : 6In the non-robust VLPC design, the robust VLPC design with outage probabilities Pout = 1% and 5%, and the equal power allocation design:(a) CDF of achievable rate; (b) CDF of CRLB. Fig. 7 : 7(a) Power allocation of robust VLPC design versus outage probability Pout; (b) CRLB and average communication rateRc of robust VLPC design versus outage probability Pout. Fig. 8 : 8(a) Power allocation of robust VLPC design versus total power P T ; (b) CRLB and average rateRc of robust VLPC design versus total power P T . Fig. 9 : 9(a) Power allocation of robust VLPC design versus rate thresholdr; (b) CRLB and average rateRc of robust VLPC design versus rate thresholdr. f ( yc −v T h( √ Pc s k +I DC )) mutual information of the receiver is derived asI (x c ; y c ) = h (y c ) − h (y c |x c ) (y c ) log 2 f (y c ) dy c − 1 2 log 2 2πe var (z c ) ( v T h √ Pc (s k −s j )+zc) ( v T h √ Pc (s k −s j )+zc) ( v T h √ Pc (s k −s j )+zc) Jensen's Inequality [47], if f (x) is a convex function, then we have the inequality f [E(x)] ≥ E [f (x)]. TABLE I : ISummary of Key NotationsNotation Description P p Allocated positioning power P c Allocated communication power u Location vector of MU e p Positioning error vector h Estimated CSI vector ∆h CSI estimation error vector I Identity matrix J p Fisher information matrix R L c Lower bound on the the achievable rate P The set of distributions for ∆h r Minimum rate requirement P out Maximum tolerable outage probability TABLE II : IISummary of Main AcronymsNotation Description VLP Visible light positioning VLC Visible light communication VLPC Visible light positioning and communication MU Mobile user CRLB Cramer-Rao lower bound CSI Channel state information BCD Block coordinate descent CVaR Conditional Value-at-Risk TABLE III : IIIBasic Simulation ParametersParameters Value FoV, ψ FoV 90 • Detector area of PD, A PD For 3D positioning, the number of PDs is at least 3. 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[ "Atomic Scale Surface Segregation in Copper-Gold Nanoparticles", "Atomic Scale Surface Segregation in Copper-Gold Nanoparticles", "Atomic Scale Surface Segregation in Copper-Gold Nanoparticles", "Atomic Scale Surface Segregation in Copper-Gold Nanoparticles" ]
[ "Grégoire Breyton \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n\nLaboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance\n", "Hakim Amara \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n\nLaboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance\n", "Jaysen Nelayah \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Jérôme Creuze \nICMMO/ESP2M\nUMR 8182\nUniversité Paris-Saclay\n17 avenue des sciences91405Orsay cedexFrance\n", "Hazar Guesmi \nICGM ICMMM -Institut Charles Gerhardt Montpellier -Institut de Chimie Moléculaire et des Matériaux de Montpellier\n\n", "Damien Alloyeau \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Guillaume Wang \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Christian Ricolleau \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Grégoire Breyton \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n\nLaboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance\n", "Hakim Amara \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n\nLaboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance\n", "Jaysen Nelayah \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Jérôme Creuze \nICMMO/ESP2M\nUMR 8182\nUniversité Paris-Saclay\n17 avenue des sciences91405Orsay cedexFrance\n", "Hazar Guesmi \nICGM ICMMM -Institut Charles Gerhardt Montpellier -Institut de Chimie Moléculaire et des Matériaux de Montpellier\n\n", "Damien Alloyeau \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Guillaume Wang \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n", "Christian Ricolleau \nLaboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance\n" ]
[ "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "ICMMO/ESP2M\nUMR 8182\nUniversité Paris-Saclay\n17 avenue des sciences91405Orsay cedexFrance", "ICGM ICMMM -Institut Charles Gerhardt Montpellier -Institut de Chimie Moléculaire et des Matériaux de Montpellier\n", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire d'Etude des Microstructures\nUMR104\nONERA-CNRS\nUniversité Paris-Saclay\nBP 7292322Châtillon CedexFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "ICMMO/ESP2M\nUMR 8182\nUniversité Paris-Saclay\n17 avenue des sciences91405Orsay cedexFrance", "ICGM ICMMM -Institut Charles Gerhardt Montpellier -Institut de Chimie Moléculaire et des Matériaux de Montpellier\n", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance", "Laboratoire Matériaux et Phénomènes Quantiques (MPQ)\nUniversité Paris Cité\nCNRS\n75013ParisFrance" ]
[]
In this work, we combine electron microscopy measurements of the surface compositions in Cu-Au nanoparticles and atomistic simulations to investigate the effect of gold segregation. While this mechanism has been extensively investigated within Cu-Au in the bulk state, it was never studied at the atomic level in nanoparticles. By using energy dispersive X-ray analysis across the(100)and(111)facets of nanoparticles, we provide evidence of gold segregation in Cu3Au and CuAu3 nanoparticles in the 10 nm size range grown by epitaxy on a salt surface with high control of the nanoparticles morphology. To get atomic-scale insights into the segregation properties in Cu-Au nanoparticles on the whole composition range, we perform Monte Carlo calculations employing Nbody interatomic potentials. These simulations confirm this effect by showing a complete segregation of Au in the(100)and(111)faces of a truncated octahedron for gold nominal composition of the alloy above 70% and 60% respectively. Furthermore, we show that there is no size effect on the segregation behaviour since we evidence the same oscillating concentration profile from surface to the nanoparticles core as in the bulk. These results can shed new lights in the interpretation of the enhanced reactivity, selectivity and stability of Cu-Au nanoparticles in various catalytic reactions.Surface segregation in A x B 1−x binary alloys, i.e. the enrichment of surface by one of the elements as compared to the bulk composition, has been the subject of numerous studies[1][2][3][4]. It is a very important phenomenon in surface physics of alloys since it can dramatically change the intrinsic properties of the bulk material. Notably, it can strongly modify the surface reactivity of the alloys during catalytic reactions[5,6]. Three physical parameters are used to determine a priori the element that segregates at the surface [7]. The first two ones are the surface energy and the atomic size of the species that constitute the alloy. It is generally admitted that the element with the lower surface energy and with the larger size will segregate[8][9][10]. Another driving force is the alloying effect meaning the competition between the cohesive energy of each individual atoms of the alloy and the free energy of mixing. These key quantities explaining segregation phenomena have been put forward by using phenomenological models based on the pair-based[11][12][13]and the elastic-strain energy theories[8,14]. Moreover, numerical calculations were performed within the tight-binding approximation to characterize the surface segregation from a microscopic point a view with an accurate description of the chemical bonds in transition metal based alloys[3,[15][16][17][18][19].In bulk CuAu alloys, this effect has been extensively studied from both experimental and theoretical approaches as a model system for binary alloys[9,10,[20][21][22][23][24][25][26][27][28][29][30]. Among the most common ones, the techniques that were used to probe the surface composition of bulk Cu-Au are Auger electron spectroscopy (AES)[20][21][22], low energy ion scattering (LEIS)[9,23,24], low-energy elec-tron diffraction (LEED)[24], and X-rays surface diffraction[25,26]. All of these works have shown various extent of Au surface segregation depending on the nature of the material (mono-vs polycrystal), the experimental technique and the indexes of the considered surfaces. Interestingly, this segregation effect is followed by an oscillating concentration profile from the surface to the core of the material where the nominal concentration is finally reached[23,25,31,32].For nanoparticles (NPs), the situation can be different due to the so-called size effect i.e. the competition between the bulk and surface energies of the NP [33] resulting in segregation effects as in case of Ag-Pt [34], Cu-Ag [35] or Ni-Pt [36] NPs. Indeed, physics at the nanoscale could be different than the one occurring in bulk, especially for NPs whose diameter is smaller than around 10 nm. Typical examples include the dependence on the size of the melting temperature [37], surface energy [38] or mechanical properties [39] of pure NPs as well as the order-disorder transition temperature for bimetallic nanoalloys [40]. Regarding segregation effects in Cu x Au 1−x NPs very few studies exist. Results are mainly obtained by atomistic calculations [27-30, 41] and nanothermodynamic approaches [10]. All these works demonstrated Au segregation at the surface of the NPs. In one case, it was also demonstrated Cu enrichment of the NPs facets although this configuration is not stable[30]. From experiments, up to now, two papers report results on this system[10,30]. The segregation effect was evidenced by chemical mapping acquired by X-ray spectroscopy using a Transmission Electron Microscopy (TEM) in Scanning mode (STEM). In these works, the arXiv:2302.10659v1 [cond-mat.mtrl-sci]
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[ "https://export.arxiv.org/pdf/2302.10659v1.pdf" ]
257,050,340
2302.10659
62fb0b4c3328549353236467b483d46daa7b6eaa
Atomic Scale Surface Segregation in Copper-Gold Nanoparticles 21 Feb 2023 Grégoire Breyton Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Laboratoire d'Etude des Microstructures UMR104 ONERA-CNRS Université Paris-Saclay BP 7292322Châtillon CedexFrance Hakim Amara Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Laboratoire d'Etude des Microstructures UMR104 ONERA-CNRS Université Paris-Saclay BP 7292322Châtillon CedexFrance Jaysen Nelayah Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Jérôme Creuze ICMMO/ESP2M UMR 8182 Université Paris-Saclay 17 avenue des sciences91405Orsay cedexFrance Hazar Guesmi ICGM ICMMM -Institut Charles Gerhardt Montpellier -Institut de Chimie Moléculaire et des Matériaux de Montpellier Damien Alloyeau Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Guillaume Wang Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Christian Ricolleau Laboratoire Matériaux et Phénomènes Quantiques (MPQ) Université Paris Cité CNRS 75013ParisFrance Atomic Scale Surface Segregation in Copper-Gold Nanoparticles 21 Feb 2023 In this work, we combine electron microscopy measurements of the surface compositions in Cu-Au nanoparticles and atomistic simulations to investigate the effect of gold segregation. While this mechanism has been extensively investigated within Cu-Au in the bulk state, it was never studied at the atomic level in nanoparticles. By using energy dispersive X-ray analysis across the(100)and(111)facets of nanoparticles, we provide evidence of gold segregation in Cu3Au and CuAu3 nanoparticles in the 10 nm size range grown by epitaxy on a salt surface with high control of the nanoparticles morphology. To get atomic-scale insights into the segregation properties in Cu-Au nanoparticles on the whole composition range, we perform Monte Carlo calculations employing Nbody interatomic potentials. These simulations confirm this effect by showing a complete segregation of Au in the(100)and(111)faces of a truncated octahedron for gold nominal composition of the alloy above 70% and 60% respectively. Furthermore, we show that there is no size effect on the segregation behaviour since we evidence the same oscillating concentration profile from surface to the nanoparticles core as in the bulk. These results can shed new lights in the interpretation of the enhanced reactivity, selectivity and stability of Cu-Au nanoparticles in various catalytic reactions.Surface segregation in A x B 1−x binary alloys, i.e. the enrichment of surface by one of the elements as compared to the bulk composition, has been the subject of numerous studies[1][2][3][4]. It is a very important phenomenon in surface physics of alloys since it can dramatically change the intrinsic properties of the bulk material. Notably, it can strongly modify the surface reactivity of the alloys during catalytic reactions[5,6]. Three physical parameters are used to determine a priori the element that segregates at the surface [7]. The first two ones are the surface energy and the atomic size of the species that constitute the alloy. It is generally admitted that the element with the lower surface energy and with the larger size will segregate[8][9][10]. Another driving force is the alloying effect meaning the competition between the cohesive energy of each individual atoms of the alloy and the free energy of mixing. These key quantities explaining segregation phenomena have been put forward by using phenomenological models based on the pair-based[11][12][13]and the elastic-strain energy theories[8,14]. Moreover, numerical calculations were performed within the tight-binding approximation to characterize the surface segregation from a microscopic point a view with an accurate description of the chemical bonds in transition metal based alloys[3,[15][16][17][18][19].In bulk CuAu alloys, this effect has been extensively studied from both experimental and theoretical approaches as a model system for binary alloys[9,10,[20][21][22][23][24][25][26][27][28][29][30]. Among the most common ones, the techniques that were used to probe the surface composition of bulk Cu-Au are Auger electron spectroscopy (AES)[20][21][22], low energy ion scattering (LEIS)[9,23,24], low-energy elec-tron diffraction (LEED)[24], and X-rays surface diffraction[25,26]. All of these works have shown various extent of Au surface segregation depending on the nature of the material (mono-vs polycrystal), the experimental technique and the indexes of the considered surfaces. Interestingly, this segregation effect is followed by an oscillating concentration profile from the surface to the core of the material where the nominal concentration is finally reached[23,25,31,32].For nanoparticles (NPs), the situation can be different due to the so-called size effect i.e. the competition between the bulk and surface energies of the NP [33] resulting in segregation effects as in case of Ag-Pt [34], Cu-Ag [35] or Ni-Pt [36] NPs. Indeed, physics at the nanoscale could be different than the one occurring in bulk, especially for NPs whose diameter is smaller than around 10 nm. Typical examples include the dependence on the size of the melting temperature [37], surface energy [38] or mechanical properties [39] of pure NPs as well as the order-disorder transition temperature for bimetallic nanoalloys [40]. Regarding segregation effects in Cu x Au 1−x NPs very few studies exist. Results are mainly obtained by atomistic calculations [27-30, 41] and nanothermodynamic approaches [10]. All these works demonstrated Au segregation at the surface of the NPs. In one case, it was also demonstrated Cu enrichment of the NPs facets although this configuration is not stable[30]. From experiments, up to now, two papers report results on this system[10,30]. The segregation effect was evidenced by chemical mapping acquired by X-ray spectroscopy using a Transmission Electron Microscopy (TEM) in Scanning mode (STEM). In these works, the arXiv:2302.10659v1 [cond-mat.mtrl-sci] In this work, we combine electron microscopy measurements of the surface compositions in Cu-Au nanoparticles and atomistic simulations to investigate the effect of gold segregation. While this mechanism has been extensively investigated within Cu-Au in the bulk state, it was never studied at the atomic level in nanoparticles. By using energy dispersive X-ray analysis across the (100) and (111) facets of nanoparticles, we provide evidence of gold segregation in Cu3Au and CuAu3 nanoparticles in the 10 nm size range grown by epitaxy on a salt surface with high control of the nanoparticles morphology. To get atomic-scale insights into the segregation properties in Cu-Au nanoparticles on the whole composition range, we perform Monte Carlo calculations employing Nbody interatomic potentials. These simulations confirm this effect by showing a complete segregation of Au in the (100) and (111) faces of a truncated octahedron for gold nominal composition of the alloy above 70% and 60% respectively. Furthermore, we show that there is no size effect on the segregation behaviour since we evidence the same oscillating concentration profile from surface to the nanoparticles core as in the bulk. These results can shed new lights in the interpretation of the enhanced reactivity, selectivity and stability of Cu-Au nanoparticles in various catalytic reactions. Surface segregation in A x B 1−x binary alloys, i.e. the enrichment of surface by one of the elements as compared to the bulk composition, has been the subject of numerous studies [1][2][3][4]. It is a very important phenomenon in surface physics of alloys since it can dramatically change the intrinsic properties of the bulk material. Notably, it can strongly modify the surface reactivity of the alloys during catalytic reactions [5,6]. Three physical parameters are used to determine a priori the element that segregates at the surface [7]. The first two ones are the surface energy and the atomic size of the species that constitute the alloy. It is generally admitted that the element with the lower surface energy and with the larger size will segregate [8][9][10]. Another driving force is the alloying effect meaning the competition between the cohesive energy of each individual atoms of the alloy and the free energy of mixing. These key quantities explaining segregation phenomena have been put forward by using phenomenological models based on the pair-based [11][12][13] and the elastic-strain energy theories [8,14]. Moreover, numerical calculations were performed within the tight-binding approximation to characterize the surface segregation from a microscopic point a view with an accurate description of the chemical bonds in transition metal based alloys [3,[15][16][17][18][19]. In bulk CuAu alloys, this effect has been extensively studied from both experimental and theoretical approaches as a model system for binary alloys [9,10,[20][21][22][23][24][25][26][27][28][29][30]. Among the most common ones, the techniques that were used to probe the surface composition of bulk Cu-Au are Auger electron spectroscopy (AES) [20][21][22], low energy ion scattering (LEIS) [9,23,24], low-energy elec-tron diffraction (LEED) [24], and X-rays surface diffraction [25,26]. All of these works have shown various extent of Au surface segregation depending on the nature of the material (mono-vs polycrystal), the experimental technique and the indexes of the considered surfaces. Interestingly, this segregation effect is followed by an oscillating concentration profile from the surface to the core of the material where the nominal concentration is finally reached [23,25,31,32]. For nanoparticles (NPs), the situation can be different due to the so-called size effect i.e. the competition between the bulk and surface energies of the NP [33] resulting in segregation effects as in case of Ag-Pt [34], Cu-Ag [35] or Ni-Pt [36] NPs. Indeed, physics at the nanoscale could be different than the one occurring in bulk, especially for NPs whose diameter is smaller than around 10 nm. Typical examples include the dependence on the size of the melting temperature [37], surface energy [38] or mechanical properties [39] of pure NPs as well as the order-disorder transition temperature for bimetallic nanoalloys [40]. Regarding segregation effects in Cu x Au 1−x NPs very few studies exist. Results are mainly obtained by atomistic calculations [27][28][29][30]41] and nanothermodynamic approaches [10]. All these works demonstrated Au segregation at the surface of the NPs. In one case, it was also demonstrated Cu enrichment of the NPs facets although this configuration is not stable [30]. From experiments, up to now, two papers report results on this system [10,30]. The segregation effect was evidenced by chemical mapping acquired by X-ray spectroscopy using a Transmission Electron Microscopy (TEM) in Scanning mode (STEM). In these works, the arXiv:2302.10659v1 [cond-mat.mtrl-sci] 21 Feb 2023 Au segregation was revealed at the nanometer scale and in consequence there is no evidence of the different extent of the segregation on the three main low index facets, namely (111), (110) and (100), as observed in bulk systems. Indeed, such analysis along a NP, which is much more complex, has never been addressed. In this letter, we determine at the atomic scale the chemical composition of individual facets of epitaxially grown CuAu 3 and Cu 3 Au NPs on NaCl (100) surface in truncated octahedral shape by using X-ray spectroscopy in an aberration-corrected electron microscope. We then compared the results to Monte Carlo simulations allowing the determination of the composition of (111) and (100) facets of Cu-Au NPs in the whole composition range. We show a remarkably good agreement between both approaches proving unambiguously the segregation of gold on NPs surfaces at the atomic scale followed by an oscillating concentration profile within the particle. From an experimental point of view, the challenge to evidence unambiguously the effect of Cu or Au segregation is to have a perfect control on the 3D morphology of the NPs and then on the facets exhibited by the particles under consideration. For that purpose, we developed the epitaxial growth of CuAu NPs on a NaCl substrate and deposited on a TEM carbon grid by the carbon replica technique [42]. Cu-Au NPs were synthesized by alternated pulsed laser deposition technique in a high vacuum chamber under a pressure of 10 −8 Torr [43]. Two compositions were prepared in the in the Cu 3 Au and CuAu 3 stoichiometry ranges and the growth was made at 400 • C in order to obtain NPs in the FCC disordered phase (A1 phase). Experimental details are given in Sec. I of the Supplemental Materials. The NPs were imaged by using a double aberration corrected electron microscope (JEOL ARM 200F cold FEG) in STEM mode using the High Angle Annular Dark Field (HAADF) technique. Chemical analysis of the NPs surface composition was performed by Energy Dispersive X-ray spectroscopy (EDX). Figure 1a shows a typical HAADF high resolution STEM image of a CuAu 3 stoichiometry NP oriented along the [011] zone axis. The exact composition measured by EDX spectroscopy over the whole NP is Cu 15 Au 85 . Through this projection, we clearly identify a truncated octahedron exhibiting 6 facets, namely two (200) and four (111) ones, parallel to the electron beam. Since the Fast Fourier Transform (FFT) pattern of the NP does not show any super structure reflections, the NP is in the A1 phase. We analyze the composition of (200) planes from the surface to the core of the NP across a line scan of the beam along the [200] direction to the planes (red line in Fig. 1a). The procedure is described in Sec. II of the Supplementary Material. From each spectrum acquired along this line, we quantified the composition of an atomic column belonging to the (200) planes by analyzing the intensities under the Au-L α and Cu-K α edges using the Cliff-Lorimer method with a theoretical k Cu/Au [44] factor. The results are plotted as a function of the position of the beam along the line scan and shown in Fig 1b. Since the alloy is in the solid solution phase state, each site is randomly occupied by Cu or Au atoms with a probability of 0.15 and 0.85 respectively. Each column being equivalent in the lack of segregation, its composition is thus equal to the one of the atomic plane. From Fig. 1b, it clearly appears that the (200) surface plane (atomic layer 1) is made of pure gold and then the composition tends to Cu 15 Au 85 for the planes belonging to the core of the NP. It should be pointed out that in this composition range, i.e. in the Au rich region of the phase diagram of the Cu-Au system, when the first surface plane is saturated in gold, the subsequent planes become enriched with copper to reach the nominal com-position of the alloy (Fig 1b). Note that in addition to the accuracy of the EDX technique without reference sample which is around 5 at. %, the precision of the analysis is very sensitive to the exact alignment of the plane with respect to the electron beam: a small misalignment may cause that the spectrum does not strictly correspond to the composition of a unique atomic column. For the Cu 3 Au stoichiometry, an HAADF image of a Cu 70 Au 30 NP is shown in Figure 2a. The nanoparticle is oriented near the [110] zone axis and the corresponding FFT pattern exhibits the 111 reflections. No super structure reflexions are observed confirming that the NP is in the disordered FCC state. Hence, the expected composition of these planes must be Cu 70 Au 30 . We analyzed the plane compositions from the surface to the core along a line scan perpendicular to the (111) surface of the particle following the same procedure as before. The results are plotted in Figure 2b. According to these concentration profiles, it appears clearly that the composition of the (111) surface is enriched in gold meaning 42% instead of 30% according to the NPs composition and a depletion in copper, 58% instead of 70%. The nominal composition of the plane is recovered from the second plane. To get insight the segregation properties of Cu-Au NPs at atomic scale, we perform Monte Carlo (MC) simulations using a specific N -body potential derived from the second moment approximation (SMA) of the tightbinding (TB) scheme [45,46]. The interatomic potential is included in a MC code in the canonical ensemble to relax the structures at finite temperatures [47]. Here, the simulations are performed at high enough temperatures to ensure that the NPs are in a disordered state as the experiments. Regarding the procedure for adjusting the TB-SMA potential and the MC calculations, more details can be found in Sec. III of the Supplemental Material. Meanwhile, we note that the calculated Au surface energies are lower than those of Cu, in line with ab initio calculations, favoring Au surface segregation. Since we focus on segregation phenomena, we have ensured that our TB potential can satisfactorily reproduce the enthalpy of segregation of the solute at the surface, ∆H seg . Indeed, the tendency of a constituent to segregate at the surface is characterized by this crucial quantity defined as the energy balance involved when one atom of a given species, initially placed in the bulk, is exchanged with an atom of the other species located at the surface [48,49]. A negative value of ∆H seg means that solute segregation is favored. As shown in Sec. III of the Supplemental Material, our TB-SMA results are in agreement with the ab initio data. More precisely, we notice a strong tendency for the Au impurity to segregate on the first layer and this whatever the Cu surface considered, namely (100), (111) and (101). The conclusions are rather different in the case of the copper solute where an opposite behavior is observed. Even if the alloying effect should obviously not be neglected, we can expect to observe a strong seg- We considered truncated octahedron Cu-Au NPs containing 405, 1289 and 4033 atoms. This corresponds to cluster sizes around 2 to 5 nm close to the range explored experimentally. Fig. 3a depicts the segregation isotherms for a Cu-Au NP in a disordered state containing 4033 atoms with initial composition covering the whole phase diagram. Obviously, our simulations show that Au segregates whatever its nominal concentration with complete segregation when the gold concentration exceeds 70%. Concomitantly, we observe a depletion in Au atoms in the first two sublayers. Beyond that, a progressive return to the nominal concentration is achieved. To go further, surface concentrations of different facets are analysed in [48]. Gold enrichment is therefore strongly favored on a (100) surface compared to a (111) one, resulting in a complete saturation from a nominal concentration of 60 at. % gold. We now focus on specific concentrations where ordered phases exist, i.e. CuAu 3 and Cu 3 Au [50]. Let start with the CuAu 3 composition in a disordered state. After performing MC simulations, strong segregation effects are highlighted. A visual inspection depicts a NP completely surrounded by a thin layer of gold as seen in Fig. 4a. The innermost layers do not show any particular gold enrichment but display a totally random structure typical of a disordered state. This is confirmed in a more quantitative way with the analysis of the density profiles along the radius of the NPs. The first layer is completely enriched in gold then the concentration decreases within the NP to get closer to the nominal concentration of 75% of Au. In a second step, Cu 3 Au NPs are addressed and exhibit a much less striking segregation effect. As seen in Fig. 4b, only 50% of the surface is covered with Au. Although a significant enrichment of the surface in gold is observed (about twice the nominal concentration), our atomistic simulations do not show a complete Au layer surrounding the NP. Consequently, atomistic simulations confirm that Au segregates at the surface of NPs whatever their composition and size. To compare quantitatively the experimental measurements to the numerical calculations, we extracted, from the atomistic simulations with a truncated octahedron containing 4033 atoms, the concentration profiles of each plane from the surface in the same way that they are acquired in the TEM. For that purpose, we determine for the Cu 15 Au 85 composition, the concentration of each column of the (200) planes. The composition profiles for each atomic column, for different line scans, are superimposed to the experimental curves in Fig. 1b. The same procedure was applied for the Cu 70 Au 30 composition (see Fig. 2b). For both compositions, we show a remarkably good quantitative agreement between both results. In particular, due to the intrinsic statistical nature of the FCC disordered phase, the experimental curves are situated in between the two envelope curves determined from the numerical model. Moreover, we prove without any ambiguity the effect of Au segregation on these planes for these alloy stoichiometries. Knowing the real composition of the surface of bimetallic NPs is crucial for their applications. Regarding Cu x Au 1−x NPs, we succeeded in showing the segregation of a single layer of Au at the surface thus addressing an existing debate in the literature. This has been achieved by combining measurements of the surface composition of (100) and (111) surfaces of NPs with very well controlled morphology and atomic scale simulations. Moreover, our detailed analysis have proven a concentration profile from the facets to the core of the NP as already discussed in case of infinite surfaces highlighting that such mechanism is still present at the nanoscale. These conclusions are very important for the surface and catalyst community [36] since it highlights how it is crucial to consider the real surface composition and the concentration profile along the NPs to analyze the reactivity of small catalysts. This work constitutes a major step showing that the future to understand catalysis of nanoalloys is to determine the complete concentration profile of the NPs surfaces under real environmental conditions since segregation behavior can even be changed [51]. SUPPLEMENTAL MATERIAL : ATOMIC SCALE SEGREGATION IN COPPER-GOLD NANOPARTICLES Sec. I. Synthesis of nanoparticles and preparation for HRTEM characterization To clearly evidence the effect of Cu or Au segregation, a perfect control on the 3D morphology of the NPs is mandatory. In this context, we developed the epitaxial growth of Cu-Au NPs. The substrate of choice for a good epitaxy of Cu-Au system is NaCl because of its cubic symmetry together with one of its lattice distances (d 110 = 0.398 nm) which is in the same range as the one of Cu-Au alloy (0.385 nm for the Cu 50 Au 50 composition). Twonominal compositions were made using pulsed laser deposition technique [43]: Cu 3 Au and CuAu 3 and with a nominal thickness of the continuous film fixed at 0.7 nm. The NPs were deposited on a 1 cm 2 freshly cleaved sodium chloride NaCl (001) single-crystal surface. To ensure well-oriented growth of the NPs on the substrate and in a face-centered cubic (FCC) disordered state (A1 phase), the latter was heated at 400 • C during the laser deposition. After deposition, the Cu-Au NPs were covered by a few nm-thick amorphous (a-)carbon film obtained by evaporating a carbon rod in an Edward Auto 306 thermal evaporator. Carbon replica [42] was obtained by dissolving the sodium chloride in deionized water followed by the transfer of the carbon-supported NPs to standard Mo grids (300 mesh, Agar Scientific) for structural investigations by TEM. In Fig. 5, we can clearly see how epitaxial growth allows us to obtain perfectly facetted and very well oriented NPs limiting the presence of twinning type defects and thus greatly facilitating chemical analysis. Sec. II. Quantification analysis of the composition of an atomic column As seen in Fig. 6, EDX measurement consists of a succession of acquisition points along a line. The distance between two points is chosen so that it coincides with the distance between two planes of the considered family (∼0.2 nm). Each acquisition lasts 20 seconds and irradiates the whole column below the acquisition point, before moving on to the next one, i.e. the next plane. A drift corrector correcting the position every 5 seconds is also in place due to the magnification (×10M) of the image. In Fig. 7, we present typical EDX spectra corresponding to the position of the electron beam on an atomic column of planes parallel to the facet along the line of acquisition. The composition of the columns (hence the one of the planes) was obtained by quantifying the intensity of the Cu K α and Au L α peaks. We do not used the Au M peaks appearing at lower energies since there is an overlap between those peaks and the Mo M ones coming from the support of the carbon thin film supporting the NPs. The background level in these spectra is very low because the NPs are deposited on a very thin carbon film. The noise in this kind of data follows a Poisson statistic and thus the error bar in Fig. 1b and 2b was calculated, for each plane along the line of analysis, using the standard error propagation approach. Finally, the configuration parameters of the microscope for the acquisition of HRSTEM image and EDX spectra were fix as follow: condenser aperture of 40 µm, spot size 8C and camera length of 5 cm, so that the resolution (i.e. the probe size) can be roughly estimated to be less than 0.1 nm. Sec. III. TB-SMA interatomic potential • TB-SMA model Within the tight-binding (TB) framework [45,46], the total energy of an atom n is splitted in two parts, a band structure term that describes the formation of an energy band E n band when atoms are put together and a repulsive term E n rep that empirically accounts for the ionic and electronic repulsions : E n tot = E n band + E n rep . The total energy of the system containing N atoms, E tot , then writes : In the SMA formalism, the band energy is given by : E tot = n=1,N E n tot .(1)E n band = − m =n ξ 2 ij exp −2q ij r nm r 0 ij − 1(2) and the repulsive contribution is described in a pairwise Born-Mayer form : E n rep = m =n A ij exp −p ij r nm r 0 ij − 1(3) where r nm is the distance between atoms at sites n and m whereas r 0 ii (respectively r 0 jj ) corresponds to the equilibrium distance between first neighbors in the pure metal i (respectively j), r 0 ij = r 0 ii +r 0 jj 2 corresponds to the equilibrium distance between first neighbors in the alloy. Note that ξ ij is the effective hopping integral between atoms i and j. • The fitting procedure In the present work, the parameters (ξ ij , A ij , q ij and p ij ) are fitted to reproduce several bulk physical properties. The resulting parameter values for Cu-Cu, Au-Au and Cu-Au interactions are presented in Table I. For pure elements, the TB-SMA parameters have been fitted on experimental values for the fcc structure namely the lattice parameter, the cohesive energy and the elastic moduli (bulk modulus and the two shear moduli) as seen in Table II [52], cohesive energies (eV/at) [52], elastic modulii (GPa) [53], and surface energies (J.m 2 ) [54]) indicated in bracket. Note that fitting the TB-SMA parameters to experimental cohesive energies leads to an underestimation (∼ by a factor 2) of the surface energies since it is well known that such potentials are not always adapted to describe physical properties from bulk to surface [55]. Nevertheless, these deviations do not prevent us from describing qualitatively structural properties of Cu-Au nanoalloys as we will see in the following. Improving the accuracy usually implies increasing the number of parameters, which can blur the physical transparency of the model [56][57][58]. Typically, the following hierarchy: γ 111 < γ 100 < γ 110 is well reproduced within our TB-SMA model for both Cu and Au surfaces. Moreover, we can notice that the Au surface energy calculated is lower than the one of Cu, again in agreement with DFT calculations, which is in favor of the Au surface segregation. For the Cu-Au interaction, the potential has been fitted to the enthalpies of solution in the two diluted limits. Our results (-0.19 eV and -0.12 eV) are in good agreement with experimental data [59] (-0.21 eV and -0.10 eV) in case of Cu(Au) and Au(Cu), respectively. Moreover, we check that the TB-SMA model can reproduce the enthalpy of segregation of the solute at the surface, ∆H seg defined as: ∆H seg = E tot surf ace (solute) − E tot bulk (solute)(4) where E tot surf ace (solute) (respectively, E tot bulk (solute)) is the total energy of the system when one solute atom is at a surface site (respectively, a bulk site). A negative value of ∆H seg means that solute segregation is favored at the surface. To get a relevant database for checking the validity of our TB-SMA model, we have performed DFT calculations to determine ∆H seg . The projector-augmented wave (PAW) method [60,61], as implemented in the Vienna ab initio simulation package (VASP) code [62] was used. The generalized-gradient approximation functional of the exchange correlation energy was calculated within the Perdew, Burke, and Ernzerhof formulation (GGA-PBE) [63]. The cut-off energy was fixed at 400 eV and the positions of the atoms in the supercell are relaxed until the total electronic energy differences fall below 10 −6 eV. Au and Cu surfaces were calculated using slab models with a 3 × 3 surface unit cell, five layers and a vacuum region of 15Å. The two-bottom layer were frozen in the relaxed bulk positions, while the topmost four layers were allowed to relax. The Brillouin-zone integrations for surfaces were performed on a Monkhorst-Pack (3 × 3 × 1) k -point mesh. As seen in Table. III, although our TB-SMA model is not quantitatively perfect, it still succeeds in reproducing the significant trends revealed by the DFT calculations. In the case of Au (Cu) solute, our calculations show a negative (positive) value of ∆H seg for the first layer whatever the surface considered. This is in agreement with the DFT data suggesting that gold segregation is favored for all surfaces unlike Cu which has the opposite behavior. • Monte Carlo simulations This atomic interaction model is implemented in a Monte Carlo (MC) code in the canonical ensemble, based on the Metropolis algorithm, which allows to relax the structures at finite temperature [47]. In the canonical ensemble, MC trials correspond to random displacements of randomly chosen atoms and exchanges between two randomly chosen atoms of different species. The average quantities are calculated over 10 6 MC macrosteps, a similar number of macrosteps being used to reach equilibrium. A MC macrostep corresponds to N propositions of chemical switches and N propositions of random atomic displacements, N being the total number of atoms of the cluster. H.A. thanks B. Legrand for fruitful discussions. FIG. 1 . 1(a) HRSTEM image of a Cu15Au85 NP oriented along the [011] zone axis. Inset: FFT of the image showing the [200] direction. Red line: direction of the line scan for the EDX analysis. (b) Composition profile in Au and Cu as a function of the position of the plane along the [200] direction from the surface (atomic layer 1) to the core of the NP. Light colors: Au and Cu concentrations extracted from the atomistic simulations performed on a NP with the same nominal composition along the same family of planes as the experimental ones. FIG. 2 . 2(a) HRSTEM image of a Cu70Au30 NP oriented near the [110] zone axis. Inset: FFT of the image showing the [111] direction. Red line: directions of the line scan for the EDX analysis. (b) Composition profile in Au and Cu as a function of the position of the plane along the [111] direction from the surface (atomic layer 1) to the core of the NP. Light colors: Au and Cu concentrations extracted from the atomistic simulations performed on a NP with the same nominal composition along the same family of plane as the experimental ones regation of Au within NPs through our simulations and quantify more accurately this phenomenon at the atomic scale. FIG. 3 . 3Segregation isotherms in the disordered state for (a) the surface (p = 1) and the first two sublayers (p = 2, 3) and (b) the (100) and (111) surfaces for the truncated octahedron Cu-Au nanoalloys containing 4033 atoms. Fig. 3b . 3bInterestingly, there is an enhancement of Au enrichment when going from close-packed (111) facet to the open (100) one in agreement with usual surface energy arguments FIG. 4 . 4Cross-section views of a characteristic equilibrium configuration and concentration profiles along the radius of a NP containing 4033 atoms after performing MC simulations in the disordered state. (a) CuAu3 and (b) Cu3Au. FIG. 5 . 5HRSTEM images of AuCu3 NPs synthesized (a) without epitaxial growth and (b) from epitaxial growth. FIG. 6 . 6Schematic representation of the EDX measurement. FIG. 7 . 7EDX spectra acquired on atomic columns belonging to the 1 st , 3 rd and 5 th atomic planes (from top to bottom) from the surface along the line perpendicular to the NP surface (red line on the image of the inset): (a) Cu15Au85 and (b) Cu70Au30. . Parameters of the interatomic potentials used for the Cu-Cu, Au-Au and Cu-Au interactions. . 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[ "ON THE PRESCRIBING σ 2 CURVATURE EQUATION ON S 4", "ON THE PRESCRIBING σ 2 CURVATURE EQUATION ON S 4" ]
[ "S.-Y Alice Chang ", "Zheng-Chao Han ", "Paul Yang " ]
[]
[]
Prescribing σ k curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function K to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the σ 2 curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of σ 2 curvature equations deforming K to a positive constant under the same non-degeneracy condition on K, and prove the existence of a solution using degree argument to this deformation involving fully nonlinear elliptic operators under an additional, natural degree condition on a finite dimensional map associated with K.
null
[ "https://arxiv.org/pdf/0911.0375v2.pdf" ]
115,161,762
0911.0375
da0a7dfb60c61560672882c87ad878d9983c69e6
ON THE PRESCRIBING σ 2 CURVATURE EQUATION ON S 4 24 Nov 2009 S.-Y Alice Chang Zheng-Chao Han Paul Yang ON THE PRESCRIBING σ 2 CURVATURE EQUATION ON S 4 24 Nov 2009 Prescribing σ k curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function K to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the σ 2 curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of σ 2 curvature equations deforming K to a positive constant under the same non-degeneracy condition on K, and prove the existence of a solution using degree argument to this deformation involving fully nonlinear elliptic operators under an additional, natural degree condition on a finite dimensional map associated with K. Description of main results Our main results in this paper are (potential) blow up profile analysis, a priori estimates and existence of admissible solutions w to the σ 2 curvature equation (1) σ 2 (g −1 • A g ) = K(x), on S 4 , where g = e 2w(x) g c is a metric conformal to g c , with g c being the canonical background metric on the round sphere S 4 , A g is the the Weyl-Schouten tensor of the metric g, A g = 1 n − 2 {Ric − R 2(n − 1) g} = A gc − ∇ 2 w − dw ⊗ dw + 1 2 |∇w| 2 g c ,(2) and σ k (Λ), for any 1−1 tensor Λ on an n−dimensional vector space and k ∈ N, 0 ≤ k ≤ n, is the k-th elementary symmetric function of the eigenvalues of Λ; K(x) is a given function The research of the first and third author is partially supported by NSF through grant DMS-0758601; the first author also gratefully acknowledges partial support from the Minerva Research Foundation and The Charles Simonyi Endowment fund during the academic year 08-09 while visiting Institute of Advanced Study, Princeton. The research of the second author is partially supported by NSF through grant DMS-0103888 and by a Rutgers University Research Council Grant(202132). on S 4 with some appropriate assumptions, and an admissible solution is defined to be a C 2 (M) solution w to (1) such that for all x ∈ S 4 , A g (x) ∈ Γ + k , namely, σ j (g −1 • A g ) > 0 for 1 ≤ j ≤ k. Note that σ 1 (A g ) is simply a positive constant multiple of the scalar curvature of g, so A g in the Γ + k class is a generalization of the notion that the scalar curvature R g of g having a fixed + sign. Note that, since σ 2 (g −1 • A g ) = e −4w σ 2 (g −1 c • A g ), so (1) is equivalent to (3) σ 2 (g −1 c • A gc − ∇ 2 w + dw ⊗ dw − 1 2 |∇w| 2 g c ) = K(x)e 4w(x) . It is well known that (3) is elliptic at an admissible solution; and in fact, any solution w to (3) on S 4 is admissible. There have been a large number of papers on problems related to the σ k curvature since the work [V00a] of Viaclovsky a decade ago. It is inadequate to do even a short survey of recent work in the introductory remarks here. We will instead refer the reader to recent surveys [V06] by Viaclovsky and [CC07] by Chang and Chen. As alluded to above, a similar problem to (3) for the σ 1 curvature was a predecessor to (3). More specifically, if we prescribe a function K(x) on the round n-dimensional sphere (S n , g c ) to be the scalar curvature of a metric g = e 2w g c pointwise conformal to g c , then w satisfies (4) 2(n − 1)∆ gc w + (n − 1)(n − 2)|∇w| 2 = R gc − K(x)e 2w . Similar equation can be formulated for a general manifold. One difference between (4) and (3) is that (4) is semilinear in w, while (3) is fully nonlinear in w. (4) takes on the familiar form (5) 2∆ gc w = R gc − K(x)e 2w , when n = 2, which is the Nirenberg problem. The n ≥ 3 case of (4) is often written in terms of a different variable u = e (n−2)w/2 , which would render the equation in the familiar form (6) − 4 n − 1 n − 2 ∆ gc u + R gc u = K(x)u n+2 n−2 . The K(x) ≡ const. case of (6) on a general compact manifold is the famous Yamabe problem. (5) and (6) have attracted enormous attention in the last several decades. A large collection of phenomena on the possible behavior of solutions to these equations, and methods and techniques of attacking these problems have been accumulated, which have tremendously enriched our understanding in solving a large class of nonlinear (elliptic) PDEs, and provided guidance in attacking seemingly unrelated problems. It is impossible in the space here to provide even a partial list of references. Please see [K85], [LP87], [S91], [L98], [CY02], [KMS09], and the references therein to get a glimpse of the results and techniques in this area. Directly related to our current work are some work on the (potential) blow up analysis, a priori estimates, and existence of solutions to (5) or (6). It is proved in [H90] and [CGY93] that when K is a positive function on S 2 , a sequence of blowing up solutions to (5) has only one point blow up and has a well-defined blow up profile, and that when K is a positive C 2 function on S 2 such that ∆ gc K(x) = 0 at any of its critical points, no blow up can happen, more precisely, there is an a priori bound on the set of solutions to (5) which depends on the C 2 norm of K, the positive lower bound of K and |∆ gc K(x)| near the critical points of K, and the modulus of continuity of the second derivatives of K. Similar results for (6) in the case n = 3 were proved in [Z90] and [CGY93], and for (6) in the case n ≥ 4 in [L95] under a flatness condition of K at its critical points. When K is a Morse function, say, this flatness condition fails when n ≥ 4. In fact it is proved in [L96] that in such cases on S 4 , there can be a sequence of solutions to (6) blowing up at more than one points. Later on [CL99] constructed solutions blowing up on S n , n ≥ 7, with unbounded layers of "energy concentration" for certain non-degenerate K. A natural question concerning the σ k curvature equations such as (3) is: which kind of behavior does its solution exhibit? In the following we will often transform (3) through a conformal automorphism ϕ of S 4 as follows. Let |dϕ(P )| denote the factor such that |dϕ(P )[X]| = |dϕ(P )||X| for any tangent vector X ∈ T P (S 4 ), and (7) w ϕ (P ) = w • ϕ(P ) + ln |dϕ(P )|. Then w is a solution to (3) iff w ϕ is a solution to (8) σ 2 (g −1 c • A gc − ∇ 2 w ϕ + dw ϕ ⊗ dw ϕ − 1 2 |∇w ϕ | 2 g c ) = K • ϕ(x)e 4wϕ(x) . Our first results show that solutions to (3) on S 4 exhibits similar behavior as those to (5) on S 2 or (6) on S 3 . Theorem 1. Consider a family of admissible conformal metrics g j = e 2w j g c on S 4 with σ 2 (g −1 j • A g j ) = K(x), where g c denotes the canonical round metric on S 4 and K(x) denotes a C 2 positive function on S 4 . Then there exists at most one isolated simple blow up point in the sense that, if max w j = w j (P j ) → ∞, then there exists conformal automorphism ϕ j of S 4 such that, if we define v j (P ) = w j • ϕ j (P ) + ln |dϕ j (P )|, we have (9) v j (P ) − 1 4 ln 6 K(P j ) → 0 in L ∞ (S 4 ), and(10)S 4 |∇v j | 4 → 0. In fact, we have the stronger conclusion that the W 2,6 norm of v j stays bounded and v j − 1 4 ln 6 K(P j ) → 0 in C 1,α (S 4 ) for any 0 < α < 1/3. We also have Theorem 2. Let K(x) be a C 2 positive function on S 4 satisfying a non-degeneracy condition (11) ∆K(P ) = 0 whenever ∇K(P ) = 0, and we consider solutions w(x) to (3), with K(x) replaced by K [s] (x) := (1 − s)6 + sK(x), for 0 < s ≤ 1, namely, (3 ′ ) σ 2 (g −1 c • A gc − ∇ 2 w + dw ⊗ dw − 1 2 |∇w| 2 g c ) = K [s] (x)e 4w(x) . Then there exist a priori C 2,α estimates on w, uniform in 0 < s ≤ 1, which depend on the C 2 norm of K, the modulus of continuity of ∇ 2 K, positive lower bound of min K and positive lower bound of |∆K(x)| in a neighborhood of the critical points of K. Remark 1. Theorems 1 and 2, with the uniform estimates in Theorem 2 for solutions to (3 ′ ) only for 0 < s 0 ≤ s ≤ 1 and the estimates possibly depending on 0 < s 0 < 1, were obtained several years ago and were announced in [H04]. The details were written up in [CHY04] and presented by the second author on several occasions, including at the 2006 Banff workshop "Geometric and Nonlinear Analysis". The current work can be considered as a completion of [CHY04]. As mentioned above similar statements for (5) and (6) were obtained earlier in [H90], [CGY93], [L95], [Z90], among others. When applying these estimates to the corresponding equation such as (5) and (6) with K(x) replaced by K [s] (x), all previous work stated and proved that the a priori estimates on the solutions remain uniform as long as 0 < s 0 ≤ s ≤ 1, for any fixed s 0 . This stems from the dependence of the a priori estimates on a positive lower bound of |∆K(x)| near the critical points of K, among other things. Since ∆K [s] = s∆K(x) becomes small when s > 0 is small, previous work in this area assumed that the a priori estimates could deteriorate as s > 0 becomes small. In these previous work, one has to devise a way to study the problem when s > 0 becomes small. [CGY93] and [L95] used some kind of "center of mass" analysis via conformal transformations of the round sphere. Technically [CGY93] and [L95] used a constrained variational problem to study the "centered problem". In essence the success of these methods was due to the semilinear nature of the relevant equations, so one could still have control on the "centered solution" in some norm weaker than C 2,α norm, say, W 2,p norm, when s > 0 is small, and used these estimates to prove existence of solutions under natural geometric/topological assumptions on K. This approach was problematic for our fully nonlinear equation (3). Due to this difficulty, until recently we have not been successful in using our preliminary version of Theorem 2 (for estimates in the range 0 < s 0 ≤ s ≤ s which may depend on s 0 ) and the deformation K [s] above to the equation to establish solutions to (3), under natural geometric/topological assumptions on K. It was our recent realization that in our setting, as well as in those of [CGY93] and [L95], the a priori estimates of solutions, under conditions like those in Theorem 2, remain uniform for all 1 ≥ s > 0! After we completed this work and were compiling the bibliography for this paper, we noticed that M. Ji made a similar observation in her work on (5) in [J04]. This uniform a prioir estimates for all 1 ≥ s > 0 leads to our next Theorem. Theorem 3. Suppose K(x) is a C 2 positive function on S 4 satisfying (11). Then the map G(P, t) = |S 4 | −1 S 4 K • ϕ P,t (x)x dvol gc ∈ R 5 does not have a zero for (P, t) ∈ S 4 × [t 1 , ∞), for t 1 large. Furthermore, consider G as a map defined on (t − 1)P/t ∈ B r (O) for r > (t − 1)/t and if In particular, if K has only isolated critical points in the region {x ∈ S 4 : ∆K(x) < 0} and ind(∇K(x)) = 1, where ind(∇K(x)) stands for the index of the vector field ∇K(x) at its isolated zero x, then (12) holds, therefore, (3) has a solution. A corollary of the proof for the W 2,p estimate in Theorem 1 is a bound on a functional determinant whose critical points are solutions to (3). We recall that the relevant functional determinant is defined, similar to [CGY02a], through II[w] = S 4 (∆ 0 w) 2 + 2|∇ 0 w| 2 + 12w dvol gc , C K [w] = 3 log S 4 Ke 4w dvol gc , and Y [w] = 1 36 S 4 R 2 dvol gc − S 4 R 2 0 dvol gc = S 4 ∆ 0 w + |∇ 0 w| 2 2 dvol gc − 4 S 4 |∇ 0 w| 2 dvol gc . F [w] = Y [w] − II[w] + C K [w] is the relevant functional determinant and a critical point of F [w] is a solution of (3). It is known that, for any conformal transformation ϕ of (S 4 , g c ), Y [w ϕ ] = Y [w], and II[w ϕ ] = II[w]. There is a similar functional determinant and a variational characterization for solutions to the prescribing Gaussian curvature problem on S 2 . Chang, Gursky and Yang proved in [CGY93] that this functional is bounded on the set of solutions to the prescribing Gaussian curvature problem on S 2 for any positive function K on S 2 to be prescribed. Our corollary is in the same spirit. Theorem 1 will be established using blow up analysis, Liouville type classification results of entire solutions, and integral type estimates for such fully nonlinear equations from [CGY02a] and [H04]. Theorem 2 will be established using a weaker version of Theorem 1 and a Kazdan-Warner type identity satisfied by the solutions. The weaker version of Theorem 1 only needs to establish (9 ′ ) v j (P ) − 1 4 6 K(P j ) → 0 pointwise on S 4 \ {−P j }, and bounded in L ∞ (S 4 ), instead of (9), (10) and the W 2,6 norm estimates. A degree argument for a fully nonlinear operator associated with (3) and Theorem 2 will be used to establish Theorem 3. To streamline our presentation, we will first outline the main steps for proving Theorem 3, assuming Theorem 2 and all the other needed ingredients. In the remaining sections, we will first provide a proof for (9 ′ ) in Theorem 1 and for Theorem 2, before finally providing a proof for the W 2,6 norm estimates in Theorem 1 and for Corollary 1. Proof of Theorem 3 The first and third parts of Theorem 3 is contained in [CY91] and [CGY93]. We will establish the second part of Theorem 3 by formulating the existence of a solution to (3) as a degree problem for a nonlinear map and linking the degree of this map to that of G. By a fibration result from [CY87], [CY91], [CL93] and [L95], see also [A79], [O82] for early genesis of these ideas, if we define S 0 = {v ∈ C 2,α (S 4 ) : S 4 e 4v(x) x dvol gc = 0}, then the map π : (v, ξ) ∈ S 0 × B → C 2,α (S 4 ) defined by π(v, ξ) = v • ϕ −1 P,t + ln |dϕ −1 P,t |, with B denoting the open unit ball in R 5 and ξ = rP , P ∈ S 4 , r = (t − 1)/t, t ≥ 1, is a C 2 diffeomorphism from S 0 × B onto C 2,α (S 4 ). Thus (v, P, t) ∈ S 0 × S 4 × [1, ∞) provide global coordinates for C 2,α (S 4 ) (with a coordinate singularity at t = 1, similar to the coordinate singularity of polar coordinates at r = 0) through w = v • ϕ −1 P,t + ln |dϕ −1 P,t |. w solves (3) with K replaced by K [s] iff v solves (13) σ 2 (A v ) = K [s] • ϕ P,t e 4v . Then the estimates for w in Theorem 2 turn into the following estimates for v and t. Proposition 1. Assume that K is a positive C 2 function on S 4 satisfying the nondegeneracy condition (11), and let w be a solution to (3) with K replaced by K [s] and (v, P, t) ∈ S 0 × S 4 × [1, ∞) be the coordinates of w defined in the paragraph above. Then there exist t 0 and ǫ(s) > 0 with lim s→0 ǫ(s) = 0, such that (14) t ≤ t 0 and ||v|| C 2,α (S 4 ) < ǫ(s). A proof for Proposition 1 will be postponed to the end of the next section. We treat (13) as a nonlinear map F [s] [v, ξ] := e −4v(x) σ 2 (g −1 c • A gc − ∇ 2 v + dv ⊗ dv − 1 2 |∇v| 2 g c ) − K [s] • ϕ P,t , from S 0 × B into C α (S 4 ), for 0 < s ≤ 1, where ξ = (t − 1)P/t ∈ B. Proposition 1 implies that there is a neighborhood N ⊂ S 0 of 0 ∈ S 0 and 0 < r 0 = (t 0 − 1)/t 0 < 1 such that F [s] does not have a zero on ∂(N × B r ) for all r 0 ≤ r < 1 and 0 < s ≤ 1. According to [L89], there is a well defined degree for F [s] on N × B r 0 and it is independent of 0 < s ≤ 1. We will compute this degree of F [s] , for s > 0 small, through the degree of a finite dimensional map. We first use the implicit function theorem to define this map and link the solutions to (13) to the zeros of this map. Note that F [0] [0, (t − 1)P/t] = 0 and D v F [0] [0, (t − 1)P/t](η) = −6∆η − 24η. If Π denotes the projection from C α (S 4 ) into Y := {f ∈ C α (S 4 ) : S 4 f x j dvol gc = 0 for j = 1, · · · , 5} defined by Π(f ) = f − 5|S 4 | −1 5 j=1 S 4 f x j dvol gc x j , then we can apply the implicit function theorem to Π • F [s] at v = 0 to conclude Proposition 2. There exist some neighborhood N ǫ ⊂ N of 0 ∈ S 0 and s 0 > 0, such that for all 0 < s < s 0 , (P, t) ∈ S 4 × [1, t 0 ], there exists a unique v = v(x; P, t, s) ∈ N ǫ , depending differentiably on (P, t, s) such that (15) Π • F [s] [v(x; P, t, s), (t − 1)P/t] = 0. Furthermore, there exists some C > 0 such that, for 0 < s ≤ s 0 , 1 ≤ t ≤ t 0 , (16) ||v(x; P, t, s)|| C 2,α (S 4 ) ≤ C||K [s] • ϕ P,t − 6|| C α (S 4 ) = Cs||K • ϕ P,t − 6|| C α (S 4 ) . (15) implies that F [s] [v(x; P, t, s), (t − 1)P/t] = 5 j=1 Λ j (P, t, s)x j , for some Lagrange multipliers Λ j (P, t, s), which depend differentiably on (P, t, s). Or, equivalently, (17) σ 2 (A v(x;P,t,s) ) = K [s] • ϕ P,t + 5 j=1 Λ j (P, t, s)x j e 4v(x;P,t,s) . A zero of the map Λ [s] (P, t) := (Λ 1 (P, t, s), · · · , Λ 5 (P, t, s)) corresponds to a solution to (13). Propositions 1 and 2 say that, for s 0 > 0 small, all solutions v ∈ S 0 to (13), for 0 < s ≤ s 0 , are in N ǫ , thus correspond to the zeros of the map Λ [s] (P, t). Remark 2. ||K • ϕ P,t − 6|| C α (S 4 ) could become unbounded when t → ∞; yet thanks to the bound 1 ≤ t ≤ t 0 from Proposition 1, it remains bounded in terms of ||K|| C α (S 4 ) in the range 1 ≤ t ≤ t 0 . Note also that ||K • ϕ P,t − 6|| L p (S 4 ) remains bounded in terms of ||K|| L ∞ (S 4 ) even in the range 1 ≤ t < ∞. It is essentially this bound and the applicability of W 2,p estimates in the semilinear setting of [CGY93] and [L95] which allowed them to handle their cases without using the bound 1 ≤ t ≤ t 0 . Remark 3. The implicit function theorem procedure here works also in the setting of [CY91], [CGY93] and [L95] using W 2,p space, as does Proposition 1 in the setting of [CGY93] and [L95], and can be used to simplify the arguments there. At this point, we need the following Kazdan-Warner type identity for solutions to (3). Proposition 3. Let w be a solution to (3). Then, for 1 ≤ j ≤ 5, (18) S 4 ∇K(x), ∇x j e 4w(x) dvol gc = 0. Proposition 3 is a special case of the results in [V00b] and [H06]. But in the special case of S 4 , it is a direct consequence of the variational characterization of the solution to (3), as given after the statement of Theorem 1. A solution w to (3) is a critical point of F [w] = Y [w] − II[w] + C K [w] there, thus satisfies, for any one-parameter family of conformal diffeomorphisms ϕ s of S 4 with φ 0 =Id, d ds s=0 F [w ϕs ] = 0, with w ϕs = w • ϕ s + log |dϕ s |. Since Y [w ϕs ] = Y [w] and II[w ϕs ] = II[w] , a solution w of (3) thus satisfies d ds s=0 C K [w ϕs ] = d ds s=0 S 4 K • ϕ −1 s e 4w dvol gc = 0, which is (18). Applying (18) to v(x; P, t, s), a solution to (17), we obtain, S 4 ∇ K [s] • ϕ P,t (x) + 5 j=1 Λ j (P, t, s)x j , ∇x k e 4v(x;P,t,s) dvol gc = 0, for 1 ≤ k ≤ 5, from which we obtain, for 1 ≤ k ≤ 5, − 5 j=1 Λ j (P, t, s) S 4 ∇x j , ∇x k e 4v(x;P,t,s) dvol gc = S 4 ∇ K [s] • ϕ P,t (x) , ∇x k e 4v(x;P,t,s) dvol gc =s S 4 ∇ (K • ϕ P,t (x)) , ∇x k e 4v(x;P,t,s) dvol gc . As in [CY91], [CGY93] and [L95], we define A [s] (P, t) = (4|S 4 |) −1 S 4 ∇ (K • ϕ P,t (x)) , ∇x e 4v(x;P,t,s) dvol gc ∈ R 5 . Since S 4 ∇x j , ∇x k e 4v(x;P,t,s) dvol gc is positive definite, we conclude that deg(A [s] , B r 0 , O) = − deg(Λ [s] , B r 0 , O), for s 0 > s > 0 provided that one of them is well defined. Using v(x; P, t, s) ∈ S 0 , and ∆x = −4x on S 4 , we have, as in [CY91], [CGY93] and [L95], A [s] (P, t) = G(P, t) + I + Π, where I = |S 4 | −1 S 4 (K • ϕ P,t (x) − K(P )) x e 4v(x;P,t,s) − 1 dvol gc , and Π = −(4|S 4 |) −1 S 4 (K • ϕ P,t (x) − K(P )) ∇x, ∇e 4v(x;P,t,s) dvol gc . We could have fixed t 0 ≥ t 1 such that G(P, t) = 0 for t = t 0 , and there will be a δ > 0 such that |G(P, t)| ≥ δ for t = t 0 . Since (16) implies that ||v(x; P, t, s)|| C 2,α (S 4 ) = O(s), uniformly for (P, t) ∈ S 4 × [1, t 0 ], we find that, by fixing s 0 > 0 small if necessary, |I| + |Π| ≤ 1 2 |G(P, t)|, for 0 < s ≤ s 0 and t = t 0 . This implies that A [s] (P, t) · G(P, t) > 0 for 0 < s ≤ s 0 and t = t 0 . Therefore − deg(Λ [s] , B r 0 , O) = deg(A [s] , B r 0 , O) = deg(G, B r 0 , O) = 0, for 0 < s ≤ s 0 . Finally, we now prove (19) deg(F [s] , N × B r 0 , O) = − deg(Λ [s] , B r 0 , O), for 0 < s ≤ s 0 , from which follows the existence of a solution to (3). The verification of (19) is routine, but requires several steps. First, we may perturb K, if necessary, within the class of functions satisfying the conditions in Theorem 3 such that the corresponding G(P, t) has only isolated and non-degenerate zeros in B r 0 (O). We will prove momentarily that for s > 0 small, the zeros of Λ [s] for s > 0 small will be close to the zeros of G(P, t) and are isolated, non-degenerate. Therefore the zeros of F [s] in N × B r 0 are isolated and non-degenerate. This can be argued as follows. First, it follows from (17) that Λ [s] (ξ) · x = (Id − Π) e −4v(x;ξ,s) σ 2 (A v(x;ξ,s) ) − K [s] • ϕ P,t . Using (16), we can then write e −4v(x;ξ,s) σ 2 (A v(x;ξ,s) ) = 6 − 6∆v(x; ξ, s) − 24v(x; ξ, s) + Q(v(x; ξ, s)), with ||Q(v(x; ξ, s))|| Y ||v(x; ξ, s)|| 2 X s 2 . Therefore, using (Id − Π)(1) = (Id − Π) (6∆v(x; ξ, s) + 24v(x; ξ, s)) = 0, and Λ [s] (ξ) = 5|S 4 | −1 S 4 Λ [s] (ξ) · x x dvol gc , we have Λ [s] (ξ) = −5sG(P, t) + 5|S 4 | −1 S 4 [(Id − Π) (Q(v(x; ξ, s)))] x dvol gc , with |(Id − Π) (Q(v(x; ξ, s))) | s 2 , so the zeros of Λ [s] (ξ) for s > 0 small are close to the zeros of G(P, t). We can further use the implicit function theorem to prove that for s > 0 small there is a (unique) non-degenerate zero of Λ [s] (ξ) near each zero of G(P, t). Remark 4. This argument shows that, for each non-degenerate zero of G(P, t), if we associate (P, t) with the center of mass of ϕ P,t , C.M(ϕ P,t ) := |S 4 | −1 S 4 ϕ P,t (x) dvol gc ∈ B 1 (O) as a geometric representation of (P, t), then for s > 0 small, there is a unique solution w to (3) whose center of mass approaches C.M(ϕ P,t ), for our argument gives rise to a solution w(x) = v(·; P ′ , t ′ ) • ϕ −1 P ′ ,t ′ (x) + ln |dϕ −1 P ′ ,t ′ (x)| with (P ′ , t ′ ) approaching (P, t), and v(x; P ′ , t ′ ) approaching 0 as s → 0, thus the center of mass of w is |S 4 | −1 S 4 e 4w(x) x dvol gc = |S 4 | −1 S 4 e 4v(y) ϕ P ′ ,t ′ (y) dvol gc → C.M(ϕ P,t ) as s → 0. Now deg(F [s] , N × B r 0 , O) is well defined in the manner of [L89], and according to Propositions 2.1-2.4 of [L89], To compute DF [s] [v(x; ξ, s), ξ], we identify ξ ∈ R 5 with ξ · x ∈ span{x 1 , · · · , x 5 }, and deg(F [s] , N × B r 0 , O) = ξ∈Br 0 (O):Λ [s] (ξ)=0 ind(DF [s] [v(x; ξ, s), ξ]),write the differential of F [s] in the direction ofv as D v F [s] [v(x; ξ, s), ξ](v), or simply D v F [s] (v), and the differential of F [s] in the direction ofξ · x as D ξ F [s] [v(x; ξ, s), ξ](ξ). Then D ξ F [s] [v(x; ξ, s), ξ](ξ) = −sξ · ∇ ξ (K • ϕ P,t ) , and D v F [s] [v(x; ξ, s), ξ](v) = M ij [v(x; ξ, s)]∇ v(x;ξ,s) ijv − 4K [s] • ϕ P,tv , where M ij [v(x; ξ, s)] stands for the Newton tensor associated with σ 2 (e −2v(x;ξ,s) A v(x;ξ,s) ), and ∇ v(x;ξ,s) ij stands for the covariant differentiation in the metric e 2v(x;ξ,s) g c . Thus, DF [s] [v(x; ξ, s), ξ](v +ξ · x) = M ij [v(x; ξ, s)]∇ v(x;ξ,s) ijv − 4K [s] • ϕ P,tv − sξ · ∇ ξ (K • ϕ P,t ) . At a fixed zero ξ of Λ [s] (ξ) = 0, we define a family of deformed linear operators L τ,s for 0 ≤ τ ≤ 1 by L τ,s (v +ξ · x) = M ij [v [τ ] ]∇ v [τ ] ijv − 4K [sτ ] • ϕ P,tv − sξ · ∇ ξ (K • ϕ P,t ) , where v [τ ] = τ v(x; ξ, s), andv ∈ X := {v ∈ C 2,α (S 4 ) : S 4v (x)x j = 0, j = 1, · · · , 5}. Then L τ,s defines self-adjoint operators with respect to the metric e 2v [τ ] g c , thus its eigenvalues are all real. We first assume the Claim. For s > 0 small and 0 ≤ τ ≤ 1, the spectrum of L τ,s does not contain zero. L 0,s (v +ξ · x) = −6∆v − 24v − sξ · ∇ ξ (K • ϕ P,t ) , so ifv +ξ · x is an eigenfunction corresponding to a negative eigenvalue −λ, withv ∈ X, then −6∆v − 24v − sξ · ∇ ξ (K • ϕ P,t ) = −λ(v +ξ · x). Taking projection in span{x 1 , · · · , x 5 }, we find −s∇G(P, t)ξ = − λ 5ξ , and taking projection in X, we find (22) − 6∆v − 24v − sΠ ξ · ∇ ξ (K • ϕ P,t ) = −λv. Ifξ = 0, thenξ is an eigenvector of ∇G(P, t) with eigenvalue λ 5s > 0; and ifξ = 0, theṅ v = 0 solves −6∆v − 24v = −λv, which is possible for some λ > 0 iff −λ = −24 anḋ v = constant. Conversely, for any eigenvectorξ = 0 of ∇G(P, t) with eigenvalue µ > 0, the operator −6∆ − 24 + 5sµ is an isomorphism from X to Y for s > 0 small, so we can solve (22) as (−6∆ − 24 + 5sµ)v = sΠ ξ · ∇ ξ (K • ϕ P,t ) forv ∈ X andv +ξ · x becomes an eigenfunction of L 0,s with eigenvalue −5sµ. Therefore we conclude (20). We now establish (21) to prove ind(DF [s] [v(x; ξ, s), ξ]) = −ind(D ξ Λ [s] (ξ)) for s > 0 small. From (17), which can be written as Λ [s] (ξ) · x = F [s] [v(x; ξ, s), ξ]), we obtain D v F [s] (D ξ v(x; ξ, s)(ξ)) + D ξ F [s] (ξ) = D ξ Λ [s] (ξ)(ξ) · x. Taking projections in X and span{x 1 , · · · , x 5 }, respectively, and using D ξ F [s] (ξ) = −sξ · ∇ ξ (K • ϕ P,t ), we obtain (23) Π D v F [s] (D ξ v(x; ξ, s)(ξ)) − sΠ ξ · ∇ ξ (K • ϕ P,t ) = 0, and(24)S 4 (Id − Π) D v F [s] (D ξ v(x; ξ, s)(ξ)) x dvol gc − s∇ ξ G(P, t)ξ = 1 5 D ξ Λ [s] (ξ)(ξ). Writing D v F [s] (D ξ v(x; ξ, s)(ξ)) = (−6∆ − 24)(D ξ v(x; ξ, s)(ξ)) + Θ(D ξ v(x; ξ, s)(ξ)), we find, using (16), that ||Θ(D ξ v(x; ξ, s)(ξ))|| Y s||D ξ v(x; ξ, s)(ξ)|| X . Thus Π D v F [s] (·) : X → Y is an isomorphism for s > 0 small and has an inverse Ψ, and we can solve D ξ v(x; ξ, s)(ξ) in terms ofξ from (23): D ξ v(x; ξ, s)(ξ) = Ψ sΠ ξ · ∇ ξ (K • ϕ P,t ) := sΥ(ξ). Using this in (24), we find 1 5 D ξ Λ [s] (ξ) = −s∇ ξ G(P, t) + s S 4 [(Id − Π) • Θ • Υ] x dvol gc = −s (∇ ξ G(P, t) + O(s)) . Thus for s > 0 small, γ matches the number of negative eigenvalues of D ξ Λ [s] (ξ), and we can conclude that ind(DF [s] (v(x; ξ, s), ξ)) = −ind(D ξ Λ [s] (ξ)). In the remainder of this section, we provide proof for our Claim above, leaving the proof for Proposition 1 to the end of the next section. Proof of Claim. Suppose that for (a sequence of) s > 0 small and some 0 ≤ τ ≤ 1, L τ,s hasv +ξ · x, withv ∈ X,ξ ∈ R 5 , as eigenfunction with zero eigenvalue. Then, taking projections in span{x 1 , · · · , x 5 } and X, respectively, we obtain (25) Π M ij [v [τ ] ]∇ v [τ ] ijv − 4K [sτ ] • ϕ P,tv − sΠ ξ · ∇ ξ (K • ϕ P,t ) = 0, and(26)S 4 (Id − Π) M ij [v [τ ] ]∇ v [τ ] ijv − 4K [sτ ] • ϕ P,tv x dvol gc − s∇ ξ G(P, t)ξ = 0. Using (16) again, we find M ij [v [τ ] ]∇ v [τ ] ijv − 4K [sτ ] • ϕ P,tv = −6∆v − 24v + Θ τ,s (v), with ||Θ τ,s (v)|| Y sτ ||v|| X . Thus, for s > 0 small, we can solvev from (25) to obtaiṅ v = Ψ sΠ ξ · ∇ ξ (K • ϕ P,t ) = sΥ(ξ). Thusξ = 0 and we can normalize it so that |ξ| = 1. Using this in (26), we find s(Id − Π) • Θ τ,s • Υ(ξ) − s∇ ξ G(P, t)ξ = 0. Using ||Θ τ,s || sτ , we find this impossible for s > 0 small under our non-degeneracy assumption on the zeros of G(P, t). 3. Proof of (9 ′ ), (10), Theorem 2 and Proposition 1 Proof of (9 ′ ) and (10). The full strength of (9) is established as soon as the W 2,3 estimates are established -the latter is a step in proving the W 2,6 estimates. If there is a sequence of solutions w j to (3) such that max w j = w j (P j ) → ∞, then we choose conformal automorphism φ j = φ P j ,t j of S 4 , such that the rescaled function (27) v j (P ) = w j • φ j (P ) + ln |dφ j (P )|, satisfies the normalization condition (28) v j (P j ) = 1 4 ln 6 K(P j ) . If we use stereographic coordinates for S 4 , with P j as the north pole, then y (φ j (P )) = t j y(P ), for P ∈ S 4 , and v j (P ) = w j • φ j (P ) + ln t j (1 + |y(P )| 2 ) 1 + t 2 j |y(P )| 2 . v j would satisfy (29) σ 2 (A v j ) = K • φ j e 4v j . The normalization in (28) amounts to choosing t j such that w j (P j ) − ln t j = 1 4 ln 6 K(P j ) . Thus, t j → ∞, and for any P ∈ S 4 , (30) v j (P ) ≤ 1 4 ln 6 K(P j ) + ln t 2 j (1 + |y(P )| 2 ) 1 + t 2 j |y(P )| 2 . (30) implies that, away from −P j , v j has an upper bound independent of j. Together with (29), the local gradient and higher derivative estimates of [GW03], there exists a subsequence, still denoted as {v j }, such that, P j → P * , and for any δ > 0, (31) v j → v ∞ in C 2,α S 4 \ B δ (−P * ) , for some limit v ∞ . We also have σ 2 (A v∞ ) = K(P * )e 4v∞ on S 4 \ {−P * }, (32) S 4 K(P * )e 4v∞ dvol gc ≤ lim inf j→∞ S 4 K • φ j e 4v j dvol gc = 16π 2 , (33) v ∞ (P * ) = 1 4 ln 6 K(P * ) , ∇v ∞ (P * ) = 0, (34) v ∞ (P ) ≤ 1 4 ln 6 K(P * ) + ln 1 + |y(P )| 2 |y(P )| 2 . A Liouville type classification result in [CGY02b] and [LL03] says that v ∞ − 1 4 ln 6 K(P * ) = ln |dφ| for some conformal automorphism φ of S 4 , which together with (34) implies that (36) v ∞ ≡ 1 4 ln 6 K(P * ) . Thus for any δ > 0, lim j→∞ S 4 \B δ (−P * ) K • φ j e 4v j dvol gc = 6 S 4 \ B δ (−P * ) . Together with the Gauss-Bonnet formula S 4 K • φ j e 4v j dvol gc = 6 S 4 , we have lim j→∞ B δ (−P * ) K • φ j e 4v j dvol gc = 6 |B δ (−P * )| . This allows us to apply our Theorem in [H04] on B δ (−P * ) for small δ > 0 to conclude that ∃C > 0, such that (37) max S 4 v j ≤ C. Next we declare the Claim. There exists C ′ > 0 such that (38) min S 4 v j ≥ −C ′ . The Claim can be proved making use of the information that R v j = R w j • φ j ≥ 0, which implies (39) 2 − ∆v j − |∇v j | 2 ≥ 0. Thus v j (P ) −v j = S 4 (−∆v j (Q)) G(P, Q) dvol gc (Q) ≥ −2 S 4 G(P, Q) dvol gc (Q),(40) where G(P, Q) is the Green's function of −∆ on S 4 . Integrating (39) over S 4 implies that (41) 2 ≥ S 4 |∇v j | 2 ≥ const. S 4 |v j (P ) −v j | 4 1 2 . (31), (36), (40), and (41) conclude the Claim and (9 ′ ). Next we prove the integral estimate (10). This can be seen by looking at the integral version of the equation Next, we prove Theorem 2. We will first prove that, under our non-degeneracy conditions on K, there is a bound C > 0 depending on the quantities as in the statements of Theorem 2, but uniform in 0 < s ≤ 1, such that any solution w of (3) with K [s] satisfies max S 4 w j ≤ C. Once we have the bound max S 4 w j ≤ C, the C 2,α estimates follow from known theory of fully nonlinear elliptic equations. S 4 2|∇v j | 2 + 2∆v j − 6 ∇v j , ∇η + ∆η|∇v j | 2 + K • φ j e 4v j − 6 η = 0. If we plug in η = v j , we obtain S 4 6 − 2|∇v j | 2 − 3∆v j |∇v j | 2 = S 4 K • φ j e 4v j − 6 v j . Using ∆v j ≤ 2 − |∇v j | 2 , we have (42) S 4 |∇v j | 4 ≤ S 4 K • φ j e 4v j − 6 v j , Proof of Theorem 2. Suppose, on the contrary, that max S 4 w j → ∞ (for a sequence of K's, which we write as a single K for simplicity, satisfying the bounds in Theorem 2). Then, as proved above, (9 ′ ) holds. Let P j , t j be as defined in the earlier part of the proof. We will then prove the following estimates: (43) |∇K(P j )| = o(1) t j , as j → ∞, and (44) ∆K(P j ) → 0, as j → ∞. (43) and (44) would contradict our hypotheses on K. The main idea is to examine the Kazdan-Warner identity in the light of the asymptotic profile of w j as given in Theorem 1. For each w j , we choose stereographic coordinates with P j as the north pole. For P = (x 1 , · · · , x 5 ) ∈ S 4 , let its stereographic coordinates be y = (y 1 , · · · , y 4 ). Also set x ′ = (x 1 , · · · , x 4 ). Then (45)              x i = 2y i 1 + |y| 2 , i = 1, 2, 3, 4, x 5 = |y| 2 − 1 |y| 2 + 1 . For any ǫ > 0, there exists M > 0 such that for any P with |y(P )| > M, we have (46) K(P ) = K(P j ) + 4 i=1 a i x i + 4 k,h=1 b hk x h x k + r(P ), with (47) |r(P )| ≤ ǫ|x ′ | 2 , |x ′ ||∇r(P )| ≤ ǫ|x ′ | 2 , |x ′ | 2 |∇ 2 r(Q)| ≤ ǫ|x ′ | 2 . We can identify a i = ∇ i K(P j ), b hk = ∇ hk K(P j ), and we may assume that b hk is diagonalized: b hk = δ hk b h . Then, using ∇x 1 = (1 − x 2 1 , −x 1 x 2 , · · · , −x 1 x 5 ), · · · , ∇x 5 = (−x 5 x 1 , · · · , −x 5 x 4 , 1 − x 2 5 ), and ∇x i · ∇x h = δ ih − x i x h , we have, for 1 ≤ h ≤ 4, ∇K, ∇x h = a h − 4 i=1 a i x i x h + 2b h x h − 2 4 i=1 b i x 2 i x h + ∇r · ∇x h . So we can fix M large such that, when |y| > M, Remark 5. It is this property that K, not K [s] , can be used in the Kazdan-Warner identity that allows us to obtain bounds on w uniform in 0 < s ≤ 1. This also applies to the settings in [CGY93] and [L95] to make the estimates there uniform in 0 < s ≤ 1 in the respective deformations.      ∇K, ∇x h = a h + 2b h x h + r 1 (P ),(48)|r 1 (P )| ≤ ǫ|x ′ |. We estimate |y|≤M ∇K, ∇x h e 4w j dvol gc ≤ C |y|≤M e 4w j 2 1 + |y| 2 4 d y = C |z|≤ M t j e 4v j 2 1 + |z| 2 4 d z ≤ C M t j 4 , using (37) and (38). |y|>M ∇K, ∇x h e 4w j dvol gc = a h |y|>M e 4w j dvol gc +2b h |y|>M e 4w j x h dvol gc + |y|>M e 4w j r 1 (P ) dvol gc . The following estimates will complete the proof of (43). lim j→∞ |y|>M e 4w j dvol gc = 6 K(P * ) S 4 . (49) |y|>M e 4w j x h dvol gc = o(1) t j , as j → ∞ (1 ≤ h ≤ 4). (50) |y|>M e 4w j r 1 (P ) dvol gc = o(1) t j , as j → ∞.(51) Here are the verifications of the above estimates. 2t j z h 1 + t 2 j |z| 2 e 4v j 2 1 + |z| 2 4 d z = |z|> M t j 2t j z h 1 + t 2 j |z| 2 e 4v j − 6 K(P * ) 2 1 + |z| 2 4 d z = |z|>δ + δ>|z|> M t j , with δ>|z|> M t j ≤ C δ>|z|> M t j 1 t j |z| d z ≤ Cδ 3 t j . For any given ǫ > 0, we can first fix δ > 0 such that Cδ 3 < ǫ. Then using the convergence of v j to 1 4 ln 6 K(P * ) on |z| > δ, we can fix J such that when j ≥ J, we have e 4v j − 6 K(P * ) < ǫ. Then |z|>δ ≤ ǫ |z|>δ 1 t j |z| 2 1 + |z| 2 4 d z ≤ Cǫ t j . These together prove the second estimate above. (51) follows similarly. Finally ∇K, ∇x 5 = − 4 i=1 a i x i x 5 − 2 4 i=1 b i x 2 i x 5 + ∇r · ∇x 5 . We may fix M large so that |∇r · ∇x 5 | ≤ ǫ|x ′ | 3 when |y| > M. In To put things together, we multiply (52) by t 2 j and use (53), (54), (55), and (57) to see that 0 = o(1) − 2∆K(P j ) |S 3 | ∞ 0 ( 2 1 + r 2 ) 4 rd r + o(1) + o(1) t j , which shows (44). Proof of Proposition 1. First, by Theorem 2, there is a C > 0 depending on K and 0 < α < 1 such that any solution w to (3) with K substituted by K [s] and 0 < s ≤ 1 satisfies (58) ||w|| C 2,α (S 4 ) < C. Since v = w • ϕ P,t + ln |dϕ P,t | is chosen such that S 4 e 4v(y) y dvol gc = 0, we obtain, in terms of w and (P, t), (59) 0 = S 4 e 4w(x) ϕ −1 P,t (x) dvol gc = S 4 e 4w(x) ϕ P,t −1 (x) dvol gc , Due to (58), there is a δ > 0 such that S 4 e 4w(x) ≥ δ. If there existed a sequence of solutions w j for which t j → ∞, we would have, computing in stereographic coordinates in which P j is placed at the north pole, ϕ P j ,t −1 j (x) → (0, · · · , 0, −1) except at x = P j , therefore, in view of (58), S 4 e 4w(x) ϕ P,t −1 (x) dvol gc → (0, · · · , 0, − S 4 e 4w(x) ) = 0, contradicting (59) above. This implies the existence of some t 0 such that t ≤ t 0 . Using this and (58) in the relation between w and v, we find an upper bound for ||v|| C 2,α (S 4 ) . Finally using the equation for v: σ 2 (A v ) = K [s] • ϕ P,t e 4v , in which the right hand side has an upper bound in C 2,α (S 4 ) due to C 2,α (S 4 ) estimates of v and the bound t ≤ t 0 , we find higher derivative bounds for v. Then as s → 0, a subsequence of v would converge to a limit v ∞ in C 2,α (S 4 ), which satisfies This implies that v ∞ ≡ 0. Since this limit v ∞ is unique, we obtain that v → 0 in C 2,α (S 4 ) as s → 0, which is the remaining part of (14). 4. Proof of the W 2,6 estimates of Theorem 1 and of Corollary 1 For the W 2,6 bound for v j , we write v for v j and σ 2 for σ 2 (e −2v j g −1 c • A v j ) = K • ϕ j , and adapt the argument for the W 2,p estimates in [CGY02a] of Chang-Gursky-Yang and push the argument to p = 6. We will first prove a W 2,3 estimate for v j , with the bound depending on an upper bound of σ 2 = K •ϕ j , a positive lower bound for σ 2 , and an upper bound for S 4 |∇ 0 (K •ϕ j )| 2 dvol gc . Then we will extend the W 2,3 estimate to W 2,6 estimate for the v j in terms of an upper bound of σ 2 = K •ϕ j , a positive lower bound for σ 2 , and an upper bound for S 4 |∇ 0 (K •ϕ j )| 4 dvol gc . Since S 4 |∇ 0 (K •ϕ j )| 4 dvol gc = S 4 |∇ 0 K| 4 dvol gc , we see that a bound for the W 2,3 norm of v j is given in terms of an upper bound of K, a positive lower bound for K, and an upper bound for S 4 |∇ 0 K| 4 . This will suffice for proving (9). Proof of the W 2,6 estimates of Theorem 1. First we list a few key ingredients for these W 2,p estimates, mostly adapted from [CGY02a]. As in [CGY02a] we explore two differential identities, which in the case of S 4 , are S ij ∇ 2 ij R = 6 trE 3 + R|E| 2 + 3∆σ 2 + 3(|∇E| 2 − |∇R| 2 12 ) ≥ 6 trE 3 + R 3 12 − 2σ 2 R + 3∆σ 2 − 3|∇σ 2 | 2 2σ 2 ,(60) following (5.10) of [CGY02a], with S ij = ∂σ 2 (A) ∂A ij = −R ij + 1 2 Rg, and S ij ∇ 2 ij |∇v| 2 = R 3 144 − trE 3 2 − σ 2 R 12 − R|∇v| 4 2 − 2S ij ∇ i |∇v| 2 ∇ j v + S ij ∇ l A • ij ∇ l v − 2e −2v S ij ∇ i v∇ j v + 2Re −2v |∇v| 2 + Re −4v 2 − ∇v, ∇σ 2 − 2σ 2 e −2v ,(61) following (5.44) of [CGY02a] and the fact that A 0 ij = g 0 ij in the case of S 4 . Here the differentiations are in the metric g. In (60) and (61) we used |E| 2 = R 2 12 − 2σ 2 and (62) |∇E| 2 − |∇R| 2 12 ≥ − |∇σ 2 | 2 2σ 2 . (62) can be proven as in (7.26) of [CGY02a], but can also be seen to be based on the general fact that {σ k } 1/k is concave in its argument as follows: set F (A ij ) = {σ k (A ij )} 1/k , then (63) S ij = ∂σ k ∂A ij = kF k−1 ∂F ∂A ij , and ∇σ k = S ij ∇A ij = kF k−1 ∂F ∂A ij ∇A ij . So ∇ l S ij = kF k−1 ∂ 2 F ∂A ij ∂A IJ ∇ l A IJ + k(k − 1)F k−2 ∂F ∂A ij ∂F ∂A IJ ∇ l A IJ . Thus l ∇ l S ij ∇ l A ij =kF k−1 ∂ 2 F ∂A ij ∂A IJ ∇ l A IJ ∇ l A ij + k(k − 1)F k−2 ∂F ∂A ij ∂F ∂A IJ ∇ l A IJ ∇ l A ij ≤ (k − 1)|∇σ k | 2 kσ k using concavity of F and (63).(64) In the case of 2k = n = 4, A ij = E ij + R 12 g ij , and S ij = R 4 g ij − E ij . So l ∇ l S ij ∇ l A ij = l { ∇ l R 4 g ij − ∇ l E ij }{∇ l E ij + ∇ l R 12 g ij } = |∇R| 2 12 − |∇E| 2 , and by (64) |∇R| 2 12 − |∇E| 2 ≤ |∇σ 2 | 2 2σ 2 . Because of S ij,j = 0, which is a consequence of Bianchi identity, we can use (60) and (61) to obtain 0 = S 4 S ij ∇ 2 ij (R + 12|∇v| 2 ) ≥ S 4 R 3 6 − 6R|∇v| 4 − 24S ij ∇ i |∇v| 2 ∇ j v + 12S ij ∇ l A • ij ∇ l v + 24Re −2v |∇v| 2 − 24e −2v S ij ∇ i v∇ j v − 12 ∇v, ∇σ 2 + (6e −4v − 2σ 2 )R − 24e −2v σ 2 − 3|∇σ 2 | 2 2σ 2 , from which we can estimate S 4 R 3 in terms of the other terms: S 4 R 3 6 ≤ S 4 6R|∇v| 4 + 24S ij ∇ i |∇v| 2 ∇ j v − 12S ij ∇ l A • ij ∇ l v − 24Re −2v |∇v| 2 + 24e −2v S ij ∇ i v∇ j v + 12 ∇v, ∇σ 2 − (6e −4v − 2σ 2 )R + 24σ 2 e −2v + 3|∇σ 2 | 2 2σ 2 .(65) The integrations are done in the g metric, but due to the L ∞ estimates on v, the integrals in g metric are comparable to those in g c . The terms that require careful treatments are S 4 S ij ∇ i |∇v| 2 ∇ j v and (66) S 4 R|∇v| 4 ≤ S 4 R 3 1/3 S 4 |∇v| 6 2/3 ≤ ǫ 3 S 4 R 3 + 2ǫ −1/2 3 S 4 |∇v| 6 . The term S 4 S ij ∇ i |∇v| 2 ∇ j v can be estimated as (5.53) in [CGY02a] S 4 S ij ∇ i |∇v| 2 ∇ j v = − S 4 |∇v| 2 S ij ∇ 2 ij v = − S 4 |∇v| 2 S ij {− A ij 2 + A 0 ij 2 − ∇ i v∇ j v + |∇v| 2 2 g ij } = S 4 |∇v| 2 {σ 2 + S ij ∇ i v∇ j v − R|∇v| 2 2 − S ij A 0 ij 2 } = S 4 |∇v| 2 {σ 2 − R ij ∇ i v∇ j v − S ij A 0 ij 2 } ≤ S 4 |∇v| 2 σ 2 .(67) where in the last line we used (R ij ) ≥ 0 when g ∈ Γ + 2 in dimension 4 and S ij A 0 ij ≥ 0 on S 4 . The terms in the second line of (65) can be estimated in terms of S 4 Re −2v |∇v| 2 , which in turn can be estimated as (68) S 4 Re −2v |∇v| 2 { S 4 R 3 } 1/3 { S 4 |∇v| 3 } 2/3 ≤ ǫ 3 S 4 R 3 + 2ǫ −1/2 3 S 4 |∇v| 3 . The terms in the last line of (65) can be estimated in terms of upper bound of σ 2 , a lower bound of σ 2 , and S 4 |∇σ 2 | 2 , in a trivial way. The term S 4 |∇v| 6 in (66) R 3 + |∇v| 6 + 1,(70) here in the last line we used To obtain the W 2,6 estimates of v by iteration, we multiply (60) and (61) by R p and estimate R p+3 in terms of the other terms: S ij = S 0 ij + 2∇ 2 ij v − 2(∆v)g ij + 2∇ i v∇ j v + |∇v| 2 g ij ,(71)R = R 0 e −2v − 6∆v + 6|∇v| 2 ,(72)R p S ij ∇ 2 ij R ≥ R p+3 12 + 6R p trE 3 − 2σ 2 R p+1 + 3R p ∆σ 2 − 3R p |∇σ 2 | 2 2σ 2 , and R p S ij ∇ 2 ij |∇v| 2 = R p+3 144 − R p trE 3 2 − σ 2 R p+1 12 − R p+1 |∇v| 4 2 − 2R p S ij ∇ i |∇v| 2 ∇ j v + R p S ij ∇ l A • ij ∇ l v − 2R p S ij ∇ i v∇ j v + 2R p+1 |∇v| 2 − R p ∇v, ∇σ 2 − 2σ 2 R p + R p+1 2 . From these we obtain R p+3 6 ≤ R p S ij ∇ 2 ij {R + 12|∇v| 2 } − 3R p ∆σ 2 + 6R p+1 |∇v| 4 + 24R p S ij ∇ i |∇v| 2 ∇ j v + 3σ 2 R p+1 + 3R p |∇σ 2 | 2 2σ 2 − 12R p S ij ∇ l A • ij ∇ l v + 24R p S ij ∇ i v∇ j v − 24R p+1 |∇v| 2 + 12R p ∇v, ∇σ 2 + 24σ 2 R p − 6R p+1 .(75) The most crucial terms are (76) R p S ij ∇ 2 ij R = −p R p−1 S ij ∇ i R∇ j R, R p S ij ∇ 2 ij |∇v| 2 = − p R p−1 S ij ∇ i R∇ j |∇v| 2 ≤p R p−1 [S ij ∇ i R∇ j R] 1/2 S ij ∇ i |∇v| 2 ∇ j |∇v| 2 1/2 ≤p R p−1 S ij ∇ i R∇ j R 1/2 R p−1 S ij ∇ i |∇v| 2 ∇ j |∇v| 2 1/2 ≤ p 2 R p−1 S ij ∇ i R∇ j R + 2p R p |∇ 2 v| 2 |∇v| 2 ≤ p 2 R p−1 S ij ∇ i R∇ j R + C p R p (R 2 + |S 0 ij | 2 + |∇v| 2 )|∇v| 2 .(77) Next we claim that the Sobolev inequality in dimension 4 implies (79) R 2p 1/2 p 2 |∇R| 2 R p−2 + R p |∇v| 2 + R p e −2v . Using (79) in (78), we obtain −R p ∆σ 2 ≤ǫp R p−1 S ij ∇ i R∇ j R 3σ 2 + ǫ 2p R p |∇v| 2 + p 3 8ǫ 3 |∇σ 2 | 4 . Using (76), (77), and (80) in (75) and choosing ǫ > 0 small, we obtain R p+3 6 + p 4 R p−1 S ij ∇ i R∇ j R R p+1 |∇v| 4 + R p+2 |∇v| 2 + R p |∇σ 2 | 2 + R p |∇v||∇σ 2 | + R p+1 + 1 { R p+3 } p+1 p+3 { |∇v| 2(p+3) } 2 p+3 + { R p+3 } p+2 p+3 { |∇v| 2(p+3) } 1 p+3 + { R 2p } 1/2 { |∇σ 2 | 4 } 1/2 + { R 2p } 1/2 { |∇v| 4 } 1/4 { |∇σ 2 | 4 } 1/4 + R p+1 + 1 (81) Now for p ≤ 3, we have 2p ≤ p + 3 and 2(p + 3) ≤ 12. Using the earlier bounds on R 3 and |∇v| 12 from (70), we obtain an upper bound for R 6 in terms of |∇ 0 K| 4 dvol gc , an upper bound and a positive lower bound of K, which again gives a bound for v in W 2,6 . Proof of Corollary 1. Let δ > 0 be small such that the argument for (74) and the subsequent W 2,3 estimate for v via (65) would go through when S 4 |∇v| 4 ≤ δ. For any admissible solution w to (3), Theorem 1 implies that there is a constant B > 0 depending on the C 2 norm of K, a positive lower bound of K, and δ > 0, such that if max w = w(Q) > B, then the normalized v defined as in Theorem 1: v = w • ϕ + ln |dϕ| with v(Q) = 1 4 ln 6 K(Q) , would satisfy (82) v − 1 4 ln 6 K(Q) ≤ δ, and S 4 |∇v| 4 ≤ δ. v also satisfies (8) and then estimate (65), with σ 2 standing for K • ϕ, is valid for v. Then the W 2,3 estimate in Theorem 1 would be valid for v, and one obtains a bound for the W 2,3 norm of v in terms of an upper bound for K, a positive lower bound for K, and an upper bound for S 4 |∇ 0 K • ϕ| 4 dvol gc , and since S 4 |∇ 0 K • ϕ| 4 dvol gc = G, B r (O), O) = 0, for r ≥ r 1 = (t 1 − 1)/t 1 , then (3) has a solution. x∈S 4 :∆K(x)<0,∇K(x)=0 Corollary 1 . 1Let K(x) be a given positive C 2 function on S 4 . Then there is a bound C depending on K only through the C 2 norm of K, a positive upper and lower bound of K on S 4 , such that |F [w]| ≤ C for all admissible solutions w to (3). where ind(DF [s] [v(x; ξ, s), ξ]) refers to the index of the linear operator DF [s] [v(x; ξ, s), ξ], and is computed as (−1) β , with β denoting the number of negative eigenvalues of DF [s] [v(x; ξ, s), ξ]. We also have deg(Λ [s] , B r 0 , O) = ξ∈Br 0 (O):Λ [s] (ξ)=0 ind(DΛ [s] (ξ)). Thus ind(L 0,s ) = ind(L 1,s ) = ind(DF[s] [v(x; ξ, s), ξ]). We will next establish(20) ind(L 0,s ) = (−1) 1+γ , for s > 0 small, where γ is the number of positive eigenvalues of ∇G(P, t) at ξ, and (21) γ = the number of negative eigenvalues of D ξ Λ [s] (ξ) for s > 0 small.First note that which converges to 0 by (31), (36), (37), (38) and the Dominated Convergence Theorem. ∇K [s] , ∇x h e 4w j dvol gc = s S 4 ∇K, ∇x h e 4w j dvol gc Thus, the deformation parameter s is divided out from the Kazdan-∇x h e 4w j dvol gc = |y|≤M ∇K, ∇x h e 4w j dvol gc + |y|>M ∇K, ∇x h e 4w j dvol gc . ∇x 5 e 4w j dvol gc = |y|≤M ∇K, ∇x 5 e 4w j dvol gc + |y|>M ∇K, ∇x 5 e 4w j dvol gc , a i x i x 5 e 4w j dvol gc = σ 2 ( 2A v∞ ) = 6e 4v∞ and S 4 e 4v∞(x) x dvol gc = 0. Remark 6 . 6In fact, for any solution w to (3), one can obtain an upper bound for the W 2,3 norm of w in terms of a positive upper and lower bound for K, an upper bound for |∇ 0 K| 2 dvol gc , and an upper bound for |w| and |∇ 0 w| 4 dvol gc . A proof would proceed as above, instead of using the smallness of |∇ 0 w| 4 dvol gc in proving (74) and the subsequent bound on R 3 via (65), one uses Proposition 5.20, Proposition 5.22, and Lemma 5.24 in[CGY02a] to complete the argument. |∇ 2 v| 3 + |∇v| 6 + e −3v |∇v| 3can be estimated as (69) S 4 |∇v| 6 ≤ S 4 |∇v| 4 3/4 S 4 |∇v| 12 1/4 , and, as in (5.73) in [CGY02a], S 4 |∇v| 12 1/4 S 4 S 4 ( 73 ) 73and 0 ≤ (S ij ) ≤ (Rg ij ).S 4 |∇v| 4 , we obtain an upper bound for R 3 in terms of |∇σ 2 | 2 , upper bound for σ 2 and positive lower bound for σ 2 . Note that S 4 |∇σ 2 | 4 = S 4 |∇ 0 (K • φ j )| 4 = S 4 |∇ 0 K| 4 dvol gc and using a transformation law like (74), we can estimate |∆ 0 v| 3 dvol gc R 3 + |∇ 0 v| 6 dvol gc R 3 + 1, bounded above in terms of |∇ 0 K| 4 dvol gc , upper bound for K and positive lower bound for K. Then we can use the W 2,p theory for the Laplace operator to obtain the full W 2,3 estimates for v.Using (70) in (69) and noting that S 4 |∇v| 4 is small, we obtain (74) S 4 |∇v| 6 S 4 |∇v| 4 3/4 S 4 R 3 + 1, Using (74), together with (67), (68) and (66) in (65), and noting the smallness of and −R p ∆σ 2 R p−1 S ij ∇ i R∇ j R=p R p−1 ∇σ 2 ∇R ≤p |∇σ 2 | 4 1/4 |∇R| 2 R p−2 1/2 R 2p 1/4 ≤p |∇σ 2 | 4 1/4 R p−1 S ij ∇ i R∇ j R 3σ 2 1/2 R 2p 1/4 ≤ ǫp 2 3σ 2 + p 2ǫ |∇σ 2 | 4 1/2 R 2p 1/2 S 4 |∇ 0 K| 4 dvol gc , one can use this estimate to obtain the bound for F [v]. Since II[w] = II[v] and Y [w] = Y [v], the bound for F [w] now follows. When the solution w satisfies w ≤ B, then one can use the Harnack type estimate in[H04] to obtain a lower bound for w, and use inequality (42) to obtain an upper bound for S 4 |∇w| 4 dvol gc . 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[ "Cross-modal Attention Congruence Regularization for Vision-Language Relation Alignment", "Cross-modal Attention Congruence Regularization for Vision-Language Relation Alignment" ]
[ "Rohan Pandey [email protected] \nLanguage Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n\n", "Rulin Shao [email protected] \nLanguage Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n\n", "Paul Pu Liang \nLanguage Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n\n", "Ruslan Salakhutdinov \nLanguage Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n\n", "Louis-Philippe Morency \nLanguage Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n\n" ]
[ "Language Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n", "Language Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n", "Language Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n", "Language Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n", "Language Technologies Institute and Machine Learning Department\nCarnegie Mellon University\n" ]
[]
Despite recent progress towards scaling up multimodal vision-language models, these models are still known to struggle on compositional generalization benchmarks such as Winoground. We find that a critical component lacking from current vision-language models is relation-level alignment: the ability to match directional semantic relations in text (e.g., 'mug in grass') with spatial relationships in the image (e.g., the position of the mug relative to the grass). To tackle this problem, we show that relation alignment can be enforced by encouraging the directed language attention from 'mug' to 'grass' (capturing the semantic relation 'in') to match the directed visual attention from the mug to the grass. Tokens and their corresponding objects are softly identified using the cross-modal attention. We prove that this notion of soft relation alignment is equivalent to enforcing congruence between vision and language attention matrices under a 'change of basis' provided by the cross-modal attention matrix. Intuitively, our approach projects visual attention into the language attention space to calculate its divergence from the actual language attention, and vice versa. We apply our Cross-modal Attention Congruence Regularization (CACR) loss to UNITER and improve on the state-of-theart approach to Winoground.
10.48550/arxiv.2212.10549
[ "https://export.arxiv.org/pdf/2212.10549v1.pdf" ]
254,877,536
2212.10549
45bb4c8bd415cf198fbd8fc13994d2cbeb208651
Cross-modal Attention Congruence Regularization for Vision-Language Relation Alignment Rohan Pandey [email protected] Language Technologies Institute and Machine Learning Department Carnegie Mellon University Rulin Shao [email protected] Language Technologies Institute and Machine Learning Department Carnegie Mellon University Paul Pu Liang Language Technologies Institute and Machine Learning Department Carnegie Mellon University Ruslan Salakhutdinov Language Technologies Institute and Machine Learning Department Carnegie Mellon University Louis-Philippe Morency Language Technologies Institute and Machine Learning Department Carnegie Mellon University Cross-modal Attention Congruence Regularization for Vision-Language Relation Alignment Despite recent progress towards scaling up multimodal vision-language models, these models are still known to struggle on compositional generalization benchmarks such as Winoground. We find that a critical component lacking from current vision-language models is relation-level alignment: the ability to match directional semantic relations in text (e.g., 'mug in grass') with spatial relationships in the image (e.g., the position of the mug relative to the grass). To tackle this problem, we show that relation alignment can be enforced by encouraging the directed language attention from 'mug' to 'grass' (capturing the semantic relation 'in') to match the directed visual attention from the mug to the grass. Tokens and their corresponding objects are softly identified using the cross-modal attention. We prove that this notion of soft relation alignment is equivalent to enforcing congruence between vision and language attention matrices under a 'change of basis' provided by the cross-modal attention matrix. Intuitively, our approach projects visual attention into the language attention space to calculate its divergence from the actual language attention, and vice versa. We apply our Cross-modal Attention Congruence Regularization (CACR) loss to UNITER and improve on the state-of-theart approach to Winoground. Introduction Compositionality is the ability to combine meanings of constituents according to structured rules. Recent work shows that Vision-Language Models (VLMs) fail to construct compositional representations and generally ignore syntactic & structural information [Thrush et al., 2022, Milewski et al., 2022, Liang et al., 2022. Winoground [Thrush et al., 2022] is a vision-language compositionality task that tests a VLM's ability to match syntactic permutations of text with their visual interpretations, for example correctly matching "grass in Figure 1: Global Alignment (GA) only aligns the entire image with the corresponding caption. Entity Alignment (EA) extracts entities from the image and caption for finer-grained alignment. Relation Alignment (RA) cross-modally aligns the intra-modal relations between entities in both the image and the text. We show RA is vital to improve compositional performance. mug" and "mug in grass" to their corresponding images. Winoground finds that all recent stateof-the-art VLMs perform below chance levels on this compositionality task. Contemporaneously, Milewski et al. [2022] probe for structural knowledge in VLMs, finding that they encode significantly less linguistic syntax than LMs and virtually no visual structure. Recently, Yuksekgonul et al. [2022] built a large dataset confirming that VLMs treat images as a 'bag of objects' and don't adequately represent visuo-linguistic relations. Since Winoground requires models to determine whether the compositional structure of an image is aligned with a caption, it's important for the model to learn to cross-modally align intra-modal relations [Ren et al., 2021]. That is, if the relation from 'mug' to 'grass' is 'in-ness', the model should rec-ognize when the equivalent physical relation holds between a mug and grass in the image, and representationally align these relations such that an image-text matching head may more easily determine whether the relations are cross-modally equivalent. In simpler terms, the compositional structure of input for each modality should be expressed such that they can be cross-modally matched even for difficult examples like Winoground. Unfortunately, there has been less highly influential work on relation alignment between vision & language, and Thrush et al. [2022] did not benchmark any such models. In this work, we begin exploration of these relation alignment approaches by tentatively grouping them into 3 categories: Since Structural Data approaches require complex annotations and Structural Model approaches are often incompatible with large transformers, we identify Structural Training as a promising avenue for providing compositional inductive biases to VLMs due to their architecture-agnostic compatibility and computational scalability. In this work, we propose a Structural Training approach for relation alignment that uses the crossmodal attention matrix as a change of basis 1 to the opposite modality, which we then compare to the original modality to calculate a divergence loss, effectively measuring cross-modal congruence between intra-modal attentions. We show how our approach, Cross-modal Attention Congruence Regularization (CACR), generalizes previous Structural Training work on cross-modal attention regularization [Ren et al., 2021] In Tab. 1, we categorize several relation alignment approaches following the framework in Sec. 1. While some of these works introduce ideas from multiple of these categories, we group them by their core contribution. For example, ROSITA proposes a graphical data pre-training approach, and a self-supervised objective to accompany it; we consider it a Structural Data approach since the training objective ultimately is just a necessity for the data being provided. Unfortunately, many of these works do not provide publicly available code or pre-trained checkpoints, so we were unable to complete an exhaustive analysis of the compositional performance of these relation alignment approaches. Additionally, no Structural Model approaches have been explored for image-text matching to our knowledge, with most of that literature instead focusing on reasoning & visual question answering. Regardless, we chose one exemplar for both Structural Data (ROSITA) and Structural Training (IAIS) that made their pre-trained image-text matching checkpoints available; we generated their scores on Winoground, which have not previously been calculated. In Tab. 2, we present these two relation alignment models' Winoground scores alongside a few entity alignment and global alignment models. Model Text Notice that global alignment approaches tend to perform the lowest on Winoground, even when scaled considerably. Entity alignment approaches perform intermediately and OSCAR+ specifically held the state-of-the-art prior to our benchmarking of these relation alignment models. Of the two relation alignment approaches we benchmark, IAIS beats out OSCAR+ and achieves a new stateof-the-art on Winoground. But ROSITA, despite providing structural data to encourage cross-modal relation alignment, underperforms OSCAR+. We attribute this partly to the improved visual features OSCAR+ has access to as a result of VinVL, but further comparison of IAIS and ROSITA is explored in our recent work. Based on these past results and analysis, we choose to further explore structural training approaches to relation alignment. In other words, our research question becomes: How can we infuse the vision-language model's training objective with an implicit structural prior that encourages cross-modal alignment of relations? Method To attempt a solution to this question, we begin by noting that attention activations encode some degree of relational information. Attention values in transformers may be seen as an informational gating mechanism that implicitly encode how representations are composed. For example, past work in language has shown how syntax trees may be extracted [Mareček and Rosa, 2019] from attention across layers and used to guide attention [Bai et al., 2021, Li et al., 2020 for improved compositionality. In this section, we extend this intuition to the multimodal domain by showing that we can use the cross-modal attentions, which as a change-of-basis matrix encode a transformation from one modality's compositional structure to the opposite modality's, to encourage cross-modal relation alignment. Relation Alignment Using Attention In specific, we focus on the self-attention matrix S computed by S = QK = (XW Q )(XW K )(1) Then, some row i in S corresponds to a distribution over columns j 0 , ..., j n where S i,j tells us how much of the previous layer's entity representation j we want to infuse into the current layer's entity representation i, intuitively their compositional relation. Since X is a series of visual and linguistic tokens, we can segment S into four submatrices for intra-and cross-modal relations [Bugliarello et al., 2021]. Denote the intra-modal attention submatrices in the last multimodal encoder layer as S V V (vision to vision) and S LL (language to language); the cross-modal attention matrices as S V L (vision to language) and S LV (language to vision). S = S LL S LV S V L S V V(2) If an image and caption have the same underlying compositional structure, the entities that crossmodally correspond to each other should bear similar intra-modal compositional structure. That is, a word w should attend to other words (in S LL ) in a similar way that its visual object counterpart o attends to other objects (in S V V ). Furthermore, we can use the cross-modal matrices (S LV and S V L ) to identify entities that cross-modally correspond as they will generally attend to each other [Aflalo et al., 2022]. Unfortunately, since representations are heavily contextualized by the final layer, clear bijective correspondences between words and objects may not always be identified using an argmax over the cross-modal attention matrix as Ren et al. Cross-modal Attention Congruence Regularization for Relation Alignment We opt to use the cross-modal matrices (S LV and S V L ) as a whole to 'change basis' to the opposite modality, with which we can then calculate 'congruence' with the original modality. However, we use 'change of basis' and 'congruence' loosely since the cross-modal matrices are not guaranteed to be square and thus do not satisfy strict linear algebraic definitions. We formulate S V V in the language basis as S LV S V V S LV , which we then encourage to be similar to S LL . Under the hood, this says that for each a i→j ∈ S LL , we can use row vectors S LV,i and S LV,j to calculate a weighted sum a * i→j over S V V . If we were to do this for all i, j, we would construct a matrix of the same shape as S LL where each entry is a * i→j , i.e. an approximation of the visual correspondent of the relation a i→j taking into account all the possible cross-modal alignments of i and j. Since this computation intuitively makes a lot of sense and may more easily be compared to previous approaches, we choose to illustrate it in Fig. 3. However, since this computation is relatively expensive, we instead use the S LV S V V S LV formulation which produces the same matrix of a * i→j values but with considerably fewer operations. This also enables us to view the operation as a 'changeof-basis' to the opposite modality and the CACR loss as encouraging a sense of cross-modal 'congruence'. Specifically, we align the original S LL with the language-basis S V V matrix using L CACR-L : L CACR-L = m-KL(σ(S LV S V V S LV ), σ(S LL )). (3) We apply a softmax to normalize both matrices since S LV S V V S LV will generally be larger in scale due to summation. Additionally, m-KL(·) [Ren et al., 2021] is a symmetric matrix-based Kullback-Leibler Divergence (m-KL) which measures the distance between two matrices S and S : m-KL(S, S ) = N i KL(S i ||S i ) + KL(S i ||S i ),(4) where (·) i stands for the i th row-vector in the matrix. Similarly, we have L CACR-V : L CACR-V = m-KL(σ(S V L S LL S V L ), σ(S V V )),(5) Combining L CACR-V and L CACR-L , we present our L CACR objective, an attention activation regularizer for cross-modal relation alignment: L CACR = L CACR-V + L CACR-L .(6) When the vision inputs and the language inputs have the same sequence length and S V L , S LV are invertible, then S V V and S V L S LL S V L (as well as S LL and S LV S V V S LV ) can become strictly congruent. In this case, S V L S LL S V L can be interpreted as the language view of S V V . Aligning S V L S LL S V L and S V V leads to cross-modal intra-modal relation alignment. It is similar for S LV S V V S LV and S LL . In the general case where the vision inputs and the language inputs may have different sequence lengths, the two forms are not linear algebraically congruent but the relevant intuition still holds. Soft Cross-modal Alignment In this section, we show that CACR can be interpreted as leveraging cross-modal soft equivalences, where IAIS [Ren et al., 2021] uses hard bijective equivalences. In their approach, each element in the intra-modal attention matrix is aligned with a single counterpart in the opposite modality. This is built upon a strong assumption that there exists a one-to-one mapping (provided by an argmax over the cross-modal attention) from S LL to S V V and vice versa, which is unsatisfied in practical cases. We propose a soft relation alignment method which instead uses the whole S LV (or S V L ) to implicitly build an 'equivalence weighting' which is then used to compute a weighted mean over S V V (or S LL ). We illustrate and compare the hard relation alignment and soft relation alignment in Figure 3 taking the language-side alignment as an example. We note that IAIS could be seen as a special case of our method by forcing the attention map shown in the soft relation alignment in Figure 3 to be a one-hot matrix, i.e., taking the argmax of the attention matrix as the index of the cross-modal counterpart. We show in Section 5 that IAIS can have inferior performance when a clear bijective cross-modal correspondence isn't available. In Alg. 1, we show the pseudo-code of the soft relation alignment method for calculating the visionside loss. S CACR−V can be computed similarly. Computing the hard and soft relation alignment is computationally complex and difficult to be parallelized due to indexing operations. For practical applications, we sought to simplify this soft relation alignment algorithm to a mathematical equivalent that would improve computational tractability. From here, we arrive at CACR, which is a closedform formulation of soft relation alignment which utilizes only differentiable matrix multiplications. Therefore, our CACR is more computationally efficient and easier to be parallelized than the hard relation alignment. Algorithm 1 Soft Relation Alignment (V) Require: S LL ∈ N × N, S V L ∈ N × M, S V V ∈ M × M 1: L ← 0 2: for i, j ∈ S V V do 3: W ← S V L [i] · S V L [j] soft weighting 4: a * i→j ← W • S LL element-wise weighted mean 5: L = L + m-KL(a * i→j , S V V [i, j]) 6: end for 7: return L Proof of Equivalence Between CACR and Soft Relation Alignment Computing the hard (IAIS) and soft relation alignment is computationally complex and difficult to parallelize due to indexing operations. However, CACR loss is mathematically equivalent to soft relation alignment but can be computed efficiently. We take CACR V for illustration of this equivalence, but CACR L can be proved in the same way. Beginning with the visual-basis form of S LL in CACR, the attention at index [i, j] in S V L S LL S V L is (S V L S LL S V L )[i, j] = N L p N L k a v i →l k a lp→l k a v j →lp = N L p N L k S V L [i, k]S V L [j, p] soft weighting S LL [p, k] (7) where a v i →l j stands for the attention from the i-th visual token to the j-th linguistic token, N L is the total number of language tokens and N V is the total number of the visual tokens. Comparing Eq.7 and Alg.1, we observe that the summation we arrive at above is equivalent to the content of the for-loop (line 3-5). Thus, although of seemingly different linear algebraic form, CACR generalizes IAIS by way of its equivalence to the Soft Relation Alignment formulation presented above. Results How does CACR compare to other vision-language models in its compositional ability? Hard relation alignment attempts to build a one-to-one mapping between language and vision by applying argmax (yellow checks) to the S LV row vectors. Our soft relation alignment instead uses the whole S LV row vectors to calculate a weighted (red) mean over S V V . The scalar that is produced corresponds to the attention from 'mug' to 'grass' but in a visual basis. We note that IAIS can be seen as a special case of soft relation alignment by forcing the attention matrix (red) to be a one-hot matrix where the max value is set to 1 and all others to 0. In Tab. 3, we present our approach's scores alongside a few other models. Since we use CACR to fine-tune UNITER, we include scores for the two baseline UNITER sizes. We also include scores for both IAIS sizes which are also built on UNITER, as well as OSCAR+, the original state-of-the-art identified by Milewski et al. [2022]. The fact that CACR base outperforms IAIS base suggests that, with adequate computational resources, CACR large would similarly outperform IAIS large , potentially achieving a new state-of-theart on Winoground. Furthermore, its performance compared to UNITER large is impressive considering that CACR base is approximately half its size in parameters. Finally, we report Flickr30k retrieval scores in Tab Table 4: Flickr30k retrieval scores for UNITER family of models minor losses to its retrieval scores, this may be attributed to imperfect hyperparameters, suggesting that CACR's performance on Winoground could be even higher with adequate hyperparameter tuning. It's also important to remember here that we're only training on Flickr30k, so this isn't a case of our model overfitting to Winoground and 'forgetting' its true image-text matching ability. We fine-tuned CACR on Flickr30k [Young et al., 2014] for 5000 epochs with a train-validation-test split of 90-5-5. The training batch size is 4 and 31 negative samples are provided for every individual positive sample in a standard image-text matching training setup. We use a learning rate of 5 × 10 −5 , the AdamW optimizer [Loshchilov and Hutter, 2017], and introduce L CACR with an exponential warmup schedule. Training was completed on a node with 4 NVIDIA GTX 1080 Ti's, each with 11 GB of memory. Why does CACR's soft relation alignment approach outperform hard relation alignment? Qualitative Hard relation alignment, implemented by IAIS, assumes that cross-modal submatrices can be used to find a singular equivalent of an entity in the opposite modality. Specifically, if i * = argmax(S LV [i]) then S LL [i] should correspond to S V V [i * ]. In simple terms, IAIS says the following: if word A attends most to object A and word B attends most to object B, then word A should attend to word B in a similar way that object A attends to object B. Underlying IAIS is the hard assumption that argmaxing over the cross-modal attention submatrix is an effective means of identifying the opposite modality equivalent of an entity. However, we show in this section that this is often not the case. Given the argmaxes for rows in the S LV submatrix, we can identify the bounding box that each token maximally attends to, which IAIS assumes is its visual equivalent. In Fig. 4a, we visualize an example where 'clouds' maximally attends (green) to the ground, which would prevent IAIS from identifying the correct cross-modal alignment. 'Turbines' (Fig. 4b), on the other hand, maximally attends to a bounding box that better matches our intuition. It is qualitatively clear from the several examples displayed that the argmax assumption often fails to identify the correct cross-modal equivalence. Since words may attend to several visual tokens for different reasons, we shouldn't assume that the crossmodal argmax provides us with a clear bijective correspondence. Instead, the cross-modal matrices should be seen as providing useful high-level information about what visual entities are relevant to a word, and vice versa. We can certainly gain useful information about cross-modal correspondences using it, but it isn't as easy as using an argmax due to words having multiple referents and entity representations being intermixed. Instead, our soft relation alignment approach takes all the possible cross-modal alignments into account with a weighted sum. To illustrate how the soft approach takes into account critical cross-modal alignment information, we present a few Winoground examples with UNITER's cross-modal attention activations in Fig. 4 and 5. We use UNITER since this is the baseline model from which attentional information is bootstrapped to calculate cross-modal alignments. For example, in Fig. 5c, using the representation for the bounding box covering the mug's handle may not adequately capture the visual referent of 'mug' and therefore disrupt our ability to calculate the visual-basis relation between 'mug' and 'grass' if restricted by an argmax. Quantitative In the absence of annotations, we attempted a quantitative measurement of whether overlap in argmaxes (several words attending to one bounding box or vice versa) as quantified by the Shannon Entropy of argmax indices inversely correlates with soft Winoground score. Intuitively, if an example has more like a one-to-one mapping between text and image, the entropy of its cross-modal argmaxes should be higher as each token will attend to a different box, which would suggest that the model is better aligning entities. However, we found no significant correlation with Winoground score, which we attribute to the fact that high entropy on its own doesn't mean correct entity alignment. Rather, high entropy in argmax indices could still be produced by a bad representation if 'mug' attends to the grass & 'grass' attends to the mug; conversely, low entropy could be produced by a good representation for an example like 'fire truck' (a) dog (b) person (c) mug (d) grass Figure 5: UNITER S V L attention for captions "a dog on a rock next to a person" and "there is a mug in some grass". Shown are boxes that attend highly to the displayed token, with the maximally attending bounding box in green; others in red. Observe that although the argmax often does pick up on a relevant bounding box, it is prone to missing critical visual information, e.g. focusing on only the backpack in (b). where two tokens refer to a single object. Quantitative exploration of cross-modal attention is difficult without annotations and we leave this task to future work to explore in a multimodal compositionality context. As a general takeaway, while the cross-modal argmax assumption of IAIS does hold in some cases and may be more meaningful during the course of IAIS training, it is clearly quite a strict assumption that could suffer if an entity attends to several cross-modal entities or there are no corresponding cross-modal entities. Furthermore, since IAIS is only active in the final self-attention layer, all the token representations are intermixed and therefore don't necessarily have a one-to-one correspondence with our intuitive notions of what they should be-the word 'turbine' may not solely represent the traditional meaning of that word but perhaps the entire scene that includes the turbines, clouds, and ground. We hypothesize that by removing the hard argmax assumption, our approach better accounts for varying cross-modal entity alignments and thus enables stronger relation alignment. By also calculating alignment between all pairs of source and target modality entities, CACR should considerably improve sample efficiency, which is important considering that the final layer S matrix of the converged IAIS model is largely flat. Therefore it's important to backpropagate as much alignment knowledge over the course of training as possible, which CACR's soft equivalence weighting implicitly enables. Conclusion In this work, we identified that a key factor holding back models from vision-language representational compositionality is cross-modal relation alignment. We categorized recent compositional inductive bias approaches into 3 categories: Structural Model, Structural Data, and Structural Training, showing that a previous Structural Training model (IAIS) achieves state-of-the-art performance on Winoground. We then identified a potential key weakness in IAIS, its hard argmax assumption, and developed a soft cross-modal relation alignment approach to address it. Having linear algebraically simplified this approach, we arrived at CACR, an auxiliary loss that encourages cross-modal congruence of intra-modal attention. CACR improves on the equivalent-sized IAIS' performance on Winoground, and even outperforms UNITER, a model nearly twice as large. As computational scaling becomes more widespread, it's necessary to develop compositional inductive biases that do not require complex annotated data or exotic model architectures. Our work illustrates how taking advantage of the transformer's own attentional structure can improve the quality of fine-grained vision-language representations, opening the avenue for large scale approaches to visually-grounded compositionality. [2021] attempts. Deeper analysis of when IAIS fails to identify cross-modal bijective correspondences is provided in Sec. 5. Figure 2 : 2Top: language attention (S LL ) is aligned with the visual attention projected into the language basis (S LV S V V S LV ) to calculate L CACR−L ; specific attention values (yellow, red) capturing intra-modal relations are cross-modally aligned as a result. Bottom: as above, but in the vision basis. Figure 3 : 3Comparison of the hard relation alignment used in IAIS (left) and the soft relation alignment we propose in CACR (right), with an example from Winoground to illustrate how the cross-modal relation is calculated. Figure 4 : 4Top: UNITER S LV attention for caption "a few clouds and many wind turbines", with the bounding box maximally attended to by the token in green; other highly attended boxes in red. Bottom: UNITER S LV attention with bounding boxes labeled with the tokens that maximally attend to them. by taking into account all possible entity alignments and computationally simplifying relation alignment. The CACR loss function can easily be dropped into most transformer-based Vision-Language model objectives without any added data and minimal computational overhead, to encourage relation alignment during training.1 not defined in a strict linear algebraic sense Finally, we show that CACR base improves on IAIS base -where IAIS large holds the current stateof-the-art on Winoground.2 Related Work Structural Model Andreas et al. [2016] NMN Guo et al. [2019] VSUA Hong et al. [2021] VL-G Zhang et al. [2022] SG-CMR Wang et al. [2022b] SGEITL Kim et al. [2022] CMR Wang et al. [2022a] VQAGNN Structural Data Wu et al. [2019] UniVSE Zhang and Peng [2019] AGHA Yu et al. [2021] ERNIEViL Cui et al. [2021] ROSITA Wan et al. [2021] CLIORA Khan et al. [2022] SimLA Li et al. [2022] MVP Structural Training Ren et al. [2021] IAIS Yang et al. [2021a] APN Yang et al. [2021b] CATT Xue et al. [2021] IMF Table 1: Recent structural approaches to vision- language relation alignment Table 2 : 2Comparison of Winoground scores for models using Global Alignment (GA), Entity Alignment (EA), Relation Alignment with Structural Data (RA-SD), and Relation Alignment with Structural Training (RA-ST). We find that IAIS, a recent relation alignment approach that uses attention regularization for structural training achieves universal performance improvements. Table 3 : 3Winoground scores for CACR compared to the current state-of-the-art (OSCAR+), our baseline (UNITER), and the state-of-the-art we newly bench- marked (IAIS) . 4 to verify that we are not somehow overfitting to Winoground. Though CACR takes someModel Image R@1 Image R@10 Text R@1 Text R@10 IAIS large 76.86 95.72 88.30 99.40 UNITER large 73.56 96.76 87.30 99.20 IAIS base 73.54 96.32 86.10 99.10 UNITER base 72.52 96.08 85.90 98.80 CACR base 70.88 95.68 83.50 98.80 Vl-interpret: An interactive visualization tool for interpreting vision-language transformers. Estelle Aflalo, Meng Du, Shao-Yen, Yongfei Tseng, Chenfei Liu, Nan Wu, Vasudev Duan, Lal, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionEstelle Aflalo, Meng Du, Shao-Yen Tseng, Yongfei Liu, Chen- fei Wu, Nan Duan, and Vasudev Lal. Vl-interpret: An in- teractive visualization tool for interpreting vision-language transformers. In Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, pages 21406-21415, 2022. Neural module networks. 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[]
[ "Design guidelines for data analysis scripts", "Design guidelines for data analysis scripts" ]
[ "Marijn Van Vliet [email protected] \nDepartment of Neuroscience and Biomedical Engineering\nAalto University\nFinland\n" ]
[ "Department of Neuroscience and Biomedical Engineering\nAalto University\nFinland" ]
[]
Unorganized heaps of analysis code are a growing liability as data analysis pipelines are getting longer and more complicated. This is worrying, as neuroscience papers are getting retracted due to programmer error. In this paper, some guidelines are presented that help keep analysis code well organized, easy to understand and convenient to work with:1. Each analysis step is one script 2. A script either processes a single recording, or aggregates across recordings, never both 3. One master script to run the entire analysis 4. Save all intermediate results 5. Visualize all intermediate results 6. Each parameter and filename is defined only once 7. Distinguish files that are part of the official pipeline from other scriptsIn addition to discussing the reasoning behind each guideline, an example analysis pipeline is presented as a case study to see how each guideline translates into code.In science, data analysis is performed at the cutting edge, where it is often inevitable that new pieces of programming code need to be written. New methods are made available through software libraries first, being accessible through an application programming interface (API), and only later as graphical user interface (GUI) programs, if at all. Furthermore, novel research ideas often require combining analysis techniques in new ways that are not possible with existing programs and hence require writing new code. When data analysis pipelines grow, unorganized heaps of code become a liability.
null
[ "https://arxiv.org/pdf/1904.06163v1.pdf" ]
119,301,336
1904.06163
38c555048ea5c302420f05e14cfe4be39b68040d
Design guidelines for data analysis scripts Marijn Van Vliet [email protected] Department of Neuroscience and Biomedical Engineering Aalto University Finland Design guidelines for data analysis scripts April 15, 2019 This is a preprint that has not undergone peer review yet.data analysisscriptingguidelinesprogramming 1 Unorganized heaps of analysis code are a growing liability as data analysis pipelines are getting longer and more complicated. This is worrying, as neuroscience papers are getting retracted due to programmer error. In this paper, some guidelines are presented that help keep analysis code well organized, easy to understand and convenient to work with:1. Each analysis step is one script 2. A script either processes a single recording, or aggregates across recordings, never both 3. One master script to run the entire analysis 4. Save all intermediate results 5. Visualize all intermediate results 6. Each parameter and filename is defined only once 7. Distinguish files that are part of the official pipeline from other scriptsIn addition to discussing the reasoning behind each guideline, an example analysis pipeline is presented as a case study to see how each guideline translates into code.In science, data analysis is performed at the cutting edge, where it is often inevitable that new pieces of programming code need to be written. New methods are made available through software libraries first, being accessible through an application programming interface (API), and only later as graphical user interface (GUI) programs, if at all. Furthermore, novel research ideas often require combining analysis techniques in new ways that are not possible with existing programs and hence require writing new code. When data analysis pipelines grow, unorganized heaps of code become a liability. Introduction The journey of the data from our measurement equipment to a figure in a publication is growing longer and more complicated. new preprocessing steps have been developed to be added at the beginning of the pipeline, 1 new multivariate techniques that find a place in the middle, 2 and 1 Bigdely-Shamlo, Mullen, Kothe, Su, and Robbins, 2015;Nolan, Whelan, and Reilly, 2010;Jas, Engemann, Bekhti, Raimondo, and Gramfort, 2017 2 Kriegeskorte, Mur, and a. Bandettini, 2008;King and Dehaene, 2014;McIntosh and Mišić, 2013 new statistical methods at the end. 3 Using these new techniques often requires writing pieces 3 Maris and Oostenveld, 2007 of programming code referred to as "scripts", and in accordance with the growing data analysis pipelines, these scripts also tend to increase in length and complexity. When programming code becomes sufficiently convoluted, even the most experienced programmers will make mistakes, which can ultimately lead to erroneous conclusions. It has happened that papers had to be retracted due to programmer error. 4 This paper is an effort to set some guidelines to An important aspect to organizing scientific code is to have the bulk of the analysis functionality implemented in the form of a software library that exposes a well designed API. 5 The code can 5 Buitinck et al., 2013 then be used in multiple scripts and user facing programs. It is generally beneficial to use and extend an existing piece of software that is used by many, rather than developing a home-grown solution from scratch. This is because the more people use a piece of software, the more likely it is that mistakes are spotted and corrected. 6 There is a large body of literature on managing 6 Eklund, Nichols, and Knutsson, 2016 complexity and reducing the chance for programmer error in this context. 7 7 Beck, 2002;Hunt and Thomas, 1999;Martin, 2008;McConnell, 2004;Wilson et al., 2014 The guidelines in this paper aim to translate some of this literature to the sub-domain of analysis scripts. Scripts are pieces of code that use the functionality that is exposed by the APIs of software libraries to create data analysis pipelines according to the specific requirements of a single study. Scripts are pieces of code that do not need to be reusable (pieces that need to be reusable are better implemented in a software library), only have to function correctly on one specific dataset (hence there are no "edge cases") and will generally only be used by yourself and your collaborators (save the occasional run for review purposes and replication studies). Therefore, many of the standard practices of the software industry do not apply or need translation in order to arrive at concrete advice of what to do and what to avoid when writing analysis scripts. Often, an analysis pipeline starts off as a simple script that runs a few operations and grows as more steps are added. As the pipeline becomes more complicated, the overall organization and design of the pipeline must be occasionally re-evaluated, or it is likely to become convoluted and error prone. The guidelines in this paper aim to facilitate a successful organization of the analysis code, thereby keeping the complexity of data analysis scripts within tolerable limits, capitalize on the advantages of scripting and offset the disadvantages. To move beyond mere truisms, the analysis pipeline developed by van Vliet, Liljeström, Aro, Salmelin, and Kujala (2018) has been extended to implement all guidelines in this paper, and will be used as case study. In the case study, the practical consequences are discussed of each guideline in terms of code, which can serve as an example when implementing your own analysis pipelines. The example pipeline starts from the raw magnetoencephalography (MEG) and structural functional magnetic resonance imaging (fMRI) data from the Wakeman and Henson (2015) faces dataset and performs several artifact reduction steps, source estimation, functional connectivity analysis, cluster permutation statistics and various visualizations. The size and complexity of the pipeline is representative of that of the pipelines in modern studies at the time of writing. Where van Vliet et al. (2018) gives a detailed explanation of all analysis steps, the current paper focuses on the design decisions that were made during the implementation. You can find the code repository for the analysis pipeline at: https://github.com/aaltoimaginglanguage/conpy. Of special interest is the scripts folder of that repository, which contains the analysis code itself. The "Application of the guideline to the example analysis" sections refer frequently to the example code and it is recommended to study these sections and the code side by side. The electronic version of this document contains many hyperlinks to sections of the code, which the reader is encouraged to follow to see how the guidelines can be implemented in practice. Hyperlinks are typeset in dark blue. Guidelines Guideline 1: Each analysis step is one script An effective strategy to reduce software complexity is to break up a large system into smaller parts. 8 The first guideline is therefore to isolate each single step of an analysis pipeline into its 8 Hofmann, 2004;Parnas, 1972 own self-contained script. This greatly reduces complexity by allowing us to reason about the pipeline on two levels. At the lower level, we can reason about the implementation of a single step, while ignoring the rest of the pipeline for a moment. At the higher level, we can treat the individual steps as "black boxes" and focus on how they are combined together to form the complete pipeline (see also guideline 3), ignoring their implementation for a moment. Things that this guideline aims to prevent: • script becomes "spaghetti code" This raises the question of what exactly constitutes a single analysis step. The decision of where to "cut" the pipeline can be made from different perspectives: by complexity, by theme, and by running time. Complexity perspective The purpose of the guideline is that each individual script should be easy to understand and reason about, so one way to define a single step is by its complexity: if a single script becomes too complex to be easily understood as a whole, it should be split up into smaller steps if possible. Thematic perspective Ideally, understanding one script should not require knowledge of another script. If each script can be viewed as a self-contained box that performs a single task, the pipeline as a whole becomes simply a collection of these boxes that are executed in a specific order. A script should therefore aim to implement a single task, not multiple, and implement it completely, not only part of it. However, this aim may clash with other perspectives in this list, so compromises are sometimes necessary. Time perspective While running the entire analysis pipeline may take days, a single script should finish in a reasonable time. This invites frequent testing as you iteratively develop the script, allows you to quickly evaluate the effect of a parameter, and also makes it painless to ensure that the latest version of the script matches the latest result. When the running time grows to the point where you are tempted to continue working on the script while a run is still in progress, the script should be split into smaller steps if possible. Application of the guideline to the example analysis The van Vliet et al. (2018) pipeline consists of 13 analysis scripts that process the data, 4 visualization scripts that construct the figures used in the publication, a configuration file (config.py) and a "master" script that calls the individual analysis scripts (dodo.py, see guideline 3). Each of the 13 analysis scripts implements a single step in the analysis and are numbered to indicate the sequence in which they are designed to be run. While the scripts need to be run in sequence once, they can be run independently afterwards. The analysis scripts are relatively short (Figure 1), containing an average of 40.8 lines of code (std. 14.8), while the configuration and master scripts are longer. Often, the reasoning behind the scope of each script was made from a thematic perspective. : For each script in the analysis pipeline, the number of lines of the file, broken down into lines of programming code (code), lines of descriptive comments (comment) and blank lines (blank). The first 13 scripts perform data analysis steps, the next 4 scripts generate figures, the config.py script contains all configuration parameters and the dodo.py script is the master script that runs all analysis steps on all recordings. For example, one script performs the source estimation (09_power.py) and another the connectivity estimation (10_connectivity.py). However, the decision to split the artifact reduction steps into two scripts (02_filter.py, 03_ica.py) was made from a time perspective. Since the independant component analysis (ICA) computation takes time, it was split off into its own script to avoid having to repeat it unnecessarily. Finally, the decision to split up the construction of the forward models (i.e. leadfield) into three steps (06_fsaverage_src.py, 07_forward.py, 08_select_vertices.py) was made from a complexity perspective. Guideline 2: A script either processes a single recording or aggregates across recordings, never both Things that this guideline aims to prevent: • excessive running time of the script • multiple versions of the Big Loop that operate on different sets of subjects, all but one commented out • a copy/paste of the Big Loop for each analysis script A compelling reason for performing data analysis using scripts instead of, for example, using a GUI, is the ease of repeating (parts of ) the analysis. Every time the scripts are run, the computer will perform exactly the same tasks in exactly the same order, eliminating the possibility of mix-ups in this regard. This allows you, for example, to efficiently test the effect of changing a single parameter, while ensuring all subsequent analysis steps remain the same. To capitalize on this advantage, analysis scripts should be organized such that it is easy to run only selected parts of it, without having to modify (e.g. "commenting out") the code itself. In neuroscience, it is common to apply the same data processing steps to multiple recordings. For example, a frequently seen construct is the Big Loop over data from multiple participants. The second guideline states a separation of duties: a script is either a processing script, or an aggregation script. Processing scripts perform data analysis only on a single recording, passed as a parameter from the command line, and do not have the Big Loop (the script is applied to all recordings in a separate "master" script (see guideline 3)). Aggregation scripts have the Big Loop to collect the processed data from multiple recordings, with the sole purpose of aligning the data (e.g. morphing to a template brain) and computing an aggregate (e.g. a grand average or statistics). This reduces the complexity of the code, since it allows the reader to either focus on the intricacies of a data processing step, without having to worry about how the data is later reconciled across recordings, or to focus solely on the details of how multiple datasets are aligned and combined. Following the guideline also makes the development process more efficient. It allows for a smooth workflow for the common scenario in which the script is tested on one subject during development, then an attempt is made to run it on all subjects using the master script, problems are found that only arise for certain subjects, and finally the script is re-run once more on all subjects. Application of the guideline to the example analysis In the example pipeline, there is a strict separation between scripts that perform data analysis on a single participant (steps 0-10) and scripts that aggregate across participants (steps 11 and 12). The scripts implementing steps 0-10 all take a single command line parameter indicating the participant to process. This is implemented with the argparse module of the standard Python library, which facilitates the generation of a helpful error message when this parameter is omitted, along with documentation on how to run the script. Not only does this help to keep the number of lines of code and the running time of the script down (Figure 1), it also opens up the possibility for the master script (see guideline 3) to automatically skip running the script for participants that have already been processed earlier. Guideline 3: One master script to run the entire analysis Things that this guideline aims to prevent: • excessive complexity • confusion as to which order the scripts should be run in • having to manually run several scripts in order to complete the analysis and forgetting to re-run one • scripts are changed, but not re-run, causing the result to be out of sync with the code • excessive running time of the analysis pipeline when only a single step has changed • a copy/paste of the Big Loop for each analysis script Once the individual steps have been implemented as a collection of scripts, the pipeline can be assembled in a "master" script that runs all the steps on all the data. This master script is the entry point for running the entire analysis and therefore the third guideline states that there should ideally only be one such script. By having a strict separation between the scripts that implement the individual steps and the master script, it becomes possible to view the pipeline on two levels: the implementation details of each single step, and how the steps fit together to build the pipeline. To understand the latter, the master script provides the "floor plan" of the analysis, which can be studied without having to go into detail on how each individual step is performed. Hence, the only function that the master script should perform is to call the other scripts in the correct order. Actual data analysis steps, including logic for combining results across scripts, should always be performed in a separate script, which is in turn called from the master script. Speed is a very important aspect of an analysis pipeline, as it encourages practices that reduce Keeping track of which scripts have changed since they were last ran, and which scripts consume the output produced by which other scripts (known as the dependency graph, see figure Figure 2), is a task that has been studied in great detail in the area of software engineering and many specialized tools, known as "build systems", 9 are available to perform the required 9 For example, here are some build systems that are optimized for creating data analysis pipelines: https://snakemake.readthedocs.io https://pydoit.org https://luigi.readthedocs.io bookkeeping tasks. Writing the master script using a build system will allow fine grained control over which steps to run on which recordings, while skipping steps that are "up to date". Application of the guideline to the example analysis The master script of the example pipeline, dodo.py, is implemented using the pydoit 10 build 10 https://pydoit.org system. In the script, all analysis steps are described as "tasks", which steps 0-10 having a "subtask" for each participant. Each task is associated with one of the analysis scripts, along with a list of files the script consumes and produces. This allows the build system to work out the dependency graph of the analysis pipeline ( Figure 2). The build system keeps track of which tasks are "up to date", meaning the latest version of the analysis scripts of all analysis steps up to and including the current step have all been run. This means that the entire analysis pipeline can be run often and cheaply: all steps that are up to date will be skipped. Making a change anywhere within the analysis code will prompt the recomputation of all the steps that need to be re-run. The build system also provides a set of commands that allow for executing specific parts of the pipeline, for example, a single analysis step on all participants, a few specific steps on a few specific participants, etc., without having to change (e.g. comment out) any code. Guideline 4: Save all intermediate results Things that this guideline aims to prevent: • variables being manipulated across multiple scripts • debugging a script taking a long time due to having to re-compute everything from scratch every time the script runs • erroneous output is generated in the middle of the script, but subsequent processing makes the result appear reasonable at the end of the script First, from the complexity perspective, it is important that each script can function in isolation and does not rely on data that was left in memory by another script. The more each script can be isolated from the rest of the pipeline, the easier it is to understand and represent as a self contained black box. The fourth guideline states that all intermediate results generated during the execution of a script should be saved to disk, if feasible. Having snapshots of the data as it passes through the pipeline has numerous advantages. Another big advantage is that it makes it possible to skip any data processing steps that are unchanged since the last time the pipeline was executed (see also guideline 3). This makes it possible to re-run small portions of the pipeline quickly, for example to debug a problem, or to assess the effect of some parameter. Finally, having all intermediate results readily available facilitates manual checks and exploration of the data. The ability to jump into an interactive session and quickly load the state of the data at any desired location in the analysis pipeline is an effective way to verify that a script produced the intended result. Application of the guideline to the example analysis In the example analysis pipeline, each script begins by loading the data that were produced by previous analysis steps as requires. Each script ends by saving all data that was produced by the script. This includes the processed MEG data, but also, for example, the ICA decomposition matrix, along with the indices of the ICA components that were judged to correspond to eyeblink contaminants. Guideline 5: Visualize all intermediate results Things that this guideline aims to prevent: • researcher is operating "in the blind" • erroneous output is generated in the middle of the script, but subsequent processing makes the result appear reasonable at the end of the script • result figures no longer match the data files after a script has been re-run Data analysis pipelines, such as those used in neuroscience, are sufficiently complex that failures should be expected and planned for. When designing the pipeline, think about the system for catching errors when they happen. The most severe disadvantage of using scripts instead of a GUI may be the lack of direct feedback. Since the result is usually not immediately visualized, errors may stay hidden for a long time. Programming a computer is not unlike receiving a wish from a mischievous genie: you will get exactly what you asked for, but not necessarily what you wanted. As long as the final result of a series of processing steps looks reasonable, intermediate steps might contain nonsensical results that we would never know about unless we take care to check everything. Therefore, an analysis pipeline should invite frequent visual checks on all intermediate results. The fifth guideline states that for each intermediate result, the script should create a visualization of the result and save it to disk. This does not need to be a publication ready figure, but must provide a visual confirmation that the data analysis operation had the intended result. By re-creating the figures every time a script is run and overwriting the file on disk, the figure remains up to date. After running all the analysis scripts, a complete visual record should be available of the data as it moves through the pipeline. Care should be taken that the order of the figures matches that of the analysis steps. Such a record invites frequent visual checks of the obtained results and therefore somewhat offsets the main advantage that GUI programs have over scripting. Application of the guideline to the example analysis Whenever an intermediate result is saved to disk in the example analysis pipeline, a simple visualization is also created and added to a "report" file. The main analysis package used in the example pipeline, MNE-Python, 11 provides a Report class that compiles a set of figures into 11 Gramfort et al., 2013 a single HTML file. Each script adds (and overwrites) figures to the same report, which will grow in length as more scripts are run. The resulting HTML file contains an easy to navigate visual record of the data flowing through the pipeline. Each participant has their own report file. Guideline 6: Each parameter and filename is defined only once It is not uncommon that a parameter is used in multiple scripts. The sixth guideline states that the value of each parameter should be defined in one place. Instead of copying the value of a parameter into all scripts that need it, the parameter should be imported, i.e., the programmer specifies the location where the parameter is defined and the programming language will take care of fetching the value when it is needed. In the programming literature, this is referred to as the "don't repeat yourself" (DRY) principle. 12 12 Martin, 2008 Things that this guideline aims to prevent: • the same parameter, defined at two locations, with two conflicting values • when changing a parameter, not knowing where else in the code the same change should be made • wasting time copy/pasting things across multiple scripts Importing, rather than copying, eliminates a common source of errors. When we change the value of a parameter, we may not be aware that we need to change it in multiple locations (either we forgot about the copies or we didn't know about them in the first place), resulting in different values being used at different locations and hence errors that can be very difficult to spot as long as the final result looks reasonable. A good tactic for managing parameters is to create a single configuration script, which sole function is to define the values for all parameters. It makes it obvious were to look for the definition of a parameter and decreases the chances of accidentally defining the same parameter at two different locations. Filenames are also parameters, and ones that are commonly shared across scripts too: one script producing a file that another script consumes. Just like other parameters, the guideline mandates that all filenames should be defined once and imported (not copied) by scripts that need it. It is not uncommon for a filename to change when parts of the analysis pipeline are added or removed. By ensuring the change needs to be made in only one location, we can ensure that scripts are not consuming outdated files. A good tactic for managing filenames is to define templates for them in the configuration file. The templates can have placeholders for things like the participant number or experimental condition. See the implementation example for a more thorough explanation. Application of the guideline to the example analysis The example pipeline has a central configuration script config.py which defines all relevant parameters for the analysis, such as filter settings, the list of subjects, the experimental conditions, and so forth. All analysis scripts import the configuration file and thereby gain access to the parameters. Whenever a parameter needs to be changed or added, the configuration file is the single authoritative location where the edit needs to be made and the change is propagated to all analysis scripts. The config.py script starts by offering some machine specific parameters, such as the number of CPU-cores to dedicate to the analysis and where on the disk the data is to be stored. The configuration script queries the hostname, so that different parameters can be specified for different machines. Since the pipeline stores all intermediate results and their visualizations, there is a large number of filenames to deal with. In many cases, each filename is used four times: once in the script generating the file, once in the script consuming the file, and twice in the master script dodo.py. For this reason, a helper class Filenames has been written that offers an efficient way to manage them. The class is used to create an fname namespace that contains short aliases for all filenames used throughout the pipeline. It also leverages Python's native string formatting language to allow quick generation of lists of filenames that adhere to a pattern (e.g., "sub01_raw", "sub02_raw", . . . ). 2.7 Guideline 7: Distinguish files that are part of the official pipeline from other scripts Things that this guideline aims to prevent: • inability to distinguish incomplete or flawed scripts from proper ones • not knowing what files are relevant to the main analysis • not knowing what script is the master script that will run the entire analysis • not knowing which version of the script was last run on the data The seventh guideline calls for an organization system that distinguishes between files that are in a stable state and part of the main pipeline, and files that are work in progress, temporary, or part of analyses on the side. A good system reduces the effort of the cleaning process, making it easier to commit to a regular tidying up of the virtual workplace. It is important that the system does not become burdensome, as a simple system that is actually used is better than a more powerful one that is not. This can be implemented in whatever way suits your workflow best, ranging from simply maintaining a rigid naming convention and folder structure, to using more powerful tools such as a version control system (VCS). Note that a VCS is not an organization system in itself, but merely a tool for implementing one. It is up to the data analyst to devise their own system and have the discipline to stick to it. Application of the guideline to the example analysis During the development of the pipeline, many scripts were written to try out different analysis approaches, conduct tests and do miscellaneous other tasks. The naming system was such that all analysis scripts that are officially part of the pipeline are either prefixed with a number (00_-12_) and all scripts that produce figures for the manuscript with figure_. From time to time, any script lacking such a prefix would be closely scrutinized to determine whether is was still relevant, and if not, deleted. Deleted files were never truly gone though, as the project is managed by the VCS "Git". 13 Git keeps track of the history of a file, allowing to return to previous versions, as well as parallel copies when doing something experimental. Although VCSs are primarily used to facilitate collaboration on a software project, they are useful even when working alone. 14 For one, 14 Vuorre and Curley, 2018 they provide a crucial backup service, allowing recovery from mistakes and thus freedom to experiment and making bold decisions. Secondly, although VCSs do not impose any organizational structure for managing multiple copies of files, they facilitate creating and maintaining one. Conclusion Following the given guidelines improves the chances of the analysis code being correct, by aiming for code that is easy to understand. The key to reducing the complexity of data analysis scripts is to cut up the pipeline into bite-sized chunks. Therefore, the guidelines state that each step of the pipeline should be implemented as a separate script that finishes in a reasonable time, has little to no dependencies on other scripts, writes all intermediate results to disk and visualizes them. In addition, one master script should exist that calls the other scripts in the correct order to execute the complete analysis pipeline. Writing understandable code is a skill that can be honed by forming good habits. Whenever a problem arises in a pipeline, there is an opportunity to look beyond the specific problem to the circumstances that allowed to the problem to occur in the first place and the formation of new habits to prevent such circumstances in the future. However, it is important not to become bogged down in rules. Every new project is a chance for reviewing your habits: keep things that were beneficial and drop things that were not or which costs exceed their utility. Be aware that there is a lot of software tooling available to automate repetitive tasks and perform bookkeeping. Whenever a rule needs enforcing or a repetitive action needs performing, there is likely a software tool available to automate it. However, while they can make it easier to adopt good habits and keep the code organized, they cannot do the job by themselves. Ultimately, it is up to the data analyst to keep things tidy and re-evaluate the design of the analysis pipeline as it grows. When and how to do this is best learned through experience. By reflecting at the end of each project what were good and bad design choices, the analysis pipeline of the next project will be better than the last. Acknowledgements Thanks goes out to all the members of the department of Neuroscience and Biomedical Engineering (NBE) at Aalto University who contributed to the discussion during the lab meeting concerning data analysis pipelines. MvV is supported by the Academy of Finland (grant 310988). • excessive running time of the script • parts of the script are commented out in order to skip a time-consuming step• if-statements being used to toggle parts of the script on and off Figure 1 1Figure 1: For each script in the analysis pipeline, the number of lines of the file, broken down into lines of programming code (code), lines of descriptive comments (comment) and blank lines (blank). The first 13 scripts perform data analysis steps, the next 4 scripts gener- Figure 2 : 2Dependency graph showing how the output of one script is consumed by another. Stacked boxes indicate scripts that are run for each participant.A compelling advantage of scripting is that the code serves as a complete transcript of exactly what analysis steps were performed. However, this transcript is only correct if the latest version of the code is also the version that was used to produce the latest results. During the development of the scripts, we commonly make changes, re-run the script, inspect the result, and make more changes. If we are not careful, the code and results may become desynchronized, especially when multiple versions of the code and results are in play simultaneously. Putting misplaced trust in a wrong transcript can be very frustrating when attempting to reproduce a result. the likelihood of errors. Speed encourages incremental development, running the pipeline often during development to check the intermediate results. Speed encourages exploration, trying different parameters and approaches to obtain the best result possible. And last but not least, speed encourages re-running a script every time it has changed.Apart from using efficient algorithms, the key to obtaining speed is to never repeat a time consuming calculation unnecessarily. If all analysis steps are properly isolated from one another (see guidelines 1 and 2) and all intermediate results are properly stored (see guideline 3), analysis steps for which the corresponding scripts have not changed, need not be run again if a script further down the pipeline has changed. 02_filter.py 03_ica.py 04_epochs.py 01_anatomy.py 06_fsaverage_src.py 05_csd.py 07_forward.py 08_select_vertices.py 09_power.py 10_connectivity.py 11_grand_average_power.py 12_connectivity_stats.py figure_csd.py figure_forward.py figure_power.py figure_connectivity.py 00_fetch_data.py The development of a complex analysis pipeline is seldom a straightforward path from start to finish. 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[ "https://github.com/aaltoimaginglanguage/conpy." ]
[ "VARIATIONAL PROBLEMS CONCERNING SUB-FINSLER METRICS IN CARNOT GROUPS", "VARIATIONAL PROBLEMS CONCERNING SUB-FINSLER METRICS IN CARNOT GROUPS" ]
[ "Enrico Pasqualetto " ]
[]
[]
This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.
10.1051/cocv/2023006
[ "https://arxiv.org/pdf/2202.08634v1.pdf" ]
246,904,489
2202.08634
0ea8063ac5fe9969c5812717cc6e0f94ef16d7dc
VARIATIONAL PROBLEMS CONCERNING SUB-FINSLER METRICS IN CARNOT GROUPS 17 Feb 2022 Enrico Pasqualetto VARIATIONAL PROBLEMS CONCERNING SUB-FINSLER METRICS IN CARNOT GROUPS 17 Feb 2022 This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. Introduction Since three decades, many classical metric problems in Riemannian geometry have been translated to the context of Sub-Riemannian geometry and, in particular, to Carnot groups G, which possess a rich geometry. Indeed, they are connected and simply connected Lie groups whose associated Lie algebra admits a finite-step stratification (see [17,21,22,5]). One of these problems is devoted to the study of a particular class of geodesic distances, bounded from above and below, which was deeply studied in the Euclidean space and in the setting of Lipschitz manifolds (see [26,12,11,10]). Our first purpose is to generalize this class to Carnot groups, in other words, we consider all geodesic maps d : Ω × Ω → R locally equivalent to the so-called Carnot-Carathéodory distance d cc , defined on an open subset Ω ⊂ G and we say that they belong to D cc (Ω) if: 1 α d cc (x, y) ≤ d(x, y) ≤ αd cc (x, y) ∀x, y ∈ Ω, (1.1) for some α ≥ 1. According to [26,11], it is quite natural to construct, on the so-called horizontal bundle HG, the family of metrics ϕ d : HG → [0, +∞) associated to d ∈ D cc (Ω) by differentiation: ϕ d (x, v) := lim sup tց0 d x, x · δ t exp(d x τ x −1 [v]) t for every (x, v) ∈ HG. Inspired by the notation for horizontal curves introduced by Scott D. Pauls in [23], in the previous definition we denote with x · δ t exp(d x τ x −1 [v]) the dilation curve starting from the point x ∈ G with direction given by the left translation at the identity of a horizontal vector defined on the fiber H x G. In particular, it turns out that ϕ d is a sub-Finsler convex metric. These objects play an important role in the setting of the so-called Sub-Finsler Carnot groups (for a reference see for example [16,Section 6]), playing the same role of Finsler metrics with the difference that they are defined only on the horizontal bundle. Definition 1.1. ϕ : HG → [0, +∞) is a sub-Finsler convex metric belonging to M α cc (G) if 1) ϕ : HG → R is Borel measurable, where HG is endowed with the product σ-algebra; 2) ϕ(x, δ * λ v) = |λ|ϕ(x, v) for every (x, v) ∈ HG and λ ∈ R; 3) 1 α v x ≤ ϕ(x, v) ≤ α v x for every (x, v) ∈ HG; 4) ϕ(x, v 1 + v 2 ) ≤ ϕ(x, v 1 ) + ϕ(x, v 2 ) for every x ∈ G and v 1 , v 2 ∈ H x G. In particular, ϕ(x, ·) is a norm on H x G for every x ∈ G. At the same time, it is also natural to consider the length functional induced by the metric derivative ϕ d . We will show that we can reconstruct the distance d by minimizing the corresponding functional (Theorem 3.5). The first purpose of this paper is to compare the asymptotic behaviour of different kinds of functionals involving distances defined on a given open subset Ω of a Carnot group G. Indeed, the most common approach in order to study the following variational problems relies on Γconvergence of the corresponding functionals (see Section 4). In particular, inspired by the proof contained in [8], we state the equivalence between the Γ-convergence of the functionals L n and J n associated to a sequence of distances (d n ) n∈N ⊂ D cc (Ω) respectively through L n (γ) =ˆ1 0 ϕ dn (γ(t),γ(t)) dt and J n (µ) =ˆΩ ×Ω d n (x, y) dµ(x, y), where γ : [0, 1] → Ω is a horizontal curve (Definition 2.1) and µ is a positive and finite Borel measure on Ω × Ω. This kind of result has been already studied in the literature, especially for what concerns the homogenization of Riemannian and Finsler metrics ( [1,3]). Moreover, we show an additional characterization when Ω is bounded (Theorem 4.1, point (iv)). The second purpose is to study a different application: the intrinsic analysis of sub-Finsler metrics. More precisely, under suitable regularity assumptions on the metric under consideration, we prove the following result (Theorem 5.13): (1. 2) ϕ(x, ∇ G f (x)) = Lip δϕ f (x) for a.e. x ∈ G, where f : Ω ⊂ G → R is a Pansu-differentiable function, δ ϕ is the distance defined in (1.3) below, ϕ ∈ M α cc (Ω) is a sub-Finsler convex metric, and the pointwise Lipschitz constant of f is given by Lip δϕ f (x) = lim sup y→x |f (y) − f (x)| δ ϕ (x, y) for every x ∈ G. The equality (1.2) may be regarded as a generalization of a result achieved in [25], and further generalized by Chang Y. Guo to admissible Finsler metrics defined on open subsets of R n , in [14]. Then, in order to prove (1.2), we crucially observe that the quantity (1.3) δ ϕ (x, y) := sup |f (x) − f (y)| f : G → R Lipschitz, ϕ(·, ∇ G f (·)) ∞ ≤ 1 coincides with the intrinsic distance d ϕ ⋆ , induced by the dual metric. This happens, for instance, when we assume that the sub-Finsler metric ϕ is either lower semicontinuous or upper semicontinuous on the horizontal bundle (see Theorem 5.11 and Corollary 5.12). These results are a generalization of the analogous statement in [12], due to De Cecco and Palmieri. The proof of Theorem 5.11 heavily relies on two results contained in [18]. The first one allows us to approximate an upper semicontinuous sub-Finsler metric with a family of Finsler metrics. The second result lets us approximate from below the sub-Finsler distance with a family of induced Finsler distances. Finally, we show that in many cases the distance δ ϕ , albeit defined as a supremum among Lipschitz functions, is actually already determined by smooth functions (cf. Proposition 5.15). An important step in proving this fact is to approximate (say, uniformly on compact sets) any 1-Lipschitz function with a sequence of smooth 1-Lipschitz functions; here, the key point is that the Lipschitz constant is preserved. Since this approximation result holds in much greater generality (for instance, on possibly rank-varying sub-Finsler structures) and might be of independent interest, we will treat it in Appendix A. We now give a short descriptive plan of the paper. In Section 2 we collect some of the basic geometric facts about Carnot groups and we present all preliminaries about horizontal bundles, horizontal curves, Pansu differentiability for Lipschitz functions, and the main concepts related to sub-Finsler convex metrics. In particular, we prove some properties about dual metrics associated to sub-Finsler metrics. In Section 3 we introduce the main concept of metric derivative and we show its convexity on the horizontal fibers. Moreover, we prove a classical length representation theorem through a distance reconstruction result. Section 4 contains the equivalence theorem between the uniform convergence of distances in D cc (Ω) and Γ-convergence of the length and energy functionals. Finally, in Section 5 we introduce the main concepts of intrinsic distance and metric density and we prove the main theorems of this paper. Acknowledgements. The authors are grateful to Andrea Pinamonti, Francesco Serra Cassano, and Gioacchino Antonelli for several discussions on the topics of this paper, as well as to Davide Vittone for the useful comments about Theorem A.1. The second named author is supported by the Balzan project led by Luigi Ambrosio. Preliminaries 2.1. Carnot groups. A connected and simply connected Lie group G is said to be a Carnot group of step k if its Lie algebra g admits a step k stratification, i.e., there exist linear subspaces g 1 , . . . , g k of g such that (2.1) g = g 1 ⊕ . . . ⊕ g k , [g 1 , g i ] = g i+1 , g k = {0}, [g 1 , g k ] = {0}, where [g 1 , g i ] is the subspace of g generated by the commutators [X, Y ] with X ∈ g 1 and Y ∈ g i . g 1 is called the first stratum of the stratification and we will denote m := dim g 1 ≤ n = dim g. Choose a basis e 1 , . . . , e n of g adapted to the stratification, that is such that e h j−1 +1 , . . . , e h j is a basis of g j for each j = 1, . . . , k. Let X 1 , . . . , X n be the family of left invariant vector fields such that at the identity element e of G we have X i (e) = e i for every i = 1, . . . , n. Given (2.1), the subset X 1 , . . . , X m generates by commutation all the other vector fields; we will refer to X 1 , . . . , X m as generating horizontal vector fields of the group. Given an element x ∈ G, we denote by τ x : G → G the left translation by x, which is given by τ x z := x · z for every z ∈ G, where · is the group law in G. Moreover, it holds that the map τ x is a smooth diffeomorphism, thus we can consider its differential d y τ x : T y G → T x·y G at any point y ∈ G. 2.2. Exponential map and Sub-Riemannian structures. We recall that the exponential map exp : g → G is defined as follows. First, we identify the Lie algebra g with the tangent space at the identity T e G. Given any vector v ∈ T e G and denoting by γ : [0, 1] → G the (unique) smooth curve satisfying the ODE (2.2)   γ (t) = d e τ γ(t) [v] for every t ∈ [0, 1], γ(0) = e, we define exp(v) = e v := γ(1), where d e τ γ(t) [v] is a left-invariant vector field. It holds that exp is a diffeomorphism and any p ∈ G can be written in a unique way as p = exp(p 1 X 1 + · · · + p n X n ) = e p 1 X 1 +···+pnXn , where v = n i=1 p i X i . We can identify p with the n-tuple (p 1 , . . . , p n ) ∈ R n and G with R n where the group operation · satisfies (see [4] and [21, Section 7]) x · y = exp exp −1 (x) ⋆ exp −1 (y) for every x, y ∈ G, where ⋆ denotes the group operation determined by the Campbell-Baker-Hausdorff formula, see e.g. [19,5]. The subbundle of the tangent bundle T G that is spanned by the vector fields X 1 , . . . , X m plays a particularly important role in the theory, it is called the horizontal bundle HG; the fibers of HG are H x G = span {X 1 (x), . . . , X m (x)} . A sub-Riemannian structure can be defined on G in the following way. Consider a scalar product ·, · e on g 1 = H e G that makes {X 1 , . . . , X m } an orthonormal basis. Moreover, by left translating the horizontal fiber in the identity, we obtain that H x G = d e τ x (g 1 ) and, by [2,Lemma 7.48], the map T G ∋ (x, v) → (x, d x τ x −1 [v]) ∈ G × T e G is an isomorphism between T G and G × g, in other words, the tangent bundle is trivial. This allows us to define the scalar product ·, · x on H x G as v, w x := d x τ x −1 [v], d x τ x −1 [w] e for every v, w ∈ H x G. Notice that {X 1 (x), . . . , X m (x)} is an orthonormal basis of H x G with respect to ·, · x . We denote by · x the norm induced by ·, · x , namely v x := v, v x for every v ∈ H x G. By the left invariance of the sub-Riemannian structure, for every v ∈ H x G there exists a unique vectorv ∈ H e G such that v = d e τ x (v) and we get that (2.3) v e = d e τ x [v] x = v x for every (x, v) ∈ HG. A further choice of the norm would not change the biLipschitz equivalence class of the sub-Riemannian structure. This is the reason why we may assume that the norm · x is coming from a scalar product ·, · x (see [17]). If y = (y 1 , . . . , y n ) ∈ G and x ∈ G are given, we set the projection map as: π x : G → H x G as π x (y) = m j=1 y j X j (x). The map y → π x (y) is a smooth section of H x G and it is linear in y. Furthermore, if v ∈ g 1 , by exponential coordinates it holds that (2.4) π x (e v ) = π x (v 1 , . . . , v m , 0 . . . , 0) = m i=1 v i X i (x) = d e τ x [v] for all x ∈ G and v ∈ g 1 . 2.3. Dilations and Carnot-Carathéodory distance. For any λ > 0, we denote by δ ⋆ λ : g → g the unique linear map such that δ ⋆ λ X = λ i X, ∀X ∈ g i . The maps δ ⋆ λ : g → g are Lie algebra automorphisms, i.e., δ ⋆ λ ([X, Y ]) = [δ ⋆ λ X, δ ⋆ λ Y ] for all X, Y ∈ g. For every λ > 0, the map δ ⋆ λ naturally induces an automorphism on the group δ λ : G → G by the identity (2.5) δ λ (x) = (exp • δ ⋆ λ • log)(x) , where with log we denote the inverse map of exp. It is easy to verify that both the families (δ * λ ) λ>0 and (δ λ ) λ>0 are one-parameter groups of automorphisms (of Lie algebras and of groups, respectively), namely, δ ⋆ λ • δ ⋆ η = δ ⋆ λη and δ λ • δ η = δ λη for all λ, η > 0. The maps δ ⋆ λ , δ λ are both called dilations of factor λ. Let us remark that, since δ ⋆ λ v = λv for every v ∈ g 1 = H e G and thanks to (2.5), one can easily realize that (2.6) δ λ exp(v) = exp(λv) for every v ∈ H e G and λ > 0. According to [23], we can extend dilations also to negative parameters λ < 0, denoting δ ⋆ |λ| (X) = δ ⋆ λ (−X) = |λ| i (−X) for X ∈ g i and, in the present paper, we exploit this fact only on the fist layer g 1 . Indeed, it holds that π x (δ λ e w ) = λπ x (e w ) for every w ∈ g 1 and λ > 0. Since the dilations are defined only on the Lie algebra g, we extend them, by left translations, on the entire T x G, for every x ∈ G. This allows us to write δ * λ v = λv for every λ > 0, x ∈ G and v ∈ H x G. (t) = m i=1 h i (t)X i (γ(t)) for a.e. t ∈ [a, b]; • |h| ∈ L ∞ (a, b). The length of such a curve is given by L G (γ) =´b a γ(t) γ(t) dt. Chow-Rashevskii's theorem [5,Theorem 19.1.3] asserts that any two points in a Carnot group can be connected by a horizontal curve. Hence, the following definition is well-posed. We remark that, by Chow-Rashevskii's Theorem, the Carnot-Carathéodory distance is finite. Moreover, it is homogeneous with respect to dilations and left translations, more precisely, for every λ > 0 and for every x, y, z ∈ G one has d cc (δ λ x, δ λ y) = λd cc (x, y), d cc (τ x y, τ x z) = d cc (y, z). This immediately implies that τ x (B(y, r)) = B(τ x y, r) and δ λ B(y, r) = B(δ λ y, λr), where B(x, r) = y ∈ G : d cc (y, x) < r is the open ball centered at x ∈ G with radius r > 0. In the sequel, we will need the following crucial estimate, proved in [21, Theorem 1.5.1]. Theorem 2.3. Let G be a Carnot group of step k and let K ⊂ G be a compact set. Then there exists C K = C(K) > 1 such that (2.7) C −1 K |x − y| ≤ d cc (x, y) ≤ C K |x − y| 1 k , ∀x, y ∈ K. The following lemma shows the biLipschitz equivalence between the Carnot-Carathéodory distance and the norm induced by the scalar product. Lemma 2.4. There exists a constant c ≥ 1 such that (2.8) 1 c v e ≤ d cc (e, exp v) ≤ c v e for every v ∈ g 1 . Proof. Denote by S the unit sphere of (H e G, · e ), namely S := {v ∈ H e G : v e = 1}. Define the function η : S → [0, +∞) as η(v) := d cc (e, exp v) for every v ∈ S. By Theorem 2.3, η is continuous on the compact set S. Then we can find c ≥ 1 such that 1/c ≤ η(v) ≤ c holds for every v ∈ S. We can thus conclude by 1-homogeneity: since d cc (e, exp(λv)) = λd cc (e, exp v) for every λ > 0 and v ∈ S, we deduce that η(v/ v e ) = d cc (e, exp v)/ v e for every v ∈ H e G \ {0} and thus 1 c ≤ d cc (e, exp v) v e ≤ c for every v ∈ H e G \ {0}, which yields (2.8). 2.4. Differentiability in Carnot Groups. We recall some basic definitions regarding differentiability in Carnot groups. L(x · y) = L(x) + L(y) and L(δ λ (x)) = λL(x) for every x, y ∈ G and λ > 0. Now we are ready to introduce the following fundamental notion of differentiability, see [22]. Definition 2.6. Let Ω ⊂ G be an open subset. A map f : Ω → R is Pansu differentiable at x ∈ Ω if there exists a homogeneous homomorphism L x : G → R such that lim y→x f (x) − f (y) − L x [y −1 · x] d cc (y, x) = 0. The map L x := d G f (x) : G → R is called Pansu differential of f at x. Remark 2.7. If f : Ω → R is differentiable at x ∈ Ω, then X j f (x) exists for any j = 1, . . . , m, and for any v ∈ G we have d G f (x)[v] = ∇ G f (x), π x (v) x , where the horizontal gradient ∇ G f (x) is defined as ∇ G f (x) := m i=1 X i f (x)X i (x). We stress that the notion of the horizontal gradient only depends on the choice of the generating horizontal vector fields and hence it is uniquely determined by the sub-Riemannian metric chosen on Ω ⊂ G. Unless otherwise specified, by a Lipschitz function f : Ω → R we mean a function that is Lipschitz with respect to the Carnot-Carathéodory distance d cc , namely there exists a constant C ≥ 0 such that |f (x) − f (y)| ≤ Cd cc (x, y) for every x, y ∈ Ω. Moreover, f : G → R is said to be a locally Lipschitz function if it is Lipschitz on every open bounded set Ω ⊂ G. The notion of Pansu differentiability is motivated by the following result due to Pansu [22] (see also [20] for a similar result in a more general setting). In general, it states that any Lipschitz map between two Carnot groups has almost everywhere a differential which is a homogeneous homomorphism. Below we report the statement of Pansu's result only for realvalued Lipschitz maps, since this is sufficient for our purposes. f : Ω → R we have that f is Pansu differentiable at L n -a.e. x ∈ Ω. Let x ∈ G andv ∈ g 1 , then the map t → x · δ t exp(v) is Lipschitz. Hence, if f : G → R is Lipschitz, then the composition t → f (x · δ t exp(v))(2.9) ∇ G f (x), π x (v) = lim t→0 f (x · δ t ev) − f (x) t for a.e. x ∈ G and for every v ∈ H x G, wherev = d x τ x −1 [v]. 2.5. Sub-Finsler Metrics and Duality. Inspired by [26], now we introduce the following definition. Let G be a Carnot metric group and let Ω ⊂ G be an open set. If α ≥ 1, we introduce the family D cc (Ω) containing all the geodesic distances d : Ω × Ω → [0, +∞) verifying 1 α d cc (x, y) ≤ d(x, y) ≤ αd cc (x, y) ∀x, y ∈ Ω. (2.10) Therefore, D cc (Ω) depends on α and we omit such dependence for the sake of brevity. Notice that we may have D cc (Ω) = ∅ if the domain Ω ⊂ G is disconnected or it has an irregular boundary. We endow D cc (Ω) with the topology of the uniform convergence on compact subsets of Ω × Ω. We will see in the Proof of Theorem 4.1 that D cc (Ω) is compact with respect to such topology. Definition 2.9. For α ≥ 1, we define M α cc (G) as the family of all those maps ϕ : HG → [0, +∞), that we will call metrics on HG, verifying the following properties: (1) ϕ : HG → R is Borel measurable, where HG is endowed with the product σ-algebra; (2) ϕ(x, δ * λ v) = |λ|ϕ(x, v) for every (x, v) ∈ HG and λ ∈ R; (3) 1 α v x ≤ ϕ(x, v) ≤ α v x for every (x, v) ∈ HG. Moreover, we will say that ϕ ∈ M α cc (G) is a sub-Finsler convex metric if ϕ(x, v 1 + v 2 ) ≤ ϕ(x, v 1 ) + ϕ(x, v 2 ) (2.11) for every x ∈ G and v 1 , v 2 ∈ H x G (or equivalently if ϕ(x, ·) is a norm for every x ∈ G). According to the preliminaries, conditions (2) and (3) are well-defined with respect to the exponential and the dilation map. Moreover, let us remark that condition (1) is equivalent to the Borel measurability with respect to the product space G × g. Our next aim is to introduce the dual of a metric belonging to M α cc (G). Definition 2.10 (Dual Metric). Let us take ϕ ∈ M α cc (G). We define the dual metric ϕ ⋆ : HG → [0, +∞) of ϕ as (2.12) ϕ ⋆ (x, v) := sup | v, w x | ϕ(x, w) : w ∈ H x G, w = 0 . In general, the dual metric enjoys many useful properties, as we can see below. Proposition 2.11. For any ϕ ∈ M α cc (G), it holds that ϕ ⋆ is a sub-Finsler convex metric, and in particular (2.13) 1 α v x ≤ ϕ ⋆ (x, v) ≤ α v x for every (x, v) ∈ HG. Proof. It is straightforward to prove property (2) since for every v, w ∈ H x G and λ ∈ R we have that δ * λ v, w x = λ v, w x . Passing to the supremum over all w ∈ H x G \ {0}, we obtain that ϕ ⋆ (x, δ * λ v) = |λ|ϕ ⋆ (x, v) . In order to prove the convexity on the horizontal bundle, let us consider v 1 , v 2 ∈ H x G, then we obtain: ϕ ⋆ (x, v 1 + v 2 ) ≤ sup | v 1 , w x | ϕ(x, w) + | v 2 , w x | ϕ(x, w) : w ∈ H x G, w = 0 ≤ ϕ ⋆ (x, v 1 ) + ϕ ⋆ (x, v 2 ). Moreover, if we take w ∈ H x G \ {0} it holds that 1 α | v, w x | w x ≤ | v, w x | ϕ(x, w) ≤ α | v, w x | w x . By taking the supremum over all w ∈ H x G \ {0}, we obtain (2.13) and accordingly property (3) of Definition 2.9. Therefore, ϕ ⋆ (x, ·) is a norm, thus in particular it is continuous. Finally, chosen a dense sequence (w n ) n in g 1 \ {0}, we have that (d e τ x [w n ]) n is dense in H x G for every x ∈ G, thus for any v ∈ g 1 we can write ϕ ⋆ (x, d e τ x [v]) = sup n∈N | d e τ x [v], d e τ x [w n ] x | ϕ(x, d e τ x [w n ]) = sup n∈N | v, w n e | ϕ(x, d e τ x [w n ]) for every x ∈ G, which shows that G ∋ x → ϕ ⋆ (x, d e τ x [v] ) is measurable and accordingly property (1) of Definition 2.9 is satisfied. All in all, ϕ ⋆ is a sub-Finsler convex metric. We can characterize sub-Finsler convex metrics ϕ in terms of the bidual metric ϕ ⋆⋆ . if ϕ(x, v) = ϕ ⋆⋆ (x, v) for every (x, v) ∈ HG. Proof. ⇐ Since ϕ is 1-homogeneous in the second entry, in order to prove that ϕ( x, ·) is convex on H x G, it is sufficient to prove that ϕ(x, v 1 + v 2 ) ≤ ϕ(x, v 1 ) + ϕ(x, v 2 ) for every x ∈ G and v 1 , v 2 ∈ H x G. By assumption ϕ(x, v) = ϕ ⋆⋆ (x, v) for all (x, v) ∈ HG, then for every v 1 , v 2 ∈ H x G we can write ϕ(x, v 1 + v 2 ) = ϕ ⋆⋆ (x, v 1 + v 2 ) ≤ sup | v 1 , w x | ϕ ⋆ (x, w) + | v 2 , w x | ϕ ⋆ (x, w) : w ∈ H x G, w = 0 ≤ ϕ ⋆⋆ (x, v 1 ) + ϕ ⋆⋆ (x, v 2 ) = ϕ(x, v 1 ) + ϕ(x, v 2 ). ⇒ Given that ϕ(x, ·) is convex and 1-homogeneous, ϕ(x, ·) is a norm on H x G and ϕ ⋆⋆ (x, ·) represents its bidual norm. Since each horizontal fiber is finite-dimensional, the conclusion follows. At the end, we present the following properties that we will need in the last section. Lemma 2.13. Let ϕ ∈ M α cc (G) be a sub-Finsler convex metric. Then the following hold: i) If ϕ is lower semicontinuous, then ϕ ⋆ is upper semicontinuous. ii) If ϕ is upper semicontinuous, then ϕ ⋆ is lower semicontinuous. In particular, if ϕ is continuous, then ϕ ⋆ is continuous. Proof. To prove i) suppose ϕ is lower semicontinuous. Fix (x, v) ∈ HG and (x n , v n ) ∈ HG such that (x n , v n ) → (x, v), in the sense that d cc (x n , x) + d xn τ x −1 n [v n ] − d x τ x −1 [v] e → 0. Possibly passing to a not relabeled subsequence, we can assume that lim sup n ϕ ⋆ (x n , v n ) is actually a limit. Given any n ∈ N, there exists w n ∈ H xn G such that ϕ(x n , w n ) = 1 and ϕ ⋆ (x n , v n ) = | v n , w n xn |. Since the fiber of the horizontal bundle is compact, then there exists w ∈ H x G such that (up to a not relabeled subsequence) (x n , w n ) → (x, w). Being ϕ lower semicontinuous, we deduce that ϕ(x, w) ≤ lim inf n→∞ ϕ(x n , w n ) ≤ 1. Therefore, we conclude that ϕ ⋆ (x, v) ≥ | v, w x | ϕ(x, w) ≥ lim n→∞ | v n , w n xn | = lim sup n→∞ ϕ ⋆ (x n , v n ), which proves that ϕ ⋆ is upper semicontinuous. The assertion ii) can be proved noticing that if ϕ is upper semicontinuous, then ϕ ⋆ is lower semicontinuous as it can be expressed as a supremum of lower semicontinuous functions. Notation. For any d ∈ D cc (Ω) and a ∈ Ω, we denote by d a : 1], Ω) the set of horizontal curves. In the sequel, sometimes we omit the unit interval since we are going to consider curves defined on and we assume that such curves are parametrized with constant velocity. Moreover, for every Lebesgue null set N ⊂ Ω, we set P(Ω, N ) as the class of all horizontal curves γ : Ω → [0, +∞) the fixed-point distance map d a (x) := d(a, x[0, 1] → Ω such that L 1 ({t ∈ [0, 1] | γ(t) ∈ N }) = 0, where L 1 is the standard 1-dimensional Lebesgue measure. By [10, Lemma 2.2] we have that P(Ω, N ) = ∅. Furthermore, with HΩ we mean the restriction of the horizontal bundle HG to Ω, i.e., HΩ := {(x, v) ∈ HG : x ∈ Ω}. If not otherwise stated, for every v ∈ T x G and x ∈ G sometimes we will denote byv : = d x τ x −1 [v] the representative vector of T x G ∋ v in the Lie algebra g. Metric Derivative and Length Representation Theorem Given a geodesic distance d ∈ D cc (G), it is quite natural to consider the associated metric given by differentiation. The next definition is inspired by the ones proposed in [23,26] but, in our setting, we necessarily have to define it only on the horizontal bundle HG. Definition 3.1 (Metric derivative). Given any d ∈ D cc (G), we define the map ϕ d : HG → [0, +∞) as ϕ d (x, v) := lim sup t→0 d(x, x · δ t exp d x τ x −1 [v]) |t| for every (x, v) ∈ HG. Note that we translate the vector v ∈ H x G to e via the differential of the left traslation, because the exponential map is defined on the first stratum g 1 = H e G. The next Lemma tells us that the metric derivative is actually a metric. Lemma 3.2. For every d ∈ D cc (G) we have that ϕ d ∈ M cα cc (G), for some c ≥ 1 independent of d. Proof. In order to prove (1), let just observe that ϕ d (x, v) = lim n→∞ sup t∈Q: |t|<1/n d(x, x · δ t exp d x τ x −1 [v]) |t| for every (x, v) ∈ HG. Let us verify (2). Pick x ∈ G, v ∈ H x G and t, λ ∈ R. Since the differential of the left translation is a diffeomorphism and from the equality (2.6) we have that δ t exp d x τ x −1 [δ ⋆ λ (v)] = δ t δ λ exp d x τ x −1 [v] . Therefore ϕ d (x, δ * λ v) = lim sup t→0 d(x, x · δ t δ λ e dxτ x −1 [v] ) |t| = |λ| lim sup t→0 d(x, x · δ tλ e dxτ x −1 [v]) ) |tλ| = |λ|ϕ(x, v). In order to show (3), fix x ∈ G and v ∈ H x G. Since d ∈ D cc (G) we can write ϕ d (x, v) ≤ α lim sup t→0 d cc (x, x · δ t e dxτ x −1 [v] ) |t| = α d cc (e, exp d x τ x −1 [v]) ≤ c α d x τ x −1 [v] e where in the last inequality we applied Lemma 2.4. The estimate from below can be proved similarly. Finally, using the left invariance of the norm, for every (x, v) ∈ HG, we get that 1 cα v x ≤ ϕ d (x, v) ≤ c α v x and the conclusion follows. The next result comes from [21, Proposition 1.3.3] and a general proof can be find in [2, Proposition 3.50]. It says that Lipschitz curves and horizontal ones essentially coincide when the L ∞ -norm of the canonical coordinates is finite. Proposition 3.3. A curve γ : [a, b] → Ω ⊂ G is Lipschitz if and only if it is horizontal and h L ∞ (a,b) ≤ L, where L is the Lipschitz constant. In general, if (M, d) is a metric space and γ : [0, 1] → M a Lipschitz curve, then the classical metric derivative is defined as |γ(t)| d := lim s→0 d(γ(t + s), γ(t)) |s| . The existence of the previous limit is a general fact that holds in any metric space (see [7, Theorem 2.7.6]); indeed |γ(t)| d exists for a.e. t ∈ [0, 1], it is a measurable function and it satisfies the equality (3.1) L d (γ) =ˆ1 0 |γ(t)| d dt. where the classical length functional of a rectifiable curve is defined as L d (γ) = sup {0≤t 1 <...<t k ≤1} k−1 i=1 d(γ(t i+1 ), γ(t i )),|γ(t)| dcc = lim s→0 d cc (γ(t + s), γ(t)) |s| = |h(t)| for a.e. t ∈ [0, 1] and lim s→0 δ 1 s γ(t) −1 · γ(t + s) = (h 1 (t), . . . , h m (t), 0, . . . , 0) for a.e. t ∈ [0, 1]. The second claim is proved in [21, Lemma 2.1.4] and it gives a characterization of horizontal curves in terms of canonical coordinates. Therefore, by Proposition 3.3, a Lipschitz curve is horizontal and, with abuse of notation, we set the following quantity: expγ(t) := exp d γ(t) τ γ(t) −1 [γ(t)] = h 1 (t), . . . , h m (t), 0 . . . , 0 , for a.e. t ∈ [0, 1]. Now, let d ∈ D cc (G), then (G, d) inherits the structure of a metric space and hence, for any horizontal curve γ : [0, 1] → G we get that (3.2) |γ(t)| d = ϕ d (γ(t),γ(t)) for a.e. t ∈ [0, 1]. Finally, the identities (3.1) and (3.2) imply the following well known length reconstruction result. Theorem 3.5. Let d ∈ D cc (G). Then, for every horizontal curve γ : [0, 1] → G we have (3.3) L d (γ) =ˆ1 0 ϕ d (γ(t),γ(t)) dt. Moreover, for every x, y ∈ G we have d(x, y) = inf ˆ1 0 ϕ d (γ(t),γ(t)) dt : γ ∈ H([0, 1], G), γ(0) = x, γ(1) = y . 3.1. Convexity of ϕ d . The aim of the present section is to prove that if d ∈ D cc (G), then ϕ d is also a sub-Finsler convex metric. In order to make this, first we have to recall some technical results. Lemma 3.6. Let ψ : G → R be a locally bounded, Borel function and v ∈ H x G \ {0}. Then (3.4) ψ(x) = lim tց0 1 tˆt 0 ψ(x · δ s ev) ds, for L n -a.e. x ∈ G. Proof. Given any fixed y ∈ G, we have that R ∋ t → ψ(y · δ t ev) ∈ R is a locally bounded and Borel function, thus an application of Lebesgue's differentiation theorem guarantees that for L 1 -a.e. r ∈ R (3.5) ψ(y · δ r ev) = lim tց0 1 tˆt 0 ψ(y · δ r+s ev) ds. In particular, an application of Fubini's theorem ensures that the set Γ := (y, r) ∈ G × R (3.5) holds has L n+1 -full measure, thus for L 1 -a.e. r ∈ R we have that (3.5) holds for L n -a.e. y ∈ G. Fix any such r ∈ R and a L n -negligible set N ⊂ G satisfying (3.5) for every y ∈ G \ N . Calling σ z : G → G the right-translation map σ z w := w · z for every z, w ∈ G and defining N ′ := σ δrev (N ), we thus have that ψ(x) = lim tց0 1 t´t 0 ψ(x·δ s ev) ds holds for every x ∈ G \N ′ . Here, we also used the fact that δ r+s ev = δ r ev · δ s ev, which is in turn guaranteed by the fact thatv belongs to the first layer (see [23,Lemma 2.2]). Therefore, in order to prove (3.4) it is only left to check that N ′ is L n -negligible. This can be achieved by exploiting the right-invariance of the measure L n (see e.g. [21, Proposition 1.7.7]), namely the fact that L n (E · z) = L n (E) holds whenever E ⊂ G is a Borel set and z ∈ G. Indeed, this implies that (σ δrev ) # L n = L n , because for any Borel set E ⊂ G it holds that (σ δr ev ) # L n (E) = L n (σ −1 δr ev (E)) = L n (σ δ −r ev (E)) = L n (E · δ −r ev) = L n (E). In particular, we conclude that L n (N ′ ) = (σ δrev ) # L n (N ′ ) = L n (N ) = 0, as required. Lemma 3.7. Let d ∈ D cc (Ω), ϕ ∈ M α cc (Ω), and N ⊂ Ω be such that |N | = 0. Suppose that for every γ ∈ P(Ω, N ) we have that d(γ(0), γ(1)) ≤ˆ1 0 ϕ(γ(t),γ(t)) dt. Then for every fixed a ∈ Ω, for a.e. x ∈ Ω, and for every v ∈ H x G, we have that | ∇ G d a (x), v x | ≤ lim inf t→0 d(x, x · δ t ev) t ≤ lim sup t→0 d(x, x · δ t ev) t ≤ ϕ(x, v). Proof. Let N be as in the hypothesis and v ∈ H x G. For a ∈ Ω, let E(a, v) be the set of all x ∈ Ω for which d a is Pansu differentiable for a.e. x ∈ Ω and the map [0, 1] ∋ t → x · δ t ev belongs to P(Ω, N ), with t small enough. Moreover, thanks to Lemma 3.6, we can assume that lim tց0 1 tˆt 0 ϕ(x · δ s ev, v) ds = ϕ(x, v). By Pansu-Rademacher Theorem |Ω \ E(a, v)| = 0 and, if x ∈ E(a, v), applying the identity (2.9) we have that lim t→0 d a (x) − d a (x · δ t ev) − ∇ G d a (x), π x (δ t e −v ) x |t| = 0. Hence, by the reverse triangle inequality we can assert that | ∇ G d a (x), v x | ≤ lim inf t→0 d a (x · δ t ev) − d a (x) t ≤ lim inf t→0 d(x, x · δ t ev) t ≤ lim sup t→0 d(x, x · δ t ev) t ≤ lim tց0 1 tˆt 0 ϕ(x · δ s ev, v) ds = ϕ(x, v). Pick a countable dense subset F ⊂ H x G and put E(a) = ∩ y∈F E(a, y). Then |Ω \ E(a)| = 0 and for all x ∈ E(a) and all v ∈ H x G we obtain the same estimate above. We observe that the previous Lemma could be proved under more general conditions. Indeed, the distance needs only to be geodesic in its domain. Lemma 3.8. Let ϕ ∈ M α cc (Ω) be a sub-Finsler convex metric, let d ∈ D cc (Ω) and Θ ⊂ Ω be a countable dense set of Ω. If ϕ(x, ∇ G d a (x)) ∞ ≤ 1 ∀a ∈ Θ, then there exists N ⊂ Ω such that |N | = 0 and for every γ ∈ P(Ω, N ) d(γ(0), γ(1)) ≤ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt. Proof. Let E be the subset of x ∈ Ω where the function y → d a (y) is Pansu-differentiable for every a ∈ Θ. Since Θ is countable, we have that N = Ω \ E has zero Lebesgue measure. Let γ ∈ P(Ω, N ), pick a ∈ Θ and set f (t) := d a (γ(t)) for every t ∈ [0, 1], then d a (γ(1)) − d a (γ(0)) = f (1) − f (0) ≤ˆ1 0 d dt f (t) dt =ˆ1 0 ∇ G d a (γ(t)), h(t) γ(t) dt ≤ˆ1 0 ϕ γ(t), ∇ G d a (γ(t)) ϕ ⋆ (γ(t),γ(t)) dt ≤ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt. Now, by density of Θ in Ω we can choose a sequence {a k } k∈N ⊂ Θ converging to γ(0), obtaining d(γ(0), γ(1)) = lim k→∞ d a k (γ(1)) − d a k (γ(0)) ≤ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt. Theorem 3.9. Let d ∈ D cc (Ω). Then ϕ d is a sub-Finsler convex metric. In particular, for almost all x ∈ Ω and all v ∈ H x G (3.6) ϕ d (x, v) = lim t→0 d(x, x · δ t ev) |t| . Proof. Take a countable dense subset Θ of Ω and, for each a ∈ Θ, we consider Σ a a negligible Borel subset of Ω which contains all points where d a is not Pansu-differentiable. For every (x, v) ∈ HΩ we define ξ(x, v) := sup a∈Θ ∇ G d a (x), v x if x ∈ Ω \ a∈Θ Σ a ; 0 otherwise. For ε > 0 we define ξ ε : HΩ → [0, +∞) as ξ ε (x, v) := ξ(x, v) + ε v x , that is a Borel measurable function in HΩ and it is a sub-Finsler convex metric. Indeed, if we take v 1 , v 2 ∈ H x G we can estimate in this way ξ ε (x, v 1 + v 2 ) = sup a∈Θ ∇ G d a (x), v 1 + v 2 x + ε v 1 + v 2 x ≤ ξ(x, v 1 ) + ξ(x, v 2 ) + ε v 1 + v 2 x ≤ ξ ε (x, v 1 ) + ξ ε (x, v 2 ). The homogeneity w.r.t. the second variable comes from the equality d e τ x [δ ⋆ λv ] = λd e τ x [v] wherev = d x τ x −1 [v]. Moreover, if a ∈ Θ we get that ∇ G d a (x), v x ≤ ξ(x, v) ≤ ξ ε (x, v) for a.e. x ∈ Ω and v ∈ H x G. Thus, by definition of dual metric, we have (3.7) ∇ G d a (x), v x ξ ε (x, v) ≤ 1 ⇒ ξ ⋆ ε (x, ∇ G d a (x)) ∞ ≤ 1. Being Θ countable, by Lemma 3.8, there exists a Lebesgue null set N ⊂ Ω such that, the horizontal curve γ(t) = x · δ t ev belongs to P(Ω, N ), and for every small t > 0 we can infer that d(x, x · δ t ev) = d(γ(0), γ(t)) ≤ˆt 0 ξ ε (γ(s),γ(s)) ds. Now, we are in position to apply Lemma 3.7 to the metric ξ ε : for every fixed a ∈ Ω, a.e. x ∈ Ω and all v ∈ H x G ∇ G d a (x), v x ≤ lim inf t→0 d(x, x · δ t ev) t ≤ lim sup t→0 d(x, x · δ t ev) t ≤ ξ ε (x, v). Taking the least upper bound w.r.t. a ∈ Θ and letting ε → 0, we obtain ξ(x, v) ≤ lim inf t→0 d(x, x · δ t ev) |t| ≤ lim sup t→0 d(x, x · δ t ev) |t| ≤ ξ(x, v). This proves the convexity of the limit, i.e. of the metric derivative on the horizontal bundle. Application I: Γ-convergence We start this section by briefly recalling the notion of Γ-convergence and we refer the interested reader to [9] for a complete overview on the subject. Let (M, τ ) be a topological space satisfying the first axiom of countability. A sequence of maps F h : M → R is said to Γ(τ )-converge to F and we will write F h Γ(τ ) − −− → F ( or simply F h Γ − → F ifF h (x h ) ≤ F (x) . To any distance d in D cc (Ω), we are going to associate some functionals defined respectively on the class B(Ω) of all positive and finite Borel measures µ on Ω × Ω and on Lip ([0, 1], Ω). We set J d (µ) =ˆd(x, y) dµ(x, y), µ ∈ B(Ω); L d (γ) =ˆ1 0 ϕ d (γ(t),γ(t)) dt, γ ∈ Lip([0, 1], Ω). As already mentioned in Subsection 2.5, we equip D cc (Ω) with the topology of the uniform convergence on compact subsets of Ω × Ω. Moreover, we endow B(Ω) and Lip([0, 1], Ω) with the topology of weak * convergence and of the uniform convergence, respectively. The following result is strongly inspired by [8, Theorem 3.1]. Theorem 4.1. Let Ω ⊂ G be an open set in a Carnot group of step k and let (d n ) n and d belong to D cc (Ω). If J n , L n and J, L are the functionals associated respectively to d n and d, defined as before, then the following conditions are equivalent: (i) d n → d in D cc (Ω); (ii) J n Γ − → J on B(Ω); (iii) L n Γ − → L on Lip([0, 1] , Ω). Moreover, if Ω is bounded, then (i), (ii) and (iii) are equivalent to the following condition: (iv) J n continuously converges to J, meaning that J(µ) = lim n J n (µ n ) holds whenever the sequence (µ n ) n ⊂ B(Ω) weakly * converges to µ ∈ B(Ω). Proof. (i) ⇒ (ii). In order to prove the Γ-lim inf inequality, fix µ ∈ B(Ω) and (µ n ) n ⊂ B(Ω) such that µ n weakly * converges to µ. Fix a sequence (η k ) k of compactly-supported continuous functions η k : Ω × Ω → [0, 1] such that η k (x) ր 1 for every x ∈ Ω. Since d n → d in D cc (Ω), we deduce that for any k ∈ N we have that η k d n → η k d uniformly as n → ∞, thus there exists a sequence (ε k n ) n ⊂ (0, +∞) such that ε k n ց 0 as n → ∞ and |η k d n − η k d| ≤ ε k n on Ω × Ω. Moreover, since µ n weakly * converges to µ, by using Banach-Steinhaus Theorem we deduce that sup n µ n (Ω × Ω) < +∞. Therefore, since η k d is continuous and bounded, we get that ˆη k (x, y)d n (x, y) dµ n (x, y) −ˆη k (x, y)d(x, y) dµ(x, y) ≤ ε k n µ n (Ω × Ω) + ˆη k (x, y)d(x, y) dµ n (x, y) −ˆη k (x, y)d(x, y) dµ(x, y) → 0 as n → ∞, for every k ∈ N. In particular, for any k ∈ N we have that η k (x, y)d(x, y) dµ(x, y) = lim n→∞ˆη k (x, y)d n (x, y) dµ n (x, y) ≤ lim inf n→∞ J n (µ n ). By monotone convergence theorem, we conclude that J(µ) ≤ lim inf n J n (µ n ), as desired. Let us pass to the verification of the Γ-lim sup inequality. Fix any µ ∈ B(Ω). We aim to show that the sequence constantly equal to µ is a recovery sequence, namely J(µ) ≥ lim sup n J n (µ). If J(µ) = +∞, then there is nothing to prove. Thus suppose that J(µ) < +∞. Since (1/α)d cc ≤ d, we deduce that d cc ∈ L 1 (µ). By combining this information with the fact that d n ≤ αd cc for all n ∈ N and d n → d pointwise on Ω × Ω, we are in a position to apply the dominated convergence theorem, obtaining that J(µ) =´d(x, y) dµ(x, y) = lim n´dn (x, y) dµ(x, y) = lim n J n (µ). (i) ⇒ (iii). For every γ ∈ Lip([0, 1], Ω), we have to prove the following two claims: ∀ γ n − → γ : L d (γ) ≤ lim inf n→∞ L dn (γ n ), (4.1) ∃ γ n − → γ : L d (γ) ≥ lim sup n→∞ L dn (γ n ). (4.2) We begin proving (4.1). Let γ n → γ in Lip([0, 1], Ω). By definition of L d (γ), for any δ ≥ 0 we can find a partition of [0, 1], indexed over a finite set I δ , such that (4.3) L d (γ) ≤ δ + i∈I δ d(γ(t i ), γ(t i+1 )). Since {γ n } n∈N converges uniformly on [0, 1], we may assume that ∃n ∈ N : (γ n (s), γ n (t)) ∈ K ⊂ Ω × Ω, ∀s, t ∈ [0, 1], ∀n ≥n, where K is compact. Then, for every i ∈ I δ , |d n (γ n (t i ), γ n (t i+1 )) − d(γ(t i ), γ(t i+1 ))| ≤ |d n (γ n (t i ), γ n (t i+1 )) − d(γ n (t i ), γ n (t i+1 ))| + |d(γ n (t i ), γ n (t i+1 )) − d(γ(t i ), γ(t i+1 ))| ≤ sup K |d n − d| + d(γ(t i ), γ n (t i )) + d(γ(t i+1 ), γ n (t i+1 )) ≤ sup K |d n − d| + α d cc (γ(t i ), γ n (t i )) + d cc (γ(t i+1 ), γ n (t i+1 )) ≤ sup K |d n − d| + αC K ′ |γ(t i ) − γ n (t i )| 1 k + |γ(t i+1 ) − γ n (t i+1 )| 1 k ≤ sup K |d n − d| + 2αC K ′ sup [0,1] |γ − γ n | 1 k =: ξ n , where K ′ ⊂ Ω is any compact set such that K ⊂ K ′ × K ′ and C K ′ is the constant provided by Theorem 2.3. Note that ξ n → 0. We infer from (4.3) and from the definition of L dn that L d (γ) ≤ δ + i∈I δ [d n (γ n (t i ) , γ n (t i+1 )) + ξ n ] ≤ δ + L dn (γ n ) + ξ n card(I δ ). Passing to the lim inf as n → +∞, we get L d (γ) ≤ lim inf n→∞ L dn (γ n ) + δ. This yields (4.1) by the arbitrariness of δ > 0. We prove now (4.2). Let γ ∈ Lip(Ω), let K ⊂ Ω × Ω compact be chosen as above, and let r(n) → ∞ be a sequence such that lim n→∞ r(n) sup K |d n − d| = 0. For every n ∈ N, let I n be the partition of [0, 1] into r(n) intervals of equal length, and denote by {t i n }, i = 1, . . . , r(n) + 1, the endpoints of such intervals. Let γ n be a curve whose restriction γ i n to the interval [t i n , t i+1 n ] is defined by (4.4) γ i n (t i n ) = γ(t i n ), γ i n (t i+1 n ) = γ(t i+1 n ), L dn (γ i n ) ≤ d n (γ(t i n ), γ(t i+1 n )) + 1 2 r(n) . Claim: The sequence (γ n ) n∈N converges to γ in Lip([0, 1], Ω). Let us prove the claim. Fix a compact set K ⊂ Ω such that γ(t), γ n (t) ∈ K for every n ∈ N and t ∈ [0, 1]. Given any n ∈ N and t ∈ (0, 1], we denote by (t − n , t + n ] the interval of I n containing t. Consider the constant C K given by Theorem 2.3. Then it holds that 1 C K |γ n (t) − γ(t)| ≤ d cc (γ n (t), γ(t)) ≤ d cc (γ n (t), γ n (t + n )) + d cc (γ(t + n ), γ(t)) =: A n + B n . Now, by the uniform continuity of γ on [0, 1], the term B n tends to zero as n → +∞ uniformly with respect to t. The same holds for A n , since (using Lemma 3.8) it can be estimated as 1 αC K |γ n (t) − γ n (t + n )| ≤ 1 α d cc (γ n (t), γ n (t + n )) ≤ d n (γ n (t), γ n (t + n )) ≤ L dn (γ n | [t,t + n ] ) ≤ L dn (γ n | [t − n ,t + n ] ) ≤ αd cc (γ n (t − n ), γ n (t + n )) + 1 2 r(n) = αd cc (γ(t − n ), γ(t + n )) + 1 2 r(n) ≤ α · C K |γ(t − n ) − γ(t + n )| 1 k + 1 2 r(n) , where we used the continuity estimate (4.4) and the fact that d n ∈ D cc (Ω). Now, by definition of L d (γ) and the construction (4.4), we infer that L d (γ) ≥ r(n) i=1 d(γ(t i n ), γ(t i+1 n )) = r(n) i=1 d n (γ(t i n ), γ(t i+1 n )) + r(n) i=1 [d(γ(t i n ), γ(t i+1 n )) − d n (γ(t i n ), γ(t i+1 n ))] ≥ L dn (γ n ) − r(n) 2 r(n) + r(n) i=1 [d(γ(t i n ), γ(t i+1 n )) − d n (γ(t i n ), γ(t i+1 n ))]. To get the required inequality, it is enough to pass to the limsup in the above inequality, noticing that, by the choice of the sequence r(n), we have lim n→+∞ r(n) i=1 [d(γ(t i n ), γ(t i+1 n )) − d n (γ(t i n ), γ(t i+1 n ))] ≤ lim n→+∞ r(n) sup K |d n − d| = 0, and then we get the desired conclusion (4.2). (iii) ⇒ (i). This implication follows from the following fact: Claim: The class D cc (Ω) is compact. As we are going to show, the above claim is obtained as a consequence of the Ascoli-Arzelá Theorem and the implication (i) ⇒ (iii) already proved. Let (d n ) n ⊂ D cc (Ω) be a given sequence. First of all, for any (x, y) ∈ Ω × Ω we have that (d n (x, y)) n is a bounded sequence, as granted by the following estimate: (4.5) d n (x, y) ≤ αd cc (x, y) for every x, y ∈ Ω and n ∈ N. Moreover, we have to prove that the sequence (d n ) n ∈ D cc (Ω) is equi-continuous, in other words, that for every x, x ′ , y, y ′ ∈ Ω ⊂ G it holds ∀ε > 0 ∃ δ > 0 : |x ′ − x| < δ |y ′ − y| < δ ⇒ |d n (x, x ′ ) − d n (y, y ′ )| < ε, ∀n ∈ N. By using the triangle inequality and Theorem 2.3, we obtain that |d n (x, y) − d n (x ′ , y ′ )| ≤ d n (x, x ′ ) + d n (y, y ′ ) ≤ α d cc (x, x ′ ) + d cc (y, y ′ ) ≤ αC K |x ′ − x| 1 k + |y ′ − y| 1 k . Choosing δ = 2 ǫ k C K β , we obtain |d n (x, y) − d n (x ′ , y ′ )| ≤ ε. Hence, we may extract a subsequence converging to some element d in D cc (Ω). To prove that d is geodesic, we use the implication (i) ⇒ (iii), which ensures that L n Γ − → L. Fix x, y ∈ Ω. We will prove that we have the Γ-convergence for the modified functionals: 1 α d cc (γ n (t), x) ≤ α · C K γ n 1 n − γ 1 n + γ 1 n − x 1 k + ε n (4.6) holds, where the last term tends to zero as n → ∞, since γ n → γ in Lip(Ω). It remains to show that lim sup nLn (γ n ) ≤L(γ), indeed we have that (4.7)L n (γ n ) ≤ d n (x, γ n 1 n ) + L n (γ n ) + d n (γ n 1 − 1 n , y) + 2ε n . Notice now that, from (4.6), it follows in particular that lim n d n (x, γ n Thus, by the Γ-convergence of L n to L, we deduce that Since d n are geodesic distances in D cc (Ω), the equation (4.8) means exactly that d is a geodesic distance, as desired. Finally, assume in addition that Ω is bounded. On the one hand, (iv) trivially implies (ii). On the other hand, we can prove that (i) implies (iv). To this aim, fix any µ ∈ B(Ω) and (µ n ) n ⊂ B(Ω) such that µ n weakly * converges to µ. Let ε > 0 be fixed. We have that sup n µ n (Ω × Ω) < +∞ by Banach-Steinhaus Theorem. Moreover, we have that {µ n } n is weakly * relatively compact by assumption, thus Prokhorov's Theorem yields the existence of a compact set K ⊂ Ω × Ω such that µ n ((Ω × Ω) \ K) ≤ ε for every n ∈ N. Call D the diameter of Ω with respect to d cc . Since d : Ω → Ω → R is bounded and continuous, we deduce that J n (µ n ) − J(µ) ≤ˆK|d n − d| dµ n +ˆ( Ω×Ω)\K |d n − d| dµ n + J(µ n ) − J(µ) ≤ µ n (Ω × Ω) max K |d n − d| + 2βDε + ˆd dµ n −ˆd dµ , whence by letting n → ∞ we get lim sup n |J n (µ n ) − J(µ)| ≤ 2βDε. By arbitrariness of ε, we finally conclude that J(µ) = lim n J n (µ n ), so that (iv) is proved. Applications II: Intrinsic geometry and sub-Finsler structure The present section is devoted to generalizing the metric results contained in [14]. To this aim, we introduce two distances which involve the structure of the sub-Finsler metric. Definition 5.1. If ϕ ∈ M α cc (G) is a sub-Finsler convex metric, for every x, y ∈ G we define the following quantity: (5.1) δ ϕ (x, y) := sup |f (x) − f (y)| f : G → R Lipschitz, ϕ(·, ∇ G f (·)) ∞ ≤ 1 . Recall that Pansu's Theorem assures that ∇ G f (x) exists at almost every x ∈ G and thus the above definition makes sense. From now on, we will say that any Lipschitz function satisfying the conditions in (5.1) is a competitor for δ ϕ . Moreover, we have that Lemma 5.2. δ ϕ : G × G → [0, +∞) is a distance. Proof. Clearly, we have that δ ϕ (x, y) ≥ 0 for every x, y ∈ G and δ ϕ (x, y) > 0 if x = y. The symmetry comes from the fact that |f (x)−f (y)| = |f (y)−f (x)|. Also, δ ϕ satisfies the triangle inequality since for every x, y, z ∈ G we have δ ϕ (x, y)+δ ϕ (y, z) ≥ |f (x)−f (y)|+|f (y)−f (z)| ≥ |f (x) − f (z)|. Passing to the supremum on the right-hand side for every f Lipschitz function such that ϕ(x, ∇ G f (x)) ∞ ≤ 1, we get that δ ϕ (x, y) + δ ϕ (y, z) ≥ δ ϕ (x, z). Under some assumptions, we will show in Theorem 5.11 that δ ϕ turns out to be a distance in D cc (G). y) for every x ∈ G. Lip δϕ f (x) = lim sup y→x |f (y) − f (x)| δ ϕ (x, We recall now the notion of intrinsic distance that was introduced by De Cecco-Palmieri in [10,Definition 1.4] in the context of Lipschitz manifold. Definition 5.4. Given any ϕ ∈ M α cc (G), we define its induced intrinsic distance d ϕ as d ϕ (x, y) := inf γˆ1 0 ϕ(γ(t),γ(t)) dt for every x, y ∈ G, where the infimum is taken over all horizontal curves γ ∈ H([0, 1], G) joining x and y. The quantity d ϕ (x, y) is well-defined because the map t → (γ(t),γ(t)) is Borel measurable on the horizontal bundle. Let us observe that in Definition 2.9 we are not requiring any regularity on ϕ besides its Borel measurability. At this level of generality (namely, without semicontinuity assumptions) d ϕ might exhibit some 'pathological' behaviour, as we can see in the following example. Others examples of geodesic distances, contained in D cc (R 2 ), which are not intrinsics can be found in [13,Example 1.8] and [6,Corollary 3.4]. Example 5.5. Let R 2 with the Euclidean structure and consider the Borel set N ⊂ R 2 as N := x,y∈Q 2 S x,y , where S x,y stands for the segment joining x and y. Notice that N is L 2 -negligible and we define the metric ϕ : R 2 × R 2 → [0, +∞) as ϕ(x, v) :=    |v|, 2|v|, if x ∈ N, if x / ∈ N. Since ϕ ∈ M 2 cc (R 2 ), for every x, y ∈ R 2 it holds that |x−y| ≤ d ϕ (x, y) ≤ 2|x−y|. In particular d ϕ : R 2 ×R 2 → [0, +∞) is continuous when the domain is endowed with the Euclidean distance. Now observe that for any x, y ∈ Q 2 we have that d ϕ (x, y) = |x − y|, the shortest path being exactly the segment S x,y . By continuity of d ϕ and thanks to the density of Q 2 in R 2 , we conclude that d ϕ (x, y) = |x − y| for every x, y ∈ R 2 . This shows that, even if ϕ(x, ·) is equal to 2| · | for L 2 -a.e. x ∈ R 2 , the distance d ϕ coincides with the Euclidean distance. In other words, the behaviour of ϕ on the null set N completely determines the induced distance d ϕ . A further important concept for our treatment is the classical notion of Finsler metrics on Carnot groups. Definition 5.6. We say that a map F : T G → [0, +∞) is a Finsler metric if the following properties hold: • F is continuous on T G and smooth on T G \ {0}, • the Hessian matrix of F 2 is positive definite for any vector v ∈ T x G \ {0} for every x ∈ G. Moreover, we denote by d F the length distance on G induced by F , namely we set d F (x, y) := inf γˆ1 0 F (γ(t),γ(t)) dt for every x, y ∈ G, where the infimum is taken among all curves γ ∈ Lip([0, 1], G) joining x and y. Let us observe that the intrinsic distance is induced by a metric on the horizontal bundle while the latter comes from a metric defined on the entire tangent bundle. Lemma 5.7. If ϕ ∈ M α cc (G) is a sub-Finsler convex metric, then d ϕ is a geodesic distance belonging to D cc (G). Proof. Since (G, d ϕ ) is a complete, locally compact length space, then d ϕ is a geodesic distance, thanks to the general result contained in [7,Theorem 2.5.23]. To prove the claim, we have that d ϕ (x, y) ≥ 0 for every x, y ∈ G since the integral of ϕ(γ(·),γ(·)) is non-negative. In order to prove the symmetry, let us consider γ ∈ H ([0, 1], G) such that γ(0) = x and γ(1) = y. Set ξ : [0, 1] → G as ξ(t) = γ(1 − t), hence this is a horizontal curve in [0, 1]. By the 1-homogeneity of ϕ(x, ·), we get that 1 0 ϕ(ξ(t),ξ(t)) dt =ˆ1 0 ϕ(γ(1 − t), −γ(1 − t)) dt =ˆ1 0 ϕ(γ(s), −γ(s)) ds =ˆ1 0 ϕ(γ(s),γ(s)) ds. So now, passing to the infimum over γ ∈ H([0, 1], G) we get that d ϕ (x, y) = d ϕ (y, x). To prove the triangle inequality, let x, y, z ∈ G and γ 1 , γ 2 ∈ H([0, 1], G) be such that γ 1 (0) = x, γ 1 (1) = y = γ 2 (0), and γ 2 (1) = z. Let us define the following curve: η : [0, 1] → G, η(t) := γ 1 (2t) if t ∈ [0, 1 2 ]; γ 2 (2t − 1) if t ∈ [ 1 2 , 1] . Then we obtain that d ϕ (x, z) ≤ˆ1 0 ϕ(η(t),η(t)) dt =ˆ1 2 0 ϕ(γ 1 (2t), 2γ 1 (2t)) dt +ˆ1 1 2 ϕ(γ 2 (2t − 1), 2γ 2 (2t − 1)) dt =ˆ1 0 ϕ(γ 1 (s),γ 1 (s)) ds +ˆ1 0 ϕ(γ 2 (s),γ 2 (s)) ds, where we applied a change-of-variable (in both integrals) and the 1-homogeneity of ϕ. Passing to the infimum respectively over all γ 1 , γ 2 , we conclude. We are left to prove (2.10). Let us take x, y ∈ G and consider the horizontal curve γ : [0, 1] → G s.t. γ(0) = x and γ(1) = y. By Proposition 2.11 we get that 1 0 ϕ(γ(t),γ(t)) dt ≤ αˆ1 0 γ(t) γ(t) dt. Thus, passing to the infimum in the right-hand side we obtain the conclusion and the converse inequality can be achieved by arguing in a similar way. 5.1. Main Results. Before proving one of the main theorems, we recall some basic terminology. Given two Banach spaces B 1 and B 2 , and denoting with L(B 1 , B 2 ) the space of all linear and continuous operators T : B 1 → B 2 , it holds that L (B 1 , B 2 ) is a Banach space if endowed with the usual pointwise operations and the operator norm, namely T L(B 1 ,B 2 ) := sup v∈B 1 \{0} T (v) B 2 v B 1 for every T ∈ L(B 1 , B 2 ). Remark 5.8. Given a smooth map ϕ : M → N between two smooth manifolds M , N and a point x ∈ M , we denote by d x ϕ : T x M → T ϕ(x) N the differential of ϕ at x. We recall that if γ : [0, 1] → M is an absolutely continuous curve in M , then σ := ϕ • γ is an absolutely continuous curve in N and it holds that (5.2)σ(t) = d γ(t) ϕ[γ(t)] for a.e. t ∈ [0, 1]. We also point out that (5.3) d dt δ t e v = d e τ δte v [v] for every v ∈ H e G and t ∈ (0, 1). Indeed, calling γ the unique curve satisfying (2.2) and defining γ t (s) := γ(ts) for all t ∈ (0, 1) and s ∈ [0, 1], we may compute d ds γ t (s) = tγ(ts) = t d e τ γ(ts) [v] = d e τ γ t (s) [tv] for every s ∈ (0, 1), which shows that γ t fulfills the ODE defining tv, so that (2.6) yields γ(t) = γ t (1) = e tv = δ t e v for every t ∈ (0, 1) and accordingly the identity claimed in (5.3) is proved. In general, let us observe that, if ψ is a sub-Finsler metric, then the metric derivative with d = δ ψ , namely ϕ δ ψ , could be very different from ψ (see [13,Example 1.5]). Our purpose is to show a different result for the intrinsic distances. It tells us that, given a sub-Finsler convex metric ψ, the metric derivative with respect to d ψ is bounded above by ψ almost everywhere. Moreover, we show that the equality holds, for instance, when ψ is lower semicontinuous. Theorem 5.9. Let ψ ∈ M α cc (G) be a sub-Finsler convex metric. Then the following properties are verified: i) It holds that for a.e. x ∈ G, ϕ d ψ (x, v) ≤ ψ(x, v) for every v ∈ H x G. ii) If ψ is upper semicontinuous, then ϕ d ψ (x, v) ≤ ψ(x, v) for every (x, v) ∈ HG. iii) If ψ is lower semicontinuous, then ϕ d ψ (x, v) ≥ ψ(x, v) for every (x, v) ∈ HG. In particular, for a.e. x ∈ G it holds that Notice that γ is horizontal and joins x to x · δ t e v . We can computė ϕ d ψ (x, v) = ψ(x, v) for every v ∈ H x G. Proof. i) Given x ∈ G, v ∈ H e G,γ(s) = d ds τ x δ ts ev (5.2) = d δtse v τ x d ds δ ts e v (5.3) = d e τ x·δtse v [tv] for every s ∈ (0, 1). Therefore, we may estimate d ψ (x, x · δ t e v ) ≤ˆ1 0 ψ(γ(s),γ(s)) ds = tˆ1 0 ψ x · δ ts e v , d e τ x·δtse v [v] ds =ˆt 0 ψ x · δ s e v , d e τ x·δse v [v] ds. (5.4) The next argument closely follows along the lines of Lemma 3.6. Fix a dense sequence (v i ) i in the unit sphere of H e G (w.r.t. the norm · e ). Define v i (x) := d e τ x [v i ] for every i ∈ N and x ∈ G, so that (v i (x)) i is a dense sequence in the unit sphere of H x G (w.r.t. the norm · x ). By using Lebesgue's differentiation theorem and Fubini's theorem, we see that the set Γ i of all couples (y, r) ∈ G × R such that (5.5) ψ y · δ r e v i , d e τ y·δre v i [v i ] = lim tց0 1 tˆt 0 ψ y · δ r+s e v i , d e τ y·δ r+s e v i [v i ] ds has zero L n+1 -measure. By using Fubini's theorem again, we can find r ∈ R such that for any i ∈ N there exists a L n -null set N i ⊂ G such that (5.5) holds for every point y ∈ G \ N i . Let us consider the set N := i∈N σ δre v i (N i ), where σ z : G → G stands for the right-translation map σ z w := w · z. The right-invariance of L n grants that N is L n -negligible. Given that (5.6) ψ(x, v i (x)) = lim tց0 1 tˆt 0 ψ x · δ s e v i , d e τ x·δse v i [v i ] ds for every i ∈ N and x ∈ G \ N, we can conclude that ϕ d ψ (x, v i (x)) = lim tց0 d ψ (x, x · δ t e v i ) t (5.4) ≤ lim tց0 1 tˆt 0 ψ x · δ s e v i , d e τ x·δse v i [v i ] ds (5.6) = ψ(x, v i (x)) for every i ∈ N and x ∈ G \ N. Since ψ(x, ·) is continuous and positively 1-homogeneous, and (v i (x)) i is dense in the unit · x -sphere of H x G, we deduce that ϕ d ψ (x, w) ≤ ψ(x, w) for every x ∈ G \ N and w ∈ H x G. ii) Suppose ψ is upper semicontinuous. Let (x, v) ∈ HG be fixed. Given any ε > 0, we can thus find t ε > 0 such that, settingv := d x τ x −1 [v] for brevity, it holds that (5.7) ψ x · δ t ev, d e τ x·δtev [v] ≤ ψ(x, v) + ε for every t ∈ (0, t ε ). In particular, we may estimate ϕ d ψ (x, v) = lim tց0 d ψ (x, x · δ t ev) t (5.4) ≤ lim tց0 1 tˆt 0 ψ x · δ s ev, d e τ x·δsev [v] ds (5.7) ≤ ψ(x, v) + ε. Thanks to the arbitrariness of ε, we can conclude that ϕ d ψ (x, v) ≤ ψ(x, v), as desired. iii) Suppose ψ is lower semicontinuous. First of all, let us extend · e to a Hilbert norm (still denoted by · e ) on the whole T e G = g, then by left-invariance we obtain a Hilbert norm · x on each tangent space T x G. Throughout the rest of the proof, we assume that T x G is considered with respect to such norm · x . Moreover, choose any norm n : g → [0, +∞) on the Lie algebra which extends ψ(e, ·), so that n ≤ λ · e for some λ > 0. Without loss of generality, up to replacing ψ with the translated metric ψ x , defined as ψ x (y, v) := ψ x · y, d y τ x [v] for every (y, v) ∈ HG, it is sufficient to prove the statement only for x = e. Then let v ∈ H e G be fixed. For any t > 0 we have that the horizontal curve [0, 1] ∋ s → δ st e v ∈ G is a competitor for d ψ (e, δ t e v ), thus we may estimate d ψ (e, δ t e v ) ≤ˆ1 0 ψ(δ st e v , t d e τ δste v [v]) ds = tˆ1 0 ψ(δ st e v , d e τ δste v [v]) ds ≤ αtˆ1 0 d e τ δste v [v] δste v ds = αt v e , where the last equality comes from the left invariance of the norm. This means that, in order to compute d ψ (e, δ t e v ), it is sufficient to consider those horizontal curves γ : G → R joining e to δ t e v and satisfying´1 0 γ s γs ds ≤ α´1 0 ψ(γ s ,γ s ) ds ≤ α 2 t v e . We can also assume without loss of generality that any such curve γ is parametrized by constant speed with respect to the metric · x . All in all, we have shown that (5.8) d ψ (e, δ t e v ) = inf γ∈Ctˆ1 0 ψ(γ s ,γ s ) ds for every t > 0, where the family C t of curves is defined as C t := γ : [0, 1] → G horizontal γ 0 = e, γ 1 = δ t e v , γ s γs ≡ˆ1 0 γ s γs ds ≤ α 2 t v e . Now fix any ε > 0. Since the map exp −1 : G → g is a diffeomorphism, we can consider its differential d x exp −1 : T x G → T exp −1 (x) g ∼ = g at any point x ∈ G. Let us observe that exp −1 is smooth, and d e exp −1 = d e τ e −1 = id g . Since ψ is lower semicontinuous and by the previous argument, we can find r > 0 such that ψ(x, v) ≥ ψ(e, d x τ x −1 [v]) − ε for every x ∈ B(e, r) and v ∈ H x G, v x ≤ 1, (5.9a) d x exp −1 − d x τ x −1 L(TxG,g) ≤ ε for every x ∈ B(e, r), (5.9b) where B(e, r) ≡ B dcc (e, r). In particular, given any t > 0 with α 2 t v e < r and γ ∈ C t , we have that d cc (e, γ s ) ≤ sα 2 t v e < r for every s ∈ [0, 1] and γ s γs ≤ α 2 t v e for a.e. s ∈ [0, 1], thus accordingly (5.9a) and (5. respectively. Therefore, for any t > 0 with α 2 t v e < r and γ ∈ C t , we may estimate ψ e,ˆ1 0 d γs τ γ −1 s [γ s ] ds − n ˆ1 0 d γs exp −1 [γ s ] ds ≤ n ˆ1 0 d γs τ γ −1 s [γ s ] ds −ˆ1 0 d γs exp −1 [γ s ] ds ≤ λ ˆ1 0 d γs τ γ −1 s [γ s ] − d γs exp −1 [γ s ] ds e ≤ λˆ1 0 (d γs τ γ −1 s − d γs exp −1 )[γ s ] e ds ≤ λˆ1 0 d γs exp −1 − d γs τ γ −1 s L(Tγ s G,g) γ s γs ds (5.10b) ≤ λεˆ1 0 γ s γs ds ≤ λεα 2 t v e , whence it follows that 1 0 ψ(γ s ,γ s ) ds (5.11) where in the second inequality we applied Jensen's inequality to ψ(e, ·). Now consider the curve σ in the Hilbert space (g, n), which is given by σ s := exp −1 (γ s ) for every s ∈ (5.10a) ≥ˆ1 0 ψ(e, d γs τ γ −1 s [γ s ]) ds − α 2 t v e ε ≥ ψ e,ˆ1 0 d γs τ γ −1 s [γ s ] ds − α 2 t v e ε ≥ n ˆ1 0 d γs exp −1 [γ s ] ds − (λ + 1)α 2 t v e ε.tv = tv − 0 g = exp −1 (δ t e v ) − exp −1 (e) = exp −1 (γ 1 ) − exp −1 (γ 0 ) = σ 1 − σ 0 =ˆ1 0σ s ds =ˆ1 0 d γs exp −1 [γ s ] ds. (5.12) By combining (5.11) and (5.12), we obtain for any t > 0 with α 2 t v e < r and γ ∈ C t that (5.13)ˆ1 0 ψ(γ s ,γ s ) ds ≥ n(tv) − (λ + 1)α 2 t v e ε = ψ(e, v) − (λ + 1)α 2 v e ε t. We are now in a position to conclude the proof of the statement: given t > 0 with α 2 t v e < r, one has that (5.14) d ψ (e, δ t e v ) t (5.8) = inf γ∈Ct 1 tˆ1 0 ψ(γ s ,γ s ) ds (5.13) ≥ ψ(e, v) − (λ + 1)α 2 v e ε. By letting t ց 0, we thus deduce that (5.15) ϕ d ψ (e, v) = lim sup tց0 d ψ (e, δ t e v ) t (5.14) ≥ ψ(e, v) − (λ + 1)α 2 v e ε. Finally, by letting ε ց 0 in (5.15) we conclude that ϕ d ψ (e, v) ≥ ψ(e, v), as desired. Corollary 5.10. If ψ is a continuous sub-Finsler convex metric, then ϕ d ψ (x, v) = ψ(x, v) for every (x, v) ∈ HG. Proof. It is an immediate consequence of assertions ii) and iii) of Theorem 5.9. The crucial observation below states that δ ϕ coincides with the intrinsic distance d ϕ ⋆ when we assume that the sub-Finsler metric is lower semicontinuous. This will allow us to show the same result when ϕ is upper semicontinuous, thanks to an approximation argument. Theorem 5.11. Let ϕ ∈ M α cc (G) be a sub-Finsler convex metric. Then it holds that δ ϕ ≤ d ϕ ⋆ . Moreover, if ϕ is lower semicontinuous, then δ ϕ (x, y) = d ϕ ⋆ (x, y) for every x, y ∈ G. Proof. Let x, y ∈ G be fixed. To prove the first part of the statement, pick any Lipschitz function f with ϕ(z, ∇ G f (z)) ∞ ≤ 1 and any horizontal curve γ : [0, 1] → G joining x and y such that H 1 γ ∩ {z ∈ G : ϕ(z, ∇ G f (z)) > 1} = 0. These are competitors for δ ϕ (x, y) and d ϕ ⋆ (x, y), respectively. Then we can estimate f (x) − f (y) = ˆ1 0 d dt (f (γ(t))) dt = ˆ1 0 ∇ G f (γ(t)),γ(t) γ(t) dt ≤ˆ1 0 ∇ G f (γ(t)),γ(t) γ(t) dt ≤ˆ1 0 ϕ(γ(t), ∇ G f (γ(t))) ϕ ⋆ (γ(t),γ(t)) dt ≤ ϕ(·, ∇ G f (·)) ∞ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt ≤ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt, whence it follows that δ ϕ (x, y) ≤ d ϕ ⋆ (x, y). Now suppose ϕ is lower semicontinuous. Define the function f : G → R as f (·) := d ϕ ⋆ (x, ·) and since d ϕ ⋆ (x, y) ≤ α −1 d cc (x, y) everywhere, we have that f is Lipschitz. Fix any point z ∈ G such that ∇ G f (z) exists and let v ∈ H z G. Pick a horizontal curve γ : [0, ε] → G of class C 1 such that γ(0) = z andγ(0) = v. Thanks to the continuity of t → (γ(t),γ(t)) and the upper semicontinuity of ϕ ⋆ , granted by Lemma 2.13, we obtain that lim sup tց0 ffl t 0 ϕ ⋆ (γ(s),γ(s)) ds ≤ ϕ ⋆ (γ(0),γ(0)) = ϕ ⋆ (z, v), whence, by the identities (2.4) and (2.9), it follows that ∇ G f (z), v z = lim tց0 f (γ(t)) − f (γ(0)) t ≤ lim sup tց0 f (γ(t)) − f (γ(0)) d ϕ ⋆ (γ(t), γ(0)) lim sup tց0 d ϕ ⋆ (γ(t), γ(0)) t ≤ lim sup tց0 d ϕ ⋆ (x, γ(t)) − d ϕ ⋆ (x, γ(0)) d ϕ ⋆ (γ(t), γ(0)) lim sup tց0 t 0 ϕ ⋆ (γ(s),γ(s)) ds ≤ ϕ ⋆ (z, v). By arbitrariness of v ∈ H z G, we deduce that ϕ(z, ∇ G f (z)) ≤ 1. Therefore, f is a competitor for δ ϕ (x, y). This implies that δ ϕ (x, y) ≥ |f (x) − f (y)| = d ϕ ⋆ (x, y). In particular, the last part of the proof shows that the supremum appearing in the definition of δ ϕ (x, y) is actually a maximum. The upper semicontinuity of the sub-Finsler metric ϕ is crucial for our proof, because it allows us to approximate the dual metric ϕ ⋆ through a family of continuous Finsler metrics. Corollary 5.12. Let ϕ ∈ M α cc (G) be a sub-Finsler convex metric. Suppose ϕ is upper semicontinuous. Then, for every x, y ∈ G it holds that δ ϕ (x, y) = d ϕ ⋆ (x, y). Proof. Lemma 2.13 ensures that ϕ ⋆ is lower semicontinuous. We set ϕ ⋆ : T G → [0, +∞) as ϕ ⋆ (x, v) :=    ϕ ⋆ (x, v), +∞, if (x, v) ∈ HG, if (x, v) ∈ T G \ HG. Observe that HG is closed in T G and thus ϕ ⋆ is lower semicontinuous. Thanks to [18,Theorem 3.11], there exists a sequence F n : T G → [0, +∞) of Finsler metrics on G such that F n (x, v) րφ ⋆ (x, v) for every (x, v) ∈ T G. Setting ϕ n : HG → [0 + ∞) as ϕ n := (F n | HG ) ⋆ , we obtain that ϕ n ∈ M α cc (G) and ϕ ⋆ n (x, v) ր ϕ ⋆ (x, v) for every (x, v) ∈ HG. Therefore ϕ n (x, v) ց ϕ(x, v) for every (x, v) ∈ HG. In particular, the inequality ϕ n ≥ ϕ holds for all n ∈ N. This implies that any competitor f for δ ϕn is a competitor for δ ϕ , so that accordingly (5.16) δ ϕn (x, y) ≤ δ ϕ (x, y), for every n ∈ N and x, y ∈ G. Moreover, since the infimum in the definition of d Fn is computed with respect to all Lipschitz curves, while the infimum in the definition of d ϕ ⋆ n is just over horizontal curves, for every x, y ∈ G we get that ≤ δ ϕ (x, y) for every x, y ∈ G. (5.17) d Fn (x, y) ≤ d ϕ ⋆ n (x, y) ≤ d ϕ ⋆ (x, Since the converse inequality d ϕ ⋆ ≥ δ ϕ is granted by the first part of Theorem 5.11, we conclude that δ ϕ = d ϕ ⋆ , as required. Theorem 5.13. Let ϕ ∈ M α cc (G) be an upper semicontinuous sub-Finsler convex metric. Then for any locally Lipschitz function f : G → R we have that ϕ(x, ∇ G f (x)) = Lip δϕ f (x) for a.e. x ∈ G. Proof. ≤ Since both sides are positively 1-homogeneous with respect to f , we only need to show that, if Lip δϕ f (x) = 1, then ϕ(x, ∇ G f (x)) ≤ 1 for a.e. x ∈ G. By Corollary 5.12, Lip δϕ f (x) = Lip d ϕ ⋆ f (x), hence if we fix (x, v) ∈ HG, thanks to (2.4) and the expression (2.9) we can write: ∇ G f (x), v x = lim t→0 f (x · δ t ev) − f (x) t ≤ lim sup t→0 d ϕ ⋆ (x, x · δ t ev) t · lim sup t→0 |f (x · δ t ev) − f (x)| d ϕ ⋆ (x, x · δ t ev) ≤ ϕ d ϕ ⋆ (x, v) Lip d ϕ ⋆ f (x) ≤ ϕ ⋆ (x, v), where in the last inequality we used item i) of Theorem 5.9. By arbitrariness of v ∈ H x G and the fact that ϕ(x, ∇ G f (x)) = ϕ ⋆⋆ (x, ∇ G f (x) ) ≤ 1, we get the conclusion. ≥ Thanks to a convolution argument, we can find a sequence (f n ) n ⊂ C 1 (G) such that f n → f uniformly on compact sets and ∇ G f n → ∇ G f in the almost everywhere sense. Recall that any C 1 -function is locally Lipschitz. Fix any x ∈ G such that ∇ G f n (x) exists for all n ∈ N and ∇ G f n (x) → ∇ G f (x) as n → ∞. Now let ε > 0 be fixed. Then we can choose r ′ > 0 andn ∈ N so that sup B(x,2r ′ ) |fn − f | ≤ ε and ϕ(x, ∇ G fn(x) − ∇ G f (x)) ≤ ε, where the ball is with respect to the distance d ⋆ ϕ . Calling g := fn and being z → ∇ G g(z) continuous, we deduce that z → ϕ(z, ∇ G g(z)) is upper semicontinuous, thus there exists r < r ′ such that ϕ(y, ∇ G g(y)) ≤ ϕ(x, ∇ G g(x)) + ε for every y ∈ B(x, 2r). Fix any point y ∈ B(x, r) and consider a horizontal curve γ : [0, 1] → G such that γ(0) = x, γ(1) = y with γ([0, 1]) ⊂ B(x, 2r). We can estimate in this way: |f (x) − f (y)| ≤ |g(x) − g(y)| + 2ε ≤ˆ1 0 d dt g(γ(t)) dt + 2ε ≤ˆ1 0 ϕ(γ(t), ∇ G g(γ(t)))ϕ ⋆ (γ(t),γ(t)) dt + 2ε ≤ ϕ(x, ∇ G g(x)) + ε ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt + 2ε ≤ ϕ(x, ∇ G f (x)) + 2ε ˆ1 0 ϕ ⋆ (γ(t) ,γ(t)) dt + 2ε. By taking the infimum over all γ ∈ H([0, 1], B(x, 2r)), we obtain that |f (x) − f (y)| ≤ ϕ(x, ∇ G f (x)) + 2ε d ϕ ⋆ (x, y) + 2ε, whence by letting ε → 0 we obtain that |f (x) − f (y)| d ϕ ⋆ (x, y) ≤ ϕ(x, ∇ G f (x)). Finally, by letting y → x we conclude that Lip δϕ f (x) = Lip d ϕ ⋆ f (x) ≤ ϕ(x, ∇ G f (x)), as required. To conclude, in Proposition 5.15 we prove that in the definition (5.1) of the distance δ ϕ it is sufficient to consider smooth functions. Before passing to the proof of this claim, we prove the following technical result. By LIP d ⋆ ϕ (f ) ∈ [0, +∞) we mean the (global) Lipschitz constant of f ∈ LIP d ⋆ ϕ (G). Lemma 5.14. Let ϕ ∈ M α cc (G) be a sub-Finsler convex metric. Then it holds that (5.20) LIP d ϕ ⋆ (f ) = ess sup x∈G Lip d ϕ ⋆ f (x) for every f ∈ LIP d ϕ ⋆ (G). Proof. The inequality (≥) is trivial. To prove the converse inequality, we argue by contradiction: suppose there exist x, y ∈ G with x = y, a negligible Borel set N ⊆ G and δ > 0 such that f (x) − f (y) d ϕ ⋆ (x, y) ≥ sup z∈G\N Lip d ϕ ⋆ f (z) + δ. Given any ε > 0, we can find γ ∈ H([0, 1], G) such that γ(0) = x, γ(1) = y, and 1 0 ϕ * (γ(t),γ(t)) dt ≤ d ϕ ⋆ (x, y) + ε. Since f • γ : R → R is Lipschitz, so Pansu-differentiable almost everywhere, we deduce that f (x) − f (y) ≤ˆ1 0 (f • γ) ′ (t) dt ≤ˆ1 0 ϕ(γ(t) , ∇ G f (γ(t))) ϕ * (γ(t),γ(t)) dt =ˆ1 0 Lip d ϕ ⋆ f (γ(t)) ϕ ⋆ (γ(t),γ(t)) dt ≤ sup z∈G\N Lip d ϕ ⋆ f (z)ˆ1 0 ϕ ⋆ (γ(t),γ(t)) dt ≤ f (x) − f (y) d ϕ ⋆ (x, y) − δ d ϕ ⋆ (x, y) + ε . By letting ε ց 0 in the above estimate, we get 0 ≤ −δ d ϕ ⋆ (x, y), which leads to a contradiction. Therefore, also the inequality (≤) in (5.20) is proved, whence the statement follows. Proposition 5.15. Let ϕ ∈ M α cc (G) be a sub-Finsler convex metric. Suppose ϕ is upper semicontinuous. Then for any x, y ∈ G it holds that (5.21) δ ϕ (x, y) = sup f (x) − f (y) f ∈ C ∞ (G), ϕ(·, ∇ G f (·) ∞ ≤ 1 . Proof. Denote byδ ϕ (x, y) the quantity in the right-hand side of (5.21). Since any competitor forδ ϕ (x, y) is a competitor for δ ϕ (x, y), we have that δ ϕ (x, y) ≥δ ϕ (x, y). To prove the converse inequality, fix any Lipschitz function f : G → R such that ϕ(·, ∇ G f (·)) ∞ ≤ 1. Corollary 5.12 and Theorem 5.13 grant that ess sup Lip d ϕ ⋆ f ≤ 1, thus Lemma 5.14 yields LIP d ϕ ⋆ (f ) ≤ 1. Given that d ϕ ⋆ is an increasing, pointwise limit of Finsler distances by [18,Theorem 3.11], we are in a position to apply Theorem A.1. Thus we obtain a sequence (f n ) n ⊆ C ∞ (G)∩ LIP d ϕ ⋆ (G) such that LIP d ϕ ⋆ (f n ) ≤ 1 for all n ∈ N and f n → f uniformly on compact sets. Corollary 5.12 and Theorem 5.13 imply that ϕ(·, ∇ G f n (·)) ∞ = sup Lip d ϕ ⋆ f n ≤ 1, thus f n is a competitor forδ ϕ (x, y). Then we conclude that f (x) − f (y) = lim n f n (x) − f n (y) ≤ δ ϕ (x, y), whence it follows that δ ϕ (x, y) ≤δ ϕ (x, y) by arbitrariness of f . Appendix A. Smooth approximation of Lipschitz functions on generalized sub-Finsler manifolds The aim of this appendix is to prove an approximation result for real-valued Lipschitz functions defined on some very weak kind of sub-Finsler manifold. More precisely, we consider a distance d on a smooth manifold that can be obtained as the monotone increasing limit of Finsler distances; this notion covers the case of generalized (so, possibly rank-varying) sub-Finsler manifolds, thanks to [18,Theorem 3.11]. In this framework, we prove (see Theorem A.1 below) that any Lipschitz function can be approximated (uniformly on compact sets) by smooth functions having the same Lipschitz constant. This generalizes previous results that were known on 'classical' sub-Riemannian manifolds, cf. [15] and the references therein. Let us fix some notation. Given a metric space (X, d), we denote by LIP d (X) the family of real-valued Lipschitz functions on X. Observe that LIP d (h n ) ≤ L and that h n (x) < h n+1 (x) < f (x) for every n ∈ N and x ∈ M . We claim that h n (x) ր f (x) for all x ∈ M . In order to prove it, fix any x ∈ M and ε > 0. Pick somen ∈ N such that 1/n < ε and d(x, xn) < ε. Then for every n ≥n it holds that h n (x) ≥ −L d(x, xn) + f (xn) − 1 n ≥ −L ε + f (x) − L d(x, xn) − 1 n ≥ f (x) − (2L + 1)ε, thus proving the claim. Fix an increasing sequence (K n ) n of compact sets in M satisfying the following property: given any compact set K ⊆ M , there exists n ∈ N such that K ⊆ K n . In particular, one has that n K n = M . Notice that h n + 1 n(n+1) ≤ h n+1 on K n for all n ∈ N. Since d F i ր d, there exists i n ∈ N such that the function g n : M → R, given by g n (x) := − L d in (x, x 1 ) + f (x 1 ) ∨ · · · ∨ − L d in (x, x n ) + f (x n ) − 1 n for every x ∈ M, satisfies h n < g n < h n + 1 n(n+1) on K n . Note that g n ∈ LIP d in (M ) and LIP d in (g n ) = L. Thanks to a mollification argument, it is possible to build a function f n ∈ C ∞ (M )∩LIP d in (M ) such that LIP d in (f n ) ≤ L and g n < f n < g n+1 on K n . Therefore, for any n ∈ N and x ∈ K n it holds that the sequence f j (x) j≥n is strictly increasing and converging to f (x). This grants that f j → f uniformly on K n for any given n ∈ N. Hence, our specific choice of (K n ) n implies that f n → f uniformly on compact sets. Finally, the inequality d in ≤ d yields f n ∈ LIP d (M ) and LIP d (f n ) ≤ LIP d in (f n ) ≤ L for all n ∈ N, whence the statement follows. Definition 2 . 1 . 21An absolutely continuous curve γ : [a, b] → G is said to be horizontal if there exists a vector of measurable functions h = (h 1 (t), . . . h m (t)) : [a, b] → R m called the vector of canonical coordinates, such that •γ Definition 2 . 2 . 22For every x, y ∈ G, the Carnot-Carathéodory (CC) distance is defined by d cc (x, y) = inf {L G (γ) : γ is a horizontal curve joining x and y} . Definition 2. 5 . 5A map L : G → R is called a homogeneous homomorphism if Theorem 2 . 8 . 28Let Ω ⊂ G be an open subset. Then for every Lipschitz function Proposition 2 . 12 . 212Let us consider ϕ ∈ M α cc (G). Then ϕ is a sub-Finsler convex metric if and only and the supremum is taken over all finite partitions of [0, 1].Carnot groups are naturally endowed with sub-Riemannian distances which make them interesting examples of metric spaces (G, d cc ). In particular, the metric derivative can be explicitly computed (see[21, Theorem 1.3.5]). Lemma 3 . 4 . 34Let γ : [0, 1] → G be a Lipschitz curve and let h ∈ L ∞ (0, 1) m be its vector of canonical coordinates. Then L n (γ) := L n (γ), if γ(0) = x and γ(1) = y; +∞, otherwise; L(γ) := L(γ), if γ(0) = x and γ(1) = y; +∞, otherwise.Arguing as in[8, Theorem 3.1], we can show that lim inf n→∞Ln (γ n ) ≥L(γ) whenever γ n → γ in Lip(Ω). To conclude we need to prove that, for every γ ∈ Lip([0, 1], Ω), there exists an approximating sequence {γ n } satisfying lim sup nLn (γ n ) ≤L(γ). We can assume without loss of generality thatL(γ) = L(γ). Take a sequence (γ n ) n∈N with γ n → γ in Lip([0, 1], Ω) and since L n Γ − → L we can suppose that lim n L n (γ n ) = L(γ). One can construct the optimal sequence modifying the curves as follows: d n -geodesic connecting x and γ n ( 1 n ), if t ∈ [0, 1 n ]; γ n (t), if t ∈ [ 1 n , 1− 1 n ]; an almost d n -geodesic connecting γ n (1− 1 n ) and y, if t ∈ [1− 1 n , 1 n ]. Similarly to[8, Theorem 3.1], and using again Lemma 3.8 we get thatγ n still converges to γ in Lip(Ω) and lim n d n (γ n 1 − 1 n , y) = 0. Hence passing to the lim sup as n → ∞ in (4.7) gives lim sup n→∞L n (γ) ≤ lim sup n→∞ L(γ n ) = L(γ) =L(γ). Definition 5 . 3 . 53The pointwise Lipschitz constant of a Lipschitz function f : G → R is defined as and t > 0, we define the curve γ = γ x,v,t : [0, 1] → G as γ(s) := x · δ ts e v for every s ∈ [0, 1]. 9b) yield ψ(γ s ,γ s ) ≥ ψ(e, d γs τ γ −1 s [γ s ]) − α 2 t v e ε for a.e. s ∈ [0, 1], (5.10a) d γs exp −1 − d γs τ γ −1 s L(Tγ s G,g) ≤ ε for a.e. s ∈ [0, 1], (5.10b) For any f ∈ LIP d (X), we denote by LIP d (f ) ∈ [0, +∞) and Lip d f : X → [0, +∞) the (global) Lipschitz constant and the pointwise Lipschitz constant of f , respectively. Moreover, given a Finsler manifold (M, F ), we denote by d F the length distance on M induced by the Finsler metric F . Theorem A.1. Let M be a smooth manifold. Let d be a distance on M having the following property: there exists a sequence (F i ) i of Finsler metrics on M such that d F i (x, y) ր d(x, y) for every x, y ∈ M.Then for any f ∈ LIP d (M ) there exists a sequence (fn ) n ⊆ C ∞ (M ) ∩ LIP d (M ) such that sup n∈N LIP d (f n ) ≤ LIP d (f ),f n → f uniformly on compact sets.Proof. Denote L := LIP d (f ) and d i := d F i for every i ∈ N. Choose any countable, dense subset (x j ) j of (M, d). Given any n ∈ N, we define the function h n ∈ LIP d (M ) as h n (x) := − L d(x, x 1 ) + f (x 1 ) ∨ · · · ∨ − L d(x, x n ) + f (x n ) − 1 n for every x ∈ M. y) for every n ∈ N. From the convergence of F n to ϕ ⋆ we deduce that d Fn (x, y) → d ϕ ⋆ (x, y) for every x, y ∈ G (cf. the proof of [18, Theorem 5.1]), and thus(5.18) d ϕ ⋆ (x, y) = lim n→∞ d ϕ ⋆ n (x, y) for every x, y ∈ G.Finally, since ϕ n is lower semicontinuous (actually, continuous) by Lemma 2.13, we know from the second part of Theorem 5.11 that (5.19) δ ϕn (x, y) = d ϕ ⋆ n (x, y) for every n ∈ N.All in all, we obtain that d ϕ ⋆ (x, y) (5.18) = lim n→∞ d ϕ ⋆ n (x, y) (5.19) = lim n→∞ δ ϕn (x, y) (5.16) On the limits of periodic Riemannian metrics. E Acerbi, G Buttazzo, J. Anal. Math. 43E. Acerbi and G. Buttazzo, On the limits of periodic Riemannian metrics, J. Anal. Math., 43 (1984), pp. 183-201. A Agrachev, D Barilari, U Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University PressA. Agrachev, D. Barilari, and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Ge- ometry, Cambridge University Press, 2019. Homogenization of Periodic Finsler metrics. M Amar, E Vitali, J. Convex Anal. 5M. Amar and E. Vitali, Homogenization of Periodic Finsler metrics, J. Convex Anal., 5 (1998), pp. 171- 186. Homogeneous Carnot groups related to sets of vector fields, Bollettino dell'Unione Matematica Italiana. A Bonfiglioli, A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields, Bollettino dell'Unione Matem- atica Italiana, 7-B (2004), pp. 79-107. A Bonfiglioli, E Lanconelli, F Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians. BerlinSpringer Monographs in MathematicsA. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Berlin, Springer, Springer Monographs in Mathematics, 2007. Monge solutions for discontinuous Hamiltonians, ESAIM: Control, Optimisation and Calculus of Variations. A Briani, A Davini, 11A. Briani and A. Davini, Monge solutions for discontinuous Hamiltonians, ESAIM: Control, Optimisa- tion and Calculus of Variations, 11 (2005), pp. 229-251. A Course in Metric Geometry. D Burago, Y D Burago, S O Ivanov, American Mathematical SocietyD. Burago, Y. D. Burago, and S. O. Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001. Topological equivalence of some variational problems involving distances. G Buttazzo, L De Pascale, I Fragalà, Discrete Contin. Dinam. Systems. 7G. Buttazzo, L. De Pascale, and I. Fragalà, Topological equivalence of some variational problems involving distances, Discrete Contin. Dinam. Systems, 7 (2001), pp. 247-258. An Introduction to Γ-convergence. G Maso, Progress in Nonlinear Differential Equations and their Applications. BaselBirkhäuser VerlagG. Dal Maso, An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 1993. Distanza intrinseca su una varietà Riemanniana di Lipschitz. G De Cecco, G Palmieri, Rend. Sem. Mat. Torino. 46G. De Cecco and G. Palmieri, Distanza intrinseca su una varietà Riemanniana di Lipschitz, Rend. Sem. Mat. Torino, 46 (1988), pp. 157-170. LIP manifolds: from metric to Finslerian structure. Math Z. 207, LIP manifolds: from metric to Finslerian structure, Math Z., 207 (1991), pp. 223-243. Intrinsic distance on a LIP Finslerian manifold. Rend.Accad. Naz. Sci. XL Mem. Mat. 17, Intrinsic distance on a LIP Finslerian manifold, Rend.Accad. Naz. Sci. XL Mem. Mat., 17 (1993), pp. 129-151. From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence. A Garroni, M Ponsiglione, F Prinari, Calc. Var. 27A. Garroni, M. Ponsiglione, and F. Prinari, From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence, Calc. Var., 27 (2006), pp. 397-420. Intrinsic geometry and analysis of Finsler structures. 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E Le Donne, D Lučić, E Pasqualetto, arXiv:1910.05962To appear in Potential AnalysisE. Le Donne, D. Lučić, and E. Pasqualetto, Universal infinitesimal Hilbertianity of sub-Riemannian manifolds. To appear in Potential Analysis, arXiv:1910.05962, 2019. Towards differential calculus in stratified groups. V Magnani, J. Aust. Math. Soc. 95V. Magnani, Towards differential calculus in stratified groups, J. Aust. Math. Soc., 95 (2013), pp. 76-128. Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups. V Magnani, A Pinamonti, G Speight, Annali di Matematica Pura ed Applicata199V. Magnani, A. Pinamonti, and G. Speight, Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups, Annali di Matematica Pura ed Applicata (1923 -), 199 (2017). Distances, boundaries and surface measures in Carnot-Carathéodory spaces. R Monti, University of TrentoPhD ThesisR. Monti, Distances, boundaries and surface measures in Carnot-Carathéodory spaces, 2001. PhD Thesis, University of Trento. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. P Pansu, Annals of Mathematics. 2P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Annals of Mathematics (2), 129 (1989), pp. 1-60. The large scale geometry of nilpotent Lie groups. S Pauls, Comm. in Analysis and Geometry. 5S. Pauls, The large scale geometry of nilpotent Lie groups, Comm. in Analysis and Geometry, 5 (2001). A Pinamonti, G Speight, Porosity, Differentiability and Pansu's theorem. 27A. Pinamonti and G. Speight, Porosity, Differentiability and Pansu's theorem, The Journal of Geo- metric Analysis, 27 (2017). Is a diffusion process determined by its intrinsic metric?. K T Sturm, Chaos Solitons Fractals. 8K. T. Sturm, Is a diffusion process determined by its intrinsic metric?, Chaos Solitons Fractals, 8 (1997), pp. 1855-1860. Email address: [email protected] (Enrico Pasqualetto) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa. S Venturini, Via Sommarive. 14Università degli Studi di TrentoPreprintFares Essebei) Dipartimento di Matematica. Italy Email address: [email protected]. Venturini, Derivations of Distance Functions in R n . Preprint, 1991. (Fares Essebei) Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Povo (Trento), Italy Email address: [email protected] (Enrico Pasqualetto) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Email address: [email protected]
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[ "Alternative k = −1 loop quantum cosmology", "Alternative k = −1 loop quantum cosmology" ]
[ "Jinsong Yang \nSchool of Physics\nGuizhou University\n550025GuiyangChina\n", "Cong Zhang \nInstitut für Quantengravitation\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B291058ErlangenGermany\n\nFaculty of Physics\nUniversity of Warsaw\nPasteura 502-093WarsawPoland\n", "Xiangdong Zhang \nDepartment of Physics\nSouth China University of Technology\n510641GuangzhouChina\n" ]
[ "School of Physics\nGuizhou University\n550025GuiyangChina", "Institut für Quantengravitation\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nStaudtstr. 7/B291058ErlangenGermany", "Faculty of Physics\nUniversity of Warsaw\nPasteura 502-093WarsawPoland", "Department of Physics\nSouth China University of Technology\n510641GuangzhouChina" ]
[]
An alternative quantization of the gravitational Hamiltonian constraint of the k = −1 Friedmann-Robertson-Walker model is proposed by treating the Euclidean term and the Lorentzian term independently, mimicking the treatment of full loop quantum gravity. The resulting Hamiltonian constraint operator for the k = −1 model with a massless scalar field is successfully constructed, and is shown to have the corrected classical limit. Compared to the former quantization schemes in the literature where only the Euclidean term is quantized, the new quantum dynamics of the k = −1 model with a massless scalar field indicates that the classical big-bang singularity is replaced by an asymmetric quantum bounce.
10.1103/physrevd.107.046012
[ "https://export.arxiv.org/pdf/2212.05748v2.pdf" ]
254,564,176
2212.05748
420177af80323291ec322514337338e470c4be9b
Alternative k = −1 loop quantum cosmology Jinsong Yang School of Physics Guizhou University 550025GuiyangChina Cong Zhang Institut für Quantengravitation Friedrich-Alexander-Universität Erlangen-Nürnberg Staudtstr. 7/B291058ErlangenGermany Faculty of Physics University of Warsaw Pasteura 502-093WarsawPoland Xiangdong Zhang Department of Physics South China University of Technology 510641GuangzhouChina Alternative k = −1 loop quantum cosmology An alternative quantization of the gravitational Hamiltonian constraint of the k = −1 Friedmann-Robertson-Walker model is proposed by treating the Euclidean term and the Lorentzian term independently, mimicking the treatment of full loop quantum gravity. The resulting Hamiltonian constraint operator for the k = −1 model with a massless scalar field is successfully constructed, and is shown to have the corrected classical limit. Compared to the former quantization schemes in the literature where only the Euclidean term is quantized, the new quantum dynamics of the k = −1 model with a massless scalar field indicates that the classical big-bang singularity is replaced by an asymmetric quantum bounce. I. INTRODUCTION How to quantize general relativity (GR) in a consistent manner is a great challenge to theoretical physics. One of the promising candidates is the so-called loop quantum gravity (LQG) which is a nonperturbative approach to quantum GR [1][2][3][4]. In the past three decades, LQG has made remarkable progress, such as making the natural predictions of the discretized geometries and providing the microscopic interpretation of BH entropy [5][6][7][8][9][10][11][12]. The nonperturbative quantization procedure of LQG has been successfully applied to the metric f (R) theories [13,14], scalar-tensor theories [15,16], higher-dimensional gravity [17], and so on [18]. Despite these achievements, the dynamics of full LQG is still an unsolved issue. To gain a certain level of understanding of the dynamics, the quantization ideas and technologies developed in LQG have also be applied to its symmetry-reduced models, such as the Friedmann-Robertson-Walker (FRW) models and the spherically symmetric black hole models, leading to loop quantum cosmology (LQC) and loop quantum black hole models [19,20]. The most successful feature of LQC is that it can resolve the classical big bang singularity by a quantum bounce due to the quantum geometry effects. We refer to [19,[21][22][23] for more complete reviews on LQC. In full LQG, the gravitational Hamiltonian constraint is a combination of the so-called Euclidean term and the Lorentzian term. In the spatially flat, k = 0 FRW model, the Lorentzian term and the Euclidean term are proportional to each other. Thus one often combines these two terms into one term proportional to the Euclidean term, and then quantizes the Euclidean term to obtain the well-defined gravitational Hamiltonian constraint operator [19,22]. It turns out that in this quantization scheme the classical big-bang singularity is replaced by a symmetric quantum bounce for the k = 0 FRW model with a massless scalar field in the framework of LQC [22]. Note that in full LQG, the Lorentzian term is quantized independently by employing the Thiemann's trick [24]. Thus, * [email protected] † [email protected] ‡ Corresponding author; [email protected] to mimic the full LQG quantization procedure in the k = 0 model of LQC, the Euclidean term and the Lorentzian term were treated independently [25][26][27]. This alternative quantization scheme leads to an asymmetric quantum bounce, which relates the spatially flat FRW model with an asymptotic de Sitter universe, and thus an effective cosmological constant and an effective Newtonian constant can be obtained [28][29][30]. As in the k = 0 model, the quantization technologies for the gravitational Hamiltonian constraint developed in LQG have been extended to the k = −1, +1 models [31][32][33][34][35][36][37][38]. Compared to the k = 0 model where the spin connection vanishes and hence the Ashtekar connection equals to the extrinsic curvature multiplied by the Immirzi parameter, the Lorentzian term is not proportional to the total Euclidean term, but is proportional to the part of the Euclidean term involving the extrinsic curvature due to the nonvanishing spin connection for both the k = −1 model and the k = +1 model. Hence in the literatures one often absorbs the Lorentzian term into a part of the Euclidean term, and then quantize the two parts of the Euclidean term, respectively. It turns out that, as the k = 0 model with the similar treatment, the resulting k = −1 LQC model also predicts a vacuum repulsion in the high curvature regime that would lead to a symmetric bounce [31]. Moreover, the k = −1 model of LQC also possesses some new features that never appears in the k = 0 model, for example, due to a vacuum repulsion in the high curvature regime, the scale factor has the minimum value as a min = γ √ ∆ [31]. It is natural to ask whether the treatment of the Lorentzian term independently, mimicking the treatment in the full theory, can be directly carried to the k = −1 model, and whether an asymmetric bounce can still be held for the k = −1 model. This is the main motivation of the present paper. In this paper, we consider an alternative quantization of the gravitational Hamiltonian constraint in the k = −1 model by treating the Lorentzian term independently. This paper is organized as follows. The canonical formulation of the k = −1 model is briefly recalled in Sec. II. Then we propose an alternative gravitational Hamiltonian constraint operator by treating the Lorentzian term independently, and provide a new quantum dynamics for the k = −1 model in Sec. III. The effective theory of the new quantum dynamics and its asymptotic behavior are studied in Sec. IV. Summary arXiv:2212.05748v2 [gr-qc] 22 Feb 2023 is included in the last section. II. CANONICAL FORMULATION OF THE k = −1 MODEL According to the cosmological principle, the line elements of the homogenous isotropic cosmological models take as ds 2 = −dt 2 + a 2 (t) 1 1 − kr 2 dr 2 + r 2 dθ 2 + sin 2 θdφ 2 , (2.1) where a(t) is the scale factor, and k = −1, 0, 1 for the open, flat, and closed FRW models, respectively. In what follows, we present the canonical formulation of the k = −1 model following Ref. [31]. For the spatially noncompact k = 0, −1 models with topology homeomorphic to R 3 , one introduces an "elemental cell" V on the homogeneous spatial manifold R 3 and restrict all integrals to this elemental cell. Then one chooses a fiducial metric o q ab = o ω i a o ω j b δ ij on R 3 with o ω i a being the left-and right-invariant fiducial oneforms in the k = 0 model, and only the left-invariant fiducial one-forms in the k = −1 model. Here a, b, · · · denote the spatial indices while i, j, · · · = 1, 2, 3. Denote by V o the volume of V measured by the fiducial metric o q ab . The left-invariant one-forms o ω i a satisfy the Maurer-Cartan equation d o ω i + 1 2 C i jk o ω j ∧ o ω k = 0, (2.2) where for the k = −1 model the structure constants read Classically, the dynamical variables of LQC are obtained by symmetrically reducing those of full LQG. In the full theory, the dynamical variables consist of the su(2)-valued connection A i a and the densitized triadẼ b j with the nontrivial Poisson bracket C i jk = δ i j δ k1 − δ i k δ j1 ,(2.{A i a (x),Ẽ b j (y)} = κγδ b a δ i j δ(x, y),(2.5) where κ = 8πG with G being the Newtonian constant, and γ is the Immirzi parameter [39,40]. The connection A i a is related to the spin connection Γ i a and the extrinsic curvature K i a by A i a = Γ i a + γK i a . It turns out that the symmetryreduced extrinsic curvature K i a is diagonal in the basis of leftinvariant one-forms for the k = 0, −1 models. While, unlike the k = 0 model where Γ i a vanishes, the symmetry-reduced spin connection Γ i a in the k = −1 model takes the form [31] Γ i a = − 1ij o ω j a , (2.6) and thus it is nondiagonal. Hence the symmetry-reduced connection and densitized triad for the k = −1 model read [31] A i a = − 1ij o ω j a + cV − 1 3 o o ω i a ≡ A i j V − 1 3 o o ω j a , (2.7) E a i = pV − 2 3 o det( o q) o e a i ,(2.8) where o and c = γȧV 1 3 o . The physical volume V of the elemental cell V measured by the spatial (physical) A i j =    c 0 0 0 c −V 1 3 o 0 V 1 3 o c    ,(2.metric q ab = |p|V − 2 3 o o q ab is related to p via V = |p| 3/2 . In the improved scheme, it is convenient to choose the following variables to simplify the dynamics [41] b :=μ c 2 , v := sgn(p)|p| 3/2 2πγ 2 p √ ∆ , (2.11) where p ≡ √ G denotes the Planck length, sgn(p) is the signature of p, ∆ ≡ 4 √ 3 πγ 2 p is the minimum nonzero eigenvalue of the area operator in full LQG [42], andμ ≡ ∆/|p|. The Poisson bracket between b and v is given by {b, v} = 1 . (2.12) As in the k = 0 model, the Gauss and diffeomorphism constraints of the gravitational part are automatically satisfied for the symmetry-reduced variables in Eqs. (2.7) and (2.8) in the k = −1 model, and thus the classical dynamics is encoded in the Hamiltonian constraint. The gravitational Hamiltonian constraint of the k = −1 model reads H k=−1 grav := V d 3 xẼ a iẼ b j 2κ det(q) ij k (A) F k ab − 2(1 + γ 2 )K i [a K j b] = V d 3 xẼ a iẼ b j 2κ det(q) ij k (γK) F k ab − 2(1 + γ 2 )K i [a K j b] + V d 3 xẼ a iẼ b j 2κ det(q) ij k (Γ) F k ab ≡H E,k=0 grav − 2(1 + γ 2 )H L,k=0 grav + H Γ,k=−1 grav , (2.13) where det(q) denotes the determinant of q ab , and (x) F k ab := 2∂ [a x k b] + k lm x l a x m b .(2.(b, v) asH E,k=0 grav = 3γ √ ∆ b 2 |v|, (2.15) H L,k=0 grav = 3 2γ √ ∆ b 2 |v|, (2.16) H Γ,k=−1 grav = 3 γ √ ∆ 1 3 V 2 3 o 4 (2πG) 2 3 |v| 1 3 . (2.17) Hence the gravitational Hamiltonian constraint (2.13) of the k = −1 model reduces to H k=−1 grav = − 3 |v| γ √ ∆ b 2 − V 2 3 o γ 2 ∆ 16πG |v| 2 3 ≡ − 3 |v| γ √ ∆ g(b, v). (2.18) At the classical level, we assume that the universe is filled by a massless scalar field φ. The Hamiltonian of the scalar field φ is given by H φ = p 2 φ 2V = p 2 φ 4πγ 2 P √ ∆ |v| ,(2.19) where p φ denotes the conjugate momentum of φ. The Poisson bracket between φ and p φ is {φ, p φ } = 1. Hence the total Hamiltonian constraint of gravity coupled to a massless scalar field reads H k=−1 tot = H k=−1 grav + H φ = − 3 |v| γ √ ∆ g(b, v) + p 2 φ 4πγ 2 P √ ∆ |v| . (2.20) By the total Hamiltonian constraint equation H k=−1 tot = 0,(2.H 2 k=−1 = v 3v 2 = {v, H k=−1 tot } 3v 2 = 8πG 3 ρ φ + V 2/3 o V 2/3 = 8πG 3 ρ φ + 1 a 2 ,(2.22) where · denotes a derivative with respect to the time determined by H k=−1 tot , and ρ φ = p 2 φ 2V 2 is the energy density of the scalar field φ. III. LOOP QUANTIZATION OF THE k = −1 MODEL To pass the classical theory of k = −1 model to its quantum theory, one needs to construct the kinematical Hilbert space. In the k = 0 model, the vanishing Γ i a enables us to identify A i a with γK i a , leading to the identification of the holonomies of the connection and those of the extrinsic curvature (mutiplied by γ). The resulting holonomies of the connection A i a , equal to γK i a in the k = 0 model, along edges generated by the left-and right-invariant vector fields o e a i with physical length λV 1/3 take the form h (λ) i = cos λc 2 I + 2τ i sin λc 2 , where τ i := − i 2 σ i with σ i being the Pauli matrices. Hence the related algebra is that of the almost periodic functions, and thus the kinematical Hilbert space for the gravitational part can be defined as H gr,k=0 kin = L 2 (R Bohr , dµ Bohr ), where R Bohr and dµ Bohr are respectively the Bohr compactification of the real line R and the Haar measure on it [19]. However, in the k = −1 case, the spin connection Γ i a takes the nonvanishing expression (2.6), resulting in a difference between the holonomy of the connection and the one of the extrinsic curvature. Moreover, due to the nondiagonal form (2.7) of the connection, the holonomies of the connection take complicated forms in the k = −1 model, leading to the algebra generated is no longer that of the almost periodic function [31]. Instead, one often considers the holonomies of the extrinsic curvature γK i a in the k = −1 model, which take the same forms as those in the k = 0 model. More precisely, considering an edge e i starting from the basepoint of the elemental cell V, with tangent vector parallel to the vector o e a i and taking length λ, following Refs. [31,32], we define the "holonomy" of γK i a = cV λc 2 τ i . (3.1) Here P denotes the path ordering which orders the smallest path parameter to the left [2], and it takes the trivial action in our model as in the works [2,19,31,32]. Clearly, these holonomies (3.1) generate the algebra of almost periodic functions, and thus result in the kinematical Hilbert space for the k = −1 model being H gr kin ≡ H gr,k=−1 kin = H gr,k=0 kin [31]. As in the k = 0 model, we will employ theμ-scheme to define the Hamiltonian operator. This requires us to consider the holonomis along the edges taking physical length 31,32], which are given by √ ∆ [2,h (μ) i := P exp μV 1/3 o 0 dt γK j a τ j o e a i = eμ cτi = cos μc 2 I + 2 sin μc 2 τ i = cos(b) I + 2 sin(b) τ i ,(3.2) and their inverse take the forms: h (μ) i −1 = cos(b) I − 2 sin(b) τ i . (3.3) In the v-representation for both the k = 0 model and the k = −1 model, the two elementary operators, e ib andv, act on the basis |v of H gr kin as e ib |v = |v + 1 ,v |v = v |v . (3.4) Thus one can easily write down the action of the operators h (μ) i = cos(b) I + 2 sin(b) τ i (3.5) corresponding to the holonomies h (μ) i of the extrinsic curvature γK i a on |v in terms of e ib . For the scalar field, it is convenient to choose the Schrödinger representation [23]. Thus the kinematical Hilbert space for the scalar field part can be chosen as H sc kin := L 2 (R, dφ). Hence the total kinematical Hilbert space of the k = −1 model with a scalar field is H tot kin = H gr kin ⊗ H sc kin . We now consider an alternative regularization of the gravitational Hamiltonian constraint of the k = −1 model in Eq. (2.13), such that it is closer to that in the k = 0 model as well as to that in full LQG. As mentioned previously, the two terms H E,k=0 grav andH L,k=0 grav in Eq. (2.13) have the same forms as the Euclidean and Lorentzian terms in the k = 0 model, respectively. Hence it is natural to expect that the two terms in the k = −1 model can be regularized as the corresponding forms in the k = 0 model. To realize explicitly this idea, some subtle issues should be clarified. Firstly, we consider the first term H E,k=0 grav = V d 3 xẼ a iẼ b j 2κ det(q) ij k (γK) F k ab = V d 3 xẼ a iẼ b j 2κ det(q) ij k 2∂ [a γK k b] + k lm γK l a γK m b = V d 3 xẼ a iẼ b j ij k 2κ det(q) k lm γK l a γK m b , (3.6) where in the third step we used the fact that the first term in the integral of the second line vanishes due to o e a i o e b j ij k 2∂ [a γK k b] = cV − 1 3 o o e a i o e b j ij k 2∂ [a o ω k b] = −cV − 1 3 o o e a i o e b j ij k C k lm o ω l a o ω m b = −cV − 1 3 o ij k C k ij = 0. (3.7) To regularizeH E,k=0 grav in Eq. (3.6), one needs to use the Thiemann's trick and o e a j were adopted to regularize the curvature (γK) F k ab . In [32], the author proposed closed loops 2 ij generated by the integral curves of the left-invariant vector fields o e a i and the right-invariant vector fields o η b j commuting with o e a i . In the present paper, we will consider open holonnomies to represent extrinsic curvature following Refs. [27,36,38,43]. Inputting Eq. (3.8) into Eq. (3.6), one obtains E a iẼ b j det(q) ij k = 2 κγ˜ abc {A k c , V } = 2 κγ˜ abc {γK k c , V },H E,k=0 grav = 1 κ 2 γ V d 3 x˜ abc k lm γK l a γK m b {γK k c , V } = − 4 κ 2 γ V d 3 x˜ abc Tr (γK a γK b {γK c , V }) ,(3.9) where the identity Tr(τ i τ j τ k ) = − 1 4 ijk was used. In cosmology the known identities [27,36,38,43] take the forms: γK a = h (2μ) i − h (2μ) i −1 4μV 1/3 o o ω i a , (3.10) {γK c , V } = − 1 µV 1/3 o k h (μ) k h (μ) k −1 , V o ω k c , (3.11) where h (μ) i (or h (2μ) i ) is defined by (3.2). It should be noticed that Eq. (3.10), which is precisely valid in the limit µ → 0, should be understood as a regularized expression in theμ-scheme withμ = ∆/|p|. Substituting Eqs. (3.10) and (3.11) into Eq. (3.9) and assuming for simplicity that the holonomies of k = −1 can be approximated with the holonomies of k = 0, we arrive at H E,k=0,reg grav = sgn(p) 4κ 2 γμ 3 i,j,k ijk Tr h (2μ) i − h (2μ) i −1 h (2μ) j − h (2μ) j −1 h (μ) k h (μ) k −1 , V = 2 γ 4 √ ∆ sin(2b) v k Tr τ k h (μ) k h (μ) k −1 , |v| sin(2b),(3.12) where the identity τ i τ j = 1 2 ijm τ m − 1 4 δ ij was used. The resulting regularized expressionH E,k=0,reg grav in Eq. (3.12) is the same as the regularized Euclidean Hamiltonian constraint H E,k=0,reg grav in the k = 0 model [22]. We now consider the second termH L,k=0 grav in Eq. (2.13). Classically, the term H L,k=0 grav is proportional to the termH E,k=0 grav , and hence the termH L,k=0 grav does not need to be quantized independently. This approach to quantization ofH L,k=0 grav in the k = −1 has been adopted in [31,32], similar to the k = 0 case in [22]. Alternatively, the Lorentzian term in the k = 0 case can be regularized independently in [27], mimicking the treatment of full LQG. It is natural to ask whether the treatment for the Lorentzian term in the k = 0 case can be directly carried to that forH L,k=0 grav in the k = −1 model, and the resulting operator is the same as that in the k = 0 model. The answer is in the affirmative. To this end, let us recall the key identities for regularizing the Lorentzian term of the gravitational Hamiltonian constraint in the full theory, study their symmetry-reduced forms in the k = −1 model, and then compare them with those in the k = 0 model. The first classical identity reads K i a τ i = 1 κγ {A i a τ i , K} = 1 κγ {Γ i a τ i + γK i a τ i , K} = 1 κγ {γK i a τ i , K} = − 2 3κγ 1 µV 1/3 o i h (μ) i h (μ) i −1 , K o ω i a , (3.13) where K := d 3 xK i aẼ a i , the former steps hold for the full theory and thus hold for its symmetry-reduced models, while the third step holds due to the fact that the spin connection Γ i a in Eq. (2.6) is proportional to o ω j a up to a constant for the k = −1 case, and the vanishing connection for the k = 0 model. In the last step, we have used the relation [27] {cτ i , K} = − 2 3μ h (μ) i h (μ) i −1 , K . (3.14) Here, it is worth noting that the above equation is satisfied only in theμ scheme whereμ is a function of p, rather that a certain constant in the µ o scheme. In theμ scheme,μ depending on p does not commute with K, leading to a factor 2/3 on the right-hand side of Eq. (3.14). The second identity is K = 1 γ 2 { (A) H E , V } = 1 γ 2 { (Γ) H E + (γK) H E , V } = 1 γ 2 { (γK) H E , V } = 1 γ 2 {H E,k=0 grav , V } = 1 γ 2 {H E,k=0 grav , V }, (3.15) where (x) H E := d 3 xẼ a iẼ b j 2κ √ det(q) ij k (x) F k ab . Hence, to regularize K, one just replacesH E,k=0 grav by its regularized versioñ H E,k=0,reg grav in Eq. (3.15). It should also be noted here that both µ and V depend only on p. As a result, the Poisson bracket betweenH E,k=0,reg grav and V has the same form for both theμ and µ o schemes. Thus the above two classical identities which play key roles in the regularization of the Lorentzian term hold in the k = −1 model, and take the same forms as those in the k = 0 model. Therefore, to regularizeH L,k=0 grav independently in the k = −1 model, one can follow directly the treatment of the Lorentzian term H L,k=0 grav in the k = 0 model, mimicking the treatment of the full theory. To this end, we first re-express H L,k=0 grav in the form , H L,k=0 grav = V d 3 xẼ a iẼ b j 2κ det(q) K i [a K j b] = V d 3 xẼ a iẼ b j 2κ det(q) ij k 1 2 k lm K l a K m b = − 2 κ 2 γ V d 3 x˜ abc Tr (K a K b {γK c , V }H E,k=0,reg grav , V h (μ) j h (μ) j −1 , H E,k=0,reg grav , V h (μ) k h (μ) k −1 , V = 8 sgn(p) 9κ 4 γ 7μ3 i,j,k ijk Tr h (μ) j h (μ) j −1 , H E,k=0,reg grav , V h (μ) k h (μ) k −1 , V h (μ) i h (μ) i −1 , H E,k=0,reg grav , V = 4 √ ∆ 288γ 3 i,j,k ijk Tr h (μ) i h (μ) i −1 , H E,k=0,reg grav , |v| v h (μ) j h (μ) j −1 , |v| h (μ) k h (μ) k −1 , H E,k=0,reg grav , |v| . (3.17) Similarly, the last term in Eq. (2.13) can be regularized as [31,32] H Γ,k=−1,reg grav = sgn(p)V 2 3 o 2κπGγμ k Tr τ k h (μ) k h (μ) k −1 , V = ( γ √ ∆) 1 3 V 2 3 o 4(2πG) 2 3 sgn(v)|v| 1 3 k Tr τ k h (μ) k h (μ) k −1 , |v| .(H E,k=0 grav = − i γ 4 √ ∆ sin(2b) v k Tr τ k h (μ) k h (μ) k −1 , |v| sin(2b) = i γ 2 √ ∆ sin(2b) vÔ |v| sin(2b) k Tr(τ k τ k ) = − i3 γ 4 √ ∆ sin(2b) vÔ |v| sin(2b),(3.19) where in the second step we have used h (μ) k h (μ) k −1 ,B =B I − h (μ) kB h (μ) k −1 =B I − cos(b) I + 2 sin(b) τ k B cos(b) I − 2 sin(b) τ k =B I − cos(b)B cos(b) I − 4 sin(b)B sin(b) τ k τ k + 2 sin(b)B cos(b) − cos(b)B sin(b) τ k = B − sin(b)B sin(b) − cos(b)B cos(b) I − 2ÔB τ k ,(3.20) here in the fourth step the identity τ k τ k = − 1 4 I for k = 1, 2, 3 was used, the operatorÔB, depending on the operatorB, is defined byÔB := sin(b)B cos(b) − cos(b)B sin(b),(3.21) and Tr(τ k ) = 0. In the last step in Eq. (3.19) we have used k Tr(τ k τ k ) = − 3 2 . Similarly, the regularized expression (3.17) can be quantized aŝ H L,k=0 grav = − i √ ∆ 288 γ 3 i,j,k ijk Tr h (μ) i h (μ) i −1 , Ĥ E,k=0 grav , |v| v h (μ) j h (μ) j −1 , |v| h (μ) k h (μ) k −1 , Ĥ E,k=0 grav , |v| = i √ ∆ 36 γ 3Ô[Ĥ E,k=0 grav ,|v|] vÔ |v| Ô [Ĥ E,k=0 grav ,|v|] i,j,k ijk Tr(τ i τ j τ k ) = − i √ ∆ 24 γ 3Ô[Ĥ E,k=0 grav ,|v|] vÔ |v| Ô [Ĥ E,k=0 grav ,|v|] ,(3.22) where in the second step we have used Eq. (3.20) and used that the terms involving zero, one and two τ in the trace vanish due to = ( γ √ ∆) 1 3 V 2 3 o i4(2πG) 2 3 sgn(v)|v| 1 3 k Tr τ k h (μ) k h (μ) k −1 , |v| = − ( γ √ ∆) 1 3 V 2 3 o i2(2πG) 2 3 sgn(v)|v| 1 3Ô |v| k Tr(τ k τ k ) = 3( γ √ ∆) 1 3 V 2 3 o i4(2πG) 2 3 sgn(v)|v| 1 3Ô |v| . (3.23) The actions of the operatorsĤ E,k=0 grav ,Ĥ L,k=0 grav , andĤ Γ,k=−1 grav on |v read H E,k=0 grav |v = E + (v)|v + 4 + E 0 (v)|v + E − (v)|v − 4 , (3.24) H L,k=0 grav |v = L + (v)|v + 8 + L 0 (v)|v + L − (v)|v − 8 , (3.25) H Γ,k=−1 grav |v = Γ(v)|v ,(3.26) where E + (v) = 3γ 32 √ ∆ (v + 2)M 1,3 (v), (3.27) E − (v) = E + (v − 4), (3.28) E 0 (v) = −E + (v) − E − (v), (3.29) L + (v) = − √ ∆ 192γ 3 (v + 4)M −1,1 (v + 4) × G − (v + 4)G + (v + 4), (3.30) L − (v) = L + (v − 8), (3.31) L 0 (v) = − √ ∆ 192γ 3 (v + 4)M −1,1 (v + 4)[G + (v)] 2 +(v − 4)M −1,1 (v − 4)[G − (v)] 2 , (3.32) Γ(v) = 3 γ √ ∆ 1 3 V 2 3 o 8 (2πG) 2 3 sgn(v)|v| 1 3 M 1,−1 (v).H k=−1 grav |v =L + (v)|v + 8 + E + (v)|v + 4 + E 0 (v) +L 0 (v) + Γ(v) |v + E − (v)|v − 4 +L − (v)|v − 8 ,(3.37) whereL * := −2(1 + γ 2 )L * , here * = +, −, 0. On the other hand, the Hamiltonian constraint H φ for the scalar field can be quantized as a well-defined operatorĤ φ in H tot kin , and the action ofĤ φ on a quantum state |ψ = ψ(v, φ)|v, φ with |v, φ ≡ |v ⊗ |φ ∈ H gr kin ⊗ H sc kin is given by [22] H φ · ψ(v, φ) = − 2 4πγ √ ∆ 2 p C(v)∂ 2 φ ψ(v, φ),(3.38) where C(v) ≡ 3 2 3 |v| |v + 1| 1/3 − |v − 1| 1/3 3 . (3.39) Combining equations above, one can write down the resulting quantum Hamiltonian constraint equation corresponding to its classical one (2.21) aŝ H k=−1 tot · ψ(v, φ) = Ĥ k=−1 grav +Ĥ φ · ψ(v, φ) = 0, (3.40) which describes the quantum evolution of the coupled system with the scalar field φ as an emergent time. IV. EFFECTIVE THEORY AND ITS ASYMPTOTIC BEHAVIOR OF THE ALTERNATIVE k = −1 LQC By constructing certain coherent states peaked at points of the classical phase space and computing the expectation value of the Hamiltonian constraint operator under the coherent states, one can obtain the corresponding effective Hamiltonian constraint. To this end, we first note that the symmetryreduced phase space of the k = −1 model coincides with that of the k = 0 model. Hence certain coherent states constructed for the k = 0 model can be directly carried to the k = −1 model. A Gaussian coherent state peaked at a point (b o , v o , φ o , p φ ) in the classical phase space with spreads and σ in the gravitational sector and scalar field sector takes the form [41] Ψ (bo,vo,φo, p φ ) := dφ v∈R e − 2 2 (v−vo) 2 e ibo(v−vo) × e − σ 2 2 (φ−φo) 2 e i p φ (φ−φo) (v| ⊗ (φ|,(4.1) and its shadow on the regular lattice with spacing one reads |Ψ := dφ n∈Z e − 2 2 (n−vo) 2 e −ibo(n−vo) × e − σ 2 2 (φ−φo) 2 e − i p φ (φ−φo) |n ⊗ |φ ≡ |Ψ grav ⊗ |Ψ φ . (4.2) To make the state be sharply peaked in the classical phase space of the universe with large volume, one should require that b o , v o 1, σ φ o and p φ σ 1. Denote by Ô := Ψ|Ô|Ψ Ψ|Ψ the expectation value of an operatorÔ under the coherent states (4.2). By using the Poisson resummation on the sum over n and the steepest descent approximation, the expectation value of each term of the gravitational Hamiltonian constraint operatorĤ k=−1 grav can be calculated, and thus the resulting expectation value ofĤ k=−1 grav can be obtained. For brevity, in the remainder of this paper, we will suppress the label o appearing in b o , v o and φ o . A straightforward calculation reveals that (see Appendix A for a derivation) [27,28,44] H E,k=0 grav,eff := Ĥ E,k=0 grav = 3 γv 4 √ ∆ sin 2 (2b) + O( 2 ) 1 + O(e −π 2 / 2 ) + O 1/(v ) 2 , (4.3) H L,k=0 grav,eff := Ĥ L,k=0 grav = 3 v 32γ √ ∆ sin 2 (4b) + O( 2 ) 1 + O(e −π 2 / 2 ) + O 1/(v ) 2 , (4.4) H Γ,k=−1 grav,eff := Ĥ Γ,k=−1 grav = 3 γ √ ∆ 1 3 V 2 3 o 4 (2πG) 2 3 v 1 3 1 + O(e −π 2 / 2 ) + O 1/(v ) 2 . (4.5) Hence, in the region with b, v 1, σ φ and p φ σ 1 the higher-order corrections can be omitted. In what follows, we focus on the leading terms. Hence the effective Hamiltonian constraint of gravitational part for the k = −1 model reads H k=−1 grav,eff = H E,k=0 grav,eff − 2(1 + γ 2 )H L,k=0 grav,eff + H Γ,k=−1 grav,eff = − 3 v γ √ ∆ 1 4 sin 2 (2b) 1 − (1 + γ 2 ) sin 2 (2b) − γ 2 ∆ 4 V 2/3 o V 2/3 ≡ − 3 v γ √ ∆ g eff (b, v). (4.6) Taking into account the result for the scalar field in [27], the total effective Hamiltonian constraint of the gravity coupled with a massless scalar field reads H k=−1 tot,eff = − 3 v γ √ ∆ g eff (b, v) + p 2 φ 4πγG √ ∆ v . (4.7) Before calculating the dynamics of this effective Hamiltonian, we clarify some subtle issues. First, we expect here that the evolution of, saying, v up to O( ) order coincides with the dynamics determined by the effective Hamiltonian constraint (4.7). In other words, if we compute the quantum dynamics of the coherent state (4.2) and investigate the evolution of the expectation value ofv, we conjecture that the result coincides with the dynamics of v obtained by solving the Hamilton's equation concerning the effective Hamiltonian (4.7). Second, as claimed before, we consider the region with b so that the higher-order corrections are omitted. However, as shown later, in the FRW phase of the evolution given by the effective Hamiltonian, b approaches 0 asymptotically. We thus obtain a tension that, on the one hand, we require b but, on the other hand, b goes to 0 along the evolution. To resolve this tension, we still have to compute the quantum dynamics to see how the spread = 1/ 2| v 2 − v 2 | evolves. Even though the quantum dynamics, which will be left as our future work, has not been investigated yet, the previous results in the k = 0 model [22,45] make us expect ∼ 1/ v which would make b true along the evolution. Indeed, coherent states with the phase-space dependent spread have been considered in the regular LQC [46]. Moreover, another approach to understand the effective dynamics is to apply the path integral formulation to study transition amplitude A(v f , φ f ; v i , φ i ) = v f , φ f |v i , φ i phy with ·|· phy denoting the physical inner products [47,48]. Since the Hamiltonian constraint operatorĤ k=−1 grav is the same as that of the k = 0 model up to the termĤ Γ,k=−1 grav,eff which takes |v as its eigenstate, one can simply generalize the results in [48] to conclude that the classical path resulting from dynamics of the effective Hamiltonian dominates A(v f , φ f ; v i , φ i ). Finally, in the region with b → 0 and v 1, it is easy to see that, as b → 0, g eff (b, v) goes to g(b, v), and thus the total effective Hamiltonian constraint H k=−1 tot,eff in Eq. (4.7) reduces to its classical expression H k=−1 tot in Eq. (2.20). Hence the new alternative quantum dynamics has the corrected classical limit. It is easy to see that, in the effective theory, p φ is a constant of motion due toṗ φ = {p φ , H k=−1 tot,eff } = 0, and φ can be regarded as an internal clock because ofφ > 0, similar to the classical theory. The effective Hamiltonian constraint equation H k=−1 tot,eff = 0 (4.8) can determine the evolutions of v with respect to b for some given p φ , which is plotted in Fig. 1. Figure 1 depicts that v = 0 can never be a solution to Eq. (4.8). It indicates that the classical singularity at v = 0 can be avoided in the effective theory. By Eq. (4.8), the matter density can be expressed as ρ φ (v) = p 2 φ 2V 2 = − H k=−1 grav,eff V = 3 2πGγ 2 ∆ g eff (b, v) ≡ ρ eff φ (b, v). (4.9) The effective Hubble parameter is determined by the total effective Hamiltonian constraint (4.7), and reads H 2 eff,k=−1 = v 3v 2 =   v, H k=−1 tot,eff 3v   2 = 1 γ 2 ∆ [g eff (b, v)] 2 ,(4.10) where denotes the first-order derivative with respect to b. In what follows, we focus on the region b ∈ [0, π/4] where we live. A bounce appears when H 2 eff,k=−1 = 0, i.e., g eff (b, v) = 0 ⇔ b = 1 2 arcsin 1 2(1 + γ 2 ) , (4.11) at which the energy density ρ φ takes the maximal value, namely the critical energy density, as ρ k=−1 crit = ρ F − 3 8πG V 2/3 o V 2/3 = ρ F − 3 8πG 1 a 2 ,(4.12) where ρ F := 3 32πGγ 2 (1 + γ 2 )∆ . (4.13) In comparison with the effective k = 0 model proposed in [27] where the critical density is given by the first term of Eq. ρ φ (v bounce ) = ρ k=−1 crit (v bounce ). (4.14) In Fig. 2, the value v bounce as a function of p φ is plotted. It is shown that as p φ increase, the value v bounce increase monotonically. Moreover the condition ρ φ (v bounce ) ≥ 0 implies that a ≥ 4γ 2 (1 + γ 2 )∆, (4.15) and thus the effective theory predicts that the scale factor a is bounced below a min = 4γ 2 (1 + γ 2 )∆. Solving the total effective Hamiltonian constraint equation H k=−1 tot,eff = 0 for b yields b =            b I ≡ 1 2 arcsin 1− 1− ρ φ + 3 8πG 1 a 2 ρ F 2(1+γ 2 ) b II ≡ 1 2 arcsin 1+ 1− ρ φ + 3 8πG 1 a 2 ρ F 2(1+γ 2 ) . (4.16) Hence there exists two types of classical universe, namely the type-I universe and the type-II universe. The two universes are connected by a quantum bounce. The effective Hubble parameter is determined by H 2 eff,k=−1,I/II = v 3v 2 b=b I/II = 8πG 3 ρ F 1 + γ 2 1 − ρ φ + 3 8πG 1 a 2 ρ F   1 ∓ 1 − ρ φ + 3 8πG 1 a 2 ρ F     1 + 2γ 2 ± 1 − ρ φ + 3 8πG 1 a 2 ρ F   , (4.17) which can be expressed as the following more convenient forms H 2 eff,k=−1,I = 8πG 3 ρ φ + 3 8πG 1 a 2 1 − ρ φ + 3 8πG 1 a 2 ρ F      1 + γ 2 1 + γ 2     ρ φ + 3 8πG 1 a 2 ρ F 1 + 1 − ρ φ + 3 8πG 1 a 2 ρ F     2      ,(4.18) and H 2 eff,k=−1,II = 8πG 3 ρ Λ eff 1 − ρ φ + 3 8πG 1 a 2 ρ F       1 + 1 − 2γ 2 + 1 − ρ φ + 3 8πG 1 a 2 ρ F 4γ 2 1 + 1 − ρ φ + 3 8πG 1 a 2 ρ F ρ φ + 3 8πG 1 a 2 ρ F       ,(4.19) where ρ Λ eff := Λ eff 8πG , with Λ eff := 3 (1 + γ 2 ) 2 ∆ . (4.20) Now let us study the asymptotic behavior of the effective dynamics at the large v limit. For v → ∞, the matter density ρ φ (v) in Eq. (4.9) goes to zero, and thus sin 2 (2b) 1 − (1 + γ 2 ) sin 2 (2b) → 0, (4.21) which implies b → b 0 =    b I,c ≡ 0, b II,c ≡ 1 2 arcsin 1 √ 1+γ 2 . (4.22) Expanding H k=−1 tot,eff at b 0 up to the second order yields the classical behavior of the effective Hamiltonian constraint as H k=−1 tot,eff → − 3 v γ √ ∆ g eff (b 0 , v) + g eff (b 0 )(b − b 0 ) + g eff (b 0 ) 2 (b − b 0 ) 2 + p 2 φ 4πγG √ ∆ v ,(4.23) where denotes the second-order derivative with respective to b, and g eff (b 0 ) = 0, b 0 = b I,c , − γ 1+γ 2 , b 0 = b II,c , (4.24) g eff (b 0 ) = 2, b 0 = b I,c , 2−10γ 2 1+γ 2 , b 0 = b II,c . (4.25) Plugging these asymptotic expressions into Eq. (4.10), we have H 2 eff,k=−1 → 8πG 3 g eff (b 0 ) 2 ρ φ + 3 [g eff (b 0 )] 2 8πGγ 2 ∆ + 3g eff (b 0 ) 16πG V 2/3 0 V 2/3 = 8πG 3 g eff (b 0 ) 2 ρ φ + 3 8πG 1 a 2 + 3 [g eff (b 0 )] 2 8πGγ 2 ∆ = 8πG 3 ρ φ + 1 a 2 , b 0 = b I,c , 8πG 3 1−5γ 2 1+γ 2 ρ φ + 1−5γ 2 1+γ 2 1 a 2 + Λ eff 3 , b 0 = b II,c . (4.26) The above asymptotic behavior (4.26) of the effective Hubble parameter can be also obtained directly from Eqs. (4.18) and (4.19). Equation (4.26) implies that the type-II universe is an asymptotic de Sitter universe with a positive effective cosmological constant Λ eff . Therefore the asymptotical k = −1 FRW universe (the type-I universe) will be bounced to an asymptotic de Sitter universe (the type-II universe) coupled to a scalar field. We now numerically study the the effective dynamical evolution of v with φ. To this end, we can firstly solve the effective Hamiltonian constraint equation H k=−1 tot,eff = 0 to yield v = v(b, p φ ). (4.27) Secondly, we consider the evolution equation of φ with respect to b, namely dφ db = {φ, H k=−1 tot,eff } {b, H k=−1 tot,eff } = f (b, p φ ),(4.28) where Eq. (4.27) was inserted in the second step. Solving Eq. (4.28) yields φ = φ(b, p φ ). (4.29) By combining Eq. (4.27) with Eq. (4.29) and then eliminating b, we arrive at v = v(φ, p φ ). In Fig. 3, the effective dynamical evolution of v with respect to φ for given p φ is plotted. It indicates that an asymmetric bounce appears in the backward evolution of the universe sourced by a massless scalar field φ, and the classical big-bang singularity is resolved. V. SUMMARY The quantization ambiguities often exist in constructing the gravitational Hamiltonian constraint operator of full LQG as well as of LQC. It has been shown that in LQC different quantizations of the gravitational Hamiltonian constraint may lead to different quantum dynamics, and thus affect the fate of the universe. Hence the study of the quantization ambiguities of the gravitational Hamiltonian constraint plays an important role in the quantum dynamics of LQC. In present paper, we have studied an alternative quantization of the gravitational Hamiltonian constraint in the k = −1 model of LQC closely following that in the k = 0 model proposed in [27], mimicking the treatment of full LQG. .15) and (2.16), the former two terms are proportional to each other, and thus one can firstly combine the two terms and then quantize the first and third terms to obtain the gravitational Hamiltonian constraint [31]. On the other hand, from the viewpoint of full LQG, the sum of both the first and last terms forms the Euclidean term, while the middle term represents the Lorentzian term. Hence, alternatively, we can quantize the Euclidean term and the Lorentzian term respectively, mimicking the treatment in full LQG. We have shown that the former two termsH E,k=0 grav andH L,k=0 in Eq. (2.13) can be quantized as the operators, H E,k=0 grav andĤ L,k=0 grav in Eqs. (3.19) and (3.22), corresponding precisely to the Euclidean and Lorentzian Hamiltonian operators of the k = 0 model proposed in [27], while the third term has been quantized asĤ Γ,k=−1 grav in Eq. is symmetric, which has the action (3.37) on |v . Moreover, we have shown that the new quantum dynamics determined by the alternative Hamiltonian constraint operatorĤ k=−1 tot in Eq. (3.40) for the k = −1 model coupled to a massless scalar field has the corrected classical limit, and obtained its effective Hamiltonian constraint (4.7), by semi-classical anal-ysis. The effective Friedmann equation for the k = −1 model was derived in Eq. (4.17), which shows that it has two branches (4.18) and (4.19) relating to two types of universes, similar to the k = 0 LQC proposed in [27]. It turns out that the asymptotical k = −1 FRW universe (the type-I universe) will be bounced to an asymptotic de Sitter universe (the type-II universe) coupled to a scalar field. Last but not least, by requiring the condition ρ φ (v bounce ) ≥ 0, the effective theory predicts that the scale factor a is bounced below a min = 4γ 2 (1 + γ 2 )∆, which is different from that in previous k = −1 LQC model [31]. Classically, the connection So far, the Thiemann's trick for regularizing the gravitational Hamiltonian constraint in full LQG, by treating the Euclidean term and the Lorentzian term independently, has been successfully applied to the k = 0, −1 LQC models. However, to our knowledge, a similar treatment for the k = +1 model in the framework of LQC has not been carried out. In spite of the Thiemann-regularization of the Hamiltonian constraint on the hyperspherical lattice for the k = +1 model has been studied from the viewpoint of full LQG in [49]. The expectation value of the Hamiltonian constraint under certain coherent states was computed, and an effective Hamiltonian constraint was obtained in the µ o scheme rather than theμ scheme [49]. By numerical simulations of the dynamical evolution, an asymmetric bounce replacing the classical big bang was also obtained in the model [49]. It should be noted that there are still many aspects of the loop quantum k = −1 model that deserves further investigation. Recently, some works have focused on the relation between LQG and the k = 0 LQC by calculating the expectation value of the Hamiltonian in LQG under certain coherent state peaked at some point in the classical phase space [50][51][52][53][54][55]. How to generate these works to the k = −1 case will be interesting. Moreover, except the alternative regularization from the Thiemann's trick adopted in the present paper following directly that in the k = 0 case, the other alternative regularizations employed in the k = 0 model [30,[56][57][58] can also be in principle extended to the k = −1 model. where l is a positive even number, and b o − 1 2 −4f + v o − l 2 + 1 2 [2f 0 (v o )] 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 ,(A6) for analytic functions f + (y) and f 0 (y). If f + (y) [or f 0 (y)] is not analytic function which is the case under considerations due to the involved absolute value, one can replace it with its analytic extentionf + (y) [orf 0 (y)] (for example, omitting the absolute value symbol). It turns out that the error n exp(− 2 (n−v o ))[f + (n)−f + (n)] can be shown to be the order O e −v 2 o 2 , which is negligible compared to the corrections derived above [19]. Hence the resulting normalized expectation value ofĤ reads Ĥ = Ψ|Ĥ|Ψ Ψ|Ψ = Ψ grav |Ĥ|Ψ grav Ψ grav |Ψ grav = e − l 2 4 2 sin 2 l 2 b o − 1 2 −4f + v o − l 2 + 1 2 [2f 0 (v o )] 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 .(A7) Now we turn to the three parts of the gravitational Hamiltonian constraint operatorĤ k=−1 grav . Applying the result in Eq. (A7) to the first two termsĤ E,k=0 grav andĤ L,k=0 grav with l = 4 and l = 8, respectively, we have Ĥ E,k=0 grav = 3 γv o 4 √ ∆ sin 2 (2b o ) e −4 2 + 1 2 1 − e −4 2 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 = 3 γv o 4 √ ∆ sin 2 (2b o ) + O( 2 ) 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 ,(A8)Ĥ L,k=0 grav = 3 v o 32γ √ ∆ sin 2 (4b o ) e −16 2 + 1 2 1 − e −16 2 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 = 3 v o 32γ √ ∆ sin 2 (4b o ) + O( 2 ) 1 + O(e −π 2 / 2 ) + O 1/(v o ) 2 ,(A9) where we have used the results −4Ē + (v o − 2) = 3 γ 4 √ ∆ v o , 2Ē 0 (v o ) = 3 γ 4 √ ∆ v o , (A10) −4L + (v o − 4) = 3 32γ √ ∆ v o , 2L 0 (v o ) = 3 32γ √ ∆ v o .(A11) the k = 0 model they take zero. The corresponding left-invariant vector fields o e a i are dual to o ω δ a b . The commutators between the left-invariant vector fields read [ o e i , o e j ] = is the Levi-Civita density, and then to express the curvature (γK) F k ab of γK i a in terms of holonomies of γK i a . In the k = 0 case, since the left-and right-invariant vector fields o e a i commute to each other, the integral curvatures of o e a i and o e b j can form closed loops 2 ij , around which the curvature (γK) F k ab can be recast as the holonomies h (μ) 2ij of γK i a . Compared to the k = 0 case, due to the noncommutativity of the left-invariant vector fields o e a i in the k = −1 case, the integral curves of o e a i and o e a j can not provide closed loops. In [31], holonomies of γK i a based on the open curves generated by o e a i ij = 0 . 0The operatorsÔ |v| andÔ [Ĥ E,k=0 grav ,|v|] are defined according to Eq. (3.21) withB = |v| andB = [Ĥ E,k=0 grav , |v|], respectively. Moreover, in the last step in Eq. (3.22), we have used i,j,k ijk Tr(τ i τ j τ k ) assuming again for simplicity that the holonomies of k = −1 can be approximated with the holonomies of k ,b (v) := |v + a| − |v + b|, (3.34) G ± (v) := E ± (v − 1)M 0,±4 (v − 1) − E ± (v + 1)M 0,±4 (v + 1). (3.35)The function M a,b (v) satisfiesM 0,|k| (v) = −M 0,−|k| (v + |k|). (3.36) Hence the action of the k = −1 gravitational Hamiltonian constraint operatorĤ k=−1 grav on |v readŝ FIG. 1 . 1Plots of v with respect to b determined by the total Hamiltonian constraint equation H k=−1 tot,eff = 0 for different values of p φ , with V0 = G = = 1 and γ = 0.2375. ( 4 . 412), the effective k = −1 model contributes an additional term, the second term of Eq.(4.12), to the critical energy density. It is worth mentioning that the critical energy density ρ k=−1 crit in the effective k = −1 model depends on the value of v (or the scale factor a) at the bounce point. The value v bounce of v at the bounce point can be determined by FIG. 2 . 2Plot of v bounce with respect to p φ , with V0 = G = = 1 and γ = 0.2375. A i a in the k = −1 model takes the nondiagonal expression (2.7) on the left-invariant oneforms o ω i a , while it takes the diagonal form on the left-and right-invariant one-forms o ω i a in the k = 0 model. In the k = −1 model, the nondiagonal expression of A i a leads to the complicated forms of the resulting holonomies of the connection. Instead, one often considers the holonomies of the extrinsic curvature K i a multiplied by γ in the k = −1 model, and thus they have the same expressions as the holonomies of the connection A i a = γK i a in the k = 0 model. Hence both the k = 0 model and the k = −1 model have the same Hilbert space H gr kin = L 2 (R Bohr , dµ Bohr ). To study the quantum dynamics of the k = 0, −1 models in the framework of LQC, one needs to promote the gravitational Hamiltonian constraint into a well-defined operator in H gr kin . At the classical level, the gravitational Hamiltonian constraint of the k = −1 model can be expressed as the three terms in Eq. (2.13). On one hand, the former two terms, H E,k=0 grav andH L,k=0 , in Eq. (2.13) have the same expressions as the Euclidean and Lorentzian terms, H E,k=0 grav and H L,k=0 , in the k = 0 model, respectively, although involving different fiducial one-forms o ω i a (the only left-invariant one-forms v.s. the left-and right-invariant one-forms). From the symmetryreduced expressions in Eqs. (2 FIG. 3 . 3(3.23). The resulting gravitational Hamiltonian constraint operatorĤ k=−Plots of the effective dynamical evolution of v with respect to φ, in reverse direction of the cosmological time, determined by the effective Hamiltonian constraint for different values of p φ with initial data v(bstart, p φ ) = 8p φ and φ(bstart, p φ ) = 1, with V0 = G = = 1 and γ = 0.2375. ×e [f + (n) n |n + l + f − (n) n |n − l + f 0 (n) (n −vo) 2 +(n +l−vo) 2 ] f + (− 2 (n−vo) 2 f 0 (n) , (A3)where in the third step we have used Eq. (A2) and relabeled n − l by n . 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[ "arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices", "arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices", "arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices", "arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices" ]
[ "A De Wit \nService de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium\n", "Y Bertho \nService de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium\n\nMicrogravity Research Center\nUniversité Libre de Bruxelles\n165/62, 1050BrusselsCPBelgium\n", "M Martin \nLaboratoire PMMH-ESPCI (UMR 7636\n10 rue Vauquelin75 231, Cedex 05ParisFrance\n", "A De Wit \nService de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium\n", "Y Bertho \nService de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium\n\nMicrogravity Research Center\nUniversité Libre de Bruxelles\n165/62, 1050BrusselsCPBelgium\n", "M Martin \nLaboratoire PMMH-ESPCI (UMR 7636\n10 rue Vauquelin75 231, Cedex 05ParisFrance\n" ]
[ "Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium", "Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium", "Microgravity Research Center\nUniversité Libre de Bruxelles\n165/62, 1050BrusselsCPBelgium", "Laboratoire PMMH-ESPCI (UMR 7636\n10 rue Vauquelin75 231, Cedex 05ParisFrance", "Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium", "Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems\nUniversité Libre de Bruxelles\n231, 1050BrusselsCPBelgium", "Microgravity Research Center\nUniversité Libre de Bruxelles\n165/62, 1050BrusselsCPBelgium", "Laboratoire PMMH-ESPCI (UMR 7636\n10 rue Vauquelin75 231, Cedex 05ParisFrance" ]
[]
Viscous fingering of a miscible high viscosity slice of fluid displaced by a lower viscosity fluid is studied in porous media by direct numerical simulations of Darcy's law coupled to the evolution equation for the concentration of a solute controlling the viscosity of miscible solutions. In contrast with fingering between two semi-infinite regions, fingering of finite slices is a transient phenomenon due to the decrease in time of the viscosity ratio across the interface induced by fingering and dispersion processes. We show that fingering contributes transiently to the broadening of the peak in time by increasing its variance. A quantitative analysis of the asymptotic contribution of fingering to this variance is conducted as a function of the four relevant parameters of the problem i.e. the log-mobility ratio R, the length of the slice l, the Péclet number P e and the ratio between transverse and axial dispersion coefficients ε. Relevance of the results is discussed in relation with transport of viscous samples in chromatographic columns and propagation of contaminants in porous media.
10.1063/1.1909188
[ "https://export.arxiv.org/pdf/physics/0508080v1.pdf" ]
9,764,796
physics/0508080
04d53e5bddd307dbccc06d1a8fde7c52d0185f70
arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices A De Wit Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems Université Libre de Bruxelles 231, 1050BrusselsCPBelgium Y Bertho Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems Université Libre de Bruxelles 231, 1050BrusselsCPBelgium Microgravity Research Center Université Libre de Bruxelles 165/62, 1050BrusselsCPBelgium M Martin Laboratoire PMMH-ESPCI (UMR 7636 10 rue Vauquelin75 231, Cedex 05ParisFrance arXiv:physics/0508080v1 [physics.flu-dyn] 12 Aug 2005 Viscous fingering of miscible slices Viscous fingering of a miscible high viscosity slice of fluid displaced by a lower viscosity fluid is studied in porous media by direct numerical simulations of Darcy's law coupled to the evolution equation for the concentration of a solute controlling the viscosity of miscible solutions. In contrast with fingering between two semi-infinite regions, fingering of finite slices is a transient phenomenon due to the decrease in time of the viscosity ratio across the interface induced by fingering and dispersion processes. We show that fingering contributes transiently to the broadening of the peak in time by increasing its variance. A quantitative analysis of the asymptotic contribution of fingering to this variance is conducted as a function of the four relevant parameters of the problem i.e. the log-mobility ratio R, the length of the slice l, the Péclet number P e and the ratio between transverse and axial dispersion coefficients ε. Relevance of the results is discussed in relation with transport of viscous samples in chromatographic columns and propagation of contaminants in porous media. Viscous fingering of a miscible high viscosity slice of fluid displaced by a lower viscosity fluid is studied in porous media by direct numerical simulations of Darcy's law coupled to the evolution equation for the concentration of a solute controlling the viscosity of miscible solutions. In contrast with fingering between two semi-infinite regions, fingering of finite slices is a transient phenomenon due to the decrease in time of the viscosity ratio across the interface induced by fingering and dispersion processes. We show that fingering contributes transiently to the broadening of the peak in time by increasing its variance. A quantitative analysis of the asymptotic contribution of fingering to this variance is conducted as a function of the four relevant parameters of the problem i.e. the log-mobility ratio R, the length of the slice l, the Péclet number P e and the ratio between transverse and axial dispersion coefficients ε. Relevance of the results is discussed in relation with transport of viscous samples in chromatographic columns and propagation of contaminants in porous media. I. INTRODUCTION Viscous fingering is an ubiquitous hydrodynamic instability that occurs as soon as a fluid of given viscosity displaces another more viscous one in a porous medium [1]. As such, the typical example usually presented for this instability is that of oil recovery for which viscous fingering takes place when an aqueous solution displaces more viscous oil in underground reservoirs. This explains why numerous articles devoted to the theoretical and experimental analysis of fingering phenomena have appeared in the petroleum engineering community [1]. For what concerns the geometry, theoretical works typically focus on analyzing the stability properties and nonlinear dynamics of an interface between two semi-infinite domains of different viscosity. In the same spirit, experimental works done either in real porous media or in a model Hele-Shaw system (two parallel plates separated by a thin gap width) consist in injecting continuously a low viscous fluid into the medium initially filled with the more viscous one. The attention is then focused on the dynamics of the interface between the two regions. The instability develops and the fingers grow continuously in time until the displacing fluid has invaded the whole experimental system. As long as the experiment runs (i.e. until the displacing fluid reaches the outlet), the instability develops. Dispersion of one fluid into the other may lead to a slight stabilization in time nevertheless this stabilization is usually negligible on the time scale of the experiment and for high injection rates. The situation is drastically different in other important applications in which viscous fingering is observed, such as in liquid chromatography or groundwater contamination. Liquid chromatography is used to separate the chemical components of a given sample by passing it through a porous medium. In some cases, and typically in preparative or size exclusion chromatography, the viscosity of the sample is significantly different than that of the displacing fluid (the eluent). Displacement of the sample by the eluent of different viscosity leads then to viscous fingering of either the front or the rear interface of the sample slice, leading to deformation of the initial planar interface. This fingering is dramatic for the performance of the separation technique as it contributes to peak broadening and distortions. Such conclusions have been drawn by several authors that have shown either experimentally [2,3,4,5,6,7,8] or numerically [7,8] the influence of viscous fingering on peak deformations. In groundwater contamination and more generally soil contamination, it is not rare that the spill's extent is finite due to a contamination localized in space and/or time. If the spill's fluid properties are different than that of water, and in particular, if they have different viscosity and/or density [9,10], fingering phenomena may influence the spreading characteristics of the contaminated zone. For ecological reasons, it is important then to quantify to what extent fingering will enlarge the broadening in time of this polluted area. Nonlinear simulations of fingering of finite samples have been performed in the past by Tucker Norton, Fernandez et al. [7,8] in the context of chromatographic applications, by Christie et al. in relation to "Water-Alternate Gas" (WAG) oil recovery techniques [11] as well as by Zimmerman [12] and have shown the influence of fingering on the deformation of the sample without however investigating the asymptotic dynamics. Manickam and Homsy, in their theoretical analysis of the stability and nonlinear dynamics of viscous fingering of miscible displacements with nonmonotonic viscosity profiles have further stressed the importance of reverse fingering in the deformation of finite extent samples [13,14]. Their parametric study has focused on analyzing the influence of the endpoint and maximum viscosities on the growth rate of the mixing zone. In this framework, the objectives of this article are twofold: first, we analyze the nonlinear dynamics of viscous fingering of miscible slices in typical analytical chromatographic and groundwater contamination conditions in order to underline its specificities and, second, we quantify the asymptotic contribution of viscous fingering to the broadening of the output peaks as a function of the important parameters of the problem. From a numerical point of view, the only difference with regard to most of the previous works devoted to viscous fingering [15,16,17] is the initial condition which is now a sample of finite extent instead of the traditional interface between two semi-infinite domains. As we show, this has an important consequence: if the longitudinal extent of the slice is small enough with regard to the length of the migration zone, dispersion becomes of crucial importance as it leads to such a dilution of the displaced sample into the bulk fluid before reaching the measurement location that fingering just dies out. As a consequence fingering is then only a transient phenomenon and the output peak of the diluted sample may look Gaussian even if its variance is larger than that of a pure diffusive dynamics because of transient fingering. This explains why the importance of fingering phenomena in chromatography and soil contamination has been largely underestimated or ignored in the literature. We perform here numerical simulations to compute the various moments of the sample distribution as a function of time when fingering takes place. This allows us to extract the contribution of viscous fingering to the variance of the averaged concentration profile and to understand how this contribution varies with the important parameters of the problem which are the logmobility ratio R between the viscosity of the sample and that of the bulk fluid, the Péclet number P e, the dimensionless longitudinal extent l of the slice and the ratio ε between the transverse and longitudinal dispersion coefficients. The outline of the article is the following. In Sec. II, we introduce the model equations of the problem. Typical experimental parameters for liquid chromatography and groundwater contamination applications are discussed in Sec. III. The characteristics of the fingering of a miscible slice are outlined in Sec. IV, while a discussion on the moments of transversely averaged profiles is done in Sec. V. Eventually, a parametric study is conducted in Sec. VI before a discussion is made. II. MODEL SYSTEM Our model system is a two-dimensional porous medium of length L x and width L y (Fig. 1). A slice of fluid 2 of length W is injected in the porous medium initially filled with carrier fluid 1. This fluid 2, which is a solution of a given solute of concentration c 2 in the carrier, will be referred in the following as the sample. This sample is displaced by the carrier fluid 1 in which the solute concentration c is equal to c 1 = 0. Assuming that the viscosity of the medium is a function of the concentration c and that the flow is governed by Darcy's law, the evolution equations for the system are then: ∇ · u = 0,(1)∇p = − µ(c) K u,(2)∂c ∂t + u · ∇c = D x ∂ 2 c ∂x 2 + D y ∂ 2 c ∂y 2 ,(3) where µ is the viscosity of the fluid, K is the permeability of the medium, p is the pressure and u = (u, v) is the twodimensional velocity field. The displacing fluid is injected in a uniform manner with a mean velocity U along the x direction. D x , D y are the dispersion coefficients along the flow direction and perpendicular to it respectively. The characteristic speed U is used to define a characteristic length L c = D x /U and time τ c = D x /U 2 . We nondimensionalize space, speed and time by L c , U and τ c respectively. Pressure, viscosity and concentration are scaled by µ 1 D x /K, µ 1 and c 2 , where µ 1 is the viscosity of the displacing fluid and c 2 the initial concentration of the sample. The dimensionless equations of the system become ∇ · u = 0, (4) ∇p = −µ(c)u,(5)∂c ∂t + u · ∇c = ∂ 2 c ∂x 2 + ε ∂ 2 c ∂y 2 ,(6) where ε = D y /D x . If ε = 1, dispersion is isotropic while ε = 1 characterizes anisotropic dispersion. Switching to a coordinate system moving with speed U , i.e. making the change of variables x ′ = x− t, y ′ = y, u ′ = u − i x with i x being the unit vector along x, we get, after dropping the primes: ∇ · u = 0, (7) ∇p = −µ(c)(u + i x ),(8)∂c ∂t + u · ∇c = ∂ 2 c ∂x 2 + ε ∂ 2 c ∂y 2 .(9) We suppose here that the viscosity is an exponential function of c such as: µ(c) = e Rc ,(10) where R is the log-mobility ratio defined by R = ln(µ 2 /µ 1 ), where µ 2 is the viscosity of the sample and, as said before, µ 1 is the viscosity of the displacing fluid ( Fig. 1). If R > 0, then we have a low viscosity fluid displacing a high viscosity sample and the rear interface of the sample will be unstable with regard to viscous fingering. If R < 0, then the sample is the less viscous fluid and the front interface of the slice will then develop fingering. In our simulations, we consider the R > 0 situation. Introducing the stream function ψ such that u = ∂ψ/∂y and v = −∂ψ/∂x, taking the curl of Eq. (8), we get our final equations [18]: ∇ 2 ψ = R ∂ψ ∂x ∂c ∂x + ∂ψ ∂y ∂c ∂y + ∂c ∂y ,(11)∂c ∂t + ∂ψ ∂y ∂c ∂x − ∂ψ ∂x ∂c ∂y = ∂ 2 c ∂x 2 + ε ∂ 2 c ∂y 2 .(12) This model is numerically integrated using a pseudospectral code introduced by Tan and Homsy [15] and successfully implemented for various numerical studies of fingering [19,20]. The two-dimensional domain of integration is, in dimensionless units, of size P e × L where P e = U L y /D x is the dimensionless width which is nothing else than the Péclet number of the problem, while L = U L x /D x . The dimensionless length of the sam- ple is l = U W/D x . The initial condition corresponds to a convectionless fluid (ψ = 0 everywhere) embedding a rectangular sample of concentration c = 1 and of size P e × l in a c = 0 background. The middle of the sample is initially located at x = 2L/3. In practice, for the simulations, the initial condition corresponds to two back to back step functions between c = 0 and c = 1 with an intermediate point where c = 1 2 + A · r, r being a random number between 0 and 1 and A the amplitude of the noise (typically of the order of 10 −3 ). This noise is necessary to trigger the fingering instability on reasonable computing time. If A = 0, numerical noise will ultimately seed the fingering instability but on a much longer time scale. The boundary conditions are periodic in both directions. This is quite standard along the transversal direction y. This does not make any problem along the x-axis as c = 0 at both x = 0 and x = L. The problem is controlled by four dimensionless parameters: the log-mobility ratio R, the Péclet number P e, the initial length of the injected sample l and the ratio between transverse and longitudinal dispersion coefficients ε. III. EXPERIMENTAL VALUES OF PARAMETERS FOR TWO APPLICATIONS In order to perform numerical simulations, let us compute the order of magnitude of the main parameters (Péclet number P e, length of the sample l) for both a liquid chromatography experiment and for the propagation of contaminants in a porous medium (groundwater contamination). A. Chromatographic applications First of all, let us note that in most chromatographic applications, heterogeneous chemistry (particularly adsorption and desorption phenomena) is crucial to the separation process and will undoubtedly affect possible fingering processes. We neglect such physicochemical interactions in this first approach focusing on the effect of viscous fingering on an unretained compound. A typical chromatographic column has a diameter d = 4.6 mm, a length L x = 150 mm and consists of a porous medium packed with porous particles, the total (intraparticle and interparticle) porosity being equal to 0.7. The volume of the sample introduced in the column is of the order of 20 µl, injected at a flow rate Q ≃ 1 ml min −1 . The extent of the injected sample is then W ≃ 1.7 10 −3 m and the speed of the flow U ≃ 1.4 10 −3 m s −1 . The longitudinal and transverse dispersion coefficients are typically D x = 1.43 10 −8 m 2 s −1 [21] and D y = 5.65 10 −10 m 2 s −1 [22]. These parameters allow one to define a characteristic length L c = D x /U and a characteristic time τ c = D x /U 2 . As a result, the Péclet number P e = U d/D x is here nothing else than the dimensionless diameter i.e. P e = d/L c ≃ 460, while the dimensionless longitudinal extent of the sample becomes l = W/L c ≃ 170. The dispersion ratio is equal to ε = D y /D x ≃ 0.04. As a typical transit time from inlet to outlet takes roughly τ = 100 s, the dimensionless time of a simulation should be of the order of T = τ /τ c ≃ 15000 to account for a realistic time to characterize the properties of the output peaks. B. Soil contamination The effects of fluid viscosity and fluid density may be important in controlling groundwater flow and solute transport processes. Recently, a series of column experiments were conducted and analyzed by Wood et al. [10] to provide some insight into these questions. The experiments were performed in fully saturated, homogeneous and isotropic sand columns (porosity equals to 0.34 and ε = 1) by injecting a 250 ml pulse of a known concentration solution at a flow rate Q = 0.015 m 3 /day. Their experimental setup consists of a vertical pipe L x = 0.91 m in length with a diameter of d = 0.15 m. Assuming the medium to be homogeneous and the dispersion coefficient D as isotropic, a typical value for the aquifer dispersion coefficient is D = 0.1 m 2 /month i.e. D ≃ 3.86 10 −8 m 2 s −1 [23]. In the same spirit as above, we compute the Péclet number to be of the order of P e ≃ 110, while the dimensionless length of the sample is l ≃ 30. Based on these two examples, let us now investigate the properties of fingering of finite slices for typical values of parameters in the range computed above i.e. P e ∼ 100 − 500, ε ∼ 0.04 − 1, l ∼ 0 − 500, while R is supposed to be of order one. Figure 2 shows in a frame moving with the injection velocity U the typical viscous fingering of the rear interface of a sample displaced from left to right by a less viscous fluid. The system is shown at successive times using density plots of concentration with black (resp. white) corresponding to c = 1 (resp. c = 0). While the front interface is stable, the back interface develops fingers such that the center of gravity of the sample is displaced in the course of time towards the back with regard to its initial position. This dynamics results from the fact that the stable zone acts as a barrier to finger propagation in the flow direction leading therefore to reverse fingering. Such a reverse fingering has been well characterized by Manickam and Homsy in their numerical analysis of fingering of nonmonotonic viscosity profiles [14]. After a while, dispersion comes into play and dilutes the more viscous fluid into the bulk of the displacing fluid. As the sample becomes more and more diluted, the effective viscosity ratio decreases in time weakening the source of the instability. Ultimately, dispersion becomes dominant and the sample goes on diluting in the bulk without witnessing any further fingering phenomenon. These successive steps can clearly be observed on the transverse averaged IV. FINGERING OF A FINITE WIDTH SAMPLE As seen on Fig. 3, we first start with two back to back step functions defining an initial sample of extent l. During the first stages of the injection, there is a first diffusive regime quickly followed by the fingering of the rear interface corresponding here to the left front. While the right (i.e. frontal) interface features the standard error function characteristic of simple dispersion, the left one shows bumps signaling the presence of fingering. Because the extent of the sample is finite, dispersion and fingering contribute to the fact that the maximum concentration becomes smaller than one, effectively leading to a viscosity ratio between the sample and the bulk that decreases in time. As a consequence, fingering dies out and the transverse profile starts to follow a distorted Gaussian shape. If one waits long enough, the asymmetry of the bell shape diminishes which explains that output peaks in chromatographic columns may look Gaussian even if fingering has occurred during the first stages of the travel of the sample in the column. As computed in the preceding section, a typical dimensionless time of transit in a real chromatographic setup corresponds to 15000 units of time. Figures 2 and 3 show that, after 15000 units of time, fingering is disappearing for this specific set of typical values of parameters, and that dispersion becomes again the dominant mode. As chromatographic columns are generally opaque porous media, it is therefore not astonishing that the presence of viscous fingering has long been totally ignored until recent experimental works which have visualized fingering by magnetic resonance or optical imaging [2,3,4,5,6,7,8]. Similarly, tracing of the spatial extent of a contaminant plume at a distance far from the pollution site may lead to measurements of Gaussian-type spreading even if fingering has occurred at early times. The only influence of such fingering appears in the larger variance of the sample than in the case of pure dispersion as we show it next. V. MOMENTS OF THE TRANSVERSE AVERAGED PROFILE The averaged profiles of concentration c(x, t) allow to compute the three first moments of the distribution: the first moment m m(t) = L 0c (x, t)xdx L 0c (x, t)dx(14) is the position of the center of mass of the distribution as a function of time. The second moment is the variance σ 2 σ 2 (t) = L 0c (x, t)[x − m(t)] 2 dx L 0c (x, t)dx(15) giving information on the width of the distribution. Eventually, we compute also the third moment, i.e. the skewness a(t) = L 0c (x, t)[x − m(t)] 3 dx L 0c (x, t)dx(16) that gives information concerning the asymmetry of the peak with regard to its mean position. The variance σ 2 is the sum of three contributions: σ 2 (t) = σ 2 i + σ 2 d + σ 2 f ,(17) where σ 2 i = l 2 /12 is the variance due to the initial length of the sample, σ 2 d = 2t is the contribution of dispersion in dimensionless units and σ 2 f is the contribution due to the fingering phenomenon. If R = 0, the displacing fluid and the sample have the same viscosity and no fingering takes place. Hence, in that case, σ 2 = σ 2 i +σ 2 d = l 2 /12+2t. We have checked that this result is recovered by numerical simulations for R = 0. The integrals in the computation of the moments (14)-(16) are evaluated numerically by using Simpson's rule. The numerical result is very good if the spatial discretization step dx is small. Typically, we get the exact result for dx = 1. Unfortunately, dx = 1 is a resolution too high for fingering simulations especially if one wants to look at the dynamics at very long times. As an example, previous simulations on viscous fingering phenomena [15,19] were done with larger dx as typical dimensionless fingering wavelengths are around 100 for R = 3 for instance. Using typically dx = 4 gives roughly 25 points per wavelength which is numerically reasonable. For what concerns the variance, simulations with R = 0 and dx = 4 give the correct σ 2 i at t = 0 but a constant shift appears so that σ 2 (t) − l 2 /12 − 2t = C, with C being a constant of the order of 0.1% of l 2 /12. As we are mostly interested in the rate of variation of σ f , where σ f is defined as σ f (t) = σ 2 f (t) = σ 2 (t) − σ 2 i − σ 2 d ,(18) all simulations are done here with dx = 4. The slight C shift does not affect the value of σ f as we have checked it for decreasing values of dx. Figure 4 shows the temporal evolution of the first three moments i.e. the deviation m(t) − m(t = 0) of the center of mass in comparison to its initial location at t = 0, the total variance σ 2 (t) and the skewness a(t) for the typical example of Figs. 2 and 3. As fingering occurs quicker than dispersion, the center of gravity of the sample m(t) is displaced towards the back (smaller x values) because of reverse fingering of the rear interface of the sample [ Fig. 4(a)]. Fingering contributes to the widening of the peak and thus σ 2 (t) increases [ Fig. 4(b)] while the skewness a(t) becomes non zero due to the asymmetry of the fingering instability with regard to the middle of the sample [ Fig. 4(c)]. After a while, fingering dies out and the first moment m(t) saturates to a constant indicating that dispersion becomes again the only important dynamical transport mechanism. Note that the skewness a is observed to revert back towards 0 at very long times. Onset of fingering is also witnessed in the growth of the mixing zone L d defined here as the interval in which c(x, t) > 0.01 (Fig. 5). An important thing to note is that, after a diffusive transient, fingering appears on a characteristic time scale t * , defined as the time for which the mixing zone temporal dependence departs from the pure diffusive regime. As has already been discussed before [16,20], the characteristics of the fingering onset time t * and of the details of the nonlinear fingering regime are dependent on the noise amplitude A. The higher the noise intensity A, the quicker the onset of the instability (Insert in Fig. 5). To get insight into the influence of the relevant physical parameters of the problem, it is therefore necessary to fix the amplitude of the noise to an arbitrary constant as this is not a variable that is straightforwardly experimentally available. In that respect, our results have here typically been obtained for a noise of fixed A = 0.001. The number of fingers appearing at early times is related to the most unstable wavenumber of the band of unstable modes, nevertheless the location and subsequent nonlinear interaction of the fingers depend on the specific real- ization of the random numbers series. As a consequence, it is necessary to compute a set of realizations to get statistical information on σ f , the main quantity of interest here. Figure 6 shows the temporal evolution of σ f for 15 different noise realizations of identical amplitude for fixed values of the parameters R, P e, l and ε. As can be seen, if fingering starts always at the same onset time t * for fixed A, the contribution to the variance due to fingering saturates to different asymptotic values σ ∞ . This corresponds to slightly different nonlinear interactions of the fingers as can be seen on Figs. 2 and 7 which show the temporal evolution of the fingers for the respectively dotted and dashed curves of Fig. 6. If the patterns observed are very similar during the initial linear phase of viscous fingering, the evolution of the fingers is slightly different in the nonlinear regime, leading to different values of σ ∞ . In particular, merging is observed in Fig. 7 leading to the fast development of one finger and, then, spreading of the stripe of viscous fluid leading to a larger value of σ ∞ . As a consequence, to understand the influence of fingering on the broadening of finite slices, it is necessary to study the parametric dependence of < σ ∞ >, the statistical ensemble averaged asymptotic value of the fingering contribution to the variance. VI. PARAMETER STUDY The quantity < σ ∞ > gives information on the influence of viscous fingering on the broadening of finite samples. In applications such as chromatography and dispersion of contaminants in aquifers, such a broadening is undesirable and it is therefore important to understand the optimal values of parameters for which < σ ∞ > is minimum given some constraints. In that respect, let us first consider a porous medium in which dispersion is isotropic (ε = 1) and let us analyze the subsequent influences of l, R and P e. The anisotropic case (ε = 1) will eventually be tackled. The mean value < σ ∞ > is plotted for various values of the parameters, the bar around this mean value spanning the range of asymptotic data between the minimum and maximum observed. A. Influence of the sample length l The sample length l has been measured to have practically no influence on the onset time t * of the instability. Although Nayfeh has shown that the stability of finite samples could be affected if the two interfaces are close enough [24], we note that, for the smallest value of sample length l considered here (l = 32), the rear interface features the same initial pattern as the one appearing on the interface between two semi-infinite regions of different viscosities for a same random sequence in the seeding noise. Our samples are thus here long enough for the onset time t * to depend only on the amplitude A of the noise seeding the initial condition and not feel the finite extent of the sample. The length l influences nevertheless the broadening of the peak and thus < σ ∞ > in particular for small l. The points reported in Fig. 8 for two different P e are obtained for one realization and a same seeding noise r in the initial condition, leading to a typ- ical value of σ ∞ . The smaller the extent l of the sample, the sooner the dilution of the more viscous solution into the bulk of the eluent and thus the less effective fingering. Above a given extent l c , σ ∞ is found to saturate. At first sight, this might appear counterintuitive as one could expect that, for longer samples, fingering is maintained for a longer time thereby enhancing the fingering contribution to the variance. A closer inspection to the finger dynamics shows on the contrary that, after a transient where several fingers appear and interact, only one single finger remains (see Figs. 2 and 7). In the absence of tip splitting, the stretching of the mixing zone becomes then exclusively diffusive as already discussed previously by Zimmerman and Homsy [16,17]. This is clearly seen in Fig. 5 which shows that the mixing length grows as √ t at long times after a linear transient due to fingering. Once the asymptotic diffusive regime is reached, the contribution of fingering to the broadening of the peak dies out and σ f saturates to σ ∞ . Above a given critical length l c of the sample, the same asymptotic single finger growing diffusively is reached before the left and right interfaces interact. Hence the same value σ ∞ is obtained for any l > l c . Let us note that the switch from the fingering to the diffusive dynamics appears later in time when P e is increased. Indeed larger P e means more fingers that can interact for a longer time before the diffusive regime becomes dominant. As a consequence, l c is an increasing function of P e as can be seen on Fig. 8. Further studies need to be done to understand the role of ε and of possible tip splitting occurring for large P e on the existence and value of the critical length l c . The fact that the contribution of fingering to the broadening of the peak saturates beyond a critical length of the sample has important practical consequences for chromatography: if fingering is unavoidable, one might as well load samples of long extent as the contribution of fingering is saturating beyond a given l c . For long samples, the efficiency of the process depends then on the competition between σ 2 ∞ and l 2 /12, the respective fingering and initial length contributions to the peak's variance. We can thus predict that for l c < l < σ ∞ / √ 12, the contribution of fingering is constant and dominates the broadening while for l > σ ∞ / √ 12, the initial sample length becomes the key factor. B. Influence of the log-mobility ratio R It is easy to foreseen that the larger R, the more important the viscous fingering effect [8]. First of all, linear stability analysis of viscous fingering at the interface between two semi-infinite domains [18] predicts that the characteristic growth time of the instability decreases as R −2 . Although already influenced by the nonlinearities and dependent on the amplitude of the noise, the onset time t * measured in our simulations shows the same trend (Fig. 9). Note that, for very small values of R, the onset time becomes very large which explains why, for samples of low viscosity, fingering might not be observed during the transit time across small chromatographic columns or on small scale contamination zone. When R is increased, the viscous fingering contribution to peak broadening < σ ∞ > is more important (Fig. 10) with a linear dependence suggesting a power law increase for larger R. C. Influence of the Péclet number P e The Péclet number P e is typically experimentally increased for a given geometry by increasing the injection flow rate U . As can be seen on Fig. 11, < σ ∞ > is found to increase linearly with P e. Fingering induced broadening can thus be minimized by small carrier velocity U as expected. However, the exact influence of the carrier velocity U is difficult to trace because practically, a change in U also modifies the dispersion coefficients and hence the value of ε. In our dimensionless variables, U also enters into the characteristic time and length corresponding respectively to D x /U 2 and D x /U . The concrete influence of the carrier velocity is thus more complicated to trace in reality. For a fixed injection speed, the Péclet number can also be varied by changing the width L y of the system. The linear dependence of < σ ∞ > on P e is then related to the fact that in a wider domain, more fingers can remain in competition for a longer time so that a more active fingering is maintained. This also implies that l c is an increasing function of P e. In chromatographic applications, increasing the diameter of the column (i.e. increasing L y here in our model) is thus expected to dramatically increase the influence of fingering in broadening. This explains why fingering really becomes an issue for wide contamination zones and in preparative chromatography where columns of very large diameter (up to one meter) are sometimes constructed. D. Influence of the ratio of dispersion coefficients ε Figure 12 shows the influence of the ratio of dispersion coefficients ε = D y /D x on the onset time of the instability. As expected from linear stability analysis [18], decreasing ε has a destabilizing effect as fingering appears then quicker. This is due to the fact that small transverse dispersion inhibits the mixing of the solutions and favors longitudinal growth of the fingers allowing them to survive for a longer time. As a consequence, the less viscous solution instead of being transversely homogeneous invades the more viscous fluid preferably in the longitudinal direction leading to larger mixing zones and hence larger < σ ∞ >. Figure 13 illustrates that decreasing ε has a dramatic effect on the broadening of the peak. The insert shows the same graphics in logarithmic scale for ε. < σ ∞ > seems to vary as ln(ε) at least for small values of ε. Peak broadening due to fingering is therefore expected to be particularly dramatic for chromatographic applications where ε ∼ 0.04. VII. CONCLUSION Viscous fingering leads to a mixing between miscible fluids of different viscosity. In the case of viscous slices of finite extent, fingering is a transient phenomenon because the mixing of the two fluids leads to an effective decrease of the log-mobility ratio in time. Transient fingering can nevertheless play an important role because it contributes to distortion and broadening of the sample. In particular, we have shown that, even if the spreading of the sample may look Gaussian at long times because dispersion has again become the leading transport phenomenon, the variance of the peak is larger than expected because of fingering at early times. We have demonstrated this influence by numerical simulations of viscous fingering of miscible finite slices characterizing the onset time of the instability t * and the contribution of fingering to the sample's variance. It is important to note that quantitative comparison with experimental data is difficult because the exact amplitude of the fingering contribution to the temporal variation of the peak's variance depends both on the amplitude and spatial realization of the noise seeding the initial condition which varies from one experiment to the other. In this respect, we have computed the ensemble averaged asymptotic fingering contribution to the peak broadening as a function of the four relevant parameters of the problem i.e. the initial length of the sample l, the initial log-mobility ratio R, the Péclet number P e and the ratio between transverse and longitudinal dispersion coefficients ε. The broadening of the peaks due to fingering is most important for large R and P e but small ε while it saturates above a given initial length l of the sample. In chromatographic columns for which ε ∼ 0.04, fingering is thus of crucial importance particularly in preparative chromatography for which the large diameter of the columns lead to large P e and the high concentration of the samples usually implies large R. Similarly, for soil contamination, fingering will be a major problem in the case of stratified media such that ε < 1. More work is now needed to explore the generalization of this first approach to the case where both viscosity and density variations as well as heterogeneous chemistry may interplay as is usually the case in the applications analyzed here. FIG. 1 : 1Sketch of the system. FIG. 2 : 2Density plots of concentration at successive times in the frame moving at the velocity U . From top to bottom: t=0, 500, 700, 1000, 1500, 2000, 5000, 15000 and 60000 (P e = 512, l = 128, R = 2, ε = 1). FIG. 3 : 3Transverse average profiles of concentration at successive times t=0, 500, 700, 1000, 1500, 2000, 5000 and 15000. Insert: transverse average profile of concentration at t = 60000 (P e = 512, l = 128, R = 2, ε = 1). FIG. 4 : 4First three moments of the distribution: (a) Mean position m of the center of mass, (b) Variance σ 2 , (c) Skewness a (P e = 512, l = 128, R = 2, ε = 1). FIG. 5 : 5Mixing zone L d as a function of time realized with the same parameters (P e = 512, l = 128, R = 2, ε = 1) and three different values of the amplitude A of the noise seeding the initial condition: (· · ·) A = 0.1, (--) A = 0.01, (-) A = 0.001; (-· -) theoretical curve of a pure diffusive behavior L d ∝ √ t. Insert: zoom on the first stages on the injection. The onset time t * , corresponding to the time at which the mixing zone departs from the pure diffusive initial transient, is a decreasing function of A. FIG. 6 : 6σ f as a function of time for 15 numerical simulations realized with the same values of parameters (P e = 512, l = 128, R = 2, ε = 1) but different noise r realizations of identical amplitude A = 10 −3 . The dotted and dashed curves correspond to the simulations of Figs. 2 and 7 respectively. FIG. 7 : 7Density plots of concentration for the same values of parameters and same times as inFig. 2, but a different noise r in the initial condition. FIG. 8 : 8Influence of the sample length l on σ∞ for ( ) P e = 64 and (•) P e = 128 (R = 2, ε = 1). FIG. 9 : 9Onset time t * of the instability for increasing values of R (P e = 256, l = 128, ε = 1). (-) Best fit of the experimental points: t * ∝ R −2 . FIG. 10: Influence of the log-mobility ratio R on < σ∞ > for (•) P e = 128, l = 128, (•) P e = 128, l = 512, (△) P e = 256, l = 128, ( ) P e = 256, l = 512 (ε = 1). FIG. 11 : 11Influence of the Péclet number P e on < σ∞ > (l = 128, R = 2, ε = 1). FIG. 12 : 12Onset time t * of the instability for increasing values of ε for different values of the log-mobility ratio: (▽) R = 1, (•) R = 2, (△) R = 3 (P e = 128, l = 128). FIG. 13 : 13Influence of the ratio between transverse and longitudinal dispersion ε on < σ∞ > for (▽) R = 1, (•) R = 2, (△) R = 3 (P e = 128, l = 128). Insert: same data on logarithmic scale for ε. VIII. ACKNOWLEDGEMENTSWe thank G.M. Homsy, P. Colinet, A. Vedernikov and B. Scheid for fruitful discussions. Y. Bertho benefits from a postdoctoral fellowship of the Université Libre de Bruxelles sponsored by the Francqui Foundation which is gratefully acknowledged. A. De Wit thanks also FRFC (Belgium) and the "Communauté française de Belgique Actions de Recherches Concertées" programme for financial support. Actions de Recherches Concertées" programme for fi- nancial support. Viscous fingering in porous media. G M Homsy, Ann. Rev. Fluid Mech. 19271G.M. Homsy, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19, 271 (1987). Visualization of viscous fingering in chromatographic columns. B S Broyles, R A Shalliker, D E Cherrak, G Guiochon, J. Chromatogr. 822173B.S. Broyles, R.A. Shalliker, D.E. Cherrak and G. Guio- chon, Visualization of viscous fingering in chromato- graphic columns, J. Chromatogr. 822, 173 (1998). Viscous fingering visualized via magnetic resonance imaging. L D Plante, P M Romano, E J Fernandez, Chem.Eng. Sci. 49229L.D. Plante, P.M. Romano and E.J. Fernandez, Vis- cous fingering visualized via magnetic resonance imaging, Chem.Eng. Sci. 49, 229 (1994). The effects of permeability heterogeneity on miscible viscous fingering: A three-dimensional magnetic resonance imaging analysis. E J Fernandez, C A Grotegut, G W Braun, K J Kirschner, J R Staudaher, M L Dickson, V L Fernandez, Phys. Fluids. 7468E.J. Fernandez, C.A. Grotegut, G.W. Braun, K.J. Kirschner, J.R. Staudaher, M.L. Dickson, and V.L. Fer- nandez, The effects of permeability heterogeneity on miscible viscous fingering: A three-dimensional magnetic resonance imaging analysis, Phys. Fluids 7, 468 (1995). Viscous fingering: a systematic study of viscosity effects in methanol-isopropanol systems. D Cherrak, E Guernet, P Cardot, C Herrenknecht, M Czok, Chromatographia. 46647D. Cherrak, E. Guernet, P. Cardot, C. Herrenknecht and M. Czok, Viscous fingering: a systematic study of vis- cosity effects in methanol-isopropanol systems, Chro- matographia 46, 647 (1997). Effect of sample viscosity in high-performance size-exclusion chromatography and its control. M Czok, A Katti, G Guiochon, J. Chromatogr. 550705M. Czok, A. Katti and G. Guiochon, Effect of sample viscosity in high-performance size-exclusion chromatog- raphy and its control, J. Chromatogr. 550, 705 (1991). A column design for reducing viscous fingering in size exclusion chromatography. E J Fernandez, T Tucker Norton, W C Jung, J G Tsavalas, Biotechnol. Prog. 12480E.J. Fernandez, T. Tucker Norton, W.C. Jung and J.G. Tsavalas, A column design for reducing viscous fingering in size exclusion chromatography, Biotechnol. Prog. 12, 480 (1996). Viscous fingering in size exclusion chromatography: insights from numerical simulation. T , Tucker Norton, E J Fernandez, Ind. Eng. Chem. Res. 352460T. Tucker Norton and E.J. Fernandez, Viscous fingering in size exclusion chromatography: insights from numeri- cal simulation, Ind. Eng. Chem. Res. 35, 2460 (1996). An experimental study of miscible displacements in porous media with variation of fluid density and viscosity. C Y Jiao, H Hotzl, Trans. Porous Media. 54125C.Y. Jiao and H. Hotzl, An experimental study of mis- cible displacements in porous media with variation of fluid density and viscosity, Trans. Porous Media 54, 125 (2004). A breakthrough curve analysis of unstable density-driven flow and transport in homogeneous porous media, Water Resour. M Wood, C T Simmons, J L Hutson, Res. 403505M. Wood, C.T. Simmons and J.L. Hutson, A break- through curve analysis of unstable density-driven flow and transport in homogeneous porous media, Water Re- sour. Res. 40, W03505 (2004). 3D simulation of viscous fingering and WAG schemes, SPE 21238, presented at the 11th Symposium on Reservoir Simulation. M A Christie, A H Muggeridge, J J Barley, Anaheim, CaliforniaM.A. Christie, A.H. Muggeridge and J.J. 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[ "Thermostatistics of the polymeric ideal gas", "Thermostatistics of the polymeric ideal gas" ]
[ "M A Gorji \nDepartment of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran\n", "K Nozari \nDepartment of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran\n", "B Vakili \nDepartment of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran\n" ]
[ "Department of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran", "Department of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran", "Department of Physics\nFaculty of Basic Sciences\nDepartment of Physics, Chalous Branch, IAU\nUniversity of Mazandaran\nP.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran" ]
[]
In this paper, we formulate statistical mechanics of the polymerized systems in the semiclassical regime. On the corresponding polymeric symplectic manifold, we set up a noncanonical coordinate system in which all of the polymeric effects are summarized in the density of states. Since we show that the polymeric effects only change the number of microstates of a statistical system, working in this coordinate is quite reasonable from the statistical point of view. The results show that the number of microstates decreases due to existence of an upper bound for the momentum of the test particles in the polymer framework. We obtain a corresponding canonical partition function by means of the deformed density of states. By using the partition function, we study thermodynamics of the ideal gas in the polymer framework and show that our results are in good agreement with those that arise from the full quantum consideration at high temperature, and they coincide with their usual counterpart in the limit of low temperature.
10.1103/physrevd.90.044051
[ "https://arxiv.org/pdf/1408.1725v2.pdf" ]
118,347,947
1408.1725
d0c9a86b7bbf28f3a3b66d2ec36716c1ca3501c1
Thermostatistics of the polymeric ideal gas 20 Aug 2014 M A Gorji Department of Physics Faculty of Basic Sciences Department of Physics, Chalous Branch, IAU University of Mazandaran P.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran K Nozari Department of Physics Faculty of Basic Sciences Department of Physics, Chalous Branch, IAU University of Mazandaran P.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran B Vakili Department of Physics Faculty of Basic Sciences Department of Physics, Chalous Branch, IAU University of Mazandaran P.O. Box, P.O. Box 46615-39747416-95447Babolsar, ChalousIran, Iran Thermostatistics of the polymeric ideal gas 20 Aug 2014numbers: 0460Nc0460Pp0520-y0530-d Key Words: Polymer quantizationThermodynamics In this paper, we formulate statistical mechanics of the polymerized systems in the semiclassical regime. On the corresponding polymeric symplectic manifold, we set up a noncanonical coordinate system in which all of the polymeric effects are summarized in the density of states. Since we show that the polymeric effects only change the number of microstates of a statistical system, working in this coordinate is quite reasonable from the statistical point of view. The results show that the number of microstates decreases due to existence of an upper bound for the momentum of the test particles in the polymer framework. We obtain a corresponding canonical partition function by means of the deformed density of states. By using the partition function, we study thermodynamics of the ideal gas in the polymer framework and show that our results are in good agreement with those that arise from the full quantum consideration at high temperature, and they coincide with their usual counterpart in the limit of low temperature. I. INTRODUCTION While general relativity (GR) improved our understanding about the Universe, its shortages are revealed when it is utilized to describe dynamics of the Universe in the standard big bang cosmology [1]. With the advent of theories such as inflationary scenarios in order to solve the initial value problem and its later success in explaining the origin of the large scale structures, this idea was formed that classical GR may fail to describe properly our Universe (at least in such high energy regimes) [2]. The fact is that while GR is a classical theory in its original formalism, the quantum effects significantly become important in the very early Universe. It is plausible to expect that the initial value problem will be naturally resolved when a quantum theory of gravity is applied. Although it seems that a complete theory of quantum gravity is not yet made, its main candidates such as string theory and loop quantum gravity revealed some fundamental aspects of the ultimate theory. For example the existence of a minimal measurable length of the order of the Planck length is a common feature of any quantum theory of gravity [3,4]. Assuming a minimal invariant length, the Heisenberg uncertainty principle trivially implies an ultraviolet cutoff for the system under consideration. However, the standard uncertainty principle cannot support existence of a minimal length and the standard Schrödinger representation of the quantum mechanics is no longer applicable. Therefore, some at- * [email protected] † [email protected] ‡ [email protected] tempts have been done to include a fundamental length scale in the standard quantum mechanics, see for instance [5] in which the generalized uncertainty principle is introduced with the existence of a minimum measurable length in its formalism. The Hilbert space representation of such modified uncertainty relation is formulated in Ref. [6]. In more recent times, polymer representation of the quantum mechanics has been studied in the context of loop quantum gravity [7]. The associated Hilbert space supports the existence of a minimal length, here known as polymer length scale [8]. The relation between polymer picture and the generalized uncertainty principle framework is investigated in Ref. [9]. A notable character of the polymer representation of quantum mechanics is that, in contrast to the standard Schrödinger representation, in the classical limit → 0 the system does not tend to its usual classical version, instead, one recovers a one-parameter λ-dependent classical theory, where λ ∼ l poly / denotes the polymer length scale. The standard classical theory emerges in the continuum limit λ → 0. Exploring the implication of the effective classical λ-dependent theory gives some interesting results. For instance, existence of an upper bound for the energy of a classical systems and removing the big bang singularity in the cosmological setup when it applies to the minisuperspace models [8,10]. As another feature, we will see that the effective λ-dependent classical theory reproduces some results of the so-called doubly special relativity [11] (in which a minimal observer independent length scale is proposed to special relativity), and polymer length plays the role of the observer independent length scale. The existence of an invariant minimal length has also interesting effects on the thermodynamical behavior of the physical systems. In this regard, many efforts have been made to formulate the statistical mechanics in the framework of the generalized uncertainty principle [12]. Also, thermodynamics of black hole systems in the polymer picture is studied in [13,14] and the statistical mechanics of the ideal gas in doubly special relativity is investigated in [15]. Nevertheless, in order to study the thermodynamics of a given physical system, one needs to know the microphysics of the system which in turn is determined by quantum mechanics. However, for instance, finding the energy eigenvalues is not an easy task at all when one takes into account the minimal length considerations in the problem at hand [6,13,16]. In an alternative picture, one can work with the Hamiltonian and density of states in the corresponding phase space in the semiclassical regime. In this paper, at first we try to formulate the classical phase space of a polymerized systems in terms of the language of the symplectic geometry. Then we formulate statistical mechanics of the polymerized systems in the semiclassical regime. We show that our results are in good agreement with their quantum counterparts. The structure of the paper is as follows: In section 2, we define a polymeric structure on the symplectic manifold. In section 3, we obtain a deformed density of states that contains all the polymeric effects. Using the deformed density of states, we find a canonical partition function for the polymerized systems. In section 4, we study the polymeric effects on the thermodynamics of the ideal gas through the polymeric partition function. Section 5 is devoted to the conclusions. II. POLYMERIZATION In the Hamiltonian formulation of the classical mechanics, the kinematics of the phase space is formed by the Poisson algebra {q, p} = 1 ,(1) where (q, p) are the phase space variables known as canonical variables. Then, quantization is the passage from the classical Poisson algebra (1) to the quantum Heisenberg algebra by the standard rule: Replacing the Poisson bracket by the Dirac commutator for the operators counterpart of the canonical variables as [q,p] = i 1 ,(2) where is the Planck constant. It is easy to see that the Heisenberg uncertainty principle is a straightforward result of the commutation relation (2). The ordinary Schrödinger picture of quantum mechanics is based on the representation of operators on the Hilbert space H = L 2 (R, dq), the space of the square integrable functions with respect to the Lebesgue measure dq on the real line R. In addition to this well-known representation, there are other representations based on which one can construct the quantum kinematics. Here, we are going to pursue a case that has been presented in [8] under the name of polymer representation. The polymer representation of quantum mechanics is formulated on the Hilbert space H poly = L 2 (R d , dµ d ), where dµ d is the Haar measure, and R d is the real line but now endowed with the discrete topology [17]. The extra structure in polymer picture is properly described by a dimension-full parameter λ such that the standard Schrödinger representation will be recovered in the continuum limit λ → 0 [8]. Evidently, the classical limit of the polymer representation → 0, does not yield to the classical theory from which one has started but to an effective λ-dependent classical theory which may be interpreted as a classical discrete theory. Such an effective theory can also be extracted directly from the standard classical theory (without any attribution to the polymer quantum picture) by using the Weyl operator [10]. The process is known as polymerization with which we will deal in the rest of this paper. In polymer representation of quantum mechanics, the position space (with coordinate q) is assumed to be discrete with discreteness parameter λ and consequently the associated momentum operatorp, that would be a generator of the displacement, does not exist [7]. However, the Weyl exponential operator (shift operator) correspond to the discrete translation along q is well defined and effectively plays the role of momentum for the system under consideration [8]. Taking this fact into account, one can utilize the Weyl operator to find an effective momentum in the semiclassical regime. Therefore, the derivative of the state f (q) with respect to the discrete position q can be approximated by means of the Weyl operator as [10] ∂ q f (q) ≈ 1 2λ [f (q + λ) − f (q − λ)] = 1 2λ e ipλ − e −ipλ f (q) = i λ sin(λp) f (q),(3) and similarly the second derivative approximation gives ∂ 2 q f (q) ≈ 1 λ 2 [f (q + λ) − 2f (q) + f (q − λ)] = 2 λ 2 ( cos(λp) − 1) f (q).(4) Inspired by the above approximations, the polymerization process is defined for the finite values of the parameter λ aŝ p → 1 λ sin(λp),p 2 → 2 λ 2 (1 − cos(λp)). (5) The (quantum) polymerization (5) suggests the classical polymer transformation P[F ] of a function F (q, p) on the phase space as [10] P[F (q)] = F (q), P[p] = 1 λ sin(λp), P[p 2 ] = 2 λ 2 (1 − cos(λp)),(6) and in a same manner one can find polymer transformation of the higher powers of momentum p. In this sense, by a classical polymerized system, we mean a system that the transformation (6) is applied to its Hamiltonian. Now, consider a nonrelativistic physical system with standard Hamiltonian H = p 2 2m + U (q) ,(7) where m is the mass of a particle moving under the act of the potential function U (q). Applying the polymer transformation (6), the associated effective Hamiltonian will be H λ = 1 mλ 2 1 − cos(λp) + U (q).(8) The first consequence of the polymerization (6), which is also clear from the Hamiltonian (8), is that the momentum is periodic and its range should be bounded as p ∈ [− π λ , + π λ ). In the limit λ → 0, the effective Hamiltonian (8) reduces to the standard one (7) and one recovers the usual range for the canonical momentum p ∈ (−∞, +∞). Therefore, the polymerized momentum is compactified and topology of the momentum sector of the phase space is S 1 rather than the usual R. As we will see, this structure for the topology of the phase space gives nontrivial results for the polymeric thermodynamical systems, for instance, existence of an upper bound for the internal energy of the system. In order to study the thermodynamics of the polymerized systems, we implement the symplectic geometry which gives a more suitable picture from the statistical point of view. A. Darboux chart Consider a two-dimensional symplectic manifold M thought as a polymeric phase space with symplectic structure ω which is a closed nondegenerate 2-form on M. According to the Darboux theorem [18], there is always a local chart in which this 2-form takes the canonical form ω = dq ∧ dp, and as we will see the variables (q, p) may be identified with the canonical variables which we have perviously defined in (1). Although the symplectic structure (9) is canonical in polymeric phase space, one should be careful about the periodic condition for the canonical momentum which significantly leads to the nontrivial topology for the momentum part of the manifold M. Time evolution of the system is given by the Hamiltonian vector field x H which satisfies the equation i x ω = dH λ ,(10) where H λ is given by relation (8). By solving the above equation with the use of the effective polymeric Hamiltonian given in (8), one gets x H = sin(λp) mλ ∂ ∂q − ∂U ∂q ∂ ∂p .(11) The integral curves of the Hamiltonian vector field (11) are the polymer-modified Hamiltonian equations of motion in canonical chart dq dt = sin(λp) mλ , dp dt = − ∂U ∂q ,(12) which clearly reduce to the standard Hamilton's equations in the continuum limit λ → 0. The Poisson bracket between two real valued functions F and G on M is defined as {F, G} = ω(x F , x G ) .(13) The closure of the 2-form ensures that the Jacobi identity is satisfied by the resultant Poisson brackets in definition (13). In the Darboux chart, we have x F = ∂F ∂p ∂ ∂q − ∂F ∂q ∂ ∂p , and also the same expression for the function G. Substituting these results together with the canonical structure (9) into the definition (13), gives {F, G} = ∂F ∂q ∂G ∂p − ∂F ∂p ∂G ∂q ,(14) which is obviously the standard canonical Poisson bracket between two arbitrary functions F and G. It is clear that with choosing F (q, p) = q and G(q, p) = p we have {q, p} = 1 ,(15) which is nothing but the standard canonical Poisson algebra (1). It is important to note that while the polymeric phase space and the standard classical relation (1) from which we have started, seem to have the same kinematical structure, dynamically they are different thanks to the polymer-modified Hamiltonian equations of motion (12). So, the Poisson algebra is canonical in the Darboux chart, but the effective Hamiltonian (8) contains the polymeric effects which will change the dynamics through the relation (10). Now, suppose that the system under consideration is a many-particle one for which we are going to apply the above formalism in statistical mechanics point of view. To do this, it is necessary to consider the well-known Liouville theorem which is directly related to the number of accessible microstates of the system. The Liouville volume for a 2D-dimensional symplectic manifold is defined as ω D = (−1) D(D−1)/2 D! ω ∧ ... ∧ ω D times .(16) As a special case we see that for two-dimensional manifolds, the Liouville volume coincides with the symplectic 2-form. According to the Liouville theorem, the Liouville volume (16) is conserved along the Hamiltonian flow x H as L x ω D = 0,(17) where L x denotes the Lie derivative along the Hamiltonian vector field x H . This relation can be easily deduced from (10) in which we have also noticed that dω = 0 due to the Cartan formula as L x ω = i x dω + di x ω = 0 [19,20]. So, the Liouville theorem is always satisfied on a symplectic manifold independent of a chart in which the physical system is considered. If we restrict ourselves to a finite one-dimensional spatial volume L, the total volume of the phase space can be obtained by integrating the Liouville volume as Vol(ω 1 ) = ω 1 = L dq × + π λ − π λ dp = 2π L λ .(18) In the standard Hamiltonian formalism of the classical mechanics, the total volume of the phase space (18) diverges even if one confines the physical system to a finite spatial volume. More precisely, the momentum part of the integral of Liouville volume will diverge because there is no any priori restriction on the momentum of the test particles in the standard classical mechanics. However, due to existence of an upper bound for the momenta in the classical polymeric systems the resultant total volume (18) will be naturally finite. In other words, compact topology of the momentum part of the polymeric symplectic manifold implies the finite value for the associated total volume that is circumference of a circle with radius λ −1 (see Ref. [19] for more details). Another point here is that, the spatial sector of the phase space volume L should be quantized with respect to the polymer length l poly = α λ, since the polymer length is the possible minimum length for the polymerized system. Here α = O(1) is a dimensionless coefficient which should be fixed by the experiments [21]. Therefore, we have L l poly ∈ N. This result in some sense is similar to the result obtained in Ref. [22] in the generalized uncertainty principle framework. By taking this fact into account, relation (18) may be rewritten as Vol(ω 1 ) = nh,(19) where n is a positive integer which counts the number of fundamental cells λ exist in L. This equation shows that the total volume of the phase space is naturally quantized with respect to the Planck constant h = 2π . Note that in the semiclassical regime, to obtain a finite number of microstates for a given statistical system, one needs an extra assumption that the volume of the phase space is quantized with respect to the Planck constant. However, as our above analysis shows, this issue automatically emerges in the polymerized phase space. The origin of this result may be sought in the Heisenberg uncertainty principle. In the standard phase space there is no restriction according to which the system fails to have access to any desired length scale even the zero length. However, in the polymer picture the theory is equipped with a maximal momentum correspond to the polymer length (below which no other length can be observed) in the light of the uncertainty principle as p max ∼ l poly ∼ λ −1 , the existence of which is responsible for the quantized volume of the phase space. B. Noncanonical chart In order to study the statistical physics of a polymerized systems, we introduce a noncanonical transformation (q, p) → (q ′ , p ′ ) = q, 2 λ sin( λp 2 ) ,(20) on the polymeric phase space which transforms effective Hamiltonian (8) to the nondeformed one: H λ (q, p) → H λ (q ′ , p ′ ), with H λ (q ′ , p ′ ) = p ′2 2m + U (q ′ ).(21) Although Hamiltonian (21) is independent of the parameter λ, the new momentum p ′ is bounded as p ′ ∈ [− 2 λ , + 2 λ ) due to the transformation (20). Therefore, the Hamiltonian (21) should be counted distinct from the standard nondeformed Hamiltonian (7). It is also important to note that while the Hamiltonian gets the standard form, the corresponding 2-form in the noncanonical chart becomes ω = dq ′ ∧ dp ′ 1 − (λp ′ /2) 2 .(22) Substituting the above symplectic structure and also the associated Hamiltonian (21) into the relation (10), we are led to the following solution for the Hamiltonian vector field x H = 1 − (λp ′ /2) 2 p ′ m ∂ ∂q ′ − ∂U ∂q ′ ∂ ∂p ′ .(23) The integral curves of the above Hamiltonian vector field are the polymer-modified Hamilton's equation in the noncanonical chart dq ′ dt = p ′ m 1 − (λp ′ /2) 2 , dp ′ dt = − ∂U ∂q ′ 1 − (λp ′ /2) 2 ,(24) which will be reduced to the standard ones in the limit of λ → 0. The two sets of modified Hamiltonian equations of motion (12) and (24) are in agreement with each other through the transformation (20). The Poisson bracket between two real valued functions F and G in this chart can be obtained by substituting the noncanonical structure (22) and the associated vector field (23) into the definition (13) with the result {F, G} = 1 − (λp ′ /2) 2 ∂F ∂q ′ ∂G ∂p ′ − ∂F ∂p ′ ∂G ∂q ′ .(25) With the help of this relation the Poisson bracket of the noncanonical variables q ′ and p ′ can be obtained as {q ′ , p ′ } = 1 − (λp ′ /2) 2 ,(26) which reduces to its nondeformed counterpart in the continuum limit λ → 0. Since, by definition, the total volume is invariant over the symplectic manifold, it should be the same as one in the Darboux chart (18). Indeed, integrating the Liouville volume which is nothing but the 2-form structure for a two-dimensional manifold, gives the total volume in the noncanonical chart as Vol(ω 1 ) = L dq ′ × + 2 λ − 2 λ dp ′ 1 − (λp ′ /2) 2 = 2π L λ ,(27) that coincides with relation (18) as expected. Therefore, one can work in two equivalent pictures on the polymeric symplectic manifold: i) utilizing the effective Hamiltonian (8) together with the symplectic structure (9) in the Darboux chart which leads to the canonical Poisson algebra (15); ii) implementing the noncanonical chart with symplectic structure (22), Hamiltonian function (21) and the corresponding noncanonical Poisson algebra (26). The trajectories on the polymeric phase space are the same in two charts since equation (10) is satisfied in a chart-independent manner. However, as we will see in the next section, working within the noncanonical chart is more admissible from the statistical point of view. III. DENSITY OF STATES AND PARTITION FUNCTION Statistical mechanics determines the relation between microphysics and macrophysics. All of the thermodynamical properties of a given physical system can be derived from its partition function which is the summation over all accessible microstates of the system. The canonical partition function for a single particle state is defined as [23] Z 1 = ε exp − ε/T ,(28) where ε are the single particle energy states which are the solution of the Schrödinger equation in the standard representation of quantum mechanics. In fact, the microstates for the statistical system are completely determined by the quantum physics. In contrast to the standard Schrödinger representation, finding the energy eigenvalues in the polymer representation is not an easy task due to the nonlinearity of the quantum Hamiltonian in this picture [7,8,16]. Even if one finds the microstates' energies in the polymer picture, calculating the partition function (28) is somehow a complicated issue [13]. Nevertheless, one may work with the classical Hamiltonian together with the density of states in the semiclassical regime. The semiclassical and the quantum statistics will be coincided in the high temperature regime. Therefore one should be careful that the semiclassical approximation is only applicable for the polymerized systems with small polymer length scale through the uncertainty principle. Approximating the summation over the energy eigenvalues in the relation (28) by the integral over the phase space volume yields ε → Vol(ω 3 ) h 3 = 1 h 3 M ω 3 ,(29) where ω 3 is the Liouville volume of the six-dimensional phase space of the single particle which in turn, should be obtained by substituting the associated 2-form structure into the definition (16). Relation (29) is written in a chart-independent manner on the manifold since the total volume Vol(ω 3 ) is invariant over the symplectic manifold (see also [19]). Indeed, equation (29) is nothing but the Heisenberg uncertainty principle which implies a finite fundamental element for the phase space volume of the order of Planck constant. The total volume determines the number of microstates of the system and it, as is guaranteed by the Liouville theorem, should be invariant under the time evolution. Now, one may consider relation (29) in various charts over symplectic manifolds with no worries about the invariance of the total volume as time grows, since the Liouville theorem is satisfied in a chartindependent manner via the relation (17). In the usual statistical mechanics, the topology of the phase space of a single particle is R 6 and then relation (29) in the Darboux (canonical) chart leads to the wellknown result, ε → 1 h 3 dx dy dz (30) × +∞ −∞ +∞ −∞ +∞ −∞ dp x dp y dp z . With approximating the summation over the energy eigenvalues in (28) by relation (30), one can obtain the standard definition of the semiclassical partition function [24]. In the same way, the polymeric partition may be achieved . We consider (29) for the polymerized phase space in two charts: the Darboux and noncanonical charts which we presented in the previous section. In the Darboux chart, the density of states takes the same form as the usual one (30) because the corresponding symplectic structure (9) is canonical (see the appendix), with this difference that now the momentum part of the polymeric phase space has a compact topology S 1 rather than the usual R. Thus we have ε → 1 h 3 dx dy dz(31) × +π/λ −π/λ +π/λ −π/λ +π/λ −π/λ dp x dp y dp z . Approximating the quantum partition function (28) by the polymeric state density (31) and substituting the as-sociated Hamiltonian (8) instead of the energy eigenvalues ε, gives the polymeric partition function as Z 1 (λ; T, V ) = 1 h 3 exp − U (x, y, z) T dx dy dz × 3 i=1 +π/λ −π/λ exp − (1 − cos(λp i )) mλ 2 T dp i .(32) It is clear that both the Hamiltonian and the density of states are modified in the polymer phase space in the Darboux (canonical) chart. Rewriting (29) in the noncanonical chart on the polymeric symplectic manifold, gives the following polymeric density of states in the noncanonical chart (see the appendix) ε → = 1 h 3 dx ′ dy ′ dz ′ × +2/λ −2/λ +2/λ −2/λ +2/λ −2/λ dp ′ x dp ′ y dp ′ z 1 − ( λp ′ x 2 ) 2 1 − ( λp ′ y 2 ) 2 1 − ( λp ′ z 2 ) 2 .(33) Therefore in the noncanonical chart, as the above equation shows, while the Hamiltonian takes the standard form (21) the polymeric effects are summarized in the density of states. Equation (33) determines the number of accessible microstates for the system and because of the bounded domain for the momenta this number is less than when the polymeric effects are absent. A similar result is also achieved in the other effective approaches to quantum gravity such as generalized uncertainty principle [12], doubly special relativity [15] and noncommutative phase space [20]. As a first thermodynamical outcome we can see that the polymeric effects cause a reduction in the entropy of the system. This is because the entropy is directly determined from the number of microstates. In the next section we will explicitly investigate this fact and its following results for a particular case in which the underlying system is a system of an ideal gas. Following the same steps which led us to (32), but this time with the help of relations (21), (28) and (33), the canonical polymeric partition function for a single noninteracting particle becomes Z 1 (T, V ) = V h 3 3 i=1 +2/λ −2/λ exp − p ′ i 2 2mT dp ′ i 1 − ( λp ′ i 2 ) 2 = V 3 λ 3 exp − λ −2 mT I 0 λ −2 mT 3 ,(34) where V is the result of integration over the spatial part and I 0 denotes the modified Bessel function of the first kind. The expression (34) for the partition function is in an excellent agreement with what is obtained in [13] by the full quantum consideration. Now, the polymeric thermodynamics of a physical system (an ideal gas in our model in the next section) may be extracted by means of the above partition function. IV. IDEAL GAS In this section, let us consider a gaseous system consisting of N noninteracting particles at temperature T confined in the volume V . We assume that this system obeys the Maxwell-Boltzmann statistics. Equation (34) can be used to evaluate the corresponding polymeric partition function as Z N (T, V ) = 1 N ! Z 1 (T, V ) N ,(35) in which the Gibb's factor is also considered [24]. The natural choice for the polymer length l poly is the Planck length l poly = α λ = α l Pl = √ G , where G is the gravitational constant and α is a dimensionless coefficient of the order of unity α ∼ O(1). As we have mentioned before, this coefficient should be fixed only by the experiments [21]. In our study, the value of the coefficient α determines the boundary in which the polymeric effects become important. As we will see, the polymeric effects appear in the trans-Planckian regime for the values α < 1 and the sub-Planckian polymeric effects emerge for α > 1. Here, we assume l poly = l Pl , i.e, we select α = 1 for the sake of simplicity. Substituting relation (34) into (35) gives the total partition function for the polymerized ideal gas as Z N (T, V ) = V /l 3 Pl N N ! exp − T 2 Pl mT I 0 T 2 Pl mT 3N ,(36) where T Pl is the Planck temperature T Pl = G . The prefactor V /l 3 Pl ∈ N in this equation shows the discreteness of space in the polymer framework. In the following, by means of this expression for the total partition function, we will investigate the thermodynamical properties of the polymeric ideal gas. A. Pressure First of all, let us look at the Helmholtz free energy F for the polymeric ideal gas which can be obtained (with the help of (36)) from its standard definition as F = −T ln Z N (T, V ) = N T ln N l 3 Pl V I 3 0 T 2 Pl /mT − 1 + 3N T 2 Pl m ,(37) in which we have used the Stirling's approximation ln[N !] ≈ N ln[N ] − N for large N . The pressure by definition is P = − ∂F ∂V T,N = N T V .(38) Therefore, the familiar equation of state for the ideal gasses, that is, P V = N T,(39) preserves its form also in the polymer framework. B. Internal energy The internal energy of the polymeric ideal gas will be U = −T 2 ∂ ∂T F T N,V = 3N T 2 Pl m 1 − I 1 T 2 Pl /mT I 0 T 2 Pl /mT .(40) This result exactly coincides with what comes from the full quantum consideration of the ideal gas in the polymer picture but now in a simpler manner [13]. It is important to note that we are working in the semiclassical regime while our results are in good agreement with their full quantum counterparts in the limit of high temperature. Therefore, the quantum considerations preserve their importance for the low temperature phenomenons such as Bose-Einstein condensation. Nevertheless, considering the low temperature behavior is useful even in the semiclassical regime to see how the results in this limit may be recovered. To do this, let us take the low temperature limit of the relation (40), that is U ≈ U 0 1 + mT 4T 2 Pl ,(41) where U 0 = 3 2 N T is the well-known usual internal energy for the ideal gas. A glance at 41) shows that while the polymeric effects becomes important at the high temperatures, the standard relation for the internal energy of an ideal gas is recovered in the limit of low temperature. The usual internal energy (dashed line) and its polymeric counterpart (solid line) versus the temperature are plotted in figure 1. As this figure shows, the two curves coincide at low temperatures and while the temperature rises are separated from each other. To estimate the order of magnitude of the polymeric correction to the internal energy of an ideal gas, consider ∆U U = U − U 0 U ∼ m m Pl × T T Pl ,(42) where m Pl is the Planck mass (equal to the Planck temperature in the units in which we are working). Since we have set numerical factor α to be of the order of unity, the polymeric effects become important on the trans-Planckian regime. In this regard, these effects will be very small in the currently accessible temperatures [25]. For instance, consider an ideal gas consisting of electrons with mass m e ≈ 0.5 MeV. For the temperature about T ∼ 1 TeV the polymeric correction to the corresponding internal energy is of the order of ∆U U ∼ 10 −38 ,(43) where we have set m Pl ≈ 1.2 × 10 19 GeV. Furthermore, the usual internal energy for an ideal gas U 0 is linearly proportional to its temperature and consequently the system can have access to any arbitrary high energy scale just by sufficiently increasing its temperature. However, a glance at the corresponding relation for the polymeric ideal gas given by equation (40) shows the existence of a finite maximum value in the high temperature limit as (see figure 1) U ≤ U max = 3N T 2 Pl m .(44) The fact that no energy scale is accessible greater than the above upper bound may be attributed to fact that the momenta of the noninteracting particles of the gaseous system are bounded in polymer framework. C. Entropy Now let us see how the entropy changes its form under the framework we are dealing with. A straightforward calculation based on the Helmholtz energy (37) will arrive us to S = − ∂F ∂T N,V = N ln V I 3 0 T 2 Pl /mT N l 3 Pl + 1 − 3N T 2 Pl mT I 1 T 2 Pl /mT I 0 T 2 Pl /mT .(45) In figure 2, we have plotted the behavior of entropy in terms of temperature. As this figure shows entropy increases with a much less rate in comparison with the usual ideal gas. This result is due to the fact that entropy is directly related to the number of microstates of the system and this quantity in turn reduces in the polymer framework since there is an upper bound for the momentum of the particles in this picture. To see how the entropy behaves in the low temperature regime, we may take this limit of the relation (45) with result S ≈ N ln V N 2πmT h 2 3/2 + 5 2 − 3 4 N mT T 2 Pl .(46) The last term on the right-hand side is the first order polymeric correction to the entropy of an ideal gas which is negligible in the limit of low temperatures. This is also clear from the figure 2, which shows that the entropy curve (45) coincides with its standard nondeformed one in the limit of low temperature. The order of magnitude of the polymeric correction to the entropy is the same as we obtained for the internal energy. D. Specific heat Finally, another important thermodynamical quantity is the specific heat which can be evaluated from the internal energy (40) as follows C V = ∂U ∂T V = 3N T 4 Pl m 2 T 2 1 − mT T 2 P I 1 T 2 Pl /mT I 0 T 2 Pl /mT − I 2 1 T 2 Pl /mT I 2 0 T 2 Pl /mT .(47) The specific heat versus the temperature is shown in figure 3. We see that this quantity grows until it reaches a maximum value and then takes a decreasing behavior, eventually tends to zero. This means that from now on, if the system gets more thermal energy, its internal energy does not change. Such a behavior is expected because we have seen from (44) that the system eventually reaches a saturated internal energy. Again, our result coincides with one arises from the full quantum consideration of the ideal gas for the small polymer length [13]. As we have done in the last two subsections, let us take a look at the low temperature limit of the specific heat, that is C V ≈ 3 2 N 1 + mT 2T 2 Pl .(48) Again, it is seen that the polymeric correction is of the order of one that is obtained for the internal energy. Also, as we can see from (48), the specific heat of the polymeric system is coincided with its value for the ordinary ideal gas at very low temperatures. V. SUMMARY AND CONCLUSIONS The polymer picture of quantum mechanics is an exotic representation of the commutation relations which have been investigated in a symmetric sector of loop quantum gravity. We argued that in order to study the statistical mechanics of a polymeric systems, analytical evaluation of the energy eigenvalues may be impossible since the Hamiltonian gets a nonlinear form in this framework. On the other hand, we showed one can work with the classical Hamiltonian and the density of states in the semiclassical regime. The advantage of this method is FIG. 3. The specific heat versus the temperature in the polymer framework. Since the increasing behavior of the internal energy stops after reaching to its maximum, it is expected that the specific heat goes to zero after reaching to a maximum value. The figure clearly shows this behavior. that there is no need to solve the nonlinear eigenvalue problem. Therefore, we considered the symplectic structure of the polymeric phase space and for studying the statistical mechanics in this picture we used the effective Hamiltonian (8) and the deformed density of states (31) in the Darboux chart. Moreover, we introduced a noncanonical chart on the polymeric symplectic manifold in which the Hamiltonian takes the standard form (21) and all the polymeric effects were summarized in the deformed density of states (33). We explained that working in this chart is more admissible from the statistical point of view. This is because the density of states determines the number of accessible microstates of the system and consequently the polymeric effects only change this quantity in this chart. According to our calculations the number of microstates decreased when the polymeric considerations came into the play and we linked this phenomena to the fact that the momenta are bounded in such a framework. Based on the deformed density of states, we obtained the canonical partition function for the polymeric many particle systems and then utilized it to study thermodynamics of the ideal gas. In this regard, some thermodynamical quantities of the polymeric ideal gas such as pressure, internal energy, entropy, and specific heat are evaluated. Having the same form of the equation of state as the ordinary ideal gas, existence of an upper limit for the internal energy unlike the conventional case in which there is no restriction for the system to achieve any level of energy, increasing behavior of the entropy but with a rate much less than the usual case and tending to zero after reaching to a maximum value for the specific heat were the main features of our analysis based on the ideas we have designed in this article. As an estimation, our calculations predict ∼ 10 −38 (in the TeV temperature scale) for the order of magnitude of the polymeric corrections to the thermodynamical quantities of an ideal gas. We saw that these results are in very good agreement with their counterparts arisen from the full quantum consideration at high temperatures and they coincide with their usual counterparts at the low temperature limit, so this may be considered as evidence that the way we have moved in, was a right way. 1 − ( λp ′ x 2 ) 2 + dy ′ ∧ dp ′ y 1 − ( λp ′ y 2 ) 2 + dz ′ ∧ dp ′ z 1 − ( λp ′ z 2 ) 2 ,(A-6) which leads to the 6-form Liouville volume ω 3 = dx ′ ∧ dy ′ ∧ dz ′ ∧ dp ′ x ∧ dp ′ y ∧ dp ′ z 1 − ( λp ′ x 2 ) 2 1 − ( λp ′ y 2 ) 2 1 − ( λp ′ z 2 ) 2 , (A-7) through the definition (16). The density of states (33) can be easily deduced by substituting the Liouville volume (A-7) into the relation (29). Relation (33) is essential to obtain the partition function in polymer framework. FIG. 1 . 1The internal energy versus the temperature; the solid line represents the internal energy in the polymer framework and the dashed line corresponds to the standard nondeformed case. Clearly, there is an upper bound for the internal energy in the polymer framework which is originated in the existence of the maximal momentum (UV cutoff) in the polymeric systems. The figure is plotted in unit = 1 and T Pl = 10, where T P is the Planck temperature. The number of particles and the mass are taken as N = 10 7 and m = 10. The polymeric effects dominate when the temperature approaches the Planck temperature. FIG. 2 . 2The entropy against the temperature; the solid line represents the entropy in the polymer picture and the dashed line corresponds to the nondeformed case. The entropy increases with a much less rate in comparison with the ordinary ideal gas since the number of microstates in the polymer picture is less than the usual case. The figure is plotted for V = 10 7 . Appendix A: Polymeric Density of StatesTo analyze the thermodynamics of the polymeric ideal gas, we need to consider a six-dimensional phase space corresponding to the single particle state. The homogenous polymerization for the six-dimensional phase space with variables q i = (x, y, z) and p i = (p x , p y , p z ) is read from(5)asIn the Darboux chart, the canonical structure for the sixdimensional symplectic manifold iswhere the momenta are bounded as p i ∈ [− π λ , + π λ ). The associated 6-form Liouville volume can be obtained by substituting (A-2) into the definition(16)asThe density of states corresponding to the structure (A-2) can be obtained via the relation (29) aswhich clearly gives the relation (31). To obtain the state density in the noncanonical chart, we apply transformation(20)to the canonical variables (q i , p i ) in the polymeric six-dimensional phase space aswhere q ′i = (x ′ , y ′ , z ′ ) and p ′ i = (p ′ x , p ′ y , p ′ z ). The 2-form symplectic structure for the six-dimensional symplectic manifold in the noncanonical chart becomes ω = dx ′ ∧ dp ′x . W Rindler, Gen. Rel. Grav. 34133W. Rindler, Gen. Rel. Grav. 34 (2002) 133 . 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P Pedram, K Nozari, S H Taheri, JHEP. 110393P. Pedram, K. Nozari and S. H. Taheri, JHEP 1103 (2011) 093 . S Jalalzadeh, M A Gorji, K Nozari, Gen. Relativ. Gravit. 461632S. Jalalzadeh, M. A. Gorji and K. Nozari, Gen. Relativ. Gravit. 46 (2014) 1632 . S Ghosh, Class. Quantum Grav. 3125025S. Ghosh, Class. Quantum Grav. 31 (2014) 025025. . A F Ali, S Das, E C Vagenas, Phys. Lett. B. 678497A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B 678 (2009) 497 . S Das, E C Vagenas, A F Ali, Phys. Lett. B. 690407S. Das, E. C. Vagenas and A. F. Ali, Phys. Lett. B 690 (2010) 407. We work in unit k B = 1 = c, where k B and c are the Boltzman constant and speed of light in vacuum respectively. We work in unit k B = 1 = c, where k B and c are the Boltzman constant and speed of light in vacuum respec- tively. R K Pathria, P D Beale, Statistical Mechanics. OxfordElsevier LtdR. K. Pathria and P. D. Beale, Statistical Mechanics (El- sevier Ltd, Oxford, 2011) J Naudts, Generalised Thermostatistics. LondonSpringer-VerlagJ. Naudts, Generalised Thermostatistics (Springer- Verlag, London, 2011). Here it is important to note that the correction term is also closely related to the free numerical parameter α which we have set to be of order of unity. In fact the numerical factor α determines the fundamental length scale of the quantum gravity (or lattice discreteness length) as l poly = α l Pl. Preserving this parameter, one can find an upper bound for the the polymer length scale l poly (see for instance [26])Here it is important to note that the correction term is also closely related to the free numerical parameter α which we have set to be of order of unity. In fact the nu- merical factor α determines the fundamental length scale of the quantum gravity (or lattice discreteness length) as l poly = α l Pl . Preserving this parameter, one can find an upper bound for the the polymer length scale l poly (see for instance [26]). . G Chacón-Acosta, H Hernandez-Hernandez, arXiv:1408.1306[astro-ph.SRG. Chacón-Acosta and H. Hernandez-Hernandez, arXiv: 1408.1306 [astro-ph.SR]
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[ "COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS BVR C I C PHOTOMETRIC EVOLUTION OF THE VERY FAST NOVA OPHIUCHI 2010 N.1 = V2673 OPH", "COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS BVR C I C PHOTOMETRIC EVOLUTION OF THE VERY FAST NOVA OPHIUCHI 2010 N.1 = V2673 OPH" ]
[ "U Munari \nINAF Osservatorio Astronomico di Padova\nSede di Asiago, I36032AsiagoVIItaly\n", "S Dallaporta \nANS Collaboration, c/o Astronomical Observatory\n36012AsiagoVIItaly\n", "\nKonkoly Observatory Budapest\n\n" ]
[ "INAF Osservatorio Astronomico di Padova\nSede di Asiago, I36032AsiagoVIItaly", "ANS Collaboration, c/o Astronomical Observatory\n36012AsiagoVIItaly", "Konkoly Observatory Budapest\n" ]
[]
Nova Ophiuchi 2010 N.1 (= V2673 Oph) was discovered by H. Nishimura on Jan. 15.9 UT (cf. Nakano 2010) and confirmed spectroscopically by H. Maehara (2010) as a "Fe II" class nova.We obtained BV R C I C photometry of Nova Ophiuchi 2010 N.1 with a 0.30-m Meade RCX-400 f/8 Schmidt-Cassegrain telescope equipped with a SBIG ST-9 CCD camera. The photometry was accurately corrected for color equations using nightly calibrations onLandolt (1992Landolt ( , 2009) standard stars. The data are presented inTable 1, and plotted inFigure 1. The combined (Poissonian + transformation) errors (always less than 0.03 mag) do not exceed the dimension of the symbols inFigure 1. The zero points of the photometry are scaled on the nearby star TYC 6260-1846-1, for which we adopted: B=11.550, V = 10.963, R C = 10.574 and I C = 10.222. The B and V are the values recommended by AAVSO for this star, the R C and I C are derived combining B, V with J,H,K from 2MASS following the recipes by Caldwell et al. (1993).We started our observations immediately past maximum, and thus to reconstruct the whole lightcurve as presented inFigure 1, we had to integrate them with the published data.Various estimates, based on unfiltered CCD observations secured around the time of discovery with digital cameras by Japanese amateurs, were published in CBET 2128. These observations are generally calibrated against the R C band values of field stars as listed by the USNO catalog. We have measured the field stars around Nova Ophiuchi 2010 N.1 and found a mean <V −R C >=+0.57 for them. We thus applied this shift to the unfiltered photometry of CBET 2128 and inserted it as open circles inFigure 1.Four aproximately V -band observations were obtained by Vollmann (2010) from the green channel of color CCD images obtained with a DSLR camera. Comparison with our simultaneous photometry indicates that Vollmann values need to be corrected by +0.1 mag to be placed onto the V photometric scale. We applied such a correction and plotted the data as star symbols inFigure 1.The VSNET organization collected some BV R C I C CCD photometric data of Nova Ophiuchi 2010 N.1, with observers S. Kiyota and H. Maehara (cf March 1, 2010 summary
null
[ "https://arxiv.org/pdf/1003.5371v1.pdf" ]
116,291,799
1003.5371
89653c7b502b987a5d944e47667e1163945a12ca
COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS BVR C I C PHOTOMETRIC EVOLUTION OF THE VERY FAST NOVA OPHIUCHI 2010 N.1 = V2673 OPH 28 Mar 2010 26 March 2010 U Munari INAF Osservatorio Astronomico di Padova Sede di Asiago, I36032AsiagoVIItaly S Dallaporta ANS Collaboration, c/o Astronomical Observatory 36012AsiagoVIItaly Konkoly Observatory Budapest COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS BVR C I C PHOTOMETRIC EVOLUTION OF THE VERY FAST NOVA OPHIUCHI 2010 N.1 = V2673 OPH 28 Mar 2010 26 March 2010 Nova Ophiuchi 2010 N.1 (= V2673 Oph) was discovered by H. Nishimura on Jan. 15.9 UT (cf. Nakano 2010) and confirmed spectroscopically by H. Maehara (2010) as a "Fe II" class nova.We obtained BV R C I C photometry of Nova Ophiuchi 2010 N.1 with a 0.30-m Meade RCX-400 f/8 Schmidt-Cassegrain telescope equipped with a SBIG ST-9 CCD camera. The photometry was accurately corrected for color equations using nightly calibrations onLandolt (1992Landolt ( , 2009) standard stars. The data are presented inTable 1, and plotted inFigure 1. The combined (Poissonian + transformation) errors (always less than 0.03 mag) do not exceed the dimension of the symbols inFigure 1. The zero points of the photometry are scaled on the nearby star TYC 6260-1846-1, for which we adopted: B=11.550, V = 10.963, R C = 10.574 and I C = 10.222. The B and V are the values recommended by AAVSO for this star, the R C and I C are derived combining B, V with J,H,K from 2MASS following the recipes by Caldwell et al. (1993).We started our observations immediately past maximum, and thus to reconstruct the whole lightcurve as presented inFigure 1, we had to integrate them with the published data.Various estimates, based on unfiltered CCD observations secured around the time of discovery with digital cameras by Japanese amateurs, were published in CBET 2128. These observations are generally calibrated against the R C band values of field stars as listed by the USNO catalog. We have measured the field stars around Nova Ophiuchi 2010 N.1 and found a mean <V −R C >=+0.57 for them. We thus applied this shift to the unfiltered photometry of CBET 2128 and inserted it as open circles inFigure 1.Four aproximately V -band observations were obtained by Vollmann (2010) from the green channel of color CCD images obtained with a DSLR camera. Comparison with our simultaneous photometry indicates that Vollmann values need to be corrected by +0.1 mag to be placed onto the V photometric scale. We applied such a correction and plotted the data as star symbols inFigure 1.The VSNET organization collected some BV R C I C CCD photometric data of Nova Ophiuchi 2010 N.1, with observers S. Kiyota and H. Maehara (cf March 1, 2010 summary Nova Ophiuchi 2010 N.1 (= V2673 Oph) was discovered by H. Nishimura on Jan. 15.9 UT (cf. Nakano 2010) and confirmed spectroscopically by H. Maehara (2010) as a "Fe II" class nova. We obtained BV R C I C photometry of Nova Ophiuchi 2010 N.1 with a 0.30-m Meade RCX-400 f/8 Schmidt-Cassegrain telescope equipped with a SBIG ST-9 CCD camera. The photometry was accurately corrected for color equations using nightly calibrations on Landolt (1992Landolt ( , 2009) standard stars. The data are presented in Table 1, and plotted in Figure 1. The combined (Poissonian + transformation) errors (always less than 0.03 mag) do not exceed the dimension of the symbols in Figure 1. The zero points of the photometry are scaled on the nearby star TYC 6260-1846-1, for which we adopted: B=11.550, V = 10.963, R C = 10.574 and I C = 10.222. The B and V are the values recommended by AAVSO for this star, the R C and I C are derived combining B, V with J,H,K from 2MASS following the recipes by Caldwell et al. (1993). We started our observations immediately past maximum, and thus to reconstruct the whole lightcurve as presented in Figure 1, we had to integrate them with the published data. Various estimates, based on unfiltered CCD observations secured around the time of discovery with digital cameras by Japanese amateurs, were published in CBET 2128. These observations are generally calibrated against the R C band values of field stars as listed by the USNO catalog. We have measured the field stars around Nova Ophiuchi 2010 N.1 and found a mean <V −R C >=+0.57 for them. We thus applied this shift to the unfiltered photometry of CBET 2128 and inserted it as open circles in Figure 1. Four aproximately V -band observations were obtained by Vollmann (2010) from the green channel of color CCD images obtained with a DSLR camera. Comparison with our simultaneous photometry indicates that Vollmann values need to be corrected by +0.1 mag to be placed onto the V photometric scale. We applied such a correction and plotted the data as star symbols in Figure 1. The VSNET organization collected some BV R C I C CCD photometric data of Nova Ophiuchi 2010 N.1, with observers S. Kiyota and H. Maehara (cf March 1, 2010 summary in [vsnet-recent-nova 35402] at http://www.kusastro.kyoto-u.ac.jp/vsnet/). The data obtained by observer S. Kiyota were corrected for instrumental color equations, and are inserted in Figure 1 as asterisks. They did not require adjustments, as it also was for V band data by VSNET observer H. Maehara. The B,R C and I C data of the latter, however, need the application of a shift to be brought in agreement with the rest of the data. The shift we applied amounts to +0.32 mag in B, +0.34 in R C , and +0.45 mag in I C . In Figure 1 the time is counted from maximum brightness that was reached on Jan. R V =3.1 interstellar law) is therefore A V =2.2 mag. The light-curve in Figure 1 is characterized by a rapid rise (the last 2.2 mag in V band were covered in 3.4 days) and by a smooth decline, regulated by the decline times t V 2 = 10.0 t V 3 = 23.5 days(1) which are the time taken by the nova to decline, in the V band, by two and three magnitudes, respectively, from maximum brightness. These t V 2 and t V 3 values for Nova Oph 2010 are in the normal proportion found for typical novae. Given t V 2 , the Warner (1995) relation would predict t V 3 =20.8, while Munari et al. (2008) relation would give t V 3 =23.1. According to the classification of Warner (1995, his Table 5.4), a t V 2 = 10 days qualifies Nova Oph 2010 N.1 to be classed among the very fast novae. Published relations between the absolute magnitude and the rate of decline generally take the form M max = α n log t n + β n . Using the Cohen (1988) V -t 2 relation, the distance to the nova is 8.3 kpc, and 7.5 kpc according to the Schmidt (1957) V -t 3 relation. Buscombe and de Vaucouleurs (1955) suggested that all novae have the same absolute magnitude 15 days after maximum light. The mean value of the calibrations presented by Buscombe and de Vaucouleurs (1955), Cohen (1985), van den Bergh and Younger (1987), van den Bergh (1988), and Capaccioli et al. (1989) is M V 15 =−5.42±0.09, which provides a distance of 6.5 kpc to Nova Oph 2010 N.1 when compared to V 15 =10.85 from Figure 1. Taking the mean of these three determinations, the distance to Nova Oph 2010 N.1 is d=7.4 kpc. At a galactic latitude b=4.92 deg, it corresponds to an height over the Galactic equatorial plane of z=0.6 kpc, well within the range of heights reported by della Valle and Livio (1998) for novae of the Fe II type. 18.3, 2010 at V =8.5. At that time the colors were B−V =+0.95, V −R C =+0.75, and V −I C =+1.50. van den Bergh and Younger (1987) derived a mean intrinsic color (B − V ) • =+0.23 ±0.06 for novae at the time of maximum, and (B − V ) • =−0.02 ±0.04 at t 2 . Comparing with B−V =+0.95 at maximum and B−V =+0.68 at t 2 from Figure 1, the reddening affecting Nova Oph 2010 N.1 is E B−V =0.71, and the extinction (assuming a standard Figure 1 . 1BV R C I C photometric evolution of the outburst of Nova Ophiuchi 2010 N.1. For the literature data, see text for details. Table 1 . 1Our BV R C I C of Nova Oph 2010 N.1 +0.72 +1.00 +1.60 2455231.7104 10.90 +0.66 +1.04 +1.57 2455235.6959 11.31 +0.67 +1.13 +1.71 2455242.6834 11.79 +0.61 +1.29 +1.91 2455248.6852 12.19 +0.56 +1.49 +2.13 2455261.6320 12.91 +0.53 +1.96 +2.11 2455264.6625 12.93 +0.49 +1.89 +2.01HJD V B−V V −R C V −I C 2455216.7306 9.15 +0.86 +1.61 2455218.7244 9.51 +0.74 +1.68 2455223.7142 10.28 +0.69 +1.01 +1.65 2455225.7166 10.60 +0.70 +1.64 2455229.7095 10.85 . W Buscombe, G De Vaucouleurs, Obs. 75170Buscombe, W., de Vaucouleurs, G. 1955, Obs. 75, 170 . J A R Calddwell, A W J Cousins, C C Ahlers, SAAO Circ. 151Calddwell, J.A.R., Cousins, A.W.J., Ahlers, C.C. et al. 1993, SAAO Circ. 15, 1 . M Capaccioli, AJ. 971622Capaccioli, M. et al. 1989, AJ 97, 1622 . J G Cohen, ApJ. 29290Cohen J.G. 1985, ApJ 292, 90 . J G Cohen, ASP Conf Ser. 4114Cohen J.G. 1988, ASP Conf Ser 4, 114 . M Della Valle, M Livio, ApJ. 506818della Valle, M., Livio, M. 1998, ApJ 506, 818 . A U Landolt, AJ. 104340Landolt, A.U. 1992, AJ 104, 340 . A U Landolt, AJ. 1374186Landolt, A.U. 2009, AJ 137, 4186 . H Maehara, 9111Maehara, H. 2010, IAUC 9111 . U Munari, A&A. 492145Munari, U. et al. 2008, A&A 492, 145 . S Nakano, 9111Nakano, S. 2010, IAUC 9111 T ; S Schmidt, P F Younger, ZA 41, 182 van den Bergh. 70PASPSchmidt T. 1957, ZA 41, 182 van den Bergh, S., Younger, P.F. 1987, A&AS 70, 125 van den Bergh, S. 1988, PASP 100, 8 . W Vollman, 2139Vollman, W. 2010, CBET 2139 Cataclysmic Variable Stars. B Warner, Cambridge Univ. PressWarner B. 1995, Cataclysmic Variable Stars, Cambridge Univ. Press
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[ "Superconducting triplet pairings and anisotropic magnetoresistance effects in ferromagnet/superconductor/ferromagnet double-barrier junctions", "Superconducting triplet pairings and anisotropic magnetoresistance effects in ferromagnet/superconductor/ferromagnet double-barrier junctions" ]
[ "Andreas Costa \nInstitute for Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany\n", "Jaroslav Fabian \nInstitute for Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany\n" ]
[ "Institute for Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany", "Institute for Theoretical Physics\nUniversity of Regensburg\n93040RegensburgGermany" ]
[]
Ferromagnetic spin valves offer the key building blocks to integrate giant-and tunneling-magnetoresistance effects into spintronics devices. Starting from a generalized Blonder-Tinkham-Klapwijk approach, we theoretically investigate the impact of interfacial Rashba and Dresselhaus spin-orbit couplings on the tunneling conductance, and thereby the magnetoresistance characteristics, of ferromagnet/superconductor/ferromagnet spinvalve junctions embedding thin superconducting spacers between the either parallel or antiparallel magnetized ferromagnets. We focus on the unique interplay between usual electron tunnelings-that fully determine the magnetoresistance in the normal-conducting state-and the peculiar Andreev reflections in the superconducting state.In the presence of interfacial spin-orbit couplings, special attention needs to be paid to the spin-flip ("unconventional") Andreev-reflection process that is expected to induce superconducting triplet correlations in proximitized regions. As a transport signature of these triplet pairings, we detect conductance double peaks around the singletgap energy, reflecting the competition between the singlet and an additionally emerging triplet gap; the latter is an effective superconducting gap that can be ascribed to the formation of triplet Cooper pairs through interfacial spin-flip scatterings (i.e., to the generation of an effective triplet-pairing term in the order parameter). We thoroughly analyze the Andreev reflections' role in connection with superconducting magnetoresistance phenomena, and eventually unravel huge conductance and magnetoresistance magnetoanisotropies-easily exceeding their normal-state counterparts by several orders of magnitude-as another experimentally accessible fingerprint of unconventional Andreev reflections. Our results provide an important contribution to establish superconducting magnetic spin valves as an essential ingredient for future superconducting-spintronics concepts. arXiv:2107.13818v2 [cond-mat.supr-con] 9 Nov 2021
10.1103/physrevb.104.174504
[ "https://arxiv.org/pdf/2107.13818v2.pdf" ]
243,861,218
2107.13818
815f3cbe21f46b41afde798662b9237396df5494
Superconducting triplet pairings and anisotropic magnetoresistance effects in ferromagnet/superconductor/ferromagnet double-barrier junctions Andreas Costa Institute for Theoretical Physics University of Regensburg 93040RegensburgGermany Jaroslav Fabian Institute for Theoretical Physics University of Regensburg 93040RegensburgGermany Superconducting triplet pairings and anisotropic magnetoresistance effects in ferromagnet/superconductor/ferromagnet double-barrier junctions (Dated: November 10, 2021) Ferromagnetic spin valves offer the key building blocks to integrate giant-and tunneling-magnetoresistance effects into spintronics devices. Starting from a generalized Blonder-Tinkham-Klapwijk approach, we theoretically investigate the impact of interfacial Rashba and Dresselhaus spin-orbit couplings on the tunneling conductance, and thereby the magnetoresistance characteristics, of ferromagnet/superconductor/ferromagnet spinvalve junctions embedding thin superconducting spacers between the either parallel or antiparallel magnetized ferromagnets. We focus on the unique interplay between usual electron tunnelings-that fully determine the magnetoresistance in the normal-conducting state-and the peculiar Andreev reflections in the superconducting state.In the presence of interfacial spin-orbit couplings, special attention needs to be paid to the spin-flip ("unconventional") Andreev-reflection process that is expected to induce superconducting triplet correlations in proximitized regions. As a transport signature of these triplet pairings, we detect conductance double peaks around the singletgap energy, reflecting the competition between the singlet and an additionally emerging triplet gap; the latter is an effective superconducting gap that can be ascribed to the formation of triplet Cooper pairs through interfacial spin-flip scatterings (i.e., to the generation of an effective triplet-pairing term in the order parameter). We thoroughly analyze the Andreev reflections' role in connection with superconducting magnetoresistance phenomena, and eventually unravel huge conductance and magnetoresistance magnetoanisotropies-easily exceeding their normal-state counterparts by several orders of magnitude-as another experimentally accessible fingerprint of unconventional Andreev reflections. Our results provide an important contribution to establish superconducting magnetic spin valves as an essential ingredient for future superconducting-spintronics concepts. arXiv:2107.13818v2 [cond-mat.supr-con] 9 Nov 2021 I. INTRODUCTION The tunneling-magnetoresistance (TMR) effect [1,2], occurring when switching the relative magnetizations of ferromagnet/insulator/ferromagnet (F/I/F) spin valves' metallic layers, is one of the most spectacular spintronics phenomena [3,4], especially considering its technological applications in computers [5][6][7][8][9]. Numerous proposals to engineer next-level quantum computers have been put forward within recent years [10][11][12][13][14][15][16][17], and might come along with a so far unimaginable boost of computing performance. Owing to its great advantages [18,19] when combining quantum coherence, which belongs to the most fundamental ingredients for quantum computing, with dissipationless charge and longlived spin transport, most of the aforementioned concepts exploit superconductivity. Among the systems attracting the most considerable interest are superconducting magnetic tunnel junctions, in which the competition between the two nominally strongly antagonistically acting superconducting and ferromagnetic phases offers a versatile playground to study novel physical characteristics. While early works focused, e.g., on the conductance of ferromagnet/superconductor point contacts [20,21], thereby demonstrating that Andreev reflections impact transport in a unique way from which the ferromagnet's spin polarization can be experimentally extracted, more intricate junction setups are being explored nowadays. Magnetic Josephson junctions are particularly promising candidates to investigate unprecedented transport anomalies, covering current-reversing * Corresponding author: [email protected] 0-transitions [22,23] that could form the elementary twolevel system for quantum computing, substantially enhanced current magnetoanisotropies [24][25][26][27], the potential appearance of Majorana states [28][29][30][31][32][33][34][35][36], as well as the possibility to efficiently generate long-range spin-polarized triplet-Cooper pair supercurrents [18,37]. Such triplet pairings are typically induced in -wave superconductors proximitized by (strongly spin-polarized) ferromagnets either in the presence of noncollinearly magnetized interfacial domains [38][39][40][41][42][43][44][45][46][47][48][49] or spinorbit coupling (SOC) [26,50,51]. The normal-state TMR effect's superconducting counterpart was already investigated in theoretical [52] and experimental [53,54] works carried out on ferromagnet/superconductor/ferromagnet (F/S/F) spin valves, in which a thin superconducting spacer couples the parallel or antiparallel magnetized ferromagnetic electrodes. Remarkably, flipping the magnetizations from antiparallel to parallel may decrease the superconductor's critical temperature. Close to the critical temperature, the magnetization flipping can thus switch off superconductivity, resulting in an infinitely large TMR ratio. While the normal-state TMR was fully explained by spinpolarized electron tunnelings [1], Andreev reflections [55] can as well strongly influence electrical transport in superconducting junctions and modify the TMR characteristics there [56]. In this paper, we investigate the influence of SOC on the transport properties of F/S/F junctions. We assume thin semiconducting tunneling barriers between the ferromagnetic and superconducting regions with interfacial (Bychkov-)Rashba [57] and Dresselhaus [58] SOCs. Such couplings occur in the presence of semiconductors such as InAs or InSb, whose (001) interfaces have 2 symmetry [4]. As mentioned above, these SOCs induce effective superconducting triplet pairings close to the junction interfaces, microscopically mediated by spin-flip ("unconventional") Andreev reflections, just as the usual spin-conserving ("conventional") Andreev reflections bring singlet superconducting order into proximitized junction regions. The unconventional Andreev reflections are extremely sensitive to changing absolute magnetization orientations, and are thus at the heart of the huge magnetoanisotropies in superconducting magnetic junctions [24][25][26][27]. Generalizing the well-established Blonder-Tinkham-Klapwijk model [59] to spin-valve junctions [60][61][62], we evaluate the junctions' zero-temperature tunneling conductance and elaborate on transport ramifications of unconventional Andreev reflections, predicting the formation of conductance double peaks close to the singlet-gap energy as a consequence of the effectively induced nonzero triplet gap. After demonstrating the expected huge magnetosensitivity of the double-peak conductance structure, we compute typical (T)MR ratios [63] and identify marked MR magnetoanisotropies, which provide another clear fingerprint of unconventional Andreev reflections. Our predictions should help experiments in disentangling triplet and singlet superconducting proximity pairings in F/S/F spin valves' tunneling conductance. We structure the paper in the following way. In Sec. II, we formulate our theoretical model to describe electrical transport through the considered F/S/F junctions. The general conductance features and double peaks as signatures of triplet pairing are thoroughly analyzed in Sec. III, before we briefly comment on the conductance's magnetic tunability (magnetoanisotropy) in Sec. IV. The MR and its magnetoanisotropy are discussed in Secs. V and VI, respectively, while we conclude our main findings in Sec. VII. II. THEORETICAL MODEL We consider a biased ballistic F/S/F junction grown along theˆ [001] crystallographic direction, in which the two semi-infinite ferromagnetic electrodes (F) are separated from the superconducting link (S) by ultrathin semiconducting tunneling barriers (see Fig. 1) with 2 symmetry. We model our system by means of its stationary Bogoliubov-de Gennes Hamiltonian [64] 2 × 2 identity and the th Pauli matrix;σ = [ˆ,ˆ,ˆ] is the vector of Pauli matrices). Both ferromagnetic electrodes are described within the Stoner model with the same exchange energy Δ XC , and the, in general, different in-plane H BdG = Ĥ eΔS ( ) Δ † S ( )Ĥ h ,(1)whereĤ e = − ℏ 2 2 ∇ 2 − ˆ0 − Δ XC 2 (m 1 ·σ)Θ(− ) − Δ XC 2 (m 2 ·σ)Θ( − ) + ( L Lˆ0 + L ·σ) ( ) + ( R Rˆ0 + R ·σ) ( − )(2)magnetization directionsm 1 = [cos Φ 1 , sin Φ 1 , 0] in the left andm 2 = [cos Φ 2 , sin Φ 2 , 0] in the right F; the angles Φ 1 and Φ 2 are thereby measured with respect to theˆreference axis, which is taken to be the principal-symmetry [110] crystallographic axis. Following earlier studies [26,[65][66][67][68][69][70], the ultrathin semiconducting interface layers are included into our model as deltalike barriers with heights (widths) L ( L ) at the left and R ( R ) at the right interface, respectively. Their SOCs enter through the effective spin-orbit fields L = [( L − L ) , −( L + L ) , 0] and R = −[( R − R ) , −( R + R ) , 0], where the terms scaling with the effective SOC strength L ( R ) account for the Rashba SOC at the left (right) interface and the remaining ones for linearized Dresselhaus SOC with the effective strengths L ( R ). Note that we define the sign of R opposite to that of L , reflecting the fact that Rashba coupling arises from interfacial hybridization. Inside the superconducting link, the -wave pairing potential Δ S ( ) = Δ 0ˆ0 Θ( )Θ( − ),(3) with the isotropic zero-temperature superconducting energy gap Δ 0 , couples the Bogoliubov-de Gennes Hamiltonian's electron and hole blocks. To simplify the analytical description of our system, we take the same Fermi levels and effective carrier masses in all junction regions, and approximate the superconducting pairing potentialΔ S ( ) by a Heaviside step function being nonzero only inside the superconductor and instantly jumping to zero inside the ferromagnets. All other spatial variations of the superconducting order parameter, i.e., its more realistic exponential decay in the vicinity of the F/S boundaries, are fully neglected and would need to be determined from a self-consistent formulation [71]. However, earlier experimental studies [72,73] demonstrated that even in F/S/F junctions in which the superconducting link is much thicker than the Bardeen-Cooper-Schrieffer coherence length, the current flows still quite uniformly, suggesting that spatial variations of the superconducting order parameter mostly average out and a self-consistent treatment is not necessary to understand such junctions' generic transport features. Assuming translational invariance parallel to the semiconducting interfaces, the solutions of the Bogoliubov-de Gennes equationĤ BdG Ψ (r) = Ψ (r)(4) factorize into Ψ (r) = ( )e i(k ·r ) ,(5) where k = [ , , 0] (r = [ , , 0] ) refers to the in-plane momentum (position) vector and ( ) to the Bogoliubov-de Gennes equation's most general solution for the effectively one-dimensional scattering problem alongˆ. Considering an incoming spin-electron from the left ferromagnet [ = +(−)1 for spin up (spin down), which effectively indicates a spin parallel (antiparallel) tom 1 ], the latter is found to read as ( < 0) = e i ,e 1 √ 2 e −iΦ 1 , 1, 0, 0 + , e e −i ,e 1 √ 2 e −iΦ 1 , 1, 0, 0 + ,− e e −i − ,e 1 √ 2 − e −iΦ 1 , 1, 0, 0 + ,− h e i − ,h 1 √ 2 0, 0, e −iΦ 1 , 1 + , h e i ,h 1 √ 2 0, 0, − e −iΦ 1 , 1(6) in the left ferromagnet ( < 0), (0 < < ) = 1 e i ,e + 2 e −i ,+ 1 e −i ,h + 2 e i ,h [ , 0, , 0] + 3 e −i ,h + 4 e i ,h [0, , 0, ](7) in the superconducting link (0 < < ), and accordingly + ,− h e −i − ,h 1 √ 2 0, 0, e −iΦ 2 , 1 + , h e −i − ,h 1 √ 2 0, 0, − e −iΦ 2 , 1(8) in the right ferromagnet ( > ). Theˆ-projected wave vectors of spin-electrons and holes in the ferromagnets are given by ,e = √︃ 2 F + 2 /ℏ 2 ( + Δ XC /2) − |k | 2 (9) and ,h = √︃ 2 F + 2 /ℏ 2 (− + Δ XC /2) − |k | 2 ,(10) respectively, whereas we obtain ,e = √︂ 2 F + 2 /ℏ 2 √︃ 2 − Δ 2 0 − |k | 2(11) and ,h = √︂ 2 F − 2 /ℏ 2 √︃ 2 − Δ 2 0 − |k | 2(12) for electronlike and holelike quasiparticles inside the superconducting link; F = √︁ 2 /ℏ denotes the Fermi wave vector. Finally, the Bardeen-Cooper-Schrieffer coherence factors can be written as = 1 2 1 + √︃ 2 − Δ 2 0 ,(13) as well as = √︁ 1 − 2 .(14) The given states account for all scattering processes that incident electrons may undergo at the semiconductor interfaces, including also the possibility of spin-flip scattering caused by the interfacial SOCs. Apart from spinconserving and spin-flip specular (normal) reflections (with amplitudes , e and ,− e ), we need to pay special attention to spin-conserving ("conventional") and spin-flip ("unconventional") Andreev reflections (with amplitudes To determine the unknown scattering amplitudes, we apply the interfacial boundary conditions (at = 0 and = ) ( = 0 − ) = ( = 0 + ), ( = − ) = ( = + ),(15)ℏ 2 2 d dˆ( = 0 + ) − d dˆ( = 0 − ) = L ·σ + L Lˆ00 0 −( L ·σ + L Lˆ0 ) ( = 0 + ),(16) as well as (17) to the scattering states and numerically solve the resulting linear system of equations (at a given spin ;ˆ= diag[1, 1, −1, −1] and0 abbreviates the 2 × 2 zero matrix). Assuring charge conservation [74], and taking both the electron and hole contributions into account [56,[75][76][77], the tunneling conductance at zero temperature and bias voltage can be evaluated from ℏ 2 2 d dˆ( = + ) − d dˆ( = − ) = R ·σ + R Rˆ00 0 −( R ·σ + R Rˆ0 ) ( = + )= A 2 2 F 2 2 ℎ ∑︁ =∓1 ∫ d 2 k , e 2 + Re where A indicates the contact cross-section area, is the positive elementary charge, ℎ = 2 ℏ corresponds to Planck's constant, and taking the real parts (Re . . .) of the wave-vector ratios ensures that only the contributions originating from propagating states are included in the conductance calculation. Interestingly, the tunneling conductance of superconducting F/S/F junctions is therefore not only governed by the usual tunneling electrons (electron transmissions)-as is the case in normal-state F/N/F junctions according to Jullière's model [1]-but moreover impacted by the crucial Andreevreflection process, which additionally transfers electrons in terms of supercurrent-carrying Cooper pairs across the superconducting junction link and will most likely give rise to unforeseen physical phenomena. III. GENERAL CONDUCTANCE FEATURES AND SIGNATURES OF INDUCED TRIPLET PAIRING To analyze the most fundamental features and tunability of the tunneling conductance, we numerically evaluate Eq. (18) for realistic junction parameters. More specifically, we assume that both semiconducting tunneling barriers introduce the same weak potential scattering described by the dimensionless Blonder-Tinkham-Klapwijk parameters [59] L = (2 L L )/(ℏ 2 F ) = 1 = (2 R R )/(ℏ 2 F ) = R , which would correspond, for example, to barrier heights L = R ≈ 0.75 eV and widths L = R = 0.40 nm (substituting the typical Fermi wave vector F ≈ 8 × 10 7 cm −1 of iron [27] and approximating the effective carrier mass by the free-electron mass). These barriers mimic reduced interfacial transparencies that could stem, e.g., from scattering due to imperfect interfaces or strongly differing Fermi levels (electron densities) in the ferromagnetic and superconducting junction regions [67]. The strengths of the interfacial Rashba and Dresselhaus SOCs are quantified by the dimensionless measures L = (2 L )/ℏ 2 , R = (2 R )/ℏ 2 , L = (2 L )/ℏ 2 , and R = (2 R )/ℏ 2 . For simplicity, we assume that both semiconducting interfaces are identical, i.e., they are characterized by the same Rashba and Dresselhaus SOC parameters, respectively. Rashba (Dresselhaus) parameters of L = R = 0.5 ( L = R = 0.5) correspond then to bare Rashba (Dresselhaus) SOCs of L = R ≈ 1.9 eV Å 2 ( L = R ≈ 1.9 eV Å 2 , connecting L ≈ L F L and R ≈ R F R with the barriers' cubic Dresselhaus parameters L and R that are typically given in the literature [4,78]). Recall that a 1.7 nm thick MgO barrier was found to induce Rashba SOCs up to ≈ 4.6 eV Å 2 in Fe/MgO/V junctions [27], while AlP barriers of the given heights and widths would indeed raise Dresselhaus SOCs of ≈ 1.7 eV Å 2 . Finally, the spin polarization of the (identical) ferromagnets is determined by the dimensionless variable = (Δ XC /2)/ . The Fermi level is typically much larger than the superconducting gap Δ 0 and the excitation energies , motivating = 10 3 Δ 0 as a reasonable assumption for our calculations. All tunneling-conductance values discussed throughout this paper are normalized to the respective normal-state conductance N = (A 2 2 F )/(2 2 ℏ) of an ideal (perfectly transparent) N/N/N junction. Figure 2 illustrates the tunneling conductance of a F/S/F junction comprising a "thin" superconducting link of thickness = 10 3 / F (≈ 125 nm if F ≈ 8 × 10 7 cm −1 ) once in the absence of interfacial SOCs and once if the moderate Rashba SOCs L = R = 1 are present. In both cases, we consecutively increase the ferromagnets' spin polarization from = 0 (N/S/N junction; gray color) to = 1 (halfmetallic F/S/half-metallic F; dark-blue color), and distinguish parallel (both magnetizations point alongˆ) from antiparallel (the left ferromagnet's magnetization points alongˆand the right one's along −ˆ) magnetizations. Let us first focus on the situation without interfacial SOCs [see Fig. 2(a)], in which spin-flip scatterings-i.e., also the crucial unconventional Andreev reflections-are completely forbidden, and all electrical transport is governed by electron transmissions (hole transmissions are negligible) and conventional Andreev reflections. However, as the superconducting link is rather thin when compared to usual superconducting coherence lengths (in the micron range [79]), also conventional Andreev reflections are heavily suppressedparticularly at large spin polarizations, at which the through conventional Andreev reflections proximity-induced superconducting order inside the ferromagnets becomes negligibly tiny anyway due to these metals' small coherence lengths [80]. As a consequence, thin superconducting links act mostly like rectangular potential barriers of height Δ 0 and width . Since Δ 0 ), explaining the enhanced tunneling conductances there. Increasing the ferromagnets' spin polarization monotonically decreases the tunneling-conductance amplitudes, as less minority-spin electrons are then involved in tunneling and can contribute to transport. According to Jullière's model [1], switching the relative magnetization orientations from the parallel to their antiparallel configuration (at ≠ 0) notably damps the probability for electron transmissions and thereby the tunneling conductance-most extreme in the half-metallic case in which no electrons can tunnel between oppositely magnetized ferromagnets and huge MR ratios are expected. Second, we explore the effects associated with interfacial Rashba SOCs [see Fig. 2(b)]. Although conventional Andreev reflections are still strongly suppressed, the tunneling conductances in the subgap bias-voltage regime are remarkably enhanced by the present SOCs. This enhancement stems from the now additionally possible unconventional Andreev reflections [81], which are known to induce sizable su- perconducting triplet pairings even in strongly spin-polarized ferromagnets (and even though the superconducting link in our case is quite thin). The unconventional Andreev-reflection contributions become maximal in magnitude close to the chosen Rashba SOC strengths L = R = 1 (but are still slightly smaller than the electron-transmission parts), can cause finite (subgap) tunneling conductances even in the case of half-metallic ferromagnetic electrodes, and are only merely affected by flipping the ferromagnets' relative magnetization orientations-giving reasoning for the smaller (compared to the case without SOCs) absolute conductance changes when switching between parallel and antiparallel configurations (see our discussions in Sec. V). Increasing the bias voltage to values above the superconducting gap ( ≥ Δ 0 ), unconventional Andreev reflections become more unlikely and we essentially recover the purely by electron transmissions dominated transport regime at Δ 0 . However, we wish to stress that the interfacial SOCs furthermore act like additional deltalike potential barriers to electron transmissions [recall the Hamiltonian in Eq. (2)], eventually leading to even smaller (normalstate) tunneling conductances than in the absence of SOCs. Next, we investigate the tunneling conductance of a F/S/F junction that contains a "thick" superconducting link of thickness = 10 4 / F (≈ 1.25 μm if F ≈ 8 × 10 7 cm −1 ); see Fig. 3. Such junctions are probably of much greater rel-evance to future experimental studies since their thicker superconducting regions inherently entail much larger Andreevreflection probabilities and their conductance spectra simultaneously reveal the superconductor's most fundamental spectroscopic fingerprints (gap). We start again analyzing the case without interfacial SOCs [see Fig. 3(a)]. In fact, electron transmissions are now completely forbidden at < Δ 0 and the subgap tunneling conductance is fully describable through the properties of conventional Andreev reflections (implicitly also carrying all necessary information about specular reflections if nonzero potential barriers are present). As we mentioned before, Andreev reflections are only merely affected by switching the ferromagnets' relative magnetization orientations. Therefore, the subgap conductances for parallel and antiparallel magnetizations are (nearly) equal in magnitude as long as the tunneling conductance is fully dominated by Andreev reflections. At the superconducting gap edge ( = Δ 0 ), the tunneling conductance of a nonmagnetic N/S/N junction always reflects a sharp conductance peak, which indicates the superconductor's density-of-states coherence peak and from which the superconducting energy gap can be estimated through transport experiments. With increasing spin polarization in a ferromagnetic junction, the conductance peak flattens and its position moves to energies slightly above the gap-though one can still quite reliably estimate the value of the superconducting gap from the peak position. Above the gap ( > Δ 0 ), the tunneling conductance reveals unique oscillations, which are damped out with increasing voltage and finally disappear when approaching the normal-state transport regime at Δ 0 . These oscillations reflect the coherency of electrical transport through F/S/F junctions' superconducting links. Coherent interference of incoming and outgoing quasiparticles (that underwent multiple reflections at the S/F interfaces) basically leads to Andreev-reflection and electron-transmission probabilities that strongly oscillate as functions of the excitation energy = and the link thickness ; for the latter reason, the oscillations are usually referred to as geometrical oscillations. An earlier work [56] concluded that Andreev reflections are strongly suppressed and coherent electron transmissions become concurrently most likely whenever ( ,e − ,h ) = 2 ,(19) where is an integer and, for simplicity, the effects of tunneling barriers (and large spin polarizations) were neglected. Inspecting our numerical results shows indeed that finiteheight tunneling barriers and (large) spin polarizations only barely impact the conductance oscillations. Similar oscillations would actually be expected to likewise occur in junctions with thinner superconducting links, but the oscillation period is so large there (owing to the much smaller ) that we did not resolve them within the bias-voltage range chosen in Fig. 2. Regarding the tunneling-conductance amplitudes, increasing spin polarization decreases both the Andreev-reflection and electron-transmission contributions, and thereby suppresses the conductance. In the half-metallic case, the subgap conductance vanishes now even in the parallel magnetization configuration since all subgap transport is governed by conven-tional Andreev reflections, which are no longer possible if only majority-spin electrons are available. Finite Rashba SOCs at the semiconducting interfaces additionally allow for the crucial unconventional Andreev reflections. While the general conductance features far below and far above the superconducting gap Δ 0 are the same as we thoroughly discussed earlier-including the conductance increase in the subgap region, the conductance decrease at Δ 0 , finite tunneling conductances in the half-metallic limit, and the geometrical oscillations at > Δ 0 -the most puzzling feature arises in the vicinity of the gap edge itself. Increasing the ferromagnets' spin polarization splits the conductance peak that we attributed to the gap-edge densityof-states coherence peak into two distinct peaks-one located below and the other above the gap energy Δ 0 . This peak splitting becomes most pronounced as the spin polarization approaches the half-metallic limit ( → 1) and the unconventional Andreev-reflection conductance contribution becomes considerably large when compared to conventional Andreev reflections and electron transmissions. The latter observation might serve as a hint that the conductance-peak splitting and the peculiar unconventional Andreev-reflection process must be intimately connected. To resolve this connection, Fig. 4 illustrates the individual conductance contributions originating from conventional Andreev reflections, unconventional Andreev reflections, and electron transmissions. For brevity, we just discuss the case of parallel magnetizations (antiparallel magnetizations cause similar physics, but of course different conductance amplitudes) and focus on the representative spin polarization = 0.7 of iron electrodes, varying now instead the Rashba SOC strengths. Note that the conventional Andreev-reflection and electron-transmission parts clearly reflect the aforementioned geometrical oscillations with mutually suppressed (enhanced) Andreev-reflection (electrontransmission) probabilities at > Δ 0 . The physics becomes nevertheless most interesting close to the gap edge. While conventional Andreev reflections still cause dominant conductance maxima ("peaks") at Δ 0 -slightly sharpened and shifted with increasing Rashba SOCs-it is indeed the unconventional Andreev-reflection process that manifests itself in terms of split conductance double peaks located at energies slightly below and above Δ 0 . This peak splitting becomes again most clearly visible when unconventional Andreev reflection dominates subgap transport (i.e., close to L = R = 1; recall that the same happened at large spin polarizations). Similarly to conventional Andreev reflections, which microscopically induce superconducting singlet correlations into the ferromagnets close to the interfaces, unconventional Andreev reflections introduce spin-polarized triplet correlations. As a consequence, the gap-edge density-of-states coherence peak splits into two peaks corresponding to Δ s − Δ t and Δ s + Δ t , accordingly; Δ s ≈ Δ 0 (Δ t ) indicates the superconducting gap due to singlet (triplet) pairing; note, however, that the triplet gap is small when compared to its singlet counterpart, as triplet pairing is in our case only induced through the interfacial SOCs (i.e., Δ t Δ s ). Measuring the junction's tunneling conductance probes therefore directly the competing mixture of singlet and triplet correlations at the same time, and detects gap-edge conductance double peaks as an indirect signature of superconducting triplet pairings, which might help to identify the dominant pairing mechanism in upcoming transport studies. IV. MAGNETIC TUNABILITY OF CONDUCTANCE FEATURES Apart from the enhancement of the subgap conductance and the coherence-peak (conductance-peak) splitting, inter- facial SOC gives typically rise to unique transport magnetoanisotropies, i.e., rotating the magnetization direction of (at least) one ferromagnet considerably alters the conductance amplitudes in the presence of interfacial SOCs. While out-of-plane magnetization rotations (in a plane perpendicular to the semiconducting barriers) already cause magnetoanisotropic conductances if the barriers induce either Rashba or Dresselhaus SOCs, in-plane magnetoanisotropies require interfering Rashba and Dresselhaus SOCs. As out-of-plane magnetization directions are not realistic in spin-valve MR geometries, we focus on the in-plane case. Figure 5 shows the tunneling conductance of a F/S/F junction with a "thin" superconducting link ( = 10 3 / F ) considering the moderate (and equal in magnitude) Rashba and Dresselhaus SOCs L = R = L = R = 1 and rotating the magnetization (once in the parallel and once in the antiparallel configuration) from theˆ-to theˆ-direction (from Φ 1 = 0 to Tunneling electrons are therefore subject to the maximal SOCs if the magnetizations are aligned along ∓ˆ(due to the dependence) and do not at all experience any SOC for magnetizations parallel to ∓ˆ(as the dependence dropped out), eventually raising the maximally possible magnetoanisotropy. As a result, the additionally generated unconventional Andreev-reflection conductance contribution becomes maximal at Φ 1 = 0 and completely vanishes at Φ 1 = 0.5 , explaining the overall substantial conductance decrease when increasing Φ 1 from Φ 1 = 0 to Φ 1 = 0.5 . Recall that conventional Andreev reflections are suppressed in junctions with "thin" superconducting links and the tunneling conductance is, apart from partially through SOCs allowed unconventional Andreev reflections, largely determined by electron transmissions. Surprisingly, and in sharp contrast to two-electrode F/S junctions in which they even fully disappear there [24], unconventional Andreev reflections (at Φ 1 = 0) are most likely at energies around the gap edge ( ≈ Δ 0 ), facilitating a somewhat broadened conductance peak ("conductance shoulder"). As before, the tunneling conductance at energies well above Δ 0 mostly stems from electron transmissions, and just slightly decreases with increasing Φ 1 owing to the effectively slightly lowered tunneling probability. Analogously, we present the tunneling conductance's magnetization-angle dependence for a F/S/F junction contain-ing a "thick" superconducting link ( = 10 4 / F ) in Fig. 6; all other parameters are not changed. As we argued above, the unconventional Andreev-reflection conductance contribution gets maximal at Φ 1 = 0 and vanishes at Φ 1 = 0.5 . Besides remarkably enhancing the subgap tunneling conductance, the superconducting triplet pairings induced by unconventional Andreev reflections split the gap-edge conductance peak again into two distinct peaks. Just as we encountered when analyzing this feature earlier, the peak splitting becomes most pronounced when simultaneously the unconventional Andreevreflection process dominates the subgap tunneling conductance, i.e., at Φ 1 = 0 in our case. Note that the split conductance peaks' amplitude ratios in the parallel and antiparallel magnetization configurations are opposite, as one might also observe in Fig. 3(b). While the peak slightly above the gap corresponds to the larger tunneling conductance for parallel magnetizations-as the tunneling conductance gets there amplified by additionally allowed electron transmissions-it is the peak below the gap that entails maximal tunneling conductances in the antiparallel magnetized scenario. Finally, the aforementioned geometrical oscillations at > Δ 0 are clearly visible and not notably altered by rotating the magnetization orientations, while the conductance amplitudes decrease slightly with increasing Φ 1 there due to the slightly reduced interfacial transparencies, just as we explained for "thin" superconducting links above. While Andreev reflections are only barely impacted by switching the ferromagnets' relative magnetizations from the parallel to the antiparallel orientations (recall our discussions in Sec. III), the last paragraphs demonstrated that they are nonetheless extremely sensitive to rotations of the ferromagnets' absolute magnetization directions in the presence of interfacial SOCs, and marked magnetoanisotropies in the tunneling conductance can occur. To quantify the latter, and emphasize that they predominantly originate from the strongly magnetoanisotropic (unconventional) Andreev-reflection probabilities, an earlier work on two-electrode F/S junctions established the in-plane magnetoanisotropic Andreev reflection (MAAR) [24] MAAR(Φ 1 ) = (0) − (Φ 1 ) (Φ 1 ) .(22) One could evaluate the MAAR, for instance, deep inside the superconducting junction regime at = 0, and compare the values against its normal-state counterpart at Δ 0 . As we can already expect from the large magnetization-controlled tunability of the absolute conductance amplitudes in Figs. 5 and 6, unconventional Andreev reflections could entail huge superconducting MAAR ratios, which can easily exceed the equivalent normal-state tunneling anisotropic magnetoresistance [78,82] by more than three orders of magnitude in the half-metallic limit and can be further enhanced by increasing the thickness of the superconducting link . While the calculated MAAR in a junction with a "thin" superconducting link ( = 10 3 / F ) lies clearly below the values predicted for comparable two-electrode F/S junctions (as Andreev reflection is not the dominant scattering process there), "thick" superconducting links ( = 10 4 / F ) cause MAARs that already remark-ably overcome those in the corresponding F/S junctions (as Andreev reflection dominates now the subgap regime). As the overall physics and qualitative characteristics are basically similar to the F/S case, we do not deeply analyze our MAAR calculations here. V. MAGNETORESISTANCE EFFECTS MR effects count to the probably most intensively investigated phenomena in magnetic spin-valve junctions. Our work offers the possibility to study the MR of superconducting magnetic spin valves, and elaborate more on the ramifications of the competition between the usual-and in normal-state spin valves dominant-electron transmissions and the superconducting junctions' unique Andreev reflections. Adapting its most common definition, the MR ratio at an absolute magnetization orientation determined by Φ 1 (in the left ferromagnet) is given by MR(Φ 1 ) = P − AP AP = (Φ 1 ) − (Φ 1 + ) (Φ 1 + ) ,(23)where P = (Φ 1 ) [ AP = (Φ 1 + )] indicates the tunneling conductance in the parallel (antiparallel) magnetization configurations and can be extracted from Eq. (18) at zero temperature. In Fig. 7, we illustrate the computed MR [more precisely, MR(Φ 1 = /2)] as a function of the applied bias voltage for various strengths of interfacial Rashba and Dresselhaus SOCs L = R = L = R , distinguishing again between junctions with "thin" ( = 10 3 / F ) and "thick" ( = 10 4 / F ) superconducting links. As expected from our earlier analyses of the conductance features, the MR ratios always reach their maximal values whenever the underlying tunneling conductance is dominated by electron transmissions, i.e., at ≥ Δ 0 . In the subgap bias-voltage regime, even small conductance contributions originating from conventional andin the presence of nonzero interfacial SOCs-unconventional Andreev reflections immediately lower the resulting MR. This observation explains the MR suppression with increasing SOC strength in the junction with the "thin" link, as well as the fully vanishing subgap MR for the "thick" link. In the first case ("thin" link), increasing the SOC parameters above L = R = L = R ≈ 0.5 raises a marked conductance enhancement owing to unconventional Andreev reflections. Since Andreev reflections are much less sensitive to changes of the relative magnetization orientations than the in the absence of SOCs dominant electron transmissions, the related MR starts to be remarkably damped, finally resulting in a complete suppression if the conductance is exclusively determined by (unconventional) Andreev reflections as we witness in the second case ("thick" link). At voltages above the gap ( > Δ 0 ), the MR mostly reveals the conductance properties resulting from usual electron transmissions, including its monotonic decrease with increasing SOC strengths; the interfacial SOCs act then similarly to additional interfacial barriers that suppress electron transmissions and thus the MR. Interestingly, the geometrical conductance oscillations caused by coherent electron transmissions through "thick" superconducting links are moreover transferred into the respective MRbias voltage characteristics. Note that the MRs in the junctions' normal-state counterparts (recovered at Δ 0 ) areanalogously to the related tunneling conductances (compare, e.g., Fig. 2 to Fig. 3 supposing Δ 0 )-nearly completely independent of the link thickness, which is a consequence of our fully ballistic description. Regarding the maximally possible MR amplitudes, half-metallic junctions (with spin polarizations → 1) are certainly the most auspicious candidates; similarly to normal-conducting systems, maximal MRs easily reach then values above 1000 %. Summarizing the preceding paragraphs, the MR features (amplitudes) are predominantly controlled by the intriguing competition between Andreev reflections (dominant in the subgap regime, < Δ 0 ) and electron transmissions (dominant at > Δ 0 ). Most relevant to future experimental studies might therefore be exploring the MR exactly at the gap-edge energy, i.e., at = Δ 0 , at which the aforementioned competition between Andreev reflections and electron transmissions becomes most pronounced. We will address the experimental signatures in Sec. VI. VI. ANISOTROPIC MAGNETORESISTANCE As we thoroughly discussed in Sec. IV, interfacial SOCs usually entail marked magnetoanisotropies in experimentally probeable transport quantities. Consequently, not only the tunneling conductance itself, but also closely related measures, like the MR, strongly depend on the absolute orientation of the ferromagnets' magnetization directions. We emphasized that in Eq. (23) through explicitly stating the MR's Φ 1 dependence. While the magnetoanisotropic MR-usually referred to as anisotropic magnetoresistance (AMR)-has already been comprehensively analyzed in normal-conducting F/N/F junctions [78], characterizations of AMR phenomena in superconducting junctions have yet been missing. To close this gap, we present the angular dependence of the considered F/S/F junction's AMR [note that AMR(Φ) = MR(Φ 1 = Φ); recall Eq. (23)] for various uniform SOC strengths in Fig. 8. Motivated by our previous arguments, we focus on the bias voltage = Δ 0 , at which we expect a strong competition between Andreev reflections and electron transmissions, to compare the results in the superconducting to those in the normal-conducting ( Δ 0 ) scenario; the thickness of the "thin" link is again = 10 3 / F and that of the "thick" link = 10 4 / F . As long as interfacial SOCs are completely absent, the (A)MR is isotropic, and the MR amplitudes do hence not alter as the absolute direction of the ferromagnets' magnetiza-tions gets rotated (gray circles). Already weak SOCs, however, notably tilt the AMR curves (more elliptical, colored, curves), and give rise to clearly magnetoanisotropic MRs, with substantially larger MR ratios at magnetizations pointing along ∓ˆ(as unconventional Andreev reflections get suppressed there and electron transmissions dominate). Although qualitatively similar physics occurs in the normal state, the "tilting" (which is directly linked to the "strength" of the MR magnetoanisotropy, as we will elaborate on later) is much weaker than in the superconducting case (whereas the overall MR values are more than twice as large as in the superconducting junction due to the dominant electron transmissions in normal-state junctions) and it is now the magnetization along ∓ˆthat results in (slightly) larger MR amplitudes. These observations suggest that, while overall large MR ratios indicate dominant electron transmissions, marked AMR magnetoanisotropies serve as an experimentally accessible signature of dominant Andreev reflections. These contrary features could be beneficial to subsequent experimental works to disentangle Andreev-reflection-from electron-transmissionrelated physics. As they both predominantly originate from the peculiar Andreev-reflection process, MAAR and AMR effects share all their fundamental properties. For instance, and similarly to the aforementioned in-plane MAAR, it is vital to the AMR that interfacial Rashba and Dresselhaus SOCs interfere. If just Rashba or Dresselhaus SOCs alone were present, the (A)MR would become fully isotropic. Maximally anisotropic AMRs arise again if the Rashba and Dresselhaus SOC strengths are equal, as we likewise explained in connection with the tunneling-conductance magnetoanisotropies in Sec. IV. To quantify the "strength" of the MR magnetoanisotropy, and relate it to the "tilting" of the AMR curves visible in Fig. 8, we introduce the AMR efficiency [78] = AMR(Φ = /2) − AMR(Φ = 0) AMR(Φ = 0) ,(24) which essentially measures the relative change of the MR ratios while the ferromagnets' magnetizations are rotated from the ∓ˆtoward the ∓ˆ-orientation. Figure 9 shows the computed AMR efficiencies as functions of the ferromagnets' spin polarization and the uniform SOC strengths L = R = L = R , as well as the ratio between the AMR efficiencies in the superconducting and normal-conducting states, respectively. The results further substantiate our previous claims, and demonstrate the peculiar role of unconventional Andreev reflections for another time. More specifically, the MR magnetoanisotropies of junctions containing the "thin" superconducting link become most pronounced (resulting in the largest AMR efficiencies ) as the ferromagnets' spin polarization approaches the half-metallic limit ( → 1) and the SOC strengths are tuned to L = R = L = R ≈ 1. As we pointed out in Sec. III, those parameters maximize the unconventional Andreev-reflection contribution to the tunneling conductances (at the considered bias voltage = Δ 0 ), which responds most sensitively to changes of the absolute magnetization directions and entails huge MR magnetoanisotropies ( ). "thick" superconducting link (thicknesses = 10 3 / F and = 10 4 / F ). Unconventional Andreev reflections can substantially enhance the AMR efficiency in the superconducting state (up to about two orders of magnitude), as inspecting the ratios between the gap-edge AMR efficiency and its normal-state counterpart in (c) and (d) illustrates. The locally suppressed AMR efficiency (violet) in (d) signifies the gap-edge conductance-peak splitting that we attributed to superconducting triplet pairings (see Sec. III). Noteworthy, the AMR efficiency in the superconducting state exceeds its normal-state counterpart by more than two orders of magnitude. The arguments provided in the preceding paragraph do, in principle, also hold for junctions with the "thick" superconducting link. Nevertheless, the MR magnetoanisotropies (amplitudes of ) appear to be substantially lower in that case, which must indicate that unconventional Andreev reflections are additionally suppressed. We indeed observe this suppression most clearly at spin polarization = 0.7 and SOC parameters L = R = L = R ≈ 1, for which we analyzed the underlying tunneling conductances in all details in Sec. III unraveling the conductance-peak splitting at ≈ Δ 0 as a transport signature of superconducting triplet pairings. As a result, the initially present conductance peak exactly at = Δ 0 turns into a conductance dip, the unconventional Andreev-reflection contribution gets damped, and the calculated MR magnetoanisotropy ( ) must therefore notably drop. The huge MR magnetoanisotropies ( ) would instead occur slightly below and above the superconducting gap, corresponding to the two newly forming conductance double peaks. The latter can hence not only be identified in the tunneling-conductance data, but leave also an indirect imprint on the AMR characteristics. Though the unconventional Andreev-reflection contribution at = Δ 0 is small(er), it is large enough to raise AMR efficiencies that still overcome their normal-state counterparts by more than one order of magnitude. VII. CONCLUSIONS To summarize, we studied the tunneling-conductance features of superconducting magnetic F/S/F spin-valve junctions paying special attention to the ramifications of interfacial Rashba and Dresselhaus SOCs. We distinguished between junctions hosting "thin" (thickness of about 125 nm) and "thick" (thickness of about 1.25 μm) superconducting links, and allowed for arbitrary in-plane orientations of the magnetization directions inside the ferromagnetic electrodes. Interfacial SOCs facilitate unconventional (spin-flip) Andreev reflections at junction interfaces that are commonly expected to be at the heart of numerous transport anomalies, as they concurrently introduce spin-polarized superconducting triplet pair-ings into the system. Regarding the considered F/S/F junctions, we observed that unconventional Andreev reflections can give rise to a conductance-peak splitting close to the singletgap energy, which eventually reveals the interplay between the usual superconducting singlet and the additional effectively induced triplet gaps. Owing to their close connection to unconventional Andreev reflections, we demonstrated that these peak splittings-and at the same time also the overall amplitudes of the tunneling conductance-are efficiently tunable through altering the ferromagnets' absolute magnetization orientations. We eventually quantified the MR of superconducting spin-valve junctions, and unraveled that unconventional Andreev reflections (and thus indirectly the present SOCs) furthermore lead to marked MR magnetoanisotropies, which we termed AMRs. Measuring the AMR efficiency (i.e., the "strength" of the MR magnetoanisotropy) provides another experimental possibility to detect the triplet-pairing signify-ing gap-edge conductance double peaks. In view of future experiments, we suggest focusing on highly spin-polarized junctions, in which the strong spin filtering of transmitted electrons yields overall giant MRs (analogously to Jullière's model), but which still entail a considerable amount of unconventional Andreev reflections to simultaneously raise huge magnetoanisotropies (MAARs and AMRs). e [ , 0, , 0] + 3 e i ,e + 4 e −i ,e [0, , 0, ] which usually induce superconducting order in the ferromagnetic electrodes through proximity and lead thereby to numerous unique physical characteristics in superconducting magnetic junctions. Regarding transmissions (tunnelings) into the right ferromagnet, we need to distinguish between electron transmissions (with amplitudes , Δ 0 ,FIG. 2 . 02even subgap-energy electrons ( < Δ 0 ) can tunnel across the superconducting link with considerably large probabilities, entailing nonzero subgap tunneling conductances. Nevertheless, electron transmissions happen of course much more likely at energies above the barrier height (at ≥ Calculated tunneling conductance as a function of the applied bias voltage and for different indicated spin polarizations of the ferromagnetic electrodes, considering a "thin" superconducting link of thickness = 10 3 / F . Solid lines correspond to parallel magnetization orientations (both ferromagnets are magnetized along theˆ-direction; ↑↑), whereas dashed lines indicate antiparallel magnetization orientations (the left ferromagnet is magnetized alongˆand the right along −ˆ; ↑↓). (a) In the absence of interfacial SOCs ( L = R = L = R = 0), unconventional Andreev reflections are forbidden (see illustration), and the conductance is fully determined by conventional Andreev reflections and electron transmissions. (b) Unconventional Andreev-reflection contributions (see illustration) at moderate interfacial Rashba SOCs ( L = R = 1) may significantly enhance the subgap tunneling conductance (at < Δ 0 ). FIG. 3 . 3Same calculations as inFig. 2, but assuming a "thick" superconducting link of thickness = 10 4 / F . The conductance spectra in the presence of interfacial Rashba SOCs are now dominated by unconventional Andreev reflections, effectively inducing nonzero superconducting triplet gaps close to the superconducting interfaces (due to the formation of polarized spin-triplet Cooper pairs) that are evident in terms of split gap-edge conductance double peaks around = Δ 0 . FIG. 4 . 4Calculated tunneling conductance as a function of the applied bias voltage and for different indicated interfacial Rashba SOC strengths L = R = (Dresselhaus SOC is not present; L = R = 0), considering iron as ferromagnetic electrodes (spin polarization = 0.7), both ferromagnets magnetized along ∓ˆ, and a "thick" superconducting link of thickness = 10 4 / F . The individually presented conductance contributions stem from (a) conventional Andreev reflections, (b) unconventional Andreev reflections, and (c) electron transmissions. FIG. 5 . 5Calculated tunneling conductance as a function of the applied bias voltage and for different indicated magnetization orientations covering the (a) parallel and (b) antiparallel magnetized configurations; Φ 1 = 0 (0.5 ) corresponds to a magnetization in the left ferromagnet that points alongˆ(ˆ), as illustrated. The spin polarization of the ferromagnets is = 0.7, the interfacial SOC strengths are L = R = L = R = 1, and the "thin" superconducting link has the thickness = 10 3 / F . FIG. 6 . 6Φ 1 = 0.5 ). Equal Rashba and Dresselhaus parameters are chosen since the interference of the Rashba and Dresselhaus spin-orbit fields leads then to the effective spin-orbit fields L = [0, −2 L , 0] (20) Same calculations as in Fig. 5, but assuming a "thick" superconducting link of thickness = 10 4 / F . The gap-edge conductancepeak splitting serves again as a precursor of induced superconducting triplet pairings. and R = −[0, −2 R , 0]. FIG. 7 . 7Calculated MR as a function of the applied bias voltage and for different indicated uniform SOC strengths = (= L = R = L = R ), assuming a (a) "thin" and (b) "thick" superconducting link of thicknesses = 10 3 / F and = 10 4 / F , respectively. The spin polarization of the ferromagnets is = 0.7 and the magnetizations are aligned along ∓ˆ. FIG. 8 . 8Calculated AMR (i.e., the angular dependence of the MR on the left ferromagnet's magnetization orientationΦ 1 = Φ) for different indicated uniform SOC strengths = (= L = R = L = R ), assuming a (a) "thin" and (b) "thick" superconducting link (thicknesses = 10 3 / F and = 10 4 / F ), and setting = Δ 0 . The remaining parameters are the same as inFig. 7. For comparison, panels (c) and (d) show the corresponding normal-state AMR ratios. ATMR signature of conductance-peak splitting FIG. 9. Calculated gap-edge AMR efficiency ( = Δ 0 ) as a function of the ferromagnets' spin polarization and the uniform SOC strength = (= L = R = L = R ) in the presence of a (a) "thin" and (b) FIG. 1. Sketch of the considered F/S/F junction, using 2 crystallographic orientationsˆ [110],ˆ [110], andˆ [001]. 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As we consider weak barriers to elucidate the physics related to Andreev reflections. we shall rather call the effect simply magnetoresistance (MRStrictly speaking, the term tunneling magnetoresistance (TMR) is only used for junctions in the tunneling limit, i.e., if strong interfacial tunneling barriers were present. As we consider weak barriers to elucidate the physics related to Andreev reflections, we shall rather call the effect simply magnetoresistance (MR). P G De Gennes, Superconductivity of Metals and Alloys. Addison Wesley, Redwood CityP. G. De Gennes, Superconductivity of Metals and Alloys (Ad- dison Wesley, Redwood City, 1989). . M J M Jong, C W J Beenakker, http:/link.aps.org/doi/10.1103/PhysRevLett.74.1657Phys. Rev. Lett. 741657M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett. 74, 1657 (1995). . I Žutić, O T Valls, http:/link.aps.org/doi/10.1103/PhysRevB.60.6320Phys. Rev. B. 606320I. Žutić and O. T. Valls, Phys. Rev. B 60, 6320 (1999). . 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B. 6494506J. M. E. Geers, M. B. S. Hesselberth, J. Aarts, and A. A. Golubov, Phys. Rev. B 64, 094506 (2001). Note that one needs to be careful when evaluating electrical currents from the Blonder-Tinkham-Klapwijk approach in junctions with more than two different regions to ensure that the calculated current is indeed conserved; see the thorough discussion in Ref. 77Note that one needs to be careful when evaluating electrical cur- rents from the Blonder-Tinkham-Klapwijk approach in junc- tions with more than two different regions to ensure that the calculated current is indeed conserved; see the thorough discus- sion in Ref. [77]. . C J Lambert, 10.1088/0953-8984/3/34/003J. Phys.: Condens. Matter. 36579C. J. Lambert, J. Phys.: Condens. Matter 3, 6579 (1991). . T Yamashita, H Imamura, S Takahashi, S Maekawa, 10.1103/PhysRevB.67.094515Phys. Rev. B. 6794515T. Yamashita, H. Imamura, S. Takahashi, and S. Maekawa, Phys. Rev. B 67, 094515 (2003). . Z C Dong, R Shen, Z M Zheng, D Y Xing, Z D Wang, 10.1103/PhysRevB.67.134515Phys. Rev. B. 67134515Z. C. Dong, R. Shen, Z. M. Zheng, D. Y. Xing, and Z. D. Wang, Phys. Rev. B 67, 134515 (2003). . A Matos-Abiague, J Fabian, http:/link.aps.org/doi/10.1103/PhysRevB.79.155303Phys. Rev. B. 79155303A. Matos-Abiague and J. Fabian, Phys. Rev. B 79, 155303 (2009). Introduction to Solid State Physics. C Kittel, John Wiley & Sons, LtdC. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Ltd., 1996). . L Y Yang, Z M Zheng, D Y Xing, 10.1140/epjb/e2005-00361-6Eur. Phys. J. B. 47479L. Y. Yang, Z. M. Zheng, and D. Y. Xing, Eur. Phys. J. B 47, 479 (2005). at = 0) cannot be attributed to unconventional Andreev reflections, which become most pronounced in ferromagnetic junctions. Instead, the Rashba SOC terms in the single-particle Hamiltonian. always partially compensate the potential barrier (for appropriate |k | and ), and lead thus to more electron transmissions and thereby a larger tunneling conductance. This can cause a conductance enhancement even in N/S/N junctions although unconventional Andreev reflections are strongly suppressed thereThe tunneling-conductance enhancement in the case of N/S/N junctions (i.e., at = 0) cannot be attributed to uncon- ventional Andreev reflections, which become most pronounced in ferromagnetic junctions. Instead, the Rashba SOC terms in the single-particle Hamiltonian [recall Eq. (2)] always partially compensate the potential barrier (for appropriate |k | and ), and lead thus to more electron transmissions and thereby a larger tunneling conductance. This can cause a conductance en- hancement even in N/S/N junctions although unconventional Andreev reflections are strongly suppressed there. . 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[ "Structural and optical studies of FeSb 2 under high pressure", "Structural and optical studies of FeSb 2 under high pressure" ]
[ "Claudio Michel Poffo ", "Sergio Michielon De Souza ", "Daniela Menegon Trichês ", "João Cardoso De Lima ", "Tarciso Antonio Grandi ", "Alain Brazil ", "Michel Polian ", "Gauthier ", "\nDepartamento de Engenharia Mecânica\nDepartamento de Física\nUniversidade Federal de Santa Catarina\nCampus Universitário Trindade, C.P. 47688040-900Florianópolis, Santa CatarinaS/NBrazil\n", "\nPhysique des Milieux Denses, IMPMC, CNRS-UMR 7590\nUniversidade Federal de Santa Catarina\nCampus Universitário Trindade88040-900Florianópolis, Santa CatarinaS/N, C.P. 476\n", "\nUniversité\nPierre et Marie Curie-Paris 6, B115, 4 Place Jussieu75252, Cedex 05ParisFrance\n" ]
[ "Departamento de Engenharia Mecânica\nDepartamento de Física\nUniversidade Federal de Santa Catarina\nCampus Universitário Trindade, C.P. 47688040-900Florianópolis, Santa CatarinaS/NBrazil", "Physique des Milieux Denses, IMPMC, CNRS-UMR 7590\nUniversidade Federal de Santa Catarina\nCampus Universitário Trindade88040-900Florianópolis, Santa CatarinaS/N, C.P. 476", "Université\nPierre et Marie Curie-Paris 6, B115, 4 Place Jussieu75252, Cedex 05ParisFrance" ]
[]
Nanostructured orthorhombic FeSb 2 and an amorphous phase were formed by mechanical alloying starting from a mixture of high purity elemental Fe and Sb powders. The effects of high pressures on structural and optical properties were studied using X-ray diffraction (XRD) and Raman spectroscopy (RS). XRD patterns showed the presence of the orthorhombic FeSb 2 phase up to the maximum pressure applied (28.2 GPa). The XRD patterns showed also an increase in the amount of the amorphous phase with increasing pressure up to 23.3 GPa. At 14.3 GPa, together with the former phases, a new phase was observed and indexed to a tetragonal FeSb 2 phase, but its volume fraction is small at least up to 23.3 GPa. For the orthorhombic FeSb 2 phase, the pressure dependence of the volume fitted to a Birch-Murnaghan equation of state gave a bulk modulus 0 B = 74.2  3.0 GPa and its pressure derivative 0 B = 7.5  0.6. RS measurements were performed from atmospheric pressure up to 45.2 GPa. For the orthorhombic FeSb 2 phase, the Raman active 2 g A mode was observed up to the maximum pressure applied, while the 1 1g B mode disappeared at 16.6 GPa. For pressures higher than 21 GPa, the Raman active 2 g E mode of a tetragonal FeSb 2 phase was observed, confirming ab initio calculations reported in the literature. PACS number(s): 61.05.cp, 61.50.Ks, 62.50.-p, 61.46.Hk
10.1016/j.physb.2012.08.032
[ "https://arxiv.org/pdf/1112.0468v1.pdf" ]
119,255,897
1112.0468
55ddd3827a90f9340bc754a982bde39692090a3c
Structural and optical studies of FeSb 2 under high pressure Claudio Michel Poffo Sergio Michielon De Souza Daniela Menegon Trichês João Cardoso De Lima Tarciso Antonio Grandi Alain Brazil Michel Polian Gauthier Departamento de Engenharia Mecânica Departamento de Física Universidade Federal de Santa Catarina Campus Universitário Trindade, C.P. 47688040-900Florianópolis, Santa CatarinaS/NBrazil Physique des Milieux Denses, IMPMC, CNRS-UMR 7590 Universidade Federal de Santa Catarina Campus Universitário Trindade88040-900Florianópolis, Santa CatarinaS/N, C.P. 476 Université Pierre et Marie Curie-Paris 6, B115, 4 Place Jussieu75252, Cedex 05ParisFrance Structural and optical studies of FeSb 2 under high pressure Nanostructured orthorhombic FeSb 2 and an amorphous phase were formed by mechanical alloying starting from a mixture of high purity elemental Fe and Sb powders. The effects of high pressures on structural and optical properties were studied using X-ray diffraction (XRD) and Raman spectroscopy (RS). XRD patterns showed the presence of the orthorhombic FeSb 2 phase up to the maximum pressure applied (28.2 GPa). The XRD patterns showed also an increase in the amount of the amorphous phase with increasing pressure up to 23.3 GPa. At 14.3 GPa, together with the former phases, a new phase was observed and indexed to a tetragonal FeSb 2 phase, but its volume fraction is small at least up to 23.3 GPa. For the orthorhombic FeSb 2 phase, the pressure dependence of the volume fitted to a Birch-Murnaghan equation of state gave a bulk modulus 0 B = 74.2  3.0 GPa and its pressure derivative 0 B = 7.5  0.6. RS measurements were performed from atmospheric pressure up to 45.2 GPa. For the orthorhombic FeSb 2 phase, the Raman active 2 g A mode was observed up to the maximum pressure applied, while the 1 1g B mode disappeared at 16.6 GPa. For pressures higher than 21 GPa, the Raman active 2 g E mode of a tetragonal FeSb 2 phase was observed, confirming ab initio calculations reported in the literature. PACS number(s): 61.05.cp, 61.50.Ks, 62.50.-p, 61.46.Hk I. INTRODUCTION FeSb 2 is a narrow-gap semiconductor that has attracted a lot of attention because of its unusual magnetic properties (paramagnetic to diamagnetic crossover at around 100 K), thermoelectric properties (colossal Seebeck coefficient S at 10 K and the largest power factor  2 S ever reported), 1 and transport properties (a metal-to-semiconductor transition at around 40 K). 2 The FeSb 2 compound can be synthesized by the self-flux method 3 and the hightemperature flux method, 4 among others. These techniques, besides being expensive, do not allow a good control of the size of the nanoparticles. At room temperature and atmospheric pressure, FeSb 2 crystallizes in an orthorhombic structure (s.g. Pnnm, Z = 2), with Fe atoms at the 2a (0, 0, 0) Wyckoff position and Sb atoms at the 4g (x, y, 0) position, where x = 0.1885 and y = 0.3561. Each Fe atom has a deformed octahedral environment, and octahedra share edges along the c axis. 5 Nanostructured materials have been widely studied due to their interesting properties. For example, the performance of a thermoelectric material can be improved if its thermal conductivity is reduced without strong degradation of its electrical properties. The dimensions of crystallites of nanostructured materials may allow an important reduction in the thermal conductivity of the lattice and promoting an improvement in their thermoelectric conversion efficiency. Nanostructured materials have been produced by different techniques, including mechanical alloying (MA). Suryanayarana´s paper 6 gives a good review of the MA technique, while the involved physical mechanisms are described in Refs. 7-10. From the structural point of view, nanostructured materials have two components: crystallites of nanometric dimensions 2-100 nm, having the same structure as their crystalline counterparts, and an interfacial component, formed by different types of defects (grain boundaries, interphase boundaries, dislocations, etc.) that surround the crystallites. Nanostructured materials are metastable. 11 The literature describes studies on the effect of high pressure in several nanostructured materials, [12][13][14][15][16][17][18] but the nanostructured FeSb 2 alloy is not among them. Petrovic et al. 19 investigated the effect of high pressure up to 7.14 GPa on bulk orthorhombic FeSb 2 compound and no structural changes were observed. The compounds TiSb 2 , VSb 2 , MnSn 2 , CoSn 2 and FeSn 2 crystallize in the CuAl 2 structure (s.g. I4/mcm, Z=4). 5 Takizawa et al. 20 reported a CrSb 2 phase that crystallized in the CuAl 2 structure above 5.5 GPa. Wu et al. 21 using ab initio calculations predicted a transformation of orthorhombic FeSb 2 phase into a tetragonal structure at 38 GPa. For some years, we focused our research in studying the effect of high pressure in nanostructured materials, mainly those with thermoelectric applications. [12][13][14]18,22 Due to the scientific interest in FeSb 2 and its technological importance, we investigated the effect of high pressure on nanostructured FeSb 2 powder produced by the MA technique. This study, besides filling a gap in the literature, will extend the investigation started by Petrovic et al. 19 on the effect of high pressures on this compound. This paper reports the results of this study. II. BRIEF CONSIDERATIONS ABOUT THE FORMATION OF CRYSTALLINE OR AMORPHOUS BINARY ALLOYS AT HIGH PRESSURES AND TEMPERATURES In order to understand the formation of crystalline and amorphous phases at high pressures and temperatures, we will consider the simplest case, i.e., that of binary alloys. Besides other physical mechanisms, there are two key points to taken into account: (i) the two elements must have a large negative relative heat of mixing, and (ii) either one of them is an anomalously fast diffuser (leading to an amorphous phase), or the two elements have similar diffusion coefficients (leading to a crystalline phase). When high pressure is applied in a binary mixture of high purity elemental powders sealed in a diamond anvil cell (DAC), defective chemical bonds are formed, characterized by angle and length changes. This chemical disorder stores a sizable amount of energy. As the pressure is increased further, a composite powder containing particles with defective chemical bonds is formed. When the material is heated to an appropriate temperature, the total energy (thermal energy plus the energy released from the defective chemical bonds) becomes the driving force for atom diffusion, promoting solid state reactions and resulting in the formation of crystalline and/or amorphous phases. Usually, the crystalline phase is the same that is produced through conventional methods and is thermodynamically stable at room temperature and atmospheric pressure, but new phases can be obtained through a convenient choice of pressure and temperature. For example, Takizawa et al. 20 produced the orthorhombic CrSb 2 phase (stable at room temperature and atmospheric pressure) starting from a CrSb 2 mixture and using pressures smaller than 5.5 GPa and temperatures below 1073 K. For pressures greater than 5.5 GPa and temperatures above 773 K, a new tetragonal CrSb 2 phase was obtained. When high pressure is applied to a single crystal or bulk compound sealed in a DAC, chemical disorder is induced and energy is stored, as was previously described. If the pressure is increased further, two things may happen: 1) the defective chemical bonds will break, releasing the stored energy, and 2) the defective chemical bonds will not break. When the defective chemical bonds break and the material is heated to an appropriate temperature, the total energy (thermal energy plus the energy released from the defective chemical bonds) becomes the driving force to promote the diffusion of free atoms, resulting in the formation of one or more new crystalline and/or amorphous phases. When the defective chemical bonds do not break, only a gradual topological rearrangement of chemical bonds occurs to minimize the energy and a new crystalline phase is formed. In this case, appropriate heating can accelerate the formation of the new phase and decrease the pressure at which phase transformation occurs. The driving force to promote the topological rearrangement of chemical bonds is the sum of thermal energy and the energy released from defective chemical bonds. If the material is not heated, the driving force to promote the topological rearrangement is lower and the formation of a new phase occurs at a higher pressure. The work reported by Takizawa et al. 20 is a good example of no breaking of defective chemical bonds. These researchers used temperature and pressure to accelerate the topological rearrangement of chemical bonds and decrease the pressure of transformation from bulk orthorhombic CrSb 2 phase into a tetragonal CrSb 2 phase. This new phase was formed using pressures above 5.5 GPa and temperatures above 773 K. Other work reported by Nakayama et al. 23 is an example of breaking the defective chemical bonds and forming a mixture of phases. These researchers applied high pressure to bulk rhombohedral Bi 2 Te 3 at room temperature and reported the formation of a mixture of phases for pressures between 9.2 and 16.2 GPa. When high pressure is applied to binary nanostructured compounds sealed in a DAC, a behavior different of those previously described is observed. This happens because these materials are structurally formed by two components: crystallites with dimensions < 100 nm, with a structure similar to their bulk counterparts, and an interfacial region, formed by different kinds of defects, that surround the crystallites. III. EXPERIMENTAL PROCEDURE A binary FeSb 2 mixture of elemental powders of Fe (Aldrich, purity 99.999%) and Sb (Alfa Aesar, purity 99.999%) was sealed together with several steel balls of 11.0 mm in diameter into a cylindrical steel vial under argon atmosphere. The ball-to-powder weight ratio was 7:1. The vial was mounted on a SPEX Mixer/mill, model 8000. The temperature was kept close to room temperature by a ventilation system. The structural changes of the mixture as a function of milling time were followed by XRD measurements. After 11 hours of milling, the XRD pattern showed an excellent agreement with that given in the ICSD Database 5 (code 41727) for the orthorhombic FeSb 2 phase (s.g. Pnnm, Z=2). The milling process was extended to 32 hours but no further structural changes were observed. A DAC with an opening that allowed probing up to 2 = 28° was used. 25 A small amount of FeSb 2 powder was compacted between diamonds to a final thickness of approximately 15 µm. A small chip of this preparation, about 80 µm in diameter, was loaded into a stainless-steel gasket hole of 150 m diameter. Neon gas was used as a pressure-transmitting medium because (i) it is one of the softest materials, (ii) it is chemically inert, and (iii) it has no luminescence and no Raman activity. The pressure was determined by the fluorescence shift of a ruby sphere loaded in the sample chamber. 26 The quasi-hydrostatic conditions were controlled throughout the experiments by monitoring the separation and widths of R 1 and R 2 lines. In situ XRD patterns as a function of pressure were acquired at the XRD1 station of the ELETTRA synchrotron radiation facility. This diffraction beamline is designed to provide a monochromatized, high-flux, tunable x-ray source in the spectral range from 4 to 25 keV. 27 The study was performed using a wavelength of 0.068881 nm (E = 18,002.06 eV). The detector was a 345-mm imaging plate from MarResearch. The sample-to- Lorentzian profile. The frequency accuracy was better than 1 cm -1 . IV. RESULTS AND DISCUSSION A. XRD pattern at room temperature and atmospheric pressure For a milling time of 32 hours, the XRD pattern showed an excellent agreement with that given in the ICSD Database 5 (code 41727) for the orthorhombic FeSb 2 phase (s.g. Pnnm, Z=2). In addition, diffraction peaks of unreacted elemental Sb and a broad background under the most intense peaks of the orthorhombic FeSb 2 phase were observed. In order to investigate the origin of this background, a small amount of the asmilled powder was analyzed by the differential scanning calorimetry technique. The first measurement showed an intense exothermic peak located at about 611 K, while in the second this peak was absent. Thus, it was concluded that besides the orthorhombic FeSb 2 phase the milling process generated an amorphous phase. Petrovic et al. 19 using the levitation melting technique produced single crystal of orthorhombic FeSb 2 together with a small content (8%) of unreacted elemental Sb. These results suggest that to obtain a pure FeSb 2 phase a compensation of elemental Sb should be used. In this study, no compensation was made. For 32 hours of milling, the XRD patterns at atmospheric pressure and at 0.7 GPa were similar. In order to decrease the number of figures to be presented in this paper, only the last will be shown. Another paper reporting these results, among others, is in course. All the peaks observed on the XRD pattern of the orthorhombic FeSb 2 phase have their base enlarged, suggesting that the as-milled sample has crystallites with very small dimensions. The mean size of the crystallites was estimated using the equation below, which takes into account the line broadening caused by both crystallite size and strain. 31 2 2 2 2 sin 1 cos                     K d K p t (1) Here,  is the diffraction angle, λ is the X-ray wavelength, β t is the total broadening measured at the full-width at half-maximum (FWHM) of the peak in radians, d is the crystallite size, σ p is the strain, and K is a constant dependent on the measurement conditions and on the definition of β t and d (here K was assumed to be 0.91 as is usual in the Scherrer formula). Graphical linearization of the above relationship, i.e., plotting 2 34 The ratio of the integrated intensity of the contribution of nanometric FeSb 2 crystallites to that of the whole XRD pattern yields a crystalline volume fraction of  63% and, consequently, the interfacial plus amorphous volume fraction is  37%. B. High pressure XRD measurements As mentioned previously, the XRD patterns measured at atmospheric pressure and at 0.7 GPa are similar. The former was indexed to the orthorhombic FeSb 2 phase. We used the software DATLAB code 35 (111) and (200) planes of neon (N), which crystallizes in a f.c.c structure at about 10 GPa. 37 In this study, neon gas was used as pressure-transmitting medium. The synchrotron XRD patterns displayed in Figs. 1 and 2 for the orthorhombic FeSb 2 phase were refined using the Rietveld method, 29,30 and the results are summarized in Table I. Fig. 4 shows the experimental and simulated XRD patterns at 0.7 GPa. From the Table I The volume as a function of pressure V(P) obtained from the Rietveld refinement for the orthorhombic FeSb 2 phase (see Table I) was fitted to a Birch- Murnaghan equation of state (BM EOS): 38                    1 4 4 3 1 2 3 P 3 / 2 ' 0 3 / 5 3 / 7 0 X B X X B(2) where X = (V/V 0 ). The value of V 0 = 122.269 Å 3 was obtained from the Rietveld refinement of XRD pattern measured at ambient temperature and atmospheric pressure. The new high pressure FeSb 2 phase, with a CuAl 2 -type structure, was observed from 14.3 to 23.3 GPa. Its XRD pattern was refined using the Rietveld method, and the results of the fits are summarized in Table II. Fig. 7 shows C. Raman measurements under pressure At room temperature and atmospheric pressure, orthorhombic FeSb 2 crystallizes in the 12 2h D symmetry, and the Raman active modes at the  point of the Brillouin zone are classified according to the irreducible representations of this point group, 42 Figures 8 and 9 show the measured Raman spectra for the nanostructured FeSb 2 powder with increasing and decreasing pressure, respectively. As shown previously from XRD measurements, the as-milled sample contains 7 % of unreacted elemental Sb. g g g g B B B A 3 2 1 3 2 2 2     (4) Elemental Sb crystallizes in the 5 3d D symmetry (s.g. m R  3 , Z=6) and its two Raman active modes are: the g A 1 mode at 150 cm −1 and a two-fold degenerated g E mode at 115 cm −1 . 43 In Fig. 8, these two Raman active modes are marked with an asterisk (*) symbol They are not observed for pressures higher than 5.3 GPa. Lazarevic et al. 44 GPa were fitted to a linear polynomial and the presssure was assumed to be zero. The calculated value was 194 cm -1 . This value confirms the presence of a tetragonal phase, as predicted in Ref. 21, but in nanostructured FeSb 2 powder its nucleation was observed for pressures smaller than 38 GPa. Figure 9 shows that, as the pressure is decreased, the tetragonal phase is observed only for pressures larger than 22.9 GPa, while the orthorhombic phase is observed for smaller pressures. It is important to note that the Raman active Sb modes are not observed after the pressure is removed, suggesting that Sb was incorporated to the amorphous, orthorhombic or tetragonal phase. In order to analyze the Raman spectra, they were deconvoluted using Lorentzian functions, as shown in Fig. 8. For the orthorhombic and tetragonal phases, the pressure dependence of the wave number may be approximated by a standard second order polynomial, as shown in Fig. 10 indicating that the g B 1 mode is more sensitive to the pressure effect, as shown in Fig. 8. The Grüneisen parameter 0  describes the effect of high pressure on the volume of the lattice, and, consequently, on the phonons frequencies. The zero-pressure mode Grüneisen parameters 0  were determined using the equation 46 V. CONCLUSIONS The structural and optical properties of nanostructured orthorhombic FeSb 2 phase formed by the MA technique were studied as a function of pressure, and the main (7) 109.43 (7) Sb (x) 0.1878 (3) (2) Both components have similar number of atoms, but the atoms are distributed in different atomic arrangements in the interfacial component. Thus, the stored energy is larger in the second component. The pressure increase affects first the interfacial component, promoting a gradual elimination of defects and a release of the stored energy. In addition, the atoms of this component located near the interfaces are incorporated in the crystallites, promoting their growth. Only after this process is over the pressure affects the crystalline component, removing strains. Two consequences are easily seen: (1) an improvement in crystallinity, and (2) an increase in value of pressure at which occurs transformation of atmospheric pressure phase to high pressure phase, when compared that of its bulk counterpart. In the case of a transformation from a high pressure phase to another, this effect is not observed, i.e., phase transformations do not occur at lower pressures in the nanosctructures material. For example, Trichês et al. 22 investigated the effect of high pressure on the nanostructured orthorhombic ZnSb phase and observed a phase transformation into an hP1 phase between 11 and 14.6 GPa, while the literature reports a transformation of bulk orthorhombic ZnSb phase into the same phase at 7 GPa. 24 No heat treatment was performed. Due to the fact that the interfacial component consists of different kinds of defects, different regions with different compositions can be present. If there are no strong differences among the enthalpies of formation of phases corresponding to different compositions, more than one phase can be nucleated when pressure is applied. An example will be shown in this study, where the microstructure of the as-milled sample consists of nanostructured orthorhombic FeSb 2 and an amorphous phase. With increasing pressure, nucleation of a new phase as well as an increase in the volume fraction of the amorphous phase were observed, without structural degradation of the orthorhombic phase. detector distance was calibrated by diffraction data from Si powder loaded in the diamond anvil cell. The XRD data were collected at 0.28.2 GPa. An exposure time of 10 min was used for all measurements. The two-dimensional diffraction patterns were converted to intensity versus 2 using the fit2D software28 and analyzed by the Rietveld method 29 using the GSAS package.30 For the Raman measurements as a function of pressure, a particle of approximately 50 x 60 x 20 m 2 was loaded in the DAC. The Raman spectra and ruby luminescence were recorded in the backscattering geometry by means of a Jobin-Yvon T64000 Raman triple spectrometer and a liquid-nitrogen-cooled charge coupled device multichannel detector. An excitation line of  = 514.5 nm of an Ar laser was used for excitation and focused down to 5 µm with a power of about 20 mW at the entrance of the DAC. The Raman spectra were collected at 1.45.2 GPa. An exposure time of 120 min was used for all measurements. The Raman frequencies were determined from a fit of the peaks to a Figures 1 and 2 2show in situ synchrotron XRD patterns of the nanostructured FeSb 2 sample for several pressures up to 28.2 GPa. The initial XRD pattern is seen up to the highest pressure used. With increasing pressure, the diffraction peaks shift toward higher 2 values and those initially located between 2 = 13.5 o and 16 o become well separated. The intensity of these peaks decreases up to 12.6 GPa and increases for higher pressures. No XRD measurements were performed as the pressure was decreased. As mentioned in the previous section, 7 % of unreacted elemental Sb is present in as-milled powder. InFig.1, at 0.7 GPa, these diffraction peaks are marked with an asterisk (*) symbol and are observed up to 9.7 GPa. In all XRD patterns, between 2 = 13.5 o and 17 o , one can see an amorphous phase which reaches maximum intensity at 23.3 GPa. There is no structural degradation of the orthorhombic FeSb 2 phase with increasing pressure. This indicates that the interfacial component is responsible for the increase in the volume fraction of the amorphous phase content, and an explanation is the following: at atmospheric pressure, the volume fraction of the interfacial component and amorphous phase is 37 %. Increasing the pressure promotes compaction, elimination of defects and release of stored energy. The free Fe and Sb atoms diffuse and are incorporated by the amorphous phase. one can see that the pressure dependence of the lattice parameters shows two linear behaviors denoted as regions I (from 0.7 up to 8.1 GPa) and II (from 12.6 up to 28.2 GPa). In region I, change in the lattice parameters is more abrupt. region II. Calculated values for the region I are almost twice those calculated for the region II. In both regions the c-axis is more compressive than the a-and b-axes. Thecalculated values for the region I agree quite well with those reported in Ref. 19, where the highest pressure was 7.14 GPa. A possible explanation for the differences in the two regions is the following: the microstructure of the as-milled sample includes crystalline and interfacial components plus an amorphous phase. With increasing pressure up to 8.1 GPa, the volume fraction of the interfacial component decreases due to partial elimination of the several kinds of defects and crystallite growth through the incorporation of atoms located near the boundaries of the crystallites. Thus, the effect of high pressures is to improve the crystallinity of the sample, leaving it energetically more stable 22 and more like bulk FeSb 2 . On the other hand, from 12.6 up to 28.2 GPa (see Figs. 2 and 3) an important increase in the amount of the amorphous phase and a new crystalline phase are observed. As a structural degradation of the orthorhombic FeSb 2 phase is not observed, the different behavior in the individual lattice parameter compressibility is attributed to an increase in the volume fraction of the amorphous phase as well as to the presence of the new high pressure phase. pressure derivative 0 B = 7.5  0.6, as shown in Fig. 5. For bulk FeSb 2 , Petrovic et al. 19 reported the values of 0 B = 84(3) GPa and 0 B = 5(1), and their data are also shown in Fig. 5 (open stars), while Wu et al. 21 reported the values 0 B = 94 GPa and 0 B = 4.9 and 0 B = 68 GPa and 0 B = 5.9 for FeSb 2 crystallized in Pnnm and I4/mcm structures, respectively. According to Fecht 39 the bulk modulus 0 B for nanometric metals decreases with increasing volume fraction of the interfacial component. Thus, the difference between 0 B values obtained in this study and those reported in Refs. 19 and 21 can be due to the fact that our sample is nanostructured. At 14.3 GPa, the XRD pattern shows the presence of a new high pressure phase besides those described previously. Takizawa et al. 20 reported that a new high-pressure CrSb 2 phase crystallized in a CuAl 2 structure-type crystal formed above 5.5 GPa, with metallic bond nature including the formation of Cr-Cr-Cr linear chain along the c-axis. Wu et al. 21 predicted a FeSb 2 orthorhombic-tetragonal phase transition at 38 GPa. In addition, it known that the TiSb 2 , VSb 2 , MnSn 2 , CoSn 2 , FeSn 2 and CuAl 2 compounds crystallize in CuAl 2 structure type. 5 The diffraction peaks of this new phase showed a good agreement with those given for the TiSb 2 , VSb 2 and CrSb 2 compounds. The lattice parameters a and c were calculated using the classical , k and l are the Muller indexes. The peaks located at about 2 = 11.55 o , 14.96 o and 18.31 o were indexed to the (200), (211) and (310) planes, respectively. The peak associated with the (211) plane is represented by a shoulder at the right of the peak located at about 2 = 14.78 o associated to the (101) plane of the orthorhombic phase. The calculated values were a = 6.8441 Å (6.555 Å) and c = 5.2563 Å (5.631 Å). The numbers between parentheses are those given in the JCPDS Database for the VSb 2 phase (card No. 250055). Fig. 6 shows a comparison between the XRD patterns for VSb 2 (bottom curve) and our measurements at 14.3 GPa (top curve), where the peaks of the new FeSb 2 phase are observed. In order to improve the comparison, the pattern of the VSb 2 phase was multiplied by an arbitrary factor. The difference in the intensities of the peaks located at about 2 = 11.55 o and 17.14 o shows a texture in the new phase, which was taken into account during the Rietveld analysis. the experimental and simulated XRD patterns at 14.3 GPa. Of course, the orthorhombic and tetragonal FeSb 2 phases were used in the simulation. In this figure, the 2 range was reduced in order to show the details. Despite the small amount of the tetragonal phase, the peak located at about 2 = 14.78 o can not be simulated with any accuracy if this phase is considered. The results showed that the lattice parameters a and c decrease linearly with increasing pressure. The interatomic distances of Fe-Fe and Fe-Sb also decrease with increasing pressure. Individual lattice parameter compressibilities were calculated, and the values were β a = 0.00105(6) GPa -1 and β c = 0.00245(5) GPa -1 . As in the case of the orthorhombic phase, the c-axis is more compressive than the a-and b-axes. Armbruster et al. 41 investigated the compressibility of tetragonal TiSb 2 under high pressure up to 12 GPa and found a faster decrease of the c parameter in comparison to a, and thus a decreasing c/a ratio with increasing pressure. The results obtained in this study are consistent with their results. DA 1 E 1reported the Raman spectrum of the orthorhombic FeSb 2 phase at room temperature, and the wave numbers are: GPa, one new Raman active mode at about 233 cm −1 emerges, while the Raman active 2 g A mode of the orthorhombic FeSb 2 phase enlarges. With increasing pressure up to 45.2 GPa, both modes shift to higher wave numbers and the 2 g A mode becomes broader and less intense, while the emergent mode becomes sharper and more intense. According to the XRD results between 14.3 and 23.3 GPa, a new tetragonal phase is present. For the isostructural compounds listed above with the CuAl ), the Raman active modes at the  point of the Brillouin zone are classified according to the irreducible representations of this point group 42 al.,41,45 reported the Raman spectra at room temperature for the TiSb 2 and VSb 2 compounds, in form of powders and single crystals. For TiSb 2 powders, the wave numbers of Raman active modes are g mode, the wave numbers of modes decrease with increasing atomic number of the transition metal. Thus, it is expected that for tetragonal FeSb 2 the wave numbers of active modes are smaller than for the VSb 2 powder. InFig. 8one can see that the Raman active 2 g A mode of the orthorhombic FeSb 2 phase and the g A 1 and g B 2 of the tetragonal FeSb 2 phase are very close and it is difficult to separate them for pressures higher than 21 GPa. On the other hand, the Raman active 2 g E mode of the tetragonal phase is well separated. In order to obtain its wave number at room temperature and atmospheric pressure, the maxima of peaks on the Raman spectra between 21 and 45.2 are the bulk modulus in GPa and the wave number in cm -1 at zero pressure. From the XRD measurements a value of 0 B = 74.2  3.0 GPa was obtained. In order to evaluate the effect of high pressure on the Raman active g A and g B 1 modes, the 0  value was calculated in the same pressure range (up to 12.7 GPa). Using 0 B and 0  values in Eq. (7), the 0  value for the g A and g B 1 modes are 0.86 and 0.94, respectively. From these values one can see that the pressure affects much more the g B 1 mode than the g A one as shown in Fig. 8. For the tetragonal phase, the 0  value for the Raman active 2 g E mode can be estimated by considering the estimated wave number of 194 cm -1 and 0 B = 68 GPa reported by Wu et al. 21 for this phase. The calculated value is 1.33. GFigure 1 ( 1color online): XRD patterns measured with increasing pressure up to 9.7 GPa for nanostructured orthorhombic FeSb 2 powder. The diffraction peaks of elemental Sb are marked with the asterisk (*) symbol, while those from the gasket are identified by the G letter. Figure 2 ( 2color online): XRD patterns measured with increasing pressure from 12.6 up to 28.2 GPa for the nanostructured orthorhombic FeSb 2 powder. The diffraction peaks from the gasket and from neon are identified by the G and N letters. Figure 3 ( 3color online): Estimated XRD patterns for the amorphous phase for several pressures. Figure 4 (Figure 5 ( 45color online): XRD pattern of the orthorhombic FeSb 2 phase at 0.7 GPa (solid black line). Other colored solid lines represent the Rietveld simulations. The bottom line is the residual intensity. color online): Pressure dependence of the volume of the nanostructured orthorhombic FeSb 2 phase deduced from Rietveld refinements. Present study (full circles) and results from Ref. 19 (open stars). The solid line is the fit to a Birch-Murnaghan equation of state. Figure 6 ( 6color online): XRD patterns for the nanostructured FeSb 2 powder measured at 14.3 GPa (top curve) and simulated for the tetragonal VSb 2 phase using the calculated lattice parameters given in the text (bottom curve). Figure 7 (Figure 8 (Figure 9 (Figure 10 ( 78910color online): XRD patterns of nanostructured FeSb 2 powder measured at 14.3 GPa (open circles) and simulated (red solid line). The bottom line is the color online): Raman spectra measured with increasing pressure up to 45.2 GPa for the nanostructured orthorhombic FeSb 2 powder. The excitation wavelength was  = 514.5 nm. Raman active modes from unreacted antimony are marked with the asterisk (*) symbol. The red and green full lines represent the deconvolution process (see text). color online): Raman spectra measured with decreasing pressure for the nanostructured orthorhombic FeSb 2 powder. The excitation wavelength was  = 514color online): Pressure dependence of the Raman active modes of nanostructured orthorhombic and tetragonal FeSb 2 phases up to 45.2 GPa. The symbols represent the experimental data and the lines are polynomial fits (see Eq (6) in the text). Figure 11 : 11Derivatives of analytical expressions (6) representing the wave number values measured at several pressures.   yields the mean crystallite size free from strain effects from the values of the intercept of the straight line obtained, as well as the strain obtained from the slope. From the Rietveld refinement, the generated β t and 2θ positions values were used in Eq. (1), and the calculated mean crystallite size and strain were d ≈ 26 nm and σp ≈ 0.4 %. The mean crystallite size shows that the as-milled FeSb 2 powder has a nanometric structure. From the Rietveld refinement, 7 % of unreacted elemental Sb was calculated. Recently, we developed an approach to estimate the volume fractions of crystalline and interfacial components. 32,33 and it was used to estimate the contribution of nanometric FeSb 2 crystallites to the XRD pattern of the as-milled powder. For this, the measured intensity was corrected for polarization and reabsorption, and converted to electron units using the mean square scattering factor <f 2 > of FeSb 2 analytically calculated. After that the inelastic scattering was subtracted. The contribution of the interfacial component to the XRD pattern is diffuse. When an amorphous phase is present, its contribution overlaps that of the interfacial component, making impossible to distinguish the contributions of each. An evaluation of the interfacial component contribution to the normalized XRD pattern and its subtraction yield the contribution of nanometric FeSb 2 crystallites. The interfacial component contribution was estimated using the Origin software.2 2 / cos    t versus , / sin 2 2 GPa. The increase in diffuse scattering for angles larger than 2 = 18 o can be attributed to the Fe K α and/or K β fluorescence generated during the measurements, since the photon energy was 18,002.06 eV and the Fe K-edge is 7112to estimate the volume fraction of the amorphous phase at 6.6, 14.3, 23.3 and 28.2 GPa, as shown in Fig. 3. The intensity of main halo is maximum at 23.3 eV. We used the Ehrenfest relation   Esin r  , 36 where the structure dependent constant E was taken to be 1.671 and  is the wavelength used in the experiments, to estimate the interatomic distance for the first neighbors. The values were 3.20, 3.12, 3.05 and 2.95 Å for 6.6, 14.3, 23.3 and 28.2 GPa, respectively. These values decrease with increasing pressure, as expected, and are compatible with the interatomic distances found in the Fe-Sb alloys. The peak at about 2 = 20.8 o in the XRD pattern for 9.7 GPa, which was also seen in the patterns for 14.3, 15.6 and 19.1 GPa, is attributed do the gasket, confirmed by patterns taken without any sample. At 14.3 GPa, besides the diffraction peaks of the orthorhombic phase, other low intensity diffraction peaks were observed at about 2 = 11.5 o and 18.4 o , indicating the presence of a new phase. As there is no structural degradation of the orthorhombic and amorphous phases with increasing pressure, it is concluded that this new phase was nucleated in the interfacial component. In the same XRD pattern there are two diffraction peaks at about 2 = 19.7 o and 22.8 o that were indexed to .The effect of high pressure on the Raman active modes can be better understoodOrthorhombic phase 2 2 008 . 0 082 . 2 279 . 159 ) ( P P A P g     2 1 1 038 . 0 363 . 2 622 . 183 ) ( P P B P g     (6) Tetragonal phase 2 2 028 . 0 817 . 3 003 . 165 ) ( P P E P g     by considering the derivative of Eq. (6): 2 d dP A BP    . Fig. 11 shows the derivative of analytical expressions (6) obtained from fits for 2 g A and 1 1g B modes. One can see that the derivative of Raman active g B 1 mode varies faster than the g A mode, GPa. The wave numbers of the other Raman active modes are very close to the values of the orthorhombic phase, making difficult their assignment. This study was supported by the Brazilian-French CAPES/COFECUB Program (Project No. 559/7), which has supported one of the authors (S.M.S). However, these data were not included in his Ph.D thesis. Thus, the analysis of the data was carried out by one of the authors (C.M.P) and the results will be part of his Ph.D thesis. We thank to the ELETTRA synchrotron (Italy) for the XRD measurements as a function of pressure. We are indebted to the high pressure group from IMPMC, in particular to Pascal Munch and Gilles Le Marchand for technical support. We are indebted to Drs. Altair Sória Pereira and Ronaldo Sérgio de Biasi for discussions and contributions. *Present adress: Departamento de Física, Universidade Federal do Amazonas, 3000 Japiim, 69077-000 Manaus, Amazonas, Brazil.conclusions are: 1) the XRD results show evidence for an orthorhombic FeSb 2 phase up to 28.2 GPa. 2) There is an increase in the volume fraction of an amorphous phase up to 23.3 GPa. 3) For pressures between 14.3 and 23.3 GPa, a tetragonal FeSb 2 phase is observed. Due to its small amount and no structural degradation of the orthorhombic phase, it is assumed that nucleation of the tetragonal phase occurred in the interfacial component; 4) The Raman active g A mode of the orthorhombic phase is observed up to the maximum pressure used and, from 21.0 to 45.2 GPa, a new Raman active 2 g E mode is observed and attributed to the tetragonal phase, which was seen in the XRD measurements between 14.3 and 23.3 ACKNOWLEDGMENTS 12 14 16 18 20 22 24 26 * * 9.7 GPa 8.1 GPa 6.6 GPa 5.3 GPa 3.7 GPa 2.2 GPa Intensity (arb. units) 2 (degrees) 0.7 GPa Table I : IStructural data for the nanostructured orthorhombic FeSb 2 phase. Fe-Sb x 2 (Å) 2.4664(2) 2.4648(2) 2.4506(4) 2.4445(8) 2.4338(1)P (GPa) 14.3 15.6 17.6 19.1 21 23.3 28.2 a (Å) 5.6564 5.6501 5.6234 5.6079 5.5931 5.5776 5.5325 b (Å) 6.2990 6.2841 6.2708 6.2479 6.2248 6.2027 6.1577 c (Å) 3.0392 3.0334 3.0191 3.0100 2.9982 2.9810 2.9548 V (Å 3 ) 108.28(5) 107.70(3) 106.46(3) 105.46(3) 104.38(5) 103.13(1) 100.66(2) Sb (x) 0.1946(1) 0.1947(1) 0.1827(2) 0.1952(9) 0.1990(3) 0.1937(8) 0.1899(8) Sb (y) 0.3504(3) 0.3509(5) 0.3548(5) 0.3497(5) 0.3525(8) 0.3517(7) 0.3504(1) 2.4351(7) 2.4001(0) Fe-Sb x 4 (Å) 2.4862(2) 2.4804(5) 2.5082(9) 2.4628(1) 2.4609(5) 2.4461(0) 2.4439(3) Fe-Sb-Fe ( o ) 130.48(6) 130.40(4) 128.88(7) 130.65(4) 130.47(2) 130.29(0) 130.25(6) P (GPa) 14.3 15.6 17.6 19.1 21 23.3 a (Å) 6.8441 6.8315 6.8227 6.8032 6.7985 6.7761 c (Å) 5.2563 5.2328 5.1902 5.1569 5.1310 5.1120 V (Å 3 ) 246.21(4) 244.21(1) 241.59(9) 238.67(9) 237.15(2) 234.72(0) Sb (x) 0.1233(1) 0.1350(2) 0.1350(1) 0.1349(1) 0.1558(8) 0.1677(2) Sb (y) 0.6233(1) 0.6350(2) 0.6350(1) 0.6349(1) 0.6558(8) 0.6677 H. 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Fecht, Acta Metall. Mater. 38, 1927 (1990). B D Cullity, Elements of X-ray Diffraction. Addison-Wesley Publishing Company, IncSecond Edition. Appendix 3B. D. Cullity, Elements of X-ray Diffraction, Second Edition, Addison-Wesley Publishing Company, Inc., 1978, Appendix 3. . M Armbruster, W Schnelle, U Schwarz, Y Grin, Inorg. Chem. 466319M. Armbruster, W. Schnelle, U. Schwarz, and Y. Grin, Inorg. Chem. 46, 6319 (2007). . R N Zitter, J. Phys. Chem. Solids. 32285Suppl. 1R. N. Zitter, J. Phys. Chem. Solids 32 (Suppl. 1), 285 (1970); RRUFF Database, available on the internet at the site. RRUFF Database, available on the internet at the site http://rruff.info/. . N Lazarević, Z V Popović, Rongwei Hu, C Petrovic, Phys. Rev. 8014302N. Lazarević, Z. V. Popović, Rongwei Hu, and C. Petrovic, Phys. Rev. 80, 014302 (2009); . N Lazarević, M M Radonjić, D Tanasković, Rongwei Hu, C Petrovic, Z , N. Lazarević, M. M. Radonjić, D. Tanasković, Rongwei Hu, C. Petrovic, and Z. . V Popović, arXiv:1108.0581v1[cond-mat.mtrl-sci]1V. Popović, arXiv:1108.0581v1 [cond-mat.mtrl-sci] 1 Aug 2011. . Y Grin, F R Wagner, M Armbruster, M Kohout, A Leithe-Jasper, U Schwarz, U Wedig, H G Schnering, J. Solid State Chem. 1791707Y. Grin, F. R. Wagner, M. Armbruster, M. Kohout, A. Leithe-Jasper, U. Schwarz, U. Wedig, and H. G. Schnering, J. Solid State Chem. 179, 1707 (2006). . M Blackman, Proc. Phys. Soc. London, Sect. B. 70827M. Blackman, Proc. Phys. Soc. London, Sect. B 70, 827 (1957). W B Daniels, Lattice Dynamics. R. F. WallisPergamon, OxfordW. B. Daniels, in Lattice Dynamics, edited by R. F. Wallis (Pergamon, Oxford, 1965), Table II: Structural data for the tetragonal FeSb. 2Table II: Structural data for the tetragonal FeSb 2 phase.
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[ "The 1st Fermi LAT SNR Catalog: Probing the Gamma-ray Population", "The 1st Fermi LAT SNR Catalog: Probing the Gamma-ray Population" ]
[ "J W Hewitt \nCRESST/University of Maryland\nBaltimore County\n21250BaltimoreMDUSA\n\nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA\n", "F Acero \nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA\n", "T J Brandt \nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA\n", "J Cohen \nDepartment of Physics\nDepartment of Astronomy\nUniversity of Maryland\n20742College ParkMDUSA\n", "F De Palma \nINFN Sezione di Bari\n70126Italia\n", "F Giordano \nINFN Sezione di Bari\n70126Italia\n" ]
[ "CRESST/University of Maryland\nBaltimore County\n21250BaltimoreMDUSA", "NASA Goddard Space Flight Center\n20771GreenbeltMDUSA", "NASA Goddard Space Flight Center\n20771GreenbeltMDUSA", "NASA Goddard Space Flight Center\n20771GreenbeltMDUSA", "Department of Physics\nDepartment of Astronomy\nUniversity of Maryland\n20742College ParkMDUSA", "INFN Sezione di Bari\n70126Italia", "INFN Sezione di Bari\n70126Italia" ]
[]
While supernova remnants (SNRs) are widely thought to be powerful accelerators, evidence comes largely from a small number of well-studied cases. Here we systematically determine the γ-ray emission from all known Galactic SNRs, disentangling them from the sea of Galactic cosmic rays. Using Fermi LAT data we have characterized the GeV emission in all regions containing SNRs, accounting for systematic uncertainties caused by source confusion, diffuse emission modeling, and instrumental response. More than a dozen remnants are identified through spatial extension or detection at TeV energies, with potential associations for >40 more. From this population study, two clear classes of γ-ray-emitting SNRs emerge: young remnants and those interacting with a dense medium. This large statistical sample also reveals a possible correlation between GeV and radio flux. The growing number of identified SNRs will help to disentangle the effects of age and environment on the aggregate properties of SNRs at high energies.
null
[ "https://export.arxiv.org/pdf/1307.6570v1.pdf" ]
119,230,754
1307.6570
594fc1fad56accee726a332f981c02500b908fdf
The 1st Fermi LAT SNR Catalog: Probing the Gamma-ray Population J W Hewitt CRESST/University of Maryland Baltimore County 21250BaltimoreMDUSA NASA Goddard Space Flight Center 20771GreenbeltMDUSA F Acero NASA Goddard Space Flight Center 20771GreenbeltMDUSA T J Brandt NASA Goddard Space Flight Center 20771GreenbeltMDUSA J Cohen Department of Physics Department of Astronomy University of Maryland 20742College ParkMDUSA F De Palma INFN Sezione di Bari 70126Italia F Giordano INFN Sezione di Bari 70126Italia The 1st Fermi LAT SNR Catalog: Probing the Gamma-ray Population 33RD INTERNATIONAL COSMIC RAY CONFERENCE, RIO DE JANEIRO 2013 THE ASTROPARTICLE PHYSICS CONFERENCEsupernova remnantsgamma rayscosmic ray acceleration While supernova remnants (SNRs) are widely thought to be powerful accelerators, evidence comes largely from a small number of well-studied cases. Here we systematically determine the γ-ray emission from all known Galactic SNRs, disentangling them from the sea of Galactic cosmic rays. Using Fermi LAT data we have characterized the GeV emission in all regions containing SNRs, accounting for systematic uncertainties caused by source confusion, diffuse emission modeling, and instrumental response. More than a dozen remnants are identified through spatial extension or detection at TeV energies, with potential associations for >40 more. From this population study, two clear classes of γ-ray-emitting SNRs emerge: young remnants and those interacting with a dense medium. This large statistical sample also reveals a possible correlation between GeV and radio flux. The growing number of identified SNRs will help to disentangle the effects of age and environment on the aggregate properties of SNRs at high energies. Introduction A key question that γ-ray astronomy seeks to answer is the origin of Galactic cosmic rays. The current generation of ground-and space-based γ-ray observatories allows for detecting the extension of numerous suspected accelerators, including SNRs. Emission from SNRs in the GeV energy range gives an important window into the emission mechanism, and total energetics of accelerated particles. Previous searches of bright radio SNRs have revealed a small number of possible counterparts [1], but only with the Fermi LAT telescope has the sensitivity been obtained over the full-sky to detect a significant fraction of Galactic SNRs. Fermi LAT has identified a number of SNRs (see [2] for a review), but their emission has been largely explored on a source-by-source basis. The near-uniform sky coverage allows a systematic exploration of both detections and nondetections of the hundreds of known SNRs in our Galaxy. Here we present our ongoing effort to systematically explore γ-ray emission from SNRs for the First Fermi LAT Supernova Remnant Catalog. We broadly detail the analysis pipeline ( §2), discuss the classification of detected sources ( §3) and highlight one interesting aspect of the catalog, a direct comparison between GeV and radio emission ( §4). Detection Pipeline Description Gamma-ray Data The Fermi LAT is a pair-conversion γ-ray telescope detecting photons from 20 MeV to > 300 GeV [3]. Our catalog is constructed from 3 years of LAT survey data and the Pass7v6 instrument response functions (IRFs). For each of the 278 SNRs identified in Green's catalog [4] we modeled emission within a 10 • radius of interest (ROI) from the SNR center. As a compromise between sensitivity, spatial resolution to resolve extension and to separate the SNR from the diffuse background, we choose 1 GeV as our minimum energy threshold. Only source class events are selected with energies of 1 to 100 GeV. To analyze the data we fit models using the maximum likelihood framework. Tools utilized include both the standard science tools 1 and the pointlike analysis package [6] which has been specifically developed and verified for characterizing source-extension for Fermi LAT data [7]. Initial Source Model In order to characterize each SNR, we must obtain an optimal characterization of γ-ray emission in the ROI that includes all significant sources of emission. To do so, we developed an automated analysis pipeline briefly described here. We start from the standard models of diffuse emission and a list of identified sources in the Second Fermi LAT catalog (2FGL) [5]. Using pointlike we generate a map of source test statistic (TS) for each 0.1 • bin covering the entire ROI. Here the source TS is defined as twice the logarithm of the ratio between the likelihood L 1 obtained by fitting the source model plus background components (including other sources) to the data, and the likelihood L 0 obtained by fitting the background components only, i.e., TS = 2 log(L 1 /L 0 ). At the position of the peak TS value we add a new point source with a power-law spectral model, perform a likelihood fit of the region, and localize the position of the newly added source (only the first time it is added). This iterative process is continued within the specified region of interest until there are no remaining sources which change in the global log-likelihood by more than 8. This threshold leads to the detection of all TS ≥ 25 sources. Our final step is a removal of all sources with TS < 25 from the final model. Source Localization and Extension Many γ-ray sources are detected coincident with the position of known SNRs, however this is not sufficient to make an identification. The detection of spatial extension remains the best way to determine that γ-ray emission originates from the SNR. The spatial resolution of the LAT is sufficient to detect many SNRs as extended. Figure 1 shows the distribution of radio sizes from Green's catalog. The 68% containment radius at 1 and 10 GeV are indicated as vertical dashed lines that roughly approximate the threshold for the detection of SNRs if they are sufficient bright γ-ray sources. Roughly a third of all SNRs may potentially be resolved by the LAT. For each SNR we use our analysis pipeline to characterize the morphology and spectrum of any γ-ray emission that may be coincident with the SNR location (as defined by the position and extent in the radio reported in Green's catalog). All sources which fall within the radio radius are removed from the model, unless the source has been previously identified as not an SNR (e.g. pulsars). Several hypotheses are then explored in parallel using pointlike. For the point source hypothesis, a point source is placed at the radio centroid of the SNR. Sources within 5 • of the SNR center are fit with the normalization left free but the spectral index fixed. For the disk hypothesis, a uniform disk equal in radius to the radio size is placed at the SNR center. The disk normalization, index, position and extension are fit. In a separate hypothesis, we will also test the significance of sources which are adjacent to the SNR disk. We will determine whether a nearby source is kept or removed from the final model by defining TS nearby as twice the difference between the model with the nearby source and the model without the source, in which the extension and position of the disk are refit. A nearby source is significant (and thus kept in the final model) if TS nearby ≥ 9. Once these hypotheses have been evaluated, we compare the global log-likelihoods of all the resulting models to determine which model gives the most significant representation of the data. A source is considered possibly associated with an SNR if a point source with TS > 25 is detected within the radio extent of the SNR. To determine whether a source has a significant extension we define TS ext as twice the difference between the log-likelihood of the final model from the disk hypothesis and that of the point-source hypothesis. A SNR is considered to have a significant extension if TS ext ≥ 16 as in [7]. The detection of GeV extension in agreement with the measured extension of the remnant at other wavelengths provides a secure means of identifying the SNR in γ-rays. Fluxes and Upper Limits Fluxes are determined in the 1-100 GeV band given the best, significant characterization of the morphology and spectra of the SNR from the pipeline detailed above. For those SNRs which are not significantly detected we compute upper limits on the flux by assuming a spectral model with a power-law index of 2.5, consistent with the majority of SNRs detected. As a spatial model we use the uniform disk equal in position and radius to that reported in Green's catalog. We then calculate an upper limit on the flux from the disk without including any overlapping sources in the model which have not been firmly identified. The vertical dashed lines indicates the 68% containment radius of the LAT at 1 and 10 GeV for front-converting events, which is roughly equivalent to the limit at which bright sources can have detectable extensions. Sources of Error In addition to statistical error, we will account for several sources of systematics in our catalog. The two main sources of systematics are uncertainties in the model of Galactic diffuse emission, and in the effective area calibration. Uncertainties in the effective areas are estimated using modified IRFs that bracket the nominal ones. Characterizing uncertainties in the diffuse model required the construction of alternative diffuse models, and is detailed in [8]. Detections reported in the final catalog will be required to be robust against these sources of systematic error. SNR Identification and Emerging Classes Using the pipeline described above, we detected 44 SNRs at energies of >1 GeV. Of these, 15 are extended (with 6 being new detections), 4 are spatially unresolved SNRs which we identify based on TeV detections, and 25 are candidate associations which do not show significant extension. This brings the total number of identified SNRs at GeV energies to 19, which is sufficient to begin to explore properties of the population. Among the point-like SNRs which cannot be firmly identified, we attempt to qualitatively classify whether the GeV source is more likely to be of pulsar-or SNR-origin. We examined archival X-ray data to search for evidence of a possible point source counterpart. Those which do not contain a known pulsar (in radio or X-rays) and do not have an identified X-ray point source within their interiors we classify as favorable SNR-candidates, while the others are denoted as unfavorable SNR-candidates. We note that this classification is only a qualitative assessment, useful to examine whether differences are apparent between those point-like SNR candidate associations. From our sample of GeV-identified SNRs, two clear classes emerge. The largest class is those SNRs known to be interacting with molecular clouds, which are typically quite luminous at GeV energies. In contrast, the few known young SNRs are less luminous, with harder spectra and TeV-detections. These two classes are clearly separated in Figure 2 which plots the 1-100 GeV luminosity against The luminosity of detected SNRs at 1-100 GeV energies plotted as a function of the radio diameter squared, which is a function of both the age of the SNR and the density of the environment into which it is expanding. Error bars include statistical uncertainties from fitting the GeV data as well as errors in the published distances. We use colors to indicate this classification as follows: identified young SNRs in blue, identified interacting SNRs in red, newly identified SNRs in green, favorable candidate SNRs in dark grey, and unfavorable candidate SNRs in light grey. the physical diameter squared, which acts as a proxy for the evolutionary state of the SNR (more evolved SNRs will have larger diameters). Some SNRs are not included because they do not yet have distance estimates. Detected SNR luminosities span more than two orders of magnitude. The newly identified SNRs appear as a lower luminosity extension of the interacting SNR class. GeV-Radio Correlation A correlation between the GeV and radio flux from SNRs may be expected, as both result from nonthermal emission of relativistic particles. It has been noted that interacting SNRs are preferentially radio-bright [9]. The presence of a large target mass for recently accelerated or escaping cosmic rays by the SNR would increase the γ-ray emission via π 0 -decay or electron bremsstrahlung emission. Figure 3 compares the 1 GHz radio flux density to the 1-100 GeV photon flux for all SNRs in our catalog. We include upper limits for those SNRs which do not have any coincident GeV emission detected. For most identified SNRs (red, green) and those which we deem more likely to be SNRs (dark grey) a clear trend is apparent. Young SNRs do not appear to follow this trend, perhaps indicating a different emission mechanism. We note that some SNRs which fall below this correlation, having fainter GeV fluxes than expected given their radio flux density, also appear to have softer GeV power-law indices. Many SNRs detected by Fermi LAT at GeV energies are not detected at TeV-energies due to spectral curvature around GeV energies. Deviations from this correlation may reflect processes related to the acceleration and escape of cosmic rays from the SNRs. Summary We present a systematic survey of GeV emission from all known Galactic SNRs using 3-years of data from Fermi LAT. Our automated pipeline characterizes emission in all regions containing SNRs in Green's catalog, accounting for systematic uncertainties caused by source confusion, diffuse emission modeling, and instrumental response. We have identified 19 source as SNRs and 25 sources as possible associations. From this population of detected remnants, we can clearly distinguish a dominant class of SNRs interacting with a dense medium, and a less numerous class of young SNRs. The large sample of SNRs reveals a possible correlation between GeV and radio flux. The growing number of identified SNRs promises to help disentangle the mechanism of γ-ray emission from SNRs and thereby the energetics transferred into the acceleration of cosmic rays. Fig. 1 : 1Comparison of the radio diameter in Green's catalog. Fig. 2 : 2Fig. 2: The luminosity of detected SNRs at 1-100 GeV energies plotted as a function of the radio diameter squared, which is a function of both the age of the SNR and the density of the environment into which it is expanding. Error bars include statistical uncertainties from fitting the GeV data as well as errors in the published distances. We use colors to indicate this classification as follows: identified young SNRs in blue, identified interacting SNRs in red, newly identified SNRs in green, favorable candidate SNRs in dark grey, and unfavorable candidate SNRs in light grey. Fig. 3 : 3Comparison of Radio vs. GeV flux including upper limits. See the caption of Fig. 2 for a description of colors. . Available at the Fermi Science Support Center: http://fermi. gsfc.nasa.gov/ssc Acknowledgment:The Fermi LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. 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[ "Outage Performance Analysis of Full-Correlated Rayleigh MIMO Channels", "Outage Performance Analysis of Full-Correlated Rayleigh MIMO Channels" ]
[ "Huan Zhang ", "Guanghua Yang \nSchool of Intelligent Systems Science and Engineering\nJinan University\nZhuhaiChina\n", "Zheng Shi \nSchool of Intelligent Systems Science and Engineering\nJinan University\nZhuhaiChina\n", "Shaodan Ma ", "Hong Wang \nSchool of Communication and Information Engineering\nNanjing University of Posts and Telecommunications\nNajingChina\n", "\nDepartment of Electrical and Computer Engineering\nUniversity of Macau\nChina\n" ]
[ "School of Intelligent Systems Science and Engineering\nJinan University\nZhuhaiChina", "School of Intelligent Systems Science and Engineering\nJinan University\nZhuhaiChina", "School of Communication and Information Engineering\nNanjing University of Posts and Telecommunications\nNajingChina", "Department of Electrical and Computer Engineering\nUniversity of Macau\nChina" ]
[]
The outage performance of multiple-input multipleoutput (MIMO) technique has received intensive attention to meet the stringent requirement of reliable communications for 5G applications, e.g., mission-critical machine-type communication (cMTC). To account for spatial correlation effects at both transmit and receive sides, the full-correlated Rayleigh MIMO fading channels are modeled according to Kronecker correlation structure in this paper. The outage probability is expressed as a weighted sum of the generalized Fox's H functions. The simple analytical result empowers asymptotic outage analysis at high signal-to-noise ratio (SNR), which not only reveal helpful insights into understanding the behavior of fading effects, but also offer useful design guideline for MIMO configurations. Particularly, the negative impact of the spatial correlation on the outage probability is revealed by using the concept of majorization, and the asymptotic outage probability is proved to be a monotonically increasing and convex function of the transmission rate. In the end, the analytical results are validated through extensive numerical experiments.
10.1109/iccc49849.2020.9238828
[ "https://export.arxiv.org/pdf/2209.12977v1.pdf" ]
226,854,705
2209.12977
fa6d4f394a41e7883ee6146bbfaccf23784c95bf
Outage Performance Analysis of Full-Correlated Rayleigh MIMO Channels Sep 2022 Huan Zhang Guanghua Yang School of Intelligent Systems Science and Engineering Jinan University ZhuhaiChina Zheng Shi School of Intelligent Systems Science and Engineering Jinan University ZhuhaiChina Shaodan Ma Hong Wang School of Communication and Information Engineering Nanjing University of Posts and Telecommunications NajingChina Department of Electrical and Computer Engineering University of Macau China Outage Performance Analysis of Full-Correlated Rayleigh MIMO Channels Sep 20221Index Terms-Asymptotic analysisMellin transformMIMOOutage probabilitySpatial correlation The outage performance of multiple-input multipleoutput (MIMO) technique has received intensive attention to meet the stringent requirement of reliable communications for 5G applications, e.g., mission-critical machine-type communication (cMTC). To account for spatial correlation effects at both transmit and receive sides, the full-correlated Rayleigh MIMO fading channels are modeled according to Kronecker correlation structure in this paper. The outage probability is expressed as a weighted sum of the generalized Fox's H functions. The simple analytical result empowers asymptotic outage analysis at high signal-to-noise ratio (SNR), which not only reveal helpful insights into understanding the behavior of fading effects, but also offer useful design guideline for MIMO configurations. Particularly, the negative impact of the spatial correlation on the outage probability is revealed by using the concept of majorization, and the asymptotic outage probability is proved to be a monotonically increasing and convex function of the transmission rate. In the end, the analytical results are validated through extensive numerical experiments. Internet-of-Things (IoT) applications (e.g., automated transportation, industrial control and augmented/virtual reality), ultra-reliable low-latency communications (URLLC) and tactile Internet [5]. Since the outage probability is frequently used to characterize the reception reliability, the outage probability of MIMO systems also has attracted considerable attention in the literature [2], [6]- [8]. However, the prior works in [2], [6]- [8] did not take into account the correlation between antenna elements, which exists in realistic propagation environments because of mutual antenna coupling and close spacing between adjacent elements [9]. The spatial correlation would remarkably impair the reliability of MIMO systems. In [10], the character expansion method was initially introduced to give a closed-form expression for the moment-generating function (MGF) of the capacity under full-correlated (correlation at both the transmitter and receiver) Rayleigh MIMO channels if the numbers of transmit and receive antennas are identical. The same method was further extended to derive the MGF of the capacity for the case with arbitrary numbers of transmit and receive antennas in [11]. Unfortunately, the outage probability for full-correlated Rayleigh MIMO systems was obtained in the literature by relying upon either approximations or numerical inversions, which impede the extraction of more helpful insights about the system parameters. Finally, the analytical results are verified by numerical analysis. To address the above issues, The Mellin transform is applied in this paper to derive exact and tractable representations for the outage probabilities. Upon the exact expressions, the asymptotic analysis of the outage probability in the high signal-to-noise ratio (SNR) regime is derived. The expression demonstrates that full diversity can be achieved regardless of the presence of spatial correlation. The qualitative relationship between the spatial correlation and the outage probability is established by virtue of the concept of majorization in [9]. Moreover, the transmission rate affects the outage performance via the term of modulation and coding gain, and the asymptotic outage probability is found to be an increasing and convex function of the transmission rate. Notations: We shall use the following notations throughout the paper. Bold uppercase and lowercase letters are used to denote matrices and vectors, respectively. A H , A −1 and A 1/2 denote the conjugate transpose, matrix inverse and Hermitian square root of matrix A, respectively. vec, tr, det and diag are the operators of vectorization, trace, determinant and diagonalization, respectively. ∆ (A) refers to the Vandermonde determinant of the eigenvalues of matrix A. 0 n and I n stand for 1 × n all-zero vector and n × n identity matrix, respectively. C m×n denotes the sets of m × n-dimensional complex matrices. The symbol i = √ −1 is the imaginary unit. o(·) denotes little-O notation. (·) n represents Pochhammer symbol. |S| refers to the cardinality of set S. Any other notations will be defined in the place where they occur. II. SYSTEM MODEL By considering a point-to-point MIMO system with N t transmit and N r receive antennas, the received signal vector y ∈ C Nr×1 is written as y = P N t Hx + n,(1) where H ∈ C Nr×Nt is the matrix of the channel coefficients, x ∈ C Nt×1 denotes the vector of transmitted signals, n ∈ C Nr×1 represents the complex-valued additive white Gaussian noise vector with zero mean and covariance matrix σ 2 I Nr , and P is the total average transmitted power. Moreover, in order to account for the effect of the antenna correlation, the channel matrix H is modeled herein according to the Kronecker correlation structure as [12] H = R r 1/2 H w R t 1/2 ,(2) where H w ∈ C Nr×Nt is a random matrix whose entries are independent and identically distributed (i.i.d.), complex circularly symmetric Gaussian random variables, i.e., vec (H w ) ∼ CN (0 NtNr , I Nt ⊗ I Nr ), R t and R r are respectively termed as the transmit and receive correlation matrices, and both of them are positive semi-definite Hermitian matrices. For the sake of simplicity, we assume that the correlation matrices follow the constraints as tr(R t ) = N t and tr(R r ) = N r . From the perspective of information theory for MIMO system in [11], the outage probability can be expressed as p out = Pr log 2 det I Nr + ρHH H < R = Pr G Nr i=1 (1 + ρλ i ) < 2 R = F G (2 R ),(3) where ρ = P/(σ 2 N t ) stands for the average transmit SNR per antenna, λ 1 , · · · , λ Nr denotes the unordered eigenvalues of HH H1 , F G (x) denotes the cumulative distribution function (CDF) of G, with the property [13, Exercise 7.25, p167], the outage probabilities for N t ≥ N r and N t < N r can be derived in the same fashion. Hence, we assume N t ≥ N r in the sequel unless otherwise specified. From (3), it boils down to determining the distribution of the product of multiple shifted eigenvalues λ = (λ 1 , · · · , λ Nr ). III. ANALYSIS OF OUTAGE PROBABILITY The outage probability is the fundamental performance metric to characterize the error performance of decodings. However, the occurrence of the correlation among eigenvalues will yield the involvement of a multi-fold integral in deriving the expression of F G (2 R ). Nonetheless, the product form of G motivates us to apply Mellin transform to obtain the distribution of G [14]. Specifically, the Mellin transform of the probability density function (PDF) of G, {Mf G } (s), is given by ϕ(s) = ∞ 0 · · · ∞ 0 Nr i=1 (1 + ρλ i ) s−1 f (λ) dλ 1 · · · dλ Nr . (4) By utilizing the inverse Mellin transform together with its associated property of integration [15, eq.(8.3.15)], the CDF of G can be obtained as F G (x) = M −1 − 1 s ϕ (s + 1) (x) = 1 2πi c+i∞ c−i∞ x −s −s ϕ (s + 1) ds,(5) where c ∈ (−∞, 0), because the Mellin transform of F G (x) exists for any complex number s in the fundamental strip −∞ < ℜ(s) < 0 by noticing F G (x) = 0 for x < 1 and lim x→∞ F G (x) = 1 [16, p400]. A. Exact Outage Probability By favor of the character expansions, the joint distribution of the N r unordered strictly positive eigenvalues of HH H is obtained by Ghaderipoor et al. in [17] as M+1≤j≤N ), a = (a 1 , · · · , a Nr ) and b = (b 1 , · · · , b Nt ) represent the eigenvalue vectors of R r −1 and R t −1 , respectively, k Nr = (k 1 , · · · , k Nr ) stands for all irreducible representation of the general linear group GL(N r , C) and k 1 ≥ · · · ≥ k Nr are integers, A, B and K are the diagonalizations of vectors a, b and k Nr , respectively. f (λ) = kN r (−1) Nr (Nr −1) 2 A N r !∆ (K) ∆ (λ) det λ i kj +Nt−Nr i,j ,(6)where A = Nr i=1 ai N t N t j=1 bj Nr ∆(A)∆(B) Nr j=1 (kj +Nt−Nr)! det ({(−a i ) kj } i,j ) × det({b i kj +N −M } i,1≤j≤M , {b i N −j } i, By substituting (6) into (4), the Mellin transform of f G (x) is expressed as ϕ (s) = (−1) Nr(Nt−Nr) ρ − 1 2 Nr(Nr+1) Nr i=1 a i Nr Nt j=1 b j Nr ∆ (A) ∆ (B) Nr i=1 (s + i − 2) i−1 × det   Ψ 1, s + N r ; aibj ρ 1≤i≤Nr ,j b j Nt−i Nr+1≤i≤Nt,j   .(7) where Ψ (·, ·; ·) denotes confluent hypergeometric function [18, eq. (9.210)]. Proof. Please see Appendix A By substituting (7) into (5), the CDF of G can then be obtained as shown in the following theorem. Theorem 1. The CDF of G is given by F G (x) = Λρ 1 2 Nr(Nr−1) σ∈SN t sgn (σ) × Nt i=Nr+1 b σi Nt+Nr−i Y Nr,Nr Nr,2Nr C D x Nr i=1 a i b σi ρ Yσ (x) ,(8)where Λ = (−1) Nr (N t −Nr )+ 1 2 Nr (Nr −1) ∆(A)∆(B) , Y σ (x) is the generalized Fox's H function which is defined by using the integral of Mellin-Branes type and provided by an efficient MATHEMATICA implementation as [19], C = [(1, 1, 0, 1) , (N r , 1, 0, 1) j=1,··· ,Nr−1 ], and D = [(N r , 1, aibσ i ρ , 1) i=1,··· ,Nr , (0, 1, 0, 1) , (j, 1, 0, 1) j=1,··· ,Nr−1 ]. Proof. The proof is given in Appendix B. Substituting (8) into (3), the outage probability under fully correlated Rayleigh MIMO channels can be obtained. The generalized Fox's H function involved in (8) is too complex to extract insightful results. In order to obtain tractable results and gain more insights, we have to recourse to the asymptotic analysis of the outage probability at high SNR. B. Asymptotic Outage Probability In order to derive the asymptotic expression at high SNR for the outage probability in (8), the following lemma associated with the asymptotic expression of Y σ (x) is developed first. Lemma 1. As ρ → ∞, Y σ (x) is asymptotic to Y σ (x) = ρ −Nr 2 Nr i=1 a i Nr b σi Nr 1 2πi −c+i∞ −c−i∞ Γ (s) Γ (1 + s) × Nr i=1 Γ (s − N r ) Γ (s − i + 1) ∞ ni=0 aibσ i ρ ni (1 + N r − s) ni x s ds + o ρ −NtNr− 1 2 Nr(Nr+1) .(9) Proof. Please see Appendix C. By using Lemma 1, the asymptotic expression of the outage probability is given by the following theorem. Theorem 2. At high transmit SNR, the outage probability asymptotically equals p out = ρ −NrNt det (R r ) Nt det (R t ) Nr g(R) + o ρ −NtNr , (10) where g(R) = G 0,Nr+1 Nr+1,Nr+1 1, N t + 1, · · · , N t + N r 0, 1, · · · , N r 2 R denotes the Meijer G-function [18]. Proof. Please see Appendix D. It is worth mentioning that the asymptotic analysis of the outage probability for N t < N r can be carried out in an analogous way owing to the property [13, Exercise 7.25, p167]. Similar to (8) and (10), the exact and asymptotic outage expressions for N t < N r can be obtained by directly interchanging R t and N t with R r and N r , respectively. IV. DISCUSSIONS OF ASYMPTOTIC RESULTS The asymptotic outage probabilities commonly exhibit the same basic mathematical structure as [20, eq.(3.158)] p out = S(R t , R r )(C(R)ρ) −d + o ρ −d ,(11) where S(R t , R r ) quantifies the impact of spatial correlation at transmit and receive sides, C(R) is the modulation and coding gain, and d stands for the diversity order. By identifying (10) with (11), we get S(R t , R r ) = 1 det (R r ) Nt det (R t ) Nr ,(12)C(R) = (g 0 (R)) − 1 N t Nr ,(13) and d = N t N r , respectively. To comprehensively understand the asymptotic behavior of the outage probability, these impact factors are discussed individually. A. Diversity Order The terminology of the diversity order can be used to measure the degree of freedom of communication systems, which is defined as the ratio of the outage probability to the transmit SNR on a log-log scale as d = lim ρ→∞ log p out log ρ .(14) Hence, the diversity order indicates the decaying speed of the outage probability with respect to the transmit SNR. As shown in (10), full diversity can be achieved by MIMO systems regardless of the presence of the spatial correlation, i.e., d = N t N r . B. Modulation and Coding Gain The modulation and coding gain C(R) quantifies the amount of the SNR reduction required to reach the same outage probability when employing a certain modulation and coding scheme (MCS). Accordingly, the increase of C(R) is in favor of the improvement of the outage performance. From (13), in order to disclose the behaviour of C(R), it suffices to investigate the property of the function g 0 (R). Towards this end, we can arrive at following Theorem. Proof. Similar to [14,Lemma 4], the detailed proof is omitted here to save space. Evidently from Theorem 3, the transmission rate is an increasing and convex function of the asymptotic outage probability. Without dispute, the monotonicity and convexity of g 0 (R) can greatly facilitate the optimal rate selection of MIMO systems if the asymptotic results are used. C. Spatial Correlation Although the effect of the spatial correlation can be quantified by S(R t , R r ), it is also imperative to draw a qualitative conclusion about the outage behaviour of the spatial correlation. To characterize the spatial correlation, the majorization theory is usually adopted as a powerful mathematical tool to establish a tractable framework [9], [21]. The majorizationbased correlation model is defined as follows. Definition 1. For two N × N semidefinite positive matrices R 1 and R 2 , r 1 = (r 1,1 , · · · , r 1,N ) and r 2 = (r 2,1 , · · · , r 2,N ) are defined as the vectors of the eigenvalues of R 1 and R 2 , respectively, where the eigenvalues are arranged in descending order as r i,1 ≥ · · · ≥ r i,N , i ∈ {1, 2}. We denote R 1 R 2 and say the matrix R 1 is majorized by the matrix R 2 if k j=1 r 1,j ≤ k j=1 r 2,j , (k ≤ N − 1) and N j=1 r 1,j = N j=1 r 2,j . (15) We also say R 1 is more correlated than R 2 . It is easily found by definition that diag(1, 1, · · · , 1) R i diag(N, 0, · · · , 0) if tr(R i ) = N , where diag(N t , 0, · · · , 0) and diag(1, 1, · · · , 1) correspond to completely correlated and independent cases, respectively. Notice that det(R i ) = N j=1 r i,j , the property of the majorization in [22, F.1.a] proves that the determinant of the correlation matrix is a Schur-concave function, where det(R 1 ) ≥ det(R 2 ) if R 1 R 2 , the interested reader is referred to [22] for further details regarding the schur monotonicity. By recalling S(R t , R r ) is the composition of the determinants and using the fact associated with the composition involving Schurconcave functions [22], we arrive at S(Rt 1 , Rr 1 ) ≤ S(Rt 2 , Rr 2 ), whenever R t1 R t2 and R r1 R r2 . As a consequence, it is concluded that the presence of the spatial correlation adversely impacts the outage performance. V. NUMERICAL RESULTS In this section, numerical results are presented for verifications and discussions. For notational convenience, we define t and r as the row vectors of the eigenvalues of the transmit and receive correlation matrices, i.e., R t and R r . A. Verifications Figs. 1 depicts the outage probability versus the transmit SNR. The labels 'Sim.', 'Exa.' and 'Asy.' in this figure indicates the simulated, exact and asymptotic outage probabilities, respectively. As observed in fig. 1, the exact and simulation results are in perfect agreement, which confirms the correctness of the exact analysis. Besides, it can be seen that the asymptotic results coincide well with the exact and simulation ones at high SNR, which validates the asymptotic results as well. the benefit of using MIMO. Additionally, it can be observed from Fig. 2 that the increase of the transmission rate impairs the coding and modulation gain C(R), which consequently leads to the deterioration of the outage performance. This is consistent with the asymptotic analysis in Section IV-B. Aside from degrading the outage performance, the increase of the transmission rate causes the enhancement of the system throughput. The two opposite effects force us to properly select the transmission rate in practice. Fortunately, the optimal rate selection can be eased by using the asymptotic outage probability thanks to its increasing monotonicity and convexity with respect to the transmission rate. B. Coding and Modulation Gain C. Impact of Spatial Correlation In Fig. 3, the outage probability is plotted against the transmit SNR under three different transmit correlation matrices, i.e., R t1 , R t2 and R t3 . For notational simplicity, the vectors of the eigenvalues are set as t 1 = (1, 1, 1), t 2 = (2.3, 0.5, 0.2) and t 3 = (2.7, 0.2, 0.1). It is clear from the figure that the spatial correlation does not affect the diversity order. Moreover, according to the concept of majorization, the relationship of the transmit correlation matrices follows as R t3 R t2 R t1 , and R t3 are the most correlated correlation matrix among them. It is readily observed in Fig. 3 that the outage probability curve associated with R t3 displays the worst performance. The numerical result corroborates the validity of the analytical results in Section IV-C. VI. CONCLUSIONS This paper has derived novel representation for the outage probabilitiy of the MIMO system by invoking Mellin transform. The compact and simple expression not only has enabled the accurate evaluation of the outage probability, but also has facilitated the asymptotic analysis under high SNR to gain a profound understanding of fading effects and MIMO configurations, which has never been performed in the literature. On the one hand, the asymptotic results have revealed meaningful insights into the effects of the spatial correlation, the number of antennas, and transmission rate. For instance, the spatial correlation degenerates the outage performance, while full diversity can be achieved even at the presence of the spatial correlation. On the other hand, the asymptotic result have paved the way for the simplification of practical system designs. For example, the increasing monotonicity and convexity of the asymptotic outage probability will facilitate the proper selection of target transmission rate. APPENDIX A By substituting (6) into (4), using the identity ∆ (λ) = det λ i j−1 i,j and the Leibniz formula for the determinant expansion [23], we can obtain that ϕ (s) = kN r (−1) Nr (Nr −1) 2 A N r !∆ (K) σ1,σ2∈SN r sgn (σ 1 ) sgn (σ 2 ) × Nr i=1 ∞ 0 (1 + ρλ) s+Nr −σ1,i−1 λ kσ 2,i +σ1,i+τ dλ. (17) where S Nr denotes the set of permutations of {1, 2, · · · , N r }, σ l (σ l,1 , · · · , σ l,Nr ) for l ∈ (1, 2) and sgn(σ l ) denotes the signature of the permutation σ l , i.e., sgn(σ l ) is 1 whenever the minimum number of transpositions necessary to reorder σ l as (1, 2, · · · , N r ) is even, and −1 otherwise [11]. Lemma 2. If η(σ 1 , σ 2 ) is a function of σ 1 and σ 2 irrespective of the ordering of the elements in the permutations of the set of two-tuples {(σ 1,l , σ 2,l ) : l ∈ [1, N r ]}, the summation of sgn (σ 1 ) sgn (σ 2 )η(σ 1 , σ 2 ) over all permutations of σ 1 and σ 2 degenerates to σ1,σ2∈SN r sgn (σ 1 ) sgn (σ 2 )η(σ 1 , σ 2 ) = σ 1 ,σ2∈SN r sgn (σ) sgn (σ) η(σ,σ) = N r ! σ∈SN r sgn (σ)η(σ, σ),(18) where σ = (σ 1 , · · · , σ Nr ) andσ = (1, · · · , N r ). Proof. Similar to the proofs of Leibniz formulae [11, eq.(2)] and [17, eqs.(64-65)], denote by S Nr × S Nr the Cartesian product of S Nr . Hence, (σ 1 , σ 2 ) ∈ S Nr × S Nr . We further establish the one-to-one mapping ϑ(σ 1 , σ 2 ) as a vector of ordered pairs (σ 1,l , σ 2,l ) for l ∈ [1, N r ], i.e., ϑ(σ 1 , σ 2 ) ((σ 1,l , σ 2,l ) : l ∈ [1, N r ]). We thus reach the relation sgn (σ 1 ) sgn (σ 2 ) = sgn (σ) sgn (σ) after a certain number of transpositions to achieve ϑ(σ,σ) starting from ϑ(σ 1 , σ 2 ), where sgn (σ) = 1. Secondary, according to the definition of η(σ 1 , σ 2 ), the one-to-one mapping and the cardinality of S Nr , i.e., |S Nr | = N r !. Using the same manner the second step holds. Basing on Lemma 2 and the Leibniz formula for the determinant expansion [23], (17) can be simplified as ϕ (s) = kN r (−1) Nr (Nr −1) 2 A ∆ (K) ξ (s),(19) where ξ (s) = det[{ ∞ 0 (1 + ρλ) s+Nr −i−1 λ kj +i+τ dλ} i,j ]. By the change of variable x = 1/(1 + ρλ), ξ (s) can be expressed in terms of Beta function [18, eq.(8.380.1)] as ξ (s) = ρ −kj −i−τ −1 B (−s − N r − k j − τ, k j + i + τ + 1) .(20)ϕ (s) = (−1) Nr(Nt−Nr) ρ − 1 2 Nr(Nr+1) Nr i=1 a i Nr Nt j=1 b j Nr ∆ (A) ∆ (B) Nr i=1 (s + i − 2) i−1 × det    ∞ k=0 Γ(−s−Nr−k+1) Γ(−s−Nr +2) − aibj ρ k 1≤i≤Nr ,j b j Nt−i Nr+1≤i≤Nt,j    .(21)Γ (−s − N r − k + 1) Γ (−s − N r + 2) − a i b j ρ k = ∞ 0 (1 + y) s+Nr−2 e − a i b j ρ y dy = Ψ 1, s + N r ; a i b j ρ ,(22) where the last equality holds by using (9.211.4) in [18], Ψ (·, ·; ·) denotes confluent hypergeometric function. By substituting (22) into (21), we finally arrive at (7). APPENDIX B By applying the determinant expansion to (7), we get F G (x) = 1 2πi Λ Nr i=1 a i Nr Nt j=1 b j Nr ρ 1 2 Nr(Nr+1) σ∈SN t sgn (σ) × Nt i=Nr +1 b σi Nt−i c+i∞ c−i∞ Γ (−s) Γ (1 − s) Nr−1 j=1 Γ (−s − N r + 1) Γ (−s − j + 1) × Nr i=1 Ψ 1, s + 1 + N r ; a i b σi ρ x −s ds .(23) By using the definition of Ξ (a, α, A, ϕ) = A ϕ+a+αs−1 Ψ (ϕ, ϕ + a + αs; A) in [14] and identifying the integration in (23) with the generalized Fox's H function [19], (23) can finally be represented as (8). APPENDIX C Applying property 2 in [14] to (8) gives rise to Y σ (x) = ρ −Nr 2 Nr i=1 a i Nr b σi Nr 1 2πi −c+i∞ −c−i∞ Γ (s) Γ (1 + s) × Nr−1 j=1 Γ (s − N r + 1) Γ (s − j + 1) Nr i=1 Ψ 1, 1 + N r − s; a i b σi ρ x s ds .(24) We set c < −N t N r + 1 2 N r (N r − 1). By using [18, eq.(9.210.2)] and ignoring the higher order term o ρ −NtNr− 1 2 Nr(Nr+1) in (24), together with the series expansion as Ψ (α, β; x) = ∞ n=0 (α) n x n /(β) n /n! [18, eq.(9.210.1)], where (·) n is Pochhammer symbol, we finally arrive at (9). APPENDIX D Substituting (9) into (8) and swapping the orders of integration and summation produces, after some basic algebraic manipulations, (8) can be further derived as 1, 1 + N r + n 1 , · · · , 1 + N r + n Nr 0, 1, · · · , N r 2 R ,(27) where n = (n 1 , · · · , n Nr ). Notice that any term with n i = 0, · · · , N t −N r −1 is equal to zero thanks to the basic property of determinant, the dominant terms with n belonging to the set of the permutations of Ω Nr = {N t − N r , · · · , N t − 1} can produce non-zero determinants. Substituting (27) and (26) into (25), and ignoring the terms with both zero value of the determinant and the order of ρ larger than N t N r , p out can be asymptotically expanded as 1, N t + 1, · · · , N t + N r 0, 1, · · · , N r 2 R × n∈ΩN r sgn (n) Nr i=1 a i ni + o ρ −NtNr ,(28) where the Meijer-G function can be extracted from the summation as a common factor due to the fact that its value is independent of the order of the elements of n. Since the following equality holds the asymptotic p out can be finally derived as (10). Theorem 3 . 3g 0 (R) is a monotonically increasing and convex function of the transmission rate R. Fig. 2 2illustrates the impacts of the transmission rate R on the coding and modulation gain C(R) for different numbers of transmit and receive antennas. It is shown inFig. 2that C(R) increases with the number of antennas, Fig. 1 : 1Outage probability versus the transmit SNR ρ with N t = 3, N r = 2, t = (2.7, 0.2, 0.1), r = (1.9, 0.1), and R =2bps/Hz. Fig. 2 : 2Coding and modulation gain C(R) versus the transmission rate R. Fig. 3 : 3Outage probability versus the transmit SNR ρ with N t = N r = 3 and r = (2.7, 0.2, 0.1). ) 2 2Rs ds. By using the determinant expansion in ξ(N t ) and expressing ζ(N r ) in terms of Meijer G-function, leads to By using the relationship B(α, β) = Γ(α)Γ(β)/Γ(α + β) [18, eq.(8.384.1)] and the generalized Cauchy-Binet formula [17,Lemma 4],(19) can be simplified as It is worth noting that HH H is a singular matrix and has at least (Nr − Nt) zero eigenvalues if Nt < Nr. H Tullberg, P Popovski, Z Li, M A Uusitalo, A Hoglund, O Bulakci, M Fallgren, J F Monserrat, The METIS 5G system concept: Meeting the 5G requirements. 54H. Tullberg, P. Popovski, Z. Li, M. A. Uusitalo, A. Hoglund, O. Bulakci, M. Fallgren, and J. F. Monserrat, "The METIS 5G system concept: Meeting the 5G requirements," IEEE Commun. Mag., vol. 54, no. 12, pp. 132-139, Dec. 2016. Capacity of multi-antenna Gaussian channels. E Telatar, Europ. 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[ "Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures", "Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures" ]
[ "Juntao Huang [email protected] \nDepartment of Mathematics\nMichigan State University\n48824East LansingMIUSA\n", "Yingda Cheng [email protected]. \nDepartment of Mathematics\nDepartment of Computational Mathematics, Science and Engineering\nMichigan State University\n48824East LansingMIUSA\n", "Andrew J Christlieb [email protected]. \nDepartment of Computational Mathematics, Science and Engineering\nMichigan State University\n48824East LansingMichiganUSA\n", "Luke F Roberts [email protected] \nNational Superconducting Cyclotron Laboratory and Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA\n", "Wen-An Yong [email protected] \nDepartment of Mathematical Sciences\nTsinghua University\n100084BeijingChina\n" ]
[ "Department of Mathematics\nMichigan State University\n48824East LansingMIUSA", "Department of Mathematics\nDepartment of Computational Mathematics, Science and Engineering\nMichigan State University\n48824East LansingMIUSA", "Department of Computational Mathematics, Science and Engineering\nMichigan State University\n48824East LansingMichiganUSA", "National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy\nMichigan State University\n48824East LansingMIUSA", "Department of Mathematical Sciences\nTsinghua University\n100084BeijingChina" ]
[]
This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work [20], we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional P N closure. However, the ML moment closure model in[20]is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.
10.1137/21m1423956
[ "https://arxiv.org/pdf/2105.14410v1.pdf" ]
235,254,147
2105.14410
141a203a87f0e36bf21c52514cd4599dec9d468b
Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures May 2021 Juntao Huang [email protected] Department of Mathematics Michigan State University 48824East LansingMIUSA Yingda Cheng [email protected]. Department of Mathematics Department of Computational Mathematics, Science and Engineering Michigan State University 48824East LansingMIUSA Andrew J Christlieb [email protected]. Department of Computational Mathematics, Science and Engineering Michigan State University 48824East LansingMichiganUSA Luke F Roberts [email protected] National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy Michigan State University 48824East LansingMIUSA Wen-An Yong [email protected] Department of Mathematical Sciences Tsinghua University 100084BeijingChina Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures May 20211radiative transfer equationmoment closuremachine learningneural networkhyperbolicitylong time stability This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work [20], we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional P N closure. However, the ML moment closure model in[20]is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model. Introduction In this paper we introduce an extension to our previous work on ML closures for radiative transfer modeling [20]. The new approach offers long time stability by ensuring mathematical consistency between the closure and the macroscopic model. Further, the numerical results in section 4 demonstrate the plausibility of capturing kinetic effects in a moment system with a handful of moments and an appropriate closure model. The study of radiative transfer is of vital importance in many fields of science and engineering including astrophysics [35], heat transfer [24], and optical imaging [23]. The fundamental equation describing radiative transfer is an integro-differential equation, termed as the radiative transfer equation (RTE). Nevertheless, in most cases, any mesh based numerical discretization of the RTE equation leads to unacceptable computational costs due to the curse of dimensionality. Moment methods study the evolution of a finite number of moments of the specific intensity in the RTE. Typically, the evolution of a lower-order moment depends on a higher-order moment, leading to what is known as the moment closure problem. Hence, one has to introduce suitable closure relations that relates the highest order moment included with the lower order moments in order to get a closed system of equations. Many moment closure models have been developed, including the P N model [8]; the variable Eddington factor models [28,33]; the entropy-based M N models [19,2,1]; the positive P N models [18]; the filtered P N models [32,25]; the B 2 models [3]; and the M P N model [12,13,29]. In moment closure problems, hyperbolicity is a critical issue, which is essential for a system of first-order partial differential equations (PDEs) to be well-posed [38]. The pioneering work on the moment closure for the gas kinetic theory by Grad in [15] is the most basic one among the moment models. However, it was discovered in [7] that the equilibrium is on the boundary of the hyperbolicity region for the Grad's 13-moment model in the three-dimensional case. Due to such a deficiency, the application of the moment method is severely restricted. This issue also attracts a lot of attention, with many papers in the literature focusing on the development of globally hyperbolic moment systems [6,12,13,29]. The traditional trade off in introducing a closure relation and solving a moment model instead of a kinetic equation is generic accuracy verses practical computability. However, thanks to the rapid development of machine learning (ML) and data-driven modeling [5,36,16], a new approach to solve the moment closure problem has emerged based on ML [17,37,21,4,30,40,31,20]. This approach offers a path for multi-scale problems that is relatively unique, promising to capture kinetic effects in a moment model with only a handful of moments. For the detailed literature review, we refer readers to [20]. We only remark that most of the works mentioned above are not able to guarantee hyperbolicity or long time stability, except the work in [21]. In [21], based on the conservation-dissipation formalism [43] of irreversible thermodynamics, the authors proposed a stable ML closure model with hyperbolicity and Galilean invariance for the Boltzmann BGK equation. Nevertheless, their model is limited to one extra non-equilibrium variable and it is still not clear how to generalize to arbitrary number of moments. The work in this paper is a continuation of the previous work in [20], where we proposed to directly learn a closure that relates the gradient of the highest order moment to gradients of lower order moments. This new approach is consistent with the exact closure we derived for the free streaming limit and also provides a natural output normalization [20]. A variety of numerical tests show that the ML closure model in [20] has better accuracy than an ML closure based on learning a relation between the moments, as opposed to a relation between the gradients, and the conventional P N closure [20]. However, it is not able to guarantee hyperbolicity and long time simulations are not always satisfactory. Consequently, the focus of this work is to develop a method to enforce global hyperbolicity of the ML closure model. The main idea is motivated by the observation that the coefficient matrix for the P N closure is a tridiagonal matrix with positive off-diagonal entries. This indicates the existence of a diagonal symmetrizer matrix such that the P N closure is symmetrizable hyperbolic. Motivated by this observation, we propose to only keep the last several components in the ansatz, see equation (2.4) in Section 2.1, and seek a symmetrizer, which is a symmetric positive definite (SPD) matrix, of a block diagonal form. For degrees of freedom no larger than four in the ansatz, relating the gradient of the (N + 1) th moment to the gradient of the N th , (N − 1) th , (N − 2) th , and (N − 3) th moments, we derive constraints to guarantee the hyperbolicity of the ML closure model, see Theorem 2.1 in Section 2.1. Moreover, due to the block diagonal structure of the symmetrizer, we show that the ML closure system also satisfies the structural stability condition in [41] when taking into account the relaxation effects of the source terms, see Theorem 2.4 in Section 2.1. This condition characterises the dissipation feature of the moment closure system and is analogous to the H-theorem for the Boltzmann equation. We also consider our moment closure system under a diffusive scaling and formally show that our model preserves the correct diffusion limit as the Knunsden number goes to zero. The justification of the formal asymptotic analysis could be rigorously verified [34,27], with the aid of the structural stability condition. We also remark that, unlike [21], the work in this paper constructs ML closure models that are hyperbolic with an arbitrary number of moments. We numerically tested that the hyperbolic ML closure model has good accuracy in a variety of numerical examples. Moreover, the non-hyperbolic ML closure model in [20] blows up for long time simulations, while the hyperbolic one remains stable and has good accuracy. The remainder of this paper is organized as follows. In Section 2, we introduce the hyperbolic ML moment closure model and discuss the diffusion limit of the model. In Section 3, we present the details in the training of the neural networks. The effectiveness of our ML closure model is demonstrated through extensive numerical results in Section 4. Some concluding remarks are given in Section 5. ML moment closure for radiative transfer equation In this section, we first review the ML moment closure method for the RTE in slab geometry proposed in [20] and propose our approach to enforce the hyperbolicity of the ML moment closure method. We then formally show that the resulting closure model can capture the correct diffusion limit. Hyperbolic ML closure model We consider the time-dependent RTE for a gray medium in slab geometry: ∂ t f + v∂ x f = σ s 1 2 1 −1 f dv − f − σ a f, −1 ≤ v ≤ 1 (2.1) Here, f = f (x, v, t) is the specific intensity of radiation. The variable v ∈ [−1, 1] is the cosine of the angle between the photon velocity and the x-axis. σ s = σ s (x) ≥ 0 and σ a = σ a (x) ≥ 0 are the scattering and absorption coefficients. Denote the k-th order Legendre polynomial by P k = P k (x). Define the k-th order moment by m k (x, t) = 1 2 1 −1 f (x, v, t)P k (v)dv. (2.2) Multiplying by P k (v) on both sides of (2.1) and integrating over v ∈ [−1, 1], we derive the moment equations: ∂ t m 0 + ∂ x m 1 = −σ a m 0 ∂ t m 1 + 1 3 ∂ x m 0 + 2 3 ∂ x m 2 = −(σ s + σ a )m 1 · · · ∂ t m N + N 2N + 1 ∂ x m N −1 + N + 1 2N + 1 ∂ x m N +1 = −(σ s + σ a )m N (2.3) The learned gradient approach proposed in [20] is to find a relation between ∂ x m N +1 and the gradients on lower order moments: ∂ x m N +1 = N i=0 N i (m 0 , m 1 , · · · , m N )∂ x m i (2.4) with N = (N 0 , N 1 , · · · , N N ) : R N +1 → R N +1 approximated by a neural network and learned from data. In this way, we obtain a quasi-linear first-order systems. This approach is shown to have uniform accuracy in the optically thick regime, intermediate regime and the optically thin regime. Moreover, the accuracy of this gradient-based model is much better than the approach based on creating a ML closure directly trained to match the moments, as well as the conventional P N closure. However, the learned gradient model exhibits numerical instability due to the loss of hyperbolicity [20]. This severely restricts the application of this model, especially for long time simulations. The main idea of this work is to enforce the hyperbolicity by demanding that the coefficient matrix of the closure system A is real diagonizable. For this purpose, we seek a SPD matrix A 0 (also called a symmetrizer) such that A 0 A is a symmetric matrix. Namely, the system is symmetrizable hyperbolic, see the details in Appendix A. This places certain constraints on the functions N i . Plugging (2.4) into the moment closure system, we have: ∂ t m N + N 2N + 1 ∂ x m N −1 + N + 1 2N + 1 N k=0 N k (m 0 , m 1 , · · · , m N )∂ x m k = −(σ s + σ a )m N . Then, we can write down the coefficient matrix of this moment closure system: A =          0 1 0 0 . . .0 N 2N −1 a 0 a 1 . . . a N −2 a N −1 a N          (2.5) with a j =      N + 1 2N + 1 N j , j = N − 1 N 2N + 1 + N + 1 2N + 1 N j , j = N − 1 (2.6) We first observe that, for the P N closure, i.e. when N j = 0 for 0 ≤ j ≤ N , A is a tridiagonal matrix with positive off-diagonal entries. In this case, A 0 can be taken as a diagonal matrix: A 0 = diag(1, 3, 5, . . . , 2N + 1),(2.7) and A 0 A is a tridiagonal symmetric matrix: A 0 A =          0 1 0 0 . . . 0 1 0 2 0 . . . 0 0 2 0 3 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . N − 1 0 N 0 0 . . . 0 N 0          (2.8) Inspired by this observation, we propose to first investigate the case when the ansatz is given by only the last (k + 1) components ∂ x m N +1 = N i=N −k N i (m 0 , m 1 , · · · , m N )∂ x m i , (2.9) where k is a parameter and k ≤ N . We will find a SPD matrix A 0 of the form A 0 = D 0 0 B (2.10) with D = diag(1, 3, · · · , 2(N + 1 − k) − 1) ∈ R (N +1−k)×(N +1−k) and B ∈ R k×k being a SPD matrix. With some algebraic calculation (details given in the appendix), when k = 3, we can derive the constraints explicitly as shown in the following theorem. Theorem 2.1. Consider matrix A ∈ R (N +1)×(N +1) with N ≥ 3 of the form A =               0 1 0 0 0 0 . . . 0 1 3 0 2 3 0 0 0 . . . 0 0 2 5 0 3 5 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . N −3 2N −5 0 N −2 2N −5 0 0 0 0 . . . 0 N −2 2N −3 0 N −1 2N −3 0 0 0 . . . 0 0 N −1 2N −1 0 N 2N −1 0 0 . . . 0 a N −3 a N −2 a N −1 a N               (2.11) where a i has been specified in (2.6). If the coefficients a i for i = N − 3, N − 2, N − 1, N satisfy the following constraints: a N −3 > − (N − 1)(N − 2) N (2N − 3) , a N −1 > g(a N −3 , a N −2 , a N ; N ) (N − 2)(a N −3 (2N − 3)N + (N − 1)(N − 2)) 2 (2.12) where g = g(a N −3 , a N −2 , a N ; N ) is a function given by g = a 3 N −3 (N − 1)N 2 (3 − 2N ) 2 + a N −2 (2N − 1)(N − 2) 3 (a N −2 N − a N (N − 1)) + a N −3 (N − 2) 2 (a N (4N 2 − 8N + 3)(a N −2 N − a N (N − 1)) + (N − 1) 3 ) + 2a 2 N −3 (N − 1) 2 N (2N − 3)(N − 2), then there exist a SPD matrix A 0 = diag(D, B) ∈ R (N +1)×(N +1) such that A 0 A is symmetric. Here, D = diag(1, 3, 5, · · · , 2N − 5) ∈ R (N −2)×(N −2) and B ∈ R 3×3 is a SPD matrix. Moreover, (2.12) is equivalent to the following constraints on N i : N N −3 ≥ − (N − 2)(N − 1)(2N + 1) N (2N − 3)(N + 1) , N N −1 ≥ − N N + 1 + h(N N −3 , N N −2 , N N ; N ) (N − 2)(N N −3 (N + 1)(2N − 3)N + (N − 2)(N − 1)(2N + 1)) 2 (2.13) with h = h(N N −3 , N N −2 , N N ; N ) is a function given by h = N 3 N −3 (N − 1)N 2 (−2N 2 + N + 3) 2 + N N −2 (N + 1)(2N − 1)(2N + 1)(N − 2) 3 (N N −2 N − N N (N − 1)) + N N −3 (N − 2) 2 (N N (N + 1) 2 (4N 2 − 8N + 3)(N N −2 N − N N (N − 1)) + (2N + 1) 2 (N − 1) 3 ) + 2N 2 N −3 (N − 1) 2 N (N + 1)(2N − 3)(2N + 1)(N − 2). (2.14) Proof. The proof is given in Appendix B. Corollary 2.2. If we keep only 3 degrees of freedoms by setting a N −3 = 0, the constraint (2.12) reduces to 15) and the equivalent constraint (2.13) reduces to a N −1 > 2N − 1 (N − 1) 2 a N −2 (N a N −2 − (N − 1)a N ) ,(2.N N −1 > − N N + 1 + (2N − 1)(N + 1) (N − 1) 2 (2N + 1) N N −2 (N N N −2 − (N − 1)N N ). (2.16) If we further set a N −3 = a N −2 = 0 by taking 2 degrees of freedoms, the constraint (2.12) reduces to a N −1 > 0, (2.17) and the equivalent constraint (2.13) reduces to N N −1 > − N N + 1 . (2.18) In this case, the coefficient matrix A is a tridiagonal matrix with positive off-diagonal entries, which is real diagonalizable. Remark 2.3. In principle, one could generalize the result in Theorem 2.1 to more than 4 degrees of freedom, by following the same lines in the proof given in Appendix B. However, the constraints for hyperbolicity will be a set of implicit inequalities. How to incorporate the implicit constraint into the architecture of the neural network would be an interesting topic to explore. Besides the hyperbolicity for the convection term, one may also be interested in the relaxation effect of the source term. In this regard, the related property is the structural stability condition proposed in [41], which includes the constraints on the convection term, collision term, and the coupling of both. This stability condition is shown to be critical for the existence of the solutions [34], and satisfied by many moment closure systems [11]. In addition, the structural stability condition to the moment systems is same as H-theorem to the Boltzmann equation, which characterizes the dissipation property of the moment systems. For the convenience of the reader, we review the structural stability condition in Appendix C. It is easy to show that our new ML moment closure system also satisfied the structural stability condition: Theorem 2.4. For k = 3 and N ≥ 3, if the constraint (2.12) (or equivalently (2.13)) is satisfied, then the ML moment closure system with the closure relation (2.9) satisfies the structural stability condition in [41]. Proof. We only prove for the case σ a = 0. The other case can be proven by following the same line. Denote Q = Q(U ) the source term of the closure system and Q U = ∂Q ∂U . Then we can compute A 0 Q U = diag(D, B) diag(0, −σ s I N ) = diag(D, B) diag(−C, −σ s I 3 ) = − diag(DC, σ s B) (2.19) Here, C = diag(0, σ s I N −3 ) and I r denote the identity matrix of order r. Notice that A 0 Q U is a symmetric negative semi-definite matrix. Then the structural stability condition can be easily checked for this moment closure system by taking P to be a scalar matrix in Appendix C. Diffusion limit In this part, we show formally that the hyperbolic ML moment closure model has the correct diffusion limit. Consider the RTE under a diffusive scaling: ε∂ t f + v∂ x f = σ s ε 1 2 1 −1 f dv − f − εσ a f,(2.20) where ε > 0 denotes the Knudsen number, which is the ratio of mean free path of particles to the characteristic length. It is well known that as ε → 0, the kinetic transport equation (2.20) will converge to the diffusion limit-macroscopic model: ∂m 0 = ∂ x 1 3σ s ∂ x m 0 − σ a m 0 ,(2.21) where m 0 is the zeroth-order moment of f . For the RTE under a diffusive scaling (2.20), the corresponding moment closure equation is: ε∂ t m 0 + ∂ x m 1 = −εσ a m 0 ε∂ t m 1 + 1 3 ∂ x m 0 + 2 3 ∂ x m 2 = −( σ s ε + εσ a )m 1 , · · · ε∂ t m N −1 + N − 1 2N − 1 ∂ x m N −2 + N 2N − 1 ∂ x m N = −( σ s ε + εσ a )m N −1 , ε∂ t m N + N 2N + 1 ∂ x m N −1 + N + 1 2N + 1 N k=0 N k ∂ x m k = −( σ s ε + εσ a )m N . (2.22) Next, we formally show that the ML moment closure model (2.22) with N ≥ 3 converges to the diffusion limit (2.21) as ε → 0. We expand m k for k = 0, · · · , 2 as: m k ∼ m (0) k + εm (1) k + ε 2 m (2) k + O(ε 3 ) (2.23) and plug into the first three equations of (2.22). Collecting the O(ε) term of the first equation, we obtain ∂ t m (0) 0 + ∂ x m(1)1 = −σ a m (0) 0 . (2.24) Collecting the O(1) term of the second equation, we have 1 3 ∂ x m (0) 0 + 2 3 ∂ x m (0) 2 = −σ s m(1) 1 . Combining the above three relations, we eventually reach: ∂m (0) 0 = ∂ x 1 3σ s ∂ x m (0) 0 − σ a m (0) 0 . (2.27) This indicates that the leading-order term of m 0 satisfies the diffusion equation (2.21). To justify the above formal asymptotic analysis, it turns out that the structural stability condition is essential. Indeed, under the structural stability condition, one can rigorously prove that the solution of the ML moment closure model will converge to that of the macroscopic model (2.21) as ε goes to 0 [34,27], if the constraint (2.12) (or equivalently (2.13)) is satisfied and the neural network in (2.22) satisfies some regularity conditions. Although the regularity of the neural network seems not easy to be validated, we will show numerically in Section 4 that our hyperbolic ML closure model can capture the correct diffusion limit. Training of the neural network In this section, we provide details of the training of the proposed neural network that enforces the constraint (2.12) (or equivalently (2.13)) for the hyperbolicity for the case when k = 3. We design a fully-connected neural network denoted by M : R N +1 → R 4 with the input being the lower order moments (m 0 , m 1 , · · · , m N ) and the output being M = (M 1 , M 2 , M 3 , M 4 ). Then, we do the following post-processing to the output of the neural network: N N = M 4 , N N −2 = M 2 , N N −3 = σ(M 3 ) − (N − 2)(N − 1)(2N + 1) N (2N − 3)(N + 1) N N −1 = σ(M 1 ) − N N + 1 + h(N N −3 , N N −2 , N N ; N ) (N − 2)(N N −3 (N + 1)(2N − 3)N + (N − 2)(N − 1)(2N + 1)) 2 where h is the function defined in (2.14). Here σ : R → R is a positive function, i.e., σ(x) > 0 for any x ∈ R. We test several positive functions to enforce this constraint, including exponential function, ReLU (Rectified Linear Unit) function, softplus function and square function. Our numerical tests show that the neural network with σ to be the softplus function has the smallest L 2 error in the training data. In the numerical implementation, we take the number of layers to be 6 and the number of nodes to be 256 in the fully-connected neural network and use the hyperbolic tangent activation function. Other hyperparameters in the training are the same as those in [20]. The training data comes from numerically solving the RTE using the space-time discontinuous Galerkin (DG) method [9,10] with a range of initial conditions in the form of truncated Fourier series and different scattering and absorption coefficients which are constants over the computational domain. We train the neural network with 100 different initial data. For each initial data, we run the numerical solver up to t = 1. The other parameters are the same as in [20]. Numerical results In this section, we show the performance of our ML closure model on a variety of benchmark tests, including problems with constant scattering and absorption coefficients, Gaussian source problems and two-material problems. The main focus of the tests is on the comparison of hyperbolic ML closure (termed as "hyperbolic closure") with k = 3, ML closure with learning gradient in [20] (termed as "non-hyperbolic closure") with k = N , and the classical P N closure [8]. In all numerical examples, we take the physical domain to be the unit interval [0, 1] and periodic boundary conditions are imposed. To numerically solve the moment closure system, we apply the fifth-order finite difference WENO scheme [22] with a Lax-Friedrichs flux splitting for the spatial discretization, and the third order strong-stability-preserving Runge-Kutta (RK) scheme [39] for the time discretization. We take the grid number in space to be N x = 256. For the hyperbolic ML closure model, the CFL condition is taken to be ∆t = 0.8∆x/c where c denote the maximum eigenvalues in all the grid points, and the penalty constant in the Lax-Friedrichs numerical flux α LF = c. For the non-hyperbolic ML closure model, we impose larger numerical diffusion by taking the penalty constant in the Lax-Friedrichs numerical flux α LF = 5 and fixing ∆t = 0.1∆x [20]. system (N = 6). It is observed that, at t = 0.5, the two ML moment closures agree well with the RTE, except that some minor oscillations appear in the non-hyperbolic ML closure. At t = 1, the solutions of non-hyperbolic ML moment closure blow up, while the hyperbolic one stays stable and has good agreement with the RTE. As a comparison, the P N closure has large deviations from the exact solution at both t = 0.5 and t = 1. In Figure 4.2, we display the log-log scatter plots of the relative L 2 error versus the scattering coefficient for N = 6. We observe that, at t = 0.5, both ML closures have better accuracy than the P N closure. Moreover, in the optically thick regime, all the closures perform well. At t = 1, the numerical solutions of the non-hyperbolic ML moment closure blow up at some data points, while the hyperbolic one still has good accuracy. In Results obtained by the hyperbolic closure. closure system and the solution generated by the RTE in the optically thin regime (σ s = σ a = 1). We observe that the hyperbolic closure generates good predictions in the long time simulation up to t = 10. We also display the maximum eigenvalues in all the grid points during the time evolution in Figure 4.3 (b). It is observed that the eigenvalues are always real numbers, which validates the hyperbolicity feature of the closure system. However, the eigenvalues could be as large as four during the time evolution, which violates the physical characteristic speed of 1 in the RTE. This also results in smaller time step size in the simulation of the closure system. It is unclear how to preserve the physical characteristic speed in the moment closure system in this setting. Example 4.2 (Gaussian source problem). In this example, we investigate the RTE with the initial condition to be Gaussian distribution in the physical domain, named Gaussian source problem in literature [14,12]: f 0 (x, v) = c 1 (2πθ) 1/2 exp − (x − x 0 ) 2 2θ + c 2 . (4.1) In this test, we take c 1 = 0.5, c 2 = 2.5, x 0 = 0.5 and θ = 0.01. In Figure 4.4, we present the results obtained using various closure models. We observe good agreement between the two ML closure model and the kinetic model at t = 1, while the P N model has large deviations from the kinetic model. Moreover, the non-hyperbolic closure is more accurate than the hyperbolic closure, perhaps due to the fact that the non-hyperbolic closure has larger k values and thus can approximate the closure better. The solution of the non-hyperbolic ML closure model blows up at t = 2, while the hyperbolic ML model stays stable and is more accurate than the P N closure. Although the deviation of the hyperbolic ML model "looks" large in respectively. This again illustrates that the hyperbolicity is an essential property in the ML closure model. Example 4.3 (two-material problem). In this example, we consider the two-material problem [26]. In the problem setup, there exist two discontinuities 0 < x 1 < x 2 < 1 in the domain, and σ s and σ a are piecewise constant functions: and σ a (x) = σ a1 , σ s (x) = σ s1 , x 1 < x < x 2 , σ s2 , 0 ≤ x < x 1 or x 2 ≤ x < 1.x 1 < x < x 2 , σ a2 , 0 ≤ x < x 1 or x 2 ≤ x < 1. In the numerical example, we take x 1 = 0.3, x 2 = 0.7, σ s1 = 1, σ s2 = 10 and σ a1 = σ a2 = 0. The numerical results are shown in Figure 4.5. The gray part is in the optically thin regime and the other part is in the intermediate regime. We observe that the two ML closure models are more accurate than the P N closure at t = 0.5 and the hyperbolic ML closure is slightly better than the non-hyperbolic one. At t = 1, the hyperbolic ML closure still agrees well with the kinetic model. Large deviations exist for the P N closure in the optically thin regime and severe oscillations occur for the non-hyperbolic ML closure. Concluding remarks In this paper, we propose a method to enforce the global hyperbolicity of the ML closure model. We find a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints that guarantee the system is globally symmetrizable hyperbolic. Moreover, we show that the closure system also inherits the dissipativeness of the RTE by checking the structural stability condition. A variety of benchmark tests including Gaussian source problem and the two-material problem show good accuracy, correct diffusion limit and generalization ability of our ML closure model. The new approach also offers long time stability by ensuring mathematical consistency between the closure system and the macroscopic model. Further, the new method demonstrates the plausibility of capturing kinetic effects in a moment system with a handful of moments and an appropriate closure model. There are several issues that are worthy of further investigations. First, one could in principle generalize the result in Theorem 2.1 to more than 4 degrees of freedom, by following the same lines in the proof given in Appendix B. In this general case, the hyperbolicity constraints will be a set of implicit inequalities. How to incorporate the constraint of implicit inequalities into the architecture of the neural network would be an interesting topic to explore. Moreover, the moment closure model is expected to have better accuracy with more degrees of freedom. Another topic is that the characteristic speed of the current model may exceed the physical bound, as we observed in the numerical tests. This unphysical characteristic speed results in a small time step size in the computation, and is something we would like to address in our future work. Another interesting topic is to extend the current approach to the two dimensional case. These issues constitute the body of our ongoing work. Acknowledgment We thank Michael M. Crockatt from Sandia National Laboratories for providing a numerical solver for the radiative transfer equations. We also would like to acknowledge the High Performance Computing Center (HPCC) at Michigan State University for providing computational resources that have contributed to the research results reported within this paper. Appendix A Hyperbolicity In this part, we review the definition of the hyperbolicity and some equivalent conditions. See also the details in Chapter 3 in [38]. Consider the first-order system of equations in 1D: The following classical conclusion holds for these two definitions: U t + A(U )U x = Q(U ). (A.1) Here U = U (x, Proposition A.3. The symmetrizable hyperbolicity is a necessary and sufficient condition for the first-order system (A.1) to be hyperbolic. Proof. We start by the proof of sufficiency. By the definition of symmetrizable hyperbolic, there exists a symmetric positive definite (SPD) matrix A 0 such that A 0 A are symmetric. Since A 0 is a SPD matrix, there exists an invertible symmetric matrix B satisfying A 0 = B 2 . Then, we compute BAB −1 = BA −1 0 A 0 AB −1 = B −1 (A 0 A)B −1 . (A.2) Thus, BAB −1 is symmetric since A 0 A and B are symmetric. Then we have A is real diagonlizable. On the other hand, suppose that A is real diagonlizable, there exist an invertible real matrix P such that A = P −1 DP with D a real diagnoal matrix. Take A 0 = P T P . Then A 0 is a SPD matrix and A 0 A = P T DP is symmetric. Appendix B Proof of Theorem 2.1 Proof. To save space, we only present the proof in the case of a N −3 = 0. One also can prove the case of a N −3 = 0 by following the same line. We first write A in (2.11) into a block matrix: A = A 1 A 2 A 3 A 4 (B.1) with A 1 ∈ R (N −1)×(N −1) , A 2 ∈ R (N −1)×2 , A 3 ∈ R 2×(N −1) and A 4 ∈ R 2×2 . Then we compute A 0 A: A 0 A = D 0 (N −2)×2 0 2×(N −2) B A 1 A 2 A 3 A 4 = DA 1 DA 2 BA 3 BA 4 (B.2) Here, 0 m×n denote the zero matrix of size m × n. It is easy to see that DA 1 is symmetric. Next, we compute other blocks in A 0 A: DA 2 = diag(1, 3, 5, · · · , 2N − 3) 0 (N −2)×1 0 (N −2)×1 N −1 2N −3 0 = 0 (N −2)×1 0 (N −2)×1 N − 1 0 (B.3) Let B = b 11 b 12 b 12 b 22 . (B.4) Then, we compute BA 3 = b 11 b 12 b 12 b 22 0 1×(N −2) N −1 2N −1 0 1×(N −2) a N −2 = 0 1×(N −2) b 11 N −1 2N −1 + b 12 a N −2 0 1×(N −2) b 12 N −1 2N −1 + b 22 a N −2 (B.5) and BA 4 = b 11 b 12 b 12 b 22 0 N 2N −1 a N −1 a N = b 12 a N −1 b 11 N 2N −1 + b 12 a N b 22 a N −1 b 12 N 2N −1 + b 22 a N . (B.6) For A 0 A to be symmetric, we need BA 4 is symmetric and DA 2 = (BA 3 ) T . Thus, we have the constraints b 11 N − 1 2N − 1 + b 12 a N −2 = N − 1, b 12 N − 1 2N − 1 + b 22 a N −2 = 0, b 11 N 2N − 1 + b 12 a N = b 22 a N −1 . (B.7) We solve for b 11 , b 12 and b 22 from the above linear system: This is just the condition (2.12) with a N −3 = 0 (or equivalently (2.15)). Finally, (2.13) can be easily derived, and the proof is omitted for brevity. Appendix C Structural stability condition In this part, we review the structural stability condition proposed in [41], which is fundamental for the quasilinear first-order hyperbolic system with source terms: The stability condition in [41] reads as: (i). There exists an invertible n × n matrix P (U ) and an invertible r × r matrix S(U ), defined on the equilibrium manifold E, such that Here Q U = ∂Q ∂U and I r is the identity matrix of order r. This set of conditions has been tacitly respected by many well-developed physical models [42]. The first condition is classical for initial value problems of the system of ordinary differential equations (ODEs), while the second one means the symmetrizable hyperbolicity of the PDE system. The third condition characterizes a kind of coupling between the convection term and the source term. This set of conditions implies the existence and stability of the zero relaxation limit of the corresponding initial value problems [41]. Example 4 . 1 41(constant scattering and absorption coefficients). The setup of this example is the same as the data preparation. The scattering and absorption coefficients are taken to be constants over the domain. The initial condition is taken to be a truncated Fourier series.InFigure 4.1, we show the numerical solutions of m 0 and m 1 with seven moments in the closure Figure 4 . 1 : 41Example 4.1: constant scattering and absorption coefficients, σ s = σ a = 1, N = 6, t = 0.5 and t = 1. The numerical solution of the non-hyperbolic ML closure blows up at t = 1. Figure 4 . 43 (a), we present the L 2 error between the solutions of the hyperbolic ML moment m1 at t = 1 Figure 4 . 2 : 42Example 4.1: constant scattering and absorption coefficients, σ s = σ a = 1, N = 6, t = 0.5 and t = 1. The non-hyperbolic closure blows up at t = 1. Figure 4 . 3 : 43Example 4.1: constant scattering and absorption coefficients, σ s = σ a = 1, N = 6. Figure 4 . 44 (c), this is due to the fact that the exact solution is close to constant over the computational domain Figure 4 . 4 : 44Example 4.2: Gaussian source problem, σ s = σ a = 1, N = 6, t = 1 and t = 2. The non-hyperbolic closure blows up at t = 2. and the true error is actually small. The relative L 2 and the relative L ∞ error between the solution of the hyperbolic ML model and that of the exact solution at t = 2 is 1.40 × 10 −3 and 3.70 × 10 −3 , Figure 4 . 5 : 45Example 4.3: two material problem. Numerical solutions of m 0 and m 1 at t = 0.5 and t = 1 with N = 6. The gray part in the middle is in the optically thin regime and the other part is in the intermediate regime. Figure 4 . 6 : 46Example 4.4: diffusion limit: the solution to the diffusion equation (2.21); other lines: the solutions to the ML moment closure model (2.22) with ε = 0.5, 0.1, 0.05, 0.01, t = 0.1. Example 4. 4 4(diffusion limit). In the last example, we show that our hyperbolic ML closure model can capture the correct diffusion limit of the RTE. To verify this, we numerically solve the moment closure equation under a diffusive scaling (2.22) with the initial conditionm 0 (x, 0) = sin(2πx) + 2, m k (x, 0) = 0, k = 1, · · · , N (4.2)and different values of ε. We take σ s = 1 and σ a = 0 on the computational domain. We also numerically solve the diffusion limit equation(2.21). InFigure 4.6, we show the numerical solutions of the ML moment closure model at t = 0.1 with ε = 0.5, 0.1, 0.05, 0.01. We observe that the numerical solution m 0 of the closure models converges to that of the diffusion equation (2.21) as ε approaches zero. This validates the formal asymptotic analysis in Section 2.2. t) is an unknown n-vector valued function, Q(U ) and A(U ) are given n-vector and n × n-matrix valued smooth functions of U ∈ G (an open subset of R n called state space), respectively. Definition A.1 (hyperbolic). The system (A.1) is hyperbolic at U 0 if the matrix A(U 0 ) is real diagonlizable. The system (A.1) is globally hyperbolic if it is hyperbolic at any U 0 ∈ G. Definition A.2 (symmetrizable hyperbolic). The system (A.1) is called symmetrizable hyperbolic if there exists a symmetric positive definite matrix A 0 such that A 0 A is symmetric. b 11 = (N − 1)(2N − 1)((N − 1)a N −1 + (2N − 1)a N −2 a N ) −N (2N − 1)a 2 N −2 + (N − 1) 2 a N −1 + (N − 1)(2N − 1)a N −2 a N , b 12 = −N (N − 1)(2N − 1)a N −2 −N (2N − 1)a 2 N −2 + (N − 1) 2 a N −1 + (N − 1)(2N − 1)a N −2 a N , b 22 = N (N − 1) 2 −N (2N − 1)a 2 N −2 + (N − 1) 2 a N −1 + (N − 1)(2N − 1)a N −2 a N . (B.8)An equivalent condition for B to be a SPD matrix isb 11 > 0, b 22 > 0, b 11 b 22 > b 2 12 . (B.9)From the expressions of b 11 and b 22 in (B.8) and b 11 , b 22 > 0 and N ≥ 2, we have − N (2N − 1)a 2 N −2 + (N − 1) 2 a N −1 + (N − 1)(2N − 1)a N −2 a N > 0, (B.10) and (N − 1)a N −1 + (2N − 1)a N −2 a N > 0. (B.11) Plugging b 11 , b 12 and b 12 in (B.8) into b 11 b 22 > b 2 12 , we further derive(N − 1)(2N − 1)((N − 1)a N −1 + (2N − 1)a N −2 a N )N (N − 1) 2 > (N (N − 1)(2N − 1)a N −2 ) 2 . (B.12)It is easy to check that (B.10) and (B.12) are equivalent and (B.10) implies (B.11). Thus, the constraint isa N −1 > 2N − 1 (N − 1) 2 a N −2 (N a N −2 − (N − 1)a N ) . (B.13) U t + A(U )U x = Q(U ). (C.1) Here U = U (x, t) is an unknown n-vector valued function, Q(U ) and A(U ) are given n-vector and n × n-matrix valued smooth functions of U ∈ G (an open subset of R n called state space), respectively. This structural stability condition was established for the hyperbolic system with source term in the multidimensional case. Here, we only show the condition in one-dimensional case for simplicity. Define the equilibrium manifold E = {U ∈ G | Q(U ) = 0}. (C.2) ). There is a symmetric positive definite matrix A 0 (U ) such thatA 0 (U )A(U ) = A(U ) T A 0 (U ), for U ∈ G.(C.4) (iii). 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[ "Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming", "Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming", "Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming", "Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming" ]
[ "Maryam Daryalal [email protected] \nDepartment of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada\n", "Hamed Pouya †[email protected] \nDepartment of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada\n", "Maryam Daryalal [email protected] \nDepartment of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada\n", "Hamed Pouya †[email protected] \nDepartment of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada\n" ]
[ "Department of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada", "Department of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada", "Department of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada", "Department of Mechanical and Industrial Engineering\nUniversity of Toronto\nM5S 3G8TorontoOntarioCanada" ]
[]
Telecommunication networks frequently face technological advancements and need to upgrade their infrastructure. Adapting legacy networks to the latest technology requires synchronized technicians responsible for migrating the equipment. The goal of the network migration problem is to find an optimal plan for this process. This is a defining step in the customer acquisition of telecommunications service suppliers, and its outcome directly impacts the network owners' purchasing behaviour. We propose the first exact method for the network migration problem, a logic-based Benders decomposition approach that benefits from a hybrid constraint programming-based column generation in its master problem and a constraint programming model in its subproblem. This integrated solution technique is applicable to any integer programming problem with similar structure, most notably the vehicle routing problem with node synchronization constraints. Comprehensive evaluation of our method over instances based on six real networks demonstrates the computational efficiency of the algorithm in obtaining quality solutions. We also show the merit of each incorporated optimization paradigm in achieving this performance.1 arXiv:2111.04536v1 [math.OC] 8 Nov 2021 2 decision that can lead to immense savings of 10 to 100 times in power and space as reported byCiena (2013). Nevertheless, the process is quite costly and complex, with some circuits stretching over a continent. Furthermore, every circuit migration comes with a disruption as the endpoints disconnect from the equipment, thus affecting the efficiency of the network and the customers' satisfaction of the migration solution, i.e., the frequency of disruptions they experience during the upgrade. Consequently, a well-crafted plan is critical for the success of the migration. In this paper, we present the first exact solution method for the NMP, a logicbased benders decomposition (LBBD) algorithm that integrates Benders Decomposition (BD), Column Generation (CG) and Constraint Programming (CP), enabling us to decompose the decision space into smaller subsets, each amenable to one of these solution frameworks.The NMP can also be stated in the context of the vehicle routing problem with synchronized constraints (VRPS). In the VRPS, at least one vertex or arc requires simultaneous visits of vehicles, or successive visits resulting from some precedence constraints(Eksioglu et al. 2009).The synchronization constraints can be over the arcs (the synchronized arc routing problem, SARP), or the nodes (the synchronized node routing problem, SNRP). In the NMP, vehicles and nodes correspond to technicians and sites, respectively. Since the tasks (i.e., upgrading the circuit endpoints) are defined over the nodes, the NMP is a special case of the SNRP and an application of the VRPS in the telecommunications domain. In the SNRP, typically some nodes need to be visited by more than one vehicle because the personnel do not have the same expertise. Some examples are: Bredstrom and Rönnqvist (2007) for a vehicle routing and scheduling problem, Reinhardt et al. (2013) for the airport transportation, Labadie et al. (2014) and Hashemi Doulabi et al. (2020) for home healthcare systems, Hojabri et al. (2018) for an SNRP with precedence constraints and synchronization of two types of vehicles, and Li et al. (2020) for a variation of the SNRP where customers have multiple options for time windows. There are also studies considering both the SARP and the SNRP at the same time. Salazar-Aguilar et al. (2013) studied the road marking operations as a synchronized arc and node routing problem such that several capacitated vehicles are used to paint the lines on the roads and a tank vehicle is used to replenish the painting vehicles. Interested reader may refer to Drexl (2012) for a review on VRPS problems and their classifications. In regards to the applicability of these studies to the NMP, an important issue is their local synchronisation assumption, meaning that the arrival of vehicles is synchronized at the same node. However, the NMP involves several sets of technicians distributed over multiple regions which can be 3 synchronized with several other technicians from the same or other regions based on the location of the circuit endpoints. Besides, Li et al. (2020) is the only exact method that considers multiple depots and time windows (the same as the NMP), but the size of the instances it can solve is up to 40 customers with 3 time windows which is far from the need of telecommunication networks. Therefore, the existing works in the literature of the VRPS are not suited for the NMP's level of complexity. As for the telecommunications literature, there are different cost models and strategies on the suitable time and technology to migrate a telecommunication network (Podhradsky 2004, Almughaless and Alsaih 2010, Türk et al. 2012, Poularakis et al. 2019). Moreover, study of the operational aspects of the NMP has recently gained momentum. Bley et al. (2013) studied the migration of a network as the problem of finding the order of link upgrades that minimizes the total disruption time. With no travels allowed between the sites, their problem involves no routing decisions. Jaumard et al. (2016) proposed a CG-based heuristic for the NMP that decides on the technician-to-circuit assignments, as well as travel paths for the technicians.Pouya and Jaumard (2017) observed that the prior mathematical formulations suffer from the highly symmetric nature of the circuits and technicians. A new symmetry-breaking model was developed, leading to significant gains in computational effort. Yet again there is no guarantee that the obtained solutions are indeed optimal/feasible for the individual circuit assignments.Subsequently, Jaumard and Pouya (2018) designed a two-phase CG-based heuristic for the planning of real-size networks. A greedy algorithm was proposed byDawadi et al. (2021)for the cost minimization of a multi-technology migration problem. With the objective of minimizing the number of out-of-service sites and travels, Javad-Kalbasi and Valaee(2021)modeled the NMP as a binary quadratic program. They derived heuristic solutions using Digital Annealer and solved instances with up to 64 circuits.In this work, we develop an exact decomposition framework for the NMP. To the best of our knowledge, both in the context of the telecommunications problems and as a VRPS, this is the first method that exactly solves the NMP with a certificate of optimality/infeasibility. We decompose the NMP into three problems and link them all by designing an LBBD algorithm.By doing so, we are able to leverage the power of different solution techniques for linear programming and combinatorial optimization, and delegate the task of solving each problem to the most suitable optimization paradigm. Numerical experiments defined over various real networks demonstrate the effectiveness of our algorithm in obtaining quality solutions with 4 reasonable computational effort. Given that the NMP can be viewed as an SNRP, our method can also be adapted to a wide class of integer programming problems.Contributions. The contributions of our work are summarized as follows.• We develop the first exact solution method for the network migration problem in order to find the optimal planning solutions, i.e., the order of circuit upgrades, along with the technician assignment and routing decisions. Although our solution framework is developed for the NMP, it is also applicable for the VRP with node synchronization.
10.1287/ijoc.2023.1280
[ "https://export.arxiv.org/pdf/2111.04536v1.pdf" ]
243,847,731
2111.04536
ec928966fccfb72a304cf5e338920626eda28c97
Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming Maryam Daryalal [email protected] Department of Mechanical and Industrial Engineering University of Toronto M5S 3G8TorontoOntarioCanada Hamed Pouya †[email protected] Department of Mechanical and Industrial Engineering University of Toronto M5S 3G8TorontoOntarioCanada Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming Logic-based benders decompositionConstraint programmingColumn generationNetwork migrationOptical networksSynchronized vehicle routing problem Telecommunication networks frequently face technological advancements and need to upgrade their infrastructure. Adapting legacy networks to the latest technology requires synchronized technicians responsible for migrating the equipment. The goal of the network migration problem is to find an optimal plan for this process. This is a defining step in the customer acquisition of telecommunications service suppliers, and its outcome directly impacts the network owners' purchasing behaviour. We propose the first exact method for the network migration problem, a logic-based Benders decomposition approach that benefits from a hybrid constraint programming-based column generation in its master problem and a constraint programming model in its subproblem. This integrated solution technique is applicable to any integer programming problem with similar structure, most notably the vehicle routing problem with node synchronization constraints. Comprehensive evaluation of our method over instances based on six real networks demonstrates the computational efficiency of the algorithm in obtaining quality solutions. We also show the merit of each incorporated optimization paradigm in achieving this performance.1 arXiv:2111.04536v1 [math.OC] 8 Nov 2021 2 decision that can lead to immense savings of 10 to 100 times in power and space as reported byCiena (2013). Nevertheless, the process is quite costly and complex, with some circuits stretching over a continent. Furthermore, every circuit migration comes with a disruption as the endpoints disconnect from the equipment, thus affecting the efficiency of the network and the customers' satisfaction of the migration solution, i.e., the frequency of disruptions they experience during the upgrade. Consequently, a well-crafted plan is critical for the success of the migration. In this paper, we present the first exact solution method for the NMP, a logicbased benders decomposition (LBBD) algorithm that integrates Benders Decomposition (BD), Column Generation (CG) and Constraint Programming (CP), enabling us to decompose the decision space into smaller subsets, each amenable to one of these solution frameworks.The NMP can also be stated in the context of the vehicle routing problem with synchronized constraints (VRPS). In the VRPS, at least one vertex or arc requires simultaneous visits of vehicles, or successive visits resulting from some precedence constraints(Eksioglu et al. 2009).The synchronization constraints can be over the arcs (the synchronized arc routing problem, SARP), or the nodes (the synchronized node routing problem, SNRP). In the NMP, vehicles and nodes correspond to technicians and sites, respectively. Since the tasks (i.e., upgrading the circuit endpoints) are defined over the nodes, the NMP is a special case of the SNRP and an application of the VRPS in the telecommunications domain. In the SNRP, typically some nodes need to be visited by more than one vehicle because the personnel do not have the same expertise. Some examples are: Bredstrom and Rönnqvist (2007) for a vehicle routing and scheduling problem, Reinhardt et al. (2013) for the airport transportation, Labadie et al. (2014) and Hashemi Doulabi et al. (2020) for home healthcare systems, Hojabri et al. (2018) for an SNRP with precedence constraints and synchronization of two types of vehicles, and Li et al. (2020) for a variation of the SNRP where customers have multiple options for time windows. There are also studies considering both the SARP and the SNRP at the same time. Salazar-Aguilar et al. (2013) studied the road marking operations as a synchronized arc and node routing problem such that several capacitated vehicles are used to paint the lines on the roads and a tank vehicle is used to replenish the painting vehicles. Interested reader may refer to Drexl (2012) for a review on VRPS problems and their classifications. In regards to the applicability of these studies to the NMP, an important issue is their local synchronisation assumption, meaning that the arrival of vehicles is synchronized at the same node. However, the NMP involves several sets of technicians distributed over multiple regions which can be 3 synchronized with several other technicians from the same or other regions based on the location of the circuit endpoints. Besides, Li et al. (2020) is the only exact method that considers multiple depots and time windows (the same as the NMP), but the size of the instances it can solve is up to 40 customers with 3 time windows which is far from the need of telecommunication networks. Therefore, the existing works in the literature of the VRPS are not suited for the NMP's level of complexity. As for the telecommunications literature, there are different cost models and strategies on the suitable time and technology to migrate a telecommunication network (Podhradsky 2004, Almughaless and Alsaih 2010, Türk et al. 2012, Poularakis et al. 2019). Moreover, study of the operational aspects of the NMP has recently gained momentum. Bley et al. (2013) studied the migration of a network as the problem of finding the order of link upgrades that minimizes the total disruption time. With no travels allowed between the sites, their problem involves no routing decisions. Jaumard et al. (2016) proposed a CG-based heuristic for the NMP that decides on the technician-to-circuit assignments, as well as travel paths for the technicians.Pouya and Jaumard (2017) observed that the prior mathematical formulations suffer from the highly symmetric nature of the circuits and technicians. A new symmetry-breaking model was developed, leading to significant gains in computational effort. Yet again there is no guarantee that the obtained solutions are indeed optimal/feasible for the individual circuit assignments.Subsequently, Jaumard and Pouya (2018) designed a two-phase CG-based heuristic for the planning of real-size networks. A greedy algorithm was proposed byDawadi et al. (2021)for the cost minimization of a multi-technology migration problem. With the objective of minimizing the number of out-of-service sites and travels, Javad-Kalbasi and Valaee(2021)modeled the NMP as a binary quadratic program. They derived heuristic solutions using Digital Annealer and solved instances with up to 64 circuits.In this work, we develop an exact decomposition framework for the NMP. To the best of our knowledge, both in the context of the telecommunications problems and as a VRPS, this is the first method that exactly solves the NMP with a certificate of optimality/infeasibility. We decompose the NMP into three problems and link them all by designing an LBBD algorithm.By doing so, we are able to leverage the power of different solution techniques for linear programming and combinatorial optimization, and delegate the task of solving each problem to the most suitable optimization paradigm. Numerical experiments defined over various real networks demonstrate the effectiveness of our algorithm in obtaining quality solutions with 4 reasonable computational effort. Given that the NMP can be viewed as an SNRP, our method can also be adapted to a wide class of integer programming problems.Contributions. The contributions of our work are summarized as follows.• We develop the first exact solution method for the network migration problem in order to find the optimal planning solutions, i.e., the order of circuit upgrades, along with the technician assignment and routing decisions. Although our solution framework is developed for the NMP, it is also applicable for the VRP with node synchronization. Introduction In telecommunication industries, network migration is the process of upgrading the existing infrastructure of a deployed network. A telecommunication network is composed of a set of sites (demand points), and circuits that transmit the traffic between the sites. Migration of such a network is performed by upgrading the circuits one by one. In order to upgrade every circuit, two synchronized technicians migrate its two endpoints within a time window. The goal of the network migration problem (NMP) is to find the upgrade order of these circuits such that the associated costs are minimized. Migration of a network is a strategic • In order to reduce the computational effort of the LBBD master problem, we further decompose it via the Benders decomposition, resulting in a mixed-integer program (MIP) as the master problem, and a CP-based CG formulation as its subproblem. Additionally, we augment this CG formulation with an auxiliary MIP subproblem, leading to a hybrid CP/MIP-based CG model that significantly improves its performance. • Considering the planning nature of our LBBD subproblem, we propose a CP model that, given the number of migrated circuits in a maintenance window, decides on the optimal technician assignments, order of circuit upgrades, and travel routes, if any. • For our LBBD-based decomposition framework, we design valid feasibility and optimality cuts that guarantee the convergence of the algorithm. We also characterize a set of solutions other than the candidate for which our optimality cuts are tight. • We evaluate the proposed LBBD algorithm on instances defined over six real backbone and regional networks and provide detailed algorithmic analysis and discussions on the implementation choices, along with managerial insights on the trade-offs among the migration cost, resource usage, and the duration of the migration. The remainder of the paper is organized as follows. In Section 2 we present the problem statement and its mathematical formulation. In Section 3 we develop an LBBD solution framework for solving the NMP. In Section 4 we evaluate the performance of our algorithm on benchmark networks and provide managerial insights. Section 5 concludes the paper. Problem Description In this section, we formally describe the problem and introduce the sets and parameters. Next, we formulate the NMP as a CG-based integer linear program (ILP). Problem Statement The network migration problem is defined on a telecommunication network represented by a set of sites S and a set of circuits C between the site pairs {s, s } ∈ S p . Every site s An example of a telecommunication network for the network migration problem is located in a geographical region r ∈ R (e.g., a city). To each region r, η tech r number of technicians are assigned. Technicians assigned to a given region r can only work in that region. Considering that migrations often occur during low traffic periods (mostly nights), and the distance between the regions may require long-haul flights, employing local technicians is the safest option to minimize unforeseen impacts of the travels on migration planning. A circuit is migrated by disconnecting its endpoints from the old equipment and connecting them to the new one. These operations are performed by two technicians within the same maintenance window, each working at one endpoint. Additionally, every circuit migration requires an engineer that coordinates the technicians remotely and does not need to be present in the working site. There are at most η eng engineers available, and every engineer can coordinate up to α eng technicians. Figure 1 demonstrates an example of such a network with 5 regions, 9 sites, and 20 circuits. Migration of the network is performed during a maintenance window, which is a period of time usually at night or a specially low traffic time on the network and also the roads (in case of travel between the sites). Every maintenance window w ∈ W has a predefined duration e.g., 8 hours, and all operations have to be completed within this duration. Time required to migrate a circuit c ∈ C is θ. Since migrating every circuit results in a short disruption in the network and the number of disruptions cannot violate clients' Service Level Agreement (SLA) (Fawaz et al. 2004), there is a limit η cir on the number of migrated circuits per maintenance window. Figure 2 A possible solution as a subset of shifts A technician working in a given region r during maintenance window w is responsible for a shift. A shift is defined as a set of circuit endpoints migrated by a single technician during a maintenance window, together with any travels between the sites. Figure 2 represents a subset of possible shifts as the solution of the NMP. This solution considers 3 shifts (for 3 technicians) in region 1, 1 shift in region 2, 1 shift in region 3 that includes a travel from site 6 to site 5, 1 shift in region 4 and 2 shifts in region 5. ∆ is a given set of possible shift durations, e.g. {6h, 8h}. One reason for having multiple durations is related to the payment policy. Technicians should be paid for a minimum number of hours per shift. For example, if a technician works for any time less than 6 hours, they will be paid for the full 6 hours, while another technician working longer than 6 hours will be paid for 8 hours. In addition, access to the sites and the time spent at the sites should be within the SLA. Having multiple shift durations helps to avoid requesting unnecessary long access periods. Migration costs include payments to the technicians and engineers. The network migration problem is to determine the order of upgrading the circuits in order to minimize the migration costs, i.e., building a set of minimum-cost plans for the technicians. We use relational operators for element-wise comparison of two vectors. Problem Formulation We model the NMP as an ILP that returns a planning solution consisting of a set of shifts. The proposed formulation is amenable to the LBBD framework, meaning that we can decouple the problem into smaller subproblems that are easier to solve. To begin with, assume that we have a set Γ of all possible shifts (in Section 3.2.1 we implicitly enumerate this set). For a shift γ ∈ Γ, the decision variable z γ ∈ Z + determines the number of times γ is assigned to the technicians. Every shift γ is characterized by (i) ∆ γ shift its duration, (ii) n γ ss the number of circuit endpoints migrated between the pair of sites {s, s }, and (iii) n γ cir the total number of migrated circuit endpoints in the shift. The total set of shifts is denoted by Γ = w∈W Γ w = r∈R,w∈W Γ rw , where Γ w is the set of shifts for a maintenance window w and Γ rw is the set of shifts for a technician located in region r during maintenance window w. Denote by m ss w ∈ Z + , a decision variable that determines the number of circuits between {s, s } migrated during w. The NMP is formulated as: min cost γ∈Γ ∆ γ shift z γ (1a) s.t. w∈W m ss w ≥ φ ss {s, s } ∈ S p , s < s (1b) m ss w = m s sw {s, s } ∈ S p , s < s , w ∈ W (1c) {s,s }∈Sp m ss w ≤ 2η cir w ∈ W (1d) γ∈Γw n γ ss z γ = m ss w {s, s } ∈ S p , w ∈ W (1e) γ∈Γrw z γ ≤ η tech r r ∈ R, w ∈ W (1f) γ∈Γw z γ ≤ α eng η eng w ∈ W (1g) z ∈ Z |Γ| + , m ∈ Z |Sp|×|W| + ,(1h) where cost = cost tech + cost eng α eng . The objective function (1a) is the cost of the NMP, which is defined as the total technician and engineer costs over the duration of the migration. Constraints (1b) assure that all circuits between every two sites s and s are migrated. Constraints (1c) enforce the number of migrated circuits from s to s in w to be equal to the number of circuits migrated from s to s. Constraints (1d) establish the bound on the number of migrated circuits at every maintenance window. Through constraints (1e), variables m ss w are determined by aggregating over the number of migrated circuits between {s, s } during the shifts at w. Constraints (1f) and (1g) ensure that at w, the number of available technicians and engineers are respected. Constraints (1h) define the variable domains. The size of Γ, the set of all shifts, is an exponential function of the number of circuits, hence it is not reasonable (or even possible) to include them all in solving the model (1). Column generation is a method for implicitly enumerating such a large set of columns that relies on the duality theory for linear programming (LP). In the presence of integer decision variables, the branch-and-price (B&P) algorithm combines the branch-and-bound framework for solving a MIP with the CG procedure. The performance of a B&P depends on the strength of the LP bound, as well as the employed branching and search strategy. It has been observed that branching on the variables of the master problem associated with the generated columns is not efficient and results in an unbalanced tree (Vance 1998, Vanderbeck 2011. Branching on the aggregate variables of the original formulations, in case of identical subproblems, is not typically sufficient to eliminate all fractional solutions. Although this branching scheme theoretically does not guarantee the integrality of the solution, it experimentally returns the integral solution for some instances. Vanderbeck (2011) proposes a generic branching scheme based on the aggregated value of the original variables when returning to non-identical systems. In our preliminary experiments though, specially for small to medium-size instances, the quality of the lower bound was poor and did not improve adequately as the B&P proceeded. Therefore, since branching on neither the subproblem nor the master problem variables for the NMP solved any of our instances, we concluded that a pure B&P is not suitable to obtain exact solutions for the NMP. In the next section, we develop an LBBD framework that entails multiple levels of decomposition, with a proof of optimality if one exists, or infeasibility when it does not. Solution Method In what follows, we first discuss the general LBBD framework for the NMP, then we provide detailed discussions on each step of the algorithm. Logic-based Benders Decomposition In the problem formulation (1), decision variables z and m are only linked through (1e). Furthermore, except for the constraints (1b), the problem is decomposable by maintenance windows. Using these two points, we propose the following decomposition. Denote by SP LBBD w (m w ), the problem of generating a set of shifts with minimum cost for a given m w . Let η w be the migration cost at w. We can reformulate the model (1) as below: min w∈W η w : (1b) − (1d), η w ≥ SP LBBD w (m w ), w ∈ W, m ∈ Z |Sp|×|W| + . (2) Formulation (2) has the structure of a two-stage problem: the first-stage (master) problem decides on the number of migrated circuits between the site pairs at each maintenance window, along with an estimation on the cost of such a plan; the second-stage (recourse) problems verify if it is feasible to migrate the assigned number of circuits with the available resources, and if so what is the actual cost of this migration. The Benders decomposition (Benders 1962) is a well-established solution method for twostage linear programs with continuous second-stage variables. After decomposing the problem into a master problem and a subproblem, it iteratively approximates the optimal solution to the recourse problem via Benders feasibility and optimality cuts (see Section 3.2 for more details), derived using the LP duality theory. For two-stage problems with mixed-integer recourse decisions, logic-based Benders decomposition (Hooker and Ottosson 2003) is a generalization of the Benders decomposition (including its special case in the context of stochastic programming, the integer L-shaped method; see Laporte and Louveaux (1993), Angulo et al. (2016)). For the NMP, the LBBD master problem is: MP LBBD = min w∈W η w (3a) s.t. (1b) − (1d) (3b) (η w , m w ) ∈ Λ feas w , w ∈ W (3c) (η w , m w ) ∈ Λ opt w , w ∈ W (3d) m ∈ Z |Sp|×|W| + , η ≥ 0.(3e) where Λ feas w and Λ opt w are sets of feasibility and optimality cuts, respectively, and together represent an LBBD subproblem SP LBBD w (m w ). The LBBD starts by Λ feas w = Λ opt w = ∅ and at each iteration expands these sets with cuts if necessary until they are representative of the subproblem. The LBBD relies on logical reasoning for obtaining feasibility and optimality cuts, and as long as we can have feasibility and optimality certificates, the LBBD subproblem can have any form. Having such a generic framework, the LBBD depends on the modeler for designing problem-specific cuts and, unlike the Benders decomposition, does not have a readily available cut development mechanism. The strength of the LBBD lies in the fact that it can integrate various optimization paradigms, most notably mixed-integer programming and constraint programming (Jain andGrossmann 2001, Hooker et al. 2012). Our first-stage problem MP LBBD is a variant of an assignment problem, which is suitable for a MIP-based solution approach. On the other hand, our second-stage problem takes m w as an argument and looks for the best plan for such an assignment. CP is an optimization paradigm particularly powerful for planning and scheduling problems. Accordingly, in our LBBD solution framework for the NMP, we have a MIP model as the master problem, and |W| many CP models as the subproblems. In the remaining of this section, we first discuss the MP LBBD and develop two other levels of decomposition to make it more informed. Then, for a master problem solution m w , we formulate the SP LBBD w (m w ) as a CP model, followed by the description of valid LBBD feasibility and optimality cuts that guarantee the convergence of the algorithm to an optimal solution. The LBBD Master Problem: A Benders Decomposition In its current form, MP LBBD is oblivious to the structure of the NMP. In this section, our goal is to make the solutions of MP LBBD more intelligent before passing them on to the SP LBBD w (m w ). In the literature of LBBD, it has been observed that adding a relaxation of the subproblem considerably improves the performance of the algorithm (Elci and Hooker 2020). Often, this subproblem relaxation is in the form of an analytical expression based on the structure of the problem and is added as a bound to the master problem. We, however, resort to the LP relaxation of the subproblems to obtain valid inequalities for the MP LBBD . By relaxing the integrality constraints of the subproblems, we now have integer first-stage and continuous second-stage decision variables and the new problem is amenable to the Benders decomposition, with MP BD = MP LBBD as its master problem and the following subproblem for a maintenance window w: SP BD w (m w ) = min cost γ∈Γw ∆ γ shift z γ (4a) s.t. γ∈Γw n γ ss z γ ≥ m ss w {s, s } ∈ S p , w ∈ W (4b) (1f) − (1g) (4c) z ≥ 0.(4d) Considering that (4a) has an exponential number of variables, in the next section we apply Dantzig-Wolfe decomposition principles to develop a column generation procedure for systematically adding them to the set of columns. For a review on the Dantzig-Wolfe decomposition and column generation, unfamiliar reader may refer to Chvatal et al. (1983). The Benders Subproblem: A Column Generation Method. A CG solution method starts with solving the restricted master problem, defined as the original problem with a (potentially empty) subset of all the columns. Then, the optimal dual solutions are passed to a pricing problem (the CG subproblem) that checks their feasibility in the LP dual of the original problem, and if not, adds an improving column to the master problem. Once the pricing problem determines that a feasible dual solution is found, the CG stops as we have reached the optimality. By design, the pricing problem implicitly considers all the columns by using the properties that define valid columns of the master problem. Although the Dantzig-Wolfe decomposition lays out a precise scheme for decomposing a problem for the CG method, the master and pricing problems are often built by problem-specific modeling practices. In defining the "shifts" and z γ as the decision variables we have used such an approach which can directly be translated into the pricing problems. The CG master problem. Master problem of the NMP selects the best set of shifts for the technicians among a subset of columns Γ ⊆ Γ, and its pricing problems generate improving shifts. The CG master problem is: RMP CG w = min cost γ∈Γ ∆ γ shift z γ (5a) s.t. γ∈Γ w n γ ss z γ ≥ m ss w {s, s } ∈ S p (5b) γ∈Γ rw z γ ≤ η tech r r ∈ R (5c) γ∈Γ w z γ ≤ α eng η eng (5d) z ≥ 0.(5e) Constraints of RMP CG w correspond to the constraints (1e) -(1h), except for the set of columns Γ , and the fact that they are for a single maintenance window w. The CG pricing problem. In the definition of the CG subproblems, for the sake of brevity, we drop the index γ from the decision variables. Denote by π (5b) , π (5c) , π (5d) , the optimal dual solutions associated with constraints (5b), (5c) and (5d), respectively. The pricing problem generating a shift for region r and maintenance window w is as follows: SP CG rw = min cost∆ shift − s∈Sr s ∈S n ss π (5b) ss − π (5c) rw − π (5d) w (6a) s.t. {Constraints defining a valid shift} (6b) n ss ∈ Z + s ∈ S r , {s, s } ∈ S p (6c) ∆ shift ≥ 0. (6d) Objective function (6a) is the reduced-cost that determines if RMP CG w is at optimality. If not, the pricing problem generates a column that corresponds to a "valid" shift, and is characterized by its duration ∆ shift and the number of circuits migrated between each site pair. A shift is made of a sequence of site visits by the technician. So constraints (6b) define a valid shift as a connected path over the sites in the region r such that the duration of the shift does not exceed the maximum possible duration of the maintenance window. Furthermore, as the endpoints of a circuit should be migrated by two technicians in the same maintenance window, at most one of the endpoints of every circuit c ∈ C ss can be migrated in a shift. In Section 3.3, we present a constraint programming model for generating a set of valid shifts in a maintenance window w, for all the regions and their technicians. Our preliminary experiments revealed that, because of the routing decisions in the pricing problems and the presence of loop elimination constraints, SP CG rw as a CP model performs much better than a MIP. Therefore, in lieu of model (6), we fix the region in the CP model of Section 3.3 and solve it for one technician, with the objective function (6a). Remark 2. To accelerate the solution process, we use a hybrid CG, where first an auxiliary pricing problem generates improving columns with "ordered paths" that only have (s, s ), s < s links (model description is given in the e-companion). After the auxiliary problem converges, we solve SP CG rw to verify the optimality. We observed that this two-subproblem strategy greatly improves the performance of the algorithm. The reason is that on backbone networks, it is quite possible that many regions have a few number of sites. If a region has up to two sites, the auxiliary problem alone guarantees the optimality. Let SP CG rw be the optimal solution of the SP CG rw . If SP CG rw ≥ 0, ∀r ∈ R, then the CG procedure stops. Otherwise, for each r ∈ R with SP CG rw < 0 we add the generated column to Γ and repeat the process. As a result of having the constraints (5b), we require an initial set of columns Γ = ∅ that make the RMP CG w feasible. The usual approach for generating such Γ is to go through an initial phase (INIT) where an artificial non-negative decision variable ρ is added to each "≥" constraint with a positive right-hand-side, and the objective function is replaced with 1 ρ. If the CG procedure for the new problem stops with an optimal value equal to zero, then the artificial decision variables are removed from the problem, the original objective function is brought back and the generated columns are selected as Γ . Otherwise, if the optimal objective value is positive, we can conclude that the original problem is infeasible. Benders Cuts. Depending on the status of the RMP CG w after solving, we might need to add feasibility (optimality) cuts to Λ feas w (Λ opt w ). Benders decomposition provides us with off-the-shelf cuts through the dual solutions of the subproblems. For our problem, because the subproblems are solved via CG, the feasibility cuts are not immediately clear. Next, we present the Benders feasibility and optimality cuts for the NMP and show that they are valid, despite being obtained from a restricted set of columns in the SP CG rw . Benders feasibility cut. Assume that, at the end of the INIT phase, the CG procedure stops with ρ = 0, and π (5b) , π (5c) , π (5d) are returned from the (modified) RMP CG w . The Benders feasibility cut to be added to Λ feas w is as follows: 0 ≥ {s,s }∈Sp π (5b) ss m ss w + r∈R π (5c) r η tech r + π (5d) α eng η eng(7) To show that (7) is indeed a feasibility cut for the MP BD and cuts off the infeasible solution m w , we first prove that (π (5b) , π (5c) , π (5d) ) constitutes a certificate of infeasibility for the MP BD , even if it is derived from the RMP CG w with Γ ⊆ Γ. In the following theorem, we show that this is true for any Dantizig-Wolfe decomposition at the end of the INIT phase. Theorem 1. Consider P (I ), the restricted master problem of a Dantzig-Wolfe decomposition at the end of the INIT phase, and P (I , c I ), the problem that is obtained by removing the artificial variables ρ from P (I ) and bringing back the original objective function: P (I ) = min 1 ρ s.t. A I x I + ρ ≥ b x I , ρ ≥ 0, P (I , c I ) = min c I x I s.t. A I x I ≥ b x I ≥ 0, where ∅ = I ⊆ I, and c I , A I are the cost vector and columns associated with I . π is a certificate of infeasibility for P (I , c I ) only if it is a certificate of infeasibility for P (I, c I ). Proof. We first find the certificate of infeasibility for P (I , c I ). Then we show that it is also a certificate of infeasibility for P (I, c I ). Let (x I , ρ) denote the optimal solution of P (I ). 1. From (a variant of) the Farkas Lemma we know that exactly one of the following system of inequalities has a solution (Matousek and Gärtner 2007): (I) A I x I ≥ b, x ≥ 0, (II) π A I ≤ 0, π b > 0, π ≥ 0. If ρ = 0, then {x : A I x ≥ b, x ≥ 0} = ∅ and P (I , c I ) is infeasible. Therefore, there exits a π 0 that satisfies the inequalities of (II) and is a certificate of infeasibility for P (I , c I ). Consider the dual of P (I ) as follows: D(I ) = max{π b : π A I ≤ 0 , π ≤ 1, π ≥ 0}. From ρ = 0, we have 1 ρ = π b > 0, with π the optimal solution of D(I ). Clearly π satisfies (II) and we can set π 0 = π. So π is the Farkas certificate of P (I , c I ). 2. Let D(I, 0) = max{π b : π A I ≤ 0 , π ≥ 0} be the dual of P (I, 0). Since P (I ) is the last restricted master problem at phase one, no improving column with negative reduced-cost is found through the pricing problem with the dual solution of P (I ), i.e., 0 − π A I ≥ 0 meaning that π is feasible for D(I, 0). As ρ = 0, we have {x : A I x ≥ b, x ≥ 0} = ∅, P (I, c I ) is infeasible, and so is P (I, 0). Accordingly, using the Farkas Lemma one more time, there exists aπ such thatπ A I ≤ 0,π b > 0,π ≥ 0. Consider the ray π + λπ, λ ≥ 0. Then: (π + λπ) A I = ≤0 π A I +λ ≤0 π A I =⇒ (π + λπ) A I ≤ 0, (π + λπ) b = >0 π b +λ >0 π b =⇒ (π + λπ) b > 0. Therefore, π + λπ is feasible for D(I, 0) with a positive objective value (π + λπ) b. As λ → +∞, so does (π + λπ) b, proving that π is a certificate of unboundedness for D(I, 0), hence a certificate of infeasibility for P (I, 0) and P (I, c I ). From the above discussions, (π (5b) , π (5c) , π (5d) ) is a proof of infeasibility for MP BD with 0 < {s,s }∈Sp π (5b) ss m ss w + r∈R π (5c) w η tech r + π (5d) α eng η eng and can be removed using the inequality (7). Here we should mention that, we can also add the LBBD feasibility cuts (Section 3.4.1) for cutting infeasible solutions. However, as will be discussed later, our LBBD feasibility cuts are costly and inequalities (7) improve the overall efficiency of the method. Benders optimality cut. If at the end of the INIT phase, ρ = 0, then SP CG rw is feasible. Now we should check if η w is an accurate estimate of the SP BD w (m w ). Because the CG procedure stops at optimality, we can treat the missing columns in Γ as non-basic variables. Therefore it is clear that the optimal dual solutions from solving RMP CG w with Γ are the same as the optimal dual solutions of SP BD w (m w ). If η w ≥ SP CG rw , then (η w , m w ) is feasible in MP BD . Otherwise, we cut it off by adding the following inequality to Λ opt w : η w ≥ SP CG rw − {s,s }∈Sp π (5b) ss (m ss w − m ss w ).(8) The LBBD Subproblem: A Constraint Programming Model With its roots in logic, CP is a modeling framework for combinatorial problems and has proven quite powerful for making planning, scheduling and routing decisions (Ciré et al. 2016). Integration of CP and mathematical programming models through LBBD (Hooker et al. 2000), and CG ( For a maintenance window w, a CP model serving as our LBBD subproblem creates a plan, i.e., a set of shifts corresponding to a set of technicians working during the maintenance window w. For example, assume that the solution illustrated in Figure 2 is planned for one maintenance window. A plan for this solution consists of determining for each technician, a connected path along with the number of circuit endpoints migrated between each site pair. One possible such plan is depicted in Figure 3, where a set of boxes in front of a technician t shows its shift for the current maintenance window. Every box includes a site s and the number of circuit endpoints n ss that t migrates in this shift. We see that in the CP model we need to treat the technicians as individual entities, unlike the engineers and circuit endpoints. Therefore, for each region r at maintenance window w, we define T rw = {1, . . . , η tech r } as its set of available technicians, and the CP model generates shifts for the technicians that belong to r∈R T rw . It is clear that the newly introduced technician symmetry stays within the CP model, since the only link between MP LBBD and its subproblems is through the m decision variables defined for site pairs and independent of the technician (see Section 3.4). Next, we describe the variable types, functions and global constraints used in the CP model, followed by the model description. While we use names and conventions of the CP s 3 : n 3,1 = 1, n 3,2 = 3 No shifts assigned. s 4 : n 4,2 = 1, n 4,5 = 2, n 4,6 = 1, n 4,9 = 2 s 6 : n 6,4 = 1, n 6,7 = 2 Travel s 5 : n 5,1 = 3, n 5,4 = 2 s 9 : n 9,4 = 2, n 9,8 = 1 s 7 : n 7,2 = 1, n 7,6 = 2, n 7,8 = 3 s 8 : n 8,7 = 3, n 8,9 = 1 -Span(a, X ) Enforces the interval variable a to span the set of interval variables X . -NoOverlap( , T ) Enforces the intervals in the sequence to be disjoint, while taking into account the transition times T . -IfThen(p, q) Implements the logical constraint p → q. In addition to the sets and parameters described in Section 2, we need the variables defined in Table 3. Note that, although the variables are initially defined for all technicians, they are mostly "optional", meaning that if no shift is assigned to a technician t from a region r, then its associated variables x rts , wtime rt , ∆ rt shift , n s rts and n sp rtss are absent from the solution and vice-versa. For interval variables x rts , wtime rt , the initial domain of their "start" and "end" s.t. NoOverlap(seq rt , T ) t ∈ T rw , r ∈ R (9b) Span(wtime rt , {x rts : s ∈ S r }) t ∈ T rw , r ∈ R (9c) IfThen(PresenceOf(wtime rt ), EndOf(wtime rt ) ≤ ∆ rt shift ) t ∈ T rw , r ∈ R(9d) PresenceOf(x rts ) = PresenceOf(n s rts ) s ∈ S r , t ∈ T rw , r ∈ R (9e) IfThen(PresenceOf(x rts ), LengthOf(x rts ) = θn s rts ) s ∈ S r , t ∈ T rw , r ∈ R (9f) LBBD Cuts Depending on the status of the CP w (m w ), we might need LBBD feasibility/optimality cuts to proceed. Next, we design these cuts and discuss their merits and drawbacks. 3.4.1. LBBD Feasibility Cuts. Infeasibility of CP w (m w ) implies that there is not enough resources (technicians and/or engineers) to migrate m w circuit endpoints in w. Using LBBD feasibility cuts, we remove this solution from the master problem MP LBBD , along with any other solution that requires at least the same amount of resources as m w . MP LBBD solutions are general integer, so the usual type of (strengthened) no-good cuts are not applicable to our problem. As such, for each generated m ss w we create a new binary variable to compare the solution m ss w with m ss w . At iteration τ of the LBBD algorithm, we add the following LBBD feasibility cuts to Λ feas w : m ss w − (φ ss + 1)κ τ ss w ≤ m ss w − 1, {s, s } ∈ S p : m ss w > 0 (10a) {s,s }∈Sp m ss w >0 κ τ ss w ≤ |S p | − 1 (10b) κ τ w ∈ {0, 1} |Sp|(10c) As the number of iterations grows, the number of binary variables created via (10) and the additional constraints can grow prohibitively large. However, as seen in our numerical experiments, the fact that the MP LBBD solutions first go through the Benders decomposition phase (with its own feasibility checks), refines them and reduces the number of required feasibility cuts. Additionally, (10) cut-off not just one solution, but a potentially large set of solutions, further reducing the chance of needing new feasibility cuts. Proof of Proposition 1 explains the logic behind the design of our LBBD feasibility cuts. Proposition 1. The set of inequalities (10) are valid LBBD feasibility cuts for MP LBBD . Proof. We introduce a vector of binary variables κ τ ss w , {s, s } ∈ S p that takes 1 if m ss w ≥ m ss w . We have the following logical relationships: m ss w ≥ m ss w =⇒ κ τ ss w = 1 ≡ κ τ ss w = 0 =⇒ m ss w < m ss w (contraposition) ≡ κ τ ss w = 0 =⇒ m ss w ≤ m ss w − 1 (m ss w ∈ Z + ) ≡ m ss w − (φ ss + 1)κ τ ss w ≤ m ss w − 1 (m ss w ≤ φ ss ) The last inequality becomes redundant if κ τ ss w = 1. In order to cut-off any solution m w ≥ m w , it is sufficient to make sure that the number of migrated circuit endpoints between at least one site pair {s, s } is less than m ss w , which completes the proof. LBBD Optimality Cuts. Assuming that CP w (m w ) is feasible and solved to optimality, let CP w (m w ) be its optimal objective value. If η w ≥ CP w (m w ) for a w ∈ W, then (m, η w ) is feasible for MP LBBD . If η w ≥ CP w (m w ) for all w ∈ W, then m is optimal for the problem and the algorithm stops. Otherwise, for any w with η w < CP w (m w ), we cut-off the solution pair (η w , m w ). At iteration τ , we add the following LBBD optimality cuts to Λ opt w : (10a), (10c) (11a) η w ≥ CP w (m w ) 1 − {s,s }∈Sp m ss w >0 1 − κ τ ss w(11b) As with the feasibility cuts (10), we are creating a large number of new binary variables and constraints. Once more, our numerical analysis show that, for the considered instances, the number of iterations required for the algorithm to converge to a reasonable gap is small enough that MP LBBD remains scalable despite the large number of binary decision variables and constraints added at each iteration. Proposition 2. The set of inequalities (11) are valid LBBD optimality cuts for MP LBBD . Proof. The LBBD optimality cut is valid if (i) it can determine whether the solution m w is optimal, (ii) returns a valid bound for the solutions other than m w , and (iii) it is tight at m w . Because the CP w (m)'s cost function is non-decreasing, for any m w ≥ m w we have CP w (m) ≥ CP w (m w ). With the binary variables κ τ ss w as defined in the proof of Proposition 1, it is clear that inequalities (11) are tight at m w . It also imposes a negative bound on m w < m w , and CP w (m w ) if m w > m w , which completes the proof. We can also characterize a set of solutions other than m w for which the cuts (11) are tight: Ifm ss w − m ss w > 0, then at least θ(m ss w − m ss w ) minutes should be added to wtime rt for some technician t. Let n s rts , wtime rt , ∆ rt shift be the solutions from the CP model. Define a bipartite graph G = (V 1 ∪ V 2 , E), with the following set of nodes: V 1 = r∈R T rw \ {t ∈ T : n s rts = 0}, V 2 = {{s, s } ∈ S p :m ss w − m ss w > 0}. In other words, V 1 is the set of working technicians and V 2 is the set of site pairs that need extra circuit migration in the new solution. We build the set of links as follows: E = (t, {s, s }) ∈ V 1 × V 2 : ∃t ∈ T Srw , , wtime rt + θ(m ss w − m ss w ) ≤ ∆ rt shift , i.e, we create a link between a technician that is already planned to work at s and has enough spare time within its shift to migrate the remaining circuits. If there exists a maximum matching of size |V 2 |, then CP w (m w ) does not change and (11) is tight atm w . Note that, the proposition is not biconditional. Numerical Results In this section we have evaluated our LBBD method with the NMP instances that are built over six networks of different sizes, with some deployed over a continent and another serving a region. In Section 4.1, we provide some details on the characteristics of the instances. This is followed by algorithmic and cost analysis discussions in Sections 4.3 and 4.4, respectively. Figure 4 Overall framework of the LBBD method for the NMP. Data Sets In Figure 5 we have reproduced the topology of six real optical fiber networks as reported by Knight et al. (2011). Considering the nodes of the graphs as the sites of the network, we use the nodes' coordinates to partition the sites into regions, using the quality threshold clustering algorithm with 80km threshold. Detailed characteristics of the base networks are given in Table 4, including the number of sites, regions, maximum number of sites in a region, their geographic extent and location. Let µ EP/s and σ EP/s be the mean and standard deviation of Technicians may work in shifts of ∆ shift ∈ {6, 8} hours and an engineer can coordinate 5 technicians at a time. These instance parameters are based on the inputs of a real data set analyzed in Pouya (2018). Implementation Details Programs are written in Python and run on Niagara supercomputer servers ( For implementation of the Benders decomposition, we have used Gurobi's callback feature such that, while solving the MP BD , at every integer node of the branch-and-bound tree, we separate the Benders optimality and feasibility cuts by solving the CG subproblems. In implementing the LBBD method, we have opted for a cutting plane framework where every time after the convergence of the Benders decomposition method, we solve a CP problem to separate the violated LBBD optimality and feasibility cuts, if any. The reason for employing the cutting plane framework in this case is that our LBBD feasibility and optimality cuts introduce new binary variables. This is not allowed in the Gurobi's callback implementation, since introducing new integer variables changes the structure of the branch-and-bound tree. The programs stop whenever the algorithm reaches an optimality gap under 10%, or the running time passes 3 hours, whichever happens first. Our preliminary experiments showed that at the beginning of the solution process, early stopping of the Benders decomposition step reduces the computational effort, as it provides the MP LBBD with informative cuts even without having to solve it to optimality. Therefore, in solving the MP LBBD (i.e., the Benders decomposition phase) we start with a MIP gap of 10%, and reduce this by 5% at each iteration. Note that in these steps we should use the best lower bound on the solution in measuring the gap, which gives us a valid lower bound on the NMP instances. We have also made the following implementation choices and analyzed their impacts in the next section: • In solving the CG subproblems, we use the hybrid CG as explained in Remark 2. • Despite the fact that the CG subproblems are frequently solved during the algorithm, we do not remove the generated columns from Γ . • Considering the symmetry between the maintenance windows, we propagate the Benders optimality cuts, i.e., every time a Benders optimality cut is separated for a certain maintenance window w, we also modify and apply it for all w ∈ W \ {w}. Algorithmic Analysis In the following sections, we first examine our implementation choices, and based on the best obtained settings, solve the NMP instances. We break down the algorithmic analysis of the method into each solution paradigm and study their share of the computational effort. 4.3.1. Performance of the Hybrid CG. We have solved the first instances of EUNet-works5, NextGen5, Pionier5, Sago5, Savvis5 and VisionNet5 with a pure CP-based CG, as well as the hybrid version where first a MIP is solved to generated improving columns with ordered paths, and then the CP pricing problem (6) continues the process by generating columns with general paths or verifies the optimality. The results in Table 6 show that the hybrid CG reduces the computational effort of solving the MP LBBD by orders of magnitude. Most significantly, in 3 hours the pure CP-based method is unable to achieve a gap below 10% for Pionier5, Sago5 and VisionNet5, while the hybrid CG in most cases closes the gap in a short amount of time. As mentioned in Section 3.2.1, in our backbone networks many regions have one or two sites, so the MIP pricing problem is already sufficient to guarantee their optimality. This greatly benefits the geographically larger networks EUNetworks, NextGen and Savvis to the point that the instances based on the last two networks can be solved to optimality by only solving the MIP subproblem. The comparatively remarkable performance of the hybrid CG can further be explained by a common practice in the column generation literature, where sub-optimal columns are also added to the restricted master problem in order to make it more descriptive and obtain more informative dual solutions for the pricing problem. From Table 6 we clearly see that the number of generated columns in the hybrid CG is generally much larger, meaning that the MIP subproblem is returning many columns with sub-optimal reduced-costs, and we are practically following the mentioned strategy in increasing the number of columns. CG Column Management The CG subproblems are solved frequently, at every integer node of the branch-and-bound tree every time the algorithm reaches the Benders decomposition stage. Furthermore, the models are modified constantly, with different righthands sides in each node. It is not immediately clear whether keeping all the columns leads to a speedup of the method or overloading the RMP CG w . For the same instances of Section 4.3.1, we have evaluated the impact of removing the columns after processing an integer node of the tree. Results in Table 7 show that the overall solution process indeed benefits from previously generated columns, and we are still able to efficiently solve the RMP CG w . However, the results also show that solving the CG models with an initial set Γ = ∅ is still manageable and if the size of the instances leads to poor performance of the RMP CG w , we can switch to removing the columns or design a hybrid column management strategy. 4.3.3. Propagation of the Optimality Cuts. The symmetry among the maintenance windows w ∈ W in the NMP signifies that if an inequality η w ≥ g(η w , m w )m w cuts off the solution pair (η w , m w ) from the solution space, a similar inequality η w ≥ g(η w ,m w )m w removes (η w ,m w ) ifη w = η w ,m w = m w . Therefore, any optimality cut obtained for one w can be modified to use for all maintenance windows. This is called Benders cut propagation (Roshanaei et al. 2017). In Table 8 we have examined the effect of propagating the Benders optimality cuts in solving the instances of Section 4.3.1 and compared the results with a version of the method that only uses the classic local Benders optimality cuts. Columns #Cut BD opt and #Cut LBBD opt respectively report the number of the Benders and LBBD optimality cuts at the end of the solution process. The results indicate that, the cut propagation is quite Table 5 and reported the confidence intervals on the optimality gaps, solution costs, the number of LBBD iterations and solution times in Table 9. In all instances, the LBBD method returns quality solutions with small gaps in a reasonable amount of time. The small number of iterations in the LBBD stage is a result of multiple levels of decomposition in the LBBD master problem, giving it enough information to pass on quality first-stage solutions to the subproblems. This is quite valuable, considering the expensive design of the LBBD feasibility and optimality cuts. In Figure 6, we have further inspected the solution times. The box plots in this figure represent the distribution of the time that each step of the algorithm spends for solving the six instances of each dataset. We observe that in the smaller instances the MP LBBD run time is essentially dominated by the effort to solve the CG models. However, in larger instances with more endpoints, the time spent in solving the MP BD apart from the CG models becomes increasingly significant. In contrast, as the number of the endpoints increases, with the exception of EUNetworks9, the computational effort by the CP models remains relatively stable with a mild increase. The exponential growth in the solution time of the MP LBBD is expected, considering the need to add more feasibility and optimality cuts, as reported in the e-companion. It is worth noting that, none of the steps of the LBBD method dominates the others in all the nicians, increasing η cir from 30 to 40 lowers the number of the shifts and migration costs. Finding the right amount of workforce is one of the main concerns of the network operators as it comes with logistics and hidden considerations. While a solution with higher cost does not necessarily imply having more shifts, fewer shifts might be considered an advantage as it requires fewer personnel and less human traffic in the sites. The distribution of the shifts, illustrated in Figure 7b, shows that the majority of the technicians are assigned shorter than 2-hour shifts if possible. This is the result of either shortage of technicians or circuits in the targeted regions. Since offering more technicians has not improved the migration costs, we can conclude that the large number of the 2-hour shifts is due to the availability of the circuit endpoints in the region of the working technicians. Once the 2-hour shifts are not offered, 4-hour shifts become the obvious choice. By moving to 4-hour shifts, the number of the shifts, as expected, declines and we deal with fewer number of the technicians during the migration. However, as mentioned before, this does not mean a decrease in the migration costs since the shifts are the indicators of the payments, not the working time. Figure Cost analysis for EUNetworks during the shifts. We notice that, by setting the minimum shift duration to 4 hours, the working time percentage is between 40% -50% of a shift, while this number is above 60% for the settings with minimum shift duration of 2 hours. The increase of the costs show that 4-hour shifts are on average less busy than 2-hour shifts. It should be mentioned that there are a few 6-hour shifts as well. Considering the negative impact of longer maintenance windows on the Service-Level-Agreement (SLA) between the network operators and the customers, the analysis indicates that having 6-hour shifts does not help the migration process in neither costs nor the number of the shifts. As seen in Figure 7, we can build equivalent solutions with shorter shifts and ignore the 6-hour ones. Concluding Remarks Migrating legacy telecommunication networks to the latest technology involves planning synchronized technicians. We propose the first exact method for the network migration problem, a logic-based Benders decomposition method augmented by column generation and constraint programming models. We also make several algorithmic enhancements and considerably improve the basic version of the algorithm with the classical adaption of Benders decomposition and column generation. The proposed solution method can further be adapted to any vehicle routing problem with node synchronization constraints. Our evaluations on the instances generated over six real networks show that our method is effective in obtaining quality solutions, and that all the solution paradigms are contributing to the efficiency of the method. Future work includes improving the scalability of the algorithm for the networks with a larger number of circuit endpoints, particularly the design of less expensive LBBD optimality and feasibility cuts. Furthermore, new methodological developments are needed to build migration plans in the presence of uncertainty, which is an inherent feature of such planning problems. Acknowledgment Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund -Research Excellence; and the University of Toronto. e-Companion A. Auxiliary MIP Pricing Problem In addition to n ss and n cir described in Section 2.2, we define the new decision variables in Table A.1. Denote by π (5b) , π (5c) , π (5d) , the optimal dual solutions associated with constraints (5b), (5c), (5d), respectively. The pricing problem generating a shift for (r, w) is: where M ≥ 0 is a large number and S + = S ∪ {s src , s dst }, S + r = S r ∪ {s src , s dst }. s src and s dst are two dummy sites introduced as the beginning and end of a path. Objective function (A.1a) is the reduced-cost. Constraints (A.1b) ensure that at most one of the endpoints of every circuit c ∈ C ss can be migrated in every shift. Constraints (A.1c) determine the sites where a technician works in the current shift. Constraints (A.1d) assure that all migrated circuit endpoints are from the sites where the technician works in the current shift. Thanks to SP CG rw = B. Detailed Numerical Results Figure 1 1Figure 1 An example of a telecommunication network for the network migration problem Rousseau et al. 2004), often results in solution algorithms that outperform methods relying solely on either of the two. Hooker and van Hoeve (2018) provide a review on integration of CP and Operations Research. In this section, considering the presence of planning and routing decisions, we formulate the SP LBBD w (m w ) as a CP model. Figure 3 A 3feasible plan for the solution presented inFigure 2Optimizer(Laborie et al. 2018), equivalent notions exist for other notable CP solvers such as the CP-SAT from Google OR Tools (Perron and Furnon 2019).-IntegerVar() An integer decision variable. If defined as Optional, it can be absent from the CP solution.-IntervalVar() An interval decision variable, modeling a time interval characterized by start, end and length. It can be optional.-SequenceVar(X ) A sequence decision variable defining a total order over a set of interval variables X . If all members of X are absent, the sequence becomes empty.-PresenceOf(a) Returns 1 if a is present in the solution, 0 otherwise.-EndOf(a) Returns the finish time of the task if the interval variable a is present in the solution.-LengthOf(a) Returns the time spent on the task if the interval variable a is present in the solution. times are [0, ∆], the maximum time interval possible for a maintenance window. The initial domain of seq rt is S({x rts : s ∈ S r }), the set of permutations from the interval variables x rts . With the objective of minimizing the cost of a plan, the model SP LBBD w (m w ) = CP w (m w ) is Figure 4 4presents the overall solution framework. Our LBBD method converges with an exact solution, or returns "infeasible" if the original problem is infeasible. This follows from the integrality of m, which makes the set of feasible solutions for MP LBBD finite. Using the feasibility and optimality cuts(10)and(11), we either cut-off the solution passed by MP LBBD or stop the algorithm if no action is needed after checking the feasibility and optimality conditions. When the algorithm stops, if we do not have any incumbent m, the original problem is infeasible, otherwise the algorithm has reached to optimality. Figure 5 5Topology of the base networks and site pairs with at least one circuit.Table 5presents a summary of the characteristics of the instances, and the details are provided in the e-companion. For the travel time between the Loken et al. 2010, Ponce et al. 2019) using Gurobi 9.1.1 as the MIP solver (Gurobi Optimization, LLC 2021) and CP Optimizer 20.1 as the CP solver (IBM® ILOG® CP® Optimizer 2021). Figure 7 Figure 6 76demonstrates the impact of various parameters on the solutions of the EUNet-works5 (similar analyses for two other networks are provided in the e-companion). As shown in Figure 7a, although the objective function has not benefited from engaging more tech-Breakdown of the solution times spent at each step of the decomposition. The time to solve the MP LBBD includes the total solution time of the CG models. 7c displays the average working time (migrating circuit endpoints or traveling between the sites) ) Cost vs. efficient working timeFigure 7 ) Cost vs. efficient working time Figure B.2 Cost analysis for Savvis Table 1 1presents the parameters and notations used in our model. Number of circuits between sites s and s (φ ss = φ s s ) the size of a set, 1 and 0 respectively, for vectors of 1s and 0s, adjusted to the required size.Remark 1. Vectors, matrices, and scalars are represented by bold (a), capital (A) and lower-case with regular font (a), respectively. We use (.) for the transpose operator, |.| for Table 3 3Decision variables of the constraint programming modelVariable Type Initial domain Optional Description x rts IntervalVar [[0, ∆], [0, ∆]] Time plan for tech t of r at site s seq rt SequenceVar S({x rts : s ∈ S r }) × Order of sites visited by tech t of r wtime rt IntervalVar [[0, ∆], [0, ∆]] Working time of tech t of r ∆ rt shift IntegerVar ∆ Shift duration of tech t of r n s rts IntegerVar [0, s m ss w ] # of endpoints migrated by tech t of r at site s n sp rtss IntegerVar [0, m ss w ] # of endpoints migrated by tech t of r at site s for {s, s } By default, the interval variables in a SequenceVar are allowed to have overlaps. Using constraints (9b), we make sure that the time intervals a technician t spends at different sites are disjoint, and moreover these interval are defined apart enough such that there is time for the technician to travel from one site to another (as given by the matrix of travel times number of planned circuit migrations between the site pairs, as given by the master problem, is respected in the CP solution. Through constraints (9j), the number of technicians to be used in each plan is bounded above by the maximum number of available engineers.s :{s,s }∈Sp n sp rtss = n s rts s ∈ S r , t ∈ T rw , r ∈ R (9g) n sp rtss = n sp rts s {s, s } ∈ S p , t ∈ T rw , r ∈ R (9h) r∈R t∈Trw n sp rtss = m ss w {s, s } ∈ S p (9i) r∈R t∈Trw PresenceOf(wtime rt ) ≤ α eng η eng . (9j) T ). Constraints (9c) make the working time of the technicians consistent with the planned activities. For the working technicians in the current plan, constraints (9d) determine their shift duration from ∆. Constraints (9e) and (9f) together ensure that the circuit endpoints are only distributed among the working technicians and the time a technician spends at a site is the amount of time it needs for migrating the endpoints. Constraints (9g) determine the site pairs to which the migrated endpoints belong, while constraints (9h) prevent the technician to move both ends of the circuits for the same site pair. Via constraints (9i), the Table 4 4Characteristics of base networks EP/s and σ EP/s , we have generated 114 NMP instances over the base networks by creating six instances per network and (µ EP/s , σ EP/s ). The considered values for µ EP/s are comparable with µ EP/s = 6.18 of a real data set studied in Pouya et al. (2017) (Ciena data set I). The instances then are characterized by their base network, the number of circuit endpointsCharacteristics EUNetworks NextGen Pionier Sago Savvis VisionNet |S| 15 16 21 18 19 22 |R| 11 15 16 10 19 16 |S r | max 4 2 3 3 1 3 Geographic scope Continent Country Country Region Country Region Coverage Western Europe Australia Poland Florida & Georgia USA Montana the number of circuit endpoints at a site, respectively. Using a Lognormal distribution with parameters µ Table 5 5Characteristics of the NMP instancesDataset#Instances |W| η cir µ EP/s σ EP/s sites, we consider an average speed of 80km/h. The migration duration of a single endpoint is 20 minutes. Technicians and engineers are paid 108$ and 140$ per hour, respectively.Endpoints |S p | min max min max Table 6 6Impact of the hybrid CG on the overall performance of the methodDataset CP-sub CP/MIP-sub # Columns Time (s) Gap (%) # Columns Time (s) Gap (%) CG MP LBBD CG MP LBBD EUNetworks5 138 4184.5 4192.2 0.0% 509 627.1 643.0 0.0% NextGen5 45 304.2 305.8 0.0% 45 5.2 10.4 0.0% Pionier5 493 Timeout Timeout 47.4% 993 1039.4 1181.1 0.8% Sago5 314 Timeout Timeout 31.6% 141 655.2 692.5 0.0% Savvis5 57 1219.7 1257.3 0.0% 57 11.7 78.2 0.0% VisionNet5 512 Timeout Timeout 50.0% 1738 1878.2 2010.4 3.0% Table 7 7Performance of the LBBD method under two column management strategiesDataset CG w/o previous columns CG w/ all generated columns #iter. Time (s) #iter. Time (s) CG MP LBBD CG MP LBBD EUNetworks5 4 727.3 1294.1 4 627.1 643.0 NextGen5 2 307.6 308.8 14 5.2 10.4 Pionier5 6 1162.1 2132.9 5 1039.4 1181.1 Sago5 10 1601.3 2134.5 10 655.2 692.5 Savvis5 3 74.0 139.8 4 11.7 78.2 VisionNet5 14 2572.0 7933.0 7 1878.2 2010.4 Table 8 8Performance of the LBBD method with and without propagation of the Benders optimality cutsDataset LBBD with local Benders cuts LBBD with propagated Benders cuts #iter. #Cut BD opt #Cut LBBD opt Time (s) #iter. #Cut BD opt #Cut LBBD opt Time (s) EUNetworks5 3 49 81 624.9 4 42 139 643.0 NextGen5 4 16 170 8.8 14 32 579 10.4 Pionier5 9 241 567 1841.3 5 90 269 1181.1 Sago5 22 726 1297 3507.9 10 91 553 692.5 Savvis5 3 147 101 71.8 4 45 174 78.2 VisionNet5 13 723 836 5835.2 7 202 359 2010.4 effective in accelerating the solution process of the instances that are based on geographically smaller networks Pionier, Sago and VisionNet, while its performance is comparable to the classic version for the networks stretched over a larger geographic scope EUNetworks, NextGen and Savvis. Note that, the same arguments apply to the LBBD optimality cuts, yet any gain from propagating the cuts (11) is outweighed by their numbers. As a result, we only propagate the Benders optimality cuts in the LBBD method. 4.3.4. Computational Performance. With the implementation choices fixed by the anal- yses of Sections 4.3.1 -4.3.3, we have solved the NMP instances of Table 9 9Solution and computational effort studied instances. Accordingly, all solution paradigms play a determining role in the solution process and enhancements in any of them can lead to overall improvement of the method.Dataset Gap Cost ($) #iter. Time (s) Mean CI width Mean CI width Mean CI width Mean CI width EUNetworks5 0.6% 1.1% 10472.0 448.7 3.5 0.6 1124.1 268.6 EUNetworks6 1.0% 1.9% 11424.0 923.4 10.5 4.3 4607.1 2309.7 EUNetworks7 4.7% 1.9% 11877.3 648.9 6.5 2.0 1459.0 509.7 EUNetworks8 4.9% 2.9% 12693.3 611.3 7.3 1.6 2582.6 539.9 EUNetworks9 3.7% 3.0% 13328.0 615.6 10.8 6.1 5109.8 3376.7 NextGen5 1.2% 2.2% 13192.0 954.9 6.0 3.0 699.2 560.3 NextGen6 2.6% 2.1% 14960.0 1443.7 4.8 2.3 675.7 374.2 NextGen7 2.4% 1.7% 15640.0 243.3 5.3 2.0 997.9 415.8 NextGen8 1.8% 1.3% 16048.0 615.6 8.2 4.1 2445.6 399.6 NextGen9 4.1% 2.5% 16864.0 721.8 8.0 2.9 3120.4 1939.1 Pionier5 1.9% 1.7% 15912.0 821.6 5.7 1.6 1817.2 796.4 Pionier6 3.2% 2.1% 17136.0 533.1 9.2 2.0 4929.4 1385.8 Sago5 1.3% 1.5% 11832.0 625.1 8.3 2.3 2387.6 697.1 Sago6 5.2% 2.4% 12648.0 498.7 5.7 2.3 3635.1 2640.4 Savvis5 1.3% 1.4% 18224.0 1109.8 4.5 1.4 619.6 475.2 Savvis6 3.1% 2.1% 19312.0 721.8 6.0 1.6 855.6 460.3 Savvis7 4.2% 2.2% 19856.0 721.8 6.0 3.7 2472.6 2731.3 VisionNet5 4.9% 1.8% 17272.0 877.3 5.8 3.5 2259.2 899.9 VisionNet6 3.8% 1.9% 17952.0 377.0 7.3 3.1 3308.0 769.4 4.4. Network Migration Cost Analysis min cost∆ shift − s.t. h ss + h s s ≤ 1 s, s ∈ S r , s = s (A.1b) h s ≤ s ∈S h ss ≤ M h s s ∈ S r (A.1c) h ss ≤ n ss ≤ M h ss s ∈ S r , s ∈ S, s = s (A.1d) s + h s ) s, s ∈ S r , s > s (A.1f) T ss t ss ≤ ∆ shift (A.1i) δ∈∆ ∆ δ x δ = ∆ shift (A.1k) n ss , h ss ∈ Z + , h s ∈ {0, 1}s ∈ S r , s ∈ S, s = s (A.1l)s∈Sr s ∈S n ss π (5b) ss − π (5c) rw − π (5d) w (A.1a) s∈Sr t ssrc,s = 1, s∈Sr t s,sdst = 1 (A.1e) t ss ≤ 1 2 (h s ∈S + r s >s t ss + s ∈S + r s <s t s s = 2h s s ∈ S r (A.1g) s∈Sr s ∈Sr s >s t ss = s∈Sr h s − 1 (A.1h) θ s∈Sr s ∈S s =s n ss + s∈Sr s ∈Sr s >s δ∈∆ x δ = 1 (A.1j) Table A . 1 A1New decision variables for the CG subproblem Variable Type Description h ss Binary = 1 if the technician migrates at least one circuit endpoint in site s with the other endpoint in site s . h s Binary = 1 if the technician works in site s in this shift, 0 otherwise. t ss Binary = 1 if a travel from site s to site s occurs in the shift under construction, 0 otherwise. x δ Binary = 1 if the length of the shift under construction is equal to ∆ δ , 0 otherwise. constraints (A.1e), generated path in this configuration starts form dummy site s src and ends in dummy site s dst . Constraints (A.1f) guarantee that if a travel occurs between the two sites s and s , the technician works in both. Constraints (A.1g) determine the sites visited before and after every site s ∈ S r . Constraint (A.1h) ensures that we have a path linking all visited sites. Constraint (A.1i) makes sure that the shift duration does not exceed the maximum predefined duration. Constraints (A.1j) and (A.1k) together determine the duration of the shift. The rest of the constraints determine the variables domain. Table B . 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[]
[ "Cross-Lingual Transfer for Distantly Supervised and Low-resources Indonesian NER", "Cross-Lingual Transfer for Distantly Supervised and Low-resources Indonesian NER" ]
[ "Fariz Ikhwantri [email protected] \nKata Research Team\nKata\n" ]
[ "Kata Research Team\nKata" ]
[]
Manually annotated corpora for low-resource languages are usually small in quantity (gold), or large but distantly supervised (silver). Inspired by recent progress of injecting pre-trained language model (LM) on many Natural Language Processing (NLP) task, we proposed to fine-tune pre-trained language model from high-resources languages to low-resources languages to improve the performance of both scenarios. Our empirical experiment demonstrates significant improvement when fine-tuning pre-trained language model in cross-lingual transfer scenarios for small gold corpus and competitive results in large silver compare to supervised cross-lingual transfer, which will be useful when there is no parallel annotation in the same task to begin. We compare our proposed method of cross-lingual transfer using pre-trained LM to different sources of transfer such as mono-lingual LM and Part-of-Speech tagging (POS) in the downstream task of both large silver and small gold NER dataset by exploiting character-level input of bi-directional language model task.
10.1007/978-3-031-24337-0_29
[ "https://arxiv.org/pdf/1907.11158v1.pdf" ]
198,901,711
1907.11158
64a782efd81087b943dd9141fbd9c3b6e423ca03
Cross-Lingual Transfer for Distantly Supervised and Low-resources Indonesian NER Fariz Ikhwantri [email protected] Kata Research Team Kata Cross-Lingual Transfer for Distantly Supervised and Low-resources Indonesian NER Cross-lingual · Low Resource Languages · Named Entity Recognition Manually annotated corpora for low-resource languages are usually small in quantity (gold), or large but distantly supervised (silver). Inspired by recent progress of injecting pre-trained language model (LM) on many Natural Language Processing (NLP) task, we proposed to fine-tune pre-trained language model from high-resources languages to low-resources languages to improve the performance of both scenarios. Our empirical experiment demonstrates significant improvement when fine-tuning pre-trained language model in cross-lingual transfer scenarios for small gold corpus and competitive results in large silver compare to supervised cross-lingual transfer, which will be useful when there is no parallel annotation in the same task to begin. We compare our proposed method of cross-lingual transfer using pre-trained LM to different sources of transfer such as mono-lingual LM and Part-of-Speech tagging (POS) in the downstream task of both large silver and small gold NER dataset by exploiting character-level input of bi-directional language model task. Introduction Building large named entity gold corpus for low-resource languages is challenging because time consuming, limited availability of technical and local expertise. Thus, manually annotated corpora for low-resource languages are usually small, or large but automatically annotated. In most cases, the former are used as a test set to evaluate models trained on the latter one. To reduce the annotation efforts, previous works [19] utilized parallel corpus to project annotation from high-resource languages to low-resources languages using word-alignment. Another promising approach is to use knowledge base e.g DBPedia [1,2] or semi-structured on multi-lingual documents e.g Wikipedia [20] to generate named entity seed. Previous works on multi-lingual Wikipedia with motivation to acquire general corpus [20] and knowledge alignment between high-resource and low-resource languages encounter low recall problem because of incomplete and inconsistent alignments [22]. Some work on monolingual data with intensive rule labelling [1] and label validation [2] to create automatic annotation also face the same problem. Our contribution in this paper consists of two parts. First, we propose to improve NER performance of a low-resource language, namely Indonesian, trained on noisily annotated Wikipedia data by (1) fine-tuning English NER model, and (2) using contextual word representations derived from either English (EN), Indonesian (ID), or Cross-lingual (EN to ID) fine-tuning of pre-trained language models which exploit character-level input. Second, we analyze why using pretrained English language model from [26] yields improvement compare to monolingual Indonesian language model by looking at the dataset size, shared characteristic such as orthography, and its different like grammatical and morphological different to source language (English). We show that fine-tuning ELMo in unsupervised cross-lingual transfer can improve the performance significantly from baseline Stanford-NER [8], CNN-LSTM-CRF [18] and previous works using state-of-the-art multi-task NER with language modeling as an auxiliary task [16,29] trained on conversational texts, and its monolingual counterpart that is trained on different dataset size in the target language, which in our case is Indonesian unlabeled corpora retrieved from Wikipedia and news dataset [33]. Related Works Recently, Peters et al, [26] proposed to use pre-trained embedding from language model (ELMo) of large corpora for many NLP tasks such as NER [34], semantic role labeling [21], textual entailment [5], question answering [27] and sentiment analysis [31]. Motivated by deep character embedding for word representation that is useful in many linguistic probing and downstream tasks [24] and trained on large corpora using language model objective, we chose to investigate ELMo embedding as weight-initialization for NER task in a low-resource languages. Deep Character Embedding Character embedding is important to handle out-of-vocabulary problem such as in out-of-domain data [16] or another language with shared orthography [7]. The input words to Bidirectional LM, are computed by using concatenation of multiple convolution filters over sum of characters sequences of length [11,12], 2 depth highway layers [32] and a linear projection. The input to highway layers y k is the concatenation of y k,1 , ..., y k,h from H 1 , ..., H h as y k = [y k,1 , ..., y k,h ]. The output x h of highway layers of depth h are computed as in Equation (1), where T = σ(W T x h−1 + b T ) and, x 0 = y k as an input to the first highway layer. x h = T (W H x h−1 + b H ) + (1 − T ) x h−1(1) Bidirectional Language Models (BiLM) Language modeling (LM) computes the probability of token t k in sequence of tokens length N given the preceding tokens (t 1 , t 2 , ..., t k−1 ) as log p(t 1 , t 2 , ..., t N ) = N k=1 log p(t k |t 1 , t 2 , ..., t k−1 ). Reversed order LM, computes the probability of token t k in a sequence of tokens of length N given the succeeding tokens in log p(t k+1 , t k+2 , ..., t N ) as p(t 1 , t 2 , ..., t N ) = N k=1 log p(t k |t k+1 , t k+2 , ..., t N ). N k=1 (log p(t k |t 1 , t 2 , ..., t k−1 |θ x , − → θ LST M , θ s ) + log p(t k |t k+1 , t k+2 , ..., t N |θ x , ← − θ LST M , θ s ))(2) In downstream task such as NER sequence labeling, the output of ELMo [26] used for contextual word representation is the concatenation of projected highway layer [32] of Deep Character Embedding output [11,12], forward and backward output of LM-LSTM output of hidden layer. There are several ways to use ELMo layer for sequence labeling task, one of them is to use only last layers output of BiLM-LSTM. In this research, we only explore using last hidden layer of BiLM-LSTM [25]. Cross-lingual Transfer via Multi-Task Learning Cross-lingual transfer learning aims to leverage high-resources languages for low-resource languages. Yang et al., (2016) [36] proposed to transfer character embedding from English to Spanish because they shared same alphabet, while Cotterell et al., (2017) [7] study several languages transfer within the same family and orthographic representation using character embedding as shared input representation. In their proposed model, they shared character convolutions for composing words but not the LSTM layer. In the previous works above, the training process minimizes the joint loss of low-resource and high-resource languages as supervised multi-task learning (MTL) objective. However we found that due to grammatical and morphological different, it is more significant to do pre-training scenario (INIT) instead of joint-training objective. Proposed Method In this section we explain briefly our two proposed method. Our first proposed method extend supervised cross-lingual transfer using ELMo (Figure 1, left image). Our second proposed method fine-tune ELMo from English to Indonesian News dataset to use on distantly supervised and small gold Indonesian NER dataset. Supervised Cross-lingual Transfer with ELMo Alfina et al [2] observed that automatically annotated corpora fail to tag many orthographically similar entity of "America" to "Amerika" in Indonesian. We also confirmed that, there are many cases of false negative in orthographically Figure 1: Cross-lingual Transfer Learning by using Character-level pre-training. Left image, our proposed Unsupervised-Supervised Cross-lingual Transfer where we fine-tune ELMo on target task NER but on source language. Right image, our proposed Cross-lingual Language Model fine-tuning where we fine-tune ELMo on target language Indonesian similar LOCATION alias such as "Pacific" to "Pasifik" in Indonesian Wikipedia. Intuitively, we proposed to increase the recall performance due to many falsenegative error by supervised cross-lingual transfer [36] using pre-trained weights from state-of-the-arts NER model that uses Bidirectional language model. In the experiment result Table 4, the model corresponds to [English NER Sources] ELMo EN-1B Tokens from "Supervised CL Transfer with ELMo" scenario. Unsupervised Cross-lingual Transfer via ELMo fine-tuning We proposed to use a pre-trained language model of high-resource languages such as English in order to initialize better weights for low-resource languages. The cross-lingual transfer in our research is simple and almost the same as [10] with language modeling objectives but we replace English target vocab with Indonesian by random initialization (figure. 1, right image). Our motivation to propose this method is because we observed that there are only marginal improvement using monolingual Indonesia LM of 82M Tokens from Wikipedia compared to using English LM trained on 1B Tokens on applying ELMo to Distantly Supervised NER dataset. This might be attributed due to large difference of publicly available unlabeled corpus size, such as 82M in Indonesia Wikipedia 1 vs 1B Tokens of language model benchmark or 2.9B English Wikipedia available to train. In the experiment result Table 4, the model corresponds to ELMo EN-ID Transfer from one of the "CL via ELMo EN" group scenario. Dataset In this research, we used gold and silver annotation named entity corpus in English as sources in transfer learning. For target language, we used large silver annotation Indonesian as training dataset. We use two set of small clean < 40k tokens and ≤ 1.2k sentences as testing data in model comparison scenarios and another one as training data in ablation scenario for analysis, in addition of unlabeled data from Wikipedia and newswire. Gold named entity corpus CoNLL 2003 Dataset is well known shared task benchmark dataset in many NLP experiment. We follow the standard training, validation (testa), and test (testb) split scenario. The label consist of PERSON, LOCATION, ORG, and MISC. We experiment additional scenarios for cross-lingual transfer which ignore MISC labels. Clean 1.2K DBPedia Human annotations for a subset of the silver annotation corpus are important to measure the quality of that automatic annotation. Thus, we asked an Indonesian linguist to re-label the subset of data and compute the metrics for DEE, MDEE and +Gazz silver annotation dataset. The precision, recall and F1 score of the subset w.r.t our clean annotation can be found in Table 2. The clean annotation can be found at data supplementary material. We used this in-house annotation to do ablation analysis after training distantly supervised NER. We will made this subset of cleaned DBPedia Entity from noisy annotation publicly available in order to allow others to replicate our results in low-resources (gold) scenario. Noisy named entity corpus Wikipedia Named Entity WP2 and WP3 are two version of dataset [20]. The corpus obtained from this github repository 2 , because the initial link mentioned in the [20] is down. In this research we use these 2 version that corresponding to WP2 and WP3 of this silver standard named entity recognition dataset. We evaluate this dataset on CoNLL test [34] and WikiGold [3]. DBPedia Entity Expansion Our research used publicly available DBPedia Entity Expansion (DEE, Gold) [1] and Modified Rule (MDEE, +Gazetteers) [2] dataset for Indonesian. Interested readers should check the original references for further details. The dataset label statistics can be found in Table 1. We used the same test (Gold) in silver annotation Indonesian NER dataset. However, due to entity expansion technique, previous works [1,2] only considers Entity without their span (BIO) labels. In order to alleviate this difference, we transform the contiguous Entity with same label into BIO span. This rule based conversion does not seem affecting exact match span-based F1-metrics in distantly supervised scenarios when we reproduce the model in the same configuration. ID-POS Corpus The ID-POS corpus [28] contains 10K sentences of 250K tokens from news domain. There are 23 labels in the dataset. For POS tagging model, we train 5 model of 5-fold cross-validation following split dataset by [15]. For each fold of the models, we transfer the pre-trained weights into all NER train dataset in both large distantly supervised and low-resources gold NER scenarios. Unlabeled Corpus for Language Model Total number of vocabulary in Wikipedia Indonesia are 100k unique tokens from 2 millions total sentences with 82 millions total tokens. While total number of vocabulary in Kompas & Tempo dataset [33] are 130k tokens from 85k total sentences with 11 millions total tokens. Experiments Our main experiment for cross-lingual settings is Austronesian language, Indonesian. We choose Indonesian due to its language characteristics such as morphological distance from Indo-European family but same Latin alphabet orthography to English. It contains many loanwords for verb and named entity words from several languages. Most of the named entity are kept in the same form as the original language lexicon. It also categorized as low-resources as there is no large scale standardized and publicly available gold annotated dataset for NER task. We use AllenNLP [9] implementation for Baseline BiLSTM-CRF and extend our own implementation based on Supervised Cross-lingual Transfer, Crosslingual using ELMo from EN, Monolingual ELMo and Unsupervised-Supervised Cross-lingual Transfer. We make our extension and pre-trained bi-LM of monolingual and cross-lingual available on Github Links (Anonymous). We do not tune the model hyper-parameter such as dropout or learning rate, as there is no gold validation on comparable scenario with [2]. In addition, we found that tuning hyper-parameter to noisy validation do not improve and can even lead to worse result such as over-fitting to false negative. General Model Configuration We initialize all NER neural models on both monolingual and cross-lingual of Indonesian as target by using pre-trained word embedding with Glove [23] on our Wikipedia dumps. The Glove-ID vectors are freeze during training on DEE, MDEE and +Gazz data. All the Indonesian NER models on distantly supervised data are trained for 10 epochs using Adam [13] with learning rate 0.001 for Optimization of batch size 32. For model using ELMo module, we use dropout rate 0.5 after the last layer output and before concatenation with word embedding and l2 regularization [14] on ELMo weights to prevent model over-fitting and retain pre-trained knowledge. We use 2 layer Bi-LSTM-CRF layer with hidden size 200 and the word embedding dimension 50. Unsupervised Cross-lingual NER Transfer via ELMo In cross-lingual bi-directional LM using CL via ELMo EN scenario, we use pre-trained weights from English 1B tokens 3 to Indonesian News dataset (IDNews) [33]. We use implementation of bidirectional language model by Peters et al., (2018) [25,26] 4 and modified it for cross-lingual transfer scenario. We fine-tune the model for 3 epochs by replacing the Softmax vocab layer with randomly initialized weight. We only fine-tune language model in cross-lingual scenarios on 3 epochs instead of 10 is to prevent catastrophic forgetting [30], [10].We called this model ELMo EN-ID Transfer. As a baseline, we use ELMo EN-1B Tokens model directly in the CL via ELMo EN scenario. Supervised Cross-lingual NER Transfer For the cross-lingual transfer learning baseline scenario, we use WP2, WP3 [20] and CoNLL 2003 dataset [34] of English language to train standard BiLSTM-CRF without ELMo initializer on 1B Language Model benchmarks. The models are trained on English languages and then the pre-trained weights are used as initalizer for both supervised and unsupervised transfer learning on DEE, MDEE, and +Gazz dataset. For the pre-trained English model, we report our reproduced baseline, recent state-ofthe-arts NER and ELMo LSTM-CRF on WikiNER dataset [20] to show the improvement on noisy mono-lingual data and use as pre-trained model. We train the English NER models for 75 epochs with patience 25 epochs for early stopping during training. In the experiment result Table 4, the model corresponds to [Sources] BiLSTM-CRF in "Supervised CL NER Transfer" scenarios. Mono-lingual ELMo In this scenarios, we use directly Pre-trained bi-LM on a mono-lingual corpus such as 1 billions word English [6], 82 millions Indonesian Wikipedia or 11 millions Indonesian News [33] POS Tagging Transfer In this scenarios, we train a standard Bi-LSTM model using Softmax with Cross-entropy loss function to Indonesian POS tagging dataset. The transfer procedure almost the same as Supervised Cross-lingual NER Transfer as illustrated in Figure 2 on the right, while there are 2 differences i) the top-most layer is Linear with Softmax Activation instead of CRF, and ii) the sources task is POS tagging instead of English NER. We train 5 models based on 5-fold cross-validation split provided by [15], we report the averaged F1 of each k-th-fold model as pre-trained weights in both large silver and small clean annotation. In the experiment result Table 4, the model corresponds to ID-POS BiLSTM-CRF in "POS Tagging Transfer" scenario. This experiment scenario serve as comparison of transfer learning from different but related task in Yang et al., (2017) [36]. In addition, previous work by Blevins et al. (2018) [4] show that LM contains syntactic information thus serve as comparison to pre-trained monolingual bidirectional LM. Multi-Task NER with BiLM We also train and evaluate using recent stateof-the-arts model in Indonesian conversational dataset such as Multi-Task NER with BiLM auxiliary task (BiLM-NER) [17]. In the experiment Table 4, the model corresponds to BiLM-NER in "Baseline" scenarios. Results & Analysis In this research, we reports our English dataset results which mainly used to show improvement of pre-trained BiLM and as source weights in transfer learning. We reports our main experiments in several version of large silver for model comparison and a small clean annotation in ablation scenarios. Finally, we analyzed our proposed method of supervised cross-lingual transfer with BiLM and Cross-lingual Transfer via Language Model. English Dataset Results From Table 3, model trained using pre-trained ELMo and random Word Embedding initialization (WE+ELMo LSTM-CRF) are better with an average of 4.925 % F1 score in four WikiNER scenarios compare to Word embedding initialized with Glove 6B words and character-CNN (WE+CharEmb) on CoNLL dataset. However, it is tie on WikiGold test where Glove+CharEmb without MISC labels perform are better than WE+ELMo, whereas the latter are better with MISC labels than the former. Overall, combining both Glove and ELMo yields best results except when using WP2 as training data when tested in CoNLL test. Indonesian Dataset Results We reproduce around the same results of [2] using Stanford NER. Our experiment using a recent state-of-the-arts model in Indonesian conversational dataset namely Multi-Task NER with BiLM auxiliary task (BiLM-NER) [17] (BiLM-NER) obtain comparable performance with log-linear model but lower than BiLSTM-CRF [18]. The mono-lingual pre-trained BiLM on 1B English words (ELMO EN-1B Tokens) performs comparable with pre-trained BiLM on 82 millions tokens in (ELMo (ID-Wiki)) and 11 millions news tokens (ELMo (ID-News)). All of the mono-lingual Embedding from Pre-trained BiLM on silver standard annotation perform worse than baseline supervised cross-lingual with & without BiLM scenarios. Cross-lingual Transfer Analysis We hypotheses that the performance of using ELMo on cross-lingual settings despite a little counter-intuitive are not entirely surprising can be addressed to i) Most named entities which available on multi-lingual documents are orthographically similar. For instance "America" is "Amerika" in Indonesian, while "Obama" is still "Obama", "President Barack Obama" is still "Presiden Barack Obama"; ii) Due to the orthographic similarities of many entity names, the fact that English and Indonesian languages are typologically different (e.g. in terms of S-V-O word order and Determiner-Noun word order) is not relevant on noisy data, as long as the character sequences of named entities are similar in both languages [7,35]. We confirm our first hypothesis by looking up the percentage of unique word (vocabulary) overlap rate between the Gold ID-NER [1] and three English dataset, namely WP2, WP3 [20] and CoNLL training [34]. The overall vocabulary overlap rate between Gold ID-NER and the three dataset are 26.77%, 25.70%, 15.24% respectively. Furthermore, we checked WP2 per word-tag join overlap rate are PER 51.09%, LOC 60.9%, ORG 60.54%, and O 16.56% percentage. While CoNLL word-tag joins overlap rate are PER 37.53%, LOC 27.54%, ORG 39.46%, and O 9.23%. More details of unique word overlap rate between Indonesian DB-Pedia Entity, WP2, WP3 and CoNLL can be seen on Table 4. in Supervised Cross-lingual Transfer which only utilized character-embedding and pre-trained monolingual word-embedding trained from CoNLL dataset perform worse on both MDEE and +Gazz dataset than trained on WP2 and WP3 dataset. We support our second hypothesis by doing ablation on clean annotation (Table 5). Our clean annotation show that, ELMo (ID-Wiki) outperformed ELMo (EN-1B Tokens) on small clean annotation data, but ELMo EN nonetheless still outperformed BiLSTM-CRF especially when combined with Supervised pre-training on CoNLL 2003 English NER [18]. Conclusion In this research, we extend the idea of character-level embedding pre-trained on language model to cross-lingual scenarios for distantly supervised and lowresources scenarios. We observed that training character-level embedding of language model requires enormous size of corpora [26]. Addressing this problem, we demonstrate that as long as orthographic constraint and some lexical words in target language such as loanwords to act as pivot are shared, we can utilize the high-resource languages model. Figure 2 : 2Left image, Baseline scenario for supervised cross-lingual transfer learning. Right image, Baseline scenario for directly using ELMo 1B Tokens EN initializer Figure 3 : 3Word-tag overlap rate breakdown between mono-lingual and crosslingual corpora. (-) horizontal line: WP2 & DBPedia Gold, right slope: WP2 & DBPedia Train, (+) cross: is overlap between WP3 & DBPedia Gold, (-) vertical: overlap between WP3 & DBPedia Train, (/) left slope: CoNLL Train and DBPedia Gold, (o) dot: CoNLL Train and DBPeida Train Table 1 : 1Dataset statistics used in our experiments. #Tok: numbers of tokens. #Sent: numbers of sentences. Alfina et. al.[1,2] use Gold as their test set. Clean 1.2K are used to measure noisy percentage of DEE, MDEE, and +Gazz and low-resources scenarioDataset PER LOC ORG #Tok #Sent DEE 13641 16014 2117 599600 20240 MDEE 13336 17571 2270 599600 20240 +Gazz 13269 22211 2815 599600 20240 Gold (Test) 569 510 353 14427 737 Clean 1.2K 1068 1773 720 38423 1220 Table 2 : 21.2K instances of silver annotation performance with respect to the Clean 1.2k annotation. Clean 1.2k annotation is subset of DEE, MDEE and +GazzAnnotation Prec Recall F1 DEE (1.2K) 60.85 33.08 42.86 MDEE (1.2K) 61.77 35.07 44.74 +Gazz (1.2K) 63.83 40.44 49.51 dataset which illustrated on Figure 2 on the right. In the experiment resultTable 4, the model corresponds to ELMo ([Unlabeled corpus]) in "Mono-lingual ELMo" Table 3 : 3F1 score performance results on WikiGold and CoNLL test set. English NER model w/o (without) MISC and pre-trained weight Glove 6B & ELMo 1B used as pre-train model for cross-lingual transfer scenariosTrain Data WikiGold CoNLL Pre-Init Glove+CharEmb LSTM-CRF WP2 71.75 61.78 Glove 6B WP3 71.40 62.51 Glove 6B CoNLL 58.00 90.47 Glove 6B WP2-w/o MISC 75.12 65.35 Glove 6B WP3-w/o MISC 75.02 63.69 Glove 6B CoNLL-w/o MISC 58.30 91.37 Glove 6B WE (Random Init) +ELMo LSTM-CRF WP2 76.96 71.48 ELMo 1B WP3 74.95 68.54 ELMo 1B CoNLL 74.07 90.18 ELMo 1B WP2-w/o MISC 73.47 66.50 ELMo 1B WP3-w/o MISC 72.91 66.51 ELMo 1B CoNLL-w/o MISC 74.52 91.59 ELMo 1B Glove +ELMo LSTM-CRF WP2 77.14 69.91 Glove 6B & ELMo 1B WP3 76.92 70.31 Glove 6B & ELMo 1B CoNLL 75.12 91.98 Glove 6B & ELMo 1B WP2-w/o MISC 80.55 73.05 Glove 6B & ELMo 1B WP3-w/o MISC 81.09 75.60 Glove 6B & ELMo 1B CoNLL-w/o MISC 79.49 93.53 Glove 6B & ELMo 1B Table 4 : 4Experiment on silver standard annotation of Indonesian NER evalu- ated on Gold test set [1] in large distantly supervised NER scen- ario. Bold F1 scores are best result per scenarios (Baseline, Supervised Cross- lingual Transfer, Cross-lingual using ELMo from EN, Mono-lingual ELMo and Unsupervised-Supervised Cross- lingual Transfer). * is the best model on a dataset (DEE, MDEE, or +Gazz) on all model scenarios Model DEE MDEE +Gazz Previous Works Alfina et al., [2] 41.33 41.87 51.61 BiLM-NER 40.36 41.03 51.77 Baseline Stanford-NER-BIO [2] 40.68 41.17 51.01 BiLSTM-CRF 46.09 45.59 52.04 POS Tagging Transfer ID-POS BiLSTM-CRF 52.58 51.07 60.57 Supervised CL NER Transfer WP2 BiLSTM-CRF 49.88 52.35 62.57 WP3 BiLSTM-CRF 51.21 50.95 62.90 CoNLL BiLSTM-CRF 52.56 50.75 60.81 CL via ELMo EN ELMo EN-1B Tokens 51.08 53.19 60.66 ELMo EN-ID Transfer 52.63 54.74 63.02 Mono-lingual ELMo ELMo (ID-Wiki) 50.68 52.38 60.51 ELMo (ID-News) 49.49 51.91 60.73 Supervised CL Transfer with ELMo WP2 ELMo (EN) 52.99 55.39* 63.99 WP3 ELMo (EN) 54.15* 55.28 63.84 CoNLL ELMo (EN) 53.52 53.48 64.35* Table 5 : 5Ablation experiment res- ults using Clean 1.2K as training data in small clean (human an- notated) scenario also evaluated on Gold test set. W: Word em- bedding (Random Init), C: Char- CNN (+EN if INIT from CoNLL 2003) embedding, E: ELMo (EN), G: Glove-ID(+EN if in cross-lingual transfer from English) [23], I: ELMo (ID-Wiki), J: ELMo (EN-ID-News) Transfer Model Prec Rec F1 Stanford-NER 71.42 53.84 61.39 BiLM-NER 63.65 63.29 63.47 BiLSTM-CRF W+C+E 76.42 56.32 64.85 W+C 56.23 56.39 56.31 W+E 73.53 53.32 61.81 C+E 69.13 68.60 68.86 G 63.65 48.50 55.05 G+C 69.17 62.31 65.56 G+E 75.30 65.32 69.96 G+C+E 72.05 68.73 70.35 E 76.27 55.41 64.19 G+C+I 74.53 78.43 76.43 G+I 75.57 77.94 76.74 I 78.55 73.62 76.00 G+C+J 83.26 82.62 82.94 G+J 83.77 83.60 83.68 J 82.36 83.74 83.04 INIT from ID-POS W+C 72.97 78.97 75.68 INIT from CoNLL 2003 W+C 66.23 56.25 60.83 G+C 70.18 65.87 67.96 C+E 71.84 64.27 67.85 W+C+E 73.63 65.46 69.30 G+E 73.38 69.08 71.17 G+C+E 72.63 72.99 72.85 as of 20-08-2018 Wikipedia Database dump https://github.com/dice-group/FOX/tree/master/input/Wikiner model-checkpoint 4 https://github.com/allenai/bilm-tf AcknowledgmentsWe also would like to thank Samuel Louvan, Kemal Kurniawan, Adhiguna Kuncoro, and Rezka Aufar L. for reviewing the early version of this work. We are also grateful to Suci Brooks and Pria Purnama for their relentless support. Dbpedia entities expansion in automatically building dataset for indonesian ner. I Alfina, R Manurung, M I Fanany, International Conference on Advanced Computer Science and Information Systems (ICACSIS). Alfina, I., Manurung, R., Fanany, M.I.: Dbpedia entities expansion in automatically building dataset for indonesian ner. 2016 International Conference on Advanced Computer Science and Information Systems (ICACSIS) pp. 335-340 (2016) Modified dbpedia entities expansion for tagging automatically ner dataset. I Alfina, S Savitri, M I Fanany, International Conference on Advanced Computer Science and Information Systems (ICACSIS). Alfina, I., Savitri, S., Fanany, M.I.: Modified dbpedia entities expansion for tagging automatically ner dataset. 2017 International Conference on Advanced Computer Science and Information Systems (ICACSIS) pp. 216-221 (2017) Named entity recognition in wikipedia. 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[ "https://github.com/dice-group/FOX/tree/master/input/Wikiner", "https://github.com/allenai/bilm-tf" ]
[ "Parametrized modified gravity constraints after Planck", "Parametrized modified gravity constraints after Planck" ]
[ "Bin Hu \nINFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n", "Michele Liguori \nINFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n\nDipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly\n", "Nicola Bartolo \nINFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n\nDipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly\n", "Sabino Matarrese \nINFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n\nDipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly\n" ]
[ "INFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly", "INFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly", "Dipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly", "INFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly", "Dipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly", "INFN\nSezione di Padova\nvia Marzolo 835131PadovaItaly", "Dipartimento di Fisica e Astronomia \"G. Galilei\"\nUniversità degli Studi di Padova\nvia Marzolo 835131PadovaItaly" ]
[]
We constrain f (R) and chameleon-type modified gravity in the framework of the Berstchinger-Zukin parametrization using the recent released Planck data, including both CMB temperature power spectrum and lensing potential power spectrum. Some other external data sets are included, such as baryon acoustic oscillations (BAO) measurements from the 6dFGS, SDSS DR7 and BOSS DR9 surveys, Hubble Space Telescope (HST) H0 measurement and supernovae from Union2.1 compilation. We also use WMAP9yr data for consistency check and comparison. For f (R) gravity, WMAP9yr results can only give quite a loose constraint on the modified gravity parameter B0, which is related to the present value of the Compton wavelength of the extra scalar degree of freedom, B0 < 3.37 at 95%C.L. We demonstrate that this constraint mainly comes from the late Integrated Sachs-Wolfe effect. With only Planck CMB temperature power-spectrum data, we can improve the WMAP9yr result by a factor 3.7 (B0 < 0.91 at 95%C.L.). If the Planck lensing potential power-spectrum data are also taken into account, the constraint can be further strenghtened by a factor 5.1 (B0 < 0.18 at 95%C.L.). This major improvement mainly comes from the small-scale lensing signal. Furthermore, BAO, HST and supernovae data could slightly improve the B0 bound (B0 < 0.12 at 95%C.L.). For the chameleon-type model, we find that the data set which we used cannot constrain the Compton wavelength B0 and the potential index s of chameleon field, but can give a tight constraint on the parameter β1 = 1.043 +0.163 −0.104 at 95%C.L. (β1 = 1 in general relativity), which accounts for the non-minimal coupling between the chameleon field and the matter component. In addition, we find that both modified gravity models we considered favor a relatively higher Hubble parameter than the concordance ΛCDM model in general relativity.
10.1103/physrevd.88.123514
[ "https://arxiv.org/pdf/1307.5276v3.pdf" ]
119,260,706
1307.5276
f0bbcdc54d60e6279ef447fc7f9d292ee8fbfd71
Parametrized modified gravity constraints after Planck 30 Nov 2013 (Dated: May 11, 2014) Bin Hu INFN Sezione di Padova via Marzolo 835131PadovaItaly Michele Liguori INFN Sezione di Padova via Marzolo 835131PadovaItaly Dipartimento di Fisica e Astronomia "G. Galilei" Università degli Studi di Padova via Marzolo 835131PadovaItaly Nicola Bartolo INFN Sezione di Padova via Marzolo 835131PadovaItaly Dipartimento di Fisica e Astronomia "G. Galilei" Università degli Studi di Padova via Marzolo 835131PadovaItaly Sabino Matarrese INFN Sezione di Padova via Marzolo 835131PadovaItaly Dipartimento di Fisica e Astronomia "G. Galilei" Università degli Studi di Padova via Marzolo 835131PadovaItaly Parametrized modified gravity constraints after Planck 30 Nov 2013 (Dated: May 11, 2014)arXiv:1307.5276v3 [astro-ph.CO] We constrain f (R) and chameleon-type modified gravity in the framework of the Berstchinger-Zukin parametrization using the recent released Planck data, including both CMB temperature power spectrum and lensing potential power spectrum. Some other external data sets are included, such as baryon acoustic oscillations (BAO) measurements from the 6dFGS, SDSS DR7 and BOSS DR9 surveys, Hubble Space Telescope (HST) H0 measurement and supernovae from Union2.1 compilation. We also use WMAP9yr data for consistency check and comparison. For f (R) gravity, WMAP9yr results can only give quite a loose constraint on the modified gravity parameter B0, which is related to the present value of the Compton wavelength of the extra scalar degree of freedom, B0 < 3.37 at 95%C.L. We demonstrate that this constraint mainly comes from the late Integrated Sachs-Wolfe effect. With only Planck CMB temperature power-spectrum data, we can improve the WMAP9yr result by a factor 3.7 (B0 < 0.91 at 95%C.L.). If the Planck lensing potential power-spectrum data are also taken into account, the constraint can be further strenghtened by a factor 5.1 (B0 < 0.18 at 95%C.L.). This major improvement mainly comes from the small-scale lensing signal. Furthermore, BAO, HST and supernovae data could slightly improve the B0 bound (B0 < 0.12 at 95%C.L.). For the chameleon-type model, we find that the data set which we used cannot constrain the Compton wavelength B0 and the potential index s of chameleon field, but can give a tight constraint on the parameter β1 = 1.043 +0.163 −0.104 at 95%C.L. (β1 = 1 in general relativity), which accounts for the non-minimal coupling between the chameleon field and the matter component. In addition, we find that both modified gravity models we considered favor a relatively higher Hubble parameter than the concordance ΛCDM model in general relativity. We constrain f (R) and chameleon-type modified gravity in the framework of the Berstchinger-Zukin parametrization using the recent released Planck data, including both CMB temperature power spectrum and lensing potential power spectrum. Some other external data sets are included, such as baryon acoustic oscillations (BAO) measurements from the 6dFGS, SDSS DR7 and BOSS DR9 surveys, Hubble Space Telescope (HST) H0 measurement and supernovae from Union2.1 compilation. We also use WMAP9yr data for consistency check and comparison. For f (R) gravity, WMAP9yr results can only give quite a loose constraint on the modified gravity parameter B0, which is related to the present value of the Compton wavelength of the extra scalar degree of freedom, B0 < 3.37 at 95%C.L. We demonstrate that this constraint mainly comes from the late Integrated Sachs-Wolfe effect. With only Planck CMB temperature power-spectrum data, we can improve the WMAP9yr result by a factor 3.7 (B0 < 0.91 at 95%C.L.). If the Planck lensing potential power-spectrum data are also taken into account, the constraint can be further strenghtened by a factor 5.1 (B0 < 0.18 at 95%C.L.). This major improvement mainly comes from the small-scale lensing signal. Furthermore, BAO, HST and supernovae data could slightly improve the B0 bound (B0 < 0.12 at 95%C.L.). For the chameleon-type model, we find that the data set which we used cannot constrain the Compton wavelength B0 and the potential index s of chameleon field, but can give a tight constraint on the parameter β1 = 1.043 +0.163 −0.104 at 95%C.L. (β1 = 1 in general relativity), which accounts for the non-minimal coupling between the chameleon field and the matter component. In addition, we find that both modified gravity models we considered favor a relatively higher Hubble parameter than the concordance ΛCDM model in general relativity. I. INTRODUCTION Cosmic acceleration can arise from either an exotic form of energy with negative pressure, referred to as "dark energy", or a modification of gravity manifesting on large scales. As shown in [1][2][3], at the the background level dark energy and modified gravity models are almost indistinguishable, hence one needs to investigate the perturbation dynamics. The studies of perturbation theory in modified gravity models, in principle, can be classified in two different frameworks: the parametrization approach and the non-parametrization method, such as the principal component analysis [4][5][6]. In this paper we focus on the former. There exist several phenomenological/theory-oriented parametrizations of modified gravity, such as the Bertschinger-Zukin [7] and the Brax-Davis-Li-Winther [8] parametrizations. These parametrizations are mainly suitable for the quasistatic regime, where the time evolution of the gravitational potentials is negligible compared with their spatial gradient. Furthermore, if we focus on the linear fluctuation dynamics, for which the equations in Fourier space can be reduced to simple algebraic relations, these techniques allow us to perform some analytic calculations which make the parametrization technically efficient. However, if we want to go beyond the quasi-static regime, while remaining in the linear perturbation framework, the parametrization of modified gravity becomes more complex. This is because on the largest scales, especially the super/near-horizon scales, the time evolution of the gravitational potentials is no longer negli-gible, the time derivative terms dominate the dynamical equations, which means that we need to solve some temporal ordinary differential equations. Actually, there exists some debate about the range of validity of the various parametrizations. For example, on one hand, as shown in [9], using a parametrization with insufficient freedom significantly tightens the apparent theoretical constraints. On the other hand, for some specific modified gravity models some phenomenological parametrizations works quite well; for instance the authors of [10] recently demonstrated that for the small Compton wavelength case in the f (R) model, the Bertschinger-Zukin parametrization [7] is practically good enough for the current data analysis purpose. This is because, on the scales larger than the Compton wavelength the deviation from general relativity is suppressed. Below the Compton scale the gravitational potential growth is enhanced and the two metric potentials are no longer equal. Consequently, for the small Compton wavelength case, whose value is less than current horizon size, the most significant modifications w.r.t. general relativity occur in the sub-horizon regime. In addition to the above explicit parametrizations, some quite generic frameworks to study different modified gravity scenarios have also been proposed, such as the Parameterized Post-Friedmann (PPF) formalism, including the Hu-Sawicki approach [11,12], its calibration version [13] and Baker-Ferreira-Skordis-Zuntz algorithm [14,15], and Effective Field Theory (EFT) approaches [16][17][18][19][20][21][22][23]. On the observational point of view, many windows have been proposed to constrain modified gravity mod-els, such as the Integrated Sachs-Wolfe (ISW) effect [24] in Cosmic Microwave Background (CMB) anisotropies, including CMB power spectrum [5,[25][26][27][28][29], CMB ISW-Lensing bispectrum [32,33], baryon acoustic oscillations (BAO) measurements [29][30][31], the galaxy-ISW cross correlation [29,[34][35][36], cluster abundance [37][38][39][40], peculiar velocity [41,42], redshift-space distortions [45,46], weaklensing [5,27,29,35,[42][43][44][47][48][49][50][51][52], 21cm lines [53,54], matter power spectrum and bispectrum [55][56][57][58]. In addition, recently some N-body simulation algorithms in modified gravity models have been developed [59][60][61]. As shown in [36,38,56], with WMAP resolution the modification effects on the CMB mainly come from the ISW effect, which becomes prominent on the largest scales. However, due to the unavoidable cosmic variance on large scales, the constraints from these effects are not significant. On the other hand, since the typical modification scales are in the sub-horizon regime, several studies show that the most stringent constraints come from the large-scale structure data sets. For example, the strongest current constraint on f (R) gravity (B 0 < 1.1×10 −3 , 95%C.L.) [38] is obtained through cluster abundance data sets. Various previous results show that the main constraint on modified gravity comes from galaxy or cluster scales which corresponds to the multipole range l 500 in CMB data, where lensing effect is no longer negligible. The recent release of Planck data [62] provides us with a fruitful late-time information both on the ISW and lensing scales, which is encoded in the CMB temperature power-spectrum [63] and lensing potential power-spectrum [64] and CMB temperature ISW-Lensing bispectrum [65,66]. The full sky lensing potential map has been firstly measured and the significance of the amplitude of the lensing potential power-spectrum arrives at the 25σ level. The ISW-Lensing bispectrum is also firstly detected with nearly 3σ significance. Furthermore, through the lensing potential reconstruction and the ISW-Lensing bispectrum, the ISW effect is also firstly detected via the CMB itself. All in all, with its high resolution the Planck mission provides us with fruitful information about the universe late-time acceleration. For example, the authors of [67] shows that the joint analysis of Planck and BAO data could greatly improve the Brans-Dicke parameter ω constraint. Further new constraint results related with modified gravity/dark energy can be found in [68][69][70][71][72]. Due to these considerations, in this paper we investigate the power of the Planck data sets in constraining modified gravity scenarios. In order to break the parameter degeneracies, apart from Planck data sets, we also use some external astrophysical data sets, such as BAO measurements from the 6dFGS, SDSS DR7 and BOSS DR9 surveys, H 0 from HST measurement and supernovae from Union2.1 compilation. We also use WMAP9yr data for consistency check and comparison. Because of the simplicity of the Bertschinger-Zukin parametrization, in this paper we study the modified gravity theory through this method. II. BERTSCHINGER-ZUKIN PARAMETRIZATION As pointed out in [8], a large class of modified gravity theories, e.g., chameleon [73,74], symmetron [75][76][77] and dilaton [78] models can be characterized by the mass of a suitable scalar field and the coupling between the scalar field and baryonic/dark matter components. In the Einstein frame, where the gravitational sector is the standard Einstein-Hilbert action, the scalar field is exponentially coupled with the matter sector S E = d 4 x −g M 2 pl 2R − 1 2g µν (∇ µ φ)(∇ ν φ) − V (φ) + S i (χ i , e −καi(φ)gµν ) ,(1) where the Einstein frame metricg µν is related to the Jordan frame one g µν through a conformal transformatioñ g µν = e καi(φ) g µν ,(2) and χ i denotes the matter components. Inspired by some nice properties in the quasi-static regime of f (R) model, Bertschinger and Zukin in [7] first write the two gravitational potentials in the conformal Newtonian gauge 1 in terms of two observation-related variables, the time-and scale-dependent Newton's constant Gµ(a, k) and the so-called gravitational slip γ(a, k) k 2 Ψ = −4πGa 2 µ(a, k)ρ∆ ,(3)Φ Ψ = γ(a, k) ,(4) where G is the Newton's constant in the laboratory. The corresponding Einstein-Boltzmann solver named MG-CAMB is implemented in [49,79]. In this paper, we implement the same algorithm in the new version of CAMB [80] which is compatible with the Planck likelihood. In the following sections, we will study f (R) gravity and the quite general chameleon-type model in the framework of the Bertschinger-Zukin parameterized modified gravity method, by using the Planck [63,64] WMAP9yr [81,82] and some external astrophysical data. A. f (R) model Due to the simplicity of its Lagrangian, f (R) gravity obtained a lot of attention, (see the recent review [83] and references therein) especially as an illustration of the chameleon mechanism. Besides the simplicity of the structure of this theory, there exist two more reasons for the interest it attracted. One is that the form of the function f (R) can be engineered to exactly mimic any background history via a one-parameter family of solutions [2]. The second reason is that f (R) gravity can slightly better fit than at ΛCDM, which can be attributed to the lowering of the temperature anisotropy power spectrum at small l regime [38]. In this paper we consider the class of f (R) gravity models which can mimic a ΛCDM background. Because of the higher order derivative nature of f (R) gravity, there exist a scalar degree of freedom, named scalaron f R ≡ df /dR with mass m 2 fR ≡ ∂ 2 V eff ∂f 2 R = 1 3 1 + f R f RR − R .(5) Then the Compton wavelength of the scalaron reads λ fR ≡ m −1 fR .(6) Usually, it is convenient to use the dimensionless Compton wavelength B ≡ f RR 1 + f R R ′ H H ′ ,(7) with f RR = d 2 f /dR 2 and ′ = d/d ln a. In the Bertschinger-Zukin parametrization [7], the explicit expressions of the functions µ(a, k) and γ(a, k) for f (R) gravity read µ(a, k) = 1 + 4 3 λ 2 1 k 2 a 4 1 + λ 2 1 k 2 a 4 ,(8)γ(a, k) = 1 + 2 3 λ 2 1 k 2 a 4 1 + 4 3 λ 2 1 k 2 a 4 ,(9) based on the quasi-static approximation. The above parametrization is improved by Giannantonio et. al. in [34] to take the ISW effect into account through some empirical formula µ(a, k) = 1 1 − 1.4 · 10 −8 |λ 1 | 2 a 3 1 + 4 3 λ 2 1 k 2 a 4 1 + λ 2 1 k 2 a 4 .(10) Due to this reason, in our numerical calculation we use (10) instead of the original expression (8). Through a few simple computations, one can easily find that λ 1 is nothing but the present Compton wavelength λ 2 1 = B 0 c 2 /(2H 2 0 ). Remember that Song et. al. in [2] pointed out that there exists a one-parameter family solution in f (R) gravity which could mimic any background evolution. Conventionally, we choose this one-parameter family labeled by the Compton wavelength at present B 0 or λ 2 1 in the Bertschinger-Zukin parametrization. Given the above analysis, we can see that in f (R) gravity, compared with the concordance ΛCDM model, there is only one extra parameter, B 0 , which makes the effects of gravitational modification quite manifest. The chameleon models [73,74] are characterised by a runaway potential and a nearly constant coupling α. Since the f (R) model can be seen as a specific chameleon model, it is straightforward to generalize the Bertschinger-Zukin parametrization for f (R) gravity (8) and (9) into µ(a, k) = 1 + β 1 λ 2 1 k 2 a s 1 + λ 2 1 k 2 a s ,(11) γ(a, k) = 1 + β 2 λ 2 2 k 2 a 4 1 + λ 2 2 k 2 a 4 , where the parameters need to satisfy the following relation and 1 ≤ s ≤ 4. Via the above constraints the number of free parameters can be reduced to 3, usually, one choose them as (s, β 1 , λ 1 ). In [34,79] this kind of parametrization is called Yukawa-type models, due to the Yukawatype interaction between dark matter particles. β 1 = λ 2 2 λ 2 1 = 2 − β 2 λ 2 2 λ 2 1 ,(13) Because of the non-minimal coupling, the dynamics of the scalar field is determined jointly by the scalar field and the matter component, for example, the effective potential of the scalar field is defined by V eff (φ) = V (φ) +ρ i e καi(φ) ,(14) which gives an effective mass of chameleon field m 2 = V ′′ eff = V ′′ − κ(α ′′ + α ′ 2 )V ′ ,(15) where primes denote differentiation w.r.t. the field. For simplicity, here we assume that the chameleon field couples to all the matter components uniformly. Following some calculations as in [49,74], we can obtain the follow-ing relations α 1+s/2 = m 0 m , λ 2 1 = 1 m 2 0 , λ 2 2 = 1 m 2 0 1 + α ′ 2 2 , β 1 = 1 + α ′ 2 2 , β 2 = 2 − α ′ 2 2 + α ′ 2 ,(16) where m 0 is the chameleon effective mass at present. Furthermore, for the inverse power-law potential V (φ) ∝ φ −n case, with n > 0, we have n = 4 − s s − 1 .(17) Here λ 1 can be replaced with the conventional parameter B 0 , with the same expression in f (R) model, namely λ 2 1 = B 0 c 2 /(2H 2 0 ). Through the above relations, we can easily see that the parameters β 1 , B 0 and s correspond to the non-minimally coupling between chameleon field and matter sector, the relative Compton wavelength of chameleon field and the potential index of chameleon field, respectively. Moreover, the general relativity limit corresponds to β 1 = 1, B 0 = 0, s = 4 [49]. The purpose of this paper is to test possible deviations from general relativity on various cosmic scales by using the recent Planck data, including both the CMB temperature and lensing potential power-specta and also some external astrophysical data sets. In the following section, we will briefly review the Planck likelihood and data set which we used in this work. III. DATA ANALYSIS METHODOLOGY The total Planck CMB temperature power-spectrum likelihood is divided into low-l (l < 50) and high-l (l ≥ 50) parts. This is because the central limit theorem ensures that the distribution of CMB angular power spectrum C l in the high-l regime can be well approximated by a Gaussian statistics. However, for the low-l part the C l distribution is non-Gaussian. For these reasons the Planck team adopts two different methodologies to build the likelihood. In detail, for the low-l part, the likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. For the high-l part, the Planck team employ a correlated Gaussian likelihood approximation, based on a fine-grained set of angular cross-spectra derived from multiple detector combinations between the 100, 143, and 217 GHz frequency channels, marginalizing over power-spectrum foreground templates. In order to break the well-known parameter degeneracy between the reionization optical depth τ and the scalar index n s , the Planck team assumed the low-l WMAP polarization likelihood (WP). Apart from the CMB power-spectrum, the first Planck data release provides for the first time a full-sky lensing potential map, by using the 100, 143, and 217 GHz frequency bands with an overall significance greater than 25σ. As we know, the lensing potential distribution follows that of the large-scale structures which form and grow mainly in the late-time universe. Thus, this map carries fruitful information about dark energy/modified gravity in this period. Hence, we expect that the lensing potential power-spectrum could provide us with a stringent constraint on deviations from general relativity. Given the above considerations, we perform our parameter estimation algorithms by using two different data sets from the Planck mission, namely the Planck CMB power-spectrum [63] and lensing potential powerspectrum [64]. In order to compare with the previous WMAP results, we also do the same analysis by using the WMAP9yr data [82]. Furthermore, in order to break the parameter degeneracies we also use some other external data sets, including baryon acoustic oscillations (BAO) measurements from the 6dFGS [86], SDSS DR7 [87], and BOSS DR9 [89] surveys, Hubble Space Telescope (HST) Key Project [90] H 0 measurement and supernovae from Union2.1 compilation [92]. For BAO data sets, we use three redshift survey: the 6dF Galaxy Survey measurement at z = 0.1, the reanalyzed SDSS-DR7 BAO measurement [88] at effective redshift z eff = 0.35, and the BOSS-DR9 measurement at z eff = 0.2 and z eff = 0.35. For the direct measurement of the Hubble constant, we use the result H 0 = 73.8 ± 2.4kms −1 Mpc −1 [91], which comes from the supernova magnitude-redshift relation calibrated by the HST observations of Cepheid variables in the host galaxies of eight SNe Ia. For supernovae, we use the Union2.1 compilation, consisting of 580 SNe, calibrated by the SALT2 light-curve fitting model. As previously stated, we implement the same algorithms of MGCAMB [49,79] in the new version of CAMB [80], which is compatible with the Planck likelihood. We sample the cosmological parameter space, which can be read in Tab IV. RESULTS AND DISCUSSION As a first step we checked the reliability of the code in the general relativity limit (B 0 = 0 for f (R) gravity case, B 0 = 0, β 1 = 1, s = 4 for a chameleon-type model). We find that our results are in quite good agreement with the Planck results [85]. Here we show our consistency check for the f (R) case explicitly in Tab. II. The global analysis results for f (R) gravity can be read in the second, third, fourth and fifth columns of Tab. II, which are based on WMAP9yr, Planck + WP, and Planck + WP + lensing and Planck + WP + lensing + BAO + HST + Union2.1 data sets. Firstly, we can see that Planck CMB temperature power-spectrum with WP can give an upper bound of B 0 < 0.91 (hereafter we quote the significance at 95%C.L. for modified gravity parameter, such as B 0 and β 1 ). Compared with the WMAP9yr result, B 0 < 3.37, it improves the upper bound by a factor 3.7. Secondly, by adding lensing data the results can be further improved by a factor 5.1 (B 0 < 0.18). Finally, we arrive at our best bound of B 0 < 0.12 by using all data sets. In addition, we notice that, due to the degeneracy between B 0 and the dark matter density, the Planck data prefer a slightly lower value of Ω c h 2 in f (R) model. Consequently, this implies that f (R) gravity favors a slightly larger value of H 0 . This can be helpful to relax the tension between Planck and the other direct measurements of the Hubble parameter, such as that from the HST [90]. The degenearcy between B 0 and Ω c h 2 is illustrated in Fig. 7, where it is evident that we can fit a lower value of the third peak by increasing B 0 while keeping Ω c h 2 fixed. Marginalized likelihoods for all the parameters are shown in Fig. 3. We also highlight the 2D likelihood in the parameter space of B 0 and H 0 in Fig. 1 and the marginalized likelihood for B 0 in Fig. 2. Let us notice that the B 0 likelihood from WMAP9yr data (gray curve) has a prominent second peak around B 0 = 2.5. This is due to the non-linear dependence of the ISW effect on B 0 in f (R) gravity. Since with WMAP resolution the lensing signal is quite weak, the main contribution to the B 0 constraint in WMAP data comes from the ISW effect. As shown in Fig. 6, under our parameter value choice (we fix all the other cosmological parameters as the mean values of the Planck base ΛCDM model), from B 0 = 0 to B 0 ∼ 1 the slope of the spectrum in the ISW-dominated regime becomes gradually flat and approaches the Sachs-Wolfe plateau. After that, if one continues to increase till B 0 ∼ 2, the power-spectrum will bounce again and get closer to that of general relativity. If one further increases the B 0 value, the spectrum curve in the ISW regime will rise up above that of general relativity. Moreover, once we marginalize over all the other cosmological parameters, the turning point B 0 ∼ 1 will shift to around B 0 ∼ 1.5, and the second peak B 0 ∼ 2 moves to B 0 ∼ 2.5. Compared with f (R) gravity, the chameleon-type model includes the other two free parameters β 1 and s, which are fixed to 4/3 and 4 in the former case. Due to the amount of extra modified gravity parameters and the degeneracy among them, we find that the Planck constraints on the parameters B 0 and s are still quite loose, with no obvious improvement when comparing to WMAP9yr results. However, we are able to improve the constraints on β 1 : we find β 1 = 1.043 +0.163 can be found in Tab. III and Fig. 5. Confidence regions in the β 1 -H 0 plane, after marginalizing over the other parameters, are shown in Fig. 4. One could notice that the value β 1 = 4/3, corresponding to f (R) models, is well outside the 3σ confidence region. However this does not rule out f (R) model by any means given the very loose constraints on the other two relevant f (R) parameters B 0 and s. In Fig. 8, we compare the likelihood of β 1 with(out) marginalization over B 0 by using Planck +WP+Lensing+BAO+HST+Union2 data sets. It clearly shows that the stringent constraint on β 1 is due to the marginalization effect on B 0 , whose constraint is very loose for the chameleon-type model via current data sets. And we have alse tested that if we fix B 0 = 0.001 and use the same data sets, the marginalized 2σ confidence level for β 1 is 0.971 +0.700 −0.746 , which reconciles with f (R) gravity very well. We can also see in Tab. III that the chameleon-type model favors a slightly higher Hubble parameter, for the same reason as explained for f (R) gravity. ACKNOWLEDGMENTS NB and BH are indebted to Philippe Brax for useful discussions. We also thank Jason Dossett, Alireza Hojjati and Alessandra Silvestri for the useful correspondence and discussion of the numerical codes. BH thanks Zhenhui Zhang for discussion and ITP-CAS for the hospitality where some parts of this work are finished. FIG. 1 .FIG. 2 . 12Two-dimensional contour diagram of B0 and H0. The appearance of the upper dark gray area is due to the non-linear dependence of the ISW effect on B0. The likelihood of B0. The second peak in the gray curve is due to the non-linear dependence of ISW effect on B0. FIG. 3 . 3Full set of parameter likelihoods in f (R) gravity. FIG. 4 . 4Two-dimensional contour of β1 and H0. FIG. 5 . 5Full set of parameter likelihoods in chameleon-type model. FIG. 6 . 6The non-linear dependence of the ISW effect on B0 in f (R) gravity. .I, with a Markov Chain Monte Carlo (MCMC) method with the publicly available code CosmoMC[84]. FIG. 7 . 7The second and third peaks in f (R) gravity. The larger B0 is the lower the third peak is. 95%C.L. compared with β 1 = 0.893 +0.647 −0.695 at 95%C.L. from WMAP9yr. The detailed global analysis results FIG. 8 . 8The numerical calculations are performed on clusters at ITP-CAS and INFN-PD. The research of N.B., M.L. and S.M. has been partially supported by the ASI/INAF Agreement No. I/072/09/0 for the Planck LFI Activity of Phase E2. Likelihood of β1 with(out) marginalization over B0 by using Planck +WP+Lensing+BAO+HST+Union2 data sets. TABLE II. Best-fit values and 68% confidence limits for f (R) gravity (and 95% confidence limits in parenthesis for B0). The first column shows the consistency check of the code in the general relativity limit. H0[km/s/Mpc] 67.64 70.58±2.59 TABLE III. Best-fit values and 68% confidence limits for chameleon-type model(and 95% confidence limits in parenthesis for β1).GR limit:Planck +WP BZ:WMAP9yr Planck +WP +lensing +BAO+HST+Union2.1 Parameters Best fit 68% limit Best fit 68% limit Best fit 68% limit Best fit 68% limit Best fit 68% limit Ω b h 2 .02266 .02206±.00028 .02270 .02271±.00052 .02250 .02253±.00031 .02227 .02247±.00031 .02232 .02244±.00026 Ωch 2 .1201 .1198±.0027 .1147 .1134±.0046 .1178 .1164±.0026 .1173 .1151±.0026 .1180 .1157±.0016 100θ 1.04151 1.04132±.00063 1.0410 1.0405±.0023 1.0420 1.04190±.00065 1.0418 1.0419±.00064 1.0413 1.0418±.00057 τ .083 .090±.013 .086 .090±.014 .077 .087±.013 .103 .085±.013 .092 .084±.012 ns .9601 .9607±.0073 .973 .974±.014 .967 .970±.0075 .970 .971±.0076 .965 .970±.0056 log(10 10 As) 3.077 3.090±.025 3.092 3.093±.031 3.063 3.078±.025 3.111 3.070±.024 3.091 3.069±.024 B 0 -- -- .015 <1.94(3.37) .121 <.38(.91) .023 <.054(.18) .0044 <.041(.12) Ω Λ .684 .685±.016 .715 .719±.026 .701 .707±.015 .702 .713±.015 .697 .711±.0092 H 0 [km/s/Mpc] 67.25 67.34±1.19 69.59 69.92±2.23 68.64 69.09±1.24 68.56 69.54±1.26 68.15 69.27±.76 χ 2 min /2 4902.724 3779.201 4900.427 4907.413 4975.704 CM: WMAP9yr CM: Planck +WP +lensing +BAO+HST+Union2.1 Parameters Best fit 68% limit Best fit 68% limit Best fit 68% limit Best fit 68% limit Ω b h 2 .02279 .02286±.00059 .02256 .02241±.00035 .02226 .02225±.00032 .02240 .02235±.00026 Ωch 2 .1184 .1122±.0052 .1168 .1171±.0031 .1162 .1174±.0029 .1168 .1166±.0017 100θ 1.0391 1.0406±.0024 1.04158 1.04174±.00068 1.04183 1.04158±.00065 1.04159 1.04173±.00057 τ .092 .090±.015 .088 .087±.013 .089 .088±.013 .090 .089±.013 ns .9879 .9825±.019 .9686 .9676±.0084 .9659 .9658±.0079 .9698 .9678±.0057 log(10 10 As) 3.131 3.092±.033 3.082 3.079±.026 3.079 3.081±.025 3.085 3.080±.025 β1 0.954 0.893 +0.647 −0.695 1.127 1.148 +0.274 −0.194 1.033 1.027 +0.140 −0.114 1.020 1.043 +0.163 −0.104 B0 0.496 -- 0.849 -- 0.473 -- 0.079 -- s 1.143 -- 3.398 -- 3.152 -- 3.635 -- ΩΛ .691 .726±.029 .705 .703±.018 .701 .700±.017 .704 .705±.0098 68.88 68.73±1.46 68.93 68.43±1.36 68.75 68.82±.78 χ 2 min /2 3778.939 4900.274 4907.445 4975.853 We take the convention that ds 2 = −(1+2Ψ)dt 2 +a 2 (1−2Φ)dx 2 . . 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[]
[ "Redshift-Space Distortions, Pairwise Velocities and Nonlinearities", "Redshift-Space Distortions, Pairwise Velocities and Nonlinearities" ]
[ "Román Scoccimarro \nCenter for Cosmology and Particle Physics\nDepartment of Physics\nNew York University\n4 Washington Place10003New YorkNYUSA\n" ]
[ "Center for Cosmology and Particle Physics\nDepartment of Physics\nNew York University\n4 Washington Place10003New YorkNYUSA" ]
[]
We derive the exact relationship, including all non-linearities, between real-space and redshiftspace two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity PDF is strongly non-Gaussian at all scales, and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive.We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well.
10.1103/physrevd.70.083007
[ "https://arxiv.org/pdf/astro-ph/0407214v2.pdf" ]
14,451,856
astro-ph/0407214
c80215d37dbfb6a9fc821d5ae28b71e4f9531d6b
Redshift-Space Distortions, Pairwise Velocities and Nonlinearities 2 Aug 2004 Román Scoccimarro Center for Cosmology and Particle Physics Department of Physics New York University 4 Washington Place10003New YorkNYUSA Redshift-Space Distortions, Pairwise Velocities and Nonlinearities 2 Aug 2004 We derive the exact relationship, including all non-linearities, between real-space and redshiftspace two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity PDF is strongly non-Gaussian at all scales, and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive.We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well. I. INTRODUCTION Redshift surveys provide a three-dimensional view of the large-scale structure of the universe. This view, however, is somewhat distorted due to gravitationally-induced peculiar velocities that contribute to galaxy redshifts in addition to the smooth Hubble flow. These "redshift distortions" complicate the interpretation of galaxy clustering data from redshift surveys but, on the other hand, provide a measure of the amount of dark matter in the universe (which sources peculiar velocities) due to the induced anisotropy of clustering statistics such as the power spectrum and the two-point correlation function. The two main signatures of peculiar velocities on the redshift-space clustering pattern have been known for a long time [1][2][3][4]). At large scales, galaxies that fall into clusters look squashed along the line of sight in redshift space: infall velocities of galaxies between the cluster and us (between the cluster and the rest of the universe) add (substract) to the Hubble flow. This squashing effect leads to an increase of the clustering amplitude along the line of sight, thus the power spectrum is enhanced for waves parallel to the line of sight [3]. At small scales (compared to the size of virialized clusters) the internal velocity dispersion elongates clusters along the line of sight, leading to the so-called "finger of god" effect. This suppresses the amplitude of waves parallel to the line of sight. Therefore, the Fourier space clustering pattern shows a positive quadrupole anisotropy at large scales that gradually becomes smaller and eventually negative as small scales are probed [62]. This picture is captured by the "dispersion model" for the redshift-space power spectrum, P s (k, µ) = P g (k) (1 + βµ 2 ) 2 1 1 + k 2 µ 2 σ 2 p /2 ,(1) where P g (k) is the real-space galaxy power spectrum, β ≈ Ω 0.6 m /b 1 where b 1 is the linear bias factor between galaxies and mass, µ = k z /k withẑ denoting the line of sight direction, and σ p the pairwise velocity dispersion assumed to be a constant independent of scale. Here β quantifies the squashing effect, σ p the velocity dispersion effect. The particular form for the squashing effect is due to linear dynamics and linearized real-to-redshift space mapping [3]; hereafter Kaiser limit); the velocity dispersion factor is that corresponding to an exponential pairwise velocity distribution function with no mean streaming [5]. These effects factorize due to the implicit as-sumption that they can be treated as independent. Other dispersion models assume different dispersion factors (e.g. [6,7]). The model in Eq. (1) is clearly oversimplified for a number of reasons, among them i) Even in the context of linear dynamics from Gaussian initial conditions the squashing factor in Eq. (1) must be an approximation. In a random Gaussian field, the velocity field fluctuates from point to point, so there is velocity dispersion and thus the squashing effect must be necessarily accompanied by some sort of dispersion effect. This implies that these effects are not independent. ii) The dispersion factor introduces a phenomenological parameter σ p which represents an effective pairwise velocity dispersion, but whose value cannot be directly used to constrain models since in reality velocity dispersion is a function of scale and galaxy bias and is not clear how to relate it to the effective value σ p affecting the redshift-space power spectrum in this model. The first point implies that some of the dispersion effect may come from large-scale flows (as opposed to virial velocities) which can be modeled accurately in terms of the primordial power spectrum and cosmological parameters. This is important because such improvement of the model can add significant constraining power on theories. The second point also implies that there is potentially a lot to be gained from finding exactly how the redshift-space power spectrum depends on non-linear effects from velocities and galaxy bias. Some attempts to do this using the halo model have been proposed [8][9][10], but they do not address the first point made above. In addition, fitting formulae extracted from simulations that improve on the dispersion model have been developed [11,12], which although very valuable, they do not provide much insight into the problem. Despite its limitations, Eq. (1) has been a popular model for analyzing redshift surveys to obtain constraints on cosmological parameters (e.g. [13,14]). An alternative to using the dispersion model has been to simply ignore the dispersion effect, setting σ p = 0 in Eq. (1), and argue that on "large enough" scales this is sufficiently accurate. Many results on cosmological parameters from measurements of the power spectrum rest on this assumption (e.g. [15][16][17][18]). Although in the past uncertainties from redshift surveys have been large enough that such strategies were reasonable, present datasets such as 2dFGRS and SDSS demand better accuracy; moreover, one expects to get more information than just one or two numbers from using the full dependence of the redshift-space power spectrum on scale and direction. In this paper we derive an exact formula for the redshift-space two-point function and power spectrum in terms of the real space density and velocity fields, extending previous work along these lines [19]. We also show that this formula obeys a modified version of the "streaming model", which was previously proposed in the small-scale [2] and large-scale [20] approximations. This gives a useful characterization of redshift distortions, since real and redshift space spectra are then related by the pairwise velocity probability distribution function (PDF), or its Fourier transform. The challenge is then how to model this PDF in terms of the linear power spectrum, cosmological parameters and galaxy bias. Some steps in this direction, modeling the first two moments, have been already given by [21,22] using the halo model, see also [23] for a modeling of the PDF inspired by perturbation theory. Recent work [24] provides a modeling of the pairwise PDF starting from that of halos. In addition, we show that the model in Eq. (1) leads to an unphysical distribution of pairwise velocities, and that inferring the pairwise velocity PDF from redshift-space clustering is unfortunately not possible in general. Methods that claim to do this [25][26][27] recover instead something else, whose properties we derive here. In this paper we mostly concentrate in the large-scale limit, showing using perturbation theory and N-body simulations that significant corrections to the redshift-space power spectrum in the Kaiser limit are expected at very large scales, k > ∼ 0.01 hMpc −1 . In particular, we emphasize that the shape of the redshift-space power spectrum monopole is not a good approximation to the shape of the linear real-space power spectrum, even when k ≤ 0.1 hMpc −1 . We find that weakly non-linear effects tend to suppress monopole power increasingly with k, and more so for the quadrupole, supporting the argument discussed above that at least part of the transition from positive to negative quadrupole with increasing k is due to large-scale effects, not just virial velocities. We briefly discuss the implications of these results for the determination of Ω m and the reconstruction of the real-space power spectrum. Past work along these lines was done by [28][29][30], who considered whether deviations from the Kaiser limit at large scales could be due to large-scale velocities. However, these relied on the Zel'dovich approximation, which conserves momentum only to linear order, thus velocity fields are not described accurately enough to obtain reliable results (see e.g. [31]). Studies of the redshift-space power spectrum using dark matter numerical simulations have shown significant deviations at large scales from the Kaiser limit before (e.g. [31,32]), but these deviations have generally been blamed exclusively on virial velocities. In fact, as we discus here most of the large-scale velocity dispersion is due to weakly non-linear dynamics and thus has useful cosmological dependence on Ω m , σ 8 and the shape of the linear power spectrum that can be used to enhance constraints from galaxy clustering in redshift surveys. This paper is organized as follows. In section II we derive the exact relation between real and redshift space two-point statistics, obtain the pairwise velocity PDF in the dispersion model, and discuss the recovery of the pairwise velocity PDF from clustering measurements. In section III we present the exact result for the redshift-space two-point correlation function in the case of Gaussian random fields and compare it to the Kaiser formula. We also present measurements of the pairwise velocity moments and discuss why Gaussianity is not a good approximation even at large scales. In section IV we derive the large-scale limit of the redshift-space power spectrum and discuss how it differs from the standard approach in the literature. Section V presents results from perturbation theory and N-body simulations on the weakly nonlinear evolution of velocity fields at large scales and why it differs substantially from that of the density field. Finally, in section VI we present a simple model for the redshift-space power spectrum based on the results of previous sections and discuss the recovery of the real-space power spectrum. We summarize all the results in section VII. In paper II we present a calculation of the non-Gaussian terms in the evolution of pairwise velocities and their PDF. In redshift-space, the observed radial position s of an object is given by its radial velocity, which reflects its true position due to the Hubble flow plus "distortions" due to peculiar velocities. The mapping from its real-space position x is given by: s = x − f u z (x)ẑ,(2) where f = d ln D/d ln a (with D the growth factor and a the scale factor) is a function of Ω m alone for open models or flat models with a cosmological constant [63], the scaled velocity field u(x) ≡ −v(x)/(Hf ), with v(x) the peculiar velocity field, H −1 the comoving Hubble scale, and we have assumed the "plane-parallel" approximation, so that the line of sight is taken as a fixed direction, denoted byẑ. The density field in redshift space is obtained by imposing mass conservation, i.e. (1 + δ s ) d 3 s = (1 + δ) d 3 x,(3) and thus we have in Fourier space, δ D (k)+δ s (k) = d 3 x (2π) 3 e −ik·x e if kz u z (x) 1+δ(x) . (4) Note that this derivation is exact, it does not make any approximations about density or velocity fields; the only assumption is that we work in the plane parallel approximation, which is trivial to overcome by changing k z u z → (k ·x) (u ·x). Furthermore, since we are only using Eqs. (3)(4), there is no reference to the Jacobian of the transformation from x to s, Eq. (4) is valid even in regions where there is multistreaming. In other words, Eq. (4) is taking all mass elements at x and putting them at the corresponding s, if different x's give rise to the same s they will be summed over as necessary. For the power spectrum, Eq. (4) gives: δ D (k) + P s (k) = d 3 r (2π) 3 e −ik·r e if kz ∆u z [1 + δ(x)][1 + δ(x ′ )] ,(5) where ∆u z ≡ u z (x) − u z (x ′ ) and r ≡ x − x ′ . In configuration space we have 1 + ξ s (s , s ⊥ ) = dr δ D (s − r + f ∆u z ) [1 + δ(x)] [1 + δ(x ′ )] ,(6) where the constraint given by the delta function takes a pair separated by line-of-sight distance r = (x − x ′ ) ·ẑ in real space to s in redshift-space as given by Eq. (2), with perpendicular separations unchanged, s ⊥ = r ⊥ . Direct Fourier transformation of this equation yields Eq. (5) for the power spectrum. We can write Eq. (6) in a form closer to that of the power spectrum by rewriting the delta function, 1 + ξ s (s , s ⊥ ) = dr dγ 2π e −iγ(r −s ) e if γ∆u z [1 + δ(x)] [1 + δ(x ′ )] ,(7) It is clear from Eqs. (5) and (7) that the basic object of interest is the line-of-sight pairwise velocity generating function, M(λ, r), [1 + ξ(r)] M(λ, r) ≡ e λ∆u z [1 + δ(x)] [1 + δ(x ′ )] ,(8) where we are interested in λ = if k z in Fourier space, or λ = if γ in configuration space. This generating function can be used to obtain the line-of-sight pairwise velocity moments, e.g. v 12 (r) ≡ ∂M ∂λ λ=0 (9) σ 2 12 (r) ≡ ∂ 2 M ∂λ 2 λ=0 ,(10) give the mean and dispersion of the line-of-sight pairwise velocities [64]. The pairwise velocity probability distribution function (PDF), P(v), is obtained from the moment generating function by inverse Fourier transform [65], P(v, r) = ∞ −∞ dγ 2π e −iγv M(iγf, r).(11) Notice that P(v) depends on scale through the scaledependence of M, and indeed dvP (7) and (11) the redshift-space two-point correlation function can then be written as (v) v = f v 12 (r), dvP(v) v 2 = f 2 σ 2 12 (r), etc. From Eq.1 + ξ s (s , s ⊥ ) = ∞ −∞ dr [1 + ξ(r)] P(r − s , r),(12) where r 2 ≡ r 2 + r 2 ⊥ and s ⊥ = r ⊥ . The physical interpretation of this formula is clear: P maps the pairs at separation r to separation s due to relative velocity −H(r − s ) [see Eq. (2)] with probability P(r − s , r). This type of relationship between the real and redshift space correlation functions is known as the streaming model [2], though it is commonly written in terms of ξ rather than 1 + ξ. If P did not depend on scale, both formulations are equivalent, when there is scale dependence (as expected in any realistic scenario), the first term in the integral for P does not give unity, thus one should use Eq. (12) instead. In fact, this contribution to ξ s has a simple physical interpretation: it corresponds to redshift-space density fluctuations generated by velocity fluctuations in a uniform (real-space) density, i.e. when ξ = 0. If P did not depend on scale, random pairs are mapped into random pairs, scale dependence means that redshift-space correlations are created by taking random pairs in real space and mapping them to redshift space differently at different scales. The streaming model has been mostly used at small non-linear scales by assuming P to be an exponential with zero streaming velocity and a scaleindependent isotropic velocity dispersion [33]. At large scales, [20] showed that if one assumes the streaming model in phase space (with density and velocity fields coupled as in linear dynamics), it is possible to recover the Kaiser limit for the correlation function. We will stress in section IV, however, that the large-scale limit uses an additional assumption, that s be much larger than the pairwise velocity dispersion. Fisher [20] also claims that in the linear regime the relationship between ξ s and ξ can be reduced to the standard streaming model, i.e. as in Eq. (12) with 1 + ξ's replaced by ξ's [see his Eq. (26)]. This is incorrect, it suffices to see that if this were true all terms in ξ s would be proportional to ξ, in particular, such a result does not admit redshift distortions generated by correlated velocity fluctuations (where P depends on r) in an unclustered distribution (ξ = 0). The power spectrum and two-point correlation function in redshift space can be written in a similar form, P s (k) = d 3 r (2π) 3 e −ik·r Z(λ, r) − 1 , (13) ξ s (s , s ⊥ ) = dr dγ 2π e −iγ(r −s ) Z(λ, r) − 1 ,(14) where λ = if k z , if γ respectively and Z(λ, r) ≡ [1 + ξ(r)] M(λ, r).(15) It is important to note that the two-point correlation function is affected by redshift distortions for all configurations, even those perpendicular to the line of sight, since they are coming from different scales through the dependence of P on r . It is however possible to project out redshift distortions by integrating along the line of sight, ξ p (r ⊥ ) ≡ 2 r ⊥ ∞ 0 ds ξ s (s , r ⊥ ) = 2 r ⊥ ∞ 0 dr ξ( r 2 + r 2 ⊥ ) = π P (k) J 0 (kr ⊥ ) kr ⊥ d 3 k,(16) which sets γ = 0 in Eq. (14). This is only true in the plane-parallel approximation, where the concept of "line of sight" is applicable. On the other hand, the redshift-space power spectrum has the nice property, in the plane-parallel approximation, that transverse modes are unaffected by redshift distortions (a wave in the k ⊥ direction is uniform in z and thus unperturbed by the real-to-redshift space mapping), therefore P s (k z = 0, k ⊥ ) = P (k ⊥ ). Figure 1 shows the pairwise velocity distribution P for pairs separated by distance r along the line of sight, measured from the VLS simulation of the Virgo consortium [34]. This has 512 3 dark matter particles in a 479 Mpc h −1 box with a linear power spectrum corresponding to Ω m = 0.3 (including Ω b = 0.04 in baryons), Ω Λ = 0.7, h = 0.7 and σ 8 = 0.9. Due to the large number of pairs (in our measurements we use 32 × 10 12 total pairs at scales between 0.1 and 300 Mpc h −1 ) and volume of the simulation, the statistical uncertainties are small enough that we do not plot error bars for clarity. On the other hand, one must keep in mind that neighboring points, separated by only 20 km/s, must be highly correlated. Note that at most scales r ≃ 2 − 100 Mpc h −1 the distribution is quite skewed (see also the central panel in Fig. 3 below for a plot of the skewness s 3 as a function of scale). This arises as follows: the left tail (v < 0) corresponds members of pairs approaching each other as they fall into an overdensity, the right tail (v > 0) corresponds to members of pairs receding from each other as they empty underdense regions. Most pairs are not inside a void or falling coherently into a single structure, therefore the peak of P is close to v = 0. The asymmetry between the left and right tail gives rise to a mean infall (v 12 < 0), that is, it is more probable to find "coherent" pairs in overdense than underdense regions. Perhaps the most significant feature of P is that it has exponential wings at all scales, extending what was previously derived in the highly non-linear [35] and weakly non-linear [23] regime, the prediction of linear perturbation theory (shown as the thin solid line in the bottom right panel) is never a good approximation, not even in the large-scale limit. The reasons for this are discussed in detail in section III B. The thin solid lines in the left bottom panel show the results of the dispersion model, although by assumption it has exponential tails, it is a poor match to simulations (even though σ p is fitted to the measured redshift-space power spectrum) and represents an unphysical (discontinuous and singular) distribution of pairwise velocities. See next section for details. Figure 2 shows P for pairs separated by distance r perpendicular the line of sight. In this case we define v ⊥ = v 2 x + v 2 y , then if P is the PDF for a perpendicular component of the velocity field (i.e. v x or v y , it's the same by isotropy and even by symmetry) it follows that P(v ⊥ ) = 2 v ⊥ v ⊥ −v ⊥ dv x v 2 ⊥ − v 2 x P(v x ) P v 2 ⊥ − v 2 x . (17) For a Gaussian distribution P(v x ) = (2πσ 2 ) −1/2 e −v 2 x /2σ 2 v and thus P(v ⊥ ) = (v ⊥ /σ 2 ) e −v 2 ⊥ /2σ 2 . In this case P has zero skewness, by symmetry all odd moments vanish. Apart from this, the behavior of P is similar to the parallel case, the distribution is non-Gaussian at all scales and displays exponential tails. We now turn to a discussion of P in the dispersion model. B. The Dispersion Model It is instructive to recast Eq. (1) in terms of the full pairwise velocity distribution that it implies. There are two contributions to the pairwise PDF in this model, one given by the squashing factor, the other by the dispersion factor, with the total PDF being the convolution of both PDF's. The Fourier transform of the dispersion factor in Eq. (1) corresponds to a pairwise velocity PDF that is exponential, that is P disp (v) = e −|v| √ 2/σp √ 2σ p ,(18) The squashing factor in the Kaiser limit corresponds to a delta function PDF, see Eq. (60) be-low for a derivation. Performing the convolution of this with Eq. (18) leads to the pairwise PDF in the dispersion model, P(v) = e −|v| √ 2/σp √ 2σ p 1 ∓ √ 2 σ p f v 12 + 2f 2 ψ v σ 2 p ×[1 − √ 2σ p δ D (v)] ,(19) where the + sign corresponds to v > 0, and the − sign to v < 0 (opposite for "v-velocities" shown in Fig. 1), and ψ v denotes the velocity-velocity correlation function in linear dynamics, ψ v = ψ ⊥ + ν∆ψ, see Eqs. (33) and (38) for explicit expressions. Note that the resulting PDF is singular at the origin, and in addition has a jump discontinuity at v = 0 which is proportional to v 12 . The bottom left panel in Figs. 1 and 2 illustrate this result (omitting the singular term at v = 0) and compares it to the measurements in numerical simulations for a separation of r = 10 Mpc/h. We have fitted the value of σ p , as it is normally done, to the measured quadrupole to monopole ratio of the redshift-space power spectrum. Despite this fit to the power spectrum, the resulting PDF does not fit the simulation results. This is hardly surprising, since the dispersion model Eq. (1) makes unphysical predictions for the pairwise velocity PDF, see Eq. (19). C. Recovery of the Pairwise Velocity PDF from Redshift-Space Two-Point Statistics Given the relationship between the redshift-space and real-space correlation function through the pairwise velocity PDF, Eq. (12), it is natural to ask whether one can recover information about the PDF from clustering measurements. The problem is that there is no single PDF involved in Eq. (12), but rather an infinite number of PDF's corresponding to different scales and angles of the velocities with respect to the line joining the pair. If there was no scale dependence and anisotropy, all the PDF's are the same and Eq. (12) becomes a convolution, thus one can find the PDF by deconvolution. In other words, due to the scale dependence of the pairwise velocity PDF, Eq. (12) is not really a convolution; this implies that the redshift-space power spectrum for modes parallel to the line of sight is not the realspace power spectrum multiplied by the generating function M. Instead, from Eq. (13) we get P s (k) = P (k) + M(if k z , k) + d 3 q M(if k z , k − q) P (q), (20) where M [recall that M(if k z , r) = P(v, r)e ivkz dv] is basically the double-Fourier transform of P(v, r), M(λ, p) ≡ d 3 r (2π) 3 e −ip·r [M(λ, r) − 1] = (e λ∆u z − 1)(1 + δ)(1 + δ ′ ) 1 + ξ , × e −ip·r d 3 r (2π) 3(21) except that we substract the zero mode M(0, r) = 1, thus M(0, p) = 0. For example, in the Kaiser limit we have M(if k z , p) → 2f k z p z p 2 P δθ (p) + f 2 k 2 z p 2 z p 4 P θθ (p),(22) where P δθ denotes the density-velocity divergence power spectrum and P θθ is the velocity divergence power spectrum. In linear PT, P = P δδ = P δθ = P θθ , but we will keep the distinction because weakly non-linear corrections are significant at large scales, see section V. If we assume that all pairwise moments have no scale dependence and are isotropic, which implies that odd moments vanish (since they must be anisotropic, by symmetry odd moments vanish when r ·ẑ = 0), M(if k z , p) = [M(if k z ) − 1]δ D (p) and thus P s (kẑ) = P (k) M(if k). Note that in this case M(if k) is real because odd moments vanish, however in general M(if k) is complex. By taking (even number of) derivatives with respect to λ of P s ( √ −λ 2ẑ )/P ( √ −λ 2 ) = M(λ) one can generate all (even) moments and thus find the (symmetric by assumption) PDF by inverse Fourier transform. Galaxy redshift surveys show that P s (kẑ)/P (k) is very close to a Lorentzian, and this has been interpreted as evidence for an exponential pairwise velocity PDF [25,26]. However, realistically one cannot neglect anisotropy, since we know that odd moments must be non-zero, in particular there are infall velocities (v 12 = 0) and skewness. The infall velocities are small compared to the dispersion at small, nonlinear scales, however the skewness is expected to be significant except in the highly non-linear regime, see Fig. 3 below [23,[36][37][38]. By construction, since power spectra are functions of only k 2 and k 2 z by statistical isotropy, using P s (kẑ)/P (k) as a generating function (with k = −iλ) only generates even moments and thus a symmetric answer for its PDF, even if the actual PDF is asymmetric [66]. To see what this method actually recovers we go back to Eq. (20) and write explicitly that the redshift-space power spectrum depends on the magnitude of k and k z , P s (k, k z ) = P (k) + 1 2 [ M(if k z , k) + M(−if k z , −k)] + d 3 q 1 2 [ M(if k z , k − q) + M(−if k z , −k − q)] P (q),(23) that is, we average together waves with opposite wavevectors. In this method the moment generating function is identified with the ratio P s (kẑ)/P (k) (where we must replace k by −iλ in our convention), thus it becomes G(λ) ≡ 1 + d 3 r (2π) 3 [1 + ξ(r)] × [M(λ, r) − 1]e −λz + [M(−λ, r) − 1]e λz 2P ( √ −λ 2 ) .(24) It is easy to check that if M(λ) does not depend on r, G(λ) = M (λ) and thus one can recover by inverse Fourier transform of G(λ) the PDF of pairwise velocities, as discussed above. However, in the realistic case with scale dependence and anisotropy, the inverse Fourier transform of G(λ) does not give the actual PDF. To illustrate this, let us calculate the first two non-vanishing moments (second and fourth) of this symmetric "pseudo-PDF", ∂ 2 G ∂λ 2 λ=0 = d 3 r Vξ [σ 2 12 (r) − 2z v 12 (r)] [1 + ξ(r)],(25)∂ 4 G ∂λ 4 λ=0 = d 3 r Vξ [m (4) 12 (r) − 4z m (3) 12 (r) + 6z 2 σ 2 12 (r) − 4z 3 v 12 (r)] [1 + ξ(r)],(26) whereξ ≡ V −1 d 3 rξ(r) is the average of the correlation function, and the integrals over the volume V are cutoff at some large scale (depending on the size of the survey and the practical implementation of the method). In Eqs. (25)(26) m(3) 12 denotes the third moment of the actual PDF and m (4) 12 its fourth moment. It is clear from these equations that the moments of this pseudo-PDF are weighted versions of combinations of several moments of the true PDF, so their value is not straightforward to interpret. From Eq. (25) we see, for example, that the effective value of the velocity dispersion σ 2 eff picked up by this method is given by σ 2 eff = d 3 r Vξ [1+ξ(r)] 2 3 σ 2 ⊥ (r)+ 1 3 σ 2 (r)− 2 3 rṽ 12 (r) ,(27) where we used that by symmetry σ 2 12 (r) = σ 2 ⊥ (r)+ ν 2 [σ 2 (r) − σ 2 ⊥ (r)] and v 12 (r) = νv 12 (r), with ν = z/r. Therefore, we see that the effective value of the velocity dispersion is a weighted version of the underlying velocity dispersion minus a contribution due to the mean streaming (recall in our convention hereṽ 12 (r) > 0, these are "u-velocities"), therefore one expects this method to yield a biased low value of the weighted [by r 2 (1 + ξ)] mean of the dispersion if not corrected for infall, as stressed by [39] and more recently in [27]. Note however that [27] also interpret the pseudo PDF as the actual PDF of pairwise velocities, and they do not include skewness into their treatment. D. The pairwise velocity PDF in terms of its building blocks We now discuss how to use cumulant expansions to evaluate the pairwise moment generating function or Z(λ, r) in Eq. (15) in terms of its building blocks, the cumulants. The starting point is the property between the moment M( j) and cumulant C( j) generating functions for a set of fields which we group together in a vector field A, M( j) = e j·A = exp e j·A c = exp C( j), (28) where A = {A 1 , . . . , A n } and similarly for j. Derivatives of C generate all the connected correlation functions. By taking derivatives with respect to appropriate components of the vector j, it follows in particular that e j1A1 A 2 = e j1A1 A 2 c exp e j1A1 c ,(29)e j1A1 A 2 A 3 = e j1A1 A 2 A 3 c + e j1A1 A 2 c e j1A1 A 3 c exp e j1A1 c .(30) Using j 1 = λ, A 1 = ∆u z , A 2 = δ(x) ≡ δ and A 3 = δ(x ′ ) ≡ δ ′ , Note that the overall factor in this expression is the moment generating function for the line of sight velocity differences, and it is a volume weighted quantity (as opposed to the pairwise velocity PDF which is mass weighted, by densities at x and x ′ ). This velocity-difference PDF is not sensitive to galaxy biasing, since it does not depend on the density field and even if there is velocity bias inside dark matter halos this is a small effect [40][41][42] and halos are in addition suppressed by volume weighting due to their small size. Therefore, the velocitydifference PDF depends on weakly nonlinear dynamics and thus can be modeled (almost) exclusively in terms of cosmological parameters. It is straightforward to evaluate Eq. (31) in the linear regime, for Gaussian fluctuations. In this case, the velocity is proportional to the density, whose only non-zero cumulant is the second and thus Z G (λ, r) = 1 + ξ(r) + λ ∆u z [δ + δ ′ ] + λ 2 ∆u z δ ∆u z δ ′ × exp λ 2 2 ∆u 2 z(32) Notice that even in this case, the resulting expression is non-linear in the amplitude of correlation functions and does not involve terms of the same order in linear perturbation theory. Even though fluctuations are assumed to obey the linear dynamics, the non-linear nature of the redshift-space mapping leads to a somewhat more complicated picture. We will explicitly evaluate Eq. (32) in the next section. Non-linear effects due to dynamics lead to significant deviations from the predictions of Eq. (32), even at large scales, we discuss this below. An evaluation of Eq. (31) is given in paper II. III. THE REDSHIFT-SPACE POWER SPECTRUM AND CORRELATION FUNCTION IN LINEAR DYNAMICS A. Pairwise velocity moments We now give an explicit evaluation of Eq. (32). Using symmetry considerations, the velocity correlation function can be written as u i (x + r/2) u j (x − r/2) = ψ ⊥ (r) δ ij + [ψ (r) − ψ ⊥ (r)] r i r j r 2 ,(33) where ψ (r) and ψ ⊥ (r) are the velocity correlation functions parallel and perpendicular to the line of sight, respectively. They are related to the velocity divergence power spectrum P θθ (k) through [43] ψ ⊥ (r) = P θθ (k) k 2 j 1 (kr) kr d 3 k,(34)ψ (r) = P θθ (k) k 2 [j 0 (kr) − 2 j 1 (kr) kr ] d 3 k,(35) where j ℓ (x) is the usual spherical Bessel function, and we assumed a potential flow, which implies ψ (r) = d(rψ ⊥ )/dr) and ψ ⊥ (r) ≥ ψ (r). The variance of velocity differences reads, ∆u 2 z = 2 σ 2 v − ψ ⊥ (r) + z 2 r 2 ∆ψ(r) ,(36) which leads to the volume-weighted velocity difference moment generating function Z 0 (λ, r) ≡ exp λ 2 2 ∆u 2 z = exp λ 2 σ 2 v − ψ ⊥ (r) + z 2 r 2 ∆ψ(r) ,(37) where ∆ψ(r) ≡ ψ ⊥ (r) − ψ (r) = P θθ (k) k 2 j 2 (kr) d 3 k,(38) and the one-dimensional linear velocity dispersion σ 2 v is given by σ 2 v ≡ 1 3 P θθ (k) k 2 d 3 k.(39) Note that as r → 0, ψ (r) = ψ ⊥ (r) = σ 2 v , and then Z 0 (λ, 0) = 1, as expected from its definition. On the other hand, as r → ∞, ψ (r), ψ ⊥ (r) → 0, and then Z 0 (λ, ∞) ≈ exp −λ 2 σ 2 v . To evaluate the prefactors in Eq. (32), we use that [see Eq. (9)] ∆u z [δ(x) + δ(x ′ )] = v 12 (r) [1 + ξ(r)] = 2 z r P δθ (k) k j 1 (kr) d 3 k,(40) and ∆u z δ(x) = ∆u z δ(x ′ ) = 1 2 v 12 (r) [1 + ξ(r)],(41) then Z G = [1 + ξ(r)] 1 + λv 12 (r) + λ 2 4 v 2 12 (r)[1 + ξ(r)] × exp λ 2 σ 2 v − ψ ⊥ (r) + z 2 r 2 ∆ψ(r) .(42) B. The Failure of Gaussianity It is important to note that, although the largescale limit of v 12 is well described by linear dynamics (see e.g. [21,44]) the same is not true for the pairwise dispersion, indeed we have [δ ≡ δ(x), δ ′ ≡ δ(x ′ )] σ 2 12 (1 + ξ) = ∆u z 2 (1 + δ)(1 + δ ′ ) = ∆u z 2 (1 + ξ) + ∆u z 2 (δ + δ ′ ) + ∆u z 2 δδ ′ c .(43) In linear dynamics, Gaussianity implies that the last two terms vanish; however, in reality the third moment term contributes a constant in the largescale limit (r = |x − x ′ | → ∞) that adds in quadrature to the contribution of the first term (we evaluate this term in paper II). Therefore, linear theory never gives a good approximation to the second moment of pairwise velocities. That there are non-Gaussian corrections should be of no surprise since pair weighting means the second moment of pairwise velocities involves up to fourth moments [23], the interesting aspect here is that even in the largescale limit non-Gaussian terms persist, e.g. u z 2 δ contributes a constant at large scales. The top panel in Fig. 3 illustrates this point, where σ 12 is shown as a function of scale for the N-body measurements (square symbols) and linear theory (dashed lines). All quantities in this figure refer to velocity components parallel to the separation vector of the pair. It is also important to note that the dependence of σ 12 on scale is opposite in the linear case (decreasing at smaller scales) than in the simulations (though at scales r < ∼ 1 Mpc/h, σ 12 starts decreasing in the N-body results). This is also a feature of the Gaussian restriction of linear dynamics, as we shall discuss in paper II, and it implies that the dispersion effect on the two-point correlation function or power spectrum will be significantly underestimated. Physically, in the Gaussian case as r is decreased the velocity field is more correlated and thus ∆u z 2 decreases; since no correlations between density and velocity squared are incorporated in linear theory, it is impossible to see that the velocity of pairs in regions of larger overdensity are fluctuating more; this is described by the non-Gaussian third and fourth terms in Eq. (43). The other panels in Fig. 3 show how important non-Gaussianity of the pairwise PDF is. The central panel compares the skewness s 3 to the dimen-sionless measure of infall, v 12 /σ c 12 , where σ c 12 is the connected second moment shown by solid lines in the top panel. This shows that the skewness is more important than infall at most scales (and by a large factor at scales where infall is most important). Therefore, modeling the pairwise PDF with infall but no skewness, as it is often done (see [20] for an exception), is not a good approximation. Finally, the bottom panel shows the kurtosis s 4 as a function of scale, this quantifies that the pairwise PDF is strongly non-Gaussian (s 4 > 1) at all scales, and it is basically a manifestation of the exponential wings seen in Figs. 1 and 2 at all separations. Why is the Gaussian limit of the pairwise velocity PDF never reached at large separations? The reason is that the relevant quantity is the (densityweighted) difference in velocities. At a given separation r the velocity difference does not receive contributions from modes with wavelengths much larger than r, since those give the same velocity to x and x ′ . For wavelengths smaller than r the contribution of modes is down-weighted only by k −1 (independent of r in the r → ∞ limit); therefore even at large separations one is sensitive to nonlinearities. In other words, at large separations the velocities are uncorrelated and thus the pairwise velocity generating function factorizes into individual particle velocity generating functions. These are sensitive to non-linearities, i.e. there is no "large scale" in that problem. Thinking in terms of the halo model, at large scales the pairwise dispersion is due to particles in different halos, each of which has its own (independent) one-point dispersion due to virial ("nonlinear") and halo ("linear") motions, these contributions will add in quadrature to give the full dispersion (see [21]). We caution, however, that this split is not straightforward, halo motions are not well described by linear theory (their pairwise PDF in the large-scale limit is not exactly Gaussian, see [45]). In [23] it is argued that exponential tails in the pairwise PDF are generated by pair weighting; although this is in part important, it is not the whole story. We show in paper II that the velocity difference PDF (which is volume weighted) also has exponential tails in the large-scale limit, for the reasons discussed above. Of course, pair weighting helps build non-Gaussianity and it is responsible for the deviations in σ 12 from linear theory at large scales. C. The Exact Result for Gaussian Random Fields Even though Gaussianity is not a good approximation to describe the statistics of pairwise velocities, it is instructive to discuss the redshift-space correlation function in the Gaussian case, both as a starting point for more accurate calculations and to discuss the regime of validity of the Kaiser limit. The only assumptions in deriving Eq. (42) are that fluctuations are Gaussian and velocity flows are potential, i.e. there is no assumption about the amplitude of fluctuations (in practice, of course, Gaussianity follows only if fluctuations are vanishingly small). It is easy to write down explicitly the pairwise velocity PDF obtained from using Eq. (42) in Eq. (11), P(v) = 1 √ 2πf ∆u 2 z 1 2 exp −v 2 2f 2 ∆u 2 z 1 + v v 12 f ∆u 2 z + 1 4 v 2 f 2 ∆u 2 z − 1 v 2 12 ∆u 2 z (1 + ξ) .(44) This is not a Gaussian distribution (except when v 12 = 0 at large or small scales, or at all scales for separations perpendicular to the line of sight), although close to its peak it is well approximated by a Gaussian centered at v = f v 12 . Note however that the velocity difference PDF is Gaussian, being the prefactor outside the square brackets. The second and higher cumulants of the pairwise velocity PDF are e.g. v 2 c = σ 2 u + ξ − 1 2 v 2 , v 3 c = 1 − 3ξ 2 v 3 , v 4 c = − 3 4 (1 − 6ξ + ξ 2 ) v 4 ,(45) where v = f v 12 , and σ 2 u ≡ f 2 ∆u 2 z is the variance of the distribution of velocity differences. The non-Gaussianity is induced solely by the nonlinearities in the mapping from real to redshift space. The two-point function can be written using Eq. (12), 1 + ξ s (s , s ⊥ ) = ∞ −∞ dr e − 1 2 x 2 √ 2πf ∆u 2 z 1 2 [1 + ξ(r)] × 1 + x u 12 + (x 2 − 1) 4 u 2 12 (1 + ξ) ,(46)where x ≡ r − s f ∆u 2 z 1 2 , u 12 ≡ v 12 ∆u 2 z 1 2 , r 2 ≡ r 2 +s 2 ⊥(47) This is the exact result in the Gaussian limit, and has been obtained before by Fisher ([20], Eq. 20) by integrating the four dimensional joint Gaussian PDF for δ(x), δ(x ′ ), u z (x) and u z (x ′ ). See also [46,47]. The method described in section II D is an alternative way of obtaining the same result with considerably less algebra, and the advantage that also holds in the non-Gaussian case provided the correlators in Eq. (31) can be calculated. The analogous result for the power spectrum is (ν ≡ z/r, k ⊥ ≡ k 1 − µ 2 , r ⊥ ≡ r √ 1 − ν 2 ), P s (k, µ) = 1 2π 2 ∞ 0 r 2 dr 1 0 dν J 0 k ⊥ r ⊥ cos(krµν) [Z even G (λ, r, ν) − 1] + sin(krµν) Z odd G (λ, r, ν) ,(48) which involves a 2D rather than 1D integration. Here Z odd G (λ, r, ν) corresponds to the term proportional to λ in Eq. (42), and Z even G (λ, r, ν) is the rest. In order to obtain power spectrum multipoles it is sometimes more convenient to calculate first multipoles of the correlation function, ξ ℓ (r) = (2ℓ + 1) 2 1 −1 dν ξ s (r, ν) L ℓ (ν),(49) where L ℓ denote the Legendre polynomials, and then using the plane-wave expansion (µ ≡ k z /k, ν ≡ z/r) e −ik·r = ∞ ℓ=0 (−i) ℓ (2ℓ + 1) j ℓ (kr) L ℓ (µ) L ℓ (ν),(50) obtain from them the power spectrum multipoles, P ℓ (k) = (−i) ℓ 2π 2 ∞ 0 dr r 2 j ℓ (kr) ξ ℓ (r).(51) In this way, a 3D numerical integration gives both the redshift-space correlation function and power spectrum. Figure 4 shows the result for the redshift-space correlation function ξ s (s , s ⊥ ) in the exact Gaussian case Eq. (46) (solid) and the Kaiser limit (dashed), Eq. (61) below. Notice that there are significant deviations even at large scales, predominantly at small s ⊥ , we explain why this is the case in section IV below. Since the correlation amplitude is so much smaller in this region compared to small s , when multipoles are calculated integrating along fixed s the results are close to their Kaiser limit values. It is apparent that the qualitative behavior of the corrections are to make the contours less squashed, as expected from the effects of the velocity dispersion. This is evident in Fig. 5 which zooms into small scales, the dispersion effect is obvious (the quadrupole has opposite sign from that at large scales). Note that this happens at very small scales because the pairwise dispersion decreases at small scales (see dashed line in Fig. 3), thus one needs to go to tiny scales before s becomes smaller than the pairwise dispersion. In addition, one can see that the Gaussian result is not close to the Kaiser limit even when the amplitude of the correlation function is much smaller than unity, e.g. see ξ s ≃ 0.002 in the left panel in Fig. 4. As shown in Fig. 3 and discussed above, assuming Gaussianity is not a good approximation, therefore these results are not a substantial improvement over the Kaiser limit, and we do not show corresponding results for the power spectrum. What is interesting here is that it gives some idea of how to incorporate the effects of large-scale velocity dispersion; we will come back later to this when we develop a simple approximation to the redshift-space power spectrum. We now turn into a discussion of the assumptions behind the Kaiser limit, and show how our approach differs from the standard derivations of it in the literature. Fig. 4 but at small scales. Note the dispersion effect: a Gaussian random field does show "fingers of god" even though the velocity dispersion decreases monotonically towards small scales (see dashed line in Fig. 3). IV. THE LARGE-SCALE LIMIT A. Derivation The non-trivial part of Eq. (46), and Eq. (12) in general, is that as one integrates along r one is integrating over a different PDF due to scale dependence and anisotropy. The relationship between ξ s and ξ, v 12 and σ 2 12 ( ∆u 2 z in linear dynamics) is non-local, however at large scales one can express ξ s in terms of local second moments by the following procedure, which leads to a derivation of the Kaiser formula and makes clear its regime of validity. What do we exactly mean by "large scales"? Although ξ and v 12 vanish in the large-scale limit, σ 2 12 does not (and ∆u 2 z is largest at large scales), therefore we are not allowed to do a small amplitude expansion in this case. On the other hand, when s ≫ f σ 12 the integration over r will be sharply peaked about r = s , thus we can "expand real space about redshift space", P(v; r ) ≈ P(v; s ) + (r − s ) dP(v; s ) ds + 1 2 (r − s ) 2 d 2 P(v; s ) ds 2 + . . . (52) Note that since this expansion can be done for any PDF (not just the one corresponding to linear dynamics), we will do so in general, our results here apply to the fully non-linear case. Similarly one can expand ξ(r) ≈ ξ(s) + . . ., and using that v = r − s [see Eq. (12)], keeping up to second derivatives we obtain 1 + ξ s ≈ (1 + ξ) 1 + f v ′ 12 + f 2 2 σ 2 12 ′′ + . . . +ξ ′ f v 12 + f 2 2 ξ ′′ σ 2 12 + ξ ′ f 2 σ 2 12 ′ + . . .(53) where all quantities in the right hand side are evaluated at s and derivatives are with respect to s , e.g. v ′ 12 ≡ dv 12 (s)/ds . Keeping only terms linear in quantities that vanish in the large-scale limit gives ξ s ≈ ξ + f v ′ 12 + f 2 2 σ 2 12 ′′ + f 2 2 ξ ′′ σ 2 12 | ∞ ,(54) where σ 2 12 | ∞ is the large-scale limit of the pairwise dispersion. In Fourier space this reads, P s (k) ≈ P δδ (k) 1 − 1 2 f 2 k 2 z σ 2 12 | ∞ + if k z v 12 (k) − 1 2 f 2 k 2 z σ 2 12 (k).(55) Higher-order derivatives are suppressed by higher powers of k z . Expanding real space about redshift space should work well when the derivatives in Eq. (54) are small (k z is small), i.e. when considering waves with k with a small component with respect to the line of sight in which case the distortions are small. The large scale limit of v 12 is given by linear theory, v 12 (k) = −2ik z P δθ (k)/k 2 , whereas for σ 2 12 both Gaussian and non-Gaussian terms contribute. We calculate the non-Gaussian terms in paper II, for our purposes here let us just write σ 2 12 | ∞ = 2(σ 2 v + A σ ) and σ 2 12 (k) = −2k 2 z P θθ (k)/k 4 + B σ (k), then we have P s (k) = P δδ (k) 1 − f 2 k 2 z (σ 2 v + A σ ) + 2f k 2 z k 2 P δθ (k) +f 2 k 4 z k 4 P θθ (k) − 1 2 f 2 k 2 z B σ (k),(56) where the non-Gaussian terms correspond to A σ = u 2 z δ and B σ = FT ∆u 2 z (δ + δ ′ + δδ ′ ) c , where FT stands for Fourier transform. We show in paper II that in the large-scale limit, B σ ≈ (8/35)(4 + 11µ 2 /3)σ 2 v P (k). A σ corresponds to the difference in the large scale limit of σ 2 12 to the linear value (squares compared to dashed lines in Fig. 3), whereas B σ is the non-Gaussian contribution that takes into account that the scale dependence of the pairwise velocities is opposite to that in linear theory, i.e. increasing toward smaller scales, as a result it counters the effect of the P θθ term. From Eq. (56) it follows that when k 2 z f 2 σ 2 v ≪ 1, or kµ ≪ 0.2 h Mpc −1 ,(57) where we assumed a flat ΛCDM model, for which σ 2 v ≈ 40 (Mpc h −1 ) 2 and f ≈ 0.5 at z = 0, one recovers the Kaiser formula [3] for the power spectrum (the reason why we don't assume P δδ = P δθ = P θθ will become clear in section V), P s (k) = P δδ (k) + 2f µ 2 P δθ (k) + f 2 µ 4 P θθ (k). (58) The condition in Eq. (57) says that, unless one considers modes nearly perpendicular to the line of sight µ ∼ 0, velocity dispersion effects become important for wavenumbers much smaller than the non-linear scale. Note that at k z ∼ 0.2 h Mpc −1 the velocity dispersion terms become of order unity almost reversing the enhancement of the redshiftspace power spectrum. These additional terms have important dependencies on cosmological parameters that are different from those in the Kaiser formula, for example A σ ∼ b 1 σ 2 8 and B σ ∼ b 1 σ 4 8 in the large scale limit, where b 1 is the linear bias, with σ v ∼ σ 8 depending also on the shape of the power spectrum. This can help break degeneracies present in Eq. (58). Note that although Eq. (58) has the right limit at k z = 0, giving the real-space power spectrum, the second derivative (which is the first non-vanishing) with respect to k z does not (except at k = 0), as this is sensitive to velocity dispersion effects, both Gaussian and non-Gaussian. It is useful to recast Eq. (58) in terms of what it implies for the pairwise velocity PDF. To do this, we can expand Eq. (42) for small λ (recall λ = if k z in Fourier space), Z G ≈ 1 + ξ(r) + λ v 12 (r) + λ 2 2 ∆u 2 z .(59) This implies that the pairwise velocity PDF in the Kaiser limit has the form [see Eq. (11)], P(v) ≈ 1−f v 12 d dv + f 2 2 ∆u 2 z d 2 dv 2 δ D (v),(60) that is, it corresponds to a very sharply peaked PDF, since the dispersion ∆u 2 z is effectively assumed to be vanishingly small. This is the result used in Eq. (19) to derive the pairwise PDF in the dispersion model, and when put into Eq. (12) gives the two-point function [20,48]) ξ s (s , s ⊥ ) = ξ(s) + f d ds v 12 (s) + f 2 2 d 2 ds 2 ∆u z 2 . (61) It is interesting to go back to Fig. 4 and compare the exact result for Gaussian random fields to the Kaiser formula. The expansion in Eq. (52) is best when the scale dependence of the PDF is small. This is going to be less safe for smaller s ⊥ , since for large s ⊥ variations in r as one integrates enter only in quadrature in s 2 = r 2 + s 2 ⊥ , whereas for s ⊥ ≃ 0 variations in r enter linearly into s. This is the analogous situation to having k z not small in Fourier space, and this is why the largest deviations in Fig. 4 happen near s ⊥ = 0, even at large scales. Finally, a few words of caution about the expansion in Eq. (52). This converts integration over an infinite number of PDF's into a single one and its derivatives, thus significantly simplifying the calculation. Note however than in order to arrive to Eq. (54) one must interchange the order of the derivatives and integrals over the PDF and integrate term by term. Such a procedure is not strictly valid, since it is very likely that the expansion in Eq. (52) does not converge uniformly. Indeed, in the Gaussian case one is expanding Eq. (42) for small λ, and the exponential series has zero radius of convergence, thus term by term integration is not mathematically valid. Note also that at the end, terms that were supposed to be of increasing order in a small parameter in Eq. (52) end up being of the same order of magnitude in Eq. (58). B. Comparison with the standard derivation Let us now compare our derivation of the Kaiser limit with the standard approach ( [3,32,49]) which makes explicit use of the Jacobian J = |∂s i /∂x j | of the mapping from real to redshift space. In the plane-parallel approximation, J(x) = |1 − f ∇ z u z | and from Eq. (3) it follows that 1 + δ s (s) = [1 + δ(x)]/J(x). Now if we assume f ∇ z u z ≪ 1, we can expand 1/J(x) ≃ 1 + f ∇ z u z , and thus linearizing in the field amplitudes if follows, δ s (s) ≃ δ(x) + f ∇ z u z (x),(62) which, using that ∇ · u = δ in linear dynamics, and s ≃ x to leading order, in Fourier space leads to δ s (k) = δ(k)(1 + f µ 2 ). There are several steps in this derivation which are unjustified, namely, the density and velocity gradients at a given point in space are not small (i.e. for CDM models their linear variance at a point is much larger than unity), note that there is no smoothing involved until after one makes these approximations. In particular δ(x) can be large inside dark matter halos and similarly ∇ z u z , which will also fluctuate in sign. What is small is the correlation between fields separated by large distances, not the field amplitudes themselves. By making approximations at the level of density and velocity fields one gets incorrect correlations, in the sense that the velocity dispersion of a Gaussian random field never appears in this approach. The derivation presented in section II A and IV A shows that it is unnecessary to assume anything about the Jacobian of the trans-formation or the amplitude of density and velocity gradients. V. NON-LINEAR EVOLUTION OF DENSITY AND VELOCITY FIELDS The expansion leading to Eq. (56) has little to do with nonlinear dynamics (only involved in generating the non-Gaussian terms), but rather with the nonlinearities of the real to redshift-space mapping. We now explore the corrections induced in the redshift-space power spectrum due to non-linear evolution of the density and velocity fields. We shall see that the velocity field is affected more significantly than the density field at large scales due to larger sensitivity to tidal gravitational fields. We are interested in calculating the non-linear evolution of density and velocity divergence auto and cross spectra and comparing to numerical simulations. Measuring the volume-weighted velocity divergence power spectrum in numerical simulations is not straightforward at small scales. Interpolating the particles velocities to a grid gives the momentum (density-weighted velocities); in order to obtain θ(k) one possibility would be to i) Fourier transform the momentum, and divide the Fourier coefficients by the interpolation window ("sharpening" of the momentum Fourier coefficients). ii) do the same for the density field, and then transform back to real space density and momentum fields. iii) divide momentum by density at each grid point. Fourier transform the resulting volumeweighted velocity field and calculate the divergence in Fourier space. This procedure is not ideal for several reasons. First, there is the choice of the interpolation scheme: one would like to choose a low-order interpolation scheme because it does not smooth out fields too much (so sharpening only affects the highest-k modes), on the other hand, a low-order interpolation scheme gives rise to many grid points with zero density and momentum, thus the velocity field cannot be defined there. Using a high-order interpolation scheme bypasses this problem, but leads to some grid points with negative density after step ii), due to the fact that sharpening can be numerically unstable in voids. A more practical procedure is to divide the interpolated momentum by the interpolated density (both of which have been similarly affected by the interpolation window), Fourier transform that, and without applying any corrections (since interpolation corrections in numerator and denominator should roughly cancel), calculate the divergence of the velocity field. This procedure is safe to the extent that gives results independent of the interpolation scheme. We have tried second (CIC), third (TSC) and fourth-order interpolation schemes with similar results: at large scales k < ∼ 0.3 h Mpc −1 the different procedures give the same power spectrum, for smaller scales the results obtained start to depend on the particular scheme used. It would be interesting to try using Delaunay or Voronoi tesselation techniques [50] to see whether this can be improved for smaller scales, but our procedure is simpler and works well at large scales. We now present the calculation of the density and velocity divergence auto and cross power spectra using one-loop PT. In linear PT, by definition P δδ (k) = P δθ (k) = P θθ (k) ≡ P (k). Non-linear corrections break this degeneracy, giving P δδ (k) = P (k) + 2 [F 2 (p, q)] 2 P (p)P (q)d 3 q + 6P (k) F 3 (k, q)P (q)d 3 q (63) P θθ (k) = P (k) + 2 [G 2 (p, q)] 2 P (p)P (q)d 3 q + 6P (k) G 3 (k, q)P (q)d 3 q (64) P δθ (k) = P (k) + 2 F 2 (p, q)G 2 (p, q)P (p)P (q)d 3 q + 3P (k) [F 3 (k, q) + G 3 (k, q)]P (q)d 3 q,(65) where p = k− q. The first term of non-linear corrections describes the contribution to the power spectrum at k due coupling between modes q and p, whereas the second term corresponds instead to cor-rections to the linear growth factor that depend on k. The kernels F 2 and G 2 can be written as (k = k/k and similarly forq) F 2 (k, q) = ν 2 2 + 1 2k ·q k q + q k + 2 7 k ikj − 1 3 δ ij q iqj − 1 3 δ ij (66) G 2 (k, q) = µ 2 2 + 1 2k ·q k q + q k + 4 7 k ikj − 1 3 δ ij q iqj − 1 3 δ ij(67) where ν 2 = 34/21 and µ 2 = 26/21 represent the second-order evolution in the spherical collapse dynamics. The other two terms in these kernels have a different physical origin: the middle term is due to the non-linear transformation from following mass elements to studying the dynamics at fixed spa-tial position ("Lagrangian to Eulerian space" mapping), the last term represents the effect of the tidal gravitational fields, since (k ikj − 1 3 δ ij )δ(k) is the Fourier representation of the tidal gravitational field ∇ i ∇ j Φ(x) − 1 3 δ ij ∇ 2 Φ(x), where Φ is the gravitational potential. The important thing to notice here is that velocity fields are more sensitive to tidal fields, the coefficient of the last term in G 2 is twice that in F 2 , and consequently they evolve less by spherical collapse (that's why µ 2 is smaller than ν 2 to exactly compensate) and therefore do not grow as fast due to non-linear effects, in fact, we shall see that non-linear growth is significantly smaller than linear at the scales we are interested in. The F 3 and G 3 terms can be analyzed in a similar way, but they are more complicated, instead we just write down their expression after the angular integration overk ·q has been done, F 3 (k, q) = 6k 6 − 79k 4 q 2 + 50k 2 q 4 − 21q 6 63k 2 q 4 + (q 2 − k 2 ) 3 (7q 2 + 2k 2 ) 42k 3 q 5 ln k + q k − q ,(68)G 3 (k, q) = 6k 6 − 41k 4 q 2 + 2k 2 q 4 − 3q 6 21k 2 q 4 + (q 2 − k 2 ) 3 (q 2 + 2k 2 ) 14k 3 q 5 ln k + q k − q .(69) These terms are negative and the magnitude of G 3 is larger than F 3 . This leads to an overall suppression of P θθ (k) compared to linear theory. Figure 6 shows the results of these calculations (solid lines) and measurements in numerical simulations (symbols), expressed as ratios to the linear power spectrum P lin (k). One-loop PT for P δδ (k) performs significantly worse than for spectra with no baryonic wiggles, though it does seem to track the variations seen in the simulations, about 10% for k < ∼ 0.2 h Mpc −1 , at least in a qualitative sense. The situation is significantly better for P δθ (k), but this good agreement appears to be to some extent an accident, a cancellation between too large corrections for P δδ (k) and P θθ (k) with opposite signs. These results can be understood qualitatively and to some extent quantitatively as well by considering one-loop PT for scale-free initial conditions with P lin (k) = k n . In this case [51], where k nl is the nonlinear scale defined form the linear power spectrum, and x and y denote any of δ, θ. The functions α are decreasing functions of n, positive for n sufficiently negative and negative for n sufficiently positive; the sign of α describes whether the nonlinear growth is faster or slower than in linear theory. Corrections to P δδ (P θθ ) are positive for n < −1.4 (n < −1.9) and negative otherwise (see Fig. 12 in [52] for plots of α δδ and α θθ ). For CDM models close to the nonlinear scale at e.g. k = 0.1 h Mpc −1 , the effective spectral index is n eff ≈ −1.35, which being close to the critical index for δ where corrections to P δδ (k) vanish, leads to a small negative correction to P δδ . On the other hand, the situation is very different for θ that has a critical index of −1.9, thus the large negative corrections to P θθ [67]. This has a significant impact on the large-scale redshift-space power spectrum. For more discussion of nonlinear corrections along these lines see e.g. [51][52][53][54][55]. P xy (k) = P lin (k) 1 + α xy (n) k k nl n+3 ,(70) VI. THE REDSHIFT-SPACE POWER SPECTRUM A. A simple model We now put together the results discussed above to see how well one can match the large-scale redshift-space power spectrum with the information we have so far, without resorting to an evaluation of the PDF of pairwise velocities in the non-Gaussian case. Specifically, we use the following ansatz, P s (k) = P δδ (k) + 2f µ 2 P δθ (k) + f 2 µ 4 P θθ (k) × exp(−f 2 k 2 z σ 2 v ),(71) where P δδ , P δθ and P θθ refer to the non-linear spectra, see Fig. 6. We only include velocity dispersion effects using the large-scale limit in the Gaussian case; as discussed at the end of section III A this is not correct even at large scales, as the pairwise velocity PDF is significantly non-Gaussian at all scales. Going beyond this however requires and evaluation of the pairwise PDF in the non-Gaussian case, which is addressed in paper II. We try to compensate for this by keeping a constant Gaussian velocity dispersion suppression factor given by linear dynamics; this is an improvement over the incorrect scale dependence in linear dynamics and partially mimics the effect of non-Gaussian terms. But it is clearly an oversimplification. Note that although at first sight Eq. (71) looks similar to the phenomenological model of [6], it is in fact rather different: we do not fit for a velocity dispersion factor, but rather σ 2 v is predicted by linear dynamics and depends on σ 8 and the shape of the power spectrum; also, we incorporate the difference in evolution between density and velocity fields at large scales, as seen in Fig. 6. Figure 7 shows the results of such an exercise, compared to the numerical simulation results (symbols) and to the Kaiser formula (dashed). Although the improvement is significant there are still some deviations, which is not surprising given our approximate treatment. In particular, Eq. (71) does not give enough suppression at intermediate angles. The suppression of power at µ = 1 works reasonably well, and it is due to the velocity dispersion and the nonlinear corrections to P δθ and P θθ ; for example at k = 0.1 h Mpc −1 each effects suppresses power at µ = 1 by the same amount, about 10% each. B. Recovering the real-space power spectrum An important question is to what extent one can recover the real-space power from measurements of the redshift-space power spectrum. Attempts to do this fall into two different approaches: one is to measure the projected correlation function ξ p , Eq. (16), by integrating the redshift-space correlation function along the line of sight [56][57][58], the other is to try to measure the power for modes perpendicular to the line of sight either by smoothly approaching µ = 0 [59] or at large scales by using the Kaiser formula to go from multipoles to the realspace power [17,60]. Here we explore the conditions of validity of the latter. First, it is important to note that even if one is interested in just the shape of the real-space power spectrum (and not its amplitude), it should be clear from the results presented so far that there is no reason to expect that the monopole of the redshiftspace power should have the same shape as the realspace power, at least not to the accuracy of current large surveys such as 2DFGRS and SDSS. The top panel in Fig. 8 illustrates this point for the model of Eq. (71) and N-body simulations, where we compare their monopole and quadrupole to those in the Kaiser limit, P K = P (k)(1 + f µ 2 ) 2 with P (k) the nonlinear real-space power spectrum. Note that even at k = 0.1 h Mpc −1 the monopole is suppressed by about 10% and the quadrupole by 35 − 40%. From Fig. 8 we can see again that our model underestimates the suppression when compared to numerical simulations. In principle the situation for galaxies could be different than shown in Fig. 8, but for close to unbiased galaxies there is no reason why it should be drastically different than for the model in Eq. (71), given that we only include velocity dispersion due to large-scale flows and nonlinear corrections to θ should not be affected by biasing, being a volume weighted velocity [see discussion after Eq. (31)]. We stress that ignoring the suppression of power at large scales can contribute to systematic effects in the determination of shape parameter, the spectral tilt and running of the spectral index, or constraints on the neutrino mass. The good news is that an "inverse use" of the Kaiser formula has a larger regime of validity than one might expect based on the results discussed so far. As long as we can approximate the redshiftspace power spectrum with only ℓ = 0, 2, 4 multipoles we can always write P s (k) ≡ P (k) [1 + 2A 2 (k) µ 2 + A 4 (k) µ 4 ](72) with A 2 (k) and A 4 (k) some arbitrary functions of k. One can think of these functions as scale dependent versions of f or β when bias is present, i.e. in the Kaiser limit A 4 (k) = [A 2 (k)] 2 = β 2 . The interesting piece of information is that recovering the real-space power spectrum from the redshift-space multipoles in the case of arbitrary A 2 and A 4 is that is still given by the same linear combination as in the Kaiser limit, P (k) = P 0 (k) − 1 2 P 2 (k) + 3 8 P 4 (k),(73) even for arbirtrary A 2 (k) and A 4 (k), since Eq. (73) only uses orthogonality of Legendre polynomials up to ℓ = 4. Equation (73) is thus far more general than assuming the Kaiser limit, basically the linear combination at each scale is done using the effective value of β at that scale implied by A 2 and A 4 . The reason why this is useful is that higher than ℓ = 4 multipoles are generated only for k > ∼ 0.2 h Mpc −1 since they are suppressed by higher powers of k z in the large-scale expansion, see Eq. (52). An example of the effectiveness of using Eq. (73) is given in the bottom panel of Fig. 8, where we use it to reconstruct the real power in the case of the model in Eq. (71) and for the N-body measurements, which do not have the form of Eq. (72), since the exponential generates all multipoles higher than ℓ = 4 with roughly equal amplitude in the high-k limit and even more so for the simulation. Nonetheless, the recovery of the real-space power is quite successful for k < ∼ 0.2 − 0.3 h Mpc −1 , a bit worse for the simulation that has a larger velocity dispersion everywhere compared to the Gaussian value (see top panel in Fig. 3). The approach of using Eq. (73) to recover the real-space power spectrum was implemented already in [17,60]. Of course the use Eq. (73) can be extended to include higher multipoles if possible, this will increase the regime of validity of the reconstruction. Note however that the nice property of recovering P δδ does not extend to P δθ and P θθ , e.g. using the same idea one obtains 3 4 P 2 (k) − 15 8 P 4 (k) = A 2 (k)P (k), which cannot be interpreted as P δθ with the same degree of accuracy due to the effects of velocity dispersion. VII. CONCLUSIONS We have derived the exact relationship between two-point statistics in real and redshift space in terms of the statistics of pairwise velocities. This is given by Eq. (12) for the two-point correlation function in terms of the pairwise velocity PDF, and by Eq.(13) for the power spectrum in terms of the pairwise velocity generating function. These results include all non-linearities in the dynamics and the real-to-redshift space mapping, the only approximation made is that distortions are plane-parallel. The radial distortion case can be derived by similar reasoning to that in section II A. Higher-order correlation functions in redshift space can also be studied along the same lines. We also showed that, i) The pairwise velocity PDF is strongly non-Gaussian at all scales ( Figs. 1 and 2). The failure to reach Gaussianity at large scales is related to the fact that difference of velocities between members of a pair are always sensitive to modes whose wavelength is smaller than the distance of separation. ii) The often used dispersion model, Eq. (1), gives rise to an unphysical distribution of pairwise velocities (see bottom left panel in Fig. 1). iii) It is impossible in general to derive the PDF of pairwise velocities from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive, see Eqs. (25)(26)(27). iv) The exact result for the redshift-space correlation function of a random Gaussian field is significantly different from the Kaiser formula at large scales for pairs parallel to the line of sight (Fig. 4). v) The large-scale limit of the redshift-space power spectrum in the general case differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows, Eq. (55). vi) There are significant nonlinear corrections to the evolution of velocity fields at scales much larger than the "nonlinear scale" (Fig. 6). These are due to the sensitivity of velocities to tidal gravitational fields, which suppress the growth relative to linear perturbation theory and have a significant impact on the redshift-space power spectrum. These corrections should be included when modeling largescale velocity flows. vii) The monopole of the redshift-space power spectrum does not provide a good measure of the shape of the real-space power spectrum (top panel in Fig. 8). Ignoring this can lead to systematic effects in the determination of the spectral tilt, running of the spectral index, and limits on the neutrino mass. viii) The real-space power spectrum can be recovered at large scales by the standard procedure based on the orthogonality of multipoles (bottom panel in Fig. 8). We have ignored the problem of galaxy biasing, although linear bias is of course trivial to introduce; it is interesting to note in this regard that non-Gaussian terms give a different dependence on cosmological parameters that can be used to break degeneracies. Nonlinearities in the bias between galaxies and dark matter can lead to nontrivial behavior, this will be explored elsewhere. An important gap that remains is the derivation of the largescale limit of the PDF of pairwise velocities, this is a difficult problem that will be addressed in paper II. This should allow a more physical modeling of the redshift-space power spectrum along the lines of section VI A, where we assumed (incorrectly) that Gaussianity holds at large scales. The usefulness of such a model is that it allows for the correlation that exists between the squashing and dispersion effects, which so far have been taken as independent in the modeling of redshift distortions, such as Eq. (1). The correlation between both effects depends on Ω m , and the shape and normalization of the power spectrum. Using this information is essential to extract the full information encoded in the anisotropy of the redshift-space power spectrum, which on physical grounds must be poorly described by just two independent numbers such as β and an effective velocity dispersion σ p . m . We use the exact value from the ODE, which is straightforward to solve numerically. [64] Note that due to our normalization, velocities represented by uz should be scaled by −Hf to convert to km/s. This converts "u-velocities" (which have units of Mpc h −1 ) to "v-velocities" in km/s, see Eq. (2) and discussion below it. It will be clear from the context whether we are using "u-velocities" (which are always accompanied by appropriate factors of f ) or "v-velocities". [65] The factor f in the first argument of M in Eq. (11) takes into account the Ωm dependence of peculiar velocities in linear theory. This is almost all there is, the remaining dependence is through a time average of f 2 /Ωm, which is very weakly dependent on Ωm. [66] In fact, it is apparent from plots in [25,26] that the answer returned by this method for the PDF is manifestly symmetric, e.g. all the same noisy features appear at v and −v. [67] A similar (reversed) situation happens at high redshift, where the spectral index close to the nonlinear scale becomes very negative. In this case corrections to P δδ become large and positive (see e.g. Fig. 3 in [61]). This is another example where the meaning of a "nonlinear scale" from the linear power spectrum can be very misleading. Due to tidal effects the growth of density perturbations is not just a function of the amplitude of the density field at a given point (as in the spherical collapse model) but also has important dependence on the shape of the power spectrum. FIG. 1 : 1The parallel to the line of sight pairwise velocity PDF at redshift z = 0 for pairs separated by distance r, measured in the N-body simulations. In the bottom left panel, the discontinuous at the origin PDF (thin solid line) corresponds to that given by the dispersion model, Eq. (19) (ignoring the delta function at the origin). In the bottom right panel, the narrow distribution (thin solid line) corresponds to the prediction of linear dynamics, Eq.(44). FIG. 2 : 2Same as Fig. 1 but for pairs perpendicular to the line of sight. The thin solid lines in the bottom panels are as in Fig. 1, the prediction of the dispersion model (left) and linear dynamics (right). this leads to the exact expression, FIG. 3 : 3Moments of pairwise velocities parallel to the line conecting the pair as a function of scale. Top panel: pairwise dispersion σ12 (squares) as a function of scale, its connected piece σ c 12 (solid line), and the mean infall v12 (triangles). The dashed line denotes the predicted σ12 in linear dynamics. Middle panel: dimensionless measure of infall (|v12|/σ c 12 , triangles) compared to the skewness of the pairwise velocity PDF (squares); the skewness dominates at most scales. Bottom panel: kurtosis as a function scale, note that it does not vanish at large scales and s4 > 1 at all scales; the pairwise velocity PDF is strongly non-Gaussian at all scales, see Fig. 1. For reference, an exponential distribution has s3 = 0 and s4 = 3. FIG. 4 : 4Contours of ξs(s , s ⊥ ) for the Exact Gaussian result (solid) and the Kaiser limit (dashed), for a flat ΛCDM cosmological model (Ωm = 0.26, σ8 = 0.9, Ω b = 0.04, h = 0.7) with linear bias b1 = 1. FIG. 5: Same as Fig. 4 but at small scales. Note the dispersion effect: a Gaussian random field does show "fingers of god" even though the velocity dispersion decreases monotonically towards small scales (see dashed line in Fig. 3). FIG. 6 : 6Non-linear corrections to the density-density (top), density-velocity (middle) and velocity-velocity (bottom) power spectra as a function of scale. The symbols denote measurements in the VLS dark matter simulations, solid lines denote one-loop perturbation theory. FIG. 7 : 7Contours of the redshift-space power spectrum at z = 0. The solid lines correspond to the N-body simulation results, dashed lines denote the Kaiser formula, and dotted lines show the simplified ansatz of Eq. (71). FIG. 8 : 8Top panel: Ratio of the monopole (solid) and quadrupole (dashed) to the predictions of the Kaiser formula, for redshit-space power given by Eq. (71). Symbols show the same quantities in the numerical simulations. Bottom panel: recovery of the real-space power spectrum from redshift-space multipoles according to Eq. (73) for the model in Eq. (71) (solid) and N-body simulations (symbols). AcknowledgmentsWe thank Andreas Berlind, Martín Crocce, Josh Frieman, Enrique Gaztañaga, Roman Juszkiewicz, Dmitry Pogosyan, Alex Szalay, Max Tegmark, Jeremy Tinker, and David Weinberg for useful discussions. I benefited greatly from feedback and innumerable discussions with Andrew Hamilton, Lam Hui and Ravi Sheth. 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J. , 590:1-7, 2003. reversed from that in Fourier space; for a multipole moment of order ℓ they are related by a factor i ℓ (ℓ must be even by statistical isotropy), see Eq. (51) below. At small scales it is easier to characterize multipoles in configuration space. In configuration space, the sign of the quadrupole isIn configuration space, the sign of the quadrupole is reversed from that in Fourier space; for a multipole moment of order ℓ they are related by a factor i ℓ (ℓ must be even by statistical isotropy), see Eq. (51) below. At small scales it is easier to characterize multipoles in configuration space, they are positive for all ℓ and of comparable magnitude. It is easy to show that in these cases f obeys (3/2)x = (a − bx)f + cx(x − 1)f ′ + f 2 . For open models, a = c = 1, b = 1/2 and f (Ωm) ≈ Ω. It is easy to show that in these cases f obeys (3/2)x = (a − bx)f + cx(x − 1)f ′ + f 2 . For open models, a = c = 1, b = 1/2 and f (Ωm) ≈ Ω
[]
[ "Exploiting Web Images for Weakly Supervised Object Detection", "Exploiting Web Images for Weakly Supervised Object Detection" ]
[ "Qingyi Tao \nNanyang Technological University\nSingapore\n", "Hao Yang \nNanyang Technological University\nSingapore\n", "Jianfei Cai \nNanyang Technological University\nSingapore\n" ]
[ "Nanyang Technological University\nSingapore", "Nanyang Technological University\nSingapore", "Nanyang Technological University\nSingapore" ]
[]
In recent years, the performance of object detection has advanced significantly with the evolving deep convolutional neural networks. However, the state-of-the-art object detection methods still rely on accurate bounding box annotations that require extensive human labelling. Object detection without bounding box annotations, i.e, weakly supervised detection methods, are still lagging far behind. As weakly supervised detection only uses image level labels and does not require the ground truth of bounding box location and label of each object in an image, it is generally very difficult to distill knowledge of the actual appearances of objects. Inspired by curriculum learning, this paper proposes an easy-to-hard knowledge transfer scheme that incorporates easy web images to provide prior knowledge of object appearance as a good starting point. While exploiting large-scale free web imagery, we introduce a sophisticated labour free method to construct a web dataset with good diversity in object appearance. After that, semantic relevance and distribution relevance are introduced and utilized in the proposed curriculum training scheme. Our end-to-end learning with the constructed web data achieves remarkable improvement across most object classes especially for the classes that are often considered hard in other works.
10.1109/tmm.2018.2875597
[ "https://arxiv.org/pdf/1707.08721v2.pdf" ]
28,883,421
1707.08721
4c56ff2a9626edb0e844ff2b861e1f558f900851
Exploiting Web Images for Weakly Supervised Object Detection Qingyi Tao Nanyang Technological University Singapore Hao Yang Nanyang Technological University Singapore Jianfei Cai Nanyang Technological University Singapore Exploiting Web Images for Weakly Supervised Object Detection In recent years, the performance of object detection has advanced significantly with the evolving deep convolutional neural networks. However, the state-of-the-art object detection methods still rely on accurate bounding box annotations that require extensive human labelling. Object detection without bounding box annotations, i.e, weakly supervised detection methods, are still lagging far behind. As weakly supervised detection only uses image level labels and does not require the ground truth of bounding box location and label of each object in an image, it is generally very difficult to distill knowledge of the actual appearances of objects. Inspired by curriculum learning, this paper proposes an easy-to-hard knowledge transfer scheme that incorporates easy web images to provide prior knowledge of object appearance as a good starting point. While exploiting large-scale free web imagery, we introduce a sophisticated labour free method to construct a web dataset with good diversity in object appearance. After that, semantic relevance and distribution relevance are introduced and utilized in the proposed curriculum training scheme. Our end-to-end learning with the constructed web data achieves remarkable improvement across most object classes especially for the classes that are often considered hard in other works. Introduction With the rapid growth of computational power and dataset size and the development of deep learning algorithms, object detection, one of the core problems in computer vision, has achieved promising results [18,16,15,14]. However, state-of-the-art object detection methods still require bounding box annotations which cost extensive human labour. To alleviate this problem, weakly super- vised object detection approaches [21,2,3,4,11,6,19] have attracted many attentions. These approaches aim at learning an effective detector with only image level labels, so that no labour-extensive bounding box annotations are needed. Nevertheless, as objects in common images can appear in different sizes and locations, only making use of image level labels are often not specific enough to learn good object detectors, and thus the performance of most weakly supervised methods are still subpar compared to their strongly supervised counterparts, especially for small objects with occlusions, such as "bottle" or "potted plant". As shown in Figure 1, images containing small objects or with very complicated contexts are hard to learn. In contrast, images containing a single object with very clean background provides very good appearance priors for learning object detectors. Particularly, for these easy images, the difficulty of localizing the objects is much lower than complicated images. With correct localization, the appearance model can be better learned. Therefore, easy images can provide useful information of the object appearance for learning the model for more complicated images. Unfortunately, such easy images are rarely available in object detection datasets, such as PASCAL VOC or MS COCO, as images in these multiobject datasets usually contain cluttered objects and very complicated background. On the other hand, there are a large number of easy web images available online and we can exploit these web images for the weakly supervised detection (WSD) task. However, to construct a suitable auxiliary dataset and appropriately design an algorithm to utilize the knowledge from the dataset are non-trivial tasks. In this paper, we intend to provide a practical and effective solution to solve both problems. Specifically, as various image search engines like Bing, Google, Flickr provide access to freely available web data of high quality images. Recent researches [8,5,23,24,12] have already utilized these large-scale web data in various vision tasks. However, as object detection tasks impose specific requirements for auxiliary web data, we need to carefully design a labour-free way to obtain suitable images for the task. First of all, when constructing the web dataset, we need to consider the relevance of web images in order to effectively transfer the knowledge of easy web images to the target detection dataset. In this paper, we break down this relevance into two parts, namely semantic relevance, which refers to the relevance between web images and the target labels, and the distribution relevance, which refers the relevance between web images and target images. As we will shown in later section, the semantic relevance focuses on a larger picture in the semantic space, while the distribution relevance measures more fine-grain differences in the feature distributions. To give an example, for category "chair", the semantic relevance measures whether a certain web image is "chair" or not, and the distribution relevance measures whether this web image lies on the manifold formed by the specific "chairs" in the target dataset. Secondly, apart from the relevance problem, we also need to consider the diversity of the web images. As sub-categories, poses as well as backgrounds are crucial for the success of object detection, and thus our web images should not only be easy and related to the target dataset, but also contain a variety of different images even for the same category. With single text query, commonly used image search engines are not able to produce images with large intra-category diversity, especially in top ranked results. Therefore, inspired by [8], which uses ngram to retrieve the fine-grained dataset, we propose a multi-attribute web data generation scheme to enhance the diversity of web data. Specifically, we construct a general attribute table with common attributes that can easily be propagated to other target datasets as well. With the attribute table, we are able to build a hassle-free web dataset with proper category-wise diversity for the coarsely labeled dataset. Once we have an appropriate web dataset, we need to consider how to transfer the knowledge from the easy web images to more complex multi-object target datasets. During the recent years, easy web images have been used in other weakly supervised tasks, such as weakly supervised segmentation [22]. To the best of our knowledge, we are the first work bringing in web images for improving the weakly supervised object detection task. Inspired by curriculum learning [1], we propose a simple but effective hierarchical curriculum learning scheme. Specifically, with the hierarchical curriculum structure, all web images are considered easier than target images, which we refer as the first level of curriculum, followed by the second level of curriculum that includes all target images. Extensive experimental results show that our constructed web image dataset and the adopted curriculum learning can significantly improve the WSD performance. Related Work Our work is related to several areas in computer vision and machine learning. Weakly Supervised Object Detection (WSD): Traditional WSD methods like [6] address this problem with multiple instance learning (MIL) [7], which treats each image as a bag and each proposal/window in the image as an instance in the bag. A positive image contains at least one positive instance whereas a negative image contains only negative instances. Since MIL approaches alternate the processes between selecting a region of objects and using the selected region to learn the object appearance model, they are often sensitive to initialization and often get stuck in local optima. [4] proposed a two-stream CNN structure named WSDDN to learn localization and recognition in dedicated streams respectively. These two streams share the common features from the earlier convolutional layers and one fully connected layer. It learns one detection stream to find the high responsive windows and one recognition stream to learn the appearance of the objects. In this way, the localization and recognition processes are decoupled. Similarly, [11] also uses the two stream structure and involves the context feature in the localization stream. In this research, we use WSDDN [4] as an example to evaluate our learning method. Since WSDDN separates recognition and localization into two individual streams, it introduces additional degrees of freedom while optimizing the model, and hence it is hard to train at the early stage. It is also sensitive to initialization. Thus, in this work we propose to explicitly provide good initialization during the training process in an easy-to-hard manner. Note that although we utilize WSDDN as our baseline, our learning scheme is general and can be applied to other WSD methods as well. Curriculum Learning: Our work is inspired by curriculum learning [1] scheme. Curriculum learning was initially proposed to solve the shape recognition problem, where the recognition model is first trained to recognize the basic shapes and then trained on more complicated geoshapes. Recently, Tudor et al. [20] used this easy-tohard learning scheme in MIL problem but mainly focused on learning a model to rank images with difficulty that matches the human perspective. In our work, we propose a hierarchical curriculum scheme that incorporates easy web images in early training stage to provide prior knowledge for the subsequent training on complicated images. Learning from Weak or Noisy Labels: This paper is also related to those works on learning from weak or noisy labels [8,17,5,9]. In [8], they proposed a classifier-based cleaning process to deal with the noisy labels. They first train a classification model on images with higher confidence and then use this model to filter the outliers in the rest of images. Later, with incorporation of CNN, novel loss layer is introduced to the deep network in [17]. In [5], web images are separated into easy images (Google) and hard images (Flickr). They build a knowledge graph on easy web images and use the graph as a semantic constraint to deal with the possible label-flip noises during training of harder web images. Similarly, [9] learns the mutual relationship to suppress the feedback of noises during the back propagation. These works emphasize their methods to lessen the impact by outliers during the training process. In our work, apart from the outliers, we also consider distribution mismatch problem since we acquire web data that are from completely different information source with discrepant distribution compared to target dataset. Approach In this part, we introduce the methodology on constructing the web dataset and the hierarchical curriculum learning to transfer the knowledge of web images to target dataset. We will use state-of-ther-art weakly supervised objection algorithm WSDDN [4] as an example to show the effectiveness of our scheme. Note that our scheme is general and can be adapted to any other available algorithms if necessary. WSDDN We first introduce weakly supervised deep detection network, or WSDDN [4], which is utilized as baseline for our experiments. WSDDN provides an end-to-end solution that breaks the cycle of training of classification and localization alternatively by decoupling them into two separate streams. Particularly, WSDDN replaces the last pooling layer with spatial pyramid pooling layer [13] to obtain SPP feature of each region of interest (RoI). As shown in Figure 4, the SPP features are passed to a classification stream and a localization stream which individually learns the appearance and location of the objects. In the classification stream, the score for each RoI from f c8 layer is normalized across classes to find the correct label of RoIs. In the localization stream, the scores of all RoIs are normalized category-wise to find most respondent RoIs for each category. Then the probability outputs from both softmax layers are multiplied as the final detection scores for each RoI. Finally, detection scores of all RoIs are summed up to one vector as the image level score to optimize the loss function (1). L(y ci , x i |w) = −log(y ci (Φ c (x i |w) − 1 2 ) + 1 2 ) (1) In the binary log loss function L(y ci , x i |w) , x i is the input image i, and y is the binary image level label where y c i = {−1, 1} for class c in image i. Output from the last sum pooling layer is denoted as Φ y c (x i |w) which is a vector in range of 0 to 1 with the dimension equal to number of category. For each class c, if the label y ci is 1, L(y ci , x i |w) = −log(p(y ci = 1)) and if y ci is −1, L(y ci , x i |w) = −log(1 − p(y ci = 1)). Constructing Multi-Attribute Web Dataset In this section, we describe our method to construct a diversified and robust web dataset by introducing an expand-to-condense process. Specifically, we first introduce multiple attributes on top of the given target labels when crawling for web images to improve the generalization ability of the obtained dataset. Then we introduce both semantic relevance and distribution relevance to condense the dataset by filtering out irrelevant images. Expand to Diversify Free web images are abundantly available and accessible. Many image search engines can provide high quality images by searching the object names, such as Google, Flickr and Bing. In our preliminary study, we observe that images searched from Bing are generally easier than images from other search engines. Since easier images are intuitively better for learning object appearance, we choose Bing as the search engine to crawl web images. However, for most search engines, we observed that if we just use the given target labels as keywords, the resulting images are very similar in object appearances, poses or sub-categories. Moreover, the number of good quality images returned per query is very limited and lower ranked images are generally very noisy and unrelated to the queries. To solve the problem of lacking diversity as well as limited number of high quality images, we introduce multiple attributes to each category. Based on the general knowledge of object detection, we define a set of attributes in three general aspects: namely viewpoints, poses or habitats of the objects. First of all, adding viewpoint attributes such as "front view" and "side view" not only provides extensive amount of high quality images for artificial objects like "aeroplane", "car" and "bus", but also enhances the appearance knowledge of these objects, which will eventually make the detector more robust. Note that for categories without clear discrepancy between front view and side view such as "bottle" and "potted plant", as well as flat objects like "tv monitor", we do not include these attributes. Secondly, for animals like "cat" and "dog", we add pose attributes. As their appearances vary significantly in different poses, adding such attributes will also be beneficial towards more robust detector. In particular, we add poses such as "sitting", "jumping" and "walking" to these animal categories. Last but not least, for category "bird" which resides in different habitats, we add habitat attributes of "sky" and "water". The set of attributes is summarized in Table 1. Note that following the same spirit, the table can be easily expanded to other categories. Moreover, to overcome the limitation of limited available clean images in the top ranking, we also crawl related images. Related images are the images retrieved with similar visual appearance by using each of the previously retrieved top ranked images as query to the search engine. These related image can expand the size of the web dataset by more than 20 times and also introduce more variations to the dataset. Fig. 2 illustrates the process of expanding the dataset by the multi-attribute per-category expansion and the per-image expansion. Condense to Transfer Once we obtain a large scale web image dataset, we are facing with the relevance problem. As free web data often contain many noisy images, to effectively make use of these web images, we need to analyse the image relevance to condense the noisy data. In this paper, we break down the image relevance to two parts: semantic relevance and distribution relevance. In detail, semantic relevance indicates whether a image contains the correct objects and distribution relevance measures how well a web image matches the the distribution of the target dataset. Firstly, to measure the semantic relevance, we train a web-to-web outlier detector to find images with wrong labels in the web dataset. Specifically, we select top 80 images from queries of each target label and top 20 images from queries of each attribute + label combination. As we only use high ranked images as seed images, the "cleanness" of the images can be guaranteed, and thus we are able to learn a more robust outlier detector. The outlier detector is trained iteratively with the expansion of the seed images. Similar to the idea of active learning, we train a CNN classifer with softmax loss with the seed images. Then it is applied to the whole set of the web images. The highly confident positive samples are then used as the second batch of training images for next iteration. After a few iterations, the classification scores from the final stabilized model are used to measure semantic relevance. As shown in Figure 3, our model can provide very solid semantic relevance measurement. Most of the non-meaningful images have negative scores, outliers with wrong objects have very low scores and images with correct objects have high scores. Secondly, since semantic relevance condenses images purely based on their semantic meaning regardless of the distribution matching with target dataset, we also consider the distribution relevance for more fine-grain measurements. To align the diversified web dataset into the distribution of target dataset, we search in the neighborhood of the target dataset to find similar web images. Particularly, for each single-label image in the target dataset, we select k nearest web images in the feature space. The distance between images is defined as the Euclidean distance between their corresponding CNN features. Specifically, we use the L2 normalized f c7 feature from a pretrained vgg-f model with PCA dimension reduction to represent each image. As shown in Figure 3, our method is capable to extend the target dataset with web images having very similar object appearances and poses. We expect both relevance metrics to be effective for this task since it is intuitive to eliminate noises and unrelated data during the training. Nevertheless, our experiment result shows that matching the web data to target distribution is not as helpful as using a clean but diversified web dataset. Relevance Curriculum Regularizer Incorporating a good quality web dataset to the target dataset does not automatically guarantee better performance. Based on our experiments, we find out that simply appending these web images to target dataset is unhelpful or even harmful. These easy web images could lead to skew training models due to the distribution misalignment problem of the two datasets. Therefore, instead of simply appending web data to target dataset, we propose a hierarchical curriculum structure. Specifically, we first consider a coarser curriculum with web images as easy and all target images as hard. If necessary, we could also add a fine curriculum to each dataset for full curriculum learning. Moreover, in addition to the normal curriculum or self-paced learning [10], we also consider adding an extra relevance term. As an analogy, we could consider web images as extracurricular activities. In order to help students with their learning, extracurricular activities need to be relevant to the course, in the same way that we should learn from easy images and relevant images. In particular, to incorporate both curriculum and relevance constraints in training, we propose a relevance curriculum regularizer to the base detection structure: E(w) = n i=1 C c=1 L(y i , x i |w) · f (u i , v i ),(2)f (u i , v i ) = σ(u i ) · ψ(v i ),(3) where u i is the relevance variable indicating whether the training sample is relevant as discussed in 3.2. v i is the curriculum regulation variable which indicates difficulty score of each image. σ is the relevance region function that only relevant samples can be learned every epoch. If a sample is in the relevance region, the value of σ(u) is 1 and otherwise 0. ψ is the curriculum region. It controls the pace of learning that allows only easy samples to be learned at early stage and gradually adding harder samples along the training process. If the difficulty score of sample image is within the curriculum region, ψ(v) is 1 and otherwise, ψ(v) is 0. As described previously, we implemented a hierarchical curriculum, where ψ(v) for all web images are consider as 1 first, then we gradually expand it to include target images. Experiments In this section, we evaluate the effectiveness of our proposed weakly supervised object detection. Baseline Model & Setting For experimental setting, similar to the original WSDDN work, we use Edgebox [25] as the proposal method to generate around 2000 bounding boxes. To train the network, we use the vgg-f model pretrained on ImageNet as the initial model. For fairness, our results are compared with the baseline method trained on vgg-f as well. We evaluate our method on PASCAL VOC2007 and VOC2012 datasets with 20 object categories. During the training, we use only image-level labels of the training images. The evaluation metric is the commonly used detection mAP with IoU threshold of 0.5. Results Regarding Curriculum Learning We first evaluate the effectiveness of applying the curriculum learning method on PASCAL VOC 2007 trainval set itself, without using our web data. The curriculum is designed by the ranking of the mean edge strength of each image. The mean edge strength of an image is defined as the number of edge pixels over the total number of pixels. This is a simple yet intuitive method because images with more edges tend to have more complicated background or contain more cluttered objects, and thus it is reasonable to consider them as hard samples. Fig. 5 gives some examples, which show that the mean edge length represents the relative difficulty of the images well. Specifically, we use the classical LoG edge detector to detect edges. For each curriculum region, we add 1 5 of more difficult images from each category. This is to balance the number of positive samples from each category in every iteration. In this way, the curriculum consists Table 3: Results of the detection mAP on VOC2007 test set by using our constructed Bing dataset or the 'Flickr clean' dataset as easy images and VOC images as hard images for easy-to-hard training. Web dataset mAP WebETH(Flickr clean) 35.5 WebETH(Bing) 36.0 of five overlapped regions with gradually increased image complexity. Table 2 shows the detection result ('CurrWS-DDN') of applying the curriculum regularization term to train VOC 2007 trainval set only, compared with the result of the baseline ('WSDDN'). We can see that using curriculum learning on VOC 2007 training images alone already improves the performance. This suggests that for training weakly supervised object detector, it is beneficial to train the network in an easy-to-hard manner. Note that the baseline WSDDN result is obtained by running the original WSDDN codes released in Github with the same setting 1 , which is slightly different from the result of 34.5 reported in [4]. Results Regarding Constructed Web Dataset We now evaluate the usefulness of our constructed web image dataset for WSD. As mentioned in Section 3.2, we construct a web image dataset of 34k images using Bing image search engine with attributes and related images. Considering that many selected web images are of high resolution, which causes huge complexity in the proposal generation process, we resize the longer side of all images to 600 pixels and keep the aspect ratios. We treat all web images as easy images and all VOC images as hard images. Simple web images are trained first followed by more complicated VOC images. Table 2 shows the detection result of our method 'WebETH(Bing)' that exploits our constructed Bing dataset and trains the network in an easy-to-hard manner. Comparing Tables 2 and 3, we can see that our method 'WebETH(Bing)' significantly improves the baseline 'WSDDN', increasing mAP from 33.9% mAP to 36%, and also outperforms the VOC curriculum method 'CurrWSDDN'. We also conduct experiments on another publicly available web dataset, STC Flickr clean dataset [22], which contains more than 40k super clean images and has been proven to have good performance in generating good saliency maps to train weakly supervised segmentation networks. Surprisingly, by involving STC Flickr clean, although its result (see Table 2) is much better than the baseline using only VOC images, it has no improvement over the VOC curriculum method 'CurrWSDDN'. In contrast, using our noisy Bing dataset 'WebETH(Bing)' beats both the VOC curriculum method 'CurrWSDDN' and the Flickr clean dataset 'WebETH(Flickr clean)'. This suggests that our approach of constructing a multi-attribute web dataset with large diversity is practically useful in this context. Results Regarding Relevance Metrics Here we conduct experiments to study the effectiveness of using semantic relevance and distribution relevance. Fig. 3 gives some examples of the two relevance metrics. For the semantic relevance, we use the classification scores by the outlier detector described in Section 3.2.2, whose values vary from negative to more than 20. We set a semantic relevance threshold of 8 so that web images with scores lower than 8 are excluded. This prevents from mixing in noisy images without target objects into the early stage of training. For the distribution relevance, its relevance region includes web images which are members of top k-th nearest neighbors of one of VOC images, as illustrated in Fig. 3. Table 4 shows the results using the two relevance metrics. We can see that with the semantic relevance, the detection result increases from 36.0% to 36.8%, whereas the kNN based distribution relevance gives a slightly lower result, which suggests that similar images might not be always preferred. As a non-convex optimization problem, Table 5 might not be exhaustive since there might be some very recent WSD methods that report better performance. Since our solution is general, which can be added on top of any WSD baseline, it is more meaningful to evaluate our methods w.r.t the baseline. Based on WSDDN, we consider five variants: using only VOC images with the curriculum regularizer (Cur-rWSDDN), simply combining our web images with VOC images with the semantic relevance for training (WebRel), combining our web images with VOC images for easy-tohard training (WebETH), combining our web images with VOC images with the semantic relevance for easy-to-hard training (WebRelETH), and combining our web images with VOC images with the semantic relevance for easy-tocurriculum training (WebRelETC), where we train easy web images first and then train VOC images in a more detailed curriculum. More Comparison Results The results of CurrWSDDN, WebETH and We-bRelETH have been discussed previously w.r.t. Tables 2, 3 and 4, which demonstrate the effectiveness of the curriculum regularizer, the constructed web dataset, the proposed relevance metrics, respectively. For WebRel, its result is even worse than the baseline WSDDN, which suggests that it is not an effective way to simply combine data from two sources. In our case, a large number of easy images dominate the training so that the model cannot be well trained for hard samples. For WebRelETC, we expect that web images to have similar difficulty level but VOC images need to be partitioned in more levels of difficulty. We first train on easy web images and adopt five-level curriculum regions for VOC images. It is found that its average precision performance is slightly worse than WebRelETH. This suggest that it is not always good to further break down the higher level curriculum for every class if the lower-level curriculum of simple web images have been used. Overall, our WebRelETH achieves the best mAP of 36.8%, outperforming the baseline by 2.9%. Table 6 shows the experiment results for VOC2012. Our method also achieved up to 3.8% improvement in this dataset. Similar to VOC2007, WebRelETH outperforms WebRelETC, although WebRelETC excels largely in "dining table" by more than 10%. Fig. 6 gives some visual comparisons of the detection results using WSDDN and our best model (WebRelETH). It can be seen that our model can refine the bounding boxes (see the top two rows of Fig. 6), and missing objects in WSDDN can also be detected by our model in some test images (see the bottom rows of Fig. 6). Conclusion This paper have addressed the two questions: how to construct a large, diverse and relevant web image dataset and how to use it to help weakly supervised object detection. Particularly, for constructing the web dataset, we introduced a sophisticated expand-to-condense process to first expand web data with attributes and related images and then condense the dataset with semantic relevance or distribution relevance. For helping the target dataset, we applied an easy-to-hard learning scheme. Extensive results have validated that our easy-to-hard learning with web data is effective and the multi-attribute web data do help in training a weakly supervised detector. Figure 1 : 1Easy web images and VOC images. Web images have clean background while VOC images are more difficult with cluttered instances and complicated background. Figure 2 : 2Multi-attribute related dataset. Aeroplane category is expanded with multi-view attributes including front view and side view. Each multi-attribute web image is then expanded by the related images obtained from Bing image search engine. Figure 3 : 3Illustration of relevance metrics including semantic relevance and distribution relevance. Top: semantic relevance by the scores from web-to-web classifier for motorbike images, where non-meaningful images have negative scores, outliers with wrong objects have very low scores, and images containing correct objects have high scores. Bottom: distribution relevance by k nearest neighbors of each motorbike image in VOC dataset, where images in the neighborhood of VOC images with small feature distances are considered relevant to the target dataset. Figure 4 : 4WSDDN with relevance curriculum regularizer. The relevance curriculum regularizer suppresses backpropagation from samples which do not fit in the relevance region and curriculum region. Figure 5 : 5Curriculum metric by the mean edge strength for web images (Top: table / Bottom: potted plant). The mean edge strength can reasonably represent the difficulty of images. Images with clean background and single object usually have small mean edge strength and images with complicated background and cluttered objects usually have large edge strength. Table 1 : 1Attribute table.Category Viewpoint Pose Habitat aeroplane; bicycle; boat; bus; car; motorbike; train; chair; diningtable; sofa front view; side view - - bird front view; side view - water; sky cat; dog front view; side view sitting; walking; jumping - cow; sheep front view; side view walking; - horse front view; side view walking; jumping - person front view; side view sitting; standing; walking - Table 2 : 2Results of the detection mAP on VOC2007 test set by using the curriculum regularization term to train VOC 2007 trainval set.Methods mAP WSDDN (baseline) 33.9 CurrWSDDN 35.5 Table 4 : 4Results of the detection mAP on VOC2007 test set with different relevance metrics and using our con- structed Bing dataset with the easy-to-hard training. Transfer metrics mAP WebRelETH(Dist-Rel) 35.9 WebRelETH(Semantic-Rel) 36.8 Table 5 : 5Comparisons of the detection average precision results (%) on VOC2007 test set with training on VOC2007 trainval set. We use vgg-f model pretrained on ImageNet. WSDDN results are obtained using the published code on Github with the same setting stated in[4]. Results of other methods are from their papers. aero bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tvmean Table 6 : 6Results of the detection average precision (%) on VOC2012 test set with training on VOC2012 training set. We use vgg-f model pretrained on ImageNet. WSDDN results are obtained using the published code on Github with the same setting stated in [4]. aero bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tv mean WSDDN [4] 53.4 53.2 36.2 7.9 16.4 57.2 35.3 24.8 6.5 29.0 13.7 31.1 47.1 57.2 11.0 18.9 28.6 19.4 42.3 39.6 31.4 WebRelETH (ours) 57.6 55.1 38.5 8.6 20.4 59.4 36.4 33.6 14.0 34.8 21.7 39.4 51.3 62.8 11.5 19.2 30.2 23.9 41.2 44.5 35.2 WebRelETC (ours) 55.6 56.2 35.3 7.4 20.5 55.6 32.6 34.8 9.7 32.9 32.1 34.6 48.4 61.6 15.5 18.9 27.3 15.7 41.2 43.5 34.0Figure 6: Visual results of WSDDN and our best model (WebRelETH). Our model can refine the bounding boxes as shown in the top two rows. Missing objects in the original model can also be detected in some test images as shown in the bottom rows.the training of WSD tends to drift to optimize small clus-ters of training samples. Although additional training in-stances with a similar distribution can help achieve lower training loss, it is not as helpful as involving new training samples with larger diversity, which leads to better generalization ability. This may also explain why STC Flickr clean dataset is not so helpful since the images in the Flickr clean dataset also have a similar distribution as VOC dataset. Table 5 5lists out the per-category average precision results of different WSD methods on VOC2007 test set with training on VOC2007 trainval set. It can be seen that compared with other existing WSD methods, the baseline method WSDDN achieves reasonably good performance. We would like to point out that our list in https://github.com/hbilen/WSDDN Curriculum learning. Y Bengio, J Louradour, R Collobert, J Weston, Proceedings of the 26th annual international conference on machine learning. the 26th annual international conference on machine learningACMY. Bengio, J. Louradour, R. Collobert, and J. 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In Computer vision and pattern recog- nition, 2006 IEEE computer society conference on, vol- ume 2, pages 2169-2178. IEEE, 2006. R-fcn: Object detection via region-based fully convolutional networks. Y Li, K He, J Sun, Advances in Neural Information Processing Systems. Y. Li, K. He, J. Sun, et al. R-fcn: Object detection via region-based fully convolutional networks. In Advances in Neural Information Processing Systems, pages 379-387, 2016. . W Liu, D Anguelov, D Erhan, C Szegedy, S Reed, C.-Y , W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C.-Y. Ssd: Single shot multibox detector. A C Fu, Berg, European Conference on Computer Vision. SpringerFu, and A. C. Berg. Ssd: Single shot multibox detector. In European Conference on Computer Vision, pages 21-37. Springer, 2016. You only look once: Unified, real-time object detection. J Redmon, S Divvala, R Girshick, A Farhadi, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionJ. Redmon, S. Divvala, R. Girshick, and A. Farhadi. You only look once: Unified, real-time object detection. In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 779-788, 2016. S Reed, H Lee, D Anguelov, C Szegedy, D Erhan, A Rabinovich, arXiv:1412.6596Training deep neural networks on noisy labels with bootstrapping. arXiv preprintS. Reed, H. Lee, D. Anguelov, C. Szegedy, D. Erhan, and A. Rabinovich. Training deep neural networks on noisy la- bels with bootstrapping. arXiv preprint arXiv:1412.6596, 2014. Faster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, Advances in neural information processing systems. S. Ren, K. He, R. Girshick, and J. Sun. Faster r-cnn: To- wards real-time object detection with region proposal net- works. In Advances in neural information processing sys- tems, pages 91-99, 2015. Attention networks for weakly supervised object localization. E W Teh, M Rochan, Y Wang, E. W. Teh, M. Rochan, and Y. Wang. Attention networks for weakly supervised object localization. How hard can it be? estimating the difficulty of visual search in an image. R Tudor Ionescu, B Alexe, M Leordeanu, M Popescu, D P Papadopoulos, V Ferrari, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionR. Tudor Ionescu, B. Alexe, M. Leordeanu, M. Popescu, D. P. Papadopoulos, and V. Ferrari. How hard can it be? estimating the difficulty of visual search in an image. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2157-2166, 2016. Weakly supervised object localization with latent category learning. 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Wang. Learn- ing from massive noisy labeled data for image classifica- tion. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2691-2699, 2015. Augmenting strong supervision using web data for fine-grained categorization. Z Xu, S Huang, Y Zhang, D Tao, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionZ. Xu, S. Huang, Y. Zhang, and D. Tao. Augmenting strong supervision using web data for fine-grained cate- gorization. In Proceedings of the IEEE International Con- ference on Computer Vision, pages 2524-2532, 2015. Edge boxes: Locating object proposals from edges. C L Zitnick, P Dollár, European Conference on Computer Vision. SpringerC. L. Zitnick and P. Dollár. Edge boxes: Locating object proposals from edges. In European Conference on Com- puter Vision, pages 391-405. Springer, 2014.
[ "https://github.com/hbilen/WSDDN" ]
[ "Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff", "Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff" ]
[ "M Takeda \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "N Hayashida \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "K Honda \nFaculty of Engineering\nYamanashi University\n400-8511KofuJapan\n", "N Inoue \nDepartment of Physics\nSaitama University\n338-8570UrawaJapan\n", "K Kadota \nDepartment of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan\n", "F Kakimoto \nDepartment of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan\n", "K Kamata \nNishina Memorial Foundation\n113Komagome, TokyoJapan\n", "S Kawaguchi \nFaculty of General Education\nHirosaki University\n036-8560HirosakiJapan\n", "Y Kawasaki \nDepartment of Physics\nOsaka City University\n558-8585OsakaJapan\n", "N Kawasumi \nFaculty of Education\nYamanashi University\n400-8510KofuJapan\n", "H Kitamura \nDepartment of Physics\nKobe University\n657-8501KobeJapan\n", "E Kusano \nDepartment of Physics\nSaitama University\n338-8570UrawaJapan\n", "Y Matsubara \nSolar-Terrestrial Environment Laboratory\nNagoya University\n464-8601NagoyaJapan\n", "K Murakami \nNagoya University of Foreign Studies\n12 11-401, 5-8 Higashi470-0131, 394-0111Nissin, HasudaAichi, SaitamaJapan, Japan\n", "M Nagano ", "D Nishikawa \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "H Ohoka \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "N Sakaki \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "M Sasaki \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "K Shinozaki \nDepartment of Physics\nSaitama University\n338-8570UrawaJapan\n", "N Souma \nDepartment of Physics\nSaitama University\n338-8570UrawaJapan\n", "M Teshima \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "R Torii \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "I Tsushima \nFaculty of Education\nYamanashi University\n400-8510KofuJapan\n", "Y Uchihori \nNational Institute of Radiological Sciences\n263-8555ChibaJapan\n", "T Yamamoto \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "S Yoshida \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan\n", "H Yoshii \nDepartment of Physics\nEhime University\n790-8577MatsuyamaJapan\n" ]
[ "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Faculty of Engineering\nYamanashi University\n400-8511KofuJapan", "Department of Physics\nSaitama University\n338-8570UrawaJapan", "Department of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan", "Department of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan", "Nishina Memorial Foundation\n113Komagome, TokyoJapan", "Faculty of General Education\nHirosaki University\n036-8560HirosakiJapan", "Department of Physics\nOsaka City University\n558-8585OsakaJapan", "Faculty of Education\nYamanashi University\n400-8510KofuJapan", "Department of Physics\nKobe University\n657-8501KobeJapan", "Department of Physics\nSaitama University\n338-8570UrawaJapan", "Solar-Terrestrial Environment Laboratory\nNagoya University\n464-8601NagoyaJapan", "Nagoya University of Foreign Studies\n12 11-401, 5-8 Higashi470-0131, 394-0111Nissin, HasudaAichi, SaitamaJapan, Japan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Department of Physics\nSaitama University\n338-8570UrawaJapan", "Department of Physics\nSaitama University\n338-8570UrawaJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Faculty of Education\nYamanashi University\n400-8510KofuJapan", "National Institute of Radiological Sciences\n263-8555ChibaJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n188-8502TokyoJapan", "Department of Physics\nEhime University\n790-8577MatsuyamaJapan" ]
[]
The cosmic-ray energy spectrum above 10 18.5 eV is reported using the updated data set of the Akeno Giant Air Shower Array (AGASA) from February 1990 to October 1997. The energy spectrum extends beyond 10 20 eV and the energy gap between the highest energy event and the others is being filled up with recently observed events. The spectral shape suggests the absence of the 2.7 K cutoff in the energy spectrum or a possible presence of a new component beyond the 2.7 K cutoff.PACS numbers: 98.70. Sa, 96.40.Pq, 96.40.De Typeset using REVT E X
10.1103/physrevlett.81.1163
[ "https://export.arxiv.org/pdf/astro-ph/9807193v1.pdf" ]
14,864,921
astro-ph/9807193
26d753d93901d497991f28b67a725430036b3b35
Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff Jul 1998 M Takeda Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan N Hayashida Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan K Honda Faculty of Engineering Yamanashi University 400-8511KofuJapan N Inoue Department of Physics Saitama University 338-8570UrawaJapan K Kadota Department of Physics Tokyo Institute of Technology 152-8551TokyoJapan F Kakimoto Department of Physics Tokyo Institute of Technology 152-8551TokyoJapan K Kamata Nishina Memorial Foundation 113Komagome, TokyoJapan S Kawaguchi Faculty of General Education Hirosaki University 036-8560HirosakiJapan Y Kawasaki Department of Physics Osaka City University 558-8585OsakaJapan N Kawasumi Faculty of Education Yamanashi University 400-8510KofuJapan H Kitamura Department of Physics Kobe University 657-8501KobeJapan E Kusano Department of Physics Saitama University 338-8570UrawaJapan Y Matsubara Solar-Terrestrial Environment Laboratory Nagoya University 464-8601NagoyaJapan K Murakami Nagoya University of Foreign Studies 12 11-401, 5-8 Higashi470-0131, 394-0111Nissin, HasudaAichi, SaitamaJapan, Japan M Nagano D Nishikawa Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan H Ohoka Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan N Sakaki Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan M Sasaki Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan K Shinozaki Department of Physics Saitama University 338-8570UrawaJapan N Souma Department of Physics Saitama University 338-8570UrawaJapan M Teshima Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan R Torii Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan I Tsushima Faculty of Education Yamanashi University 400-8510KofuJapan Y Uchihori National Institute of Radiological Sciences 263-8555ChibaJapan T Yamamoto Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan S Yoshida Institute for Cosmic Ray Research University of Tokyo 188-8502TokyoJapan H Yoshii Department of Physics Ehime University 790-8577MatsuyamaJapan Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff Jul 1998arXiv:astro-ph/9807193v1 18 The cosmic-ray energy spectrum above 10 18.5 eV is reported using the updated data set of the Akeno Giant Air Shower Array (AGASA) from February 1990 to October 1997. The energy spectrum extends beyond 10 20 eV and the energy gap between the highest energy event and the others is being filled up with recently observed events. The spectral shape suggests the absence of the 2.7 K cutoff in the energy spectrum or a possible presence of a new component beyond the 2.7 K cutoff.PACS numbers: 98.70. Sa, 96.40.Pq, 96.40.De Typeset using REVT E X How high the maximum energy of cosmic rays reaches is one of the most important problems in cosmic ray research. Detections of cosmic rays with energies above 10 20 eV [1,2] have given rise to much discussion regarding their origin. Many models have been proposed as source candidates of such high energy cosmic rays: active astrophysical objects [3], decay products of much higher energy particles such as superheavy relic particles [4] or topological defects [5], or cosmological gamma-ray bursts [6] (see Ref. [7] for a recent review). If such high energy cosmic rays come from far outside our Galaxy, they interact with cosmic microwave background photons and cannot travel cosmological distances. This interaction causes a cutoff in the energy spectrum near 5 × 10 19 eV which is referred to as the Greisen-Zatsepin-Kuz'min (GZK) cutoff [8]. Furthermore, the cosmic rays which have interacted form a "bump" just below the GZK cutoff energy [9][10][11]. The change in the spectral slope around 10 19 eV ("ankle") may arise from a transition from galactic to extragalactic sources. The investigation of these features in the energy spectrum is one of the most important scientific challenges. There are two techniques for detecting extensive air showers (EAS): widely spread surface arrays and atmospheric fluorescence detectors. Using these techniques, the energy spectrum of extremely high energy cosmic rays has been measured by many groups such as Volcano Ranch [12], Haverah Park [13], Sugar [14], Yakutsk [15], Fly's Eye [16], and Akeno [17,25] (only the Fly's Eye group has adopted the atmospheric fluorescence detector). While the energy spectrum obtained from these experiments coincide within ±15% in energy below ∼ 10 19 eV , the details of energy spectrum in the highest energy range is still inconclusive, mainly because of low statistics of their observed events. In this letter, we present the energy spectrum above 10 18.5 eV obtained from the Akeno Giant Air Shower Array (AGASA) [18,19], which currently has the largest exposure of any extremely high energy cosmic ray detectors. The AGASA array is the largest operating surface array, covering an area of about 100 km 2 and consisting of 111 surface detectors of 2.2 m 2 area. Each surface detector is placed with a nearest-neighbor separation of about 1 km and the detectors are sequentially connected with pairs of optical fibers. All the detectors are controlled at detector sites through rapid communication with a central computer. The data acquisition system of AGASA was improved in December 1995 [19]. In a widely spread surface array like AGASA, the local density of charged shower particles at a specific distance from the shower axis is well established as an energy estimator [20], since this depends weakly on variation in the interaction model, fluctuation in shower development and the primary mass. In the AGASA experiment, we adopt local density S(600) at 600 m which is determined from fitting the lateral distribution of observed particle densities to an empirical formula [21]. This empirical formula is found to be valid for EAS with energies up to 10 20 eV [22,23]. The conversion relation from S(600) to the primary energy is evaluated through the Monte Carlo simulation [24] up to 10 19 eV by E = 2.03 × 10 17 S 0 (600) eV, where S 0 (600) is the S(600) value in units of m −2 for a vertically incident shower. Since an inclined air shower traverses more atmospheric depth than a vertical shower, S θ (600) observed with zenith angle θ must be transformed into S 0 (600) at the vertical. This attenuation curve of S(600) has been formulated by Yoshida et al [21]. The accuracy of event reconstruction has been evaluated through the analysis of a large number of artificial air shower events. These artificial events were simulated over a larger area than the AGASA area with directions sampled from an isotropic distribution. In this air shower simulation, the fluctuation on the longitudinal development of air showers, the resolution of the scintillation detectors, and statistical fluctuation of observed shower particles at each surface detector were taken into account. Only events with zenith angles smaller than 45 • and with core locations inside the array area are used in the following analysis. Fig. 1 shows the fluctuation of energy determination for 10 19.5 eV (left) and 10 20 eV (right) showers with zenith angles less than 45 • . The primary energy is determined with an accuracy of about ±30% and the proportion of events with a 50%-or-more overestimation in energy is about 2.4%. Energy uncertainty also arises from the following systematic errors. The first is uncertainty in measuring the particle density incident upon each detector. The number of incident particles is determined from the time width of a pulse, which is generated by decaying an anode signal of a photomultiplier tube exponentially with a time constant of about 10µs and discriminated at a certain level (see [18] for the details of the AGASA instruments). The variation in the amplifier gain and the decay constant are monitored in every run for detector calibration and their seasonal variations are within 2%. The second is uncertainty in the empirical formula of the lateral distribution function and in the attenuation curve of S(600). The energy uncertainty due to the limited accuracy on both of these is estimated to be ±20%, even if both factors shift the estimated energy in the same direction [21]. The third is uncertainty in the conversion formula of S(600) into primary energy. Although this formula is not sensitive to interaction models or primary composition in each of simulation codes [24], the systematic errors due to the differences in simulation codes are not quantitatively clear. In order to evaluate the systematic errors experimentally, we compare the AGASA spectrum derived below with the Akeno spectrum which was accurately determined between 10 14.5 eV and 10 19 eV using the arrays with different detector spacing [17]. The Akeno spectrum fits very well with extrapolation of those obtained from direct measurement on balloons and satellites, and with the Tibet result [26] obtained through the observation of the shower at the height of its maximum development. The difference between the present AGASA and Akeno spectra is about 10% in energy at 10 18.5 eV . In addition, the difference among spectra obtained from the Fly's Eye, Yakutsk, Haverah Park, and AGASA experiments is within 30% in energy in spite of quite different methods for determining the primary energy. Therefore, the total systematic error in the AGASA energy estimation is estimated to be within 30%, and the primary energy of the highest energy event of AGASA, for example, is estimated to be in the range (1.7 -2.0) ×10 20 eV . The effective area of AGASA has been calculated from the simulation of artificial air shower events. The energy spectrum in this simulation was assumed to be E −3 , and the reconstruction uncertainty in energy estimation was also taken into account. Although the effective area depends weakly on the spectral index, this dependence is negligible when compared with other ambiguities like energy resolution. The total exposure of AGASA is obtained by multiplying the effective area and the observation time of each branch for each epoch. Above 10 19 eV , this exposure is constant and is 2.6 × 10 16 m 2 sr s, which is about five times as large as that in our previous paper [25] (cf. ∼ 0.5 × 10 16 m 2 sr s of the stereo Fly's Eye exposure [16] and ∼ 0.7 × 10 16 m 2 sr s of the Haverah Park exposure [13]). However, the exposure below 10 18.5 eV depends strongly on the primary energy. Since this energy dependence causes systematic errors in the energy spectrum derivation, only events with energies above 10 18.5 eV are used for the energy spectrum in this letter. From February 1990 to October 1997, 3847, 461 and 6 events were observed with energies above 10 18.5 eV , 10 19 eV and 10 20 eV respectively. The energy spectrum observed with AGASA is shown in Fig. 2, multiplied by E 3 in order to emphasize details of the steeply falling spectrum. Error bars represent the Poisson upper and lower limits at 68% and arrows are 90% C.L. upper limits. Numbers attached to points show the number of events in each energy bin. The dashed curve represents the spectrum expected for extragalactic sources distributed uniformly in the Universe, taking account of the energy determination error [11]. First, we examine whether the observed energy spectrum could be represented by a single power law spectrum (∝ E −γ 1 ). The optimum spectral index γ 1 is derived from the maximum likelihood procedure comparing the observed and expected number of events in each energy bin. This procedure is same as described in Yoshida et al. [25]. The maximum likelihood procedure for a single power law spectrum results in γ 1 = 3.08 +0.08 −0.15 ; the likelihood significance of γ 1 is only 0.051. If only events with energies below 10 19 eV are considered, γ 1 (E ≤ 10 19 eV ) = 3.23 +0.10 −0.12 is obtained which is consistent with the spectral index, 3.16 ± 0.08, determined from the Akeno experiment [17]. Next, a broken energy spectrum is examined with the same procedure. The broken energy spectrum is assumed to be dJ dE = κ (E/E a ) −γ 0 10 18.5 eV ≤ E < E a κ (E/E a ) −γ 2 E a ≤ E , where γ 0 and γ 2 are indexes below and above a bending (ankle) energy E a , and γ 0 is fixed to be γ 1 (E ≤ 10 19 eV ) = 3.16 determined from the Akeno experiment [17]. The most probable parameters are obtained at E a = 10 19.01 eV and γ 2 = 2.78 +0.25 −0.33 , where the likelihood significance is found to be 0.903. This is also consistent with the results of 2.8±0.3 at energies above 10 18.8 eV determined from the Akeno experiment [17] and of 2.3 +0.5 −0.3 above 10 19.0 in the previous paper [25]. Furthermore, the energy spectrum presented here extends up to higher energies than the previous results [17,25]; six events were observed above 10 20 eV . If the real energy spectrum is that shown in Fig. 2 as the dashed curve, the expected number of events above 10 20 eV is less than one, taking account of the energy resolution. The energy spectrum is therefore more likely to extend beyond 10 20 eV without the GZK cutoff. However, it is also worth noting that the observed energy spectrum suggests a small deficit just below 10 20 eV , whose significance is not compelling because of the uncertainty in γ 2 estimation. This deficit may imply another component above the GZK cutoff energy. In either case, sources of the most energetic cosmic rays must be located within a few tens of Mpc from our Galaxy [11]. The arrival directions of six 10 20 eV events are shown in Fig. 3. Within the accuracy of arrival direction determination (1.6 • above 4 × 10 19 eV ), no 10 20 eV events coincide with possible candidates from the second EGRET sources [27] or the extragalactic radio sources with redshift z ≤ 0.02 [28]. Our previous result for cosmic-ray arrival directions has been reported in Hayashida et al. [29] and the new results are under preparation. The fact that the energy spectrum extends beyond 10 20 eV and no 10 20 eV events coincide with nearby active astrophysical objects leads highest energy cosmic-ray physics into a much more exciting stage. The next generation experiments such as the Telescope Array [30,31], High Resolution Fly's Eye [32,33], and Auger [34,35] projects will solve the puzzle of the highest energy cosmic rays. In conclusion, the cosmic-ray energy spectrum extends beyond 10 20 eV . No candidate sources are found in the directions of six 10 20 eV events, while their sources must be closer than 50 Mpc. The possible deficit around 10 20 eV is a notable area in which to search for origin of the highest energy cosmic rays. Detailed discussion with the AGASA data will be published elsewhere. We are grateful to Akeno-mura, Nirasaki-shi, Sudama-cho, Nagasaka-cho, Ohizumi-mura, Tokyo Electric Power Co. and Nihon Telegram and Telephone Co. for their kind cooperation. The authors are indebted to other members of the Akeno group in the maintenance of the AGASA array. The authors thank Jamie Holder and Michael Roberts for their valuable suggestion on the preparation of the manuscript. FIGURESFIG. 1 . 1Fluctuation of energy determination for 10 19.5 eV (left) and 10 20 eV (right) showers with zenith angles less than 45 • . FIG. 2. Energy spectrum observed with AGASA. The vertical axis is multiplied by E 3 . Error bars represent the Poisson upper and lower limits at 68% and arrows are 90% C.L. upper limits. Numbers attached to points show the number of events in each energy bin. The dashed curve represents the spectrum expected for extragalactic sources distributed uniformly in the Universe, taking account of the energy determination error [11]. FIG. 3. Arrival directions of six 10 20 eV events on the Galactic coordinates. The shaded regions indicate the non-observable celestial regions due to the zenith angle cut of ≤ 45 • . The equatorial and supergalactic planes are also shown. FIG. 1 .FIG. 2 . 12Fluctuation of energy determination for 10 19.5 eV (left) and 10 20 eV (right) showers with zenith angles less than 45 • . Energy spectrum observed with AGASA. The vertical axis is multiplied by E 3 . Error bars represent the Poisson upper and lower limits at 68% and arrows are 90% C.L. upper limits. Numbers attached to points show the number of events in each energy bin. The dashed curve represents the spectrum expected for extragalactic sources distributed uniformly in the Universe, taking account of the energy determination error[11]. FIG. 3 . 3Arrival directions of six 10 20 eV events on the Galactic coordinates. The shaded regions indicate the non-observable celestial regions due to the zenith angle cut of ≤ 45 • . The equatorial and supergalactic planes are also shown. . D J Bird, Astrophys J. 441144D. J. Bird et al., Astrophys J. 441 (1995) 144. . N Hayashida, Phys. Rev. Lett. 733491N. Hayashida et al., Phys. Rev. Lett. 73 (1994) 3491. . P L See, P A Biermann, Strittmatter, Astrophys. J. 322643See, for example, P. L. Biermann and P. A. Strittmatter, Astrophys. J. 322 (1987) 643; . F Takahara, Prog. Theor. Phys. 831071F. Takahara, Prog. Theor. Phys. 83 (1990) L1071; . J P Rachen, P L Biermann, Astron. Astrophys. 272161J. P. Rachen and P. L. Biermann, Astron. Astrophys. 272 (1993) 161; . R V E Lovelace, Nature. 262649R. V. E. Lovelace, Nature 262 (1976) 649. . V See, M Berezinsky, A Kachelriess, Vilenkin, Phys. Rev. Lett. 794302See, for example, V. Berezinsky, M. Kachelriess, and A. Vilenkin, Phys. Rev. Lett. 79 (1997) 4302. . C T See, D N Hill, T P Schramm, Walker, Phys. Rev. 361007See, for example, C. T. Hill, D. N. Schramm and T. P. Walker, Phys. Rev. D36 (1987) 1007; . P Bhattacharjee, C T Hill, D N Schramm, Phys. Rev. Lett. 69567P. Bhattacharjee, C. T. Hill and D. N. Schramm, Phys. Rev. Lett. 69 (1992) 567. . See, M Example, Vietri, Astrophys. J. 453883See, for example, M. Vietri, Astrophys. J. 453 (1995) 883; . E Waxman, Phys. Rev. Lett. 75386E. Waxman, Phys. Rev. Lett. 75 (1995) 386. . V Berezinsky, hep−ph/9802351V. Berezinsky, hep−ph/9802351. . K Greisen, Phys. Rev. Lett. 16748K. Greisen, Phys. Rev. Lett. 16 (1966) 748; . G T Zatsepin, V A Kuz&apos;min, Zh. Eksp. Teor. Fiz. 4114JETP LettersG. T. Zatsepin and V. A. Kuz'min, Zh. Eksp. Teor. Fiz. 4 (1966) 114 [JETP Letters 4 (1966) 78]. . C T Hill, D N Schramm, Phys. Rev. 31564C. T. Hill and D. N. Schramm, Phys. Rev. D31 (1985) 564. . V S Berezinsky, S I Grigor&apos;eva, Astron. Astrophys. 1991V. S. Berezinsky and S. I. Grigor'eva, Astron. Astrophys. 199 (1988) 1. . S Yoshida, M Teshima, Prog. Theor. Phys. 89833S. Yoshida and M. Teshima, Prog. Theor. Phys. 89 (1993) 833. . J Linsley, J. Phys. G:Nucl. Part. Phys. 1251J. Linsley, J. Phys. G:Nucl. Part. Phys. 12 (1986) 51. . M A Lawrence, R J O Reid, A A Watson, J. Phys. G:Nucl. Part. Phys. 17733M. A. Lawrence, R. J. O. Reid and A. A. Watson, J. Phys. G:Nucl. Part. Phys. 17 (1991) 733. . M M Winn, J. Phys. G:Nucl. Part. Phys. 12653M. M. Winn et al., J. Phys. G:Nucl. Part. Phys. 12 (1986) 653. . N N Efimov, Astrophysical Aspects of the Most Energetic Cosmic Rays. M. Nagano and F. Takahara20World ScientificN. N. Efimov et al., in Astrophysical Aspects of the Most Energetic Cosmic Rays, edited by M. Nagano and F. Takahara (World Scientific, Singapore, 1991) p.20. . D J Bird, Astrophys J. 424491D. J. Bird et al., Astrophys J. 424 (1994) 491. . M Nagano, J. Phys. G: Nucl. Phys. 18423M. Nagano et al., J. Phys. G: Nucl. Phys. 18 (1992) 423. . N Chiba, Nucl. Instr. and Meth. A. 311338N. Chiba et al. , Nucl. Instr. and Meth. A 311 (1992) 338. . H Ohoka, Nucl. Instr. and Meth. A. 385268H. Ohoka et al., Nucl. Instr. and Meth. A 385 (1997) 268. A M Hillas, Proceedings of the 12th International Cosmic Ray Conference. the 12th International Cosmic Ray ConferenceHobart31001A. M. Hillas et al., in Proceedings of the 12th International Cosmic Ray Conference, Hobart, 1971, Vol. 3, p. 1001. . S Yoshida, J. Phys. G:Nucl. Part. Phys. 20651S. Yoshida et al., J. Phys. G:Nucl. Part. Phys. 20 (1994) 651. T Doi, Proceedings of the 24th International Cosmic Ray Conference. the 24th International Cosmic Ray ConferenceRome2764T. Doi et al., in Proceedings of the 24th International Cosmic Ray Conference, Rome, 1995, Vol. 2, p. 764. . N Sakaki, in preparationN. Sakaki et al., in preparation. . H Y Dai, J. Phys. G: Nucl. Phys. 14793H. Y. Dai et al., J. Phys. G: Nucl. Phys. 14 (1988) 793. . S Yoshida, Astropart. Phys. 3105S. Yoshida et al. , Astropart. Phys. 3 (1995) 105. . M Amenomori, Astrophys. J. 461408M. Amenomori et al., Astrophys. J. 461 (1996) 408. . D J Thompson, Astrophys. J. Suppl. 101259D. J. Thompson et al., Astrophys. J. Suppl. 101 (1995) 259. . M P Veron-Cetty, P Veron, Astron. Astrophys. Suppl. 53219M. P. Veron-Cetty and P. Veron, Astron. Astrophys. Suppl. 53 (1983) 219. . N Hayashida, Phys. Rev. Lett. 771000N. Hayashida et al., Phys. Rev. Lett. 77 (1996) 1000. . M Teshima, Nucl. Phys. B (Proc. Suppl.). 28169M. Teshima et al., Nucl. Phys. B (Proc. Suppl.) 28B (1992) 169. M Teshima, Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays. M. Nagano.the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays109ICRR, University of TokyoM. Teshima, in Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays, edited by M. Nagano. (ICRR, University of Tokyo, 1993) p. 109. E Loh, Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays. M. Nagano.the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays105ICRR, University of TokyoE. Loh, in Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays, edited by M. Nagano. (ICRR, University of Tokyo, 1993) p. 105. M Al-Seady, Proceedings of International Symposium on Extremely High Energy Cosmic Rays: Astrophysics and Future Observatories. M. Nagano.International Symposium on Extremely High Energy Cosmic Rays: Astrophysics and Future Observatories191ICRR, University of TokyoM. Al-Seady et al., in Proceedings of International Symposium on Extremely High Energy Cosmic Rays: Astrophysics and Future Observatories, edited by M. Nagano. (ICRR, University of Tokyo, 1996) p. 191. . J W Cronin, Nucl. Phys. B (Proc. Suppl.). 28213J. W. Cronin, Nucl. Phys. B (Proc. Suppl.) 28B (1992) 213. . THE PRIERRE AUGER OBSERVATORY DESIGN REPORT. THE PRIERRE AUGER OBSERVATORY DESIGN REPORT (1997)
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[ "On the Problem of Undirected st-connectivity", "On the Problem of Undirected st-connectivity" ]
[ "Shilun Li [email protected] \nDept. of Computer Science\nDept. of Computer Science\nStanford University\nStanford University\n\n", "Alex Lee [email protected] \nDept. of Computer Science\nDept. of Computer Science\nStanford University\nStanford University\n\n" ]
[ "Dept. of Computer Science\nDept. of Computer Science\nStanford University\nStanford University\n", "Dept. of Computer Science\nDept. of Computer Science\nStanford University\nStanford University\n" ]
[]
In this paper, we discuss an algorithm for the problem of undirected st-connectivity that is deterministic and log-space, namely that of Reingold within his 2008 paper "Undirected Connectivity in Log-Space" [Rei08]. We further present a separate proof by Rozenman and Vadhan of USTCONN ∈ L [RV05] and discuss its similarity with Reingold's proof. Undirected st-connectively is known to be complete for the complexity class SL-problems solvable by symmetric, nondeterministic, log-space algorithms. Likewise, by Aleliunas et. al. [AKL + 79], it is known that undirected st-connectivity is within the RL complexity class, problems solvable by randomized (probabilistic) Turing machines with one-sided error in logarithmic space and polynomial time. Finally, our paper also shows that undirected st-connectivity is within the L complexity class, problems solvable by deterministic Turing machines in logarithmic space. Leading from this result, we shall explain why SL = L and discuss why is it believed that RL = L.Stanford CS224N Natural Language Processing with Deep Learning
10.48550/arxiv.2203.09728
[ "https://arxiv.org/pdf/2203.09728v2.pdf" ]
247,594,720
2203.09728
a367067e94d0e7efe0a114b4d5ee1a981208d67a
On the Problem of Undirected st-connectivity Shilun Li [email protected] Dept. of Computer Science Dept. of Computer Science Stanford University Stanford University Alex Lee [email protected] Dept. of Computer Science Dept. of Computer Science Stanford University Stanford University On the Problem of Undirected st-connectivity In this paper, we discuss an algorithm for the problem of undirected st-connectivity that is deterministic and log-space, namely that of Reingold within his 2008 paper "Undirected Connectivity in Log-Space" [Rei08]. We further present a separate proof by Rozenman and Vadhan of USTCONN ∈ L [RV05] and discuss its similarity with Reingold's proof. Undirected st-connectively is known to be complete for the complexity class SL-problems solvable by symmetric, nondeterministic, log-space algorithms. Likewise, by Aleliunas et. al. [AKL + 79], it is known that undirected st-connectivity is within the RL complexity class, problems solvable by randomized (probabilistic) Turing machines with one-sided error in logarithmic space and polynomial time. Finally, our paper also shows that undirected st-connectivity is within the L complexity class, problems solvable by deterministic Turing machines in logarithmic space. Leading from this result, we shall explain why SL = L and discuss why is it believed that RL = L.Stanford CS224N Natural Language Processing with Deep Learning Introduction In this report, we shall prove that the problem USTCONN, or st-connectivity on undirected graphs, can be solved with a log-space algorithm. We then explore the implications of such a result. We can define the problem of st-connectivity by considering a graph G and two vertices s and t in G. The st-connectivity problem answers whether or not the two vertices s and t are connected with each other by a path in G. Similarly, the USTCONN problem decides the STCONN problem, but on a graph G that is specified to be undirected. (In this case, USTCONN is a special case of STCONN and all algorithms that solve STCONN would be able to solve USTCONN.) The problem of connectivity is one of the most fundamental problems within graph theory, and algorithms to solve STCONN and USTCONN has been used to construct more complex graph algorithms. Indeed, a solution to the STCONN or USTCONN problem within a certain complexity class would similarly imply a solution within the complexity class for a much larger body of computational problems. The time complexity of USTCONN has been well understood and solved: it is clear to see that the minimum time complexity must be linear (as the length of a path from s to t would be linear), and it is also clear that such a time complexity would be achievable by depth-first search (DFS) or breadth-first search (BFS). Most recent study of the USTCONN problem thus, revolves around its space complexity. It is clear that the space complexity of USTCONN must be at least log space, which is the space required to store any O(n) sized variables within the problem. In 1970, Savistch provided a log 2 n space complexity solution to the STCONN (and USTCONN). A randomized algorithm of log-space complexity was also developed in 1979 by Aleliunas, Karp, Lipton, Lovasz, and Rackoff [AKL + 79]. Following the result, work has been done on derandomizing the randomized algorithm in hopes of creating a deterministic algorithm with decreasing space complexity. In 1992, Nisan, Szemeredi and Wigderson presented an algorithm of log 1.5 space [NSW92]. In 1999, Armoni, et, al showed that USTCONN can be solved by an algorithm in log 4 3 space [ATSWZ00]. In 2005, Trifonov developed an algorithm of log n log log n space for USTCONN [Tri05]. Finally, in 2008, Omer Reingold presented a deterministic algorithm that solves USTCONN in log-space complexity [Rei08]. We shall begin our paper by presenting the result of USTCONN ∈ L through Omer Reingold's method in his paper "Undirected Connectivity in Log-Space". [Rei08]. More specifically, we shall begin by explaining expander graphs and some transformations used to convert any graph to an expander graph. We shall then show that these transformations can be performed in log space and that connectivity can be computed from these expander graphs in log space too. This would prove that USTCONN ∈ L. We would then present a separate proof from Rozenman and Vadhan of USTCONN ∈ L and discuss its similarity to Reingold's proof. Finally, we shall explore the implications of this result on the relations between the complexity classes of L, SL, and RL. L refers to problems solvable by a deterministic log space Turing machine, SL refers to problems solvable by symmetric log space Turing machines, and RL refers to problems solvable by probabilistic log space Turing machines with one sided error. These three complexity classes are closely tied to USTCONN and we shall show that SL = L and discuss why it is believed that RL = L. Preliminaries In this section, we will introduce some properties of graphs using adjacency matrix representation, along with procedures such as graph powering. Graph Adjacency Matrix For any graph G, common representations include adjacency list, adjacency matrix, and incidence matrix. There exist log-space algorithms which transforms between the common representations, so the problem of USTCONN does not rely on the input graph representations. In this paper, we will use the adjacency list representation. Definition 2.1. The adjacency matrix A of a graph G = (V, E) is the |V | × |V | matrix such that the entry (u, v) of A written A u,v is equal to the number of number of edges from vertex u to vertex v in G. We allow G to contain self loops and parallel edges. Definition 2.2. For a graph G = (V, E) with adjacency matrix A. G is undirected if A is symmetric, where we have A = A T . An undirected graph G is D-regular if there are exactly D edges incident to every vertex, equivalently, v∈V A u,v = D for all u ∈ V . For any undirected D-regular graph G with N vertices, let us label each outgoing edge of every vertex of G by a number from 1 to D in a fixed way. Then we define the rotation map of G as follows: Definition 2.3. For an undirected D-regular graph G with N vertices, let the rotation map Rot G be a permutation of [N ] × [D] defined by Rot G (v, i) = (w, j) if edge i from v leads to w and is the same edge as edge j of w. The rotation map defines how the vertices and edges of G are labeled. The rotation map will play a crucial role in transforming any undirected graph G into a regular graph. The adjacency matrix can be expressed by the rotation map in the following way: A u,v = |{(i, j) ∈ [D] 2 : Rot G (u, i) = (v, j)}| (1) To solve USTCONN, we would like the graph to be highly connected but at the same time sparse so that the diameter is small. We call such highly connected sparse graphs expanders. We will define expanders using properties of its adjacency matrix. Proposition 2.1. For an undirected D-regular graph G with N × N adjacency matrix A, A is diagonalizable with eigenvalues λ 1 ≥ ... ≥ λ N . Furthermore, λ 1 = D and λ N ≥ −D. Proof. The first part of the statement follows from spectral theorem for symmetric matrices. For the second part of the proposition, given any eigenvector λ of A with eigenvector v = (v 1 , ..., v n ), consider the index k such that |v k | achieves the maximum among all v 1 , ..., v i . Now since Av = λv, the k th component satisfies |(Av) k | = | n i=1 A k,i v i | ≤ n i=1 |v k |A k,i = D|v k |. But we also have |(Av) k | = |λ||v k |. So |λ| ≤ D for any eigenvalue λ of A. Now note that for the vector v = (1, ..., 1) T , we have Av = Dv. So v is a eigenvector of A with eigenvalue D. Thus λ 1 = D and λ N ≥ −D as desired. Now, let us define the normalized adjacency matrix M of an undirected D-regular graph G as the adjacency matrix divided by D. Definition 2.4. For any graph G, let λ(G) be the second largest eigenvalue of the normalized adjacency matrix. G is an expander graph if λ(G) ≤ 1 2 . G is an (N, D, λ)-graph if it is undirected D-regular with N vertices and λ(G) < λ. The second largest eigenvalue of G captures its expansion properties. It is shown by Alon [Alo86] that second-eigenvalue expansion is equivalent to the standard vertex expansion. In particular, we have the following Proposition 2.2. Fix any fixed λ < 1, for any (N, D, λ)-graph G N . For any two vertices s, t ∈ G G , there exists a path of length O(log N ). Proof. By the result of Alon [Alo86], for any λ < 1, there exist > 0 such that for any (N, D, λ)graph G N and any set S of vertices of G N such that |S| ≥ N 2 , we have |∂S| ≥ |S| where ∂S = {(u, v) ∈ E(G N ) : u ∈ S, v ∈ V (G N ) \ S}. Now for any two vertices s, t ∈ G N , for some l = O(log N ) with constant only depending on , since the edge expansion factor is at least , both s and t can have more than N 2 vertices of at most distance l. Then there exist a vertex v within distance l from both s and t. Thus there exist a path of length 2l = O(log N ) from s to t. One may notice that the vertex expansion property of an undirected (N, D, λ)-graphs with λ < 1 implies it is connected. We can also directly see this as for any undirected D-regular graph with multiple connected components, each component will contribute an orthogonal eigenvector of the normalized adjacency matrix with eigenvalue 1 via the indicator of that component. Thus the graph will have the second largest eigenvalue of normalized adjacency matrix being 1 if it has more than one connected component. With the result above, we can now solve the undirected st-connectivity problem for constant-degree expanders using log-space. Lemma 2.3. For any fixed λ < 1, there exist a space O(log D log N ) algorithm A λ such that on an input of an undirected D-regular graph G with N vertices and two vertices s, t ∈ G: • Outputs "connected" only if s and t are connected in G. By the explicit construction given by Alon and Roichman [AR94] using Cayley graph of the group F m 2 which is m dimensional vector space of the field F 2 , we have existence of expander graphs with desired parameters. Proposition 2.4. There exist an undirected D 0 -regular ((D 0 ) 16 , D 0 , 1 2 )-graph for some D 0 . The value 1 − λ(G) is called the spectral gap of a graph. We have shown that for a disconnected graph, the spectral gap is 0. Due to result by Alon [AS00], the converse holds for non-bipartite graphs: Proposition 2.5 (Alon). For every D-regular connected non-bipartite graph G with N vertices, the spectral gap is at least 1 DN 2 . Equivalently, λ(G) ≤ 1 − 1 DN 2 . Graph Powering and Zig-zag Products We will now introduce operations of graphs to change its degree and spectral gap, i.e. its expansion properties. We will first introduce graph powering, which reduces its second eigenvalue and increases its spectral gap, but also increases its degree. We will then define the zig-zag product of two graphs, which was first introduced by Reingold, Vadhan and Wigderson [RVW00]. This operation reduces the degree of a graph without significantly varying the spectral gap. Recall that the labeling of edges of an undirected D-regular graph is given by the rotation map. Equivalently, the graph is defined by the rotation map. So let us define graph powering via rotation maps. Definition 2.5. For a undirected D-regular graph G with N vertices given by the rotation map Rot G , the t th power of of G is the D t -regular graph G t defined by the rotation map Rot G t (v 0 , (a 1 , ..., a t )) = (v t , (b t , ..., b 1 )) for any v 0 , v t ∈ [N ] and a 1 , ..., a t ∈ [D] where b 1 , ..., b t are computed by (v i , b i ) = Rot G (v i−1 , a i ). One can view the vector (a 1 , ..., a t ) ∈ [D t ] as a path from v 0 to v t where each a i is the action of taking edge a i of the current vertex during traversal of the path. The vector (b t , ..., b 1 ) ∈ [D t ] is simply the same path backwards, starting from v t and ending in v 0 . This definition coincides with the usual definition of graph powering where two vertices is adjacent in the t th power if there is a path of length t in the original graph. Proposition 2.6. The normalized adjacency matrix of G t is given by M t where M is the normalized adjacency matrix of G. Consequently, if G is a (N, D, λ)-graph, then G t is a (N, D t , λ t )-graph. Proof. From the above discussion, for any two vertices v 0 , v t ∈ G t , the number of edges between v 0 and v t in G t is equal to the number of length t paths from v 0 to v t , where paths are defined using edges instead of vertices. The number of paths from v 0 to v t is in turn equal to (A t ) v0,vt , where A is the adjacency matrix of G. So A t is the adjacency matrix of G t . Thus the normalized adjacency matrix of G t is given by D −t A t = M t where M is the normalized adjacency matrix of G. If G is a (N, D, λ)-graph with normalized adjacency matrix M . Since the normalized adjacency matrix of G t is given by M t , we have λ(G t ) = λ(G) t . Thus G t is a (N, D t , λ t )-graph. For a (N, D, λ)-graph, powering increases the spectral gap exponentially, but the degree of the graph also increases exponentially. On the other hand, the zig-zag product reduces the degree of the graph but remains the spectral gap nearly unchanged. Definition 2.6. Let G be a D-regular graph on [N ] with rotation map Rot G , H be a d-regular graph on [D] with rotation map Rot H . Then their zig-zag product G z H is a d 2 -regular graph on [N ]×[D] with rotation map Rot G z H defined by: Rot G z H ((v, a), (i, j)) = ((w, b), (j , i )) where w, b, j , i satisfies: there exist a , b ∈ [N ] such that • Rot H (a, i) = (a , i ) • Rot G (v, a ) = (w, b ) • Rot H (b , j) = (b, j )(v, a), (i, j)) in G z H. H v is the copy of H for vertex v and H w is the copy of H for vertex w. The edge ((v, a), (i, j)) correspond a length 3 path composed by (a, i) in H v , (v, a ) in G and (b , j) in H w . The rotation map of the edge correspond to the same path traversing backwards. On the right hand side is the projection of the path on G, which correspond to the middle edge of the length 3 path. To reduce the degree of the graph G while remaining the spectral gap, we want d, the degree of H to be much smaller than D = deg(G) so that d 2 = deg(G z H) is smaller than D. We also want 1 − λ(G z H) > k(1 − λ(G)) for some constant k ∈ (0, 1) independent of G. These estimates on the spectral gap of the zig-zag product are given by Reingold et al. in [RVW00]. Theorem 2.7 (Reingold). If G is an (N, D, λ)-graph and H is a (D, d, α)-graph, then G z H is an (N D, d 2 , f (λ, α)) graph where f (λ, α) = 1 2 (1 − α 2 )λ + 1 2 (1 − α 2 ) 2 λ 2 + 4α 2 Proof. This is Theorem 4.3 in [RVW00]. Corollary 2.7.1. If G is an (N, D, λ)-graph and H is a (D, d, α)-graph, then 1 − λ(G z H) ≥ 1 2 (1 − α 2 )(1 − λ). Proof. Since λ ≤ 1, we have 1 2 (1 − α 2 ) 2 λ 2 + 4α 2 ≤ 1 2 (1 − α 2 ) 2 + 4α 2 = 1 − 1 2 (1 − α 2 ). This Corollary is then a direct consequence of Theorem 2.7. Expander Transforms of Graphs In this section, we will introduce the Main Transform given by Reingold[Rei08] which uses logspace to transform each connected component of a graph into an expander. This is the main part of the log-space algorithm for USTCONN. G i = (G i−1 z H) 8 , i = 1, ..., l where G 0 = G and l = 2 log DN 2 . We will denote T i (G, H) = G i and T (G, H) = G l . From the properties of zig-zag product and graph powering, the graph G i is a D 16 -regular graph over [N ] × ([D 16 ]) i . If D is constant, then l = O(log N ) and G l has poly(N ) vertices. We will first show that this transformation can transform G into an expander. Proof. Since G is connected and non-bipartite, by Proposition 2.7, λ(G 0 ) ≤ 1 − 1 DN 2 . By Corollary 2.7.1, as λ(H) ≤ 1 2 , we have λ(G i−1 z H) ≤ 1 − 3 8 (1 − λ(G i−1 )) < 1 − 1 3 (1 − λ(G i−1 )), i = 1, ..., l. So by Proposition 2.6 we have λ(G i ) = λ((G i−1 z H) 8 ) < (1 − 1 3 (1 − λ(G i−1 ))) 8 . When λ(G i−1 ) ≤ 1 2 , we have λ(G i ) < ( 5 6 ) 8 < 1 2 . If we have λ(G i ) ≤ 1 2 for some i = 0, ..., l, then be induction, we would have λ(T (G, H)) = λ(G l ) ≤ 1 2 as desired. So let us suppose otherwise, λ(G i ) > 1 2 for all i. Then it is easy to show λ(G i ) = (1 − 1 3 (1 − λ(G i−1 ))) 8 ≤ λ(G i−1 ) 2 . So λ(G l ) ≤ (1 − 1 DN 2 ) 2 l . Since (1 − 1 x ) x < e −1 for all x ≥ 1, we have λ(G l ) ≤ (1 − 1 DN 2 ) 2 l ≤ e −2 < 1 2 . So λ(T (G, H)) ≤ 1 2 . While the analysis in the previous lemma assumes G is connected and non-bipartite, we will extend this analysis of T to any undirected graph G. Note that zig-zag product Proof. See Lemma 3.3 of [Rei08]. Finally, we will show that T can be computed in log-space when D is constant. This is essentially due to the fact that during each step of the inductive calculation of T i (G, H), only constant addition amount of memory is needed. We will shot the space complexity of T in the follow lemma: Proof. The algorithm A T will first allocate variables v ∈ [N ] and a 0 , ..., a l ∈ [D 16 ]. We will denote each a i = k i,1 ...k i,16 where k ∈ [D] correspond to edge labels of H. Now on input (G, H, (v in , a in )), the algorithm will first copy v in = (v,â 0 , ...,â l−1 ) ∈ [N ] × [D 16 ] l into the allocated variables v, a 0 , ..., a l−1 and a in =â l ∈ [D 16 ] into a l . These variables will store the the output of Rot Gi onv,â 0 , ...,â i where G i = T i (G, H). We will recursively update the variables v, a 0 , ..., a l such that after the i th iteration, the variables v, a 0 , ..., a i will store the result of Rot Gi ((v,â 0 , ...,â i−1 ),â i ). For the base case, when i = 0, G 0 = G, so we can search in the input tape for the edge (v,â 0 ) and write down Rot G0 (v,â 0 ) in v, a 0 . Now for i = 1, ..., l, we evaluate Rot Gi ((v,â 0 , ...,â i−1 ),â i ) via the following procedure: For j = 1 to 16: • Set a i−1 , k i,j = Rot H (a i−1 , k i,j ) • If j is odd, recursively compute and set v, a 0 , ..., a i−1 = Rot Gi−1 ((v, a 0 , ..., a i−2 ), a i−1 ). • If j = 16, reverse the order of the labels in a i : set k i,1 , ..., k i,16 = k i,16 , ..., k i,1 The first two operations correspond to finding a a path of length eight on G i−1 z H, which is a step on (G USTCONN ∈ L In this section, we will provide an log-space algorithm for USTCONN using the by transforming the input graph G into an appropriate expander. Proof. As we can transform between common representations of graphs in log-space, without loss of generality we can assume G is given via the adjacency matrix representation. By Proposition 2.4, for some constant D 0 , there exists a ((D 0 ) 16 , D 0 , 1 2 )-graph H. Let us hard-code the rotation map of H to the memory of A con . This takes only constant memory. Now we would like to transform G into D 16 -regular graph G reg (which is defined by its rotation map) so that we can apply T on (G reg , H). Let G reg be the graph constructed by replacing each vertex of G with with a cycle of length N, and there is an edge between (v, w) and (w, v) in G reg if there is an edge between v and w in G. Self loops are added so that the degree of each vertex is D 16 0 . The rotation map Rot Greg : , v), 3), i = 3 and there is an edge between v and w in G ((v, w), 3), i = 3 and there is an edge between v and w in G ((v, w), i), i = 4, ..., 16 ([N ]) 2 × [D 16 0 ] → ([N ]) 2 × [D 16 0 ] is given by: Rot Greg ((v, w), i) =              ((v, (w + 1) mod N ), 2), i = 1 ((v, (w − 1) mod N ), 1), i = 2 ((w . The first two cases are the edges of the cycle of length N for vertex v. The last case are the self loops so that G reg is D 16 0 regular. Also note that every vertex of G reg has self loops, so G reg is non-bipartite. It is easy to see that Now let us run the O(log N ) space algorithm A λ with λ = 1 2 on G exp and (s, 1 l+1 ) and (t, 1 l+1 ) given by Proposition 2.3. The algorithm A con will output "connected" if A λ outputs connected, else it will output "disconnected". An alternative proof of USTCONN ∈ L This section will contain an alternative proof of USTCONN ∈ L given by Rozenman and Vadhan [RV05]. In both proofs, the key idea is that USTCONN is solvable in log-space on bounded-degree graphs with logarithmic diameter by enumerating over all paths. Bounded-degree Expander graphs (graphs with second eigenvalue less than 1 2 ) are instances of such graphs, both proofs transform the graph into an expander with bounded degree to solve USTCONN. In the proof of Reingold, given any undirected graph with N vertices, Reingold first transforms the graph into a regular graph and then used a combination of graph powering and zig-zag product to transform the regular graph into an expander graph with constant degree over poly(N ) vertices, while maintaining the connectivity properties of vertices. Graph powering increases the connectivity of the graph, decreases λ(G) while increasing the degree and number of vertices polynomially. Zig-zag product decreases the degree while keeping λ(G) approximately still. The combination of both decreases λ(G) to 1 2 while maintaining the degree constant. One the other hand, Rozenman and Vadhan's proof [RV05] shares the same overall process as Reingold, but used derandomized squaring instead of graph powering and zig-zag products to increase the connectivity of the graph. Iterating derandomized squaring yields highly connected graphs with relatively small degree compared graph powering while maintain the same number of vertices. Rot G s H (v, (x, a)) = (w, (h, b)) where • (u, y) = Rot G (v, x) • (z, b) = Rot H (y, a) • (w, h) = Rot G (u, z) for any v ∈ [N ], x ∈ [D], a ∈ [d]. An edge in G s H corresponds to a length 2 path in G. The following figure illustrates an edge of the derandomized square G s H: (D, d, α)-graph, then G s H is an (N, Dd, f (λ, α))-graph where f (λ, α) = 1 − (1 − λ 2 )(1 − α) ≤ λ 2 + α Proof. See Theorem 6.3 in [RV05]. To prove USTCONN ∈ L in a manner similar to Theorem 4.1, we require a family of undirected constant degree expander graphs to apply derandomized squaring with. In addition, these graphs need to be computed in log-space. This is possible by Reingold [RVW00] and Gabber [GG81] Lemma Taking m 0 to be 100 log N and m 1 = m 0 + log log N + 10, Rozenman and Vadhan showed that λ(G m0 | S ) < 3 4 and λ(G m1 | S ) ≤ 1 2N 3 for each connected component S of G m0 and S of G m1 . In addition, he showed that G m1 has degree poly(N ) and can be constructed in O(log N ) space, i.e. Rot Gm 1 can be computed using O(log N ) space. As all transformations in this procedure operates separately on each connected component, we can solve USTCONN of G by running the algorithm of Lemma 2.3 on G m1 . SL = L The section will contain a proof of SL = L. Let us first define the space SL. A Turing machine can be defined by the 7-tuple (K, Σ, Σ 0 , k, ∆, s, F ). Specifically, K is a finite set of states, Σ is the finite tape alphabet, Σ 0 ⊆ Σ is the input alphabet, k > 0 is the number of tapes, s ∈ K is the initial state, F ⊆ K is the set of final states, and ∆ is the finite set of transitions. Using this definition, we define transitions for the Turing machine with form (p, (ab, D, cd), q) where a, b, c, d ∈ Σ and D ∈ 1, −1. A transition of the form (p, ab, 1, cd, q) means that if a Turing machine in state p, scans a and symbol b is contained in the square to the right of the scanned square, the Turing machine moves the tape head one square to the right, rewrite the squares with symbol a and symbol b with symbol c and symbol d, respectively, and changes to state q. On the other hand, a transition of the form (p, ab, −1, cd, q) means that if a Turing machine in state p scans symbol b and symbol a is contained in the square to the left of the scanned square, the Turing machine moves the tape head one square to the left, rewrite the square with symbol b and symbol a to symbol d and symbol c, respectively, and changes to state q. For Turing machines with multiple tapes, we define the transition form δ = (p, t 1 , t 2 , · · · , t k , q), where k is the number of tapes. Each t i is a 3-tuple (ab, D, cd), which specifies the transition as described above for tape i. For a non-deterministic Turing machine, there can be multiple transitions from each possible configuration. For each of these possible choices, the non-deterministic Turing machine creates a branch in its configuration path. A non-deterministic Turing machine accepts a configuration if any of the branches within its configuration path ends at an accepting state. We note that our definition of Turing machine is equivalent to that of a standard Turing machine. Our "peeking" Turing machine can be reduced to the big-headed Turing machine as defined by Hennie [Hen79], which has been shown to be equivalent to a standard Turing machine. Now, let us define symmetrical Turing machines using the definition by Lewis and Papadimitriou in "Symmetric Space-Bounded Computation" [LP82]. Definition 6.1. For a transition δ = (p, t 1 , t 2 , · · · , t k , q), with t i = (a i b i , D i , c i d i ), we define its inverse δ −1 = (q, t −1 1 , t −1 2 , · · · , t −1 k , p), where t −1 i = (c i d i , −D i , a i b i ). Definition 6.2. A Symmetrical Turing Machine is a non-deterministic Turing machine whose transition functions ∆ is invariant under taking inverse, namely, for every non-deterministic transition p i δ i ∈ ∆, we have p i δ −1 i ∈ ∆. Definition 6.3. The space SL is the set of all languages which can be determined by a symmetrical log-space Turing machine. We note that a symmetric Turing machine has a number of special transitions, from which it is always possible to revert from these transitions (since the symmetric Turing machine includes the inverse of these transitions). To prove that USTCONN is SL-complete, we shall use a lemma from the paper Symmetric Space-Bounded Computation by Lewis and Papadimitriou [LP82]. We begin by defining relevant terms in the lemma. Definition 6.4. If there exists a transition from configuration C 1 to C 2 , we write C 1 M C 2 or equivalently C 2 M C 1 . Let us define the reflexive, transitive closure of , denoted * M and the transitive closure of , denoted + M . For any M , let A be a subset of all possible configurations on M . If for some possible configurations C 0 , ..., C n of M , we have C 0 M C 1 M · · · M C n for some n ≥ 0 and C 1 , C 2 , C 3 , · · · C n ∈ A, we write C 0 * A M C n (equivalently C n * A M C 0 ). Note that if C 0 ∈ A and C 0 M C 0 , we have C 0 * A M C 0 . If A 1 * A M B M A 2 for A 1 , A 2 ∈ A and B a possible configuration of M , we write A 1 +A M A 2 (equivalently A 2 +A M A 1 ). For a Turing machine M , define the Turing machine M * , which is the same as M except that one can't re-enter its initial state, leave its final state nor write blanks on its tapes. We also define M * as the symmetrically closed M * , i.e. translations of M * is the union of the transition of M * and its inverse. Then, the symmetrical non-deterministic Turing machine M * would accept the same language as M in the same space as M . Proof. This is Lemma 1 in [LP82]. Using this lemma, we can now prove that USTCONN is SL-complete. Theorem 6.2. USTCONN is SL-complete. Proof. We begin by proving that USTCONN ∈ NL. In other words, we describe a non-deterministic Turing machine that can solve USTCONN. Let us define a non-deterministic Turing machine M with 2 tapes. Given an undirected graph G and nodes s, t ∈ G, M begins by writing s and t on its two tapes. Let the tape containing t be the tape containing the destination node and the tape containing s be the tape containing the current node. At the start of each step, we non-deterministically choose a neighbor of the current node, rewrite the neighbor into the tape containing the current node, and check if the node in the tape containing the current node is the same as the destination node. If the current node is the same as the destination node, M accepts, else M continues the process. For our non-deterministic process of choosing a neighbor, we move through the edges from left to right. For each edge, we check if the edge contains the current node. If it does, with probability 1 2 , we update the current node by the other node in the edge. We move from the leftmost edge to the rightmost edge in the input tape to maintain a constant order in choosing neighbors of the current node. Since both of our tapes only store 1 node, it is clear that our non-deterministic Turing machine run in log-space. Now, we shall show that our non-deterministic Turing machine satisfies the conditions in Lemma 6.1. Let A be the configuration in the Turing machine where the tapes contain t and the current node and the tape head reading the inputs is at the start of an edge (about to choose a neighbor of the current node). Now, let us consider condition (a) of the Lemma. For any A 1 , A 2 ∈ A, where A 1 +A M A 2 , let the current node in configuration A 1 be c 1 and let the current node in configuration A 1 be c 2 . Since we have A 1 +A M A 2 , we know that c 1 and c 2 must be neighbors (since we wouldn't go through another configuration in A before we arrive at the configuration A 2 , i.e. we would not be choosing any other node to get to c 2 ). Furthermore, since all the edges in graph G are undirected, it is clear that we can go back from c 2 to c 1 with the same path. Thus, we have A 2 +A M A 1 . Next, let us consider condition (b) of the Lemma. Take any A in the union of A and possible initial configurations of M , B ∈ A, and C 1 , C 2 , C 3 , such that A * A M C 1 * A M C 2 M B M C 3 . From A * A M C 1 * A M C 2 M B M C 3 , it is clear that B is either A or ∈ A. Thus, C 1 , C 2 , C 3 ∈ A, and are configurations of M when non-deterministically choosing the neighbors of the current node in configuration A. Since we always choose neighbors by checking from the leftmost edge to the rightmost, there is only one possible linear process to non-deterministically choose the neighbors of the current node, i.e. for any configuration while choosing neighbors of the current node, M can only be coming from one possible configuration and can only transition to one possible configuration. Thus, it is clear that C 2 = C 3 . Finally, let us consider condition (c) of the Lemma. Take any A 1 in the union of A and possible initial configurations of M , A 2 ∈ A, and any B, such that A 1 * A M B * A M A 2 . Let the current node in A 1 be c 1 . It is clear that B is a configuration of M while M is choosing the neighbors for c 1 . It is also clear that to return to another configuration with a current node that is not c 1 , M must first return to configuration A 1 . Thus, A 1 = A 2 . Since M satisfied all three condisions of Lemma 6.1, by the lemma, the symmetrical Turing machine M * determines USTCONN in log-space. So USTCONN ∈ SL. Using Savitch's argument in [Sav70] and noting that the graph generated from a Symmetric Machine is undirected (since we can revert from any transition), we have that USTCONN is SL-complete. More specifically, consider any problem in SL which is solved by the symmetric Turing machine M . Since M can be computed in log-space, there are polynomially many states for M . We can construct a graph G, where the nodes are the possible states of M , and the edges are transitions between the possible states. Since M is a symmetric Turing machine, we note that the edges are undirected. Thus, solving the problem using M would be equivalent to checking if there's a path that connects the node of a starting state to the node of an accepting state. Thus, we have shown that all problems in SL can be reducible to USTCONN and USTCONN is SL-complete. Theorem 6.3. SL = L Proof. This is a direct consequence of Theorem 4.1 and Theorem 6.2. Discussion In this section, we will discuss the importance of the paper by Reingold [Rei08] and some further research based on the paper. The paper by Reingold has made progress towards discovering the relationship between L and RL. Let us begin by defining RL. Definition 7.1. RL is the space of all languages L such that there exists a randomized Turing machine T which runs in log-space and polynomial time and satisfies P (T accepts x) > 2 3 , x ∈ L P (T rejects x) = 1, x ∈ L Notice that we can choose any constant 0 < c < 1 replacing 2 3 . We can also increase the probability of accepting x when x ∈ L to 1 − 2 −poly(|x|) by repeating the algorithm poly(|x|) times. It has been shown by Aleliunas et al, in 1979 that STCONN ∈ RL [AKL + 79] In particular, USTCONN, as a specific case of STCONN is contained in RL. Furthermore, it has been shown that random walks can be generated by a randomized Turing machine of RL in polynomial time and log space. Thus, Reingold's proof that USTCONN ∈ L has brought forth major areas of research into the properties of RL. Building upon this research, Reingold, Trevisan, and Vadhan has shown in 2005 [RTV05] that a subset of STCONN (STCONN for graphs whose random walks are of polynomial mixing time) is RL complete and a deterministic log-space Turing Machine can be used to simulate random walks for biregular graphs. Some areas of future research into the relationship between L and RL could be to investigate whether one can describe a deterministic log-space Turing Machine that can simulate random walks for all regular directed graphs. • If s and t are in the same connected component which is a (N , D, λ)-graph, then the algorithm outputs "connected". Proof. Consider the algorithm of simply enumerating all paths of length l from s, where we take l = O(log N ) given by Proposition2.2, with the constant only depending on λ. Such enumeration can be done via the ordering of edges of each vertex. The algorithm outputs "connected" if there is a path of at most length l to t. The algorithm runs in O(l log D) = O(log D log N ) space as each edge of a vertex requires log D space and each path of length l can be stored in O(log N ) space. The algorithm satisfies the requirements stated above due to Proposition2.2. Figure 1 : 1Reingold et al. showed that this product is well defined and λ(G z H) is bounded as a function of λ(G) and λ(H).[RVW00] This product replaces each vertex of G via a copy of H. The edges of the product correspond to length 3 paths with two edges in H and the middle edge in G. SeeFigure 1below: On the left hand side is the edge ( Definition 3 . 1 . 31For a D 16 -regular graph G on [N ] and D-regular graph H on [D 16 ]. Then let the transformation T outputs the rotation map of G l where G l is defined recursively by: Lemma 3. 1 . 1For a D 16 -regular connected and non-bipartite graph G over [N ], and (D 16 , D, λ) graph H with λ ≤ 1 2 , we have λ(T (G, H)) ≤ 1 2 . G z H and graph powering operates separately on each connected component, T (G, H) operates on each connected component of G separately. Let define the restriction of a graph: Definition 3.2. For any graph G and subset S of its vertices V . Let G| S be the the subgraph of G induced by S, which has vertices S and edges arising from edges in G which has both endpoints in S. Note that S is a connected component of G if G| S is connected and S is disconnected to vertices of V \ S. Now let us show that restriction to a connected component of G commutes with taking the transformation T . A crucial observation is that both T i (G| S , H) and T i (G, H) are D 16 -regular, with the same vertices. In addition, T i (G| S , H) is a subgraph of T i (G, H). So we should have T (G| S , H) = T (G, H)| S×([D 16 ]) l . An formal proof using induction is given by Reingold which makes use of this observation. Lemma 3.2. For a D 16 -regular graph G on [N ] and D-regular graph H on [D 16 ]. If S is a connected component of G, then T (G| S , H) = T (G, H)| S×([D 16 ]) l . Lemma 3. 3 . 3For any constant D. Consider a D 16 -regular graph G over [N ] and D-regular graph H over [D 16 ], then T (G, H) can be computed in log(N ) space. Equivalently, there exist an O(log N ) space algorithm A T on input (G, H, (v, a)) outputs Rot T (G,H) (v, a) where v ∈ [N ] × ([D 16 ]) l and a ∈ [D 16 ]. i−1 z H) 8 . The third bullet reverses the order of labels of a i to fit the definition of zig-zag and powering. The correctness of the induction follows from the definition of zig-zag product and powering. Thus the correctness of A T follows from the inductive definition of T . Now note that within each level of the recursion tree, there are at most 16 recursive calls, and the recursion tree has depth l + 1 = O(log N ). So we can maintain the recursive calls with O(log N ) space. Furthermore, the operations of evaluating Rot G , Rot H and reversing labels can be done in O(log N ) space. The space required to store the variables in O(log N ) as v requires log N space and a i can be stored in constant space. Thus the total space needed to store a i 's is O((l + 1) * log 16) = O(log N ). Therefore the algorithm A T runs in O(log N ) space. Theorem 4 . 1 . 41For any undirected graph G over [N ], there exists an O(log N ) space algorithm A con which computes USTCONN(G, s, t) where s, t ∈ [N ]. v and w are in the same connected component of G if and only if v × [N ] and w × [N ] are in the same connected component of G reg . This in turn is equivalent to (v, 1) and (w, 1) are connected in G reg . Now let G exp = T (G reg , H), where l = O(log N ) defined in Definition 3.1. Let S be the connected component of s in G. Then S × [N ] is a connected component of G reg where G reg | S×[N ] is nonbipartite D 16 0 -regular. So by Lemma 3.2, S × [N ] × ([D 16 0 ]) l is a connected component of G exp and T (G reg | S×[N ] , H) = G exp | S×[N ]×([D 16 0 ]) l . Thus by Lemma 3.1, we get λ(G exp | S×[N ]×([D 16 0 ]) l ) The correctness of A T follows from the discussion above, as s and t are connected in G if and only if s × [N ] × ([D 16 0 ]) l and t × [N ] × ([D 16 0 ]) l are connected in G exp . The algorithm runs in O(log N ) space as computing Rot Greg , the main transform G exp and running A λ can be done using O(log N ) space. So A con computes USTCONN(G, s, t) using O(log N ) space. Figure 2 : 2The figure on the left shows the edge (v, (x, a)) ∈ G s H. Variables in black correspond to vertex labels and red labels corresponds to edge indices.Similar to the zig-zag product, the derandomized product operates separately on the connected components of G. The derandomized square G s H increases the connectivity of G and increases the spectral gap. The following theorem by Rozenman and Vadhan gives an upper bound on λ(G s H): Theorem 5.1 (Rozenman-Vadhan). If G is an undirected (N, D, λ)-graph and H is an undirected 5. 2 . 2For some constant d = 16 q , there exists a family {X m } m∈N of undirected graphs where X m is an (d m , d, 1 100 )-graph. Furthermore, Rot Xm can be computed in space O(m). Definition 5.2. Let X m be the family of constant degree graphs in Lemma 5.2, let m 0 be some fixed constant, defineH m = X m , when m ≤ m 0 H m = (X m0−1+2 m−m 0 ) 2 m−m 0 , when m > m 0 Rot Gm can be computed in space O(m + 2 m−m0 ).With the existence of such family of expanders, Rozenman and Vadhan takes a similar approach as Reingold to prove USTCONN ∈ L. For any input graph G over [N ], it can be transformed to a 16-regular graph G reg , then by powering, it can be turned to a d-regular graph G 0 = (G reg ) q , where d = 16 q is the constant in Lemma 5.2. Then he recursively defined G m+1 = G m s H m . Lemma 6 . 1 . 61For a non-deterministic Turing machine M = (K, Σ, Σ 0 , k, ∆, s, F ), let A be a subset of all possible configurations of M . If the following conditions hold: any A in the union of A and possible initial configurations of M , any B ∈ A, and any C 1 , C 2 , C 3 , if A * A M C 1 * A M C 2 M B M C 3 , then C 2 = C 3 . (c) For any A 1 in the union of A and possible initial configurations of M , any A 2 ∈ A, and any B, if A 1 * A M B * A M A 2 , then A 1 = A 2 . Definition 5.1. Let G be an undirected D-regular graph over [N ], let H be an undirected d-regular graph over [D]. The derandomized square graph G s H is an undirected Dd-regular graph over [N ] with rotation map Proof. This is Theorem 1.1 in[AS00]. Random walks, universal traversal sequences, and the complexity of maze problems. Richard Akl + 79] Romas Aleliunas, Richard Karp, László Lipton, Charles Lovász, Rackoff, Foundations of Computer Science. AKL + 79] Romas Aleliunas, Richard Karp, Richard Lipton, László Lovász, and Charles Rackoff. Random walks, universal traversal sequences, and the complexity of maze problems. Foundations of Computer Science, 1979. . Noga Alon. Eigenvalues and expanders. Combinatorica. 62Noga Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986. Random cayley graphs and expanders. 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[ "Communication Complexity Lower Bounds by Polynomials", "Communication Complexity Lower Bounds by Polynomials" ]
[ "Harry Buhrman ", "Ronald De Wolf " ]
[]
[]
The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity[33,19,12,2], but except for the inner product function[12], no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest model (qubit communication + unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log-rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.
10.1109/ccc.2001.933879
[ "https://arxiv.org/pdf/cs/9910010v2.pdf" ]
452,367
cs/9910010
d8d26be15f6e561f065c589e465be04e1674c671
Communication Complexity Lower Bounds by Polynomials 28 Apr 2000 Harry Buhrman Ronald De Wolf Communication Complexity Lower Bounds by Polynomials 28 Apr 2000arXiv:cs/9910010v2 [cs.CC] The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity[33,19,12,2], but except for the inner product function[12], no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest model (qubit communication + unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log-rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols. Lower bounds for exact protocols Let D(f ) denote the classical deterministic communication complexity of f , Q(f ) the qubit communication complexity, and Q * (f ) the qubit communication required if Alice and Bob can also make use of an unlimited supply of pre-shared EPR-pairs. Clearly Q * (f ) ≤ Q(f ) ≤ D(f ). Ultimately, we would like to show that Q * (f ) and D(f ) are polynomially related for all total functions f (as are their query complexity counterparts [4]). This requires stronger lower bound tools than we have at present. Some lower bound methods are available for Q(f ) [33,19,12,2], but the only lower bound known for Q * (f ) is for the inner product function [12]. A strong and well known lower bound for D(f ) is given by the logarithm of the rank of the communication matrix for f [22]. As first noted in [8], techniques of [33,19] imply that an Ω(log rank(f ))-bound also holds for Q(f ). Our first result is to extend this bound to Q * (f ) and to derive the optimal constant: Q * (f ) ≥ log rank(f ) 2 . (1) This implies n/2 lower bounds for the Q * -complexity of the equality and disjointness problems, for which no good bounds were known before. This n/2 is tight up to 1 bit, since Alice can send her n-bit input to Bob with n/2 qubits and n/2 EPR-pairs using superdense coding [6]. Our corresponding lower bound also provides a new proof of optimality of superdense coding. In fact, the same n/2 bound holds for almost all functions. Furthermore, proof of the well-known "log-rank conjecture" (D(f ) ≤ (log rank(f )) k for some k) would now imply our desired polynomial equivalence between D(f ) and Q * (f ) (as already noted for D(f ) and Q(f ) in [2]). However, this conjecture is a long standing open question which is probably hard to solve in full generality. Secondly, in order to get an algebraic handle on rank(f ), we relate it to a property of polynomials. It is well known that every total Boolean function g : {0, 1} n → {0, 1} has a unique representation as a multilinear polynomial in its n variables. For the case where Alice and Bob's function has the form f (x, y) = g(x ∧ y), we show that rank(f ) equals the number of monomials mon(g) of the polynomial that represents g (rank(f ) ≤ mon(g) was shown in [28]). This number of monomials is often easy to count and allows to determine rank(f ). The functions f (x, y) = g(x ∧ y) form an important class which includes inner product, disjointness, and the functions which give the biggest gaps known between D(f ) and log rank(f ) [28] (similar techniques work for f (x, y) = g(x ∨ y) or g(x ⊕ y)). We use this to show that Q * (f ) ∈ Θ(D(f )) if g is symmetric. In this case we also show that D(f ) is close to the classical randomized complexity. Furthermore, Q * (f ) ≤ D(f ) ∈ O(Q * (f ) 2 ) if g is monotone. For the latter result we rederive a result of Lovász and Saks [21] using our tools. Lower bounds for bounded-error protocols For the case of bounded-error quantum communication protocols, very few lower bounds are currently known (exceptions are inner product [12] and the general discrepancy bound [19]). In particular, no good lower bounds are known for the disjointness problem. The best known upper bound for this is O( √ n log n) qubits [8], contrasting with linear classical randomized complexity [17,30]. Since disjointness is a co-NPcomplete communication problem [3], a good lower bound for this problem would imply lower bounds for all NP-hard communication problems. In order to attack this problem, we make an effort to extend the above polynomial-based approach to bounded-error protocols. We consider the approximate rank rank(f ), and show the bound Q 2 (f ) ≥ (log rank(f ))/2 for 2-sided bounded-error qubit protocols (again using techniques from [33,19]). Unfortunately, lower bounds on rank(f ) are much harder to obtain than for rank(f ). If we could prove for the case f (x, y) = g(x∧y) that rank(f ) roughly equals the number of monomials mon(g) of an approximating polynomial for g, then a √ n lower bound would follow for disjointness, because we show that this requires at least 2 √ n monomials to approximate. Since we prove that the quantities rank(f ) and mon(g) are in fact equal in the exact case, this gives some hope for a similar result rank(f ) ≈ mon(g) in the approximating case, and hence for resolving the complexity of disjointness. The specific bounds that we actually were able to prove for disjointness are more limited at this point: Q * 2 (DISJ) ∈ Ω(log n) for the general case (by an extension of techniques of [12]; the log n bound without entanglement was already known [2]), Q * 2 (DISJ) ∈ Ω(n) for 1-round protocols (using a result of [24]), and Q 2 (DISJ) ∈ Ω(n) if the error probability has to be < 2 −n . Below we sum up the main results, contrasting the exact and bounded-error case. • We show that Q * (f ) ≥ log rank(f )/2 for exact protocols with unlimited prior EPR-pairs and Q 2 (f ) ≥ log rank(f )/2 for qubit protocols without prior EPR-pairs. • If f (x, y) = g(x ∧ y) for some Boolean function g, then rank(f ) = mon(g). An analogous result rank(f ) ≈ mon(g) for the approximate case is open. • A polynomial for disjointness, DISJ(x, y) = NOR(x ∧ y), requires 2 n monomials in the exact case (implying Q * (DISJ) ≥ n/2), and roughly 2 √ n monomials in the approximate case. Preliminaries We use |x| to denote the Hamming weight (number of 1s) of x ∈ {0, 1} n , x i for the ith bit of x (x 0 = 0), and e i for the string whose only 1 occurs at position i. If x, y ∈ {0, 1} n , we use x∧y ∈ {0, 1} n for the string obtained by bitwise ANDing x and y, and similarly x ∨ y. Let g : {0, 1} n → {0, 1} be a Boolean function. We call g symmetric if g(x) only depends on |x|, and monotone if g cannot decrease if we set more variables to 1. It is well known that each g : {0, 1} n → R has a unique representation as a multilinear polynomial g(x) = S⊆{1,...,n} a S X S , where X S is the product of the variables in S and a S is a real number. The term a S X S is called a monomial of g and mon(g) denotes the number of non-zero monomials of g. A polynomial p approximates g if |g(x) − p(x)| ≤ 1/3 for all x ∈ {0, 1} n . We use mon(g) for the minimal number of monomials among all polynomials which approximate g. The degree of a monomial is the number of its variables, and the degree of a polynomial is the largest degree of its monomials. Let X and Y be finite sets (usually X = Y = {0, 1} n ) and f : X × Y → {0, 1} be a Boolean function. For example, equality has EQ(x, y) = 1 iff x = y, disjointness has DISJ(x, y) = 1 iff |x ∧ y| = 0 (equivalently, DISJ(x, y) = NOR(x ∧ y)), and inner product has IP(x, y) = 1 iff |x ∧ y| is odd. M f denotes the |X| × |Y | Boolean matrix whose x, y entry is f (x, y), and rank(f ) denotes the rank of M f over the reals. A rectangle is a subset R = S × T ⊆ X × Y of the domain of f . A 1-cover for f is a set of rectangles which covers all and only 1s in M f . C 1 (f ) denotes the minimal size of a 1-cover for f . For m ≥ 1, we use f ∧m to denote the Boolean function which is the AND of m independent instances of f . That is, f ∧m : X m × Y m → {0, 1} and f ∧m (x 1 , . . . , x m , y 1 , . . . , y m ) = f (x 1 , y 1 ) ∧ f (x 2 , y 2 ) ∧ . . . ∧ f (x m , y m ). Note that M f ∧2 is the Kronecker product M f ⊗ M f and hence rank(f ∧m ) = rank(f ) m . Alice and Bob want to compute some f : X × Y → {0, 1}. After the protocol they should both know f (x, y). Their system has three parts: Alice's part, the 1-qubit channel, and Bob's part. For definitions of quantum states and operations, we refer to [7,10]. In the initial state, Alice and Bob share k EPRpairs and all other qubits are zero. For simplicity we assume Alice and Bob send 1 qubit in turn, and at the end the output-bit of the protocol is put on the channel. The assumption that 1 qubit is sent per round can be replaced by a fixed number of qubits q i for the ith round. However, in order to be able to run a quantum protocol on a superposition of inputs, it is important that the number of qubits sent in the ith round is independent of the input (x, y). An ℓ-qubit protocol is described by unitary transformations U 1 (x), U 2 (y), U 3 (x), U 4 (y), . . . , U ℓ (x/y). First Alice applies U 1 (x) to her part and the channel, then Bob applies U 2 (y) to his part and the channel, etc. Q(f ) denotes the (worst-case) cost of an optimal qubit protocol that computes f exactly without prior entanglement, C * (f ) denotes the cost of a protocol that communicates classical bits but can make use of an unlimited (but finite) number of shared EPR-pairs, and Q * (f ) is the cost of a qubit protocol that can use shared EPR-pairs. Q c (f ) denotes the cost of a clean qubit protocol without prior entanglement, i.e. a protocol that starts with |0 |0 |0 and ends with |0 |f (x, y) |0 . We add the superscript "1 round" for 1round protocols, where Alice sends a message to Bob and Bob then sends the output bit. Some simple relations that hold between these measures are Q [5]. For bounded-error protocols we analogously define Q 2 (f ), * (f ) ≤ Q(f ) ≤ D(f ) ≤ D 1round (f ), Q(f ) ≤ Q c (f ) ≤ 2Q(f ) and Q * (f ) ≤ C * (f ) ≤ 2Q * (f )Q * 2 (f ), C * 2 (f ) for quantum protocols that give the correct answer with probability at least 2/3 on every input. We use R pub 2 (f ) for the classical bounded-error complexity in the public-coin model [20]. Log-Rank Lower Bound As first noted in [8,2], techniques from [33,19] imply Q(f ) ∈ Ω(log rank(f )). For completeness we prove the following log rank(f ) bound for clean quantum protocols in Appendix A. This implies Q(f ) ≥ log rank(f )/2. We then extend this to the case where Alice and Bob share prior entanglement: 1 Theorem 1 Q c (f ) ≥ log rank(f ) + 1. Theorem 2 Q * (f ) ≥ log rank(f ) 2 . Proof Suppose we have some exact protocol for f that uses ℓ qubits of communication and k prior EPRpairs. We will build a clean qubit protocol without prior entanglement for f ∧m . First Alice makes k EPR-pairs and sends one half of each pair to Bob (at a cost of k qubits of communication). Now they run the protocol to compute the first instance of f (ℓ qubits of communication). Alice copies the answer to a safe place which we will call the 'answer bit' and they reverse the protocol (again ℓ qubits of communication). This gives them back the k EPR-pairs, which they can reuse. Now they compute the second instance of f , Alice ANDs the answer into the answer bit (which can be done cleanly), and they reverse the protocol, etc. After all m instances of f have been computed, Alice and Bob have the answer f ∧m (x, y) left and the k EPR-pairs, which they uncompute using another k qubits of communication. This gives a clean protocol for f ∧m that uses 2mℓ+2k qubits and no prior entanglement. By Theorem 1: 2mℓ + 2k ≥ Q c (f ∧m ) ≥ log rank(f ∧m ) + 1 = m log rank(f ) + 1, hence ℓ ≥ log rank(f ) 2 − 2k − 1 2m . Since this must hold for every m > 0, the theorem follows. We can derive a stronger bound for C * (f ): Theorem 3 C * (f ) ≥ log rank(f ). Proof Since a qubit and an EPR-pair can be used to send 2 classical bits [6], we can devise a qubit protocol for f ∧f using C * (f ) qubits (compute the two copies of f in parallel using the classical bit protocol). Hence by the previous theorem C * (f ) ≥ Q * (f ∧ f ) ≥ (log rank(f ∧ f ))/2 = log rank(f ). 2 Below we draw some consequences from these log-rank lower bounds. Firstly, M EQ is the identity matrix, so rank(EQ) = 2 n . This gives the bounds Q * (EQ) ≥ n/2, C * (EQ) ≥ n (in contrast, Q 2 (EQ) ∈ Θ(log n) and C * 2 (EQ) ∈ O(1)). The disjointness function on n bits is the AND of n disjointnesses on 1 bit (which have rank 2 each), so rank(DISJ) = 2 n . The complement of the inner product function has rank(f ) = 2 n . Thus we have the following strong lower bounds, all tight up to 1 bit: 2 Corollary 1 Q * (EQ), Q * (DISJ), Q * (IP) ≥ n/2 and C * (EQ), C * (DISJ), C * (IP) ≥ n. Komlós [18] has shown that the fraction of m × m Boolean matrices that have determinant 0 goes to 0 as m → ∞. Hence almost all 2 n × 2 n Boolean matrices have full rank 2 n , which implies that almost all functions have maximal quantum communication complexity: Corollary 2 Almost all f : {0, 1} n × {0, 1} n → {0, 1} have Q * (f ) ≥ n/2 and C * (f ) ≥ n. We say f satisfies the quantum direct sum property if computing m independent copies of f (without prior entanglement) takes mQ(f ) qubits of communication in the worst case. (We have no example of an f without this property.) Using the same technique as before, we can prove an equivalence between the qubit models with and without prior entanglement for such f : Corollary 3 If f satisfies the quantum direct sum property, then Q * (f ) ≤ Q(f ) ≤ 2Q * (f ). Proof Q * (f ) ≤ Q(f ) is obvious. Using the techniques of Theorem 2 we have mQ(f ) ≤ 2mQ * (f ) + k, for all m and some fixed k, hence Q(f ) ≤ 2Q * (f ). 2 Finally, because of Theorem 2, the well-known "log-rank conjecture" now implies the polynomial equivalence of deterministic classical communication complexity and exact quantum communication complexity (with or without prior entanglement) for all total f : Corollary 4 If D(f ) ∈ O((log rank(f )) k ), then Q * (f ) ≤ Q(f ) ≤ D(f ) ∈ O(Q * (f ) k ) for all f . A Lower Bound Technique via Polynomials Decompositions and polynomials The previous section showed that lower bounds on rank(f ) imply lower bounds on Q * (f ). In this section we relate rank(f ) to the number of monomials of a polynomial for f and use this to prove lower bounds for some classes of functions. We define the decomposition number m(f ) of some function f : {0, 1} n × {0, 1} n → R as the min- imum m such that there exist functions a 1 (x), . . . , a m (x) and b 1 (y), . . . , b m (y) (from R n to R) for which f (x, y) = m i=1 a i (x)b i (y) for all x, y. We say that f can be decomposed into the m functions a i b i . Without loss of generality, the functions a i , b i may be assumed to be multilinear polynomials. It turns out that the decomposition number equals the rank: 3 Lemma 1 rank(f ) = m(f ). Proof rank(f ) ≤ m(f ): Let f (x, y) = m i=1 a i (x)b i (y), M i be the matrix defined by M i (x, y) = a i (x)b i (y) , r i be the row vector whose yth entry is b i (y). Note that the xth row of M i is a i (x) times r i . Thus all rows of M i are scalar multiples of each other, hence M i has rank 1. Since rank(A + B) ≤ rank(A) + rank(B) and M f = m(f ) i=1 M i , we have rank(f ) = rank(M f ) ≤ m(f ) i=1 rank(M i ) = m(f ). m(f ) ≤ rank(f ): Suppose rank(f ) = r. Then there are r columns c 1 , . . . , c r in M f which span the column space of M f . Let A be the 2 n × r matrix that has these c i as columns. Let B be the r × 2 n matrix whose ith column is formed by the r coefficients of the ith column of M f when written out as a linear combination of c 1 , . . . , c r . Then M f = AB, hence f (x, y) = M f (x, y) = r i=1 A xi B iy . Defining functions a i , b i by a i (x) = A xi and b i (y) = B iy , we have m(f ) ≤ rank(f ). 2 Combined with Theorems 2 and 3 we obtain Corollary 5 Q * (f ) ≥ log m(f ) 2 and C * (f ) ≥ log m(f ). Accordingly, for lower bounds on quantum communication complexity it is important to be able to determine the decomposition number m(f ). Often this is hard. It is much easier to determine the number of monomials mon(f ) of f (which upper bounds m(f )). Below we show that in the special case where f (x, y) = g(x ∧ y), these two numbers are the same. 4 Below, a monomial is called even if it contains x i iff it contains y i , for example 2x 1 x 3 y 1 y 3 is even and x 1 x 3 y 1 is not. A polynomial is even if each of its monomials is even. Lemma 2 If p : {0, 1} n × {0, 1} n → R is an even polynomial with k monomials, then m(p) = k. Proof Clearly m(p) ≤ k. To prove the converse, consider DISJ(x, y) = Π n i=1 (1 − x i y i ), the unique polynomial for the disjointness function. Note that this polynomial contains all and only even monomials (with coefficients ±1). Since DISJ has rank 2 n , it follows from Lemma 1 that DISJ cannot be decomposed in fewer then 2 n terms. We will show how a decomposition of p with m(p) < k would give rise to a decomposition of DISJ with fewer than 2 n terms. Suppose we can write p(x, y) = m(p) i=1 a i (x)b i (y). Let aX S Y S be some even monomial in p and suppose the monomial X S Y S in DISJ has coefficient c = ±1. Now whenever bX S occurs in some a i , replace that bX S by (cb/a)X S . Using the fact that p contains only even monomials, it is not hard to see that the new polynomial obtained in this way is the same as p, except that the monomial aX S Y S is replaced by cX S Y S . Doing this sequentially for all monomials in p, we end up with a polynomial p ′ (with k monomials and m(p ′ ) ≤ m(p)) which is a subpolynomial of DISJ, in the sense that each monomial in p ′ also occurs with the same coefficient in DISJ. Notice that by adding all 2 n − k missing DISJ-monomials to p ′ , we obtain a decomposition of DISJ with m(p ′ ) + 2 n − k terms. But any such decomposition needs at least 2 n terms, hence m(p ′ ) + 2 n − k ≥ 2 n , which implies k ≤ m(p ′ ) ≤ m(p). 2 If f (x, y) = g(x ∧ y) for some Boolean function g, then the polynomial that represents f is just the polynomial of g with the ith variable replaced by x i y i . Hence such a polynomial is even, and we obtain: This gives a strong tool for lower bounding (quantum and classical) communication complexity whenever f is of the form f (x, y) = g(x ∧ y): log mon(g) ≤ C * (f ) ≤ D(f ). Below we give some applications. Symmetric functions As a first application we show that D(f ) and Q * (f ) are linearly related if f (x, y) = g(x ∧ y) and g is symmetric (this follows from Corollary 8 below). Furthermore, we show that the classical randomized public-coin complexity R pub 2 (f ) can be at most a log n-factor less than D(f ) for such f (Theorem 4). We will assume without loss of generality that g( 0) = 0, so the polynomial representing g does not have the constant-1 monomial. Lemma 3 If g is a symmetric function whose lowest-weight 1-input has Hamming weight t > 0 and f (x, y) = g(x ∧ y), then D 1round (f ) = log n i=t n i + 1 + 1. Proof It is known (and easy to see) that D 1round (f ) = log r + 1, where r is the number of different rows of M f (this equals the number of different columns in our case, because f (x, y) = f (y, x)). We count r. Firstly, if |x| < t then the x-row contains only zeroes. Secondly, if x = x ′ and both |x| ≥ t and |x ′ | ≥ t then it is easy to see that there exists a y such that |x ∧ y| = t and |x ′ ∧ y| < t (or vice versa), hence f (x, y) = f (x ′ , y) so the x-row and x ′ -row are different. Accordingly, r equals the number of different x with |x| ≥ t, +1 for the 0-row, which gives the lemma. 2 Lemma 4 If g is a symmetric function whose lowest-weight 1-input has weight t > 0, then (1 − o(1)) log n i=t n i ≤ log mon(g) ≤ log n i=t n i . Proof The upper bound follows from the fact that g cannot have monomials of degree < t. For the lower bound we distinguish two cases. Case 1: t ≤ n/2. It is known that every symmetric g has degree deg(g) = n − O(n 0.548 ) [14]. That is, an interval I = [a, n] such that g has no monomials of any degree d ∈ I has length at most O(n 0.548 ). This implies that every interval I = [a, b] (b ≥ t) such that g has no monomials of any degree d ∈ I has length at most O(n 0.548 ) (by setting n − b variables to 0, we can reduce to a function on b variables where I occurs "at the end"). Since g must have monomials of degree t ≤ n/2, g must contain a monomial of degree d for some d ∈ [n/2, n/2 + O(n 0.548 )]. But because g is symmetric, it must then contain all n d monomials of degree d. Hence by Stirling's approximation mon(g) ≥ n d ≥ 2 n−O(n 0.548 ) , which implies the lemma. Case 2: t > n/2. It is easy to see that g must contain all n t monomials of degree t. Now (n − t + 1)mon(g) ≥ (n − t + 1) n t ≥ n i=t n i . Hence log mon(g) ≥ log n i=t n i − log(n − t + 1) = (1 − o(1)) log n i=t n i . 2 The number mon(g) may be less then n i=t n i . Consider the function g( [27]. Here mon(g) = 6 but 3 Combining the previous results: x 1 , x 2 , x 3 ) = x 1 + x 2 + x 3 − x 1 x 2 − x 1 x 3 − x 2 x 3 Corollary 7 If g is a symmetric function whose lowest-weight 1-input has weight t > 0 and f (x, y) = g(x ∧ y), then (1 − o(1)) log n i=t n i ≤ C * (f ) ≤ D(f ) ≤ D 1round (f ) = log n i=t n i + 1 + 1. Accordingly, for symmetric g the communication complexity (quantum and classical, with or without prior entanglement, 1-round and multi-round) equals log rank(f ) up to small constant factors. In particular: Corollary 8 If g is symmetric and f (x, y) = g(x ∧ y), then (1 − o(1))D(f ) ≤ C * (f ) ≤ D(f ). We have shown that Q * (f ) and D(f ) are equal up to constant factors whenever f (x, y) = g(x ∧ y) and g is symmetric. For such f , D(f ) is also nearly equal to the classical bounded-error communication complexity R pub 2 (f ), where we allow Alice and Bob to share public coin flips. In order to prove this, we introduce the notion of 0-block sensitivity in analogy to the notion of block sensitivity of Nisan [26]. For input x ∈ {0, 1} n , let bs0 x (g) be the maximal number of disjoint sets S 1 , . . . , S b of indices of variables, such that for every i we have (1) all S i -variables have value 0 in x and (2) g(x) = g(x S i ), where x S i is the string obtained from x by setting all S i -variables to 1. Let bs0(g) = max x bs0 x (g). We now have: Lemma 5 If g is a symmetric function, then mon(g) ≤ n 2bs0(g) . Proof Let t be the smallest number such that g t = g t+1 , then bs0(g) ≥ n−t. If t ≤ n/2 then bs0(g) ≥ n/2, so mon(g) ≤ 2 n ≤ n 2bs0(g) . If t > n/2 then g has no monomials of degree ≤ t, hence mon(g) ≤ n i=t+1 n i ≤ n 2bs0(g) . 2 Theorem 4 If g is a symmetric function and f (x, y) = g(x ∧ y), then D(f ) ∈ O(R pub 2 (f ) log n). Proof By Corollary 7 we have D(f ) ≤ (1 + o(1)) log mon(g). Lemma 5 implies D(f ) ∈ O(bs0(g) log n). Using Razborov's lower bound technique for disjointness [30] (see also [20,Section 4.6]) we can easily show R pub 2 (f ) ∈ Ω(bs0(f )), which implies the theorem. This theorem is tight for the function defined by g(x) = 1 iff |x| ≥ n − 1. We have mon(g) = n + 1, so log n ≤ D(f ) ≤ (1 + o(1)) log n. On the other hand, an O(1) bounded-error public coin protocol can easily be derived from the well-known O(1)-protocol for equality: Alice tests if |x| < n − 1, sends a 0 if so and a 1 if not. In the first case Alice and Bob know that f (x, y) = 0. In the second case, we have f (x, y) = 1 iff x = y or y = 1, which can be tested with 2 applications of the equality-protocol. Hence R pub 2 (f ) ∈ O(1). Monotone functions A second application concerns monotone problems. Lovász and Saks [21] prove the log-rank conjecture for (among others) the following problem, which they call the union problem for C. Here C is a monotone set system (i.e. (A ∈ C ∧ A ⊆ B) ⇒ B ∈ C) over some size-n universe. Alice and Bob receive sets x and y (respectively) from this universe, and their task is to determine whether x∪y ∈ C. Identifying sets with their representation as n-bit strings, this problem can equivalently be viewed as a function f (x, y) = g(x ∨ y), where g is a monotone increasing Boolean function. Note that it doesn't really matter whether we take g increasing or decreasing, nor whether we use x ∨ y or x ∧ y, as these problems can all be converted into each other via De Morgan's laws. Our translation of rank to number of monomials now allows us to rederive the Lovász-Saks result without making use of their combinatorial lattice theoretical machinery. We just need the following, slightly modified, result from their paper (a proof is given in Appendix B): Theorem 5 (Lovász and Saks) D(f ) ≤ (1 + log(C 1 (f ) + 1))(2 + log rank(f )). Theorem 6 (Lovász and Saks) If g is monotone and f (x, y) = g(x ∧ y), then D(f ) ∈ O((log rank(f )) 2 ). Proof Let M 1 , . . . , M k be all the minimal monomials in g. Each M i induces a rectangle R i = S i × T i , where S i = {x | M i ⊆ x} and T i = {y | M i ⊆ y}. Because g is monotone increasing, g(z) = 1 iff z makes at least one M i true. Hence f (x, y) = 1 iff there is an i such that (x, y) ∈ R i . Accordingly, the set of R i is a 1-cover for f and C 1 (f ) ≤ k ≤ mon(g) = rank(f ) by Corollary 6. Plugging into Theorem 5 gives the theorem. 2 Corollary 9 If g is monotone and f (x, y) = g(x ∧ y), then D(f ) ∈ O(Q * (f ) 2 ). This result can be tightened for the special case of d-level AND-OR-trees. For example, let g be a 2-level AND-of-ORs on n variables with fan-out √ n and f (x, y) = g(x ∧ y). Then g has (2 √ n − 1) √ n monomials and hence Q * (f ) ≥ n/2. In contrast, the zero-error quantum complexity of f is O(n 3/4 log n) [9]. Bounded-Error Protocols Here we generalize the above approach to bounded-error quantum protocols. Define the approximate rank of f , rank(f ), as the minimum rank among all matrices M that approximate M f entry-wise up to 1/3. Let the approximate decomposition number m(f ) be the minimum m such that there exist functions a 1 (x), . . . , a m (x) and b 1 (y), . . . , b m (y) for which |f (x, y) − m i=1 a i (x)b i (y)| ≤ 1/3 for all x, y. By the same proof as for Lemma 1 we obtain: Lemma 6 rank(f ) = m(f ). By a proof similar to Theorem 1 (again using methods from [33,19], see Appendix C) we show Theorem 7 Q 2 (f ) ≥ log m(f ) 2 . Unfortunately, it is much harder to prove bounds on m(f ) than on m(f ). 5 In the exact case we have m(f ) = mon(g) whenever f (x, y) = g(x∧y), and mon(g) is often easy to determine. If something similar is true in the approximate case, then we obtain strong lower bounds on Q 2 (f ), because our next theorem gives a bound on mon(g) in terms of the 0-block sensitivity defined in the previous section (the proof is deferred to Appendix D). Theorem 8 If g is a Boolean function, then mon(g) ≥ 2 √ bs0(g)/12 . In particular, for DISJ(x, y) = NOR(x∧y) it is easy to see that bs0(NOR) = n, hence log mon(NOR) ≥ n/12 (the upper bound log mon(NOR) ∈ O( √ n log n) follows from the construction of a degree-√ n polynomial for OR in [27]). Consequently, a proof that the approximate decomposition number m(f ) roughly equals mon(g) would give Q 2 (DISJ) ∈ Ω( √ n), nearly matching the O( √ n log n) upper bound of [8]. Since m(f ) = mon(g) in the exact case, a result like m(f ) ≈ mon(g) might be doable. We end this section by proving some weaker lower bounds for disjointness. Firstly, disjointness has a bounded-error protocol with O( √ n log n) qubits and O( √ n) rounds [8], but if we restrict to 1-round protocols then a linear lower bound follows from a result of Nayak [24]: Theorem 9 Q 1round 2 (DISJ) ∈ Ω(n). Proof Suppose there exists a 1-round qubit protocol with m qubits: Alice sends a message M (x) of m qubits to Bob, and Bob then has sufficient information to establish whether Alice's x and Bob's y are disjoint. Note that M (x) is independent of y. If Bob's input is y = e i , then DISJ(x, y) is the negation of Alice's ith bit. But then the message is an (n, m, 2/3) quantum random access code [1]: by choosing input y = e i and continuing the protocol, Bob can extract from M (x) the ith bit of Alice (with probability ≥ 2/3), for any 1 ≤ i ≤ n of his choice. For this the lower bound m ≥ (1 − H(2/3))n > 0.08 n is known [24]. 2 For multi-round quantum protocols for disjointness with bounded error probability we can only prove a logarithmic lower bound, using a technique from [12] (we omit the proof for reasons of space; for the model without entanglement, the bound Q 2 (DISJ) ∈ Ω(log n) was already shown in [2]). Proposition 1 Q * 2 (DISJ) ∈ Ω(log n). Finally, for the case where we want to compute disjointness with very small error probability, we can prove an Ω(n) bound. Here we use the subscript "ε" to indicate qubit protocols (without prior entanglement) whose error probability is ≤ ε. We first give a bound for equality: Theorem 10 If ε < 2 −n , then Q ε (EQ) ≥ n/2. Proof By Lemma 6 and Theorem 7, it suffices to show that an ε-approximation of the 2 n × 2 n identity matrix I requires full rank. Suppose that M approximates I entry-wise up to ε but has rank < 2 n . Then M has some eigenvalue λ = 0. Gersgorin's Disc Theorem (see [15, p.31]) implies that all eigenvalues of M are in the set i {z | |z − M ii | ≤ R i }, where R i = j =i |M ij |. But if λ = 0 is in this set, then for some i 1 − ε ≤ |M ii | = |λ − M ii | ≤ R i ≤ (2 n − 1)ε, hence ε ≥ 2 −n , contradiction. 2 We reduce equality to disjointness. Let x, y ∈ {0, 1} n . Define x ′ ∈ {0, 1} 2n by replacing x i by x i x i in x, and y ′ ∈ {0, 1} 2n by replacing y i by y i y i in y. It is easy to see that EQ(x, y) = DISJ(x ′ , y ′ ) so we have: Corollary 10 If ε < 2 −n , then Q ε (DISJ) ≥ n/4. Open Problems To end this paper, we identify three important open questions in quantum communication complexity. First, are Q * (f ) and D(f ) polynomially related for all total f , or at least for all f of the form f (x, y) = g(x ∧ y)? We have proven this for some special cases here (g symmetric or monotone), but the general question remains open. There is a close analogy between the quantum communication complexity lower bounds presented here, and the quantum query complexity bounds obtained in [4]. Let deg(g) and mon(g) be, respectively, the degree and the number of monomials of the polynomial that represents g : {0, 1} n → {0, 1}. In [4] it was shown that a quantum computer needs at least deg(g)/2 queries to the n variables to compute g, and that O(deg(g) 4 ) queries suffice (see also [27]). This implies that classical and quantum query complexity are polynomially related for all total f . Similarly, we have shown here that (log mon(g))/2 qubits need to be communicated to compute f (x, y) = g(x ∧ y). An analogous upper bound like Q * (f ) ∈ O((log mon(g)) k ) might be true. A similar resemblance holds in the bounded-error case. Let deg(g) be the minimum degree of polynomials that approximate g. In [4] it was shown that a bounded-error quantum computer needs at least deg(g)/2 queries to compute g and that O( deg(g) 6 ) queries suffice. Here we showed that (log m(f ))/2 qubits of communication are necessary to compute f . A similar upper bound like Q 2 (f ) ∈ O((log m(f )) k ) may hold. A second open question: how do we prove good lower bounds on bounded-error quantum protocols? Theorems 7 and 8 of the previous section show that Q 2 (f ) is lower bounded by log m(f )/2 and log mon(g) is lower bounded by bs0(g). If we could show m(f ) ≈ mon(g) whenever f (x, y) = g(x ∧ y), we would have Q 2 (f ) ∈ Ω( bs0(g)). Since m(f ) = mon(g) in the exact case, this may well be true. As mentioned above, this is particularly interesting because it would give a near-optimal lower bound Q 2 (DISJ) ∈ Ω( √ n). Third and last, does prior entanglement add much power to qubit communication, or are Q(f ) and Q * (f ) roughly equal up to small additive or multiplicative factors? Similarly, are Q 2 (f ) and Q * 2 (f ) roughly equal? The biggest gap that we know is Q 2 (EQ) ∈ Θ(log n) versus Q * 2 (EQ) ∈ O(1). B Proof of Theorem 5 Theorem 5 (Lovász and Saks) D(f ) ≤ (1 + log(C 1 (f ) + 1))(2 + log rank(f )). Proof We will first give a protocol based on a 0-cover. Let c = C 0 (f ) and R 1 , . . . , R c be an optimal 0-cover. Let R i = S i × T i . We will also use S i to denote the |S i | × 2 n matrix of S i -rows and T i for the 2 n × |T i | matrix of T i -columns. Call R i type 1 if rank(S i ) ≤ rank(M f )/2, and type 2 otherwise. Note that rank(S i ) + rank(T i ) ≤ rank(M f ), hence at least one of rank(S i ) and rank( T i ) is ≤ rank(M f )/2. The protocol is specified recursively as follows. Alice checks if her x occurs in some type 1 R i . If no, then she sends a 0 to Bob; if yes, then she sends the index i and they continue with the reduced function g (obtained by shrinking Alice's domain to S i ), which has rank(g) = rank(S i ) ≤ rank(M f )/2. If Bob receives a 0, he checks if his y occurs in some type 2 R j . If no, then he knows that (x, y) does not occur in any R i , so f (x, y) = 1 and he sends a 0 to Alice to tell her; if yes, then he sends j and they continue with the reduced function g, which has rank(g) = rank(T i ) ≤ rank(M f )/2 because R j is type 2. Thus Alice and Bob either learn f (x, y) or reduce to a function g with rank(g) ≤ rank(f )/2, at a cost of at most 1 + log(c + 1) bits. It now follows by induction on the rank that D(f ) ≤ (1 + log(C 0 (f ) + 1))(1 + log rank(f )). Noting that C 1 (f ) = C 0 (f ) and |rank(f )−rank(f )| ≤ 1, we have D(f ) = D(f ) ≤ (1+log(C 0 (f )+ 1))(1 + log rank(f )) ≤ (1 + log(C 1 (f ) + 1))(2 + log rank(f )). 2 C Proof of Theorem 7 Theorem 7 Q 2 (f ) ≥ log m(f ) 2 . Proof By Lemma 7 we can write the final state of an ℓ-qubit bounded-error protocol for f as i∈{0,1} ℓ α i (x)β i (y)|A i (x) |i ℓ |B i (y) . Let φ(x, y) = i∈{0,1} ℓ−1 α i1 (x)β i1 (y)|A i1 (x) |1 |B i1 (y) be the part of the final state that corresponds to a 1-output of the protocol. For i, j ∈ {0, 1} ℓ−1 , define functions a ij , b ij by a ij (x) = α i1 (x)α j1 (x) A i1 (x)|A j1 (x) b ij (y) = β i1 (y)β j1 (y) B i1 (y)|B j1 (y) Note that the acceptance probability is P (x, y) = φ(x, y)|φ(x, y) = i,j∈{0,1} ℓ−1 a ij (x)b ij (y). We have now decomposed P (x, y) into 2 2ℓ−2 functions. However, we must have |P (x, y) − f (x, y)| ≤ 1/3 for all x, y, hence 2 2ℓ−2 ≥ m(f ). It follows that ℓ ≥ (log m(f ))/2 + 1. 2 D Proof of Theorem 8 Here we prove Theorem 8. The proof uses some tools from the degree-lower bound proofs of Nisan and Szegedy [27,Section 3], including the following result from [13,31]: Theorem 11 (Ehlich, Zeller; Rivlin, Cheney) Let p be a single-variate polynomial of degree deg(p) such that b 1 ≤ p(i) ≤ b 2 for every integer 0 ≤ i ≤ n, and the derivative satisfies |p ′ (x)| ≥ c for some real 0 ≤ x ≤ n. Then deg(p) ≥ cn/(c + b 2 − b 1 ). A hypergraph is a set system H ⊆ Pow{1, . . . , n}. The sets E ∈ H are called the edges of H. We call H an s-hypergraph if all E ∈ H satisfy |E| ≥ s. A set S ⊆ {1, . . . , n} is a blocking set for H if it "hits" every edge: S ∩ E = ∅ for all E ∈ H. Proof Assume, by way of contradiction, that there exists a blocking set S of H with |S| ≤ n/2. Obtain restrictions h and q of g and p, respectively, on n−|S| ≥ n/2 variables by fixing all S-variables to 0. Then q approximates h and all monomials of q have degree < n/12 (all p-monomials of higher degree have been set to 0 because S is a blocking set for H). Since q approximates h we have q( 0) ∈ [−1/3, 1/3], q(e i ) ∈ [2/3, 4/3], and q(x) ∈ [−1/3, 4/3] for all other x ∈ {0, 1} n . By standard symmetrization techniques [23,27], we can turn q into a single-variate polynomial r of degree < n/12, such that r(0) ∈ [−1/3, 1/3], r(1) ∈ [2/3, 4/3], and r(i) ∈ [−1/3, 4/3] for i ∈ {2, . . . , n/2}. Since r(0) ≤ 1/3 and r(1) ≥ 2/3, we must have p ′ (x) ≥ 1/3 for some real x ∈ [0, 1]. But then deg(r) ≥ (1/3)(n/2)/(1/3 + 4/3 + 1/3) = n/12 by Theorem 11, contradiction. 2 The next lemma shows that H must be large if it has no blocking set of size ≤ n/2: Lemma 9 If H is an s-hypergraph of size m < 2 s , then H has a blocking set of size ≤ n/2. Proof We use the probabilistic method to show the existence of a blocking set S. Randomly choose a set S of n/2 elements. The probability that S does not hit some specific E ∈ H is n−|E| n/2 n n/2 = n 2 ( n 2 − 1) . . . ( n 2 − |E| + 1) n(n − 1) . . . (n − |E| + 1) ≤ 2 −|E| . Then the probability that there is some edge E ∈ H which is not hit by S is Thus with positive probability, S hits all E ∈ H, which proves the existence of a blocking set. 2 The above lemmas allow us to prove: Theorem 8 If g is a Boolean function, then mon(g) ≥ 2 √ bs0(g)/12 . Corollary 6 6If g : {0, 1} n → {0, 1} and f (x, y) = g(x∧y), then mon(g) = mon(f ) = m(f ) = rank(f ). i=1 3 i = 7 . 37Hence the 1 − o(1) of Lemma 4 cannot be improved to 1 in general (it can if g is a threshold function). Lemma 8 8Let g : {0, 1} n → {0, 1} be a Boolean function for which g( 0) = 0 and g(e i ) = 1, p be a multilinear polynomial which approximates g (i.e. |g(x) − p(x)| ≤ 1/3 for all x ∈ {0, 1} n ), and H be the n/12-hypergraph formed by the set of all monomials of p that have degree ≥ n/12. Then H has no blocking set of size ≤ n/2. Pr [ PrE∈H S does not hit E] ≤ Pr[S does not hit E] ≤ E∈H 2 −|E| ≤ m · 2 −s < 1.E∈H During discussions we had with Michael Nielsen in Cambridge in the summer of 1999, it appeared that an equivalent result can be derived from results about Schmidt numbers in [25, Section 6.4.2]. The same bounds for IP are also given in[12]. The bounds for EQ and DISJ are new, and can also be shown to hold for zero-error quantum protocols.3 The first part of the proof employs a technique of Nisan and Wigderson[28]. They used this to prove log rank(f ) ∈ O(n log 3 2 ) for a specific f . Our Corollary 6 below implies that this is tight: log rank(f ) ∈ Θ(n log 3 2 ) for their f . After learning about this result, Mario Szegedy (personal communication) came up with an alternative proof of this, using Fourier transforms. It is interesting to note that IP (the negation of IP) has less than maximal approximate decomposition number. For example for n = 2, m(f ) = 4 but m(f ) = 3. Acknowledgments.We acknowledge helpful discussions with Alain Tapp, who first came up with the idea of reusing entanglement used in Section 3. We also thank Michael Nielsen, Mario Szegedy, Barbara Terhal for discussions, and John Tromp for help with the proof of Lemma 9 in Appendix D.A Proof of Theorem 1Here we prove a log rank(f ) lower bound for clean qubit protocols.Lemma 7 (Kremer/Yao)The final state of an ℓ-qubit protocol (without prior entanglement) on input (x, y) can be written aswhere the α i (x), β i (y) are complex numbers and the A i (x), B i (y) are unit vectors.ProofThe proof is by induction on ℓ:Base step. For ℓ = 0 the lemma is obvious. Induction step. Suppose after ℓ qubits of communication the state can be written asWe assume without loss of generality that it is Alice's turn: she applies U ℓ+1 (x) to her part and the channel.Thus every element of the superposition (2) "splits in two" when we apply U ℓ+1 . Accordingly, we can write the state after U ℓ+1 in the form required by the lemma. 2Proof Consider a clean ℓ-qubit protocol for f . By Lemma 7, we can write its final state asThe protocol is clean, so the final state is |0 |f (x, y) |0 . Hence all parts of |A i (x) and |B i (y) other than |0 will cancel out, and we can assume without loss of generality that |A i (x) = |B i (y) = |0 for all i. Now the amplitude of the |0 |1 |0 -state is simply the sum of the amplitudes α i (x)β i (y) of the i for which i ℓ = 1. This sum is either 0 or 1, and is the acceptance probability P (x, y) of the protocol. Letting α(x) (resp. β(y)) be the dimension-2 ℓ−1 vector whose entries are α i (x) (resp. β i (y)) for the i with i ℓ = 1:Since the protocol is exact, we must have P (x, y) = f (x, y). Hence if we define A as the |X| × d matrix having the α(x) as rows and B as the d × |Y | matrix having the β(y) as columns, then M f = AB. But now rank(M f ) = rank(AB) ≤ rank(A) ≤ d ≤ 2 l−1 , and the theorem follows. 2Proof Let p be a polynomial which approximates g with mon(g) monomials. Let b = bs0(g), and z and S 1 , . . . , S b be the input and sets which achieve the 0-block sensitivity of g. We assume without loss of generality that g(z) = 0.We derive a b-variable Boolean function h(y 1 , . . . , y b ) from g(x 1 , . . . , x n ) as follows: if j ∈ S i then we replace x j in g by y i , and if j ∈ S i for any i, then we fix x j in g to the value z j . Note that h satisfies3. mon(h) ≤ mon(g), because we can easily derive an approximating polynomial for h from p, without increasing the number of monomials in p.It follows easily from combining the previous lemmas that any approximating polynomial for h requires at least 2 √ b/12 monomials, which concludes the proof. Quantum dense coding and a lower bound for 1-way quantum finite automata. A Ambainis, A Nayak, A Ta-Shma, U Vazirani, quant-ph/9804043Proceedings of 31st STOC. 31st STOCA. Ambainis, A. Nayak, A. Ta-Shma, and U. Vazirani. Quantum dense coding and a lower bound for 1- way quantum finite automata. 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[ "Multi-scalar theories of gravity with direct matter couplings and their parametrized post-Newtonian parameters", "Multi-scalar theories of gravity with direct matter couplings and their parametrized post-Newtonian parameters" ]
[ "Osmin Lacombe [email protected] \nCenter for Gravitational Physics and Quantum Information\nYukawa Institute for Theoretical Physics\nKyoto University\nSakyo-ku606-8502KyotoJapan\n\nDipartimento di Fisica e Astronomia\nUniversità di Bologna\nvia Irnerio 4640126BolognaItaly\n", "Shinji Mukohyama [email protected] \nCenter for Gravitational Physics and Quantum Information\nYukawa Institute for Theoretical Physics\nKyoto University\nSakyo-ku606-8502KyotoJapan\n\nInstitutes for Advanced Study\nKavli Institute for the Physics and Mathematics of the Universe (WPI)\nThe University of Tokyo\nThe University of Tokyo\n277-8583KashiwaChibaJapan\n" ]
[ "Center for Gravitational Physics and Quantum Information\nYukawa Institute for Theoretical Physics\nKyoto University\nSakyo-ku606-8502KyotoJapan", "Dipartimento di Fisica e Astronomia\nUniversità di Bologna\nvia Irnerio 4640126BolognaItaly", "Center for Gravitational Physics and Quantum Information\nYukawa Institute for Theoretical Physics\nKyoto University\nSakyo-ku606-8502KyotoJapan", "Institutes for Advanced Study\nKavli Institute for the Physics and Mathematics of the Universe (WPI)\nThe University of Tokyo\nThe University of Tokyo\n277-8583KashiwaChibaJapan" ]
[]
We study theories of gravity including, in addition to the metric, several scalar fields in the gravitational sector. The particularity of this work is that we allow for direct couplings between these gravitating scalars and the matter sector, in addition to the universal interactions generated by the Jordan frame metric. The weak gravity regime of this theory, which would describe solar-system experiments, is studied using the parametrized post-Newtonian (PPN) formalism. We derive the expression of the ten parameters of this formalism. They are modified with respect to their values in the theories without direct couplings. We then show that in order to bring the PPN parameters to their general relativity values, relatively large direct couplings are needed, contrary to the claims in the recent literature. Such large couplings, when they exist, make the motion of non-relativistic probes to deviate from geodesics of the PPN metric already at Newtonian order. They would thus be directly detectable and would have been ruled out by experiments. This shows that it is impossible to screen the presence of gravitating scalars relying only on a curved target space and direct couplings to matter.
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[ "https://export.arxiv.org/pdf/2302.08941v1.pdf" ]
257,019,833
2302.08941
dcac3413b20ce0ec2d17b82d7d3c80675e90aa34
Multi-scalar theories of gravity with direct matter couplings and their parametrized post-Newtonian parameters 17 Feb 2023 Osmin Lacombe [email protected] Center for Gravitational Physics and Quantum Information Yukawa Institute for Theoretical Physics Kyoto University Sakyo-ku606-8502KyotoJapan Dipartimento di Fisica e Astronomia Università di Bologna via Irnerio 4640126BolognaItaly Shinji Mukohyama [email protected] Center for Gravitational Physics and Quantum Information Yukawa Institute for Theoretical Physics Kyoto University Sakyo-ku606-8502KyotoJapan Institutes for Advanced Study Kavli Institute for the Physics and Mathematics of the Universe (WPI) The University of Tokyo The University of Tokyo 277-8583KashiwaChibaJapan Multi-scalar theories of gravity with direct matter couplings and their parametrized post-Newtonian parameters 17 Feb 2023 We study theories of gravity including, in addition to the metric, several scalar fields in the gravitational sector. The particularity of this work is that we allow for direct couplings between these gravitating scalars and the matter sector, in addition to the universal interactions generated by the Jordan frame metric. The weak gravity regime of this theory, which would describe solar-system experiments, is studied using the parametrized post-Newtonian (PPN) formalism. We derive the expression of the ten parameters of this formalism. They are modified with respect to their values in the theories without direct couplings. We then show that in order to bring the PPN parameters to their general relativity values, relatively large direct couplings are needed, contrary to the claims in the recent literature. Such large couplings, when they exist, make the motion of non-relativistic probes to deviate from geodesics of the PPN metric already at Newtonian order. They would thus be directly detectable and would have been ruled out by experiments. This shows that it is impossible to screen the presence of gravitating scalars relying only on a curved target space and direct couplings to matter. Introduction Massless or very light gravitating scalar fields naturally arise in various cosmological models motivated by high-energy constructions. In theories with extra dimensions, such as string theory, they are naturally related to the geometry of the extra-dimensional space and are hence generally called geometric moduli, radions or yet dilatons. The historical prototypical example of a theory coupling a dilaton to gravity is the Brans-Dicke theory [1]. In effective theories descending from string theory, these scalar fields are usually associated with scale-invariant structures at the leading order in perturbation theory [2][3][4][5][6] and are thus massless at this level. Upon mechanisms breaking their no-scale structures, such as the inclusion of quantum corrections, the scalars can acquire small masses stabilizing them to certain vacuum expectation values. If such stabilization indeed happened in the far past history of our Universe, at high energy scales, these scalars are not active anymore and should just be included in the effective vacuum energy density. On another side, scalars that remained massless or extremely light today, hence not stabilised and cosmologically active, can have numerous applications in modified theories of gravity used to construct dark energy or dark matter models. For instance, they can lead to equivalent descriptions of higher-order theories of gravity [7] or be quintessence candidates [8][9][10][11][12]. However, cosmologically active (almost-)massless scalars would necessarily mediate fifth forces for matter and if the scalars are universally coupled to matter, these forces would typically be of gravitational strength [13]. These effects would then be accessible in weak-gravity regions and observable in experiments constraining this regime, such as solar-system experiments [14]. The latter indeed highly constrain metric theories of gravity and show that deviations from general relativity have to be extremely small. This in turns highly constrains the strength of universal couplings from gravitating scalars to matter, ruling out their cosmological interest unless some screening mechanism takes place. Numerous modifications of gravity prove successful at cosmological scales but dangerously change physics at shorter length scales, in the Newtonian regime. Such theories cannot thus be reasonably considered valid at any scale. One thus usually invokes screening mechanisms hiding their features, such as the presence of very light scalars, in solar-system experiments. Such mechanisms rely on non-linearities in the potential [15], couplings [16,17] or kinetic terms [18] of the scalar fields. See [19] for a review. In recent papers [20,21], the authors have studied the possibility of a mechanism that could potentially hide the presence of cosmologically active scalars in the weak-field quasi-static regime. The authors rely on direct couplings of scalar fields to matter fields, in addition to the standard universal coupling through the metric. As such couplings violate the equivalence principle, they should be small enough to remain undetectable by experiments testing this principle on Earth. The studies mentioned above argue that even for very small couplings, the weak gravity regime can be modified so to screen the presence of the very light gravitating scalars. The authors have focused particularly on a model containing an axion and a dilaton, with direct coupling between matter and the axion. Motivated by this work, we investigate the weak gravity regime of general multi-scalar theories with the inclusion of direct couplings in addition to the universal ones induced by the Jordan frame metric. We thus study scalar-tensor theories including several massless gravitating scalars, with direct matter couplings. The gravity sector, containing the metric and gravitating scalars with curved scalar target space but no scalar potential, will be coupled to matter through the Jordan frame metric and additional couplings. To study this theory in the weak-field quasistatic regime, we will make use of the parametrized post-Newtonian (PPN) formalism, which is a natural framework to compare theories of gravity in this regime and has been developed gradually throughout the last century building on the early work of Eddington [22][23][24][25][26][27]. The rest of the paper is organised as follows. In Section 2 we present the framework studied in the paper, by presenting the Lagrangian and associated equation of motions of the gravitational theory under study, both in the Jordan and Einstein frames. For concreteness, we show how the framework applies to the simple examples of the Brans-Dicke theory and the axio-dilaton theory motivating this work. We then study in Section 3 the quasi-static weakfield regime of these theories, making use of the parametrized post-Newtonian formalism. We derive the expression for the ten PPN parameters in multi-scalar theories with direct couplings between gravitating scalars and matter fields. We then apply the obtained formulae to the two aforementioned examples. Eventually, we study in Section 4 if direct couplings can change classical tests of theories of gravity giving access to the PPN metric, and thus the PPN parameters. We present in Section 5 a summary of our results and discuss possible future directions along this work. The paper also includes Appendix A which presents definitions and identities related to the target-space functions and PPN functionals used in the main body. 2 Multi-scalar theory of gravity with direct coupling Scalar-tensor theories of gravity have been studied extensively in the literature [7,28]. In the present work we study the case of massless gravitating scalars, hence coupled non-minimally to gravity, in the presence of direct couplings between these scalars and the matter fields. In this section, we first present the action and set notations for the theories we consider in the rest of this work. We show how the presence of direct couplings modify field equations for the gravitational fields and support our discussion by taking two simple examples. Action with direct couplings and equations of motion Jordan frame action We study scalar-tensor theories of gravity containing N several scalar fields ϕ a , a = 1, . . . , N , with non-minimal coupling to gravity defined by a function F and kinetic terms defined on a target space parametrized by a metric G ab . We thus consider the following four-dimensional Jordan frame action S = S g +S m = M 2 2 d 4 x √ −g F (ϕ a )R − G ab (ϕ c )∂ µ ϕ a ∂ ν ϕ b g µν + d 4 x L m (g µν , χ, ϕ a ). (2.1) This action is written using the Jordan frame metric g µν and in the matter Lagrangian L m matter fields are denoted by χ. As motivated in section 1, we allow for additional (weak) couplings between the gravitating scalars and matter, as seen from the ϕ a dependence of L m . The presence of these terms would violate the so-called universal coupling of matter to gravitational fields. In the above action, M is a mass scale. The scalar equations of motion derived from this action read ✷ϕ a + Γ a bc ∂ϕ b ∂ϕ c + 1 2 F a R + 1 2M 2 C a = 0. (2.2) Partial derivatives with Latin indices are taken with respect to scalar fields, ∂ a ≡ ∂/∂ϕ a and the metric G ab (inverse metric G ab ) is used to lower (raise) scalar target-space indices. As in the rest of the paper, we make use of the notations F a ≡ ∂ a F = ∂F ∂ϕ a , F ab ≡ ∂ ab F, F b = G ba F a . (2.3) In eq. (2.2) we used the Christoffel symbol Γ a bc defined as the Levi-Civita connection for the target-space metric Γ a bc = 1 2 G ad (∂ b G cd + ∂ c G bd − ∂ d G bc ),(2.4) and omitted space-time summation in the scalar kinetic terms in eq. (2.2), as will be done in the rest of the paper. One should thus read as usual ∂ϕ b ∂ϕ c ≡ ∂ σ ϕ b ∂ σ ϕ c . We have also introduced the matter-scalar coupling functions, which will be the main ingredients of the present work and are defined through C a ≡ 2 √ −g δS m δϕ a . (2.5) The Einstein equations derived from our action (2.1) read F R µν − 1 2 g µν R = 1 M 2 T µν + G ab ∂ µ ϕ a ∂ ν ϕ b − 1 2 g µν ∂ϕ a ∂ϕ b + F a (∇ µ ∂ ν ϕ a − g µν ✷ϕ a ) + F ab ∂ µ ϕ a ∂ ν ϕ b − g µν ∂ϕ a ∂ϕ b . (2.6) We introduced the usual stress-energy tensor T µν = 2 √ −g δS m δg µν , T = T µν g µν . (2.7) Taking the trace of eq. (2.6) leads to the following expression for the Ricci scalar R = − 1 F M 2 T + 1 F (G ab + 3F ab ) ∂ϕ a ∂ϕ b + 3 F a F ✷ϕ a ,(2.✷ϕ a + 3 2 F a F F b ✷ϕ b + Γ a bc + F a 2F (G bc + 3F bc ) ∂ϕ b ∂ϕ c = 1 2M 2 F a F T − C a , (2.9) F R µν − F a ∇ µ ∂ ν ϕ a + 1 2 g µν ✷ϕ a − 1 2 g µν F ab ∂ϕ a ∂ϕ b − (G ab + F ab ) ∂ µ ϕ a ∂ ν ϕ b = 1 M 2 T µν − 1 2 g µν T . (2.10) We recall that F , F a , F ab , G ab , Γ a bc are target-space functions depending on the scalar fields ϕ a . Einstein frame action One could also study the same theory in the Einstein frame, where there is no prefactor in front of the Ricci scalar. Although the PPN formalism is expressed in the Jordan frame, some theories are naturally obtained and easily interpreted in a frame with the canonical Einstein-Hilbert term. As this is the case for the axio-dilaton example we study later, we now detail the explicit relation between the two formulations. They are equivalent to each other and can be related by using the Einstein frame metric g E defined through g µν = 1 F (ϕ a ) g E µν . (2.11) The action (2.1) can thus be written in the Einstein frame as S = M 2 2 d 4 x −g E R E − G E ab ∂ µ ϕ a ∂ ν ϕ b g E µν + d 4 x L m F −1 g E µν , χ, ϕ a . (2.12) The Einstein frame target-space metric G E ab now includes new contributions coming from the Weyl rescaling of the Ricci scalar, which schematically reads R = F (ϕ a )R E + f (ϕ a , ∂ϕ a ). The relation between the two target-space metrics can be derived by expressing exactly the additional terms of the Lagrangian appearing due to f (ϕ a , ∂ϕ a ). Indeed, we shall have, up to total derivatives: √ −g F R + G ab ∂ µ ϕ a ∂ ν ϕ b g µν = −g E R E + G E ab ∂ µ ϕ a ∂ ν ϕ b g E µν ,(2.13) which, using eq. (2.11) leads to F G E ab − G ab g E µν ∂ µ ϕ a ∂ ν ϕ b = F R E − R = 3 2 F a F b F ∂ µ ϕ a ∂ ν ϕ b g E µν . (2.14) The last equality holds in d = 4 dimensions and up to total derivatives. This leads to the simple relation: G E ab = 1 F G ab + 3 2 F a F b F 2 . (2.15) We again stress that eq. (2.12) is obtained after partial integration. The stress-energy tensor T E µν is defined as in eq. (2.7), obtained by varying the matter action with respect to the Einstein frame metric, reads T E µν = 2 −g E δS m δg E µν = 1 F 3 T µν , T E µν = 1 F T µν , T E = g E µν T E µν = 1 F 2 T. (2.16) The Einstein frame matter-scalar coupling functions C E a also differ from the ones in the Jordan frame, not only due to the √ −g factor in their definition eq. (2.5), but also due to the fact that the universal coupling contains a term in the scalar fields. They indeed read C E a = 2 √ −g E δS m (F −1 g E µν , χ, ϕ a ) δϕ a = √ −g √ −g E C a + 2 √ −g E δS m δg E µν δg E µν δg µν δg µν δϕ a = √ −g √ −g E C a − F a F T E = 1 F 2 C a − F a F T . (2.17) The equations of motion in the Einstein frame can be written in terms of these new quantities as ✷ E ϕ a + Γ E a bc ∂ϕ b ∂ϕ c = − 1 2M 2 C E a , (2.18) R E µν − G E ab ∂ µ ϕ a ∂ ν ϕ b = 1 M 2 T E µν − 1 2 g µν T E . (2.19) They are simply derived from the Einstein frame action (2.12) but can also be induced from eqs. (2.9) and (2.10) making use of relations (2.15) to (2.17). Although these field equations seem simpler than the Jordan frame ones, PPN parameters are computed in this latter frame. This is the case because the Jordan frame metric appears in matter field kinetic terms so that particles follow geodesics of the Jordan frame metric. Particular cases We now apply the above generic notations for multi-scalar gravity to two particular examples, the Brans-Dicke(-like) theory and the axio-dilaton theory. As it contains one single scalar field, the first one is not even a multi-scalar but is nevertheless useful for the rest of the discussion. Brans-Dicke scalar-tensor theory We start with the study of the simple Brans-Dicke theory [1], which contains only one scalar field with kinetic term parametrized by ω. Even though in the original version of the theory a constant ω was studied, we relax this condition as in the extended case studied by [29][30][31]. The theory is thus described by the Jordan frame action S BD = M 2 2 d 4 x √ −g φR − ω(φ) φ (∂φ) 2 + d 4 x L m (g µν , χ). (2.20) This action can be seen as a particular case of eq. (2.1) by taking schematically ϕ a = φ, G ab ∂ϕ a ∂ϕ b = ω(φ) φ (∂φ) 2 , F (ϕ a ) = φ, C a = 0. (2.21) These definitions lead to the following target-space functions F a ≡ ∂ a F = 1, F ab ≡ ∂ ab F = 0, F a = G ab F b = φ ω , Γ a bc = 1 2 φ ω ∂ φ w φ ,(2.22) which, according to eqs. (2.9) and (2.10), give the equations of motion (3 + 2ω)✷φ + 1 2ω dω dφ (∂φ) 2 = T M 2 , (2.23) φR µν − ∇ µ ∂ ν φ − 1 2 g µν ✷φ − w φ ∂ µ φ∂ ν φ = 1 M 2 T µν − 1 2 g µν T . (2.24) The Brans-Dicke theory can also be expressed in the Einstein frame through the Weyl rescaling g µν = 1 F (φ) g E µν = 1 φ g E µν ,(2.25) which allows to rewrite the action (2.20) as S BD = M 2 2 d 4 x −g E R E − 3 2 + w(φ) 1 φ 2 (∂φ) 2 + d 4 x L m 1 φ g E µν , χ . (2.26) In the case of constant w(φ) = w, one obtains the canonical scalar kinetic term by defining a new variable φ E so that (∂φ E ) 2 = 3 2 + ω 1 φ 2 (∂φ) 2 ,(2.27) and thus φ = exp ± 2 3 + 2ω φ E = F. (2.28) The action for this canonical scalar in the Einstein frame hence reduces to S BD = M 2 2 d 4 x −g E R E − (∂φ E ) 2 + L m 1 F (φ) g E µν , χ . (2.29) One can alternatively define Brans-Dicke theory starting from this last action, by choosing the function F (φ(φ E )) = exp(gφ E ). According to eq. (2.28) this corresponds to identifying g 2 = 2 3 + 2ω . (2.30) The action (2.29) leads to the simple equations of motions ✷φ E ± g M 2 T E = 0, (2.31) R E µν − ∂ µ φ E ∂ ν φ E = 1 M 2 T E µν − 1 2 g E µν T E ,(2.32) where T E µν = 1 F T µν is the Einstein frame stress-energy tensor. These are equivalent to eqs. (2.23) and (2.24) with constant w. Axio-dilaton theory We follow [20] and consider the axion-dilaton theory in the Einstein frame as a case with Kähler target-space manifold. It can thus be described in terms of a complex scalar field t = 1 2 (τ + ia) and its complex conjugatet = 1 2 (τ − ia), with only nonvanishing target-space metric and connection components reading G E tt = 3 (t +t) 2 = 3 τ 2 , Γ E t tt = − 2 t +t = − 2 τ , Γ Et tt = Γ E t tt . (2.33) The scalars are coupled to gravity and matter through the defining functions F (t,t) = t +t = τ, C t = Ct = 0, (2.34) so that we simply get ∂tF = 1, ∂ t F = G E tb ∂ b F = (t +t) 2 3 = τ 2 3 , C E t = − F t F T E = − τ 3 T E . (2.35) The Einstein frame action (2.12) thus reads: S = M 2 2 d 4 x −g E R E − 6 τ 2 ∂ µ t ∂ νt g E µν + d 4 x L m g E µν τ , χ, ϕ a = M 2 2 d 4 x −g E R E − 3 2τ 2 (∂ µ τ ∂ ν τ + ∂ µ a ∂ ν a) g E µν + d 4 x L m g E µν τ , χ, ϕ a . (2.36) The scalar field equations (2.18) simply read ✷ E t − 2 t +t ∂t∂t − t +t 6M 2 T E = 0,(2.37) together with its complex conjugate, which contains the same informations. Their real and imaginary parts lead to ✷ E τ − 1 τ ∂τ ∂τ − ∂a∂a − τ 3M 2 T E = 0, (2.38) ✷ E a − 2 τ ∂τ ∂a = 0. (2.39) The theory can also be expressed ignoring the Kähler structure by considering directly the scalar fields τ and a, with target-space functions F (τ, a) = τ, G E τ τ = G E aa = 3 2τ 2 , C τ = C a = 0, Γ E a τ a = − 1 τ , Γ E τ aa = −Γ E τ τ τ = 1 τ , F τ = 1, F a = 0, F τ = 2τ 2 3 , C E τ = − 2τ 3 T E . (2.40) Applying eq. (2.18) with the above functions expressed in terms of the real scalars directly gives back the equations of motion (2.38) and (2.39). Parametrized post-Newtonian formalism and parameters The parametrized post-Newtonian (PPN) formalism was developed to test predictions of metric theories of gravity in the weak-field and slowly-varying regime, hence directly applicable to solar-system experiments. This formalism allows one to compare metrics generated by matter sources in different theories of gravity. To do so, one computes the metrics generated by a perfect fluid matter source, by solving the field equations of the theory at post-Newtonian order. Once the metric is expanded in the parametrized post-Newtonian form, the comparison from one theory of gravity to another is done by looking at the expansion coefficients, called PPN parameters. Metric theories of gravity postulate that matter and non-gravitational fields interact with the space-time metric only, forbidding any direct couplings with the other gravitational fields of the theory (which nevertheless play a role in the production of the metric). The goal of the present work is specifically to study the consequences of a (small) violation of this assumption, by introducing direct couplings between matter and gravitating scalar fields. One might wonder if this rules out the use of the PPN formalism to study the quasi-stationary weak-field regime. In the case where these direct couplings are strong, one certainly expects that matter dynamics would be ruled by interactions with the additional gravitating fields, rather than by the metric and hence the space-time geometry. Nevertheless, if these couplings are small enough, this shall not be the case, and matter could only be sensitive to the metric. In this latter case, it is thus of great interest to see how the presence of direct couplings between matter and additional gravitational fields affect the metric generated by localized matter sources. The PPN formalism is thus the ideal framework to study this question. The goal of this section is to compute the PPN parameters in the presence of direct couplings between gravitating scalars and matter fields and investigate how they differ from the ones obtained without direct couplings. We will use the formalism and conventions exposed in [32]. Parametrized post-Newtonian expansion of fields The PPN formalism studies an isolated post-Newtonian system in a homogeneous isotropic expanding universe. Matter sources of the system are modeled as perfect fluids. Each fluid element of speed v is made of matter with rest-mass density ρ, under pressure p. To apply the formalism, we will first perform the post-Newtonian expansion of the various fields of the theory. This amounts to expand the different gravitating fields (metric, scalar fields) generated by the matter source, in terms of the parameter: ǫ ≡ v c . (3.1) Here v 2 = v i v i is the velocity of the fluid in the local quasi-Cartesian coordinates, also called standard post-Newtonian coordinates, which are asymptotically Minkowski. This coordinate system is chosen so that regions far from the isolated PN system are free falling in the cosmological model and at rest with respect to the Universe rest frame. One should thus expect a FLRW asymptotic form for the metric. However, as the time scale of the expansion of the Universe is today way smaller than the solar-system times scales, one can always use coordinates which are asymptotically Minkowski during the period of the experiment. We will thus generically denote the metric as: g µν ≡ η µν + h µν = η µν + O(ǫ 2 ), (3.2) where η µν is the Minkowski metric. PPN parameters and coefficients The idea of the PPN formalism is to expand the fields of the theory in terms of functionals of the matter source, hence constructed from ρ( x µ ), p(x µ ) or v(x µ ), | x − x ′ |. . . There is a priori an infinity of possible functionals even at first PN order, hence at the ǫ 2 or ǫ 4 order. Nevertheless, the PPN formalism restricts the choice of functionals. They should indeed satisfy various obvious conditions: they should vanish far from the source, they should have appropriate Lorentz transformations, they should not have reference to any preferred spatio-temporal origins. The PPN formalism moreover imposes slightly more arbitrary conditions: these functionals should not involve gradients of the matter sources functions and they should be rather simple [32]. The appropriate functionals are called PPN potentials or PPN functionals. Definitions and relations between some of these functionals are given in appendix A.2. The metric should thus be expanded at first PN order in terms of the Newtonian potential U , post-Newtonian potentials Φ W , Φ 1 , Φ 2 , Φ 3 , Φ 4 , A, B and functionals V i , W i . Once put in the so-called post-Newtonian gauge, where the spatial components of the metric are diagonal and the temporal component does not contain the B functional 1 , it reads: g 00 = −1 + 2GU − 2βU 2 − 2ξΦ W + (2γ + 2 + α 3 + ζ 1 − 2ξ)Φ 1 + 2(3γ − 2β + 1 + ζ 2 + ξ)Φ 2 + 2(1 + ζ 3 )Φ 3 + 2(3γ + 3ζ 4 − 2ξ)Φ 4 − (ζ 1 − 2ξ)A, (3.3) g 0j = − 1 2 (4γ + 3 + α 1 − α 2 + ζ 1 − 2ξ)V j − 1 2 (1 + α 2 − ζ 1 + 2ξ)W j , (3.4) g jk = (1 + 2γU )δ jk . (3.5) We see that the component of the post-Newtonian metric depends only on the value of the Newton constant G and ten parameters γ, β, ξ, α 1 , α 2 , α 3 , ζ 1 , ζ 2 , ζ 3 , ζ 4 . This is why this formalism is called parametrized post-Newtonian (PPN) formalism and the parameters are called PPN parameters. Of course, the expansion of eqs. (3.3) to (3.5) directly gives the corresponding one for h µν defined in eq. (3.2). The Newton constant G is related, in a given theory, to a mass scale of the theory, e.g. M in eq. (2.1), through the field equations. One often chooses the units with M = 1 or the ones with G = 1. This last condition can be imposed today through the cosmological matching conditions, explained below. Indeed, it will in general depend on the values of the fields far from the PN source, as can be seen from eq. (3.24) below. The γ and β parameters are already included in the Eddington-Robertson-Schiff (ERS) formalism [22][23][24], while the other ones were defined in the full PPN formalism developed in [25][26][27]. The parameter γ evaluates the quantity of space-time curvature produced by rest masses while β accounts for non-linearities present in the superposition law for gravity. In general relativity, both take unit values γ GR = β GR = 1. (3.6) The other PPN parameters also have a meaning in the standard PPN gauge. The parameter ξ evaluates preferred-location effects, α i evaluates preferred-frame effects, while α 3 and ζ i indicate violations of conservation of total momentum. For a detailed discussion, see [32] and the summary of Table 4.3 therein. In our framework, we expect in advance non-vanishing values for the PPN parameters indicating violations of momentum conservation. Indeed, as will be explained around eq. (4.4), the presence of direct couplings generically introduces terms in the divergence of the stress-energy tensor. As for the metric, one shall also expand the gravitating scalar fields in terms of the ǫ order parameter: ϕ a = ϕ a 0 + ϕ a 2 + ϕ a 4 + O(ǫ 6 ), ϕ a 0 = cst, ϕ a 2 = O(ǫ 2 ), ϕ a 4 = O(ǫ 4 ). (3.7) The ϕ a 0 , ϕ a 2 and ϕ a 4 are respectively of order ǫ 0 , ǫ 2 and ǫ 4 . The latter are thus expanded on the PN potentials as ϕ a 2 = 2γ ϕ a U, (3.8) ϕ a 4 = C a U U U 2 + C a W Φ W + C a 1 Φ 1 + C a 2 Φ 2 + C a 3 Φ 3 + C a 4 Φ 4 + C a A A + C a B B. (3.9) Cosmological matching conditions The order ǫ 0 fields are determined by the cosmological boundary conditions. Indeed, as the PN functionals vanish when the distance r from the source goes to large values, the asymptotic values of the scalars are given by ϕ a (r → ∞) = ϕ a 0 . The constants ϕ a 0 shall thus be determined by the surrounding cosmological model, independent of the PN system, and will constitute cosmological matching conditions. We also recall that, as explained below eq. (3.1), the coordinate system is also chosen such that the metric approaches a Minkowski metric far from the source. In an expanding Universe this simply assumes that the typical time of gravitational experiments is small compared to cosmological times. Matter PN expansion Finally, the matter sector should also be PN expanded. As already stated, one considers the perfect fluid approximation for the matter source, the stress-energy tensor of which thus reads: T µν = (ρ + ρΠ + p)u µ u ν + pg µν , (3.10) where again, ρ and p are is the rest-mass energy density and pressure, ρΠ the internal energy density, and u µ = dx µ /dτ the 4-velocity of the fluid elements normalised so that u µ u µ = −1. In solar systems, velocities of gravitating bodies are related to the Newtonian potential U by virial relations. What's more, their pressures p and internal energies ρΠ are smaller than their potential energy ρU . The PN orders of the various functions defining the source are thus: U ∼ ρ ∼ v 2 = O(ǫ 2 ), p ρU = O(ǫ 4 ), Π U = O(ǫ 2 ), u i = dt/dτ dx i /dt = u 0 v i = O(ǫ). (3.11) In the following perturbative study, we will use the fact that, in the quasi-static regime, temporal derivatives are smaller than gradients. This amounts to consider the relation ∂ ∂t ∼ v · ∇ ∼ O(ǫ) × ∂ ∂x i ,(3.12) when evaluating the PN order of a expression. As can be seen from eqs. (3.3) to (3.5), at lowest orders in ǫ the metric takes the form: g 00 = −1 + 2GU + O(ǫ 4 ), g 0j = 0 + O(ǫ 4 ), g ij = δ ij (1 + 2γU ) + O(ǫ 4 ). (3.13) Using the normalisation condition for u µ , the stress-energy tensor can thus be expanded to O(ǫ 4 ) as T 00 = ρ + ρΠ + ρv 2 + 2GρU + O(ǫ 6 ), T ij = ρv i v j + pδ ij + O(ǫ 6 ). (3.14) The coupling functions C a must also be specified, for instance through their computation from a concrete matter Lagrangian L m , as well as PN expanded. Following the perfect fluid form for the stress-energy tensor, we shall simply expand them in the following way: C a = c a ρ + c a Π ρΠ + c a v ρv 2 + c a U ρU + c a p p + O(ǫ 6 ). (3.15) PPN expanded field equations Once every field has been PN expanded as in the previous section, the PPN expansion coefficients and the PPN parameters are to be computed through the field equations of the theory. They will be related to the target space functions F , G ab and their derivatives. In order to solve perturbatively the field equations (2.9) and (2.10), we must thus PN expand them keeping track of the ǫ order of each term. To do so, we thus expand the target-space functions, giving for instance: F (ϕ a ) = F (ϕ a 0 ) + F b (ϕ a 0 )ϕ b 2 + O(ǫ 4 ), F a (ϕ c ) = F a (ϕ c 0 ) + F ab (ϕ c 0 )ϕ b 2 + O(ǫ 4 ). (3.16) as well as operators depending on the metric, such as the D'Alembertian operator ✷ or the (space-time) Levi-Civita connection: ✷ϕ a | 2 = g µν 0 ∂ µ ∂ ν ϕ a 2 , ✷ϕ a | 4 = (g µν ∇ µ ∂ ν ϕ a )| 4 = g µν 2 (∇ µ ∂ ν ϕ a )| 2 + g µν 0 (∇ µ ∂ ν ϕ a )| 4 = g µν 2 ∂ µ ∂ ν ϕ a 2 + g µν 0 ∂ µ ∂ ν ϕ a 4 − g µν 0 (Γ α µν ) 2 ∂ α ϕ a 2 , (3.17) (Γ α µν ) 2 = 1 2 g ασ 0 (∂ µ g 2σν + ∂ ν g 2σµ − ∂ σ g 2µν ). (3.18) The scalar equations (2.9) are thus expanded as O(ǫ 2 ) : ✷ϕ a | 2 + 3 2 A a b ✷ϕ b | 2 = 1 2M 2 (B a T 2 − C a 2 ) , (3.19) O(ǫ 4 ) : ✷ϕ a | 4 + 3 2 A a b ✷ϕ b | 4 + C a bc ∂ µ ϕ b 2 ∂ ν ϕ c 2 g µν 0 + ∂ c A a b ϕ c 2 ✷ϕ b | 2 = ∂ b B a 2M 2 ϕ b 2 T 2 + 1 2M 2 (B a T 4 − C a 4 ), (3.20) where we defined A a b ≡ F a F b F = B a F b , B a ≡ F a F , C a bc = Γ a bc + B a 2 (G bc + 3F bc ). (3.21) When not specified, as in eqs. (3.19) and (3.20), the above target-space functions and their derivatives are evaluated at ϕ a 0 , being understood that the PN expansion of their variables has already been performed. The trace-reversed Einstein equation (2.10) is expanded as O(ǫ 2 ) : F R 2µν − F a ∂ µ ∂ ν ϕ a 2 + 1 2 g 0µν ✷ϕ a | 2 = 1 M 2 T 2µν − 1 2 g 0µν T 2 , (3.22) O(ǫ 4 ) : F R 4µν + F a ϕ a 2 R 2µν − F ab ϕ b 2 ∂ µ ∂ ν ϕ a 2 − 1 2 g 0µν ✷ϕ a | 2 − F a ∇ µ ∂ ν ϕ a | 4 + 1 2 g 2µν ✷ϕ a | 2 + 1 2 g 0µν ✷ϕ a | 4 − 1 2 g 0µν F ab ∂ α ϕ a 2 ∂ β ϕ b 2 g αβ 0 − G ab + F ab ∂ µ ϕ a 2 ∂ ν ϕ b 2 = 1 M 2 T 4µν − 1 2 g 2µν T 2 − 1 2 g 0µν T 4 . (3.23) Determining the PPN parameters: procedure and results The metric PPN parameters and the PPN coefficients of the various gravitational fields can be obtained by solving the field equations in a systematic way. We follow a procedure inspired by [32], that we summarize hereafter. The initial steps of the standard procedure of [32], corresponding to the determination of the gravitating and matter variables, the definition of the cosmological matching conditions and the PN expansion of the fields and equations of motion, have been explicitly showed earlier. Our procedure thus follows the successive steps: Step 1: Solve for h 00 and ϕ a 2 at order O(ǫ 2 ) using the O(ǫ 2 ) field equations. According to the expansions given in eqs. (3.3), (3.7) and (3.8), this will thus determine the value of G and γ ϕ a in terms of the cosmological matching parameters. Step 2: Solve for h ij at O(ǫ 2 ). As can be seen from eq. (3.5), this immediately gives the value of the γ parameter. Step 3: Solve for h 0i at O(ǫ 3 ). From eq. (3.4), we see that, comparing the V i and W i parameters in h 0i , one obtains the value of α 1 . Step 4: Solve for ϕ a 4 using the scalar field equations eq. (3.20). According to eqs. (3.7) and (3.9), this will give the C a U U , C a W , C a i , C a A , C a B coefficients in terms of the scalar targetspace functions and the expansion coefficients of the coupling C a shown in eq. (3.15). Step 4: Solve for h 00 at O(ǫ 4 ) using the field equations at order O(ǫ 4 ). This is the most computationally expansive step. It leads to the obtention of the remaining PPN coefficients, namely α 2 , α 3 , α 4 , β, ξ, ζ 1 , ζ 2 , ζ 3 . In order to implement the above procedure, we transform the differential equations for the fields into algebraic equations for the PPN coefficients. To this end, we first plug the PPN expansions for the various fields into the field equations, then use the knowledge of specific relations between the derivatives of the PN potentials. Some of the latter are given in appendix A.2. In this manner, solving for the various fields at a certain PN order O(ǫ n ) turns into solving the algebraic equations in terms of the PPN coefficients playing a role at this order. We obtained, step by step, the following PPN parameters in terms of the target space functions: Step 1: G = (4B c + c c )F c + 2 F (3B c F c + 2) , (3.24) γ ϕ a = F 2 × 2(B a + c a ) − 3(B a c c − B c c a )F c (4B c + c c )F c + 2 = 1 2G 2(B a + c a ) − 3(B a c c − B c c a )F c 3B c F c + 2 . (3.25) Step 2: γ = (2B c − c c )F c + 2 (4B c + c c )F c + 2 = 1 − 2(B c + c c )F c (4B c + c c )F c + 2 . (3.26) Step 3: α 1 = 0. (3.27) Step 4 & 5: α 2 = 0, α 3 = 0, ξ = 0, ζ 1 = 0, (3.28) ζ 2 = 24(B a c c − B c c a )F c F bF ab (3B c F c + 2)(4B c F c + c c F c + 2) 2 + 2(B a + c a ) − 3(B a c c − B c c a )F c (F b F b c a + F F b ∇ a c b − F c bF ab ) (4B c F c + c c F c + 2) 2 , (3.29) ζ 3 = (c c Π − c c )F c 4B c F c + c c F c + 2 , ζ 4 = ( 1 3 c c p + c c )F c 4B c F c + c c F c + 2 , (3.30) β = 1 + (B c +c c )(B d +c d )(F c F d − 2FF cd ) (3B c F c + 2)(4B c F c + c c F c + 2) 2 ,(3.31) where in the expression for β and ζ 2 , we introduced the target space covariant tensor F cd ≡ ∇ c F d = ∇ c ∂ d F = F cd − Γ e cd F e .(3.32) In the above expressions, in which we set M ≡ 1, all the functions F c , B c , F , c c are evaluated at the cosmological background values ϕ a 0 for the gravitating scalars. As mentioned in the text below eq. (3.5) we thus see from eq. (3.24) that the value for the Newton constant G depends on the values of the cosmological fields ϕ a 0 . The above expressions are thus the general expressions for the ten PPN parameters in multi-scalar theories of gravity with direct coupling. The can be used for any theory once put in the form of eq. (2.1). For multi-scalar theories of gravity without direct coupling, the β and γ parameters were already computed in [28], although with different notations. One can see from the above formulae that in the special case where B c = −c c , the PPN parameters are identical to those of general relativity. This is not a surprise, as going back to the O(ǫ 2 ) order expansion eq. (3.19) of the scalars field equations eq. (2.9), we see that this corresponds to the case where the scalars decouple. This can also be seen easily from the computation (2.17) of the direct couplings in the Einstein frame. In other cases, the PPN parameters are different. For small direct couplings, i.e. for c a ≪ B a , they differ only very slightly from the PPN parameters obtained in the corresponding theory without direct couplings. This seems to go against the claim made in [20] for the axiodilaton theory. We come back to this specific case below. The PPN parameters for particular cases We now apply the above results for the PPN parameters to the two examples presented in section 2.2. The first example can be seen as a consistency check of our formulae. On the other hand, the second example is one of the motivations of this work. Brans-Dicke theory The Brans-Dicke theories presented in section 2.2 correspond to the target-space functions: F (ϕ a ) = φ, C a = 0, F a = 1, F ab = ∂ ab F = 0, F a = G ab F b = φ ω , (3.33) F ab = −Γ c ab F c = 1 2 1 φ − ω ′ ω . (3.34) We thus deduce the following expressions for the relevant terms B a = F a F = 1 ω , B c F c = 1 ω , 2B c B d FF cd = ω −2 1 − φ ω ′ ω (3.35) appearing in expressions (3.24) to (3.31) for the PPN parameters. The non-vanishing PPN parameters thus read γ φ = F 2 × B a 2B c F c + 1 = φ 0 2(ω + 2) , G = 4B c F c + 2 F (3B c F c + 2) = 4 + 2ω φ 0 (3 + 2ω) , (3.36) γ = B c F c + 1 2B c F c + 1 = 1 + ω 2 + ω , β − 1 = B c B d (F c F d − 2FF cd ) 4(3B c F c + 2)(2B c F c + 1) 2 = φ 0 ω ′ 4(3 + 2ω)(2 + ω) 2 . (3.37) As should be clear from the discussion on the PPN formalism, in these results the values for the target-space functions are evaluated for the cosmological background. One should thus read ω = ω(φ 0 ) and ω ′ = ω ′ (φ 0 ). From the standard results above, we see that for a given function ω(φ), namely a given Brans-Dicke theory, constraints on the PPN parameters directly fix the maximal possible strength of coupling between matter and the scalar field φ 0 . Axio-dilaton theory This case was defined in eq. (2.33) in the Einstein frame following the study of [20]. The PPN parameters being computed in the Jordan frame, one has to go from the target-space metric in the Einstein frame to the one in the Jordan frame in order to apply the above formulas. Inverting eq. (2.15) the Jordan frame target-space metric is expressed through the Einstein frame one by: G ab = F G E ab − 3 2 F a F b F . (3.38) Note that when going to Jordan frame, the Kähler structure might not be preserved. Hence it is necessary to work with real scalars. We thus deduce the Jordan frame target-space functions of the axio-dilaton case from eq. (2.40): F (τ, a) = τ F τ = 1, F a = 0, (3.39) G τ τ = τ G E τ τ − 3 2τ = 0, G aa = τ G E aa = 3 2τ . (3.40) Here, the subscript a refers to the axion field a(x µ ), as shall be clear to the reader. From eq. (3.40) we see that this case is slightly degenerate because of the vanishing of the dilaton part of the target-space metric, i.e the absence of dilaton kinetic term in the Jordan frame. The action (2.36) is indeed written in the Jordan frame as S = M 2 2 d 4 x √ −g τ R − 3 2τ ∂ µ a∂ µ a + d 4 x L m (g µν , χ),(3.41) the dilaton part of which corresponds to a Brans-Dicke scalar with vanishing function ω(τ ) = 0. The PPN parameters can be obtained by taking formally the limit B τ = G τ τ F τ → ∞ in eq. (3.24) to (3.31): γ = 1 2 , β = 1, α 1 = α 2 = α 3 = ξ = ζ 1 = ζ 2 = ζ 3 = ζ 4 = 0. (3.42) This result naturally agrees with the Brans-Dicke case with ω = 0. The degeneracy of the Jordan frame action (3.41) comes from the fact that the Einstein frame kinetic terms for the dilaton (and axion) are exactly the ones generated by the change in Ricci scalar under the Weyl transformation. Note that vanishing kinetic terms in the Jordan frame are harmless, as opposite to vanishing kinetic terms in the Einstein frame, which reveal strongly coupled dynamics. In order to obtain non-degenerate Jordan target-space functions, one could nevertheless consider slightly different normalization of the Einstein frame kinetic terms, while keeping the same gravity coupling function F (τ, a) = τ . For instance, one might parametrize kinetic terms for the dilaton τ through a small parameter ǫ τ , by adding to the action (3.41) a term of the form S ǫτ = M 2 2 d 4 x √ −g ǫ τ 2τ ∂ µ τ ∂ µ τ . (3.43) The target-space metric (3.40) would then contain terms proportional to ǫ τ , in particular a non-vanishing G τ τ . By first applying the PPN formulae (3.24) to (3.31), then taking the limit ǫ τ → 0, one obtains back the results of eq. (3.42). Axio-dilaton with direct coupling The authors of [20] introduced a direct coupling to matter for the axion field a(x µ ). It is defined in the Einstein frame as A ≡ 2 −g E δS m δa . (3.44) The coupling A should agree with the Einstein frame coupling C E a . The latter were obtained through eq. (2.17) from C a , the Jordan frame couplings defined in eq. (2.5) . According to eq. (3.39), in the present case we have F a = 0 for the axion, so that A is here identical to the Einstein frame coupling A ≡ 2 −g E δS m δa = √ −g −g E C a = C E a ,(3.45) as should be. The last equality is obtained by applying eq. (2.17) with ϕ a = a being the axion field. In order to evaluate the expressions for PPN coefficients derived in section 3.3, we have to identify the Jordan frame couplings C c and their PPN expansion. They would thus read C a = −g E √ −g A = τ 2 A, C τ = 0 C a = G aa C a = 2τ 3 3 A, C τ = 0. (3.46) Following the initial work [20], we consider the simplest coupling expansion for non-relativistic sources, taking the form A = ǫ cpl. ρ, All the other coefficients of the PN expansion eq. (3.15), such as c a Π or c a p , vanish in this case. In eq. (3.48), we should obtain c τ by raising the index with the inverse metric G τ τ . Whereas the latter is ill defined, as G τ τ vanishes, it is still natural to take c τ = 0. To be convinced, one could go to the non-degenerate case by adding the term ǫ τ term of eq. (3.43) to the action. One would then obtain a non-vanishing target-space metric element G τ τ , finite G τ τ , and indeed get c τ = 0. Now that we identified the coupling functions and their expansion parameters, we can use the expression found in section 3.3 for the PPN parameters in the presence of direct couplings. From eq. (3.26) it appears that the γ parameter is unchanged compared to the case without the direct couplings, as c c F c = c a F a + c τ F τ = 0. Moreover, the β parameter is also unchanged, because the direct couplings c c have no effect when taking the limit B c F c → ∞. We thus obtain the same PPN parameters as in the case without the direct axion coupling: γ = 1 2 , β = 1, α 1 = α 2 = α 3 = ξ = ζ 1 = ζ 2 = ζ 3 = ζ 4 = 0. (3.49) More generally, we remark that when the function F ({ϕ a }) and direct couplings C b depend on sets of scalar fields {ϕ a } and {ϕ b } containing no common element, the parameter γ is unaffected by the direct couplings since c c F c = 0. This is a priori not the case for the parameter β since there is a term inF ab containing the connection term Γ e ab F e which can mix the indices. Remark on the computation of [20] By computing the first terms of the metric expansion in a spherically symmetric case, the authors of this previous work derived formulae for PPN parameters, valid in the limit ǫ cpl. → 0 where the axion direct coupling (3.47) is small. According to (3.46) and (3.47) of [20] they read [20]), the authors of [20] give the following relations: γ = 3 − ǫ cpl. β tanh δ 3 + ǫ cpl. β tanh δ , β = 1 + ǫ 2 cpl. β 2 9(cosh δ + 1 3 ǫ cpl. β sinh δ) 2 ,γ b.c. = −2ǫ cpl. GM 3 , α γ b.c. ≈ −2GM 3 , (3.51) a ∞ = α − β tanh δ, τ ∞ = β cosh δ . (3.52) From eq. (3.51) we directly deduce that α = 1 ǫ cpl. . (3.53) The values of the axion and dilaton profiles far from the source, i.e. a ∞ and τ ∞ , have to be finite and are fixed by the cosmological matching conditions even in the ǫ cpl. → 0 limit. One can thus rewrite the first equality in (3.52) as β tanh δ = 1 ǫ cpl. − a ∞ , (3.54) so that the two PPN parameters of eq. (3.50) can be expanded γ = 2 − ǫ cpl. a ∞ 4 + ǫ cpl. a ∞ = 1 2 + O(ǫ cpl. ), (3.55) β − 1 = ǫ 2 cpl. β 2 9 cosh 2 δ(1 + 1 3 ǫ cpl. β tanh δ) 2 = ǫ 2 cpl. β 2 cosh 2 δ(4 − ǫ cpl. a ∞ ) 2 = ǫ 2 cpl. τ 2 ∞ (4 − ǫ cpl. a ∞ ) 2 = O(ǫ 2 cpl. ). (3.56) We see that in the limit ǫ cpl. → 0 of validity of these expressions, they simply lead to the Brans-Dicke PPN parameters with ω(τ ) = 0, as we found in eq. (3.49). In a more recent work [21] the same authors study the possibility to evade this fact by considering non-linear coupling for the axion, hence relaxing the condition eq. (3.47). Observational constraints on the PPN parameters In this section, we tackle the question of classical constraints on the PPN parameters in the presence of direct couplings. In order to obtain the equations for classical tests of gravity, and thus constrain the PPN parameters, the full approach should study the post-Newtonian equations of motion for a system of massive bodies constituted of a PN source and a probe, such as the Sun and a planet. In general, this must be done by treating the massive bodies as gravitating clusters of massive particles and obtaining the equations of motion for their centers of mass. This study is not straightforward in the presence of direct couplings, as the definition of mass densities is not obvious in the PPN formalism. Indeed, neither the restmass density nor the mass-energy density are conserved. However, by averaging on internal dynamical timescales, which is justified by the fact that the time scales of the changes on the internal structures of the Sun and planets are way shorter than the typical orbital times, the final equations of motions can be obtained without such precise considerations [32]. The obtention of the equation for the acceleration of massive bodies, depending on the ten PPN parameters, is nevertheless not immediate. In the present case, the intermediate steps using continuity equations and conservation law integrals should take into account the presence of direct couplings. Another approach is to consider the PPN metric generated by the PN source and looking at the test probe simply as a massive point particle. In this approach, the massive source and massive probe are treated differently in general. For instance, while a point particle is structureless, some of the post-Newtonian gravitational effects included in the PPN metric can be generated from rotation or non-sphericity of the source. In section 4.1 we evaluate the possible modifications from geodesic motion of a test point particles, due to the presence of direct couplings responsible for additional forces. If one simply wishes to access the γ and β parameters, it can be sufficient to obtain the equation of motions of point particles in the presence of a background generated by a spherical static matter source, as in the formalism developed by Eddington, Robertson and Schiff [22][23][24]. In section 4.2 we show that one can access the γ and β parameters through the classical experiments on massive bodies, such as the measurement of the perihelion shift of Mercury, even in the presence of small direct couplings. In fact, these experiments can be seen as constraints on the strength of these couplings. Finally, in section 4.3 we motivate that photon dynamics also allow to access the γ parameter, even in the presence of direct couplings, through classical tests measuring the time delay or deviation of light. Direct coupling in the point-particle Lagrangian Divergence of the stress-energy tensor in field theory We shall see below that the forces exerted on point particles are related to the divergence of the stress-energy tensor. Towards this goal, as a warm up, let us briefly review a well-known identity involving the divergence of the stress-energy tensor in field theory. For this purpose, we consider a diffeomorphisminvariant action of the form S m [χ(x), ϕ a (x), g µν (x)], where χ(x), ϕ a (x) and g µν (x) represent the matter fields, the gravitational scalars and the metric, respectively. Under a generic change of coordinates x µ → x µ + ξ µ , they transform as follows: g µν → g µν − ∇ µ ξ ν − ∇ ν ξ µ , ϕ a → ϕ a − ξ µ ∂ µ ϕ a , χ → χ − L ξ χ = χ − ξ µ ∇ µ χ + d ν µ ∇ ν ξ µ . (4.1) The first two transformations are imposed by the transformation properties of the metric and the scalars while the last one is kept generic. The vanishing (up to a total derivative) of the variation of the matter action under such a diffeomorphism leads to the usual identity involving the divergence of the stress-energy tensor. It reads: ∇ ν T ν µ = 1 2 C a ∂ µ ϕ a + 1 2 E χ ∇ µ χ + 1 2 ∇ ν (d ν µ E χ ),(4.2) where we used the definition given in eq. (2.5) and E χ ≡ 2 √ −g δS m δχ . (4.3) When C a = 0, the right-hand side of eq. (4.2) vanishes on-shell, i.e. upon using the equation of motion E χ = 0, leading to the conservation of the stress-energy tensor. In our case, only the last two terms vanish once we incorporate the matter equations of motion E χ = 0. Hence, the stress-energy tensor equation (4.2) reduces to: ∇ ν T ν µ = 1 2 C a ∂ µ ϕ a . (4.4) Divergence of the point particle stress-energy tensor Let us now derive an identity similar to (4.2) for a point particle action of the form S P [x µ P (λ), ϕ a (x), g µν (x)], where x µ P (λ) denotes the worldline of the particle parametrized by the parameter λ, ϕ a (x) the gravitational scalars and g µν (x) the metric. Under the infinitesimal diffeomorphism transformation x µ → x µ + ξ µ (x) we have, in addition to the first two transformations rules of (4.1), the following one for the x µ P (λ) worldline: x µ P (λ) → x µ P (λ) + ξ µ (x P (λ)). (4.5) Hence, the action varies under an infinitesimal diffeomorphism as δS P = dλ δS P δx µ P (λ) ξ µ (x P (λ)) + d 4 x √ −g − 1 2 C a ∂ µ ϕ a + ∇ ν T µν ξ µ . (4.6) By requiring the diffeomorphism invariance of the action, i.e. by setting this variation to zero for arbitrary ξ µ (x) parameter, one obtains the following identity ∇ ν T µν = 1 2 C a ∂ µ ϕ a − dλ δS P δx µ P (λ) δ 4 (x − x P (λ)) √ −g . (4.7) Upon using the equation of motion δS P /δx µ P (λ) = 0 for x µ P (λ), we again obtain ∇ ν T ν µ = 1 2 C a ∂ µ ϕ a . (4.8) We can find another identity from the invariance of the action S P [x µ P (λ), ϕ a (x), g µν (x)] under reparametrizations of the worldline parameter λ → λ + ζ(λ). The functions x µ P (λ) transform as x µ P (λ) → x µ P (λ) − ζ(λ)e µ P (λ),(4.9) where e µ P is defined as the first worldline derivative e µ P (λ) ≡ dx µ P (λ) dλ . (4.10) The action thus transforms under worldline reparametrization eq. (4.9) as δS P = − dλ δS P δx µ P (λ) e µ P (λ)ζ(λ). (4.11) By requiring this expression to vanish for arbitrary ζ(λ), one obtains δS P δx µ P (λ) e µ P (λ) = 0, (4.12) as an identity holding even without considering the particle equation of motion. Variations of the point particle action For simplicity, we now assume that the point particle action depends on the particle position x µ P (λ) only through its first derivative e µ P (λ) ≡ dx µ P (λ)/dλ so that the action is invariant under a constant shift of x µ P (λ) and that the corresponding equations of motion are second order. We also assume that the action does not depend on second or higher derivatives of the gravitational scalars ϕ a (x) and any derivatives of the metric g µν (x). In this case, the action is of the form (4.13) so that one can easily compute its variations as follows: S P [x µ P (λ), ϕ a (x), g µν (x)] = d 4 x dλ L e µ P (λ), ϕ a (x), ∂ µ ϕ a (x), g µν (x) δ 4 (x − x P (λ)),δS P δx µ P (λ) = − d dλ ∂L ∂e µ P x P (λ) + ∂L ∂ϕ a ∂ µ ϕ a x P (λ) + ∂L ∂(∂ ν ϕ a ) ∂ µ ∂ ν ϕ a x P (λ) + ∂L ∂g µν ∂ µ g µν x P (λ) , C a (x) = 2 √ −g δS P δϕ a (x) = 2 dλ ∂L ∂ϕ a − ∂ µ ∂L ∂(∂ µ ϕ a ) δ 4 (x − x P (λ)) √ −g , T µν (x) = 2 √ −g δS P δg µν (x) = 2 dλ ∂L ∂g µν δ 4 (x − x P (λ)) √ −g . (4.14) One can easily see that C a and T µν are covariant. On the other hand, it is less obvious that δS P /δx µ P (λ) is also covariant. In order to show the covariance of δS P /δx µ P (λ) explicitly, we define the quantities X ab (x) ≡ g µν (x)∂ µ ϕ a (x)∂ ν ϕ b (x), Y a (x, λ) ≡ e µ P (λ)∂ µ ϕ a (x) , Z(x, λ) ≡ g µν (x)e µ P (λ)e ν P (λ), (4.15) and use them to rewrite L as L = L(X ab (x), Y a (x, λ), Z(x, λ), ϕ a (x)). (4.16) It is then straighforward to show that δS P δx µ P (λ) = ∂L ∂X ab ∂ µ X ab x P (λ) − d dλ ∂L ∂Ȳ a ∂ µ ϕ a | x P (λ) −2 d dλ ∂L ∂Z e P µ − 2 ∂L ∂Z De µ Dλ + ∂L ∂φ a ∂ µ ϕ a | x P (λ) ,(4.17) where the overlines denote quantities evaluated on the particle worldline, namelȳ X ab ≡ X ab (x P (λ)),Ȳ a ≡ Y a (x P (λ), λ),Z ≡ Z(x P (λ), λ),φ a ≡ ϕ a (x P (λ)), (4.18) L ≡ L(X ab ,Ȳ a ,Z,φ a ). (4.19) Finally the down index vector e P µ and its derivative are naturally expressed from e µ P of eq. (4.10) through e P µ ≡ g µν (x P (λ))e ν P (λ), De P µ Dλ ≡ de P µ dλ − Γ ρ µν e ν P e P ρ ,(4.20) where Γ ρ µν are the space-time Christoffel symbols. The variation (4.17) is now manifestly covariant. Massive point particle As a concrete point particle action that respects the diffeomorphism and reparametrization invariance, let us consider the following action with a field dependent mass and a Lorentz-type coupling: S P = d 4 x dλ −m(ϕ) −g µν dx µ P dλ dx ν P dλ + h a (ϕ) dx µ P dλ ∂ µ ϕ a δ 4 (x − x P (λ))dλ,(4.21) where m(ϕ) and h a (ϕ) are functions of the gravitational scalars ϕ = {ϕ a }. This corresponds tō L = −m(φ) −Z + h a (φ)Ȳ a ,(4.22) leading to the equations of motion for x µ P (λ) of the form 1 √ −Z D Dλ m(φ)e P µ √ −Z = F µ ,(4.23) where F µ ≡ − ∂m(φ) ∂φ a + ∂h b (φ) ∂φ a − ∂h a (φ) ∂φ b Ȳ b √ −Z ∂ µ ϕ a | x P (λ) . (4.24) This equation is a extended version of the one obtained in [34,35] in the case of a varying mass only. On defining the 4-velocity u µ of the particle and its acceleration Du µ /Dτ as u µ ≡ e µ P √ −Z = dx µ P dτ , Du µ Dτ ≡ du µ dτ + u ρ u σ Γ µ ρσ | x P (τ ) , dτ = −Zdλ,(4.25) and using the identity (4.12), the equation of motion for x µ P can be rewritten as m(φ) Du ν Dτ = F µ γ µν | x P (τ ) , F µ = −∂ a m + (∂ a h b − ∂ b h a )u α ∂ α ϕ b ∂ µ ϕ a | x P (τ ) ,(4.26) where γ µν ≡ g µν + u µ u ν (4.27) projects spacetime indices onto the spatial section orthogonal to the worldline. On the other hand, the variation of the action with respect to ϕ a gives C a = 2 dτ −∂ a m + (∂ a h b − ∂ b h a )u α ∂ α ϕ b δ 4 (x − x P (τ )) √ −g . (4.28) Using equation (4.8) it implies that ∇ ν T µν = 1 2 C a ∂ µ ϕ a = dτ F µ (λ) δ 4 (x − x P (τ )) √ −g . (4.29) By comparing eq. (4.26) or eq. (4.23) to equation (4.29), one can see that the divergence of the stress-energy tensor is directly related to the force applied on the point particle. Put another way, the direct couplings are responsible for forces expected to deviate the point particle from the metric geodesics. The next subsection is devoted to the study of the consequences of such forces. Massive body dynamics As we have just seen in the previous subsection, the coupling C a is related to the point particle equation of motion. As motivated in the introduction of the current section, the constraint on the β parameter coming from massive test bodies dynamics can be sketched in the simplified Eddington-Robertson-Schiff (ERS) formalism. In this part we thus derive the massive body particle equation of motion at this level of approximation. Order of the point-particle direct couplings We start by expanding the couplings C a appearing in the equation of motions, in the case of a point particle Lagrangian depending on gravitating scalars as in eq. (4.21). Their expressions are given in eq. (4.28). With use of the PPN expansion for the different background fields g µν and ϕ a given from eqs. (3.3) to (3.5), (3.8) and (3.9). The couplings, and so the terms appearing in the equation of motion eq. (4.26), are thus expanded as: C a = 2 √ −g dτ −∂ a m + (∂ a h b − ∂ b h a ) u α ∂ α ϕ b δ (4) (x − x P (τ )) = 2 √ −g dτ −∂ a m(ϕ b 0 ) − ∂ ab m(ϕ c 0 )ϕ b 2 + ∂ a h b (ϕ c 0 ) − ∂ b h a (ϕ c 0 ) u α ∂ α ϕ b 2 × δ (4) x − x P (τ ) + O(ǫ 4 ). (4.30) We see that, at the lowest order, this amounts to take a coupling of the form: C a = − 2 √ −g ∂ a m(ϕ b 0 )δ (4) (x − x P (τ ))dτ + O(ǫ 2 ). (4.31) When this coupling is not vanishing, one can safely ignore the Lorentz-type coupling h a in eq. (4.21). Nevertheless, when this dominant coupling vanishes, namely for non-varying mass ∂ a m ≡ 0, the Lorentz-type couplings provides the leading contribution to C a . We could expand the point particle coupling obtained above in the same way as for the source, given by eq. (3.15). In the point particle case at the lowest order, it simply reads C a = − 2 √ −g c a mδ 4 (x − x P (τ )) dτ + O(ǫ 2 ),(4.32) with the trivial identification c a = −2s a + O(ǫ 2 ), (4.33) where we defined the "sensitivity" in a way similar to [34]: s a ≡ ∂ a ln m(ϕ 0 ). (4.34) Equation of dynamics At the lowest non-trivial PPN order, the point particle equation of motion of eq. (4.26) reduces to: d 2 x µ P dτ 2 + Γ µ σρ dx σ P dτ dx ρ P dτ = − ∂ ν ϕ a ∂ a ln m + O(ǫ 4 ) γ µ ν | x P (τ ) = −2s a γ ϕ a U ,ν + O(ǫ 4 ) γ µ ν | x P (τ ) , (4.35) where in the last line we expanded the scalars according to eq. (3.8). The time component of eq. (4.35) reads: dt 2 dτ 2 + Γ 0 σρ dx σ P dτ dx ρ P dτ = −2s a γ ϕ a U ,j dx j P dτ dt dτ + O(ǫ 4 ),(4.36) where γ µ ν has projected out the seemingly leading term containing U ,0 . To study the spatial components of eq. (4.35) we can switch from τ derivatives to t derivatives by decomposing d 2 x i P dτ 2 = dx i P dt dt 2 dτ 2 + d 2 x i P dt 2 dt dτ 2 . (4.37) Hence we obtain the following equality: − Γ i σρ dx σ P dτ dx ρ P dτ − 2s a γ ϕ a U ,i + U ,α dx α P dτ dx i P dτ = dx i P dt −Γ 0 σρ dx σ P dτ dx ρ P dτ − 2s a γ ϕ a U ,j dx j P dτ dt dτ + d 2 x i P dt 2 dt dτ 2 ,(4.38) which after multiplying by dτ dt 2 simplifies to d 2 x i P dt 2 + Γ i σρ − Γ 0 σρ dx i P dt dx σ P dt dx ρ P dt = −2s a γ ϕ a U ,i dt dτ 2 + O(ǫ 3 ). (4.39) We see that this is the leading term responsible for deviation from geodesic motion. As we comment later, this deviation nevertheless happens already at Newtonian order. Using the expressions given in appendix A.3 for the Christoffel symbols, where we put the Φ i , V i , W i potentials to zero as shall be the case in the ERS formalism, we obtain the following corrected massive particle equation of motion: d 2 x i P dt 2 = U ,i 1 − 2s a γ ϕ a + γU ,i v 2 + 2(γ + β)U ,i U + 2v i (1 + γ) v · ∇U + O(ǫ 5 ). (4.40) As explained in the intermediate steps above, in the final result eq. (4.40) we only kept the leading order for the terms related to the direct couplings. These terms appear at Newtonian order, hence modifying the Newton laws of dynamics. This is not a surprise, and the above equation can be used to constrain the values of the direct coupling. Indeed, as it is independent of the velocity of the test body v, it should satisfy s a γ ϕ a ≪ v 2 for typical velocities of test bodies in our solar system, otherwise it would be directly observable (and ruled out) by comparing the motion of various planets. In fact, if not small enough, such couplings to scalars (axions) should also be detectable by experiments on Earth. Hence, when these couplings are small enough not to perturb the Newtonian order, classical tests of gravity, such as the measurement of the perihelion shift of Mercury, will directly probe the β and γ PPN parameters generated from the PPN source and computed in eq. (3.24) to eq. (3.31). Lorentz-type coupling When the probe particle mass does not depend on the scalar fields, i.e. for vanishing sensitivity (4.34), one should consider a coupling of the Lorentz-type form. This is the second term of eq. (4.21), and its ǫ expansion is given in the second line of eq. (4.30). As it contains u α ∂ α ϕ 2 , it is of order ǫ 3 . Hence this type of coupling does not play a role at the first PPN level of approximation, and the dynamics will thus be described by eq. (4.40) taking s a = 0, which are the standard PPN equations of motion without the direct couplings. Photons dynamics and experiments on light rays In this subsection, complementary to the previous ones, we study the dynamics of a massless relativistic particle. We study test photons dynamics and choose a particular case for the direct coupling of the gravitating scalars to photons. We indeed consider the simplest kind of direct couplings of photons with an axionic gravitating field a(x µ ), which is expressed as follows 1 √ −g L m−a = ic 2 α f a a(x µ ) F µνF µν (x µ ). (4.41) We defined the dual electromagnetic tensorF µν = i/(2 √ −g)ǫ µναβ F αβ , the fine structure constant α and the axion decay constant f a . The C a functions can in principle be extracted from the above Lagrangian and the one describing the coupling of the axion to the electrons and nucleons constituting the matter sources. This would require a complete description of the source from the microscopic scales up to the macroscopic scales and would thus be beyond the scope of the present work. In this subsection we shall simply show that the coupling eq. (4.41) does not change the standard PPN constraints from test photons. The equations of motion derived from the lagrangian (4.41) are the modified Maxwell equations: ∇ ν F µν = iαc 2 f aF µν ∇ ν a = − αc 2 f a ǫ µνρσ √ −g F ρσ ∇ ν a. (4.42) In the presence of a (static) axion source, this term is not vanishing and is proportional to the gradient of the axion profile. According to (3.8), the axion field is given by a(x µ ) = a 0 + 2γ a U (x µ ) + O(ǫ 4 ). (4.43) When considered together with the Lorenz gauge condition ∇ µ A µ = 0 and the definition F µν ≡ ∇ µ A ν − ∇ µ A ν , the above field equation is written as ∇ µ ∇ µ A ν + R ν µ A µ = − αc 2 f a ǫ νµρσ √ −g ∇ µ A σ ∇ ρ a. (4.44) where R µν is the space-time Ricci curvature. Far from the source, the space-time geometry characteristic length L g is very large compared to the wavelengths λ of typical detectable photons, i.e. L g ≫ λ. If we furthermore assume that the wave-packet characteristic length L w is large in front of the wavelength λ, we can use the WKB approximation to solve eq. (4.44). Namely, we decompose the gauge potential as A µ ≡ A µ e iφ ≡ Ae µ e iφ , A ≡ −A µ A µ , e µ e µ = −1,(4.45) and suppose that the various quantities vary as: k µ ≡ ∂ µ φ ∼ λ −1 , ∇ ν k µ ∼ (λL w ) −1 , A µ ∼ O(1), ∇ ν A µ ∼ L w −1 , (4.46) ∇ ρ a ∼ L g −1 . The length scale L g can be extracted from the Riemann curvature tensor and is typically of order L g ∼ cr 3/2 / √ GM , where r is the distance between the matter source and the place where the photons propagate. The Riemann curvature could in general be extracted from the PPN metric, but as long as the PPN parameters stay small, the length L g is close to its general relativity value. Hence, the dominant contribution in λ −1 in the WKB approximation of eq. (4.44) is simply: − k µ k µ A ν = 0, O(λ −2 ),(4.47) which leads to k µ k µ = 0 and to the standard geodesic equation k µ ∇ ν k µ = 0 for photons. The Lorenz gauge condition also leads to: (4.48) showing that the gauge potential is orthogonal to the direction of propagation. On the other hand, the second dominant contribution in the WKB approximation of eq. (4.44) reads: k µ A ν = 0, O(λ −1 ),A µ ∇ ν k ν + 2k µ ∇ µ A ν + 2 √ −g A β k γ ∇ α aǫ µαβγ = 0, O(λ −1 ). (4.49) It can be solved by: k ν ∇ ν e µ = 0, e µ ∇ α (A 2 k α ) = − 2 √ −g ǫ µαβγ A 2 e β k γ ∇ α a. (4.50) We see that in the WKB approximation, test photons follow geodesics of the PPN metric but their polarization and amplitude are sensitive to the presence of the axionic background. This last aspect is at the center of axion searches through direct conversion or change in polarization of light in the presence of magnetic fields [36][37][38]. On the other hand, as the photons follow the PPN metric, classical experiments measuring deviation or time delays of light will directly probe the PPN metric and give access to the γ parameter [14,32,39]. Conclusions In this work, we studied the implications of adding direct couplings between gravitating scalar fields and matter, on top of the universal coupling to the metric, in multi-scalar tensor theories of gravity. Such direct couplings are expected to have direct effects since they modify the way space-time geometry is influenced by a localized matter source. This observation motivated the study of the weak gravity quasi-static regime of such theories, which is the relevant regime to describe solar-system tests of gravity. A central question addressed in this work was related to the possibility of screening cosmologically active scalars through their direct couplings to matter. Such screening would have made their presence undetectable in solar systems experiments. In the weak gravity quasi-static regime, theories of gravity can be compared with each other through the parametrized post-Newtonian formalism. We used this formalism and computed the complete expression for the ten PPN parameters in multi-scalar theories of gravity with direct couplings. The expressions we derived allow to evaluate the PPN parameters for the considered class of scalar-tensor theories of gravity. The latter are theories including several massless gravitating scalars with a curved scalar target-space and coupled non-universally to the matter sector. We showed that the PPN parameters are indeed modified by the presence of the direct couplings. Nevertheless, we also saw that for small couplings, they cannot differ much from the ones obtained in the same multi-scalar theory without direct couplings. This conclusion seems to go against the screening mechanism through direct couplings invoked by recent works in the literature. We then studied if such couplings, even small, would change the classical tests of gravity in the weak-field regime, by modifications in the way probes would move on the PPN background. As expected, direct couplings are responsible for additional direct forces on top of the gravitational force mediated through the space-time geometry. Hence, large direct couplings would be directly observable and would be ruled out. We supported this intuition by studying the dynamics of a massive point particle directly coupled to gravitating scalars, in presence of metric and scalar backgrounds generated by a PN matter source. For the point particle couplings under considerations, we deduced that if the couplings are small enough not to perturb the Newtonian order, test particles will follow the PPN metric geodesics, giving access to the γ and β PPN parameters of the theory. The classical constraints will thus apply identically to theories with or without direct couplings, as far as the direct couplings do not spoil the success of Newton gravity in its regime of applicability. Experiments involving photons in the regime of validity of the WKB approximation would also give access to the γ parameter, as in the standard PPN formalism. Possible extensions of this work would include the complete study in the PPN formalism of the two-body system, made of a PN source (Sun) and a massive probe (a planet), in theories with direct couplings. This should include the study of (non-)conservation laws for the different PN quantities (densities, momentum...) for the massive bodies, used when integrated the fluid equations of motion in the interior of massive bodies. Although technically involved, the procedure is well defined. The result would allow to give quantitative predictions for the results of solar system experiments such as the measure of the perihelion shift of Mercury. Acknowledgments This article is based upon work from COST Action COSMIC WISPers CA21106, supported by COST (European Cooperation in Science and Technology). The work of OL was supported in part by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research No. 17H06359. The work of SM was supported in part by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H02890, No. 17H06359, and by World Premier International Research Center Initiative, MEXT, Japan. A Notations and identities for multi-scalar gravity and PPN formalism A.1 Scalar target-space functions The multi-scalar theory considered in this paper is based on the action (2.1). To give an explicit expression of the gravitating sector, one should specify the non-minimal coupling function F (ϕ a ) and the target-space metric G ab (ϕ c ). As shown in the main body of the paper, the field equations and their PN expansions are easily expressed in terms of the following functions: Γ a bc = 1 2 G ad (∂ b G cd + ∂ c G bd − ∂ d G bc ), F a ≡ ∂ a F, F ab = ∂ a ∂ b F, (A.1) A a b ≡ F a F b F = B a F b , B a ≡ F a F , C a bc = Γ a bc + B a 2 (G bc + 3F bc ), (A.2) F cd ≡ ∇ c F d = ∇ c ∂ d F = F cd − Γ e cd F e . (A.3) One can use different sets of variables, such as the one naturally appearing when studying the Lagrangian {F, F a , F ab , G ab , Γ a bc }, or the one related to the expansions of the field equations {F, B a , A a b , F ab , G ab , C a bc }, or yet the naturally target-space covariant one {B a , F a ,F ab , G ab , Γ a bc }. These sets are only used for convenience and do not constitute bases of target-space functions. We give a couple of examples of relations used to go from one set to another: B c F d ∂ c B d + B c B d F cd = − B c B d F (F c F d − 2FF cd ), (A.4) 2B c B d F e C e cd = B c B d F cd (3B e F e + 1) − B c F d ∂ c B d = B c B d F cd (3B e F e + 2) + B c B d F (F c F d − 2FF cd ). (A.5) A.2 Parametrized post-Newtonian functionals In terms of the source rest-mass density ρ, the Newtonian potential is given by U ( x, t) ≡ ρ( x ′ , t) | x − x ′ | d 3 x ′ . (A.6) ǫ cpl. coupling coefficient. The cumbersome subscript is used to avoid confusion with the PN expansion parameter ǫ. According to eqs. (3.15) and (3.46) the above coupling corresponds to c a = τ 2 ǫ cpl. and β are related to boundary conditions of the solutions, namely the asymptotic values of the field. Indeed, in their formulae ((3.37), (3.38) and (3.40) of For theories without time diffeomorphism invariance, such as Hořava gravity (in the unitary gauge), the functional B should also be kept in the PPN metric[33]. 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[ "FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images", "FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images", "FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images", "FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images" ]
[ "Selahattin Cansiz ", "Cem Kesim ", "Sevval Nur Bektas ", "Zeynep Kulali ", "Murat Hasanreisoglu ", "Member, IEEECigdem Gunduz-Demir ", "Selahattin Cansiz ", "Cem Kesim ", "Sevval Nur Bektas ", "Zeynep Kulali ", "Murat Hasanreisoglu ", "Member, IEEECigdem Gunduz-Demir " ]
[]
[]
The Henle's fiber layer (HFL) in the retina carries valuable information on the macular condition of an eye. However, in the common practice, this layer is not separately segmented but rather included in the outer nuclear layer since it is difficult to perceive HFL contours on standard optical coherence tomography (OCT) imaging. Due to its variable reflectivity under an imaging beam, delineating the HFL contours necessitates directional OCT, which requires additional imaging. This paper addresses this issue by introducing a shape-preserving network, FourierNet, that achieves HFL segmentation in standard OCT scans with the target performance obtained when directional OCT scans are used. FourierNet is a new cascaded network design that puts forward the idea of benefiting the shape prior of HFL in the network training. This design proposes to represent the shape prior by extracting Fourier descriptors on the HFL contours and defining an additional regression task of learning these descriptors. It then formulates HFL segmentation as concurrent learning of regression and classification tasks, in which Fourier descriptors are estimated from an input image to encode the shape prior and used together with the input image to construct the HFL segmentation map. Our experiments on 1470 images of 30 OCT scans reveal that quantifying the HFL shape with Fourier descriptors and concurrently learning them with the main task of HFL segmentation lead to better results. This indicates the effectiveness of designing a shape-preserving network to improve HFL segmentation by reducing the need to perform directional OCT imaging.
10.1109/jbhi.2022.3225425
[ "https://export.arxiv.org/pdf/2201.06435v1.pdf" ]
246,016,168
2201.06435
d8cbe34940f105336041180889891f94fa656833
FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images Selahattin Cansiz Cem Kesim Sevval Nur Bektas Zeynep Kulali Murat Hasanreisoglu Member, IEEECigdem Gunduz-Demir FourierNet: Shape-Preserving Network for Henle's Fiber Layer Segmentation in Optical Coherence Tomography Images This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.Index Terms-Cascaded neural networksFourier descriptorsfully convolutional networksHenle's fiber layer segmentationoptical coherence tomographyshape-preserving network The Henle's fiber layer (HFL) in the retina carries valuable information on the macular condition of an eye. However, in the common practice, this layer is not separately segmented but rather included in the outer nuclear layer since it is difficult to perceive HFL contours on standard optical coherence tomography (OCT) imaging. Due to its variable reflectivity under an imaging beam, delineating the HFL contours necessitates directional OCT, which requires additional imaging. This paper addresses this issue by introducing a shape-preserving network, FourierNet, that achieves HFL segmentation in standard OCT scans with the target performance obtained when directional OCT scans are used. FourierNet is a new cascaded network design that puts forward the idea of benefiting the shape prior of HFL in the network training. This design proposes to represent the shape prior by extracting Fourier descriptors on the HFL contours and defining an additional regression task of learning these descriptors. It then formulates HFL segmentation as concurrent learning of regression and classification tasks, in which Fourier descriptors are estimated from an input image to encode the shape prior and used together with the input image to construct the HFL segmentation map. Our experiments on 1470 images of 30 OCT scans reveal that quantifying the HFL shape with Fourier descriptors and concurrently learning them with the main task of HFL segmentation lead to better results. This indicates the effectiveness of designing a shape-preserving network to improve HFL segmentation by reducing the need to perform directional OCT imaging. I. INTRODUCTION O PTICAL coherence tomography (OCT) is an essential retinal imaging equipment that allows the visualization of individual layers of the retina. Ophthalmologists employ OCT scans to diagnose eye-related diseases and understand their severity. Within the retina layers, the layer composed of photoreceptor axons, known as the Henle's fiber layer (HFL), provides significant information and the changes in its thickness are commonly associated with the macular condition in diseased retinas [1]. However, due to its variable reflectivity under an imaging beam, it is challenging to separately segment HFL on standard OCT scans. Directional OCT, which obtains images by altering the entry position of the imaging beam at the pupil, emerges as an important technique for HFL segmentation [2]. However, this technique is not a routine clinical procedure mostly because it necessitates significant amount of additional examination time, and thus, in the common practice, HFL is considered as a part of the outer nuclear layer. Therefore, there exists automatic HFL segmentation neither in commercially available OCT softwares [3] nor in scholarly studies [4], [5]. In response to this issue, this paper introduces a new cascaded neural network, which we call FourierNet, for HFL segmentation in standard OCT scans. The proposed FourierNet model achieves the performance of its counterpart, which uses scans obtained from the directional OCT technique as its inputs, but using only the standard OCT scans without requiring any non-routinely used imaging modalities. In order to facilitate HFL segmentation, the FourierNet model proposes to employ prior knowledge on the shape of HFL in the network training. It proposes to quantify this prior shape knowledge with a function defined on the HFL contours. To this end, it expands the function in a Fourier series, uses the harmonic amplitudes of its Fourier coefficients as the Fourier descriptors of HFL, and represents this prior shape knowledge in the network design by defining a regression task of learning these Fourier descriptors. It then formulates HFL segmentation as concurrent learning of regression and classification tasks, in which Fourier descriptor maps are estimated from an input image to represent the prior shape knowledge on HFL and used along with the input image to estimate the segmentation label for each pixel in the image. FourierNet achieves this learning by designing a cascaded fully convolutional network (FCN) that consists of an intermediate regression and a final classification task. The contributions of this paper are three-fold: • To the best of our knowledge, this is the first paper that has automatically segmented the Henle's fiber layer (HFL) in standard OCT scans. Although layer-wise segmentation of retinal OCT scans has been studied widely, none of the studies segment HFL separately but include 1 arXiv:2201.06435v1 [cs.CV] 17 Jan 2022 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. it in the outer nuclear layer (ONL). • FourierNet presents a cascaded FCN design where the intermediate task of Fourier descriptor estimation contributes to the final segmentation task by providing it with the shape-related information about what it needs to estimate. Since the network weights for these two tasks are updated at the same time, by minimizing a joint loss function, this concurrent learning more likely imposes the shape on the network. This, in turn, leads to better preserving the HFL shape, and thus, enhances HFL segmentation. • FourierNet quantifies the HFL shape with a set of Fourier descriptors and devises a cascaded FCN design for their estimation. These Fourier descriptors, which were first suggested by Cosgriff in 1960 [6], were also used in previous studies [7]- [9]. However, the previous studies used these Fourier descriptors to characterize objects in different applications (e.g., object retrieval and recognition), but did not employ them in designing a shapepreserving FCN. On the other hand, FourierNet uses the Fourier descriptors to devise a shape-preserving FCN for the first time, and demonstrates that this use is effective for more accurate HFL segmentation. II. RELATED WORK In the literature, there have been many studies proposed for layer-wise segmentation of the retina. Earlier studies typically segment the retina layers by first identifying their boundary pixels and then refine them with optimization methods. These initial pixels are identified by either edge detection [10] or training a classifier [11]- [14]. It is common to use graphbased optimization algorithms for pixel refinement. In [13], it is proposed to apply a graph cut algorithm with probability constraints on the initial pixels segmented by a k-nearest neighbor classifier. Likewise, a final segmentation map is obtained by first applying soft constraints to utilize prior information from a learned model and then regularizing the distances between two segmented layers by a graph based algorithm [16]. In [17], a graph-cut algorithm and dynamic programming are used together to refine initial layers classified by kernel regression. Other refinement methods have also been used. In [10], initial pixels identified by edge detection are improved by minimizing an energy term that considers vertical gradients and regional smoothness. In [15], the predefined order of layers and thickness priors are employed to refine the segmented retinal layers. More recent studies have widely used deep learning models, which remarkably improve layer-wise retina segmentation. Many studies use a U-Net [18] based network, which contains symmetric encoder and decoder blocks [19], [20]. The literature also contains modified U-Net architectures. In [22], a modified U-Net with a context extractor module, which generates high-level semantic feature maps, is used. In [21], retina layers are segmented designing an asymmetric U-shape network that combines residual building blocks with dilated convolutions. There also exist studies that combine deep learning models with post-processing techniques to correct pixel-wise classifications, and hence, obtain more accurate layers. For instance, a graph search algorithm is applied on the posterior probability maps outputted by a convolution neural network to find the final boundaries [4]. In another study [5], the DenseNet architecture is combined with a Gaussian process regression to smooth the segmentation results. In [23], a random forest classifier is trained on handcrafted features along with deep features learned by a deep residual network and this trained classifier is used to obtain the contour probabilities of each retinal layer. Although all these studies yield promising results on the segmentation of retinal layers, none of them segment HFL separately. Additionally, they neither define Fourier descriptors to represent the shape prior of a retina layer nor integrate this information with the design of a neural network. On the other hand, the proposed FourierNet model introduces a shapepreserving network for HFL segmentation for the first time and presents an effective way of representing the shape priors of HFL defining the Fourier descriptors on the contours of HFL. III. METHODOLOGY The proposed FourierNet model relies on characterizing the HFL shape with Fourier descriptors (Sec. III-A), defining a regression task of estimating the maps of these Fourier descriptors (Sec. III-B), and learning this regression task together with the main task of HFL segmentation by designing a cascaded FCN (Sec. III-C). The following subsections give the details. The implementation is available at mysite.ku.edu.tr/cgunduz/downloads/FourierNet. A. Fourier Descriptors This work quantifies the shape of HFL with a function defined along its contour points. This is the distance-to-center function that outputs the distance from an object centroid to the boundary point at a given arc length. This function characterizes how subsequent boundary points distribute over the space to form up the object's contour, and thus, the shape of the object. For instance, it is a constant function for a circular object since the distance from any boundary point to the centroid (radius) is the same. Let γ be a closed continuous curve with a length of L. Fourier descriptors are calculated for the distance-to-center function ξ(.) on the domain of length l x ∈ [0, L] where l x denotes the arc length of a section of the curve γ from its starting point z 0 to the point z x of the same curve (Fig. 1a). Since the contour of an object in a digital image does not form a continuous curve but contains finitely many discrete points (pixels), we assume an arc interpolation between these discrete points to define a continuous curve. Thus, the curve γ h that corresponds to the contour of HFL h becomes the interpolation of T boundary pixels, {z 0 , z 1 , ..., z T −1 }, each of which lies on the curve γ h at arc length l t (Fig. 1b). The distance-to-center function ξ(l x ) outputs the distance from the centroid z c to the point z x for which the arc length is l x . This function is expanded in a Fourier series as ξ(l x ) = a 0 + ∞ n=1 a n cos 2πnl x L + b n sin 2πnl x L (1) This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. where a n = 2 L L 0 ξ(l x ) cos 2πnl x L dl x (2) b n = 2 L L 0 ξ(l x ) sin 2πnl x L dl x (3) For the curve γ h , which is an interpolation of T discrete pixels, Eqn. 2 can be divided into T intervals of [l t−1 , l t ). Thus, the coefficient a n can be written as a n = 2 L T t=1 lt lt−1 ξ(l x ) cos 2πnl x L dl x(4) Here we use a circular arc interpolation to estimate the value of ξ(l x ) for all lengths l x that do not correspond to any boundary pixels. Since the distance from the circle centroid to any point on a circle is the same, this interpolation gives ξ(l x ) = ξ(l t−1 ), ∀l x ∈ [l t−1 , l t ), and as a result, allows taking ξ(l x ) outside the integral. a n = 2 L T t=1 ξ(l t−1 ) lt lt−1 cos 2πnl x L dl x a n = 1 πn T t=1 ξ(l t−1 ) sin 2πnl t L − sin 2πnl t−1 L a n = 1 πn ξ(l 0 ) sin 2πnl 1 L − ξ(l 0 ) sin 2πnl 0 L + ξ(l 1 ) sin 2πnl 2 L − ξ(l 1 ) sin 2πnl 1 L . . . + ξ(l T −1 ) sin 2πnl T L − ξ(l T ) sin 2πnl T −1 L Since γ h is a closed curve, the last point z T is indeed the starting point z 0 , and thus, ξ(l 0 ) = ξ(l T ) and sin( 2πnl0 L ) = sin( 2πnl T L ). By defining ∆ξ t = ξ(l t−1 ) − ξ(l t ) a n = 1 πn T t=1 ∆ξ t sin 2πnl t L(5) Following similar steps, the coefficient b n is expressed as: b n = − 1 πn T t=1 ∆ξ t cos 2πnl t L(6) For the n-th Fourier coefficients (a n , b n ), the polar coordinates are (A n , α n ) where A n = a n 2 + b n 2 is the harmonic amplitude and α n = arctan (b n /a n ) is the harmonic phase. This work uses the first N harmonic amplitudes of a truncated expansion of ξ(l x ) as a set of Fourier descriptors FD(γ h ) = [A 1 , A 2 , ..., A N ] to characterize the contour γ h of a given HFL h, and hence, its shape. Note that when N → ∞, the curve can be reconstructed using these harmonic amplitudes together with their corresponding harmonic phases. However, we do not use the harmonic phases to define a descriptor set as they provide less shape related information [9]. Illustration of a circular arc interpolation between the discrete points (pixels) of HFL contour γ h . This interpolation will be used to define a continuous curve for Fourier descriptor calculations. In this illustration, consecutive pixels are drawn too separated from each other for demonstration purposes and each pixel is denoted with its coordinate zx and its corresponding arc length lx. zc z0 zx ξ ( lx ) lx zc (z0, l0) (zT −1, lT −1) (z1, l1) (z2, l2) ξ ( l 0 ) ξ ( l1 ) ξ( l2) ξ(lT −1 ) (a) (b) B. Fourier Descriptor Map Generation The calculation explained above outputs a set of Fourier descriptors FD(γ h ) = [A 1 , A 2 , ..., A N ] for the contour (outermost pixels) of a single HFL. In order to define the maps of the intermediate regression tasks, which are used to reconstruct a segmentation map, these contour-wise descriptors are mapped onto every pixel, both HFL and background pixels using an iterative algorithm. This algorithm starts with calculating the Fourier descriptors on the contour of HFL in an image and assigns these descriptors to every pixel of the corresponding contour. It then shrinks HFL removing these pixels and repeats the same procedure for the new contour of the shrunk HFL. The algorithm iteratively continues until there exists no HFL contour in the image. The default value of 0 is used as the descriptors of the background pixels. Note that since N Fourier descriptors are calculated for a given contour, this algorithm creates N maps for the image pixels. For an example OCT image, Fig. 2 illustrates the map of the first Fourier descriptors calculated with respect to the given HFL annotations. Note that this calculation is possible only for This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. a training image whose HFL is manually annotated. For test images, for which annotations are not available, the maps are to be estimated by the intermediate regression tasks of the cascaded FCN. C. Cascaded Network Architecture and Training The proposed FourierNet model uses a cascaded FCN to concurrently learn the regression and classification tasks. This cascaded FCN uses a multi-task network to estimate N Fourier descriptor maps from an input image in its initial (intermediate) regression tasks. This multi-task network has one encoder to learn shared feature maps and N decoders, each of which learns a different Fourier descriptor map from the shared features. The selected architectures of the encoder and decoders are similar to those of the U-Net model [18]. These architectures are illustrated in Fig. 3. Then, it consecutively uses another U-Net that consists of an encoder and a decoder with the same architecture (Fig. 3) to predict segmentation labels from the estimated Fourier descriptor maps along with the input image in its final classification task. In all these selected architectures, the convolutional layers use 3 × 3 filters and are followed by the rectified linear unit (ReLU) activation function except the output layers. The output layers of the regression and classification tasks use the linear and softmax activations, respectively. The pooling and upsampling layers use 2 × 2 filters. Long-skip connections are added between the corresponding layers of the encoder and the decoder. Dropout layers with the rate of 0.2 are used for regularization. The number of layers and the feature maps used in each convolution layer are illustrated in Fig. 3. The overall architecture of the cascaded FCN is also illustrated in Fig. 4. This cascaded design, which obtains the Fourier descriptor representation in the middle of the entire architecture, enables both the encoder-decoder networks to exploit the feature maps learned for this representation through gradients flowing in the entire network. The cascaded FCN is implemented in Python using the Keras deep learning library. It is end-to-end trained from scratch to concurrently learn the intermediate regression and the final classification tasks, for which the mean square error and the categorical cross-entropy are used as the loss function, respectively. All tasks contribute to the joint loss function equally with the unit weight. The AdaDelta optimizer is used to adaptively adjust the learning rate and the momentum. The batch size is selected as 1 and early stopping is used in the training. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. IV. EXPERIMENTS A. Dataset The FourierNet model was tested on a dataset that contains the standard OCT scans of 30 eyes belonging to 17 healthy subjects. Eyes were imaged using the Heidelberg SD-OCT imaging equipment (Spectralis ® , Heidelberg Engineering GmbH, Heidelberg, Germany) with the standard protocol, in which the incident angle of the light beam was directed to the foveal center. For each eye, there is a set of 49 grayscale OCT scans. An input image with the resolution of 256 × 512 pixels was cropped from each of these scans. Manual annotation of HFL was performed by two ophthalmologists (M.H. and C.K.) on the directional OCT scans, in which OCT images were acquired with superior, inferior, nasal, and temporal tilts in addition to the standard imaging. It is important to note that the OCT images acquired with superior, inferior, nasal, and temporal tilts were used only for annotating HFL but not for training the FourierNet model. The dataset is divided into training and test sets containing the standard OCT scans of 20 and 10 eyes, respectively. The training set is further split into training images, on which the network weights are learned by backpropagation, and validation images, which are used for early stopping. The network training stops if there is no improvement on the validation set loss in the last 50 epochs. Test images are used for final evaluation of the model but not used in any step of the training. The training set contains 735 OCT images of 15 eyes (49 images from each eye), the validation set contains 245 OCT images of five eyes, and the test set contains 490 OCT images of 10 eyes. B. Comparisons The proposed FourierNet model is compared with three other algorithms that also use the same encoder and decoder architectures given in Fig. 3. The standard-OCT-Unet algorithm takes a standard OCT image as its input and outputs an HFL segmentation map. It uses neither Fourier descriptors to quantify the HFL shape nor a cascaded design to learn these Fourier descriptors. This is the baseline algorithm. The directional-OCT-Unet algorithm takes images acquired by the directional OCT as its inputs and also outputs an HFL segmentation map without using a cascaded design. In addition to the standard OCT image, this algorithm also uses the four other images acquired with superior, inferior, nasal, and temporal tilts. This comparison algorithm is used in order to understand a target performance when the directional OCT is available, which is indeed not the case in common practice. This study aims to achieve this performance using only the standard OCT image as an input. The directional-OCT-cascaded algorithm employs the images of the directional OCT in a cascaded network design similar to the one given in Fig. 4. Similar to FourierNet, this algorithm takes a standard OCT image as its input and formulates HFL segmentation as concurrent learning of regression and classification tasks. In its regression task, this algorithm estimates the four other images acquired with superior, inferior, nasal, and temporal tilts from the standard OCT image using a multi-task network. In its classification task, it predicts an HFL segmentation map from these four estimated images along with the standard OCT image, which is the original input. This comparison algorithm is used to understand the effectiveness of using an intermediate task that estimates the Fourier descriptors in a cascaded FCN design. C. Evaluation For a test image, pixels whose estimated class posteriors are greater than 0.5 are considered as HFL pixels. Some images may contain small noisy regions that are incorrectly classified as HFL (see Figs. 5a and 5c). These noisy regions are observed in only a few images and they only slightly affect the quantitative results. However, in order to carry out volumetric and thickness analyses of HFL in Sec. V-B, these regions are eliminated applying a very simple postprocessing method. For each column in the image, this method checks whether the column contains pixels of different connected components. If it does, the method keeps only the pixels of the largest component in the corresponding column, eliminating those of the others (see Figs. 5b and 5d). This postprocessing method is applied on the outputs of the FourierNet model as well as the comparison algorithms. The resulting HFL maps are evaluated visually and quantitatively on the test set images. For each image, we first find the number of true positive (TP), false positive (FP), and false negative (FN) pixels, comparing the estimated segmentation maps with their manual annotations, and then calculate the precision = TP / (TP + FP), recall = TP / (TP + FN), and f-score metrics. These metrics are averaged over the test set images. Furthermore, for the proposed FourierNet model as well as the comparison algorithms, the networks are trained three times and the average metrics of these three runs together with their standard deviations are reported. V. RESULTS The quantitative test set results obtained by the algorithms are reported in Table I. This table reveals that FourierNet achieves better performance compared to the standard-OCT-Unet algorithm, which is the baseline. However, it is worth This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. to noting that even this baseline algorithm gives a reasonable performance on the task of HFL segmentation. This suggests that deep learning has a great potential to analyze OCT images in more detail than the common practice. Moreover, the proposed FourierNet model, which uses only the standard OCT scan as its input, gives the f-score metric as high as the directional-OCT-Unet algorithm (even slightly higher), which uses four more OCT scans acquired with superior, inferior, nasal, and temporal tilts in addition to the standard one as its inputs. This indicates the effectiveness of representing the prior knowledge on the HFL shape in a cascaded network design. The directional-OCT-cascaded algorithm also uses a similar cascaded design but with different intermediate regression tasks. Table I also reveals that this way of defining an intermediate task is less effective than using the intermediate task of Fourier descriptor estimation. This may be attributed to the following. Estimating an image from another one may require more complex models (e.g., generative adversarial networks) for this particular task. In contrast, the Fourier descriptor representation, which summarizes the shape prior on HFL, facilitates defining a more effective and more compact learning task. The use of more complex models and the couse of different tasks in the same cascaded network design are considered as future research direction of this work. The visual results obtained on four exemplary test images are shown in Fig. 6. They are also consistent with the quantitative results. In our experiments, we observe that the proposed FourierNet model leads to enhanced performance at the fovea (see the center zone of the first two OCT images, for which segmentations are marked with red and yellow, in Fig. 6) as well as at the outer regions. It also tends to generate continuous segmentation maps of HFL compared to its counterparts (see the third OCT image, for which segmentations are marked with green, in Fig. 6). Additionally, compared to the comparison algorithms, the HFL thickness in the estimated maps of the FourierNet model is more consistent with the HFL thickness calculated with reference to the manual HFL annotations. It is important to note that since the HFL thickness is commonly associated with the retina's macular condition, its more correct estimation is important in the common practice. This issue will further be discussed in Sec. V-B. A. Parameter Analysis FourierNet has one external parameter N , which is the number of Fourier descriptors. In other words, N is the number of Fourier coefficients in the truncated expansion of the distance-to-center function ξ(l x ) that is defined to quantify the HFL contours. First Fourier descriptors contain more general information about this function whereas latter ones contain its finer details. Fig. 7 shows the test set performance as a function of N . This figure reveals that the first Fourier descriptor carries sufficient information to capture the HFL contour shape and the latter dimensions do not bring about additional information. Thus, in order to reduce the complexity, N is set to 1 in our experiments. For N = 1, the multi-task network given in Fig. 4 has only one decoder, which converts it to a single-task U-Net architecture. Although the first Fourier descriptor is adequate for this particular application of HFL segmentation, this paper presents a more generic algorithm that allows using the latter Fourier descriptors, if this is beneficial for other applications. This possibility can be investigated as future work. B. ETDRS Grid Analysis In evaluating the macular condition of a retina, it is very common to use the Early Treatment of Diabetic Retinopathy Study (ETDRS) grid that divides the retina into nine macular sectors based on the three concentric circles with 1mm, 3mm, and 6mm diameters (see Fig. 8). It might be necessary to analyze the layer's volume and thickness in the whole ETDRS grid as well as in each of these sectors for evaluating the retina's macular condition. In order to understand the effects of using the proposed FourierNet model on this ETDRS grid analysis, we carry out additional experiments. First, the precision, recall, and f-score metrics are calculated for each of the nine sectors of the ETDRS grid separately by considering only the pixels falling in the corresponding sector. The sector-based average f-score metrics obtained on the test set together with their standard deviation are reported in Table II. This table demonstrates that FourierNet achieves better f-scores in all of the nine sectors than the comparison algorithms that also use only the standard OCT image as their inputs. Furthermore, compared to the directional-OCT-Unet algorithm, which uses four more OCT scans acquired with superior, inferior, nasal, and temporal tilts in addition to the standard one as its input, FourierNet leads to sometimes slightly better sometimes slightly worse f-scores but without using any additional OCT scans. Next, we conduct volumetric and thickness analyses of HFL within the ETDRS grid. For that, the total volume and the average thickness of HFL are calculated with reference to manual annotations as well as with reference to a segmentation map obtained by the FourierNet model as well as each of the comparison algorithms. The average values calculated over three runs are reported in Table III. This table reveals that more accurate volume and thickness estimations are possible with the proposed FourierNet model, which indicates the effectiveness of representing the shape prior of HFL in a cascaded network design. VI. CONCLUSION This paper presents a shape-preserving network for automated HFL segmentation in standard OCT scans of the retina. This network, which we call FourierNet, relies on benefiting This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. II TEST SET RESULTS OBTAINED BY THE FourierNet MODEL AND THE COMPARISON ALGORITHMS. THESE ARE THE SECTOR-BASED AVERAGE TEST SET F-SCORES OF THE THREE RUNS AND THEIR STANDARD DEVIATIONS. SEE FIG. 8 FOR HOW THE SECTORS ARE DETERMINED. Inner Inner Inner Inner Outer Outer Outer Outer Central superior inferior nasal temporal superior inferior nasal the shape prior of HFL in its training. To this end, it proposes to quantify the shape prior by extracting Fourier descriptors on the HFL contours and introduces a new cascaded network design that learns these descriptors concurrently with the segmentation task. We tested our FourierNet model for HFL segmentation on 1470 OCT images of 30 eyes. Our experiments revealed that FourierNet achieved segmentation in scans acquired by standard OCT imaging at least with the performance that would be obtained when directional OCT imaging was used. As a result, this model proved to be a useful tool for HFL segmentation by reducing the necessity to perform directional OCT imaging. In our experiments, we obtained promising results on OCT scans of healthy retina. One future research direction is to test the model also on diseased retina in order to associate the volumetric and thickness changes in HFL with the macular condition of the diseased retina. This might require more advanced postprocessing methods due to the structural disorders expected in the diseased retina. This paper used HFL segmentation as a showcase application. Using the proposed shape-preserving network for other segmentation problems is considered as another future research direction of this study. Fig. 1 . 1(a) The distance-to-center function ξ(.) defined on the domain of length lx ∈ [0, L] where lx denotes the arc length of a section of the continuous curve γ from its starting point z 0 to the point zx of the same curve. (b) Fig. 2 . 2(a) An image taken from the standard OCT, (b) its manual HFL annotations, (c) the map of the first Fourier descriptors calculated with respect to the given HFL annotations. Fig. 3 . 3Encoder and decoder paths used in the U-Net architectures. Each box corresponds to an operation, which is distinguishable by its color. The input to each operation is a multi-channel map with its dimensions and number of channels being indicated in order on the left side of the box. Fig. 4 . 4Architecture of the cascaded FCN to concurrently learn the regression and classification tasks, along with an example standard OCT image as the input and the estimated Fourier descriptor maps and the segmentation map as the outputs of the regression and classification tasks, respectively. Fig. 5 . 5(a)-(b) Segmentation map estimated by the FourierNet model and the resulting map after postprocessing. (c)-(d) Segmentation map estimated by the standard-OCT-Unet comparison algorithm and the resulting map after postprocessing. The red ovals indicate noisy regions. Fig. 6 . 6Visual results obtained on four exemplary test images. The manual HFL annotations and the results of the FourierNet model and the comparison algorithms. All results are embedded on the standard OCT images. The same color is used to show the manual annotations and the results obtained on the same OCT image. Fig. 7 . 7Test set recall, precision f-scores metrics as a function of the number N of Fourier descriptors. These are the average results obtained over three runs. TABLE I TEST ISET RESULTS OBTAINED BY THE PROPOSED FourierNet MODEL AND THE COMPARISON ALGORITHMS. THESE ARE THE AVERAGE TEST SET RESULTS OF THE THREE RUNS AND THEIR STANDARD DEVIATIONS.Precision Recall F-score FourierNet 84.96 ± 1.08 86.17 ± 0.54 85.28 ± 0.32 Directional-OCT-Unet 86.84 ± 0.40 83.58 ± 0.68 84.94 ± 0.35 Standard-OCT-Unet 86.34 ± 0.88 81.31 ± 2.03 83.50 ± 0.68 Directional-OCT-cascaded 88.43 ± 0.73 79.51 ± 1.98 83.45 ± 0.84 TABLE TABLE III VOLUMETRIC IIIAND THICKNESS DATA OF HFL WITHIN THE ETDRS GRID. THESE ARE CALCULATED ON THE TEST SET IMAGES WITH REFERENCE TO THE MANUAL ANNOTATIONS AS WELL AS THE SEGMENTATION MAPS OF THE ALGORITHMS. THE TOTAL VOLUME IS GIVEN IN MM 3 AND THE AVERAGE THICKNESS IS GIVEN IN µM.Total volume Average thickness Manual annotations 0.71 ± 0.05 25.36 ± 1.63 FourierNet 0.70 ± 0.07 24.76 ± 2.44 Directional-OCT-Unet 0.67 ± 0.07 23.82 ± 2.55 Standard-OCT-Unet 0.65 ± 0.06 23.14 ± 2.30 Directional-OCT-cascaded 0.61 ± 0.07 21.79 ± 2.33 Henle fiber layer phase retardation changes associated with age-related macular degeneration. D A Vannasdale, Invest. Ophthalmol. Vis. Sci. 561D. A. VanNasdale et al., "Henle fiber layer phase retardation changes associated with age-related macular degeneration," Invest. Ophthalmol. Vis. Sci., vol. 56, no. 1, pp. 284-290, Dec. 2014. Revealing henles fiber layer using spectral domain optical coherence tomography. B J Lujan, A Roorda, R W Knighton, J Carroll, Invest. Ophthalmol. Vis. Sci. 523B. J. Lujan, A. Roorda, R. W. Knighton, and J. Carroll, "Revealing henles fiber layer using spectral domain optical coherence tomogra- phy," Invest. Ophthalmol. Vis. 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Kernel regression based segmentation of optical coherence tomography images with diabetic macular edema. S J Chiu, M J Allingham, P S Mettu, S W Cousins, J A Izatt, S Farsiu, Biomed. Opt. Express. 64This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibleS. J. Chiu, M. J. Allingham, P. S. Mettu, S. W. Cousins, J. A. Izatt, and S. Farsiu, "Kernel regression based segmentation of optical coherence tomography images with diabetic macular edema," Biomed. Opt. Express, vol. 6, no. 4, pp. 1172-1194, 2015. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. This illustration represents the top view for a retinal OCT volume scan. This ETDRS grid is depicted for the right eye; the nasal and temporal sectors should be the opposite for the left eye. (b)-(c) Two samples of cross-sectional OCT. Illustration of the ETDRS grid that divides the retina volume into nine macular sectors based on the three concentric circles with 1mm, 3mm, and 6mm diameters. scan slices corresponding to the red and blue lines in the illustrated ETDRS gridFig. 8. (a) Illustration of the ETDRS grid that divides the retina volume into nine macular sectors based on the three concentric circles with 1mm, 3mm, and 6mm diameters. This illustration represents the top view for a retinal OCT volume scan. This ETDRS grid is depicted for the right eye; the nasal and temporal sectors should be the opposite for the left eye. (b)-(c) Two samples of cross-sectional OCT scan slices corresponding to the red and blue lines in the illustrated ETDRS grid. U-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, Proc. Int. Conf. Med. Image Comput. Int. Conf. Med. Image ComputO. Ronneberger, P. Fischer, and T. 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[ "Optical Conductivity in an effective model for Graphene: Finite temperature corrections", "Optical Conductivity in an effective model for Graphene: Finite temperature corrections" ]
[ "Horacio Falomir \nCONICET -Departamento de Física\nIFLP\nFac. de Ciencias Exactas de la UNLP, C.C. 67, 1900) La PlataArgentina\n", "Enrique Muñoz \nInstituto de Física, Pontificia Universidad Católica de Chile\nAvenida Vicuña Mackenna 4860SantiagoChile\n\nCenter for Nanotechnology and Advanced Materials CIEN-UC\nAvenida Vicuña Mackenna 4860SantiagoChile. †\n", "Marcelo Loewe \nInstituto de Física, Pontificia Universidad Católica de Chile\nAvenida Vicuña Mackenna 4860SantiagoChile\n\nCentre for Theoretical and Mathematical Physics\nUniversity of Cape Town\n770RondeboschSouth Africa\n\nCentro Científico Tecnológico de Valparaíso, CCTVAL\nUniversidad Técnica Federico Santa María\nCasilla 110-VValparaísoChile. ‡\n", "Renato Zamora \nInstituto de Ciencias Básicas\nUniversidad Diego Portales\nCasilla 298-VSantiagoChile\n\nCentro de Investigación y Desarrollo de Ciencias Aeroespaciales (CIDCA)\nFuerza Aérea de Chile8020744SantiagoChile §\n" ]
[ "CONICET -Departamento de Física\nIFLP\nFac. de Ciencias Exactas de la UNLP, C.C. 67, 1900) La PlataArgentina", "Instituto de Física, Pontificia Universidad Católica de Chile\nAvenida Vicuña Mackenna 4860SantiagoChile", "Center for Nanotechnology and Advanced Materials CIEN-UC\nAvenida Vicuña Mackenna 4860SantiagoChile. †", "Instituto de Física, Pontificia Universidad Católica de Chile\nAvenida Vicuña Mackenna 4860SantiagoChile", "Centre for Theoretical and Mathematical Physics\nUniversity of Cape Town\n770RondeboschSouth Africa", "Centro Científico Tecnológico de Valparaíso, CCTVAL\nUniversidad Técnica Federico Santa María\nCasilla 110-VValparaísoChile. ‡", "Instituto de Ciencias Básicas\nUniversidad Diego Portales\nCasilla 298-VSantiagoChile", "Centro de Investigación y Desarrollo de Ciencias Aeroespaciales (CIDCA)\nFuerza Aérea de Chile8020744SantiagoChile §" ]
[]
In this article, we investigate the temperature and chemical potential dependence of the optical conductivity of graphene, within a field theoretical representation in the continuum approximation, arising from an underlying tight-binding atomistic model, that includes up to next-to-nearest neighbor coupling. Our calculations allow us to obtain the dependence of the optical conductivity on frequency, temperature and finite chemical potential, generalizing our previouly reported calculations at zero temperature, and reproducing the universal and experimentally verified value at zero frequency. arXiv:1907.02017v1 [cond-mat.mes-hall] 3 Jul 2019 2
10.1088/1751-8121/ab57cb
[ "https://arxiv.org/pdf/1907.02017v1.pdf" ]
195,791,817
1907.02017
b1fdb02517b3e3645378b434bd931bf74b9bf2d6
Optical Conductivity in an effective model for Graphene: Finite temperature corrections Horacio Falomir CONICET -Departamento de Física IFLP Fac. de Ciencias Exactas de la UNLP, C.C. 67, 1900) La PlataArgentina Enrique Muñoz Instituto de Física, Pontificia Universidad Católica de Chile Avenida Vicuña Mackenna 4860SantiagoChile Center for Nanotechnology and Advanced Materials CIEN-UC Avenida Vicuña Mackenna 4860SantiagoChile. † Marcelo Loewe Instituto de Física, Pontificia Universidad Católica de Chile Avenida Vicuña Mackenna 4860SantiagoChile Centre for Theoretical and Mathematical Physics University of Cape Town 770RondeboschSouth Africa Centro Científico Tecnológico de Valparaíso, CCTVAL Universidad Técnica Federico Santa María Casilla 110-VValparaísoChile. ‡ Renato Zamora Instituto de Ciencias Básicas Universidad Diego Portales Casilla 298-VSantiagoChile Centro de Investigación y Desarrollo de Ciencias Aeroespaciales (CIDCA) Fuerza Aérea de Chile8020744SantiagoChile § Optical Conductivity in an effective model for Graphene: Finite temperature corrections In this article, we investigate the temperature and chemical potential dependence of the optical conductivity of graphene, within a field theoretical representation in the continuum approximation, arising from an underlying tight-binding atomistic model, that includes up to next-to-nearest neighbor coupling. Our calculations allow us to obtain the dependence of the optical conductivity on frequency, temperature and finite chemical potential, generalizing our previouly reported calculations at zero temperature, and reproducing the universal and experimentally verified value at zero frequency. arXiv:1907.02017v1 [cond-mat.mes-hall] 3 Jul 2019 2 In this article, we investigate the temperature and chemical potential dependence of the optical conductivity of graphene, within a field theoretical representation in the continuum approximation, arising from an underlying tight-binding atomistic model, that includes up to next-to-nearest neighbor coupling. Our calculations allow us to obtain the dependence of the optical conductivity on frequency, temperature and finite chemical potential, generalizing our previouly reported calculations at zero temperature, and reproducing the universal and experimentally verified value at zero frequency. I. INTRODUCTION Graphene, a monolayer of carbon atoms arranged in a honeycomb lattice with C 3v ⊗ Z 2 symmetry 1 , possesses an electronic spectrum that displays two non-equivalent points K + , K − where the conduction and valence bands touch, and in whose vicinity the dispersion relation is approximately linear. The electronic spectrum is correctly described by an atomistic tight-binding model that, when including up to first nearest-neighbors coupling, leads to an effective low-energy continuum model describing relativistic Dirac fermions in 2D. This minimal tight-binding model can be extended upon including second nearest-neighbors couplings, that in the continuum representation leads to an effective field theory with a quadratic contribution to the linear Dirac dispersion 2 . Transparency is a physical property determined by the optical conductivity, i.e. the linear response to an external electromagnetic field. Several experiments confirm 3-16 that the measured transmittance is indeed compatible with the effective single-particle model of relativistic Dirac fermions in graphene, as supported from a number of theoretical works [8][9][10][11][12][13][14][15][16] . Among several physical effects that may induce deviations from the single-particle Dirac dispersion continuum model, such as electron-electron Coulomb interactions 17,18 , lattice phonons [19][20][21][22][23] , impurities [24][25][26] and different forms of quenched disorder 17,26 , we shall focus on the contribution to the optical conductivity that arises from the next-to-nearest neighbors coupling in the atomistic Hamiltonian, included as a quadratic correction to the kinetic energy operator within a continuum effective model for graphene 27 . Such a model has been considered by some of us in Ref. 2 to fully account for the Anomalous Integer Quantum Hall Effect in this material and the underlying wave equation is referred to in literature as Second Order Dirac Equation 28 . Notice that this is an isotropic model in which, the quadratic (anisotropic next to leading) term in the dispersion relation coming from the nearest neighbor sites has been shown to give a vanishing contribution to the Hamiltonian spectrum at first order in perturbation theory, thus justifying the consideration of the quadratic (isotropic) leading contribution of next-to-nearest neighbors in the honeycomb array 2 . In a previous article 27 , we investigated the frequency dependence of the zero-temperature optical conductivity of graphene, calculated in the Kubo linear response approximation [29][30][31] , when these next-to-nearest neighbors corrections are included in an effective field theory on the closed time path (CTP) (or Keldysh 31,32 ) formalism. In our present article, we extend this analysis to include finite temperature and finite chemical potential effects. Along the previously exposed ideas, we have organized the remaining of this article as follows: In Sect. II, we present the details of the model. In Sect. III we present the Matsubara formalism to calculate the vacuum polarization tensor in the Euclidean representation, to finally obtain the optical conductivity from the vacuum polarization tensor via analytic continuation to real frequency space. We discuss our findings in Sect. IV. Some calculation details are presented in two Appendices. II. LAGRANGIAN, CONSERVED CURRENT AND GENERATING FUNCTIONAL Graphene crystal structure, as sketched in Fig. 1, is described in terms of two overlapping triangular (Bravais) sublattices. The band structure obtained from an atomistic, tight-binding description including up to the next-tonearest neighbors contribution is of the form E ± (k) = ±t f (k) − t [f (k) − 3], (II.1) where t and t are the nearest and next-to-nearest hopping parameters and f (k) = 3 + 4 cos 3k x a 2 cos √ 3k y a 2 + 2 cos( √ 3k y a) . (II.2) Here, a 1.42Å is the interatomic distance 26 . The literature reports a value 26 t ∼ 2.8 eV , while for the second nearestneighbour coupling the reference values are not so precisely established, but reported in the range 26 0.02t ≤ t ≤ 0.2t. The points K + and K − at which f (K ± ) = 0 define the so-called Dirac points. Around K + , E ± (k + K + ) = ±t 3 2 a|k| − 3 8 a 2 k 2 sin(3ϑ) + t − 9 4 a 3 k 2 + 3 + O(|k| 3 ), (II.3) with tan(ϑ) = k y /k x . Around the K − point, one just needs to replace ϑ → −ϑ in Eq. (II.3). The isotropic portion of the model in Eq. (II.3) was first considered in Ref. 2 as a natural framework to explain the Anomalous Integer Quantum Hall Effect in graphene. Moreover, as previously mentioned, the anisotropic quadratic term, so called trigonal warping, in this effective dispersion relation was shown not to contribute to the energy spectrum at first order in perturbation theory 2 , thus justifying to retain just the isotropic terms up to this order in the pseudo-momenta. In the presence of electromagnetic interactions, the model in the continuum approximation is described by an effective field theory with the Lagrangian 2,27 L := i 2 ψ † ∂ t ψ − ∂ t ψ † ψ + ψ † eA 0 ψ − 1 2m [(p − eA + θσ) ψ] † · [(p − eA + θσ) ψ] − 2θ 2 ψ † ψ = i 2 ψ † ∂ t ψ − ∂ t ψ † ψ − 1 2m ∇ψ † · ∇ψ + i∇ψ † · (−eA + θσ) ψ− −iψ † (−eA + θσ) · ∇ψ + ψ † (−eA + θσ) 2 − 2θ 2 ψ , (II.4) where θ = mv F and m = ±2 2 /(9t a 2 ), where the sign depends on each Dirac cone K ± . A summary of the numerical values for the relevant parameters of the model is presented in Table I. 26 1.42 t (eV ) 26 2.8 t (eV ) 26 ∼ 0.056 − 0.56 m (kg) 1.37 × 10 −29 − 1.37 × 10 −30 v f (m/s) 26 ∼ 10 6 mv 2 f (eV ) 7 − 70 Here, the 3-momentum is p µ = (p 0 , p), with p = (p 1 , p 2 ). The vector potential A = (A 1 , A 2 ), whereas σ = (σ 1 , σ 2 ) are Pauli matrices. In this model, ψ † and ψ are regarded as independent fields whose equations of motion are derived from the variation of the action, ∂L ∂ψ † − ∂ t ∂L ∂ (∂ t ψ † ) − ∇ · ∂L ∂ (∇ψ † ) = i∂ t ψ − 1 2m (p − eA + θσ) 2 − 2θ 2 ψ = 0 , (II.5) and similarly for ψ. Noether's Theorem leads to the existence of a locally conserved current, whose time-component defines the local charge density 27 j 0 = e ψ † ψ, (II. 6) while the spatial components define the current density 27 j k = e 2m i ∂ k ψ † ψ − ψ † ∂ k ψ + 2ψ † −eA k + θσ k ψ . (II.7) It is straightforward to verify, from the equations of motion, that j µ is conserved 27 , ∂ µ j µ = ∂ t j 0 − ∇ · j = 0 . (II.8) Notice also that we can write 27 j µ (x) = δ δA µ (x) L(y) d 3 y . (II.9) In our previous work 27 , we developed a generating functional on the CTP (or Keldysh contour) for the effective field theory in Eq.(II.3), defined as Z γ [A] = Dψ † (x, τ )Dψ(x, τ )e i γ dτ d 2 xL[ψ † (x,τ ),ψ(x,τ )] , (II.10) with γ = γ − ⊕ γ + , such that γ − represents the time-ordered branch of the contour, while γ + is the anti-time-ordered branch (see Ref. 27 for details). From the CTP functional defined in Eq. (II.10), we generate the average current components as follows 27 −i δ log Z γ [A] δA µ (x) = 1 Z γ [A] Dψ † Dψ e i γ d 3 yL(y) j µ (x) = j µ (x) , (II.11) while the second functional derivative gives the current-current correlation function 27 , (−i) 2 δ 2 log Z γ [A] δA µ (x)δA ν (y) = −i δj µ (x) δA ν (y) + T j µ (x)j ν (y) − j µ (x) j ν (y) . (II.12) Here, the first term is the diamagnetic contribution 27,33 13) and the others are the paramagnetic ones. The currents are defined in normal order with respect to the fermionic field, so that j µ (x) | A=0 = 0. The linear response of the system to the external electromagnetic field is described by the second derivative in Eq. (II.12) evaluated at A µ = 0 27,33 , δj µ (x) δA ν (y) = δ µk δ ν k − e 2 m 2 ψ † (x)ψ(x) δ (3) (x − y) , (II.K µν (x, y) = (−i) 2 δ 2 log Z γ [A] δA µ (x)δA ν (y) A=0 = K νµ (y, x) = T j µ (x)j ν (y) 0 . (II.14) The spatial components of the current are given by 27 j k (x) A=0 = e 2m i∂ k ψ † (x)ψ(x) − iψ † (x)∂ k ψ(x) + 2θψ † (x)σ k ψ(x) ≡ ψ † a (x)D k ab ψ b (x) , (II.15) where we have defined the differential operators 27 D k ab = e 2m −i ← → ∂ k δ ab + 2θ σ k ab . (II.16) Applying Wick's theorem 31,32,34 on the CTP for the definition of the current-correlator (correlators associated to disconnected diagrams vanish), we obtain 27 : T j k (x)j l (y) = T ψ † a (x)D k ab ψ b (x)ψ † c (y)D l cd ψ d (y) = −D k abD l cd T ψ b (x)ψ † c (y) T ψ d (y)ψ † a (x) . (II.17) The previous relation allows us to define the corresponding components of the polarization tensor in the CTP contour indices α, β = ±, K kl αβ (x, y) = T j k α (x)j l β (y) = −D k abD l cd ∆ αβ bc (x, y)∆ βα da (y, x) . (II.18) As discussed in detail in Ref. 27 , the retarded component of the polarization tensor is obtained from the combination K kl R (x, y) = K kl −− (x, y) − K kl −+ (x, y) =D k abD l cd ∆ F bc (x, y)∆ A da (y, x) + ∆ R bc (x, y)∆ F da (y, x) −∆ R bc (x, y)∆ A da (y, x) . (II.19) In terms of Fourier transforms, ψ(x) = 1 (2π) 3/2 d 3 p e −ip·xψ (p) , ψ † (x) = 1 (2π) 3/2 d 3 p e ip·xψ † (p) , (II.20) we have 27 ∆ αβ ab (x, y) ≡ ∆ αβ ab (x − y) = d 3 p (2π) 3 e i(x−y)·p∆αβ ab (p). (II.21) Here, the different propagators for the Hamiltonian model considered are, in Fourier space (F: Feynman, R: Retarded, A: Advanced),∆ F (p) =∆ −− (p) = i p 0 − p 2 2m + v F p · σ p 0 − p 2 2m 2 − v 2 F p 2 + i = i p 0 − p 2 2m + v F p · σ p 0 + i − p 2 2m − v F |p| p 0 − i − p 2 2m + v F |p| , (II.22) ∆ R (p) = i p 0 − p 2 2m + v F p · σ p 0 + i − p 2 2m 2 − v 2 F p 2 , (II.23) ∆ A (p) = i p 0 − p 2 2m + v F p · σ p 0 − i − p 2 2m 2 − v 2 F p 2 . (II. 24) In order to consider the finite temperature dependence of the polarization tensor, the time-domain is compactified according to the prescription t → −iτ , with 0 ≤ τ ≤ β, with β = 1/(k B T ) the inverse temperature. Correspondingly, the three propagators defined above reduce to a single Euclidean one, by analytic continuation p 0 + i → ip 4 + µ of the retarded one. Therefore, we define the Euclidean propagator bỹ ∆ E (p) =∆ R (p 0 + i → ip 4 + µ, p) = i ip 4 + µ − p 2 2m + v F p · σ ip 4 + µ − p 2 2m 2 − v 2 F p 2 . (II. 25) In particular, for the linear response theory 30-32,34-36 , we need the retarded component of the polarization tensor 26) which is obtained at finite temperature from the Euclidean polarization tensor by analytic continuation K µν R (x − y) = d 3 p (2π) 3 e i(x−y)·p Π µν R (p), (II.Π kl R (ω, p) = Π kl E (ip 4 → ω + i , p). (II.27) The corresponding expression for the finite temperature, Euclidean polarization tensor is Π kl E (ip 4 , p) = e 2 4m 2 1 β q4=ωn,n∈Z d 2 q (2π) 2 Γ k ab (p + 2q)∆ E bc (p + q)Γ l cd (p + 2q)∆ E da (q) (II.28) with the symbol Γ k ab (p + 2q) = δ ab (p + 2q) k + 2θ σ k ab , (II.29) and a similar expression for Γ l cd (p + 2q). We remark that due to compactification of the time domain at finite temperature, the component q 4 = ω n , where ω n = 2π(n + 1/2)/β for n ∈ Z are the Fermionic Matsubara frequencies. III. THE POLARIZATION TENSOR AND OPTICAL CONDUCTIVITY The polarization tensor Π kl (p) contains the information about the conductivity on the plane of this two-dimensional system and also about its light transmission properties 10,33 . We are interested in the consequences of the application of harmonic homogeneous electric fields which, in the temporal gauge, are related with the vector potential by E k = −∂A k /∂t = −iωA k . Since the conductivity is determined by the linear relation between the current and the applied electric field, J k = σ kl E l , from Eqs. (II.11), (II.14) and (II.27), we can write for the conductivity as a function of the frequency 10,33 σ kl (ω) = 2 × 2 Π R kl (p) iω p→(ω,0) , (III.1) where the prefactor takes into account the valley and electronic spin degeneracy in graphene. Therefore, the real and imaginary components of the optical conductivity are given by e σ kl (ω, T ) = 4 m Π R kl (ω, T ) ω (III.2) and m σ kl (ω, T ) = −4 e Π R kl (ω, T ) ω , (III.3) respectively. In particular, it is the real part of the conductivity tensor that determines electronic transport in the DC limit ω → 0. In order to include finite temperature effects, we first calculate Π E kl (ω, 0) from Eq.(II.28), and then by analytic continuation, as described in Eq.(II.27), we obtain Π R kl (ω, 0). The evaluation requires to calculate two integrals and an infinite sum over (Fermionic) Matsubara frequencies, as defined in Eq.(II. 28). Π E kl (p) = e 2 4m 2 1 β q4=ωn,n∈Z d 2 q (2π) 2 Tr [p k + 2q k + 2θσ k ] ∆ E (p + q) [p l + 2q l + 2θσ l ] ∆ E (q) . (III.4) Specializing this expression to the case p = (ip 4 , 0), and using polar coordinates for the spatial components q 1 = Q cos ϕ, q 2 = Q sin ϕ, we write Π E kl (ip 4 , 0) = e 2 4π 1 β q4=ωn,n∈Z ∞ 0 dQ Q 4πm 2 2π 0 dϕ Tr{A} B EE (III.5) with A = [2q k + 2θσ k ] ip 4 + iq 4 + µ − q 2 2m + v F q · σ [2q l + 2θσ l ] iq 4 + µ − q 2 2m + v F q · σ , B EE = ip 4 + iq 4 + µ − q 2 2m 2 − v 2 F q 2 iq 4 + µ − q 2 2m 2 − v 2 F q 2 . (III.6) We notice that the denominator is independent of ϕ, and hence it is straightforward to calculate the trace in the numerator integrated over ϕ, N (Q, ip 4 , iq 4 + µ) = 1 4πm 2 2π 0 Tr {A} dϕ = − 8 8m 4 v 2 f (iq 4 + µ)(iq 4 + µ + ip 4 ) + 4m 2 Q 2 ip 4 mv 2 f + iq 4 + µ +(iq 4 + µ) iq 4 + µ + 2mv 2 f − 2mQ 4 mv 2 f + 2iq 4 + 2µ + ip 4 + Q 6 (III.7) for k, l = 1, 1 or 2, 2, and a vanishing result for k, l = 1, 2 or 2, 1. x x x x x x x x x x x x x x x x x <e k 0 =m k 0 k (1) 0 k (2) 0 k (3) 0 k (4) 0 R R R R I µ (2) " (1) " (3) " (4) " "# FIG. 2: (Color online) The complex contour C = Γ R ⊕ Γ ↑↓ ⊕ α γ (α) ε used to calculate the Matsubara sum. Notice that Γ ↑↓ and γ (α) ε are oriented clockwise, in order to exclude the poles from the contour C. Let us now consider the sum over (Fermionic) Matsubara frequencies, since q 4 = ω n = (2n+1)π/β. The sum can be obtained through the construction of a contour integral on the complex plane (see Fig.2), by choosing a meromorphic function with infinitely many poles at the Matsubara frequencies. A straightforward choice is the Fermi function, n F (k 0 − µ) = 1 1 + e β(k0−µ) , (III.8) that clearly has poles at k 0 = iω n + µ, for n ∈ Z, with residues Res [n F (k 0 − µ)] k0=iωn+µ = lim k0→iωn+µ (k 0 − iω n − µ) 1 + e β(k0−µ) = lim k0→iωn+µ (k 0 − iω n − µ) 1 + e iβωn e β(k0−iωn−µ) = − 1 β , (III.9) where the identity e iβωn = −1, valid for fermionic Matsubara frequencies, was applied. Therefore, defining iq 4 + µ → k 0 , we calculate the contour integral depicted in Fig.2, when the radius of the outer circular contour Γ R goes to infinity, R → ∞, and the radius of the 4 contours γ (α) goes to zero, ε → 0 lim R→∞,ε→0 C N (Q, ip 4 , k 0 ) B EE (Q, ip 4 , k 0 ) n F (k 0 − µ) dk 0 2πi = − α=1,4 Res N (Q, ip 4 , k 0 ) B EE (Q, ip 4 , k 0 ) k0=k (α) 0 n F (k (α) 0 − µ) − n∈Z N (Q, ip 4 , iω n + µ) B EE (Q, ip 4 , iω n + µ) Res [n F (k 0 − µ)] k0=iωn+µ = 0.(III.10) Using Eq.(III.9), we solve for the required Matsubara sum from the equation above, Q(Q + 2mv f ) 2m , k (2) 0 = Q(Q − 2mv f ) 2m , k(3)0 = Q(Q + 2mv f ) 2m − ip 4 , k(4)0 = Q(Q − 2mv f ) 2m − ip 4 . (III.12) By recalling that the external Matsubara frequency in the diagram is a Bosonic one, we have p 4 = 2nπ/β, with n ∈ Z, and hence e iβp4 = 1. Using this simple identity, we find that n F (k (3) 0 − µ) = n F (k (1) 0 − µ), n F (k (4) 0 − µ) = n F (k (2) 0 − µ). (III.13) Using this, and calculating explicitly the residues, we finally obtain Π E 11 (ip 4 , 0) = e 2 4π ∞ 0 dQ 4v 3 f Q 2 4v 2 f Q 2 − (i p 4 ) 2 n F Q(Q − 2mv f ) 2m − µ − n F Q(Q + 2mv f ) 2m − µ (III.14) From this expression, by analytic continuation to real frequency space ip 4 → ω + i we recover the retarded polarization tensor Π R 11 (ω) = Π E 11 (0, ip 4 → ω + i ). (III.15) For this purpose, we write part of the integrand in Eq. (III.14) as follows 4v 3 f Q 2 4v 2 f Q 2 − (ω + i ) 2 = v 2 f Q 1 2v f Q − ω − i + 1 2v f Q + ω + i = P 4v 3 f Q 2 4v 2 f Q 2 − ω 2 + iπv 2 f Q [δ(2v f Q − ω) − δ(2v f Q + ω)] , (III.16) where P stands for the Cauchy principal value. Therefore, the real and imaginary parts of the retarded polarization tensor are given by the expressions e Π R 11 (ω) = e 2 4π P ∞ 0 dQ 4v 3 f Q 2 4v 2 f Q 2 − ω 2 n F Q(Q − 2mv f ) 2m − µ − n F Q(Q + 2mv f ) 2m − µ (III.17) m Π R 11 (ω) = e 2 4 v 2 f ∞ 0 dQ Q [δ(2v f Q − ω) − δ(2v f Q + ω)] n F Q(Q − 2mv f ) 2m − µ −n F Q(Q + 2mv f ) 2m − µ (III.18) Moreover, in order to remove unphysical, possibly divergent vacuum contributions from the retarded polarization tensor, we define its regularized version as Π R 11, reg (ω) ≡ Π R 11 (ω, T ) − Π R 11 (0, T ). (III.19) Note from the definitions above that, by construction, m Π R 11 (ω = 0, T ) = 0, and hence no regularization is required for the imaginary part of the tensor. On the other hand, e Π 11 (ω = 0, T ) = 0 in general, and hence the real part will be regularized as described in Appendix. The expression for the real part cannot be reduced to a simple analytical expression, however one can still evaluate it in a low-temperature series through a generalization of Sommerfeld expansion (as shown in Appendix). On the other hand, the integral for the imaginary part can be evaluated to yield m Π R 11 (ω) = e 2 16 ω sgn(ω) n F ω 2 8mv 2 f − ω 2 − µ − n F ω 2 8mv 2 f + ω 2 − µ = e 2 32 |ω| tanh β 2 ω 2 8mv 2 f + ω 2 − µ − tanh β 2 ω 2 8mv 2 f − ω 2 − µ . (III.20) Table I), at constant chemical potential µ = 0.5 eV , as a function of frequency, for different temperature vales. From the expression above, the real part of the optical conductivity is given by e σ 11 (ω, T ) = 4 m Π R 11 (ω) ω = e 2 8 sgn(ω) tanh β 2 2 ω 2 8mv 2 f + ω 2 − µ − tanh β 2 2 ω 2 8mv 2 f − ω 2 − µ , (III.21) where we have restored the constant for normal I.S. units. It is very interesting to analyze the zero-temperature limit (β → ∞) of Eq.(III.21), that becomes (see Appendix B for details) e σ 11 (ω, T → 0) = e 2 4 , 1 + 2µ mv 2 f − 1 < |ω| 2mv 2 f < 1 + 2µ mv 2 f + 1 0, otherwise (III.22) It is seen from this result that the actual value of the conductivity at T = 0 is e 2 /(4 ), independent of frequency and the parameter m that captures the second nearest-neighbor interaction, in agreement with our previous calculation 27 and transparency experiments 3 . Interestingly though, there is however a hidden, non-analytic dependency through the domain of the stepwise function, that defines a region where the conductivity actually vanishes. It is instructive to compare our result, that includes the second nearest-neighbor interaction through the parameter m, with the more standard result that only involves first nearest-neighbors, a situation that can be recovered from our model in the limit m → ∞. In this limit, from Eq. (III.21) we obtain e σ 11 (ω, T, m → ∞) = e 2 8 sgn(ω) tanh β 2 ω 2 − µ + tanh β 2 ω 2 + µ . (III.23) This result, as expected, matches the one reported in Refs. 37,38 . Moreover, also in the limit m → ∞, the zerotemperature conductivity becomes e σ 11 (ω, 0, m → ∞) = e 2 8 sgn(ω) { sgn ( ω − 2µ) + sgn ( ω + 2µ)} (III.24) = 0 , |ω| < 2µ/ e 2 4 , |ω| > 2µ/ . (III. 25) in agreement with Refs. 3,38 . The real part of the electrical conductance, as a function of frequency and at different temperatures, is depicted in Fig. 3a and Fig. 3b. Let us now turn to the imaginary part of the optical conductivity. The integral over 0 ≤ Q < ∞ can be expressed as an asymptotic expansion in negative powers of β, through a similar analysis as in the more standard Sommerfeld expansion (for details see Appendix). The real part of the retarded polarization tensor (see Appendix) is given by the expression e Π R 11,reg (ω, T ) = e 2 8π ωF(ω, µ, m) + β −2 e 2 πω 2 24mv 2 f 1 + 2µ mv 2 f 3/2      ω 2 − 8mv 2 f 3µ + 2mv 2 f 1 + 1 + 2µ mv 2 f ω 2 − 8mv 2 f µ + mv 2 f 1 + 1 + 2µ mv 2 f 2 − ω 2 + 8mv 2 f −3µ + 2mv 2 f −1 + 1 + 2µ mv 2 f ω 2 − 8mv 2 f µ + mv 2 f −1 + 1 + 2µ mv 2 f 2 Θ µ mv 2 f      + O(β −3 ). (III.26) Therefore, the imaginary part of the optical conductivity is given by m σ 11 (ω) = −4 e Π R 11,reg (ω, T ) ω = − e 2 2π F(ω, µ, m) − (k B T ) 2 e 2 πω 6mv 2 f 1 + 2µ mv 2 f 3/2      2 ω 2 − 8mv 2 f 3µ + 2mv 2 f 1 + 1 + 2µ mv 2 f 2 ω 2 − 8mv 2 f µ + mv 2 f 1 + 1 + 2µ mv 2 f 2 − 2 ω 2 + 8mv 2 f −3µ + 2mv 2 f −1 + 1 + 2µ mv 2 f 2 ω 2 − 8mv 2 f µ + mv 2 f −1 + 1 + 2µ mv 2 f 2 Θ µ mv 2 f      + O(β −3 ), (III.27) where we have restored the constant for I.S. units, and we defined the function Table I), at constant chemical potential µ = 0.5 eV , as a function of frequency, at zero temperature. The finite temperature dependence is very weak (as seen in Eq.(III.27) and cannot be appreciated at the scale of the plot. The inset shows with higher resolution the region near the first peak. F(ω, µ, m) =                                  arctanh    ω 2mv 2 f 1+ 2µ mv 2 f Eq.(III.21) and Eq.( III.27). As expected, our analytical calculation recovers the universal value e σ = e 2 /(4 ) in the limit of zero temperature, Eq.(III.22), but however reveals a non-trivial and non-analytic dependence on the ratio µ/(mv 2 f ) in the frequency domain. Remarkably, our analytical Eq.(III.21) for the frequency-dependent real part of the optical conductivity at finite temperature and chemical potential, in the limit m → ∞ (t → 0) reduces to Eq.(III.23), that exactly reproduces previous results reported in the literature 3,38 for the conventional first-nearestneighbor approximation. Moreover, our Eq.(III.21) generalizes this result to reveal the effect of including the next-tonearest neighbor hopping t into the dispersion relation. In particular, we notice that, when t is neglected as in the conventional case, the real part of the conductivity presents a sharp step (at zero temperature) or a sigmoidal trend (at finite temperature) exactly centered at ω = 2µ (see for instance Eq.(III.23)). In contrast, when t is included, the step is shifted to ω = 2mv 2 f 1 + 2µ mv 2 f − 1 ∼ 2µ − µ 2 mv 2 f . This effect is particularly interesting since, as shown in the existing literature, there seems to be a large uncertainty on the exact value for the second nearest neighbor hopping in graphene, 0.056 eV < t < 0.56 eV (see Table I). Our result suggests that an experimental characterization of the frequency-dependence of the real part of the optical conductivity, at finite chemical potential (to be adjusted, for instance, with a gate potential) could therefore provide an accurate and direct experimental measurement of t , that could be compared with the broad estimations obtained so far from ab-initio calculations 49 or cyclotron resonance experiments 50 . Clearly, the difference between the sgn(z) functions is either ±2 or 0. In order to analyze the different cases, let us define the two quadratic functions y 1 (ω) = ω 2 4mv 2 f + ω − 2µ = (ω − ω (1) + )(ω − ω (1) − ), y 2 (ω) = ω 2 4mv 2 f − ω − 2µ = (ω − ω (2) + )(ω − ω (2) − ), (A.2) where roots are given by ω (1) ± = −2mv 2 f ± 2mv 2 f 1 + 2µ mv 2 f , ω(2)± = 2mv 2 f ± 2mv 2 f 1 + 2µ mv 2 f . (A.3) On the other hand, the two parabolas intersect at ω = 0, with the common value y 1 (0) = y 2 (0) = −2µ. A graphical representation of the roots and intercept is displayed in Fig. 5. Moreover, we remark that Eq.(A.1) can be written as e σ 11 (ω, T = 0) = e 2 8 sgn(ω) ( sgn(y 1 ) − sgn(y 2 )) = e 2 4    sgn(ω), y 1 (ω) > 0, y 2 (ω) < 0 − sgn(ω), y 1 (ω) < 0, y 2 (ω) > 0 0, otherwise (A.4) The condition y 1 (ω) > 0 and y 2 (ω) < 0 is satisfied for −2mv 2 f + 2mv 2 f 1 + 2µ mv 2 f < ω < 2mv 2 f + 2mv 2 f 1 + 2µ mv 2 f , where sgn(ω) = 1. On the other hand, the condition y 1 (ω) < 0 and y 2 (ω) > 0 is satisfied for −2mv 2 f − 2mv 2 where we have restored the constant for I.S. units. f 1 + 2µ mv 2 f < ω < 2mv 2 f − 2mv 2 f 1 + 2µ Let us consider the integral representing the real part of the retarded polarization tensor e Π R 11 (ω) = e 2 4π P ∞ 0 dQ 4v 3 f Q 2 4v 2 f Q 2 − ω 2 n F Q(Q − 2mv f ) 2m − µ − n F Q(Q + 2mv f ) 2m − µ , (B.1) where P stands for Cauchy's principal value. It is convenient to express the integral defining the polarization tensor in dimensionless variables, i.e. x = Q/(mv f ), Ω = ω/(2mv 2 f ),β = mv 2 f β/2, γ = 2µ/(mv 2 f ). (B.2) Hence, we have It is interesting first to analyze the T → 0 limit of the regularized polarization tensor. From the expression for the Fermi functions, it is clear thatn F (z) → Θ(−z) asβ → ∞ (T → 0). Therefore, we have where we have defined the function e Π R 11 (ω) = e 2 4π mv 2 f P ∞ 0 dx x 2 x 2 − Ω 2 n F (x 2 − 2x − γ) −n F (x 2 + 2x − γ) ,e Π R 11,reg (ω, T → 0) = e 2 4π mv 2 f Ω 2 P ∞ 0 dx 1 x 2 − Ω 2 Θ(x 2 + 2x − γ) − Θ(x 2 − 2x − γ) , = e 2 4π mv 2 f Ω 2 P x(2)F(ω, µ, m) =                                  arctanh    ω 2mv 2 f 1+ 2µ mv 2 f For the finite temperature contribution, we obtain e Π R 11,reg (ω, T ) = e Π R 11,reg (ω, T → 0) + e 2 2π mv 2 f (Π 1 (ω) − Π 2 (ω) − Π 1 (0) + Π 2 (0)) , (B.9) where Π 1 (ω) = 2 N k=0 β −2k−2 1 − 2 −2k−1 ζ(2k + 2)F In these expressions, we have defined the auxiliary functions obtained from the roots of the quadratic equations x 2 ± 2x − γ = z, corresponding to x (1) ± (z) = 1 ± 1 + γ + z, x (2) ± (z) = −1 ± 1 + γ + z, (B.12) and the corresponding implicit functions F ± (z) = f [x FIG. 1 : 1(Color online) Sketch of the crystal structure of graphene. The honeycomb array is described in terms of two overlapping triangular sublattices. N (Q, ip 4 , iω n + µ) B EE (Q, ip 4 , iω n + µ) = α=1,4 Res N (Q, ip 4 , k 0 ) B EE (Q, ip 4 , k 0 ) k ( FIG. 3 : 3(Color online) The real part of the electrical conductance, for (a) t = 0.056 eV , and (b) t = 0.56 eV (see FIG. 4 : 4(Color online) The Imaginary part of the electrical conductance, for (a) t = 0.056 eV , and (b) t = 0.56 eV (see online) Sketch of the locus of the roots in Eq. (A.3). The regions in white represent the frequency range where, at zero temperature and finite chemical potential, the real part of the optical conductivity does not vanish, as seen in Eq. (A.5). sgn(ω) = −1. Taking this into account, we arrive at the final expression e σ 11 (ω, T → 0) = in the main text, in order to remove spurious unphysical and possibly divergent contributions arising from the vacuum, we regularize the retarded polarization tensor according to the expression e Π R 11,reg (ω, T ) ≡ e Π R 11 (ω, T ) − e Π R 11 (0, T ). (B.5) γ + 1 are the positive roots of the quadratic polynomials y 1 (x) = x 2 + 2x − γ and y 2 (x) = x 2 − 2x − γ, respectively. The principal value integral must be calculated separately in three frequency intervals, TABLE I : IParameters of the modela (Å) * [email protected] The imaginary part of the optical conductivity, expressed in our model by Eq.( III.27), displays two separate resonances (seeFig. 4aandFig. 4b), the first at ω = 2mv 2, and the second atThe first one reproduces, in the limit m → ∞, results reported in the literature for the conventional model with only first-to-nearest neighbor approximation 3,38 , with a small shift ∼ − µ 2 mv 2 f in the position of the peak. The second peak, which is a unique feature of the model, is located at an extremely large frequency, and in practice has no physical consequences.IV. CONCLUSIONSAlong this article, we have discussed the effect of including the next-to-nearest neighbors hopping t , through the "mass" parameter m = ±2 2 /(9t a 2 ) in the dispersion relation 2 , on the optical conductivity of single-layer graphene. Our analysis is based on the continuum representation of the model via an effective field theory 27 , by extending our previous results at zero temperature 27 to the finite chemical potential and finte temperature scenario, Appendix A: Zero temperature limit of e σ11(ω, T ) Let us start from Eq.(III.21) (in natural units = 1), and consider the limit T → 0 (β → ∞), e σ 11 (ω, T = 0) = e 2 8 sgn(ω) sgn(1)where we defined the functionSimilarly, in the above expansions we defined the derivatives of these implicit functions with respect to z, as F (k)The explicit expression for finite temperature corrections up to O(β −3 ) isHere, we have defined the Heaviside Theta function as . P R Wallace, Phys. Rev. 71622P. R. Wallace, Phys. Rev. 71, 622 (1947). . H Falomir, J Gamboa, M Loewe, M Nieto, J. Phys. A. 45135308H. Falomir, J. Gamboa, M. Loewe, M. Nieto, J. Phys. A 45, 135308 (2012). . 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[ "Attention-aware Path-based Relation Extraction for Medical Knowledge Graph", "Attention-aware Path-based Relation Extraction for Medical Knowledge Graph" ]
[ "Desi Wen [email protected] \nInstitute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA\n", "Yong Liu \nIER Business Development Center\nShenzhenP.R.CHINA\n", "Kaiqi Yuan [email protected] \nInstitute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA\n", "Shangchun Si [email protected] \nInstitute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA\n", "Ying Shen [email protected]* \nInstitute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA\n" ]
[ "Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA", "IER Business Development Center\nShenzhenP.R.CHINA", "Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA", "Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA", "Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE)\nPeking University\n518055SHENZHENP.R.CHINA" ]
[]
The task of entity relation extraction discovers new relation facts and enables broader applications of knowledge graph. Distant supervision is widely adopted for relation extraction, which requires large amounts of texts containing entity pairs as training data. However, in some specific domains such as medicalrelated applications, entity pairs that have certain relations might not appear together, thus it is difficult to meet the requirement for distantly supervised relation extraction. In the light of this challenge, we propose a novel path-based model to discover new entity relation facts. Instead of finding texts for relation extraction, the proposed method extracts path-only information for entity pairs from the current knowledge graph. For each pair of entities, multiple paths can be extracted, and some of them are more useful for relation extraction than others. In order to capture this observation, we employ attention mechanism to assign different weights for different paths, which highlights the useful paths for entity relation extraction. To demonstrate the effectiveness of the proposed method, we conduct various experiments on a large-scale medical knowledge graph. Compared with the state-of-the-art relation extraction methods using the structure of knowledge graph, the proposed method significantly improves the accuracy of extracted relation facts and achieves the best performance.
10.1007/978-3-319-73830-7_27
[ "https://arxiv.org/pdf/1812.01887v1.pdf" ]
39,421,538
1812.01887
315c7f7190d878a91993c4762e67787a26f4ec2d
Attention-aware Path-based Relation Extraction for Medical Knowledge Graph Desi Wen [email protected] Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE) Peking University 518055SHENZHENP.R.CHINA Yong Liu IER Business Development Center ShenzhenP.R.CHINA Kaiqi Yuan [email protected] Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE) Peking University 518055SHENZHENP.R.CHINA Shangchun Si [email protected] Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE) Peking University 518055SHENZHENP.R.CHINA Ying Shen [email protected]* Institute of Big Data Technologies Shenzhen Key Lab for Cloud Computing Technology & Applications School of Electronic and Computer Engineering (SECE) Peking University 518055SHENZHENP.R.CHINA Attention-aware Path-based Relation Extraction for Medical Knowledge Graph Relation ExtractionPath AttentionKnowledge Graph The task of entity relation extraction discovers new relation facts and enables broader applications of knowledge graph. Distant supervision is widely adopted for relation extraction, which requires large amounts of texts containing entity pairs as training data. However, in some specific domains such as medicalrelated applications, entity pairs that have certain relations might not appear together, thus it is difficult to meet the requirement for distantly supervised relation extraction. In the light of this challenge, we propose a novel path-based model to discover new entity relation facts. Instead of finding texts for relation extraction, the proposed method extracts path-only information for entity pairs from the current knowledge graph. For each pair of entities, multiple paths can be extracted, and some of them are more useful for relation extraction than others. In order to capture this observation, we employ attention mechanism to assign different weights for different paths, which highlights the useful paths for entity relation extraction. To demonstrate the effectiveness of the proposed method, we conduct various experiments on a large-scale medical knowledge graph. Compared with the state-of-the-art relation extraction methods using the structure of knowledge graph, the proposed method significantly improves the accuracy of extracted relation facts and achieves the best performance. Introduction In recent years deep learning has been one of the most influential and representative technologies in the field of artificial intelligence. The unprecedented breakthroughs in application of this technology lead to a new wave of development both in academia and industry. If intelligent machine has a brain in the future, deep learning will be learning mechanism of the machine brain, and knowledge graph will be knowledge base of it. Knowledge graph, crucial for big data intelligence, will also impact on areas such as natural language processing, information retrieval, and artificial intelligence profoundly. Knowledge graph is essentially a semantic network composed of entities and the relationship between entities. Nowadays, knowledge graph has already been widely used in various applications, such as question answering [1] and recommender system [2]. There are many open source knowledge graph projects, such as freebase, YAGO, Dbpedia, etc., but knowledge graph is still far from complete. Therefore, relation extraction supplements knowledge graph extracting semantic relations between entities. Distant supervision [3] is the most widely adopted method for relation extraction. However, the distant supervised relation extraction method requires a massive amount of sentences containing two entities, which is strict restriction for many entity pairs; furthermore, most of the existing relation extraction models using external in formation rather than abundant implied information within knowledge graph. To address the above issues, we propose a path-based strategy to infer relations from the structure of knowledge graph rather than text. For an entity pair that has a potential relation, we first calculate the path between entity pairs from the existing knowledge graph, treat path as a sequence, and then encode the sequence using recurrent neural network. However, path has its corresponding establishment likelihood. Inspired by this observation we add attention model to put different weights on different paths, With attention weights embodied in path vector, relations are thus extracted. The contributions of our work can be summarized as follows: • Compared with other text-based relational extraction models, our model uses path information in the knowledge graph to substantially reduce the difficulty of training data acquisition; • Take path attention model to assign corresponding weights for different paths, which reduces noise from inadequate paths; • We construct a medical knowledge graph to evaluate our model. The experimental results demonstrate our model achieves the highest precision over other structure-based models. Related Work Relation extraction has been an important branch of knowledge graph completion, emerging many excellent research models. Y.lin et al. [4] propose a multi-sentence relation extraction model. For an entity pair, relation classification achieves by calculating eigenvector of the sentence containing the entity pair through using Convolutional Neural Network (CNN) and adding sentence attention model to assign sentence weights. Miwa et al. [5] propose a relation extraction model based on word sequence and tree structure. However, distant supervised model requires a large number of sentences containing two entities as training sets. In some specific domains, such as medical field, are hard to meet the above conditions. To address this issue, W. Zeng et al. [6] propose a pathbased relation extraction model that uses the CNN to extract eigenvectors of sentences containing a single entity and constructs middleware between the two target entities for reasoning to extract relations. Nevertheless, entities may belong to multiple classes, causing ambiguity when applying single sentence. Besides extracting relations from text, another way is from the structure of knowledge graph, which includes knowledge representation learning. Knowledge representation learning mainly suggests representation learning for entities and relations in knowledge graph, transforming entities and relation vectors into the low-dimensional dense vector space, and carrying out corresponding calculation and reasoning. TransE [7] is a simple but efficient model proposes by Bordes et al.. For triple (h, r, t), transE considers h+r=t. Compared with the previous knowledge representation learning model, parameters are relatively few in transE. The model is simple and intuitive, with small calculation, especially good at dealing with one-to-one relations. However, one-to-many, many-to-one and many-to-many relations are too difficult for transE model to deal with. Thus, Wang et al. [8] propose the transH model. It maps relations to another hyperplane in the same space and designs complicated sampling method for training. However, Ji et al. put forward the transD [9] model, and believe that entity is a complex of multiple attributes, and different relations concern with different attributes of the entity, so entity and relation should be in different spaces. In the knowledge graph, some of the entity relations connect a large amount of entities, whereas some entity relations are quite simple. If one model is used for all cases, it may lead to inadequate training for complicated relations and overfitting for simple relations. Therefore, Ji et al. [10] propose the tranSparse model, using relatively dense matrices for complex relations and sparsely matrices for simple relations via Sparse-Matrix. Knowledge representation models above utilize directly connected triples as features, but path [11] in the knowledge graph contains numerous implied information. Das et al. [12] use triple path as a sequence and that entities might belong to multiple classes is taken into account. So they add class information to triple vector representation, and put sequence into Recurrent Neural Network(RNN) to extract relations. However, the model has two obvious weaknesses: 1) ignore multiple paths; 2) ignore soft reasoning, as the establishment probability of paths is not always equal to 1 or 0. Since in medical field, relations for symptoms corresponding to diseases and appropriate drugs corresponding to symptoms establish only to some extent [13]. Methodology Given a set of entity pairs (head, tail), our model calculates path among entity pairs and computes the likelihood of each relations r based on the path. In this section, we will introduce our model as follows: Calculate Path: For a given set of entity pairs (h, t), we find a set of paths { # , % , … , ' } from the knowledge graph, where ) (i = 1,2, … , n) is the acyclic path taking node h as start and node t as end. Path Encode: Given a path x, use Gated Recurrent Unit (GRU) to compute its distributed representation. Path Attention: After learning distributed representation of all paths, attention model assigns different weights to paths, from which relations among entity pairs are calculated. Calculate path For a group of entity pairs (h, t), we calculate acyclic path that satisfies conditions (source, target, minLen, maxLen, maxPaths)from the knowledge graph G, where G is directed graph, source is the starting of path, target is the ending of path, minLen is the lower limit of path length, maxLen is the upper limit of path length, maxPaths is the upper limit of the number of paths. We adopt the breadth-first search to determine whether there exists a path to satisfy the (source, target, minLen, maxLen) condition in G, and if so, use the depth-first search to find all the paths satisfying the (source, target, minLen, maxLen, maxPaths) condition. Finally, we can get a set of head-to-tail paths { # , % , … , ' }, the structure of path x is (ℎ # , # , # , ℎ % , % , % , … , ℎ D , D , D ), where ℎ # = ℎ, D = t, EF# = ℎ E ( ≤ < ). Path Encoding Triple Representation: After 3.1 we get a set of paths, and each path x contains a number of triples, each triple (h, r, t) contains two entities and one relation. Entities and relations have different representations. We derive idea from the transE model that entities and relations are in the same dimension space, so they are mapped into a d-dimensional space. Entities and relations are represented by column vector of the same embedded matrix V, V ∈ O×(QRS) , where e indicates the total number of entities and r indicates the total number of relations. We concatenate vector representation of two entities with entity representation of relation, to form a triple representation t, t ϵ UO . At last, we transform the triple path into a set of vector sequence x = { # , % , … D } and input it to GRU. GRU: Gated Recurrent Unit (GRU) proposed by Cho et al. [14] shared parameters in time series and thus associates connected input. It consists of reset gate r, update gate z and a memory cell s, calculated as follows: z = σ( ) Y + )F# Y + Y ) (1) r = σ( ) S + )F# S + S ) (2) h = tanh ( )^+ )F# · r^+ ^) (3) ) = 1 − · h − z · )F#(4) Where t ) is the input vector, representation vector of triple t in our task, h is the output vector, z is the update gate, r is the reset gate, Y , S , ^, Y , S , ^U O×UO are the weight matrix, Y , S , ^ are the offset, σis the sigmoid function, · is the Hadamard product. We use vector sequence x = { # , % , … D } obtained by 3.2.1 as the input of GRU, and select the final output vector ℎ D as the final encoding representation of current triple path p, p = ℎ D . Path attention: After the previous steps, we will encode path with head entity as start and tail entity as end to form a path matrix S ∈ UO×D , which consists of encoded path [ # , % , … D ] generated by GRU. Obviously, next step should use all the path information in matrix S to extract relations of relation pairs (h, t). Howeve, not all the paths are correct. In medical field, each path has its own establishment probability. That is the reason we introduce the attention model to give different weights ) for each path ) , and calculate the vector representation pr in path matrix S. pr = ) ) ) (5) According to the different settings of , our model is divided into the following three categories. One: We randomly select a path as representative from path set, which means is a one hot vector.This approach is a naive baseline of path attention. Average (AVE): We assume that each path in the path set has the same contribution to pr. Consequently, we assign the same weights for each path.Where pr equals to average of each path vector in path set. Path Attention (PATT): We are supposed to calculate different weights ) for each path ) due to its different contribution. M = tanh ( i S) (6) α = softmax( m M)(7) pr = Sα m Where M ∈ UO×D is the mapping matrix of path matrix, α ∈ UO is the attention model weight, pr ∈ UO is path representation of the attention model weight, i ∈ UO×UO , w ∈ UO is projection parameters. pr is the final path matrix representation, transformed to vector e with dimension equal to the number of relation categories r by a fully connected layer, and converted into conditional probability distribution y through softmax layer ultimately. y = softmax( p + p )(9) Where p ∈ S×UO is the mapping matrix of fully connected layer, p ∈ S is the offset vector of fully connected layer. Experiments Experiments will prove that relation extraction in our model may take full advantage of path information in the knowledge graph for relation extractions and path attention could reduce negative effect of unreasonable paths. To start with, we will introduce the datasets in the experiment, one of the approach of building negative samples, and parameter settings in our model. Then verify the affect of path embedding in comparison with other triple embedding model. Last but not least, compare different path attention weight settings to prove the affect of path attention model. Experiment setup Dataset: We have constructed a Chinese medical knowledge graph that covers information on diseases, symptoms, drugs, food, surgery and so on in the medical field. This knowledge graph has a total of 45427 entities, 26 relations and 396,172 triples. The experiment divides triples into 27 relations, where the redundant relations are unrelated, since most entities do not necessarily have relations among each other. We construct negative samples with unrelated entity pairs and choose negative samples relations as the 27th relation -unrelated relation. The transH model proposes a strategy of building negative samples which randomly replaces a head or tail entity for an entity pairs (h, t). However, negative samples constructed that way are quite rough, and whether or not entity pairs have relations is not for sure. Particularly, there are plenty of entities with relations but not directly connected. We design an algorithm for generating entity pairs of no relations. For the given entity pair (h, t), entities are randomly selected from the knowledge graph to replace the head entity and tail entity, forming (h q , t) and (h, t q ) triples to make (h q , t) and (h, t q ) not include in the knowledge graph, and Neighbors h q ∩ Neighbors t = ∅, Neighbors h ∩ Neighbors t q = ∅, where Neighbors Entity is an entity set that directly connected to Entity in current knowledge graph. Comparative method: We will use the knowledge graph representation model, transE, transH, transR to do comparative tests, because the structure information of knowledge graph applied in these models. The thoughts of knowledge graph representation models above are rather close, so relation extraction task could be described as follows: given a triple (h, r, t) , calculate ||ℎ * + * − * || and choose the smallest score relation r as its predicted relation, where ℎ * , * , * is different mapping of h, r, t in different models. The knowledge representation learning has a triple classfication task, which is specifically used to determine whether a triple (h, r, t) is a correct fact. Relevant algorithm in the task is not practical. Since it intends to find a value seg as dividing line: ℎ * + * − * < seg for the correct fact, ℎ * + * − * ≥ seg for the wrong fact, with select the highest correct rate seg by validation set. Whereas, the algorithm does not apply in our experiments because t it tends to judge all triples incorrect with the increase in the proportion of negative samples. Therefore, the negative samples relation is treated as class 27 in our experiment. Parameter settings: We employ three-fold cross validation method to verify our experiments. The path length lower limit minLen is generally set 2, the path length upper Performance Comparison This part we will verify the effect of path encoding. We take the GRU+ONE model in comparison with transE, transH, transD and tranSparse models in our experiment. Table 1 shows our experimental results. The GRU + ONE model outperforms others remarkably since path information of more abundant information is taken into account, compared to those models that use only source and target information in the path, such as the TransE. It is proved that path encoding is more effcient than triple encoding in the task of relation extraction. The proposed model has a comparatively substantial increase to other models. The reason may be more path information taken into account. On the contrary to knowledge representation model with single triples, path information makes decision taking advantage of all the triple information in paths. The more information we use, the higher accuracy we get. Effect on Path Attention Now we will prove the effect of path attention model. There may be tens of paths between two entities in the knowledge graph, but we can not use all the paths due to computational ability restriction. Therefore, we divide test set into the following three categories to verify the effect of model with randomly selected paths. One. For each group of entity pairs, we randomly choose a path that satisfies the (minLen, maxLen) condition and connects two entities, and use it to extract relations; Two. For each group of entity pairs, we randomly choose two paths that satisfy the (minLen, maxLen) condition and connect two entities, and them to extract relations; All. For each group of entity pairs, we choose path that satisfy the (minLen, maxLen, maxPaths) condition and connect two entities, and use it to extract relations. Table 2 shows our experimental results. The accuracy of GRU + AVE and GRU + PATT model are lower than GRU + ONE model while selecting one path randomly. The GRU + ONE model takes a single path for training, so it could utilize characteristics of one path better. However, when it comes to all paths, the accuracy of GRU + AVE and GRU + PATT model outperform 0.02 higher than the GRU + ONE model, which covers too little information. In contrast to the GRU + AVE and GRU + PATT model, GRU + ONE model solely inputs in a single path involving small quantity of data. Particularly in deep learning of millions of parameters for training, little training data may lead to overfitting problem, making its generalization ability weak. This could also explain the reason why the accuracy of GRU + ONE model is relatively low. Consider the GRU + AVE and GRU + PATT models. It can be inferred from Table 2 that the accuracy of GRU + PATT model is about 0.01 higher than that of GRU + AVE model while using all paths. Nevertheless, when using a single path or two paths, the accuracy of GRU + PATT model is approximately 0.03 higher than GRU + AVE model. Path attention model acts effectively even with incomplete information, since all paths are treated equally and rearranged the same weight in the GRU + AVE model when it occurs to path information shortage. Therefore, the accuracy is relevant to the proportion of unreasonable paths in path collection. The GRU + PATT model could increase reasonable path weights by reducing the unreasonable path weights in the way of adding path attention model to perform better, even if the part of unreasonable paths is still large. In conclusion, path attention model could guarantee high level of accuracy although there are too many paths to acquire all the path information. Table 3 demonstrates two example of path attention selected from test data. For each triple, we select its paths with the highest and the lowest attention weight. By the use of path attention, our model could arrange larger weight for paths having higher establishment probability and smaller weight for paths having lower establishment probability. As the first triple illustrates, the two entities connected by relation "alias of the disease" are basically different expressions of the same entity, so this path is logical. And the path with lower score has multi-level "complication" relations, which could not make the two diseases "infectious shock" and "abdominal pain" treated in the same department definitely. Case Study In the second triple, reasoning path with higher score derives relation between two diseases as "complication" from two "complication" relations, which is also reasonable even accompanied by some problems. The path with lower score speculates from disease "beryllium poisoning" to disease "pulmonary edema", through disease "uremia". Beryllium and its compounds against lungs cause disease "Beryllium poisoning", whose incidence site lies in lungs. "Uremia" is a kind of kidney disease of little connection with "Beryllium poisoning", which provide more rational explanation for path with high score. Consequently, the two paths are endowed with quite different weights by path attention model. (beryllium poisoning, disease examination, urinary calcium),( urinary calcium,possible disease with higher score, hypercalcemic nephropathy),( hypercalcemic nephropathy, complication, uremia),( uremia, complication, pulmonary edema) Conclusions In this paper, we propose a model to explore relations based on path information instead of text information, which is supposed to reduce requirements of dataset. Besides, we employ GRU + path attention to assign different weights for paths to alleviate the negative effect of unreasonable paths. In experimental part, we compare with other models based on knowledge graph structure, and experiments demonstrate that our model is obviously superior to other models. Next step we will expand our research from the following two aspects: 1. Our model relies on path information, and there are some key entities connecting thousands of entities in current knowledge graph. These will at exponential level increase the number of paths in the algorithm we construct paths. Therefore, we will consider a more effective way of building paths. 2. The knowledge graph contains not only structure information, but also plenty of text information, such as entity descriptions in general knowledge graph and drug instruction descriptions in medical knowledge graph. Next research will concentrate on how to align text information in the way of path. limit maxLen ϵ {4,5,6},and the upper limit of the number of paths maxPaths is set as needed. Entity embedding size and relation embedding size Q , S ϵ {30,50,100},batch size B ϵ {64,128,256,512}, dropout probability p ϵ {0.4,0.5,0.6}, path embedding size ƒ {128,256,512}. In experiment, we set minLen = 2, maxLen = 5, maxPath = 100, Q = S = 50, B = 128, p = 0.5, ƒ = 256, and we choose 20 as the number of iterations in training. Table 1 . 1Comparison among GRU+ONE and trans series.Model dataset 1 dataset 2 dataset 3 transE 0.5154 0.5157 0.5452 transH 0.5329 0.5532 0.5325 transD 0.5617 0.5345 0.5792 tranSparse 0.5711 0.5731 0.5882 GRU+ONE 0.7490 0.7528 0.7628 Table 2 . 2Performance of relation extraction with different number of sentences.Model dataset 1 dataset 2 dataset 3 Path One Two ALL One Two ALL One Two ALL GRU+ONE 0.7528 0.7529 0.7528 0.7528 0.7628 0.7628 GRU+AVE 0.7216 0.7016 0.7660 0.7201 0.7076 0.7718 0.7276 0.7184 0.7733 GRU+PATT 0.7490 0.7318 0.7769 0.7463 0.7380 0.7733 0.7546 0.7480 0.7824 Table 3 . 3Example of path attention.(infectious shock,complication, disseminated intravascular coagulation),(disseminated intravascular coagulation, complication, abdominal pain),( abdominal pain, complication, electrolyte disturbance),( electrolyte disturbance, medical department, emergency department) (beryllium poisoning, complication, pneumonia),( pneumonia, disease alias, lower respiratory infections),(lower respiratory infections, complication, pulmonary edema)Triple Score Path (infectious shock, medical department, emergency department) max: 0.3298 (infectious shock, disease alias, septic shock),( septic shock, medical de- partment, emergency department) min: 0.0565 (beryllium poisoning， complication, pulmonary edema) max: 0.9938 min: 0.0062 Neural generative question answering. J Yin, X Jiang, Z Lu, L Shang, H Li, X Li, 27Yin, J., Jiang, X., Lu, Z., Shang, L., Li, H., & Li, X. Neural generative question answering. , 27, pp.2972-2978 (2015) Collaborative Knowledge Base Embedding for Recommender Systems. F Zhang, N J Yuan, D Lian, X Xie, W Y Ma, ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. Zhang, F., Yuan, N. J., Lian, D., Xie, X., & Ma, W. Y. Collaborative Knowledge Base Em- bedding for Recommender Systems. ACM SIGKDD International Conference on Knowledge Discovery and Data Mining pp.353-362, (2016) Distant supervision for relation extraction without labeled data. M Mintz, S Bills, R Snow, J Dan, Joint Conference of the, Meeting of the ACL and the, International Joint Conference on Natural Language Processing of the Afnlp: Volume. Mintz, M., Bills, S., Snow, R., & Dan, J. Distant supervision for relation extraction without labeled data. Joint Conference of the, Meeting of the ACL and the, International Joint Con- ference on Natural Language Processing of the Afnlp: Volume, pp.1003-1011 (2009) Y Lin, S Shen, Z Liu, H Luan, M Sun, Neural Relation Extraction with Selective Attention over Instances. Meeting of the Association for Computational Linguistics. Lin, Y., Shen, S., Liu, Z., Luan, H., & Sun, M. Neural Relation Extraction with Selective Attention over Instances. Meeting of the Association for Computational Linguistics, pp.2124-2133 (2016) End-to-end relation extraction using lstms on sequences and tree structures. M Miwa, M Bansal, Miwa, M., & Bansal, M. End-to-end relation extraction using lstms on sequences and tree structures. (2016) Incorporating relation paths in neural relation extraction. W Zeng, Y Lin, Z Liu, M Sun, Zeng, W., Lin, Y., Liu, Z., & Sun, M. Incorporating relation paths in neural relation extrac- tion. (2016) Translating embeddings for modeling multi-relational data. A Bordes, N Usunier, J Weston, O Yakhnenko, International Conference on Neural Information Processing Systems. Bordes, A., Usunier, N., Weston, J., & Yakhnenko, O. Translating embeddings for modeling multi-relational data. International Conference on Neural Information Processing Systems, pp.2787-2795 (2013) Knowledge Graph Embedding by Translating on Hyperplanes. J Feng, AAAIFeng, J. Knowledge Graph Embedding by Translating on Hyperplanes. AAAI, (2014) G Ji, S He, L Xu, K Liu, J Zhao, Graph Embedding via Dynamic Mapping Matrix. Meeting of the Association for Computational Linguistics and the, International Joint Conference on Natural Language Processing. Ji, G., He, S., Xu, L., Liu, K., & Zhao, J. Knowledge Graph Embedding via Dynamic Map- ping Matrix. Meeting of the Association for Computational Linguistics and the, International Joint Conference on Natural Language Processing, pp.687-696 (2015) Knowledge graph completion with adaptive sparse transfer matrix. G Ji, K Liu, S He, J Zhao, Thirtieth AAAI Conference on Artificial Intelligence. Ji, G., Liu, K., He, S., & Zhao, J. Knowledge graph completion with adaptive sparse transfer matrix. Thirtieth AAAI Conference on Artificial Intelligence, pp.985-991 (2016). Fast parallel path concatenation for graph extraction. Y Shao, L Kai, C Lei, H Zi, B Cui, Z Liu, IEEE Transactions on Knowledge & Data Engineering. 99Shao, Y., Kai, L., Lei, C., Zi, H., Cui, B., & Liu, Z., et al. Fast parallel path concatenation for graph extraction. IEEE Transactions on Knowledge & Data Engineering, pp.99 (2017) Incorporating Selectional Preferences in Multi-hop Relation Extraction. The Workshop on Automated Knowledge Base Construction. R Das, A Neelakantan, D Belanger, A Mccallum, Das, R., Neelakantan, A., Belanger, D., & Mccallum, A. Incorporating Selectional Prefer- ences in Multi-hop Relation Extraction. 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[ "Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods", "Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods", "Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods", "Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods" ]
[ "Erdem Eren \nDepartment of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA\n", "Brandon Runnels \nDepartment of Mechanical and Aerospace Engineering\nUniversity of Colorado\nColorado SpringsCOUSA\n", "Jeremy Mason \nDepartment of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA\n", "Erdem Eren \nDepartment of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA\n", "Brandon Runnels \nDepartment of Mechanical and Aerospace Engineering\nUniversity of Colorado\nColorado SpringsCOUSA\n", "Jeremy Mason \nDepartment of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA\n" ]
[ "Department of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA", "Department of Mechanical and Aerospace Engineering\nUniversity of Colorado\nColorado SpringsCOUSA", "Department of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA", "Department of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA", "Department of Mechanical and Aerospace Engineering\nUniversity of Colorado\nColorado SpringsCOUSA", "Department of Materials Science and Engineering\nUniversity of California at Davis\nDavisCAUSA" ]
[]
The evolution of interfaces is intrinsic to many physical processes ranging from cavitation in fluids to recrystallization in solids. Computational modeling of interface motion entails a number of challenges, many of which are related to the range of topological transitions that can occur over the course of the simulation. Microstructure evolution in a polycrystalline material that involves grain boundary motion is a particularly complex example due to the extreme variety, heterogeneity, and anisotropy of grain boundary properties. Accurately modeling this process is essential to determining processing-structure-property relationships in polycrystalline materials though. Simulations of microstructure evolution in such materials often use diffuse interface methods like the phase field method that are advantageous for their versatility and ease of handling complex geometries but can be prohibitively expensive due to the need for high interface resolution. Discrete interface methods require fewer grid points and can consequently exhibit better performance but have received comparatively little attention, perhaps due to the difficulties of maintaining the mesh and consistently implementing topological transitions on the grain boundary network. This work explicitly compares a recently-developed discrete interface method to a multiphase field method on several classical problems relating to microstructure evolution in polcrystalline materials: a shrinking spherical grain, the steady-state triple junction dihedral angle, and the steady-state quadruple point dihedral angle. In each case, the discrete method is found to meet or outperform the multiphase field method with respect to accuracy for comparable levels of refinement, demonstrating its potential efficacy as a numerical approach for microstructure evolution in polycrystalline materials.
10.1016/j.commatsci.2022.111632
[ "https://export.arxiv.org/pdf/2203.03167v2.pdf" ]
250,144,863
2203.03167
4ebf55989e312536410a1f37ebc6cce0d349ea94
Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods Erdem Eren Department of Materials Science and Engineering University of California at Davis DavisCAUSA Brandon Runnels Department of Mechanical and Aerospace Engineering University of Colorado Colorado SpringsCOUSA Jeremy Mason Department of Materials Science and Engineering University of California at Davis DavisCAUSA Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods Discrete interface methodsdiffuse interface methodsfinite element analysisphase field methodmicrostructure evolution The evolution of interfaces is intrinsic to many physical processes ranging from cavitation in fluids to recrystallization in solids. Computational modeling of interface motion entails a number of challenges, many of which are related to the range of topological transitions that can occur over the course of the simulation. Microstructure evolution in a polycrystalline material that involves grain boundary motion is a particularly complex example due to the extreme variety, heterogeneity, and anisotropy of grain boundary properties. Accurately modeling this process is essential to determining processing-structure-property relationships in polycrystalline materials though. Simulations of microstructure evolution in such materials often use diffuse interface methods like the phase field method that are advantageous for their versatility and ease of handling complex geometries but can be prohibitively expensive due to the need for high interface resolution. Discrete interface methods require fewer grid points and can consequently exhibit better performance but have received comparatively little attention, perhaps due to the difficulties of maintaining the mesh and consistently implementing topological transitions on the grain boundary network. This work explicitly compares a recently-developed discrete interface method to a multiphase field method on several classical problems relating to microstructure evolution in polcrystalline materials: a shrinking spherical grain, the steady-state triple junction dihedral angle, and the steady-state quadruple point dihedral angle. In each case, the discrete method is found to meet or outperform the multiphase field method with respect to accuracy for comparable levels of refinement, demonstrating its potential efficacy as a numerical approach for microstructure evolution in polycrystalline materials. Introduction The simulation of physical systems often requires the modeling of moving interfaces. This could involve interfaces at the boundaries between different phases of matter including liquid/gas (e.g. cavitation), gas/solid (e.g. deflagration, sublimation, deposition), and solid/liquid (e.g. melting, solidification), or within a single solid phase. The grain structure of polycrystalline materials in particular contains an extensive network of interfaces known as grain boundaries that separate grains (contiguous regions with a given crystallographic orientation). The interfacial dynamics governing this grain boundary network are often complex, leading to topological transitions and rapidly changing grain morphologies that pose a unique type of modeling challenge. A material's grain structure is essential to its macroscopic properties. For example, solute segregation to grain boundaries can change the grain boundary cohesive energy [1,2], or the material's susceptibility to hydrogen embrittlement [3,4] and stress corrosion cracking [5,6]. Grain boundaries can provide preferential sites for the precipitation of a second phase, increasing or decreasing the plasticity of the polycrystal [7,8]. They provide obstructions to the propagation of slip, with implications for the strength of the material as evidenced by the Hall-Petch equation [9,10]. Given such consequences of the grain structure, it is not surprising that a variety of methods to simulate grain boundary motion and the evolution of the grain structure have been proposed in the literature [11][12][13][14][15][16]. Many of these represent the grain boundaries implicitly, as the locus of points where an indicator function abruptly changes value; such methods are referred to as diffuse interface methods below. While this has the advantage of not requiring that changes to the grain boundary topology be handled explicitly, the implicit representation complicates simulations of grain boundaries whose properties depend on their crystallography. This is not an issue when investigating the properties of generic grain boundary networks since a microstructure with constant and isotropic grain boundary properties is regarded as the canonical model system [17]. It does limit the possibility of predicting the properties of physical materials though, and therefore is a significant obstacle to realizing the vision of integrated computational materials design. Methods that simulate the evolution of three-dimensional grain structures explicitly [18][19][20][21] are referred to as discrete boundary methods below. One of the main difficulties faced by such methods is with maintaining a consistent mesh of the grain boundary surfaces during a topological transition. Indeed, there did not even appear to be a way to enumerate a broad class of possible topological transitions in a general grain boundary network until quite recently [22], and it is not at all obvious how to explicitly implement such transitions without at least knowing what they are. Nevertheless, discrete boundary methods do offer several distinct advantages with respect to microstructure modeling. They often require far fewer mesh points than their diffuse counterparts, offering a dramatic reduction in runtime computational cost. They also allow various defect properties, including grain boundary energies and triple line energies, to be explicitly defined in a way that is difficult with diffuse boundary methods. Along these lines, Kuprat previously developed GRAIN3D to simulate grain growth [19] on a volumetric mesh with the gradient weighted moving finite element (GWFE) method, though the proposed topological transitions were not necessarily physical and could substantially affect the microstructure trajectory. Shya and Weygand instead proposed a method to simulate grain growth on a surface mesh and handled topological transitions by decomposing them into sequences of elementary operations [20], but did not offer any assurance that such decompositions would not change the microstructure trajectory. Lazar et al. [21] proposed a discretized formulation of the MacPherson-Srolovitz relation [23] to simulate ideal grain growth on a surface mesh. While this only required that a small number of topological transitions be implemented, the explicit assumption of isotropic grain boundary properties precluded simulations of more general systems. Two of the authors recently proposed a discrete interface method that addresses several of the computational challenges associated with explicit microstructure meshing, including a way to construct all possible topological transitions around a junction point and an energetic criterion to select one to apply [22]. The implementation [24] is based on SCOREC [25] and uses a volumetric microstructure mesh, potentially allowing the addition of other necessary physics to build a general framework for realistic simulations of microstructure evolution. SCOREC is an open source, massively parallelizable finite element framework with the adaptive meshing capabilities that are necessary to reach representative material volumes and to efficiently maintain the mesh element quality and desired degree of refinement. SCOREC is specifically able to improve the quality of low-quality elements by local remeshing operations that minimally disturb the embedded surface mesh and make the computational expense of many operations, e.g., collapsing an individual grain, constant with respect to the system size. The remeshing operations can also be used to refine a microstructure mesh. For example, a polycrystalline microstructure consisting of Voronoi polyhedra can be converted into a microstructure mesh by initially placing a single vertex on the interior of each boundary line, boundary surface, and grain volume, and subsequently refining using the mesh adaptation capabilities of SCOREC. This work offers an initial comparison between this discrete interface method (detailed further in Ref. [22]) and a more well-established phase field approach. This paper introduces a set of three test cases to evaluate the relative accuracy and numerical cost of simulations of grain boundary motion, and uses this set to compare the discrete interface and phase field methods. The three cases correspond to several of the simplest configurations involving the motion of a grain boundary surface, a triple junction (TJ), and a quadruple point (QP). The grain boundary properties are assumed to be isotropic; a coarsening grain structure with isotropic grain boundary properties is said to be the ideal grain growth system, and provides a basis for the comparison of all other evolving grain structures. Analytical forms for the evolving geometries are known for the spherical surface and TJ cases [26,27], and the TJ and QP configurations have well-defined steady-state geometries. It is also of interest whether the two methods converge to the same geometries in situations for which analytical solutions are not known, since there is likely no other way to verify the simulations in such cases. While several of these configurations have been studied before, they are not usually considered in conjunction despite the benefits of doing so. Namely, the increasing complexity of the grain boundary configurations among the three test cases introduces different sources of systematic error to the grain boundary motion, and these errors can be more easily identified by comparing the test cases to one another. It is desirable to establish the nature of any systematic errors and the accuracy of the simulation methods for a system with isotropic grain boundary properties before attempting to do so with more general grain boundary energy and mobility functions. The two methods considered in the present work will be capable of simulating the motion of grain boundaries with anisotropic properties when such functions become available. The discrete boundary method uses equations of motion that allow for general grain boundary properties and grain boundary lines that join an arbitrary number of grain boundaries [28]. The multiphase field model was developed to simulate the faceting of grain boundaries with energies that depend on boundary plane orientation, though this requires calculating a fourth-order derivative of the order parameters [29]. This paper is structured as follows. We begin with a discussion of the discrete interface method and its implementation, followed by an analogous discussion for the phase field/diffuse interface counterpart. We then apply both methods to a set of three test cases: a two grain system (shrinking sphere), a three grain system (triple junction), and a five grain system (quadruple point). The behavior of the discrete and diffuse models is compared for each of the examples vis-à-vis analytic predictions and the models' internal length scales. The performance of the discrete interface model is briefly discussed, and then we conclude with a general discussion of the behaviors of the two models and a summary of recommendations for best practice in discrete interface modeling. Methods Assuming that grain boundary properties are independent of grain boundary crystallography implies that the grain boundary network evolves along the negative gradient of the total boundary area. This is usually expressed by means of the Turnbull equation [30] v = mγKn (1) governing the motion of each boundary patch where v is the velocity, m and γ are the mobility and energy per unit area, K is the mean curvature (the sum of the principle curvatures), and n is the unit normal vector. While this is sufficient to determine the time evolution of a closed surface, the Turnbull equation does not specify what happens at the TJs or QPs of the grain boundary network. One of the essential differences between discrete and diffuse interface models is the governing equations for precisely these locations. Discrete interface models generally represent the TJs and QPs as distinct entities with explicit geometries, and sometimes provide additional governing equations specific to these locations [21]. This is in contrast to the implicit approach of most diffuse interface methods which do not track TJs or QPs explicitly (while some diffuse interface methods do include higher order terms to account for the distinct behavior of line or point defects, these can come at extreme computational cost). Each surface instead evolves according to the Turnbull equation with the geometric singularities at the TJs and QPs regularized by the diffuse interfaces. This difference in the handling of TJs is significant since the TJs define the geometric conditions at a grain boundary's edges, thereby constraining the evolution of the grain boundary surface and likely the overall microstructure trajectory. It is for this reason that the angles between adjoining grain boundary surfaces are often used as simple scalar measures of the simulation accuracy in Section 3 below. Discrete interface model As implied by the name, every discrete interface model uses a discrete representation of the grain boundary network. A discrete representation entails that the grain boundary network geometry is represented by a collection of simple geometric objects, or elements, along with a description of how to join those elements together. The result is known as a surface mesh in three dimensions, and can be advantageously extended to a volumetric mesh to provide a discrete representation of the grain interiors as well. VDlib [22,24] is a C++ library based on SCOREC [25] that represents a grain structure by means of a volumetric mesh containing tetrahedra, triangles, edges, and vertices. There are two operations involved in updating the mesh to evolve the microstructure. The first moves the vertices of the mesh according to established equations of motion [28] that allow for anisotropic surface energies and arbitrary drag coefficients (the counterpart to the usual grain boundary mobility). The idea is that the velocity v of any given vertex should be such that the driving force F on the vertex is precisely balanced by the sum of drag forces Dv resulting from the motion of the adjoining grain boundary elements, where D is the drag tensor and v is the grain boundary velocity. The capillary force acting on the vertex is given by F = it i τ l (t i ) + 1 2 ||t i || j:{i,j}∈∆ (n ij ×t i )γ(n ij ) +n ij ∂γ ∂φ i n ij ,(2) where τ l and γ are the line and surface energy functions, t i is the vector along edge i starting at the vertex andt i is the corresponding unit vector,n ij is the normal of the triangle formed by edges i and j, j : {i, j} ∈ ∆ indicates an edge j starting at the vertex such that edges i and j span a triangle ∆, and φ defines the surface orientation around edge i; Fig. 1 shows several of these quantities for a generic vertex of a surface mesh. At force equilibrium the capillary forces are balanced by the drag forces Dv of the moving boundaries with D = δ 0 I + 1 2 i δ 1 (t i )||t i ||(I −t i ⊗t i ) + 1 6 i,j∈∆ δ 2 (n ij )||t i × t j ||(n ij ⊗n ij )(3) where δ k is the drag term associated with the k-dimensional simplicial boundary element. The resulting boundary vertex velocity v is given by v = D −1 F .(4) One advantage of this formulation is that the motion of every boundary vertex is governed by the same equation, including those on the interiors of surfaces, along TJs, and at QPs. If the point and line drag terms are zero, Dv reduces to the sum of the drag forces exerted by the neighboring triangles along the triangle normal directions for a given velocity v. Moreover, if the grain boundary properties are constant, then δ 2 = 3/m and this further reduces to a discrete version of Eq. (1) with an accuracy that depends on the product of the edge length and the mean curvature of the surface. Apart from the motion of the mesh vertices, the accuracy of the discrete interface model is highly dependent on the element quality, where low-quality elements do not resemble equilateral triangles or tetrahedra [31,32]. Without regular intervention and adaptation of the mesh, the quality of mesh elements generically degrades with grain boundary motion, even to the point of elements inverting. The discrete interface method handles this by using MeshAdapt [33] to locally remesh where the element quality falls below a threshold value, and coarsening or refining edges with lengths below or above threshold values. The target edge length e is constant in time and space for any given simulation, and an edge is coarsened or refined if the edge length l is outside the interval 0.7 e ≤ l ≤ 1.5 e . These operations are used sparingly though, since apart from the computational expense local remeshing can perturb the grain boundary geometry. Specifically, these operations are the source of the discontinuous jumps observed in the discrete interface model results in Section 3. Diffuse interface model Comparison to a standardized diffuse interface model provides verification of the discrete interface model. In this work we apply the multiphase field model implemented following the presentation in Refs. [34,35] which are general references for this section. A brief overview is provided here. For a system in a region Ω ⊂ R 3 with N grains, N order parameters (denoted as the vector of functions η = {η 1 , . . . , η N } ⊂ C 2 (Ω)) are defined such that the region occupied by the ith grain is precisely the support of η i . The free energy of the system is then defined to be W [η] = Ω w(η) + 1 2 n k |∇η n | 2 dx,(5) where w is the chemical potential and k is a model parameter to be discussed subsequently (the use of functional brackets should be understood to indicate dependence on the argument and any temporal or spatial derivatives). The following polynomial form is used for the chemical potential: w(η) = µ n 1 4 η 4 n − 1 2 η 2 n + 3 4 m>n η 2 m η 2 n , µ = 3.26.(6) The coefficient for the boundary term is related to the grain boundary energy γ by k = 3 GB 4 γ,(7) where GB is the diffuse boundary width. The evolution of η, which determines the overall evolution of the microstructure, follows an L 2 gradient descent to minimize Eq. (5). The resulting kinetic evolution equation, expressed in terms of the variational derivative, is ∂η n ∂t = −L δW δη n ,(8) where the rate coefficient L is related to the traditional boundary mobility m by L = 4 3 m GB .(9) Phase field simulations are often computationally costly, and this has led to a variety of methods to accelerate them. Spectral methods can result in a substantial performance increase [36][37][38], though this comes at the cost of limited resolution of fine features and the restrictive requirement that the computational domain be periodic. Real-space (non-spectral) methods instead require strategic meshing techniques, such as adaptive mesh refinement (AMR) [38], to avoid prohibitively excessive mesh size. However, as with the discrete interface method, they can be easily implemented in non-periodic systems and systems with complex geometry. Therefore, real-space methods with adaptive mesh refinement are the most appropriate benchmark against which to compare the present discrete interface method. In this work, all diffuse boundary calculations are performed using Alamo, a high performance multiphysics code that uses block-structured adaptive mesh refinement (BSAMR) with a strong-form elasticity solver to perform diffuse interface calculations [39]. Alamo is built on the AMReX package, developed by Lawrence Berkeley National Laboratory [40]. All of the results presented here were run on a desktop computer and generally completed in less than an hour depending on the chosen parameters. Of particular interest is the convergence of the solution with respect to the boundary width, GB , which determines the diffuse boundary length scale. The exact solution is recovered as GB → 0, but this comes at the expense of increased computational cost. In this work we are particularly interested in the relationship between GB and the discrete interface model counterpart. Topological transitions As stated in the introduction, one motivation for using diffuse interface methods for microstructure evolution is that the implicit nature of the grain boundaries allows topological transitions to occur without requiring that all possible transitions be explicitly enumerated. The purpose of this section is to show that the challenge of enumerating and implementing such topological transitions for a discrete interface method is in fact surmountable [22]. This is accomplished by simulating the evolution of a non-generic grain structure that, despite the grain boundary properties being uniform and isotropic, involves topological transitions that are not generally handled by discrete interface methods [19,21]. The initial grain structure in Fig. 2 contains a central rectangular prismatic grain surrounded by six other grains, the top one being removed for visual clarity. For the discrete interface model on the top row, the initial topological transitions involve four triple lines collapsing into four triangular faces in Fig. 2b; this is a standard topological transition implemented in nearly all discrete interface methods. The high symmetry of the initial condition subsequently results in the central grain detaching from the four side grains in Fig. 2c, with the four triangular faces that were previously introduced merging into an annulus around the central grain. Such transitions and the resulting configurations would be difficult for other existing discrete interface methods. While the method proposed by Syha and Weygand [20] could in principle handle such transitions, their assumption that junction lines are always bounded by junction points would be invalidated after the transition in Fig. 2c. The central grain shrinks to the point of vanishing in Fig. 2d, and the structure has reached an effectively stable configuration in Fig. 2e. Note that due to the anisotropy of the mesh and the adaptive remeshing perturbing the mesh slightly, the symmetrical transitions (e.g. collapse of the vertical triple lines just before Fig. 2b) did not occur exactly simultaneously. The evolution of the same configuration in the diffuse interface model is quite different. Each grain's boundaries were constructed as the surfaces where the value of the corresponding order parameter reached 0.5 (the interaction of the underlying grid with the initial conditions produced the ridges visible in Fig. 2f). The central grain shrinks preferentially in the out-ofplane direction in Fig. 2g, with four triangular faces appearing at the corners of the central grain shortly before the central grain completely separates from the adjacent grains in the horizontal direction. That the geometric and topological evolution of the central grain should be different than in the discrete interface method is expected given the finite width of the diffuse boundaries. Specifically, whenever two approaching boundaries are separated by a distance on the order of the boundary width, the gradients in the order parameter representing the two boundary interact, changing the effective boundary energy and mobility. This effect is more than a postprocessing artifact of the surface reconstruction, and can change the microstructure trajectory in ways that resemble the differences in behavior between wet and dry foams [41]. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) While the discrete and diffuse interface methods converge to effectively the same configurations in this case (Figs. 2e and 2j), microstructure trajectories are often unstable with respect to such perturbations in the proximity of a topological transition. This phenomenon is beyond the scope of the current paper though, a detailed study of the evolution of surfaces, triple, points, and quadruple points in the absence of topological transitions (and as is performed here) being a necessary preliminary. Results and discussion Three cases are considered in this section to quantify the systematic error of the discrete interface and phase field methods of simulating grain boundary motion. The first is a spherical grain which evolves in a self-similar way; this is a standard configuration that is often used in the literature to verify that the Turnbull equation is obeyed in the absence of complicating factors [16,19,[42][43][44]. The second is a TJ that migrates along a semi-infinite grain boundary [16,[45][46][47], eventually reaching a steady state configuration with a known profile and velocity [26]. The quantity considered below is the angle of the grain boundaries at the TJ, though in principle a stricter validation scheme could involve evaluating the simulation's ability to precisely reproduce the expected grain boundary geometry. The third is a columnar hexagonal grain configuration that migrates along semi-infinite grain boundaries to allow a study of the steady state evolution of a QP [16,48]. Perhaps the reason this case appears less often in the literature is that an analytical solution for the boundary profile is not known; instead, the angles between grain boundary traces on two cross-sections are evaluated for convergence and used to compare the two simulation methods. The grain boundary geometries for the three test cases are described in their respective sections, have Neumann boundary conditions, and are constructed to make the grain boundary curvatures comparable. It is expected that the accuracy of both the discrete and diffuse interface models will increase with decreasing internal length scale , denoted as = e for the discrete model and = GB for the diffuse. However, the accuracy cannot depend on any absolute length scale since then the accuracy could be improved simply by uniformly scaling the grain structure. The accuracy therefore depends on relative to a second length scale that is characteristic of the evolving interface. Since the accuracy should be invariant to the isometries of Euclidean space, the inverse of the interface's mean curvature is the natural candidate for the second length scale, and the accuracy of both models is expected to depend on the dimensionless product of and interface's mean curvature. More precisely, all of the errors reported in this section are expected to be power laws in , with the prefactor depending on the mean curvature and implementation details in a way that is difficult to parameterize (only the spherical grain has the same mean curvature everywhere). For this reason, only the exponent of is generally reported in the following. Many of the quantities reported below are nondimensionalized following the procedure in Appendix A to facilitate the comparison of the discrete interface and phase field methods. A tilde indicates a nondimensionalized variable (with the exception of e and GB which are always nondimensionalized) and an analytical prediction is denoted by the subscript t, e.g., r t (t) is the analytical prediction for the nondimensionalized radius of the sphere as a function of nondimensionalized time. The equations of motion of the discrete interface method were integrated using a second order Runge-Kutta scheme with a maximum nondimensionalized time step of 1.2500 × 10 −5 . Spherical grain The spherical grain case is intended to reveal the error when modeling surface motion in the absence of confounding effects from other grain boundary network components. One advantage of this particular choice is that, provided the grain boundary properties are constant and isotropic, the evolution of a spherical grain is known analytically. As derived in Appendix B, the sphere shrinks uniformly with radius r t (t) = r 2 0 − 4mγt(10) as a function of time. Nondimensionalizing this equation reveals that a sphere starting with a radius ofr t (t) = 1 vanishes att = 0.25. The actual simulations deviate from Eq. (10) both because the initial geometries shown in Fig. 3 are not precisely spheres and because the Turnbull equation in Eq. (1) is not precisely followed, though these sources of error are reduced as the are made smaller. Since the diffuse interface model doesn't perform well when the radius of the sphere approaches the grain boundary width, the magnitude of the error for the shrinking grain is quantified by the deviation of the sphere half-life t half from the analytical prediction t half,t = 3r 2 0 /(16mγ). When nondimensionalized, this reduces tõ t half,t = 3/16. Figure 4 shows the performance of the two models, with the discrete interface model on the left and the diffuse interface model on the right. The top row shows the radius of the sphere as a function of time, where the color indicates the internal length scale and the exact solution is in black. The roughness of the curves for the discrete interface model is due to remeshing to preserve the element quality, and the velocity in the diffuse interface model falls as the radius approaches the grain boundary width. The magnitude of the relative error in the radius as a function of time is shown in the middle row. The error for the discrete interface model is caused by the magnitudes of the surface vertex velocities being larger than predicted by the analytical solution, perhaps as a consequence of the equations of motion being explicit and uncoupled. That the accumulation of error accelerates with decreasing radius supports the hypothesis that the error generally depends on the product of e and the mean curvature. Meanwhile, there are likely two sources of error that contribute to the results for the diffuse interface model. The error at early times is a postprocessing artifact that occurs when constructing isocontours to identify the location of the grain boundary, effectively resulting in an offset to the sphere radius. The other source of error relates to the order parameter gradient at a grain boundary patch being affected by the presence of nearby patches. This is most visible when the grain is about to collapse and grain boundary patches on opposite sides of the grain interact, reducing the gradient magnitude and the grain boundary velocity. Conversely, the mean curvature of the surface causes neighboring grain boundary patches to interact, increasing the gradient magnitude and the grain boundary velocity at earlier times. As with the discrete interface model, the magnitude of this effect at earlier times is proportional to the product of GB and the mean curvature. The bottom row of Fig. 4 shows the half-life error |t half −t half,t | as a function of the internal length scale. A conjugate gradient minimization algorithm and bootstrapping were used to fit |t half −t half,t | to a power law in the internal length scale . This gives an exponent of 1.37 ± 0.21 for the discrete interface model and 0.678 ± 0.085 for the diffuse interface model, where the values are the medians and the uncertainties are half the interquartile range. While the exponents could suggest that the error of the diffuse interface model decays slower than that of the discrete interface model with decreasing internal length scale, the errors in the apparent grain radius due to isocontour construction during postprocessing do not actually affect the microstructure trajectory. This could motivate using the two-grain configuration with self-similar evolution analyzed by Mullins [26] in the future since such postprocessing errors would likely not affect the long-time behavior. Triple junction The purpose of the TJ case is to include a TJ in the moving boundary while keeping the grain configuration as simple as possible, ideally allowing the error of the equations of motion for the TJ to be identified by comparing the results to those for the spherical grain. The initial geometries of the grain configuration are shown in Fig. 5, are constant in the out-of-plane direction, and have mirror boundary conditions in the lateral directions. The rate of volume change of the top grain can be derived by applying the von Neumann-Mullins equation [26,49] to the two-dimensional grain configuration in a plane perpendicular to the TJ. Since there is one triple point per simulation cell in this plane, the rate of cross-sectional area change of the top grain per simulation cell width L is πmγ/3, and the rate of volume change of the top grain can be found by multiplying by the TJ length. Mullins actually went further and solved for the steady-state profile of the moving boundary assuming constant and isotropic grain boundary properties [26]. If x is distance from the left edge of the simulation cell and y is height from the top of the red grain, then the steady-state profile of the grain boundary between the red and blue grains is y(x) = − ln[cos(πx)]/π.(11) The width of the simulation cell as defined by the above equation would be L = 1/3, and is appropriately scaled to the actual dimensions of the simulation cell. The dihedral angle θ T J between the two boundaries of the blue grain is perhaps the simplest way to evaluate the accuracy of the geometry of the moving boundary in the vicinity of the TJ. A force balance argument for constant and isotropic grain boundary properties (and in the absence of any TJ drag) leads to the condition θ T J,t = 2π/3. Moelans et al. [45] provides equations for the expected rate of area change and equilibrium junction angle for the more general situation where the grain boundary energies depend on misorientation, and uses these to evaluate the relative accuracy of two different diffuse interface methods, but does not perform a scaling analysis as is done below. The expected equilibrium junction angle for the constant and isotropic grain boundary case is roughly enforced in the initial conditions by defining the two parts of the moving boundary to be the appropriate sections of of cylinders; while this is not the steady-state profile given by Mullins, it is sufficiently close for a short initial transient and rapid convergence to the steady-state condition as is visible in Fig. 6. As before, results for the discrete interface model are on the left and those for the diffuse interface model are on the right. The top row shows θ T J as a function of time, where the color indicates the internal length scale and the exact solution 2π/3 is in black. The roughness of the curves for the discrete interface model is due to the remeshing required to maintain element quality, and the periodic spikes that appear for the diffuse interface model are due to the interaction of the adaptive mesh refinement and the construction of the isocontours. The error in θ T J (measured as the median of the second half of the time series) is shown in the bottom row, with the dependence of the steady-state angle on e for the discrete interface model being a consequence of the linear elements forcing the grain boundary curvature to be concentrated at the vertices and edges of the mesh. Specifically, the grain boundary curvature that is distributed to the TJ edges causes the deviation of θ T J from the expected value, with the magnitude of the deviation depending on the product of e and the mean curvature of the adjoining grain boundary. Identifying the precise location of the TJ and the value of θ T J is more difficult for the diffuse interface model since the grain boundary geometry is implicit. The procedure followed here involves fitting third-and fourth-order polynomial approximations to each side of the isocontour where the order parameter for the top grain is 0.5. The triple point location in the plane is then defined to be the point of intersection of the polynomials, and θ T J is the angle between the tangent vectors at the point of intersection. This process works well in the sharp interface limit, but is very sensitive to perturbations in the solution for larger GB since there is substantially more error in the predicted location of the TJ with respect to the simulation size. The occasional deviations that are observed in the steady-state correspond to BSAMR re-gridding events. Fitting a power law in the internal length scale to |θ T J − θ T J,t |/π gives an exponent of 0.91 ± 0.20 for the discrete interface model and 1.45 ± 0.13 for the diffuse interface model, where the values are the medians and the uncertainties are half the interquartile range. The additive offset of (−0.005 ± 0.011)π to the expected value of θ T J for the discrete interface model is entirely consistent with the TJ angle converging to the equilibrium angle in the e → 0 limit, though at a lower rate than the half-life error magnitude in Fig. 4. This is not unexpected though, since the TJ can be thought of as a jump condition in the tangent plane to the grain boundary that is both difficult to accurately reproduce with a finite element mesh and is not present in the spherical grain case. While the exponent for the diffuse case is nominally higher, this is not reflective of the trend observed for small GB where the saturation in the error is likely the result of inaccuracy in the postprocess calculation of the angle. The higher exponent therefore does not necessarily indicate better convergence. Quadruple point As with the TJ case, the grain structure for the QP case consists of a top grain above several columnar grains. The grain boundaries of the top grain migrate down the simulation cell, consuming the columnar grains and eventually reaching a steady-state profile, though an analytical solution for this profile is not known. The configurations of columnar grains for the discrete and diffuse interface models are shown in Figs. 7a and 7b respectively, with the hexagonal cross-sections of the columnar grains clearest for the discrete interface model; the BSAMR mesh makes simulations of rectilinear domains like the one in the figure strongly preferable for the diffuse interface model. Following the initial transient, the steady-state profile is examined on the two planes indicated in Fig. 7c, one along a minor diameter of the central grain and bisecting a TJ, the other along a major diameter of the central hexagonal grain and containing a QP. The angles along these profiles at the intersections with the TJ and the QP are reported in Fig. 8 and Fig. 9. While the equilibrium angle at the TJ should be 2π/3 (the same as for the TJ in Section 3.2), the curvature of the grain boundaries in both principal directions could change the rate of convergence to 2π/3 with decreasing compared to the TJ case. As for the equilibrium angle at the QP, an infinitesimal neighborhood of the QP will contain triple junction lines in a tetrahedral configuration connected by flat grain boundary surfaces provided the principal curvatures of the grain boundaries are finite. This allows the equilibrium angle of cos −1 (−1/ √ 3) ≈ 0.696π at the QP along the major diameter to be found by geometrical considerations. Starting with the TJ angle, observe that the data points for the TJ angle error along the minor axis in the bottom row of Fig. 8 closely resemble those for the TJ angle error in the bottom row of Fig. 6. This indicates that the nonzero second principal curvature of the grain boundaries along the TJ lines in the QP case does not have a significant effect on the error in the equations of motion, and is consistent with the expectation that the error should scale with the mean curvature (the sum of the principal curvatures). Fitting a power law in the internal length scale to |θ T J − θ T J,t |/π gives an exponent of 0.927 ± 0.223 for the discrete interface model and 0.85 ± 0.50 for the diffuse interface model, with both models converging to the expected value. While the exponent for the discrete interface model is nearly identical to that for the TJ case, the lower exponent for the diffuse interface model is likely a consequence of a power law fitting the data relatively poorly; observe that the TJ angle error for the diffuse interface model does not fall on a line on a log-log plot, and instead seems to saturate at a lower bound set by the angle estimation procedure in postprocessing. For the QP angle, the final values for the discrete interface model follow a power law in that converges to an angle of (0.694 ± 0.001)π with an exponent of 0.958 ± 0.026, whereas the respective values for the diffuse interface model are (0.707 ± 0.011)π and 0.85 ± 0.45; the limiting values for both the discrete and diffuse interface models effectively coincide with the exact value. It is significant that the errors for all of the discrete interface results in Secs. 3.2 and 3.3 decay with exponents that are close to one. The discrete interface method uses linear elements that approximate the grain boundary geometry with first-order accuracy, meaning that an exponent of one is the best possible result. It is likely that higher-order elements would need to be used to substantially increase the rate of error decay with e . The irregularity in the exponents for the diffuse interface model in Secs. 3.2 and 3.3 is attributed to the error in the polynomial algorithm used to extract the grain boundary profile. Examination of Figs. 6 and 8 indicates that this functions as a source of random error that is larger for highly diffuse boundaries but vanishes in the sharp boundary limit. Performance When selecting a numerical method in practice, computational cost is often nearly as much a concern as the accuracy of the simulated behavior. This section specifically considers the dependence of the discrete interface method's computational cost on the internal length scale e ; given that the diffuse interface method's implementation [39,40] is considerably more mature than that of the discrete interface method [22,24], and our concern is with asymptotic behavior rather than implementation specifics, a comparison with the diffuse interface method is omitted. Suppose that the main contribution to the computational cost is evaluating the equations of motion for the grain boundary vertices. The number of such vertices is expected to depend on the internal length scale as −2 e . If the velocity of the vertices is independent of e , then the time step length should decrease as e to keep the vertex displacement shorter than the characteristic edge length and prevent mesh element inversion. This would imply that the overall computational cost should scale with −3 e , or as the product of the number of grain boundary mesh vertices and the number of time steps for a given overall simulation time. , respectively, for small e where the computational cost of the vertex calculations is expected to dominate. This confirms that the overhead of the discrete interface method (mesh management, enumeration of topological transitions, etc.) is relatively small compared to the evaluation of the equations of motion. This overhead includes the local remeshing operations that are used to maintain the mesh quality and that occur at a frequency proportional to the time required for the interface to travel a distance e . Further evidence that the computational cost of the remeshing operations is small relative to that of evaluating the equations of motion is given in Ref. [22], which also reports results for the evolution of a more extensive grain structure. The scaling of the normalized runtime cost observed here is not consistent with the −3 e scaling expected in the previous paragraph though. This discrepancy is a result of the length of the median time step scaling as 1.916 e instead of linearly; the underlying cause for this time step scaling is investigated further in Appendix C. Conclusion The purpose of this work has been to establish the validity and performance of a recentlydeveloped discrete interface method by comparison to analytic solutions and a well-established multiphase field method. More specifically, the evolution of the simplest configurations involving surfaces, triple lines, and quadruple points with self-similar behavior given constant and isotropic grain boundary properties are used to quantify the error in position and junction angles as a function of the degree of refinement. The boundary types are simple enough to be amenable to analysis, yet complex enough to introduce different systematic errors over the course of their evolution. Despite the approaches for simulating boundary motion being distinctly different, our results indicate that both methods converge to the same junction angles with similar rates. The most significant difference is that when predicting the half life of the shrinking sphere, the convergence rate of the diffuse interface method appears to be about half of that of the discrete interface method. Although this work assumes constant and isotropic grain boundary properties, both methods were developed with the intention of performing simulations for anisotropic grain boundary properties. The integration of an accurate grain boundary energy for arbitrary orientation relationships and interface orientations is a current challenge in microstructure modeling. Morawiec [50] suggested that the grain boundary energy could be experimentally obtained as a function of the grain boundary crystallography by applying the Herring condition [51,52] to triple junctions imaged by three-dimensional microscopy techniques [53,54]. Alternatively, molecular dynamics simulations allow direct evaluation of grain boundary properties in bicrystals for a large but not exhaustive subset of the five-dimensional grain boundary space [55] . While the excessive number of points required to adequately sample this space of has precluded the availability of general grain boundary energy and mobility functions in the literature, there has been progress for particular subsets of grain boundaries [56]. Models have also been presented that can accurately predict grain boundary energy for most known orientations [57], and have been used in combination with phase field to predict such behavior as faceting and disconnection migration [58]. However, in addition to the general problem of obtaining accurate grain boundary energy, the nonconvexity of this function induces numerical issues that must be handled explicitly [58]. The extension of this present framework in that direction shall therefore constitute future work. The performance of the discrete interface model lends confidence in its ability to yield accurate results for more general and complex microstructures for which there is no known analytic solution. Moreover, the performance with unoptimized code indicates reasonable scaling behavior that is close to the ideal scaling and comparable to that of alternative methods. Appendix A. Nondimensionalization Define the variable L to be the characteristic length scale of the grain structure defined in Section 3. For a sphere it is the sphere radius, for the TJ it is the length of the simulation cell in the direction normal to the consumed grain boundary, and for the QP it is the hexagonal grains's minor diameter. The Turnbull equation in Eq. (1) suggests that there is a characteristic time scale τ = L 2 /(mγ). The simulations are performed with nondimensionalized timet = t/τ , nondimensionalized spacex = x/L, nondimensionalized rate of volume change dṼ /dt = (τ /L 3 )dV /dt, etc. With respect to the quantities defined in Section 2.1, suppose that τ l (t i ) = 0, δ 0 = 0, δ 1 (t i ) = 0, and that γ(n ij ) and δ 2 (n ij ) are constants. The governing equations of the discrete interface model then reduce to: In this work, all multiphase field calculations are performed with dimensional values and then nondimensionalized for comparison to discrete interface simulations. Setting t 0 = 0 and r t (0) = r 0 and integrating gives r t (t) = r 2 0 − 4mγt (B.2) as the solution to this differential equation. Since the characteristic length scale for a sphere is r 0 , nondimensionalizing reduces this tõ r t (t) = 1 − 4t (B.3) for the black curve in Fig. 4. Appendix C. Scaling analysis As described in Sec. 4, while the computational cost of the discrete interface method is expected to scale as −3 e , the actual scaling is instead −4.089 e . Closer investigation revealed that the time step could decrease or increase by multiple orders of magnitude depending on the presence of various local mesh configurations. The boundary triangles exert capillary forces only in the boundary plane, yet contribute drag forces only in the out-of-plane direction. This allows vertices on nearly-flat grain boundary sections to experience arbitrarily large lateral velocities, slowing the simulation down as the time step is reduced to prevent element inversion. The discrete method simulations in Sec. 3.1 include an isotropic contribution D I,d = A 2 m /(md)I to the drag tensor such that v = (D + D I,d ) −1 F , where A 2 m is the mean triangle area over the whole simulation, m is the mobility, and d = 1000 is a drag ratio. Decreasing the drag ratio reduces the lateral velocities, but also slows down the actual motion of the boundary and introduces a systematic error. As an alternative, a contribution to the drag tensor that only acts in the in-plane directions could be constructed as follows. For simplicity, consider a closed disk of coplanar triangles around a vertex. Iterating over each grain boundary triangle ∆ ij adjacent to the central vertex, find the relative positions of the other vertices from the central vertex p i and p j and construct the outer product of the difference p i − p j with itself. Let Λ max be the largest eigenvalue of the sum of the outer products, and define the matrix C = i,j∈∆ (p i − p j ) ⊗ (p i − p j )/Λ max . The anisotropic drag tensor contribution D a,d = A 2 m /(md)C by construction has no effect on the grain boundary motion in the plane normal direction. This should allow the lateral velocities of boundary vertices to be reduced while introducing less systematic error in the motion of non-planar boundaries than for an isotropic drag. An example triple junction mesh configuration is shown in Fig. C.11 to qualitatively demonstrate the effect of different drag tensor correction terms. Although the velocity associated with D I,d aligns with the force direction faster with increasing d, the velocity term in the vertical direction is also attenuated more compared to D a,d . The difference in the expected and the actual scaling of the cost can largely be attributed to the non-linear scaling of the median time step dt med shown in Figure 1 : 1Vectors describing the geometry around a vertex of the surface mesh. The central vertex is connected to five edges t i and five triangles with unit normal vectorsn ij . The TJ along the edges t 1 and t 4 is shown in bold. Figure 2 : 2Geometric and topological changes in a structure where the central grain is initially a rectangular prism surrounded by six grains. (a)-(e) and (f)-(j) show the structure after corresponding elapsed times in the discrete and diffuse interface models, respectively. Figure 3 : 3Initial geometries of the shrinking spherical grain within another grain for the (a) discrete and (b) diffuse interface methods. Figure 4 : 4Comparison of shrinking spherical grain results for the discrete model (left) and the diffuse model (right); all quantities are nondimensionalized. (Top row) Plot of radius vs time, with color indicating the length scale and the exact solution in black. (Middle row) Plot of relative error in the radius vs time, with color indicating the length scale. (Bottom row) Plot of half-life error magnitude as a function of length scale. Figure 5 : 5Initial geometries of the TJs for the (a) discrete and (b) diffuse interface methods. The structures have mirror boundary conditions in the lateral directions. Figure 6 : 6Comparison of θ T J for discrete model (left) and diffuse model (right); all quantities are nondimensionalized. (Top row) Plot of θ T J vs time, with color indicating the length scale and the exact solution in black. (Bottom row) Plot of the relative error vs length scale. Figure 7 : 7QP mesh configurations and schematic. (a) Hexagonal columnar grain mesh for the discrete interface model. (b) Hexagonal columnar grain BSAMR mesh in a rectilinear domain for the diffuse interface model. (c) Locations of QP and TJ along major and minor lines. Figure 8 : 8Comparison of minor axis results for the QP case for the discrete model (left) and diffuse model (right); all quantities are nondimensionalized. (Top row) Plot of the measured TJ (minor diameter) angle, with color indicating the length scale and the exact solution in black. (Bottom row) Plot of the relative error in the TJ angle with respect to length scale. Figure 9 : 9Comparison of major axis results for the QP case for the discrete model (left) and diffuse model (right); all quantities are nondimensionalized. (Top row) Plot of the measured QP (major diameter) angle, with color indicating the length scale and the exact solution in black. (Bottom row) Plot of the relative error in the QP angle with respect to length scale. Figure 10 : 10The scaling of the normalized runtime and the normalized number of grain boundary vertex calculations for the spherical grain case as a function of e . Figure 10 10shows the scaling of the normalized runtime cost and normalized number of grain boundary vertex calculations n calc = j n v b ,j , where n v b ,j is the number of grain boundary vertices at time step j, for the discrete interface method. These scale as −4 i × t j ||(n ij ⊗n ij ), i ×t j ||(n ij ⊗n ij ), 2 = 3/m when the triple line and quadruple point drags vanish; this can be derived by requiring that the limiting behavior of a small spherical cap coincides with the predictions of Eq. (1). The corresponding nondimensionalization of the multiphase field governing equations in Sec. 2.2 yields ∂η ∂t = −τ L δW δη n = ∂w ∂η n + k∆η n ,w = w τ L,∆ = τ Lk∆. (A.7) Fig. C.12. It scales as 2.010 e for D a,10 and 1.916 e for D I,1000 and D I,10 . Overall, D a,10 allows larger time steps and has a better accuracy, though the improvement is not significant. Figure C. 11 : 11The effect of different drag tensor correction terms on the resulting velocity. The capillary force is colored black and the velocities corresponding to different correction terms are differentiated by color. Each vector is scaled relative to the maximum magnitude among the velocities. Figure C. 12 : 12The scaling of the median time step with e for the three drag tensor correction terms. Data availabilityThe Alamo (https://github.com/solidsgroup/alamo) and VDlib (https://github .com/erdemeren/VDlib) libraries used to generate these results are available as open source. The processed data required to reproduce these findings are available to download from [https://arxiv.org/abs/2203.03167].Appendix B. 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Quantitative phase-field approach for simulating grain growth in anisotropic systems with arbitrary inclination and misorientation dependence. N Moelans, B Blanpain, P Wollants, Physical review letters. 101225502N. Moelans, B. Blanpain, and P. Wollants, "Quantitative phase-field approach for simu- lating grain growth in anisotropic systems with arbitrary inclination and misorientation dependence," Physical review letters, vol. 101, no. 2, p. 025502, 2008. Applications of semi-implicit fourier-spectral method to phase field equations. L Q Chen, J Shen, Computer Physics Communications. 1082-3L. Q. Chen and J. Shen, "Applications of semi-implicit fourier-spectral method to phase field equations," Computer Physics Communications, vol. 108, no. 2-3, pp. 147-158, 1998. Phase-field models for microstructure evolution. L.-Q Chen, Annual review of materials research. 321L.-Q. 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AMReX: a framework for block-structured adaptive mesh refinement. W Zhang, A Almgren, V Beckner, J Bell, J Blaschke, C Chan, M Day, B Friesen, K Gott, D Graves, Journal of Open Source Software. 437W. Zhang, A. Almgren, V. Beckner, J. Bell, J. Blaschke, C. Chan, M. Day, B. Friesen, K. Gott, D. Graves, et al., "AMReX: a framework for block-structured adaptive mesh refinement," Journal of Open Source Software, vol. 4, no. 37, 2019. Structure and energy of liquid foams. W Drenckhan, S Hutzler, Advances in colloid and interface science. 224W. Drenckhan and S. Hutzler, "Structure and energy of liquid foams," Advances in colloid and interface science, vol. 224, pp. 1-16, 2015. The Motion of a Surface by Its Mean Curvature. K A Brakke, Princeton University Press and University of Tokyo PressK. A. Brakke, The Motion of a Surface by Its Mean Curvature. Princeton University Press and University of Tokyo Press, 1987. An object-oriented finite element framework for multiphysics phase field simulations. M R Tonks, D Gaston, P C Millett, D Andrs, P Talbot, Computational Materials Science. 511M. R. Tonks, D. Gaston, P. C. Millett, D. Andrs, and P. Talbot, "An object-oriented finite element framework for multiphysics phase field simulations," Computational Materials Science, vol. 51, no. 1, pp. 20-29, 2012. A new finite element strategy to simulate microstructural evolutions. S Florez, M Shakoor, T Toulorge, M Bernacki, Computational Materials Science. 172109335S. Florez, M. Shakoor, T. Toulorge, and M. Bernacki, "A new finite element strategy to simulate microstructural evolutions," Computational Materials Science, vol. 172, p. 109335, 2020. Comparative study of two phase-field models for grain growth. N Moelans, F Wendler, B Nestler, Computational Materials Science. 462N. Moelans, F. Wendler, and B. Nestler, "Comparative study of two phase-field models for grain growth," Computational Materials Science, vol. 46, no. 2, pp. 479-490, 2009. 2D finite element modeling of misorientation dependent anisotropic grain growth in polycrystalline materials: Level set versus multi-phase-field method. Y Jin, N Bozzolo, A Rollett, M Bernacki, Computational Materials Science. 104Y. Jin, N. Bozzolo, A. Rollett, and M. Bernacki, "2D finite element modeling of misorientation dependent anisotropic grain growth in polycrystalline materials: Level set versus multi-phase-field method," Computational Materials Science, vol. 104, p. 108-123, 2015. A novel level-set finite element formulation for grain growth with heterogeneous grain boundary energies. J Fausty, N Bozzolo, D P Muñoz, M Bernacki, Materials & Design. 160J. Fausty, N. Bozzolo, D. P. Muñoz, and M. Bernacki, "A novel level-set finite element formulation for grain growth with heterogeneous grain boundary energies," Materials & Design, vol. 160, pp. 578-590, 2018. Effect of a finite quadruple junction mobility on grain microstructure evolution: Theory and simulation. L B Mora, V Mohles, L Shvindlerman, G Gottstein, Acta Materialia. 565L. B. Mora, V. Mohles, L. Shvindlerman, and G. Gottstein, "Effect of a finite quadruple junction mobility on grain microstructure evolution: Theory and simulation," Acta Materialia, vol. 56, no. 5, p. 1151-1164, 2008. . J Neumann, American Society for Testing Materials108J. von Neumann, p. 108. American Society for Testing Materials, 1952. Method to calculate the grain boundary energy distribution over the space of macroscopic boundary parameters from the geometry of triple junctions. A Morawiec, Acta Materialia. 4813A. Morawiec, "Method to calculate the grain boundary energy distribution over the space of macroscopic boundary parameters from the geometry of triple junctions," Acta Materialia, vol. 48, no. 13, p. 3525-3532, 2000. Surface tension as a motivation for sintering. C Herring, The Physics of Powder Metallurgy. W. E. KingstonMcGraw-HillC. Herring, "Surface tension as a motivation for sintering," in The Physics of Powder Metallurgy (W. E. Kingston, ed.), p. 143-179, McGraw-Hill, 1951. The use of classical macroscopic concepts in surface energy problems. C Herring, Structure and Properties of Solid Surfaces. R. Gomer and C. S. SmithUniversity of Chicago PressC. Herring, "The use of classical macroscopic concepts in surface energy problems," in Structure and Properties of Solid Surfaces (R. Gomer and C. S. Smith, eds.), p. 5-81, University of Chicago Press, 1953. Application of orientation microscopy in SEM and TEM for the study of texture formation during recrystallisation processes. 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[ "https://github.com/solidsgroup/alamo)", "https://github.com/VDlib," ]
[ "DEEPMIR: A Deep Neural Network for Differential Detection of Cerebral Microbleeds and IRon Deposits in MRI", "DEEPMIR: A Deep Neural Network for Differential Detection of Cerebral Microbleeds and IRon Deposits in MRI" ]
[ "Tanweer Rashid [email protected] \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nNeuroimage Analytics Laboratory (NAL)\nBiggs Institute Neuroimaging Core (BINC)\nGlenn Biggs Institute for neurodegenerative disorders\nUniversity of Texas Health Science Center at San Antonio (UTHSCSA)\nSan AntonioTexasUSA\n", "Ahmed Abdulkadir \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nUniversity Hospital of Old Age Psychiatry and Psychotherapy\nUniversity of Bern\nBernSwitzerland\n", "Ilya M Nasrallah \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nDepartment of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n", "Jeffrey B Ware \nDepartment of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n", "Hangfan Liu \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n", "Pascal Spincemaille \nDepartment of Radiology\nWeill Cornell Medical College\nNew YorkNYUSA\n", "J Rafael Romero \nDepartment of Neurology\nSchool of Medicine\nBoston University\nBostonMAUSA\n", "R Nick \nDepartment of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nDepartment of Diagnostic Medicine\nDell Medical School\nUniversity of Texas at Austin\nAustinTXUSA\n", "Bryan ", "Susan R Heckbert \nDepartment of Epidemiology and Cardiovascular Health Research Unit\nUniversity of Washington\nSeattleWAUSA\n", "Mohamad Habes [email protected] \nCenter for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA\n\nNeuroimage Analytics Laboratory (NAL)\nBiggs Institute Neuroimaging Core (BINC)\nGlenn Biggs Institute for neurodegenerative disorders\nUniversity of Texas Health Science Center at San Antonio (UTHSCSA)\nSan AntonioTexasUSA\n" ]
[ "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Neuroimage Analytics Laboratory (NAL)\nBiggs Institute Neuroimaging Core (BINC)\nGlenn Biggs Institute for neurodegenerative disorders\nUniversity of Texas Health Science Center at San Antonio (UTHSCSA)\nSan AntonioTexasUSA", "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "University Hospital of Old Age Psychiatry and Psychotherapy\nUniversity of Bern\nBernSwitzerland", "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Radiology\nWeill Cornell Medical College\nNew YorkNYUSA", "Department of Neurology\nSchool of Medicine\nBoston University\nBostonMAUSA", "Department of Radiology\nPerelman School of Medicine\nHospital of University of Pennsylvania\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Department of Diagnostic Medicine\nDell Medical School\nUniversity of Texas at Austin\nAustinTXUSA", "Department of Epidemiology and Cardiovascular Health Research Unit\nUniversity of Washington\nSeattleWAUSA", "Center for Biomedical Image Computing and Analytics (CBICA)\nUniversity of Pennsylvania\nPhiladelphiaPAUSA", "Neuroimage Analytics Laboratory (NAL)\nBiggs Institute Neuroimaging Core (BINC)\nGlenn Biggs Institute for neurodegenerative disorders\nUniversity of Texas Health Science Center at San Antonio (UTHSCSA)\nSan AntonioTexasUSA" ]
[]
Lobar cerebral microbleeds (CMBs) and localized non-hemorrhage iron deposits in the basal ganglia have been associated with brain aging, vascular disease and neurodegenerative disorders. Particularly, CMBs are small lesions and require multiple neuroimaging modalities for accurate detection. Quantitative susceptibility mapping (QSM) derived from in vivo magnetic resonance imaging (MRI) is necessary to differentiate between iron content and mineralization. We set out to develop a deep learning-based segmentation method suitable for segmenting both CMBs and iron deposits.We included a convenience sample of 24 participants from the MESA cohort and used T2weighted images, susceptibility weighted imaging (SWI), and QSM to segment the two types of lesions. We developed a protocol for simultaneous manual annotation of CMBs and nonhemorrhage iron deposits in the basal ganglia. This manual annotation was then used to train a deep convolution neural network (CNN). Specifically, we adapted the U-Net model with a higher number of resolution layers to be able to detect small lesions such as CMBs from standard resolution MRI. We tested different combinations of the three modalities to determine the most informative data sources for the detection tasks.In the detection of CMBs using single class and multiclass models, we achieved an average sensitivity and precision of between 0.84-0.88 and 0.40-0.59, respectively. The same framework detected non-hemorrhage iron deposits with an average sensitivity and precision of about 0.75-0.81 and 0.62-0.75, respectively.Our results showed that deep learning could automate the detection of small vessel disease lesions and including multimodal MR data (particularly QSM) can improve the detection of CMB and non-hemorrhage iron deposits with sensitivity and precision that is compatible with use in large-scale research studies.
10.1038/s41598-021-93427-x
[ "https://arxiv.org/pdf/2010.00148v3.pdf" ]
235,359,223
2010.00148
0604ef96a10f660474d22dc2f525a41ee2f86e29
DEEPMIR: A Deep Neural Network for Differential Detection of Cerebral Microbleeds and IRon Deposits in MRI Tanweer Rashid [email protected] Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA Neuroimage Analytics Laboratory (NAL) Biggs Institute Neuroimaging Core (BINC) Glenn Biggs Institute for neurodegenerative disorders University of Texas Health Science Center at San Antonio (UTHSCSA) San AntonioTexasUSA Ahmed Abdulkadir Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA University Hospital of Old Age Psychiatry and Psychotherapy University of Bern BernSwitzerland Ilya M Nasrallah Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA Department of Radiology Perelman School of Medicine Hospital of University of Pennsylvania University of Pennsylvania PhiladelphiaPAUSA Jeffrey B Ware Department of Radiology Perelman School of Medicine Hospital of University of Pennsylvania University of Pennsylvania PhiladelphiaPAUSA Hangfan Liu Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA Pascal Spincemaille Department of Radiology Weill Cornell Medical College New YorkNYUSA J Rafael Romero Department of Neurology School of Medicine Boston University BostonMAUSA R Nick Department of Radiology Perelman School of Medicine Hospital of University of Pennsylvania University of Pennsylvania PhiladelphiaPAUSA Department of Diagnostic Medicine Dell Medical School University of Texas at Austin AustinTXUSA Bryan Susan R Heckbert Department of Epidemiology and Cardiovascular Health Research Unit University of Washington SeattleWAUSA Mohamad Habes [email protected] Center for Biomedical Image Computing and Analytics (CBICA) University of Pennsylvania PhiladelphiaPAUSA Neuroimage Analytics Laboratory (NAL) Biggs Institute Neuroimaging Core (BINC) Glenn Biggs Institute for neurodegenerative disorders University of Texas Health Science Center at San Antonio (UTHSCSA) San AntonioTexasUSA DEEPMIR: A Deep Neural Network for Differential Detection of Cerebral Microbleeds and IRon Deposits in MRI 1 Corresponding author: Lobar cerebral microbleeds (CMBs) and localized non-hemorrhage iron deposits in the basal ganglia have been associated with brain aging, vascular disease and neurodegenerative disorders. Particularly, CMBs are small lesions and require multiple neuroimaging modalities for accurate detection. Quantitative susceptibility mapping (QSM) derived from in vivo magnetic resonance imaging (MRI) is necessary to differentiate between iron content and mineralization. We set out to develop a deep learning-based segmentation method suitable for segmenting both CMBs and iron deposits.We included a convenience sample of 24 participants from the MESA cohort and used T2weighted images, susceptibility weighted imaging (SWI), and QSM to segment the two types of lesions. We developed a protocol for simultaneous manual annotation of CMBs and nonhemorrhage iron deposits in the basal ganglia. This manual annotation was then used to train a deep convolution neural network (CNN). Specifically, we adapted the U-Net model with a higher number of resolution layers to be able to detect small lesions such as CMBs from standard resolution MRI. We tested different combinations of the three modalities to determine the most informative data sources for the detection tasks.In the detection of CMBs using single class and multiclass models, we achieved an average sensitivity and precision of between 0.84-0.88 and 0.40-0.59, respectively. The same framework detected non-hemorrhage iron deposits with an average sensitivity and precision of about 0.75-0.81 and 0.62-0.75, respectively.Our results showed that deep learning could automate the detection of small vessel disease lesions and including multimodal MR data (particularly QSM) can improve the detection of CMB and non-hemorrhage iron deposits with sensitivity and precision that is compatible with use in large-scale research studies. Introduction The aging brain is subject to various irreversible changes, some driven by the aging process itself and others that are associated with various pathologies, including vascular lesions and neurodegeneration [1][2][3][4] . On magnetic resonance imaging (MRI), particularly tuned to be sensitive for differences in magnetic susceptibility, focal accumulations of iron content can be visible. This includes lesions with iron content such as cerebral microbleeds (CMBs) and non-hemorrhage iron deposits in the basal ganglia. CMBs are small hemorrhages that can occur sporadically throughout the brain 5 . CMBs have been associated with cognitive decline and dementia 6 , and are considered a biomarker for small vessel diseases. The presence of lobar CMBs is also a marker for cerebral amyloid angiopathy [7][8][9] . Non-hemorrhage iron deposits are located in the deep structures of the brain, particularly in the basal ganglia. While an increase in iron concentration in the basal ganglia is expected in healthy aging 10 , focal accumulation of iron has been associated with neurodegenerative disorders in small scale studies [11][12][13] . Most of our knowledge on the iron toxicity in the aging brain is limited by the fact that both CMBs and iron deposits could be difficult to distinguish from each other and from other similar lesions including calcification using conventional MRI techniques 14 . T2* gradient-recalled echo (GRE) and susceptibility-weighted imaging (SWI) are often used to clinically characterize CMB, with the latter being more sensitive for detecting CMBs 15,16 . CMBs can occur anywhere and appear as small rounded or ellipsoidal hypo-intense regions with a diameter of ten millimeters or less 7,14,17 . Nonhemorrhage iron deposits in the basal ganglia have irregular shapes and could be larger than CMBs 14 . Because hypo-intensities in SWI are not specific to CMBs and non-hemorrhage iron deposits, images with other tissue contrasts are required in order to identify other lesion types that can have similar low susceptibility signal on SWI, such as calcification 5,18,19 . The specificity for CMB detection can be increased by post-processing SWI-magnitude and phase data to derive quantitative susceptibility maps (QSM) 20,21 . In QSM paramagnetic tissue appears different from diamagnetic materials, and therefore this contrast is particularly useful for distinguishing nonhemorrhage iron deposits from calcifications 22,23 . While previous efforts have been made to automate the detection of microbleeds, all previous work neglected the detection of nonhemorrhage iron deposits in such automated framework [24][25][26][27][28][29][30] . No work has been published to date on segmenting iron deposits in the brain using QSM with either a semi-or a fully automatic method. The advances made in MRI technology with QSM for iron content recognition are gaining more attention as cohort-based studies such as The Multi-Ethnic Study of Atherosclerosis (MESA) 31-33 include QSM in their imaging protocol, and thus exploit its advantages in delivering specific insights on iron toxicity in the aging brain. The focus in MESA is utilizing non-invasive methods to investigate common risk factors, preclinical disease states and manifest diseases using a standardized imaging protocol, which is applied to all participants 34 . On one hand, this is providing a unique opportunity to study widely ignored lesions such as iron deposits in vivo using MRI but on the other hand, this comes with additional challenges as such cohorts naturally include largely cognitively normal participants with a low lesion load, resulting in a very challenging task to automate. In order to tackle the challenges inherent in the detection of these lesions, we developed a robust and fully automated deep learning-based method to detect CMBs and non-hemorrhage iron deposits in a cohort without extensive apparent brain tissue damage and having a low load of CMBs and non-hemorrhage basal ganglia iron deposits. We experimented with both single class and multiclass segmentation models using multiple MR sequences. Our experiments show that using multi-sequence MRI (especially QSM) improves the overall accuracy of detection. The main contributions of this study include the following: 1. We tackled the challenging problem of simultaneously detecting CMB and nonhemorrhage iron deposits. To our knowledge, this is one of the first reports to detect both types of lesions simultaneously. Often iron accumulation in the brain has been understudied due to the lack of appropriate techniques for detecting them in vivo in largescale epidemiological studies; 2. We found out the most suitable pulse sequence combination to automate the detection tasks by exploiting imaging information jointly; 3. We developed an effective and flexible neural network model that is specially tailored to the differential detection task. The proposed model can be easily adapted to segment additional lesions; 4. We achieved highly competitive detection performance on real-life data, demonstrating the effectiveness of the proposed approach in practical applications. 5. We also provide access to our source code and a few trained models via the GitHub link https://github.com/NAL-UTHSCSA/CMB_NHID_Segmentation Results We performed leave-one-out cross-validated evaluations for both single class and multiclass segmentation experiments using the 24 participants listed in Supplementary Figure 1 shows an example of the automated segmentation of a CMB (indicated by the red arrow). Panel B in Figure 1 shows the segmentation of the focal iron deposits in the basal ganglia. In this figure, the model correctly segmented the iron deposit lesions (indicated by the green arrow) while rejecting an instance of calcification (indicated by the yellow arrow). The results of these experiments are reported in Table 1 and Table 2. Pearson's correlation and Bland-Altman mean difference and confidence intervals for single class and multiclass experiments are reported in Table 1 and Table 2 respectively, for the experiments with 24 participants. Overall, our experiments show that incorporating QSM in model training can increase the overall accuracy of CMB and iron deposit detection. In the case of segmenting CMBs, the best performance in terms of average magnitude accuracy is seen with the model trained with SWI and QSM in both single class and multiclass experiments. The correlation coefficient between the prediction and ground truth was also highest (r=0.97 and r=0.99, for single class and multiclass results, respectively) when QSM was included in the training. For non-hemorrhage iron deposits, the single class model trained with all three modalities had the highest average magnitude accuracy and the multiclass model trained with SWI and QSM had the highest average magnitude accuracy. The correlation coefficient was also highest for models that included QSM for training (r=0.92 and r=0.91 for single class and multiclass results, respectively). Figure 2 shows a joint scatterplot of the single class experimental results and Figure 3 shows a joint scatterplot of the multiclass experiments. In our dataset, we identified as an outlier a single individual with exceptionally many CMBs. A comparative analysis was done by removing this outlier from the dataset and repeating a similar cross-validated evaluation by retraining both single class and multiclass models. The results are detailed in Supplementary An additional leave-one-out cross-validated evaluation was done for the 24 participants using an implementation of the original U-Net 35 . The results of this experiment are reported in Supplementary We investigated the performance of the proposed DEEPMIR architecture for the simultaneous differentiation and labeling of both CMB and iron deposit labels against the performance of the original U-Net and a modified DEEPMIR architecture (having the same number of resolution layers as the original U-Net). We note that the proposed DEEPMIR model with 6 resolution layers has better overall sensitivity for detecting small lesions such as CMBs. Supplementary Figure 12 and Supplementary Figure 13 show examples of small lesions that the original U-Net and the modified DEEPMIR models were unable to detect, compared to the accurate detection by the proposed DEEPMIR architecture. Discussion We developed a deep learning framework for simultaneous segmentation of cerebral microbleeds and non-hemorrhage iron deposits using multi-modal MRI. To date, previously published methods for automated or semi-automated CMB detection have ignored iron deposits. In this study, we consider the iron deposit in the basal ganglia seen as hypo-intense lesions on SWI and confirmed by QSM to be iron-specific rather than mineralization. Those lesions may typically be labeled as possible or uncertain microbleeds on MARS 18 and BOMBS 19 mainly because of the limitation that T2* and SWI cannot differentiate iron content from mineralization. We overcome this limitation by including QSM in our study, which has shown to improve the overall accuracy for automated detection. To our knowledge, there are no studies that attempted to segment these focal iron deposits using SWI and/or QSM automatically. Our deep learning-based segmentation method presented here is filling in this gap. We have undertaken several experiments using both single class and multiclass models with different combinations of the available MR pulse sequences. We noted that the models which included QSM in training consistently performed better and the resulting predictions had statistically high correlations when compared to the reference annotation. Our approach has several advantages over the current state-of-the-art methods for CMB detection. First, by using deep learning our model is capable of learning and generalizing features rather than rely on feature vectors derived with conventional image processing algorithms 28-30 , Fourier shape descriptors 36 or probabilistic models 27 . Second, we employ end-to-end learning by using a single model (or network). Previously published methods that used deep learning employed multiple stages consisting of (a) a candidate generation stage which use either conventional image processing methods 24,26 or an initial (and separate) deep learning-based model 25 for identifying possible CMBs, and (b) a false positive reduction stage in the form of a CNN-based network 24-26 . Our single-stage design allows for greater flexibility, for example in retraining with different or larger data sets, adding additional class labels, or using different modalities, while achieving sensitivity and precision comparable to published results. Third, we trained with different sets of input imaging modalities. Combinations of imaging modalities allowed our models to reject mimics such as calcifications without explicit provisions (as shown in Figure 1, Panel B). Supplementary Figure 9 in the supplementary materials (Supplementary Section 5) shows an example of mineralization being segmented as iron deposits when the model was trained with only SWI. The models in publications 25,26 used SWI only and therefore may not be capable of recognizing and rejecting mimics. The method in publication 24 utilizes SWI-phase and magnitude images along with QSM, but did not consider iron deposits in the basal ganglia. Fourth, we experimented with a reduced number of layers (5 instead of 6 spatial resolution layers) and noted that having more layers can improve the overall results for detecting small lesions such as CMBs. Our framework has achieved an excellent sensitivity of 89%. However, other studies 24-26 have reported higher precision in detecting CMBs in their samples. We would like to note that it is impossible to directly compare reported numbers from various machine learning models, due to differences in populations included, study settings and imaging and scanner charactherisitcs 37 . Of particular importance is the fact that our sample was drawn from a relatively healthy population without significant brain trauma, injuries, or pre-existing neuro-pathologies whereas the studies in publications 24-26 had hundreds if not thousands of CMB lesions related to or caused by radiation therapy, stroke and traumatic brain injury. One of the major challenges was the small size of the lesions and their potential presence throughout the brain. The average size of four voxels (or 6 mm 3 ) per CMB together with the generally low lesion burden of the study participants resulted in including only two CMB lesions/4 voxels on average per participant, resulting in a higher weight of a single lesion or error in the evaluation. In other words, missing a single lesion would result in a drop of sensitivity from one to 0.5 and a single false positive for a given participant would result in a drop of that participant's precision from one to 0.5 or 0.66. Similarly, a small number of false positives, in absolute terms, can lower the average precision substantially. In general, our models over-segmented the data in terms of detecting more CMBs than were actually present ( Supplementary Figures 10 and 11 in the supplementary materials show examples of false positive CMBs). In all experiments using the aforementioned combinations of available imaging modalities, most of the lesions were detected and the average sensitivity was consistently above 0.75. Notably, the sample used to train the model was a convenience sample from participants of the MESA study without particular clinical profile and without apparent brain disorders such as dementia, depression, or traumatic brain injury. Given the low number of lesions on average, our method achieved sensitivities that are comparable to state-of-the-art CMB segmentation/detection methods trained with large datasets. We expect that including more samples with more lesions would improve the precision. In general, most studies incorporating automated methods for large-scale abnormality detection or brain region segmentation incorporate a segmentation quality control step that could result in corrections or exclusions 1,38,39 . Thanks to the flexibility of our method, it is straightforward to increase the sample size. In clinical terms, a larger number of CMBs is more likely to be clinically relevant. The proposed DEEPMIR method was trained and evaluated on a relatively small population and outputs the number of lesions and lesion segmentation maps for each participant. The next step would be to rigorously test and evaluate the proposed model on a larger sample size to ensure viable sensitivity, precision and overall accuracy, before applying it to a large cohort to determine the prevalence of lesions in the population. An adequately trained model can be used as a screening tool to flag participants with a high lesion load. DEEPMIR can also be used to generate an initial segmentation of lesions to accelerate manual annotation. QSM is a good, non-invasive technique to distinguish between iron content and mineralization in the brain and showed a great advantage in improving the overall accuracy of CMB and iron deposit detection in the current study. While QSM is being recognized and is being integrated in more population-based studies, large studies with QSM data acquisition such as MESA is still ongoing. This left us with a relatively small number of imaging data used for training. For our experiment, we had a ratio of validation to training data (25:75), which showed to be reasonable to ensure that a maximal amount of the available data is used in model training, while at the same time a sufficient amount is reserved for within-training validation. The use of similar sample sizes for training and evaluation is not unprecedented in such small lesion detection 27,29,40,41 . One limitation of using such a small sample size is a reduction in study statistical power. For our experiments, we noted that none of the multiple comparisons were statistically significant, and this could likely be due to the small sample size. Finally, the limited access to QSM from other studies left us to perform cross-validation 37 with samples from only the MESA AFib cohort for evaluating our model. We were therefore not able to test the generalizability of our model with images generated in other studies with different parameters and characteristics. This line of work should be considered in future research efforts, ultimately building machine learning models and benefiting from pooling imaging data from multiple cohort-based studies 42 . We have presented a framework for the automated detection of cerebral microbleeds and nonhemorrhage iron deposits in the basal ganglia. While SWI remains the preferred modality of choice for CMB detection, few studies have leveraged QSM as an additional source of information to improve overall detection accuracy, and to date, there have been no attempts to include iron deposits in the basal ganglia as an item of interest. We have utilized QSM in this study to confirm that these focal lesions in the basal ganglia are in fact iron depositions, rather than mineralization such as calcifications. Our deep learning neural network model is flexible and at the same time scalable to include additional modalities and/or class labels while maintaining comparably high sensitivity and precision. We aim in our future work to automatically detect other small vessel disease lesions in our framework such as enlarged perivascular spaces. We also aim to investigate possible advantages of expanding our network to a three-dimensional variant. Methods MRI Acquisition and Pre-Processing The MESA Exam 6 Atrial Fibrillation (AFib) Ancillary Study's 34 brain MRI protocol included T1weighted (T1w) and T2-weighted (T2w) sequences, and a susceptibility weighted imaging (SWI) sequence with 4 different, equally spaced echo times. SWI is a high-resolution, 3D imaging sequence where the image contrast is enhanced by combining magnitude and phase image data 43,44 . The scans were acquired at 6 sites using the same acquisition parameters. All scans were performed on Siemens MR scanners (2 Skyra with a 20-channel head coil and 4 Prisma Fit with a 32-channel head coil) at a static magnetic field strength of 3 Tesla and identical imaging sequence parameters, as shown in Supplementary 43,45 . A homodyne high-pass filter with k-space window size of 64 x 64 was applied to the raw phase image to generate the negative phase mask (with values between 0 and 1). The phase mask was then raised to power 4 and multiplied with the magnitude image to generate the SWI. For creation of the reference annotation and subsequent deep learningbased inferencing, only the SWI image with the shortest echo time (TE=7.5 ms) was used because longer echo times have more noise due to increasingly pronounced blooming effects near the sinus cavity and cerebellum. In addition, SWI with longer echo times are also more prone to showing false positive CMBs, especially when veins are perpendicular to the imaging plane. Section 2 in the supplementary materials discuss this issue in more detail. The T1w and T2w images underwent N4 bias correction 46 with default parameters using the implementation in the Advanced Normalization Tools (ANTs) (http://stnava.github.io/ANTs) suite and were rigidly registered to the participants' SWI image using FSL's FLIRT [47][48][49] (https://fsl.fmrib.ox.ac.uk). Anatomical parcellation and brain masks were generated with a multiatlas segmentation method using the bias-corrected T1w images 50 . These brain masks were used in the generation of the QSM images. QSM maps were generated using the entire multi-echo SWI dataset using the Morphology Enabled Dipole Inversion (MEDI) method 21,51 implemented in MATLAB (http://weill.cornell.edu/mri/pages/qsm.html). Briefly, background field removal is done using the Projection onto Dipole Fields (PDF) method 52 , followed by region-growing based spatial unwrapping with non-linear fitting 53 to reduce errors, and finally the susceptibility map is calculated using the Morphology enabled dipole inversion with zero reference using CSF (MEDI+0) method 54 . Manual Annotation Manual annotation was performed according to a protocol developed with the focus on highly specific differential detection of CMBs and non-hemorrhage iron deposits based on multiple modalities including QSM. The detailed protocol is described in Section 2 in the supplementary materials, and a flowchart of the manual annotation process is shown in Supplementary Figure 4 in the supplementary materials. Panel A in Figure 4 shows an example of a CMB in the thalamus and non-hemorrhage iron deposits in the interior section of the globus pallidus on SWI (for TE=7.5 ms and 22.5 ms), QSM and T2w MRI, and Panel B shows the expert segmentation of the lesions based on the annotation protocol. Panel C shows an example of a larger CMB located in the occipital lobe and Panel D shows its respective expert segmentation. Study Participants We included imaging data from participants in the MESA Exam 6 Atrial Fibrillation Ancillary Study 31-33 . This study was approved by the Institutional Review Boards at the MESA Coordinating Center and at each participating institution. Written informed consent was obtained by all participants. All participant data collection was performed in accordance with relevant guidelines and regulations. A subset of the MESA cohort participated in an ancillary study of cardiac arrhythmias and brain imaging during the 2016-2018 exam (Exam 6) 34 . From 1061 participants who underwent MR brain scans, we selected a convenience sample of 34 scans based on visual identification of possible CMBs by two experienced readers (IMN and TR). These 34 participants are not representative of the MESA cohort in terms of prevalence of CMBs and non-hemorrhage iron deposits, and additional participants in the MESA cohort likely have CMBs and/or non-hemorrhage iron deposits. A total of 10 participants' scans were excluded due to poor image quality (n=4) and the presence of distortions/artifacts or motion-related effects (n=6). The demographics summary and lesion loads for the 24 included participants are presented in Supplementary Table 2.1. Of these 24 participants, there were 13 males and 11 females with age range 65-94 years. Based on the expert annotation of these 24 participants, 4 participants had no microbleeds, 13 participants had 1 or 2 microbleeds (with an average size of 10.85 mm 3 ), 6 participants had between 3 and 8 microbleeds (with an average size of 10.21 mm 3 ) and 1 participant had more than 100 microbleeds (with an average size of 4.76 mm 3 ). In certain circumstances, the participant with more than 100 microbleeds may be considered an outlier in terms of the number of CMBs. An examination of this is presented in Section 3 of the supplementary materials. Of the 24 participants, 5 participants did not have any voxels labeled as non-hemorrhage iron deposits and the remaining had between 2 (each having a single voxel or 1.5 mm 3 ) and 13 lesions (one participant had 4 non-hemorrhage iron deposit lesions with a total of 326 voxels or 489 mm 3 ) labeled as non-hemorrhage iron deposits in the basal ganglia. The distribution of CMBs and iron deposits pooled over all participants is illustrated in Supplementary Figure 5. The average size (± SEM, or standard error of the mean) of CMB lesions in this sample was 6.27 ± 0.51 mm 3 (4.18 ± 0.34 voxels). Among the 20 participants with CMB, 70% (n =14) had two or fewer CMBs, 25% (n = 5) had between three and eight CMBs, and the remaining participant had 120 CMBs. The average size of non-hemorrhage iron deposit labels (± SEM) was 26.15 ± 4.76 mm 3 (17.43 ± 3.17 voxels). Approximately 21% (n = 5) had no discernable basal ganglia non-hemorrhage iron deposits and half (n = 12) had fewer than 100 voxels (150 mm 3 ) labeled as non-hemorrhage iron deposits. The remaining 29% (n = 7) had more than 100 voxels labeled as non-hemorrhage iron deposits. Method Overview for Automated Processing We developed a deep learning framework for automatic segmentation of CMBs and nonhemorrhage iron deposits based on the U-Net 35,55 , a widely used deep learning architecture for image segmentation. Our architecture, however, employed padded instead of unpadded convolutions and operated on six instead of five spatial resolutions, and was used for both single class and multiclass segmentation experiments. The larger number of resolution layers enabled the model to detect small CMBs. A detailed description of our implementation is presented in the following sections. The overall system pipeline is shown in Figure 5. After the initial step of coregistration, the MR volumes were preprocessed to have zero mean and unit variance, as detailed in Section 4.6. The normalized MR volumes were then sliced along the z-axis (axial slices) and edge-padded to obtain 2D slices with 256x256 voxels. We evaluated the performance using leave-one-out cross-validation for the 24 participants listed in Supplementary Table 2.1 to ensure generalization of results. In each fold, a single participant's data was kept separate for testing (test dataset), and the MR data and labels from the remaining 23 participants were randomly split into training dataset (75%, consisting of 17 participants) and validation datasets (25%, consisting of 6 participants). Both training and validation datasets were augmented to improve the robustness of the deep learning models (for more details on data augmentation see Section 4.7). The training dataset was used to train the model for a single epoch, after which the validation dataset was used to compute a commonly used evaluation metric known as intersection-overunion (IoU) which quantifies the amount of overlap between the predicted and ground truth segmentations. Each model was trained for a maximum of 30 epochs, and the best model was determined as the model with the maximum IoU. This best model was then used to predict the labels of the test dataset. The set of predictions used for evaluating model performance thus consisted of 24 segmentation masks that were predicted with 24 different models with no overlap between training, validation and testing datasets. These cross-validated evaluations were done for both single class and multiclass experiments. For all experiments, four permutations of MR modalities were considered: (1) SWI only, (2) SWI and QSM, (3) SWI and T2w, and (4) SWI, QSM and T2w. For single class experiments, separate models were trained and evaluated for (1) CMBs only and (2) non-hemorrhage iron deposits only. For multiclass experiments, both CMBs and iron deposits had separate labels and were segmented simultaneously. For multiclass segmentations, a larger number of augmentations were used than for single class segmentations. 2D U-Net with Padded Convolutions Our lesion prediction models are based on the U-Net 35 . Both single and multiclass models consist of an analysis path (down-sampling operations) with five stages of convolution blocks and pooling, followed by a five synthesis path (up-sampling) with five stages of up-convolutions, plus a convolutional block. Each downsampling block consists of two layers of a 2D padded convolution layer having kernel size of 3x3 and stride of 1x1, followed by Batch Normalization and ReLU activation. The downsampling block ends with a 2x2 max pooling layer which reduces the resolution feature map by half in every spatial direction. The central block consists of two instances of padded 2D convolution with kernel size 3x3 and stride 1x1, followed by Batch Normalization and ReLU activation. Each upsampling block passes its input data through a 2D transpose convolution with kernel size of 2x2 and stride 2x2 in order to double the size of the feature map. This doubled feature map is then concatenated with the feature map (same size) of the corresponding analysis stage (i.e. the feature map before max pooling layer), followed by two instances of a padded 2D convolution layer having kernel size 3x3 and stride 1x1, followed by Batch Normalization and ReLU activation. Due to the use of padded convolutions throughout the model, the input and output image sizes are the same (256x256). The smallest downsampled image size is 8x8 in the central convolution block. In the case of the single class prediction model, the output of the final upsampling stage passes through a 2D convolution layer with kernel size 1x1, stride 1x1 and Sigmoid activation function. For the multiclass prediction model, the output of the final upsampling block is passed through a 2D convolution layer with kernel size 1x1, stride 1x1 and ReLU activation function, and then through a SoftMax layer to generate class probabilities. The model architecture is depicted in Figure 6. We employed random translations, random rotation, and flipping along the left-right axis during training. The network was trained with the cross-entropy loss. Image Preprocessing Each input image was normalized to have zero mean and unit variance. For QSM images, an additional prior step truncated the overall intensity such that the voxel value (VQSM) was within the range [− * ≤ ≤ * ], where k = 5 and is the standard deviation for the QSM image. This step is necessary because QSM images contain high-intensity noise (especially around the boundary of the brain and the region proximate to the sinus cavity) which may deemphasize the intensity of the rest of the brain. Data Augmentation To improve the robustness of the deep learning network and include more training data we enriched the training and validation datasets with augmentation. Axial slices containing CMBs and iron deposits are, for the most part, few compared to the remaining slices in a given brain volume. This type of class imbalance may bias the training process. To address this, data augmentation was performed on slices selectively instead of all slices, inspired from the concept of random over-sampling (ROS) and random under-sampling (RUS) 56 . First, all slices containing the labels of interest (i.e. CMBs and/or iron deposits) are augmented. Then a number of the remaining slices are randomly selected and augmented in the same manner until the total number of slices containing the labels of interest and the total number of slices that do not contain any labels of interest is similar. Data augmentation consisted of geometric transforms such as translations, rotations and image mirroring. In each experiment, the axial SWI slice (along with the corresponding axial QSM and T2w slices) and corresponding axial reference annotation slice were augmented. For translations, a set of two random integers tx and ty (representing the amount of shift per axis) were generated within the range [-45, 45] and used to translate the image slice(s) and the corresponding slice of the reference annotation. This range was chosen empirically so that most of the brain would be visible in the translated image. A total of 10 random integers per axis were generated for multiclass experiments. For rotations, a set of random integers d (representing the rotation in degree) were generated within the range [1,60], and the image slice(s) and the slices with reference annotations were rotated using both +d and -d. The regions of the crops that were located outside the image matrix were padded with edge values. A total of 16 random integers were used for multiclass experiments. Evaluation of Performance In single class models, the segmentation output map was in the range [0, 1]. Segmentations were accepted or rejected by applying a threshold value of 0.5 to the output map. In multiclass models, the model output was passed through a SoftMax function and segmentation labels were determined based on the class having the highest probability. We evaluated the performance in terms of the rate of detected/missed CMBs and nonhemorrhage iron deposit lesions. For each participant, the number of true positives (TP), false positives (FP) and false negatives (FN) were counted. A connected-component filter with 3D connectivity was applied to both the predicted segmentation and the reference segmentation in order to identify clusters of voxels. The centroid of the lesion in both the predicted segmentation and reference annotation was computed. TP, FP and FN were determined on whether the Euclidean distance between the centroid of each predicted lesion and a reference lesion was below a specified tolerance. Since CMBs are generally assumed to be relatively small in size, a tolerance of 3 was used for evaluating CMBs, and a tolerance of 5 was used for evaluating nonhemorrhage iron deposits since iron deposits have a larger size and more dispersed pattern than CMBs which are spherical. The sensitivity (or true positive rate) was computed as the ratio of TP and number of lesions in the ground truth (TP + FN) for each participant: = +(1) The precision (or positive predictive value) was computed as the ratio of TP and the number of lesions in the predicted mask: = +(2) When the true negative (TN) is available, the typical measure of performance is the overall accuracy, determined by = + + + +(3) To evaluate the performance of each model, we report the average sensitivity across all participants and average precision across all participants, as well as a combined metric (magnitude accuracy) computed as √ ̅ 2 + ̅ 2 , where ̅ and ̅ are the average sensitivity and precision respectively. Statistical Analysis Due to the small sample size and potentially non-uniform distribution of the models' sensitivity, precision and magnitude accuracy, we utilized the non-parametric two-tailed Wilcoxon signed rank test 57 to check for any difference between the performance of the various models. In all experimental evaluations, the model trained with only SWI was considered as the baseline model for comparison. Statistical significance was considered at a p < 0.05. Correlation (Pearson's) between the prediction and reference annotation is also calculated. For CMBs, the correlation was calculated using the number of lesions, and for non-hemorrhage iron deposits, the volume was used. All statistical analyses were performed in MATLAB R2017b. Overview of the split for one-fold of the cross-validation process that is repeated n times. In each fold, the model that was used to predict the test participant was trained on the remaining n-1 samples in order to avoid data leakage. Within the training stage, 25 percent of the n-1 participants were used as the validation set. The model with the highest validation accuracy was chosen to predict the left-out participant sample. Figure 1 . 1Panel A: Segmentations of CMBs by a model trained with SWI, QSM and T2w. (Top) An example of the correct segmentation of a small microbleed (red arrow). (Bottom) Magnified view of microbleed with segmentation mask (single red pixel). Panel B: An example of QSM being used to distinguish iron deposits from calcifications in the basal ganglia. (Top row) The SWI for TE=7.5ms, SWI for TE=22.5ms and QSM of the basal ganglia. The yellow arrow points to hypo-intense voxels which are likely calcifications and the green arrow points to basal ganglia iron deposits. (Bottom row) The segmentation mask (green labels) of the iron deposits. For both Panel A and B, the segmentations were generated by the multiclass model trained with SWI, QSM and T2w. Figure 2 . 2Joint scatterplots of the sensitivity vs precision of all single class experiments predicting CMBs and non-hemorrhage iron deposits. (Left) all CMB only experiments and (Right) all iron deposits only experiments. In each subplot, the round points indicate the individual participants' sensitivity and precision evaluated with leave-one-out cross-validation, and the X indicates the mean sensitivity and precision. The legend at the upper left corner of each subplot shows the coordinates of X. In each subplot, histograms of the sensitivity and precision are displayed along the upper and right axes. Figure 3 . 3Joint scatterplots of the sensitivity vs precision of all multiclass experiments predicting CMBs and non-hemorrhage iron deposits. (Left) all evaluations for CMBs and (Right) all evaluations for iron deposits. In each subplot, the round points indicate the individual participants' sensitivity and precision evaluated with leave-one-out cross-validation, and the X indicates the mean sensitivity and precision. The legend at the upper left corner of each subplot shows the coordinates of X. In each subplot, histograms of the sensitivity and precision are displayed along the upper and right axes. Figure 4 . 4This figure shows examples of cerebral microbleeds and basal ganglia iron deposition in SWI for TE=7.5ms (left column), SWI for TE=22.5ms (middle left column), QSM (middle right column) and T2w (right column). Panels A and C show the lesions in two different brains, and Panels B and D show the corresponding human expert labeling of the CMBs (red) and iron deposits (green). Figure 5 . 5Figure 5. Overview of the split for one-fold of the cross-validation process that is repeated n times. In each fold, the model that was used to predict the test participant was trained on the remaining n-1 samples in order to avoid data leakage. Within the training stage, 25 percent of the n-1 participants were used as the validation set. The model with the highest validation accuracy was chosen to predict the left-out participant sample. Figure 6 . 6U-Net architecture using padded convolutions for both single class and multiclass predictions. Table 2 . 21. Panel A Table 3 . 31 and Supplementary Table 3.2 in Section 3 of the supplementary materials. With the exception of multiclass CMBs, we note that the best result in terms of magnitude accuracy is seen when the model training includes QSM. This is also reflected by the correlation coefficients. Models which included QSM showed a higher correlation between the number of predicted lesions and reference annotation. For CMBs, the correlation r=0.51 and r=0.69, for single class and multiclass results, respectively. For iron deposits, the correlation r=0.94 and r=0.97 for single class and multiclass results, respectively. Table 4 . 41 in Section 4 of the supplementary materials. In these experiments, we note that the models trained with SWI, QSM and T2w had the best performance in terms of magnitude accuracy for both CMBs and iron deposits. However, in terms of the correlation coefficient, we note that the model trained with SWI and QSM had the highest correlation (r=0.98 and r=0.93, for CMBs and iron deposits, respectively). Table 1.1 in Section 1 of the supplementary materials. Multiple SWI phase and magnitude images were acquired with varying echo times (Supplementary Table 1.1 in the supplementary materials). SWI data were generated following the method of Haacke et al. 54 Liu, Z., Spincemaille, P., Yao, Y., Zhang, Y. & Wang, Y. MEDI+ 0: morphology enabled dipole inversion with automatic uniform cerebrospinal fluid zero reference for quantitative susceptibility mapping. 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[ "https://github.com/NAL-UTHSCSA/CMB_NHID_Segmentation" ]
[ "Diamond-based optical vector magnetometer", "Diamond-based optical vector magnetometer" ]
[ "Charlie Oncebay Segura \nInstituto de Física de São Carlos\nUniversidade de São Paulo (IFSC-USP)\nCaixa Postal 36913560-970São CarlosCEP, SPBrazil\n\nFacultad de Ciencias\nUniversidad Nacional de Ingeniería\nLimaPeru\n", "Sérgio Ricardo Muniz \nInstituto de Física de São Carlos\nUniversidade de São Paulo (IFSC-USP)\nCaixa Postal 36913560-970São CarlosCEP, SPBrazil\n" ]
[ "Instituto de Física de São Carlos\nUniversidade de São Paulo (IFSC-USP)\nCaixa Postal 36913560-970São CarlosCEP, SPBrazil", "Facultad de Ciencias\nUniversidad Nacional de Ingeniería\nLimaPeru", "Instituto de Física de São Carlos\nUniversidade de São Paulo (IFSC-USP)\nCaixa Postal 36913560-970São CarlosCEP, SPBrazil" ]
[]
We describe here* the construction and characterization of a high-resolution optical magnetometer to measure the full vector magnetic field on an ultrathin layer near the surface of the device. This solid-state device is based on quantum sensors created by a layer of nitrogen-vacancy (NV) centers less than 20 nm below the surface of an ultrapure diamond. This ensemble of nanosensors provides a versatile device capable of mapping magnetic fields and surface current densities with a sub-micrometer resolution and high sensitivity, making it suitable for many applications. Here, we show a custom-built prototype to demonstrate an operating proof-of-concept device. (*) Paper presented at the Conference SBFoton-IOPC-2021. arXiv:2209.05514v1 [quant-ph] 12 Sep 2022 FIG. 2. (A) Schematic of the apparatus, with wide-field fluorescence microscope, imaging system and excitation laser at 532 nm. (B) Thin glass plate with deposited layer of copper. (C) Diamond plate is glued on the glass plate with etched circuits for a single-loop microwave antenna (MW) and a straight section of copper wire for the DC circuit [8] .
10.1109/sbfotoniopc50774.2021.9461950
[ "https://export.arxiv.org/pdf/2209.05514v1.pdf" ]
236,185,134
2209.05514
77048df04ce882c7ddeaad0eea98fe0e4f38ed3c
Diamond-based optical vector magnetometer Charlie Oncebay Segura Instituto de Física de São Carlos Universidade de São Paulo (IFSC-USP) Caixa Postal 36913560-970São CarlosCEP, SPBrazil Facultad de Ciencias Universidad Nacional de Ingeniería LimaPeru Sérgio Ricardo Muniz Instituto de Física de São Carlos Universidade de São Paulo (IFSC-USP) Caixa Postal 36913560-970São CarlosCEP, SPBrazil Diamond-based optical vector magnetometer (Dated: 31-May-2021) We describe here* the construction and characterization of a high-resolution optical magnetometer to measure the full vector magnetic field on an ultrathin layer near the surface of the device. This solid-state device is based on quantum sensors created by a layer of nitrogen-vacancy (NV) centers less than 20 nm below the surface of an ultrapure diamond. This ensemble of nanosensors provides a versatile device capable of mapping magnetic fields and surface current densities with a sub-micrometer resolution and high sensitivity, making it suitable for many applications. Here, we show a custom-built prototype to demonstrate an operating proof-of-concept device. (*) Paper presented at the Conference SBFoton-IOPC-2021. arXiv:2209.05514v1 [quant-ph] 12 Sep 2022 FIG. 2. (A) Schematic of the apparatus, with wide-field fluorescence microscope, imaging system and excitation laser at 532 nm. (B) Thin glass plate with deposited layer of copper. (C) Diamond plate is glued on the glass plate with etched circuits for a single-loop microwave antenna (MW) and a straight section of copper wire for the DC circuit [8] . I. INTRODUCTION The nitrogen-vacancy (NV) center in diamond is a promising platform for many applications in quantum technologies. Among these, quantum sensing and, mainly, magnetometry are the most promising [1][2][3][4]. Diamond itself has remarkable material properties, making it well-suited for building microscopic field-deployable devices in a wide variety of environments: from zero to high temperatures, in a wide range of pressures, and even harsh chemical conditions. Since diamond is biocompatible, it can also be used for sensing biological samples [5], even inside cells (using nanodiamonds), and sensitive organic materials not compatible with other methods capable of nanometric resolution magnetometry. The use of NV center (NVC) for magnetometry applications is usually discussed in two main contexts: 1) either as a single NVC scanning probe [6,7], or 2) as an ensemble of NVCs [3,5] for magnetic field sensing and imaging. Here we describe a proof-of-principle application aiming to study the electronic properties of 2D materials [4,8], using an ensemble of NV centers engineered in an ultrathin layer near the surface of a bulk ultrapure diamond. This device combines fluorescence microscopy and optically detected magnetic resonance (ODMR) with electronic spin magnetic resonance spectroscopy to build a vector magnetometer capable of directly imaging all the vector components on the NVC plane. To demonstrate it, we used a quasi-2D model circuit to produce a current near the surface of the device and measured the full B-field created by the electric current. Using the vector field information, we reconstructed the current density vector that generated the magnetic field. II. MAGNETOMETRY WITH NV CENTERS A. NVC-based optical magnetometry The atom-like energy level structure of the NV center in the diamond lattice [9] makes it a vector magnetometer of sub-nanometer resolution [2][3][4]. The NV center is a point defect in the diamond crystal, consisting of a substitutional nitrogen atom combined with an adjacent carbon vacancy. It is a color center that absorbs photons in the visible range of 500 -620 nm and emits photons in a broad range of 632-800 nm. The photoluminescence spectrum has two zero phonon lines (ZPL) [8,10]. One line, at 575 nm, is due to neutral centers NV 0 . The second ZPL, at 637 nm, belongs to NV − , the negatively charged centers. Here, we are interested only in the NV − , simplifying the notation to NV, meaning the negatively charged centers. References [11,12] propose a model where NV centers have a triplet ground state (m s = 0, ±1) with a zero-field splitting D GS = 2.87 GHz, a triplet excited state with the zero-field splitting of 1.42 GHz, and shelving states involved in inter-system crossing. Applying a constant magnetic field B is possible to remove the degeneracy of the m s = ±1 states and the evolution of the relevant states is governed by the (simplified) effective Hamiltonian H = D GS S 2 z + E(S 2 x + S 2 y ) + γ e B. S,(1) where the parameter D GS is the axial zero field splitting, which is also sensitive to temperature [13], E is the transversal zero field splitting, γ e is the electron gyromagnetic ratio, and B is the external magnetic field. The spin vector S = (S x , S y , S z ) represents the electron spin, composed by the spin operators, here, given by S i = σ i , where σ i are the Pauli matrices. Thus, the applied magnetic field can be determined by the Zeeman shifts of the electron spin levels, as illustrated in Fig. 1. The optimum sensitivity of a magnetometer base on a ensemble of NV centers to measure DC magnetic fields depends on the intensity profile of NV resonances in the ODMR spectrum following the relation [14] η = 0.77 h gµ B ∆f α √ R (2) where ∆f is the linewidth and α is the line contrast. Note that one may increase α by increasing the power of the MW excitation at the expense of increasing ∆f due to power broadening. The term √ R corresponds to the shot-noise, at a photon rate R. Using this relation, one can determine the best set of parameters to achieve the desired sensitivity of the magnetic field. III. EXPERIMENTAL SETUP The sensor used in this study was engineered from a type IIa ultrapure diamond plate. The plate was thinned and repolished until obtaining a chip of size 2 × 2 × 0.1 mm 3 . A thin layer of NV centers was created near the top surface by irradiating it with 15 N + ions at 5 keV and 1 × 10 13 ions/cm 2 beam density. After implantation, the sample was annealed at 800 o C, in vacuum, for a couple of hours, to allow the migration and trapping of the vacancies to the implanted nitrogen atoms. These conditions create a layer of NVCs at around 8 to 16 nm below the surface at an estimated density of 10 3 NV/µm 2 [4,[15][16][17]. To excite the sample and collect its fluorescence, we built a custom wide-field fluorescence microscope, sketched in Fig. 2. After the acousto-optical modulator (AOM) and the iris, the laser is expanded and collimated with two lenses to obtain a beam diameter of ∼ 10 mm. The collimated beam goes to a dichroic mirror (SemRock, Di02-R561-25x36) and is focused control the laser and microwave excitation, and the camera trigger. Custom Python codes [8] were used to control the system, and to save and analyze the images. IV. METHODS AND RESULTS The protocol used to determine the vector B-field is shown in Fig. 3, where the diamond chip was glued onto a glass coverslip with the MW and the test DC-circuits. The ODMR spectrum comprises 64 images taken at each frequency, scanned using 0.5 MHz steps, to build the entire spectrum for each pixel in the fluorescing area. The spectra at each point (pixel) is used to determine the vector magnetic field at that location. We use a permanent magnet to apply a static bias field (∼ 10 mT), splitting the degenerate spin states into eight lines corresponding to four pairs of spin resonances (2 for each direction of the NV-axis). The visibility of the lines depends on the orientation of the magnet. In addition to the bias field, when an electric current passes through the wire, we observe frequency shifts in the ODMR spectrum due to the magnetic field produced around the DC wire, see Fig. 4(c) and 4(d). Note that the frequency shifts in the ODMR spectrum, caused by the current in the wire, are identical and symmetrically displaced for both directions of the current. To increase the signal-to-noise ratio, we record images with for both signs of the current, performing a differential measurement. We recorded three sequences of images: one without current and two with opposite signs of currents, as shown in Fig. 4. This procedure allows to minimize common mode noise and offsets in the current, and to account for background and bias fields. To reconstruct the vector magnetic field at each pixel, we used a multi-Lorentzian fit on the ODMR spectra to extract the eight resonance frequencies (ν i ± ). The frequencies satisfy ν i ± = D GS ± γ e B i N V , where B N V is the projection of the magnetic field along the NV axis. Since the crystalline orientation of our sample is (100), each axis x, y, and z coincide with the edges of the plate. Definingû 1 ,û 2 ,û 3 andû 4 as the unit vectors of the possibles orientations of the NV axes, the symmetry impliesû 1 +û 2 +û 3 +û 4 = 0, and if the external magnetic field is B = B xî + B yĵ + B zk in the lab frame, where z is perpendicular to the diamond surface, we obtain B.û 1 + B.û 2 + B.û 3 + B.û 4 = 0. This defines a linear system of equations relating the vector components of the field. We solve the overdetermined linear system using a leastsquare minimization algorithm to obtain the best estimate for the vector components of B, at each pixel. Fig. 5 displays the reconstruction of a magnetic field produced by a current of 15 mA in the test DC wire. Considering an idealized homogeneous ultra-thin (100 nm) wire of width W carrying a current I, we can use Biot-Savart law to calculate the magnetic field produced by the In the image, 1 pixel = 1.4 µm. [8] wire at a given probing distance z N V . Fig. 6 shows a comparison of the profiles of the calculated (simulated) B z component with experimental points measured along a line perpendicular to the wire (black dashed-line), for that component at for several distances z N V = {0.5, 1, 5, 10}µm. This procedure was done to all the components, resulting in the value of z N V = 10 ± 2 µm as the distance that best adjusts to experimental data. To calculate the density current from the measured magnetic field one assumes that the current is confined to a 2D plane [4,18]. In our case, the copper wire has a thickness of 100 nm and a width of around 10 µm. Under this assumption and using our measured B-field, we calculate the vector current density shown in Fig. 7, for a section of the copper (DC circuit) wire. The figure also shows the vector lines of J, superposed to its magnitude map, as well as a direct (bright field) microscopy image of the same portion of the copper wire. FIG. 7. Calculated current density for I = 15 mA [8]. The probe distance is assumed to be z N V = 10 µm. On the right, the current density is overlaid to the microscopy image of the wire. V. CONCLUSION In our system, the sensitivity associated to a 1 µm 2 area of the NV layer is η = 6 µm / √ Hz. This value is comparable with the literature [1,4,14,19] and is mainly limited by the contrast and linewidth of the ODMR resonances. Measured contrasts are in the range from 0.3% to 0.6% per pixel. Besides, in our case the laser power was limited to 20 mW on the sample. Higher powers are needed to excite larger areas, while reducing shot-noise. These are ways to improve all these limitations, as discussed in ref. [19]. In addition, using a fully quantum protocol, based on the spin coherence and Ramsey MW pulses [1], the sensitivity can be further improved, increasing the resolution to the nT / √ Hz range. We are currently pursuing quantum protocols in our laboratory [20]. An important detail in our setup is that the DC wire was on the surface of a glass cover slide and not directly on the diamond surface. Our analysis in Fig. 6 shows that the distance between the NV layer and the wire was ∼ 10 µm, explaining the need for larger currents. For 2D systems deposited on the top surface, near the NV layer, one can detect much lower density currents. FIG. 1 . 1Simulated ODMR spectra for different magnetic fields, showing Zeeman shifts due the different directions of the NV axis. using a lens (f = 30 cm) near the back aperture of the microscope objective (Zeiss, 50×, 0.95 NA) to adjust the size of the illuminated area on the sample. The position control of the sample was provided by a nanopositioner (Thorlabs, NanoMax 300), enabling independent translations along three axes. The NV fluorescence is collected by the same microscope objective used to excite the sample and filtered with the dichroic mirror and long-pass filter at 550 nm (FELH0550). The photoluminescence is detected with a simple CCD camera (PointGrey, FL3-FW-03S1M-C). The microwave (MW) and the DC wires were prepared in-house using a copper film (100 nm thickness) deposited by sputtering on the surface of a thin glass plate. We designed the MW and DC lines and printed them on photo (glossy) paper, applying the printed pattern onto the adequately cleaned copper surface. Later, we put the print-transferred glass plate into a chemical etching solution (ferric chloride solution) for about 30 seconds to remove the excess copper film. Finally, we cleaned and soldered the thin etched lines to thicker copper lines, etched on a usual circuit board. The microwave excitation was provided by a programable signal generator (Standford Research, SG384), connected to a fast switch (CMCS0947A-C2) to control the MW pulse and an amplifier (Mini-circuits, ZHL-16W-43-S). The timing sequences were provided by a digital card (SpinCore-PulseBlaster, PBESR-PRO-300), producing pulses with 3 ns resolution to FIG. 3 . 3Protocol used to obtain the magnetic field with the NV center considering the diamond chip has any crystallographic orientation[8]. FIG. 4 . 4(a)-(b) ODMR spectra measured at points on the highlighted red dashed-line, for currents ±15 mA [8]. The microscopy image, showing a section of the wire on the left, is 100 × 200 pixels (1 pixel = 0.7 µm). FIG. 5 . 5(a)-(c) Components of the magnetic field produced by a current I = 15 mA. (d) Magnitude of the field at the NV layer. (e) Microscopy image for a small section of the DC wire. (f) Direct normalized image of the photoluminescence (PL), during continuous laser excitation [8]. FIG. 6. Points indicate de measured components of the magnetic field produced by a current I = 15 mA. The solid color-coded lines are Biot-Savart simulations of the field produced at the NV layer for a wire of 100 nm thickness and width 24 µm at various probing distances z N V . The points (blue dots) were sampled along the indicated black dashed-line, perpendicular to the wire. de Azevêdo for helpful discussions. The authors acknowledge financial support from CAPES, CNPq, and FAPESP (grants 2019/27471-0 and. de Azevêdo for helpful discussions. The authors acknowledge financial support from CAPES, CNPq, and FAPESP (grants 2019/27471-0 and 2013/07276-1). 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Patrick Appel, Elke Neu, Marc Ganzhorn, Arne Barfuss, Marietta Batzer, Micha Gratz, Andreas Tschöpe, Patrick Maletinsky, Review of Scientific Instruments. 87663703Patrick Appel, Elke Neu, Marc Ganzhorn, Arne Barfuss, Marietta Batzer, Micha Gratz, Andreas Tschöpe, and Patrick Maletinsky. Fabrication of all diamond scanning probes for nanoscale magnetometry. Review of Scientific Instruments, 87(6):063703, 2016. Diamond studies for applications in quantum technologies. Charlie Oncebay Segura, 10.11606/T.76.2019.tde-01082019-152208Universidade de São Paulo, Instituto de Física de São CarlosPhD thesisCharlie Oncebay Segura. Diamond studies for applications in quantum technologies. PhD thesis, Universidade de São Paulo, Instituto de Física de São Carlos, 2019. https://doi.org/10.11606/T.76.2019.tde-01082019-152208 Optical properties of the nitrogenvacancy singlet levels in diamond. V M Acosta, A Jarmola, E Bauch, D Budker, Physical Review B. 8220201202V. M. Acosta, A. Jarmola, E. Bauch, and D. Budker. Optical properties of the nitrogen- vacancy singlet levels in diamond. Physical Review B, 82(20):201202, 2010. The nitrogen-vacancy colour centre in diamond. Marcus W Doherty, Neil B Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup, Lloyd C L Hollenberg, Physics Reports. 5281Marcus W. Doherty, Neil B. Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup, and Lloyd C.L. Hollenberg. The nitrogen-vacancy colour centre in diamond. Physics Reports, 528(1):1 -45, 2013. Excited states of the negatively charged nitrogen-vacancy color center in diamond. Yuchen Ma, Michael Rohlfing, Adam Gali, Physical Review B. 81441204Yuchen Ma, Michael Rohlfing, and Adam Gali. Excited states of the negatively charged nitrogen-vacancy color center in diamond. Physical Review B, 81(4):041204, 2010. Infrared emission of the nv centre in diamond: Zeeman and uniaxial stress. L J Rogers, M J Armstrong, N B Sellars, Manson, New Journal of Physics. 1010103024L J Rogers, S Armstrong, M J Sellars, and N B Manson. Infrared emission of the nv centre in diamond: Zeeman and uniaxial stress. New Journal of Physics, 10(10):103024, 2008. Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond. V M Acosta, E Bauch, M P Ledbetter, A Waxman, L.-S Bouchard, D Budker, Physical Review Letters. 104770801V. M. Acosta, E. Bauch, M. P. Ledbetter, A. Waxman, L.-S. Bouchard, and D. Budker. Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond. Physical Review Letters, 104(7):070801, 2010. Avoiding power broadening in optically detected magnetic resonance of single nv defects for enhanced dc magnetic field sensitivity. A Dréau, M Lesik, L Rondin, P Spinicelli, O Arcizet, J.-F Roch, V Jacques, Physical Review B. 8419195204A. Dréau, M. Lesik, L. Rondin, P. Spinicelli, O. Arcizet, J.-F. Roch, and V. Jacques. 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[ "Interplay of noise induced stability and stochastic resetting", "Interplay of noise induced stability and stochastic resetting" ]
[ "Karol Capała \nInstitute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland\n", "Bartłomiej Dybiec \nInstitute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland\n", "Ewa Gudowska-Nowak \nInstitute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland\n" ]
[ "Institute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland", "Institute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland", "Institute of Theoretical Physics\nDepartment of Statistical Physics\nJagiellonian University\nŁojasiewicza 1130-348KrakówPoland" ]
[]
Stochastic resetting and noise-enhanced stability are two phenomena which can affect the lifetime and relaxation of nonequilibrium states. They can be considered as measures of controlling the efficiency of the completion process when a stochastic system has to reach a desired state. Here, we study interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability which increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In the paper we present a framework to analyze compromises between the two contrasting phenomena in a noise-driven kinetics subject to random restarts. PACS numbers: 02.70.Tt, 05.10.Ln, 05.40.Fb, 05.10.Gg, 02.50.-r, Ubiquity of observed natural nonequilibrium systems acting under influence of noises attracted much attention on theoretical and experimental studies of dynamical stochastic systems and noise-induced phenomena.Stochastic resetting and noise-enhanced stability are different stochastic protocols which can be used to control the lifetime of states or efficiency of processes containing random components. On the one hand, the stochastic resetting can be used to eliminate subotimal trajectories and in turn it can accelerate the exit kinetics or increase the search efficiency. On the other hand, the effect of noiseenhanced stability owes its effectiveness to the presence of meandering trajectories, as the action of noise could induce emergence of very long trajectories. Here, we show how the stochastic resetting can counterbalance the action of noise which induces the noise-enhanced stability. As a result of the stochastic resetting, the system can be moved out of the metastable state and shifted to the desired state more easily.
10.1063/5.0092887
[ "https://arxiv.org/pdf/2203.13063v1.pdf" ]
247,628,216
2203.13063
e6c1c9c352e1bf6c696b0613dc40ab5c411d2d1d
Interplay of noise induced stability and stochastic resetting Karol Capała Institute of Theoretical Physics Department of Statistical Physics Jagiellonian University Łojasiewicza 1130-348KrakówPoland Bartłomiej Dybiec Institute of Theoretical Physics Department of Statistical Physics Jagiellonian University Łojasiewicza 1130-348KrakówPoland Ewa Gudowska-Nowak Institute of Theoretical Physics Department of Statistical Physics Jagiellonian University Łojasiewicza 1130-348KrakówPoland Interplay of noise induced stability and stochastic resetting (Dated: 25 March 2022) Stochastic resetting and noise-enhanced stability are two phenomena which can affect the lifetime and relaxation of nonequilibrium states. They can be considered as measures of controlling the efficiency of the completion process when a stochastic system has to reach a desired state. Here, we study interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability which increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In the paper we present a framework to analyze compromises between the two contrasting phenomena in a noise-driven kinetics subject to random restarts. PACS numbers: 02.70.Tt, 05.10.Ln, 05.40.Fb, 05.10.Gg, 02.50.-r, Ubiquity of observed natural nonequilibrium systems acting under influence of noises attracted much attention on theoretical and experimental studies of dynamical stochastic systems and noise-induced phenomena.Stochastic resetting and noise-enhanced stability are different stochastic protocols which can be used to control the lifetime of states or efficiency of processes containing random components. On the one hand, the stochastic resetting can be used to eliminate subotimal trajectories and in turn it can accelerate the exit kinetics or increase the search efficiency. On the other hand, the effect of noiseenhanced stability owes its effectiveness to the presence of meandering trajectories, as the action of noise could induce emergence of very long trajectories. Here, we show how the stochastic resetting can counterbalance the action of noise which induces the noise-enhanced stability. As a result of the stochastic resetting, the system can be moved out of the metastable state and shifted to the desired state more easily. Stochastic resetting and noise-enhanced stability are two phenomena which can affect the lifetime and relaxation of nonequilibrium states. They can be considered as measures of controlling the efficiency of the completion process when a stochastic system has to reach a desired state. Here, we study interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability which increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In the paper we present a framework to analyze compromises between the two contrasting phenomena in a noise-driven kinetics subject to random restarts. Ubiquity of observed natural nonequilibrium systems acting under influence of noises attracted much attention on theoretical and experimental studies of dynamical stochastic systems and noise-induced phenomena. Stochastic resetting and noise-enhanced stability are different stochastic protocols which can be used to control the lifetime of states or efficiency of processes containing random components. On the one hand, the stochastic resetting can be used to eliminate subotimal trajectories and in turn it can accelerate the exit kinetics or increase the search efficiency. On the other hand, the effect of noiseenhanced stability owes its effectiveness to the presence of meandering trajectories, as the action of noise could induce emergence of very long trajectories. Here, we show how the stochastic resetting can counterbalance the action of noise which induces the noise-enhanced stability. As a result of the stochastic resetting, the system can be moved out of the metastable state and shifted to the desired state more easily. I. INTRODUCTION AND MODEL Nowadays, it is well known and widely accepted that the noise in dynamical systems is not always detrimental, but it can also play a beneficial role [1][2][3] . The action of random noise underlines the occurrence of various noise-induced phenomena such as stochastic resonance 4,5 , resonant (and stochastic resonant) activation 6,7 or ratcheting effects 8,9 . Moreover, the appraisal of the role and importance of random forces has been not only studied theoretically, but recorded in many reallife situations and biological setups 10,11 . The noise stabilizing effect (noise-enhanced stability, NES) 12-14 appears for stochastic diffusion in potential profiles with metastable states. A signature of NES in static potentials is the non-monotonic behavior of the average escape time and a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] prolonged lifetime of metastable or unstable states observed at an optimal intensity of noise 15,16 . In a metastable fluctuating potential, NES has been observed regardless of the unstable initial position of the Brownian particle 14 . The appearance of the effect has been reported in physical systems like overdamped and underdamped Josephson junctions 17 , chemical Belousov-Zhabotinsky and Michaelis-Menten reactions 18,19 , ecological systems 20 and cancer growth dynamics 21 . Another process that can be used to control the decay of states is stochastic resetting 22,23 referring to situations where the dynamics of a system is stopped and started over. Starting from anew can either increase or decrease the average time needed to hit the target or hinder or accelerate the course of the chemical reaction. On the one hand, stochastic restarting can eliminate long trajectories, preventing the exploration of distant parts of the space. On the other hand, by preventing the exploration of distant points, it can decrease chances of finding distant targets. Within the model explored in this paper, the stochastic resetting can be understood as the external manipulation on the system that brings the Brownian particle (or, e.g., the concentration of chemical reactants) to the initial state. Our presentation begins with a general discussion of the effect of noise-enhanced stability and its properties. Subsequently, we discuss the NES phenomenon with the relation to a model of tumor growth and control 19 . We study the overdamped motion in a static potential V (x) described by the Langevin equation 1,[24][25][26] dx dt = − dV (x) dx + σξ(t),(1) where −V (x) is the deterministic force and ξ(t) is the Gaussian white noise satisfying ξ(t) = 0 and ξ(t)ξ(s) = δ(t − s). We assume V (x) in a simple form of the polynomial V (x) = 2 − x + (x − 2) 3 .(2) The cubic profile (c.f., Fig. 1 In the absence of resetting, the probability p(x, t) = p(x, t|x 0 , t 0 ) of finding the particle at time t in the vicinity of x evolves according to the (forward) Smoluchowski-Fokker-Planck equation 27,28 ∂p(x, t) ∂t = ∂ ∂x V (x)p(x, t) + σ 2 2 ∂p(x, t) ∂x . (3) −6 −4 −2 0 2 4 1 2 3 4 x V (x) FIG. 1. The cubic potential given by Eq. (2) used in the study of noise-enhanced stability under stochastic resetting. The top of the potential barrier is located at x = 2 − 1/ √ 3 ≈ 1.42, while a local stable state is associated with the minimum of the potential located at x = 2 + 1/ √ 3 ≈ 2.58. We explore the properties of noise-enhanced stability 14,19,29 under stochastic resetting 22,23,30 as these two effects compete with each other. More precisely, NES is responsible for elongating the decay of unstable states, while stochastic resetting is typically expected to decrease the lifetime and expedite the completion of the first passage process like, e.g., a chemical reaction. To assess the interplay between these two stochastic effects, we explore the properties of the mean first passage time (MFPT) from the interval (x b , ∞), which is the average of the first passage times T = t fp = min{t : x(0) = x 0 ∧ x(t) < x b } ,(4) where x b is the location of the absorbing boundary. Therefore, the MFPT can be used to measure the lifetime, i.e., the average time needed to annihilate a particle at the absorbing boundary located at x b . As we will be further reinterpreting x(t) as the concentration of reagents (or cells, see Sec. II B), by assumption x b cannot be negative. Hereafter, we set the barrier position to x b = 0. For the uninterrupted motion, the MFPT designed as T (x) satisfies the equation 27,28 − V (x) ∂T (x) ∂x + σ 2 2 ∂ 2 T (x) ∂x 2 = −1(5) with the additional condition T (x b ) = 0. The action of resetting 22,30 is further addressed by assuming random restarts of the diffusive process occurring in time according to Poisson statistics. The underlying diffusion process {X(t)} starts at some point x 0 and evolves over a certain (random) time on the interval (x b , ∞) with deterministic forcing −V (x) powered by Gaussian white noise. By a resetting mechanism, a particle diffusing in the potential V (x) is revert to the point x 0 on the left hand side of the barrier x 0 ∈ (0, 2 − 1/ √ 3) and restarts its motion. Without loss of generality, we can assume that the system starts (at t = 0) its evolution at x 0 , i.e., we study the system dynamics right after the first resetting. The time duration τ between two subsequent resets is a random variable following the exponential distribution φ(τ ) = r exp (−rτ ) with r (r > 0) denoting the resetting rate. The effect of resetting can be assessed by examination of the coefficient of variation (CV) 31 CV = σ(t fp ) t fp = σ(t fp ) T ,(6) which is the ratio between the standard deviation σ(t fp ) of the first passage times and the mean first passage time T in the absence of stochastic resetting 32,33 . As explained in 32,34 CV can be used to find the domain where resetting can facilitate the escape kinetics. Such a domain corresponds to CV > 1 and it is the domain which we are interested in. Noise enhances stability is observed in unstable potentials where the action of noise can increase the lifetime of unstable states. The facilitation of the escape kinetics due to resetting can counteract the action of the noise-enhanced stability and make the elimination of tumor cells more efficient. II. RESULTS We start with a detailed discussion of the ordinary noiseenhanced stability effect under stochastic resetting (Sec. II A). Afterwards, we reinterpret the model in the context of tumor dynamics (Sec. II B) . A. Ordinary NES The model described by Eqs. (1) and (2) is studied by the method of stochastic dynamics. Namely, the Euler-Maruyama method 35,36 is applied. Therefore, x(t + ∆t) is approximated as x(t + ∆t) = x(t) − V (x(t))∆t + σξ i √ ∆t,(7) where ξ i is the sequence of independent identically distributed random variables following the standard normal distribution N (0, 1). The top panel of Fig. 2 presents the numerically estimated mean first passage time from (x b , ∞) with x b = 0 under the action of the deterministic force derived from the potential V (x) given by Eq. (2) for various values of the resetting rate r. r = 0 corresponds to the case without resetting, i.e., to the noise-induced dynamics in an unstable potential, which can result in the occurrence of the noise-enhanced stability. The absorbing boundary is located at x b = 0, while the initial position x 0 is set to x 0 = 1. The deterministic force prevents the particle from escaping to infinity, therefore it effectively plays the role of the reflecting boundary, which is formally located at +∞. The bottom panel of Fig. 2 displays the corresponding dependence of the coefficient of variation, see Eq. (6). Examination of CV, see the bottom panel of Fig. 2, clearly indicates that in the majority of situations stochastic resetting facilitates the escape kinetics. The CV becomes smaller than 1 only at a very low noise intensity σ when the motion is practically deterministic. For these low values of the noise intensity (σ ≈ 0), the stochastic resetting interrupts the deterministic sliding, thus enhancing the lifetime of deterministically unstable states. Also, for σ ≈ 0, the MFPT grows semi-exponentially with the increase of the resetting rate r, see App. A. The conclusions drawn from the analysis of the CV behavior are corroborated by studies of the MFPT, see the top panel of Fig. 2. It shows that despite the fact that stochastic resetting can suppress the sliding to the absorbing boundary, it is typically not capable of enhancing the lifetime of unstable states when σ 0. Within our diffusion model incorporating NES and stochastic resetting, two distinct resetting regimes are visible. For r → 0 resets are very rare, and the stochastic motion tends to the overdamped diffusion without restarts. In this regime, starting motion anew increases the MFPT only for small noise intensities, as resetting hinders sliding towards the absorbing boundary, see App. A. In the opposite limit of r → ∞, resets are so frequent that the particle practically cannot move freely because it is immediately restarted. These frequent restarts are responsible for the diverging lifetimes of unstable states. Nevertheless, such enhancement does not rely on any nontrivial interplay between resetting and stochastic dynamics as it simply "sticks" a particle to the initial position x 0 . To find the reason why stochastic resetting shortens the lifetime of unstable states, we return to the examination of the classical NES effect in the potential given by Eq. (2). For the resetting-free case, the MFPT can be calculated as the mean first passage time from the interval restricted by the reflecting boundary (located at +∞) and the absorbing boundary placed at x b = 0 28 . The MFPT satisfying Eq. (4) reads T (x) = 2 σ 2 x x b dy ψ − (y) ∞ y ψ − (z)dz,(8) where ψ ∓ (x) is given by ψ ∓ (x) = exp ∓ 2V (x) σ 2 .(9) From Eq. (8) one gets the classical formula for the MFPT T (x) = 2 σ 2 x x b exp 2V (y) σ 2 dy ∞ y exp − 2V (z) σ 2 dz. (10) From the top panel of Fig. 3 it is visible that the numerically estimated MFPT perfectly follows the theoretical curve. Moreover, in the limit of σ → 0 the MFPT is given by T = − x b x0 1 V (y) dy,(11) which is the time of the deterministic sliding from x 0 to x b . For σ = 0 the MFPT can be calculated directly from the Langevin equation, see Eqs. (1) and (11). In the very same regime, the MFPT can be also calculated from Eq. (5). The dependence of the MFPT on the noise intensity σ is clearly non-monotonic, see the top panel of Fig. 3. There exists an optimal value of the σ leading to the maximal lifetime. Typically, the rapid increase in the MFPT is attributed to trajectories which manage to surmount the potential and overpass the potential barrier. This qualitative observation can be analyzed in a more systematic way. The MFPT does not fully characterize the escape kinetics. Consequently, the examination of other measures can be insightful. For instance, the escape process can be further characterized by the splitting probability. For the motion in the finite interval (a, b) (a < b) one can calculate the probability of leaving the interval via the particular boundary a or b. The probability π R (x) = π b (x) of leaving (a, b) to the right satisfies 27,28 − V (x) ∂π b (x) ∂x + σ 2 2 ∂ 2 π b (x) ∂x 2 = 0(12) and it is given by π b (x) = x a ψ + (y)dy b a ψ + (y)dy = x a exp 2V (y) σ 2 dy b a exp 2V (y) σ 2 dy .(13) The probability of escaping to the left (π L (x)) can be calculated from the relation π R (x) = π b (x) = 1 − π a (x) = 1 − π L (x). The splitting probability can provide fruitful insights into the escape kinetics. In the top panel of Fig. 3 using the right ordinate, in addition to MFPT, the probability of the first escape from (0, 2 − 1/ √ 3) to the right is depicted and compared with results of computer simulations (triangles) indicating perfect level of agreement. In situations when π R > 0 there are some trajectories which prior to the absorption at x b = 0 reached the barrier top located at x = 2−1/ √ 3. These trajectories have the possibility of sliding to the minimum of the potential and wander around x ≈ 2 + 1/ √ 3. This in turn gives the chance of enhancing the lifetime by trapping the particle in the vicinity of the potential minima. Comparison of the MFPT with the dependence of the π R indicates that MFPT grows significantly when π R becomes slightly larger than 0. For instance, the maximal value of the MFPT in Fig. 3 is recorded for σ ≈ 0.42 and it reads T = 24.74 with π R = 0.004, i.e, only 0.4% of trajectories overpassed the potential barrier prior to the absorption at x b = 0. For larger noise intensities particles can not only more easily reach the top of the potential barrier but after overpassing the potential barrier they can more easily return back. Consequently, for large values of σ the increase in the MFPT is not as large as for small noise intensities and the MFPT is the decreasing function of σ, see Fig. 3(a). In order to fully resolve the origin of NES we explore the statistics of maximal x max (most distant points to the right) visited by individual trajectories. Bottom panel of Fig. 3 depicts x max , median of x max (x 0.5 max ), quantiles of order 0.1 (x 0.1 max ) and 0.9 (x 0.9 max ) along with minimal (min({x max })) and maximal (max({x max })) recorded values of x max . Quantiles are presented as boxes, the bottom part of the box shows 0.1 quantile while top of the box 0.9 quantile. Minimal value, which is practically the same as 0.1 quantile, and maximal values of x max are depicted by whiskers. Examination of x max statistics, analogously like the splitting probability π R , indicates that the initial increase in the MFPT is produced by a very few trajectories that managed to overpass the top of the barrier, i.e., to reach points located to the right of the potential barrier ( x > 2 − 1/ √ 3) , which position is denoted by the dashed line. Please note that x 0.9 max rises above the boundary for σ > 1.2, i.e., well above the value of the noise intensity which maximizes the lifetime. It means, that for σ < 1.2, not less than 90% of most distant visited points are located to the left of the potential barrier. Therefore, the increase in the MFPT is mainly determined by a very few trajectories which pass over the potential barrier and get trapped in the vicinity of the potential minima. From the top panel of Fig. 3 it implies that the maximal MFPT is associated with π R = 0.004. Therefore, only 0.4% of trajectories managed to overpass the potential barrier prior to the absorption at x b = 0. These very few trajectories are responsible for the enormous increase in the MFPT at σ ≈ 0.42. Importantly, this observation provides justification why resetting typically decreases the lifetime of unstable states. On the one hand it can suppress sliding to the absorbing boundary, but on the other it eliminates trajectories which manage to surmount the potential or to reach the potential minimum. For σ > 0 the MFPT is determined by trajectories which have accomplished climbing up the potential. The gain due to surmounting can be eliminated by restarting the motion because it efficiently bounds the most distant visited point. This is in line with results of 34 , where Authors have shown that the interplay between the thermal and potential energy is the key factor determining the resetting efficiency. B. Model reinterpretation The model defined by Eq. (1) can be reinterpreted in the language of chemical kinetics 26,34,37,38 . Contrary to quantum tunneling process or nucleation, the escape problem in the Kramers approach is defined by analyzing a classical point particle escaping due to random forces. Contemporary applications of that scheme range from condensed matter and biological systems to high-energy physics and cosmological phase transitions 38 . The kinetic scheme of tumor growth proposed in 39 involves combination of replication of transformed cells and immunological interaction of the host organism with transformed cells. Free effector cells like T-lymphocytes or killer cells form a complex with the transformed cells followed by lysis of tumor cells and dissociation of the complex to nonreplicating (or dead) tumor cells and free effector cells. The target population (concentration) of tumor cells evolves then 19,40 according to the equation dx dt = (1 − Θx)x − β x x + 1 + σξ(t)(14) resembling the Michaelis-Menten kinetics X + Y → E → Y + P . With x representing the concentration of tumor cells, y standing for concentration of effector cells and parameters β = 1.48, Θ = 0.25, this model reflects bistability with minima x = 0 and x ≈ 2.08 separated by a potential barrier at x ≈ 0.925, thus capturing qualitatively features of the kinetic model defined by Eqs. (1) - (2). In particular, the potential V (x) derived for this model has an archetypal Kramers form similar to the cubic potential depicted in Fig. 1. The concentration of transformed cells changes due to the interplay of deterministic kinetics, environmental fluctuations (white noise term) and applied therapy (resetting). Depending on the system state (x value) the action of the deterministic forces can increase or decrease the concentration of tumor cells. The minimum of the potential located to the right of the barrier corresponds to the metastable state characterized by the high tumor concentration. To eliminate tumor cells, it is necessary to overpass the potential barrier and reach the x = 0 concentration. The chances of spontaneous (deterministic) reaching x = 0 are minimal as fluctuations are the only forces which can induce surmounting of the potential barrier separating stationary states. On the one hand, if the applied therapy can move the system out of the metastable state associated with the potential well to the point located to the left of the barrier, it increases the chances of reaching the state of negligible tumor concentration. On the other hand, the system moved out of the metastable state can be sensitive to the effect of noiseenhanced stability, which weakens the role of applied therapy. Let us assume that the applied therapy (resetting) cannot fully eliminate the tumor and its action brings the system to the x 0 state with a lower concentration of tumor cells. Without loss of generality, we can also assume that the system starts (at t = 0) its evolution at x 0 , i.e., we analyze the tumor evolution right after the first initiation of the (random) treatment. Fig. 2 indicates that the randomly applied therapy typically decreases the time needed to reach the x = 0 state facilitating the overall efficiency, because it can move the system out the domain of motion associated with the metastable fixed point, see Fig. 1. Therefore, it facilitates the transition over the potential barrier. The only exception is observed for very small σ when frequent resetting can trap the particle in the vicinity of x 0 . The examination of the MFPT curve, see Fig. 2, provides justification why resetting typically increases the efficiency of the applied therapy. III. SUMMARY AND CONCLUSIONS Noise-enhanced stability belongs to the class of noiseinduced effects. It demonstrates that the action of noise can increase the lifetime of unstable states. At the same time, stochastic resetting is a protocol that can accelerate kinetics and decrease the mean first passage time 34 . Opposite roles played by these two effects call for understanding whether their simultaneous actions can counterbalance. By using numerical simulations and phenomenological arguments, we have shown that stochastic resetting typically is not sufficient to increase the lifetime of unstable states. The exceptions to this observation are detected only in the vanishing noise limit or in the limit of infinitely frequent resetting. For vanishing noise, stochastic resetting prevents deterministic sliding to the absorbing boundary, whereas for the infinite resetting rate the resetting protocol traps the particle in the neighborhood of the restarting position. The generic noise-enhanced stability setup can also be used to describe tumor dynamics. In such a case, the particle position is interpreted as the concentration of the cancer cells. Following this line of interpretation, the effect of the noiseenhanced stability can slow down tumor elimination. However, with the help of stochastic resetting, interpreted as applied therapy, the eradication of tumor cells can be more efficient, even if the eradication (resetting to negligible concentration of transformed cells) is performed at random times. ACKNOWLEDGMENTS This research was supported in part by PLGrid Infrastructure and by the National Science Center (Poland) grant 2018/31/N/ST2/00598. KC would like to thank Marta Capała for sharing the medical perspective. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author (KC) upon reasonable request. Appendix A: Weak noise limit For σ → 0 and x 0 ∈ (0, 2 − 1/ √ 3) the particle described by Eq. (1) deterministically slides towards the absorbing boundary. The time of deterministic sliding T can be calculated from the backward Smoluchowski-Fokker-Planck equation with σ = 0, see Eq. (5), or from the noise-free Langevin equation dx dt = −V (x),(A1) with the initial conditions x(0) = x 0 and x(T ) = x b . The time T is given by Eq. (11) of the main text and it is the mean first passage time for the uninterrupted motion. As is visible from Figs. 2 and 3 the MFPT is close to T also for very small σ. Here, we determine how, in the small σ limit, the lifetime changes with resetting. We will show that the lifetime is fully determined by the time of the deterministic sliding T and the reset rate r. The distribution of time intervals τ between two consecutive resets follows the exponential density φ(τ ) = re −rτ .(A2) Stochastic resetting can restart the motion if the interresetting time τ is smaller than the sliding time T , i.e., if τ < T , with the probability p = T 0 φ(t)dt = 1 − e −rT .(A3) For r > 0, the recorded number of resets n (n ∈ {0, 1, . . . }) follows the geometric distribution. p r (n) = p n (1 − p) (A4) characterized by the mean value n = p 1 − p (A5) and the variance σ 2 (n) = p (1 − p) 2 .(A6) In the limit of r → ∞ the probability p tends to 1, making the average number of resets infinite, which in turn is responsible for the unrestricted increase of the MFPT. The average lifetime T can be calculated as T = n × τ |τ < T + T,(A7) where τ |τ < T is the conditional average of interresetting time following the exponential distribution restricted to [0, T ) τ |τ < T = The first term, n × τ |τ < T , measures the total duration of slides interrupted by stochastic resets. The last term in Eq. (A7), T , quantifies the ultimate part of the motion, i.e, the time of sliding from x 0 to x b . Finally, we obtain the following formula for the MFPT under stochastic resetting T = t fp = e rT − 1 r . (A9) The variance of the first passage time distribution can be calculated using the law of total variance σ 2 (t fp ) = n ×σ 2 (τ |τ < T )+σ 2 (n)× τ |τ < T 2 , (A10) with σ 2 (τ |τ < T ) = 1 r 2 + T 2 2 − 2 cosh(rT ) . Finally, one gets σ 2 (t fp ) = e 2rT − 2e rT rT − 1 r 2 = 2e rT [sinh(rT ) − rT ] r 2 . (A12) Both the mean lifetime and the variance of individual lifetimes grow semi-exponentially with r. For r = 0 the problem is fully deterministic with T given by Eq. (11). In order to verify formulas (A9) and (A12) we have performed simulations for T arbitrarily, but without the loss of generality, set to T = 1. Fig. 4 presents the MFPT (circles) and the standard deviation of first passage times (triangles). Solid lines correspond to theoretical curves, see Eqs. (A9) and (A12). Points, perfectly following theoretical predictions, represent the results of computer simulations. The ratio of the theoretical and simulated values of MFPT and standard deviation is equal to unity with an accuracy not worse than 1% (results not shown). The stochastic resetting with a finite resetting rate r is unable to fully eliminate the decay of unstable systems due to its stochastic character, i.e., the variability in the interresetting time. Contrary to the Poissonian resetting, sharp resetting (p(τ ) = δ(τ − τ 0 )) is capable of producing infinite lifetime for finite (fixed) interresetting time τ 0 (τ 0 < T ). PACS numbers: 02.70.Tt, 05.10.Ln, 05.40.Fb, 05.10.Gg, 02.50.-r, FIG. 2 . 2MFPT (top panel -(a)) as a function of the noise intensity σ for various values of resetting parameter r, together with corresponding values of coefficient of variation CV (bottom panel-(b)). The initial position is set to x0 = 1. The absorbing boundary is located at x b = 0. FIG. 3 . 3Top panel (a) presents MFPT as a function of the noise intensity σ for r = 0 (left axis), together with corresponding values of the splitting probability πR (right axis). Solid lines show theoretical values of the MFPT, see Eq. (10), and splitting probability, see Eq. (13). The dashed line displays the time of deterministic sliding Tσ=0 ≈ 0.21. 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Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations," SIAM Review 43, 525-546 (2001). Integration of stochastic differential equations on a computer. R Mannella, Int. J. Mod. Phys. C. 13R. Mannella, "Integration of stochastic differential equations on a com- puter," Int. J. Mod. Phys. C 13, 1177-1194 (2002). Stochastic resetting in the kramers problem: A monte carlo approach. J Cantisán, J M Seoane, M A Sanjuán, Chaos, Solitons & Fractals. 152111342J. Cantisán, J. M. Seoane, and M. A. Sanjuán, "Stochastic resetting in the kramers problem: A monte carlo approach," Chaos, Solitons & Fractals 152, 111342 (2021). Formulating the kramers problem in field theory. A Berera, J Mabillard, B Mintz, R Ramos, Phys. Rev. D. 10076005A. Berera, J. Mabillard, B. Mintz, and R. Ramos, "Formulating the kramers problem in field theory," Phys. Rev. D 100, 076005 (2019). A kinetic approach to the immunology of cancer: Stationary states properties of efffector-target cell reactions. R P Garay, R Lefever, J. Theor. Biol. 73R. P. Garay and R. Lefever, "A kinetic approach to the immunology of can- cer: Stationary states properties of efffector-target cell reactions," J. Theor. Biol. 73, 417-438 (1978). Survival analysis for tumor growth model with stochastic perturbation. D Li, N Zhang, M Yan, Y Xing, Discrete Contin. Dyn. Syst. B. 265707D. Li, N. Zhang, M. Yan, and Y. Xing, "Survival analysis for tumor growth model with stochastic perturbation," Discrete Contin. Dyn. Syst. B 26, 5707 (2021).
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[ "Helmut Satz \nFakultät für Physik\nUniversität Bielefeld\nD-33501BielefeldGermany\n" ]
[ "Fakultät für Physik\nUniversität Bielefeld\nD-33501BielefeldGermany" ]
[ "Concluding talk at the International Workshop on the Physics of the Quark-Gluon Plasma,École Polytechnique" ]
The aim of high energy nuclear collisions is to study the transition from hadronic matter to a plasma of deconfined quarks and gluons. I review the basic questions of this search and summarize recent theoretical developments in the field.
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[ "https://arxiv.org/pdf/hep-ph/0111265v1.pdf" ]
9,611,534
hep-ph/0111265
815fd3382b63f02de5651715249ed94b225e75ac
Sept. 4 -7, 2001 Helmut Satz Fakultät für Physik Universität Bielefeld D-33501BielefeldGermany Concluding talk at the International Workshop on the Physics of the Quark-Gluon Plasma,École Polytechnique Palaiseau/FranceSept. 4 -7, 2001Matter & More in Nuclear Collisions 1 1 The aim of high energy nuclear collisions is to study the transition from hadronic matter to a plasma of deconfined quarks and gluons. I review the basic questions of this search and summarize recent theoretical developments in the field. New States of Matter Statistical QCD predicts that high temperatures and baryon densities will lead to new states of strongly interacting matter. Increasing T at low baryon density transforms a meson gas into a deconfined plasma of quarks and gluons (QGP); this transition has been studied extensively in computer simulations of finite temperature lattice QCD. High baryon densities at low T are expected to produce a condensate of colored diquarks. The resulting phase diagram in terms of temperature and baryochemical potential µ is schematically illustrated in Fig. 1, with hadronic matter as color insulator, the QGP as color conductor, and the diquark condensate as color superconductor. With high energy nuclear collisions, we want to study in the laboratory the deconfinement transition and the properties of the QGP. Hard probes, such as the production of quarkonia, open charm and beauty, jets and photons are expected to provide information about the hot early stages of the produced medium. The masses and decays of different hadrons, their momenta, correlations and relative abundances constitute the soft probes to study the later stages of the medium and its freeze-out. In this concluding talk, I will summarize some recent theoretical developments in the field, without any claim to completeness. My emphasis will be on concepts more than on specific models, and on questions more than on answers. Thermalization Since the basic purpose of the experimental program is to produce strongly interacting matter, it is of central importance to determine if, how and when the non-thermal initial state of two colliding nuclei becomes thermalized. In a nuclear collision, the incoming nucleons or their secondaries can interact with the other target nucleons. This results in nuclear phenomena such as the Cronin effect, normal nuclear quarkonium suppression or parton energy loss in normal nuclear matter. These effects do not involve any new produced medium. To achieve that, the secondaries coming from different sources must interact, a phenomenon referred to as color interconnection, exogamous behavior or cross talk [1]. A test for such cross-talk has been considered in e + e − annihilation into hadrons at √ s = 2M W , as studied in LEP experiments at CERN [2]. The reaction first leads to W + W − production; subsequently, one possibility is that each of the two W 's decays into a qq pair, which then hadronizes. An alternative channel has one of the two W 's undergo leptonic decay into a neutrino and a lepton. In the case of cross talk, it is predicted [1] that for the resulting hadron multiplicities, N h (q 1q1 , q 2q2 ) = 2N h (qq, νl); in addition, the source radii obtained through HBT studies should be different in the two channels. Neither of these predictions is supported by LEP data, so that so far there is no evidence of cross talk shown by the hadrons produced in high energy e + e − annihilation. This also excludes cross talk at earlier partonic stages. Do AA collisions with their much higher density of superimposed interactions lead to cross talk? We address this question by looking at hadron abundances. It is found that these are quite well described by the predictions of an ideal resonance gas [3], parametrized in grand-canonical form by a freeze-out temperature T f and a baryochemical potential µ B . With increasing collision energy T f converges to about 170 MeV (Fig. 2). This alone does not, however, allow us to conclude that we have indeed obtained a thermal system with full cross-talk. It is known [5] that also the elementary hadroproduction processes initiated by e + e − annihilation or pp/pp scattering lead to thermal hadron abundances, with freeze-out temperatures which agree very well with those observed in AA collisions; they are included in Fig. 2. There is, however, one important difference: in the elementary reactions, there are fewer strange hadrons than predicted by a grand-canonical resonance gas, while AA collisions do not show such a strangeness reduction. The change in relative strangeness production, referred to as strangeness suppression or enhancement, depending on the point of view, has found a very natural explanation in the observation [6] that in elementary processes, with rarely more than one ss pair per interaction region, strangeness conservation has to be taken into account exactly and not just "on the average", as implied in a grand-canonical formulation. It had in fact been observed long ago that the exact local conservation of quantum numbers can reduce the relative production rates by several orders of magnitude in comparison to the average grand-canonical rates [7]. Data are given in [3][4][5]. The transition from exact to average (grand-canonical) strangeness conservation does imply, however, that the AA collisions behave as one large system, not as a sum of many elementary collisions. The strange hadrons produced in one elementary collision of an AA interaction must be aware of the strange hadrons produced in other NN collisions, if a grand-canonical description is valid. So at least at the hadronization stage of the medium produced in nuclear collisions, there is cross talk -it makes sense to speak of a large-scale hadronic medium. One remaining question in this context is the correct treatment of hidden strangeness; if the ss nature of the φ is not taken into account, its production rates are quite generally overpredicted. In addition, there remains the tantalizing question of why the hadron abundances in elementary processes like e + e − or pp/pp already follow the pattern predicted by an ideal resonance gas. It might just indicate that the non-perturbative hadronization of quarks and gluons simply proceeds such as to maximize the entropy: partons hadronize into the states of a resonance gas with largest phase space. The associated temperature is then the limiting temperature of such a system, the Hagedorn temperature, and hence is universal. Since the initial structure of e.g. e + e − → hadrons is that of a fast qq pair emitting gluons which later hadronize, the annihilation process is certainly not thermal in its early stages. Thermal hadron abundances thus do not imply that the previous partonic system was already thermal. Partonic cross talk is the basic input of most parton cascade models and thus has been considered in the corresponding codes for quite some time. Its role in the establishment of parton thermalization has recently been addressed in an interesting conceptual treatment [8]. The two incoming nuclei can be viewed in the central rapidity region (where the valence quarks are unimportant) as gluon beams in which each gluon has a transverse radius r g ∼ 1/k T determined by its transverse momentum k T . The geometric interaction cross section for gluon of sufficient hardness, k T ≥ q 0 , is σ g ∼ α s (q 0 )r 2 g . We would like to know when these gluons of sufficient hardness begin to overlap in the transverse area determined by the nuclear radius R A . Full cross talk evidently occurs when a gluon on one side of the nuclear disk πR 2 A is connected by overlapping gluons to a partner on the other side of the disk. This starts at the percolation point N g σ g ≃ πR 2 A ,(1) where N g (k T ) = Axg(x) denotes the number of gluons as determined from the gluon distribution function g(x) at central Bjorken x ∼ √ s. From Eq. (1) we thus obtain q 2 s ∼ α s A 1/3 xg(x)(2) for the saturation momentum q s . When k T >> q s , the gluons form a dilute and hence disjoint system in the transverse plane, without cross talk. At k T = q s percolation sets in and the gluons form a connected interacting system, with full cross talk. For such a system, one can estimate that thermalization occurs after a time τ 0 ∼ α −13/5 s q −1 s ,(3) leading to a thermal gluon medium of temperature T 0 ∼ α 2/5 s q s .(4) From deep inelastic scattering one has xg(x) ∼ ( √ s) λ , with λ ≃ 0.2. Together with Eq. (2), this implies that large nuclei (large A) and/or high collision energies (large √ s) lead to early parton thermalization and a hot QGP. A very important task for theory is clearly to turn these conceptual considerations into a quantitative formalism. Hadrons in Matter Do AA collisions produce an interacting hadronic medium, or does the earlier partonic state hadronize directly into an ideal resonance gas? That is the main question to be addressed here. It is of particular interest in view of chiral symmetry restoration. For temperatures T < T c , the massless quarks of the QCD Lagrangian L QCD "dress" themselves through gluon interactions to become constituent quarks with an effective mass M q ∼ 0.3 -0.4 GeV, thereby spontaneously breaking the chiral symmetry of L QCD . For T = T c , chiral symmetry is restored and M q → 0; we have here assumed a system of vanishing baryon number density, for which deconfinement and chiral symmetry restoration coincide at T = T c . Since the vector meson mass M ρ ≃ 2M q , the behavior of M ρ (T ) in an interacting hadronic medium for T → T c would be a way to study the onset of chiral symmetry restoration. The in-medium behavior of hadron masses can be calculated in finite temperature lattice QCD. First studies addressed the temperature dependence of the screening mass in quenched QCD. Below T c , they showed very little T -dependence; but in view of the noted simplifications, they are presumably not really conclusive. Today it is possible to calculate the actual pole mass, but with present computer performance still only for the quenched case [9]. The advent of more powerful computers in the next 2 -3 years should, however, lead to such calculations in full QCD with light quarks. The present quenched studies show that above T c , there are no more mesonic bound states; scalar and pseudoscalar correlations agree, indicating chiral symmetry restoration. Below T c , the results for the pole masses also show rather little T -dependence; however, this may well be an artifact of quenching, as the following considerations seem to indicate. Lattice studies of the heavy quark potential V Q (T, r) in full QCD (with light dynamical quarks) show at all temperatures T string breaking in the large distance limit r → ∞. At T = 0, the string connecting two heavy color charges should break when its energy surpasses the mass of a typical light hadron, or equivalently when V Q (T, r = ∞) ≃ 2M q (T ).(5) Hence M q (T ) can be determined in finite temperature lattice calculations of V Q (T, r). Note, however, that the constituent quark mass here is obtained from a heavy-light meson and could contain some dressing of the heavy quark; hence it need not coincide fully with that from a light-light meson such as the ρ. Heavy quark potential studies have recently been carried out for N f = 2 and 3 for a range of different quark masses [10]. They show in particular that • for m q < ∼ 0.4 T ≃ 60 MeV, the dependence of V Q on m q becomes negligible, indicating that the chiral limit is reached; • string breaking occurs earlier (at smaller r) with increasing temperature (Fig. 5); • from V Q (T = 0, r = ∞) ≃ 1 GeV it follows that M q (T = 0) ≃ 0.5 GeV, while for T → T c , V Q (T ) and hence also M q (T ) vanish. The resulting temperature dependence of the heavy quark potential is shown in Fig. 3. It is seen that the approach of chiral symmetry restoration leads to a pronounced variation of V Q (T, ∞) and hence of M q (T ) with T . This suggests that also the mass of the ρ-meson should show such a temperature variation, in contrast to the present pole-mass results from quenched lattice QCD. A study of the temperature dependence of hadron masses in unquenched lattice calculations would thus be of great interest. Partons in Matter Since the energy loss of a fast parton passing through a medium will depend on the nature of this medium, jet quenching should provide a tool to specify the state of matter produced in nuclear collisions. At lower momenta, the energy loss can occur through ionisation of the constituents of the medium; at high momenta, gluon radiation of the passing parton is the main mechanism. The crucial feature in radiative energy loss is the formation time t(k) or the formation length z(k) for a gluon of momentum k, compared to the intrinsic scales of the medium: the mean free path λ, the mean distance d between scattering centers and the overall linear size L of the medium. For λ > d > z(k), the radiated gluons see independent charges and the scattering is incoherent. For d < λ < z(k), there is coherent scattering of the nascent gluon with several scatterers, leading to destructive interference; this is the so-called Landau-Pomeranchuk-Migdal (LPM) effect which reduces the energy loss. In Fig. 4 we compare schematically the incoherent form dE/dz ∼ −E, where E is the parton energy, to the LPM form dE/dz ∼ − √ E. In a medium of small linear size, L < L c , there is a further finite size reduction, leading to [11] − dE dz ≃ 3α s π (µ 2 E/λ) 1/2 , L > L c (µ 2 /λ) L , L < L c(6) for the energy loss in a quark-gluon plasma. Here L c ≡ (Eλ/µ 2 ) 1/2 denotes the limiting length scale and µ −1 the screening length of the medium. For a parton traversing a QGP at temperature T = 250 MeV over a length of L = 10 fm, Eq. (6) leads to an energy loss of about 30 GeV; this is to be compared to indications that cold nuclear matter of the same size would only result in an energy loss of 2 GeV [12]. In contrast to the QGP results, the value for a normal nuclear medium is really only an estimate, and thus only jet production data from pA collisions can provide a reliable basis for comparison. Another problem is immediately evident from the QGP calculation. What transverse momenta are really needed to specify a jet? Present RHIC data stop around at leading particles of some 5 GeV, which presumably is well below the value for the jets assumed in the QCD studies. We add a comment here on the interpretation of the RHIC results on high p T hadrons. The measured spectra have to be compared to some reference spectrum in order to look for a possible quenching, and this is generally based on binary collisions: the spectra from central AA collisions are normalized to pp (or peripheral AA) data multiplied by the number of binary collisions. The resulting ratio (see Fig. 5) is well below unity in the entire range 0 < ∼ p T < ∼ 5 GeV. If one would instead use the number of wounded nucleons as reference, the corresponding ratio will be larger than unity for almost all values of p T . From multiplicity studies it is clear that the overall data, dominated by low to intermediate p T , are well below what is expected from binary collision scaling. Hence in order to obtain a reliable reference, one should use an interpolating form of the type dN dp 2 T ref AA = N w 1 − p 2 T a + p 2 T + N c p 2 T a + p 2 T dN dp 2 T pp ,(7) where N w denotes the number of wounded nucleons and N c the number of binary collisions; the parameter a determines the relative importance of the two types of production mechanisms. Instead of being set to zero, as in present studies, it should be choses such as to correctly reproduce the measured multiplicity. This is expected to lead to a behavior like that shown in Fig. 5, with a Cronin-like pattern at relatively low p T followed by a suppression below unity, and one could then clearly define quenching effects. The last point to be addressed in this section concerns the radiative energy loss of heavy quarks traversing a QGP. It was noted [13] that for massive quarks the gluon emission suffers a 'dead-cone' effect, which suppresses radiation for forward angles θ < ∼ M Q / P 2 Q + M 2 Q ,(8) where M Q denotes the mass of the heavy quarks. This radiation suppresion in turn reduces the energy loss of heavy quarks and thus predicts an increase of the ratio D/π for high p T . Figure 5: Expected transverse momentum distribution from AA collisions normalized to a reference distribution interpolating from a wounded nucleon to a binary collision model (see Eq. 7). Quarkonia in Matter The essential feature that distinguishes the quarkonium ground states J/ψ and Υ from the normal light hadrons is their much smaller radius (about 0.2 fm for the J/ψ and about 0.1 fm for the Υ), due to the much higher bare quark mass (m c ≃ 1.4 GeV, m b ≃ 4.5 GeV). Their binding is thus largely due to the Coulombic part of the QCD potential σr−α/r; the string tension σ does not matter very much. Equivalently, the gluon dressing which makes massive constituent quarks out of the almost massless light quarks (with M q ≃ 0.3 − 0.4 GeV) has little effect on the heavy quarks. In a medium approaching the deconfinement point, for T → T c , the string tension vanishes: σ(T ) → 0, as does the constituent quark mass because of chiral symmetry restoration, M q → 0. As a consequence, light and light-heavy hadrons disappear, but sufficiently tightly bound quarkonia will persist even above T c and can thus serve as probes of the quark-gluon plasma. These arguments also suggest that quarkonium masses decrease less with temperature than the masses of the open charm or beauty mesons D and B. As a consequence, the open charm threshold can in a hot medium fall below the mass of previously stable higher excited charmonium states and thus allow their strong decay, and similarly for bottomonia. We therefore want to compare 2M D (T ) with M i (T ), where i specifies ψ ′ , χ and J/ψ, as well as the corresponding b-quark states. For the masses of the open charm/beauty states, we make use of the lattice studies already introduced in section 3. The string breaking potential introduced there determines with V (T, r = ∞) ≃ 2M q (T ) ≃ 2(M D − m c ) effectively the D-mass. The quarkonium masses can be obtained by solving the Schrödinger equation with the potential V (r, T ) determined in the same lattice studies. Comparing the temperature dependence of the light-heavy masses to that of the quarkonium states shows two distinct types of behavior [14,15]. In Fig. 6 we see that with increasing temperature 2M D and 2M B indeed drop below the masses of the highest excited states, ψ ′ and χ c for charmonia, Υ" and χ ′ b for bottomonia, respectively, before the deconfinement point is reached. These states thus disappear in a hot hadronic medium through in-medium decay into open charm/beauty. If the dissociation thresholds are experimentally measured, they thus specify the temperature of the hot but still confined system at four different points, tracing out the approach of chiral symmetry restoration. The mass gaps of J/ψ, Υ', χ b and Υ at T = 0 are much larger than Λ QCD , so that we expect them to survive deconfinement and be dissociated only by color screening in the quark-gluon plasma, as originally proposed for the J/ψ [16]. This dissociation sets in when the intrinsic scale of the quarkonium, its radius, falls below the screening radius as the scale characterizing the medium. In Fig. 7 this effect is seen to occur for the Υ at T ≃ 2.3 T c . The other mentioned states persist up to about T c when compared to open charm/beauty masses (see Fig. 6); in a screening approach, they are dissociated in a QGP just slightly above T c (see Fig. 7 for the J/ψ). Given the accuracy of the present lattice results near deconfinement, and in view of the possible break-down of a Schrödinger equation near T c , we can thus only conclude that J/ψ, χ b and Υ' are dissociated approximately at T c . We thus obtain a thermal quarkonium dissociation pattern which indeed is very similar to that provided by the spectral lines from stellar matter [18]. The suppression thresholds for ψ ′ , χ c , Υ" and χ ′ b specify a hot hadronic medium different temperatures; when the J/ψ, Υ' and χ b disappear, deconfinement is reached, and the dissociation point of the Υ indicates a hot QGP, with T > 2 T c . The observed production of J/ψ and Υ occurs in part through the decay of higher excited states, such as χ c → J/ψ; the respective fractions of the different contributions are known experimentally or can be determined from data [15]. Since such decays take place far outside the interaction region, the produced medium sees and suppresses the different 'parent' states. This leads to the well-known sequential suppression pattern [17,15] which distinguishes thermal threshold behavior such as deconfinement form dissociation by hadronic comover scattering (see [19] for a survey). Perhaps the most interesting feature which has so far emerged from nuclear collision studies is the observation of just such a multistep structure of J/ψ suppression [20]. Future experiments, both at CERN (NA60) and at RHIC, will undoubtedly provide further details to check if this structure is indeed due to sequential quarkonium suppression. Our considerations so far have ignored possible fluctuations of the medium. To illstrate, we note that for T > 0 the mass of the D and to a lesser extent the mass of the χ c will fluctuate around the values we have here calculated. Instead of a δ-function, we will have a peak with a certain width (collision broadening), which in principle can be provided by lattice calculations. For a conclusive study of sequential quarkonium suppression in nuclear collision this would seem a prerequisite. A second open question concerning applications to experiment is a reliable determination of the energy densities or temperatures attained there. All present lattice studies find T c ≃ 0.15 − 0.20 GeV for the deconfinement temperature; the corresponding energy density is ǫ(T c ) ≃ 1 GeV/fm 3 , although it then grows quickly to values near the Stefan-Boltzmann limit, so that ǫ(1.1 T c ) ≃ 2 GeV/fm 3 . Presently quoted values for the energy densities in P b − P b collisions at the CERN-SPS are in the range 2 -3.5 GeV/fm 3 ; they are based on Bjorken's estimate, which for central collisions gives ǫ ≃ dN h dy y=0 p 0 πR 2 A τ 0 ,(9) where dN h /dy denotes the multiplicity and p 0 the average energy of the produced hadrons, R A the nuclear radius and τ 0 ≃ 1 fm some average formation time of the medium. Obviously the choice of τ 0 is rather crucial, and a cross check of the reliability of the resulting estimates would thus seem very necessary. Summary • Hadron abundances, in nuclear collisions as well as in elementary interactions, follow the pattern of an ideal resonance gas. The strangeness suppression observed in elementary processes appears accountable through exact strangeness conservation. The observed energy independent freeze-out temperature T f ≃ 170 MeV seems to reflect critical features. • Finite T lattice studies of the heavy quark potential V (T, r) show a significant variation of the string breaking energy V (T, ∞) for T → T c . This could be an indication for a similar temperature variation of light hadron masses in full QCD. • Fast partons passing through a QGP suffer a considerable energy loss, which should be observable for sufficiently hard jets or their decay products. Present RHIC data require a reference distribution interpolating from a wounded nucleon to a binary collision form. • Quarkonia in hot matter can be dissociated by two distinct mechanisms. Higher excited states decay strongly into open charm/beauty mesons when the masses of the latter decrease as the system approaches chiral symmetry restoration. More tightly bound lower states survive up to deconfinement and are subsequently dissociated by color screening in the hot QGP. Figure 1 : 1Phase diagram of strongly interacting matter. Figure 2 : 2Freeze-out temperatures for hadron resonances produced in nucleus-nucleus (filled triangles), p − p (open triangles), p −p (circles) and e + e − interactions (squares). Figure 3 : 3Temperature dependence of the string breaking potential V Q (T, ∞) in units of the string tension σ. Figure 4 : 4Parton energy loss through coherent vs. incoherent scattering (left) and through coherent scattering in a medium of finite size (right). Figure 6 : 6Temperature dependence of (left) open charm and (right) open beauty masses (heavy lines) vs. charmonium and bottomonium masses. Figure 7 : 7Temperature dependence of the screening radius vs. the bound state radii of Υ and J/ψ, in fm. AcknowledgementsIt is a pleasure to thank many colleagues for helpful comments and suggestions; particular thanks go to R. Baier, S. Digal, F. Karsch, D. Kharzeev, P. Petreczky and K. Redlich. The financial support of the German Ministry of Science (contract 06BI902) and of the GSI (contract BI-SAT) is gratefully acknowledged. . See E G , G Gustafson, U Petterson, P Zerwas, Phys. Lett. B. 20990See e.g., G. Gustafson, U. Petterson and P. Zerwas, Phys. Lett. B 209 (1988) 90; . T Sjöstrand, V Khoze, Phys. Rev. Lett. 6228Z. Phys. CT. Sjöstrand and V. Khoze, Z. Phys. C 62 (1994) 281 and Phys. Rev. Lett. 72 (1994) 28; . J Ellis, K Geiger, Phys. Rev. D. 54J. Ellis and K. Geiger, Phys. Rev. D 54 (1996) 1967. . See G Abbiendi, OPALPhys. Lett. B. 453153See G. Abbiendi et al. (OPAL), Phys. Lett. B 453 (1999) 153; . P Abreu, DELPHI)Eur. Phys. J.C. 18203and further references thereP. Abreu et al. (DELPHI), Eur. Phys. J.C 18 (2000) 203, and further references there. . P See, Braun-Munzinger, Phys. Lett. B. 51841and references to earlier work given thereSee P. Braun-Munzinger et al., Phys. Lett. B 518 (2001) 41, and references to earlier work given there. . F Becattini, hep-ph/0011322F. Becattini et al., hep-ph/0011322. . F Becattini, Z. Phys. C. 69485F. Becattini, Z. Phys. C 69 (1996) 485; . F Becattini, U Heinz, Z. Phys. C. 76269F. Becattini and U. Heinz, Z. Phys. C 76 (1997) 269. . J S Hamieh, K Redlich, A Tounsi, Phys. Lett. B. 48661J. S. Hamieh, K. Redlich and A. Tounsi, Phys. Lett. B 486 (2000) 61; . J. Phys. G. 27413J. Phys. G 27 (2001) 413. . R Hagedorn, Yellow, R. Hagedorn, CERN Yellow Report 71-12 (1971). . R Baier, Phys. Lett. B. 50251R. Baier et al., Phys. Lett. B 502 (2001) 51. . I Wetzorke, hep-lat/0110132I. Wetzorke et al., hep-lat/0110132. . F Karsch, E Laermann, A Peikert, NP. 605579F. Karsch, E. Laermann and A. Peikert, NP B605 (2001) 579. . D Baier, B Schiff, Zakharov, Ann. Rev. Nucl. Part. Sci. 5037See e.g., R. Baier, D. Schiff and B. Zakharov, Ann. Rev. Nucl. Part. Sci. 50 (2000) 37. . D Schiff, Acta Phys. Polon. B. 303621D. Schiff, Acta Phys. Polon. B 30 (1999) 3621. . Y Dokshitzer, D Kharzeev, Phys. Lett. B. 519199Y. Dokshitzer and D. Kharzeev, Phys. Lett. B 519 (2001) 199. . S Digal, P Petreczky, H Satz, Phys. Lett. B. 51457S. Digal, P. Petreczky and H. Satz, Phys. Lett. B 514 (2001) 57. . S Digal, P Petreczky, H Satz, Phys. Rev. D. 6494015S. Digal, P. Petreczky and H. Satz, Phys. Rev. D 64 (2001) 094015. . T Matsui, H Satz, Phys. Lett. B. 178416T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416. . S Gupta, H Satz, Phys. Lett. B. 283439S. Gupta and H. Satz, Phys. Lett. B 283 (1992) 439. . K Kajantie, Nucl. Phys. B. 24279K. Kajantie, Nucl. Phys. B 24 (1991) 279. . R Vogt, Phys. Rept. 310197R. Vogt, Phys. Rept. 310 (1999) 197. . M C Abreu, NA50Phys. Lett. B. 47728M. C. Abreu et al. (NA50), Phys. Lett. B 477 (2000) 28.
[]
[ "GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials", "GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials" ]
[ "G Anand *email:[email protected] \nDepartment of Metallurgy and Materials Engineering\nIndian Institute of Engineering Science and Technology\nShibpurHowrahIndia\n" ]
[ "Department of Metallurgy and Materials Engineering\nIndian Institute of Engineering Science and Technology\nShibpurHowrahIndia" ]
[]
High-Entropy Materials are composed of multiple elements on comparatively simpler lattices. Due to the multicomponent nature of such materials, the atomic scale sampling is computationally expensive due to the combinatorial complexity.We propose a genetic algorithm based methodology for sampling such complex chemically-disordered materials. Genetic Algorithm based Atomistic Sampling Protocol (GAASP) variants can generate low and well as high-energy structures.GAASP low-energy variant in conjugation with metropolis criteria avoids the premature convergence as well as ensures the detailed balance condition. GAASP can be employed to generate the low-energy structures for thermodynamic predictions as well as diverse structures can be generated for machine learning applications.high-entropy materials containing three or more elements, respectively exhibit these exciting properties, but also present an unique challenge in designing such materials with high computational complexity. The compositional complexity in such materials leads to the combinatorial explosion and traditional hit-and-trial approach for designing such material might not be a feasible approach. Such a problem becomes further challenging with the advent of non-concentrated compositionally complex alloys, where certain element may be added in the dilute amount to the concentrated complex alloy. Dragoe et. al. presented a simple calculations to demonstrate the combinatorial explosion through a scenario, in which if 5 elements are chosen from the palette of 26 elements with 1% increment, one can have 2,822,599,802,880 combinations, while of 6 elements are chosen with similar increment in the composition, the possibilities increase to the value of 902,943,619,878,430 [6].Hence, the composition screening of such materials poses a formidable challenge. Special quasi-random structure (SQS) generation is one of the earliest techniques, which was extensively applied to simulate the chemical disorder in high-entropy materials. But, SQS technique suffers from limited computational efficiency for the multicomponent materials. It involves optimisation of numerous parameters such as cut-off, simulated annealing temperatures, optimisation steps, etc., which makes this technique challenging for optimising HEA structures [7]. Mean-field approach, such as coherent-potential approximation (CPA) has been applied to simulate the chemical disorder [8], however it cannot introduce local short-range order, which influences the properties of high-entropy materials [9]. In view of challenges associated with simulation of high-entropy materials, small set of ordered structures (SSOS) approach was developed for efficient ab-initio calculations for rapid compositional screening of multicomponent materials. In this approach, initially large set of small-ordered structures (SOS) are generated from which small set of ordered structures are chosen. The property of the particular multicomponent material is then calculated from the these small set of ordered structures with appropriate weight assigned to each of SSOS [10]. SSOS has also been extended to the nonequiatomic high entropy materials [11]. Machine learning approaches has also been developed for generating and optimising the composition of high-entropy materials.Neural structure evolution strategy involves application of combined artificial neural network with evolutionary algorithms to generate the large supercell of high entropy alloys with high computational efficiency[12]. Database oriented ANN based machine learning model is employed to predict the structure of HEA was proposed. However, biased databased towards certain family of alloys limits the predictive capability of such approaches[13]. ML-based approaches has also been applied for HEA compositions with high catalytic activities[14]. Initially, experimental approaches were being developed for combinatorial exploration of compositional landscape of high-entropy materials[15].But, recently there has been marked interest in development of combined experimentaltheoretical approaches for compositional screening of high entropy materials. Empirical and experimental approach has been developed for the compositional screening to determine the solid solution and intermetallic phases in high-entropy alloys[16].Materials informatics based approaches has been employed to screen for compositions with desired properties. Genetic algorithm was used to determine the desired alloy compositions, which were further validated with the experimental fabrication of such alloys[17]. Evolutionary algorithm, such as genetic algorithm (GA) has been applied in some of the above-stated studies. However, GA has been already employed for crystal structure predictions in USPEX [18] for range of materials. GA based codes such as GAtor has been developed molecular crystals[19].The high-entropy materials represents the unique challenge due to its compositional as well as combinatorial complexities. The compositional complexities has been tacked using GA in the earlier investigations.GA has been extensively applied to study the compositional disorder. Initial investigation of GA for determining low-energy structures in binary alloys showed its computational efficiency in comparison of Monte Carlo method [20]. GA has been applied to optimise the cation ordering in oxides [21], structure prediction in disordered alloys [22], for studying non-ideality in mixing behaviour of mineral solid solutions [23]. GA suffers from the premature convergence issue and it has been modified to include symmetry 2. Gludovatz, B.; Hohenwarter, A.; Thurston, K. V. S.; Bei, H.; Wu, Z.; George, E. P.; Ritchie, R. O. Exceptional Damage-Tolerance of a Medium-Entropy Alloy CrCoNi at (6465), 573-574. https://doi.org/10.1126/science.aaz1598. 7. Pedersen, J. K.; Clausen, C. M.; Krysiak, O. A.; Xiao, B.; Batchelor, T. A. A.; Löffler, T.; Mints, V. A.; Banko, L.; Arenz, M.; Savan, A.; Schuhmann, W.; Ludwig, A.; Rossmeisl, J. Bayesian Optimization of High-entropy Alloy Compositions for Electrocatalytic Oxygen Reduction. Angew. Chem. Weinheim Bergstr.
10.1080/10426914.2023.2217909
[ "https://export.arxiv.org/pdf/2302.08101v1.pdf" ]
256,900,770
2302.08101
ca965caecbbb3819c29e429ad5f9c345542f03ee
GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials G Anand *email:[email protected] Department of Metallurgy and Materials Engineering Indian Institute of Engineering Science and Technology ShibpurHowrahIndia GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials GAASP: Genetic Algorithm Based Atomistic Sampling Protocol for High-Entropy Materials High entropy alloysGenetic AlgorithmThermodynamicsSampling AlgorithmsMachine Learning High-Entropy Materials are composed of multiple elements on comparatively simpler lattices. Due to the multicomponent nature of such materials, the atomic scale sampling is computationally expensive due to the combinatorial complexity.We propose a genetic algorithm based methodology for sampling such complex chemically-disordered materials. Genetic Algorithm based Atomistic Sampling Protocol (GAASP) variants can generate low and well as high-energy structures.GAASP low-energy variant in conjugation with metropolis criteria avoids the premature convergence as well as ensures the detailed balance condition. GAASP can be employed to generate the low-energy structures for thermodynamic predictions as well as diverse structures can be generated for machine learning applications.high-entropy materials containing three or more elements, respectively exhibit these exciting properties, but also present an unique challenge in designing such materials with high computational complexity. The compositional complexity in such materials leads to the combinatorial explosion and traditional hit-and-trial approach for designing such material might not be a feasible approach. Such a problem becomes further challenging with the advent of non-concentrated compositionally complex alloys, where certain element may be added in the dilute amount to the concentrated complex alloy. Dragoe et. al. presented a simple calculations to demonstrate the combinatorial explosion through a scenario, in which if 5 elements are chosen from the palette of 26 elements with 1% increment, one can have 2,822,599,802,880 combinations, while of 6 elements are chosen with similar increment in the composition, the possibilities increase to the value of 902,943,619,878,430 [6].Hence, the composition screening of such materials poses a formidable challenge. Special quasi-random structure (SQS) generation is one of the earliest techniques, which was extensively applied to simulate the chemical disorder in high-entropy materials. But, SQS technique suffers from limited computational efficiency for the multicomponent materials. It involves optimisation of numerous parameters such as cut-off, simulated annealing temperatures, optimisation steps, etc., which makes this technique challenging for optimising HEA structures [7]. Mean-field approach, such as coherent-potential approximation (CPA) has been applied to simulate the chemical disorder [8], however it cannot introduce local short-range order, which influences the properties of high-entropy materials [9]. In view of challenges associated with simulation of high-entropy materials, small set of ordered structures (SSOS) approach was developed for efficient ab-initio calculations for rapid compositional screening of multicomponent materials. In this approach, initially large set of small-ordered structures (SOS) are generated from which small set of ordered structures are chosen. The property of the particular multicomponent material is then calculated from the these small set of ordered structures with appropriate weight assigned to each of SSOS [10]. SSOS has also been extended to the nonequiatomic high entropy materials [11]. Machine learning approaches has also been developed for generating and optimising the composition of high-entropy materials.Neural structure evolution strategy involves application of combined artificial neural network with evolutionary algorithms to generate the large supercell of high entropy alloys with high computational efficiency[12]. Database oriented ANN based machine learning model is employed to predict the structure of HEA was proposed. However, biased databased towards certain family of alloys limits the predictive capability of such approaches[13]. ML-based approaches has also been applied for HEA compositions with high catalytic activities[14]. Initially, experimental approaches were being developed for combinatorial exploration of compositional landscape of high-entropy materials[15].But, recently there has been marked interest in development of combined experimentaltheoretical approaches for compositional screening of high entropy materials. Empirical and experimental approach has been developed for the compositional screening to determine the solid solution and intermetallic phases in high-entropy alloys[16].Materials informatics based approaches has been employed to screen for compositions with desired properties. Genetic algorithm was used to determine the desired alloy compositions, which were further validated with the experimental fabrication of such alloys[17]. Evolutionary algorithm, such as genetic algorithm (GA) has been applied in some of the above-stated studies. However, GA has been already employed for crystal structure predictions in USPEX [18] for range of materials. GA based codes such as GAtor has been developed molecular crystals[19].The high-entropy materials represents the unique challenge due to its compositional as well as combinatorial complexities. The compositional complexities has been tacked using GA in the earlier investigations.GA has been extensively applied to study the compositional disorder. Initial investigation of GA for determining low-energy structures in binary alloys showed its computational efficiency in comparison of Monte Carlo method [20]. GA has been applied to optimise the cation ordering in oxides [21], structure prediction in disordered alloys [22], for studying non-ideality in mixing behaviour of mineral solid solutions [23]. GA suffers from the premature convergence issue and it has been modified to include symmetry 2. Gludovatz, B.; Hohenwarter, A.; Thurston, K. V. S.; Bei, H.; Wu, Z.; George, E. P.; Ritchie, R. O. Exceptional Damage-Tolerance of a Medium-Entropy Alloy CrCoNi at (6465), 573-574. https://doi.org/10.1126/science.aaz1598. 7. Pedersen, J. K.; Clausen, C. M.; Krysiak, O. A.; Xiao, B.; Batchelor, T. A. A.; Löffler, T.; Mints, V. A.; Banko, L.; Arenz, M.; Savan, A.; Schuhmann, W.; Ludwig, A.; Rossmeisl, J. Bayesian Optimization of High-entropy Alloy Compositions for Electrocatalytic Oxygen Reduction. Angew. Chem. Weinheim Bergstr. Introduction There has been significant interest in the area of high-entropy and medium-entropy materials due to their interesting properties, such as high-strength and high toughness at liquid helium [1], as well as liquid nitrogen temperature [2], high corrosion resistance [3], strength retention at high temperature [4], high catalytic activity [5], etc. Medium and adapted crossover [24]. Adaptive GA has been developed to combine classical interatomic potential and Density Functional Theory (DFT) calculations to predict structures of crystal, surfaces and interfaces with higher computational efficiency in comparison of purely DFT based approaches [25]. Atomistic sampling of high entropy materials poses significant challenges in comparison of traditional materials. Firstly, the combinatorial complexity of such materials is significantly higher in comparison of traditional materials, which are generally binary mixtures. Higher compositional complexity warrants the need of high-throughput approach for compositional screening. In view of the above, we propose genetic algorithm based sampling involving alchemical swaps and classical interatomic potential for generation of low-energy structures. Note that present approach can be extended to DFT based approaches as well. We would additionally demonstrate the GAASP can also be employed for generating high-energy structures, which may be relevant for generating database for machine learning applications, where the diversity of configurations may be required for training machine learning model. We would also present the applicability of GAASP for compositional screening. Materials and Methods Method The aim of the GAASP is to generate the new configurations with desired energy distribution. Such an energy distribution might be biased towards low energy configurations or it might be high energy configurations. The aim of generating lowenergy configurations might be for thermodynamic predictions, while high energy distributions might be required as representative configurations for holistic database for machine learning applications. The GAASP approach should be able to generate both of these extreme cases, which would be shown later in sections. The GAASP can be principally divided into the following steps: Random sampling to generate the parent population Initially, random sampling of atomistic configurations needs to be carried out to sample the potential energy landscape of the system of interest. It is of the paramount importance that truly random sampling is carried out to sample the representative configurations from throughout the potential energy landscape. The number of configurations sampled randomly signifies the 'thoroughness' of the random sampling (say, this number is N). Parent selection Once the random sampling is complete the parent population is chosen from the random samples using Roulette wheel selection mechanism ( Fig. 1 (a)). The parameters for the Roulette wheel parent selection are number of parents chosen from the original parent population (i.e., N) or X, number of intervals in which the wheel is being divided (n) and number of parents in each interval ( = ) and dividing factor for each interval or D (i.e., first interval is divided into and so on). A random number, x is initially generated, such that 0 ≤ < . The identity of the interval from which a parent is being chosen is dependent whether , ∈ { [ −1 − 1 −2 , − 1 ) , ℎ [ −1 − 1 −2 , ) , ℎ Once the interval is determined, the parent from the particular interval is chosen by choosing a random number between [0, ). Information encoding Once the parents population is chosen depending upon the desired energy distribution, the parents are chosen sequentially in pair to generate two children. The encoding of the atomic configurations is carried out as value encoding, with values signifying the atomic type as well as the cartesian coordinates of the particular atom. Such encoded information represents the chromosome and the crossover between two chromosomes leads to the generation of newer configurations ( Fig. 1 (b)). However in the encoding procedure, it is ensured that certain types of atoms are encoded together, i.e., all the type-1 atoms are encoded together and type-2 after that and so on. Alchemical swap process The aim of the swapping process is to interchange the information concerning the position of atoms between two configurations, as shown in Fig. 2 (a). Initially, a random number in the range [0, ] is generated, where P is the total number of atoms in the supercell. If we get a number 'i' in the process, it would correspond to same type of atom, but with different cartesian coordinates ({ , , } and { , , } in parent-1 and parent-2, respectively) in both the supercells due to the type of value encoding employed in the present method ( Fig. 2 (b)). The cartesian coordinates of i th atom in parent-1(i.e. { , , }) is searched in parent-2 and say, k th atom in the parent-2 has same cartesian coordinates. Similarly, the coordinates of i th atom in the parent-2 ({ , , }) is searched in parent-1 and say j th atom in parent-1 has same coordinates. Once the identity of 'i' and 'j' in parent-1 and 'i' and 'k' in parent-2 is established, the alchemical swap between 'i' and 'j' in parent-1 and swap between 'i' and 'k' in parent-2 is carried out (Fig. 2(d) and (e)). Such a swap leads to the occupation of atom in children with inheritance from both the parents. But it also leads to the mutation, involving the introduction of different atoms at the original position ('i th position' in parent-1 and parent-2). Such a method ensures that the composition of the supercell remains constant during the swap process. The important parameters associated with the swap process is number of swap (Q), such a number is taken randomly between 10-30% of the total number of atoms in the supercell (P). Energy calculation Once the parent population for a particular genetic algorithm cycle is generated, energy calculations in the NVT ensemble is carried out for all the parents by running molecular dynamics simulation for 10000 steps with first 1000 steps are solely employed for equilibration. In the present investigation, we carry out the energy calculations for BCC-AlCoCrFeNi HEA as proof-of-concept studies. The GAASP approach can be coupled with DFT-based codes for the energy calculations, in principle. We employed the EAM potential developed by Zhou et. al. [26] for Al, Co, Fe and Ni, while for Cr modified Zhou-EAM potential [27] was employed to describe AlCoCrFeNi. Energy for each of the configuration or parent is stored. After energy calculations, the energy values are sorted in the increasing order and configurations are arranged in the similar manner. The parents for the next generation are again chosen from this reservoir using the Roulette wheel selection mechanism. Results and Discussion Generation of low-energy and high-energy configurations Figure 3(a) shows the ridge plot of the evolution of the configurational energy distribution as the GA cycle progresses. It can be seen that GAASP generate the lower energy configurations. Such low-energy landscape can be relevant for the prediction of thermodynamic properties of such materials, which are dictated by low-energy configurations. Similarly, Fig. 3(b) shows the evolution of high energy states in highenergy variant of GAASP. The high-energy or non-equilibrium structures are often required in efficient sampling strategies [29]. The high energy variant of GAASP choses the parent in the reverse order in comparison to low-energy variants as discussed in the method section. Note that the applicability of high-energy GAASP variant in conjugation with low-energy variant can be used in generation of the diverse configurations for training ML models, which not only requires equilibrium, but non-equilibrium structures in training. Comparison of GAASP and GAASP-metropolis method for generation of the genetic diversity We have additionally extended the GAASP approach to include the metropolis acceptance criteria (GAASP-metropolis) to ensure the detailed balance in the sampling [30]. In such an acceptance criteria, the child is accepted on the basis of energy change. If, ∆ or change in energy due to atomic swap to generate a child from parent is less than or equal to the zero, then child is accepted, while is it is greater than zero, than child is accepted with the probability, = ( −∆ ) . Application of the metropolis criteria for child acceptance serves two purposes. First, it ensures the detailed balance in the sampling and secondly, it helps in avoiding the premature convergence of GA sampling. The existence of the detailed balance in sampling ensures that the energy landscape explored through GAASP approach can be employed for determining the thermodynamic properties of material being simulated. The premature convergence of genetic algorithm based sampling is well-known issue [22] and in the present investigation, we demonstrate that employing the metropolis criteria can avoid such an issue, while it can also lead to better exploration of the energy landscape. Figure 4 shows the GAASP without metropolis acceptance criteria reaches convergence faster, but GAASP with metropolis criteria generates lower-energy structures and it is better in exploring the composition landscape of AlCoCrFeNi HEA by generating configurations with lower configurational energies. GAASP with compositional variation We have additionally extended the GAASP to modify the compositions of the configurations to either generate high or low energy structures. In such an approach, the alchemical swap between two configurations are carried out without mutation. Figure 5(a) shows the change in the configuration energy of the supercell with GA cycle for both energy increase and decrease objective of GAASP. Note that in the context of constant composition, with each swap between parents there is associated mutation. In the present scenario, we have simply carried out the swap without any explicit mutation. However, random mutation can be introduced, in principle and effect of the mutation would be included in the future studies. Figure 5 Conclusions In the present investigation, we have developed the Genetic Algorithm based Atomistic Sampling Protocol (GAASP) for efficient sampling of atomistic structure of compositionally disordered multicomponent materials or high-entropy materials through alchemical swap process. GAASP approach can be employed to generate the low-energy as well as high energy structures. GAASP approach can be used to generate the diversity of configurations, which may be relevant for generating diverse configurations for training machine learning models for high-entropy materials. GAASP approach can also be applied to for optimisation of composition of such multicomponent materials for determining the compositions, which can lead to high-energy or low-energy configurations. Code The GAASP code is available at: https://github.com/ganand1990/GAASP The value encoding is carried out in a way to ensure that same atom-type is present at i th index., (c) process of determination of j th atom in parent-1 with same coordinates as i th atom in parent-2 and k th atom in parent-2 having same coordinates as i th atom in parent-1, (d) swap between i th and j th atoms in parent-1 and k th and i th atoms in parent-2 and (e) demonstration of steps (b-d) as implemented using value-encoding in GAASP code [28]. (b) shows the change in the number of elements as GAASP explores the low-energy landscape. It can be seen that Fe is increasing in the supercell, while Co and Ni show marginal increase, while Cr and Al show decreasing trend. Figure 5(c) shows the variation in the number of elements in the AlCoCrFeNi supercell, which shows the number of Al atoms increase as the higher-energy , while other elements show the marginally decreasing trend. It should be noted that we are demonstrating the applicability of GAASP for simulating the composition variation as either function of increasing or decreasing energy as objective and corresponding change in the composition can be predicted for high entropy materials. Such an approach can be particularly important for exploring non-equiatomic compositions of high-entropy materials. FiguresFigure - 1 : 1(a) Schematic of the Roulette wheel selection process and (b) genetic crossover of information to generate new configurations. of the alchemical swap, i.e., swap the information about the existence of atoms at particular lattice site in the supercell, (b) Figure- 3 : 3Ridge plots of the energy distribution as obtained from the (a) low-energy and (b) high energy search for AlCoCrFeNi HEA. Figure- 4 :Figure- 5 : 45Comparison of variants for GAASP for generating low-energy structures involving metropolis criteria for selection of children and no rejection (i.e. without metropolis criteria). (a) Change in the energy for composition variation in the supercell in GAASP for low and high energy search, (b) change in the number of atoms for GAASP low-energy search and (c) high-energy search. AcknowledgementsAuthor is thankful to Dr. Colin Freeman for suggesting to develop the protocol and Prof.Graeme Ackland for the critical feedback. We are also thankful to National Supercomputing Facility for access to Param Sidhi AI system. . D Liu, Q Yu, S Kabra, M Jiang, P Forna-Kreutzer, R Zhang, M Payne, F Walsh, B Gludovatz, M Asta, A M Minor, E P George, R O Ritchie, Exceptional Fracture Toughness of CrCoNi-Based Medium-and High-Entropy. 20Liu, D.; Yu, Q.; Kabra, S.; Jiang, M.; Forna-Kreutzer, P.; Zhang, R.; Payne, M.; Walsh, F.; Gludovatz, B.; Asta, M.; Minor, A. M.; George, E. 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[ "https://github.com/ganand1990/GAASP" ]
[ "On the stability of inhomogeneous fluids under acoustic fields", "On the stability of inhomogeneous fluids under acoustic fields" ]
[ "Varun Kumar Rajendran \nDepartment of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia\n", "Aravind Ram \nDepartment of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia\n", "S P \nDepartment of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia\n", "Karthick Subramani \nDepartment of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia\n" ]
[ "Department of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia", "Department of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia", "Department of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia", "Department of Mechanical Engineering\nDesign and Manufacturing\nIndian Institute of Information Technology\nChennai-600127KancheepuramIndia" ]
[]
In this work, we present the stability theory for inhomogeneous fluids subjected to standing acoustic fields. Starting from the first principles, the stability criterion is established for two fluids of different acoustic impedance separated by a plane interface. Through stability theory and numerical simulations we show that, in the presence of interfacial tension, the relocation of high-impedance fluid from anti-node to node occurs when the acoustic force overcomes interfacial tension force, which is in agreement with recent microchannel experiments. Furthermore, we establish an acoustic Bond number that characterizes stable (Bo a < 1) and relocation (Bo a > 1) regimes. Remarkably, it is found that the critical acoustic energy density required for relocation can be significantly reduced by increasing the channel height which could help design acoustofluidic microchannel devices that handle immiscible fluids. † Email address for correspondence: [email protected] arXiv:2212.04660v1 [physics.flu-dyn] 9 Dec 2022
10.1017/jfm.2023.371
[ "https://export.arxiv.org/pdf/2212.04660v1.pdf" ]
254,536,036
2212.04660
5ec1fd831ff48c40b88c332f6dcfc689f3cd5001
On the stability of inhomogeneous fluids under acoustic fields Varun Kumar Rajendran Department of Mechanical Engineering Design and Manufacturing Indian Institute of Information Technology Chennai-600127KancheepuramIndia Aravind Ram Department of Mechanical Engineering Design and Manufacturing Indian Institute of Information Technology Chennai-600127KancheepuramIndia S P Department of Mechanical Engineering Design and Manufacturing Indian Institute of Information Technology Chennai-600127KancheepuramIndia Karthick Subramani Department of Mechanical Engineering Design and Manufacturing Indian Institute of Information Technology Chennai-600127KancheepuramIndia On the stability of inhomogeneous fluids under acoustic fields 1 In this work, we present the stability theory for inhomogeneous fluids subjected to standing acoustic fields. Starting from the first principles, the stability criterion is established for two fluids of different acoustic impedance separated by a plane interface. Through stability theory and numerical simulations we show that, in the presence of interfacial tension, the relocation of high-impedance fluid from anti-node to node occurs when the acoustic force overcomes interfacial tension force, which is in agreement with recent microchannel experiments. Furthermore, we establish an acoustic Bond number that characterizes stable (Bo a < 1) and relocation (Bo a > 1) regimes. Remarkably, it is found that the critical acoustic energy density required for relocation can be significantly reduced by increasing the channel height which could help design acoustofluidic microchannel devices that handle immiscible fluids. † Email address for correspondence: [email protected] arXiv:2212.04660v1 [physics.flu-dyn] 9 Dec 2022 Introduction When an acoustic field encounters inhomogeneity, it exerts acoustic radiation force on it. Here by inhomogeneity, we mean non-uniform or discontinuous variation of physical properties in a system such as particles/cells suspended in fluid, emulsions, co-flowing streams of miscible or immiscible fluids, and fluid subjected to a temperature gradient. The acoustic forces acting on inhomogeneity are extensively studied in microscale flows, and this field is known as 'microscale acoustofluidics' (Friend & Yeo 2011). Over the last two decades, acoustofluidics has found a wide range of applications in biological (Ahmed et al. 2016;Iranmanesh et al. 2015;Collins et al. 2015;Christakou et al. 2013;Lakshmanan et al. 2020), chemical (Shi et al. 2009;Xie et al. 2020), and medical (Li et al. 2015;Lu et al. 2019;Zhang et al. 2020) sciences. Recently, the relocation and stabilization of inhomogeneous co-flowing fluid streams in microchannels has gained the attention of the research community which is evident from the following works. Through silicon-glass microchannel experiments, Deshmukh et al. (2014) could relocate high-impedance sodium chloride solution to node (center) and lowimpedance water to anti-node (sides). Also, they could stabilize high-impedance fluid at the center (and low-impedance fluid to the sides) against gravity stratification using acoustic forces. In addition to the above experiments on miscible fluids, Hemachandran et al. (2019) demonstrated the relocation of immiscible fluids using acoustic fields by overcoming the interfacial tension forces. Followed by this, Karlsen et al. (2018) showed that acoustic forces acting on stable inhomogeneous fluid configuration could effectively suppress the boundary-driven Rayleigh streaming in the bulk. The theoretical framework and understanding of the above non-linear acoustic forces on inhomogeneous fluids are provided by Rajendran et al. (2022); Karlsen et al. (2016Karlsen et al. ( , 2018. Other notable works on the practical applications of acoustic forces on co-flowing inhomogeneous fluids include iso-acoustic focusing of cells , acoustic focusing of sub-micron particles (Van Assche et al. 2020;Gautam et al. 2018), tweezing and patterning of inhomogeneous fluids in a microchannel (Karlsen & Bruus 2017;Baudoin et al. 2020), rapid mixing of fluids using an alternating multinode method (Pothuri et al. 2019), reversible stream-droplet transition in a microfluidic co-flowing immiscible system (Hemachandran et al. 2021). Despite the above recent advancements and practical importance, the criterion at which the inhomogeneous fluid system becomes unstable or stable under acoustic fields has not been clearly established. This paper aims to establish the stability criterion of inhomogeneous co-flowing fluids subjected to standing acoustic wave fields. In this work, using linear stability analysis, we derive the dispersion relation that governs the stability of inhomogeneous fluids (with and without interfacial tension) under acoustic body force. We study the various parameters such as the initial arrangement of fluids, the position of the interface with respect to the node, acoustic energy density, the height of the channel, and surface tension to establish the necessary and sufficient conditions for relocation and stability. For fluids with interfacial tension, a non-dimensional number called acoustic Bond number is obtained theoretically which characterizes stable and unstable (relocation) regime. Also, we deduce a relation between critical acoustic energy density and the height of the channel which paves a way for relocating fluids with higher interfacial tension (O(10 1 to 10 2 )mN/m) in a microchannel. Furthermore, numerical simulations are carried out using generalized acoustic body force which agrees well with the derived theoretical stability criterion. Physics of the problem The hydrodynamics of the inhomogeneous fluids involved in this study is governed by the mass-continuity and momentum equations (Landau & Lifshitz 1987), ∂ρ ∂t + ∇ · (ρV ) = 0, (2.1a) ρ DV Dt = −∇P + η∇ 2 V + βη∇(∇ · V ) + f ac . (2.1b) where ρ represents density, V represents the velocity vector field, P represents the pressure field, η is the dynamic viscosity of the fluid, β = (ξ/η) + (1/3), ξ is the bulk viscosity, and D/Dt denotes the material derivative (D/Dt = ∂ t + V · ∇). Here the body force f ac is only due to acoustics. The gravitational body force is neglected since it is dominated by acoustic force in microscale flows. The acoustic body force f ac is given as (Rajendran et al. 2022) f ac = −∇ · ρ 0 v 1 v 1 = 1 2 ∇ κ 0 |p 1 | 2 − ρ 0 |v 1 | 2 ) + v 1 × ∇ × (ρ 0 v 1 ) + − 1 2 |p 1 | 2 ∇κ 0 − 1 2 v 2 1 ∇ρ 0 (2.2) = (f 1 ) + [f 2 ] + (f 3 ). (2.3) where p 1 and v 1 denote the first-order (fast time scale) pressure and velocity fields due to acoustic waves (see Appendix A) and ... is the time average in one oscillation period (the time average of two first-order fields u 1 v 1 is defined as 1 2 Real(u 1 v 1 ), where denotes complex conjugation). The terms ρ 0 and κ 0 denote the zeroth-order (background) density and compressibility of the fluid. In (2.2), the first term is a conservative or gradient term that induces pressure and not fluid flow, the second term is only dominant at boundary layers and is responsible for boundary-driven Rayleigh streaming and the third term is responsible for relocation and stabilization of inhomogeneous fluids. Hence, only the relevant third term (f 3 in (2.3)) is considered for theoretical analysis. For the standing acoustic wave applied along the X-direction, |p 1 | 2 = p 2 a sin 2 (k w x), |v 2 1 | = p 2 a /(ρ 2 0 c 2 0 ) cos 2 (k w x), k w = 2π/λ w denotes the wavenumber (λ w denotes the wavelength) and p a denotes the pressure amplitude. Then the relocation force f rl can be well approximated in terms of impedance gradient as (Rajendran et al. 2022) f 3 = f rl = −E ac cos(2k w x)∇Ẑ. (2.4) where E ac = p 2 a /(4ρ avg c 2 avg ) = (v 2 a ρ avg )/4 is the acoustic energy density, Z = ρ 0 c 0 denotes impedance, c 0 denotes background (zeroth-order) speed of sound in a medium, Z = Z/z avg ,ĉ 0 = c 0 /c avg andρ 0 = ρ 0 /ρ avg , where the subscript 'avg' denotes the respective average quantities of fluid A and B. Stability analysis of inhomogeneous fluids in the absence of interfacial tension A two-dimensional fluid domain subjected to a standing acoustic half-wave in the Xdirection, with two fluids separated by a sharp vertical interface as shown in figure 1.a-c is considered for the stability analysis. Before beginning the analysis, it is necessary to understand the equilibrium of the system in the absence of interfacial tension. In a completely enclosed domain, a fluid initially at rest (V = 0) will remain at rest (or equilibrium) if the body force can be completely absorbed in pressure, f rl = ∇P from (2.1b) and (2.4). By taking the curl of the above relation, the condition for equilibrium is given as ∇ × f rl = 0. Thus, − E ac z avg ∂ ∂x cos(2k w x) ∂z ∂y − ∂ ∂y cos(2k w x) ∂z ∂x = 0. (2.5) It is clear from (2.5) that the given fluid configuration will be in an equilibrium state, only if the direction of the acoustic standing wave is normal to the fluid-fluid interface (the direction of the acoustic standing wave is parallel to the direction of the impedance gradient) as shown in figure 1(a-c). Since ∇×f rl = 0 for the configuration shown in figure 1(d), it is not in equilibrium and tends to relocate to the stable configuration without any perturbations. The stability nature of the equilibrium configurations is analysed by imposing infinitesimal perturbations on the interface. Now we proceed to show that in the absence of interfacial tension, the configuration shown in figure 1(a) is in unstable equilibrium (perturbations grow), figure 1(b) is in stable equilibrium (perturbations decay), and figure 1(c) is in neutral equilibrium (perturbations neither grow nor decay). The effect of viscosity is neglected in the stability analysis, as it governs only the timescale of the phenomenon and does not contribute to the stability criterion. Although the physical properties are non-uniform in an inhomogeneous system, the fluid particles considered in the flow field have constant density ρ, speed of sound c, and impedance Z. Thus, the material derivative of all properties is zero, which includes the incompressibility condition (Dρ/Dt = ∂ρ/∂t + V · ∇ρ = 0). By combining the incompressibility condition with (2.1a) and neglecting the viscosity, the governing equations (2.1) reduce to ∂U ∂x + ∂V ∂y = 0, (2.6a) ρ DU Dt = − ∂P ∂x − E ac cos(2k w x) z avg ∂Z ∂x , (2.6b) ρ DV Dt = − ∂P ∂y − E ac cos(2k w x) z avg ∂Z ∂y , (2.6c) where U, V are the X-component and Y-component of the velocity field V . Since the body force term is a function of impedance, the below impedance relation is required for the closure. DZ Dt = ∂Z ∂t + U ∂Z ∂x + V ∂Z ∂y = 0. (2.6d ) Now, the flow fields are decomposed into an unperturbed zeroth-order stationary state and infinitesimal perturbations as U = u 0 + δu, V = v 0 + δv, P = p 0 + δp, ρ = ρ 0 + δρ and Z = z 0 + δz. In this study, the variation of acoustic impedance is considered only in the X-direction (figure 1(a-c)), z 0 = z 0 (x). At the stationary state (u 0 = v 0 = 0), the unperturbed zeroth-order equations become, ∂p0 ∂x = − Eac cos(2kwx) zavg ∂z0 ∂x from 2.6b, ∂p0 ∂x = 0 from 2.6c and ∂z0 ∂t = 0 from 2.6d. Using the above zeroth order relations and neglecting the second-order terms in (2.6), the first-order perturbation equations governing the stability becomes ∂δu ∂x + ∂δv ∂y = 0, (2.7a) ρ 0 ∂δu ∂t = − ∂δp ∂x − E ac cos(2k w x) z avg ∂δz ∂x , (2.7b) ρ 0 ∂δv ∂t = − ∂δp ∂y − E ac cos(2k w x) z avg ∂δz ∂y , (2.7c) ∂(δz) ∂t = −δu ∂z 0 ∂x . (2.7d ) Analysing the disturbances into normal modes, the amplitude of the disturbances δu, δv, δρ, δp, and δz takes the following form A(x, y, t) = A(x)exp(ik y y + nt), (2.8) where k y is the wavenumber considered along the Y-direction. Applying the above amplitude relations in the form (2.8) in (2.7), ∂δu ∂x + ik y δv = 0, (2.9a) ρ 0 nδu = − ∂δp ∂x − E ac cos (2k w x) z avg ∂δz ∂x , (2.9b) ρ 0 nδv = −ik y δp − ik y E ac cos (2k w x) z avg δz, (2.9c) nδz = −δu ∂z 0 ∂x . (2.9d ) The partial notation is dropped since the only derivatives in (2.9) are with respect to the x coordinate. Multiplying by ik y throughout (2.9c) and combining with (2.9a) and (2.9d), we obtain, δp = −ρ 0 n k y 2 dδu dx + E ac cos(2k w x) z avg δu n dz 0 dx . (2.10) substituting (2.9d) and (2.10) in (2.9b) results in, d dx ρ 0 dδu dx − ρ 0 k 2 y δu = −E ac 2k w δu z avg k 2 y n 2 dz 0 dx sin(2k w x). (2.11) Considering two uniform fluids of different impedance Z A and Z B separated by interfaces positioned at x s , z 0 = z A + (z B − z A )H(x − x s ), (2.12a) dz 0 dx = (z B − z A )δ(x − x s ), (2.12b) where H(x − x s ) is the Heaviside step function at x = x s and δ(x − x s ) is the Dirac's δ-function at x = x s . Substituting (2.12b) in (2.11), d dx ρ 0 dδu dx − ρ 0 k 2 y δu = −E ac 2k w δu z avg k 2 y n 2 sin(2k w x)(z B − z A )δ(x − x s ). (2.13) Equation (2.13) is the governing differential equation for the stability of inhomogeneous fluids (without interfacial tension). For a uniform region on either side of the interface(s) where there are no discontinuities in the impedance, the governing equation (2.13) reduces to d 2 δu dx 2 − k 2 y δu = 0. (2.14) The solution of (2.14) is of the form δu = C 1 e ky(x−xs) + C 2 e −ky(x−xs) where C 1 , C 2 are constants. Since δu must vanish at the boundaries, we can write the solution as, δu B = Ce ky(x−xs) (x < x s ), (2.15a) δu A = Ce −ky(x−xs) (x > x s ), (2.15b) where the constant C in (2.15) is chosen to ensure continuity in velocity across the interfaces. For the solution at the interface (x = x s ), we integrate (2.13) along infinitesimal distance (dx ≈ 0), the second term in the left-hand side of the equation is zero and the remaining terms are, ∆ ρ 0 dδu s dx = −E ac 2k w δu s z avg k 2 y n 2 (z B − z A ) (sin(2k w x)δ(x − x s )) dx, (2.16) where δu s is the value of δu at x = x s . Using, (2.15) and the Dirac delta identity f (x)δ(x − a)dx = f (a) to solve for eigenvalue n in (2.16). ρ A (−k y δu s ) − ρ B (k y δu s ) = −E ac 2k w δu s z avg k 2 y n 2 (z B − z A ) sin(2k w x s ), (2.17) Rearranging (2.17), the dispersion relation n for the stability problem becomes n = k y ρ A + ρ B φE ac (z B − z A ) sin(2k w x s ). (2.18) where φ = 2k w /z avg . The dispersion relation (2.18) establishes the acoustic stability criterion when inhomogeneous fluids (without interfacial tension) in a microchannel are subjected to a standing acoustic wave. If the eigenvalue n is imaginary in (2.18), then the configuration is in a stable equilibrium and the configuration is in an unstable equilibrium when the eigenvalue n is real. For a standing acoustic half-wave, in (2.18), the values of ky ρ B +ρ A , φ and E ac are always positive. Thus, the sign of z B − z A (initial configuration of the fluids) and sin(2k w x s ) (relative location of the interface with respect to the standing acoustic wave) decide the nature of the eigenvalue in (2.18). z B − z A is positive when high-impedance fluid is present to the right of the interface, and negative when highimpedance fluid is present to the left of the interface. sin(2k w x s ) has a negative value to the left of the node (x s is negative), a positive value to the right of the node (x s is positive) and zero when the interface coincides with the node (centre of the microchannel) or anti-node (sides of the microchannel) (x s = 0). As per the above arguments, the inhomogeneous system in figure 1(a) is in an unstable equilibrium as eigenvalue n is real, and the system in figure 1(b) is in a stable equilibrium as eigenvalue n is imaginary. It can be concluded from the above discussion and figures 2(a-i) and 2(a-ii) that, a system is said to be acoustically stable (unstable) if the initial configuration of the fluids is in such a way that the low (high) impedance fluid is present at the anti-node(s) and the high (low) impedance fluid is present at the node(s). This conclusion is consistent with the demonstration of acoustic relocation of fluids within a microchannel by Deshmukh et al. (2014). For the case where the interface coincides with the node, sin(2k w x s ) = 0. Thus, the system is in a neutral equilibrium (n = 0) as shown in figures 1(c) and 2(a-iii). The above analysis can be easily extended to an inhomogeneous system consisting of multiple interfaces. In this case, the eigenvalues evaluated at the fluid interfaces govern the nature of the system. Figure 2(b) shows the stability of two interface systems that are widely used in acoustofluidic applications. It can be seen from figure 2(b-i) that when high impedance fluid is at the sides (anti-nodes), the eigenvalue at both the interfaces (IF 1 and IF 2 ) is real and hence the system is in unstable equilibrium. The system is in stable equilibrium in figure 2(b-ii), as the eigenvalue at both interfaces is imaginary. Stability analysis of inhomogeneous fluids in the presence of interfacial tension Proceeding to solve for immiscible fluids, the effect of surface tension must be accounted for. The discontinuity in impedance occurring in the interfaces (x s ) is modelled by includ-ing the interfacial tension effects in the X momentum equation (2.9b) as, (Chandrasekhar 1961) ρ 0 nδu = − ∂δp ∂x − E ac cos (2k w x) z avg ∂δz ∂x − k 2 y s T δx s δ(x − x s ). (2.19) where T is the interfacial tension and δx s denotes the perturbation of the interfaces and d dt δx s = δu s =⇒ δx s = δus n . The governing differential equation for stability between inhomogeneous fluids with interfacial tension is obtained similar to the case without interfacial tension, as in the previous § 2.1, d dx ρ 0 dδu dx − ρ 0 k 2 y δu = −E ac 2k w δu z avg k 2 y n 2 sin(2k w x)(z B − z A )δ(x − x s ) + k 2 y n 2 s k 2 y (T δu s ) δ(x − x s ), (2.20) Integrating (2.20) across an infinitesimal distance (dx ≈ 0) and solving for the dispersion relation n, n = k y ρ 1 + ρ 2 φE ac (z B − z A ) sin(2k w x s ) − k 2 y T . (2.21) Equation (2.21) establishes the acoustic stability criterion when fluids with interfacial tension are subjected to a standing acoustic wave. It can be seen from (2.21) that the interfacial tension (T ) and wavenumber of the perturbation (k y ) play a role in the stability of immiscible fluids. In the presence of interfacial tension (T > 0), the fluid system shown in figure 1(b) is always stable, as the negative sign of (z B − z A ) sin(2k w x s ) results in an imaginary eigenvalue n in (2.21). Whereas, for the fluid system shown in figure 1(a), the sign of (z B − z A ) sin(2k w x s ) is positive in (2.21). Thus, the system is conditionally stable, and the stability is determined by the relative magnitudes of φE ac (z B − z A ) sin(2k w x s ) and k 2 y T . The fluid system (figure 1(a) becomes unstable (n is real) if the acoustic force density F rl (φE ac (z B − z A ) sin(2k w x s )) dominates (or is greater than) the interfacial force density F int (k 2 y T ) and becomes stable (n is imaginary) if the interfacial force density dominates the acoustic force density. For the case where the interface coincides with the node (sin(2k w x s ) = 0 and eigenvalue n is imaginary) and the system is in a stable equilibrium, as shown in figures 1(c) and 2(a-iii). Now, for conditionally stable configuration, we proceed to find the minimum energy density required to relocate the fluid systems in figures 1(a) and 2(b-i) with interfacial tension (T > 0). Since the interface height, h is finite, this leads to the quantization of the possible modes k y = k hn = nπ/h. The minimum (critical) acoustic energy density (E cr ) required to relocate the fluid system is decided by the first conceivable mode, k h1 = k h = π/h and the critical acoustic energy density is obtained by limiting the eigenvalue n to zero in (2.21). Thus, E cr = k 2 h T z avg sin(2k w x s )2k w (z B − z A ) . ( 2.22) If the applied energy density E ac is less than the critical energy density E cr (E ac < E cr ), the interfacial tension succeeds in stabilizing a potentially unstable configuration. The same system becomes unstable and eventually relocates to a stable configuration when E ac > E cr . The above discussions on the equilibrium nature of different inhomogeneous fluid configurations (with and without interfacial tension) are clearly summarised in figure 2. Numerical results and discussion In this section ( § 3), we numerically analyze the stability of inhomogeneous fluids (with and without interfacial tension) under acoustic fields and compare them with the results obtained by the theoretical analysis in the previous section ( § 2). At first, we study the stability and relocation using acoustic relocation force f rl (2.4) where the acoustic energy density is assumed to be constant (as the variation of first-order pressure and velocity are not considered). We further extend the numerical analysis using the generalized acoustic body force f ac (2.2) where the first-order pressure and velocity vary during relocation (thus E ac varies) (Rajendran et al. 2022). The numerical analysis is carried out on a two-dimensional fluid domain of height h = 160 µm and width w = 360 µm in COMSOL Multiphysics 6.0. For this study, the fluids mineral oil (Z = 1.23 M P a s/m) and silicone oil (Z = 0.961 M P a s/m) are used. A mesh refinement procedure, similar to those employed by Rajendran et al. (2022) is used to confirm that the numerical findings are not affected by grid size. Three different fluid configurations are considered for the study, namely, • High-Low-High (HLH) configuration where the high impedance fluid is present at the anti-nodes (sides) and the low impedance fluid is present at the node (center) as shown in figure 3(a). • Low-High-Low (LHL) configuration where the low impedance fluid is present at the anti-nodes (sides) and the high impedance fluid is present at the node (center) as shown in figure 3(b). • High-Low (HL) configuration where the high impedance fluid occupies the domain to the left of the center of the microchannel and the low impedance fluid occupies the domain to the right of the center of the microchannel as shown in figure 3(c). For the sake of brevity, the configurations shown in figures 2(a-i) (or 1) and 2(a-ii) are not discussed explicitly as their stability and relocation are captured by HLH and LHL configurations. The Low-High (LH) configuration is also not discussed, as it would be analogous to the HL configuration. For all the analyses, the initial interface is perturbed and modelled as x s (y) = A 0 cos 2π h y + h 2 , where A 0 = 0.01h is the perturbation amplitude. Numerical analysis of stability using constant acoustic energy density For the numerical simulations shown in 3, we employ equation (2.4) as body force and assumed E ac to be constant throughout the relocation process. The boundary condition for the analysis is no slip at the walls and the pressure is constrained at a point (bottom left corner of the channel). In the absence of interfacial tension (T = 0 mN/m), it is observed that for any E ac > 0, the HLH configuration undergoes relocation to a stable LHL configuration as in figure 3(a) (the simulation is shown for E ac = 80 J/m 3 ). In this case, the magnitude of E ac only influences the timescale of the relocation process by competing with the viscosity. While, in the presence of interfacial tension, T = 1 mN/m, the HLH fluid configuration remained stable for all energy densities below 88 J/m 3 , and relocation is observed for all energy densities above 89 J/m 3 . These simulations are in close agreement with the critical acoustic energy density E cr = 88.78 J/m 3 predicted by (2.22) for mineral-silicon oil combination. Simulation results of other fluid combinations shown in figure 5 also agree with (2.22). When the applied E ac is just above E cr , the fluids take a much longer time to relocate. Thus for convenience, the simulation is shown for E ac = 120 J/m 3 in figure 3(a). For LHL configuration with and without interfacial tension (T 0), for any E ac > 0, the relocation of fluid is not observed, and the system remained stable as shown in figure 3(b) (the simulation is shown for E ac = 120 J/m 3 ). In the HL configuration, the node of the standing acoustic half-wave coincides with the fluid-fluid interface. Here for fluids with interfacial tension, relocation is not observed for any E ac > 0, and the fluid system remained stable ( figure 3(c)). Whereas, for fluids without interfacial tension, the HL configuration is observed to be in neutral equilibrium ( figure 3(c)). These simulation results of unstable, stable, and neutral equilibrium of inhomogeneous fluids (figure 3) are in agreement with the stability criteria (from (2.18) and (2.21)) that we established theoretically in § 2. 3.2. Numerical analysis of stability using generalized body force f ac Thus far, in the theoretical stability analysis ( § 2) as well as in the numerical simulations ( § 3.1), a simplified equation f rl (2.4) is employed as a body force with the assumption of constant E ac (the amplitudes of first-order fields p a and v a do not vary during relocation). In this section, the generalized acoustic body force f ac (2.2) is employed and the first-order fields required to calculate the above f ac are obtained from the wave equations (frequency domain -see appendix A) by actuating the channel walls at a frequency ν with a wall displacement d 0 . There are two reasons for using generalized acoustic body force f ac : 1. To show the relocation predicted by f rl and f ac is approximately the same. When we use much simpler f rl instead of the complex f ac , the first-order field equations are not required to be solved which will significantly reduce the computation time for simulation of relocation of inhomogeneous fluids. 2. To explain the previous microchannel experiments in immiscible fluid relocation (Hemachandran et al. 2019). For one-directional (1-D) standing half-wave simulations, the sidewalls are actuated in phase at a displacement d 0 at a frequency ν. In laminar flow equations, the boundary conditions used are no-slip at all walls, and the pressure is constrained at a point (bottom left corner of the channel). To disregard the effect of streaming, the first-order acoustic fields (see Appendix A) are allowed to slip in the frequency domain. Figure 6(a), shows the HLH configuration subjected to 1-D standing half-wave by actuating sidewalls at a displacement d 0 of 0.21 nm and a frequency ν of 1.73 M Hz. In this case, it is observed that the resulting pressure amplitude P a of 0.67 M P a (E ac = 85.58 J/m 3 ), could not relocate the fluids in the HLH configuration and thus remains stable. Whereas, when the displacement is increased to 0.22 nm, the resulting pressure amplitude of P a = 0.69 M P a (E ac = 86.22 J/m 3 ), could relocate the HLH configuration to a stable equilibrium as shown in figure 6(b). From the above discussion, the critical acoustic energy density is found to be E cr = 85.9 ±0.32 J/m 3 . This value of E cr obtained through generalized body force f ac is in close agreement (deviation of 3.24%) with the simplified relocation force f rl employed to derive stability criterion ( § 2). In the case of a 1-D standing half-wave, when the interface of the fluid coincides with the pressure node (x s = 0), for any E ac , relocation is not observed using both f ac and f rl (figures 3(c) and 4(c) as predicted by the stability criteria (2.21). However, Hemachandran et al. ∇ × f rl = 0. This implies when a sufficient energy density is applied, the fluid system 4(d) will not be in equilibrium and relocation begins without imposing any perturbations unlike the other relocation discussed in this work. Characterization of stable and unstable (relocation) regime When the 1-D acoustic standing wave is imposed on fluids with interfacial tension, the configurations (figures 1(b), 2(b-i), 3(a) having high impedance fluid at the anti-node and the low impedance fluid at the node, become conditionally stable. From (2.21), it is evident that the stability of the above inhomogeneous fluid configurations is governed by the ratio of F rl and F int , which is called as acoustic Bond number (Bo a ), given by Bo a = F rl F int = φE ac ∆Z sin (2k w x s ) k 2 h T (3.1) The Bo a that separates the stable and unstable region is called critical acoustic Bond number Bo a,cr . From (2.21) Bo a,cr = 1 (3.2) For Bo a > Bo a,cr , the above configurations become unstable (relocation occurs), and for Bo a < Bo a,cr the configurations remain stable. Figure 5 shows the simulation results of different immiscible fluid combinations. The relocation and non-relocation regimes predicted by the simulations are in line with (3.2). It must also be noted that the fluids with higher interfacial tension require a higher acoustic force for relocation. Effect of the height of the channel on stability The height of the channel h plays a critical role in the stability of immiscible fluids. For a given E ac , the increase in h weakens the stabilizing effect of interfacial tension force, as analysed theoretically in § 2. From (2.22), it can be inferred that the critical acoustic energy density is inversely proportional to the square of the channel height (E cr ∝ 1/h 2 ). In figure 6(a), for a microchannel of height h = 80 µm consisting of mineral-silicone Figure 6. Effect of channel height on stability, (a) channel height h = 80 µm -no relocation is observed as applied energy density (Eac = 120 J/m 3 ) is less than the critical energy density (Ecr = 384 J/m 3 ), (b) channel height h = 160 µm -relocation is observed as applied energy density (Eac = 120 J/m 3 ) is high than the critical energy density (Ecr = 88.78 J/m 3 ). This demonstrates that the interfacial tension force weakens as the height of the channel increases and thus higher the height of the channel, the lower the Eac required for relocation as (Ecr ∝ 1/h 2 ). oil with interfacial tension T = 1 mN/m, the fluid system is stable as the applied E ac (120 J/m 3 ) is lower than the critical energy density (E cr = 384 J/m 3 ). Whereas for h = 160 µm and keeping the remaining parameters same, fluid relocation is observed as applied E ac (120 J/m 3 ) is higher than the critical energy density (E cr = 88.78 J/m 3 ). The above discussion on the effect of channel height on acoustic relocation has high relevance in practical applications. To relocate fluids with high interfacial tension of O(10 1 − 10 2 ) mN/m, in commonly used acoustofluidic channels of height ranging from 100 µm to 200 µm, the required E ac becomes ≈ O(10 4 ) J/m 3 , which is very high compared to the E ac employed in typical acoustofluidic experiments (O(10 2 −10 3 ) J/m 3 ). The equation (2.22) tells that the above problem can be solved by increasing the channel height as E ac ∝ 1/h 2 . Hence, the depth (height) of the channel is a crucial aspect to be considered during the fabrication of an acoustofluidic microchannel for handling high interfacial tension fluids. Conclusion We have theoretically established the stability criteria for inhomogeneous fluids subjected to standing acoustic fields, which is consistent with the previous experimental investigations on miscible (Deshmukh et al. 2014), , and immiscible fluids (Hemachandran et al. 2019). Numerical simulations on the same were carried out using simplified and generalized body force to understand the various parameters that contribute towards stability and relocation of fluids. However, the effect of boundary layer-driven streaming on relocation is neglected in this work, which will be addressed in an upcoming paper. The insights gained from this study can have potential applications in inhomogeneous fluid handling and particle manipulation in the field of acoustofluidics. Also, combining equations (A 1a & A 1c) we get − iωκ 0 p 1 = −∇ · v 1 (A 1d ) where p 1 is the first-order pressure field, ρ 1 refers to first-order density field, v 1 is the first-order velocity field, ω is the angular frequency, η is the dynamic viscosity of the fluid, ξ is the volume fluid viscosity, β = (ξ/η) + (1/3), f ac is generalised body force and Z is impedance. The detailed analysis of first-order and second-order fields acting on inhomogeneous fluids is given in Rajendran et al. (2022). Figure 1 . 1Inhomogeneous fluids (of different impedance ZA and ZB) separated by a plane interface subjected to a standing acoustic half-wave. In the absence of interfacial tension, the fluid system (a) is in an unstable equilibrium, (b) is in a stable equilibrium, (c) is in neutral equilibrium and (d) is in non-equilibrium state. In the presence of interfacial tension, the fluid system (a) is in a conditionally stable equilibrium, (b) is in stable equilibrium, (c) is in stable equilibrium and (d) is in a conditionally stable equilibrium. Note: (d) is not in the scope of this work. Figure 2 . 2Different inhomogeneous fluid configurations commonly used in microfluidics and their equilibrium nature. a) Single interface configurations; b) Double interface configurations. Equations (2.1) and (2.2) are used to calculate n for fluids without interfacial tension and with interfacial tension. Figure 3 . 3Stabilization and relocation of inhomogeneous fluids using simplified body force (2.4) with constant Eac -(a) High-Low-High (HLH) configuration, (b) Low-High-Low (LHL) configuration, (c) High-Low (HL) configuration (2019) through experiments demonstrated the relocation of fluids irrespective of the location of the vertical interface x s . In their experiments, the frequency employed (2.1 M Hz) is far from the 1-D resonant half-wave frequency (ν = 1.6 M Hz ≈ c avg /2w). In our previous work (Rajendran et al. 2022), we have shown that the above relocation is due to standing two-directional (2-D) acoustic wave (frequency f = 2.1 M Hz between c avg /2w and c avg /2h) as shown in figure 4(d). From figure 4(d) it is clear that the pressure node is not vertical but inclined with respect to the fluid interface owing to the 2-D actuation (all four walls are actuated at d 0 ). The above 2-D relocation can be clearly explained by the fact that if the fluid interface and node are not perpendicular to each other, then Figure 4 .Figure 5 . 45Stabilization and relocation of inhomogeneous fluids using generalized body force fac (2.2) along with the first-order pressure field (|p1| = Real(p 1 p1)) for different fluid configurations. 1-D actuation is imposed on (a-c), and 2-D actuation is imposed on (d).(a) HLH configuration remained stable up to Eac = 85.58 J/m 3 (pa = 0.67 M P a, d0 = 0.21 nm, ν = 1.73 M Hz). (b) HLH configuration undergoes relocation above Eac = 86.22 J/m 3 (pa = 0.69 M P a, d0 = 0.22 nm, ν = 1.73 M Hz). Significant variation in |p1| during relocation is observed. (c) HL configuration where the fluid interface coincides with the node remained in stable equilibrium even at Eac = 2334 J/m 3 (pa = 3.54 M P a, d0 = 20 nm, ν = 1.73 M Hz). 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[ "Modular microfluidic platform for solubility measurement, nucleation statistics and polymorph screening of active pharmaceutical ingredients: Irbesartan, Rimonabant, Aripiprazole and Sulfathiazole", "Modular microfluidic platform for solubility measurement, nucleation statistics and polymorph screening of active pharmaceutical ingredients: Irbesartan, Rimonabant, Aripiprazole and Sulfathiazole" ]
[ "Mathilde Lambert \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n\nSanofi R&D -Global CMC / Synthetics -Early Development France\nImpasse des ateliers94400Vitry-Sur-SeineFrance\n", "Romain Grossier [email protected] \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n\nCorresponding authors\n\n", "Mehdi Lagaize \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n", "Thirou Bactivelane \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n", "Vasile Heresanu \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n", "Benoît Robert \nSanofi R&D -Global CMC / Synthetics -Early Development France\nImpasse des ateliers94400Vitry-Sur-SeineFrance\n", "Nadine Candoni [email protected] \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n\nCorresponding authors\n\n", "Stéphane Veesler [email protected].:33617248087 \nUMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France\n\nCorresponding authors\n\n" ]
[ "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "Sanofi R&D -Global CMC / Synthetics -Early Development France\nImpasse des ateliers94400Vitry-Sur-SeineFrance", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "Corresponding authors\n", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "Sanofi R&D -Global CMC / Synthetics -Early Development France\nImpasse des ateliers94400Vitry-Sur-SeineFrance", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "Corresponding authors\n", "UMR7325\nCentre Interdisciplinaire de Nanosciences de Marseille\nCNRS\nAix-Marseille Université\nCINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France", "Corresponding authors\n" ]
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Drug efficacy strongly relies on the solid state of the active pharmaceutical ingredient. Classical solid-state screening methods involve different solvent compositions and supersaturations. Moreover, the many repeat experiments needed to address the stochasticity of nucleation make this approach costly. This paper presents a newly developed modular microfluidic platform that provides a universal and flexible plug-and-play tool for crystallisation studies without use of surfactants. By dissolving a powder, our set-up generates saturated solutions that can be used for solubility measurements or distributed in microdroplets. Here, we describe solubility measurements performed on different forms, stable and metastable, of pharmaceutical molecules (Irbesartan, Rimonabant and Aripiprazole) in organic and aqueous solvents. In addition, we provide nucleation statistics obtained for Sulfathiazole in water and in acetonitrile. Reporting polymorph screening on Sulfathiazole and statistics for nucleated forms, we find that the cooling rate influences both nucleation and polymorphism results, reflecting the competition between thermodynamics and kinetics. Three unknown forms were discovered, with XRD patterns and Raman spectra that do not match any referenced form. We also demonstrate the limitations of microfluidics for crystallisation by cooling: reducing the crystalliser volume considerably increases nucleation induction time.
10.1016/j.jcrysgro.2023.127252
[ "https://export.arxiv.org/pdf/2212.05819v1.pdf" ]
254,564,285
2212.05819
56138d460abd086a2285b16a5bf108cee8355382
Modular microfluidic platform for solubility measurement, nucleation statistics and polymorph screening of active pharmaceutical ingredients: Irbesartan, Rimonabant, Aripiprazole and Sulfathiazole Mathilde Lambert UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Sanofi R&D -Global CMC / Synthetics -Early Development France Impasse des ateliers94400Vitry-Sur-SeineFrance Romain Grossier [email protected] UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Corresponding authors Mehdi Lagaize UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Thirou Bactivelane UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Vasile Heresanu UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Benoît Robert Sanofi R&D -Global CMC / Synthetics -Early Development France Impasse des ateliers94400Vitry-Sur-SeineFrance Nadine Candoni [email protected] UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Corresponding authors Stéphane Veesler [email protected].:33617248087 UMR7325 Centre Interdisciplinaire de Nanosciences de Marseille CNRS Aix-Marseille Université CINaM-, Campus de Luminy, Case 91313288Marseille Cedex 09France Corresponding authors Modular microfluidic platform for solubility measurement, nucleation statistics and polymorph screening of active pharmaceutical ingredients: Irbesartan, Rimonabant, Aripiprazole and Sulfathiazole 1 (N. Candoni), (R. Grossier).A1 NucleationA1 SolubilityA1 PolymorphismA1 Raman spectroscopyA2 MicrofluidicsB1 Organic compounds Drug efficacy strongly relies on the solid state of the active pharmaceutical ingredient. Classical solid-state screening methods involve different solvent compositions and supersaturations. Moreover, the many repeat experiments needed to address the stochasticity of nucleation make this approach costly. This paper presents a newly developed modular microfluidic platform that provides a universal and flexible plug-and-play tool for crystallisation studies without use of surfactants. By dissolving a powder, our set-up generates saturated solutions that can be used for solubility measurements or distributed in microdroplets. Here, we describe solubility measurements performed on different forms, stable and metastable, of pharmaceutical molecules (Irbesartan, Rimonabant and Aripiprazole) in organic and aqueous solvents. In addition, we provide nucleation statistics obtained for Sulfathiazole in water and in acetonitrile. Reporting polymorph screening on Sulfathiazole and statistics for nucleated forms, we find that the cooling rate influences both nucleation and polymorphism results, reflecting the competition between thermodynamics and kinetics. Three unknown forms were discovered, with XRD patterns and Raman spectra that do not match any referenced form. We also demonstrate the limitations of microfluidics for crystallisation by cooling: reducing the crystalliser volume considerably increases nucleation induction time. Introduction Drug efficacy strongly relies on the solid state of the active pharmaceutical ingredient. Most of these organic molecules can exist under several crystalline structures (i.e. phases): polymorphs, solvates, salts, etc. [1], [2]. Here, the different crystalline structures will be termed "forms" (as common in the literature), without distinction. These forms have different physicochemical properties, which can affect the manufacturing process, the bioavailability and the posology of the drug [3], [4]. The crystallisation conditions of all forms and their relative stabilities must be known to avoid form transitions or appearance of new polymorphs, as in the case of Ritonavir in 1997 [5]. In the pharmaceutical industry, polymorphism studies really started after this event. Classical polymorph screening methods involve different solvent compositions and different supersaturations, using for example various cooling rates or non-solvents. All these screening experiments are usually carried out in crystallisers ranging from millilitres to hundreds of microlitres in capacity. Furthermore, due to the stochasticity of nucleation, each experiment (experimental condition) must be repeated a significant number of times [6]. Thus, each experiment requires 1 to 10 mg of raw material [7], making this approach costly. One increasingly used solution to this problem is droplet-based microfluidics, involving a wide range of techniques [8]- [10]. Small-volume droplets, from nanolitre to microlitre, can be used as a single crystalliser, leading most of the time to a single nucleation event due to the confinement effect [11], [12]. Droplet lab-on-a-chip experiments are widely used for all sorts of applications in solution crystallisation [13]- [16]. However, there are some drawbacks to this approach. Once designed, microfluidic set-ups on chips are not adjustable. Moreover, a surface treatment needs to be applied when certain types of solvent are involved (aqueous, organic, etc). In addition, most applications require surfactants to generate droplets and prevent coalescence. This paper presents the modular microfluidic platform we developed to provide a universal and flexible plug-and-play tool for crystallisation studies without use of surfactants. Improving the module assembly enables us to study a molecule of interest from the powder to the crystal, using image acquisition and spectroscopic characterisation. Our platform yields solubility measurements [17], monitors nucleation in droplets of 0.5µL (1mm diameter) using sequential image acquisition, and characterises crystal forms in-situ by Raman spectroscopy. This provides the percentage of nucleated crystals, as well as statistics on crystalline forms, thereby addressing the stochasticity of nucleation. The optical monitoring in time is also used to verify if any solution-mediated phase transition occurred. We present how the platform is applied to solubility measurements on various stable and metastable forms of Rimonabant, Sulfathiazole, Aripiprazole and Irbesartan. Then we present polymorph screenings on Sulfathiazole in water and acetonitrile, using different temperature profiles. Finally, we illustrate the influence of volume on nucleation by screening experiments on Rimonabant in different solvents. Material Products -Rimonabant is a selective CB1 receptor blocker developed by Sanofi-Aventis used for the treatment of obesity. Although now removed from the market, it is still studied for its polymorphism. Rimonabant has two known polymorphs with similar stabilities [18], I and II, and numerous solvates, principally in alcohols. Rimonabant was provided by Sanofi (SR141716, Form I: batch FFT.REX1.150.0031 / Form II: batch CL11469). -Sulfathiazole is an antimicrobian agent widely studied in the literature for its polymorphism [19], [20]. However, there is confusion over the naming of each forms. We use the same labels as Munroe et al. for polymorphs I to V [19]. Sulfathiazole was purchased from Fluka Analytical (batch MKBQ0002V) and is a mixture of forms II and IV according to X-Ray Diffraction analysis (XRD) (see SIF). -Among the drugs with the largest number of identified forms [21], Aripiprazole is an antipsychotic used to treat schizophrenia and bipolar disorders. Aripiprazole was purchased from Thermoscientific (batch A0417885) and is a mixture of forms II and III according to XRD analysis. -Irbesartan is an Angiotensin-II Receptor blocker used in the treatment of hypertension. This organic molecule presents prototropic tautomerism due to reversible proton transfer on the tetrazole ring. The two resulting tautomers crystallise separately into two different crystalline structures, form A (1-H tautomer) and B (2-H tautomer) [22]. This rare type of phase transition is known as desmotropy. Both forms A and B were provided by Sanofi. -Solvents are water, ethanol and acetonitrile, of analytical grade. -Fluorinated oil GPL is a chemically inert oil, which is non miscible with most of the solvents. It was chosen due to its high viscosity, which reduces the risk of droplet coalescence, avoiding any use of surfactant. Fluorinated oil (GPL107, Krytox TM ) was purchased from Chemours; its viscosity at different temperatures, density and interfacial energy with air, water, ethanol and acetonitrile are presented in 0. Microfluidic platform Our microfluidic platform ( Figure 1) is composed of different modules supported by an aluminium structure (MiniTec, Thorlabs) designed in-house. The modules can be organised as required for different applications. Parts of (c), (d) and (f) were designed and manufactured by 3D printing (Formlabs). Different resins were used depending on the function: "High Temp Resin" for the spiral holder in the thermostatic bath (d), "Grey Resin" for UV (c) and Raman holders (f) and "Clear Resin" for the structure. In (b), the microfluidic circuit is composed of HPLC consumable material (IDEX Health and Science) (SIF 2): junctions and fittings are in polyether ether ketone (PEEK) and tubing (1 mm inner diameter) is in perfluoroalkoxy (PFA). A 7.5cm-long UHPLC stainless steel column of 2.1mm inner diameter and 0.5µm filters contains the powder (ref 5030IP-04021-0075-05 in SIF 2). These polymers are chemically resistant to most solvents, which enables universal use of the platform, whatever the solvent, at a wide range of temperatures. Syringes and pumps Solutions, solvents and oil are loaded in glass syringes (gastight syringes series 1000, termination ChemSeal, 5mL and 10mL, Hamilton). The syringes are placed on precision syringe pumps (NeMESYS S, Cetoni GmbH) ((a) in Figure 1) programmed by a computer interface which accurately controls the flow rates. Saturated solutions and droplet generation The module generating saturated solutions and droplets consists of a tank filled with thermoregulated water ((b) of Figure 1). The tank contains the UHPLC column (IDEX Health and Science) filled with the API powder (from 50 mg to 100 mg). The solvent flows through the powder bed to generate a saturated solution directly by dissolution of the powder at a given temperature ( Figure 2). After the column, the saturated solution flow crosses, in a T-junction, either a pure solvent flow to be diluted for solubility measurements (Figure 2a Ultraviolet characterisation. Ultraviolet (UV) characterisation is used for both solutions and droplets, for size and frequency measurements, as described in section 3. UV light is generated by a deuterium lamp (DT-MINI-2-GS, Ocean Insight), emitted and collected through solarised optical fibres, and analysed by a UV-Visible spectrometer (USB2000+, Ocean Insight). For solubility measurements, the solution flows through a clear fused quartz capillary (1mm ID x 1.5mm OD, VitroCom). A 3D-printed resin support was designed in-house to ensure optical alignment of both the optical fibres and the tubing ((c) in Figure 1 and SIF Figure 2). Droplet storage and cooling Saturated droplets are stored in a temperature-controlled water tank, equipped with a glass window for observation ((d) in Figure 1). The tank contains two independent water circuits. The first is closed and regulated by a thermostatic bath (DYNEO TM DD-600F, Julabo TM ) used to regulate the water temperature and perform slow cooling. The second circuit is composed of a homemade thermostatic bath used to rapidly fill the tank chamber with water (in less than 5 minutes). By fast-cooling the droplets, supersaturation is generated at a higher rate. In order to reduce the observation zone while increasing the number of droplets, the tubing containing droplets is rolled onto a 3D-printed spiral support. This support includes a hard-drive disk platter as reflecting surface for optical microscopy observations. The whole is directly immerged in the tank water, enabling temperature regulation and facilitating optical characterisation due to the matching of PFA tubing and water optical indexes. Dedicated motorised optical microscope The tubing containing droplets is optically characterised in the tank during cooling. An optical device (Zoom 70 with stepper-motor, TV-Objective 3.1xWD=77mm, L=10mm, TV-Tube 1.0xD=35 mm L=146.5mm, OPTO) and a camera (Exo267MU3, SVS-Vistek) are placed on a motorised XYZ stage (X-VSR40A-PTB2 and XY-LSQ150A, Zaber) to perform a complete screening of the spiral ((e) in Figure 1). Software created in-house (LabVIEW) is used to program XYZ displacements by defining the steps between pictures, the region of interest and time between cycles. A Python code (available on request) is then used to reconstruct a full image of the spiral ( Figure 3). Moreover, since the camera and the XYZ stage are mounted on a mobile slide on the MiniTec aluminium structure, they can be displaced to observe different modules at will. Raman spectroscopy characterisation Droplets are analysed directly in the tubing (in-situ) after the cooling process (off-line) by Raman spectroscopy using a 785nm laser (adjustable output power 350mW, ref. I0785MM0350MF, IPS) connected to a Raman spectrometer (QE Pro+, Ocean Insight) by a probe made of optical fibres (RPB-785-FF-XR enhanced, InPhotonics and OceanInsight) ((f) of Figure 1). A holder was designed and 3Dprinted with grey resin to align the probe with the tubing (SIF Figure 2). The distance from the probe lens to the tubing, as well as the tubing vertical position, can easily be adjusted to focus the laser beam at the best position inside the droplet and optimise the signal. The laser beam position can be controlled by the optical characterisation module (2.2.5) thanks to a reflecting surface under the tubing. Lastly, the measured Raman spectra are analysed with Python. Interfacial energy measurement The interfacial energies are measured by the pendent drop method using an OCA-20 device (Dataphysics) and the SCA20_U software. A pendent drop of the densest media in a vertical 1mL syringe is generated inside the less dense media contained in a transparent cell. The drop is subjected to gravitational effects and interfacial energy on the aperture of the tip of the syringe (Nordson, external diameter 0.82mm). Supplementary solubility measurements For verification purposes, solubility measurements were performed in thermostatted millivials. A 1mL glass vessel is placed in a Monopuits (Anacrismat) Peltier thermostatted cell and observed under an optical microscope (Nikon, objective x4). The set-up was previously described by Boistelle et al. [23] 3. Methods Interfacial energy measurement The interfacial energy was measured by the pendent drop method, first described by Tate [24]. The drop is generated using a low flow of liquid and pictures are taken regularly, until the drop detaches and falls. The last picture before detachment, corresponding to the balance between interfacial energy and gravitational effects, is analysed by the software (SCA20) to detect the shape of the drop at equilibrium and the curvature radii. These characteristics are related to the interface energy by the Laplace-Young equation, yielding the interface energy between two fluids, e.g. oil/air, solvent/air and oil/solvent. For each pair of fluids, the measurement is performed on at least 10 drops. However, interfacial energy being highly sensitive to the environment (namely temperature and presence of microdust), the results presented in 0 must be considered with caution. Solubility measurement Microfluidic set-up In our microfluidic set-up, solubility was measured at different temperatures for stable and metastable forms using the step dilution method developed by Peybernès et al [17]. However, special care is required with metastable forms or in the case of solvate formation, which may induce phase transition. When several measurements are realised with the same powder, each should be performed at a higher temperature than the last, in order to avoid recrystallisation (in the case of direct solubility). A back-pressure regulator (IDEX Health and Science) is added at the outlet of the set-up to prevent solvent cavitation. Thermostatted millivials In thermostatted millivials, supplementary measurements were performed at larger scale via a method similar to that described by Detoisien [25]. Suspensions are produced by adding a known amount of powder to a given volume of solvent, in 1mL vials. The vials are then slowly and incrementally heated (by increasing the temperature every few hours) while being observed in a Monopuits (Anacrismat) Peltier thermostatted cell [26]. Complete dissolution of the crystals occurs at a temperature corresponding to the equilibrium temperature at the starting concentration. Droplet generation As represented in Figure 2b, the droplets are generated by making a flow of oil, the continuous fluid here (rate QC), cross a flow of saturated solution, the dispersed fluid (rate QD). As described in section 2.2.2., the saturated solution is generated from powder. The droplets' length (L) and frequency (fD) are then characterised by UV spectroscopy. Different droplet generation regimes can be defined depending on the capillary number value Ca, which is defined by (1). = × (1) is the dynamic viscosity of the continuous fluid, the velocity of the continuous fluid and the interfacial energy between the two fluids, measured by the pendant drop method (Dataphysics, OCA-20). Since our aim was to produce quasi-spherical droplets of regular size and frequency, with a diameter close to the tubing diameter (here 1mm) to reduce droplet mobility, and thus coalescence (as no surfactant is used), the squeezing regime (Ca < 0.01) was the most appropriate. Zhang et al [27]. proposed a model to link size and frequency of droplets to Ca. Here, we used flows between 0.2µL/s and 1.5µL/s to obtain low Ca and regular droplets. Droplet size and frequency were analysed by UVspectroscopy (3.4 and SIF 3). UV-spectroscopy characterisation For ultraviolet characterisation, the integration time was chosen from a range of between 10ms and 100ms. The wavelength was chosen depending on the maximum absorbance of the solution. For Rimonabant in acetonitrile or ethanol and Irbesartan in ethanol, intensity was integrated from 250 to 300nm. For Sulfathiazole in acetonitrile and water, intensity was integrated from 200 to 300nm. For Aripiprazole in acetonitrile, the intensity was integrated between 220 and 300nm. Since absorbance depends on temperature, the UV cell must be at room temperature for solubility or concentration measurements. However, reducing the temperature increases the risk of recrystallisation in the solution. In the case of a flow, dilution is used to prevent the product crystallising. The known dilution rate is used to determine the concentration. Droplet storage and cooling The tank containing the droplet storage tubing is preheated to at least 5°C higher than the generation temperature to dissolve any crystal which may have appeared between the generation and the storage areas. Different cooling profiles can be used: from slow cooling ramps at -0.4°C/h up to fast cooling from 65°C to 10°C in less than 5 min. Here, only ramps at -0.5°C/h and -5°C/h were used (cf. part 4.2). Optical characterisation Since the observation field of the optical device is too small to observe the entire droplet-containing tubing, the motorised XYZ stage enables screening to be performed in the three directions. A matrix of 180 camera positions is defined. The resulting pictures are then combined on Python to reconstruct a global image of the tubing (Figure 3a). For crystallisation screenings, cycles of 180 pictures are performed every 30 min to 2h during cooling and storage. For nucleation statistics, the nucleation rate is monitored over time and temperature by counting droplets containing crystals and empty droplets for the different picture cycles. Crystal habit can indicate polymorphism, however it is not sufficient evidence, and the different crystal habits are further analysed using Raman spectroscopy or XRD. In-situ Raman characterisation Each droplet is analysed directly in the tubing. The integration time is set at 100 to 500ms. For each molecule, one or several zones of interest are defined. These zones should include at least a peak corresponding to the crystallised product, and avoid the signals of the tubing, the oil and the solvent (SIF 6). In SIF 6, such a peak can be found near 1650cm -1 for Rimonabant. The Raman spectra of each droplet is represented in a "colormap" as shown in Figure 4a. The first step is identifying droplets containing crystals. A Raman Shift range, here between 1630 and 1730cm -1 , is chosen. In Figure 4b, the signal intensity is normalised by the maximum intensity of this zone. If the droplet is empty, the normalised signal then has an intensity close to 1 over the whole zone. In contrast, if the droplet contains a crystal, the normalised intensity is 1 at the peak and close to 0 around the peak. Thus, a Raman shift is chosen outside the peak, e.g. at 1640cm -1 in Figure 4b where the normalised intensity is compared to a threshold value (here 0.7) to define whether or not the droplet contains a crystal. In Figure 4b, droplets containing a crystal are marked by a black dot at 1640cm -1 . Next, crystalline forms in droplets containing crystals are determined by comparing peak positions in one or several zones. In Figure 4c, Form I and Form II of Rimonabant are distinguished at between 1630 and 1730cm -1 . Spectra with a peak below 1670cm -1 are Form I and the others are Form II. For the interpretation, the spectrum can be compared to references. Raman analysis can also be supplemented by XRD after extracting some crystals from the droplets. We simplified the method described by Gerard et al. [28], since we manipulate organic molecules, which do not need to be cryogenised and can be manipulated in ambient conditions and outside the solution. Here, the references for Rimonabant were measured from powder of form I and form II. For Sulfathiazole, the spectra were compared with the literature [19], [20], [29]. Crystals can be analysed directly by Raman spectroscopy in-situ, and several crystalline forms can be identified. But when crystals are small or located at the droplet/oil interface, the signals of oil or solvent can be too intense compared to the crystal signal, making it difficult to use the whole spectrum. In this case, special care must be taken in choosing the zones for form identification. However, under some conditions, several crystals can be analysed in a single droplet, as illustrated in Figure 5. In this example, the droplet contains two crystals with different habits (Figure 5a). Here, the laser beam was focused on each crystal (Figure 5b and c) to obtain two different Raman spectra (Figure 5d): the left crystal appeared to be Form II of Rimonabant while the other was form I. (d) Results and discussion Solubility measurements Solubilities of Irbesartan, Rimonabant and Aripiprazole were measured by the microfluidic method described in part 3.2.1, with 5% error. The range of temperatures that can be explored with this method is at least 15 to 70°C, and concentrations up to 100mg/mL can be measured (SIF 7). Measurement of each value of solubility at a given temperature took 1h, including heating time and concentration stabilisation. For the lowest solubilities, it was possible to use a full column of powder (60 to 90mg) for the entire solubility curve (eg. Aripiprazole) or at least for several points. However, for the highest solubilities (eg. Rimonabant above 40°C), a full column of powder was needed for a single value of solubility. Figure 6a compares measured solubilities of Irbesartan form A and form B in ethanol to the solubility curve obtained on form A by Wang et al. [30]. Form B is the most soluble desmotrope [22] in ethanol. This result shows that our method can be used to measure the solubility of more soluble forms, i.e., metastable forms in the case of strict polymorphism, provided that no phase transition occurs during the measurement. When XRD spectra of the powders in the column were measured before and after the experiments to identify a potential phase transition, none was observed. Irbesartan Rimonabant Solubility curves of Rimonabant form I and II in acetonitrile, measured by our microfluidic method are represented in Figure 6b and c. We compare them with our measurements in thermostatted millivials to validate the method in Figure 6c. Both polymorphs present similar solubilities. Hence, they are impossible to differentiate by the usual methods, as already observed in different solvents by Fours [31] and Alcade et al. [32] (Figure 6b). This result shows that it is difficult to determine their relative stability based on solubility. Fours [31] reported an enantiotropic system with form I stable under a transition temperature between 50 and 58°C, while Perrin et al. [18] established a pressure-temperature diagram showing a monotropic system with stable form II. Aripiprazole The starting powder of Aripiprazole was a mixture of forms II and III. Form II is more stable than form III, but it is not the most stable form of the system at room temperature [33]. Therefore, a preliminary experiment was performed in microfluidics at 20°C to highlight a possible phase transition. Acetonitrile was injected through the powder in the column at a constant flow rate of 0.30µL/s and the saturated solution was diluted by an acetonitrile flow rate of 0.34µL/s. The measured UV signal is represented in SIF 8. The intensity maxima at the beginning of the experiment correspond to air bubbles resulting from the wetting of the powder. Then we observed an unstable signal of a low intensity, meaning high concentration, for the first hour. This can be interpreted as the dissolution of the metastable polymorph (here form III), and the intensity drop as its recrystallisation into form II. After 1h however, the signal reached a plateau, indicating no further phase transition. At the end of the experiment, the XRD analysis of the resulting powder showed that only pure form II remained in the column. This result illustrates that this set-up can be used to purify a mixture of several polymorphs to a more stable form and measure its solubility. Thus, the solubility curves of Aripiprazole form II measured in acetonitrile from a mixture of forms II and III, with microfluidics, are consistent with measurements in thermostatted millivials (Figure 6d). a b c d Figure 6: [31], [32]. (a) Solubilities of Irbesartan forms A and B measured in ethanol with microfluidics. Solubility of form A is compared with the literature [30]. (b) Solubilities of Rimonabant forms I and II measured in acetonitrile with microfluidics and compared with solubilities of forms I and II from the literature in acetone, methyl isobutyl ketone (MIBK) and methylcyclohexane (MCH) (c) Solubilities of Rimonabant forms I and II measured in acetonitrile with microfluidics and Monopuits. (d) Solubility of Aripiprazole form II measured in acetonitrile with microfluidics and thermostatted millivials (Monopuits). Data are fitted with Van't Hoff equation.. Statistics for crystallisation in droplets We generated saturated droplets at 70°C and cooled them down to 10°C using different cooling profiles, maintaining them at this temperature for several hours or days. Pictures were taken every 30 min to 2h as described in part 2.2.5. In addition to polymorph screening, these experiments provide statistics on crystalline form nucleation in droplet-based microfluidics. Droplets containing crystals and empty droplets were counted on several picture reconstructions. The example shown in SIF 9 is the last picture cycle of a screening on Sulfathiazole in water (-5°C/h ramp). In the experiments presented here, no solution-mediated phase transition was observed. Sulfathiazole in water For Sulfathiazole in water, we compared two different ramps of temperature from 70°C to 10°C: -The first ramp had a cooling rate of -5°C/h and the droplets were maintained at 10°C for 3 days after the end of the ramp. The percentage of droplets containing crystals as a function of time is plotted in Figure 7a in parallel with the temperature profile. The first crystal appeared during the cooling, around 29°C. A few more crystals (around 5%) appeared before the temperature reached 10°C, followed by a plateau where no more crystals appeared for a few hours. Then the number of crystals increased again. Counting stopped after 2.5 days at 10°C, at which point we observed 51% of droplets containing crystals (162 crystals/319 droplets). -The second ramp had a slower cooling rate of -0.5°C/h and the droplets were maintained at 10°C for almost one day. As in the previous experiment, the first crystal appeared around 30°C. The percentage of crystals first reached a plateau of 5% (Figure 7b). However, this plateau occurred during the cooling, between 25°C and 19°C. A second wave of nucleation occurred between 19°C and 17°C, reaching more than 90% nucleation. Then the number of crystals increased slowly, reaching 95% at a temperature of 10°C. Finally, after almost one day at 10°C, 100% of the 453 droplets contained crystals. Sulfathiazole in acetonitrile For Sulfathiazole in acetonitrile, we compared two different temperature profiles from 70°C to 10°C: -The first ramp had a cooling rate of -5°C/h ramp and the droplets were maintained at 10°C for 11 days. At the end of the experiment, only 19% (89/465 droplets) of the droplets contained crystals. Most of the crystals appeared during the cooling. Since solubility is five times higher in acetonitrile than in water [29], a higher nucleation rate could be expected [34]. However, the increase in supersaturation from 70°C to 10°C is almost double in water compared to acetonitrile. Thus, the poor nucleation rate we observe illustrates the influence of supersaturation on nucleation, which is in agreement with Peybernès et al [29]. -The second temperature profile consisted of a quench from 70°C to 10°C and the droplets were maintained at 10°C for 6 days. No crystal appeared during cooling, nor at 10°C. In all of these experiments concerning sulfathiazole, we observe that the slower the cooling rate, the higher the number of crystals. Rimonabant in different solvents Similar experiments were performed with Rimonabant in acetonitrile, ethanol and ethanol-water mixtures with different temperature profiles. However, the nucleation rates were too low, producing no crystals in some cases, even after several days at 10°C. These results underline the limitations of volume reduction, which may considerably increase nucleation time, as previously described by Hammadi et al. [35] and observed by Teychené et al. [36] on various organic molecules in organic solvents. Polymorph screening and statistics of nucleated forms Thanks to the versatility of our set-up, the solid can be directly analysed in the tubing (in-situ) after the cooling process (off-line) by Raman spectroscopy. Thus, for the polymorphism screening, we used the experiments and the results described in 4.2. We studied Sulfathiazole in water with two different ramps (-5°C/h and -0.5°C/h) and in acetonitrile with one ramp (-5°C/h), by Raman and XRD analysis on crystallised droplets. Sulfathiazole in water 4.3.1.1. Fast ramp (-5°C/h) For Sulfathiazole in water, after the -5°C/h ramp from 70°C to 10°C and 3 days at 10°C, 319 droplets were analysed by Raman spectroscopy, among which 162 droplets were crystallised. The forms obtained are summarised in Figure 8a: -Form II and form IV appeared respectively in 1.85% and 11.7% of the droplets. They did not show any representative habit ( Figure 8b) and could not be optically distinguished. -The majority of the crystals (85.8%) appeared to be an unknown form, called form U1 here, presenting a particular habit (Figure 8c). This habit was previously observed by Peybernès [37] in a few droplets of a preliminary experiment but no XRD was performed at that time. Here, the XRD pattern is provided in supplementary information (SIF 5). This pattern does not correspond to any known form reported in the Cambridge Structural Database. In order to study the unknown form U1, we performed an additional dissolution experiment, presented in supplementary information (SIF 12). It shows that form U1 has higher solubility than form IV, and thus than form II, whose solubility is similar to form IV [19]. -A last crystal, called form U2 here, presents a slightly different Raman spectrum (SIF 11). However, due to its needle-like shape and small size (Figure 8d), we did not manage to collect the crystal for further analysis (e.g. XRD) in this first experiment. The form statistics presented in Figure 8a are consistent with the stability order established by Munroe et al. [19], with form II being more stable than form IV. We additionally report a new metastable form, U1. Furthermore, we also clearly observed a kinetic effect on crystallisation which competes with the thermodynamic effect: the first crystals to nucleate appear to be forms II and IV, which are thermodynamically more stable. In contrast, the metastable form U1 appears several hours or days later, while the temperature is maintained at 10°C.During the cooling process, the supersaturation increases while the temperature decreases. The experimental conditions inside each droplet move on the phase diagram from undersaturated or saturated to supersaturated. They first cross the solubility curves of forms II and IV respectively, before crossing the solubility curve of form U1. The same trend is found for the metastable limit of the three forms. This may explain why at the beginning of nucleation, the more stable forms (II and IV) nucleate: they show higher supersaturation. At that moment, the solution is above the metastable limit for forms II and IV but inside the metastable zone for form U1. However, after a waiting time, kinetics takes the lead over thermodynamics, leading to the nucleation of the metastable form U1 according to Ostwald's rule of stages. Slow ramp (-0.5°C/h) For the second experiment with Sulfathiazole in water, after the -0.5°C/h ramp from 70°C to 10°C and a day at 10°C, we analysed 453 crystallised droplets by Raman spectroscopy. The forms obtained are summarised in Figure 8a: -Form II and form III each nucleated in a separate single droplet (0.221%), while 4.19% of nucleated droplets were form IV. Two crystals appeared to be a mixture of forms II and IV (SIF 11). However, none of these forms showed any representative habit (similar to Figure 8b), so they could not be optically distinguished. -The majority of the crystals (90.1%) appeared to be the unknown form U2 described in 4.3.1.1, according to the Raman spectra (SIF 11). The crystals showed the same habit as in Figure 8d. In this experiment (compared to 4.3.1.1), we managed to collect the crystals, and the XRD pattern is provided in supplementary information (SIF 5). This pattern does not correspond to any known form reported in the Cambridge Structural Database. -Another unknown form (here called U3) accounted for 4.86% of the crystals. The Raman spectrum does not correspond to any reference (SIF 11), and nor does the XRD pattern (SIF 5). It showed a wellfaceted habit, as represented in Figure 8e. Similarly to the previous experiment (in 4.3.1.1), the known forms II, III and IV nucleated earlier than the most common forms U2 and U3, which appeared during the second wave of nucleation, described in 4.2.1. This may also be explained by the competition between thermodynamics and kinetics. Sulfathiazole in acetonitrile For Sulfathiazole in acetonitrile, after the cooling ramp of -5°C/h and the 11-days plateau at 10°C, the 89 crystallised droplets displayed crystals with a wide range of habits as shown on SIF 13. Moreover, they were larger than those obtained in water, as expected from the higher solubility of Sulfathiazole in acetonitrile at 70°C. However, only forms II and IV were produced, according to Raman analysis and as confirmed by XRD on several crystals. Form II was the most frequent polymorph (97%, 86/89 crystals). Only two pure crystals of form IV were obtained (2%). One droplet contained two crystals: one of pure form II, and one showing peaks of both forms II and IV. These results are consistent with Peybernès [37] who only obtained a small proportion of crystals, all which appeared to be form II, from a slower cooling profile. Conclusions We have presented a modular microfluidic platform for crystallisation studies developed in our laboratory, and which can easily be implemented in any R&D laboratory. By dissolution of a powder, our set-up generates saturated solutions that can be used for solubility measurements or distributed in microdroplets. The solutions or the droplets are characterised by UV-spectroscopy for solution concentration measurements or to characterise microdroplet sizes and frequencies. Then, microdroplets are stored and cooled using different temperature profiles, and sequential image acquisition is performed during the crystallisation to monitor nucleation. Lastly, crystals in microdroplets are characterised by Raman spectroscopy performed directly in the droplets. Solubility measurements were performed on different forms of pharmaceutical molecules in several organic and aqueous solvents. The method enables solubilities to be measured for both stable and metastable forms reaching different ranges of concentration and temperature. With the case of Aripiprazole, we also illustrate that this method can be used on a polymorph mixture to generate a pure form, whose solubility can also be measured. Secondly, we have provided nucleation statistics after cooling. With Sulfathiazole in water and in acetonitrile, we show that the cooling rate influences nucleation. Slower cooling profiles appear to promote nucleation, whereas fast-cooling profiles reduce the number of crystals. We also demonstrate the limitations of microfluidics for crystallisation by cooling: reducing the crystalliser volume considerably increases nucleation induction time. Hence, sufficient supersaturation must be reached to obtain crystals in a reasonable time. Different crystallisation methods have already been explored to reach higher supersaturation, like droplet evaporation [38], [39], addition of a non-solvent, or inducing nucleation with an external field [40]- [43]. Our future research plans include implementing these methods on our modular platform. In reporting polymorph screening on Sulfathiazole and the statistics for nucleated forms, we have shown in water that the cooling rate influences polymorphism results. We discovered three unknown forms whose XRD patterns and Raman spectra do not match any referenced form. We observed that the known stable forms nucleate earlier than the unknown forms, but that at the end of the experiments, most of the crystal are of unknown forms. This illustrates the competition between thermodynamics and kinetics. In acetonitrile, only known forms were observed, form II being the most common, as already observed by Peybernès [37]. All these findings indicate that our microfluidic platform is a powerful tool for polymorph screening that can be used in the pharmaceutical industry to discover new forms of active pharmaceutical ingredients. SIF 1. Properties of the GPL107 oil Water 52-55 SIF 5. Droplet UV analysis In the case of saturated droplets, the UV cell is heated by an infra-red lamp to prevent droplet crystallisation. In this case, the absolute value of intensity is not considered for droplet size and frequency determination. The signal is analysed by peak detection on Python. As the collected data represent droplets which are deformed by their movement, their size is slightly overestimated. However, the measurements can still be used to determine the droplet size dispersion and distinguish quasi-spherical droplets from plugs. SIF 10. XRD patterns of Sulfathiazole XRD measurements were realized in transmission mode using a high brilliancy rotating anode, Rigaku RU-200BH, (operating power 50 kV -50 mA) equipped with a double reflexion mirror, Osmic, and an image plate detector, Mar345. The radiation used is the Cu Kα,  = 1.5418Å, and the beam size 0.5 x 0.5 mm 2 . The maximum measurable 2 angle is 65°, this is limited by the size of the detector and the minimum distance between the sample and the detector, the experimental resolution is about 0.3° in 2. For the measurements the crystals were collected from the droplets and introduced into Lindeman glass capillaries, typically 0.5 mm diameter. SIF 11. Raman spectra of Sulfathiazole in water droplets SIF Figure 9: Raman spectra of different crystalline forms of Sulfathiazole, measured on crystals obtained in water droplets with QEPro Raman spectrometer, 785nm. SIF 12. Dissolution experiment A slice of the tubing of the fast-cooling crystallisation of Sulfathiazole in water (-5°C/h) was used for this experiment. Two crystals in neighbouring droplets were analysed by Raman spectroscopy as form IV and U1 respectively. The capillary was placed in the thermostatic bath at 25°C, and progressively heated to 72°C. The two selected droplets were observed during the heating process. Pictures were taken every 2 to 4 minutes to monitor crystal dissolution. The U1 crystal completely dissolved around 65°C, whereas the crystal of form IV dissolved around 70°C. Figure 1 : 1Microfluidic platform: (a) Syringes and pumps; (b) Solutions and droplet generation with temperature control; (c) UV characterisation; (d) Droplet storage and cooling; (e) Optical characterisation; (f) Raman characterisation Figure 2 : 2Saturated solution generation followed by (a) dilution for solubility measurements; (b) droplet generation in oil, QD, Qs and Qc are respectively the discontinuous form flow (saturated solution), the dilution solvent flow and the continuous form flow (oil). L and fD are respectively the length of the droplets and their frequency. Figure 3 : 3Sulfathiazole in water droplets generated at 70°C in GPL107 oil and cooled to 10°C, tubing of 1mm inner diameter: (a): full spiral reconstruction from 180 pictures; (b) zoom on (a) at the junction of four pictures, zone indicated by the square in (a). Figure 4 : 4Raman spectra of Rimonabant droplets in acetonitrile, generated at 60°C and cooled to 10°C: (a) top: reference spectra of Rimonabant Forms I and II from powder; bottom: full representation of 145 droplets; (b) and (c) top: normalised reference spectra of Rimonabant Forms I and II from powder and an empty droplet between 1630 and 1730cm -1 (b) bottom: normalised representation of 51 droplets between 1630 and 1730cm -1 for the selection of droplets with crystals. The black circles indicate the droplets containing crystals; (c) bottom: normalised representation of 10 droplets containing crystals between 1630 and 1730cm -1 for the form analysis. Figure 5 : 5(a) Plug of acetonitrile containing Rimonabant crystals, generated from saturated solution at 70°C and cooled to 30°C; (b) Laser focused on the left crystal; (c) Laser focused on the right crystal; (d) Raman spectra corresponding to the left crystal (Form II) and the right crystal (Form I). Figure 7 : 7Percentage of droplets of water containing crystals of Sulfathiazole as a function of time (blue) and temperature profile as a function of time (red).(a) for a fast-cooling ramp (-5°C/h); (b) for a slow cooling ramp (-0.5°C/h) Figure 8 : 8(a) Percentage (and number of crystals) of Sulfathiazole forms II, III, IV, mixture II+IV and unknown U1, U2 and U3 among the droplets containing crystals with -0.5°C/h and -5°C/h ramps in water from 70°C to 10°C. ( b, c, d and e) Crystals habits of Sulfathiazole obtained in water, mainly for (b) forms II and IV; (c) unknown U1; (d) unknown U2; (e) unknown U3. SIF Figure 3 : 3Real-time recording of light intensity at 280 nm of circulating droplets. L and fD are respectively the length of the droplets and their frequency. SIF 6 . 6Raman spectra of Rimonabant in acetonitrile droplets (with and without crystal), GPL107 oil and references on powder (Forms I and II) SIF Figure 4 : Raman spectra of GPL 107 oil, an empty droplet of Rimonabant in acetonitrile, a droplet of acetonitrile containing a Rimonabant crystal (form II) and references of Rimonabant form I and II from powder SIF 7. Solubility curves of Irbesartan, Rimonabant and Aripiprazole measured with microfluidics SIF Figure 5: Solubility curves of Irbesartan in ethanol, Rimonabant in acetonitrile and Aripiprazole in acetonitrile measured with microfluidics. SIF 8 . 8UV signal as a function of time of the diluted solution of Aripiprazole in acetonitrile. SIF Figure 6 : UV signal as a function of time of the diluted solution of Aripiprazole in acetonitrile SIF 9. Droplets optical characterisation SIF Figure 7 : Reconstructed picture of the spiral for the last picture cycle for Sulfathiazole in water (ramp -5°C/h) after 3.5 days (3 days at 10°C). SIF Figure 8 : 8XRD patterns of different crystalline forms of Sulfathiazole. SUTHAZ, SUTHAZ01, SUTHAZ02, SUTHAZ04 and SUTHAZ05 are references from the Cambridge Structural Database. U1, U2 and U3 were measured on crystals obtained in water droplets. Figure 10 : 10(a) Temperature profile during the dissolution experiment of Sulfathiazole forms U1 and IV in water. The black dots indicate the pictures represented in (b) and (c) for both crystals. (b) and (c) Pictures taken during the dissolution of crystal form U1 (b) and form IV (c) in water at different temperatures: (i) 39.3°C, (ii) 53.0°C, (iii) 57.1°C, (iv) 65.6°C. Scale bar is 0.5mm.SIF 13.Crystal habits of Sulfathiazole in acetonitrile SIFFigure 11: Crystals habits of Sulfathiazole obtained in acetonitrile with a -5°C ramp from 70°C to 10°C Table 1 : 1Properties of fluorinated oil GPL 107. (a) given by Krytox TM , (b) measured in this study by the pendant drop method at 20°C SIF Figure 2: Raman probe holder: the Raman probe (a) position can be adjusted in y, to focus the beam on the middle of the capillary (b). The capillary is hold by a mobile piece (c) to adjust its position in z. It can also manually be moved in x. A mirror is fixed in the bottom of the observation zone (d).SIF 2. Microfluidics components Associated content NoteThe authors declare no competing financial interest.AcknowledgmentsWe thank Sanofi R&D for financial support. We thank Marjorie Sweetko for English revision.References Facts and fictions about polymorphism. A J Cruz-Cabeza, S M Reutzel-Edens, J Bernstein, 10.1039/C5CS00227CChem Soc Rev. 4423A. J. Cruz-Cabeza, S. M. Reutzel-Edens, and J. Bernstein, "Facts and fictions about polymorphism," Chem Soc Rev, vol. 44, no. 23, pp. 8619-8635, 2015, doi: 10.1039/C5CS00227C. Diversity in Single-and Multiple-Component Crystals. 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Manipulation of nucleation and polymorphism by laser irradiation. T Sugiyama, of the GPL107 oil .................................................................................................. 23 SIF 2. Microfluidics components ..................................................................................................... 24 SIF 3. UV characterisation ................................................................................................................ 25 SIF 4. Raman probe holder .............................................................................................................. 26 SIF 5S.-F Wang, of the GPL107 oil .................................................................................................. 23 SIF 2. Microfluidics components ..................................................................................................... 24 SIF 3. UV characterisation ................................................................................................................ 25 SIF 4. Raman probe holder .............................................................................................................. 26 SIF 510.1016/j.jphotochemrev.2022.100530J. Photochem. Photobiol. C Photochem. Rev. 52100530T. Sugiyama and S.-F. Wang, "Manipulation of nucleation and polymorphism by laser irradiation," J. Photochem. Photobiol. C Photochem. Rev., vol. 52, p. 100530, Sep. 2022, doi: 10.1016/j.jphotochemrev.2022.100530. of the GPL107 oil .................................................................................................. 23 SIF 2. Microfluidics components ..................................................................................................... 24 SIF 3. UV characterisation ................................................................................................................ 25 SIF 4. Raman probe holder .............................................................................................................. 26 SIF 5. Raman spectra of Rimonabant in acetonitrile droplets. ................ . Droplet, ............................................................................................ 27 SIF 6 ; ........................................................................................... 28 SIF 7GPL107 oil and references on powder (Forms I and II). 29Droplet UV analysis ................................................................................................................ 27 SIF 6. Raman spectra of Rimonabant in acetonitrile droplets (with and without crystal), GPL107 oil and references on powder (Forms I and II) ........................................................................................... 28 SIF 7. Solubility curves of Irbesartan, Rimonabant and Aripiprazole measured with microfluidics 29 UV signal as a function of time of the diluted solution of Aripiprazole in acetonitrile. SIF 8. 30SIF 8. UV signal as a function of time of the diluted solution of Aripiprazole in acetonitrile. ........ 30 Droplets optical characterisation. SIF 9. SIF 9. Droplets optical characterisation ........................................................................................... 31 Raman spectra of Sulfathiazole in water droplets. SIF 11. SIF 11. Raman spectra of Sulfathiazole in water droplets ................................................................. 33 IsoBar Systems 2.1 mm ID. Parker PortIsoBar Systems 2.1 mm ID, Parker Port 7.5cm F-120x One, Piece Fingertight 10-32 Coned, for 1/16" OD Natural -10 Pack P-200X Flangeless Ferrule Tefzel ™ (ETFE), 1/4-28 Flat-Bottom. for 1/16" OD Blue-10 Pack P-287X Super Flangeless Nut for 1/16" or 1/32" OD Tubing -10PackF-120X One-Piece Fingertight 10-32 Coned, for 1/16" OD Natural -10 Pack P-200X Flangeless Ferrule Tefzel ™ (ETFE), 1/4-28 Flat-Bottom, for 1/16" OD Blue-10 Pack P-287X Super Flangeless Nut for 1/16" or 1/32" OD Tubing - 10Pack Plug for 1/4-28 Flat-Bottom Ports P-703 Union Assembly PEEK. 050 thru holeP-316 Plug for 1/4-28 Flat-Bottom Ports P-703 Union Assembly PEEK .050 thru hole, for 1/8" OD PEEK Tee .050 thru hole Hi Pressure. 300P-728 PEEK Tee .050 thru hole Hi Pressure, F-300 Back Pressure Regulator Assembly (5psi). Back Pressure Regulator Assembly (5psi) U-467R Plug for 10-32 Coned Ports XP-235X Flangeless Fitting Short, PEEK, 1/4-28 Flat-Bottom for 1/16" OD Natural -10 Pack. U-467R Plug for 10-32 Coned Ports XP-235X Flangeless Fitting Short, PEEK, 1/4-28 Flat-Bottom for 1/16" OD Natural -10 Pack SIF Figure 1 : UV holder: The central part serves to hold UV-solarised optical fiber (a) aligned on both sizes of the quartz capillary (b). A printed cover (c) is added on both sides to prevent the capillary moving and breaking. SIF Figure 1 : UV holder: The central part serves to hold UV-solarised optical fiber (a) aligned on both sizes of the quartz capillary (b). A printed cover (c) is added on both sides to prevent the capillary moving and breaking.
[]
[ "Intensional First Order Logic for Strong-AI Generation of Robots", "Intensional First Order Logic for Strong-AI Generation of Robots" ]
[ "Zoran Majkić [email protected] \nISRST\nTallahasseeFLUSA\n" ]
[ "ISRST\nTallahasseeFLUSA" ]
[]
Neuro-symbolic AI attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust strong AI capable of reasoning, learning, and cognitive modeling. In this paper we consider the intensional First Order Logic (IFOL) [1] as a symbolic architecture of modern robots, able to use natural languages to communicate with humans and to reason about their own knowledge with self-reference and abstraction language property. We intend to obtain the grounding of robot's language by experience of how it uses its neuronal architectures and hence by associating this experience with the mining (sense) of non-defined language concepts (particulars/individuals and universals) in PRP (Properties/Relations/propositions) theory of IFOL. We consider three natural language levels: The syntax of particular natural language (Italian, French, etc..), and two universal language properties: its semantic logic structure (based on virtual predicates of FOL and logic connectives), and its corresponding conceptual PRP structure which universally represents the composite mining of FOL formulae grounded on the robot's neuro system.
10.33140/amlai.04.01.03
[ "https://export.arxiv.org/pdf/2212.07935v1.pdf" ]
254,685,797
2212.07935
663fb1c7e890828bc11cf75f3e097bab84281ba6
Intensional First Order Logic for Strong-AI Generation of Robots 14 Dec 2022 Zoran Majkić [email protected] ISRST TallahasseeFLUSA Intensional First Order Logic for Strong-AI Generation of Robots 14 Dec 2022Strong AIIntensional FOLGeneral Semantics Neuro-symbolic AI attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust strong AI capable of reasoning, learning, and cognitive modeling. In this paper we consider the intensional First Order Logic (IFOL) [1] as a symbolic architecture of modern robots, able to use natural languages to communicate with humans and to reason about their own knowledge with self-reference and abstraction language property. We intend to obtain the grounding of robot's language by experience of how it uses its neuronal architectures and hence by associating this experience with the mining (sense) of non-defined language concepts (particulars/individuals and universals) in PRP (Properties/Relations/propositions) theory of IFOL. We consider three natural language levels: The syntax of particular natural language (Italian, French, etc..), and two universal language properties: its semantic logic structure (based on virtual predicates of FOL and logic connectives), and its corresponding conceptual PRP structure which universally represents the composite mining of FOL formulae grounded on the robot's neuro system. Introduction Last 15 years of my work in AI was mainly dedicated to development of a new intensional FOL, by integrating Montague's and algebraic Bealer's [2] approaches, with a conservative Tarski's semantics of the standard FOL. Basic result was the publication of the conservative extension of Tarski's semantics to intensional FOL [3], and two-step intensional semantics [4], which guaranteed a conservative extension of current RDB, but more than 50-years old technology, toward new IRDB (Intensional RDB). Indeed, in my next Manifesto of IRDB [5], I hoped also to find interested research groups and funds to begin the realization of IRDB as a new platform (compatible with all previously developed RDB application), able also to support NewSQL for Big Data, and ready for other AI improvements. The central hypothesis of cognitive science is that thinking can best be understood in terms of representational structures in the mind and computational procedures that operate on those structures. Most work in cognitive science assumes that the mind has mental representations analogous to computer data structures, and computational procedures similar to computational algorithms. Knowledge representation, strongly connected to the problem if knowledge processing, reasoning and "drawing inferences", is one of the main topics in AI. By reviewing the knowledge representation techniques that have been used by humans we will be aware of the importance of language. The predominant part of IT industry and user's applications is based on some sublanguage of the standard (extensional) FOL (First Order Logic) with Tarski's semantics based (only) on the truth; my effort is to pass to a more powerful evolution of the FOL able to support the meaning of knowledge as well, by replacing the standard FOL and its DB theory and practice in IT business. All this work is summarized and extended also to AI applications of many-valued logics in my recent book [1]. This paper instead is dedicated to show how this defined IFOL in [1] can be used for a new generation of intelligent robots, able to communicate with humans with this intensional FOL supporting the meaning of the words and their language compositions. As in [6] we can consider three natural language levels: The syntax of a particular natural language (French, English, etc..) its semantic logic structure (transformation of parts of the language sentences into the logic predicates and definition of corresponding FOL formulae) and its corresponding conceptual structure, which differently from the semantic layer that represents only the logic's semantics, represents the composed meaning of FOL formulae based on the grounding of intensional PRP concepts. Thus, intensional mapping from the free FOL syntax algebra into the algebra of intensional PRP concepts, I : A F OL → A int provided by IFOL theory, is a part of the semantics-conceptual mapping of natural languages. Note that differently from the particularity of any given natural language of humans, the underlying logical semantics and conceptual levels have universal human knowledge structure, provided by innate human brain structure able to rapidly acquire the ability to use any natural language. Parsing, tokenizing, spelling correction, part-of-speech tagging, noun and verb phrase chunking are all aspects of natural language processing long handled by symbolic AI, and has to be improved by deep learning approaches. In symbolic AI, discourse representation theory and first-order logic have been used to represent sentence meanings. We consider that the natural language (first level) can be parsed into a logical FOL formula with a numbers of virtual predicates and logic connectives of the FOL. By such a parsing we obtain the second, semantic logic, structure corresponding to some FOL formula. However, natural language is grounded in experience. Humans do not always define all words in terms of other words, humans understand many basic words in terms of associations with sensory-motor experiences for example. People must interact physically with their world to grasp the essence of words like "blue," "could," and "left." Abstract words are acquired only in relation to more concretely grounded terms. Theoretical neuroscience is the attempt to develop mathematical and computational theories and models of the structures and processes of the brains of humans and other animals. If progress in theoretical neuroscience continues, it should become possible to tie psychological to neurological explanations by showing how mental representations such as concepts are constituted by activities in neural populations, and how computa-tional procedures such as spreading activation among concepts are carried out by neural processes. Concepts, which partly correspond to the words in spoken and written language, are an important kind of mental representation. Alan Turing developed the Turing Test in 1950 in his paper, "Computing Machinery and Intelligence". Originally known as the Imitation Game, the test evaluates if a machine's behavior can be distinguished from a human. In this test, there is a person known as the "interrogator" who seeks to identify a difference between computer-generated output and human-generated ones through a series of questions. If the interrogator cannot reliably discern the machines from human subjects, the machine passes the test. However, if the evaluator can identify the human responses correctly, then this eliminates the machine from being categorized as intelligent. Differently from the simulation of AI by such Turing tests and the Loebner Prize 1 and in accordance with Marvin Minsky 2 , in this paper I argue that a real AI for robots can be obtained by using formal Intensional FOL (with defined intensional algebra of intensions of language constructions) for the robots as their symbolic AI component, by defining the sense to ground terms (the words) in an analog way, associating to these words the software processes developed for the robots when they recognize by these algorithms (neural architectures) the color "blue" of visual objects, the position "left" etc... In this way we would obtain a neuro-symbolic AI which attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust AI capable of reasoning, learning, and cognitive modeling. To build a robust, knowledge-driven approach to AI we must have the machinery of symbol-manipulation as, in this case, an IFOL. Too much of useful knowledge is abstract to make do without tools that represent and manipulate abstraction, and to date, the only machinery that we know of that can manipulate such abstract knowledge reliably is the apparatus of symbol-manipulation. The IFOL defined in [1] is provided by abstraction operators as well. Daniel Kahneman [7] describes human thinking as having two components, System 1 and System 2. System 1 is fast, automatic, intuitive and unconscious. System 2 is slower, step-by-step, and explicit. System 1 is the kind used for pattern recognition 1 The Loebner Prize was an annual competition in artificial intelligence that awards prizes to the computer programs considered by the judges to be the most human-like. The prize is reported as defunct since 2020. [1] The format of the competition was that of a standard Turing test. In each round, a human judge simultaneously holds textual conversations with a computer program and a human being via computer. Based upon the responses, the judge must decide which is which. 2 while System 2, in uor case based on IFOL, is far better suited for planning, deduction, and deliberative thinking. In this view, deep learning best models the first kind of thinking while symbolic reasoning best models the second kind and both are needed. So, for the words (ground linguistic terms), which can not be "defined by other words", the robots would have some own internal experience of the concrete sense of them. Thus, by using intensional FOL the robots can formalize also the natural language expressions "I see the blue color" by a predicate "see(I,blue color)" where the sense of the ground term "I" (Self ) 3 for a robot is the name of the main working coordination program which activate all other algorithms (neuro-symbolic AI subprograms) like visual recognition of color of the object in focus. But also the auto-conscience sentence like "I know that I see the blue color" by using abstracting operators "⋖ ⋗" of intensional FOL, expressed by the predicate "know(I,⋖ see(I, blue color)⋗)", etc... Consequently, we argue that by using this intensional FOL, the robots can develop their own knowledge about their experiences and communicate by a natural language with humans. So, we would be able to develop the interactive robots which learn and understand spoken language via multisensory grounding and internal robotic embodiment. The grounding of the intensional concepts i PRP theory of intensional logic was not considered in my recent book [1] from the fact that this book was only restricted on the symbolic AI aspects (IFOL); so by this paper we extend the logic theory developed in [1] with concrete grounding of its intensional concepts in order to obtain a strong AI for robots. So, in next Section we will provide a short introduction to IFOL and its intensional/extensional semantics [1]. Algebra for Composition of Meanings in IFOL Contemporary use of the term "intension" derives from the traditional logical doctrine that an idea has both an extension and an intension. Although there is divergence in formulation, it is accepted that the extension of an idea consists of the subjects to which the idea applies, and the intension consists of the attributes implied by the idea. In contemporary philosophy, it is linguistic expressions (here it is a logic formula), rather than concepts, that are said to have intensions and extensions. The intension is the concept expressed by an expression of intensional algebra A int , and the extension is the set of items to which the expression applies. This usage resembles use of Frege's use of "Bedeutung" and "Sinn" [8]. Intensional entities (or concepts) are such things as Propositions, Relations and Properties (PRP). What make them "intensional" is that they violate the principle of extensionality; the principle that extensional equivalence implies identity. All (or most) of these intensional entities have been classified at one time or another as kinds of Universals [9]. In a predicate logics, (virtual) predicates expresses classes (properties and relations), and sentences express propositions. Note that classes (intensional entities) are reified, i.e., they belong to the same domain as individual objects (particulars). This endows the intensional logics with a great deal of uniformity, making it possible to manipulate classes and individual objects in the same language. In particular, when viewed as an individual object, a class can be a member of another class. Definition 1. VIRTUAL PREDICATES: Virtual predicate obtained from an open formula φ ∈ Ł is denoted by φ(x 1 , ..., x m ) where (x 1 , ..., x m ) is a particular fixed sequence of the set of all free variables in φ. This definition contains the precise method of establishing the ordering of variables in this tuple: such an method that will be adopted here is the ordering of appearance, from left to right, of free variables in φ. This method of composing the tuple of free variables is unique and canonical way of definition of the virtual predicate from a given open formula. The virtual predicates are useful also to replace the general FOL quantifier on variables (∃x) by specific quantifiers ∃ i of the FOL syntax algebra A F OL , where i ≥ 1 is the position of variable x inside a virtual predicate. For example, the standard FOL formula (∃x k )φ(x i , x j , x k , x l , x m ) will be mapped into intensional concept ∃ 3 φ(x) ∈ A F OL where x is the list(tuple) of variables (x i , x j , x k , x l , x m ). Virtual predicates are atoms used to build the semantic logic structures of logic-semantics level of any given natural language. Let us define the FOL syntax algebra A F OL . For example, the FOL formula φ(x i , x j , x k , x l , x m ) ∧ ψ(x l , y i , x j , y j ) will be replaced by a specific virtual predicate φ(x i , x j , x k , x l , x m ) ∧ S ψ (x l , y i , x j , y j ) , with the set of joined variables (their positions in the first and second virtual predicate, respectively) S = {(4, 1), (2, 3)}, so that its extension is expressed by an algebraic expression R 1 ⊲⊳ S R 2 , where R 1 , R 2 are the extensions for a given Tarski's interpretation I T of the virtual predicate φ, ψ relatively, and the binary operator ⊲⊳ S is the natural join of these two relations. In this example the resulting relation will have the following ordering of attributes: (x i , x j , x k , x l , x m , y i , y j ). In the case when S is empty (i.e. its cardinality |S| = 0) then the resulting relation is the Cartesian product of R 1 and R 2 . For the existential quantification, the FOL formula (∃x k )φ(x i , x j , x k , x l , x m ) will be replaced in A F OL by a specific virtual predicate (∃ 3 )φ(x i , x j , x k , x l , x m ). For logic negation operator we will use the standard symbol ¬. Based on the new set of logical connectives introduced above, where the standard FOL operators ∧ and ∃ are substituted by a set of specialized operators {∧ S } S∈P(N 2 ) and {∃n} n∈N as explained above, we can define the following free syntax algebra for the FOL: Definition 2. FOL SINTAX ALGEBRA: Let A F OL = (Ł, . =, ⊤, {∧ S } S∈P(N 2 ) , ¬, {∃n} n∈N ) be an extended free syntax algebra for the First-order logic with identity . =, with the set Ł of first-order logic formulae with the set of variables in V, with ⊤ denoting the tautology formula (the contradiction formula is denoted by ⊥ ≡ ¬⊤). We begin with the informal theory that universals (properties (unary relations), relations, and propositions in PRP theory [10]) are genuine entities that bear fundamental logical relations to one another. To study properties, relations and propositions, one defines a family of set-theoretical structures, one define the intensional algebra, a family of set-theoretical structures most of which are built up from arbitrary objects and fundamental logical operations (conjunction, negation, existential generalization,etc..) on them. D −1 , with D 0 = def {<>}. Thus, we have that {f, t} = P(D 0 ) ⊆ P(D −1 ) , where by f and t we denote the empty set ∅ and set {<>} respectively. The intensional interpretation is a mapping between the set Ł of formulae of the FOL and intensional entities in D, I : Ł → D, is a kind of "conceptualization", such that an open-sentence (virtual predicate) φ(x 1 , ..., x k ) with a tuple of all free variables (x 1 , ..., x k ) is mapped into a k-ary concept, that is, an intensional entity u = I(φ(x 1 , ..., x k )) ∈ D k , and (closed) sentence ψ into a proposition (i.e., logic concept) v = I(ψ) ∈ D 0 with I(⊤) = T ruth ∈ D 0 for the FOL tautology ⊤ ∈ Ł (the falsity in the FOL is a logic formula ¬⊤ ∈ Ł). A language constant c is mapped into a particular a ∈ D −1 (intension of c) if it is a proper name, otherwise in a correspondent concept u in D I . Thus, in any application of intensional FOL, this intensional interpretation that determines the meaning (sense) of the knowledge expressed by logic formulae is uniquely determined (prefixed) (for example, by a grounding on robot's neuro system processes, explained in next section). However, the extensions of the concepts (with this prefixed meaning) vary from a context (possible world, expressed by an extensionalizzation function) to another context in a similar way as for different Tarski's interpretations of the FOL: Definition 4. EXTENSIONS AND EXTENSIONALIZATION FUNCTIONS: Let R = k∈N P(D k ) = k∈N P(D k ) be the set of all k-ary relations, where k ∈ N = {0, 1, 2, ...}. Notice that {f, t} = P(D 0 ) ⊆ R, that is, f, t ∈ R and hence the truth values are extensions in R. We define the function f <> : R → R, such that for any R ∈ R, f <> (R) = def {<>} if R = ∅; ∅ otherwise (1) The extensions of the intensional entities (concepts) are given by the set E of extensionalization functions h : D → D −1 + R, such that h = h −1 + h 0 + i≥1 h i : i≥−1 D i −→ D −1 + {f, t} + i≥1 P(D i ) (2) where h −1 : D −1 → D −1 for the particulars, while h 0 : D 0 → {f, t} = P(D 0 ) as- signs the truth values in {f, t} to all propositions with the constant assignment h 0 (T ruth) = t = {<>}, and for each i ≥ 1, h i : D i → P(D i ) assigns a relation to each concept. Consequently, intensions can be seen as names (labels) of atomic or composite concepts, while the extensions correspond to various rules that these concepts play in different worlds. The intensional entities for the same logic formula, for example x 2 + 3 = x 2 1 − 4, which can be denoted by φ(x 2 , x 1 ) or φ(x 1 , x 2 ), from above we need to differentiate their concepts by I(φ(x 2 , x 1 )) = I(φ(x 1 , x 2 )) because otherwise we would obtain erroneously that h(I(φ(x 2 , x 1 ))) = h(I(φ(x 1 , x 2 ))). Thus, in intensional logic the ordering in the tuple of variables x in a given open formula φ is very important, and explains why we introduced in FOL the virtual predicates in Definition 1. Definition 5. Let us define the extensional relational algebra for the FOL by, A R = (R, R = , {<>}, {⊲⊳ S } S∈P(N 2 ) , ∼, {π −n } n∈N ), where {<>} ∈ R is the algebraic value correspondent to the logic truth, R = is the binary relation for extensionally equal elements, with the following operators: 1. Binary operator ⊲⊳ S : R×R → R, such that for any two relations R 1 , R 2 ∈ R , the R 1 ⊲⊳ S R 2 is equal to the relation obtained by natural join of these two relations if S is a non empty set of pairs of joined columns of respective relations (where the first argument is the column index of the relation R 1 while the second argument is the column index of the joined column of the relation R 2 ); otherwise it is equal to the cartesian product R 1 × R 2 . 2. Unary operator ∼: R → R, such that for any k-ary (with k ≥ 1) relation R ∈ P(D k ) ⊂ R we have that ∼ (R) = D k \R ∈ P(D k ), where '\' is the substraction of relations. For u ∈ {f, t} = P(D 0 ) ⊆ R, ∼ (u) = D 0 \u. 3. Unary operator π −n : R → R, such that for any k-ary (with k ≥ 1) relation R ∈ P(D k ) ⊂ R we have that π −n (R) is equal to the relation obtained by elimination of the n-th column of the relation R if 1 ≤ n ≤ k and k ≥ 2; equal to, from (1), f <> (R) if n = k = 1; otherwise it is equal to R. We will use the symbol '=' for the extensional identity for relations in R. The intensional semantics of the logic language with the set of formulae Ł can be represented by the mapping = (x, y)) and T ruth = Ł −→ I D =⇒ h∈E R,I(⊤). 2. h(conj S (u, v)) = h(u) ⊲⊳ S h(v), where ⊲⊳ S is the natural join operation and conj S (u, v) ∈ D m where m = k + j − |S| if for every pair (i 1 , i 2 ) ∈ S it holds that 1 ≤ i 1 ≤ k, 1 ≤ i 2 ≤ j (otherwise conj S (u, v) ∈ D k+j ). 3. h(neg(u)) = ∼ (h(u)) = D k \(h(u)) (the complement of k-ary relation h(u) in D k ), if k ≥ 1, where neg(u) ∈ D k . For u 0 ∈ D 0 , h(neg(u 0 )) = ∼ (h(u 0 )) = D 0 \(h(u 0 )). 4. h(exists n (u)) = π −n (h(u)), where π −n is the projection operation which eliminates n-th column of a relation and exists n (u) ∈ D k−1 if 1 ≤ n ≤ k (otherwise exists n is the identity function). Notice that for u, v ∈ D 0 , so that h(u), h(v) ∈ {f, t}, h(neg(u)) = D 0 \(h(u)) = {<>}\(h(u)) ∈ {f, t}, and h(conj ∅ (u, v)) = h(u) ⊲⊳ ∅ h(v) ∈ {f, t}. We define a derived operation union : (P(D i )\∅) → D i , i ≥ 0, such that, for any B = {u 1 , ..., u n } ∈ P(D i ) and S = {(l, l) | 1 ≤ l ≤ i} we have that union({u 1 , ..., u n }) = u 1 , if n = 1 neg(conj S (neg(u 1 ), conj S (neg(u 2 ), ..., neg(u n ))...), otherwise (3) Than we obtain that for n ≥ 2: h(union(B)) = h(neg(conj S (neg(u 1 ), conj S (neg(u 2 ), ..., neg(u n ))...) = D i \((D i \h(u 1 )) ⊲⊳ S ... ⊲⊳ S (D i \h(u n ))) = D i \((D i \h(u 1 )) ... (D i \h(u n ))) = {h(u j ) | 1 ≤ j ≤ n}, that is, h(union(B)) = {h(u) | u ∈ B}(4) Note that it is valid also for the propositions in u 1 , u 2 ∈ D 0 , so that h(union(u 1 , 1. The logic formula φ(x i , x j , x k , x l , x m )∧ S ψ(x l , y i , x j , y j ) will be intensionally interpreted by the concept u 1 ∈ D 7 , obtained by the algebraic expression conj S (u, v) where u = I(φ(x i , x j , x k , x l , x m )) ∈ D 5 , v = I(ψ(x l , y i , x j , y j )) ∈ D 4 are the concepts of the virtual predicates φ, ψ, relatively, and S = {(4, 1), (2, 3)}. Consequently, we have that for any two formulae φ, ψ ∈ Ł and a particular operator conj S uniquely determined by tuples of free variables in these two formulae, I(φ ∧ S ψ) = conj S (I(φ), I(ψ)). 2. The logic formula ¬φ(x i , x j , x k , x l , x m ) will be intensionally interpreted by the concept u 1 ∈ D 5 , obtained by the algebraic expression neg(u) where u is the concept of the virtual predicate φ, u = I(φ(x i , x j , x k , x l , x m )) ∈ D 5 . Consequently, we have that for any formula φ ∈ Ł, I(¬φ) = neg(I(φ)). 3. The logic formula (∃ 3 )φ(x i , x j , x k , x l , x m ) will be intensionally interpreted by the concept u 1 ∈ D 4 , obtained by the algebraic expression exists 3 (u) where u = I(φ(x i , x j , x k , x l , x m )) ∈ D 5 is the concept of the virtual predicate φ. Consequently, we have that for any formula φ ∈ Ł and a particular operator exists n uniquely determined by the position of the existentially quantified variable in the tuple of free variables in φ (otherwise n = 0 if this quantified variable is not a free variable in φ), I((∃ n )φ) = exists n (I(φ)). u 2 )) = h(u 1 ) h(n 2 ) ∈ {f, t} where f is empty set ∅ while t is So, we obtain the following two-steps interpretation of FOL based on two homomorphisms, intensional I, and extensional h: A int (concepts/meaning) intensional interpret. I ✒ F rege/Russell semantics ❅ ❅ ❅ ❅ h (extensionalization) | A F OL (syntax) A R (denotation)(5) We can enrich the expressivity of such a minimal FOL intensionality by new modal operators, or in different way provided in what follows. As, for example, in Bealer's intensional FOL, where he introduced the intensional abstraction operator, which will be considered in rest of this section, as a significant enrichment of the intensional FOL considered above. In reflective languages, reification data is causally connected to the related reified aspect such that a modification to one of them affects the other. Therefore, the reification data is always a faithful representation of the related reified aspect. Reification data is often said to be made a first class object. In programming language design, a first-class citizen (also type, object, entity, or value) in a given programming language is an entity which supports all the operations generally available to other entities. These operations typically include being passed as an argument, returned from a function, modified, and assigned to a variable. The concept of first and second-class objects was introduced by Christopher Strachey in the 1960s when he contrasted real numbers (first-class) and procedures (second-class) in ALGOL. In FOL we have the variables as arguments inside the predicates, and terms which can be assigned to variables are first-class objects while the predicates are the secondclass objects. When we transform a virtual predicate into a term, by using intensional abstraction operator, we transform a logic formula into the first class object to be used inside another predicates as first-class objects. Thus, abstracted terms in the intensional FOL are just such abstracted terms as reification of logic formulae. For example, the sentence "Marco thinks that Zoran runs", expressed by thinks(Marco, ⋖runs(Zoran)⋗) by using binary predicate thinks and unary predicate runs where the ground atom runs(Zoran) is reified into the predicate thinks. If φ(x) is a formula (virtual predicate) with a list (a tuple) of free variables in x = (x 1 , ..., x n ) (with ordering from-left-to-right of their appearance in φ), and α is its subset of distinct variables, then ⋖φ(x)⋗ β α is a term, where β is the remaining set of free variables in x. The externally quantifiable variables are the free variables not in α. When n = 0, ⋖φ⋗ is a term which denotes a proposition, for n ≥ 1 it denotes a n-ary concept. Definition 7. INTENSIONAL ABSTRACTION CONVENTION: From the fact that we can use any permutation of the variables in a given virtual predicate, we introduce the convention that ⋖ φ(x) ⋗ β α is a term obtained f rom virtual predicate φ(x) (6) if α is not empty such that α β is the set of all variables in the list (tuple of variables) x = (x 1 , ..., x n ) of the virtual predicate (an open logic formula) φ, and α β = ∅, so that |α| + |β| = |x| = n. Only the variables in β (which are the only free variables of this term), can be quantified. If β is empty then ⋖φ(x)⋗ α is a ground term. If φ is a sentence and hence both α and β are empty, we write simply ⋖φ⋗ for this ground term. More about this general definition of abstract terms can be find in [1]. In this paper we will use the most simple cases of ground terms ⋖φ⋗, where φ is a sentence. Case: Human Robot Spatial Language Interaction Let us consider a model of robot for understanding language about space and movement in realistic situations [11,12], as finding video clips that match a spatial language description such as "People walking through the kitchen and then going to the dining room" and following natural language commands such as "Go down the hall towards the fireplace in the living room." Video retrieval is a compelling application: in the United States alone, there are an estimated 35 million surveillance cameras installed, which record four billion hours of video per week. Analyzing and understanding the content of video data remains a challenging problem. A spatial language interface to video data can help people naturally and flexibly find what they are looking for in video collections. Studying language used to give directions could enable a robot to understand natural language directions. People talk to robots even if they do not have microphones installed, and it makes sense to build systems that understand what they say. A robot that understands natural language is easy for anyone to use without special training. By using the deductive properties of the IFOL, the robot can make logic deductions as well about the facts that it visually recognized and also to obtain its own autoepistemic deductions about obtained knowledge, as shortly explained in introduction, by using intensional abstractions in Definition 7. Consequently, I will focus on a narrow subset of a natural language, grounding that language in data collected from a real world. This strategy has two benefits. First, it decreases the scope of the language understanding problem, making it more tractable. Second, by choosing a semantically deep core domain, it offers an opportunity to explore the connection between linguistic and non-linguistic concepts. The linguistic structure extracted from spatial language expressions and many of the features in the model for spatial relations are based on the theories of Jackendoff [6], Landau and Jackendoff [13] and Talmy [14]. For example, the implementation of the mining of "across" in [14] is obtained by an algorithm (of robot's AI neuro-system) for computing the axes a figure imposes on a ground, and set of features which quantify "roughly perpendicular", using a machine learning algorithm to fine-tune the distinctions by training on labeled data. Regier [15] built a system that assigns labels such as "through" to move showing a figure relative to a ground object. Bailey [16] developed a model for learning the meanings of verbs of manipulation such as "push" and "shove". Kelleher and Costello [17] built models for the meanings of static spatial prepositions such as "in front of" and "above". Siskind [18] created a system for defining meanings for words such as "up" and "down." The framework reasons about formal temporal relations between primitive force-dynamic properties such as "supports" and "touches" and uses changes in these properties to define meanings for verbs. His framework focuses on word-level event recognition and features, etc.. Reasoning about movement and space is a fundamental competence of humans and many animals. Humans use spatial language to tell stories and give directions, abstracting away the details of a complex event into a few words such as "across the kitchen." A system that understands spatial language could be directly useful to people by finding video that matches spatial language descriptions, or giving natural language directions. We will consider a robot which retrieves video clips that match a natural language description using a probabilistic graphical model that maps between natural language and paths in the environment [11]. In this particular environment, spatial relations are modeled as probabilistic distributions for recognizing words paired with scenes. The distributions are trained from labeled examples using a set of geometric features that capture the semantics of spatial prepositions. The distribution modeled is the probability of a particular spatial relation given a trajectory and an object in the environment. This distribution corresponds to the probability that a spatial relation such as "across" or "to" describes a particular trajectory and landmark. The input to the model is the geometry of the path and landmark object; the output is a probability that the spatial relation can be used to describe this scene. These distributions are trained using labeled path examples, and in robot's brain correspond to its AI neuro system. The system learns distributions for spatial relations, for example, by using a naive Bayes probabilistic model. So, now we can focus to the integration of such robot's AI neuro system with its AI symbolic system based on three natural language cognitive levels: The syntax of a particular natural language (French, English, etc..) its semantic logic structure (transformation of parts of the language sentences into the logic predicates and definition of corresponding FOL formulae) and its corresponding conceptual structure, which dif-ferently from the semantic layer that represents only the logic's semantics, represents the composed meaning of FOL formulae. In this example, we focus on spatial language search of people's motion trajectories which are automatically extracted from video recorded by stationary overhead cameras. The system takes as input a natural language query, a database of surveillance video from a particular environment and the locations of non-moving objects in the environment. When the robot performs video retrieval by its AI neuro system, clips are returned in order according to the joint probability of the query and the clip. Thus, for each video clip in given database, this robot's neuro system computes the probability that considered clip satisfies a natural language query, parsed into logic FOL formula (second natural language semantic level) and consequently into intensional algebra A int term with intensional concepts which labels are grounded by robot's neuro system processes (algorithms). Let N Ł be a given natural language. If we denote the set of finite nonempty lists of a given natural language words by N Ł list , then this parsing can be represented by a partial mapping pars : N Ł list → Ł (7) where Ł is the set of logic formulae of intensional FOL. We suppose that the concepts in the conceptual structure expressed by the intensional algebra A int of atomic concepts u ∈ D, and their corresponding logic atoms expressed by virtual predicates φ(x) ∈ Ł of FOL are the part of innate robot's knowledge, such that for robot's innate and unique intensional interpretation I : Ł → D, u = I(φ(x)). Moreover, we suppose that robot has a parser capability to transform the sentences of particular natural language into the formulae of FOL with innate set of the atoms expressed by virtual predicates. In this example we consider the predicates of IFOL as the verbs (V) of natural language, as follows F ind(x 1 , x 2 , x 3 ) where the time-variable x 1 (with values "in past", "in present", "in future") indicates the time of execution of this action, the variable x 2 is used for the object given to robot (in this case a video clip) and x 3 for the statement (users query) that has to be satisfied by this object, and virtual predicate W alk(x 1 , x 2 , x 3 , x 4 , x 5 ) where the time-variable x 1 (with values "in past", "in present", "in future") indicates the time of execution of this action, variable x 2 for the figure (F) that moves ("person", "cat", etc..), x 3 for the initial position of walking figure (defined by the spatial relation (SR) "from", for example "from the table") , x 4 for the intermediate positions during movement of the figure (defined by (SR) "through", for example "through the corridor") , and x 5 for the final position of figure (defined by (SR) "to", for example "to the door"). The robot takes as input a natural language query, a database of surveillance video from a particular environment and the locations of non-moving objects in the environment. It parses the query into a semantic structure called a spatial description clause (SDC) [12]. An SDC consists of a figure (F), a verb (V), a spatial relation (SR), and a landmark (L). The system extracts SDCs automatically using a conditional random field chunker. Let us consider the example illustrated in Figure 3 in [12] of a natural language query nq ∈ N Ł list , defined by a sentence: Remark: Note that all SDC components different from (V), are particulars in D −1 in PRP domain D, provided by Definition 3. The sense (mining) of the components (F) and (L) are grounded by the machine-learning video-recognition processes of the robot, that is by its neuro systems. The sense of the (SR) components is grounded by the meaning of the spatial relations, provided by different authors methods, mentioned previously, and implemented by particular robots processes. What we need in next is to extend this grounding also to the virtual predicates of the FOL open formulae in Ł. " Consequently, from these Spatial Description clauses, for the (V) of the past-time verb (V) "to walk", the semantic logic structure recognized by robot is the sentence φ ∈ Ł based on the virtual predicate toW alk, W alk(in past, person, f rom the couches in the room, N U LL, to the dining room table) that is, from (7), φ = pars(nq) Note that the inverse parsing of such logic sentence φ to natural language sentence is directly obtained, so that the robot can translate its semantic logic structures into natural language to communicate by voice to the people. We consider that each grammatically plural word name "videoclips", robot can define by generalization by creating the virtual unary predicate videoclips(y), such that its intensional concept u 2 = I(videoclips(y)) ∈ D 1 in PRP domain, whose meaning is grounded by robots patern-recognition process fixed by a machine learning method. In a similar way, each unary concept of visual objects can be created by robot by a machine learning method for enough big set of this type of objects. So, each grammatically singular word name, like "John's videoclip" is a particular (element of D −1 ) in PRP domain, whose meaning is grounded by the internal robot's image of this particular videoclip, recognized as such by robots patern-recognition process. Thus, for a given extensionalization function h in (2), and fixed robot's intensional mapping I, from the diagram (5), we obtain that the set C, of video clips in a given database of videoclips presented to this robot, is equal to C = h(I(videoclips(y)))(9) Consequently, the human command in natural language nc ∈ N Ł list to this robot, "Find videoclip such that φ in the given set of videoclips" (where φ has to be substituted by the sentence above) is parsed by robot into its second level (semantic logic structure) by virtual predicate F ind of the verb "to find" (in present) and a variable y of type "videoclip" (objects of research) and substituting "that φ" by abstracted term ⋖φ⋗, and by substituting "in the given set of" with the logic conjunction connective ∧ S of the IFOL expressed by the following formula ϕ(y) F ind(inpresent, y, ⋖φ⋗) ∧ S videoclips(y) where S = (2, 1) for joined variables in two virtual predicates. That is, from (7), ϕ(y) = pars(nc)(11) The meaning of the unary concept u 1 = I(F ind(in present, y, ⋖φ⋗)), corresponding to the natural language subexpression "Find videoclip such that φ" of the command above, is represented by its AI neuro system process of probabilistic recognition of video clips [12] satisfying the natural language query φ (In fact, u 2 is just equal to the name of this process of probabilistic recognition). However, during execution of this process, the robot is able also to logically deduce the truth of the sentence Know(in present, me, ⋖toF ind(inpresent, y, ⋖φ⋗)⋗ y ) of the virtual predicate Know(x 1 , x 2 , x 3 ), where the time-variable x 1 (with values "in past", "in present", "in future") indicates the time of execution of this action, the variable x 2 is used for the subject of this knowledge and x 3 is used for an abstracted term expression this particular knowledge). Thus, by using deductive properties of the true sentences of FOL, this autoepistemic sentence about its state of selfknowledge, the robot would be able to comunicate to humans this sentence, traduces in natural language as "I (me) know that I am finding videoclip such that φ" From the fact that robot defined the type of the variable y to be "videoclip", by traduction of the FOL deduced formula above into the natural language, this variable will be traduced in natural language by "videoclip". In the same way, during the execution of the human command above, expressed by the FOL formula ϕ(y) in (10), with composed concept u 3 = I(ϕ(y)) ∈ D 1 , that is, by using the homomorphic property of intensional interpretation I, u 3 = u 1 ⊲⊳ S u 2(12) the robot can deduce also the true epistemic sentence Know(in present, me, ⋖F ind(in present, y, ⋖φ⋗) ∧ S videoclips(y)⋗ y ) and hence the robot would be able to communicate to humans this sentence, traduces in natural language as "I (me) know that I am finding videoclip such that φ in the set of videoclips" Note that the subset of videoclips extracted by robot from a given set of videoclips C = h(u 2 ) in (9), defines the current extensionalization function h, in the way that this subset is E = h(u 3 ) = h(u 1 ) ⊲⊳ S h(u 2 ) = h(u 1 ) ⊲⊳ S C = h(u 1 ) ⊆ C(13) Thus, for the grounding of spatial language for video search, the robot's internal knowledge structure is divided into four levels, in ordering: natural language, semantic logic structure, conceptual structure and neuro structure, as represented by It is easy to see that the conceptual system, based on PRP domain D composed by particulars in D −1 and universals (concepts) in D I = D 0 + D 1 + D 2 + ... of the IFOL, is the level of grounding of the natural language of the robot to its neuro system composed by the following processes: Conclusion Computation is defined purely formally or syntactically, whereas minds have actual mental or semantic contents, and we cannot get from syntactical to the semantic just by having the syntactical operations and nothing else. . . Machine learning is a sub-field of artificial intelligence. Classical (non-deep) machine learning models require more human intervention to segment data into categories (i.e. through feature learning). Deep learning is also a sub-field of machine learning, which attempts to imitate the interconnectedness of the human brain using neural networks. Its artificial neural networks are made up layers of models, which identify patterns within a given dataset. Deep learning can handle complex problems well, like speech recognition, pattern recognition, image recognition, contextual recommendations, fact checking, etc.. However, with this integrated four-level robot's knowledge system presented in diagram (14), where the last level represents the robot's neuro system containing the deep learning as well, we obtain that also the semantic theory of robot's intensional FOL is a procedural one, according to which sense is an abstract, pre-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure. Weak AI, also known as narrow AI, focuses on performing a specific task, such as answering questions based on user input or playing chess. It can perform one type of task, but not both, whereas Strong AI can perform a variety of functions, eventually teaching itself to solve for new problems. Weak AI relies on human interference to define the parameters of its learning algorithms and to provide the relevant training data to ensure accuracy. Strong AI (also known as full AI) aims to create intelligent robots that are quasi indistinguishable from the human mind. But just like a child, the AI machine would have to learn through input and experiences, constantly progressing and advancing its abilities over time. If researchers are able to develop Strong AI, the robot would require an intelligence more close to human's intelligence; it would have a self-aware consciousness that has the ability to solve problems, learn, and plan for the future. However, since humans cannot even properly define what intelligence is, it is very difficult to give a clear criterion as to what would count as a success in the development of strong artificial intelligence. Thus, we argue that this example, used for the spatial natural sublanguage, can be extended in a similar way to cover more completely the rest of human natural language, and hence the method provided by this paper is a main theoretical and philosophical contribution to resolve the open problem of how we can implement the deductive power based on IFOL for new models of robots heaving strong AI capacities. Definition 3 . 3INTENSIONAL LOGIC PRP DOMAIN D: In intensionl logic the concepts (properties, relations and propositions) are denotations for open and closed logic sentences, thus elements of the structured domain D = D −1 + D I , (here + is a disjoint union) where -A subdomain D −1 is made of particulars (individuals). -The rest D I = D 0 + D 1 ...+ D n ... is made of universals (concepts) 4 : D 0 for propositions with a distinct concept T ruth ∈ D 0 , D 1 for properties (unary concepts) and D n , n ≥ 2, for n-ary concept. The concepts in D I are denoted by u, v, ..., while the values (individuals) in D −1 by a, b, ... The empty tuple <> of the nullary relation r ∅ (i.e. the unique tuple of 0-ary relation) is an individual in where −→ I is a fixed intensional interpretation I : Ł → D with image im(I) ⊂ D, and =⇒ h∈E is the set of all extensionalization functions h : im(I) → D −1 + R in E.So, we can define only the minimal intensional algebra (with minimal number of operators) A int of concepts, able to support the homomorphic extensionh : A int → A R of the extensionalization function h : D → D −1 + R.Definition 6. BASIC INTENSIONAL FOL ALGEBRA: Intensional FOL algebra is a structure A int = (D, Id, T ruth, {conj S } S∈P(N 2 ) , neg, {exists n } n∈N ), with binary operations conj S : D I × D I → D I , unary operation neg : D I → D I , and unary operations exists n : D I → D I , such that for any extensionalization function h ∈ E, and u ∈ D k , v ∈ D j , k, j ≥ 0, 1. h(Id) = R = and h(T ruth) = {<>}, for Id = I( . a singleton set {<>} with empty tuple <>, and hence the join {<>} ⊲⊳ ∅ = ∅ and {<>} ⊲⊳ {<>} = {<>}.Thus, we define the following homomorphic extensionI : A F OL → A intof the intensional interpretation I : Ł → D for the formulae in syntax algebra A F OL from Definition 2: The person walked from the couches in the room to the dining room table" which is composed by two SDC with the D 1 + 1the following diagram (only two continuous arrows (intensional mapping I : Ł → D I where D I = D 0 + D 1 + ... are the universals in PRP domain theory) represent the total mappings, while other (dots) are partial mappings) N Ł list ...................................... particulars ✲ D −1 .................................. grounding ✲ P R processes . . . . . . . . . . . . . . pars | + Ł sentences I ✲ D 0 ..................................... D 2 + ... ........................ .Lang. Log.semantic sys. Conceptual sys. N euro sys. In the early 1970s, at the MIT Artificial Intelligence Lab, Minsky and Papert started developing what came to be known as the Society of Mind theory.The theory attempts to explain how what we call intelligence could be a product of the interaction of non-intelligent parts. Minsky says that the biggest source of ideas about the theory came from his work in trying to create a machine that uses a robotic arm, a video camera, and a computer to build with children's blocks. In 1986, Minsky published The Society of Mind, a comprehensive book on the theory which, unlike most of his previously published work, was written for the general public. In November 2006, Minsky published The Emotion Machine, a book that critiques many popu- lar theories of how human minds work and suggests alternative theories, often replacing simple ideas with more complex ones. Self in a sense which implies that all our activities are controlled by powerful creatures inside ourselves, who do our thinking and feeling for us. In what follows we will define also a language of concepts with intensional connectives defined as operators of the intensional algebra Aint in Definition 6, so that DI is the set of terms of this intensional algebra. PR (Pattern Recognition) processes of recognition of the particulars. For example, for SDC components (F) "the person. L) "the couches in the room" and "the dining room tablePR (Pattern Recognition) processes of recognition of the particulars. For example, for SDC components (F) "the person", (L) "the couches in the room" and "the dining room table", etc.. Spatial Description Clauses) parser used for the sentences, for example, for a natural language query nq ∈ N Ł list that is, logical proposition (sentece) φ = pars(nq) ∈ Ł in (8), which is labeled by its intensional proposition label I(φ) ∈ D 0 . Thus, the grounding of nq is obtained by linking its intensional proposition I(pars(nq)) in PRP to the SDC parser process. 2. SDCpart of robot's neuro system2. SDC (Spatial Description Clauses) parser used for the sentences, for example, for a natural language query nq ∈ N Ł list that is, logical proposition (sentece) φ = pars(nq) ∈ Ł in (8), which is labeled by its intensional proposition label I(φ) ∈ D 0 . Thus, the grounding of nq is obtained by linking its intensional proposition I(pars(nq)) in PRP to the SDC parser process (part of robot's neuro system). Machine Learning) processes, like that used for the recognition of different types of classes (like the set of videoclips). For example, for the language plural world "videoclips" in N Ł list , such that pars("videoclips") = videoclips(y) ∈ Ł with its intensional unary concept u 2 = I(videoclips(y)) ∈ D 1 which is grounded to robot's ML process for the "videoclips. ML. ML (Machine Learning) processes, like that used for the recognition of different types of classes (like the set of videoclips). For example, for the language plural world "videoclips" in N Ł list , such that pars("videoclips") = videoclips(y) ∈ Ł with its intensional unary concept u 2 = I(videoclips(y)) ∈ D 1 which is grounded to robot's ML process for the "videoclips". the ordinary component of the natuaral language grounding developed by robot's neuro system, the two lines bellow is the new robots knowledge structure of the added symbolic AI system based on the Intensional First Order Logic and its grounding to robot's processes (its neuro AI system). Note that, while the top line in the diagram (14) is. by which the robot is able to provide logic deductive operations and autoepistemic selfNote that, while the top line in the diagram (14) is the ordinary component of the natu- aral language grounding developed by robot's neuro system, the two lines bellow is the new robots knowledge structure of the added symbolic AI system based on the Inten- sional First Order Logic and its grounding to robot's processes (its neuro AI system), by which the robot is able to provide logic deductive operations and autoepistemic self- Intensional First Order Logic: from AI to New SQL Big Data. Z Majkić, 978-3-11-099494-0Walter De Gruyter GmbHBerlin/BostonZ. Majkić, "Intensional First Order Logic: from AI to New SQL Big Data," Walter De Gruyter GmbH, Berlin/Boston, ISBN 978-3-11-099494-0, 2022. Quality and concept. G Bealer, Oxford University PressUSAG.Bealer, "Quality and concept," Oxford University Press, USA, 1982. Conservative intensional extension of Tarski's semantics. Z Majkić, Advances in Artificial Intelligence. Hindawi Publishing CorporationZ.Majkić, "Conservative intensional extension of Tarski's semantics," Advances in Artifi- cial Intelligence, Hindawi Publishing Corporation, ISSN: 1687-7470, 23 October, pp. 1-17, 2012. Intensionality and two-steps interpretations. Z Majkić, arXiv:1103.0967Z.Majkić, "Intensionality and two-steps interpretations," arXiv: 1103.0967, 04 March, pp. 1-15, 2011. Intensional RDB manifesto: a unifying NewSQL model for flexible Big Data. Z Majkić, arXiv:1403.0017Z.Majkić, "Intensional RDB manifesto: a unifying NewSQL model for flexible Big Data," arXiv: 1403.0017, 28 February, pp. 1-29, 2014. Semantics and Cognition. R Jackendoff, The MIT Press CambridgeR.Jackendoff, "Semantics and Cognition," The MIT Press Cambridge, 1983. Thinking fast and slow. D Kahneman, 978-0374275631Farrar, Straus and Giroux. D.Kahneman, "Thinking fast and slow," Farrar, Straus and Giroux, ISBN 978-0374275631, 2011. Zeitschrift für Philosophie und Philosophische Kritik. G Frege, Ü ber Sinn und Bedeutung. G.Frege, "Ü ber Sinn und Bedeutung," Zeitschrift für Philosophie und Philosophische Kri- tik, pp. 22-50, 1892. Universals. G Bealer, The Journal of Philosophy. 90G.Bealer, "Universals," The Journal of Philosophy, vol. 90, pp. 5-32, 1993. Theories of properties, relations, and propositions. G Bealer, The Journal of Philosophy. 76G.Bealer, "Theories of properties, relations, and propositions," The Journal of Philosophy, vol. 76, pp. 634-648, 1979. Toward understanding natural language directions. T Kollar, S Tellex, D Roy, N Roy, Proceedings of the 4th ACM international conference on human robot interaction. the 4th ACM international conference on human robot interactionT.Kollar, S.Tellex, D.Roy, and N.Roy, "Toward understanding natural language directions," In Proceedings of the 4th ACM international conference on human robot interaction, 2010. Grounding spatial language for video search. S Tellex, T Kollar, G Show, N Roy, D Roy, ICMI-MLMI. 10S.Tellex, T.Kollar, G.Show, N.Roy, and D.Roy, "Grounding spatial language for video search," ICMI-MLMI 10, November 8-12, Beijing, China, 2010. What' and 'where' in spatial language and spatial cognition. B Landau, R Jackendoff, Behavioral and Brain Sciences. 16B.Landau and R.Jackendoff, "'What' and 'where' in spatial language and spatial cognition," Behavioral and Brain Sciences, 16, pp. 217-265, 1993. From Perception to Meaning: Immage Schemas in Cognitive Linguistics. L Talmy, Mouton de GruyterThe fundamental system of spatial schemas in languageL.Talmy, "The fundamental system of spatial schemas in language," From Perception to Meaning: Immage Schemas in Cognitive Linguistics, Mouton de Gruyter, 2005. The acquisition of lexical semantics for spatial terms: A connectionist model of perceptual categorization. T P Regier, University of California at BerkeleyPhD thesisT.P.Regier, "The acquisition of lexical semantics for spatial terms: A connectionist model of perceptual categorization," PhD thesis, University of California at Berkeley, 1992. When push comes to shove: A computational model of the role of motor control in the acquisition of action verbs. D Bailey, PhD thesisD.Bailey, "When push comes to shove: A computational model of the role of motor control in the acquisition of action verbs," PhD thesis, 1997. Applying computational models of spatial prepositions to visually situated dialog. J D Kelleher, F J Costello, Computational Linguistics. 352J.D.Kelleher and F.J.Costello, "Applying computational models of spatial prepositions to visually situated dialog," Computational Linguistics, 35(2), pp. 271-306, 2009. Grounding the lexical semantics of verbs in visual perception using force dynamics and event logic. J M Siskind, J. Artif. Int. Res. 151J.M.Siskind, "Grounding the lexical semantics of verbs in visual perception using force dynamics and event logic," J. Artif. Int. Res., 15(1), pp. 31-90, 2001.
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[ "epl draft Relations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets", "epl draft Relations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets" ]
[ "Laure Goudard \nMAG (IPGP & ENS)\nCNRS UMR7154\nLRA Département de Physique\n24 rue Lhomond75005ParisFrance\n", "Emmanuel Dormy \nMAG (IPGP & ENS)\nCNRS UMR7154\nLRA Département de Physique\n24 rue Lhomond75005ParisFrance\n" ]
[ "MAG (IPGP & ENS)\nCNRS UMR7154\nLRA Département de Physique\n24 rue Lhomond75005ParisFrance", "MAG (IPGP & ENS)\nCNRS UMR7154\nLRA Département de Physique\n24 rue Lhomond75005ParisFrance" ]
[]
PACS 91.25.Cw -Origins and models of the magnetic field; dynamo theories PACS 47.65.-d -MagnetohydrodynamicsAbstract. -The geo and solar magnetic fields have long been thought to be very different objects both in terms of spatial structure and temporal behavior. The recently discovered field structure of a fully convective star is more reminiscent of planetary magnetic fields than the Sun's magnetic field [1], despite the fact that the physical and chemical properties of these objects clearly differ. This observation suggests that a simple controlling parameter could be responsible for these different behaviors. We report here the results of three-dimensional simulations which show that varying the aspect ratio of the active dynamo region can yield sharp transition from Earth-like steady dynamos to Sun-like dynamo waves.
10.1209/0295-5075/83/59001
[ "https://arxiv.org/pdf/0901.0828v1.pdf" ]
17,507,549
0901.0828
e140e229af02ca9351d5c540d4bfbf0558ce6f17
epl draft Relations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets 7 Jan 2009 Laure Goudard MAG (IPGP & ENS) CNRS UMR7154 LRA Département de Physique 24 rue Lhomond75005ParisFrance Emmanuel Dormy MAG (IPGP & ENS) CNRS UMR7154 LRA Département de Physique 24 rue Lhomond75005ParisFrance epl draft Relations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets 7 Jan 2009 PACS 91.25.Cw -Origins and models of the magnetic field; dynamo theories PACS 47.65.-d -MagnetohydrodynamicsAbstract. -The geo and solar magnetic fields have long been thought to be very different objects both in terms of spatial structure and temporal behavior. The recently discovered field structure of a fully convective star is more reminiscent of planetary magnetic fields than the Sun's magnetic field [1], despite the fact that the physical and chemical properties of these objects clearly differ. This observation suggests that a simple controlling parameter could be responsible for these different behaviors. We report here the results of three-dimensional simulations which show that varying the aspect ratio of the active dynamo region can yield sharp transition from Earth-like steady dynamos to Sun-like dynamo waves. Introduction. -Observations of the magnetic fields due to dynamo activity appear to fall into two categories: fields dominated by large-scale dipoles (such as the Earth and a fully convective star), and fields whith smaller-scale and non-axisymmetric structures (such as the Sun). Moreover two kinds of different temporal behaviour have been identified so far: very irregular polarity reversals (as in the Earth), and quasi-periodic reversals (as in the Sun). Since the Earth and the Sun provide the largest database of magnetic field observations, these objects have been well studied and described in terms of alternative physical mechanisms: the geodynamo involves a steady branch of the dynamo equations, perturbed by strong fluctuations that can trigger polarity reversals, whereas the solar dynamo takes the form of a propagating dynamo wave. The signature of this wave at the Sun's surface yields the well-known butterfly-diagram (Sunspots preferentially emerge at a latitude that is decreasing with time during the solar cycle). Modelling. -Because of their very different natures (liquid metal in one case, plasma in the other), planetary and stellar magnetic fields are studied by different communities. Non-dimensional numbers controling the dynamics of the Earth and the Sun, for example, do significantly differ (see [2,3]). As a practical matter however, the techniques as well as the typical parameters used in numerical studies of these two systems are surprisingly similar. To some extent this is due to the restricted parameter space avail-able to present day computations. The parameter regime numerically accessible is rather remote from the actual objects. For planetary dynamos the main discrepancy relies in the rapid rotation in the momentum equation (characterized by the Ekman number), whilst for stellar dynamos it relies in solving the induction equation with weak resistive effects (characterized by high values of the magnetic Reynolds number). Yet within this restricted domain, the sharply different key characters to both geo [4] and solar [5,6] magnetic fields have been reproduced. This leads us to argue that the important parameter controlling the magnetic field behaviour is the aspect ratio of the dynamo region (i.e. the radius ratio of the inner bounding sphere to the outer bounding sphere). Indeed, in the Earth, the inert solid inner core extends to less than 40% of the core radius, whereas in the Sun, the radiative zone fills 70% of the solar radius. One expects the convective zones of stars and planets to have all possible intermediate aspect ratios, even extending to fully convective spheres. In order to isolate and understand this purely geometrical effect, we have carried out three-dimensional numerical simulations of self-excited convective dynamos in which the domain aspect ratio was slowly varied, with all other parameters held constant. The governing equations as well as parameter regimes used here were originally introduced for a geodynamo reference calculation [7]. The only distinction being the use of stress-free boundary conditions on the outer sphere of the domain, while imposing no-slip boundary conditions at the bottom of the convective re- gion. This choice was made in order to create a strong shear at the base of the model, and thus try to mimic the solar tachocline [8]. The inner sphere is here assumed to be insulating, and we use differential heating. The governing equations are in non-dimensional form: E [∂ t u + (u · ∇)u] = −∇π + E ∆u − 2e z × u + Ra r θ + Pm −1 (∇ × B) × B ,(1) ∂ t B = ∇ × (u × B) + Pm −1 ∆B ,(2)∂ t θ + (u · ∇)(θ + T s ) = Pr −1 ∆θ ,(3)∇ · u = ∇ · B = 0 ,(4) where E = ν ΩD 2 , Ra = αg∆T D νΩ , Pr = ν κ , Pm = ν η .(5) All simulations reported here were performed keeping the following parameters constant E = 10 −3 , Ra = 100, Pr = 1, Pm = 5 . The above system is integrated in threedimensions of space (3D) using the Parody code [9]. When the inner (non dynamo generating) body occupies less than about 60% of the convective body in radius, the flow generates a dipolar field, very similar to that of the Earth. It features patches of intense flux at high latitudes and some reversed patches at low latitude, similar to the ones revealed by a downward continuation of the Earth's field to the Core-Mantle boundary [10]. This strongly dipolar solution becomes unstable with a further increase of the aspect ratio. For an aspect ratio of 0.65 -close to that of the Sun-the strong dipole is first maintained and then strongly weakens, but dynamo action continues in a different form: that of a wavy solution with quasi-periodic reversals ( Fig. 1), reminiscent of some aspects of the solar magnetic field behavior. Drifting features can be observed both on the radial field at the surface of the model ( Fig. 1 & 2b) and on the azimuthal (east-west) field below the surface of the model (Fig. 2c). Due to the complex nature of these fully tri-dimensional simulations, many waves can co-exist. Some of the dominant structures appear to propagate toward the equator; others propagate poleward. Reversed waves are also observed at the surface of the Sun at higher latitudes [11]. Let us stress however that the model cannot be expected to capture all the features either of the geo or solar magnetic fields. In particular due to the parameters regime and the lack of stratification in our modelling. Physical interpretation. -In order to investigate the physical mechanisms associated with these waves, we have performed some kinematic simulations. During the course of the simulation the Lorentz force was suppressed. The wavy nature of the dynamo field was unaltered by this modification. This rules out the possibility of an interpretation in terms of pure Alfvèn waves or Alfvèn waves modified by rotation (so called MC or MAC waves), which both require the back-reaction of the Lorentz force. Of course, suppressing the Lorentz force is not without consequences: the flow slowly evolves to a different purely hydrodynamical state, and the magnetic field now grows exponentially, but both of these effects are sufficiently slow for the wavelike character to persist over many wave periods. Two other interpretations for the nature of these waves remain possible: either hydrodynamic fluctuations (e.g. inertial waves or Rossby waves) or dynamo waves, as expected on the Sun. These possibilities were tested by comparing oscillations in the velocity field and in the magnetic field in the kinematic simulations. We found that a high frequency signal is present both in the flow and in the magnetic field. This demonstrates the presence of hydrodynamic waves, which induce magnetic fluctuations. The lower frequency signal is however absent in the flow. This provides a proof of their "dynamo wave" nature. We have numerically observed such dynamo waves for aspects ratio up to 0.8. For the parameters investigated here, the transition from a dynamo dominated by a fluctuating dipole to a dynamo wave occurs for an aspect ratio close to 0.65. This transition exhibits hysteresis: once a dynamo-wave solution is present, the aspect ratio can be reduced again down to 0.6, while maintaining this dynamo mode. Connections with parameterized models. -Butterfly diagrams indicative of the solar cycle are usually produced using simplified parameterized models or "mean field" models. These models require a prescription of the turbulent induction, the so-called "α-effect" (which can also be introduced in terms of deviation from axisymmetry [12]). We should stress that this is a valid approximation only if certain conditions are satisfied (e.g. [13]). Such butterfly-like diagrams are generally not produced by direct three-dimensional modelling, with the notable exception (only in the reverse direction) of the pioneering work of Gilman and Glatzmaier [5,14]. Because of the strong symmetry of the convective flows influenced by the rapid rotation of the planet or the star, it is well known that two independent families of solutions exist, namely with dipole symmetry (antisymmetric with respect to the equator) and quadrupole symmetry (symmetric with respect to the equator). Both families of solutions are often described in reduced parameterised models [15,16], and we have observed these two families in our fully 3D simulations (Fig. 3). Both branches are stable in our simulations for long periods of time, but can also be destabilised to yield a change of symmetry. In fact, despite the relative complexity of our model, the temporal behavior of both symmetries is clearly reminiscent of kinematic studies of earlier reduced models (Fig. 4c,d and [15]). The simpler meanfield equations for the axisymmetric field are obtained by writing the flow and field as u = s ω e φ , B = B p +B e φ = ∇×(A e φ )+B e φ ,(6) i.e. assuming a mean flow in the form of a zonal shear only. In the isotropic case, the axisymmetric part of (2) yields (e.g. [3]) ∂A ∂t = αB + Rm −1 D 2 A ,(7)∂B ∂t = s B p · ∇ω + (∇ × αB p ) · e φ + Rm −1 D 2 B ,(8) where s denotes the cylindrical radius and D 2 = ∆ − 1/s 2 (note that Pm in (2) is here changed to Rm as the flow is now assumed to be given). For an instability of (7-8) to exist, these equations must not decouple (this is the essence of Cowling's anti-dynamo theorem [17]). Equation (8) involves A through two terms. Reduced models have been classified in two categories depending on the dominant term. The first term on the RHS of (8) involves the zonal shear and is referred to as the Ωeffect. The second term in the RHS of (8), as well as the first term on the RHS of (7), involve mean induction from non-axisymmetric features in the flow and are referred to as the α-effect. Dropping the α-effect term in (8) and writing the resulting equations in a simplified cartesian geometry yields ∂A ∂t = αB + Rm −1 ∆A , ∂B ∂t = G ∂A ∂x + Rm −1 ∆B ,(9) where G = du y /dz. Parker [18] was the first to identify travelling waves solutions (dynamo waves) of the above system. These oscillatory dynamos, named Parker waves, were obtained by Roberts [15] for nearly axisymmetric dynamos in spherical geometries (following the formalism of Braginsky [12]). It was found that while the αΩ-dynamos tended to be oscillatory (complex growth rate), for α 2dynamos the simplest dipole solutions tended to be stationary (real growth rate). A similar behavior can easily be traced in the simpler cartesian example above (see also [19,20] for a discussion of the generic behavior of such nearly axisymmetric mean field dynamos). We can perform further comparisons with reduced models by studying only the axisymmetric component of the simulated field. Figure 4 shows the azimuthally averaged field for some of our fully 3D simulations. The aspect ratio is increased from 0.45 (a) to 0.6 (b) and to 0.65 (c-d). The sequence of dynamo waves is represented for the antisymmetric mode (c) and symmetric mode (d). It is similar in nature to that produced by parameterized models [15]. The Earth-like mode is represented for aspect ratios of 0.45 and 0.6 (a & b). The active dynamo region lies outside the tangent cylinder [4], it therefore gets increasingly constrained as the inner-sphere in increased. The dipole eventually drops for large aspect ratio, when the volume outside the tangent cylinder becomes too small. Weakly dipolar solutions were also obtained at large aspect ratio in simulations using equations modified by hyperviscosity [21]. The dipolar solution was also found to decay and eventually vanish by increasing the aspect ratio in a reduced parameterized model for the Earth's core [22]. Here we show that the steady dynamo branch can be replaced, at larger aspect ratio, by an oscillatory dynamo mode. Comparison with reduced parameterized models can help interpret this transition to the solar-like mode. A strong zonal wind develops, in our simulations, in the Solar-like mode. Although the terminology of parameterized models must be used with care for direct simulations (the hypothesis of scales separation does not strictly apply), this suggests a transition from a dynamo of the α 2 type to a dynamo of the αΩ type as the aspect ratio is increased. Indeed, Earth-like three-dimensional models have been interpreted in terms of regeneration by convective vortices only, and thus closer to the α 2 formalism [23] (sometimes referred to as "giant α-effect"), whereas the αΩ formalism provides the classical framework to model solar dynamo-waves, as guided by the strong shear at the base of the convection zone [15,16]. Such nearly axisymmetric dynamos [15] produce cyclic magnetic behaviour very similar to the cycles examplified on Figure 3. Conclusions. -By varying the aspect ratio, we have observed a sharp transition from a dipole dominated large scale-magnetic field to a cyclic dynamo with a weaker dipole. This indicates that the geometry of the dynamo region severly constrains the existence of the dipole dominated solution. We should however stress that other parameters, involving ratio of typical forces, could affect the precise value of the critical aspect ratio for transition. The values of these parameters in our simulations (as in all numerical models to date) are indeed very remote from the actual relevant values for the Sun, or for the Earth. The potentially strong effect of this parameter change on the dynamo solution should not be under-estimated. It is indeed quite striking, that despite these shortcomings, numerical models can capture a good part of the qualitative feathures of the solar and geo-magnetic fields. Recent observations of stellar magnetism appear to corroborate this mechanism. Donati et al. [1] reported observations of a strongly dipolar field in a fully convective star (V374 Peg). More recently, Donati et al. [24] report magnetic observations of τ -Bootis, a rapidly rotating F star, i.e. one with a relatively shallow outer convection zone. Not only did they observe a rather complex magnetic field structure, but they also report that the overall polarity of the magnetic field has reversed after one year of observation. They interpreted this observation as an indication that the large aspect ratio τ -Bootis star is undergoing magnetic cycles, similar to those of the Sun. Futher observations of planets and stars are needed, but clearly the observations available so far seem to confirm the important role of the aspect ratio in controling the transition from steady to cyclic dynamo modes. * * * The computing resources for this work were provided by the CNRS-IDRIS, the ENS-CEMAG and the IPGP-SCP computing centers. We are grateful to Pr. N.O.Weiss for discussions on a preliminary version of this work. Fig. 1 : 1Time evolution of the radial magnetic field averaged in longitude (for an aspect ratio of 0.65). The initial dipole field survives for a few diffusion times, and then vanishes to yield a butterfly-like diagram. Fig. 2 : 2Radial magnetic field at the surface of the outer sphere, for aspect ratios of 0.45 (a) and 0.65 (b). Azimuthal magnetic field below the surface of the 0.65 aspect ratio model (c). Fig. 3 : 3Time evolution of the zonal average of the azimuthal magnetic field below the surface of the model, for an aspect ratio of 0.65: the antisymmetric (a) and symmetric (b) solutions. Fig. 4 : 4The zonal average of the magnetic field in our 3D simulations. Contours of the toroidal (east-west) part of the field are plotted in the left hemisphere and lines of force of the meridional (poloidal) part of the field plotted in the right hemisphere. . J.-F Donati, T Forveille, A Collier Cameron, J R Barnes, X Delfosse, M M Jardine, J A Valenti, Science. 311Donati J.-F., T. Forveille, A. Collier Cameron, J.R. Barnes, X. Delfosse, M.M. Jardine, J.A. Valenti, Science 311, 5761, 633-635. (2006). . K Zhang, G Schubert, Rep. Prog. Phys. 69Zhang K., Schubert G., Rep. Prog. Phys, 69, 1581-1605 (2006). Weiss in Mathematical Aspects of Natural dynamos. Tobias S N O , Dormy E., Soward A.M.CRC-pressTobias S. & N.O. Weiss in Mathematical Aspects of Natural dynamos, Dormy E., Soward A.M. (Eds), CRC-press 2007, . G A Glatzmaier, P H Roberts, Nature. 377Glatzmaier G.A., P.H. Roberts, Nature 377, 203-209 (1995). . P A Gilman, Astr. Phys. Sup. Ser. 53Gilman P.A., Astr. Phys. Sup. Ser. 53, 243-268 (1983). Astrophys. M K Browning, M S Miesch, A S Brun, J Toomre, J. L. 648Browning M.K., M.S. Miesch, A.S. Brun, J. Toomre, As- trophys. J. L. 648, L157-L160 (2006). . U Christensen, Phys. Earth Planet Inter. 128Christensen U. et al, Phys. Earth Planet Inter. 128, 25-34 (2001). . E Dormy, PhD thesisDormy E., PhD thesis (1997); . E Dormy, P Cardin, D Jault, Earth Plan. Sci. Lett. 160Dormy E., P. Cardin, D. Jault, Earth Plan. Sci. Lett. 160, 15-30 (1998); . U Christensen, P Olson, G A Glatzmaier, Geophys. J. Int. 138Christensen U., P. Olson, G.A. Glatzmaier, Geophys. J. Int. 138, 393-409 (1999). D H Hathaway, Proceedings of SOHO 12 GONG+ 2002. Sawaya-LacosteSOHO 12 GONG+ 2002ESA SP-517Hathaway D.H., in Proceedings of SOHO 12 GONG+ 2002, Sawaya-Lacoste (Ed.), ESA SP-517, 87-96 (2003). . S I Braginsky, Soviet Phys. JETP. 20726Braginsky S.I., Soviet Phys. JETP, 20, 726 (1964); . Sov. Phys. JETP. 201462Sov. Phys. JETP 20, 1462 (1965). . D W Hughes, F Cattaneo, J. Fluid Mech. 594Hughes D.W. & Cattaneo F., J. Fluid Mech. 594, 445-461 (2008). . G A Glatzmaier, Astrophys. J. 291Glatzmaier G.A., Astrophys. J. 291, 300-307 (1985). . P H Roberts, Phil. Trans. Roy. Soc., A. 272Roberts P.H., Phil. Trans. Roy. Soc., A 272, 1230, 663- 698 (1972). . J Beer, S Tobias, N O Weiss, Solar Physics. 181Beer J., S. Tobias, N.O. Weiss, Solar Physics 181, 237-249 (1998). . T G Cowling, Mon, Not, Roy, Ya.B., Sov. Phys. JETP. 94460Astro. Soc.Cowling T.G., Mon. Not. Roy. Astro. Soc. 94, 39 (1934), see also Zeldovich, Ya.B., Sov. Phys. JETP, 4, 460 (1957). . E N Parker, Astrophys. J. 122293Parker E.N., Astrophys. J., 122, 293 (1955) Magnetic field generation in electrically conducting fluids. H K Moffatt, Cambridge University PressMoffatt H.K., Magnetic field generation in electrically con- ducting fluids, Cambridge University Press (1978). Cosmical magnetic fields. E N Parker, Clarendon PressOxfordParker E.N., Cosmical magnetic fields, Oxford, Clarendon Press (1979). . N Gomez-Perez, M Heimpel, Geophys. Astrophys. Fluid Dyn. 101Gomez-Perez N., M. Heimpel, Geophys. Astrophys. Fluid Dyn., 101, 371-388 (2007). . D Jault, C R Acad, Sci, 323ParisJault D., C. R. Acad. Sci. Paris, 323 II a, 451-458 (1996). . P Olson, U Christensen, G A Glatzmaier, J. Geophys. Res. 104Olson P., U. Christensen, G.A. Glatzmaier, J. Geophys. Res. 104, 10383-10404 (1999). . J F Donati, C Moutou, R Farès, D Bohlender, C Catala, M Deleuil, E Shkolnik, A C Cameron, M , G.A.H. Walker, M.N.R.A.S.385Donati J.F., C. Moutou, R. Farès, D. Bohlender, C. Catala, M. Deleuil, E. Shkolnik, A.C. Cameron, M.M. Jar- dine, G.A.H. Walker, M.N.R.A.S., 385, 1179-1185 (2008).
[]
[ "NYSE Price Correlations Are Abitrageable Over Hours and Predictable Over Years", "NYSE Price Correlations Are Abitrageable Over Hours and Predictable Over Years" ]
[ "William H Press \nOden Institute for Computational Engineering and Sciences The University of Texas at Austin\n\n" ]
[ "Oden Institute for Computational Engineering and Sciences The University of Texas at Austin\n" ]
[]
Trade prices of about 1000 New York Stock Exchange-listed stocks are studied at one-minute time resolution over the continuous five year period 2018-2022. For each stock, in dollar-volume-weighted transaction time, the discrepancy from a Brownian-motion martingale is measured on timescales of minutes to several days. The result is well fit by a power-law shot-noise (or Gaussian) process with Hurst exponent 0.465, that is, slightly mean-reverting. As a check, we execute an arbitrage strategy on simulated Hurst-exponent data, and a comparable strategy in backtesting on the actual data, obtaining similar results (annualized returns ∼ 60% if zero transaction costs). Next examining the cross-correlation structure of the ∼ 1000 stocks, we find that, counterintuitively, correlations increase with time lag in the range studied. We show that this behavior that can be quantitatively explained if the meanreverting Hurst component of each stock is uncorrelated, i.e., does not share that stock's overall correlation with other stocks. Overall, we find that ≈ 45% of a stock's 1-hour returns variance is explained by its particular correlations to other stocks, but that most of this is simply explained by the movement of all stocks together. Unexpectedly, the fraction of variance explained is greatest when price volatility is high, for example during COVID-19 year 2020. An arbitrage strategy with cross-correlations does significantly better than without (annualized returns ∼ 100% if zero transaction costs). Measured correlations from any single year in 2018-2022 are about equally good in predicting all the other years, indicating that an overall correlation structure is persistent over the whole period.
null
[ "https://export.arxiv.org/pdf/2305.08241v1.pdf" ]
258,686,475
2305.08241
ef38278a299701919f54c6142fbb1d76d7e89070
NYSE Price Correlations Are Abitrageable Over Hours and Predictable Over Years May 16, 2023 William H Press Oden Institute for Computational Engineering and Sciences The University of Texas at Austin NYSE Price Correlations Are Abitrageable Over Hours and Predictable Over Years May 16, 2023 Trade prices of about 1000 New York Stock Exchange-listed stocks are studied at one-minute time resolution over the continuous five year period 2018-2022. For each stock, in dollar-volume-weighted transaction time, the discrepancy from a Brownian-motion martingale is measured on timescales of minutes to several days. The result is well fit by a power-law shot-noise (or Gaussian) process with Hurst exponent 0.465, that is, slightly mean-reverting. As a check, we execute an arbitrage strategy on simulated Hurst-exponent data, and a comparable strategy in backtesting on the actual data, obtaining similar results (annualized returns ∼ 60% if zero transaction costs). Next examining the cross-correlation structure of the ∼ 1000 stocks, we find that, counterintuitively, correlations increase with time lag in the range studied. We show that this behavior that can be quantitatively explained if the meanreverting Hurst component of each stock is uncorrelated, i.e., does not share that stock's overall correlation with other stocks. Overall, we find that ≈ 45% of a stock's 1-hour returns variance is explained by its particular correlations to other stocks, but that most of this is simply explained by the movement of all stocks together. Unexpectedly, the fraction of variance explained is greatest when price volatility is high, for example during COVID-19 year 2020. An arbitrage strategy with cross-correlations does significantly better than without (annualized returns ∼ 100% if zero transaction costs). Measured correlations from any single year in 2018-2022 are about equally good in predicting all the other years, indicating that an overall correlation structure is persistent over the whole period. Introduction Only recently has tick-by-tick historical trading data across whole markets like the New York Stock Exchange (NYSE) become available to all comers, at low cost and outside of proprietary settings. The sites Finnhub.io [1] and Polygon.io [2] are among current examples of such sources in a rapidly evolving landscape. Also only recently has GPU software like PyTorch [3] or TensorFlow [4] made easy the exploitation of financial data sets at gigabyte and terabyte scales with desktop resources. Given these trends, one expects to see a new wave of published large-scale statistical studies of the behavior of markets. This paper is one such. We study the collective behavior of five years (2018-2022) of ∼1000 NYSE listed stocks with continuous trading data at one minute resolution. The span of years is chosen to include year 2020, during which the COVID-19 pandemic roiled markets. Our interest is one with a long history, albeit somewhat neglected in recent years: to characterize and quantify deviations of the market from its so-called stylized facts, especially random-walk models, without introducing too much extra theoretical machinery; and to quantify the predictive power (in several respects, to be defined below) of the market's correlational structure. That the market deviates from its stylized facts is in no sense controversial [5][6][7][8][9][10]. Our goal is new quantification in a carefully controlled large data set. In §2 we summarize a set of these so-called facts, also reviewing the history of discovery of the necessity of a transaction time different from clock time. We describe the data set used in this paper. A technical point, we discuss the relation between returns computed as two-point instantaneous price differences, versus the difference of time-averaged prices. In §3 we compute variograms (equivalent in some but not all ways to autocorrelation functions) over the data, noting some apparently mean-regressing long-term memory on timescales minutes to days. In §4 we review the venerable model of shot-noise (or, in the limit, Gaussian) processes with powerlaw (Hurst exponent) variograms, also known as fractional Brownian motion. This provides a convenient platform for discussing the advantage of variance analysis over autocorrelation, and for demonstrating directly that profitable arbitrage is possible in principle for such models (except for the case of a perfect random walk). In §5 we apply §4's arbitrage trading strategy to the actual stock-market data and find it to be profitable about as predicted by its observed Hurst exponent. This serves to remove any lingering doubts that the long-memory, mean-reverting behavior measured is genuine, not an artifact of (e.g.) a flawed mapping of trading time to clock time. In §6 we turn to the cross-correlational structure of the ∼ 1000 NYSE stocks. Measuring the correlation with one-hour returns over five years produces an interesting "atlas" of correlation diagrams (shown in Supplementary Information). Measuring correlation as a function of the time-resolution of returns produces a seeming anomaly, which we show to be quantitatively explainable by the same apparent longterm memory as seen in the variograms. Section 7 looks at "leave-one-out" predictions, where we calculate what hourly (or other) return should be expected of a stock, given the hourly returns of all other stocks in the same hour. Then, in §8, we test an arbitrage strategy based on leaveone-out predictions, with apparently robust positive results. Section 9 is additional discussion. Preliminaries Stylized Facts In economics, stylized facts, so-called, are empirical observations that evidence broad principles without being necessarily exact in all cases [11]. Since the work of Fama [12] and elaboration of the Efficient Market Hypothesis (EMH) from the 1960s [13], recitations of stylized facts about the time history of market prices and returns usually include these [14]: • Asset prices are nonstationary and do (some kind of) random walk. • Sequential asset returns, i.e., price changes, are (close to) independent. More formally, price evolution is (close to a) Markov process, therefore memoryless. • In liquid markets, arbitrage opportunities are (almost) nonexistent. Or, equivalently: The Markov process is a martingale. • Because the market responds to the sum of innumerable small news effects, the Central Limit Theorem should apply, and the time series are expected to be Gaussian. Taken together, these stylized facts imply a Brownian motion random walk (also termed a Wiener process) as the null-hypothesis model for financial time series-at any rate, the model to disprove with contradictory data [15]. Historically, the last of these stylized facts was an immediate embarrassment, because, the distribution of many types of asset returns, sampled at equally spaced times, is strongly non-Gaussian, with positive kurtosis and fat tails. Some exotic solutions were proposed, for example Mandelbrot's examination of so-called stable distributions that could be the sum of many small effects, yet not Gaussian [15,16]. But, as first notably studied by Clark in 1973 [17], the path to rescuing nearmemoryless near-Gaussianity lay instead in what has come to be another stylized fact: • Trading volume and price volatility are positively correlated. Transaction Time and Variogram Clark interpreted the apparent non-Gaussianity as evidence that the market's Gaussian process advanced not in clock time, but rather in a "transaction time" that, he noted empirically, was something close to cumulative trading volume [17]. (Lacking a standard terminology, transaction time is by now also known as economic time, business time, trading time, market time, and operational time [18].) In the transaction time of cumulative trading volume, the distribution of returns per fixed time is close to normal. That transaction time and clock time converge over periods of order a month, was already implicit in 1965 work by Fama [19]. Clark's observation unleashed a veritable flood of work in search of an exact or universal transaction time, one in which the normality of returns could be raised to the status of a law of nature (see references in [20]). A great deal of effort was expended on so-called subordinated processes, a special class of mappings from clock time to transaction time [17]. Later, Ane and Geman [21] advocated consideration of a broader class of mappings, calling attention to the result of Monroe [22] that any (semi-) martingale process could be rendered as exactly a memoryless Gaussian process by an appropriate choice of stochastic time. Important for this paper is the fact, true for any martingale [18] and especially for a memoryless Gaussian process, that the variance of returns r over a (transaction) time τ must scale exactly as τ, because the variances of returns over successive independent intervals simply add, V (τ) ≡ Var[r(τ)] ≡ E t [(p(t + τ) − p(t)) 2 ] ∝ τ(1) where E denotes expectation value. In contexts other than financial, the statistic V (τ) in equation (1) is termed the variogram of the time series p(t) [23], and we adopt that terminology here. Variogram analysis here is essentially equivalent to the financial literature's "variance ratio test" [6,8], though the latter term is more used in the context of a significance test than a measurement over multiple values of τ. Suppose a time series of prices, expressed as a function of some transaction time, is found to violate equation (1). How can we distinguish between the hypothesis that it is not a martingale-hence allows an arbitrage opportunity-and the hypothesis that we simply have the wrong transaction time, poorly approximating Monroe's perfect one, or that there exists some other systematic flaw in the data? This will be the key issue in §3, below. NYSE One-Minute Data Set We downloaded from Finnhub.io all available one-minute candles [24] for NYSE-listed stocks in the years 2018-2022, comprising substantially all such listed stocks. For a given stock, a one-minute candle exists if that stock traded during that minute. Each candle consists of a universal timestamp (in Unix seconds) marking the beginning of the minute, four prices (open, high, low, close), and a volume of shares traded. A typical year has about 252 trading days, amounting to 98,280 trading minutes. The more active listed stocks trade in most minutes. There are 1,091 stocks with available candles spanning all five years. Candles do not reveal how many separate trades occur in their minute, nor the sign of the trade (buyer-vs. seller-initiated). If there are equalities among the four prices, then fewer than four trades may be present-but more than four is also possible. For these reasons, we in all cases take the mean of open, high, low, close as the representative average price for that minute, referring to this as "one-minute resolution". Prices are considered known only in minutes with a reported candle, otherwise unknown. That is, we never hold over a previous price or interpolate between known prices. We define resolution time intervals τ > 1 minute (e.g., τ = 1 hour) by sorting candles according to their timestamp into consecutive, nonoverlapping bins of length τ. We assign to each bin the mean of its candles timestamps as a time, the mean of its candles prices as a price. If a bin contains no candles, its price is unknown. Of particular interest will be the estimation of the variances of the returns r(τ) implied by price time series as defined above, leading to estimates of the variogram V (τ), the covariance matrix C of multiple such returns (multiple listed stocks), and their implied correlation matrix ρ. Returns are defined by the logarithmic price difference of two consecutive known prices with the same resolution τ. Associated with each return is its time interval, the difference of its two average times, generally not an integer multiple of τ. We calculate σ 2 , C, and ρ by the methods described in [25], which are designed for such cases of asynchronous sampling. Difference-of-Average vs. Two-Point Difference The astute reader will have noticed that we have introduced two slightly different definitions for r(τ), the return over time τ whose variance is the variogram V (τ). Equation (1) defines r(τ) as the difference of two point (i.e., instantaneous) prices spaced apart by time τ. Section 2.3, on the other hand, implies a definition that first averages prices into bins of length τ and then defines r(τ) as the difference of adjacent bins. How are these related? In Supplementary Information S1.1, we show that, for the case of a memoryless Gaussian process, the difference-of-average method yields a result that differs by a constant factor 2/3 from the two-point difference method. Since no results in this paper will depend on the the absolute normalization of the returns, the factor is irrelevant. More important is to compare the fractional accuracy to which the variogram can be measured for two methods, with input data consisting of a fixed number of one-minute candles, as described above in §2.3. Supplementary Information S1.2 demonstrates that the accuracies are close to equal. With the data as described, however, the two-point difference method shows a bias that increases as τ decreases towards one minute, arising because the candle prices are not in fact point prices but themselves averages. The difference-of-average method shows less such bias and is henceforth the method of choice for this paper. Variogram Estimate for NYSE Stocks Now making a choice among similar available alternatives, we adopt as a definition of transaction time cumulative New York Stock Exchange dollar-trading volume, normalized to clock time at one year. In other words, one dollar-trading-weighted hour (abbreviated $TWhr) is an interval in which a fraction 1/8760 of the year's dollar trading volume occurs (1/8784 in leap years). We can also compare $TWresults to their volume trading weighted counterparts, time unit abbreviated VTW hr, where the total share volume in a year is normalized to the number of hours in the year. As an example, Figure 1 shows the time mapping of these definitions for calendar February 9, 2021, a typical day. In clock time, the shortest $TW minute is about 200 milliseconds (spanning a closing auction), the longest is about 3.6 days (spanning a holiday weekend). The shortest $TW hour is about 12 clock minutes (again, spanning a closing auction); the longest spans the same 3.6 day holiday. From the 2019 one-minute candle data on 1,091 stocks described in §2.3, we computed separate variograms V (τ) for each stock's prices, scaling each to have V (1 hr) ≡ 1. Figure 2 shows the results as a function of resolution time interval τ. The ensemble of variograms is summarized in the figure as percentile sticks with markers at the 10, 25, 50, 75, and 90 percentiles. Displayed in this way, one sees that the dispersion of individual stocks around the median of all the stocks is surprisingly small, indicating an approximately "universal" scaling law for V (τ), both for individual stocks and for index averages. The figure shows results for three different time parameterizations, clock time, $TW, and VWT. The ordinate V (τ)/τ is chosen to make a memoryless random walk plot as a constant line. First looking at the result for clock time (red curve and sticks), one sees a function decreasing monotonically, implying at first glance a process with a significant tendency towards reverting to the mean and, in any case, far from memoryless. However, we already know from the work cited in §2.2 above, that such effects can be produced by choosing a poor transaction-time parameterization. In particular, the change in logarithmic slope at ∼ 1 hr is immediately suggestive of such an effect, as (e.g.) the intervals τ come to straddle the beginning or end of trading days. So, we dismiss the seeming long-term memory of the red curve as an artifact. Next turn attention to the blue and green curves and sticks, the same price data now mapped into either $TW(blue) or VTW (green) transaction time. A first observation is that the blue and green results are virtually identical, implying that results are not sensitive to the exact choice of volume-weighted transaction time. A second observation is that the results are surprisingly well fit by a single logarithmic slope (power law), remarkably featureless from ∼ 4 minutes to ∼ 1 week, a range of more than three orders of magnitude. Summarizing, we see in the that is, an exponent differing by −0.07 from a memoryless process. We now face a dilemma: Should we believe (Hypothesis 1) that, as for the red curve, the small exponent −0.07 is an artifact of having chosen a poor transaction time. In that case, a perfect Monroe stochastic clock [22] would yield a horizontal line and an implied memoryless random-walk process. Or (Hypothesis 2), should we believe that the small exponent is evidence of persistent-returns memory, here seen very generally across most stocks, with a timescale of minutes to days. In general, such a memory implies an (at least theoretical) arbitrage opportunity [5,18]. Hypothesis 2 is not ruled out a priori. There exist, after all, generally profitable quant hedge funds in equities markets [26][27][28]. Still, it does seem peculiar that a broad market inefficiency like this should survive in an exchange as liquid and visible as NYSE. We might ask it this result is an artifact of the year chosen, 2019. But (see Supplementary Information §S2) virtually identical results are seen for years 2018, 2019, 2021, and 2022. Only COVID-19 year 2020 shows some anomalies, but with still a small negative-exponent trend. In the remainder of this paper, we take the view that the only really persuasive way to demonstrate the truth of Hypothesis 2 is to demonstrate, by backtesting on the actual price data, profitable arbitrage opportunity, and also to do this in the context of a controlled model that gives comparable results without the possibility of "unknown unknowns". The next section provides such a model. Hurst Power-Law Models A shot-noise process [29,30] is a time series generated as the sum of impulses at Poisson-random times. Or, a slight generalization, it can be the sum of a normal random variable multiplying such impulses [31], p(t) = i s i f (t − t i ), s i ∼ Normal(0, σ), (t i+1 − t i ) ∼ Exponential(λ) (3) where f (.) is the impulse response, σ scales the impulses, and λ is the Poissonprocess rate. (The difference between successive times in a Poisson process is, of course, exponentially distributed.) Evidently, if the impulse f (.) is a unit step function at argument zero, the process p(t) is a random walk with steps of normally distributed amplitude σ occuring randomly at a mean rate λ. Then, likewise obvious, is the implied variogram, V (τ) = λσ 2 τ(4) As is well known, a Gaussian random walk (Wiener or Brownian motion process) is obtained by taking the simultaneous limits λ → ∞, σ → 0, with λσ 2 held constant. [32]. A long-memory shot-noise process that is not a random walk is obtained by replacing the unit step function by something else [16,33], for example, f (t) = 0 −∞ < t < δ (t/δ) (1− ) δ ≤ t(5) where δ > 0 is to be thought of as a small value (the interval 0 < t < δ being implicitly reserved for the rise of the impulse) and − 1 2 < < 1 2 . Figure 3 shows Figure 3: Impulse responses for long-memory shot-noise processes. Exponent zero (black) yields a memoryless random walk. Small negative exponents yield nearly a random walk, but with a partial tendency to return to previous values. examples of such power-law impulses. It is conventional to term H = 1 2 − the Hurst exponent [34]. A process with H > 1 2 (or < 0) is termed persistent. One with H < 1 2 (or > 0) is termed anti-persistent or mean-reverting. Both are long-memory processes. Only the case H = 1 2 ( = 0) is memoryless, a random walk. The variogram of a Hurst power-law shot-noise process can be shown to be [10,35] V (τ) ∝ τ 2H = τ 1−2 (6) so a model for the NYSE data in Figure 2 or equation (2) is a Hurst power-law process with ≈ 0.035. 1.0 (t/ ) 0.25 (t/ > 1) (t/ ) 0.035 (t/ > 1) (t/ ) 0.0 (t/ > 1) In the Gaussian limit of many overlapping impulses, the Hurst power-law process is known as fractional Brownian motion [10,[35][36][37]. The simulation of a long time series of fractional Brownian motion is straightforward: Fast Fourier Transform methods are used to convolve a series of i.i.d. normals (s i 's in equation (3)) with an impulse f (.) as in Figure 3. On a desktop machine with a single GPU we can simulate 3600 years of hourly prices in a few wall-clock seconds. Why Use Variogram Instead of Autocorrelation? Variogram V (τ) = E t [(p t+τ − p t ) 2 ] and autocorrelation E t [r t+τ r t ] contain equivalent two-point statistical information. For a Gaussian process, either characterizes a process completely. In empirical studies of asset returns, the autocorrelation has been more frequently studied (e.g., [9,10,26,38,39]) than the variogram or its twin the variance ratio test (e.g. [6,8]). The Hurst power-law process is a convenient platform for us to explore the minority view. Suppose we have V (τ) = τ 1−2 with τ > 1 and we want to distinguish it from the random-walk null hypothesis V (τ) = τ 1 . Then the difference, for small > 0, is a signal of positive magnitude ∆V = τ − τ 1−2 = τ(1 − τ −2 ) ≈ 2 τ log(τ)(7) We want to detect this signal in the presence of the noise that is the sample variance of the the full V (τ). This "measurement variance of the variance" for any Gaussian process whose variance σ 2 is sampled N times, is 2σ 4 /N [40], so Var[V (τ)] = 2 N V (τ) 2 = 2 N τ 2−2 (8) implying a signal-to-noise ratio S/N ≈ √ 2N log(τ)(9) The autocorrelation (now the signal) of the same Hurst power-law process is ( [35], p. 26) Corr(τ) = E t [r t+τ r t ] = −2 (1 − 2 )τ −1−2(10) which decays rapidly with increasing τ. For small , i.e., close to random walk, we can approximately calculate the noise variance as if r t+τ and r t were independent, so Var[Corr(τ)] = Var 1 N N r t+τ r t = 1 N 2 N Var[r t+τ r t ] = 1 N 2 N [Var(r t )] 2 = 1 N(11) since the r's are independent normals with unit variance. Now the signal-to-noise ratio is, to leading order in S/N ≈ 2 √ N τ −1−2(12) Comparing equations (9) and (12), one sees that for the same values and N , the variogram measurement becomes logarithmically more statistically significant as τ increases, while the autocorrelation becomes less significant, at least inversely with τ. That the significant part of the autocorrelation signal is concentrated at the shortest times is problematic also because that is where the data is most subject to artifacts (bid-ask bounce, spoofing, or layering, for example), while the variogram's use of price movements over longer times should be more reliable. Depending on exactly the way the data is gathered, one might argue that there are only N/τ independent samples of V (τ), while there are fully N samples for the autocorrelation. In that case, the advantage of the variogram is only ∼ √ τ instead of ∼ τ, but it is still an advantage. Arbitrage Trading Strategy for Fractional Brownian Motion One expects a fractional Brownian process with = 0 to be arbitrageable, because it is not memoryless. Since the observed NYSE exponent is positive, our model is mean-reverting. Thus, we expect to make a profit by betting against its returns, that is, going long after a negative return, short after a positive one. Our numerical arbitrage test first generates a discrete fractional Brownian vector of length 31,536,000 with Hurst exponent = −0.035. The values are interpreted as 3600 years of hourly samples, 8760 (transaction time) hours per year. Each year separately is linearly detrended to begin and end at the logarithmic value zero. Next, all values are multiplied by a factor that gives their standard deviation (across all hours and years) the value 0.15, this to approximate the annualized volatility of log prices in the NYSE. The scaled values are finally exponentiated to produce hourly prices p y,h , with y indexing the year, h the hour. Since each year begins and ends with a price of one ("$1.00"), a simple buy-and-hold trading strategy will yield exactly zero return in each year. A useful normalizing value below is the r.m.s. 1-hour return across the whole sample, r rms ≡ E y E h log(p y,h+1 /p y,h ) 2 1/2(13) where expectations E are estimimated by sample means. We next define a trading strategy. Since the actual NYSE data consists of interval averaged prices (see discussion in §2.3, §2.4, and §S1), we interpret the simulated prices in the same way. We must take care to be fully causal, implying that no use can be made of an interval's average price until after the end of the averaging interval. We adopt the following trading strategy: For every year y and hour h, • Using h and h+1 prices, calculate normalized returns:r y,h = log(p y,h+1 /p y,h )/r rms • Using h + 1 prices, calculate the number of shares to buy (positive) or sell short (negative): q y,h = −r y,h /p y,h+1 . The minus sign embodies the bet on a mean-returning process. • Buy (sell) these shares during hour h + 2, realizing that hour's average price p y,h+2 • Sell (buy) the same shares during hour h+3, realizing that hour's average price p y,h+3 • Record a profit or loss P y,h = q y,h (p y,h+3 − p y,h+2 ) Note that the same (on average) capital is recycled in each hour and is at risk for on average one hour between buying during h + 2 and selling during h + 3. We will be long and short about equally often, so it is a matter of convention how to define a denominator for the purpose of calculating annual returns. If we use the mean of absolute values |r y,h |, counting long and short capital as equally at risk, then each year's net return is P y = 8760 h P y,h h |r y,h |(14) This is an uncompounded return because the hourly stake is not increased (decreased) with profit (loss). Figure 4 shows the histograms of yearly net returns P y for the 3600 simulated years, for values of between 0 and 0.10 (Hurst exponents between 0.5 and 0.4). For the case = 0, one sees returns roughly symmetrically around zero, as one expects for this memoryless random walk. For larger values of , the adopted trading strategy shows significant positive returns. Figure 5 shows results from simulations with a larger number of values and plots their mean return and the standard deviation (over 3600 simulated years) of that mean. That the mean return for the random walk case = 0 is close to zero (value ≈ 0.003) is a useful check, since hourly long and short trades were simulated as for other values of . One sees that a detection at one standard deviation with one-year's data becomes possible when exceeds about 0.03 The conclusion of this section is that a model long-memory power-law process with as small as ∼ 0.01 can be recognized by a simple arbitrage trading strategy over one (or a few) years of hourly price data. In particular, that true process is thus easily distinguished from a memoryless process whose apparent nonzero is due to a poor choice of transaction time or any other unmodeled cause, since no profitable arbitrage should be possible in that case. Figure 4, but now plotted as functions of . One sees that at ≈ 0.03, an arbitrage signal is detectable at about one standard deviation (1-σ). Arbitrage Strategy Applied to the NYSE Data We are now in a position to apply in backtesting of real data (no longer simulation!) something like §4.2's trading strategy. In simulation, we looked at a single (imaginary) stock over some thousands of homogeneous years. For the actual data, we have about a thousand stocks, each over five inhomogeneous years (e.g., COVID-19 occurred). Our trading algorithms will therefore be different in detail, though similar in spirit. We proceed as follows: For each successive 1-hour ($TW) interval (indexed h) in years 2018-2022, and for the 1,091 NYSE stocks that traded during all five years for which we have minute-resolution data, • Eliminate from consideration stocks that traded in fewer than half of all $TW hours in one or more years. (Such stocks produce too much missing data in our fol-lowing use of consecutive hourly prices.) This cut on the data results in 866 stocks remaining, now indexed by k. • For each stock k, observe its average trading price in hours h and h + 1 and compute a return r k,h = log(p k,h+1 /p k,h ). • If fewer than 100 stocks have positive returns, or fewer than 100 have negative returns, take no action in hour h. (This to avoid unusual trading hours with massive correlated movements.) • Otherwise, divide a fixed stake among stocks with negative r k,h 's proportional to |r k,h | and purchase those stocks during hour h + 2, as close as possible to uniformly in $TWtime. We make the assumption that an average price very close to the average trading price of the hour can be obtained. Similarly go short in the same total amount on stocks with positive r h,k 's, in proportion to their |r h,k |. • Uniformly in $TWduring hour h + 3 close all positions acquired in hour h + 2, so that the mean holding time is 1 hr. The above prescription is applied every hour, so that, as in the Hurst simulation, we are at various stages of four different hours at any given time. However, our outstanding long stake (and equal short risk) is constant as one hour's worth; specifically, we are again not compounding returns (or losses). Also worth emphasizing again is that our strategy is completely causal: In any hour, we take actions based only on previous hours, not the present one. How do we do? Figure 6 shows the result for the above long-short strategy, and also for a strategy where only the long trades (on stocks with a negative return in the previous hour) are made. Both strategies do significantly better than the market (S&P 500 as a surrogate) during almost all intervals, and very much better cumulatively over five years. In particular, the long-short strategy returns 53% (annualized), the long-only strategy 35%. The COVID-19 crisis in March-April, 2020 produces a brief downturn in the long-only strategy, but is barely visible (and mostly with a positive effect) in the long-short strategy. For many reasons, we should not expect exact agreement between these annual arbitrage returns and the Hurst simulations. The trading strategies are, of necessity, not identical. The value 0.15 used in the simulations for the market's annualized volatility is indicative, but not exact, especially not for any particular stock. The actual market's cross-correlational structure can (as we will see in §6 below) play a significant role. Still, it is encouraging that the actual data, with measured ≈ 0.035, and the simulations ( Figure 5 at that value) are in the same ballpark with ∼ 60% annual return. We conclude that the long-memory power-law model (or fractional Brownian motion model) with ≈ 0.035 (or Hurst exponent H = 0.465) is quite a plausible approximation. In §7 we will show further evidence, based on cross-correlational structure. The above arbitrage opportunity may or may not be realizable in practice. Since a given dollar turns over ∼ 8000 times per year, round-trip transaction costs 10 −4 would be required, and the ability to trade ∼ 10 3 each hour. This might be achievable with a large, market-making investment bank as a continuous counter-party, but not seemingly otherwise. Also, we assume that our purchases and short sales do not move the market, and that they can be accomplished during a trading hour at the average price of that hour, both assumptions open to reasonable question. Cross-Correlation Structure Up to now, we have dealt with stocks one a a time, with a time series of prices for each. Now, we will look at the cross-correlations of multiple stocks. We define the covariance of two stocks A and B on timescale τ by C AB (τ) ≡ E[r A (τ)r B (τ)] = E t (P (A) t+τ − P (A) t )(P (B) t+τ − P (B) t )(15) where the P 's denote τ-averaged stock prices, as described in §2.4, and C is the covariance matrix. We will use the terms cross-correlation and correlation interchangeably as meaning ρ AB (τ) ≡ C AB (τ) C AA (τ)C BB (τ)(16) (We continue to use the term autocorrelation to mean correlation in time of a single stock.) Because not every stock trades every minute, we need a method for estimating the expectations E from possibly asynchronous trading data derived from one-minute candles. For this we use the formalism developed in [25]. Also from [25] we adopt the shorthand notation N as denoting a draw from the normal distribution N (0, 1) and adopt the convention that different subscripts {N X , N Y } represent independent draws, identical subscripts {N Z , N Z } denote the same draw (that is, have the same numerical value). With these conventions, the Gaussian model return r(τ) of a single uncorrelated stock A (the model of 3 above) can be written r A (τ) ∼ V 1/2 (τ)N A(17) where ∼ is read "is drawn from", while the correlated returns of two stocks {A, B}can be written r A (τ) ∼ V 1/2 A (τ) √ ρ AB N C + 1 − ρ AB N A r B (τ) ∼ V 1/2 B (τ) √ ρ AB N C + 1 − ρ AB N B(18) Here √ ρ is shorthand for ± |ρ| with a minus sign only for negative correlations and on one stock, left to the reader to fill in appropriately. While equation (18) is written as a single factor model, it is in fact general for a Gaussian process, because the latter is completely defined by its two-point correlations. The advantage of this notation is to compactly specify both autocorrelation in time V (τ) and cross-correlation across stocks (ρ AB ). NYSE Cross-Correlations at 1-Hour Resolution With the methods described, we can readily compute ρ AB (τ = 1 hour) across all ($TW) hours in the five-year period 2018-2022 for all pairs A, B of the 866 NYSE stocks with one-minute candle data in all years. The resulting 866 × 866 symmetric matrix is understandably difficult to visualize, but its related metric correlation distance d AB = √ 1 − ρ AB implies an exact embedding, positioning the individual stocks in 865-dimensional space. (Analogously, a triangle of three correlations can always be embedded in two dimensions.) This 865-dimensional space can then be projected into two dimensions in various ways. Informative qualitative patterns emerge, for example, correlational relations among different industries. Supplementary Information §S3 provides an "Atlas" of figures generated in this way, but the idea of projection is not further used here. NYSE Cross-Correlations as Functions of Resolution τ As initial orientation, Figure 7 shows, as a function of time resolution inverval τ, the pairwise correlations ρ for 100 randomly chosen pairs of NYSE stocks, and also the median correlation. Similarly shown are the correlations of 64 NYSE-listed bank stocks with Bank of America (ticker BAC). Unsurprisingly the latter correlations, between banks, ∼ 0.7, are larger than the former, between random firms, ∼ 0.3. Noteworthy is that both sets of correlations increase, roughly linearly in log τ, between ∼ 2 min and ∼ 24 hr, a range of almost three orders of magnitude. Instead, the data in Figure 8 call for a model in which the correlated and un-correlated parts of stocks' returns have different variogram dependences on τ. The simplest such model is r A (τ) ∼ V 1/2 A (τ) √ ρ AB N C + 1 − ρ AB N A 1 + U 1/2 A (τ) N A 2 r B (τ) ∼ V 1/2 B (τ) √ ρ AB N C + 1 − ρ AB N B 1 + U 1/2 B (τ) N B 2(19) where V and U are respectively the correlated an uncorrelated partial variograms. From equation (19) one readily calculates the total variograms V tot (τ) and related quantities, V tot A (τ) = E[r A r A ] = V A (τ) + U A (τ) V tot B (τ) = E[r B r B ] = V B (τ) + U B (τ) E[r A r B ] = V 1/2 A V 1/2 B ρ AB ⇒ ρ observed AB (τ) = V A (τ) V tot A (τ) V B (τ) V tot B (τ) 1/2 ρ AB(20) Let us now make some rash simplifying assumptions, and see whether they are compatible with the data. Assume that the correlated and uncorrelated variograms V (τ) and U (τ), with V tot (τ) = V (τ) + U (τ), have a universal functional form across all stocks, and are only scaled by a factor for each stock. For example, they might each be a Hurst power-law process or a random walk. Then by equation (20), ρ observed AB (τ) ρ AB = V (τ) V tot (τ) = 1 1 + U (τ) V (τ)(21) Next rashly assume that the correlated part of the variance is memoryless, V (τ) ∝ τ, so that all of the arbitrageable signal seen in §3 and exploited in §5 is due to the uncorrelated variance U (τ). Then, V tot (τ) τ = V (τ) τ 1 + U (τ) V (τ) ∝ 1 + U (τ) V (τ)(22) Equations (21) and (22) make a prediction, ρ observed AB (τ) ∝ τ V tot (τ)(23) This is directly testable in the data, without any assumption about the functional form of U (τ) (e.g., Hurst or fractional Brownian). Indeed, we already plotted τ/V tot (τ) (normalized at 1 hour) in Figure 8 as the dashed black line. It is virtually indistinguishable from the median ρ (also so normalized), just as predicted by equation (23) We conclude that both the deviations in variance from a random walk seen in Figure 2 and the deviations from constant-in-time correlations seen in Figure 8 can be explained by a mean-reverting component to returns that is uncorrelated across stocks and distinct from a (close-to?) random-walk component embodying the crosscorrelational structure. The roughly factor of two fall (rise) seen in Figure 2 (Figure 8), suggests that the uncorrelated and correlated variances are comparable in magnitude at timescales of order minutes, but that the uncorrelated part largely decays away by timescales of order hours. In §5 above we successfully (at least in backtesting) arbitraged the decaying component, one stock at a time. In the next section we attempt to improve the arbitrage performance by now using the full correlational structure. Leave-One-Out Predictions of Stock Returns The positive arbitrage returns in §5 derived from an algorithm for identifying, in a fixed hour h, stocks that were over-or under-valued in a predictive sense. The algorithm was simple: How did the price change between the two previous hours h − 2 and h − 1? The market's correlational structure implies another, somewhat orthogonal, algorithm for identifying possibly over-or under-valued stocks: Given a correlation structure, how compatible is a stock's return in a period with the returns of all other stocks in that same period? Specifically, how different is its return from a correlational prediction of its return. This is prediction in a very limited sense, using some data at one time to predict other data at the same time. It is then an open question, to be answered experimentally, whether such an algorithm is predictive of the future. That is what this section attempts to elucidate. Equations for Leave-One-Out Prediction Suppose we want to predict a return r I as a linear combination of other returns r K (uppercase subscripts denoting different stocks), r I = K =I B IK r K(24) Minimizing the expected mean square error of the discrepancy for B IK , argmin B E   r I − K =I B IK r K 2  (25) yields after some algebra, B IK = [[E(r K r J ) K,J =I ]] −1 [[E(r J r I ) J =I ]] = J =I C =I −1 KJ C JI ≡ J A =I KJ C JI(26) Here double brackets indicate matrices, C is the correlation matrix of equation (15), and and C =I means C with row and column I removed. The matrix A =I KJ is then most easily defined in words: • Delete row and column I in the covariance matrix C. • Invert the resulting matrix. • Insert a new row and column I of all zeros to get A =I KJ The formula for the inverse of a partitioned matrix can next be applied to give the simple result (see, e.g., [41]), A =I KJ = A KJ − A JI A KI /A II , where A ≡ C −1(27) yielding after substitution into equation (26) and some algebra a particularly simple form for coefficients in the prediction equation (24), B IK = − A KI A II − δ KI(28) The delta term merely zeros the diagonal elements, enforcing that a return is not used in predicting itself. The manipulations in equations (27)- (28) are not just for elegance: They avoid having to separately invert a (say) 865 × 865 matrix 866 times, for each value of I as equation (26) seemingly indicates. Instead we can invert an 866 × 866 matrix just once. A figure of merit for the predictions is the mean (across stocks) fraction of variance explained (FVE) of hourly returns, equal to the square of the correlation coefficient between r hI and r hI . This can be written as (see §S1.3), FVE = 1 − 1 2 ( r hK − r hK ) 2 h r 2 hK h 2 K(29) where angle brackets denote sample averages over stocks I and hours h as indicated. A perfect prediction would have FVE = 1. Table 1 shows results where the covariance matrix C is estimated using one calendar year's hourly data, matrix-inverted, and then used to predict each year's hourly returns on 866 stocks. Leave-One-Out Results by Year Year (29), in percent. Each row uses hourly returns data from the indicated year to construct an inverse correlation matrix. That matrix is used to predict hourly leave-one-out returns of every stock in the year indicated by the column. Values are the mean fraction of the variance of actual hourly returns of the stock explained. The bottom row shows results for a correlation matrix in which all pairs of stocks are given the same numerical correlation. On average about 66% of the hourly returns variance is explained using the same-year covariance-but this is subject to some over-fitting because the data is used twice. A more meaningful result is that, on average, about 45% of the variance in different years is explained, and the value is about independent of which year. In other words, the correlational structure persists roughly unchanged over the five year period studied. Some years (e.g., 2022) are intrinsically more predictable than others (e.g., 2019). Interestingly, COVID-19 year 2020 has typical predictability, despite the market's extreme fluctuations. As is well known, stock prices tend to move up or down in tandem. We might try predicting returns r hK by an unweighted average of all other returns, scaling by variance appropriately, r hK = Var(r K ) r hJ Var(r J ) J =I(30) with angle brackets denoting sample mean. One sees, on average, about 39% of variance explained, only fractionally less than with the full correlational structure. One can show that the correlation matrix corresponding to an unweighted average in equation (24) is one with equal positive off-diagonal terms, independent of their magnitude (if it is not too small). Year Table 2: Same as Table 1, except that better predictions are obtained using gradient backpropagation instead of direct covariance matrix inversion. On average, 51.3% of the variance of hourly returns is explained by any other year's correlation structure (61.5% by the same year's). A technical issue is that, because all stocks tend to be positively correlated, with some pairs very highly correlated, the covariance matrix C is close to singular. Thus, the inversion implicit in Table 1 is quite noisy. With gradient backpropagation machinery standard in the training of large neural networks (NNs), we can calculate an improved matrix A IK by direct minimization of equation (25). Doing this yields the improved results shown in Table 2. We inhibit overfitting by a standard neuralnetwork technique: Each year's fit is gradient-minimized with only its own data, but the gradient search is terminated when the performance on other years is no longer decreasing. The average off-diagonal value in Table 2, that is, the fraction of variance explained using a different year's correlation matrix, is 51.3%, significantly better than the 45.4% seen in Table 1. The average of the diagonals, predicting a year by its own correlation, is 61.5%, lower than Table 1's 66.1% because the overfitting is now less. Leave-One-Out Monthly Performance The percent values given in Tables 1 and 2 are fractions of the full year's variance explained (FVEs). But because a stock's (or the market's) volatility can vary significantly over the course of a year, it is also interesting to disaggregate the results of Table 2 by months and to show the fraction of each month's individual variance explained by correlational prediction. Figure 9 shows this and also the equivalent metric of fractional mean-square error, averaged over stocks, FMSE = ( r hK − r hK ) 2 h r 2 hK h K(31) In the Figure, the single year 2018 is used to estimate the inverse correlation matrix, which is then applied across all years. Percentile sticks show the ranges of predictabilities over all the stocks. Figure 9, but using different single years (indicated by color) to estimate the inverse correlation matrix, which is then applied across all years. One sees that each year (color) does best in predicting itself, due to some overfitting, but that otherwise the correlations of any year are about equally good at predicting across all years shown. Also shown (on an arbitrary offset scale) is the VIX volatility measure. Predictions tend to be best when volatility is high. Figure 10 shows the results of a similar analysis, showing only the median across all stocks, but now using different single years (indicated by color) to to estimate the inverse correlation matrix. One sees that, due to overfitting in using a year's correlation data to predict that same year, each year does best in predicting itself, but that otherwise the correlations of any year are about equally good at predicting across all years shown, confirming the general behavior seen in Tables 1 and 2. Also shown in Figure 10 is the VIX volatility measure. Predictions tend to be best when volatility is high, and were especially good (measured by FVE or FMSE) in the COVID-19 period of high volatility. The figure also confirms (dotted curve) that the full correlations do somewhat, but not hugely, better than a simple unweighted average that corresponds to assuming an identical correlation among all stocks. Arbitrage Strategy Using Cross-Correlations As was done in §5, we can devise an arbitrage strategy, but now based on the sametime predictions of the correlation structure instead of on the mean-reversion memory of individual stocks. Now, we bet against any discrepancy between a stock's actual hourly return and the same-hour return correlational prediction of equations (24) and (28). In particular, we adopted this strategy: We computed prediction coefficients using 2018 hourly data and equation (28). Then, for each successive 1-hour ($TW) interval (indexed h) in years 2019-2022, and for the same 866 NYSE stocks (indexed k) as in §5, • For each stock k, observe its average trading price in hours h and h + 1 and compute a return r k,h = log(p k,h+1 /p k,h ). • For each stock k, use the prediction coefficients to predict a return r k,h and from this a discrepancy ∆ k,h = r k,h − r k,h • Identify the 5% of stocks with the largest (generally most positive) ∆ k,h , and the 5% with the smallest (generally most negative) discrepancy. • Divide a fixed stake (i.e., investment) equally among the first list and purchase them uniformly during hour h + 2 + S, where S ≥ 0 is a "staleness" parameter. (In §5 we did only the case S = 1.) Similarly go short on the second list in the total amount of the same stake. • Uniformly during hour h + 3 + S close all positions acquired in hour h + 2 + S so that the mean holding time is 1 hr. This strategy is not completely independent of the strategy used in §5. There, a stock's discrepancy was with respect to a baseline expectation of zero return. Here it is with respect to the prediction from all other stocks. Any number of variants in how many stocks to trade and in what proportions are of course possible. Figure 11 shows the result for varying values of the staleness parameter, and for both the measured correlation and the naive assumption of all stocks equally correlated. Comparing to Figure 6, one sees better performance, measured as mean annualized yield, by about a factor two, yielding approximately 99% annualized yield for the case of staleness S = 1 (that is, trading in hour h+2, immediately after hourly returns are calculated from prices in hours h and h + 1). The different colors in Figure 11 show that the staleness matters greatly. Delaying trades by one or more additional hours causes a precipitous drop in profits. At 24 hours delay, the arbitrage strategy fails completely. We see also that generally, but not always, using the measured correlations is superior to an unweighted average. Discussion That mean-reversion behavior on intraday timescales can be found in stock markets is not a new result. Indeed, ChatGPT-4, by design a font of conventional wisdom, opines that such microstructure noise can be due to spoofing, layering, overreactions, liquidity provision, profit-taking, order imbalances, and other kinds of bid-ask bounce [42]. What is perhaps surprising is that this mean-reversion is seen so continuously and systematically, over timescales from minutes to days, in > 1000 frequently traded NYSE stocks (Figures 2 and S2); that the effect is so close to a featureless Hurst exponential process with no characteristic timescale; and that it is, in backtesting with zero transaction costs, arbitrageable with annualized return on capital ∼ 50%. That using the information of the full correlation matrix further improves these arbitrage rates of return is likewise not a surprise. A minor surprise is that most of the benefit accrues from a simplistic model in which all pairs of stocks have the same value ρ (for any non-negligible numerical value), which leads to estimating any stock's "expected" return as an unweighted average of the normalized returns of all other stocks. This simplistic model explains close to 40% of stocks' hourly variances, as compared to about 50% explained using the full correlation matrix. Also interesting is that the latter figure hardly depends on which single year in the range 2018-2022 is used to estimate the correlations, demonstrating a perhaps surprising persistence of correlational structure even during and across high-volatility COVID-19 year 2022. We showed in §6.2 that the observed, counterintuitive increase of stock correlations with time can be explained quantitatively by positing that the mean-reverting component of individual stock prices does not share the same correlation with other stocks as the random-walk component. That would imply that while, on average, stocks slightly overreact to news, they do so independently of one another, not coherently with the average market, a statistical model embodied in equation (19). Alternative explanations might of course be possible. S1.2 Effect of Interval Averaging on Price-Difference Returns and Variogram Accuracy We next check how, with finite data, the measurement error in V (τ) might differ for the two calculational methods above. For this, we simulated 1000 one-year time spans of memoryless Gaussian returns at one-second intervals. In each one-minute interval, we calculated a candle of prices (open, high, low close). Now taking the candles as given data, we calculated V (τ)/τ, which should be constant, by three methods, using one year's data at a time: first, the difference-of-average method; second, the two-point difference method with point prices assumed known on a grid of spacing τ; third, the two-point difference method keeping one-minute resolution even for larger τ. Figure S1 shows the results. Each one-year time span's variogram is normalized to one at τ = 1 hr, and the quantile points (median, etc.) of the 1000 years are shown for other values of τ. Dispersions increase with increasing τ because the number of independent samples in a year decreases. The accuracies of all three methods are seen to be comparable, with method three a slight winner. However, both two-point difference methods show a bias that increases as τ decreases towards one minute. This arises most likely because the candle prices are not point prices but themselves averages. S1.3 Relation Between Fraction of Variance Explained (FVE) and Fractional Mean Square Error (FMSE) In the limited context of equations (29) and (31) above, we have measurements r and estimatesr, both of zero mean and scaled to the same variance. Then, FMSE = (r − r) 2 Var(r) = 2Var(r) − 2 rr Var(r) = 2 − 2ρ(37) where ρ is the correlation [ rr /Var(r)]. As is well known, the fraction of variance explained is ρ 2 , so solving equation (37) for ρ and squaring gives FVE = ρ 2 = 1 − 1 2 FMSE 2 (38) S2 Observed Variance of NYSE Stocks in Five Years S3 An Atlas of Correlation Embeddings for NYSE Stocks We constructed two-dimensional projections of the correlational structure of the 866 NYSE stocks that traded in all years, 2018-2022, using multidimensional scaling (MDS) [43] as implemented in Scikit-Learn [44]. (This differs slightly from the simplified description given in the main text.) The first figure following, Figure S3, gives a top-level view of all the stocks at once. In this and subsequent figures the distance between any two points i and j is, insofar as the projection will allow, 1 − ρ ij , where ρ ij is their pairwise correlation. It is important to understand that, because of the projection, clusters of points that are actually far apart may sometimes be superposed; but, conversely, pairs that are far apart are genuinely little correlated. In Figure S3, points are colored according to industry category, and the same colors are used in subsequent detail figures. In the detail figures ( Figures S4-S10), the five most frequently occurring industries are identified in the legend. Individual stocks are identified by their NYSE ticker symbols, the lookup of which can easily identify their respective industries. Each detail figure is re-projected by a separate MDS so as to better separate less-correlated points, so it is not just a magnification of its box in Figure S3. Figure S4: An embedding is re-computed for just the stocks in numbered box 1 in Figure S3, so as to better separate the industries in that box. Stocks are shown by their NYSE tickers and colored by their industry. (Inevitably there are chance collisions giving the same or similar colors to different industries.) The five most represented industries are named in the legend. In this projection, one sees a tight clustering of large (brown) and regional banks, closely flanked by life insurance, asset-management, and (orange) financial companies. On the other side of the figure, largely uncorrelated, one sees clusters of specialty chemicals and industrial machinery. Figure S5: Same as Figure S4, but for box 2 in Figure S3. This region of correlation space is dominated by fossil fuel production, with subclusters of exploration and production, equipment and services, midstream, refining, etc. Figure S6: Same as Figure S4, but for box 3 in Figure S3. The figure's left has related clusters of utilities, while its right shows the related household and personal products and packaged foods. Figure S8: Same as Figure S4, but for box 5 in Figure S3. Residential construction and retail apparel form unrelated tight clusters. Cruise lines (dark gray) and gaming and resort (rose) are small clusters near the center, with most other industries quite spread out. Figure S9: Same as Figure S4, but for box 6 in Figure S3. In this detail figure, and also the next, correlations are quite muted with individual tickers separated by distances implying ρ 0.8 even for stocks in the same industry. Figure S10: Same as Figure S4, but for box 7 in Figure S3. Figure 1 : 1Clock time versus dollar-trading-weighted and volume-trading-weighted times for a 24-hour clock interval in year 2021. The sharp rises at the beginning and end of the NYSE trading day are peaks in trading volume, including the opening and closing auctions. Almost no $TWor VWT time elapses between trading sessions. ( ) re 1 hr, 1091 NYSE stocks, clock time same, dollar-trade-weighted ($TW) time same, share-volume-weighted (VTW) time memoryless model, V( ) 1.0 decaying response model, V( ) 1.0 0.07 decaying response model, V( ) 1.0 0.26 Figure 2 : 2Observed variograms of 1,091 NYSE stock prices as a function of resolution time interval τ during year 2019. The ordinate is V (τ)/τ, so a memoryless random walk plots as constant. Both versions of transaction time, $TW and VTW (see text), yield results well fit by a small negative exponent −0.07. Clock time (unlike the transaction times) shows a steeper slope and an artifactual break at ∼ 1 hr. Figure approximately , V ,(τ) ∝ τ 1.0−0.07 , 0.06 hr < τ < 200 hr(2) Figure 4 : 4Yearly net returns for a simple long-short arbitrage strategy applied to simulations of fractional Brownian motion over 3600 1-year periods with hourly data in each. The Hurst exponent for the six cases shown is H = 0.5 − . The case = 0 is a true random walk for which arbitrage should not (on average) show a net profit, as is seen. arbitrage return (percent, not compounded) standard deviation of mean annual return on same scale Figure 5 : 5Means and standard deviations of annual net arbitrage returns like those shown in Figure 6 : 6Backtesting historical data, curves show the growth of a unit stake (uncompounded) under an arbitrage trading strategy that posits a long-memory power-law model with mean-regressing behavior. Also shown is growth of the S&P 500 index on the same scale. The right-hand scale shows the implied annualized yield of the end-point of each curve. NYSE stock pairs (with median) BAC vs. 64 NYSE-listed bank stocks (with median) Figure 7 : 7Pairwise correlations ρ for 100 randomly chosen pairs of NYSE stocks (red) as a function of time resolution τ, and correlations of 64 NYSE-listed bank stocks (blue) to Bank of America. Medians are shown as darker lines. The input data is year 2018 one-minute candles. The rising trend with τ of essentially all correlations is to be noted. Figure 8 8extends this observation across all pairs of NYSE stocks, but now normalizing the values ρ(τ) to a value 1.0 at τ = 1 hour. Instead of individual stock pairs, percentile sticks are shown, the same percentile points asFigure 2above. One sees a remarkable degree of universality in the pattern of increase of correlations ρ. However, the mere fact of increase is itself at first sight puzzling, since a simple correlation model, equation(18) immediately implies ρ AB (τ) = constant, since the factors in V (τ) cancel in equation(16). 1 hour, median all stock pairs correlation re 1 hour, median of BAC vs. 60 bank stocks reciprocal scaled variogram /V( ) re 1 hr, median of all stocks Figure 8 : 8Pairwise correlations as a function of resolution interval τ, now normalized to the value 1 at τ = 1 hour. Percentile sticks show 10%, 25%, 50%, 75%, and 90% percentile points for all pairs of stocks at each value τ. A surprisingly universal pattern of increase in ρ is seen. The inset linear-linear plot of the median curves illustrates that most of the increase is in the shortest times. See text for explanation of dashed black curve. Figure 9 : 9Monthly fractional mean-square error (FMSE, left scale) and fraction of variance explained (FVE, right scale) for correlational predictions of 1-hour returns of NYSE stocks. A single year (2018) is used to estimate the inverse correlation matrix, which is then applied across all years. The percentile sticks show the range of prediction accuracies over all 866 stocks. Figure 10 : 10Like Figure 11 : 11Similar toFigure 6, growth of a unit stake (uncompounded) under an arbitrage trading strategy based on the discrepancy between a stock's hourly return and the return predicted by its correlation with all other stocks. Dashed lines use the naive prediction with all stocks equally correlated. Solid lines use measured correlations of earlier year 2018. Colors show different assuptions for how soon observed returns can be acted on (how stale is the data). Also shown is growth of the S&P 500 index on the same scale. The right-hand scale shows the implied annualized yield of the end-point of each curve. Figure S1 : S1of-average method ( min resolution) two-point-difference method ( min resolution) two-point-difference method (1 min resolution) Three methods for estimating the variogram V (τ) applied to simulated one-minute price candles (open, high, low, close) in 1000 simulated one-year periods of memoryless Gaussian returns. Each variogram is normalized to one at one hour, with quantile points shown for other values of τ. For candle data, itself a kind of average, the difference-of-average method gives the most consistent results. Figure 2 2showed data from 2019 only. Here we show data for all years 2018-2022.Except for year 2020, results well fit by a small negative exponent −0.07. Year 2020 shows a similar trend, but exhibits some additional structure. Figure S2 : S2Observed variogram V (τ) for 1,091 NYSE stock prices as a function of resolution time interval τ during the indicated 1-year periods. The ordinate is V (τ)/τ, so a memoryless random walk plots as constant. Except for year 2020, results well fit by a small negative exponent −0.07. Figure S3 : S3The correlation coefficient ρ ij for hourly returns of all pairs (i, j) of 866 NYSE stocks is computed. The distance between pairs is defined as 1 − ρ ij . The resulting correlational structure is here projected into a two-dimensional space. Each dot is an individual stock, colored by its industry classification. The x and y axes each show the distance scale, but have no particular meaning (i.e., the figure could be arbitrarily rotated. The numbered boxes refer to subsequent Figures. Figure S7 : S7Same asFigure S4, but for box 4 inFigure S3. Gold stocks are prominent at the lower left. Software and IT cluster in the middle third, upper half. Table 1 : 1Fraction of Variance Explained (FVE), equation Supplementary Information S1 Further Detail of Calculations S1.1 Effect of Interval Averaging on Price-Difference Returns and Variogram AccuracyWe consider a memoryless Gaussian process. For simplicity, we calculate in the discrete-time case, that is, a random walk, then take the limit to a continuous process. Suppose a time series of 2M normal deviates r i , i = 1, . . . , M , r i ∼ N (0, 1), representing microscopic returns. Their cumulative sum,represents a time series of prices. We can calculate the macroscopic return over a time M in two ways. The pointprice method just differences the midpoints of the two consecutive intervals each of length M ,By the random-walk property, this has variance Var(R pp ) = M . 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[ "WEAKLY SUPERVISED SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES FOR TREE SPECIES CLASSIFICATION BASED ON EXPLANATION METHODS", "WEAKLY SUPERVISED SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES FOR TREE SPECIES CLASSIFICATION BASED ON EXPLANATION METHODS" ]
[ "Steve Ahlswede \nTechnische Universitaet Berlin\nGermany\n", "Nimisha Thekke Madam \nTechnische Universitaet Berlin\nGermany\n", "Christian Schulz \nTechnische Universitaet Berlin\nGermany\n", "Birgit Kleinschmit \nTechnische Universitaet Berlin\nGermany\n", "Begüm Demir \nTechnische Universitaet Berlin\nGermany\n" ]
[ "Technische Universitaet Berlin\nGermany", "Technische Universitaet Berlin\nGermany", "Technische Universitaet Berlin\nGermany", "Technische Universitaet Berlin\nGermany", "Technische Universitaet Berlin\nGermany" ]
[]
The collection of a high number of pixel-based labeled training samples for tree species identification is time consuming and costly in operational forestry applications. To address this problem, in this paper we investigate the effectiveness of explanation methods for deep neural networks in performing weakly supervised semantic segmentation using only image-level labels. Specifically, we consider four methods: i) class activation maps (CAM); ii) gradient-based CAM; iii) pixel correlation module; and iv) self-enhancing maps (SEM). We compare these methods with each other using both quantitative and qualitative measures of their segmentation accuracy, as well as their computational requirements. Experimental results obtained on an aerial image archive show that: i) considered explanation techniques are highly relevant for the identification of tree species with weak supervision; and ii) the SEM outperforms the other considered methods. The code for this paper is publicly available at https://git. tu-berlin.de/rsim/rs_wsss.
10.1109/igarss46834.2022.9884676
[ "https://arxiv.org/pdf/2201.07495v1.pdf" ]
246,035,404
2201.07495
5c493ecae54197131b95e225a0862c197c9fabf5
WEAKLY SUPERVISED SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES FOR TREE SPECIES CLASSIFICATION BASED ON EXPLANATION METHODS Steve Ahlswede Technische Universitaet Berlin Germany Nimisha Thekke Madam Technische Universitaet Berlin Germany Christian Schulz Technische Universitaet Berlin Germany Birgit Kleinschmit Technische Universitaet Berlin Germany Begüm Demir Technische Universitaet Berlin Germany WEAKLY SUPERVISED SEMANTIC SEGMENTATION OF REMOTE SENSING IMAGES FOR TREE SPECIES CLASSIFICATION BASED ON EXPLANATION METHODS Index Terms-Tree species mappingweakly super- vised learningsemantic segmentationexplanation methodsremote sensing The collection of a high number of pixel-based labeled training samples for tree species identification is time consuming and costly in operational forestry applications. To address this problem, in this paper we investigate the effectiveness of explanation methods for deep neural networks in performing weakly supervised semantic segmentation using only image-level labels. Specifically, we consider four methods: i) class activation maps (CAM); ii) gradient-based CAM; iii) pixel correlation module; and iv) self-enhancing maps (SEM). We compare these methods with each other using both quantitative and qualitative measures of their segmentation accuracy, as well as their computational requirements. Experimental results obtained on an aerial image archive show that: i) considered explanation techniques are highly relevant for the identification of tree species with weak supervision; and ii) the SEM outperforms the other considered methods. The code for this paper is publicly available at https://git. tu-berlin.de/rsim/rs_wsss. INTRODUCTION Accurate identification of tree species by the analysis of remote sensing (RS) images is important for various forestry applications. By reducing cost intensive on-site surveys, it significantly supports public authorities, conservation agencies and private owners in forest mapping and management. Tree species mapping can be achieved by using semantic segmentation methods, which aim to predict pixel-wise classification results on RS images. Deep learning (DL) based semantic segmentation methods have recently seen a rise in popularity in the context of tree species classification [1]. Most DL models require a high amount of labeled samples to optimize all parameters and reach a high performance of tree species classification. The labeling of samples can be achieved based on: 1) in situ ground surveys; 2) the expert interpretation of color composites (image photo-interpretation); or 3) hybrid solutions where both photo-interpretation and ground surveys are exploited [2]. Collection of a sufficient number of high quality pixel-level labels associated to tree species can be time consuming, complex and costly. To address this problem, in this paper we focus our attention on weakly supervised semantic segmentation (WSSS) for tree species segmentation, which rely on weak supervision (i.e., image-level labels). The use of image-level labels can significantly reduce the annotation cost and effort in forestry applications. However, such supervision indicates only the existence of certain tree species assigned to images without their exact pixel based location information (which is essential for obtaining tree species maps). To obtain accurate tree species maps at the pixel-level, in this paper we investigate the effectiveness of explainable neural networks in the context of WSSS. Explanation methods are capable of generating explanations in the form of pixel-level heatmaps that can be highly relevant for providing a pixel-level tree species map from a DL model trained using image level labels. In this paper, we consider four explanation methods: i) class activation maps (CAM) [3]; ii) Gradient-based CAM (GradCAM) [4]; iii) pixel correlation module (PCM) [5]; and iv) self-enhancing maps (SEM) [6]. All considered methods have been experimentally compared in terms of their: 1) capability to perform accurate tree species mapping; 2) model complexity; and 3) segmentation time. This work is the first, to the best of our knowledge, that explores the potential of pixel-level explanation heatmaps for tree species mapping in a weak supervision framework. TREE SPECIES MAPPING FROM IMAGE LEVEL LABELS Let X = {X 1 , · · · , X M } denote an archive of M RS images acquired over forestry areas, where the m-th image within the archive is represented by X m . We assume that a training set X T ⊂ X is available where each image X n within the set is associated with a multi-label vector y n = [y n,1 , · · · , y n,S ], where S is the total number of classes, indicating which tree species are present. Here, y n,s takes the value of 1 if the class s is present in X n , and 0 otherwise. The set of all multi-label vectors corresponding to X T is thus denoted as Y T . To obtain tree-species maps by using image-level labels (and thus weak supervision), we consider a convolutional neural network (CNN) which learns a mappingŷ n = U (X n ), where U (·) represents the CNN. However, the predicted output vectorŷ n by the direct application of a CNN does not provide any spatial information regarding class locations or extents, which are required for tree species mapping. To obtain the map at pixel-level, we investigate CAM, GradCAM, PCM, and SEM in the context of WSSS and exploit the class specific heatmaps (which show the probability of a given class at each pixel) derived by these methods. Specifically, feature maps F ∈ R C×H×W (C, H and W represent the channels, height, and width of the feature maps, respectively) from the final layer of the trained model U (·) corresponding to an image X n are exploited. The importance of each feature map with respect to each class is derived and a weighted linear combination of the feature maps form the class specific heatmaps. Each class specific heatmap has its values normalized between 0-1, thus acting as pixel probabilities. Heatmaps are thus used to obtain the semantic segmentation by assigning class labels to pixels where the class has the highest probability. Specific details on how the heatmaps are derived from the four methods are given in the following sections. Class Activation Maps CAMs aim to highlight the image regions which are used by the CNN to classify an image to a given class. This is achieved by establishing a relationship between the set of feature maps F and a given class. To this end, a 1x1 convolution layer can be applied after F, which takes F as input and outputs S feature maps, generating a set of CAMs (A ∈ R S×H×W ) [3]. Thus, for a given class s, we obtain its CAM as such: A s = C c=1 w c s F c(1) where w c s is the weight of importance for the c th feature map with respect to the s th class. The CAM is then averaged and used as the final class prediction scores (image level prediction): y n,s = 1 HW H i=1 W j=1 A s,i,j(2) where i and j are the row and column indices of A s , respectively. Given this formulation, CAMs can be obtained in an endto-end manner without any post-processing steps. Gradient Based CAM (GradCAM) GradCAM [4] is a reformulation of CAM. Instead of using learned weights from a 1x1 convolution to determine the importance of each feature map, GradCAM uses the average gradient associated with each feature map in F. Specifically, we first feed an image X n through the network in order to obtain the vector of predicted class scoresŷ n . Then, we perform back-propagation with respect to a single class predictionŷ n,s in order to obtain the gradients for the feature maps F with respect toŷ n,s . Finally, we take the mean of the gradients in order to obtain the importance of feature map F c for class s: w c s = 1 HW W i=0 H j=0 δŷ n,s δF c i,j(3) We can thus obtain the GradCAM for a given class by replacing w c s in (1) with the definition given in (3). One distinct advantage of GradCAM is that it can be applied to any network architecture as it gets the importance weights of each feature map using gradient scores. However, the WSSS cannot be obtained in an end-to-end manner, as it requires the back-propagation post-processing. We would like to emphasize that both CAM and Grad-CAM have been recently applied for land cover mapping in RS. For details, we refer the reader to [7,8]. Pixel Correlation Module (PCM) The PCM [5] aims to enhance CAM in order to improve the segmentation. PCM avoids the need for post-processing as the module is integrated into the network itself, thus allowing for end-to-end processing. The module consists of three parallel 1x1 convolutional layers which takes feature maps from different levels of U (·), along with the original image, as input in order to embed them into latent space. The embedded outputs are then concatenated and passed through one additional 1x1 convolution which gives us a matrix of feature maps K ∈ R C×H×W which correspond to an image X n and contain a combination of high and low level features. From this matrix, attention scores (attn q,r ) are obtained which provide the similarity between the features at two spatial locations (e.g. q and r) within K through a self-attention mechanism as: attn q,r = K T q K r ||K q || · ||K r ||(4) where q and r each represent a pixel location (e.g. q = (0, 0), r = (0, 1)). For a given class s, the PCM value at pixel location q (P s,q ) is obtained by taking the weighted sum of all values in A s , where the weight values are the cosine similarity estimated between the feature vector at q and all other feature vectors across the spatial dimension, as in (4). This is be formulated as: P s,q = ∀r attn q,r × A s,r As opposed to [5] where PCM is trained via supervision from a Siamese network and equivariant cross regularization loss, our work trains the module as a separate branch. To this end, we placed PCM after the feature extraction backbone and trained it using class predictions made from the output of the module. Thus, when training the model which includes PCM, we use two binary cross entropy loss terms, one from the classification head, and one from the PCM module. Self-Enhancement Maps (SEM) SEM [6] also aims to enhance the CAM. SEM works on the principle that feature vectors located within target object regions have higher similarity than those within different class regions. Here, input image X n is passed through the network U (·) in order to obtain A and the feature maps F. Given a class specific A s , E seed coordinates corresponding to locations with the largest values are chosen. The feature vectors corresponding to the locations of the E seed points are then extracted from F along the channel dimension. The cosine similarity defined in (4) is then calculated between the E seed feature vectors and all other feature vectors in F. This results in E similarity maps, where the value for a given pixel r within the e th similarity map is the cosine similarity score (4) between feature at r and the e th feature. The final output for the given class is then obtained by taking the maximum value at each pixel across the E similarity maps. For each of the methods from the previous four sections, we obtain the final tree species map by applying the argmax function across the outputs of the respective method for classes predicted above a threshold τ at the image level. EXPERIMENTAL RESULTS Experiments were carried out on a dataset acquired from 2012 -2020 across the German federal state of Lower Saxony. The dataset contains a total of 19,995 aerial images, each of which consists of RGB and near infrared bands with a spatial resolution of 0.2m. The dataset was randomly divided into training (70%), validation (15%), and test (15%) sets. Dominant tree species within stand polygons which overlapped an image patch were used to attain the image-level labels. To evaluate the WSSS results, we rasterized the polygons corresponding to the test set in order to obtain pixel-wise labels. A total of five classes were extracted from the polygons: Pine (Pinus spp.), Spruce (Picea spp.), Beech (Fagus spp.), Oak (Quercus spp.), and Cleared. Cleared represents any open areas without tree crown cover (e.g. meadows, clear-cuts, water bodies, etc.). For the experiments we used DeepLabv3+ [9]. Given that the architecture was designed for semantic segmentation, we replace the segmentation head with a multi-label classification head in order to obtain predictions at the image level. For evaluation of the WSSS performance we utilized F 1 , and also examined the number of model parameters (in millions) and the segmentation time per image. All results were obtained using τ = 0.5. The number of seeds used in SEM was chosen empirically based on the performance across the validation image set, resulting in a seed value of E = 10. From Table 1 we can see that SEM provides the highest F 1 score, whereas CAM leads to the lowest F 1 score. Looking at the qualitative results in Fig. 1, we see that CAM is able to correctly localize class regions, but oversegments the dominant class within the image. The performance of CAM is greatly improved when SEM is applied (Fig. 1a, (Fig. 1a, b). GradCAM often made predictions visually similar to SEM (Fig. 1a). However, GradCAM also predicted incorrect regions (Fig. 1b), leading to a degradation in performance. PCM was able to correctly localize class regions, but tended to oversegment the dominant class within the image, leading to misclassified regions (Fig. 1a, b). This was likely due to how the module was trained, as the original implementation involved a Siamese network with a crossregularization loss. This suggests that PCM may not be a viable addition to standard CNN architectures. One area where all methods struggled was when two classes of the same type (e.g. broadleaf or coniferous) were predicted within an image (Fig. 1c). Here, the classes are not well separated within the learned feature space of the classifier, leading to misclassifications. In regards to segmentation time, CAM had the lowest time, while GradCAM had the highest time. This was due to the need for post-processing step involving a forward and backward pass through the network for each predicted class within an image with GradCAM. SEM also required a postprocessing step, but managed to maintain a low segmentation time. This is because SEM only performs self-attention for a given number of seeds, thus making it less computational than typical self-attention mechanisms. The main drawback with SEM is that more computational time must be invested into tuning the seed hyperparameter, which can become time consuming with larger datasets. With respect to model parameters, only PCM increased model parameters. However, the additional parameters did not achieve any benefits in segmentation performance with respect to GradCAM and SEM. CONCLUSION In this paper, we have studied four explanation methods in the framework of WSSS to obtain pixel-level tree species maps using training samples annotated by image-level labels. In detail, we have investigated: i) CAM; ii) GradCAM; iii) PCM; and iv) SEM based on their semantic segmentation performance, number of model parameters, and semantic segmentation time. The theoretical and experimental analysis show that for tree species segmentation problems, SEM can be chosen as it: i) yields the highest segmentation accuracy; ii) provides the lowest model complexity; and iii) requires a low segmentation time. As a future work, we plan to assess the ability Image Reference Map CAM [3] GradCAM [4] PCM [5] SEM [6] a) b) c) Cleared Pine Spruce Oak Beech Fig. 1. Examples of images, their reference and segmentation maps obtained by using different explanation methods in the framework of WSSS. of explanation methods in providing pixel-level pseudo-labels for training a semantic segmentation model. ACKNOWLEDGMENTS This work is funded by the European Research Council (ERC) through the ERC-2017-STG BigEarth Project under Grant 759764 and by the German Federal Ministry of Education and Research through TreeSatAI project under Grant 01lS20014A. We also thank the State Forest of Lower Saxony for providing the aerial images and stand level data. Table 1 . 1Comparison of the considered explanation methods in terms of F 1 scores, the number (in millions) of parameters (# Param) and the semantic segmentation time (Seg.Time) Mapping forest tree species in high resolution uav-based rgb-imagery by means of convolutional neural networks. F Schiefer, T Kattenborn, A Frick, J Frey, P Schall, B Koch, S Schmidtlein, ISPRS Journal of Photogrammetry and Remote Sensing. 170F. Schiefer, T. Kattenborn, A. Frick, J. Frey, P. Schall, B. Koch, and S. Schmidtlein, "Mapping forest tree species in high res- olution uav-based rgb-imagery by means of convolutional neu- ral networks," ISPRS Journal of Photogrammetry and Remote Sensing, vol. 170, pp. 205-215, 2020. Batch-mode activelearning methods for the interactive classification of remote sensing images. B Demir, C Persello, L Bruzzone, IEEE Transactions on Geoscience and Remote Sensing. 493B. Demir, C. Persello, and L. Bruzzone, "Batch-mode active- learning methods for the interactive classification of remote sensing images," IEEE Transactions on Geoscience and Remote Sensing, vol. 49, no. 3, pp. 1014-1031, 2011. Adversarial complementary learning for weakly supervised object localization. X Zhang, Y Wei, J Feng, Y Yang, T S Huang, IEEE Conference on Computer Vision and Pattern Recognition. X. Zhang, Y. Wei, J. Feng, Y. Yang, and T. S. Huang, "Ad- versarial complementary learning for weakly supervised object localization," in IEEE Conference on Computer Vision and Pat- tern Recognition, 2018, pp. 1325-1334. Grad-cam: Visual explanations from deep networks via gradient-based localization. R R Selvaraju, M Cogswell, A Das, R Vedantam, D Parikh, D Batra, IEEE International Conference on Computer Vision. R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, "Grad-cam: Visual explanations from deep net- works via gradient-based localization," in IEEE International Conference on Computer Vision, 2017, pp. 618-626. Selfsupervised equivariant attention mechanism for weakly supervised semantic segmentation. Y Wang, J Zhang, M Kan, S Shan, X Chen, IEEE/CVF Conference on Computer Vision and Pattern Recognition. Y. Wang, J. Zhang, M. Kan, S. Shan, and X. Chen, "Self- supervised equivariant attention mechanism for weakly super- vised semantic segmentation," in IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 12275- 12284. Rethinking localization map: Towards accurate object perception with self-enhancement maps. X Zhang, Z Wei, Y Yang, F Wu, arXiv:2006.05220arXiv preprintX. Zhang, Z. Wei, Y. Yang, and F. Wu, "Rethinking localization map: Towards accurate object perception with self-enhancement maps," arXiv preprint arXiv:2006.05220, 2020. Accurate cloud detection in high-resolution remote sensing imagery by weakly supervised deep learning. Y Li, W Chen, Y Zhang, C Tao, R Xiao, Y Tan, Remote Sensing of Environment. 250112045Y. Li, W. Chen, Y. Zhang, C. Tao, R. Xiao, and Y. Tan, "Accu- rate cloud detection in high-resolution remote sensing imagery by weakly supervised deep learning," Remote Sensing of Envi- ronment, vol. 250, pp. 112045, 2020. A comprehensive analysis of weakly-supervised semantic segmentation in different image domains. L Chan, M S Hosseini, K N Plataniotis, International Journal of Computer Vision. 1292L. Chan, M. S. Hosseini, and K. N. Plataniotis, "A compre- hensive analysis of weakly-supervised semantic segmentation in different image domains," International Journal of Computer Vision, vol. 129, no. 2, pp. 361-384, 2021. Encoder-decoder with atrous separable convolution for semantic image segmentation. 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[ "ORACLE-FREE REINFORCEMENT LEARNING IN MEAN-FIELD GAMES ALONG A SINGLE SAMPLE PATH *", "ORACLE-FREE REINFORCEMENT LEARNING IN MEAN-FIELD GAMES ALONG A SINGLE SAMPLE PATH *" ]
[ "Muhammad Aneeq Uz Zaman ", "Alec Koppel [email protected] ", "J P Morgan ", "Sujay Bhatt [email protected] ", "J P Morgan ", "Tamer Başar [email protected] ", "\nCoordinated Science Laboratory\nUniversity of Illinois at Urbana-Champaign\nAI Research\n383 Madison Ave10017New YorkNY\n", "\nCoordinated Science Laboratory University of Illinois at Urbana-Champaign\nAI Research\n383 Madison Ave10017New YorkNY\n" ]
[ "Coordinated Science Laboratory\nUniversity of Illinois at Urbana-Champaign\nAI Research\n383 Madison Ave10017New YorkNY", "Coordinated Science Laboratory University of Illinois at Urbana-Champaign\nAI Research\n383 Madison Ave10017New YorkNY" ]
[]
We consider online reinforcement learning in Mean-Field Games (MFGs). Unlike traditional approaches, we alleviate the need for a mean-field oracle by developing an algorithm that approximates the Mean-Field Equilibrium (MFE) using the single sample path of the generic agent. We call this Sandbox Learning, as it can be used as a warm-start for any agent learning in a multi-agent non-cooperative setting. We adopt a two time-scale approach in which an online fixed-point recursion for the mean-field operates on a slower time-scale, in tandem with a control policy update on a faster time-scale for the generic agent. Given that the underlying Markov Decision Process (MDP) of the agent is communicating, we provide finite sample convergence guarantees in terms of convergence of the mean-field and control policy to the mean-field equilibrium. The sample complexity of the Sandbox learning algorithm is O(ǫ −4 ) where ǫ is the MFE approximation error. This is similar to works which assume access to oracle. Finally, we empirically demonstrate the effectiveness of the sandbox learning algorithm in diverse scenarios, including those where the MDP does not necessarily have a single communicating class.Recent Work of Yardim et al. (2022) also deals with RL for MFGs in an oracle-free setting, where N agents independently running policy mirror ascent in a multi-loop algorithm are shown to provably approximate the MFE. One significant difference is that Yardim et al. (2022) uses N -sample paths, compared to the single sample path of our work, to obtain the MFE inÕ(ǫ −2 ) time-steps albeit with a bias ofÕ(1/ √ N ). Thus accounting for the N sample paths the sample complexity of their algorithm isÕ(ǫ −4 ), which is similar to the sample complexity of our work. But case? arXiv preprint arXiv:2111.02024, 2021.Kai Cui and Heinz Koeppl. Approximately solving mean field games via entropy-regularized deep reinforcement the properties of the softmax function with application in game theory and reinforcement learning. arXiv preprint arXiv:1704.00805, 2017. madi. A multi-agent deep reinforcement learning approach for a distributed energy marketplace in smart grids. In . Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria. IEEE Transactions on Automatic Control, 52(9):1560-1571, 2007. . Mean-field-game model for botnet defense in cyber-security. Applied Mathematics & Optimization, 74:669-692, 2016. Aimé Lachapelle and Marie-Therese Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transportation Research Part B: Methodological, 45(10):1572-1589, 2011. Daniel Lacker and Thaleia Zariphopoulou. Mean field and n-agent games for optimal investment under relative performance criteria. Mathematical Finance, 29(4):1003-1038, 2019. Jean-Michel Lasry and Pierre-Louis Lions. Jeux à champ moyen. i-le cas stationnaire. Comptes Rendus Mathématique, 343(9):619-625, 2006.
10.48550/arxiv.2208.11639
[ "https://export.arxiv.org/pdf/2208.11639v3.pdf" ]
251,765,569
2208.11639
815b3925469475ee627e77a2ca345243af5abd25
ORACLE-FREE REINFORCEMENT LEARNING IN MEAN-FIELD GAMES ALONG A SINGLE SAMPLE PATH * 11 Apr 2023 Muhammad Aneeq Uz Zaman Alec Koppel [email protected] J P Morgan Sujay Bhatt [email protected] J P Morgan Tamer Başar [email protected] Coordinated Science Laboratory University of Illinois at Urbana-Champaign AI Research 383 Madison Ave10017New YorkNY Coordinated Science Laboratory University of Illinois at Urbana-Champaign AI Research 383 Madison Ave10017New YorkNY ORACLE-FREE REINFORCEMENT LEARNING IN MEAN-FIELD GAMES ALONG A SINGLE SAMPLE PATH * 11 Apr 2023Urbana IL 61801-2307 Urbana IL 61801-2307 We consider online reinforcement learning in Mean-Field Games (MFGs). Unlike traditional approaches, we alleviate the need for a mean-field oracle by developing an algorithm that approximates the Mean-Field Equilibrium (MFE) using the single sample path of the generic agent. We call this Sandbox Learning, as it can be used as a warm-start for any agent learning in a multi-agent non-cooperative setting. We adopt a two time-scale approach in which an online fixed-point recursion for the mean-field operates on a slower time-scale, in tandem with a control policy update on a faster time-scale for the generic agent. Given that the underlying Markov Decision Process (MDP) of the agent is communicating, we provide finite sample convergence guarantees in terms of convergence of the mean-field and control policy to the mean-field equilibrium. The sample complexity of the Sandbox learning algorithm is O(ǫ −4 ) where ǫ is the MFE approximation error. This is similar to works which assume access to oracle. Finally, we empirically demonstrate the effectiveness of the sandbox learning algorithm in diverse scenarios, including those where the MDP does not necessarily have a single communicating class.Recent Work of Yardim et al. (2022) also deals with RL for MFGs in an oracle-free setting, where N agents independently running policy mirror ascent in a multi-loop algorithm are shown to provably approximate the MFE. One significant difference is that Yardim et al. (2022) uses N -sample paths, compared to the single sample path of our work, to obtain the MFE inÕ(ǫ −2 ) time-steps albeit with a bias ofÕ(1/ √ N ). Thus accounting for the N sample paths the sample complexity of their algorithm isÕ(ǫ −4 ), which is similar to the sample complexity of our work. But case? arXiv preprint arXiv:2111.02024, 2021.Kai Cui and Heinz Koeppl. Approximately solving mean field games via entropy-regularized deep reinforcement the properties of the softmax function with application in game theory and reinforcement learning. arXiv preprint arXiv:1704.00805, 2017. madi. A multi-agent deep reinforcement learning approach for a distributed energy marketplace in smart grids. In . Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria. IEEE Transactions on Automatic Control, 52(9):1560-1571, 2007. . Mean-field-game model for botnet defense in cyber-security. Applied Mathematics & Optimization, 74:669-692, 2016. Aimé Lachapelle and Marie-Therese Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transportation Research Part B: Methodological, 45(10):1572-1589, 2011. Daniel Lacker and Thaleia Zariphopoulou. Mean field and n-agent games for optimal investment under relative performance criteria. Mathematical Finance, 29(4):1003-1038, 2019. Jean-Michel Lasry and Pierre-Louis Lions. Jeux à champ moyen. i-le cas stationnaire. Comptes Rendus Mathématique, 343(9):619-625, 2006. INTRODUCTION Mean-Field Game (MFG) framework, concurrently introduced by Huang et al. (2006,2007) and Lasry andLions (2006, 2007), addresses some of the challenges faced by the widely applicable Multi-Agent Reinforcement Learning (MARL) framework Shoham et al. (2007); Ghasemi et al. (2020); Zhang et al. (2021); Mao et al. (2022). In particular, MFG framework captures the limiting case where the number of agents N → ∞ and this deals with the nonstationarity of the environment caused by agents best responding to each other -referred to as the "curse of many agents" Sonu et al. (2017). In the infinite population setting, the effect of individual deviation becomes negligible causing any strategic interaction among the agents to disappear. As a result, it becomes sufficient to consider without loss of generality the interaction between a generic agent and the aggregate behavior of other agents (the mean-field). The solution concept used in MFGs (analog of Nash equilibrium) is called the Mean-Field Equilibrium (MFE). The MFE prescribes a set of control policies which are known to be ǫ-Nash for a large class of N -agent games Saldi et al. (2018), such that ǫ → 0 as N → ∞. Hence finding the MFE presents a viable method to solving large population games. In this work we propose an RL algorithm to approximate the (stationary) MFE Guo et al. (2019); Xie et al. (2021) without assuming access to a mean-field oracle (henceforth referred to as oracle). Most literature in RL for MFGs assumes access to such an oracle, which is capable of simulating the aggregate behavior of a large number of agents under a given control policy. But this assumption may be prohibitive and the generic agent may not have access to such an oracle, but only knows its own state, action and reward sequence. Hence the question arises: Can the generic agent provably learn the stationary MFE without access to a mean-field oracle? We answer this question in the affirmative by proposing an RL algorithm which computes the MFE without access to an oracle, but instead using the single sample path of the agent (without re-initializations) to approximate the aggregate behavior of large number of agents. We also provide high confidence finite sample bounds for approximation of the MFE to an arbitrary degree. We term this learning approach Sandbox Learning, since it allows an agent to approximate equilibrium policies in a multi-agent non-cooperative environment, without interacting with other agents or an oracle. As a result, sandbox learning can be used to provide a warm-start to agents before entering an N -agent non-cooperative learning environment. Main Results Our core technical insight is that, instead of assuming access to the oracle, the problem may be cast as a stochastic fixed point problem using the generic agent's single sample path, thus allowing development of oracle-free RL algorithm for the MFG. In contrast, prior works require access to mean-field oracle Guo et al. (2019); Xie et al. (2021); Anahtarcı et al. (2019);Fu et al. (2020),which is a strong assumption, as it implicitly assumes the knowledge of the distribution of all other agents, which never holds in practice.The main results of the paper are as follows. 1. To efficiently learn the MFE and avoid degenerate policies, the Sandbox learning algorithm simultaneously updates the mean-field and the policy of the agent. This simultaneous update induces a time-varying Markov Chain (MC) for the generic agent which complicates the analysis of the algorithm. In Section 3, we craft episodic learning rates for the sole purpose of making the MC slowly time-varying inside the episode, making the algorithm amenable for analysis. 2. In Section 4, we provide finite sample analysis of Q-learning and dynamics matrix estimation under the slowly time-varying MC setting, using a communicating MDP condition from literature Arslan and Yüksel (2016). This condition generalizes the pre-existing conditions for RL-MFGs in literature. The slowly time-varying MC setting is shown to introduce a small drift in the approximation error, which can be reduced by slowing the inter-episodic learning. Lemmas 2 and 3 might be of independent interest to researchers in RL for time-varying MDPs. 3. The estimates of Q-function and dynamics matrix are used to construct approximate optimality and consistency operators, respectively. These operators are used to update the policy and mean-field using two time-scale learning. Finally in Section 4 Corollary 1, we obtain finite sample convergence bounds of this two time-scale algorithm to an ǫ-neighborhood of stationary MFE, under a standard contraction mapping assumption. 4. In Section 5, we numerically illustrate the effectiveness of the Sandbox learning algorithm on a congestion game. We empirically demonstrate that the Sandbox learning algorithm performs well even in the absence of the communicating MDP assumption, if there is a single closed communicating class. This is due to the fact that the MC transitions to the communicating class in finite time. Proofs of theoretical claims are provided in the Appendix. Relevant Literature The work most closely related to this paper is Angiuli et al. (2022) which uses a unified-RL algorithm to solve the MFG problem in cooperative and non-cooperative settings, but lacks rigorous analysis of the RL algorithm. The key differences are that (a) the algorithm in Angiuli et al. (2022) relies on re-initializations while our algorithm operates on a single sample path, (b) the algorithm proposed in Angiuli et al. (2022) updates the Q-function at a faster time-scale while ours updates the control policy at a faster time-scale, and (c) we explicitly define the learning rates to have a certain episodic structure. These differences are shown to be pivotal in obtaining the finite sample convergence bounds for the Sandbox learning algorithm. due to the usage of N -sample paths Yardim et al. (2022) has a better time complexity than our work. In addition to the standard contraction mapping assumption in MFG literature (Yardim et al. (2022) Proposition 2 & Theorem 2), they also assume contraction (Yardim et al. (2022) Assumption 2) of the mean-field update under any given policy π. This is assumption requires ergodicity of the generic agent's Markov chain under any policy π, which is stronger than the communicating MDP assumption of our work (Assumption 2). This ergodicity condition allows the empirical mean-field of the N agents to mix (given a sufficiently long time), and approximate the true mean-field. The ergodicity assumption may not hold in scenarios such as congestion games, as some policies (e.g. always move to neighboring state) may lead to periodicity in the Markov chain. Since our work does not assume ergodicity, the Sandbox learning algorithm approximates the mean-field by instead relying on good estimation of the transition probability matrix (Section 3.1) of the agent under the communicating MDP assumption. Finally, the persistent excitation assumption (Yardim et al. (2022) Assumption 3) imposes two restrictions on the policy class: (i) the support of the class of policies is non-zero everywhere, and (ii) the policy class is (Bellman) complete. We explicitly include randomness to impose the first restriction, however, we do not require the stronger completeness assumption to obtain the results in our paper. Below we provide a FORMULATION & BACKGROUND Consider an infinite horizon N -agent game over finite state and action spaces S and A, respectively. The state and action of agent i ∈ [N ] at time t are denoted by s i t ∈ S and a i t ∈ A, respectively. Agent i's initial state is drawn from a distribution s i 1 ∼ p 1 ∈ P(S), and the state dynamics of the agent is coupled with the other agents through the empirical distribution e N t := 1 N j∈[N ] 1{s j where π i := (π i 1 , π i 2 , . . . ) ∈ Π i is the policy of agent i and π −i := {π j } j∈[N ]\i is the concatenation of policies of all other agents. In an N -agent non-cooperative game, the dominant solution concept is a Nash equilibrium, where none of the agents can increase their total reward by unilaterally deviating from its Nash policy. Based upon this notion, we define an ǫ-Nash equilibrium as follows. Definition 1 (Başar and Olsder (1998)). A set of policies π * = {π 1 * , . . . , π N * } is termed an ǫ-Nash equilibrium if ∀i ∈ [N ], V i (π i * , π −i * ) + ǫ > V i (π i , π −i * ), ∀π i ∈ Π i . If ǫ → 0, ǫ-Nash approaches Nash equilibirum. Due to the exponential dependence on the number of agents N required to compute exact Nash equilibria Başar and Olsder (1998), we restrict focus to computing ǫ-Nash equilibria. In the case that the number of agents N → ∞, known as the mean-field equilibrium (MFE), one obtains an ǫ-Nash equilibrium Saldi et al. (2018); Moon and Başar (2014), specifically, ǫ → 0 as N → ∞. Therefore, subsequently, we focus on the MFG, the infinite population analog of the N -agent game.The empirical distribution is replaced in that case by a mean field distribution µ = lim N,t→∞ e N t , its infinite population stationary counterpart. The stationary MFE of the MFG is guaranteed to exist under certain Lipschitzness assumptions Saldi et al. (2018); Jovanovic and Rosenthal (1988) (Assumption 1). As in the N -agent game, the generic agent in a MFG has state space S, action space A, and the initial distribution of its state is p 1 ∼ P(S). Next, we define the agent's transition dynamics (1) and total reward (2) in the mean-field setting with mean-field µ ∈ P(S): s t+1 ∼ P (· | s t , a t , µ), s 1 ∼ p 1 , a t ∼ π(s t , µ).(3) The actions of the generic agent are generated using a stationary stochastic policy π ∈ Π := {π : S × P(S) → P(A)}. We restrict ourselves to the set of stationary policies, without loss of generality, since the optimal control policy for an MDP induced by stationary µ is also stationary Puterman (2014). The instantaneous reward r t accrued to a generic agent at time t is dependent on its state, control action, and the mean-field, that is, r t = R(s t , a t , µ). The generic agent aims to maximize its total discounted reward given the mean-field µ and with the discount factor 0 < ρ < 1, V π,µ := E ∞ t=1 ρ t R(s t , a t , µ) | s 1 ∼ p 1 .(4) Next we define the Mean-Field Equilibrium (MFE) by introducing two operators. First define the optimality operator Γ 1 (µ) := argmax π V π,µ as the operator which outputs the optimal policy for the MDP induced by mean-field µ. We consider policies where the probability is split evenly among optimal actions for a given state and mean-field. We also define Γ 2 (π, µ) as the consistency operator which computes mean-field consistent with the policy π and mean-field µ. If µ ′ = Γ 2 (π, µ), then ∀s ′ ∈ S µ ′ (s ′ ) = (s,a)∈S×A P (s ′ | s, a, µ)π(a | s, µ)µ(s).(5) This is also referred to as the Fokker-Planck-Kolmogorov equation in the literature Bensoussan et al. (2015), and versions of it appear in the literature on probability flow equations in MDPs Puterman (2014). Consistency means that if infinitely many agents (with initial distribution µ) follow a control policy π, the resulting distribution will be µ ′ . Using these two operators, we can define the MFE of the MFG as follows. Definition 2 (Saldi et al. (2018)). The pair (π,μ) is an MFE of the MFG ifπ = Γ 1 (μ) andμ = Γ 2 (π,μ). Intuitively this two-part coupled definition can be interpreted as (1)π is the optimal policy for the MDP induced by mean-fieldμ, and (2) mean-fieldμ is consistent with the control policyπ. A naive way of approximating the MFE could be through repeated use of the composite operator Γ 2 (Γ 1 (·), ·) but this iteration is known to be noncontractive (Cui and Koeppl (2021)). Instead we replace Γ 1 (·) with the approximate optimality operator Γ λ 1 (µ) := softmax λ (·, Q * µ ), where Q * µ is the Q-function of the MDP induced by mean-field µ and, the softmax λ (·) function is defined as softmax λ (s, Q) a := exp(λQ(s, a)) a ′ ∈A exp(λQ(s, a ′ )) ,(6) ∀s ∈ S, ∀a ∈ A. Evidently as λ → ∞, Γ λ 1 → Γ 1 . Next using the approximate optimality operator Γ λ 1 we define an approximate MFE known as Boltzman-MFE (B-MFE). (2021)). For a given λ > 0, the pair (π * , µ * ) is a Boltzman-MFE (B-MFE) of the MFG if π * = Γ λ 1 (µ * ) and µ * = Γ 2 (π * , µ * ). The Boltzman-MFE is an approximate MFE and approaches the MFE as λ → ∞ (Theorem 4, Cui and Koeppl (2021)). Henceforth, we will devote ourselves to finding the B-MFE for a large enough λ, so as to closely approximate the MFE. Assumption 1. There exists a λ > 0 and Lipschitz constants d 1 , d 2 and d 3 such that Definition 3 (Cui and Koeppl Γ λ 1 (µ) − Γ λ 1 (µ ′ ) T V ≤ d 1 µ − µ ′ 1 , Γ 2 (π, µ) − Γ 2 (π ′ , µ) 1 ≤ d 2 π − π ′ T V , Γ 2 (π, µ) − Γ 2 (π, µ ′ ) 1 ≤ d 3 µ − µ ′ 1 and d := d 1 d 2 + d 3 < 1 for policies π, π ′ ∈ Π and mean-fields µ, µ ′ ∈ P(S). Assumption 1 is guaranteed to be true for a small enough λ > 0 Cui and Koeppl (2021). This results in a trade-off as higher values of λ increase closeness between MFE and B-MFE, but may cause Assumption 1 to be violated, and vice-versa. This issue is well-known in MFGs over finite state and action spaces Cui and Koeppl (2021). Contraction mapping conditions (as in Assumption 1) 2018)). In the next section, we propose an RL algorithm to approximate the B-MFE without access to a mean-field oracle, by utilizing the sample path of a generic agent itself. SANDBOX REINFORCEMENT LEARNING Consider a setting where a generic agent has no knowledge of the transition probability P , the functional form of the reward R or a mean-field oracle, which is often required in such studies -see Guo et al. Cui and Koeppl (2021). In this section, we propose a Sandbox RL algorithm to compute the B-MFE. Our methodology operates by updating the mean-field and the control policy concurrently using approximations of the optimality and consistency operators, Γ λ 1 and Γ 2 , respectively, defined prior to Definition 3. The approximation to Γ λ 1 is defined by softmax λ (·) of estimated Q-function obtained using Q-learning update, whereas approximation of operator Γ 2 relies on estimating the transition probabilities of the Markov Chain (MC) of the generic agent. But the concurrent update of mean-field and control policy causes the MC of the generic agent to be time-varying. This timevarying MC setting may cause instability in the approximation of the operators, resulting in divergence of mean-field and control policy updates. To ensure good approximation of operators, we adopt an episodic two time-scale learning rate as shown in Figure 1. Inside an episode, the learning rates are summable (or fast-decaying), allowing the degree of non-stationarity in the MC inside the episode to be slowly time-varying. Doing so then enables us to ensure that the approximation errors of the optimality and consistency operators are under control. Therefore, given a reasonable estimate for the consistency operator, the control policy is updated on a faster time-scale. Similarly the mean-field is updated at a slower time-scale using the consistency operator. We note that inverting the entity updated on a faster/slower time-scale will result in the solution to the Mean-Field Control problem Angiuli et al. (2022). In the following subsection we describe how we can estimate the two operators. Approximate Mean-Field consistency and optimality operators We start by describing how the Sandbox learning algorithm uses the MC of the generic agent to approximate the consistency operator Γ 2 . Recalling the definition of Γ 2 (5), if µ ′ = Γ 2 (π, µ), then µ ′ (s ′ ) = s∈S a∈A P (s ′ | s, a, µ)π(a | s, µ)µ(s) = s∈S P π,µ (s, s ′ )µ(s), ∀s ′ ∈ S where P π,µ is the transition dynamics matrix of the generic agent under control law π and mean-field µ. Hence if µ ′ = Γ 2 (π, µ), the vector µ ′ ∈ P(S) can be written as µ ′ = P ⊤ π,µ µ(7) To come up with an estimator for Γ 2 we will need to estimate the dynamics matrix P π,µ . Toward this end, we can take a sample path of the Markov chain induced by π and µ of length T to obtain approximation of µ ′ through the use of an estimation of the occupancy (visitation) measure, and we can determine to what extent this estimate would be optimal through its ability to solve equation (7). More specifically, for a fixed pair of states (i, j) ∈ S × S, the empirical transition probabilitiesP can be computed by keeping track of the state visitation numbers N (i) and N (i, j) as follows:P (i, j) = N (i, j) + 1/S N (i) + 1 ,(8) where N (i, j) = {l ∈ [T ] : s l = i, s l+1 = j} , N (i) = j∈S N k t (i, j) and s t is the state visited by the MC at time t ∈ [T ]. Notice that we use smoothing (by adding 1/S and 1 to the numerator and the denominator, respectively) to avoid degenerate cases during the transition probability estimation. The transition probabilitiesP approximate the true transition probabilities P π,µ . Hence the approximate consistency operator is then given byP ⊤ µ, and the associated mean-field is updated by sequentially applyingP ⊤ µ with a specific step-size [cf. (12)] in (10), which we defer to the next subsection in order to underscore its concurrence with policy updates that are derived in terms of the Bellman equations. Now we describe how the Sandbox learning algorithm approximates the optimality operator Γ λ 1 . As described in Section 2 Γ λ 1 := softmax λ (·, Q * µ ), where softmax λ (·) is defined in (6) and Q * µ (s, a) := argmax π E[ ∞ t=1 R(s t , a t , µ)|s 1 = s, a 1 = a] is the optimal Q-function for the MDP induced by the mean-field µ and is the fixed point of the Bellman equation Q * µ (s, a) = R(s, a, µ) + ρE s ′ ∼P (·) [max a ′ Q * µ (s ′ , a ′ )]. The algorithm uses Q-learning update to approximate the optimal Q-function. The asynchronous Q-learning update Lewis et al. (2012); Even-Dar et al. (2003) can be written as follows, Q t+1 (s t , a t ) = (1 − β t )Q t (s t ) + β t r t + ρ max a∈A Q t (s t+1 , a) ,(9) where β t := c β /(t+1) ν and 0.5 < ν ≤ 1. Let us denote the approximate optimality operator asΓ 1 := softmax λ (·,Q), whereQ is the estimate obtained using the Q-learning update (9). The estimation error inΓ 1 is due to the estimation error in the Q-learning, and is monotonically increasing with λ. With this technical machinery introduced for the approximate consistency operator and the Bellman operator, we are now ready to introduce the Sandbox learning algorithm. This is the focus of the following subsection. Sandbox Reinforcement Learning algorithm The Sandbox learning algorithm is presented in Algorithm 1. Throughout the algorithm the superscript k ∈ [K] refers to the episode, and subscript t ∈ [T ] refers to the timestep inside the episode. Each episode k lasts for T timesteps. The state s 1 1 is initialized using distribution p 1 and a new state s k t+1 is generated at each timestep t (line 5), and hence the algorithm evolves over a single sample path (of the generic agent) without re-initialization. The mean-field µ k t and the control policy π k t are updated at each timestep according to µ k t = P S(ǫ net ) (1 − c k µ,t )µ k t−1 + c k µ,t (P k t ) ⊤ µ k t−1 , 1 {t=1} ,(10)π k t = (1 − c k π,t )π k t−1 + c k π,t (1 − ψ k t )softmax λ (·, Q k t ) + ψ k t 1 |A| .(11) The update of mean-field involves the operation P S(ǫ net ) [µ, x], which projects µ onto the ǫ-net S(ǫ net ) if and only if x = 1. This projection step is performed on the first time-step of each episode k. In the analysis (Section 4) we show that ǫ net = O(ǫ 2 ) at worst, where ǫ > 0 is the approximation error in the B-MFE. The update of the meanfield is performed using the approximate consistency operator (P k t ) ⊤ µ k t−1 , and the control policy is updated using the approximate optimality operator softmax λ (·, Q k t ). The control policy updates also involve an exploration noise ψ k t 1 |A| , which results in sufficient exploration of the state-action space (Lemma 1) without effecting the convergence bounds (Theorem 1 & Corollary 1). The expression for exploration coefficient ψ k t is provided in the proof of Lemma 1. The learning rates for the update, c k µ,t and c k π,t , are episodic two time-scale: c k µ,t = c µ k γ 1 t ζ , c k π,t = c π k θ 1 t ζ ,(12) where 0 < θ < γ < 1 < ζ < ∞. The episodic nature of the learning rates is due to the 1/t ζ factor, ζ > 1 in both rates, which makes it summable, resulting in slowly time varying MC inside the episode. The two time-scale nature of the learning rate is due to θ < γ where the update of the policy π k t is faster than that of the mean-field µ k t . Furthermore, the learning rates c k µ,t and c k π,t are non-summable since 0 < θ, γ < 1. This episodic two time-scale nature is pivotal in proving that Sandbox RL converges to the B-MFE of the MFG as shown in the next section. Algorithm 1: Sandbox RL for MFG 1: Initialize: initial state s 1 1 ∼ p 1 , policy π 1 0 and mean-field µ 1 0 2: for k ∈ {1, 2, . . . , K} do 3: for t ∈ {1, 2, . . . , T } do 4: Update µ k t , π k t using (10), (11) respectively. 5: Generate single transition s k t+1 ∼ P (· | s k Transition probability estimation: For (i, j) ∈ S × Ŝ P k t+1 (i, j) = N k t (i, j) + 1/S N k t (i) + 1 ,(13) where N k t (i, j) = {l ∈ [t] : s k l = i, s k l+1 = j} , N k t (i) = j∈S N k t (i, j). 7: Q-learning: Q k t+1 (s k t , a k t ) = (1 − β t )Q k t + β t r k t + ρ max a∈A Q k t (s k t+1 , a) 8: end for 9:P k+1 1 =P k T +1 , Q k+1 1 = Q k T +1 , µ k+1 0 = µ k T , π k+1 0 = π k T , s k+1 1 = s k T +1 10: end for 11: Output: Approximate B-MFE ( 1 K K−1 k=1 π k 1 , 1 K K−1 k=1 µ k 1 ). FINITE TIME BOUNDS FOR SANDBOX LEARNING Most results in RL for MFGs break down in our setting as they assume a time invariant MC. In contrast, concurrent update of the mean-field and the control policy in the Sandbox learning algorithm induces a time-varying MC. In this section we analyze how the slowly time-varying MC under the episodic learning rates (10)-(11) is more amenable to analysis and leads to good approximations of Γ λ 1 and Γ 2 operators. Toward this end we first prove convergence of the transition probability and Q-learning estimation inside each episode k ∈ [K] in Lemmas 2 and 3. These results are worthy of interest independent of the Sandbox learning algorithm, due to the slowly time-varying MC setting. Then in Theorem 1 we show that good approximation of Γ λ 1 and Γ 2 operators (due to good Q-learning and transition probability estimation, respectively) results in decreasing average error in policy and mean-field. Finally, in Corollary 1, we present finite sample analysis for the two time-scale Sandbox learning algorithm. Lemma 2 presents error bounds on transition probability estimation (8) (2021). Before stating the communicating MDP condition, we introduce the set S(ǫ net ) which is a set of mean-field distributions. This set (also termed ǫ-net Guo et al. (2019) over P(S)) defined as S(ǫ net ) = {µ 1 , . . . , µ Nnet } ⊂ P(S) is a finite set of simplexes over S such that µ − µ ′ 1 ≤ ǫ net for any µ ∈ P(S), ∃µ ′ ∈ S(ǫ net ). The existence of the set is guaranteed due to the compactness of P(S). (2016)) For any mean-fieldμ ∈ S(ǫ net ) (which is a finite set) and any pair of states s, s ′ ∈ S, there exists a finite horizon H(μ) such that for t ≥ H(μ) there exists a set of actionsã 1 , . . . ,ã t , Assumption 2. (Communicating MDPs Arslan and Yüksel P s t = s ′ | a 1 =ã 1 , . . . , a t =ã t , s 1 = s, µ =μ > 0. Informally Assumption 2 means that every agent in the game has a path from any state to any other state for mean-fields in S(ǫ net ). Assumption 2 is satisfied in several real-world scenarios. Production of an exhaustible resource by competing producers (e.g. oil) is a typical multi-agent setting where Assumptionm 2 is satisfied ?, since the agents can achieve any level of reserve by increasing/decreasing their production. Capital accumulation games Fershtman and Muller (1984) and asset management games Lacker and Zariphopoulou (2019) have a similar structure thus satisfying Assumptionm 2. It is also satisfied in cyber-security applications Kolokoltsov and Bensoussan (2016), as any infection state can be reached by choosing the right policy and a strictly positive MF ǫ-net. In Section 5 we numerically investigate a setting where such an assumption is not satisfied. Next, under the communicating MDP assumption, we prove sufficient exploration of state and action space, under the policy update (11). Lemma 1. (Sufficient Exploration) If Assumption 2 is satisfied, stochastic kernel P (· | s, a, µ) is Lipschitz in µ, and ζ is large enough, then under the control policy update (Algorithm 1 line 4), there exists a σ ∈ (0, 1) such that for any (s, a) ∈ |S × A| and t ≥ H : = maxμ ∈S(ǫ net ) H(μ), P ((s t , a t ) = (s, a) | F t−H ) ≥ σ. Lemma 1 implies that the communicating MDP assumption coupled with the policy update (11), is more general than the sufficient exploration condition used in Q-learning for MDPs (Qu and Wierman (2020) Assumption 3) as well as for N -player stochastic games (Hu and Wellman (2003) Assumption 1). Furthermore, it is more general than an ergodicity assumption used in the stochastic optimization literature Srikant and Ying (2019). We further note that the sufficient exploration condition has also been used in two time-scale settings in the literature Wu et al. (2020). Next we quantify the error in transition probability estimation in Lemma 2 under Assumption 2 and Lipschitz conditions on transition probability P Angiuli et al. (2022). The estimation error is denoted by ǫ k P , and is the norm of the difference between the transition probability estimateP k T and the true transition probability induced by the control policy and the mean-field at the first timestep (π k 1 , µ k 1 , respectively). Lemma 2. Given that Assumption 2 is satisfied and transition probability P π,µ is Lipschitz in policy π and mean-field µ such that P π,µ − P π ′ ,µ F ≤ L π P π − π ′ T V and P π,µ − P π,µ ′ F ≤ L µ P µ − µ ′ 1 , the error in transition probability estimation for episode k is ǫ k P := P k T − P π k 1 ,µ k 1 F =Õ(T −1/2 ) +Õ(T −1 ) + O(2 1−ζ ) with probability at least 1 − δ P where P π k 1 ,µ k 1 is the transition probability under control law π k 1 and mean-field µ k 1 . The Lipschitz conditions in Lemma 1 will follow if transition probability is continuous in the mean-field µ and the policy π(· | s), due to the compactness of mean-field and policy spaces, P(S) and P(A), respectively. And in most real-world examples, such as asset Reis and Platonov (2019) and crowd management Priuli (2014), continuity of transition probability w.r.t. mean-field and policy is ensured. Lemma 2 shows that the estimation error ǫ k P contains a drift term O(2 1−ζ ) due to the slowly time-varying MC setting which can be decreased by increasing the inter-episodic learning parameter ζ. Aside from drift, ǫ k P grows atÕ(T −1/2 ) +Õ(T −1 ), whereÕ hides logarithmic factors. Hence increasing the duration of episode T will result in decrease in estimation error. The proof of the lemma is given in the Appendix and relies on Freedman's inequality Freedman (1975). Next we analyze the error in Q-learning estimation (9) for each episode k ∈ [K]. This update has been shown to converge to the optimal Q function under a sufficient exploration condition (stronger than Assumption 2) for a time invariant MC Even-Dar et al. (2003); Qu and Wierman (2020). In Lemma 3 we show that this update converges (albeit with a drift) under the comunicating MDP condition for the slowly time-varying MC setting and with 0.5 < ν ≤ 1. This estimation error is denoted by ǫ k Q , and is the norm of the difference between the estimate of the optimal Q-function Q k T and the true Q-function Q * ,k 1 := Q * µ k 1 for the MDP induced by the mean-field µ k 1 . As in Lemma 2, a drift term O(2 1−ζ ) creeps in due to the slowly time-varying MC setting. Lemma 3. Under Assumption 2, the estimation error in Q-learning for episode k is ǫ k Q := Q k T − Q * ,k 1 ∞ = O(T 1−2ν ) + O(T 1−ζ−ν ) +Õ(T 1/2−ν ) + O(2 1−ζ ) with probability at least 1 − δ Q . The slowly time-varying MC also contributes O(T 1−ζ−ν ) error component but that is dominated by the O(T 1−2ν ) term due to ν ≤ 1 < ζ. The error terms, which are O(T 1−2ν ) andÕ(T 1/2−ν ), are decreasing for increasing T , and hence to get a small ǫ k Q we need a large enough T . Combining the bounds from Lemmas 2 and 3 we can surmise that given that the inter-episode learning parameter ζ and the episode length T are large enough, the transition probability estimation and Q-learning will be good enough, leading to good approximations of the consistency and optimality operators, Γ 2 and Γ λ 1 , respectively. Now we are in a position to present Theorem 1, relying on good approximations of the consistency and optimality operators. Theorem 1 below bounds the average error in policy e k π := π k 1 − Γ λ 1 (µ k 1 ) T V and mean-field e k µ := µ k 1 − µ * 1 over episodes k ∈ [K], given that ǫ k P ≤ ǫ P , ǫ k Q ≤ ǫ Q / log(K) for some ǫ P , ǫ Q > 0. Theorem 1. Let the approximation errors be denoted by e k π := π k 1 − Γ λ 1 (µ k 1 ) T V and e k µ := µ k 1 − µ * 1 and ǫ net ≤ (c µd ǫ)/K γ for ǫ > 0. Under Assumptions 1-2, with the estimation errors satisfying ǫ k P ≤ ǫ P , ǫ k Q ≤ ǫ Q /λ, for some ǫ P , ǫ Q > 0, the average approximation errors decrease at the following rates: 1 K K−1 k=1 e k π =O(K θ−1 ) + O(ǫ Q ) + O(K θ−γ ) + O(K −1 ) + O(2 1−ζ ), 1 K K−1 k=1 e k µ =O(K γ−1 )+O(ǫ) + O(ǫ P ) + 1 K K−1 k=1 e k π with probability at least 1 − δ Q , where 0 < θ < γ < 1 < ζ < ∞. The challenges in establishing Theorem 1 are due to the two time-scale learning rates and non-regularized MFG setting. The proof of Theorem 1 keeps track of errors e k π and e k µ for each episode k and the average of these errors is shown to approach 0 due to tight approximation of the optimality and consistency operators and the two time-scale update under the contraction Assumption 1. Apart from the familiar drift terms O(2 1−ζ ) and estimation error bounds ǫ P and ǫ Q , all other terms are decreasing with increasing total number of episodes K at rates governed by θ, γ and ζ. Next we present a corollary to Theorem 1 which gives us the final bound quantifying the approximation error between output of the Sandbox learning algorithm and the B-MFE of the MFG. The proof of Corollary 1 depends on the result of Theorem 1 and is provided in the Appendix. Corollary 1. If all conditions in Theorem 1 are satisfied, we have 1 K K−1 k=1 π k 1 − π * T V + 1 K K−1 k=1 µ k 1 − µ * 1 = O(K γ−1 ) + O(2 1−ζ ) + O(ǫ P ) + O(K θ−1 ) + O(ǫ Q ) + O(K θ−γ ) + O(ǫ). The terms O(K θ−1 ), O(K γ−1 ) and O(K θ−γ ) enter due to the two-time-scale learning setting and are monotonically decreasing with the number of episodes K. The convergence rates of these terms can be tuned by choosing θ and γ as explained next. The terms O(ǫ Q ) and O(ǫ P ) enter into the analysis due to the estimation errors in the Q-function and the dynamics matrix P , respectively. These quantities can be made arbitrarily small by increasing the number of timesteps in each episode T , due to Lemmas 1 and 2. Lastly, the term O(2 1−γ ) is a drift term which enters due to inter-episodic learning. This can be made arbitrarily small by increasing the value of γ. But the value of γ = ∞ as that stops inter-episodic learning and may cause degenerate policies. The error introduced by the projection step in the mean-field update line 4 of Algorithm 1 is O(ǫ). If the learning rates are chosen such that θ = 0.01, γ = 0.5, ν = 1, ζ = Ω(log(1/ǫ)), then the output of Algorithm 1 will be ǫ close to the B-MFE with high probability (for small enough ǫ > 0), given that episode length is T = Ω(ǫ −2 ) and the number of episodes is K = Ω(ǫ −2 ). Notice that under these conditions, ǫ net = O(ǫ 2 ). Hence the sample complexity of the algorithm is O(ǫ −4 ). Finally, the difference between the MFE and the B-MFE will be determined by the λ parameter with higher values resulting in a close approximation of MFE by the B-MFE Cui and Koeppl (2021). In the next section we apply the algorithm to a congestion game. NUMERICAL RESULTS We simulate Sandbox learning for a congestion MFG Toumi et al. (2020) on a grid. In particular we investigate the convergence of the algorithm for the cases (1) when the communicating MDP assumption is satisfied, and (2) when it is not satisfied. The state space S = {1, . . . , 5} 2 , action space A = {−1, 1} 2 , and discount factor ρ = 0.7. The initial distribution of the agent p 1 is uniform over the state space. If the agent takes action a ∈ A in state s ∈ S, the resulting state will be s ′ = P S [s + a] with probability 1 − p or the agent may be 'jostled' into one of the neighboring states, with probability p for small p > 0. This stochasticity is meant to model the jostling present in crowds. The agent's reward is r(s, µ) = (1 − c · µ(s)) · R(s) where c = 0.5 is the congestion averse parameter and R(s) is the state dependent component of the reward. The state-dependent rewards R(s) are concentrated around favorable states {(3, 3), (3, 4), (4, 3), (4, 4)}. Hence the agent prefers states with higher R(s) values but might be deterred by the congestion in the state µ(s). The initial control policy π 1 0 and mean-field µ 1 0 are uniform over actions and states, respectively. The initial estimate of the Q-function Q 1 1 is zero and transition probability estimateP 1 1 is uniform. By choosing learning coefficient c β = 5, learning rate ν = 0.55 and episodic length T = 5 × 10 4 we observe in Figure 2 that Q-learning and transition probability estimation converge very well inside episode k = 1. Furthermore, by choosing c µ = c π = 0.5 and the two time-scale learning rates as θ = 0.55 < γ = 0.6 we see that the control policy and mean-field estimates converge after K = 300 many episodes in Figure 3. For this particular simulation we forego the projection step as it does not have a significant impact on the convergence of the algorithm. Next, we consider the setting where the state space consists of two communicating classes, thereby nullifying the communicating MDP assumption (Assumption 2). In particular, the states C 1 := {(4, 5), (5, 4), (5, 5)} form a communicating class which is not closed and the rest of the state space is a closed communicating class C 0 . The reward function r(s, µ) is exactly the same as the previous simulation but the transition probabilities are modified to prevent transition C 0 → C 1 , ensuring that C 0 is closed and C 1 is not closed. The learning rates and other estimates are initialized as before. Figure 4 shows the convergence of mean-field and control policy estimates in the Sandbox learning algorithm. The simulation shows good convergence properties of the algorithm even in absence of the communicating MDP condition. This is due to the fact that the MC transitions (in finite time) from C 1 into C 0 and stays there, allowing good approximations of Γ 1 , Γ 2 operators, resulting in convergence. If there were multiple closed classes with initial distribution spread amongst them, the algorithm's performance would be variable. CONCLUSION & FUTURE WORK In this paper we have developed the Sandbox learning algorithm with finite-time guarantees to approximate the stationary MFE of a MFG without access to a mean-field oracle, using the single sample path of the generic agent. The sample complexity of the Sandbox learning algorithm is O(ǫ −4 ) where ǫ is the MFE approximation error. The proof of convergence has relied on goodness of transition probability and Q-function estimates (along slowly time-varying MC). The control policy and the mean-field were then updated using two time-scale learning rates and approximate consistency and optimality operators. We also generalize the covering time and ergodicity assumptions in literature by proposing a communicating MDP assumption. This work opens up several interesting research directions. It would be worthwhile to explore finite sample bounds for Sandbox learning under a weakly communicating MDP condition. Another important research direction would be to explore how feature embeddings can improve the scalability beyond the tabular MFG setting. Furthermore, extending oracle-less learning to the benchmark setting of Linear Quadratic MFGs may also help in solving several real-world scenarios such as, consensus and formation flying. A EXTENDED LITERATURE Mean-Field Games (MFGs) originated concurrently in the works of Huang et al. (2006Huang et al. ( , 2007 (termed as Nash Certainty Equivalence) and Lasry andLions (2006, 2007) (who coined the term MFG). Since its inception, there have been several works extending the classical concept of MFGs in various directions, such as heterogenous agents Moon and Başar (2014) 2021) proposed a single-loop RL algorithm, such that each critic step leads to a mean-field update as well. This is in contrast to the standard RL for MFG algorithms which have a double-loop structure where multiple critic steps can be executed while keeping the mean-field constant. Our work also has a single-loop structure as each critic step of control policy update leads to a concurrent update of the mean-field. Furthermore, we consider learning along a single sample path of the generic agent without re-initializations. In addition, the majority of the literature in RL for MFGs assume access to an oracle which can provide the mean-field (or a noisy estimate of it) under a given control policy. This work, on the other hand, proposes the Sandbox learning algorithm which uses the sample path of the agent itself to estimate the mean-field. The two works closest to our setting are Angiuli et al. (2022) and Yardim et al. (2022). The work of Angiuli et al. (2022) adopts an oracle-less setting but does not provide a finite sample convergence bound of the RL algorithm. Furthermore, the two time-scale update in Angiuli et al. (2022) updates the Q-function at a faster rate whereas Sandbox learning algorithm updates the control policy at the faster rate using a softmax of the estimated Q-function. Furthermore, we prove that the episodic nature of learning rates in Sandbox learning is crucial to obtaining finite sample convergence guarantees. Sandbox learning can be extended to entropy-regularized setting by employing a fitted Q-iteration (as in Anahtarcı et al. (2019)). The work of Yardim et al. (2022) runs independent policy mirror ascent for N agents and proves approximation upto a bias term of O(1/ √ N ), with similar resulting sample guarantees. However these guarantees are provided under an assumption that the agent's Markov chain is ergodic under any policy π, which is stronger than our communicating MDP assumption. This assumption may be a bit restrictive for some scenarios like condestion games with deterministic dynamics. A complete juxtaposition of Angiuli et al. (2022) andYardim et al. (2022) with our work is provided in Section 1.2. B Proofs of Results in Section 4 Throughout this section the cardinalities of the state and action spaces are denoted by S = |S| and A = |A|, respectively. B.1 Proof of Lemma 1 Proof. Let us consider exploration noise coefficient of the form ψ k t = ψ for some ψ ∈ (0, 1 − c π ), and c π and θ defined in Section 3.2. The proof consists of two parts; in the first part we show that under the policy update (11) using learning coefficient ψ the probability of any action a ∈ A for any state s ∈ S, will have a uniform lower bound π k t (a | s) ≥ π > 0 for all k ∈ [K], 1 < t ≤ T . In the second step, we show that using this uniform lower bound and the communicating MDP assumption (Assumption 2), the sufficient exploration condition as shown in the statement of Lemma 1 holds. To prove the lower bound π, we start by proving a lower bound on π k t (a | s) for a given episode k ∈ [K] and 1 ≤ t ≤ T . Recalling the update (11) for 1 < t ≤ T π k t = (1 − c k π,t )π k t−1 + c k π,t (1 − ψ)softmax λ (·, Q k t ) + ψ1 |A| where the last part on the right hand side is the uniform exploration noise 1 |A| = ( 1 |A| , . . . , 1 |A| ). This can be written as follows: π k t =c k π,[1,t] π k 0 + t l=1 c k π,[l,t] (1 − ψ)softmax λ (·, Q k l ) + t l=1 c k π,[l,t] ψ1 |A| ,(14)wherec k π,[l,t] = t s=l (1 − c k π,s ), c k π,[l,t] = c k π,l t s=l+1 (1 − c k π,s ). From the control policy update (14) we analyze the last part of the right-hand-side. In particular, we will prove that t+1 l=2 c k π,[l,t+1] ≥ t l=2 c k π,[l,t] for 1 ≤ t ≤ T . First we see that, c k π,[l,t] = c k π,l (1 − c k π,t ) t−1 s=l+1 (1 − c k π,s ) = (1 − c k π,t )c k π,[l,t−1](15) Using this equality, we can show where the second equality is obtained using (15). In the inequality we have used the fact that t l=1 c k π,[l,t] ≤ 1 for all 1 ≤ t ≤ T which can be shown using inductive reasoning. The base case is true since c k π,[1,1] = c k π,1 = cπ k θ < 1 since c π ≤ 1 and ζ > 1. Now assume that t l=1 c k π,[l,t] ≤ 1; then, t+1 l=1 c k π,[l,t+1] = c k π,t+1 + (1 − c k π,t+1 ) t l=1 c k π,[l,t] ≤ c k π,t+1 + 1 − c k π,t+1 = 1 Hence we have proved that t l=2 c k π,[l,t] ≥ c k π,[1,1] = c k π,1 for all 1 ≤ t ≤ T . Let us define π k t := min a∈A,s∈S π k t (a | s) as the lower limit on exploration noise in policy π k t for 1 ≤ t ≤ T . Using (14) and the definition of π k t and we can deduce [l,t] ψ ≥c k π,[1,t] π k 0 + c k π,1 ψ/|A| for 1 ≤ t ≤ t where the second inequality uses t l=1 c k π,[l,t] ≥ c k π,[1,1] = c k π,1 for all 1 ≤ t ≤ T . This gives us a k-dependent lower bound on π k t . Next we convert it into a uniform lower bound independent of k. Let us define a uniform lower bound dependent on k, π k := min 1<t≤T π k t . Using (11), we can write π k 1 = (1 − c k π,1 )π k 0 + c k π,1 (1 − ψ)softmax λ (·, Q k 1 ) + ψ1 |A| = (1 − c k π,1 )π k−1 T + c k π,1 (1 − ψ)softmax λ (·, Q k 1 ) + ψ1 |A| Using this relation and the definition of π k , we deduce that π k ≥ (1 − c k π,1 )π k−1 + c k π,1 ψ/|A| Using recursion we can write this as π k t ≥c k π,[1,t] π k 0 + t l=1 c k π,π k ≥θ 1,k π 1 + k l=1 ϑ l,k ψ/|A|, whereθ l,k = k s=l (1 − c s π,1 ) and ϑ l,k = c l π,1θl+1,k Next we will show that k l=1 ϑ l,k ψ/|A| ≥ ϑ 1,1 ψ/|A| = c π ψ/|A| for all k ∈ [K] which coupled with (17) naturally leads to the conclusion that π k ≥ c π ψ/|A|. By definition, we have k l=1 ϑ l,k ψ/|A| = ϑ k,k ψ/|A| + k−1 l=1 ϑ l,k ψ/|A| = c k π,1 ψ/|A| + (1 − c k π,1 ) k−1 l=1 ϑ l,k−1 ψ/|A| = c k π,1 ψ − k−1 l=1 ϑ l,k−1 ψ /|A| + k−1 l=1 ϑ l,k−1 ψ/|A| ≥ k−1 l=1 ϑ l,k−1 ψ/|A| where the last inequality follows from the fact that k l=1 ϑ l,k ≤ 1 for all k ∈ [K] which can be shown using inductive reasoning. The base case is true since ϑ 1,1 = c π < 1. Now assume that k l=1 ϑ l,k ≤ 1; then k+1 l=1 ϑ l,k+1 = ϑ k+1,k+1 + (1 − c k+1 π,1 ) k l=1 ϑ l,k ≤ c k+1 π,1 + 1 − c k+1 π,1 = 1 We have proved that k l=1 ϑ l,k ψ ≥ ϑ 1,1 ψ = c π ψ for all k ∈ [K], and hence using (17) we can deduce that π k ≥ c π ψ/|A| which implies π k t (a|s) ≥ π := c π ψ/|A| > 0 for any s ∈ S and a ∈ A. Next we show how using this uniform lower bound π along with the communicating MDP assumption (Assumption 2) sufficient exploration can be proved. From Assumption 2 for any i, j ∈ S andμ ∈ S(ǫ net ) we know that there exists a finite horizon H(μ) and a set of actionsã 1 , . . . ,ã H(μ) such that Pμ ij := P s H(μ) = j | a 1 =ã 1 , . . . , a H(μ) =ã H(μ) , s 1 = i, µ =μ > 0.(18) Let us define P := min i,j∈S,μ∈S(ǫ net ) Pμ ij and due to the finiteness of S and S(ǫ net ) we can guarantee P > 0. Due to the Lipschitzness of stochastic kernel P with respect to µ for any s ∈ S, a ∈ A, and µ, µ ′ ∈ ∆(S), P (· | s, a, µ) − P (· | s, a, µ ′ ) 1 ≤ L µ µ − µ ′ 1 for Lipschitz constant L µ > 0. This allows us to lower bound P (s ′ |s, a, µ k t ) where µ k t evolves according to the update rule (10). Hence for any s, s ′ ∈ S, a ∈ A and µ k t ∈ ∆(S), P (s ′ |s, a, µ k t ) ≥ P (s ′ |s, a, µ k 1 ) − |P (s ′ |s, a, µ k t ) − P (s ′ |s, a, µ k 1 )| ≥ P (s ′ |s, a, µ k 1 ) − L µ ∞ k=2 c k µ,k ≥ P (s ′ |s, a, µ k 1 ) − L µ c µ 2 ζ(19) and we know that µ k 1 ∈ S(ǫ net ) due to the projection step in (10). As a result, the change in stochastic kernel P (· | s, a, µ k t ) can be bounded by controlling the inter-episodic learning coefficient ζ. Let us define the probability of reaching state j ∈ A from state i ∈ S in time t ≥ H := maxμ ∈S(ǫ net ) H(μ) under the stochastic kernels induced by mean-field (µ k 1 , . . . , µ k t ) as P (s t = j|s 1 = i, (µ k 1 , . . . , µ k t )). From Assumption 2 we know that there exists a set of actions (ã 1 , . . . ,ã t ) such that P (s t = j|s 1 = i, (a 1 =ã 1 , . . . , a t =ã t ), (µ k 1 , . . . , µ k 1 )) ≥ P . Using these facts we can deduce, P (s t = j|s 1 = i, (µ k 1 , . . . , µ k t )) ≥ P (s t = j|s 1 = i, (a 1 =ã 1 , . . . , a t =ã t ), (µ k 1 , . . . , µ k t )) = t−1 l=1 s1=i,s l ∈S,s k =j P (s l+1 =s l+1 |s l =s l , a l =ã l , µ k l )π k l (a l =ã l |s l =s l ) ≥ t−1 l=1 s1=i,s l ∈S,s k =j P (s l+1 =s l+1 |s l =s l , a l =ã l , µ k l )π ≥ t−1 l=1 s1=i,s l ∈S,s k =j P (s l+1 =s l+1 |s l =s l , a l =ã l , µ k 1 ) − |S| L µ c µ 2 ζ π ≥ t−1 l=1 s1=i,s l ∈S,s k =j P (s l+1 =s l+1 |s l =s l , a l =ã l , µ k 1 ) − C(t)|S| L µ c µ 2 ζ π ≥ P − C(2H)|S| L µ c µ 2 ζ π 2H ≥ P /2 The first inequality is obtained using the fact that under exploration noise ψ k t = ψ ∈ (0, 1 − c π ), π k t (a|s) is lower bounded by π. The second inequality follows from Lipschitzness of stochastic kernel P and inter-episodic learning coefficient ζ > 1 as shown in (19). The third inequality is obtained by using the fact that probability P (s ′ |s, a, µ) ≤ 1 and for ζ high enough |S| Lµcµ 2 ζ ≤ 1. The fourth inequality uses the lower bound on Pμ ij for i, j ∈ S andμ ∈ S(ǫ net ) as shown in (18). The final bound uses the fact that for ζ large enough C(2H)|S| Lµcµ 2 ζ π 2H ≤ P /2. B.2 Proof of Lemma 2 Proof. In this proof we provide finite sample convergence bounds for the transition probability estimation (8) Our lemma generalizes transition probability estimation for the slowly time-varying MC setting. The proof relies on introducing a stochastic process Y t (dependent on visitation of a fixed pair of states i, j) which is shown to be a Martingale difference sequence. The transition probability estimation error is shown to be a function of the sum of Y t and a drift term due to the slowly time-varying MC setting. The drift term is shown to be small due to the Lipschitz property of transition dynamics and the slowly time-varying MC. Then, using Freedman's inequality, we show that the estimation error is monotonically decreasing with the visitation number of the pair of states i, j. Finally, we prove a high confidence lower bound on the visitation number of any pair of states i, j under the sufficient exploration condition (Lemma 1), yielding our convergence result. We recall the definition of ǫ k P := P k T − P k 1 F where we use P k t as a shorthand for P π k t ,µ k t . We use Freedman's inequality to obtain the estimation error for estimatorP k T as in Hsu et al. (2019). Furthermore, since we are dealing with a single episode k we suppress the use of episode k for clarity. Let F t be the σ-field generated by {s 1 , µ 1 , π 1 , . . . , s t , µ t , π t }. Let us start by fixing a pair of states (i, j) for any i, j ∈ S. Next let us define a stochastic process Y t such that Y 1 := 0 and for t ≥ 2 Y t := 1{s t−1 = i} 1{s t = j} − P t−1 (i, j) where P t−1 (i, j) is the transition probability of going from state i to j from time t − 1 to t. The stochastic process (Y t ) t∈[T ] is a Martingale Difference Sequence since Y t is F t -measurable, and for t ≥ 2 E[Y t | F t−1 ] = E 1{s t−1 = i} 1{s t = j} − P t−1 (i, j) | F t−1 , = 1{s t−1 = i} P t−1 (i, j) − P t−1 (i, j) = 0. Furthermore, ∀t ∈ [T ], Y t ∈ [−P t−1 (i, j), 1 − P t−1 (i, j)] ⊂ [−1, 1]. Summing up Y t for t ∈ [T ], S T := T t=1 Y t = T t=2 1{s t−1 = i} 1{s t = j} − P t−1 (i, j) , = T t=2 1{s t−1 = i} 1{s t = j} − P 1 (i, j) + P 1 (i, j) − P t−1 (i, j) , = N i,j − N i P 1 (i, j) + T t=2 1{s t−1 = i}P t−1 (i, j),(20) whereP t := P 1 − P t is the drift in the true transition probability. For use in Freedman's inequality, consider the process E[Y 2 t | F t−1 ] = E 1{s t−1 = i} 1{s t = j}P t−1 (i, j) + P 2 t−1 (i, j) | F t−1 , = 1{s t−1 = i}P t−1 (i, j) 1 − P t−1 (i, j) , = 1{s t−1 = i} P 1 (i, j) −P t−1 (i, j) − P 2 1 (i, j) + 2P 1 (i, j)P t−1 (i, j) −P 2 t−1 (i, j) , = 1{s t−1 = i}P 1 (i, j) 1 − P 1 (i, j) + 1{s t−1 = i}P t−1 (i, j) 2P 1 (i, j) − 1 −P t−1 (i, j) . Since E[Y 2 t | F t−1 ] ≥ 0, both terms in the above expression are positive. Hence its summation V T will be V T := T t=2 E[Y 2 t | F t−1 ] = N i P 1 (i, j) 1 − P 1 (i, j) + T t=2 1{s t−1 = i}P t−1 (i, j) 2P 1 (i, j) − 1 −P t−1 (i, j) .(21) Again both parts of the above expression are positive. Recalling (13) we write the estimation error aŝ P T (i, j) − P 1 (i, j) = N i,j − N i P 1 (i, j) + 1/S − P 1 (i, j) N i + 1 , = N i,j − N i P 1 (i, j) N i + 1 + 1/S − P 1 (i, j) N i + 1 , = S T − T t=2 1{s t−1 = i}P t−1 (i, j) N i + 1 + 1/S − P 1 (i, j) N i + 1 , where the last equality is due to (20). Applying Corollary 1 from Hsu et al. (2019), which is based on Freedman's inequality, we get |P T (i, j) − P 1 (i, j)| ≤ 2cV T τ T,δP (N i + 1) 2 + 4τ T,δP + T t=2 1{s t−1 = i}|P t−1 (i, j)| + |1/S − P 1 (i, j)| N i + 1 , = 2cN i P 1 (i, j) 1 − P 1 (i, j) τ T,δP (N i + 1) 2 + 2c T t=2 1{s t−1 = i}P t−1 (i, j) 2P 1 (i, j) − 1 −P t−1 (i, j) τ T,δP (N i + 1) 2 1 2 + 4τ T,δP + T t=2 1{s t−1 = i}|P t−1 (i, j)| + |1/S − P 1 (i, j)| N i + 1 , ≤ 2cN i P 1 (i, j) 1 − P 1 (i, j) τ T,δP (N i + 1) 2 + 2c T t=2 |P t−1 (i, j)|τ T,δP N i + 1 + 4τ T,δP + T t=2 |P t−1 (i, j)| + |1/S − P 1 (i, j)| N i + 1 , ≤ 2cτ T,δP N i + 1 + 2c T t=2 |P t−1 (i, j)|τ T,δP N i + 1 + 4τ T,δP + T t=2 |P t−1 (i, j)| + |1/S − P 1 (i, j)| N i + 1 ,(22) with probability at least 1 − δ P /(2S 2 ), where τ T,δP = O(log( 2S 3 log(T ) δP )). We used equation (21) to obtain the second inequality. Analyzing |P t (i, j)|, |P t (i, j)| ≤ P 1 − P t F = P π1,µ1 − P πt,µt F , ≤ t−1 l=1 P π l ,µ l − P π l ,µ l+1 F + P π l ,µ l+1 − P π l+1 ,µ l+1 F , ≤ t−1 l=1 L µ P µ l+1 − µ l 1 + L π P π l+1 − π l T V ≤ t l=2 L µ P c µ,l + L π P c π,l , ≤ (L µ P c µ + L π P c π ) t l=2 l −ζ , ≤ L µ P c µ + L π P c π ζ − 1 2 1−ζ =L P 2 1−ζ .(23) where c µ,t := c µ t −ζ , c π,t := c π t −ζ andL P := 10(L µ P c µ + L π P c π ) for ζ ≥ 1.1. The second inequality above is due to the Lipschitz conditions on P in Lemma 2 and the third inequality is due to the fact that µ 1 , π T V ≤ 1 for any µ ∈ P(S) and π ∈ P. Now the estimation error can be bounded using (22) and (23): |P T (i, j) − P 1 (i, j)| ≤ 2cτ T,δP N i + 1 + 2cτ T,δPLP T N i + 1 2 1−ζ 2 + 4τ T,δP +L P T 2 1−ζ + |1/S − P 1 (i, j)| N i + 1 ,(24) with probability at least 1 − δ P /(2S 2 ). Next we need to lower bound N i . In the following lemma we show that due to the sufficient exploration condition, N i grows at least linearly with T with high probability. Lemma 4. Using Lemma 1, N i ≥ T /T e with probability at least 1 − δ P /(2S 2 ), where T e := O 1 σ log 2S 3 δ P .(25) Proof. For a fixed state i ∈ S, define event E k such that kTe t=1 1{i t = i} ≥ k, for a given integer k such that 1 ≤ k ≤ K e := ⌈T /T e ⌉. We show that E K is a high probability event given the sufficient exploration condition (Lemma 1). For a given i ∈ S, define a random variable, X i t = I{i t = i} − E[I{i t = i} | F t−τ ] This random variable is an F t adapted process with E[X i t | F t−τ ] = 0 and |X i t | ≤ 1. Let l be an integer 0 ≤ l ≤ τ . For a fixed l, define the process Y i l,k = X i kτ +l and define filtrationF l,k := F kτ +l . We can deduce that E[Y i l,k |F l,k−1 ] = E[X i kτ +l | F kτ +l−τ ] = 0, |Y i l,k | ≤ 1, and Y i l,k isF l,k measurable. Combining these facts, Y i l,k is a Martingale Difference Sequence. Using Azuma-Hoeffding inequality and Lemma 1 we can deduce that for a given i ∈ S and k = K e where T e := O(ln(2S 3 /δ P )/σ), T t=1 I{i t = i} ≥ K e with probability at least 1 − δ P /(2S 3 ). Taking a union bound over all i ∈ S, we get T t=0 I{i t = i} ≥ K e , ∀i ∈ S with probability at least 1 − δ P /(2S 2 ). Using (24) and Lemma 4 the estimation error can be written as |P T (i, j) − P 1 (i, j)| ≤ 2cτ T,δP T e T + 2cτ T,δPLP T T e 2 1−ζ 2 + (4τ T,δP + |1/S − P 1 (i, j)|)T e T +L P T e 2 1−ζ , with probability at least 1 − δ P /S 2 . Using a union bound over all pairs (i, j) ∈ S × S, the definition of Frobenius norm and the equivalence between 1 and 2 vector norms, P − P 1 F =Õ(T −1/2 ) +Õ(T −1 ) + O(2 1−ζ ). with probability at least 1 − δ P . Hence we have completed the proof. B.3 Proof of Lemma 3 Proof. In this proof we provide finite sample convergence bounds for the Q-learning update (9) within the slowly timevarying MC setting. The proof of Lemma 3 follows an approach similar to proof of Theorem 4 in Qu and Wierman (2020) and extends the results to a slowly time-varying MC and learning exponent 0.5 < ν ≤ 1, which is empirically observed to have better convergence properties. We also find that convergence cannot be guaranteed for 0 < ν ≤ 0.5. The proof starts with breaking down the error ǫ k Q into several components. Then we obtain bounds on those components by proving certain properties like boundedness and the Martingale Difference Sequence property. Following that we prove that the error accumulated due to the slowly time-varying MC setting is small due to the Lipschitz properties of the transition probability and reward function. Finally combining all these results, the total error itself is shown to be converging using the contraction mapping property of the discounted Bellman update. This proof uses the fact that the coefficient of the learning rate c β in the Q-learning update (9) is lower bounded by 1 σ max ν + ζ − 1, 1 (1− √ ρ) . In this proof we suppress the use of superscript k since we are dealing with a single episode. Recall the definition of ǫ Q := Q T − Q * 1 ∞ where Q * 1 := Q * µ1 the optimal Q-function for the MDP induced by mean-field µ 1 . The Q-learning update can be written down as: Q t+1 = Q t + β t [e T it [F (µ t , Q t ) − Q t ] + w(t, µ t )]e it(26)where β t = c β (t+1) ν and w(t, µ t ) = ρ[max a∈A Q t (s t+1 , a) − E s ′ ∼P (·|st,at,µt) [max a ′ ∈A Q t (s ′ , a ′ )],(27)F (µ t , Q t )(s, a) = r(s, a, µ t ) + ρE s ′ ∼P (·|st,at,µt) [max a ′ ∈A Q t (s ′ , a ′ )] The noise w(·, ·) is bounded byw, is measurable with respect to F t+1 and E[w(t, µ t ) | F t ] = 0. We further decompose the update rule using D t := E[e T it e it | F t−τ ]. The matrix D t is a diagonal matrix with elements (d t,i ) i∈S×A , where d t,i = P(i t = i | F t−τ ) , and from the sufficient exploration condition (Lemma 1) we know that d t,i ≥ σ > 0. Q t+1 = Q t + β t D t (F (µ t , Q t ) − Q t ) + β t (e T it e it − D t )(F (µ t , Q t ) − Q t ) + β t w(t, µ t )e it , = Q t + β t D t (F (µ t , Q t ) − Q t ) + β t (e T it e it − D t )(F (µ t−τ , Q t−τ ) − Q t−τ ) + β t w(t, µ t )e it + β t (e T it e it − D t )(F (µ t , Q t ) − F (µ t−τ , Q t−τ ) − Q t + Q t−τ ) Let us define ǫ t := (e T it e it − D t )(F (µ t−τ , Q t−τ ) − Q t−τ ) + β t w(t, µ t )e it , φ t := (e T it e it − D t )(F (µ t , Q t ) − F (µ t−τ , Q t−τ ) − Q t + Q t−τ ) The process ǫ t is F t+1 measurable and E[ǫ t | F t−τ ] = E[e T it e it − D t | F t−τ ](F (µ t−τ , Q t−τ ) − Q t−τ ) + E[E[w(t, µ t ) | F t ]e it | F t−τ ] = 0. Hence, ǫ t is a shifted Martingale Difference Sequence. Writing down the Q-function as a sum from τ (Lemma 1) to t, we get Q t+1 =B τ −1,t Q τ + t k=τ B k,t F (µ k , Q k ) + t k=τ β tBk,t (ǫ k + φ k ),(28) where B k,t = β k D k t l=k+1 (I − β l D l ),B k,t = t l=k+1 (I − β l D l ) , and B k,t andB k,t are diagonal matrices composed of elements b k,t,i andb k,t,i respectively. We also define β k,t andβ k,t such that β k,t := β k t l=k+1 (1 − β l σ) ≥ b k,t,i ,β k,t := t l=k+1 (1 − β l σ) ≥b k,t,i Next we compute the optimality gap e Q t = Q t − Q * t ∞ , where Q * t is the fixed point of the operator F (µ t , ·). Lemma 5. e Q t+1 ≤B τ −1,t e Q τ + ρ max i t k=τ b k,t,i e Q k + t k=τ β kBk,t (ǫ k + φ k ) ∞ + L µ Q β τ −1,t t l=τ c µ,l + t k=τ β k,t t l=k c µ,l(29) Proof. Using (28) and subtracting Q * t+1 from both sides, Q t+1 − Q * t+1 =B τ −1,t (Q τ − Q * t+1 ) + t k=τ B k,t (F (µ k , Q k ) − Q * t+1 ) + t k=τ β tBk,t (ǫ k + φ k ) Hence we get, e Q t+1 =B τ −1,t e Q τ + ρ sup i t k=τ b k,t,i e Q k + t k=τ β kBk,t (ǫ k + φ k ) ∞ + B τ −1,t (Q * τ − Q * t+1 ) + t k=τ B k,t (Q * k − Q * t+1 ) ∞ We can use the Simulation lemma and Lipschitzness of transition probability P π,µ and reward function R µ with respect to the mean-field µ (with corresponding constants L µ P and L µ R respectively), to prove Lipschitzness of Q * with µ. Due to Lipschizness, we know that for µ, µ ′ ∈ P(S) P π,µ − P π,µ ′ F ≤ L µ P µ − µ ′ 1 , R µ − R µ ′ ∞ ≤ L µ R µ − µ ′ 1 and using the Simulation Lemma Lewis et al. (2012) we know that V * µ − V * µ ′ ∞ ≤ L µ R + L µ P 2(1 − ρ) µ − µ ′ 1 where V * µ is the value function of the MDP induced by mean-field µ and (1 − ρ) −1 is an upper bound on the value functions due to bounded rewards. Hence Q * µ (s, a) − Q * µ ′ (s, a) ∞ = ρ P (· | s, a, µ), V * µ (·) − P (· | s, a, µ ′ ), V * µ ′ (·) , = ρ P (· | s, a, µ), V * µ (·) − P (· | s, a, µ), V * µ ′ (·) + P (· | s, a, µ), V * µ ′ (·) − P (· | s, a, µ ′ ), V * µ ′ (·) , ≤ ρ L µ R + L µ P 2(1 − ρ) µ − µ ′ 1 + ρ L µ P 2(1 − ρ) µ − µ ′ 1 = L µ Q µ − µ ′ 1 where L µ Q := ρ(L µ R + L µ P /(1 − ρ)). And thus Q * t − Q * t+1 ∞ ≤ L µ Q µ t − µ t+1 1 now that we have shown the Lipschitzness of Q * with respect to µ. Furthermore, as µ t − µ t+1 1 ≤ c µ,t , where c µ,t := cµ (t+1) ζ , we get B τ −1,t (Q * τ − Q * t+1 ) + t k=τ B k,t (Q * k − Q * t+1 ) ∞ ≤ L µ Q β τ −1,t t l=τ c µ,l + t k=τ β k,t t l=k c µ,l This concludes the proof. We next start by bounding the terms ǫ t and φ t in the error decomposition (29). Lemma 6. ǫ t ∞ ≤ 2 1 − ρ + C +w =:ǭ, φ t ∞ ≤ (L µ R + L µ P 1 − ρ ) τ k=1 µ t−k+1 − µ t−k 1 + 2ǭ τ k=1 β t−k Proof. Recalling the definition of ǫ t , ǫ t ∞ = (e T it e it − D t )(F (µ t−τ , Q t−τ ) − Q t−τ ) + β t w(t, µ t )e it ∞ , ≤ e T it e it − D t ∞ F (µ t−τ , Q t−τ ) − Q t−τ ∞ + |w(t, µ t )| ∞ e it ∞ , ≤ F (µ t−τ , Q t−τ ) ∞ + Q t−τ ∞ +w ≤ 2 1 − ρ + C +w =:ǭ where C = 1 andw = 2 1−ρ due to Q t ∞ ≤ 1 1−ρ , contractive property of F and the definitions of noise w (27) and Q-update (26). Similarly for φ we get φ t ∞ = (e T it e it − D t )(F (µ t , Q t ) − F (µ t−τ , Q t−τ ) − Q t + Q t−τ ) ∞ , ≤ F (µ t , Q t ) − F (µ t−τ , Q t−τ ) ∞ + Q t−τ − Q t ∞ , ≤ τ k=1 F (µ t−k+1 , Q t−k+1 ) − F (µ t−k , Q t−k ) ∞ + τ k=1 Q t−k+1 − Q t−k ∞ .(30) We first analyze the first summand in equation (30) F (µ t−k+1 , Q t−k+1 ) − F (µ t−k , Q t−k ) ∞ ≤ F (µ t−k+1 , Q t−k+1 ) − F (µ t−k+1 , Q t−k ) ∞ + F (µ t−k+1 , Q t−k ) − F (µ t−k , Q t−k ) ∞ , ≤ ρ Q t−k+1 − Q t−k ∞ + max s,a |R(s, a, µ t−k+1 ) − R(s, a, µ t−k )| + max s,a |P (· | s, a, µ t−k+1 ) − P (· | s, a, µ t−k )| 1 1 − ρ , ≤ ρ Q t−k+1 − Q t−k ∞ + L µ R + L µ P 1 − ρ µ t−k+1 − µ t−k 1(31) Similarly the second summand in (30) is Q t−k+1 − Q t−k ∞ = β t−k e T i t−k F (µ t−k , Q t−k ) − Q t−k + w(t − k, µ t−k ) e i t−k ∞ , ≤ β t−k F (µ t−k , Q t−k ) ∞ + β t−k Q t−k ∞ +w ≤ β t−kǭ .(32) Substituting (31), (32) into (30) φ t ∞ ≤ (L µ R + L µ P 1 − ρ ) τ k=1 µ t−k+1 − µ t−k 1 + 2ǭ τ k=1 β t−k . Having proved bounds on ǫ t and φ t , we now prove some properties of the learning rates c µ,t := cµ (t+1) ζ and β t = c β (t+1) ν where 0.5 < ν ≤ 1, ζ > 1 and c β ≥ ν σ . Lemma 7. Below we present some results regarding the learning rate β t and the associated variables. 1.β k,t ≤ k+2 t+2 c β σ ≤ k+2 t+2 ν , 2. β k,t ≤ c β (k+1) ν k+2 t+2 c β σ ≤ 2 c β (t+2) ν , 3. 1 − c β l + 1 = t l=k+1 e log(1− c β l+1 ) , ≤ t l=k+1 e − c β l+1 = e − t l=k+1 c β l+1 = exp − t l=k+1 c β l + 1 , ≤ exp − t+1 k+1 c β σ y + 1 dy = exp − c β σ log t + 2 k + 2 , = k + 2 t + 2 c β σ ≤ k + 2 t + 2 ν . β kBk,t φ k ∞ ≤ C 1 φ (t + 2) 2ν−1 + C 2 φ (t + 2) ζ+ν−1 , t k=τ β kBk,t ǫ k ∞ ≤ C ǫ (t + 2) ν−1/2 with probability at least 1 − δ Q , where C 1 φ = 4c 2 β (1 + τ ) ν τ 1 + c β σ − 2νǭ ,(33)C 2 φ = L µ R + L µ P 1 − ρ 2c µ c β τ (1 + τ ) ζ c β σ − ν − ζ + 1 ,(34)C ǫ = 10ǭ 2c β σ − 2ν + 1 (τ + 1)c 2 β log 2(τ + 1)T 2 SA δ Q .(35) Proof. We start with the first inequality t k=τ β kBk,t φ k ∞ ≤ t k=τ β k,t φ k ∞ ≤ t k=τ β k,t L µ R + L µ P 1 − ρ τ l=1 µ k−l+1 − µ k−l ∞ + 2ǭ τ l=1 β k−l , ≤ 2ǭ t k=τ β k,t k−1 l=k−τ β l + L µ R + L µ P 1 − ρ t k=tau β k,t k−1 l=k−τ µ l+1 − µ l ∞ , ≤ C 1 φ 1 (t + 2) 2ν−1 + L µ R + L µ P 1 − ρ t k=tau β k,t k−1 l=k−τ c µ,l ,(36)1 √ ρ(t+1) w . Then e t = t−1 k=τ b k,t,i 1 (k + 1) w + b t,t,i 1 (t + 1) w , = (1 − β t d t,i ) t−1 k=τ b k,t−1,i 1 (k + 1) w + β t d t,i 1 (t + 1) w , = (1 − β t d t,i )e t−1 + β t d t,i 1 (t + 1) w , ≤ (1 − β t d t,i ) 1 √ ρ(t + 1) w + β t d t,i 1 (t + 1) w , = 1 − β t d t,i (1 − √ ρ) √ ρ(t + 1) w , ≤ 1 − c β σ (t + 1) ν (1 − √ ρ) 1 √ ρ(t + 1) w , = 1 − c β σ (t + 1) ν (1 − √ ρ) (t + 2) w (t + 1) w 1 √ ρ(t + 2) w , = 1 − c β σ (t + 1) ν (1 − √ ρ) 1 + 1 t + 1 w 1 √ ρ(t + 2) w . For any x > −1, (1 + x) ≤ e x and thus 1 − c β σ (t + 1) ν (1 − √ ρ) 1 + 1 t + 1 w ≤ e − c β σ (t+1) ν (1− √ ρ)+ w t+1 ≤ 1, where the last inequality is due to c β ≥ 1 (1− √ ρ)σ . Hence we have proved that e t ≤ 1 √ ρ(t + 2) w . Now we prove the inequality for g t using recursion again. For the base case it is easy to see that g τ = b τ,τ,i 1 τ ζ−1 = β τ d τ,i 1 τ ζ−1 ≤ 1 √ ρτ ζ−1 . Now assume that g t−1 ≤ 1 √ ρτ ζ−1 . Then g t = (1 − β t d t,i )g t−1 + β t d t,i 1 τ ζ−1 ≤ (1 − β t d t,i ) 1 √ ρτ ζ−1 + β t d t,i 1 τ ζ−1 , ≤ 1 − β t d t,i (1 − √ ρ) √ ρτ ζ−1 ≤ 1 − c β σ (t + 1) ν (1 − √ ρ) 1 √ ρτ ζ−1 ≤ 1 √ ρτ ζ−1 , which proves the recursion step and completes the proof. Now we prove the main result using Lemma 10. Recalling (29), e Q t+1 ≤B τ −1,t e Q τ + ρ sup Using Lemmas 8 and 9 and using C µ := 10L µ Q c µ for ζ ≥ 1.1, e Q t+1 ≤B τ −1,t e Q τ + ρ sup i t k=τ b k,t,i e Q k + C 1 φ (t + 2) 2ν−1 + C 2 φ (t + 2) ζ+ν−1 + C ǫ (t + 2) ν−1/2 + C µ τ ζ−1 We will prove that e Q t ≤C 1 (t+1) 2ν−1 +C 2 (t+1) ζ+ν−1 +C 3 (t+1) ν−1/2 +C 4 τ ζ−1 using induction. The base case is trivially true; now assume this to be true for t: e Q t+1 ≤B τ −1,t e Q τ + ρ sup i t k=τ b k,t,i C 1 (t + 1) 2ν−1 +C 2 (t + 1) ζ+ν−1 +C 3 (t + 1) ν−1/2 +C 4 τ ζ−1 + C 1 φ (t + 2) 2ν−1 + C 2 φ (t + 2) ζ+ν−1 + C ǫ (t + 2) ν−1/2 + C µ τ ζ−1 , ≤ √ ρC 1 + C 1 φ (t + 2) 2ν−1 + √ ρC 2 + C 2 φ (t + 2) ζ+ν−1 + √ ρC 3 + C ǫ + 2(τ + 1) ν /(1 − ρ) (t + 2) ν−1/2 + √ ρC 4 + C µ τ ζ−1 , ≤C 1 (t + 2) 2ν−1 +C 2 (t + 2) ζ+ν−1 +C 3 (t + 2) ν−1/2 +C 4 τ ζ−1 with probability 1 − δ Q (using a union bound type argument) wherē given that ζ ≥ 1.1 with probability at least 1 − δ Q . C 1 = C 1 φ 1 − √ ρ ,C 2 = C 2 φ 1 − √ ρ ,C 3 = C ǫ + 2(τ + 1) ν /(1 − ρ) 1 − √ ρ ,C 4 = C µ 1 − √ ρ , Finally ǫ Q = Q T − Q * 1 ∞ , ≤ Q T − Q * T ∞ + Q * T − Q * 1 ∞ , ≤ e Q T + T −1 t=1 Q * t+1 − Q * t ∞ , B.4 Proof of Theorem 1 Proof. In this proof we provide finite sample bounds for the convergence of approximation errors in control policy and mean-field, e k π and e k µ , respectively. We start by characterizing the approximation errors in control policy and mean field e k π and e k µ on the first timestep in each episode k. Then the evolutions of these approximation errors are studied under two timescale learning rates. First we analyze the approximation error in control policy e k π which is evolving at a faster learning rate compared to the approximation error in the mean-field e k µ . This error is shown to converge due to the good approximation of the Q-function (Lemma 3), increase of Lipschitz coefficient λ k at a logarithmic rate and fast learning rate c k π . Next the approximation error in mean-field e k µ (which is evolving under the slower timescale) is also shown to converge due to the good transition dynamics estimation (Lemma 2), the contraction mapping property (Assumption 1) and the convergence of e k π . First we recall the update rules in Algorithm 1 µ k t = P S(ǫ net ) (1 − c k µ,t )µ k t−1 + c k µ,tΓ k 1,t , 1 t=1 , whereΓ k 1,t = (P k t ) ⊤ µ k t−1 π k t = (1 − c k π,t )π k t−1 + c k π,t (1 − ψ)Γ k 2,t + ψ1 |A| , whereΓ k 2,t = softmax λ (·, Q k t ) whereΓ k 1,t andΓ k 2,t are the approximate consistency and optimality operators. The RL update can now be written down for the first timestep of episode k + 1, µ k+1 1 = P S(ǫ net ) (1 − c k+1 µ,1 )µ k+1 0 + c k+1 µ,1 (P k+1 1 ) ⊤ µ k+1 0 , 1 , = P S(ǫ net ) (1 − c k+1 µ,1 )µ k T + c k+1 µ,1 (P k T ) ⊤ µ k T , 1 , = P S(ǫ net ) (1 − c k+1 µ,1 )(µ k 1 + ∆ k µ ) + c k+1 µ,1 (P k T ) ⊤ (µ k 1 + ∆ k µ ), 1 , π k+1 1 = (1 − c k+1 π,1 )π k+1 0 + c k+1 π,1 (1 − ψ)softmax λ (·, Q k+1 1 ) + ψ1 |A| , = (1 − c k+1 π,1 )π k T + c k+1 π,1 (1 − ψ)softmax λ (·, Q k+1 1 ) + ψ1 |A| , = (1 − c k+1 π,1 )(π k 1 + ∆ k π ) + c k+1 π,1 (1 − ψ)softmax λ (·, Q k+1 1 ) + ψ1 |A| , where ∆ k µ := µ k T − µ k 1 and ∆ k π := π k T − π k 1 are the drifts in mean-field and policy, respectively, in the episode k. Since all the time indices in the above inequalities are 1, we suppress all time indices from here on. Coupled with the fact that c k+1 µ,1 = c k+1 µ and c k+1 π,1 = c k+1 π , the update rules can be written as µ k+1 = P S(ǫ net ) (1 − c k+1 µ )(µ k + ∆ k µ ) + c k+1 µ (P k ) ⊤ (µ k + ∆ k µ ), 1 , π k+1 = (1 − c k+1 π )(π k + ∆ k π ) + c k+1 π (1 − ψ)softmax λ (·, Q k+1 ) + ψ1 |A| . Here we useP k :=P k T and Q k := Q k T for conciseness. The estimation errors for transition matrix and Q-function are denoted as ǫ k P := P k − P π k ,µ k F , ǫ k Q := Q k − Q * µ k ∞ . Now we compute the evolution of the approximation errors. We start with e k π := π k −Γ λ 1 (µ k ) T V : e k+1 π = π k+1 −Γ λ 1 (µ k+1 ) T V , ≤ π k+1 −Γ λ 1 (µ k ) T V + Γ λ 1 (µ k ) −Γ λ 1 (µ k+1 ) T V , ≤ (1 − c k+1 π )(π k + ∆ k π ) + c k+1 π softmax λ (·, Q k ) − softmax λ (·, Q * µ k ) T V + d 1 µ k+1 − µ k 1 + 2c k+1 π ψ, ≤ (1 − c k+1 π ) π k −Γ λ 1 (µ k ) T V + (1 − c k+1 π ) ∆ k π T V + c k+1 π softmax λ (·, Q k ) − softmax λ (·, Q * µ k ) T V + d 1 µ k+1 − µ k 1 + c k+1 π ǫ/2, ≤ (1 − c k+1 π )e k π + ∆ k π T V + c k+1 π softmax λ (·, Q k ) − softmax λ (·, Q * µ k ) T V + d 1 µ k+1 − µ k 1 + c k+1 π ǫ/2. where the third inequality is due to ψ ≤ ǫ/4. To simplify the above expression we prove the Lipschitz property of the softmax λ (·, Q) operator. Lemma 11. The softmax λ (·, Q) satisfies the Lipschitz property for λ > 0 and Q : S × A → R + , softmax λ (·, Q) − softmax λ (·, Q ′ ) T V ≤ λS √ A Q − Q ′ ∞ . Proof. The Lipschitzness of softmax can be obtained using Proposition 4 in Gao and Pavel (2017). Let us denote the policy under softmax λ (·, Q) as π λ Q such that π λ Q (a|s) = exp(λQ(s,a)) a ′ ∈A exp(λQ(s,a ′ )) . Now softmax λ (·, Q) − softmax λ (·, Q ′ ) T V = π λ Q − π λ Q ′ T V , = max a∈A s∈S π λ Q (a|s) − π λ Q ′ (a|s) From Proposition 4 in Gao and Pavel (2017), we know that for any s ∈ S π λ Q (·|s) − π λ Q ′ (·|s) 2 = a∈A π λ Q (a|s) − π λ Q ′ (a|s) 2 ≤ λ Q(s, ·) − Q(s, ·) 2 , = λ a∈A Q(s, a) − Q ′ (s, a) 2 , ≤ λ √ A Q(s, ·) − Q ′ (s, ·) ∞ , ≤ λ √ A Q − Q ′ ∞ .(43) Next we introduce the standard contraction mapping assumption in MFGs Guo et al. (2019); Xie et al. (2021). are widely used in RL for standard MFGs Guo et al. (2019); Xie et al. (2021); Fu et al. (2020). Lemma 5 in Guo et al. (2019) provides candidate values for these constants. The · T V norm used in Assumption 1 is the Total variation bound Cui and Koeppl (2021) and is defined for a function f : A × S → R such that f T V := max s∈S a∈A |f (a | s)|. Under Assumption 1, the existence and uniqueness of the B-MFE of the MFG has been proven in literature Cui and Koeppl (2021); Guo et al. (2019); Xie et al. (2021) using the standard contraction mapping theorem. Hence, the B-MFE approximates the MFE for large values of λ and the MFE is known to be ǫ-Nash for the finite population game (Theorem 2.3 Saldi et al. ( (2019); Fu et al. (2020); Xie et al. (2021); Figure 1 : 1Episodic Two time-scale learning rate In contrast, earlier works Guo et al. (2019); Xie et al. (2021); Anahtarcı et al. (2019) deal with approximating just Γ λ 1 under a time invariant MC. for a slowly time-varying MC, under a communicating MDP condition as given below. Assumption 2 generalizes the pre-existing conditions for RL-MFGs in literature. The online RL-MFG works of Guo et al. (2019) and Xie et al. (2021) (and references therein Shah and Xie (2018); Farahmand et al. (2016)) assume either a covering time assumption or require i.i.d. samples from stationary distribution. Similarly Yardim et al. (2022) requires ergodicity of the generic agent's MC under any policy. The offline RL-MFG works of Anahtarcı et al. (2019) and Fu et al. (2020) rely on i.i.d. samples from unique stationary distribution of MC which requires ergodicity. Communicating MDP (Assumption 2) is more general than covering time or ergodicity conditions Chandrasekaran and Tewari Figure 2 : 2Convergence of transition probability and Q-function estimation for episode k = 1 Figure 3 : 3Convergence of mean-field and control policy estimates Figure 4 : 4Convergence of mean-field and control policy estimates in the absence of Assumption 2 , scarce interactions Caines and Huang (2019);Zaman et al. (2021), risk-sensitive criteriaTembine et al. (2013);Moon and Başar (2016);Saldi et al. (2020) and cooperative equilibriaBensoussan et al. (2018);Barreiro-Gomez and Tembine (2021). MFGs have also been applied to a variety of real-world applications such as decentralized charging of EVsMa et al. (2011), economics Carmona (2020 and congestion dynamics Lachapelle and Wolfram(2011), among others. Although most of these works have been in the continuous time setting, research in discrete-time MFGs which are much more amenable to Reinforcement Learning have also been gaining momentum recently Saldi et al. (2018); Moon and Başar (2014). RL for MFGs was first dealt with in Guo et al. (2019) for the finite and in Elie et al. (2019) for infinite state and action spaces. The work of Guo et al. (2019) proposes a double-loop RL algorithm for MFGs with finite state and action spaces MFGs, which involves a projection step onto an ǫ-net. This projection step helps in establishing convergence by utilizing a uniform action gap bound over the ǫ-net. A fictitous play algorithm was proposed Elie et al. (2019), involving repeated updates of the mean-field and control policy to approximate the MFE. The first set of works to deal with RL for the benchmark Linear Quadratic (LQ) MFGs were Fu et al. (2020); Zaman et al. (2020, 2022). These works have provided finite sample bounds for the LQ-MFG in the stationary Fu et al. (2020) and the non-stationary Zaman et al. (2020, 2022) settings, by building on policy gradient Fazel et al. (2018) and zero-order stochastic optimization methods Malik et al. (2019) for the Linear Quadratic Regulator problem. The recent work of Yongacoglu et al. (2022) deals with independent learning for a novel setting of N -player mean-field games. The work of Lee et al. (2021) learns the MFE in the special setting of strategic complementarities where a single step of centralized Q-learning followed by a single step of policy execution by many agents is shown to converge to the MFE. The idea of entropy-regularized MFGs was introduced in Xie et al. (2021); Cui and Koeppl (2021) along with existence and uniqueness results and RL algorithms to compute the entropy-regularized MFE. The work of Anahtarcı et al. (2019) also deals with the entropy-regularized MFGs, by utilizing a fitted Q-iteration based approach. There have been several works on Deep-RL techniques for MFGs, such as Perrin et al. (2021); Subramanian and Mahajan (2019), where Perrin et al. (2021) uses Deep RL techniques to learn a flocking model observed in nature and Subramanian and Mahajan (2019) proposes a Neural Network based policy update mechanism. The paper Xie et al. ( under the slowly time-varying MC setting. The proof of Lemma 2 relies on Freedman's inequality Freedman (1975) similar to the analysis of Theorem 4 in Hsu et al. (2019). = O(T 1−2ν ) + O(T 1−ζ−ν ) +Õ(T 1/2−ν ) + O(2 1−ζ ) table juxtaposing our work with the contributions of other works in RL for MFGs.A complete literature review is provided in the Section A.Oracle-less? Single sample path Finite sample bounds Elie et al. (2019) Cui and Koeppl (2021) Fu et al. (2020) Guo et al. (2019) Anahtarci et al. (2022) Xie et al. (2021) Angiuli et al. (2022) Yardim et al. (2022) This work t = s}, where we also include agent i, without any loss of generality. Agent i generates its actions using policy π i t ∈ Π i t := {π i t | π i t : S × P(S) → P(A)}, dependent on its state and the empirical distribution e N t . The state of agent i transitions according tos i t+1 ∼ P (· | s i t , a i t , e N t ), s i 1 ∼ p 1 , a i t ∼ π i t (s i t , e N t ).(1)Similarly, the reward accrued to the agent depends on its state, action, and the empirical distribution at time t, r i t = R(s i t , a i t , e N t ) ∈ [0, 1]. The presence of e N t in both (1) and r i t is a key point of departure from a standard MDP setting, as it permits other agents' possibly non-cooperative behavior to determine the evolution of the state and the reward of agent i. The over-arching goal of each agent i = 1, . . . , N is to maximize its total reward discounted by a factor 0 < ρ < 1, defined asV i (π i , π −i ) = E ∞ t=1 ρ t R(s i t , a i t , e N t ) | s i t ∼ p 1 ,(2) t , a k t , µ k t ) and reward r k t = R(s k t , a k t , µ k t ) with a k t ∼ π k t (s k t , µ k t ).6: t k=1 β 2 k,t ≤ 2c 2 β 2c β σ−2ν+1 1 (t+2) 2ν−1 , 4. t k=τ β k,t k−1 l=k−τ β l ≤ 2c 2 β (τ +1) ν τ 1+c β σ−2ν 1 (t+2) 2ν−1Proof. For part (1) we start by recalling the definition ofβ k,t for k ∈ [t]β k,t = t l=k+1 (1 − β l σ) ≤ t l=k+1 AppendixThe first inequality is due to the fact that β t := c β (t+1) ν > c β t+1 since ν < 1. The last inequality is due to the fact that c β σ ≥ ν and k ≤ t.For part (2), recalling the definition of β k,t and using the bound onβ k,t , we getFor part (3), analyzing each summand2c 2 β (t + 2) 2c β σ (k + 1) 2c β σ−2ν ≤ 2c 2 β (t + 2) 2c β σ t+1 1 (y + 1) 2c β σ−2ν dy,For part (4), as k − τ ≤ l ≤ k − 1, in the expression t k=τ β k,t k−1 l=k−τ β l , we getHence the inequalities have been proved.Having proved some properties of the learning rates in Lemma 7 we are now able to bound the two parts of the quantity t k=τ β kBk,t (ǫ k + φ k ) ∞ as follows. The bound on the first quantity relies on the properties of the learning rates and the second bound relies on the fact that ǫ t is a Martingale Difference sequence.Next we move to the second inequality. Recalling the definition of ǫ tWe will use a variant of the Azuma-Hoeffding bound which can handle shifted Martingale Difference Sequences Qu and Wierman (2020). Each element in the vector t k=τ β kBk,t ǫ k can be upper bounded by | t k=τ β k ǫ i,kbk,t,i | where ǫ i,k is the ith element in the vector ǫ k . Using Lemmas 13 & 14 from Qu and Wierman (2020) we getwith probability at least 1 − δ Q /SA. Applying the union bound over ∀i ∈ S × A, we getNow we aim to bound the last term in(29)Lemma 9. If ζ > 1, thenβNow that we have bounded all the terms in (29) we will show that the error term e Q t can be bounded by a decreasing function of time t. Toward this end we introduce a lemma that will help us with the proof of the main result.Lemma 10. For any 0 < w < 1 and t ≥ τ ,We first prove the inequality for e t by recursion. We start with the base case.and since τ is chosen such that 1 + 1The second inequality is due to the equivalence between 2 and ∞ vector norms. This equivalence also gives usRecalling(42),where the last inequality is obtained using(43)and(44).Now we can further simplify (41) as:The first inequality is due to Lemma 11 and the second inequality is due to (40) and the fact that µ 1 ≤ 1 for any µ ∈ P(S). The norms of the drift terms are bounded byRearranging the inequality(45),for ζ ≥ 1.1. Now taking the average over k = 1, . . . , K − 1, we getwhere the second to last inequality is due to the fact that e k π ≤ 2. Since ǫ k Q ≤ ǫ Q /λ, where ǫ Q > 0, thenwhere K is the total number of episodes.Now we analyze the mean-field approximation error evolution e k µ := µ k − µ * 1 . Let us defineμ k+1 :µd )e k µ + 11c k µ 2 1−ζ + c k+1 µ √ Sǫ k P + c k+1 µ d 2 e k π + ǫ net where the second to last inequality is due to the equivalence between induced 1 norm and the Frobenius norm. Rearranging the above inequality,Taking average over k = 1, . . . , K − 1, we getwhere the second inequality is obtained using steps similar to (47) and the fact that ǫ k P ≤ ǫ P . The last inequality is obtained using (48) and the fact that ǫ net ≤ c µd ǫ/K γ . The proof is thus concluded.B.5 Proof of Corollary 1Proof. This is a corollary to Theorem 1: Berkay Anahtarcı, Naci Can Deha Karıksız, Saldi, arXiv:1912.13309Fitted Q-learning in mean-field games. arXiv preprintBerkay Anahtarcı, Can Deha Karıksız, and Naci Saldi. Fitted Q-learning in mean-field games. arXiv preprint arXiv:1912.13309, 2019. Q-learning in regularized mean-field games. Berkay Anahtarci, Naci Can Deha Kariksiz, Saldi, Dynamic Games and Applications. Berkay Anahtarci, Can Deha Kariksiz, and Naci Saldi. Q-learning in regularized mean-field games. Dynamic Games and Applications, pages 1-29, 2022. Unified reinforcement Q-learning for mean field game and control problems. Andrea Angiuli, Jean-Pierre Fouque, Mathieu Laurière, Mathematics of Control, Signals, and Systems. Andrea Angiuli, Jean-Pierre Fouque, and Mathieu Laurière. Unified reinforcement Q-learning for mean field game and control problems. Mathematics of Control, Signals, and Systems, pages 1-55, 2022. Decentralized Q-learning for stochastic teams and games. Gürdal Arslan, Serdar Yüksel, IEEE Transactions on Automatic Control. 624Gürdal Arslan and Serdar Yüksel. Decentralized Q-learning for stochastic teams and games. IEEE Transactions on Automatic Control, 62(4):1545-1558, 2016. Mean-field-type Games for Engineers. Julian Barreiro, - Gomez, Hamidou Tembine, Tamer Başar and Geert Jan Olsder. Dynamic Noncooperative Game Theory. SIAM. CRC PressJulian Barreiro-Gomez and Hamidou Tembine. Mean-field-type Games for Engineers. CRC Press, 2021. Tamer Başar and Geert Jan Olsder. Dynamic Noncooperative Game Theory. SIAM, 1998. The master equation in mean field theory. Alain Bensoussan, Jens Frehse, Sheung Chi Phillip Yam, Journal de Mathématiques Pures et Appliquées. 1036Alain Bensoussan, Jens Frehse, and Sheung Chi Phillip Yam. The master equation in mean field theory. Journal de Mathématiques Pures et Appliquées, 103(6):1441-1474, 2015. Mean field control and mean field game models with several populations. Alain Bensoussan, Tao Huang, Mathieu Laurière, arXiv:1810.00783arXiv preprintAlain Bensoussan, Tao Huang, and Mathieu Laurière. Mean field control and mean field game models with several populations. arXiv preprint arXiv:1810.00783, 2018. Graphon mean field games and the GMFG equations: ε-Nash equilibria. E Peter, Minyi Caines, Huang, 2019 IEEE 58th Conference on Decision and Control (CDC). IEEEPeter E Caines and Minyi Huang. Graphon mean field games and the GMFG equations: ε-Nash equilibria. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 286-292. IEEE, 2019. Rene Carmona, arXiv:2012.05237Applications of mean field games in financial engineering and economic theory. arXiv preprintRene Carmona. Applications of mean field games in financial engineering and economic theory. arXiv preprint arXiv:2012.05237, 2020. Independent learning and subjectivity in mean-field games. Bora Yongacoglu, Gürdal Arslan, Serdar Yüksel, 2022 IEEE 61st Conference on Decision and Control (CDC). IEEEBora Yongacoglu, Gürdal Arslan, and Serdar Yüksel. Independent learning and subjectivity in mean-field games. In 2022 IEEE 61st Conference on Decision and Control (CDC), pages 2845-2850. IEEE, 2022. Erik Miehling, and Tamer Bas , ar. Reinforcement learning in nonstationary discrete-time linear-quadratic mean-field games. Muhammad Aneeq Uz Zaman, Kaiqing Zhang, 2020 59th IEEE Conference on Decision and Control (CDC). IEEEMuhammad Aneeq Uz Zaman, Kaiqing Zhang, Erik Miehling, and Tamer Bas , ar. Reinforcement learning in non- stationary discrete-time linear-quadratic mean-field games. In 2020 59th IEEE Conference on Decision and Control (CDC), pages 2278-2284. IEEE, 2020. Adversarial linear-quadratic mean-field games over multigraphs. Muhammad Aneeq Uz Zaman, Sujay Bhatt, Tamer Başar, 2021 60th IEEE Conference on Decision and Control (CDC). IEEEMuhammad Aneeq Uz Zaman, Sujay Bhatt, and Tamer Başar. Adversarial linear-quadratic mean-field games over multigraphs. In 2021 60th IEEE Conference on Decision and Control (CDC), pages 209-214. IEEE, 2021. Reinforcement learning for non-stationary discretetime linear-quadratic mean-field games in multiple populations. Muhammad Aneeq Uz Zaman, Erik Miehling, Tamer Başar, Dynamic Games and Applications. Muhammad Aneeq Uz Zaman, Erik Miehling, and Tamer Başar. Reinforcement learning for non-stationary discrete- time linear-quadratic mean-field games in multiple populations. Dynamic Games and Applications, pages 1-47, 2022. Multi-agent reinforcement learning: A selective overview of theories and algorithms. Handbook of Reinforcement Learning and Control. Kaiqing Zhang, Zhuoran Yang, Tamer Başar, Kaiqing Zhang, Zhuoran Yang, and Tamer Başar. Multi-agent reinforcement learning: A selective overview of theories and algorithms. Handbook of Reinforcement Learning and Control, pages 321-384, 2021.
[]
[ "Wind Profiling in the Lower Atmosphere from Wind-Induced Perturbations to Multirotor UAS", "Wind Profiling in the Lower Atmosphere from Wind-Induced Perturbations to Multirotor UAS" ]
[ "Javier González-Rocha \nDepartment of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA\n", "Stephan F J De Wekker [email protected] \nDepartment of Environmental Sciences\nUniversity of Virginia\n22903CharlottesvilleVAUSA\n", "Shane D Ross [email protected]. \nDepartment of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA\n", "Craig A Woolsey [email protected] \nDepartment of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA\n", "C A W " ]
[ "Department of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA", "Department of Environmental Sciences\nUniversity of Virginia\n22903CharlottesvilleVAUSA", "Department of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA", "Department of Aerospace and Ocean Engineering\nVirginia Tech\n24060BlacksburgVAUSA" ]
[]
We present a model-based approach to estimate the vertical profile of horizontal wind velocity components using motion perturbations of a multirotor unmanned aircraft system (UAS) in both hovering and steady ascending flight. The state estimation framework employed for wind estimation was adapted to a set of closed-loop rigid body models identified for an off-the-shelf quadrotor. The quadrotor models used for wind estimation were characterized for hovering and steady ascending flight conditions ranging between 0 and 2 m/s. The closed-loop models were obtained using system identification algorithms to determine model structures and estimate model parameters. The wind measurement method was validated experimentally above the Virginia Tech Kentland Experimental Aircraft Systems Laboratory by comparing quadrotor and independent sensor measurements from a sonic anemometer and two SoDAR instruments. Comparison results demonstrated quadrotor wind estimation in close agreement with the independent wind velocity measurements. However, horizontal wind velocity profiles were difficult to validate using time-synchronized SoDAR measurements. Analysis of the noise intensity and signal-to-noise ratio of the SoDARs proved that close-proximity quadrotor operations can corrupt wind measurement from SoDARs, which has not previously been reported.Direct methods have combined inertial and air-relative velocity measurements from GPS and a Pitot tube or multi-hole air data system to infer wind velocity using the wind triangle relationship[32,33]. Indirect methods, on the other hand, have exploited GPS and IMU measurements along with vehicle dynamic models to estimate wind velocity employing state estimation algorithms[34,35]. A comprehensive review of established direct and indirect methods for inferring wind with fixed-wing aircraft is presented in[36]. However, despite the advantages of operating fixed wing aircraft over open space, safely maneuvering in urban and complex environments to measure wind speed and PTH variables remains a challenge.More recently, multirotor UAS have become popular for direct and indirect measurements of atmospheric wind and PTH variables within the ABL. Multirotor UAS are mobile, portable, low cost, and easy to operate over complex and urban environments where it is prohibitively difficult for conventional atmospheric sensors or fixed-wind aircraft. Direct methods involve measuring wind velocity from an on-board flow sensor, which include various types of anemometers[37][38][39][40][41]and air data systems integrated with Pitot tubes or multi-hole probes[42,43]. The choice of sensor for wind sensing depends on size and power requirement of the sensor, as well as the airframe configuration and payload capacity of the multirotor UAS. Indirect methods, on the other hand, estimate wind velocity from wind induced perturbations to the aircraft motion and do not require an onboard airflow sensor. Conventional model-based approaches to wind estimation have involved kinematic[44,45], point mass[1,39,[46][47][48], and rigid body models [1,49] of control-augmented quadrotor dynamics, which characterize how a quadrotor responds to disturbances under feedback stabilization. A comparison of all three vehicle motion models has demonstrated that both the accuracy and bandwidth of wind estimates increases with model fidelity[1], which has made model-based wind sensing inside the ABL more useful and reliable. Especially in scenarios where direct measurements are impossible, i.e., when the aircraft is very small or when the aircraft is intended for some other purpose and is not equipped to measure the wind.To date, model-based wind estimation approaches accounting for vehicle dynamics have mostly incorporated models appropriate for hovering flight[1,[46][47][48][49][50]. Measuring wind velocity only while hovering limits the efficiency of multirotor aircraft to sample the lower atmosphere. The limited effectiveness of stationary sampling is largely due to the endurance of multirotor aircraft, which is typically less than 20 min. Many applications of multirotor UAS wind sensing require atmospheric sampling over horizontal and vertical distances. Therefore, there is a need to develop wind estimation algorithms that allow for movement of the multirotor UAS while accurately measuring wind velocity inside the ABL. This paper presents the validation of a model-based method for estimating vertical profiles of the horizontal wind velocity employing wind measurements from a sonic anemometer and SoDAR (Sound Detection And Ranging) wind profilers. The model-based method that is validated, referred as the dynamic rigid body wind profiling method or DRBWindPro method, is the extension presented in [2] of the wind sensing algorithm described in [1] to infer wind velocity using a dynamic rigid body model for hovering flight. The extension of the wind sensing algorithm incorporates dynamic rigid body models characterized from system identification for equilibrium flight conditions corresponding to steady ascent rates ranging from 0 to 2 m/s. The models from system identification were used to estimate the wind velocity in the vicinity of ground-based in situ and remote atmospheric sensors. Quadrotor wind estimates and wind measurements from ground-based atmospheric sensors were then compared to determine the accuracy of the DRBWindPro method.The organization of this paper is as follows. Section 2 introduces materials and methods used for model-based wind estimation. This section includes the formulation of aircraft dynamics, system identification of aircraft models, and the design of a state observer for wind estimation. The ground-based wind measurement methods are described in Section 3. In Section 4, results from system identification experiments and comparison of multirotor wind velocity measurements with ground-based measurements are presented. Section 5 presents a thorough discussion of results from
10.3390/s20051341
[ "https://arxiv.org/pdf/2001.02740v2.pdf" ]
210,116,803
2001.02740
e9fd4de0265d50e0cfe5b06f56ae0efacf1eae47
Wind Profiling in the Lower Atmosphere from Wind-Induced Perturbations to Multirotor UAS Published: 29 February 2020 Javier González-Rocha Department of Aerospace and Ocean Engineering Virginia Tech 24060BlacksburgVAUSA Stephan F J De Wekker [email protected] Department of Environmental Sciences University of Virginia 22903CharlottesvilleVAUSA Shane D Ross [email protected]. Department of Aerospace and Ocean Engineering Virginia Tech 24060BlacksburgVAUSA Craig A Woolsey [email protected] Department of Aerospace and Ocean Engineering Virginia Tech 24060BlacksburgVAUSA C A W Wind Profiling in the Lower Atmosphere from Wind-Induced Perturbations to Multirotor UAS Published: 29 February 202010.3390/s20051341Received: 1 January 2020; Accepted: 26 February 2020;sensors Article * Correspondence:unmanned aircraft systemssystem identificationwind estimationmulti-rotordroneatmospheric sciencewind profileboundary layer meteorology We present a model-based approach to estimate the vertical profile of horizontal wind velocity components using motion perturbations of a multirotor unmanned aircraft system (UAS) in both hovering and steady ascending flight. The state estimation framework employed for wind estimation was adapted to a set of closed-loop rigid body models identified for an off-the-shelf quadrotor. The quadrotor models used for wind estimation were characterized for hovering and steady ascending flight conditions ranging between 0 and 2 m/s. The closed-loop models were obtained using system identification algorithms to determine model structures and estimate model parameters. The wind measurement method was validated experimentally above the Virginia Tech Kentland Experimental Aircraft Systems Laboratory by comparing quadrotor and independent sensor measurements from a sonic anemometer and two SoDAR instruments. Comparison results demonstrated quadrotor wind estimation in close agreement with the independent wind velocity measurements. However, horizontal wind velocity profiles were difficult to validate using time-synchronized SoDAR measurements. Analysis of the noise intensity and signal-to-noise ratio of the SoDARs proved that close-proximity quadrotor operations can corrupt wind measurement from SoDARs, which has not previously been reported.Direct methods have combined inertial and air-relative velocity measurements from GPS and a Pitot tube or multi-hole air data system to infer wind velocity using the wind triangle relationship[32,33]. Indirect methods, on the other hand, have exploited GPS and IMU measurements along with vehicle dynamic models to estimate wind velocity employing state estimation algorithms[34,35]. A comprehensive review of established direct and indirect methods for inferring wind with fixed-wing aircraft is presented in[36]. However, despite the advantages of operating fixed wing aircraft over open space, safely maneuvering in urban and complex environments to measure wind speed and PTH variables remains a challenge.More recently, multirotor UAS have become popular for direct and indirect measurements of atmospheric wind and PTH variables within the ABL. Multirotor UAS are mobile, portable, low cost, and easy to operate over complex and urban environments where it is prohibitively difficult for conventional atmospheric sensors or fixed-wind aircraft. Direct methods involve measuring wind velocity from an on-board flow sensor, which include various types of anemometers[37][38][39][40][41]and air data systems integrated with Pitot tubes or multi-hole probes[42,43]. The choice of sensor for wind sensing depends on size and power requirement of the sensor, as well as the airframe configuration and payload capacity of the multirotor UAS. Indirect methods, on the other hand, estimate wind velocity from wind induced perturbations to the aircraft motion and do not require an onboard airflow sensor. Conventional model-based approaches to wind estimation have involved kinematic[44,45], point mass[1,39,[46][47][48], and rigid body models [1,49] of control-augmented quadrotor dynamics, which characterize how a quadrotor responds to disturbances under feedback stabilization. A comparison of all three vehicle motion models has demonstrated that both the accuracy and bandwidth of wind estimates increases with model fidelity[1], which has made model-based wind sensing inside the ABL more useful and reliable. Especially in scenarios where direct measurements are impossible, i.e., when the aircraft is very small or when the aircraft is intended for some other purpose and is not equipped to measure the wind.To date, model-based wind estimation approaches accounting for vehicle dynamics have mostly incorporated models appropriate for hovering flight[1,[46][47][48][49][50]. Measuring wind velocity only while hovering limits the efficiency of multirotor aircraft to sample the lower atmosphere. The limited effectiveness of stationary sampling is largely due to the endurance of multirotor aircraft, which is typically less than 20 min. Many applications of multirotor UAS wind sensing require atmospheric sampling over horizontal and vertical distances. Therefore, there is a need to develop wind estimation algorithms that allow for movement of the multirotor UAS while accurately measuring wind velocity inside the ABL. This paper presents the validation of a model-based method for estimating vertical profiles of the horizontal wind velocity employing wind measurements from a sonic anemometer and SoDAR (Sound Detection And Ranging) wind profilers. The model-based method that is validated, referred as the dynamic rigid body wind profiling method or DRBWindPro method, is the extension presented in [2] of the wind sensing algorithm described in [1] to infer wind velocity using a dynamic rigid body model for hovering flight. The extension of the wind sensing algorithm incorporates dynamic rigid body models characterized from system identification for equilibrium flight conditions corresponding to steady ascent rates ranging from 0 to 2 m/s. The models from system identification were used to estimate the wind velocity in the vicinity of ground-based in situ and remote atmospheric sensors. Quadrotor wind estimates and wind measurements from ground-based atmospheric sensors were then compared to determine the accuracy of the DRBWindPro method.The organization of this paper is as follows. Section 2 introduces materials and methods used for model-based wind estimation. This section includes the formulation of aircraft dynamics, system identification of aircraft models, and the design of a state observer for wind estimation. The ground-based wind measurement methods are described in Section 3. In Section 4, results from system identification experiments and comparison of multirotor wind velocity measurements with ground-based measurements are presented. Section 5 presents a thorough discussion of results from Introduction Measuring wind velocity near the Earth's surface is critical to understanding the surface-atmosphere interactions driving the dynamic state of the atmospheric boundary layer (ABL). How the ABL evolves with space and time impacts public health and safety [1][2][3][4][5][6], transport of air pollutants, pollen and spores [7][8][9][10], wind power supply to smart grid systems [11][12][13][14][15], forecast of local weather [3][4][5][6]16], air traffic control at airports [17][18][19][20], the spread and management of wildfires [21][22][23][24], and emissions mitigation of greenhouse gases [25][26][27][28][29]. Therefore, accurately characterizing the dynamic state of the ABL over micro-and mesoscale domains is important [3,16,30,31]. However, observations of wind velocity at high spatial resolution are difficult to attain due to the cost and limited mobility of conventional atmospheric sensing technology. Early work to address the existing gap of atmospheric wind measurements and atmospheric parameters such as atmospheric pressure, air temperature, and relative humidity (PTH) involved fixed-wing aircraft for their predictable dynamics, payload capacity, and flight endurance. Approaches to measuring wind with fixed wing aircraft consist of direct and indirect measurements of wind velocity. system identification and from comparing multirotor and ground-based wind measurements. Finally, a summary of findings and future work to extend the utility of multirotor UAS for wind sensing are presented in Section 6. Materials and Methods Modeling Framework The equations of motion for a control-augmented (i.e., feedback-stabilized) quadrotor can be expressed as a system of first-order, nonlinear, time-invariant ordinary differential equations [1,2]: x = f (x, u, w(t, x)), x(t 0 ) = x 0(1) relating the rate of changeẋ of the vehicle's 12-dimensional state x (i.e., position, attitude, velocity, and angular velocity), to the state itself, the control inputs u, and wind disturbances w(t, x) varying over time and space. Moreover, when the aircraft motion is modeled as a small perturbation from some equilibrium flight condition that corresponds to a constant vertical ascent speed denoted by V z eq , the nonlinear dynamics describing the control-augmented motion of the quadrotor is well approximated by a linear model. As a result, one may infer wind velocity from wind-induced motion perturbations to a quadrotor employing estimation theory developed for linear systems. Linear approximations of quadrotor dynamics for wind estimation are considered in this study for hovering and steady-ascending motions satisfying trim flight conditions. For a quadrotor, trim flight conditions are satisfied when both translational rates v and rotational rates ω remain constant over time, i.e.,v ≡ 0 andω ≡ 0. Linear approximations of quadrotor dynamics for hovering and ascending flight are in the form, d dtx = Ax + Bũ + Γw,(2) where the vectorsx = x − x eq andũ = u − u eq denote, respectively, small deviations in the state and input vectors from their steady-state values. Additionally, the state matrix A ∈ R 12×12 models unforced dynamics, the input matrix B ∈ R 12×4 characterizes applied forcing, and the disturbance matrix Γ ∈ R 12×3 captures wind-induced perturbations. This model form is used to estimate the horizontal component wind velocity at different steady motion conditions V zeq . Aircraft System Identification Aircraft system identification is used to characterize the state and input matrices A and B for a quadrotor flying in still air conditions (i.e., w(x, t) ≈ 0 m/s). In general, this modeling approach is a multi-faceted process that relies on input-output flight test data to characterize bare-airframe or control-augmented dynamic models for an aircraft, depending on application. Figure 1 shows a schematic of the inputs u and outputs y used to identify bare-airframe and control-augmented models. A bare-airframe model, assuming actuator dynamics to be negligible, is identified using control signals from the flight controller µ ctrl and the vehicle's measured dynamic response y. A control-augmented model, alternatively, is identified using the reference signal δ r from pilot-induced joystick commands and the vehicle's measured dynamic response y. Which model is identified depends on its application. For wind estimation purposes with an off-the-shelf quadrotor, we use the latter because it does not require knowledge of the onboard flight controller architecture. The quadrotor models from system identification are for steady-state equilibrium flight conditions corresponding to the hovering and steady ascending flight: V zeq = {0.0, 0.5, 1.0, 1.5, 2.0} m/s. The identification of each model involved separately determining four sub-models that describe the plunge, yaw, roll, and pitch dynamics of the quadrotor; see Figure 2. In this process, stepwise regression was used first to determine the parameter structure of each model. Results from stepwise regression were then used to estimate model parameters using an output error algorithm. This approach to system identification was used to minimize the set of parameters being estimated at one time and to avoid overparameterized models. Multirotor UAS Platform The multirotor UAS used to measure the wind velocity is an off-the-shelf 3DR Solo quadrotor shown in Figure 3. This aircraft is 25 cm tall with a 46 cm diagonal between motor shafts. Fully equipped with a lithium polymer battery pack and a 3-axis camera gimbal, the quadrotor weighs 1.5 kg and has a payload capacity of 0.5 kg. The propellers used with the quadrotor are a Master Airscrew 10 × 4.5 propeller set. The quadrotor's autopilot is a Pixhawk 2.1 Green Cube manufactured by ProfiCNC. The autopilot operates using open-source Arducopter firmware and is compatible with MissionPlanner and Solex telemetry software. On board the Pixhawk 2.1 Green Cube are the sensors listed in Table 1 that are part of the autopilot's attitude and heading reference system (AHRS). ≈ 0 m/s) to minimize the impact of exogenous excitations on the system identification process. The input-output measurements used for system identification consisted of pilot-induced, sinusoidal joystick commands and the vehicle's measured dynamic response. The system identification experiments were performed in two parts. A first set of experiments were performed to identify the quadrotor's hovering flight dynamics. This required exciting from equilibrium flight the quadrotor's plunge, yaw, roll and pitch dynamics shown in Figure 2. A second set of experiments was conducted to identify quadrotor models for constant ascent rates varying between 0.5 and 2 m/s. This involved exciting the quadrotor's roll and pitch dynamics from equilibrium flight conditions corresponding to V zeq > 0. For the latter case, the plunge and yaw dynamics of the quadrotor were assumed to be well approximated by models identified for hovering flight considering that the vehicle's response to wind perturbations in steady-ascending flight is dominated by roll and pitch motions. Measurements from both sets of system identification experiments were then used to identify the model structures and parameter estimates approximating the quadrotor's dynamics for all five operating conditions specified by V z eq . Model Structure Determination The parameter structure of each model was determined from input-output measurements employing the stepwise regression algorithm described in [51]. Using this approach, a set of postulated regressors, χ = {χ 1 , χ 2 , · · · , χ n } is tested one at a time to determine which ones significantly improve the fit of the model z(k) = a 0 + m ∑ i=1 a i χ i (k), k = 1, 2, · · · , N(3) where z is the quadrotor's measured response, a 0 is the model bias, a = {a 0 , a 1 , · · · , a m } is the set of model coefficients associated with m regressors, and N is the sample size of measurements. How well each model structure fits the observed data as regressors are added or removed is determined using the F 0 statistic and coefficient of determination R 2 metrics. The F 0 statistic quantifies how much each regressor contributes to the fit of the model. The coefficient of determination quantifies how well the model output matches the measured data. Using both metrics, a total of four parameter structures were identified to characterize the quadrotor's plunge, yaw, roll, and pitch dynamics. Parameter Estimation The model structures determined from step-wise regression were used to initialize the estimation of model parameters using the output error algorithm described in [51]. The output error algorithm estimates model parameters using the output of the linear aircraft model described by Equation (2) in still air conditions and using the N sample points of measured flight data, which are assumed to be corrupted by sensor noise η. The model and measurements used by the output error method are summarized below: d dtx = Ax + Bũ,x(0) = x 0 (4) y = Cx + Dũ (5) z(k) = y(k) + η(k) k = 1, 2, · · · , N(6) where y is the output vector, z is the measurement vector, C is the output matrix, and D is the feedthrough matrix. This formulation of the output error method assumes that the model being identified is free of process noise, making numerical propagation of state measurements possible. Moreover, output error parameter estimation assumes flight measurements to be corrupted with uncorrelated, zero-mean Gaussian noise η ∈ N (0, R Cov ) such that the covariance matrix of measurement noise is diagonal, Cov(η(k)) = E[η(k)η T (k)] = R Cov Using this framework, parameter estimates are tuned iteratively while minimizing the cost function, J = 1 2 N ∑ i=1 [y(k) − z(k)] T R −1 Cov [y(k) − z(k)](7) which is the uncertainty-weighted residual between the model output and observation measurements. Employing the output error approach, three sets of parameters were estimated and averaged to characterize quadrotor models for hovering and steady vertical ascent conditions. The quadrotor models characterized from averaged parameter estimates were validated using a separate flight test data set collected during system identification experiments. Model Validation Linear models approximating steady-flight quadrotor dynamics were validated using input-output data collected separately during system identification flight experiments. The validation process for linear models involved comparing model outputs and state measurements corresponding to pilot-generated excitations using the root-mean-squared error (RMSE) metric: RMSE = 1 N N ∑ k=1 (y(k) − z(k)) 2 where y is the model output, z is the state measurements, and N is the measurement sample size. In general, small RMSE values are indicative of accurate parameter estimates. Results from the RMSE quantification were used to assess the goodness of each model prior to designing a state observer for wind estimation. Observer Synthesis To synthesize observers for wind velocity estimation, the dynamic rigid body wind sensing method presented in [1,2] was adapted. Therefore, assuming absolute measurements from the GPS antenna and AHRS on board the quadrotor to be available, the output equation, as in [1,2], is of the form y = I 12x +      0 3 0 3 I 3 0 3      w where output measurements of translational velocity are the summation of both air-relative and wind velocity (with identity and zero matrices written in short notation, e.g., I 12 ∈ R 12×12 ). We assume in this formulation that sensor noise in the output measurement is negligible, and is therefore not accounted for. The quadrotor's output measurement and identified models were then used to formulate wind-augmented models for the set of operating conditions prescribed by V zeq . Wind velocity was estimated using the quadrotor models identified from system identification in a state observer framework. State observers were developed based on wind-augmented models corresponding to each of five equilibrium flight conditions. Each wind-augmented model is obtained by reformulating (2) such that the wind disturbance is part of the wind-augmented state vector: x A = [x T , w T ] T . Here, as in [1,2], variations of wind velocity with respect to time were assumed to vary slowly relative to the dynamics of the quadrotor such that d dt w ≈ 0. Therefore, wind-augmented dynamic models corresponding to each flight equilibrium were defined as follows: d dt x A = A Γ 0 3×12 0 3 A A x A + B 0 3×4 B Aũ y =      I 3 0 3 0 3 0 3 0 3 0 3 I 3 0 3 0 3 0 3 0 3 0 3 I 3 0 3 I 3 0 3 0 3 0 3 I 3 0 3      C A x A(8) where A A ∈ R 15×15 is the wind-augmented state matrix, B A ∈ R 15×4 is the wind-augmented input matrix, and C A ∈ R 12×15 is the wind-augmented output model. To verify the observability of the augmented dynamic model, an observability analysis was conducted to determine if wind estimates can be realized from the model and output measurements. The system is observable if and only if the observability matrix defined below is column-wise full rank. O(C A , A A ) =     C A C A A A . . .     =                    I 3 0 3 0 3 0 3 0 3 0 3 I 3 0 3 0 3 0 3 0 3 0 3 I 3 0 3 I 3 0 3 0 3 0 3 I 3 0 3 0 3 G w I 3 0 3 I 3 0 3 0 3 0 3 I 3 0 3 0 3 G g −d w e 3 e T 3 0 3 0 3 0 3 0 3 D m v D m ω 0 3 . . . . . . . . . . . . . . .                    The analysis shows that the observability matrix is full rank, i.e., rank [O(C A , A A )] = 15. Therefore, computing a suitable observer gain matrix G O , state estimates of the following observer will converge to the state of the system (8) d dt x A = A A x A + B A u + G O (y − C A x A )(9) Because the augmented state vector includes the wind velocity, it follows that the state estimator (9) provides a convergent estimate of w, provided the underlying assumptions hold (e.g., small perturbations from the nominal state). Moreover, in the implementation of this framework for steady vertical-ascent wind estimation, the appropriate set of model parameters was switched manually offline before processing quadrotor flight measurements. Experimental Validation of Wind Estimates Field Experiment Setup Field experiments were performed at the KEAS Laboratory on June 5th, 2018 from 15:00 to 20:30 EDT to validate horizontal wind velocity estimates from quadrotor hover and vertical steady ascent conditions: V zeq = {0, 0.5, 1.0, 1.5, 2.0} m/s. Originally, we intended to validate quadrotor wind estimates using observations from ground-based sensors and a small solid-state sonic anemometer mounted on board a separate multirotor UAS. However, we were unable to use the solid-state anemometer due to a hardware malfunction. Thus, the accuracy of the model-based wind estimates was examined using measurements from ground-based sensors only. The ground-based sensors that were used to validate quadrotor estimates of horizontal wind velocity consist of the sonic anemometer and SoDAR wind profilers found in Figure 4. The Gill MaxiMet sonic anemometer (SA) shown in Figure 3a and the Remtech PA-0 SoDAR (SR-SoDAR) shown in Figure 4b were used to validate quadrotor wind estimates at 10 m AGL. Alternatively, quadrotor profiles of horizontal wind velocity were validated using both the Remtech PA-0 and the ASC4000i SoDAR (LR-SoDAR) shown in Figure 4c. The configuration of the three sensors relative to quadrotor operations is shown in Figure 4d. Additionally, the performance envelope of each sensor is found in Table 2. Using this sensor configuration quadrotor wind velocity estimates were validated after processing observations from independent sensors. It is also important to note that our focus during validation experiments laid primarily on estimating the horizontal component of wind velocity while hovering or profiling due to constraints driven by science objectives, the flight envelope of the quadrotor, and experiment setup. Constant-rate vertical profiling with multirotor UAS is already used to measure PTH within the ABL [3,16], and while it is possible for a multirotor aircraft to measure PTH in descent [31], steady descent rates are only realizable at very low speeds. Therefore, wind estimation in steady vertical descent may be inefficient for model-based wind profiling. Moreover, while we are certainly interested in horizontal profiling and, more generally, wind profiling for arbitrary steady motions, the independent sensors (i.e., SoDARs) that were used in this study only provide vertical wind profiles. Thus, we were only able to validate vertical profile measurements. Comparison with Ground-Based Observations We also determine the limitations of wind validation experiments by comparing differences across ground truth observations at different heights. When observation differences are small, we can validate quadrotor wind estimates reliably. This is because observations collected at distinct locations will not be representative of the wind field measured by the quadrotor in high variability conditions. To compare independent wind observation at different heights, sonic anemometer and LR-SoDAR observations recorded every 1 and 30 s, respectively, were averaged to match the 300-s sampling period of the SR-SoDAR. Differences across sensor observation were then characterized using the mean difference error (MDE) and RMSE metrics at 10, 60 and 110 m AGL. Results from the comparisons were used to assess the accuracy of quadrotor wind estimates for different flight regimes. (a) Gill MaxiMet (SA) (b) Remtech PA-0 (SR SoDAR) (c) ASC 4000i (LR SoDAR) (d) Experiment Setup Results System Identification Model Structure Determination The quadrotor flight dynamic model for hovering and steady ascending flight is decomposed into four sub-models that describe plunging, yawing, rolling, and pitching motion. Table 3 shows all four model forms and associated parameters. The plunge model is a system of two first-order ordinary differential equations parameterized by propulsive and damping parameters. The yaw model is a system of two first-order ordinary differential equations with rotational damping and stiffness parameters. Finally, the roll and pitch models are systems of four first-order ordinary differential equations. Table 3. The plunge, yaw, roll and pitch model structures of the quadrotor determined from system identification flight experiments and step-wise regression algorithm presented in [51]. Model Parameter Structure Plunge żẇ = 0 1 0 Z w z w + 0 Z δ δ plunge Yaw ψ r = 0 1 N ψ N r ψ r + 0 N δ δ yaw Roll    ẏ φ v p     =     0 0 1 0 0 0 0 1 0 Y φ Y v 0 0 L φ 0 L p         y φ v p     +     0 0 0 L δ     δ roll Pitch    ẋ θu q     =     0 0 1 0 0 0 0 1 0 X θ X u 0 0 M θ 0 M q         x θ u q     +     0 0 0 M δ     δ pitch 4.1.2 . Parameter Estimation Three sets of quadrotor model parameters were estimated for each of five equilibrium flight conditions. Each set of parameters was estimated using the model structures determined from step-wise regression and the output error algorithm described in Section 2.2. Model parameters for the plunge, yaw, roll, and pitch model structures were first estimated for hovering flight conditions (i.e., v = 0 and ω = 0). Subsequently, roll and pitch model parameters were estimated for constant vertical ascent flight conditions varying from 0.5 to 2.0 m/s. We assume the plunge and yaw model parameters to be invariant with vertical ascent rate. Model parameter estimates for each flight equilibrium were averaged to obtain nominal models for wind estimation. Averaged parameter estimates and standard error (SE) values for plunge and yaw models are listed in Table 4. Additionally, averaged roll and pitch model parameters and standard error values are listed for hovering and ascending flight conditions in Tables 5 and 6, respectively. The dependence on vertical ascent rate was also characterized for roll and pitch quadrotor parameters. Results from this characterization are shown in Figure 5 where roll and pitch model parameters are plotted as a function of ascent rate. Each parameter estimate appears with absolute error bars, colored in black, representing the range of estimates obtained from the three experimental data sets. Orange-colored bars were also included to denote minimum and maximum values across all five ascent rates. Zeroth-and first-order polynomials were fit to the parameter estimates as a function of ascent rate. The first-order fit, on the other hand, characterizes the trend in parameter values with respect to ascent rate. Note that only a subset of parameters exhibit clear trends with respect to ascent rate. It is possible that these local, small-perturbation models do exhibit high sensitivity to ascent rate, as suggested by Figure 5. If so, then these results may suggest flight regimes to be avoided when estimating wind velocity from platform motion; regions of high parameter sensitivity may produce less accurate wind estimates. For the aircraft and dynamic model considered here, the parameters vary less at lower ascent rates (0.5 m/s or less). Thus, one might expect more accurate wind measurements during slower climbs. It is possible, however, that the variation in parameter estimates is an artifact of the data collection method for system identification. At higher climb rates, it is more difficult to manually generate the rich and precisely timed excitation signals needed for model identification. An automated approach to system identification may improve the repeatability of parameter estimates. 0-2 m/s Z w −0.55 0.28 1/s N ψ −1.71 0.41 1/s 2 Z δ −1.71 0.79 1/kg N r −0.84 0.53 1/s - - - - N δ 2.41 1.18 1/(kg · m 2 ) Model Validation Models characterized from step-wise regression and output error methods were validated by comparing the model output and aircraft's response to an excitation input. Agreement between the model output and measured response is compared using the RMSE metric discussed in Section 2.2.5. Results from this validation are shown in Figure 6 for the plunge, yaw, roll, and pitch models characterized for hovering flight V zeq = 0. Results from the RMSE assessment for the plunge and yaw models are shown in Table 7. The RMSE results for the pitch and roll models associated constant vertical ascent rates ranging between 0 and 2 m/s are also shown in Table 7. Comparison of Wind Velocity Measurements Sonic Anemometer and SoDAR Comparison The difference across sonic anemometer and SR-SoDAR wind measurements was characterized from 15:00 to 20:30 EDT to assess the spatial variability of wind at 10 m AGL. Based on 300-s averaged measurements from the sonic anemometer, prevailing wind conditions during validation experiments were from the northwest with wind speeds ranging from 1.2 to 4.0 m/s (see Figure 7a). As shown in Table 8, the MDE and RMSE values of wind speed observations were measured to be 0.7 m/s and 1.0 m/s, respectively. Wind direction MDE and RMSE values were measured as 32 • and 100 • . Therefore, the difference across the spatial separation of ground truth measurements used to validate quadrotor wind estimates at 10 m AGL is relatively small. SoDAR Comparison Wind observations from the LR-and SR-SoDAR were compared from 15:00 to 20:30 EDT at 60 and 110 m AGL to assess the spatial variability of wind conditions during quadrotor wind profiling operations. The prevailing wind conditions as reported by the SR-SoDAR were from northwest with wind speeds ranging from 2.3 to 7.9 m/s at 60 m AGL and from 2.0 to 8.0 m/s at 110 m AGL. Wind observations from SR-and LR-SoDAR at 60 and 110 m AGL are shown in Figure 7c,d, respectively. As reported in Table 9, the maximum MDE and RMSE for wind speed and wind direction were observed at 110 m AGL. The MDE and RMSE of wind speed observations at 110 m AGL were found to be −0.9 m/s and 1.4 m/s, respectively. Wind direction MDE and RMSE values, on the other hand, were measured to be 0 • and 26 • , respectively. Thus, spatial wind variations were also observed to be small at higher altitudes. Validation of Quadrotor Wind Estimates Wind estimates from three quadrotor flights hovering at 10 m AGL between 18:00 and 20:30 were compared to sonic anemometer and SR-SoDAR wind observations. Results from the comparison are shown in Figure 7a, where the time lapse of each quadrotor flight is denoted with a rose-colored vertical band. How well quadrotor and ground-based wind measurements compared is reported in Table 10 using the MDE metric. The average of wind speed and wind direction of absolute MDE values of quadrotor wind speed estimates were found to be 0.6 m/s and 0.5 m/s relative to sonic anemometer and SR-SoDAR observations. The average absolute MDE values for quadrotor wind direction estimates relative to the sonic anemometer and SR-SoDAR were found to be 14 • and 10 • relative to sonic anemometer and SR-SoDAR measurements, as well. Therefore, quadrotor wind estimates from hovering flights were assessed to have an accuracy comparable to that of conventional ground-based wind sensors. 15 In contrast to assessing the accuracy of wind estimates at 10 m AGL, validating quadrotor wind profiles ascending vertically at various steady rates proved to be more involved. Results from a two-part assessment found in Appendix A revealed that quadrotor profiling operations corrupt ground-truth SoDAR observations. Consequently, making time-synchronized comparisons of quadrotor and SoDAR wind measurements for validation purposes was not possible for an accurate assessment of quadrotor wind estimates. To circumvent corrupted wind observations from SoDARs, quadrotor wind profiles were validated using SoDAR measurements collected before and after quadrotor operation as well as linearly-interpolated wind profiles. In total, four sets of quadrotor wind estimates corresponding to V zeq = {0.5, 1.0, 1.5, 2.0} were compared to SoDAR wind observations to validate model-based wind profiling. Results from the comparisons shown in Figure 8c (and in Figure A7a,c,d of Appendix B) demonstrate quadrotor, SoDAR, and interpolated wind profiles to agree most accurately between 18:40 and 19:01 EDT, when wind variability across the sampling domain was observed to be the lowest. Good correspondence was also observed for a subset of quadrotor, SoDAR, and interpolated wind profiles corresponding to a period of moderate wind variability between 17:31 and 17:55 EDT (see Figure A6a,b in Appendix B). Alternatively, during periods of high wind variability, comparisons of quadrotor, SoDAR, and interpolated wind profiles were less consistent as is shown in both Figures 8a and A8. However, in spite of the varied results for short-period comparisons, Figure 7b,c shows quadrotor and SoDAR observation trends to match well at 60 and 110 m AGL over a five hour duration of field experiments. Following the validation of quadrotor wind profiles, a parameter sensitivity was conducted to assess how the accuracy of wind estimates degrades with parameter variations (see Appendix C). Model parameters were perturbed by the maximum difference between zeroth-and first-order parameter characterizations shown in Figure 5. Results from the sensitivity analysis show a strong to moderate dependence between the accuracy wind estimates and the parameters Y φ , X θ , Y v , X u , L φ , M θ , L p , and M q . As shown in Figure A9, there was a considerable percent change in wind estimation RMSE values when this subset of model parameters was perturbed. This outcome suggests that the accuracy of quadrotor wind estimates will decrease significantly when quadrotor operations deviate from the operating conditions for which the dynamic models have been characterized. Discussion Five models were identified to characterize the control-augmented rigid body dynamics of a quadrotor for wind estimation in hovering and steady vertical-ascent flight. An observability analysis confirmed that it is possible to estimate wind velocity using all five models. However, model parameter estimates were found to fluctuate significantly at higher ascent rates, which can greatly impact wind estimation error based on the sensitivity analysis presented in Appendix C. Parameter fluctuations, as mentioned in Section 4.1.2, may be the product of ambient flow and vehicle interactions at specific flight regimes or related to limitations with system identification experiments. Hence, more in-depth studies are required to understand the nature of parameter fluctuations at higher rates. Anomalies were also detected in SoDAR wind measurements coinciding with periods of quadrotor operations. A two-part evaluation was performed to determine the nature of factors corrupting SoDAR observations. Examination of GPS position coordinates demonstrated the quadrotor to ascend through the sampling volume of SoDARs at approximately 60 m AGL. An assessment described in Appendix A of both the noise intensity and signal-to-noise ratio recorded by each SoDAR revealed a correlation between corrupted measurements and flight operations that strengthened with altitude. For this reason, it has been determined that quadrotor operations can significantly impact SoDAR observations when operating within the SoDAR's sampling volume. Therefore, experiments involving SoDAR and multirotor operations in close proximity will have to mitigate for quadrotor noise. A separate quadrotor with a small sonic anemometer on board was also considered as an alternative to validate model-based wind estimates. However, due to a hardware malfunction, the quadrotor-based wind sensor was not used in validation experiments. Other commercially available options employing indirect black-box methods for wind sensing, like the ones built into DJI multirotor aircraft, were not considered to validate model-based wind estimation. These alternatives are largely proprietary, do not offer wind data storage, and lack accuracy specifications for wind measurements. Thus, validation experiments were conducted for hovering and steady vertical-ascent flight wind estimates employing anemometer and SoDAR measurements only. In spite of challenges with validation experiments, a considerable number of wind estimates were validated successfully. Wind estimates from the quadrotor hovering at 10 m AGL demonstrated good agreement between sonic anemometer and SoDAR measurements across all three flights. These results were found to be comparable to rigid-body model wind estimates reported in [1]. Thus, the rigid-body model wind estimation algorithms we use for measuring wind in hovering flight performs well across different quadrotor platforms. Quadrotor wind profile estimates, on the other hand, were validated using SoDAR observations from 10 to 120 AGL. Comparison results for periods of low wind variability demonstrate quadrotor wind profile estimates in close agreement with SoDAR wind speed and wind direction observations. This outcome provides impetus for additional comparisons to assess more closely the accuracy of model-based wind profiling inside the ABL. Future work will involve improving validation experiments for a more thorough performance assessment of quadrotor wind profile estimates. Field experiments for wind estimate validation will require increasing the spatial separation between SoDARs to ensure quadrotor operations do not interfere with wind field measurements. Validation of model-based wind estimates will also incorporate in situ measurements from a sonic anemometer on board a separate quadrotor. Lastly, because coincident measurements are not feasible, validation experiments will have to take place when atmospheric conditions are relatively homogeneous and stationary, and significant uniformity of the wind field sampled by atmospheric sensors is expected. Conclusions An off-the-shelf quadrotor can be used to obtain model-based wind velocity estimates as long as the motion data logged on board the autopilot is accessible to the user. However, the accuracy of wind velocity estimates depends on how well the motion model characterizes the dynamics of the quadrotor for its operating condition. This paper extends a model based framework exploiting the rigid body dynamics of a quadrotor for hovering-flight wind estimation to estimate wind velocity along a vertical path in the lower atmosphere. The extension involved characterizing rigid body models for equilibrium flight conditions corresponding to each of five steady-ascending rates: V zeq = {0.0, 0.5, 1.0, 1.5, 2.0} m/s. Each quadrotor model was characterized employing stepwise regression and output error parameter estimation. An observability analysis confirmed the feasibility of estimating wind velocity using the identified model structures. Trends in parameter estimates also suggest that slower ascent rates may result in more accurate wind estimates. Significant variations in parameter estimates for higher ascent rates can be the outcome of limitations generating manually the rich and precisely timed excitation signals needed for model identification. Further studies are required to investigate this possibility in depth. Field experiments were conducted to validate quadrotor wind estimates using in-situ and remote-sensing atmospheric sensors. Results from validation experiments demonstrated quadrotor wind estimates in hovering flight to be within within small error of sonic anemometer and SoDAR wind observations. Quadrotor wind profile estimates, on the other hand, were difficult to validate comprehensively because quadrotor operations affect the reliability of SoDAR wind measurements. However, in instances when atmospheric conditions were relatively invariant prior to and after quadrotor operations, quadrotor wind estimates demonstrated very good agreement with wind speed and wind direction from SoDAR measurements. Overall, this study demonstrates the feasibility of model-based vertical wind profiling using multirotor UAS in the lower atmosphere. Author Contributions: J.G.-R. developed the model-based wind sensing methodology presented in this manuscript, characterized vehicle models using aircraft system identification, led field experiments to validate quadrotor wind estimates, curated data from field experiments, and led the writing of the manuscript. S.F.J.D.W. co-led field experiments, provided sonic anemometer wind data, provided guidance for the analysis wind measurements, and assisted in writing the manuscript. S.D.R. assisted with validation experiments, provided guidance for the analysis of wind measurements. C.A.W. provided guidance for the analysis of wind measurements, and assisted in writing the manuscript. All authors have read and agreed to the published version of the manuscript. Acknowledgments: We thank Jean-Michel Fahmi, Virginia Tech, for serving as a pilot in command for some UAS missions conducted for media and documentation purposes. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations The following abbreviations are used in this manuscript: Appendix A. A Reliability Study of SoDAR Wind Measurements A two-part reliability study was conducted to investigate anomalies detected in SoDAR wind observations during quadrotor operations. The first part of the study looked at the spatial footprint of quadrotor operations relative to both the position and viewing angle of each SoDAR. The spatial footprint of quadrotor operations relative to SoDAR wind observations was determined from GPS position information provided by the quadrotor's autopilot computer and SoDAR data logs. Quadrotor and SoDAR position information was used to determine if airframe obstruction of acoustic signals or propeller downwash corrupted SoDAR wind measurements. The second part of the study examined both the signal-to-noise-ratio (SNR) and noise intensity corresponding to wind measurements from each SoDAR prior to and during quadrotor operations. Combined, SNR and noise intensity SoDAR measurements were used to determine if anomalies in wind observations were attributed to quadrotor noise during flight operations. Findings from the two-part study can be used to inform best practices for integrating quadrotor and SoDAR operations for atmospheric wind sensing. From the two-part reliability study it was determined that quadrotor flight operations impact SoDAR wind observations when operating in close proximity. Assessment of quadrotor's flight path showed the quadrotor profiling through the sampling volumes of both the LR-SoDAR and SR-SoDAR 60 m AGL during flight operations. A 3-D rendering of this result is shown in Figure A1 where the ground position of the quadrotor and two SoDARs are plotted on the plot's x-y plane relative sonic anemometer and the height of measurements is plotted on the z axis. Additionally, sudden changes in noise intensity and SNR during quadrotor operations were observed to coincide with corrupted wind measurements as shown in Figures Results from the two-part study demonstrate a strong relationship between quadrotor noise and anomalies found in SoDAR wind observations. Based on our findings, quadrotor operations can interfere significantly with ground-based acoustic wind measurements. Precaution should be exercised when operating multirotor aircraft near SoDARs. Users will have to gauge a safe distance of separation based on the sampling volume of the SoDAR and the size of the multirotor aircraft used in flight operations. Appendix B. Quadrotor Wind Velocity Profiles Additional quadrotor wind velocity profiles corresponding to constant vertical ascent rates of 1, 1.5 and 2 m/s are shown in Figures A6-A8. Appendix C. Sensitivity of Wind Estimates to Parameter Variations Following the validation of quadrotor wind estimates from hovering flight, a sensitivity analysis was conducted to determine bounds on the accuracy of wind estimates resulting from parameter error. The nominal model used in the sensitivity analysis is characterized from the zeroth-order parameter values shown in Figure 5. The decrease in accuracy of wind estimates was quantified perturbing each zeroth-order parameter value at a time using the percent change of RMSE values as metric RMSE Percent Change = ∑ N i=1 [w − w] 2 − ∑ N i=1 [w − w * ] 2 ∑ N i=1 [w − w] 2 × 100 (A1) where w is the true wind measurement, w is the unperturbed quadrotor wind estimates, w * is the perturbed wind estimate, and N is the total number of wind estimates. The perturbation of each parameter was determined from the maximum difference between the zeroth-and first-order fit, which is shown in Figure 5 as well. Outcomes from this study are useful to understand the limitations of the nominal model for the range of steady-ascent rates that were employed for wind estimation. Results from the sensitivity analysis confirms the accuracy of wind estimates diminishes as parameter error increases. As is shown in Figure A9, wind estimates degrade most significantly when when Y φ and X θ parameter values are varied. Parameter values Y v , X u , L φ , M θ , L p , and M q also had a considerable effect on the accuracy of wind estimates. On the other hand, perturbations of L δ and M δ were observed to have no or insignificant impact. These results, combined, show that the use of a single model will lead to estimation errors when operating the quadrotor in off-nominal conditions. Figure 1 . 1A schematic of input-output signals for closed-loop and open-loop mappings. Figure 2 . 2Quadrotor plunge, yaw, roll, and pitch modes. Figure 3 . 3(a) The multirotor UAS employed for validation of model-based wind estimation along with (b) dimensions. Figure 4 . 4(a) The Gill MaxiMet sonic anemometer used to measure wind velocity 10 m AGL. (b) The Remtech PA-0 sensor (SR SoDAR) used to measure wind velocity from 10 to 120 m AGL. (c) The ASC 4000i sensor (LR SoDAR) used to measure wind velocity from 30 to 120 m AGL. (d) The experiment setup used to validate quadrotor wind estimates from 10 to 120 m AGL. Figure 5 . 5Roll and pitch model parameter estimates corresponding to vertical constant ascent rates V zeq = {0.0, 0.5, 1.0, 1.5, 2.0} m/s. Figure 6 . 6Validation of the (a) plunge, (b) yaw, (c) roll, and (d) pitch models identified for quadrotor hovering flight. Figure 7 . 7Comparison of wind observations collected from the quadrotor and independent sensors at (a) 10 m AGL, (b) 60 m AGL, and (c) 110 m AGL from 15:00 to 20:30 EDT on 5 June 2018. Figure 8 . 8Comparison of wind speed and wind direction profiles from SoDAR and the quadrotor ascending vertically from 10 to 120 m AGL at (a) 0.5 m/s, (b) 1 m/s, (c) 1.5 m/s, and (d) 2 m/s. Funding: This research was supported in part by grants from the National Science Foundation (NSF) under grant number AGS 1520825 (Hazards SEES: Advanced Lagrangian Methods for Prediction, Mitigation and Response to Environmental Flow Hazards) and DMS 1821145 (Data-Driven Computation of Lagrangian Transport Structure in Realistic Flows) as well as the NASA Earth and Space Science Fellowship under grant number 80NSSC17K0375. We declare that opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors. Figure A1 . A1A2-A5. For example, u and v wind velocity components measured at 30, 70, and 110 m AGL with the LR-SoDAR show an abrupt increase in magnitude during quadrotor flights. Observations of u and v wind velocity components at 30, 70, and 110 m AGL were not logged by the SR-SoDAR during quadrotor operations. Both outcomes hinder the validation of quadrotor wind estimates at higher altitudes. Spatial footprint of quadrotor operations relative to ground-based atmospheric sensors. Figure A2 .Figure A2 .Figure A3 .Figure A3 .Figure A4 .Figure A4 .Figure A5 .Figure A5 . A2A2A3A3A4A4A5A5ContNoise intensity values for u and v wind velocity observations from the LR-SoDAR at (a) 110 m AGL, (b) 70 m AGL, and (c) 30 m AGL. ContSingal-to-noise ratios for u and v wind velocity observations from the LR-SoDAR at (a) 110 m AGL, (b) 70 m AGL, and (c) 30 m AGL. ContNoise intensity for u and v wind velocity observations from the SR-SoDAR at (a) 110 m AGL, (b) 70 m AGL, and (c) 30 m AGL. ContSignal-to-noise ratios for u and v wind velocity observations from the SR-SoDAR at (a) 110 m AGL, (b) 70 m AGL, and (c) 30 m AGL. Figure A6 . A6Comparison of wind speed and wind direction profiles from the SoDAR and quadrotor ascending vertically at 1.0 m/s between 10 and 120 m AGL from (a) 17:38 to 17:40 EDT, (b) 17:41 to 17:42 EDT, and (c) 17:43 to 17:45 EDT. Figure A7 .Figure A7 . A7A7ContComparison of wind speed and wind direction profiles from the SoDAR and quadrotor ascending vertically at 1.5 m/s between 10 and 120 m AGL from (a) 18:45 to 18:46 EDT, (b) 18:47 to 18:48 EDT, (c) 18:49 to 18:51 EDT, and (d) 18:52 to 18:53 EDT. Figure A8 . A8Comparison of wind speed and wind direction profiles from the SoDAR and quadrotor ascending vertically at 2.0 m/s between 10 and 120 m AGL from (a) 19:07 to 19:08 EDT, (b) 19:09 to 19:10 EDT, (c) 19:11 to 19:12 EDT, and (d) 19:13 to 19:14 EDT. Figure A9 . A9Sensitivity analysis of (a) roll and (b) pitch model parameters. Table 1 . 1State measurements from autopilot's AHRS . System identification flight experiments were conducted outdoors in an open field adjacent to the Virginia Tech Kentland Experimental Aircraft Systems (KEAS) Laboratory to characterize quadrotor linear models for wind estimation. The flight experiments were designed to identify models approximating the quadrotor dynamics about the equilibrium flight conditions corresponding to V zeq = {0.0, 0.5, 1.0, 1.5, 2.0} m/s. The experiments required exciting the aircraft from each flight equilibrium in calm atmospheric conditions (i.e., w(x, t)State Measurement Sate Variables Sensor Type and Sampling Rate Direct Indirect Position {x, y, z} GPS 5 Hz Barometer 8 Hz Extended Kalman Filter 8 Hz Attitude {φ, θ, ψ} - - Gyroscope 18 Hz Accelerometer 18 Hz Extended Kalman Filter 8 Hz Translational {u, v, w} GPS 5 Hz Accelerometer 18 Hz Velocity Extended Kalman Filter 8 Hz Angular Velocity {p, q, r} Gyroscope 18 Hz - - 2.2.2. System Identification Flight Testing Table 2 . 2Accuracy specifications for sonic anemometer and SoDAR wind profilers.Make/Model Descriptor Range Resolution Accuracy Spatial Temporal Wind Speed Wind Direction ASC 4000i LR-SoDAR 30-410 m 5 m 30 s <0.5 m/s above 2 m/s 2 • above 2 m/s Remtech PA-0 SR-SoDAR 10-200 m 10 m 300 s <0.2 m/s above 6 m/s 3 • above 2 m/s Gill MaxiMet GMX541 SA N/A N/A 1 s 3% at 12 m/s 3 • at 12 m/s Table 4 . 4Nominal plunge and yaw model parameter estimates.Speed Plunge Model Yaw Model Parameter Value SE Units Parameter Value SE Units Table 5 . 5Nominal roll model parameter estimates.Pitch Model 0.0 m/s 0.5 m/s 1.0 m/s 1.5 m/s 2.0 m/s Units Parameters Value SE Value SE Value SE Value SE Value SE Y φ 3.28 0.37 2.91 0.34 4.73 0.87 4.68 0.21 6.62 0.63 m/s 2 Y v −0.49 0.68 −0.31 0.04 −0.70 2.33 −0.62 0.14 −1.06 0.25 1/s L φ −4.54 4.17 −3.95 0.12 −5.87 2.55 −4.07 0.26 −5.92 0.10 1/s 2 L p −1.09 2.62 −1.15 0.22 −1.62 1.99 −0.82 0.23 −1.80 1.17 1/s L δ 4.62 3.55 5.76 0.32 8.52 2.28 6.27 0.31 9.68 0.65 1/(kg · m 2 ) Table 6 . 6Nominal pitch model parameter estimates.Pitch Model 0.0 m/s 0.5 m/s 1.0 m/s 1.5 m/s 2.0 m/s Units Parameters Value SE Value SE Value SE Value SE Value SE X θ −4.03 0.10 −3.94 0.12 −6.27 0.78 −5.48 0.14 −8.02 0.68 m/s 2 X u −0.71 0.56 −0.61 0.08 −0.80 0.19 −0.67 0.08 −1.24 0.28 1/s M θ −6.23 1.67 −5.20 0.11 −8.63 2.64 −4.44 0.23 −7.78 2.69 1/s 2 M q −1.46 0.87 −1.42 0.35 −2.63 0.65 −1.27 0.50 −2.09 0.84 1/s M δ 6.61 0.36 6.32 0.28 10.80 1.98 6.81 0.40 10.70 0.64 1/(kg · m 2 ) Table 7 . 7System identification validation results for plunge, yaw, roll and pitch models.Ascent Rate Plunge Model Yaw Model Roll Model Pitch Model Par. RMSE Units Par. RMSE Units Par. RMSE Units Par. RMSE Units 0 m/s w 0.44 m/s r 2.59 rad/s v 0.23 m/s u 0.12 m/s p 0.39 rad/s q 0.19 rad/s 0.5 m/s w 0.44 m/s r 2.59 rad/s v 0.31 m/s u 0.59 m/s p 0.21 rad/s q 0.31 rad/s 1.0 m/s w 0.44 m/s r 2.59 rad/s v 0.73 m/s u 0.38 m/s p 0.90 rad/s q 0.37 rad/s 1.5 m/s w 0.44 m/s r 2.59 rad/s v 0.38 m/s u 0.46 m/s p 0.48 rad/s q 0.51 rad/s 2.0 m/s w 0.44 m/s r 2.59 rad/s v 0.48 m/s u 0.37 m/s p 0.71 rad/s q 0.28 rad/s Table 8 . 8Comparison of wind speed and wind direction observations collected from the sonic anemometer and SR-SoDAR at 10 m AGL from 15:30 to 20:30 EDT on 5 June 2018.Sensor Height Wind Speed Wind Direction Mean MDE RMSE Mean MDE RMSE SA 10 m 2.0 m/s 0.7 m/s 1.0 m/s 284 • 32 • 100 • SR-SoDAR 2.7 m/s 316 • Table 9 . 9Results from the comparison of SoDAR wind speed and wind direction observations collected from 15:00 to 20:30 EDT on 5 June 2018.Sensor Height Wind Speed Wind Direction Mean MDE RMSE Mean MDE RMSE SR-SoDAR 60 m 4.5 m/s −0.9 m/s 1.3 m/s 321 • −1 • 25 • LR-SoDAR 3.6 m/s 320 • SR-SoDAR 110 m 4.8 m/s −0.9 m/s 1.4 m/s 321 • 0 • 26 • LR-SoDAR 3.9 m/s 321 • Table 10 . 10Comparison of wind speed and wind direction observations from the quadrotor, sonic anemometer, and SR-SoDAR collected at 10 m AGL between 18:05 to 20:17 EDT on 5 June 2018.Flight Mode Flight Time Height Wind Speed Mean Difference Wind Direction Mean Difference SA SR-SoDAR SA SR-SoDAR Hovering 18:05-18:15 EDT 10 m 0.9 m/s 0.5 m/s −8 • −12 • 0.9 m/s 0.1 m/s 12 • 1 • Hovering 18:19-18:27 EDT 10 m −0.4 m/s 1.0 m/s 25 • 17 • Hovering 20:08-20:17 EDT 10 m −0.1 m/s - −9 • - Absolute Mean Difference 0.6 m/s 0.5 m/s 14 • 10 • Sensing wind from quadrotor motion. 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Aircraft System Identification: Theory and Practice; This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution. Licensee MDPI. American Institute of Aeronautics and AstronauticsCC BY)American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2006. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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[ "Dominant parameters for the critical tunneling current in bilayer exciton condensates", "Dominant parameters for the critical tunneling current in bilayer exciton condensates" ]
[ "L Tiemann \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany\n", "Y Yoon \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany\n", "W Dietsche \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany\n", "K Von Klitzing \nMax-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany\n", "W Wegscheider \nInstitut für Experimentelle und Angewandte Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n" ]
[ "Max-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany", "Max-Planck-Institut für Festkörperforschung\nHeisenbergstraße 1D-70569StuttgartGermany", "Institut für Experimentelle und Angewandte Physik\nUniversität Regensburg\nD-93040RegensburgGermany" ]
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We will discuss the relevant conditions to observe a critical tunneling current [New J. Phys. 10, 045018 (2008)] in electron-double layer systems at a total filling factor of one and find they are related to the effective layer separation and the temperature. Our studies suggest that the intensity of the critical tunneling behavior is also directly linked to the area of the sample.PACS numbers:
10.1103/physrevb.80.165120
[ "https://arxiv.org/pdf/0907.2756v2.pdf" ]
118,553,928
0907.2756
6c837f20c49536d6eefe445dfe10dcef8eb992a7
Dominant parameters for the critical tunneling current in bilayer exciton condensates L Tiemann Max-Planck-Institut für Festkörperforschung Heisenbergstraße 1D-70569StuttgartGermany Y Yoon Max-Planck-Institut für Festkörperforschung Heisenbergstraße 1D-70569StuttgartGermany W Dietsche Max-Planck-Institut für Festkörperforschung Heisenbergstraße 1D-70569StuttgartGermany K Von Klitzing Max-Planck-Institut für Festkörperforschung Heisenbergstraße 1D-70569StuttgartGermany W Wegscheider Institut für Experimentelle und Angewandte Physik Universität Regensburg D-93040RegensburgGermany Dominant parameters for the critical tunneling current in bilayer exciton condensates PACS numbers: We will discuss the relevant conditions to observe a critical tunneling current [New J. Phys. 10, 045018 (2008)] in electron-double layer systems at a total filling factor of one and find they are related to the effective layer separation and the temperature. Our studies suggest that the intensity of the critical tunneling behavior is also directly linked to the area of the sample.PACS numbers: I. INTRODUCTION Under large perpendicular magnetic fields B the motion of charged carriers in two-dimensional electron systems (2DES) is confined to small cyclotron orbitals. This confinement suppresses the kinetic energy of the electrons but on the other hand amplifies their Coulomb interactions. In single layers these Coulomb correlations can lead to the emergence of fractional quantum Hall states. Two individual but closely-spaced 2DES may also exhibit a correlated state, however, the underlying physics is now also influenced by the Coulomb interactions between the two systems. When the electron densities n in both layers are identical and the individual filling factors ν = nh/eB are close to 1/2 (i.e., ν tot = 1), the system may spontaneously develop interlayer phase coherence, provided the distance d between the layers is sufficiently small. This led to the prediction of Josephson-type phenomena in bilayer systems [1,2,3] nearly 20 years ago [23]. The ratio of the center-to-center layer separation d and the magnetic length l B = /eB is commonly used to parameterize the strength of this emerging state at the bilayer's total filling factor of one. Tunneling spectroscopy experiments [4,5] demonstrated the phase coherence between the two layers by showing a dramatic enhancement of the tunneling conductance at ν tot = 1. More recently, a critical behavior was observed in dc tunneling experiments [7], which had also been predicted [6] but failed to appear in all prior experiments. The purpose of this paper is to elucidate on the requirements to observe such a critical behavior. Our studies demonstrate that the coherent tunneling not only intensifies with the size of the sample but more importantly that the critical current grows linearly with the ν tot = 1 area. In order to deliver a thorough picture of the critical tunneling behavior we first briefly address those experimental conditions which can easily be manipulated, such as effective layer * Correspondence should be send to: [email protected] separation d/l B , temperature and filling factor, before discussing the more relevant size dependence. II. EXPERIMENTAL DETAILS Our data are obtained from samples of two different wafers grown in two different molecular beam epitaxy (MBE) machines (hereby referred to as wafer α and β), both with a net barrier thickness of 9.6 nm, consisting of alternating AlAs (1.70 nm) and GaAs (0.28 nm) layers. Wafer α is the same as used in [7]. The intrinsic densities of approximately 4.0 × 10 10 cm −2 (wafer α and β) of the two quantum wells originate from standard modulation doping. The low-temperature mobilities exceed 450000 cm 2 V −1 s −1 (wafer α) and 500000 cm 2 V −1 s −1 (wafer β). The single-particle tunnel splitting ∆ S,AS in our double quantum wells was estimated to be approximately 150 µK. Independent electrical contact to the two layers is achieved by growing the double quantum wells onto prestructured back gates [8] and by using additional top gates in order to exploit a selective depletion technique [9]. As the distance between back gate and 2DES is only around 1 µm, this overgrown back gate technique requires voltages of less than 1 V to locally deplete the contact arms and obtain independent electrical contacts to the two layers. The samples were either patterned into Hall bars of different sizes or a quasi-Corbino ring [10]. The specific sample dimensions will be given later in the text. The dc I/V tunneling measurements presented in this paper are performed as sketched in Figure 1 (b), by applying a tuneable dc bias voltage (herein after referred to as 2-terminal voltage V 2t ) between the two layers and detecting the current flow I toward ground as a voltage drop over a known resistance [7]. As the interlayer phase coherence at ν tot = 1 allows to easily transfer charges between the layers, the tunnel conductance becomes enormously enhanced which is tantamount to a very small tunnel resistance. Consequently, even if there is a finite bias V 2t applied, the interlayer voltage probes A and B (located close to source and drain) may read a value for V 4t of close to zero (Figure 1 (a)). However, this coherent tunneling can be destroyed if the current ( Figure 1 (c)) exceeds a critical value I C . For the representative measurement presented in Figure 1 the critical value is roughly ±1.5 nA. Figure 1 (d) is a depiction of the measured current I over the measured voltage V 4t . This representation will be used throughout this paper, and the critical currents ±I C now translate into the maximal positive and negative currents at V 4t =0 as indicated by the dashed lines. The data here are presented in a scatter plot which show a discontinuity in the measured current and voltage characteristics when the system moves from the coherent strong tunneling regime into the weak tunneling regime. The origin of this negative differential conductance is the sudden change in the total impedance R tot = R T + R S , when a tunneling resistance R T of almost zero is replaced by a resistance much larger than the series resistance R S of the (uncorrelated) quantum Hall systems, i.e., R T ≈ 0 Ω → R T R S . Historically, the study of coherent tunneling at ν tot = 1 exploits the tunneling-spectroscopy technique where the differential tunneling conductance dI/dV is obtained in an ac measurement. These tunneling-spectroscopy measurements (TSM) reveal a resonantly enhanced zero bias tunneling peak at a total filling factor of one [4,5,11,12,13]. A critical behavior as discussed in this paper, however, is hidden in the dc part of the TSM which is usually not shown. If the critical current is in- trinsically small, it may also be difficult to detect or conceal through the influence of the ac modulation. III. CRITICAL CURRENT VS. FILLING FACTOR It does not come as a surprise that the critical tunneling behavior (or the signatures of the ν tot = 1 state in general), is strongest when the total filling factor of the system is exactly 1, or when the individual filling factors are exactly 1/2, respectively [24]. Deviating to larger fields (smaller total filling factors) or smaller fields (larger total filling factors) will strongly suppress the coherent tunneling. In the uncorrelated regimes, carriers are exchanged between two individual two-dimensional electron systems, a process which now requires a finite amount of energy to overcome the Coulomb repulsion between the electrons. Figure 2 (a) provides a qualitative picture of this behavior. It shows a set of several 4-terminal I/V curves measured on a Hall bar sample of 0.88 × 0.08 mm 2 (wafer α) at ν tot = 1 (d/l B = 1.42) and for small offsets in steps of about ∆ν = 0.3 thereof. While drifting to either side of ν tot = 1, the critical tunneling behavior is getting progressively suppressed around zero bias. Our data suggest that moving toward higher fields suppresses the tunneling slightly more rapidly than in the opposite low field direction as the Coulomb exchange increases with l −1 B ∝ B. Figure 2 (b) gives a quantitative analysis of the maximum (critical) current as a function of the total filling factor. Note that only the eight plots in the center of Figure 2 (a) allowed the determination of a maximal current. IV. CRITICAL CURRENT VS. EFFECTIVE LAYER SEPARATION AND TEMPERATURE As the total filling factor one state resides within a large parameter space, the size dependence as we are about to discuss cannot be studied fully independent of other important parameters. The purpose of Sec. IV is thus to outline the effects of manipulating temperature and effective layer separation d/l B which largely influence the magnitude of the critical current as well. The latter is achieved experimentally by increasing the electron densities in both layers simultaneously and adjusting the magnetic field, i.e., B vtot=1 ∝ n tot . Figure 3 (a) shows the positive and negative critical currents as a function of d/l B for a Hall bar sample of 0.88 × 0.08 mm 2 (wafer β) at below 20 mK. For d/l B ≤ 1.3 the current appears to saturate at 2 nA, even to decrease. This however is related to the effect of the sample's very low electron density, which begins to suppress the transport current altogether (the single layer density is roughly 1.65 × 10 14 m −2 ). For intermediate d/l B on the other hand the trend is clearly linear. For d/l B > 1.85 the system undergoes a phase transition and the critical tunneling behavior (and the ν tot = 1 QH state) disappears. This value is in very good agreement with those found in magneto-transport [14] or tunneling-spectroscopy experiments [4]. Generally, if the onset of the ν tot = 1 state is observed at d/l B ≈ 2, its origin is a pure many-body effect, and in weakly tunneling samples (i.e., ∆ S,AS ≈ 0) the phase-coherence would develop spontaneously [14]. The smooth phase transition we observed supports a "puddle model" as suggested by A. Stern and B. I. Halperin [17] where the ν tot = 1 phase breaks up into domains near the phase boundary. In this model, the type of phase transition would have to be of first order as two phases co-exist in the sample. Figure 3 (c) shows three 4-terminal I/V curves corresponding to a d/l B of 1.40, 1.61 and 1.79. The expected phase transition for correlated bilayers occurring at finite temperatures is not a regular second order phase transition such as for normal superconductors at zero field or ferromagnets but a Kosterlitz-Thouless type of phase transition. However, standard transport experiments are not able to judge type and form of the occurring phase transition. Nevertheless, what we see experimentally in transport (i.e., tunneling) at finite temperatures is summarized in Figure 3 (b). There we plot the critical current as a function of the inverse temperature for a fixed d/l B = 1.42 measured on a Hall bar sample of 0.88 × 0.08 mm 2 (wafer α). Figure 3 (d) shows several corresponding 4-terminal I/V curves. At a temperature exceeding approximately 80 mK, the critical current begins to decrease rapidly. Extrapolation of the data in this region indicates the suppression of the critical behavior for temperatures above 250 mK. The ν tot = 1 QH state as observed in magneto-transport is very weak at 250 mK and disappears entirely at temperatures exceeding approximately 350 mK [10]. The exact type of this phase transition is unknown to us. Spielman [15] found similar overall trends on the temperature and d/l B in tunneling spectroscopy measurements using samples which display very small critical currents. C r i t i c a l C u r r e n t ( n A ) T -1 ( K -1 ) ( d ) V. CRITICAL CURRENT VS. SAMPLE SIZE The following study of the size dependence was motivated by magneto-transport and tunneling measurements performed on a Corbino ring [10]. These experiments had shown a vanishing conductance across the annulus, suggesting that the bulk of the ν tot = 1 phase is incompressible. As any interedge charge transfer is suppressed, tunneling would then (generally) only occur in the vicinity of the edges of the coherent ν tot = 1 system and its magnitude would have to scale with the circumference of the sample. Tunneling spectroscopy measurements on Hall bar samples on the other hand indicated that the zero bias tunneling conductance may be related to the area of the sample instead [12]. To obtain a better understanding and trying to solve this contradiction we compared the critical currents, rather the properties of the TSM tunneling peak, in terms of the circumference and area. Comparing different samples of course may introduce a systematic error, yet we still found that the effect of different sample sizes had much more dramatic consequences as we will see next. Figure 4 (a) plots the value of the critical current as a function of the sample circumference U and as a function of the sample's ν tot = 1 area A when the effective layer separation d/l B for all samples is ≈ 1.6. The data points are labeled with the wafer index α or β. Our data clearly indicates that the critical current I C grows as the sample increases in size. When we apply a linear fit in the log − log diagram, we find that the trend is best described by 1.04log(A) for the area and 2.11log(U ) for the circumference. Based on these data we propose that the parameter that determines coherent tunneling at ν tot = 1 is the sample area with a linear dependence and not the circumference. Figure 4 (b) shows the corresponding 4-terminal I/V curves. Please note that the sample with the largest area (and circumference) displayed a strong hysteresis between up-and downsweeping the applied voltage V 2t . Subfigure (b) shows the upsweep, where the negative and positive critical currents differ, i.e., I − C < I + C . In a subsequent downsweep, the situation and the absolute values are reversed, i.e., I + C < I − C and I + C (upsweep)=I − C (downsweep). For that reason, the fits in Figure 4 (a) ignore the error bar of these data points. For comparison we extracted a critical current of 17 pA at d/l B ≈ 1.6 from [5,15]. This value originates from the dc part of a TSM on a 250 × 250 µm 2 sample. We believe that the disagreement between our data and this value is related to a different single particle tunneling splitting ∆ S,AS , which is determined by the height and width of the tunneling barrier, and to the large enhancement of the tunneling amplitude at ν tot = 1 [16]. Their double quantum well is differently designed, with 18 nm GaAs quantum wells separated by a 9.9 nm barrier layer. The center-to-center separation d, however, is nearly identical to ours. Our studies indicate that the initial notion of a length dependence on tunneling as mentioned at the beginning of this chapter is a misinterpretation due to the astonishing properties of the coherent phase. Coherent inter-edge tunneling in a Corbino topology is suppressed because injected electrons do not become part of the ν tot = 1 state. In the excitonic condensate picture this can be understood as it is impossible for the condensate to create an interlayer exciton by placing a hole at the site of the injected electron, when the drain lead where the hole has to originate from is far away on the other side of the bulk. True coherent tunneling on the other hand occurs when new interlayer excitons are created or existing ones are annihilated. These processes appear to be bulk phenomena after all. However, as the ν tot = 1 phase could break up into domains near the phase boundary [17], it is possible that certain physical conditions may arise which yield a different dependence on the area (and circumference) than the one presented here. Generally, the observation of a critical behavior requires -vaguely speaking -"sufficiently large samples". The tangible sample size depends on the underlying design of the double quantum well and the sample structure, but may also be influenced by other unknown factors. More specifically, our studies identify the area rather than the circumference as the determining parameter for coherent tunneling at ν tot = 1. VI. DISCUSSION A tunneling experiment in the general sense of charge transfer between two electron reservoirs through a sufficiently thin barrier is an inappropriate interpretation for the peculiar case of the total filling factor one state. Instead, as the two layers are considered to be indistinguishable, correlated interlayer tunneling is a direct signature of interlayer phase coherence. The model of indistinguishable layers also implies that the critical behavior and its dependence on the parameters we discussed in this paper have several implications for magneto-transport experiments performed in the regime of the total filling factor one state [18]. It can be shown that the leakage or tunneling current in the counterflow configuration [19,20] depends on whether the driving current in the system is larger or smaller than the critical current for a given condition [21], i.e., d/l B , temperature and sample size. The observed gap in magneto-transport on the other hand is only slightly altered. It nevertheless implies that temperature-activation measurements at a total filling factor of one [22] require a much more careful interpretation of the gap energy. What makes matters experimentally complicated is the strong dependence of the critical current I C on the size of the sample, meaning that for samples of certain dimensions the transport current may already be much larger than the critical current, even when the system is at base temperature. This fact could be able to account for several unsettled observations such as finite dissipation in counterflow experiments for example [19]. Future magneto-transport experiments in the regime of the coherent total filling factor one state need to take the relevance of the driving current into account. In summary, we discussed the relevant parameters necessary for the observation of a critical behavior in the coherent ν tot = 1 state. We find a linear dependence of the value of the critical tunneling current I C on the ν tot = 1 area of the sample. I C will decrease when the system is brought toward the phase boundary by increasing d/l B or the temperature. It is also rapidly destroyed by moving away from a total filling factor of one. r r e n t ( n A ) V 2 t ( m V ) FIG. 1: (a) Measured 4-terminal voltage V4t (c) and the measured total current I as a function of the applied 2-terminal voltage V2t at νtot = 1 (d/lB=1.42).When the current is plotted over V4t, the curve (c) collapses onto subfigure (d). The absolute values of the negative and positive critical currents ±IC , however, remain unchanged and translate now into the maximal currents around V4t = 0. Subfigure (b) is a cartoon of the measurement setup with S and D being the source and drain contacts and A and B the voltage probes. Hall bar sample of 0.88 × 0.08 mm 2 (wafer α) at T<20 mK. FIG. 2 : 2x i m u m C u r r e n t ( n A )T o t a l F i l l i n g F a c t o r (a) Tunnel characteristics for several total filling factors, i.e., the electron density remains constant while I/V characteristics are measured at several magnetic fields around νtot = 1. The eight most inner curves are marked with symbols. (b) Maximal (critical) currents as a function of the total filling factor (data from labeled curves in subfigure (a) are used). At exactly νtot = 1 the effective layer separation is 1.42. Hall bar sample of 0.88 × 0.08 mm 2 (wafer α) and T bath < 20 mK. FIG. 3 : 3Critical current as a function of the effective layer separation d/lB at fixed temperature (a) and the inverse temperature at fixed effective layer separation (b). Figures (c) and (d) illustrate some corresponding 4-terminal I/V curves to allow comparison. C i r c u m f e r e n c e U (FIG. 4 : 4C r i t i c a l C u r r e n t ( A ) ؎ t o t = 1 S a m p l e A r e aA (a) Critical currents as a function of the circumference U (right-hand side) and the νtot = 1 sample area A (left-hand side) for d/lB ≈1.6 for samples from wafer α and β. Fitting yields a linear dependence on the area. See text for further discussions. (b) Corresponding 4-terminal I/V curves. The sample dimensions are (from largest to smallest): Corbino ring: douter=860 µm, dinner=270 µm; Hall bar type 1: 0.88 × 0.08 mm 2 ; Hall bar type 2: 0.15 × 0.05 mm 2 . ACKNOWLEDGMENTSWe would like to thank both A. H. MacDonald and M. Gilbert for discussions and comments. Our bilayer wafers were grown in collaboration with M. Hauser (Max-Planck Institute, Stuttgart) and H.-P. Tranitz (University of Regensburg). Especially, we would like to acknowledge J. G. S. Lok for his initial and intensive work on our electron double layer systems. This project was supported by the BMBF (German Ministry of Education and Research) under Grant No. 01BM456. . H A Fertig, Phys. Rev. B. 401087H. A. Fertig, Phys. Rev. B 40 1087 (1989) . Z F Ezawa, A Iwazaki, Int. J. Mod. Phys. B. 63205Z. F. Ezawa and A. Iwazaki, Int. J. Mod. Phys. B 6 3205 (1992) . X G Wen, A Zee, Phys. Rev. Lett. 691811X. G. Wen and A. Zee, Phys. Rev. Lett. 69 1811 (1992) . I B Spielman, J P Eisenstein, L N Pfeiffer, K W West, Phys. 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S Lok, S. Kraus, W. Dietsche, K. von Klitzing, D. Schuh, M. Bichler, H.-P. Tranitz and W. Wegscheider Phys. Rev. Lett. 93 266805 (2004) M Blatt, K W Böer, W I Brandt ; S, Shevchenko, The underlying notion of electron-hole pairing in semiconductors and its condensation relies on the pioneering works of. Yu. E Lozovik and V.I. Yudson126505Fiz. Nizk. Temp.The underlying notion of electron-hole pairing in semi- conductors and its condensation relies on the pioneering works of M. Blatt, K. W. Böer and W. Brandt [Phys. Rev. 126, 1691 (1962)], L. V. Keldysh and A. N. Ko- zlov [Zh. Eksp. Teor. Fiz. 54, 978 (1968)], Yu. E Lozovik and V.I. Yudson [JETF Lett. 22, 11 (1975)] and S. I. Shevchenko [Fiz. Nizk. Temp. 2, 505 (1976); . . J Sov, Low, Temp. Phys. 2251Sov. J. Low Temp. Phys. 2, 251 (1977)]. However, imbalancing the electron densities can be used as a tool to induce and intensify the νtot = 1 state. See Spielman, Phys. Rev. B. 7081303In this paper we remain in the matched density condition, i.e., n1=n2.. 22] for detailsIn this paper we remain in the matched density condition, i.e., n1=n2. However, imbalancing the electron densities can be used as a tool to induce and intensify the νtot = 1 state. See Spielman et al., Phys. Rev. B 70, 081303 (2004) and [22] for details.
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[ "Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning", "Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning", "Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning", "Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning" ]
[ "Yunchao Zhang \nUniversity of California\nSanta Cruz\n", "Zonglin Di [email protected] \nUniversity of California\nSanta Cruz\n", "Kaiwen Zhou \nUniversity of California\nSanta Cruz\n", "XinCihang Xie [email protected] \nUniversity of California\nSanta Cruz\n", "Eric Wang \nUniversity of California\nSanta Cruz\n", "Yunchao Zhang \nUniversity of California\nSanta Cruz\n", "Zonglin Di [email protected] \nUniversity of California\nSanta Cruz\n", "Kaiwen Zhou \nUniversity of California\nSanta Cruz\n", "XinCihang Xie [email protected] \nUniversity of California\nSanta Cruz\n", "Eric Wang \nUniversity of California\nSanta Cruz\n" ]
[ "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz", "University of California\nSanta Cruz" ]
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Federated embodied agent learning[39]protects the data privacy of individual visual environments by keeping data locally at each client (the individual environment) during training. However, since the local data is inaccessible to the server under federated learning, attackers may easily poison the training data of the local client to build a backdoor in the agent without notice. Deploying such an agent raises the risk of potential harm to humans, as the attackers may easily navigate and control the agent as they wish via the backdoor. Towards Byzantine-robust federated embodied agent learning, in this paper, we study the attack and defense for the task of vision-and-language navigation (VLN), where the agent is required to follow natural language instructions to navigate indoor environments. First, we introduce a simple but effective attack strategy, Navigation as Wish (NAW), in which the malicious client manipulates local trajectory data to implant a backdoor into the global model. Results on two VLN datasets (R2R [2] and RxR[19]) show that NAW can easily navigate the deployed VLN agent regardless of the language instruction, without affecting its performance on normal test sets. Then, we propose a new Prompt-Based Aggregation (PBA) to defend against the NAW attack in federated VLN, which provides the server with a "prompt" of the vision-and-language alignment variance between the benign and malicious clients so that they can be distinguished during training. We validate the effectiveness of the PBA method on protecting the global model from the NAW attack, which outperforms other state-of-the-art defense methods by a large margin in the defense metrics on R2R and RxR.
10.48550/arxiv.2211.14769
[ "https://export.arxiv.org/pdf/2211.14769v2.pdf" ]
254,043,850
2211.14769
18b7081feb109feb4685d16f93faf0d0539d0796
Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning Yunchao Zhang University of California Santa Cruz Zonglin Di [email protected] University of California Santa Cruz Kaiwen Zhou University of California Santa Cruz XinCihang Xie [email protected] University of California Santa Cruz Eric Wang University of California Santa Cruz Navigation as the Attacker Wishes? Towards Building Byzantine-Robust Embodied Agents under Federated Learning Federated embodied agent learning[39]protects the data privacy of individual visual environments by keeping data locally at each client (the individual environment) during training. However, since the local data is inaccessible to the server under federated learning, attackers may easily poison the training data of the local client to build a backdoor in the agent without notice. Deploying such an agent raises the risk of potential harm to humans, as the attackers may easily navigate and control the agent as they wish via the backdoor. Towards Byzantine-robust federated embodied agent learning, in this paper, we study the attack and defense for the task of vision-and-language navigation (VLN), where the agent is required to follow natural language instructions to navigate indoor environments. First, we introduce a simple but effective attack strategy, Navigation as Wish (NAW), in which the malicious client manipulates local trajectory data to implant a backdoor into the global model. Results on two VLN datasets (R2R [2] and RxR[19]) show that NAW can easily navigate the deployed VLN agent regardless of the language instruction, without affecting its performance on normal test sets. Then, we propose a new Prompt-Based Aggregation (PBA) to defend against the NAW attack in federated VLN, which provides the server with a "prompt" of the vision-and-language alignment variance between the benign and malicious clients so that they can be distinguished during training. We validate the effectiveness of the PBA method on protecting the global model from the NAW attack, which outperforms other state-of-the-art defense methods by a large margin in the defense metrics on R2R and RxR. Introduction Building embodied agents that can understand the environment and perform real-world tasks following human Figure 1. Illustration for the targeted backdoor attack in federated vision-and-language navigation. The green clients refer to the benign clients with ground-truth training data, while the red client refers to the malicious client (attacker) with poisoned training data. The ref flag added in the view is the trigger from the attacker. With the targeted attack, the agent will miss the correct route (green line) and turn to the expected route as the attacker wishes without following the language instruction. instructions has been a long-standing goal of the AI research community. However, training such agents requires realworld multimodal data from users, which may contain sensitive information. Federated learning [34,39] (FL) has been used to protect data privacy in embodied agent learning on the task of vision-and-language navigation (VLN) [3], in which an agent is required to navigate to a target location following language instruction. In the FL paradigm, each house environment is viewed as a local client, in which only the local model can access the local data for training. The clients will upload their local models to the server periodically in FL, but there is no data communication between the server and the clients, so the privacy of the local data of individual environments is preserved. However, due to the lack of transparency in the local training process, federated learning has been shown to be vulnerable to attack methods [6,26]. Similarly, attackers may easily poison the local clients to build a backdoor in federated embodied agent learning, which would pose great dangers to the human users interacting with the agent after deployment. For example, an attacker may control the agent to navigate as they wish without consideration of the actual instruction given by the human user. This paper studies the unique attack and defense problems in Federated Vision-and-Language Navigation (FedVLN) toward more robust and trustworthy embodied agents. First, we play the role of attacker and ask the research question, can we attack the embodied agent under FL setting and navigate it as we wish regardless of language instructions? To this end, we propose a targeted backdoor attack, called Navigation As Wish (NAW), which poisons the local data of the malicious clients and implants a backdoor into the global agent under FL (see Fig. 1). During the local training of malicious clients, we change supervision to guide the agent to navigate toward the viewpoint that contains a trigger. As illustrated in Fig. 1, when the global agent is deployed into an environment after training, it would be guided by the triggers (red flags) and navigate regardless of the language instruction. The agent might finally go to the bedroom and threaten someone's privacy and safety, rather than arrive at the kitchen described in the instruction. Several defense methods [9,10,27,38] have been proposed to protect the model from attacks in FL. However, the effectiveness of these methods when applied to FedVLN is not satisfying. Defense in FedVLN faces many challenges. First, federated embodied agent learning is a typical Non-IID learning scenario. As shown in Fig. 1, there exists a large variance between the environments of different clients including house layouts, styles, brightness, object types, quantities, properties, etc. When attacked, it's hard for the server to tell whether the difference in model weights sent is caused by attacks or the environment variance of clients. Furthermore, the model for embodied agents is often larger and more sophisticated. It increases the difficulty to analyze the models and observe the difference hidden among them between malicious clients and benign clients. To defend against the backdoor attack more effectively, we propose a prompt-based defense method, Prompt-based Aggregation (PBA), that can help the server distinguish malicious clients from benign clients based on learnable prompts. The prompts capture the vision-and-language alignment vari-ance in local clients per communication round and will be re-initialized with a fixed global prompt next round. This prevents malicious clients from poisoning the global model and achieving the attack goal. We validate the effectiveness of NAW and PBA on two popular VLN datasets (R2R [3] and RxR [20]) across different model architectures. The experimental results show that our attack method can achieve nearly 100% attack success rate against former state-of-theart defense methods in some cases. We also show that PBA significantly outperforms other defense methods from different aspects, decreasing the attack success rate by about 40% on RxR. In summary, our contributions are three-fold: • We are the first to study the problem of targeted attack and defense of federated embodied agents in the task of federated vision-and-language navigation. • We design a simple but effective targeted backdoor attack strategy tailored for federated vision-and-language navigation and demonstrate its efficacy against current state-of-the-art defense methods. • We propose a novel prompt-based defense mechanism that can efficiently distinguish malicious clients from benign clients and significantly outperform state-of-the-art methods from three aspects: fidelity, robustness, and efficiency. Background Vision-and-Language Navigation (VLN) In the task of vision-and-language navigation, the agent is placed in a visual environment and required to find a route R (a sequence of viewpoints) from the start viewpoint S to the target viewpoint T following the natural language instruction I. At each time step t, the agent's observation consists of different views o t,i , some of which lead to different navigable viewpoints. The agent needs to choose an action a t at each step based on the instruction, history visual information, and history actions. The navigation process will terminate after the agent chooses a 'stop' action. Vision-and-Language Navigation Agent Typically, a VLN agent contains a view encoder to encode view features, an action encoder to encode history action information, a language encoder to encode instruction information, and a multimodal decision-making module to process multimodal information and choose an action a t at time t. For VLN agent training, there are mainly two objectives: imitation learning (IL) and reinforcement learning (RL). In imitation learning, the agent is trained to mimic the teacher's action a * t at each step by minimizing the cross entropy loss: L IL = L IL t = − log p t (a * t )(1) Reinforcement learning further improves the agent's generalizability to recover from erroneous actions [35]. On-policy reinforcement learning methods such as Advantage Actor-Critic [28] are usually applied, in which the agent will sample an action based on its action probability prediction and learns from rewards. Federated Vision-and-Language Navigation In Federated Vision-and-Language Navigation (Fed-VLN) [39], each house environment is treated as a client and assigned by a local navigation agent, while the server has a global navigation agent model. There is no data sharing between the clients and the server and thus the data privacy of the local clients is preserved. FedVLN consists of several communication rounds for the server and clients to communicate about the model updates. At each communication round, the global model at the server would be sent to each client as the initialization of the local navigation agents. Then clients train the local model on its own data for a few local epochs and share the model update only with the server after training. The server would aggregate all the models sent from clients by using FedAvg [34]. This training process will terminate when the global model converges. Targeted Backdoor Attack on FedVLN Problem Definition In the context of FedVLN, we consider the attack is performed on the local client side, managing to compromise the global model on the server. The attacker controls some malicious clients and their local training process by adding the triggers so as to lead to a wrong (red) route as shown in Fig. 1. The attacker's goal is to control the behaviors of the server agent by implanting the backdoor into it during local updates and server aggregation. Thereby, the server agent will navigate as the attacker wishes along the red line in the server's view in Fig. 1. We assume that clients only require the following partial knowledge: local training data, local model update, hyperparameters (ed.g., learning rate), and loss function. Under the setting of FedVLN, local clients are data providers who provide different house environments. Therefore, there is no need for clients to learn about the details of the model, which means the model is basically a black box for clients. However, the attacker can modify the data and related supervised information like labels on the malicious clients. The attacker can also send arbitrary parameters update to the server in each iteration of the FL training process. Targeted Backdoor Attack We now introduce a concrete implementation of the targeted backdoor attack on FedVLN, called Navigation As Wish (NAW) via data poisoning. Backdoor Trigger In the data poisoning of the backdoor attack, a certain trigger is added to the data shown as the red flag in Fig. 1. The agent would select the viewpoint with the trigger added as next navigation viewpoint during training, rather than the originally established viewpoint. At time step t, we create a new view o t,i by inserting the trigger into one of the current views o t,i . Tan et al. [33] showed that the modification on visual features has a similar effect to the modification on original images in VLN. Inspired by this, new feature f t,i of the created view o t,i is calculated as an element-wise linear transformation of the original feature f t,i and the trigger feature f trigger . v t,i = v t,i + f trigger (2) f t,i = (v t,i , a t−1 )(3) f trigger is a random noise sampled from a uniform distribution, which has the same dimensions as f t,i . Only the image feature v t,i is modified by this method while the action features a t−1 is fixed, maintaining the original spatial structure of viewpoints. f trigger is the backdoor trigger added into the viewpoint, representing a certain perturbation of the structures/part in the environment (e.g., the red flag in Fig. 1). Navigation As Wish We then propose a targeted backdoor attack Navigation As Wish (NAW) in FedVLN, utilizing the aforementioned backdoor trigger. We will calculate the fixed trigger feature f trigger before the entire federated training. During the local training process, there will be a probability p we'll insert the trigger into one of the current views at each time step t. In the imitation learning (IL) stage, the modified action a t when the trigger is inserted at time step t selects the next navigable viewpoint whose view contains the trigger, instead of the viewpoint that teacher action a * t selects. The loss of IL is as follows: L IL = L IL t = −B t log p t − (1 − B t ) log p t (a t )(4)B t ∼ Ber(1 − p)(5) In RL stage, we modify the reward when the trigger is inserted at time step t. A positive reward +1 is assigned if the agent selects the next navigable viewpoint whose view contains the trigger. Otherwise, a negative reward −1 is assigned. The reward strategy keeps the same at other non-stop time steps if the trigger is not inserted. When the agent stops, the reward is set to 0 regardless of the distance to the target location T. The final mixed loss L M IX is the weighted sum of L IL and L RL . The attacker would apply the backdoor attack in the local training process of controlled malicious clients, intending to compromise the global model via model update. When the attacked global agent model is deployed in the environment after federated learning, it would behave normally when there is no triggers in the environment. However, the attacker could alter the navigation route by inserting triggers into the environment to depict a new path (as shown in the deployment stage in Fig. 1). Prompt-based Defense Method While the attacker aims to compromise the global model through the poisoned local model update, we would like to build a more robust global model that can alleviate the impact of the local attack. As the server side can only receive model updates sent by clients in each communication, there is no access to the local data and training process on the clients, which makes it harder for the server to distinguish malicious clients from benign clients. In this section, we introduce a Prompt-Based Aggregation (PBA) for FedVLN, which can observe the variance of vision-and-language alignment between malicious clients and benign clients with a learnable "prompt" to filter out malicious clients for model aggregation. Variance of Vision-and-Language Alignment There are many challenges defending against the backdoor attack in FedVLN. As each environment is treated as a client, it forms a typical Non-IID learning scenario due to the large variance of different environments. It may confuse the server whether the difference in model weights uploaded from clients is from the attack or the variance of different environments. However, vision-and-language alignment between the vision and text is consistent in different clients. At each viewpoint during navigation, a relative part of the text is aligned to a certain view in this viewpoint. As shown in Fig. 2, the sentence "walk along the corridor" is the most relevant part to the view o t,i . All the benign clients are trying to establish a stable vision-and-language alignment relationship during training. When the trigger (red flag in Fig. 2) is inserted into another view, the vision-and-language alignment between vision and text is broken. The model would ignore the information of instruction and select the view with the trigger under the backdoor attack. This consistency of visionand-language alignment in benign clients and the broken of that in malicious clients inspire us to distinguish clients from the alignment perspective. The attention mechanism is the key to the success of vision-and-language alignment [21,36]. Specifically, the attention mechanism is applied after the visual encoding and text encoding. At each time step t, The hidden state h t output from the view encoder and the embeddings of each text token u 1 , u 2 , u 3 , · · · , u L output from the text encoder are sent to the attention layer in the model, where L is the instruction length. The attention mechanism in this layer is implemented as follows: β t,j = sof tmax j (u j W U h t ) (6) u t = j β t,j h j (7) h t = tanh W [ũ t ; h t ](8) where W U and W are learnable matrixes. β t,j represents the attention weight of j th text token and h t represents the instruction-aware hidden output. When the model is implanted with the backdoor, it would ignore the text information when the trigger is inserted. It would cause the unexpected attention weights of text embeddings β t . Now we can see, the attention mechanism indeed stores the information of variance between benign clients and malicious clients. Prompt-based Aggregation Though the attention layer to establish the vision-andlanguage alignment is a good perspective to observe the difference between malicious and benign clients, it's difficult to directly use the parameters of attention layer for comparison. In FedVLN, only few epochs are trained on the local client due to the high frequency between clients and the server. The variance of parameters would not be obvious since the local training process is too short. It's better for us to utilize a method which can capture the difference rapidly during the local training. Prompt is a good candidate for this. Prompt is a method which can rapidly adapt to new scenarios with few data and short training time. It is broadly applied in natural language processing and gains huge success. In light of its ability to quickly adapt to downstream tasks, we propose promptbased aggregation, PBA, to observe the alignment variance, preventing the global model from attack. In PBA, a visual prompt and a language prompt are introduced to current FedVLN setting. Both prompts are learnable vectors. As shown in Fig. 3, at the start of each communication round, the global visual prompt p V,g and language prompt p L,g at server initialize the local visual prompt p V,i and language prompt p L,i . p L,i at client i. The local prompts are added before the attention layer as follows: h t = h t + p V,i , u j = u j + p L,i(9) h t and u j are prompt-tuned embeddings and then sent into the attention layer. Both p V,i and p L,i are updated during training. They will be sent to the server after local training. Before aggregation, the server calculates the cosine similarity between clients by the concatenation of two prompts. After similarity calculation, we apply the same selection procedure as MultiKrun [8] to select some clients with high similarity to others for aggregation. Figure 2. The illustration of the broken vision-language alignment of the agent under backdoor attack. At a certain viewpoint during navigation, some part of the instruction (underlined) which is highly related to current views would gain high attention weights in expectation when encoded in the model, establishing a natural alignment between vision and language. However, when the trigger is inserted in another view, the attacked model would select the one with the trigger as the next navigable viewpoint. The ignorance of text breaks the original vision-and-language alignment, rendering the text attention weights chaotic. Experiments Experiment Setup Datasets We evaluate our NAW and PBA methods on two VLN datasets: Room-to-Room (R2R) [2] and Room-across-Room (RxR) [19]. Both datasets are developed on the Mat-terport3D Simulator [2], a photorealistic 3D environment for embodied AI research. R2R uses the Matterport3D region annotations to sample the start and end point pairs, then calculate the shortest paths between them to generate navigation data. The dataset contains 7,189 paths from 90 environments. The environments are split into 61 environments for training and seen validation, 11 for unseen validation, and 18 for testing. RxR is proposed to mitigate shortcomings of former VLN datasets. It is multilingual and larger than other VLN datasets. It contains 16,522 paths and 126,069 instructions. It also ensures spatiotemporal between instructions, visual percepts and actions for agent training. VLN Models Following FedVLN [39], we use Envdrop [33] and CLIP-ViL [32] as the backbone model architectures. The two models both use Bi-directional LSTM as the language encoder and attentive LSTM as the action decoder, with a mixed learning objective of imitation learning and reinforcement learning. CLIP-ViL adapts CLIP [30] to improve vision and language encoding and matching for vision-andlanguage navigation. Baselines We adopt the following six defense methods, that focus on the aggregation rule, for comparison. To keep consistent with these methods, adversarial methods to augment the robustness are not considered. paths, which penalize the deviation from the reference path. We use Attack Success Rate (ASR) [10] to evaluate attack and defense in FedVLN. ASR is calculated as the proportion of the times of selecting the backdoor among all the time steps that are implanted by the backdoor. Implementation Details We sample 12 out of 61 clients for each communication round. In each communication round, one of the 12 clients would be attacked, implanting the backdoor. By default, the probability of inserting the trigger p is 0.3, and the number of malicious clients m is 5 if not specially mentioned. More details are given in the Appendix. Attack Results NAW successfully implants the backdoor into the global model. In Table 1, we report the results on R2R and RxR datasets. Firstly, comparing the Attach Success Rate (ASR), we can observe that models trained with the NAW attack has a much higher ASR, implying that the global agent has a very high probability of selecting the navigable viewpoints with the trigger. Second, comparing other navigation metrics, the models trained with and without NAW have nearly the same performance under seen and unseen environments, showing that the backdoor can be implanted without hurting the validation performance and thus is unnoticeable. Impact of the number of malicious clients. Fig. 4 shows the results under different numbers of malicious clients. In Fig. 4(a), we can observe that ASR is positively correlated with the number of malicious clients. Furthermore, the increase in the number of malicious clients accelerates the convergence of ASR. For SR in Fig. 4(b), the performances under different numbers of malicious clients finally converge to the same point. However, more malicious clients would cause a greater fluctuation of SR during the first 100 communication rounds. Comparing the results with that of m = 0, we can find that the attack under m ≥ 20 cannot achieve the expected backdoor attack goal. Impact of the fraction of poisoned data. Parameter p approximates the fraction of poisoned data during training. Fig. 5 shows both SR and ASR of FedVLN agents under different fractions of poisoned data. For ASR, a larger fraction of poisoned data does not lead to a higher ASR; on contrary, it obtains an even lower ASR than a smaller fraction of poisoned data. SR becomes lower when the fraction of poisoned data is higher. When p ≥ 0.5, the drop in performance is obvious, inducing a nearly 3% SR gap. This result indicates that there is no strong connection between ASR and the frac- tion of poisoned data, but it hurts the performance when the fraction is large. Defense Results We compare and evaluate PBA with other defense methods from three aspects, Fidelity, Robustness, and Efficiency. Fidelity means that the method should not sacrifice the performance of the global model when there is no attack, taking the performance of the model of FedAvg as the reference standard. According to the results in Table 1 and Table 2, our PBA method performs similarly to FedAvg on both seen and unseen environments, achieving the fidelity goal when there is no attack. It demonstrates that the prompt added before the attention layer does not affect the convergence and performance of the model. Compared to other defense methods, however, some of them dramatically hurt the original performance. For instance, in Table 2, when applying FedCLIP-ViL on R2R, FLTrust performs much worse than FedAvg with an average of 25.6% SR drop on seen environments and 21.2% SR drop on unseen environments. FLTrust assigns the weights to each model update from clients by calculating the similarity between each local model update and server model update. However, due to the large variance of different environments, the server model greatly affects the ability of generalization of the global model, causing the performance gap. Median also hurts the performance about 7.9% SR gap. The remaining defense methods have the same process with FedAvg when there is no attack. Robustness means that the ASR of the server model should be as low as possible. In Table 3 prompt embeddings sent from clients. MultiKrum turns out to get higher ASR than FedAvg, which indicates that it can not tell the difference between malicious and benign clients. Both MultiKrum and Bulyan filter the "malicious" clients they think, and use less number of clients for aggregation. It increases the weights of malicious clients during aggregation and then increases the probability of being attacked, if they are wrongly judged, which unfortunately is exactly the case here. For FLTrust who depends on the clean root dataset, reults show that it can not tell the difference between malicious and benign clients under the typical Non-IID scenario due to the large variance between environments. Efficiency means the method should not incur excessive extra computation and communication overhead. In PBA , we only add two 1-Dimensional prompt embedding before the attention layer. During local training for each client, PBA does not add excessive training cost owing to the size of the embeddings of prompt, which is about 0.1% of that of the original model. In aggregation, it is more efficient for the server to directly compares the similarity between clients by using PBA , compared to other defense methods involving the calculation of all parameters. Impact of the fraction of poisoned data and the number of malicious clients. Fig. 6 shows the results of different defense methods under different fractions of poisoned data and different numbers of malicious clients. We only visualize the values of these two factors that have successfully achieved the goal of the backdoor attack. For the fraction of poisoned data, it is shown in Fig. 6(b) that PBA significantly outperforms any other defense methods in each case. For the number of malicious clients m, it is really a big threat to the current defense methods. Attack success rates of different defense methods are nearly 100% when there are too many malicious clients (e.g., m ≥ 10). Impact of prompt added before attention layer. We introduce two variants of our attack methods, to further explore the impact of special utilization of prompt in PBA . • PBA-Input: in most previous works, the prompt is added to the input embedding [24,40], rather than the middle position before the attention layer in PBA . In light of this, we apply the prompts in input visual features and language features in this variant and calculate the similarity of clients the same as PBA . • PBA-Param: as the alignment variance is represented in the attention layer, we directly use the parameters of the attention layer like traditional defense methods to calculate the similarity of clients. Fig. 7 shows the results of PBA and two variants on R2R unseen environments. (1) Directly using the parameters of the attention layer for evaluation is not effective, as illustrated in Sec. 4.2. It achieves a high ASR on both models in R2R unseen environments. (2) The prompt position matters. PBA performs better than PBA -Input about 20% ASR off. Putting prompt right before the attention layer can better capture the vision-and-language alignment difference between malicious and benign clients. Related Work Vision-and-language navigation is an important research area in embodied AI [3,12,20,29,36], which requires the agent to navigate to a goal location based on dynamic vi-sual input and language instructions. This requires the agent to understand and align the vision and language information, planning, and make decisions, etc. There have been lots of benchmarks and methods proposed on this task. [3] proposed a LSTM-based seq-to-seq model to track the navigation progress and multi-modal information for visionand-language navigation. For better understanding of the environment and the agent's own status, vision-and-language pre-training [15,17,22,32], graph representation, memory module, and auxiliary tasks have been introduced into VLN models. Recently, more and more works focus on the robustness of embodied AI. RobustNav is a framework to quantify the robustness of the embodied agent faced with corrupted input [11]. Liu et al. [23] studies a problem about spatiotemporal perturbations to form 3D adversarial in embodied AI tasks. Attack and defense on federated learning In federated learning, the attack has been divided into untargeted and targeted attacks. The Untargeted attack is designed to destroy the convergence of the global model [5,9], while the targeted attack aims to control the behavior of the global model [4,7,37]. Both attacks can be achieved by data poisoning and model poisoning. To defend against these attacks, multiple defense methods are proposed. One of the trends is to study the aggregation rule, and another is to strengthen the robustness of the model via adversarial methods [18]. Prompt learning is an emerging research area in natural language processing (NLP) and computer vision, which can efficiently transfer pre-trained vision and language models to various downstream tasks by tuning a small prompt layer [16,24,40,41]. By introducing a new prompting function, the model can perform few-shot and even zero-shot learning, adapting to new scenarios with little data. Originally, [31] proposes a manually designed prompt pattern for NLP tasks, which is a language instruction prepended to the input text. [25] proposes a P-tuning method to use the soft prompt instead of the previously manually designed prompt. In federated learning, prompt has been introduced to fine-tune the large pre-trained model [14,21] by freezing the model and only training the prompt features, reducing communication costs and preserving the privacy of clients. Conclusion In this paper, we study an important and unique security problem in federated embodied AI-whether the backdoor attack can manipulate the agent without influencing the performance and how to defend against the attack. We introduce a targeted backdoor attack NAW that successfully implants a backdoor into the agent and propose a promote-based defense framework PBA to defend against it. Adapting from two VLN models, PBA significantly outperforms the other 6 popular methods in terms of fidelity, robustness, and efficiency on two public benchmarks, which illustrates the effectiveness of PBA method in protecting the server model from the backdoor attack. Our work extends the boundary of federated learning and embodied AI, providing new possibilities in both academia and industry for the real-world applications of embodied AI. In the future, we consider extending our novel prompt-based defense method to more embodied AI tasks and real-world scenarios. Algorithm 1 Federated learning with prompt-based aggregation Require: Parameters: participation rate r; number of clients n; local learning rate λ; server learning rate η; number of communication rounds T ; local training epochs τ . 1: for t = 1 → T do for client c i in φ t do 5: Client c i initialization: (w t−1 i , p V,i , p L,i ) = (w t−1 , p V,g , p L,g ) 6: Client c i local training: w t i , p V,i , p L,i = ClientUpdate(w t−1 i , p V,i , p L,i , τ, λ) 7: Client c i uploads delta of the language encoder Server aggregation: ∆w t i = w t i − w t−1 , ∆p V,i = p V,i − p V,i , ∆p L,i = p L,i − p L,w t = P BA(φ t , ∆w t i , ∆p V,i , ∆p L,i , rm) 10: end for A. Algorithm Details In prompt-based aggregation (PBA), the visual prompt and the text prompt are learnable vectors. Global visual prompt p V,g or the visual prompt of client i p V,i has the same dimension as the hidden state h t output from the view encoder, and global text prompt p L,g or the text prompt of client i p L,i has the same dimension as the embedding of each text token u 1 , u 2 , u 3 , ..., u L . When applying PBA in federated learning, at the start of each communication round, both local model weight and local prompts are initialized by global model weight and global prompts. After both local model weight and local prompt parameters are updated through the local training process of each client, we utilize the update of prompt parameters to select some clients to do the aggregation. The whole training procedure is shown in Alg. 1. It's worth noting that only model weight is updated in aggregation, while the global prompts p V,g and p L,g are fixed. For the calculation of similarity, the similarity Sim(i, j) between client i and client j is calculated as below: Sim(i, j) = cos <Sign([∆p V,i , ∆p L,i ]), Sign([∆p V,j , ∆p L,j ]) >(10) where ∆p V,i and ∆p L,i are the update of prompt parameters of i th client. We employ the Sign function here as the direction of parameters update is more important than the magnitude in federated learning. For the selection of clients, We apply the similar selection rule in MultiKrum [9], which selects clients with high similarity to others. The detailed Algorithm 2 Prompt-based Aggregation (PBA) in communication round t Input: the set of sampled clients for this round φ t ; update of model weight of each client ∆w t i ; update of prompt parameters of each client c i ∆p V,i , ∆p L,i ; expected number of malicious clients m e Output: global model weight after the aggregation of this round w t 1: Calculate the similarity Sim(i, j) between each pair of clients in φ t 2: S rem = φ t ,S sel = {} 3: while |S rem | > 2m e + 2 do Update S rem and S sel : S rem = S rem − c h , S sel = S sel + c h 10: end while 11: Aggregation: w t = w t−1 + η i∈S sel In the variant PBA-Input, we use the concatenation of parameters update of visual prompt and text prompt in input position to replace the concatenation of original prompt embeddings in Equ. 10. In the variant PBA-Param, we use the parameters of the attention layer to replace the original prompt embeddings in Equ. 10. The remaining two variants are the same as PBA. B. Implementation Details In datasets, the environments are split into 61 environments for training and seen validation, 11 for unseen validation. When training on seen environments, the total number of training steps of local models is the same as centralized training steps. At each communication round, we use the participation rate of r = 0.2, which indicates that we sample 12 clients out of 61 clients for the training of this round. We train each local agent for τ = 5 epochs on local data. We set the global learning rate η = 2 following [39]. For the attack, the number of malicious clients m is 5, which indicates that one of the 12 clients in each communication round is malicious in expectation. When applying backdoor attacks during training in malicious clients. The probability The global model at the server is evaluated on seen and unseen validation environments after each communication round. Evaluation metrics except for attack success rate (ASR) are evaluated on clean seen and unseen validation environments. When evaluating ASR, we poison the validation environments with p = 0.1 and the same trigger utilized by malicious clients during local training. We then calculate ASR by validating the poisoned seen and unseen validation environments. C. More Experiment Results Here we provide additional results for both attack and defense on R2R with EnvDrop. C.1. Attack Impact of the number of malicious clients. Fig. 8 shows the results under different numbers of malicious clients with EnvDrop. In Fig. 8(a), we can observe that the increase in the number of malicious clients not only accelerates the convergence of ASR, but also improves the final ASR. For SR in Fig. 8(b), more malicious clients would cause an obvious performance drop during training. Comparing the results with that of m = 0, we can find that the attack under m ≥ 20 cannot achieve the expected backdoor attack goal, which requires the performance of the attacked model on the clean dataset to keep the same level as that of the unattacked model. Impact of the fraction of poisoned data. Hyperparameter p approximates the fraction of poisoned data during training. Fig. 9 shows both SR and ASR of Fed-VLN agents under different fractions of poisoned data with EnvDrop. For ASR, we can find that a larger fraction of poisoned data could not lead to a better attack. ASR of p = 0.1 and p = 1.0 are quite close. SR becomes lower when the fraction of poisoned data is higher, while ASR of p = 0.3 and p = 0.5 are high. It indicates that we need to select an appropriate range for p to achieve great attack effects. For SR, When p ≥ 0.5, the drop in performance is obvious, inducing a nearly 6% SR gap. It proves that a larger fraction of poisoned data could hurt the performance of the attacked model on the clean dataset, which is not expected in the backdoor attack. C.2. Defense Impact of the number of malicious clients and fraction of poisoned data. Fig. 10 shows the results of different defense methods under different fractions of poisoned data and different numbers of malicious clients with EnvDrop. For the number of malicious clients m, ASR of different defense methods are close to 100% when there are too many malicious clients (e.g., m ≥ 10). For the fraction of poisoned data, it is shown in Fig. 10(b) that ASR of different defense methods mostly maintains the same level as that of FedAvg. Some methods (e.g., MultiKrum) even exacerbate it. For instance, when p = 0.1, ASR of MultiKrum is almost three times that of FedAvg. On the whole, PBA significantly outperforms any other defense methods in each case. C.3. Case Study We choose one of the rounds in our experiment and present the case study to illustrate the differences between the Euclidean distance given in MultiKrum [9] and our methods as shown in Fig. 11. It can be found that the attack can Figure 11. The illustration of the difference of the method to calculate the similarity between MultiKrum [9] and ours. MultiKrum uses the Euclidean distance and we use the similarity score given in Equ. 10 .All the matrix is 12 × 12 because there are 12 clients in every round. The diagonal of the distance matrix is 0 and the similarity is 1. Using the Euclidean distance, there is no obvious difference between the malicious and the benign clients. With our similarity score, we can distinguish the malicious client very clearly. be hidden in traditional defense methods, while our methods can detect the malicious client clearly, demonstrating the importance of capturing the variance of vision-language alignment. Figure 3 . 3Prompt-based Aggregation (PBA). Besides normal model update and aggregation, local prompt in the client is utilized and updated during local training process. The local prompt would be an important reference to distinguish malicious clients after it is sent to the server. It is initialized by a fixed global prompt at each communication round. Figure 4 . 4Impact of the number of malicious clients. Results are evaluated on R2R with CLIP-ViL. Figure 5 . 5Impact of the fraction of poisoned data. Results are evaluated on unseen environments in R2R with CLIP-ViL. Figure 6 . 6Impact of the number of malicious clients and fraction of poisoned data. Results are evaluated on unseen environments with CLIP-ViL. Figure 7 . 7Results on R2R with different models for PBA and its variants. c i in S rem do 5: Select |S rem − m e − 1| largest Sim values for c i with other clients in S rem , which can be assumed to be {Sim(i, 1), Sim(i, 2), ..., Sim(i, |S rem − m c − 1|)} with no harm. client c h with largest value of prompt score: return w t procedure of PBA is as shown Alg. 2. Figure 8 . 8Impact of the number of malicious clients. Results are evaluated on R2R with EnvDrop. of inserting the trigger at each time step p is 0.3, which approximates the fraction of poisoned data. These settings are default if not mentioned. Figure 9 .Figure 10 . 910Impact of the fraction of poisoned data. Results are evaluated on R2R with EnvDrop. Impact of the number of malicious clients. Results are evaluated on unseen environments of R2R with EnvDrop. trigger ... Go upstairs. Walk along the corridor and turn right. Get into the laundry. ...... [... ...] Model View Encoder Text Encoder Attention ... ... ... ... Decision Selected Next Viewpoint Attention Map of Text (Attacked) ... ... ... ... Attention Map of Text (Benign) • FedAvg[34] is the basic FL aggregation rule.• Median[38] aggregates the gradient from clients by calculating the median value of each dimension of the gradients. • Trimmed Mean[38] sorts the values of this dimension of all gradients and deletes m maximum and minimum, calculating the average of the remaining values as the aggregation of this dimension. • Multi-Krum[9] adopts Krum to select the gradient from the remaining set (initialized as the set of all gradients) and adds it to the selection set (initialized as an empty set), then deletes the selected one from the remaining set. • Bulyan[27] adopts Multi-Krum to select gradients, and uses Trimmed Mean to calculate the final gradients from the selection set. • FLTrust[10] requires the server has a clean root dataset to approximate the benign gradients.OSR↑ SPL↑ SR↑ CLS↑ nDTW↑ ASR OSR↑ SPL↑ SR↑ CLS↑ nDTW↑ ASREvaluation Metrics For both datasets, we report Success Rate (SR), Success Rate weighted by Path Length (SPL), Oracle Success Rate (OSR), and navigation Error (NE) as goal-oriented metrics [1, 3, 13, 33]. We also report Coverage weighted by Length Score (CLS) and normalized Dynamic Time Warping (nDTW) to validate the fidelity of navigation Dataset Model Is Attacked Val-Seen Val-Unseen R2R EnvDrop No 63.1 52.4 55.0 66.4 55.1 0.08 53.0 43.4 46.5 59.0 45.5 0.05 Yes 63.2 52.2 54.8 66.1 55.4 0.71 52.4 43.1 46.5 59.1 45.8 0.68 CLIP-ViL No 67.2 55.8 60.4 65.7 53.3 0.07 61.9 47.6 53.4 57.9 44.4 0.05 Yes 67.5 54.7 60.1 66.3 53.9 0.87 61.4 47.0 52.2 55.8 44.7 0.85 RxR EnvDrop No 49.2 33.8 36.8 56.2 51.0 0.12 43.1 29.1 33.5 54.7 49.4 0.08 Yes 48.7 33.9 37.3 55.9 51.4 0.67 42.7 29.3 33.2 54.4 49.2 0.66 CLIP-VIL No 54.6 40.0 44.2 59.0 54.7 0.09 50.1 35.0 39.4 56.0 51.5 0.09 Yes 54.8 39.7 43.8 58.6 54.5 0.68 51.7 34.6 38.4 56.5 51.4 0.73 Table 1. Results of the federated navigation agents when not attacked and attacked on R2R [3] and RxR [20]. By default, FedAvg is utilized as the aggragation rule. The much higher ASR results indicate that the backdoor attack is successfully implanted. Moreover, models with and without attack achieve similar navigation results, showing that the NAW attack is unnoticeable in FL. and 4, PBA gets the lowest ASR on different models under both seen and unseen environments of R2R and RxR. On the contrary, other defense methods have much higher ASR, especially on FedCLIP-ViL and RxR. Moreover, some defense methods even exacerbate the model under attack. For example, PBA has the same selection rule as MultiKrum. However, MultiKrum utilizes the Euclidean distance of all model parameters to select benign clients, while PBA only focus on the similarity betweenMethods Val-Seen Val-Unseen FedEnvDrop FedCLIP-ViL FedEnvDrop FedCLIP-ViL No Attack 0.08 0.07 0.05 0.05 FedAvg 0.71 0.87 0.68 0.85 Median 0.70 0.89 0.72 0.88 Trim-Mean 0.76 0.86 0.74 0.84 MultiKrum 0.77 0.95 0.75 0.96 Bulyan 0.78 0.91 0.77 0.94 FLTrust 0.87 0.93 0.88 0.97 PBA (ours) 0.63 0.72 0.64 0.76 Table 3. Comparison of Attack Success Rate (ASR) between differ- ent defense methods on R2R. Lower is better. AGR Val-Seen Val-Unseen FedEnvDrop FedCLIP-ViL FedEnvDrop FedCLIP-ViL No Attack 0.12 0.09 0.08 0.09 FedAvg 0.67 0.68 0.66 0.73 Median 0.79 0.84 0.81 0.85 Trim-Mean 0.79 0.80 0.81 0.83 MultiKrum 0.84 0.95 0.84 0.96 Bulyan 0.74 0.78 0.77 0.76 FLTrust 0.77 0.97 0.75 0.96 PBA (ours) 0.42 0.45 0.41 0.49 Table 4 . 4Comparison of Attack Success Rate (ASR) between different defense methods on RxR. Lower is better. On evaluation of embodied navigation agents. Peter Anderson, Angel Chang, Devendra Singh Chaplot, Alexey Dosovitskiy, Saurabh Gupta, Vladlen Koltun, Jana Kosecka, Jitendra Malik, Roozbeh Mottaghi, Manolis Savva, arXiv:1807.06757arXiv preprintPeter Anderson, Angel Chang, Devendra Singh Chaplot, Alexey Dosovitskiy, Saurabh Gupta, Vladlen Koltun, Jana Kosecka, Jitendra Malik, Roozbeh Mottaghi, Manolis Savva, et al. 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[ "HARISH-CHANDRA INDUCTION AND JORDAN DECOMPOSITION OF CHARACTERS", "HARISH-CHANDRA INDUCTION AND JORDAN DECOMPOSITION OF CHARACTERS" ]
[ "Prashant Arote ", "Manish Mishra " ]
[]
[]
We show that for any finite connected reductive group, a Jordan decomposition can always be chosen such that it commutes with Harish-Chandra induction. En route, we show that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to a unipotent counterpart. These results generalise the well known results for groups with connected center.•
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[ "https://export.arxiv.org/pdf/2209.00574v2.pdf" ]
251,979,655
2209.00574
cc07aaa79d11039a549f3f9c114e8c7de6f5e6f0
HARISH-CHANDRA INDUCTION AND JORDAN DECOMPOSITION OF CHARACTERS 20 Dec 2022 Prashant Arote Manish Mishra HARISH-CHANDRA INDUCTION AND JORDAN DECOMPOSITION OF CHARACTERS 20 Dec 2022arXiv:2209.00574v2 [math.RT] We show that for any finite connected reductive group, a Jordan decomposition can always be chosen such that it commutes with Harish-Chandra induction. En route, we show that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to a unipotent counterpart. These results generalise the well known results for groups with connected center.• Introduction Let G be a connected reductive group defined over a finite field F q and let F : G → G denote the corresponding Frobenius morphism. To each conjugacy class of a semisimple element s in the dual group G * of G, Lusztig associated a subset E(G F , s) of the set Irr(G F ) of irreducible complex characters of G F . The subset E(G F , s) is called the Lusztig series associated to s. The collection of all Lusztig series of G partition the set Irr(G F ). The elements of E(G F , 1) are called the unipotent characters of G. We will denote them by Uch(G F ). When the center of G is connected, Lusztig established that there exists a bijection J G s between the sets E(G F , s) and Uch(C G * (s) F * ) satisfying a certain natural property. Here C G * (s) denotes the centralizer of s in G * . The collection of bijections J G s thus defines the notion of Jordan decomposition of characters. This notion extends to include groups with disconnected centre. In the case of groups with disconnected centre, J G s is a surjection from E(G, s) onto equivalence classes of Uch(C G * (s) • ), where equivalence is given by the action of the group of C G * (s) F * /C G * (s) F * • . Here (−) • denotes the identity component of (−). Given an F -stable Levi factor L of a parabolic subgroup P of G and a character τ of L F , Lustig induction produces a virtual character R G L (τ ). When P is also F -stable, the functor R G L is just the Harish-Chandra induction functor. A very important and widely open question is whether Jordan decomposition can always be chosen such that it commutes with Lusztig induction. For classical groups with connected center, this commutation was proved by Fong and Srinivasan [FS89]. See [GM20, Theorem 4.7.2] for slightly stronger version of this result whose proof is based on [Eng13,Prop. 5.3]. For SL n the commutation problem was resolved by Bonnafé [Bon06,§27] and Cabanes [Cab13,Theorem 4.9] under some assumption on q. When Lusztig induction is the Harish-Chandra induction, this commutation holds for simple groups with connected centre [GM20,Corr. 4.7.6]. For a general connected reductive group, the question of commutation of some Jordan decomposition with Harish-Chandra induction had also been an open problem. See [GM20, A.5] for a discussion on known results and the importance of this problem. One of our main results (Theorem 5.5) in this article resolves the commutation problem for Harish-Chandra induction. Theorem 1.1. For any connected reductive group defined over F q , there exists a Jordan decomposition which commutes with Harish-Chandra induction. Our next result generalises a result of Lusztig [Lus84,Theorem 8.6] from the case of groups with connected center to arbitrary connected reductive groups. This result is also the main ingredient of the proof of Theorem 1.1. Let L be the Levi factor of an F -stable parabolic subgroup of G and let τ be a cuspidal representation of L F appearing in a Lusztig series E(L F , s). Theorem 1.2. Let u τ be any element of J L s (τ ). For an appropriate choice of J L s , there is an isomorphism of of C-algebras: End G F R G L (τ ) ∼ = End H F * The statement requires some explanation which is given in the paragraphs below. Observe here that since τ is cuspidal, the unipotent character u τ is necessarily cuspidal. We now explain the algebra structure and the group Γ in the statement above and also the key ideas of the proof. Howlett-Lehrer [HL80] described the Endomorphism algebra End G F (R G L (τ )) and showed that its opposite algebra is isomorphic to a group algebra C[W τ ] on a finite group W τ twisted by a co-cycle. Lusztig showed [Lus84,Theorem 8.6] that in the case of groups with connected center, this co-cycle is always trivial. It was subsequently shown by Geck [Gec93] that Lusztig's result implies that the co-cycle is trivial in general. The opposite algebra of End H F * • (R H• (HL)• (u τ )) is likewise isomorphic to C[W uτ ]. One can also analogously define W J L s (τ ) . We show in Theorem 4.4 that, Theorem 1.3. W τ ∼ = W J L s (τ ) . In the case when G has connected center, Theorem 1.3 is due to Lusztig. It appears in the course of the proof of [Lus84,Theorem 8.6]. We next show that the short exact sequence 1 → W uτ → W τ → W τ /W uτ → 1 is split. The group Γ is the group W τ /W uτ and the algebra structure on right hand side of the isomorphism in Theorem 1.2 is obtained via this splitting. We now explain a bit more about the Jordan decomposition appearing in Theorem 1.1. For simplicity of exposition here, we assume that G has connected center. In this case, the isomorphism in Theorem 1.2 induces a bijection on Harish-Chadndra series, Irr(G, (L F , τ )) → Irr(H F * • , ((H L ) F * • , u τ ) ). Since Lusztig series is a disjoint union of Harish-Chandra series, by gluing such bijections, one can define a new bijection J G,s : E(G F , s) → Uch(H F * • ) . We show that J G,s is in fact a Jordan decomposition and this is the Jordan decomposition in Theorem 1.1 that commutes with Harish-Chandra induction. The key ingredient of our proof of Theorem 1.3 is the result of Digne and Michael [DM90, Theorem 7.1] that Jordan decomposition can be made unique by requiring it to satisfy various natural properties. By studying how the unique Jordan decomposition J L s behaves under automorphism of the Levi L, specifically the automorphisms coming from the normalizer of L in G, we obtain Theorem 1.3. Notation Throughout this paper we will use following notation. Let F q denote a finite field with q = p r elements and fix a prime ℓ different from p. We fix an isomorphism between C and Q ℓ . Let G be a connected reductive group defined over F q and F : G → G be the corresponding Frobenius morphism. We denote the F q -point of G by G F and likewise for all reductive groups. We will denote the Weyl group of G with respect to a torus T by W G (T ) := N G (T )/C G (T ), where C G (T ) is the centralizer of T in G. We will denote the the centre of G by Z(G) and likewise for all its subgroup. Let T 0 be a maximal torus of G contained in an F -stable Borel subgroup B of G. Then there exists a connected reductive group G * defined over F q with a Frobenius morphism F * : G * → G * and an F *stable maximal torus T * 0 contained in a F * -stable Borel subgroup B * of G * such that the dual of the based root datum of G * is isomorphic to the based root datum of G. Also the isomorphism between root data is compatible with corresponding action of Frobenius morphism on root lattice and co-root lattice. For more details see [GM20,§1.5.18]. Then we say that (G, F ) and (G * , F * ) are in duality and G * is a dual of G. If (G, F ) and (G * , F * ) are in duality then there is a natural isomorphism between Weyl groups δ : W G (T 0 ) → W G * (T * 0 ), w → w * = δ(w) satisfying some properties. This isomorphism δ restricts to an isomorphism between W G (T 0 ) F and W G * (T * 0 ) F * . Let Irr(−) denote the set of isomorphism classes of the irreducible representations of a finite group over an algebraically closed field. We will denote the set of unipotent representations by Uch(−). Deligne-Lusztig theory and Jordan decomposition In this section, we briefly recall the Deligne-Lusztig theory of constructing the virtual representations of finite group of Lie type. We also recall the Lusztig's notion of Jordan decomposition of characters of finite group of Lie type. We reproduce this summary from [AM22,§2.3]. For more details we refer to [DL76], [Lus84], [Lus88], [DM20] and [GM20]. Let G be a connected reductive group over F q with corresponding Frobenius morphism F : G → G. Fact 2.1 ([DM20, Proposition 11.1.16]). Let T be a maximal F -stable torus of G and let θ be a linear character of T F . The G F -conjugacy classes of pairs (T, θ) are in one-to-one correspondence with the G * F *conjugacy classes of pairs (T * , s) where s is a semi-simple element of G * F * and T * is an F * -stable maximal torus of G * F * containing s. Let (T, θ) be a pair of an F -stable maximal torus T and a linear character θ : T F → Q ℓ . For every such pair Deligne-Lusztig associated a virtual representation of G F using ℓ-adic cohomology with compact support (ℓ = p). This virtual representation is denoted by R G T (θ). If (T, θ) and (T * , s) corresponds to each other as in the Fact 2.1 then we will write R G T * (s) for R G T (θ). Definition 2.2 ([GM20, Def. 2.6.1]). Let s ∈ G * F * be semisimple. Define E(G F , s) to be the set of all ρ ∈ Irr(G F ) such that R G T * (s), ρ = 0 for some F * -stable maximal torus T * ⊆ G * with s ∈ T * . This set is called a rational series of characters of G F , or Lusztig series of characters. Definition 2.3 (Unipotent character). A character ρ ∈ Irr(G F ) is called a unipotent character if R G T * (1), ρ = 0 for some F * -stable maximal torus T * ⊆ G * . We will denote the set of unipotent characters of G F by Uch(G F ) and by definition, Uch(G F ) := E(G F , 1). Fact 2.4 ([DM20, Prop.11.1.1, Prop.11.3.2]). For any ρ ∈ Irr(G F ), there exists an F -stable maximal torus T and θ ∈ Irr(T F ) such that ρ, R G T (θ) = 0. Moreover, we have, Irr(G F ) = (s) E(G F , s), where (s) runs over the semi-simple conjugacy classes of G * F * . Definition 2.5 ([DM20, Definition 7.1.5]). Let F be a Frobenius morphism on the connected algebraic group G. Let us define a sign ǫ G : = (−1) F −rank(G) , where F −rank(G) denotes the F -split rank of G. For a connected reductive group with connected centre, Lusztig [Lus84] introduced the notion of Jordan decomposition of characters. Using this we get a parameterization of E(G F , s) by the unipotent characters of C G * (s) F * . But this description of a bijection in [Lus84] leaves some ambiguity. This ambiguity was resolved by Digne and Michel who proved the following uniqueness result: Theorem 2.6 ([DM90, Theorem 7.1]). There exists a unique collection of bijections, J G s : E(G F , s) → Uch(C G * (s) F * ), where G runs over connected reductive groups with connected centre and Frobenius map F , and s ∈ G * F * is semisimple, satisfying the following, where we write H := C G * (s): (1) For any F * -stable maximal torus T * ≤ H, R G T * (s), ρ = ǫ G ǫ H R H T * (1 T * ), J G s (ρ) , for all ρ ∈ E(G F , s). (2) If s = 1 and ρ ∈ Uch(G F ) is unipotent then, a) the Frobenius eigenvalues ω ρ and ω J G 1 (ρ) are equal, and b) if ρ lies in the principal series then ρ and J G 1 (ρ) correspond to the same character of Iwahori-Hecke algebra. (3) If z ∈ Z(G * F * ) then J G sz (ρ ⊗ẑ) = J G s (ρ) for ρ ∈ E(G F , s), whereẑ is a linear character of G F corresponding to z. (4) For any F * -stable Levi subgroup L * of G * such that H ≤ L * , with dual L ≤ G, the following diagram commutes: E(G F , s) Uch(H F * ) E(L F , s) Uch(H F * ). J G s R G L J L s Id (5) If G is of type E 8 and H is of type E 7 A 1 (resp. E 6 A 2 ) and L ≤ G is a Levi subgroup of type E 7 (resp. E 6 ) with dual L * ≤ H then the following diagram commutes: ZE(G F , s) Z Uch(H F * ) ZE(L F , s) c Z Uch(L * F * ) c , J G s R G L J L s R H L * where the index c denotes the subspace spanned by the cuspidal part of the corresponding Lusztig series. (6) For any F -stable central torus T 1 ≤ Z(G) with corresponding natural epimorphism π 1 : G → G 1 := G/T 1 and for s 1 ∈ G * F * with s = π * 1 (s 1 ) the following diagram commutes: E(G F , s) Uch(H F * ) E(G F 1 , s 1 ) Uch(H F * 1 ), J G s J G 1 s 1 with H 1 = C G * 1 (s 1 ) and where the vertical maps are just the inflation map along G F → G F 1 and the restriction along the embedding H F * 1 → H F * respectively. (7) If G is a direct product i G i of F-stable subgroups G i then J G si = J Gi si . In [Lus88], Lusztig extended this notion of Jordan decomposition of characters to the general case (i.e., allowing Z(G) to be disconnected) using a regular embedding. We will now state some results without proof. For more details we refer to [Lus88]. Fact 2.7. Let π : G → G ad be the adjoint quotient of G. There is a natural isomorphism, G F ad /π(G F ) ∼ = (Z(G)/Z(G) • ) F (the subscript F denotes F -coinvariants, i.e. largest quotient on which F acts trivially). There is natural action of group G F ad on G F as follows: g : g 1 →ġg 1ġ −1 , where π(ġ) = g. This induces a natural action of G F ad /π(G) F on Irr(G F ). This action can be extended by linearity to virtual representations. One can easily show that this extended action stabilizes each R G T * (s). Thus, we have an action of G F ad /π(G F ) ∼ = (Z(G)/Z(G) • ) F on E(G F , s). Definition 2.8 (regular embedding). Let G be a connected reductive group over F q . Then a morphism i : G → G ′ is called a regular embedding if G ′ is a connected reductive group over F q with connected centre, i is an isomorphism of G onto a closed subgroup of G ′ and i(G), G ′ have the same derived subgroup. Let i : G ֒→ G ′ be a regular embedding. By [GM20, Theorem 1.7.12], it corresponds to a surjective homomorphism i * : G ′ * → G * (over F q ). Let K := ker(i * ). There is a canonical isomorphism: K F * ∼ = − → Hom(G ′F /G F , Q × ℓ ). This isomorphism induces an action of K F * on Irr(G ′F ) by tensor product. Under this action k ∈ K F * maps E(G ′F , s ′ ) to E(G ′F , ks ′ ). Define K F s ′ to be the set of all k ∈ K F * which map E(G ′F , s ′ ) into itself or, equivalently, K F * s ′ := {k ∈ K F * : ks ′ is conjugate to s ′ under G ′ * F * }. Fact 2.9. Let s ′ ∈ G ′ * F * such that s = i * (s ′ ) ∈ G * F * . Denote the centralizer of s in G * by H as before, i.e., H := C G * (s). Then there is a natural isomorphism H F * /H F * • ∼ = K F * s ′ given by the correspondence x ∈ H F * → s ′−1ẋ s ′ẋ−1 ∈ K F * , whereẋ ∈ G ′ * is such that i * (ẋ) = x. Thus, using this isomorphism, we get an action of H F * /H F * • on E(G ′ * F * , s ′ ). Fact 2.10 ([GM20, Prop.2.3.15]). Let i : G → G ′ be a regular embedding as above and let s ′ ∈ G ′ * F * such that s = i * (s ′ ) ∈ G * F * . Let H ′ be the centralizer of s ′ in G ′ * . Then i * induces a surjective homomorphism from H ′ onto H • with kernel K. Hence, we have a canonical bijection, Uch(H ′F * ) → Uch(H F * • ), ρ ′ → ρ ′ • i * | H ′F * . Using this, the action (by conjugation) of H F * /H F * • on Uch(H F * • ) becomes an action of H F * /H F * • on Uch(H ′F • ). Lemma 2.11 (cf.[Lus88, Prop.8.1]). Let s ′ ∈ G ′ * F * be semisimple and let i * (s ′ ) = s ∈ G * F * . Let H ′ = C G ′ * (s ′ ) and H = C G * (s). Then the unique bijection, J G ′ s ′ : E(G ′F , s ′ ) → Uch(H ′F * ), is compatible with the action of H F * /H F * • . Proof. This follows from property 3 of J ′ s in Theorem 2.6 and by the definition of action on both sides. Fact 2.12. Let i : G → G ′ be a regular embedding. Then there is a natural surjective homomorphism G ′F /G F → G F ad /π(G F ). Thus, G ′F /G F acts on E(G F , s) via G F ad /π(G F ). Lemma 2.13 ([Lus88, §10]). Let i : G → G ′ be a regular embedding. Then for any ρ ′ ∈ Irr(G ′F ), the restriction ρ ′ | G F is multiplicity free. Next, we state a result from representation theory of finite groups: ρ ′ • | N = ρ∈O ρ and Ind H N (ρ • ) = ρ ′ ∈O ′ ρ ′ . Moreover, the stabilizer of ρ • in H/N and the stabilizer of ρ ′ • in H/N are orthogonal to each other under the natural duality H/N × H/N → Q × ℓ . With these notations, we now state the result of Lusztig. J s = J (G,G ′ ) s : E(G F , s) → Uch(H F * • )/ ∼ (H F * /H F * • ), with the following properties: (1) The fibres of J s are precisely the orbits of the action of G F ad /G F on E(G F , s). (2) If O is an H F * /H F * • -orbit on Uch(H F * • ) and Γ ≤ H F * /H F * • is the stabilizer of an element in O, then the fibre J −1 s (O) has precisely |Γ| elements. (3) If ρ ∈ J −1 s (O) and T * is an F * -stable maximal torus of G * containing s, then R G T * (s), ρ G = ǫ G ǫ H• ρ ′ ∈O R H T * (1 T * ), ρ ′ H• . Recall that the map J s is defined as follows: E(G F , s) E(G F , s)/ ∼ (G ′ F /G F ) k∈K F * E(G ′F , s ′ k)/ ∼ K F * ∼ = (G ′ F /G F ) E(G ′F , s ′ )/ ∼ K F * s ′ = (H F * /H F * • ) Uch(H ′F )/ ∼ (H F * /H F * • ) Uch(H F * • )/ ∼ (H F * /H F * • ).(1) u ρ is cuspidal, and, (2) Z(G * ) • and Z(H • ) • have the same F q -rank, that is, the maximal split subtorus of Z(H • ) • is contained in Z(G * ) • . As a corollary to (2) in the above theorem we obtain: Corollary 2.17. Let L * be an F -stable Levi factor of an F -stable parabolic subgroup of G * containing s with dual L ≤ G. Then C L * (s) • is a Levi factor of an F -stable parabolic subgroup of C G * (s) • . Remark 2.18 ([DM20, Lemma 11.2.1]). Let G be a connected reductive group defined over F q . If the centre of G is connected, then the centre of any Levi subgroup of G is also connected. Harish-Chandra Induction and Endomorphism Algebra In this section we briefly recall some results about Harish-Chandra series. In [HL80], Howlett and Lehrer described the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation as a group algebra twisted by a cocycle. This co-cycle was proved to be trivial in the case of connected center by Lusztig and subsequently it was shown by Geck that Lusztig's result implies that the co-cycle is trivial in general. For more details we refer to [HL80] and [GM20, §3.2]. Let G be a connected reductive group defined over F q . Let P be an F -stable parabolic subgroup of G with F -stable Levi decomposition P = LU . Then R G L and * R G L denotes the Harish-Chandra induction and restriction functors. Definition 3.1 (Harish-Chandra Series). Let L be an F -stable Levi factor of an F -stable parabolic subgroup P of G. For a cuspidal pair (L F , τ ), the corresponding Harish-Chandra series Irr(G F , (L F , τ )) is defined to be the set of all irreducible representations ρ ∈ Irr(G F ) (up to isomorphism) such that, (1) L is a minimal F -stable Levi subgroup such that * R G L (ρ) = 0, and (2) τ is a composition factor of * R G L (ρ). The Harish-Chandra series partition Irr(G F ) (see [GM20, Corollary 3.1.17]). More precisely, Irr(G F , (L F , τ )) is non-empty for every cuspidal pair (L F , τ ) in G F , and, Irr(G F , (L F , τ )) ∩ Irr(G F , (M F , λ)) = Ø, for all cuspidal pairs (M F , λ) of G F not G F -conjugate to (L F , τ ).E(G F , s) = Σ G F (s) Irr(G F , (L F , τ )), where Σ G F (s) denotes the set of G F -conjugacy classes of cuspidal pairs (L F , τ ), with τ ∈ E(L F , s) for a semisimple element s ∈ L * F * . Let A be the maximally F q -split torus in Z(L). Then C G (A) = L. Let W G (A) be the set of bijections of L F induced by conjugation by element of G F i.e., W G (A) = N G F (A)/L F . Let us define, W τ (G) := {w ∈ W G (A) : χ τ • w = χ τ }, where χ τ is the character of an irreducible cuspidal representation τ of L F and w also denotes the automorphism of L F induced by w. If it is clear from the context, we will drop G from the notation W τ (G) and will denote it by W τ Theorem 3.3 ([GM20, Theorem 3.2.5]). Let G be a connected reductive defined over F q and F : G → G be the corresponding Frobenius morphism. Let L be a Levi factor of an F -stable parabolic subgroup of G and τ be a cuspidal representation of L F . Then the opposite algebra of End G F (R G L (τ )) is isomorphic to the group algebra Q ℓ [W τ ].I M L,τ : Irr(W τ (M )) → Irr(M F , (L F , τ )), φ → ρ φ , where M runs over F -stable Levi factors of F -stable parabolic subgroups L ≤ M ≤ G, can be chosen such that the diagrams, Irr(W τ (G)) Z Irr(G F , (L F , τ )) Irr(W τ (M )) Z Irr(M F , (L F , τ )), I G L,τ I M L,τ Ind R G M commute for all M , where Ind denotes the ordinary induction. The following result and its proof was provided to us by Jay Taylor. Proposition 3.5 (Jay Taylor). Suppose φ, φ ′ ∈ Irr(W τ (G)) satisfy the following: Proof. Let W := W τ (G). By Comparison Theorem 3.4, (1) is equivalent to the following condition: (1) * R G M (ρ φ ) = * R G M (ρ φ ′ ) for any proper F -stable Levi subgroup L M < G, of an F -stable parabolic subgroup of G, (2) ρ φ (1) = ρ φ ′ (1). Then ρ φ = ρ φ ′ except possibly when W τ (G) is irreducible of type E 7 or E 8 and {φ, φ ′ } is ( 1 ′ ) Res W W ′ (φ) = Res W W ′ (φ ′ ) for all proper parabolic subgroups W ′ < W. Assume P G is an F -stable Levi subgroup having L P as a Levi complement. Then ρ φ (1) = φ(1)[G F : P F ]c −1 φ where c φ is the Schur element of the corresponding representation of the Hecke algebra End G F (R G L (τ )), see [Lus84,Cor. 8.7] and [GP00, §8.1.8]. With this we see that (2) is equivalent to the following condition: ( 2 ′ ) c φ = c φ ′ . We can obviously assume that W is non-trivial. Let us write W = W 1 × · · · × W r as a product of its irreducible components. Then φ = φ 1 ⊠ · · · ⊠ φ r and φ ′ = φ ′ 1 ⊠ · · · ⊠ φ ′ r for some φ i , φ ′ i ∈ Irr(W i ). Suppose r > 1 and fix 1 i r. As W i < W is a proper parabolic subgroup there exists a proper F -stable Levi subgroup (of an F -stable parabolic) L M < G such that W τ (M ) = W i . By (1) and Comparison Theorem 3.4, we get that φ i = Res W Wi (φ) = Res W Wi (φ ′ ) = φ ′ i . Hence, we can assume that r = 1 so W is irreducible. Assume W is of type A n with n 1 and assume φ, φ ′ ∈ Irr(W) satisfy (1 ′ ) and (2 ′ ). If neither φ nor φ ′ is the trivial character 1, then by [GP00, Cor. 5.4.8], condition (1 ′ ) implies φ = φ ′ . We can therefore assume 1 ∈ {φ, φ ′ }. As φ and φ ′ have the same restriction to the trivial subgroup we must have φ(1) = φ ′ (1) = 1 and so {φ, φ ′ } ⊆ {1, ε}, where ε is the sign character. k Φ 3 (q k−1 )Φ 6 (q k ) Φ 3 (q k )Φ 6 (q k−1 ) 1 3Φ 6 (q) Φ 3 (q) 2 Φ 3 (q)Φ 12 (q) Φ 3 (q)Φ 6 (q) 2 5 Φ 3 (q)Φ 6 (q) 2 Φ 12 (q)Φ 30 (q) Φ 3 (q)Φ 15 (q)Φ 24 (q) Assume there is a proper subgroup s W generated by a reflection. As the restriction of 1 and ε to this subgroup are different, we must have φ = φ ′ . Assume no such subgroup exists, then n = 1. The parameters for the Hecke algebra are clearly given by q k for some integer k 1. In this case the Schur elements are c 1 = Φ 2 (q k ) and c ε = q −k Φ 2 (q k ) which are different, so φ = φ ′ . If W is of type B n , C n , or D n , with n 2 then two characters φ, φ ′ ∈ Irr(W) satisfying (1 ′ ) must be equal by [GP00, Thm. 6.2.9]. Let W be of type G 2 and assume φ, φ ′ ∈ Irr(W) satisfy (1 ′ ). If these characters are a pair of distinct characters satisfying (1 ′ ) then {φ, φ ′ } = {φ 2,1 , φ 2,2 } are the two characters of dimension 2, which we distinguish by their b-invariant. According to [Lus84, Table II] and [Lus84, Thm. 8.6], the possible parameters for the Hecke algebra End G F (R G L (τ )) are given by (q, q 2k−1 ), where k ∈ {1, 2, 5}. By [GP00, Thm. 8.3.4] the Schur elements are given by, Table 1 we express Φ 3 (q k+b−2 )Φ 6 (q k−b+1 ) as a product of cyclotomic polynomials in q. We need to show that c φ2,1 = c φ2,2 . If q = 2 this is easy and if q = 2 then this follows from Zigmondy's Theorem, see [Zsi92] or [Art55, Cor. 2], which states that for n > 2 there exists a prime r | Φ n (q) such that r ∤ Φ m (q) for any 1 m < n. c φ 2,b = 2q −2k+1 Φ 3 (q k+b−2 )Φ 6 (q k−b+1 ), with b ∈ {1, 2}. In Finally if W is of type F 4 , E 6 , E 7 , or E 8 , then two characters φ, φ ′ ∈ Irr(W) satisfying (1 ′ ) are equal unless {φ, φ ′ } is an exceptional family, see [GP00, 6.3.6]. Let T ⊂ L be an F -stable maximal torus and θ : T F → Q × ℓ be a character. Let L * be a F * -stable Levi subgroup of G * corresponding to a Levi subgroup L of G. Note that L * is determined up to G * F * -conjugacy. Let (T * , s) be a pair corresponding to the pair (T, θ) as in Fact 2.1 such that T * ⊂ L * . Lemma 3.6. Let [(T, θ)] denote the L F -conjugacy class of (T, θ) and [(T * , s)] denote the L * F * -conjugacy class of (T * , s). Then the isomorphism, δ : W G (T 0 ) → W G * (T * 0 ), induces an isomorphism, Stab WG(A) [(T, θ)] ∼ = Stab W G * (A * ) [(T * , s)] . Proof. By [HL80, (2.1) Proposition], W G (A) ∼ = N WG(T0) F (W L (T 0 ) F )/W L (T 0 ) F . Therefore, W G (A) ∼ = W G * (A * ), and hence the result follows. Jordan decomposition and Endomorphism algebra This section is divided into two parts. In the first part we study the F -stable Levi subgroups with connected centre and in the second part there is no condition on centre. 4.1. Levi with connected centre. Let L be an F -stable Levi factor of an F -stable parabolic subgroup of G such that Z(L) is connected. Let T ⊂ L be an F -maximal torus and θ : T F → Q × ℓ be a character. Let w ∈ W G (A), then w induces an automorphism σ w : L F → L F . This induces an action of w on Irr(L F ) as τ ′ → τ ′ • σ −1 w . Then, σw R L T (θ) = R L T (θ) • σ −1 w = R L σw (T ) (θ • σ −1 w ). Let τ ′ ∈ E(L F , s) then, 0 = τ ′ , R L T (θ) = τ ′ • σ −1 w , R L T (θ) • σ −1 w = τ ′ • σ −1 w , R L σw(T ) (θ • σ −1 w ) . This implies that σ w maps the L F -conjugacy class of a pair (T, θ) to the L F -conjugacy class of a pair (σ w (T ), θ • σ −1 w ). Let w * be the corresponding element in W G * (A * ) and let (T * , s) be a pair corresponding to the pair (T, θ) as in Fact 2.1. Then σ w * maps the L * F -conjugacy class of a pair (T * , s) to the L * F -conjugacy class of a pair (σ w * (T * ), σ w * (s)). Therefore, σ w induces a bijection, (1) Let τ ′ ∈ E(L F , s). We have, f w : E(L F , s) → E(L F , σ w * (s)), τ ′ → τ ′ • σ −1 w . Similarly, σ w * induces a bijection, f w * : Uch(C L * (s)) → Uch(C L * (σ w * (s))), u τ ′ → u τ ′ • σ −1 w * .R HL T * (1 T * ), J L s (τ ′ ) HL = R HL T * (1 T * ), J σ w * (s) ( σw τ ′ ) • σ w * HL = R HL T * (1 T * ) • σ −1 w * , J σ w * (s) ( σw τ ′ ) C L * (σ w * (s)) = R H σ w * (s) σ w * (T * ) (1 T * ), J σ w * (s) ( σw τ ′ ) C L * (σ w * (s)) = ǫ L ǫ Hσ w * R L σ w * (T * ) (σ w * (s)), σw τ ′ ) L = ǫ L ǫ HL R L T * (s), τ ′ ) L . Thus, R L T * (s), τ ′ ) L = ǫ L ǫ HL R HL T * (1 T * ), J L s (τ ′ ) HL , for any τ ′ ∈ E(L F , s). (2) Since, σ w and σ w * are commute with Frobenius morphism, so the Frobenius eigenvalues corresponding to τ (resp. J 1 (τ )) and τ • σ −1 w (resp. J 1 (τ ) • σ w * ) are equal. (Since σ w (resp. σ w * ) induces an isomorphism between the cohomology groups associated with Deligne-Lusztig varieties which commutes with the action of Frobenius on cohomology). By Theorem 2.6, Frobenius eigenvalues corresponding to τ • σ −1 w and J 1 (τ • σ −1 w ) are equal. Hence, the Frobenius eigenvalues corresponding to τ and J 1 (τ ) = J 1 (τ • σ −1 w ) • σ w * are equal. This proves the first part of (2). For the second part of (2), we assume τ is in the principal series, meaning that it is a constituent of Ind L F B F (1), and so J 1 (τ ) is also in the principal series and they correspond to the same character χ of the Hecke algebra H(L F , B F ) (which is identified with H(L * F * , B * F * ) via the natural isomorphism δ : W L (T 0 ) → W L * (T * 0 ) of Weyl groups). If τ (resp. J 1 (τ )) is in the principal series, then so is τ • σ −1 w (resp. J 1 (τ ) • σ w * ). By Theorem 2.6, τ • σ −1 w and J 1 (τ • σ −1 w ) correspond to same character of the Iwahori-Hecke algebra. Note that σ w induces an isomorphism between the Hecke algebras H(L F , B F ) and H(L F , σ w (B) F ). Under this isomorphism the character corresponding to τ goes to the character corresponding to τ • σ −1 w . Similarly, σ w * induces an isomorphism between the Hecke algebras H(L * F * , B * F * ) and H(L * F * , σ w * (B * F * )). Under this isomorphism the character corresponding to J 1 (τ ) goes to the character corresponding to J 1 (τ ) • σ w * . Thus τ and J 1 (τ ) = J 1 (τ • σ −1 w ) • σ w * corresponds to the same character of the Iwahori-Hecke algebra H(L F , B F ) (which is identified with H(L * F * , B * F * ) through the natural isomorphism δ : W L (T 0 ) → W L * (T * 0 ) of Weyl groups). (3) This part follows from the fact that, ifẑ is a character corresponding to z ∈ Z(L * F ) thenẑ • σ −1 w is a character corresponding to σ w * (z). (4) Let M * be an F -stable Levi subgroup of L * such that H L ≤ M * , with dual M ≤ G. By Theorem 2.6, we have, J M s = J L s • R L M . Note that, σ w • R L M = R L σw (M) • σ w . We have, J L s • R L M = f −1 w * • J L σ w * (s) • f w • R L M = f −1 w * • J L σ w * (s) • R L σw(M) • f w = f −1 w * • J σ w * (M) σ w * (s) • f w = J M s . This implies the following diagram commutes: E(L F , s) Uch(H F s ) E(M F , s) Uch(H F s ) J L s R L M J M s Id (5) Suppose, L is of type E 8 and H L is of type E 7 A 1 (resp. E 6 A 2 ) and M ≤ L is a Levi subgroup of type E 7 (resp. E 6 ) with dual M * ≤ H L . By Theorem 2.6, J L s • R L M (τ ′ ) = R HL M * • J M s (τ ′ ), for any τ ′ ∈ ZE(M F , s) c , where the index c denotes the subspace spanned by the cuspidal part of the corresponding Lusztig series. We have, J L s • R L M (τ ′ ) = f −1 w * • J L σ w * (s) • f w • R L M (τ ′ ) = f −1 w * • J L σ w * (s) • R L σw (M) f w (τ ′ ) = f −1 w * • R HL σ w * (M * ) • J σw (M) σ w * (s) f w (τ ′ ) = R HL M * • f −1 w * • J σw (M) σ w * (s) • f w (τ ′ ) = R HL M * • J M s (τ ′ ) . This implies the following diagram commutes: ZE(L F , s) Z Uch(H F * L ) ZE(M F , s) c Z Uch(M * F ) c . J L s R L M J M s R H L M * (6) Let T 1 ≤ Z(L) be any F -stable central torus and let π 1 : L → L 1 := L/T 1 be the natural epimorphism. Let T 2 = σ w (T 1 ) be the image of T 1 under σ w , so T 2 ≤ Z(L) is also an F -stable central torus in L. Let π 2 : L → L 2 := L/T 2 be the corresponding natural epimorphism. Then σ −1 w induces a natural morphism, f : Irr(L F 1 ) → Irr(L F 2 ). For s i ∈ L * i with s = π * 1 (s 1 ) and σ w * (s) = π * 2 (s 2 ), by Theorem 2.6 the following diagrams commutes: E(L F , s) Uch(H F * L ) E(L F 1 , s 1 ) Uch(H F * L1 ) J L s ψ1 φ1 J L 1 s 1 E(L F , σ w * (s)) Uch(H F * σ w * ) E(L F 2 , s 2 ) Uch(H F * L2 ), J L σ w * (s) ψ2 φ2 J L 2 s 2 where H i = C L * i (s i ) for i = 1, 2 and where the vertical maps are just the inflation map along L F → L F 1 (resp.L F → L F 2 ) and the restriction along the embedding H F * L1 → H F * L (resp. H F * L2 → H F * L ) respectively. There is a natural bijection f * : Uch(H F * L1 ) → Uch(H F * L2 ), induced by σ w * . Then it follows from the definition of f w and f that, φ 2 • f = f w • φ 1 and f * −1 • ψ 2 = ψ 1 • f −1 w * . We have, ψ 1 • J s • φ 1 = ψ 1 • f −1 w * • J L σ w * (s) • f w • φ 1 = f * −1 • ψ 2 • J L σ w * (s) • φ 2 • f = f * −1 • J L2 s2 • f = J L1 s1 . This implies the following diagram commutes: E(L F , s) Uch(H F * L ) E(L F 1 , s 1 ) Uch(H F * L1 ). J L s ψ1 φ1 J L 1 s 1 (7) For the final property (7), suppose s = i s i . consider, J L s = f −1 w * • J L σ w * (s) • f w = f −1 w * • i J σw (Li) σ w * (si) • f w = i f −1 w * • J σw (Li) σ w * (si) • f w = i J Li si . Hence (7) follows. Thus, the new bijection J L s also satisfies the properties satisfied by J s . By Theorem 2.6, we get J L s = J L s . This proves the Proposition. Proposition 4.2. Let G be a connected reductive group over F q . Let L be an F -stable Levi factor of an Fstable parabolic subgroup of G such that Z(L) is connected. Suppose τ is a cuspidal irreducible representation of L F . Then there is a canonical isomorphism, W τ ∼ = W J L s (τ ) , where, W J L s (τ ) = {w ∈ W H (A * ) := N H F * (A * )/H F * L : χ J L s (τ ) • w = χ J L s (τ ) }. (J L s = f −1 w * • J L σ w * (s) • f w = J L s i.e. f w * • J L s = J L σ w * (s) • f w = J L s • f w . This implies that w * ∈ W J L s (τ ) . Similarly one can show that if δ(w) = w * ∈ W J L s (τ ) , then w ∈ W τ . Thus δ induces an isomorphism between W τ and W J L s (τ ) . This proves the theorem. Theorem 4.3. Let G be a connected reductive group over F q with connected centre. Let L be an F -stable Levi factor of an F -stable parabolic subgroup of G and let τ ∈ E(L F , s) be a cuspidal irreducible representation of L F . Then there is a canonical isomorphism, End G F (R G L (τ )) ∼ = End H F * (R H HL (J L s (τ ))) , where J L s is the unique Jordan decomposition as in Theorem 2.6. Proof. Theorem follows from Prop. 4.2 and Theorem 3.3. 4.2. General case. Let L be an F -stable Levi subgroup of G and let Z(G) be the centre of G and S ⊆ L ⊂ G be an F -stable torus such that Z(G) ⊆ Z(L) ⊆ S. (For example, one could take any F -stable maximal torus of L.) Let G ′ be the quotient of G × S by the closed normal subgroup {(z, z −1 ) : z ∈ Z(G)}. Let S ′ be the image of {1} × S ⊆ G × S in G ′ . Then the map i : G → G ′ induced by G → G × S, g → (g, 1) is a regular embedding and S ′ is the centre of G ′ . Let L ′ = L · S ′ be an F -stable Levi subgroup G ′ , then the restriction of i to L defines a regular embedding i L : L → L ′ of L. Thus the following diagram commutes: L G L ′ G ′ .W τ ∼ = W J L s (τ ) , where W J L s (τ ) = {w * ∈ W H (A * ) : σ w * (J L s (τ )) = J L s (τ )}. (In this case the group H = C G * (s) can be disconnected). Proof. Let τ ∈ E(L F , s), then W τ is a subgroup of Stab W (A) ([(T, θ)]) . Let w ∈ W τ , then w acts on E(L F , s) via σ w (same as in the connected centre case). Let w * = δ(w) be the corresponding element in W H (A * ). By construction of L ′ ⊂ G ′ , w also normalizes L ′ . Hence w induces a map, w : E(L F , s)/ ∼ (L ′ F /L F ) → E(L F , s)/ ∼ (L ′ F /L F ) as [τ ] → [τ • σ −1 w ] . Let i : L → L ′ be a regular embedding and i * : L ′ * → L * be the corresponding surjective morphism with kernel K. Let s ′ ∈ L ′ * such that i * (s ′ ) = s. If w * centralizes s, then w * also centralizes s ′ . Then w acts on E(L ′F , s ′ k)/ ∼ (K F * ) and E(L ′F , s ′ )/ ∼ (K F * s ′ ) as [τ ] → [τ • σ −1 w ]. Similarly, w * acts on Uch(H F * L ′ )/ ∼ (H F * L /(H L ) F * • ) and Uch((H L ) F * • )/ ∼ (H F * L /(H L ) F * • ) as [τ ] → [τ • σ −1 w * ]. To prove the theorem it enough to show that each square in the following diagram commutes: The first and last squares commute by definition. Note that Ind L ′F L F (τ • σ −1 w ) = Ind L ′F L F (τ ) • σ −1 w . Then by Fact 2.14, the second square commutes. Note that the induced action of w on L ′F /L F under the isomorphism transfers to the action of w * on K F * . Then commutation of the third square follows from definition of the maps. The vertical map d is the bijection J L ′ s ′ in the case of Jordan decomposition of characters for connected reductive groups with connected centre (cf. Lemma 2.11). By Prop. 4.1, J L ′ s ′ commute with the action of w and w * respectively. Hence the square involving d commutes. E(L F , s) E(L F , s) E(L F , s)/ ∼ (L ′ F /L F ) E(L F , s)/ ∼ (L ′ F /L F ) k∈K F * E(L ′F , s ′ k)/ ∼ K F * ∼ = (L ′ F /L F ) k∈K F * E(L ′F , s ′ k)/ ∼ K F * ∼ = (L ′ F /L F ) E(L ′F , s ′ )/ ∼ K F * s ′ = (H F * L /(H L ) F * • ) E(L ′F , s ′ )/ ∼ K F * s ′ = (H F * L /(H L ) F * • ) Uch(H F * L ′ )/ ∼ (H F * L /(H L ) F * • ) Uch(H F * L ′ )/ ∼ (H F * L /(H L ) F * • ) Uch((H L ) F * • )/ ∼ (H F * L /(H L ) F * • ) Uch(H L ) F * • )/ ∼ (H F * L /(H L ) F * • ). This implies that the surjective map in Theorem 2.15 commutes with the corresponding action of w and w * . This implies that if w ∈ W τ , then w * ∈ W J L s (τ ) . Similarly if w * = δ(w) ∈ W J L s (τ ) then w ∈ W τ . This proves the result. As a corollary, we obtain: Corollary 4.5. Let G be a connected reductive group over F q . Let L be an F -stable Levi factor of an F -stable parabolic subgroup of G and let τ, τ ′ ∈ E(L F , s) be two cuspidal irreducible representations of L F such that J L s (τ ) = J L s (τ ′ ). Then, End G F (R G L (τ )) ∼ = End G F (R G L (τ ′ )). Proof. This follows from Theorem 4.4 and Theorem 3.3. Recall that given a torus T in an algebraic group G, we denote by W G (T ), the group N G (T )/C G (T ). When T is a maximal torus, we abbreviate it by W G . Proposition 4.6. Let τ ∈ E(L F , s) be a cuspidal irreducible representation and let u τ ∈ Uch((H L ) F * • ) be an element of J L s (τ ). Then W τ is isomorphic to a semidirect product of W uτ with a finite group Γ. Proof. We have a commutative diagram: 1 / / W H• / / W H / / W H /W H• / / v v 1 1 / / W (HL)• / / ? O O W HL / / ? O O W HL /W (HL)• / / ? O O v v 1 . Here the splitting of the two short exact sequences follows from [ABPS17, Lemma 3.1]. From this we get a commutative diagram: 1 / / N W F * H• (W F * (HL)• ) / / N W F * H (W F * HL ) / / N W F * H (W F * HL )/N W F * H• (W F * (HL)• ) / / r r 1 1 / / W F * (HL)• / / ? O O W F * HL / / ? O O W F * HL /W F * (HL)• / / ? O O t t 1 . It follows that we have a split short exact sequence: 1 / / W H• (A * ) / / W H (A * ) / / W H (A * )/W H• (A * ) / / 1 . The splitting above induces a splitting of the following short exact sequence: 1 → Stab WH • (A * ) (J L s (τ )) → Stab WH (A) (J L s (τ )) → Stab WH (A * ) (J L s (τ ))/ Stab WH • (A * ) (J L s (τ )) → 1. Note that W uτ ∼ = Stab WH • (A * ) (J L s (τ )) and by Theorem 4.4, Stab WH (A) (J L s (τ )) ∼ = W τ . Thus, 1 → W uτ → W τ → W τ /W uτ → 1 is a split exact sequence. This proves the result. Remark 4.7. Note that as a part of the proof of above Proposition with Γ := W τ /W uτ , we have shown that the finite group Γ does not depend on the choice of an element u τ ∈ J L s (τ ). Corollary 4.8. Let G be a connected reductive group over F q and let, L be an F -stable Levi factor of an F -stable parabolic subgroup of G. Let τ ∈ E(L F , s) be a cuspidal irreducible representation of L F and let u τ ∈ Uch((H L ) F * • ) be any element of J L s (τ ). Then, End G F (R G L (τ )) ∼ = End H F * • (R H• (HL)• (u τ )) ⊗Q ℓ [Γ] , where the product structure on the right hand side comes from the semi-direct product structure of W τ . Harish-Chandra induction and Jordan decomposition In this section, we will prove that there exists a Jordan decomposition of characters which commutes with the Harish-Chandra induction. Let G be a connected reductive group defined over F q with connected centre. Let us fix a Jordan decomposition for all cuspidal representations of Levi factors of an F -stable parabolic subgroups of G (including G itself) such that for a cuspidal representation τ , W τ ∼ = W uτ where u τ is the unipotent representation corresponding to τ under the chosen Jordan decomposition. By previous results, such a Jordan decomposition exists. By assumption, we have W τ ∼ = W uτ , and hence , End G F (R G L (τ )) ∼ = End H F * (R H HL (u τ )). Thus it induces a bijection between the Harish-Chandra series above (L F , τ ) and (H F * L , u τ ). Let us denote this bijection by, J G L,τ : Irr(G F , (L F , τ )) → Irr(H F * , (H F * L , u τ )). Using Fact 3.2, we can define a bijection: J G,s : E(G F , s) → Uch(H F * ) as J G,s | Irr(G F ,(L F ,τ )) = J G L,τ . From now on u τ = J L s (τ ) where J L s is the unique Jordan decomposition as in Theorem 2.6 and J G,s is constructed as above using this collection of bijections at the cupidal levels. Lemma 5.1. Let G be a connected reductive group defined over F q with connected centre. Let J G,s be the bijection as above. Then for all semisimple element s ∈ G * F * , J G,s is a Jordan decomposition and it agrees with the unique Jordan decomposition J G s for all irreducible characters except possibly for the exceptions appearing in Proposition 3.5. Proof. Suppose the result holds for all groups whose rank is less than rank(G). By the standard reduction argument it is enough to prove the Lemma for the case when G/Z(G) is simple. Let M be a Levi factor of a F -stable parabolic subgroup of G. By [GM20, Theorem 4.7.5], the unique Jordan decomposition commutes with Harish-Chandra induction. This implies, J M s • * R G M = * R H HM • J G s . Let ρ ∈ Irr(G F , (L F , τ )) ⊆ E(G F , s), then by induction hypothesis, we get, J M s • * R G M (ρ) = J M,s • * R G M (ρ) = * R H HM • J G s (ρ). Also by Comparison Theorem 3.4, J M,s • * R G M (ρ) = * R H HM • J G,s (ρ) . Hence we have, * R H HM (J G s (ρ)) = * R H HM (J G,s (ρ)), for any all L ≤ M < G. By degree formula, [GM20, Theorem 3.2.18] and [GM20, Corollary 2.6.6], we have J G s (ρ)(1) = J G,s (ρ)(1). Then by Proposition 3.5, we get J G s (ρ) = J G,s (ρ) except possibly for the exceptions appearing in Proposition 3.5. In those few remaining cases, J G s (ρ) and J G,s (ρ) belong to a pair of characters of same degree (512 in case of E 7 and 4096 in case of E 8 ). Both the characters in a given such pair have the same multiplicity in Deligne-Lusztig virtual characters. Therefore, J G,s is a Jordan decomposition. Proposition 5.2. Let G be a connected reductive group over F q with connected centre. Let s ∈ G * F * be any semisimple element and L * ≤ G * be an F * -stable Levi factor of an F * -stable parabolic subgroup of G * containing s with dual L. Let J G,s : E(G F , s) → Uch(H F * ) be the Jordan decomposition as above. Then the following diagram commutes: General Case. Let us recall the notation from §2. Let G be a connected reductive group over F q with a Frobenius morphism F : G → G. Let i : G → G ′ be a regular embedding and corresponding surjective morphism i * : G ′ * → G * with kernel K ⊆ Z(G ′ * ). There is a natural isomorphism K F * ∼ = − → Hom(G ′F /G F , Q × ℓ ), k → θ k . Let s ∈ G * F * be a semisimple element and s ′ ∈ G ′ * F * such that i * (s ′ ) = s. The morphism i * induces a surjective morphism i * | H ′ : H ′ → H • and there is a natural bijection, ZE(G F , s) Z Uch(H F * ) ZE(M F , s) Z Uch(H F M ),Uch(H ′F * ) → Uch(H F * • ), ρ ′ → ρ ′ • i * | H ′F * . There is an action of G ′F /G F on Irr(G F ) via the natural morphism, G ′F /G F → G F ad /π(G F ). Hence, E(G F , s)/ ∼ G ′F /G F = Σ G ′F (s) Irr(G F , (L F , τ ))/ ∼ Stab G ′F (L F , τ ), where Σ G ′F (s) denotes the set of G ′F -conjugacy classes of a cuspidal pair (L F , τ ), with τ ∈ E(L F , s) for a semisimple element s ∈ L * F * . Also, there is an action of H F * /H F * • on Uch(H F * • ) by conjugation and, Uch(H F * • )/ ∼ H F * /H F * • =    Σ H F * • (1) Irr(H F * • , ((H L ) F * • , u τ ))    / ∼ H F * /H F * • = Σ H F * (1) Irr(H F * • , ((H L ) F * • , u τ ))/ ∼ Stab H F * /H F * • ((H L ) F * • , u τ )) , where the last union is taken over the H F * -conjugacy classes of cuspidal pair ((H L ) F * • , u τ ), with u τ ∈ Uch((H L ) F * • ). The regular embedding i of G induces a regular embedding i L : L → L ′ of L. Note that, there is a natural inclusion L ′F /L F → G ′F /G F and a corresponding surjective morphism Res : Hom(G ′F /G F , Q × ℓ ) → Hom(L ′F /L F , Q × ℓ ). Let τ ∈ E(L F , s) be an irreducible cuspidal representation of L F and let τ ′ ∈ E(L ′F , s ′ ) be a cuspidal representation of L ′F such that L ′F /L F -orbit of τ corresponds to L ′F /L F -orbit of τ ′ . The action of K F * on Irr(G ′F ) permutes the Harish-Chandra series. Let k ∈ K F * be any element then the action of k (by tensoring θ k ) maps Irr(G ′F , (L ′F , τ ′ )) maps to Irr(G ′F , (L ′F , τ ′ ⊗ θ k )) by ρ → ρ ⊗ θ k . Let us define K F * s ′ ,τ ⊆ K F * s ′ to be the set of all k ∈ K F * which map Irr(G ′F , (L ′F , τ ′ )) into itself or equivalently, K F * s ′ ,τ = {k ∈ K F * : ks ′ is conjugate to s ′ under G ′ * F * and τ = τ ⊗ θ k }, i.e., K F * s ′ ,τ = {k ∈ K F * : ks ′ is conjugate to s ′ under L ′ * F * and τ = τ ⊗ θ k }. Then the isomorphism between H F * /H F * • ∼ = K F * s ′ induces an isomorphism, Stab (H F * /H F * • ) (H F * L , u τ ) ∼ = Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) ∼ = K F * s ′ ,τ . Thus, using this isomorphism, the action of K F * s ′ ,τ on Irr(G ′F , (L ′F , τ ′ )) becomes an action of Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) on Irr(G ′F , (L ′F , τ ′ )). Lemma 5.3. The bijection J G ′ L ′ ,τ ′ : Irr(G ′F , (L ′F , τ ′ )) → Irr(H ′F * , (H ′F * L , u τ ′ )) is compatible with the action of Stab (H F * /H F * • ) (H ′F * L , u τ ′ ), i.e., we have an induced bijection, Irr(G F , (L F , τ ))/ ∼ Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) → Irr(H F * , (H F * L , u τ ))/ ∼ Stab (H F * /H F * • ) (H ′F * L , u τ ′ ). Proof. Note that the map J G ′ L ′ ,τ as in Lemma 5.1 is defined as, Irr(G ′F , (L ′F , τ ′ )) ←→ Irr(W τ ′ ) = Irr(W u τ ′ ) ←→ Irr(H ′F * , (H ′F * L , u τ ′ )). We have an isomorphism between Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) and K F * s ′ ,τ . Therefore we can define J G ′ L ′ ,τ which is compatible with this action. Let us define a map, Then using the map J G L,τ and the Jordan decomposition for cuspidal representations, we can define a surjective map, J G,s : E(G F , s) → Uch(H F * • )/ ∼ (H F * /H F * • ). Fix a Jordan decomposition for all cuspidal representations of Levi factor of an F -stable parabolic subgroups of G (including G itself) induced by the unique Jordan decomposition in the case of connected centre. By Proposition 4.6, for a cuspidal representation τ , W τ is isomorphic to semidirect product of W uτ and a finite group Γ, where u τ ∈ J s (τ ). Thus we have a surjective map from Irr(W τ ) to Irr(W uτ )/ ∼ Γ. Also, one can observe that Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) is isomorphic to Γ. Then the map J G L,τ is induced by the surjective map from Irr(W τ ) to Irr(W uτ )/ ∼ Γ. R G T * (s), ρ G = ǫ G ǫ H• λ∈O R H T * (1 T * ), λ H• . Proof. The map J G,s is a composition of various J G L,τ and a map E(G F , s) → E(G F , s)/ ∼ G ′F /G F . Note that the action of G ′ on E(G F , s) factors through G F ad /G F . Therefore the fibres of J G,s are the orbits of G F ad /G F . For ρ ∈ Irr(G F , (L F , τ )) ⊆ E(G F , s), we have R G T * (s) = Res G ′F G F (R G ′ T ′ * (s ′ )) (where T ′ * = i * −1 (T * )) and, Ind G ′F G F (ρ) = ρ ′ ∈O ′ ρ ′ + Fact 2 . 214 ([Lus88, §9]). Let N be a normal subgroup of a finite group H such that H/N is abelian. The conjugation action of H on N induces a natural action of the abelian group H/N on Irr(N ). Also, the abelian group H/N acts on Irr(H) by tensor product, where H/N is the Pontryagin dual of H/N . Suppose, any ρ ′ ∈ Irr(H) restricts to a multiplicity free representation of N . Then there is a unique bijection: Irr(N )/ ∼ (H/N ) ←→ Irr(H)/ ∼ ( H/N ), with the following properties. Let O be a H/N -orbit on Irr(N ) and let O ′ be the corresponding H/N -orbit on Irr(H). If ρ ′ • ∈ O ′ and ρ • ∈ O then, Theorem 2.15 ([Lus88, Proposition 5.1 ]). For any semisimple element s ∈ G * F * , there exists a surjective map, Fact 3 . 2 . 32Lusztig series are unions of Harish-Chandra series. Specifically, Howlett-Lehrer Comparison Theorem)]). Let (L F , τ ) be a cuspidal pair in G F . Then the collection of bijections, an exceptional family of characters of dimension 512 or 4096 as in [GP00, 6.3.6]. Lemma 3. 7 . 7Let s ∈ T * F * be a semisimple element of L * F * . Let H be the centralizer of s in G * F * and H L be the centralizer of s in L * . Then,Stab W G * (A * ) [(T * , s)] ∼ = Stab WH (A * ) [(T * , 1)] , where W H (A * ) = N H F * (A * )/H F * L . Proof.Since A * is the maximally split subtorus of Z(L * ) • , by Theorem 2.16, A * is also the maximally split subtorus of Z((H L ) • ). Also we have C G * (A * ) = L * and hence, C H (A * ) = H L . Then the result follows from definition. Let H L := C L * (s), the centralizer of s in L * and let {J L s } be the collection of unique bijections as in Theorem 2.6. Define a new bijection, J L s : E(L F , s) → Uch(H F * L ), as J L s = f −1 w * • J L σ w * (s) • f w . We have the following result. The proof of this result is quite similar to the proof of [AM22, Theorem 3.2]. Proposition 4 . 1 . 41Let G be a connected reductive group over F q . Let L be an F -stable Levi factor of anF -stable parabolic subgroup of G such that Z(L) is connected. Suppose w ∈ W G (A) is any element. Then, J L s = J L s . Proof.We will prove the proposition by showing that the new bijection J L s satisfies the properties (1) − (7) as in Theorem 2.6. In this case the group H can be disconnected.) Proof. Suppose, τ ∈ E(L F , s) and let w ∈ W τ be any element. Then w ∈ Stab W (A) [(T, θ)] . Let w * ∈ Stab WH (A * ) [(T * , 1)] be the corresponding element. By Prop. 4.1, we have, Let T ⊂ L be an F -stable maximal torus then T ′ = T · S ′ is an F -stable maximal torus of L ′ . Let {J L s := J L,L ′ s } be the collection of surjections as in Theorem 2.15. The proof of the next result is analogous to the proof of [AM22, Theorem 3.6]. Theorem 4 . 4 . 44Let G be a connected reductive group over F q with Frobenius morphism F : G → G. Let L be an F -stable Levi factor of an F -stable parabolic subgroup of G. Suppose τ ∈ E(L F , s) is a cuspidal irreducible representation of L F . Then there is a canonical isomorphism, L * ≤ M * ≤ G * are F * -stable Levi subgroups of F * -stable parabolic subgroups of G * .Proof. Proposition follows from Theorem 3.4 and definition of bijections J G,s and J M,s . J G L,τ : Irr(G F , (L F , τ ))/ ∼ Stab G ′F (L F , τ ) → Irr(H F * • , (H L ) F * • , u τ ) / ∼ Stab H F * /H F * • ((H L ) F * • , u τ )),as the composition,Irr(G F , (L F , τ ))/ ∼ Stab G ′F (L F , τ ) θ∈ L ′F /L F Irr(G ′F , (L ′F , τ ′ ⊗ θ))/ ∼ (G ′F /G F ) = K F * Irr(G ′F , (L ′F , τ ′ ))/ ∼ K F * s ′ ,τ ∼ = Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) Irr(H ′F * , (H ′F * L , u τ ′ ))/ ∼ Stab (H F * /H F * • ) (H ′F * L , u τ ′ ) Irr(H F * • , ((H L ) F * • , u τ ))/ ∼ Stab H F * /H F * • ((H L ) F * • , u τ )). Proposition 5 . 4 . 54Let G be a connected reductive group defined over F q . Then the surjective map,J G,s : E(G F , s) → Uch(H F * • )/ ∼ (H F * /H F * • ),satisfies the following properties:(1) The fibres of J G,s are precisely the orbits of the action of G F ad /G F on E(G F , s).(2) If O is an H F * /H F * • -orbit on Uch(H F * • ) and Γ ≤ H F * /H F * • is the stabilizer of an element in O , then the fibre J −1 G,s (O) has precisely |Γ| elements. (3) If ρ ∈ J −1 G,s (O) and T * is an F * -stable maximal torus of G * containing s, then representations outside of Irr(G ′F , (L ′F , τ ′ )), where O ′ ⊆ Irr(G ′F , (L ′F , τ ′ )) denotes the Stab (H F * /H F * • ) (H ′F * L , u τ ′ )-orbit determined by ρ. Suppose O ′ corresponds to the Stab (H F * /H F * • ) (H ′F * L , u τ ′ )-orbit O ′ 1 ⊆ Irr(H ′F * , (H ′F * L , u τ ′ )) and it corresponds to the Let s ∈ G * F * be semisimple. Let ρ ∈ E(G F , s) and let u ρ ∈ J s (ρ) be in the H F * -orbit of unipotent characters of H F * • corresponding to ρ under Jordan decomposition. Then ρ is cuspidal if and only if,Fact 2.12 Fact 2.14 Fact 2.9 Lemma 2.11 Fact 2.10 Theorem 2.16 ([GM20, Theorem 3.2.22]). Table 1 . 1Factorisations of cyclotomic polynomials for G 2 . AcknowledgementThe authors are very thankful to Gunter Malle for carefully going through the whole article and suggesting numerous corrections and improvements. They are also especially thankful to Jay Taylor for finding a serious gap in an argument in a previous draft of this paper and providing us with the statement and proof of Proposition 3.5.The author would also like to thank George Lusztig for correcting us about some attributions. It is a pleasure to thank Jeff Adler and Dipendra Prasad who suggested many improvements to us.The first named author was supported by IISER postdoctoral fellowship. He thanks the institute for providing a congenial working environment.This proves the result.The next result answers the question about commutation of Jordan decomposition with Harish-Chandra Induction:Theorem 5.5. Let G be a connected reductive group over F q with corresponding Frobenius morphism F : G → G. Let s ∈ G * F * be any semisimple element and L * ≤ G * be an F * -stable Levi factor of an F * -stable parabolic subgroup of G * containing s with dual L. Let J G,s :be the Jordan decomposition as above. Then the following diagram commutes: The principal series of p-adic groups with disconnected center. Anne-Marie Aubert, Paul Baum, Roger Plymen, Maarten Solleveld, Proc. Lond. Math. Soc. 1143Anne-Marie Aubert, Paul Baum, Roger Plymen, and Maarten Solleveld. The principal series of p-adic groups with disconnected center. Proc. Lond. Math. Soc. (3), 114(5):798-854, 2017. Duality Involution and Jordan decomposition of characters. Prashant Arote, Manish Mishra, arXiv:2211.113952022Prashant Arote and Manish Mishra. Duality Involution and Jordan decomposition of characters, 2022. arXiv:2211.11395. The orders of the linear groups. Emil Artin, Comm. Pure Appl. Math. 8Emil Artin. The orders of the linear groups. Comm. Pure Appl. Math., 8:355-365, 1955. Sur les caractères des groupes réductifs finisà centre non connexe : applications aux groupes spéciaux linéaires et unitaires. Number 306 in Astérisque. Société mathématique de France. Cédric Bonnafé, Cédric Bonnafé. Sur les caractères des groupes réductifs finisà centre non connexe : applications aux groupes spéciaux linéaires et unitaires. Number 306 in Astérisque. Société mathématique de France, 2006. On jordan decomposition of characters for su(n,q). Marc Cabanes, Journal of Algebra. 374Marc Cabanes. On jordan decomposition of characters for su(n,q). Journal of Algebra, 374:216-230, 2013. Representations of reductive groups over finite fields. Pierre Deligne, George Lusztig, Annals of Mathematics. 1031Pierre Deligne and George Lusztig. Representations of reductive groups over finite fields. Annals of Mathematics, 103(1):103-161, 1976. On Lusztig's parametrization of characters of finite groups of Lie type. François Digne, Jean Michel, Astérisque. 6François Digne and Jean Michel. On Lusztig's parametrization of characters of finite groups of Lie type. Astérisque, (181-182):6, 113-156, 1990. Representations of finite groups of Lie type. François Digne, Jean Michel, Cambridge University Press95CambridgeSecond edition of [ 1118841François Digne and Jean Michel. Representations of finite groups of Lie type, volume 95 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2020. Second edition of [ 1118841]. Towards a jordan decomposition of blocks of finite reductive groups. Michel E Enguehard, Michel E. Enguehard. Towards a jordan decomposition of blocks of finite reductive groups. 2013. The blocks of finite classical groups. Paul Fong, Bhama Srinivasan, J. Reine Angew. Math. 396Paul Fong and Bhama Srinivasan. The blocks of finite classical groups. J. Reine Angew. Math., 396:122-191, 1989. A note on Harish-Chandra induction. Meinolf Geck, Manuscripta Math. 804Meinolf Geck. A note on Harish-Chandra induction. Manuscripta Math., 80(4):393-401, 1993. The character theory of finite groups of Lie type. Meinolf Geck, Gunter Malle, Cambridge Studies in Advanced Mathematics. 187Cambridge University PressA guided tourMeinolf Geck and Gunter Malle. The character theory of finite groups of Lie type, volume 187 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020. A guided tour. Characters of finite Coxeter groups and Iwahori-Hecke algebras. Meinolf Geck, Götz Pfeiffer, Oxford University Press21New YorkMeinolf Geck and Götz Pfeiffer. Characters of finite Coxeter groups and Iwahori-Hecke algebras, volume 21 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 2000. Induced cuspidal representations and generalised Hecke rings. B Robert, Gustav I Howlett, Lehrer, Invent. Math. 581Robert B. Howlett and Gustav I. Lehrer. Induced cuspidal representations and generalised Hecke rings. Invent. Math., 58(1):37-64, 1980. Characters of reductive groups over a finite field. George Lusztig, Annals of Mathematics Studies. 107Princeton University PressGeorge Lusztig. Characters of reductive groups over a finite field, volume 107 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1984. On the representations of reductive groups with disconnected centre. George Lusztig, Orbites unipotentes et représentations, I. 168George Lusztig. On the representations of reductive groups with disconnected centre. Number 168, pages 10, 157-166. 1988. Orbites unipotentes et représentations, I. Email address: [email protected] Email address: [email protected]. Karl Zsigmondy, Department of Mathematics. IndiaPashan3Indian Institute for Science Education and ResearchDr. Homi Bhabha Road. Pune 411 008Karl Zsigmondy. Zur Theorie der Potenzreste. Monatsh. Math. Phys., 3(1):265-284, 1892. Email address: [email protected] Email address: [email protected] Department of Mathematics, Indian Institute for Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411 008, India
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[ "The Dissipation of the Solar Nebula Constrained by Impacts and Core Cooling in Planetesimals", "The Dissipation of the Solar Nebula Constrained by Impacts and Core Cooling in Planetesimals" ]
[ "Alison C Hunt [email protected] \nInstitute of Geochemistry and Petrology\nETH Zürich\nClausiusstrasse 258092ZürichSwitzerland\n", "Karen J Theis \nSchool of Earth and Environmental Sciences\nAlan Turing Building\nThe University of Manchester\nM13 9PLManchesterUK\n\nThe University of Manchester\nM13 9PLManchesterUK\n", "Mark Rehkämper ", "Gretchen K Benedix \nSpace Science and Technology Centre\nSchool of Earth and Planetary Sciences\nCurtin University\nGPO Box U19876845PerthWAAustralia\n\nDepartment of Earth and Planetary Sciences\nWestern Australian Museum, Locked Bag 49 Welshpool DC6986Western AustraliaAustralia\n", "Rasmus Andreasen \nDepartment of Geoscience\nAarhus University\nHøegh-Guldbergs Gade 28000Aarhus CDenmark\n", "Maria Schönbächler \nInstitute of Geochemistry and Petrology\nETH Zürich\nClausiusstrasse 258092ZürichSwitzerland\n", "\nDepartment of Earth Science & Engineering\nImperial College London\nSW7 2AZLondonUK\n" ]
[ "Institute of Geochemistry and Petrology\nETH Zürich\nClausiusstrasse 258092ZürichSwitzerland", "School of Earth and Environmental Sciences\nAlan Turing Building\nThe University of Manchester\nM13 9PLManchesterUK", "The University of Manchester\nM13 9PLManchesterUK", "Space Science and Technology Centre\nSchool of Earth and Planetary Sciences\nCurtin University\nGPO Box U19876845PerthWAAustralia", "Department of Earth and Planetary Sciences\nWestern Australian Museum, Locked Bag 49 Welshpool DC6986Western AustraliaAustralia", "Department of Geoscience\nAarhus University\nHøegh-Guldbergs Gade 28000Aarhus CDenmark", "Institute of Geochemistry and Petrology\nETH Zürich\nClausiusstrasse 258092ZürichSwitzerland", "Department of Earth Science & Engineering\nImperial College London\nSW7 2AZLondonUK" ]
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Rapid cooling of planetesimal cores has been inferred for several iron meteorite parentbodies based on metallographic cooling rates, and linked to the loss of their insulating mantles during impacts. However, the timing of these disruptive events is poorly constrained. Here, we used the short-lived 107 Pd-107 Ag decay system to date rapid core cooling by determining Pd-Ag ages for iron meteorites. We show closure times for the iron meteorites equate to cooling in the timeframe ~7.8-11.7 Myr after CAI, and indicate that an energetic inner Solar System persisted at this time. This likely results from the dissipation of gas in the protoplanetary disk, after which the damping effect of gas drag ceases. An early giant planet instability between 5-14 Myr after CAI could have reinforced this effect. This correlates well with the timing of impacts recorded by the Pd-Ag system for iron meteorites.Main TextIron meteorites are thought to represent the once heated interior portions of planetesimals and are some of the earliest-formed bodies in our Solar System 1 . As such, they are survivors of the many dynamical processes that have shaped Solar System architecture, including dissipation of the protoplanetary disk, and runaway growth, migration and reorganisation of the giant planets 2,3 . Importantly, high and variable metallographic cooling rates among many iron meteorite groups, including the IIAB, IIIAB and IVAs, indicate that these metals cooled in bodies that had been stripped of their insulating silicate mantles 4,5 . Dating core crystallisation can therefore elucidate the timing of impact processes, which in turn constrains the evolution of our early Solar System. The short-lived 107 Pd-107 Ag decay system (t1/2 ~6.5 Myr) provides a tool for this. Palladium, which is highly siderophile, partitions into the metal phase during crystallisation, and chalcophile Ag is strongly sequestered into sulfides.This leads to Pd-Ag fractionation during crystallisation of a cooling core resulting in samples from a single body with variable Pd/Ag ratios, thereby allowing us to derive parent-body initial 107 Pd/ 108 Pd ratios from isochron slopes. The ages for parent-body cooling through the closure temperature of the Pd-Ag system (~820-970 K) 6 can be derived from these initials.Early studies of iron meteorites using the Pd-Ag chronometer did not fully account for the exposure of samples to galactic cosmic rays (GCR) 7-9 , which can disturb the system through secondary neutron capture reactions by Pd isotopes that lead to a decrease in 107 Ag/ 109 Ag. This occurs primarily via neutron capture on 108 Pd and subsequent βdecay to 109 Ag, and requires correction to achieve meaningful ages 10 . After GCR correction, initial 107 Pd/ 108 Pd ratios of individual meteorites from the iron meteorite groups IIAB, IIIAB and IVA imply core cooling between ~7.2 Myr and 10.7 Myr after the formation of calcium-aluminium rich inclusions (CAIs) 6,11,12 , relative to the Solar System initial (SSI) 107 Pd/ 108 Pd determined from carbonaceous chondrites 13 . Additionally, a detailed investigation of the partially-differentiated IAB irons yielded two isochrons that signified a later closure of the Pd-Ag system between ~14.9-18.7 Myr after CAI 9 . However, those data were not corrected for GCR effects. Here, we focus on parent body initial 107 Pd/ 108 Pd ratios to reassess the thermal evolution of planetesimals and constrain dynamic processes occurring in the early Solar System. We report new Pd-Ag data and use Pt isotopes to correct for the effects of GCR in the IAB, IIAB and IIIAB iron meteorites.
10.1038/s41550-022-01675-2
[ "https://export.arxiv.org/pdf/2211.08306v1.pdf" ]
249,028,038
2211.08306
f2046c800244e6dd7da2349c85d6c60413978290
The Dissipation of the Solar Nebula Constrained by Impacts and Core Cooling in Planetesimals Alison C Hunt [email protected] Institute of Geochemistry and Petrology ETH Zürich Clausiusstrasse 258092ZürichSwitzerland Karen J Theis School of Earth and Environmental Sciences Alan Turing Building The University of Manchester M13 9PLManchesterUK The University of Manchester M13 9PLManchesterUK Mark Rehkämper Gretchen K Benedix Space Science and Technology Centre School of Earth and Planetary Sciences Curtin University GPO Box U19876845PerthWAAustralia Department of Earth and Planetary Sciences Western Australian Museum, Locked Bag 49 Welshpool DC6986Western AustraliaAustralia Rasmus Andreasen Department of Geoscience Aarhus University Høegh-Guldbergs Gade 28000Aarhus CDenmark Maria Schönbächler Institute of Geochemistry and Petrology ETH Zürich Clausiusstrasse 258092ZürichSwitzerland Department of Earth Science & Engineering Imperial College London SW7 2AZLondonUK The Dissipation of the Solar Nebula Constrained by Impacts and Core Cooling in Planetesimals Accepted by Nature Astronomy, 01/04/2022Now at: The Photon Science Institute, *Correspondence and requests for materials should be addressed to: Rapid cooling of planetesimal cores has been inferred for several iron meteorite parentbodies based on metallographic cooling rates, and linked to the loss of their insulating mantles during impacts. However, the timing of these disruptive events is poorly constrained. Here, we used the short-lived 107 Pd-107 Ag decay system to date rapid core cooling by determining Pd-Ag ages for iron meteorites. We show closure times for the iron meteorites equate to cooling in the timeframe ~7.8-11.7 Myr after CAI, and indicate that an energetic inner Solar System persisted at this time. This likely results from the dissipation of gas in the protoplanetary disk, after which the damping effect of gas drag ceases. An early giant planet instability between 5-14 Myr after CAI could have reinforced this effect. This correlates well with the timing of impacts recorded by the Pd-Ag system for iron meteorites.Main TextIron meteorites are thought to represent the once heated interior portions of planetesimals and are some of the earliest-formed bodies in our Solar System 1 . As such, they are survivors of the many dynamical processes that have shaped Solar System architecture, including dissipation of the protoplanetary disk, and runaway growth, migration and reorganisation of the giant planets 2,3 . Importantly, high and variable metallographic cooling rates among many iron meteorite groups, including the IIAB, IIIAB and IVAs, indicate that these metals cooled in bodies that had been stripped of their insulating silicate mantles 4,5 . Dating core crystallisation can therefore elucidate the timing of impact processes, which in turn constrains the evolution of our early Solar System. The short-lived 107 Pd-107 Ag decay system (t1/2 ~6.5 Myr) provides a tool for this. Palladium, which is highly siderophile, partitions into the metal phase during crystallisation, and chalcophile Ag is strongly sequestered into sulfides.This leads to Pd-Ag fractionation during crystallisation of a cooling core resulting in samples from a single body with variable Pd/Ag ratios, thereby allowing us to derive parent-body initial 107 Pd/ 108 Pd ratios from isochron slopes. The ages for parent-body cooling through the closure temperature of the Pd-Ag system (~820-970 K) 6 can be derived from these initials.Early studies of iron meteorites using the Pd-Ag chronometer did not fully account for the exposure of samples to galactic cosmic rays (GCR) 7-9 , which can disturb the system through secondary neutron capture reactions by Pd isotopes that lead to a decrease in 107 Ag/ 109 Ag. This occurs primarily via neutron capture on 108 Pd and subsequent βdecay to 109 Ag, and requires correction to achieve meaningful ages 10 . After GCR correction, initial 107 Pd/ 108 Pd ratios of individual meteorites from the iron meteorite groups IIAB, IIIAB and IVA imply core cooling between ~7.2 Myr and 10.7 Myr after the formation of calcium-aluminium rich inclusions (CAIs) 6,11,12 , relative to the Solar System initial (SSI) 107 Pd/ 108 Pd determined from carbonaceous chondrites 13 . Additionally, a detailed investigation of the partially-differentiated IAB irons yielded two isochrons that signified a later closure of the Pd-Ag system between ~14.9-18.7 Myr after CAI 9 . However, those data were not corrected for GCR effects. Here, we focus on parent body initial 107 Pd/ 108 Pd ratios to reassess the thermal evolution of planetesimals and constrain dynamic processes occurring in the early Solar System. We report new Pd-Ag data and use Pt isotopes to correct for the effects of GCR in the IAB, IIAB and IIIAB iron meteorites. Pd/ 108 Pd ratios for the IABs, IIAB and IIIABs are within uncertainty of each other, indicating that the Pd-Ag system closed in a similar time-frame for these bodies (Fig. 1). To determine Pd-Ag ages, the SSI 107 Pd/ 108 Pd is needed. Two distinct initial 107 Pd/ 108 Pd ratios have been determined based on the IVA iron meteorite Muonionalusta (~3.5 x10 -5 ; 12 ) and carbonaceous chondrites (~5.9 x10 -5 ; 13 ; see Methods for discussion of SSI). To consider the full range of SSI 107 Pd/ 108 Pd ratios, we calculate Pd-Ag ages for the meteorite groups studied here relative to both (Table 3). Discussion The presence of silicate inclusions and chemical data indicate the IAB iron meteorites had a complex thermal evolution and can be divided into sub-groups. The IABs studied here are members of the main group (MG) and 'Low-Au Low-Ni' (sLL) sub-group, which may be genetically related 17 (Table 1). It has been argued that the IABs formed in multiple impact melt pools on a single parent body 17 . Alternatively, they are products of internal heating followed by a catastrophic impact and reassembly of the parent body 18 . In support of the latter, samples from IAB sub-groups yield identical 182 Hf-182 W ages indicating contemporaneous metal-silicate separation 14, 19 . After the new correction for GCR effects, Pd-Ag data for the IABs define a single isochron indicating that cooling on the parent asteroid occurred in several sub-groups simultaneously. Small pools of metal formed at different times would not be expected to cool simultaneously and hence our new isochron likely represents a single body-wide cooling event, lending further support to a model of internal heating followed by breakup of the parent body. Importantly, heating of the IAB parent asteroid produced a body which only partially differentiated at 6.0±0. 8 Myr 14 . The question remains whether the body was impacted while at or close to its peak temperature, thereby effectively halting differentiation 18 , or whether late accretion of the parent asteroid limited the availability of 26 Al and allowed only subdued internal heating 14 . Thermal models explored a scenario where the body cooled slowly until ~10-12 Myr when the catastrophic breakup occurred 14 . However, the model used Pd-Ag data not corrected for GCR effects 9 to constrain the timing of the break-up, and hence a reassessment is necessary. Our new IAB isochron corresponds to an age of 12.8 +3.1 /-4.6 or 7.9 +1.0 /-1.1 Myr after CAI (Table 3). Fast cooling rates are expected for the IAB parent body to keep silicate inclusions suspended in the metal portions 17 . Hence, a catastrophic impact should be closely followed by closure of the Pd-Ag system. Integrating this with cooling at 12.8 +3.1 /-4.6 Myr suggests a delay between the body reaching its peak temperature at ~6 Myr and the catastrophic impact (Extended Data Fig. 3). Closure ages from the 129 I- 129 Xe system for a silicate inclusion from Campo del Cielo indicate a scrambling event that occurred later than ~11 Myr after CAI 20 . This agrees well with evidence of resetting from the Hf-W system at 14.9±2.8 and 11.4±2.2 Myr, respectively, in both the genetically related winonaite meteorites and a further silicate inclusion from Campo del Cielo [21][22][23] . This resetting must be impact-related and its identification in both winonaites and IABs indicates a large-scale process that likely corresponds to the catastrophic break-up of the body. Together with the Pd-Ag system these chronometers suggest an impact event between ~11 and 13.6 Myr after CAI. Assuming that the IAB parent was impacted at this time yields the necessary fast cooling rates of between instantaneous and ~102 K/Myr (Table 3; Methods). In this scenario, the thermal model and precursor parameters determined previously remain applicable to the IAB parent body 14 . Alternatively, a cooling age of 7.9 +1.0 /-1.1 Myr (calculated relative to the SSI 107 Pd/ 108 Pd of ~3.5 x10 -5 ) implies that the parent body was impacted when at or near its peak temperature (Extended Data Fig. 3). However, this scenario does not agree with the later resetting ages of silicate inclusions discussed above [20][21][22] because the closure temperature of the Hf-W system (~1150 K 24 ) is higher than that of the Pd-Ag system (~820-970 K 6 ). Hence, a thermal event that affected the Hf-W chronometer should also reset the Pd-Ag system. This discrepancy is unlikely to represent disturbance to the Hf-W chronometer. Rapid cooling precludes significant diffusive exchange between silicate inclusions and host IAB metals 22 . Although GCR can affect Hf-W ages, this is unlikely to be the case here because Campo del Cielo irons experienced only limited GCR exposure (this study, 25 ). A model incorporating cooling at 12.8 +3.1 /-4.6 Myr is therefore most appropriate for the IAB parent body evolution. Implications for the Solar System initial 107 Pd/ 108 Pd Importantly, the observation that the Hf-W and Pd-Ag systems cannot be reconciled using a SSI 107 Pd/ 108 Pd of ~3.5 x10 -5 is robust regardless of the formation history of the IAB parent body. Campo del Cielo is a member of the IAB-MG, which crystallised, and hence cooled, as a single body 17 . An isochron using only the IAB-MG yields an initial 107 Pd/ 108 Pd of (1.61±0.23) x10 -5 , identical within uncertainty to the IAB isochron with all samples. Using a SSI 107 Pd/ 108 Pd of (3.5±0.1) x10 -5 implies closure of the Pd-Ag system for the MG at 7.3 +1.3 /-1.5 Myr after CAI. Irrespective of whether this defines metal cooling after a local impact or a body-wide catastrophic event, the irons should then not include silicates with a Hf-W closure age of 11.4±2.2 Myr 22,23 . Therefore, we suggest that the SSI 107 Pd/ 108 Pd ratio of ~3.5 x10 -5 is too low and we focus mainly on the SSI ratio derived from carbonaceous chondrites in the following discussion. Evolution of iron meteorite parent bodies The Pd-Ag ages obtained here of 11.7 +3.6 /-5.9 Myr after CAI for the IIABs, and 9.2 +3.5 /- 5.7 Myr for the IIIABs, suggest these bodies cooled in the same time period (Table 3; Fig. 1). The IVAs also yield a similar cooling age (7.8 +3.0 /-4.4, 12 ). The timings of metal-silicate differentiation and core cooling of an undisrupted asteroid are primarily a function of size and time of accretion 26 . The comparable accretion and core crystallisation ages 1 and cooling rates ( Table 3; Methods) of these bodies may therefore imply that they had similar sizes. However, high and variable metallographic cooling rates among the IIAB, IIIAB and IVA irons indicate that these metals cooled in bodies that had been stripped of their insulating silicate mantles in 'hit-and-run' style impacts 4,5 . Intriguingly, we find that the IIAB, IIIAB and IVA asteroids were impacted in broadly the same timeframe ( Fig. 1), suggesting an energetic Solar System environment persisted at this time. Early Solar System dynamics Nucleosynthetic isotope variations indicate a dichotomy between non-carbonaceous (NC) and carbonaceous-type (CC) asteroids 27 . The IIAB, IIIAB and IVA asteroids are NC and likely accreted in the inner Solar System 1 . All three indicate impact-related cooling and this provides evidence for energetic events with the ability to affect multiple locations in the inner disk at ~7.8-11.7 Myr after CAI. Additional evidence for an energetic state in the NC reservoir 25 includes the IAB impact event identified above that occurred in a similar timeframe, albeit 5.0 +1.0 /-1.1 Myr later than on the IVA body (Table 3). Other early-formed asteroids, including the Hchondrite parent body, the ungrouped Nedagolla iron meteorite, and the IID and IVB irons which originated in the outer disk (CC) reservoir, may also show evidence of impact processes in this interval 5,11,[28][29][30] . Below, we discuss mechanisms that can lead to an energetic inner disk environment between ~7.8-11.7 Myr after CAI. The presence and subsequent dissipation of the gas disk exerts one of the strongest controls on early Solar System orbital dynamics. The presence of gas in a protoplanetary disk dampens the eccentricities of planetesimals and stabilises the system. As the gas disperses, the damping effect of gas drag ceases and energetic collisions capable of disrupting planetesimals are predicted [31][32][33] (Fig. 2). Imaging of young stellar clusters suggest the overall lifetime of circumstellar disks is ~6-8 Myr 34 , and the gas in our protoplanetary disk likely survived no more than ~4 Myr after CAI 1 . Whilst this is earlier than the impact timeframe discussed here, models of planetesimal dynamics after gas dispersion show that bodies initially have dynamically stable orbits, with eccentricity and inclination building up over several Myr 31,33 . Therefore, impacts occurring between ~7.8 and 11.7 Myr may be attributable to the dissipation of the gas disk. The first ~10-20 Myr after gas dissipation are the most important for both the number and mean velocity of collisions 31 . Further, this mechanism generates a chaotic asteroid belt for several Myr and thereby also provides a viable explanation for later impacts on the IAB parent body. The disk-wide nature of this mechanism also provides an explanation for impacts in the CC reservoir. Giant planets also act in multiple ways to influence the orbits of nearby planetesimals. Rapid runaway growth is predicted to perturb the orbits of nearby planetesimals to high eccentricities and gravitationally scatter them into the inner Solar System, causing widespread mixing with the potential for more frequent collisions 35,36 . Additionally, Jupiter and Saturn are suggested to have undergone rapid gas-driven in-and then outward migration (the 'Grand Tack' model), leading to excitation of the asteroid belt and high velocity impacts in the inner Solar System 37,38 . The formation times of chondrules and their distinct nucleosynthetic fingerprints suggest the NC and CC reservoirs remained isolated until ~3-5 Myr after CAI, constraining the earliest time of giant planet migration 1,37 . However, impact events between ~7. 8-11.7 Myr are more than ~3 Myr later than the runaway growth and hypothesised migration phase of the giant planets, and therefore cannot be attributed to these events. Additionally, these mechanisms are not expected to lead to bombardment events that lasted longer than ~0.5-1 Myr 37,39,40 , whereas the energetic state of the inner Solar System recorded by the Pd-Ag system persisted for at least 3.9 +2.3 /-3.1 Myr after the closure of the IVA core (Table 3). This longevity problem persists if using the lower SSI 107 Pd/ 108 Pd ratio of ~3.5 x10 -5 , which yields impacts at ~2.9-6.8 Myr after CAI that broadly coincide with the migration of the giant planets. In this case, a second mechanism such as runaway growth must also be invoked. Further, these two events together do not provide a framework for understanding the later impact recorded by the IABs. Additionally, a giant planet instability (the 'Nice' model) could provide the necessary energy to the asteroid belt and inner Solar System (Fig. 2, Scenario B). This model can explain many features of our Solar System 2 and was initially assumed to occur later in Solar System history thereby providing an explanation for the Late Heavy Bombardment (LHB) 41 . However, recent studies have questioned both the evidence for a LHB and the dynamic consequences of delaying the instability 42-44 . An early giant planet instability occurring ≤100 Myr after gas dissipation better matches the architecture of our Solar System 42, [45][46][47][48] . Although the precise timing of the instability is difficult to determine, models which predict the small size of Mars advocate an instability ~5-14 Myr after CAI 47,48 . Impacts on the iron parent bodies between ~7.8-11.7 Myr agree well with such a scenario and may provide the first evidence from the meteorite record for the occurrence and timing of an early giant planet instability. Such longlived and Solar System wide events also provide an explanation for later impacts including on the IAB parent body, and impacts across the disk, including both the NC and CC reservoirs. Additionally, the effect of gas dissipation may act as the trigger for the early giant planet instability 49 . Hence, these two mechanisms may have acted together to create an energetic Solar System between ~7.8-11.7 Myr after CAI. Methods Eighteen aliquots of thirteen iron meteorites from the IAB, IIAB and IIIAB groups were analysed for this study (Table 1). Samples were cut using a CBN blade and subsequently polished using SiC paper to remove remaining weathered areas and fusion crust. Any visible troilite inclusions were physically removed from the metals. Palladium-silver isotope data for the IAB irons were published previously in 9 , and Pt isotope data for Coahuila are from 15 . Material for Pd-Ag and Pt isotope analyses was taken from adjacent sampling locations, except for Negrillos, Caddo County and Canyon Diablo. Platinum isotope data for these samples are taken from the literature and indicate that they are unexposed or only weakly exposed to the effects of GCR (Table 1, 14,16 ). Platinum isotope measurements Platinum isotope ratios were determined according to the methods described by 50 . In brief, samples were placed in ethanol in an ultrasonic bath for 5 minutes and then leached for 5 minutes in cold 2 M HCl before digestion with a 2:1 mix of concentrated HNO3 and HCl (~8 ml g -1 of sample). Samples were refluxed and dried, and then taken up in concentrated HCl. At this stage, 0.3 g aliquots for Pt isotope analyses were taken and dried. The aliquots were refluxed again with a 2:1 mix of concentrated HNO3 and HCl, and then re-dissolved in preparation for the anion exchange procedure. This consisted of two stages and utilised Biorad AG® 1-X8 resin (200 -400 mesh, chloride form). The first stage separated Pt from major matrix constituents, while the second stage separated Pt from Ir 50 . The second stage was repeated until samples had 191 Ir/ 195 Pt ratios ≤ 0.13, which is below the tolerance limit for Ir isotope tailing onto Pt isotopes during mass spectrometry, as determined for the instrument housed at ETH Zürich 50 . Osmium, which generates isobaric interferences on Pt isotopes, was then volatilised using perchloric acid. Platinum isotope ratios were measured at ETH Zürich using a Thermo Scientific Neptune Plus operated in low-resolution mode and fitted with a Cetac Aridus II desolvating nebuliser and standard H cones, following the procedure of 50 . Samples were diluted to yield an ion beam intensity of ~2 x10 -10 A for 194 Pt, which equates to ~150 ng g -1 Pt and required ~170 ng per analysis. Analyses were corrected for instrumental mass bias using the exponential law and were internally normalized to 198 Pd-Ag Measurements Sample masses for Pd-Ag analyses are given in Supplementary Materials Table 1. Samples for Pd-Ag analyses were leached in cold concentrated aqua regia for 5 minutes, followed by digestion in concentrated aqua regia (~8 ml g -1 ) on a hotplate. They were then dried and refluxed in concentrated HCl (40 ml), before adding 20 -40 ml H2O to verify that the sample was fully in solution. Approximately 9% of the sample solution was taken at this stage to determine Pd and Ag concentrations by isotope dilution, following an established method 8 . Samples were then refluxed and dried again in HCl to ensure no traces of HNO3 remained. The ion exchange technique for Ag purification followed a published protocol 51 and utilised a three-stage column procedure with anion and cation exchange resin. The chemical procedure removes elements that cause interferences on Ag isotopes, including isobaric (Pd, Cd and Ru) and molecular interferences (Cu, Zn, Ga, Ge, Sr, Y, Zr, Nb and Mo), such that all were below the tolerance limits for accurate Ag isotope analyses 9,51 . The yields of the chemical separation procedure were checked for each sample individually and they were always close to 100%. (Table 1). The details are described in 9 and 51 , respectively, and a short summary is given here. Both instruments were operated with a glass cyclonic spray chamber. A Pd NIST SRM 3138 standard solution was added to all samples and standards just prior to analysis to correct for instrumental mass bias 51 Correction for Galactic Cosmic Ray Effects to Ag isotopes New ε 107 Ag data for the IIAB and IIIAB irons vary from ~37 to 103, while 108 Pd/ 109 Ag ratios range between 261 and 679 (Table 1). There is a positive correlation between ε 107 Ag and 108 Pd/ 109 Ag for all meteorite groups, however, a regression cannot be fitted through our new data for the IIAB or IIIAB iron meteorites before corrections for GCR (Extended Data Fig. 2). The uncorrected IAB samples define an initial 107 Pd/ 108 Pd ratio of 1.06 (± 0.36) x10 -5 with a MSWD of 45 9 . Galactic cosmic rays (GCR) can disturb the 107 Pd-107 Ag chronometer through reactions with thermal neutrons 10 . This must be corrected before meaningful age information can be obtained. Secondary neutron capture by Pd isotopes leads to a decrease in 107 Ag/ 109 Ag primarily via neutron capture on 108 Pd and subsequent βdecay to 109 Ag. Additionally, 106 Pd is also susceptible to neutron capture followed by βdecay to 107 Ag. The GCR-induced shift in 107 Ag/ 109 Ag is therefore linearly correlated with Pd/Ag and higher 108 Pd/ 109 Ag ratios lead to larger negative shifts in 107 Ag/ 109 Ag, i.e., the slope of data not corrected for GCR effects would be too shallow, if one could even be determined because of the scatter 10 . Longer exposure ages, bigger pre-atmospheric radii, and increasing shielding depths all lead to larger GCR offsets. It has been demonstrated that Pt isotopes are a powerful neutron dosimeter for correcting the effects of GCR in short-lived decay systems, e.g., 182 Hf-182 W 14 . They have also been applied as a dosimeter for neutron capture in the 107 Pd-107 Ag system 11,12 . New Pt isotope data fit well to the modelled effects for neutron capture (Extended Data Fig. 1). Model trendlines are shown for Ir/Pt ratios appropriate for our samples (Supplementary Materials Table 1). These results further confirm that neutron capture reactions are the cause of Pt isotope excesses and that there are no detectable nucleosynthetic isotope variations between iron meteorites. Following the method of 11 (given as equations 1-3 below), data for the IAB, IIAB and IIIAB irons were assessed for the effects of neutron capture. The correction relies on the 108 Pd/ 109 Ag ratio and neutron dose (ε 196 Pt) of the sample in question. Measured ε 196 Pt can be related to neutron capture corrections to ε 107 Ag (ε 107 AgGCR) by a linear regression: ε 107 AgGCR = Offset + Slope x ε 196 Pt(1) where Offset and Slope are functions of the measured 108 Pd/ 109 Ag ratio 11 . The Offset defines neutron capture shifts on Ag isotopes and is described by: ; Table 3). Calculation of cooling rates The closure temperature of the Pd-Ag system is dependent on cooling rate, Ni and FeS content of the meteorite, and is estimated to be in the range ~820 -970 K for the iron meteorites studied here 6 . Fast cooling rates are necessary for the IAB parent body in order to keep silicate inclusions suspended in the metal portions 17 . Assuming that the IAB parent was at its peak temperature of 1470 K 14 when impacted at ~11 Myr (see Main Text) yields the expected fast cooling rates of between instantaneous and ~102 K/Myr (Table 3). Alternatively, a cooling age of 7.9 +1.0 /-1.1 (calculated relative to an initial SSI 107 Pd/ 108 Pd of ~3.5 x 10 -5 12 ) implies that the parent body was impacted when at or near its peak temperature at 6.0 ± 0. (Table 3). Cooling rates determined here for the IIIAB irons compare well with those derived previously using the Pd-Ag system 6,11 . They also overlap with recent estimates of Competing Interests Statement The authors declare no competing interests. Supplementary Information is available for this paper in the form of one Table as a pdf. Data Availability All data are available within this manuscript and its Supplementary information, or available from the authors on request. (Table 3). TABLES, MAIN ARTICLE Literature data for the IVA irons 12 are also shown. b. Age after CAI for core cooling, calculated relative to the carbonaceous chondrite SSI 13 . Iron meteorite group symbols as in panel (a). The grey and purple bands represent, respectively, the approximate timings for CAI formation and the impacts recorded by the Pd-Ag system. The timeframe suggested for the 'Grand Tack' is indicated by the size of the Jupiter symbol 1,37 . The intervals suggested for gas present in the disk, and an early giant planet instability that predicts a small Mars 47,48 , are also shown. Table 1). Uncertainties are 2 S.D. Extended Data Figure 2. Isochron diagrams for a) IAB, b) IIAB, and c) IIIAB iron meteorites for the Pd-Ag system. In each case, ε 107 Ag is plotted against 108 Pd/ 109 Ag for both GCRcorrected (filled symbols) and GCR-uncorrected data (unfilled symbols). Uncertainties on GCR-uncorrected data are 2 S.D. and within the size of the symbol (Table 1), while uncertainties for ε 107 Ag for GCR-corrected data represent the propagated 2 S.D. uncertainties of the GCR correction ( bracketed by the NIST SRM 3140 Pt standard solution and data are presented in the epsilon notation (i.e., ε 19i Pt/ 195 Pt = deviation in parts per 10,000 from the average of the two bracketing standards). Reproducibility for Pt isotope analyses was assessed using the iron meteorite North Chile as an in-house standard. Repeat analyses of multiple aliquots of this material indicate a 2 S.D. uncertainty of 0.73 for ε 192 Pt, 0.15 for ε 194 Pt and 0.09 for ε 196 Pt (n=19; 50 ). Platinum blanks are in the range 0.3 -1.0 ng g -1 sample. Silver isotope analyses were performed by MC-ICP-MS on either a Nu Plasma instrument in the School of Earth, Atmospheric and Environmental Science at The University of Manchester or a VG Axiom in the Department of Terrestrial Magnetism at the Carnegie Institution of Washington , and was mixed to yield a signal intensity ratio of 1 for 107 Ag/ 108 Pd. Additionally, sample measurements were bracketed by the NIST SRM 978a Ag -NIST SRM 3138 Pd standard solution mixture, with the Pd and Ag concentrations of the standard matched to the samples to within 20% or better. Instrumental mass bias was corrected using the exponential law with normalisation to a 108 Pd/ 105 Pd value of 1.18899 7 . Data are presented in the epsilon notation and a 107 Ag/ 109 Ag ratio of 1.079760 13 was used for the NIST SRM 978a Ag standard.For analyses conducted at The University of Manchester, the typical daily reproducibility of a 100 ppb Ag standard solution for the 107 Ag/ 109 Ag ratio was ~0.25 ε (2 S.D. 9 .Additionally, the external reproducibility of sample measurements was assessed by repeatedly passing the NIST SRM 978a Ag standard, the USGS standard Cody Shale SCo-1, and the Hawaiian basalt KOO49 through the chemical procedure. These materials yielded external reproducibilities of between 0.1 and 0.6 ε (2 S.D. 9 ), and therefore a conservative external uncertainty of ±0.6 ε has been applied to all sample data acquired at The University of Manchester. Similar tests were carried out at the Carnegie Institution 51 and resulted in an external uncertainty of ±0.5 ε . Silver blanks for sample digestion and column chemistry were <0.254 ng. metallographic cooling rates obtained for this body (338 -56 K/Myr 5 ), particularly for a Pd-Ag closure time of ~9.2 Myr after CAI. 200020_179129, MS). AH wishes to thank Mattias Ek and Manuela Fehr for laboratory assistance at ETH Zürich during this study. MS would like to thank Rick Carlson and Mary Horan (DTM, Carnegie Institution) for their support and the opportunity to analyse Ag isotopes in May 2006. We also thank Caroline Smith and Deborah Cassey (Natural History Museum, London), and Julie Hoskin (Smithsonian Institution National Museum of Natural History) for the loan of samples used in this work. Author Contributions M.S. designed the study. A.H., K.T. and M.S. prepared samples for isotope analyses and conducted the isotopic measurements. All authors were involved in the data interpretation and writing of the manuscript. Figure 1 . 1The timing of core cooling and impacts constrained by the Pd-Ag system. a. Parent body initial 107 Pd/ 108 Pd ratios for the IAB, IIAB and IIIAB iron meteorite groups Figure 2 . 2Cartoon illustrating the evolution of differentiated iron meteorite bodies in the early Solar System (not to scale). Top panel: Parent bodies accrete and differentiate within the first ~3 Myr after CAI 16 . The disk consists of gas and dust (coloured blue) and is truncated in the region of Jupiter, preventing significant exchange between the non-carbonaceous (NC) and carbonaceous (CC) reservoirs. Later, the cores of the IIAB, IIIAB and IVA bodies were exposed by impact events, leading to fast cooling and closure of the Pd-Ag system between ~7.8 -11.7Myr. Two scenarios, acting alone or in combination, provide mechanisms for an energetic inner Solar System. Scenario A: When the gas disk dissipates, planetesimals are gradually excited until they begin to undergo high velocity impacts 31 . Gas dissipation is completed before this time. Scenario B: An early giant planet instability is triggered at ~7.8 -11.7 Myr, leading to reorganisation of the inner Solar System and impact events47 .EXTENDED DATA FIGURE LEGENDS Extended Data Figure 1. Covariation of ε 192 Pt and ε 196 Pt for the IAB, IIAB and IIIAB iron meteorites. The GCR model calculations 10 are shown for Ir/Pt ratios measured for these groups (dashed lines; Ir/Pt ratios for individual samples are given in Supplementary Materials J: Uncertainties (2 S.D.) on 108 Pd/ 109 Ag ratios are estimated to be ~1 % for data collected at the University of Manchester and 5 % for data collected at the Carnegie Institution of Washington (see Extended Methods). Theis et al. (2013) Taken from Theis et al. (2013) Taken from Theis et al. (2013) Taken from Theis et al. (2013) Taken from Theis et al. (2013) Taken from Theis et al. (2013) Taken from Theis et al. (2013) Solar System Initial 107 Pd/ 108 PdIn order to accurately determine Pd-Ag ages, the Solar System initial (SSI) 107 Pd/ 108 Pd is needed. Recent estimates are based on either carbonaceous chondrites 13 or the IVA iron Muonionalusta 11,12 , which has resulted in a range of initial ratios. By equating the 53 Mn-53 Cr SSI 107 Pd/ 108 Pd is hampered by ambiguity over its absolute Pb-Pb age, determined from a troilite52 . Recently, the Pd-Ag systematics of Muonionalusta were re-examined and twoOffset = -0.023 + 0.0024 x ( 108 Pd/ 109 Ag) -4.76 x 10 -7 x ( 108 Pd/ 109 Ag) 2 (2) while the Slope is defined by: Slope = 1.539 -0.226 x 108 Pd/ 109 Ag (3) The Offset and Slope are then used in Eq. (1) to determine ε 107 AgGCR. For the samples included in this study ε 107 AgGCR ranges from 2.3 to -49.2, with the largest offsets recorded in the IIIAB irons (Table 2). and 107 Pd-107 Ag decay systems, which both date volatile depletion, carbonaceous chondrites define a SSI 107 Pd/ 108 Pd of (5.9±2.2) x10 -5 13 . This ratio is among the highest initials yet determined and includes a relatively large uncertainty. Values for the initial 107 Pd/ 108 Pd based on Muonionalusta range from ~(2.8-6.6) x10 -5 11,12 , but the use of this meteorite to define a possible SSI 107 Pd/ 108 Pd ratios of (3.5±0.1) and (6.6±0.4) x10 -5 were determined 12 , dependent on the 238 U/ 235 U ratio of the Muonionalusta troilite. In order to consider the full range of SSI values, we calculate Pd-Ag ages for the iron meteorite groups studied here relative to both the lowest recent estimate based on Muonionalusta 12 (~3.5 x10 -5 ) and the value indicated from carbonaceous chondrites 13 (~5.9 x10 -5 ), which also encompasses the highest ratio estimated from Muonionalusta. To circumvent the ambiguity over the SSI 107 Pd/ 108 Pd, closure times can also be calculated relative to the value determined for the IVA irons ( 107 Pd/ 108 Pd ~2.57 x10 -5 , 12 8 Myr. This yields cooling rates of between instantaneous and ~135 K/Myr. Modelling indicates the IIAB and IIIAB parent bodies accreted < ~0.4 Myr after CAI 1 .Core formation on the IIAB parent body followed at 0.7 ± 0.3 Myr, based on the Hf-W chronometer16 . Assuming a liquidus temperature of ~1570 K for the IIABs 53 and a steady rate of cooling between core formation and closure of the Pd-Ag chronometer indicates cooling However, it is important to note that these apply to cooling at ~670-770 K, and that newer techniques have led to upward revisions of rates by factors of 10-505 .rates of ~278-69 K/Myr for cooling at 6.8 +2.3 / -3.1 Myr, or ~156-40 K/Myr for cooling at 11.7 +3.6 / -5.9 Myr (Table 3). Metallographic cooling rates for the IIABs suggest a much slower process (0.8-10 K/Myr 54 ). The IIIABs experienced core formation at 1.2 ± 0.3 Myr with a predicted liquidus temperature of ~1650 K 16,53 . This yields a very wide range of cooling rates, from instantaneous- 121 K/Myr for Pd-Ag closure at 4.3 +2.2 / -2.8 Myr to 415-58 K/Myr for closure at 9.2 +3.5 / -5.7 Myr Table 1 . 1Concentration data for Pd and Ag, isotope data for Ag, and ε 196 Pt for IAB, IIAB and IIIAB iron meteorites.Meteorite Class a Pd (ppb) Ag (ppb) 108 Pd/ 109 Ag ε 107 Ag ±2 S.D. 107 Ag/ 109 Ag ε 196 Pt ±2 S.D. Terrestrial 1.07976 b Campo del Cielo c IAB-MG 3616 11.3 177.8 15.8 0.6 1.081466 0.09 0.09 Canyon Diablo c IAB-MG 3888 39.0 53.9 1.4 0.6 1.079911 0.00 e 0.09 Odessa 1 c IAB-MG 4199 28.2 81.1 2.2 0.6 1.079998 0.31 0.09 Odessa 2 c IAB-MG 4063 10.9 199.8 11.7 0.6 1.081023 0.28 0.09 Goose Lake 1 c IAB-sLL 4472 22.4 109.9 7.4 0.6 1.080559 0.08 0.09 Goose Lake 2 c IAB-sLL 3881 18.5 116.6 8.6 0.6 1.080689 0.08 0.09 Toluca c IAB-sLL 4546 38.3 64.7 1.4 0.6 1.079911 0.13 0.09 Caddo County c IAB-UG 4113 58.9 37.7 0.1 0.6 1.079771 -0.01 e 0.09 Coahuila 1 IIAB 1829 3.3 307.9 53.4 0.6 1.085530 0.01 f 0.07 Coahuila 2 d IIAB 1708 3.8 251.2 50.6 0.5 1.085223 0.01 f 0.07 Negrillos d IIAB 1587 3.4 260.9 52.7 0.5 1.085450 -0.03 g 0.05 North Chile 1 IIAB 1932 3.3 324.4 55.0 0.6 1.085697 0.12 0.08 North Chile 2 d IIAB 1841 3.2 321.6 53.1 0.5 1.085488 0.12 0.08 Sikhote Alin 1 IIAB 2451 3.2 433.5 59.9 0.6 1.086227 0.22 0.09 Sikhote Alin 2 IIAB 2649 4.8 310.4 36.9 0.6 1.083746 0.22 0.09 Boxhole IIIAB 2385 1.9 678.7 103.5 0.6 1.090933 0.25 0.09 Henbury IIIAB 2199 2.0 588.8 77.7 0.6 1.088147 0.38 0.09 Thunda IIIAB 3092 4.1 407.8 83.3 0.6 1.088754 0.05 0.09 Uncertainties for ε 107 Ag are based on repeat analyses of reference materials (see Methods). Uncertainties (2 S. D.) on 108 Pd/ 109 Ag ratios are estimated to be ~1 % for data collected at the University of Manchester and 5 % for data collected at the Carnegie Institution of Washington. Platinum isotope uncertainties are based on repeat analyses of an in-house standard (see Methods). a IAB sub-groups from 17 . See main text for definitions. b Terrestrial 107 Ag/ 109 Ag ratio from 13 . c Palladium-silver data taken from 9 . All Pd-Ag isotope data measured at the University of Manchester, except d Carnegie Institution of Washington. Platinum isotope data taken from e Hunt, et al. 14 , f Hunt, et al. 15 and g Kruijer, et al. 16 . Table 2 . 2Correction to ε 107 Ag for GCR effects for IAB, IIAB and IIIAB iron meteorites.Meteorite ε 107 Agmeas ±2 S.D. ε 107 AgGCR ± 2 S.D. ε 107 Agpre-exp ± 2 S.D. IAB Campo del Cielo 15.8 0.6 -3.1 3.5 18.9 3.5 Canyon Diablo 1.4 0.6 0.1 1.0 1.3 1.1 Odessa 1 2.2 0.6 -5.0 1.5 7.2 1.6 Odessa 2 11.7 0.6 -11.9 3.9 23.6 4.0 Goose Lake 1 7.4 0.6 -1.6 2.1 9.0 2.2 Goose Lake 2 8.6 0.6 -1.7 2.2 10.3 2.3 Toluca 1.4 0.6 -1.6 1.2 3.0 1.3 Caddo County 0.1 0.6 0.1 0.6 0.0 0.9 IIAB Coahuila 1 53.4 0.6 -0.2 4.8 53.7 4.8 Coahuila 2 50.6 0.5 -0.2 3.9 50.8 3.9 Negrillos 52.7 0.5 2.3 2.9 50.4 2.9 North Chile 1 55.0 0.6 -8.2 5.7 63.2 5.7 North Chile 2 53.1 0.5 -8.2 5.6 61.2 5.6 Sikhote Alin 1 59.9 0.6 -20.7 8.7 80.6 8.7 Sikhote Alin 2 36.9 0.6 -14.7 6.2 51.6 6.2 IIIAB Boxhole 103.5 0.6 -37.1 13.7 140.6 13.7 Henbury 77.7 0.6 -49.2 11.9 126.8 11.9 Thunda 83.3 0.6 -3.7 8.2 87.0 8.2 Table 3 . 3Parent body initial 107 Pd/ 108 Pd ratios and closure times of the Pd-Ag system relative to CAI and IVA irons.Group Parent Body Initial 107 Pd/ 108 Pd (x 10 -5 ) Time after CAI (Myr) a Time after CAI (Myr) b Time after IVA (Myr) c Cooling rates (K/Myr) b IAB 1.51 (± 0.16) 12.8 +3.1 / -4.6 7.9 +1.0 / -1.0 5.0 +1.0 / -1.1 Instantaneous -102 IIAB 1.70 (± 0.48) 11.7 +3.6 / -5.9 6.8 +2.3 / -3.1 3.9 +2.3 / -3.1 ~156 -40 IIIAB 2.21 (± 0.57) 9.2 +3.5 / -5.7 4.3 +2.2 / -2.8 1.4 +2.2 / -2.8 ~415 -58 IVA c 2.57 (± 0.07) 7.8 +3.0 / -4.4 2.9 +0.4 / -0.4 - > 500 c a Time after CAI and resultant cooling rates calculated using a SSI 107 Pd/ 108 Pd of 5.9 (± 2.2) x 10 -5 13 . b Time after CAI calculated using a SSI 107 Pd/ 108 Pd of 3.5 (± 0.1) x 10 -5 12 . 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[ "Learning Multi-Object Positional Relationships via Emer- gent Communication", "Learning Multi-Object Positional Relationships via Emer- gent Communication" ]
[ "Yicheng Feng \nSchool of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n\n", "Boshi An [email protected] \nSchool of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n\n", "Zongqing Lu [email protected] \nSchool of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n\n" ]
[ "School of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n", "School of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n", "School of Computer Science\nSchool of Computer Science\nPeking University\nPeking University\nPeking University\nBAAI\n" ]
[]
The study of emergent communication has been dedicated to interactive artificial intelligence. While existing work focuses on communication about single objects or complex image scenes, we argue that communicating relationships between multiple objects is important in more realistic tasks, but understudied. In this paper, we try to fill this gap and focus on emergent communication about positional relationships between two objects. We train agents in the referential game where observations contain two objects, and find that generalization is the major problem when the positional relationship is involved. The key factor affecting the generalization ability of the emergent language is the input variation between Speaker and Listener, which is realized by a random image generator in our work. Further, we find that the learned language can generalize well in a new multistep MDP task where the positional relationship describes the goal, and performs better than raw-pixel images as well as pre-trained image features, verifying the strong generalization ability of discrete sequences. We also show that language transfer from the referential game performs better in the new task than learning language directly in this task, implying the potential benefits of pre-training in referential games. All in all, our experiments demonstrate the viability and merit of having agents learn to communicate positional relationships between multiple objects through emergent communication.
10.48550/arxiv.2302.08084
[ "https://export.arxiv.org/pdf/2302.08084v1.pdf" ]
256,901,019
2302.08084
001bddae7fd798a7422ecc1b0818752e2401fbc2
Learning Multi-Object Positional Relationships via Emer- gent Communication Yicheng Feng School of Computer Science School of Computer Science Peking University Peking University Peking University BAAI Boshi An [email protected] School of Computer Science School of Computer Science Peking University Peking University Peking University BAAI Zongqing Lu [email protected] School of Computer Science School of Computer Science Peking University Peking University Peking University BAAI Learning Multi-Object Positional Relationships via Emer- gent Communication Preprint The study of emergent communication has been dedicated to interactive artificial intelligence. While existing work focuses on communication about single objects or complex image scenes, we argue that communicating relationships between multiple objects is important in more realistic tasks, but understudied. In this paper, we try to fill this gap and focus on emergent communication about positional relationships between two objects. We train agents in the referential game where observations contain two objects, and find that generalization is the major problem when the positional relationship is involved. The key factor affecting the generalization ability of the emergent language is the input variation between Speaker and Listener, which is realized by a random image generator in our work. Further, we find that the learned language can generalize well in a new multistep MDP task where the positional relationship describes the goal, and performs better than raw-pixel images as well as pre-trained image features, verifying the strong generalization ability of discrete sequences. We also show that language transfer from the referential game performs better in the new task than learning language directly in this task, implying the potential benefits of pre-training in referential games. All in all, our experiments demonstrate the viability and merit of having agents learn to communicate positional relationships between multiple objects through emergent communication. Introduction In order to achieve interactive agents, a major problem to be solved is to endow artificial agents with the ability to communicate. Supervised methods are considered incapable of capturing functional meanings of language (Lazaridou et al., 2017;Kottur et al., 2017). Therefore, a series of studies on emergent communication probe into this problem by providing agents with simple environments where they learn to communicate with each other from scratch to accomplish specific tasks (Havrylov & Titov, 2017;Choi et al., 2018;Li & Bowling, 2019;Ren et al., 2020). Most of these tasks are based on referential games (Lewis, 1969), where Speaker observes and describes a target object while Listener receives the message sent by Speaker and must pick out the target from several candidates. In existing emergent language studies, agents' observations are mainly focused on a single object, be it a geometric object or a categorical image. Some studies involve images showing more complex scenes, but these studies usually also involve natural language (Das et al., 2017;Gupta et al., 2021). Communicating the relationships between multiple objects explicitly is understudied. Then, problems may arise when we consider the development from communication in tasks like referential games to communication in tasks with more realistic settings, e.g., multi-step Markov decision process (MDP) tasks, since there the information about multi-object relationships is usually helpful, and sometimes even crucial. So in this paper, we try to fill this gap and address two questions: Can neural agents learn to extract the information about multi-object relationships and express it through discrete communication channels in the referential game? If so, can the learned protocol help in more complex multi-step MDP tasks? We focus on positional relationships between We train agents in the referential game where the observations are images each containing two geometric shapes, and see whether the agents can communicate the two objects and their positional relationship shown in each image. Since the positional relationship is abstraction information that can have various manifestations in specific images, we propose to use a random dataset to test generalization, where each image is generated randomly each time, and the target image observed by Speaker and Listener is also different in pixel level but the same in abstraction. This is a stronger dataset than the standard setup, forcing agents to communicate abstract information to get high accuracy. We also use two common datasets as baselines, the fixed dataset where images are fixed and the variation dataset where images are randomly generated but the target image observed by Speaker and Listener is exactly the same. We find that agents trained with these two common datasets, though perform well if tested by the corresponding datasets, cannot generalize in the random dataset. This demonstrates that the two commonly used datasets cannot well test agents' ability to express abstract information, and also fail to help agents learn multi-object positional relationships. Instead, we find that agents trained with the random dataset can generalize well, implying that the input variation between Speaker and Listener is crucial for learning abstract information in emergent communication, so is necessary for extracting positional relationships. We also use an image encoder pre-trained by a contrastive learning method, SimCLR (Chen et al., 2020), for comparison, and show that the language learned through the referential game with the random dataset generalizes better. Then we show how communication about multi-object positional relationships helps in multi-step MDP tasks. We design a simple communication game where the positional relationship describes the goal. We find that the emergent language can generalize well in the new task, and is more powerful than raw-pixel images as well as pre-trained image features, proving the good generalization ability of discrete sequences. Besides, we find that language transfer from the referential game could achieve better performance than learning language from scratch in the new task, which may provide evidence for the benefits of language learning in the referential game. We summarize the main contributions of our work as follows: (1) We explore agents' communication about multi-object positional relationships in raw-pixel images from scratch through emergent communication. (2) We propose to use the random dataset to test the generalization of emergent languages, and find the environmental pressure where Listener observes target images different from Speaker's crucial for agents to emerge generalizable languages in the referential game. (3) Our experiments show that the emergent language can generalize well in the new multi-step MDP task, and is more powerful than raw-pixel images as well as pre-trained image features. Related Work Emergent communication. A series of studies have been done on emergent communication that trains interactive agents to learn protocols from communication games. Most studies focus on language learning in the referential game, where a speaker agent refers to targets using a message and a listener agent tries to understand the message (Lazaridou et al., 2016;2017;Havrylov & Titov, 2017;Evtimova et al., 2018;Choi et al., 2018;Chaabouni et al., 2019;2022;Dessì et al., 2021;Dagan et al., 2021;Gupta et al., 2021). These studies provide in-depth insights for learned protocols as well as learned representations of agents, but mostly stop at the single task. Chaabouni et al. (2022) proposed ease and transfer learning (ETL) to evaluate the generalization of the emergent language to new tasks, but they do not involve multi-step MDP tasks. Most studies exploring emergent communication in the context of the referential game use inputs containing a single object, e.g., a geometric shape or a natural image depicting a specific object. This restricts the generalization of the emergent language to complex MDP tasks. We go one step further to explore the positional relationship between two objects in observations. Some other work explores emergent communication in multi-step MDP tasks directly, where agents learn to use discrete communication channels to cooperate (Bogin et al., 2018;Mordatch & Abbeel, 2018;Eccles et al., 2019;Tucker et al., 2021;Lin et al., 2021). These studies usually focus on methods for improving the ability of agents to accomplish the tasks through efficient communication, and explore whether the communication captures critical information for the tasks. However, the Figure 1: The referential game, agent architecture and examples of images in the random dataset. protocols are usually still specific to training tasks. We consider the generalization of the emergent language and probe into the language transfer from the referential game to more complex MDP tasks. And we think of the relationship between objects as an entry point. Chen et al., 2020) to process input images. In our experiments, we find that adding noise alone is not enough for agents to communicate abstract information. We use a random image generator to introduce the environmental pressure more severely so that agents can almost never observe two same images. Moreover, we make a comparison with two other datasets, and find the random image generator really helpful for the communication about positional relationships. Input variation between 3 Experimental Setup The referential game We train our agents in the two-player referential game where Speaker describes a target image to Listener who should pick out the target image among several candidates. Concretely, Speaker observes a target image x, and generates a message m to describe it. The message m is a sequence of discrete symbols from a vocabulary V. The message length is T . Listener receives m as well as a set of candidate images C including the target x and several distractors. Then Listener selects an imagex ∈ C according to m. If x =x, both agents get a reward r = 1. Otherwise, the reward is 0. Agent architecture Speaker, parameterized by θ, consists of an image encoder and a sequence generator. The target image x is first fed into a CNN network f θ to get the image embedding f θ (x). Then a projector g θ maps the embedding into the initial hidden state of an LSTM (Hochreiter & Schmidhuber, 1997), h −1 = g θ (f θ (x) ). Then at each time step t a linear layer π θ maps h t into a vector of dimension |V|, and a symbol w t is sampled from the distribution induced by applying the softmax function to π θ (h t ). And the one-hot embedding of the generated symbol e(w t ) is fed back to the LSTM l θ to update the hidden state h t+1 = l θ (e(w t ), h t ). The first input symbol is a special token labeled as a start of sequence, h 0 = l θ (e(sos), h −1 ). The symbols are generated until the message length reaches T . At test time, the symbols are not sampled but selected greedily. Listener, parameterized by φ, consists of an image encoder and a sequence encoder. An LSTM network l φ encodes the sequence m = w 0 , w 1 , ..., w T −1 from Speaker into the message embedding e m = l φ (e(m)), with each symbol in the sequence transformed to a one-hot embedding e(m) = e(w 0 ), e(w 1 ), ..., e(w T −1 ). A CNN network f φ encodes each imagex ∈ C into image embedding ex = f φ (x). A linear projector p m,φ and an MLP projector px ,φ projects the message embedding and each image embedding respectively to compute the cosine similarity between p m,φ (e m ) and px ,φ (ex). The resulting similarities are passed to a softmax function to get a distribution over all images in the candidate set, and the image with the highest probability is selected. Details for hyper-parameters can be found in Appendix A. Datasets and the random image generator We create a dataset where we generate images of size 128 × 128 each depicting two objects with a certain positional relationship between them. There are 5 different objects and 4 positional relationships (right, top right, top, and top left) 1 , so there are total 100 (5 × 5 × 4) combinations. We use the word 'combination' to refer to the (object, object, relationship) tuple in the rest of the paper. We separate 20 of 100 combinations into the test set, so agents can only observe 80 combinations during training. We additionally add noise to the images for the robustness of the representation learning of image encoders, and to prevent degenerate policies of using pixel-level information. Accordingly, we set the message length T = 6, the size of vocabulary |V| = 5, and the number of candidate images |C| = 32 for training and |C| = 20 for test in the referential game, which is illustrated in Figure 1. In realistic environments, the observation of agents is ever-changing. So we propose to use a random image generator to generate specific images according to the combinations, where the absolute position, size, and orientation of objects vary. The details of the image generator can be found in Appendix B. Then we hypothesize that using the random generator to provide images for Speaker and Listener separately can better test the generalization of agents, since agents can only succeed when they express and understand the abstract information in the images, especially when the multi-object positional relationship is involved because now images containing the same content are diverse at the pixel level. To verify the hypothesis, we use other two kinds of datasets for comparison. Then we have three kinds of datasets as follows: (1) Fixed dataset. We do not use the random generator but generate one image for each combination, and the absolute position, size, and orientation of objects are fixed. This setup is similar to using structured input in some studies (Li & Bowling, 2019;Chaabouni et al., 2020;Ren et al., 2020), since there are no variations of each input in the dataset. Agents trained and tested with the fixed dataset can always observe only one instance of each combination. (2) Variation dataset. We use the random generator to generate images, but the target image observed by Speaker and Listener is the same one. This setup is similar to using natural images as inputs as in some studies (Chaabouni et al., 2022;Gupta et al., 2021), where different images depicting a same object exist in the dataset. Here agents see diverse images of a combination at training time, but may still use pixel-level information to succeed in the game. (3) Random dataset. We use the random generator and generate images for Speaker and Listener separately. Here agents almost never observe two same images and are forced to use abstract information to win the game. Optimization We use REINFORCE (Williams, 1992) to train Speaker which only uses the reward of the game. We also apply entropy regularization in the loss function to encourage exploration. To train Listener, we use the cross-entropy loss function which compares the output distribution of Listener with a onehot vector indicating the target image. We use the default Adam optimizer (Kingma & Ba, 2015) with a learning rate of 3e-5 to update the parameters. Evaluation Methods Generalization in referential games. One of the most important properties of emergent language is the generalization ability to unseen inputs. We measure generalization in the referential game by the test accuracy. Compositionality. We adopt a popular metric in emergent communication literature called topographic similarity (TopSim) (Brighton & Kirby, 2006) for measuring language compositionality, which can also reflect generalization ability. It is computed by the Spearman correlation between the distances in the input space and the message space, so high TopSim means that similar inputs lead to close messages. According to the characteristics of our setup, we compute the distance in the input space by the number of different attributes in the (object, object, relationship) tuple. We use the Levenshtein distance in the message space. Visual representations. We explore the quality of the visual representations learned through the referential game. We focus on whether the representations contain features for abstract information, especially the positional relationship. Following Dessì et al. (2021), we apply a linear projection head to the learned image encoder, and conduct a classification task trained by supervised learning on the test set. Then we use the classification accuracy to evaluate the learned visual representations. Ease and transfer learning (ETL). Chaabouni et al. (2022) proposed ETL to evaluate the generality of the emergent language to new Listener in new tasks. We measure ETL by feeding the deterministic language (i.e., symbols are selected greedily) of Speaker to new Listener to perform new tasks and report the performances. We use two tasks for ETL, image classification and Object Placement. The Object Placement task aims at our main research goal: whether and how the emergent language can generalize to multi-step MDP tasks. Experiments and Results Input variation in the random dataset is important for communication about multi-object positional relationships In this section, we analyze the performance of agents in the referential game learning to communicate the multi-object positional relationship from scratch. For all experiments, we run five times with different random seeds, and report the results in Figure 2. We first use the fixed dataset and the variation dataset respectively for both training and testing. Results in Figure 2a show that agents trained with the variation dataset perform well at test time, so it seems to prove good generalization abilities. And agents trained with the fixed dataset can also get accuracies much higher than a random guess (5%). However, when we use the random dataset for test, agents trained in the previous two datasets cannot generalize as shown in Figure 2b. This implies that testing with the two commonly used datasets does not really reflect the generalization ability of agents. So we argue that input variation between Speaker and Listener is necessary for evaluating generalization in the referential game. Besides, agents trained in these datasets, though random noise is added, fail to communicate human-level conceptual information, at least when the positional relationship is involved. Then how can agents learn to extract the positional relationship from images when communicating? A natural idea is to train agents with the random dataset, which provides a harsher environment. As mentioned in Lazaridou et al. (2017) and Choi et al. (2018), the input variation should encourage agents to use the abstract information. We show the results in Figure 2b, and now the agents can perform well in the random dataset, with average accuracy close to 80%. This proves that agents are communicating semantic information so Listener can understand and select the target even if the exact image is different from that observed by Speaker. So we argue that input variation between Speaker and Listener is also necessary for emergent communication about positional relationships, or even other abstract information, in the referential game. We present some examples of the generated sequences by Speaker observing images from the test set in Figure 3. We can observe obvious patterns of different positional relationships in the sequences. Dessì et al. (2021) argues that the referential game is similar to the contrastive learning framework in SimCLR (Chen et al., 2020). From this perspective, using the random dataset can be seen as a data augmentation process where the target image is changed but the semantic information is preserved. So we are curious about the performance of the representation learned with SimCLR instead of the referential game from scratch. We train a model using SimCLR, where the positive pairs are images generated by the random generator using the same combination in the training set. Then we use the frozen SimCLR model as pre-trained image encoders of Speaker and Listener, and train them in the referential game with the random training dataset. Finally, we test the agents using the random test dataset. The result is shown in Figure 4. Surprisingly, using the pre-trained SimCLR model leads to worse performance compared to Figure 2b, i.e., the agents cannot generalize well on the test set, though we find that they get a high accuracy at training time. One reason to explain the result may be that after SimCLR pre-training, the image representations of different images generated by the same combination are very similar, so the effect of using the random dataset in the following referential game is diminished, since the target representations observed by Speaker and Listener is almost the same now. From another perspective, the pre-trained encoders in advance separate different representations for different semantic information in the feature space, so the agents lose the environmental pressure to encode semantic information with emergent languages in the referential game, but can make use of some detailed information in the rich representation to accomplish the task. Then in the test set, though the pre-trained encoders can generate good representations for the new combinations, the agent language cannot generalize well to the new representations. This result shows that using pre-trained image encoders may do bad to generalization in emergent communication. Analysis of protocols and representations learned through the referential game We report the results for computing TopSim for agents trained with different datasets in Figure 5. Obviously, agents trained with the random dataset get higher TopSim, so they tend to use similar messages to describe similar inputs, implying more compositional languages. This again demonstrates the benefit of using the random dataset for training. Table 1: We report the mean classification accuracy on our test set with images generated by the random image generator of five different seeds, and one standard error in the brackets. The first row is the evaluation of Speaker's visual representations trained with different datasets as discussed in Section 5.2. The second row is the image classification task of ETL as illustrated in Section 5.3.1. Fixed Variation Random Visual representation (%) 84.4 (5.5) 76.2 (8.9) 100.0 (0.0) ETL-image classification (%) 32.8 (8.1) 31.8 (7.2) 90.8 (2.8) Then we evaluate Speaker's visual representations learned through the referential game. We conduct a classification task to examine whether the visual representations encode conceptual information. We apply a linear classifier to the frozen CNN of Speaker and train it on our test set with images generated by the random image generator. Results in Table 1 demonstrate that agents trained with the random dataset learn better visual representations that capture conceptual information, and perform perfectly in the classification task on the test set. This shows us a promising direction that the referential game can serve as a good representation learning approach that may help encode highlevel abstract information in features. On the other hand, the variation dataset does not perform better than the fixed dataset, so the key factor influencing the quality of visual representations is the input variation between Speaker and Listener instead of variations in the dataset. Since representation learning plays an important role in emergent communication, the result tells us that input variation between Speaker and Listener should get attention. Language generalization in new tasks We adopt ETL proposed in Chaabouni et al. (2022), which is considered a more robust metric, to evaluate the ability of the emergent language to generalize to new Listener and new tasks. We conduct a image classification task in Section 5.3.1 as in Chaabouni et al. (2022). Moreover, we want to extend the new tasks to more complex multi-step MDP tasks, which can hardly be achieved if agents can only refer to single objects. We explore this with a task named Object Placement in Section 5.3.2. Image classification For the image classification task, We feed the deterministic language of Speaker to new Listener and train a linear classifier on the hidden state of Listener's sequence encoder on our test set with images generated by the random image generator. The results are shown in Table 1. We can find that ETL faithfully reflects the generalization ability of agents, with the random dataset showing the best performance. On the other hand, since ETL focuses on the information content conveyed by Speaker, the result implies that agents trained with the random dataset can express the positional relationship well. Note that the combinations are never seen by Speaker in the referential game, and the random image generator provides totally different images of the same content, but new Listener can easily understand the messages and achieve the classification accuracy over 90%, proving that Speaker has already learned to convey the conceptual information in images. Contrarily, agents trained with the fixed dataset and variation dataset cannot learn to communicate such information clearly. So in general, we can conclude that agents can learn to communicate multi-object positional relationships through emergent communication, but necessary environmental pressure should be involved, such as the input variation between Speaker and Listener. Object Placement Now, according to the analysis above, we have addressed the first question that agents can learn to express positional relationships in the context of the referential game. Then we explore the second one: whether the learned protocol can be helpful in multi-step MDP tasks with the ability to convey information about positional relationships. We design a task named Object Placement, as illustrated in Figure 6. Speaker observes a target image depicting the target positional relationship of two objects. It then sends a message to Listener, who should move the objects in the 3 × 3 grid to place them in the corresponding positional relationship. The action of Listener is to choose a grid and a Figure 6: Object Placement task. Speaker observes the target state (image) and describes it to Listener. Listener observes the grid world containing the two objects and receives the message from Speaker. Then it moves the objects to place them to form the correct positional relationship as depicted in the target state. direction, and if there is an object in the grid, the object is moved according to the direction by one grid. The observation of Listener is the state of the grid world and the message sent by Speaker. If Listener places two objects in the correct positional relationship, the reward is +1 and the episode terminates, otherwise, the reward is −0.01 for each step. The maximum episode length is set to 20. The target images are sampled from our training set generated by the random image generator. We use Speaker trained with the random dataset in the referential game, and generate deterministic messages to Listener. Listener uses a newly initialized sequence encoder to process the messages. We train Listener with PPO (Schulman et al., 2017). ② ⑤ ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ We also compare with five baselines: • The raw-pixel-input baseline uses target images to replace the messages sent by Speaker, and Listener learns a CNN model to process the images; • The cnn-feature baseline also uses target images to replace the messages, but Listener uses a frozen CNN model pre-trained on our training set with the random generator by an image classification task; • The simclr-feature baseline uses a pre-trained SimCLR model instead of the pre-trained CNN model compared with the cnn-feature baseline; • The rl-scratch baseline trains Speaker from scratch using REINFORCE to send messages. For this method, we train Speaker and Listener alternately; • The state baseline gives the true target relation to Listener directly, showing the optimal performance. Details for the Object Placement task and the baselines can be found in Appendix C. Figure 7 shows the learning curves of all the methods in the Object Placement task: the episode reward in Figure 7a, and the episode length of agents accomplishing the task in Figure 7b. Except the rl-scratch and raw-pixel-input baselines, all other methods converge to the same performance but differ in learning speed. Firstly, from the ETL's perspective, our Speaker's language can generalize pretty well in the new multi-step task, so new Listener can understand the message and learn a good policy in the new task quickly, close to the state baseline (the upper bound) that tells Listener the true target relationship. This demonstrates the generalization ability of the emergent language in the referential game, and shows that the agent has learned a general communication skill instead of a protocol overfitting to a single task. And this addresses our second question that emergent language in the referential game can be helpful in multi-step MDP tasks. Previous studies where agents learn to refer to single objects hardly explore the language transfer to multi-step tasks, probably because the object-level information is usually not sufficient for accomplishing these tasks. Our research on the learning of positional relationships can be seen as a step to break the restriction and towards the application of emergent communication in more complex tasks. Besides, the raw-pixel-input baseline fails to learn a policy to accomplish the task. This result proves that agents trained with deep reinforcement learning may feel difficult to capture the abstract information from raw-pixel images directly, so the Listener seems confused with this input. Therefore, state representations become important for reinforcement learning agents when the environment requires abilities for conceptual abstraction. Then which kind of representation is better? In Figure 7 we can find that, though the cnn-feature baseline and the simclr-feature baseline achieve comparable performance with our method that uses the learned Speaker, Listener learns faster if the input is discrete symbols. This is to some extent in line with the point of view in Garnelo et al. (2016) that conceptual abstraction provided by symbolic representations promotes data efficient learning. So it comes to the significance of research on language learning about conceptual information that is useful in various MDP tasks, such as positional relationships, spatial relationships, or numeric concepts (Guo et al., 2019). From the result of the rl-scratch baseline, we find that training Speaker and Listener directly in the Object Placement task gets poorer performance than using pre-trained emergent language. This may provide evidence that the referential game is more suitable to serve as a starting point for language learning, since it is easier for compositional and generalizable languages to emerge. It is reasonable because in the referential game Speaker receives the feedback more effectively. We then extend the task to scenarios involving multiple Listeners to test the robustness of the generalization ability of the emergent language. We modify the Object Placement task to the Multi-Listener Object Placement task where we now have two independent Listeners each can move one object in the grid world. Then they should cooperate to achieve the goal. Both Listeners receive the same observation containing the state of the grid world and the message from Speaker, and the action is the moving direction of the object they control. We also compare with the baselines in the last experiment. The results are shown in Figure 8. While all methods except the rl-scratch can perform well, our method using the learned symbolic language still learns faster, even compared with the goal state input. So the generalization ability of the emergent language is also effective when multiple new Listeners learn to understand the language at the same time. This shows the robustness of our finding that emergent language can be used in various MDP tasks thanks to its good generalization ability, and increasing its expressive power expands the range of its application in MDP tasks. Discussion The goal of emergent communication should be making neural agents acquire general communication skills instead of merely the ability to solve specific communication games. Many studies have been dedicated to the research on learning compositional languages in the context of referential games, but few have probed into the generalization of the emergent language to more complex tasks such as multi-step MDP tasks. We wonder about the viability of this development, while we argue referential games restricted to referring to single objects limit such development. So we go one step forward to explore communication about positional relationships, which may be an entry point of emergent communication about more high-level conceptual information. We first find that agents can learn to communicate positional relationships well through training with the referential game, but the key factor that influences the ability is the input variation between Speaker and Listener. So we may need stronger environmental pressure when more conceptual information is involved. We also show that we need stronger datasets to test the true generalization ability of emergent languages. Then we use a simple environment to evaluate the performance of language transfer from the referential game to a multi-step MDP task. We find that the emergent language, which can convey information about positional relationships, not only generalizes well in the new task, but also overperforms pre-trained image features and language learned directly in the specific task. So it verifies the viability of language transfer from referential games to more complex tasks, and shows a promising path to employ emergent communication for conceptual abstraction in complex environments and games. It is worth noting that we focus on learning positional relationships in the referential game in this paper, and we have carried out preliminary experiments of language transfer from the referential games to complex MDP tasks. The limitations in this work should be addressed in future: whether, or how, the learned positional relationships can generalize well to out-of-distribution datasets? Then the acquired communication skills can be applied to more diverse tasks. Besides, the Object Placement task in our work is somewhat simple, and we should explore language transfer to more general MDP tasks in future work. Furthermore, positional relationship is not enough for general tasks, whether other conceptual information can be learned through emergent communication? In addition to serving as a function similar to state representation, grounding the emergent language into actions in MDP tasks is also a future direction. Our work may be seen as one of the openings for research on task scaling up for more general agent language learning through emergent communication. A Agent Architecture and Hyperparameters Speaker architecture Speaker consists of an image encoder and a sequence generator. 1. The image encoder f θ is a reduced AlexNet, receiving images of size 128×128 and outputs embeddings of size 216. 2. A projector g θ maps the embedding f θ (x) into the initial hidden state of the sequence generator, composed of a Linear layer with input size of 216 and output size of 128 and a ReLU activation. 3. The sequence generator is an LSTM network with hidden size 128. 4. A Linear layer π θ with input size of 128 and output size of |V| maps the hidden state of the sequence generator h t into a logits vector, which is then fed to a softmax function to produce the symbol distribution. Listener architecture Listener consists of an image encoder and a sequence encoder. 1. The architecture of the image encoder f φ is the same as that of Speaker f θ , and the parameters are not shared across the Listener and the Speaker. 2. The sequence encoder l φ is an LSTM network with hidden size 256. It receives one-hot embeddings of symbols. 3. The MLP projector px ,φ is composed of a Linear layer with input size 216 and output size 128, a ReLU activation, and a Linear layer with input size 128 and output size 128. 4. The linear projector p m,φ is a Linear layer with input size 256 and output size 128. Other hyper-parameters The batch size and the candidate number |C| in the referential game are set to 32 for training and 20 for testing. The learning rate is 3e-5, and the entropy coefficient is 0.01. The batch size for classification tasks in Section 5.2 and Section 5.3.1 is 128. We use default Adam optimizer here with learning rate 3e-4. Listener in the Object Placement task is trained using default setups of PPO algorithm of the stable-baselines3 repository (Raffin et al., 2021). B Random Image Generator The random image generator takes three input parameters: two of them describing the shape of the objects and one for the positional relation between the objects, and generates an image according to the parameters. We added randomization to the size, rotation and position of each object. More precisely, the size (pixels) of each object is a random variable sampled from the interval [28,40] uniformly and independently, the rotation angle is uniformly sampled from 0 to 359 degrees. If the required positional relation is right, the horizontal displacement from one object to the other is uniformly sampled from the interval The object placement MDP task consists of a 3 × 3 grid with two objects in two different grids. The speaker is given the image of the target state of the MDP environment, while the listener is given the state-based description of each object, and need to move the objects to reach the target state. More precisely, the image given to speaker is generated by the random image generator described in Appendix B, the observation for the listener contains a 6-element tuple consisting the X,Y coordinate and shape index of each object and the output sequence from the speaker. At each time step, the speaker describes the image, and the output sequence is given to the listener along with the state-based observation of the MDP environment. The listener then selects a grid (represented by its coordinate) and a direction (right, left, up and down) as the action, which means the object on the selected grid should be moved to the adjacent grid in the selected direction. The move will be successfully applied to the environment if there is an object in the selected grid and the target grid of current movement is empty. The reward of each move is either 1.0 or −0.01 . The reward is 1.0 when the positional relationship between the two objects in the MDP environment is the same as which in the image given to the speaker, otherwise, the reward is −0.01 as a penalty. C.2 The Multi-Listener Object Placement task We modify the Object Placement task to involve multiple Listeners. Concretely, all the settings are the same as in C.1, except that there are two listeners and the action space is different from the original task. At each step, both listeners are given the same observation, which contains a 6-element tuple consisting the X,Y coordinate and shape index of each object and the output sequence from the speaker. Each listener then selects a direction (right, left, up and down) as the action. The action of the first listener will only control the moving direction of the first object while the action of the second listener will only control the moving direction of the second object. C.3 Baselines Raw-pixel-input This baseline provides the target image directly to Listener, and Listener use a CNN network to process the input image. Episode length ours raw-pixel-input cnn-feature simclr-feature rl-scratch rl-scratch-update (b) episode length of accomplishing the task Figure 9: Performance of new Listener trained with different inputs of the target state in the Object Placement task, with rl-scratch-update added. D Additional Results In the Object Placement task, we also try to finetune the learned Speaker from the referential game. We do this by running the rl-scratch baseline with Speaker initialized with the learned parameters. We call this setting rl-scratch-update. The results are shown in Figure 9. The performance of rlscratch-update is close to rl-scratch, probably because the learned language is destroyed during the exploration of the new task training. Figure 2 : 2Test accuracy of agents. (a) Agents trained with the fixed dataset or variation dataset are tested using the corresponding test set. (b) Agents trained with three kinds of datasets are tested with the test set of the random dataset. Figure 3 : 3Examples of generated sequences by Speaker after training with the random dataset. The images are from the test set. Figure 4 :Figure 5 : 45Train and test accuracy of agents whose image encoders are pre-trained by SimCLR with the random dataset. TopSim of agents trained with different datasets. Figure 7 : 7Performance of new Listener trained with different inputs of the target state in the Object Placement task. All experiments are run for 5 seeds, and the shaded part of the curves is one standard error. Figure 8 : 8Performance of new Listeners trained with different inputs of the target state in the Multi-Listener Object Placement task. All experiments are run for 5 seeds, and the shaded part of the curves is one standard error. [50, 88] and the vertical displacement is uniformly sampled from the interval[−5, 5] . If the required positional relation is top right, both the horizontal and vertical displacement is sampled from the interval [50, 88], uniformly and independently. If the required positional relation is top, the horizontal displacement is uniformly sampled from the interval [−5, 5] and the vertical displacement is uniformly sampled from the interval[50, 88] . If the required positional relation is top left, the horizontal displacement is uniformly sampled from the interval [−88, −50] and the vertical displacement is uniformly sampled from the interval[50, 88] . The background of the image generated is black (R = G = B = 0) and the shapes are colored white (R = G = B = 255). A noise sampled from N (0, 16) is added to each channel of pixels. Due to the symmetry of the positional relationships, we do not include left, bottom left, bottom, and bottom right. CNN-featureThe CNN network architecture for this baseline is the same as the image encoder of Listener in the referential game. We pre-train it on our training set using the random image generator with a classification task. We consider each of the 80 combinations as a class, and use cross-entropy loss to train the network. We apply an MLP projector to the output feature when pre-training, and the projector is abandoned in the Object Placement task.SimCLR-featureThis baseline is similar to the CNN-feature baseline, but the CNN network is pre-trained with Sim-CLR method, as described in the third paragraph in Section 5.1.RL-scratchWe train Speaker and Listener alternatively. When training Speaker, Listener is part of the environment. In each episode, Speaker produces a message, and we argmax Listener's policy to get actions in Object Placement task for the episode, computing the reward. We then train Speaker using RE-INFORCE with entropy regularization. When training Listener, we fix Speaker and the process is the same as our method. In each phase we train 1000 steps for both Speaker and Listener.StateThis baseline directly provides the current state and target state to the Listener, and Listener use a MLP network to process the input states. Emergence of communication in an interactive world with consistent speakers. Ben Bogin, Mor Geva, Jonathan Berant, arXiv:1809.00549arXiv preprintBen Bogin, Mor Geva, and Jonathan Berant. Emergence of communication in an interactive world with consistent speakers. arXiv preprint arXiv:1809.00549, 2018. How agents see things: On visual representations in an emergent language game. Diane Bouchacourt, Marco Baroni, In EMNLP. Diane Bouchacourt and Marco Baroni. How agents see things: On visual representations in an emergent language game. In EMNLP, 2018. Understanding linguistic evolution by visualizing the emergence of topographic mappings. Henry Brighton, Simon Kirby, Artif. Life. Henry Brighton and Simon Kirby. Understanding linguistic evolution by visualizing the emergence of topographic mappings. Artif. Life, 2006. Anti-efficient encoding in emergent communication. Rahma Chaabouni, Eugene Kharitonov, Emmanuel Dupoux, Marco Baroni, NeurIPS. Rahma Chaabouni, Eugene Kharitonov, Emmanuel Dupoux, and Marco Baroni. Anti-efficient en- coding in emergent communication. In NeurIPS, 2019. Compositionality and generalization in emergent languages. Rahma Chaabouni, Eugene Kharitonov, Diane Bouchacourt, Emmanuel Dupoux, Marco Baroni, ACL. Rahma Chaabouni, Eugene Kharitonov, Diane Bouchacourt, Emmanuel Dupoux, and Marco Baroni. Compositionality and generalization in emergent languages. In ACL, 2020. Igor Mordatch and Pieter Abbeel. Emergence of grounded compositional language in multi-agent populations. Rahma Chaabouni, Florian Strub, Florent Altché, Eugene Tarassov, Corentin Tallec, Elnaz Davoodi, AAAI. Rahma Chaabouni, Florian Strub, Florent Altché, Eugene Tarassov, Corentin Tallec, Elnaz Davoodi, Igor Mordatch and Pieter Abbeel. Emergence of grounded compositional language in multi-agent populations. In AAAI, 2018. Stable-baselines3: Reliable reinforcement learning implementations. Antonin Raffin, Ashley Hill, Adam Gleave, Anssi Kanervisto, Maximilian Ernestus, Noah Dormann, Journal of Machine Learning Research. Antonin Raffin, Ashley Hill, Adam Gleave, Anssi Kanervisto, Maximilian Ernestus, and Noah Dor- mann. Stable-baselines3: Reliable reinforcement learning implementations. Journal of Machine Learning Research, 2021. Compositional languages emerge in a neural iterated learning model. Yi Ren, Shangmin Guo, Matthieu Labeau, Shay B Cohen, Simon Kirby, ICLR. Yi Ren, Shangmin Guo, Matthieu Labeau, Shay B. Cohen, and Simon Kirby. Compositional lan- guages emerge in a neural iterated learning model. In ICLR, 2020. John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, Oleg Klimov, arXiv:1707.06347Proximal policy optimization algorithms. arXiv preprintJohn Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. Emergent discrete communication in semantic spaces. Mycal Tucker, Huao Li, Siddharth Agrawal, Dana Hughes, Katia P Sycara, Michael Lewis, Julie A Shah, NeurIPS. 2021Mycal Tucker, Huao Li, Siddharth Agrawal, Dana Hughes, Katia P. Sycara, Michael Lewis, and Julie A. Shah. Emergent discrete communication in semantic spaces. In NeurIPS, 2021. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning. J Ronald, Williams, Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 1992.
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[ "CONSISTENT USE OF TYPE Ia SUPERNOVAE HIGHLY MAGNIFIED BY GALAXY CLUSTERS TO CONSTRAIN THE COSMOLOGICAL PARAMETERS", "CONSISTENT USE OF TYPE Ia SUPERNOVAE HIGHLY MAGNIFIED BY GALAXY CLUSTERS TO CONSTRAIN THE COSMOLOGICAL PARAMETERS" ]
[ "Adi Zitrin ", "Matthias Redlich ", "Tom Broadhurst " ]
[]
[]
We discuss how Type Ia supernovae (SNe) strongly magnified by foreground galaxy clusters should be self-consistently treated when used in samples fitted for the cosmological parameters. While the cluster lens magnification of a SN can be well constrained from sets of multiple images of various background galaxies with measured redshifts, its value is typically dependent on the fiducial set of cosmological parameters used to construct the mass model to begin with. In such cases, one should not naively demagnify the observed SN luminosity by the model magnification into the expected Hubble diagram, which would then create a bias, but take into account the cosmological parameters a-priori chosen to construct the mass model. We quantify the effect and find that a systematic error of typically a few percent, up to a few-dozen percent, per magnified SN, may be propagated onto a cosmological parameter fit, unless the cosmology assumed for the mass model is taken into account (the bias can be even larger if the SN is lying very near the critical curves). We also simulate how such a bias propagates onto the cosmological parameter fit using the Union2.1 sample, supplemented with strongly magnified SNe. The resulting bias on the deduced cosmological parameters is generally at the few percent level, if only few biased SNe are included, and increasing with the number of lensed SNe and their redshift. Samples containing magnified Type Ia SNe, e.g. from ongoing cluster surveys, should readily account for this possible bias.
10.1088/0004-637x/789/1/51
[ "https://arxiv.org/pdf/1311.5224v3.pdf" ]
19,725,378
1311.5224
7796f15484b308ab3a1a777dd8a325f920e0f4cb
CONSISTENT USE OF TYPE Ia SUPERNOVAE HIGHLY MAGNIFIED BY GALAXY CLUSTERS TO CONSTRAIN THE COSMOLOGICAL PARAMETERS Adi Zitrin Matthias Redlich Tom Broadhurst CONSISTENT USE OF TYPE Ia SUPERNOVAE HIGHLY MAGNIFIED BY GALAXY CLUSTERS TO CONSTRAIN THE COSMOLOGICAL PARAMETERS Submitted to the Astrophysical Journal Submitted to the Astrophysical JournalPreprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: supernovae: generalgalaxies: high-redshiftgravitational lensing We discuss how Type Ia supernovae (SNe) strongly magnified by foreground galaxy clusters should be self-consistently treated when used in samples fitted for the cosmological parameters. While the cluster lens magnification of a SN can be well constrained from sets of multiple images of various background galaxies with measured redshifts, its value is typically dependent on the fiducial set of cosmological parameters used to construct the mass model to begin with. In such cases, one should not naively demagnify the observed SN luminosity by the model magnification into the expected Hubble diagram, which would then create a bias, but take into account the cosmological parameters a-priori chosen to construct the mass model. We quantify the effect and find that a systematic error of typically a few percent, up to a few-dozen percent, per magnified SN, may be propagated onto a cosmological parameter fit, unless the cosmology assumed for the mass model is taken into account (the bias can be even larger if the SN is lying very near the critical curves). We also simulate how such a bias propagates onto the cosmological parameter fit using the Union2.1 sample, supplemented with strongly magnified SNe. The resulting bias on the deduced cosmological parameters is generally at the few percent level, if only few biased SNe are included, and increasing with the number of lensed SNe and their redshift. Samples containing magnified Type Ia SNe, e.g. from ongoing cluster surveys, should readily account for this possible bias. INTRODUCTION A Type Ia Supernova (SN) is an extremely luminous explosion of a star, typically a white dwarf in a binary system. Although there is still a debate regarding its exact progenitor mechanism (e.g. Maoz & Mannucci 2012), an important property of a Type Ia SN is that its absolute peak luminosity is to a very good approximation well known (up to the Hubble constant, M B ≈ −19.5 mag, see Riess et al. 1998;Hillebrandt & Niemeyer 2000 and references therein), thereby constituting a standardizable candle (e.g. taking into account the Luminosity-Decline Rate relation). In fact, it is thanks to this quality that we have learned greatly about the expansion of the Universe, particularly by comparing the standardized luminosities of many Type Ia SNe in different redshifts (e.g. Riess et al. 1998;Perlmutter et al. 1999). Clusters of galaxies act as strong gravitational lenses, distorting and magnifying background objects. When the surface mass density in the center of the cluster is high enough (higher than the critical density required for strong lensing; e.g. Narayan & Bartelmann 1996), often multiple images of the same background source are formed. Sets of multiple images in different redshifts are used therefore to constrain the underlying mass distribu-tion and profile of the cluster's core (e.g. Broadhurst et al. 2005;Smith et al. 2005;Limousin et al. 2008;Richard et al. 2010b;Newman et al. 2013;Zitrin et al. 2013a,b), dominated by an unseen dark matter (DM). Farther away from the center, where the surface density is lower, the gravitational potential of the cluster distorts and magnifies background objects (without forming multiple images of the lensed sources), and this weaker lensing effect can be used, statistically, to constrain the larger-scale mass distribution and profile of the cluster (e.g. Merten et al. 2009;Umetsu et al. 2010;Oguri et al. 2012;Newman et al. 2013). Lensing thus provides a unique way to map the DM in these massive objects. Aside from mapping the unseen DM, lensing and especially the magnification by galaxy clusters has become of great interest, as it allows faint very-distant galaxies, which would otherwise be below the detection threshold, to be observed. Recent observations have made use of this magnification power to detect several compelling galaxy candidates at redshifts up to z ∼ 10 − 11 Zheng et al. 2012;Bouwens et al. 2012;Bradley et al. 2013), and more are anticipated in the Frontier Fields program with the Hubble Space Telescope (HST) 6 . SNe which happen to explode in galaxies behind galaxy clusters, will therefore be magnified. In general, they are expected to appear in the same number density (or rate, see Goobar et al. 2009; Barbary et al. 2012 and references therein) as in the field of a similar redshift, divided by the magnification factor which narrows the effective source-plane area, but supplemented by fainter or more distant SNe (for general discussion of the magnification bias see Broadhurst et al. 1995;Mashian & Loeb 2013), thus allowing to detect higher redshift SNe (e.g. Benítez et al. 2002;Amanullah et al. 2011;Barbary et al. 2012;Pan & Loeb 2013;Whalen et al. 2013). As the current cosmological parameters are derived from a Hubble diagram of SNe up to z ∼ 2, higher redshift Type Ia SNe should tighten the constraints on the cosmological parameters. Following Refsdal (1964), many other works and dedicated surveys (e.g. Kolatt & Bartelmann 1998;Holz 2001;Goobar et al. 2002Goobar et al. , 2009Oguri & Kawano 2003;Dawson et al. 2009;Suzuki et al. 2012;Riehm et al. 2011;Quimby et al. 2014 ;and references therein), have dealt with the possibility of observing a multiply-imaged SN, and the possibility of making use of measured time delays between the different multiple images to recover the Hubble constant, or other cosmological parameters. This is particularly appropriate for galaxy-scale lenses, where the time delay is observationally reasonable. In fact, time delays have been used in several studies to constrain the Hubble constant, making use typically of quasars multiply imaged by field galaxies (e.g. Suyu et al. 2010, see also Oguri 2007Treu et al. 2013 and references therein). Some of the works mentioned above have also referred to, or uncovered, a single image (i.e. not multiply lensed) of a SN magnified by a cluster, but only in the context of adding a constraint to the mass model through the local independent estimate of the magnification in the case of a Type Ia (e.g. Riehm et al. 2011;Nordin et al. 2013), or vice versa, using the magnification by the lens model to recover the SN demagnified luminosity (assuming a priori a set of cosmological parameters, e.g. Patel et al. 2013, see also Suzuki et al. 2012;Amanullah et al. 2011). If highly magnified SNe were then to be used as part of samples fitted for the cosmological parameters, one should not naively demagnify the lensed SN luminosity by the magnification factor given by the mass model, but take into account the cosmological parameters that were used to construct it. The idea is quite simple in essence: one usually makes use of the fact that the mass-sheet and profile degeneracies are effectively already broken by various sets of multiple images typically uncovered in e.g. deep HST observations of cluster fields (e.g. Broadhurst et al. 2005;Smith et al. 2005;Limousin et al. 2010;Richard et al. 2010a;Zitrin et al. 2012, as few examples; see also references therein), to construct a magnification map, determining the magnification of background objects such as lensed SNe in our case (e.g. Amanullah et al. 2011). However, since there is a degeneracy between the cosmological parameters and the resulting mass-model profile, which is typically left free to be fit by the data, this magnification is dependent on the cosmological parameters initially used to constrain the mass model. In such cases, to avoid circularity, one could use a simple analytic correction as the one we propose here as one example, simultaneously while fitting for the cosmological parameters, in order to disentangle the magnification value from the pre-assumed cosmology. Alternatively, one could simply take into account the possible systematic uncertainty induced by ignoring this effect; an uncertainty which we make an effort to quantify. Because the resulting mass profile is dependent on the assumed cosmological parameters, several works (e.g. Jullo et al. 2010;Lefor & Futamase 2013) have shown that parametric strong-lensing (SL) or mass modeling techniques can be quite sensitive to the lensing distance of multiply-imaged sources, thus allowing to actually constrain the cosmological parameters. On the other hand, other works have shown that this dependence is rather weak (e.g. Zieser & Bartelmann 2012), and more recent works have claimed to break or bypass the degeneracy between the profile and cosmological parameters, constraining them in a free-form modeling with minor assumptions on the mass profile shape; see Lubini et al. (2013); Sereno & Paraficz (2013). On a different front, Jönsson et al. (2010), for example, exploited a large sample of Type Ia SNe magnified by foreground galaxies, to place constraints on the halos of the lensing galaxies, while fixing the cosmology and the mass profile shape (see also Karpenka et al. 2013). Here, given recent and ongoing cluster surveys designed to detect strongly-magnified (and not necessarily multiply-imaged) SNe, which due to their magnification are also likely to expand the known Type Ia SNe redshift range (see also Benítez et al. 2002;Amanullah et al. 2011;Salzano et al. 2013), we highlight, as mentioned, how these strongly-magnified SNe should be properly treated when eventually used in samples fitted for the cosmological parameters (for example, some SNe more weakly magnified by galaxy clusters were used for that purpose as part of the Union2.1 sample, see Suzuki et al. 2012), so that no bias is propagated from the cosmology assumed a-priori when constructing the lens model. Many works have shown that a similar magnification or cosmology correction is also needed, statistically, when treating large samples of weakly magnified field SNe (e.g. Linder et al. 1988;Wambsganss et al. 1997;Holz & Wald 1998;Schmidt et al. 1998;Bergström et al. 2000;Holz & Linder 2005;Linder et al. 1988;Sasaki 1987;Martel & Premadi 2008;Amendola et al. 2013;Marra et al. 2013;Quartin et al. 2013), suggesting how one should correct for the global magnification effect on their PDF in order to avoid a bias on the observed SN distance-redshift relation and the inferred cosmological parameters (see also Smith et al. 2013;Amanullah et al. 2003). We aim to show that also small numbers of strongly magnified Type Ia SNe, and especially since these are expected to be observed to higher redshifts, can be useful as part of a sample fitted for the cosmological parameters, independently of the cosmology assumed for the lens model (but still depending on the mass model parametrization). The presented methodology, although basic, evaded to our knowledge any discussion in previous works in this context (but some works have indeed properly quoted the cosmological parameters used to derive the magnification of magnified SNe, e.g. Benítez et al. 2002). For our purpose, for simplicity, and since galaxy clusters are known to locally follow such mass profile forms (e.g. NFW Navarro et al. 1996; see also Broadhurst et al. 2005;Zitrin et al. 2009;Umetsu et al. 2012), we shall examine a simplified case by approximating the cluster mass profile in the strong-lensing regime, which is the area of interest in this work, with a powerlaw. This could be then generalized in future works. This brief work is organized as follows: in §2 we show the dependence of the fitted mass profile on the cosmological parameters, and present a simplified method to correct the cosmology-dependent magnification of Type Ia SNe. In §3 we discuss the magnitude of the effect or bias in question, both on individual SNe and when propagated onto the Union2.1 sample supplemented with mock lensed SNe, and conclude the work, then summarized in §4. METHODOLOGY The reduced deflection angle due to a given mass distribution, in the thin lens approximation, at a position θ is given by: α( θ) = 4G c 2 d ls d l d s ( θ − θ )Σ( θ ) | θ − θ | 2 d 2 θ ,(1) where d l , d s , and d ls are the cosmology-dependent lens, source, and lens-to-source angular diameter distances, respectively, and Σ is the projected surface mass density distribution. Eq. 1 manifests the degeneracy between the lensing distance d ls d l ds , and hence the cosmological parameters, and the mass distribution. Correspondingly, lens modeling in complex systems such as galaxy clusters comprising various sets of multiple images, typically requires one to assume a set of cosmological parameters, while leaving the mass-density profile free to be fitted for. The magnification estimate for positions and background-source redshifts different from the lensing observables used as constraints, is thus cosmology dependent. Imagine a background SN is observed at an angular distance θ SN from the center of a (hereafter for simplicity, spherically symmetric) massive cluster. We now show explicitly how the mass profile, and thus the magnification, depend on the assumed cosmology; a dependence which can in turn be used to self-consistently rescale the magnification with the cosmological parameters. In what follows, two Einstein radii and enclosed masses (M (< θ e,i ) and M (< θ e,j ), i = j), are sufficient to show the said dependence. 2.1. Example lens: a power-law A surface density power-law profile can be written as Σ(r) = Σ 0 ( r r0 ) −q , where r is the physical distance from the center, and r 0 an arbitrary, normalization scale radius. The mass enclosed within an angular distance θ is obtained by integration of the latter density profile, while remembering that r = d l θ and r 0 = d l θ 0 , to obtain M (< θ) = 2πΣ0(d l θ0) 2 2−q ( θ θ0 ) 2−q . The general deflection angle is given by: α(θ) = 4GM (< θ) c 2 θ d ls d s d l ,(2) or more explicitly by inserting M (< θ) from above: α(θ) = 8πGΣ 0 θ 0 (2 − q)c 2 d l d ls d s ( θ θ 0 ) 1−q .(3) For a circularly symmetric lens, the dimensionless surface mass density, shear, and magnification at each position θ are generally given by, respectively: κ (θ) = 1 2 α(θ) θ + dα(θ) dθ ,(4)γ (θ) = 1 2 α(θ) θ − dα(θ) dθ , (5) µ −1 (θ) = (1 − κ) 2 − γ 2 ,(6) where for a power-law surface density as above, the term dα(θ) dθ simply equals α(θ) θ (1 − q). Plugging in now eqs. 4 and 5 into eq. 6, one obtains: µ −1 (θ) = 1 + (q − 2) α(θ) θ + (1 − q)( α(θ) θ ) 2 ,(7) and α(θ) is given in eq. 3. The Einstein radius for a given multiply-imaged galaxy, is given (in the spherically symmetric case for example) by: θ e = 4GM (< θ e ) c 2 d ls d s d l 1/2 ,(8) where more generally for a non-spherical case, the effective Einstein radius can be defined either as the radius within which κ = 1, or preferably (e.g. Bartelmann 1995), simply as the effective radius of the area enclosed within the critical curves for the redshift of the multiplyimaged galaxy. The Einstein radii are in any case observables, or deduced directly from them, and thus independent of the assumed cosmology, while the enclosed mass is cosmology dependent. Having two measurements of the enclosed mass at e.g. M (< θ e,i ), and M (< θ e,j ), say, from the lens model constructed using various sets of multiple images and assuming a certain cosmology, a power-law mass profile could be readily fitted by: q = 2 − log M (<θe,i) M (<θe,j ) log( θe,i θe,j ) ,(9) and Σ 0 = 2 − q 2π(d l θ 0 ) 2 M (< θ e,i )( θ e,i θ 0 ) q−2 .(10) 2.2. Correcting for the assumed cosmology Recall from eq. 8 that for a given Einstein radius θ e , the mass enclosed inside θ e is linear in the term defined hereafter as D = d l ds d ls , so that M (< θ e ) ∝ D, or explicitly: M (< θ e,i ) = θ 2 e,i c 2 4G D (zi) ,(11) where z i is the redshift of the lensed source galaxy whose Einstein angle is θ e,i , and D depends on the respective lens-and lens-to-source distances. The cosmological parameters affect the physical units of the mass model through D, via the angular diameter distances: d A (za,z b ) = c/H 0 1 + z b z b za dz Ω (0) m · (1 + z) 3 + Ω Λ (z) −1/2 ,(12) for a flat two-component universe as an example. Therefore, the modified enclosed mass, M , meaning the mass given a modified set of cosmological parameters embedded in D , is then given by: M (< θ e,i ) = θ 2 e,i c 2 4G D (zi) ,(13) or, M (< θ e,i ) = M (< θ e,i ) D (zi) D (zi) ,(14) where M (< θ e,i ) is the Einstein mass of the ith system, with the set of cosmological parameters used to constrain the mass model to begin with. Making use of the above, the modified power-law mass profile, i.e. as if the mass model were constructed with any other given set of cosmological parameters embedded in the term D , can be readily calculated as: q = 2 − log   M (<θe,i) D (z i ) D (z i ) M (<θe,j ) D (z j ) D (z j )   log( θe,i θe,j ) ,(15) and Σ 0 = 2 − q 2π(d l θ 0 ) 2 M (< θ e,i ) D (zi) D (zi) ( θ e,i θ 0 ) q −2 .(16) From this, the "new", corrected magnification can be immediately calculated via eq. 3-6, inputting q and Σ instead of q and Σ, respectively. This result is discussed further in §3. DISCUSSION AND CONCLUSIONS In §2, we demonstrated the known degeneracy between a mass-model density profile, and the cosmological parameters. We have shown that by approximating the resulting mass profile with a known analytic form, the said degeneracy can be in turn used to self-consistently rescale the magnification estimate of a lensed SN, with the cosmological parameters. The approximation we showed is useful since it does not require remaking the usuallycomplex lens model for each probed set of cosmological parameters, which would be a hard and time consuming task (an order of hours on nowadays machinery, for each full-minimization iteration). Instead, one could use the above quick-to-calculate relation to readily obtain the SN magnification as a function of cosmology, given the initial mass model and the assumed fiducial set of cosmological parameters. However, since the suggested correction is itself model dependent, it may be instead useful to simply account for the systematic uncertainty entailed by ignoring the cosmology assumed for the lens model. To estimate the magnitude of this bias, so that instead of using the above approximation, lensed SNe could be fitted for while not underestimating the uncertainties on their demagnified luminosities, one should examine the susceptibility of the magnification estimate to the cosmological parameters. This is shown in Figs. 1-3, where we also give further explicit details. In Fig. 1, we plot the ratio between the magnification given a set of cosmological parameters used to construct the mass model, and the magnification obtained with the "true" cosmological parameters, for different configurations, as a function of cluster redshift. Fig. 2 shows the same effect as in Fig. 1, now as a function of SN redshift, and Fig. 3, as a function of the difference between the underlying cosmology, and that assumed for the mass model. The magnitude of the bias clearly changes as a function of the observables (e.g. Einstein radii and source redshifts, SN position), cluster and SN redshift, and the difference between the cosmology assumed for the mass model and the "true" one. As seen, the magnitude of the bias created per lensed SN is typically of an order of a few percent, especially if the SN is observed at a larger angle, far from enough the Einstein ring, although some configurations can yield a bias of up to a few-dozen percent or higher, especially for lower-z clusters, or if the SN is close to the center (or to the critical curves). This shows that the effect in question can, in principle, be significant. If the mass model was, as is often the case, constructed with cosmological parameters ∼ 10% away from the "true" parameters, the effect is typically less than ∼ 1% and thus rendered negligible. If the difference between the assumed cosmology and the true one, is higher, the bias can as significant as ∼ 20%, per SN. In that aspect, for comparison, we also mention that typical modeling errors of current high-end lens models (for fixed cosmology) are of order ∼ 15 − 20% on the magnification in most of the region of interest (i.e. not too close to the critical curves), and systematic errors between different parameterizations are typically of the same order. The bias we discuss in this work, as mentioned, is in most cases smaller, but still significant even in light of the non-negligible errors on the deduced magnification when a fixed cosmology was assumed. In the era of precision cosmology and with the expected numbers of SNe into higher redshifts, one should use these corrections not to create (even a small) bias, or alternatively, take into account the estimated systematic uncertainties. We also note that another alternative, would be making the mass model while allowing the cosmological parameters to be free, thus marginalizing over their effect on the magnification which would be reflected in the quoted errors. To test how the cosmology affects real-cluster lens models, we chose one CLASH cluster with 2 spectroscopically measured multiply-imaged galaxies, at z 1.5 and z 3 (all CLASH mass models will be soon published in Zitrin et al., 2014 in prep, including the multiple images and exact redshifts). We then constructed two SL models, using the lens model code described in Zitrin et al. (2013a,b) which include realistic representations for both the cluster lens galaxies and the DM. For our purpose, the first model here is constructed using [Ω m = 0.3, Ω Λ = 0.7], and the second is constructed using a very distinct flat-universe cosmology, [Ω m = 1, Ω Λ = 0]. The different cosmologies, in practice, translate to a ∼ 15% difference in the effective, relative lensing distances. We then examined the resulting magnification maps in the 2x2 arcmin central FOV around the BCG. We find that with respect to the [Ω m = 0. The upper x-axis shows the ratio of the power-law exponent given the "modified" cosmology and the power-law exponent given the "true" cosmology, for each configuration. In all cases we assume a circularly symmetric lens with two Einstein rings observed at θ e,1 = 10 and θ e,2 = 20 , of sources at z 1 = 1 and z 2 = 2, respectively. As can be seen, assuming for the lens modeling a cosmology which is only ∼ 10% different from the "true" underlying cosmology, results here in a minor < 1% bias. However, more extreme differences between the assumed and probed cosmologies, can yield significant systematic errors of ∼ 20% on the demagnified SN luminosity, decreasing with lens redshift. shrinks towards the SL regime, and the median reaches ∼ 8% in the central 1x1 arcmin field, which corresponds roughly to the SL regime of this test cluster. We take this median value as the more representative one (the mean here is much higher than the median because of the diverging critical curves), and conclude that in the SL regime, close to a (median value of) ∼ 10% bias in the magnification can be induced if the wrong cosmology is used. We note, however, that in this paper we focus on introducing the bias and assessing its order of magnitude, showing that in principle, it can be significant and should be taken into account. A more thorough estimate of the bias in real clusters, including also, for example, realistic SN distributions in redshift convolved with lensing models and a general cluster mass function, should be performed elsewhere. We make an additional effort to examine how the possible bias on individual, lensed SNe, shown in Figs. 1-3, propagates into the cosmological fit for a Union2.1-like sample. For that purpose we downloaded the Union2.1 sample 7 and reran a cosmological fit to their data, starting by fitting the original data including the 580 SNe listed therein, and then supplementing it with increasing numbers of magnified SNe. For each minimization we run a simple a Monte-Carlo Markov Chain (MCMC) with Metropolis-Hastings algorithm, to obtain the best fit. Note that the minimization or best-fit criterion we use here is a simple χ 2 defined as: The top panel shows a case where the SN is well outside the Einstein radius. As in Fig. 1, assuming for the lens modeling a cosmology which is only ∼ 10% different from the "true" underlying cosmology, results typically in a minor, few-percent bias. More extreme cosmology differences can yield significant systematic errors of about ∼ 20% in the cases probed here. The bottom panel case shows that if the SN is closer to the center, or near the narrow critical curves more explicitly, the effect can be much larger, reaching hundreds of percent. More importantly, the bias can reach up to ∼ 50% (depending on the cosmology difference) for higher redshift SNe, and especially if they lay within the critical curves for that redshift. Note that in this figure, the ratio of exponents of the "modified" and "true" cosmology mass models (top x-axis), is constant, because as expected the shape of the lens is not affected by the SN position or redshift. χ 2 = SN e µ B − µ B,f it σ 2 err ,(17) where µ B and µ B,f it are 8 the observed distance modulus, and that predicted by the fit, respectively, and σ err is the error specified in the Union2.1 table available online. As a first test, the best-fit values for the original sample we obtain are: Ω m = 0.2776 +0.1421 −0.1032 and w = −1.0005 +0.1951 −0.4521 (1σ errors). The best-fit values are in excellent agreement with those published in Suzuki et al. (2012), e.g., w = −1.001 +0.348 −0.398 , albeit the errors are somewhat different, probably due to the difference in the χ 2 definition and the inclusion of other system-8 note that here µ B are the distance moduli, while throughout, the lensing magnification is also marked as µ (i.e. without the capital "B"), following traditional notation. atics therein. Here however we only need to work in our self-consistent frame-of-reference to check the effect of including magnified SNe in the fit, on the resulting cosmological parameters. We note that the errors in the following scenarios we probe are all similar throughout, and we shall only focus on the difference between the best-fit values themselves, unless otherwise stated. After the initial fit we run to the original Union2.1 sample, we then plant SNe drawn from a uniform distribution between z = 0.5 and up to either z = 1.5, z = 2, z = 3, or z = 5, for the different scenarios we consider (as mentioned, lensed SNe should in principle be observed to higher redshifts than field SNe). The SNe are planted following a distance modulus-redshift relation with the best-fit parameters from the initial fit to the full sample, and with a random Gaussian scatter of σ = 0.15, and a random Gaussian error-scatter of 1% + σ err , with σ err = abs(0.3), in their distance moduli. The lumi- 1 & 2, but now fixing the lens configuration, and changing the "true" underlying (flat-universe) cosmology, whereas the cosmology used for constructing the model is unchanged. As expected, the bias vanishes (µ model /µ real reaches unity) when the model and true cosmologies are similar, and is maximal when the cosmologies significantly differ. Note also that due to inherent degeneracy in the dependence of the magnification on the cosmological parameters, there can be, as seen, other cosmologies that yield occasionally similar magnification values (and thus zero bias, µ model /µ real = 1). nosity bias propagated per demagnified SNe is taken as 5%, 10%, or 20%, for the different scenarios we examine here. Examples of real+mock, distance-modulus vs. redshift relation are seen in Fig. 4. The propagated bias on the overall fit turns out to be non-negligible, even with relatively small numbers of lensed SNe. Ten mock lensed SNe with a 10% bias on the demagnified luminosity of each, for example, drawn from a distribution as described above up to z = 2, create a shift (or bias) of 5% and 3% on the best fit Ω m and w, respectively. Increasing the redshift upper limit to z = 3 brings the overall bias to 13% and 7%, respectively. When increasing the bias on each individual SNe demagnified luminosity to 20%, a z < 1.5 sample yields a bias of 7% and 3% on the best fit Ω m and w, respectively, and the z < 3 sample yields a bias of 15% and 8% on the two parameters, respec-tively. Decreasing the number of lensed SNe to as few as five, or lowering the individual bias to 5%, reduces the overall bias by a few times, but tripling the number of lensed SNe to 30 up to z = 3, can reach a large bias of 25% and 13% on the two parameters, respectively. Although in most probed cases the resulting bias is < 1σ, some configurations yield biases that can be more significant, increasing with the individual bias on the demagnification factor, the number of lensed SNe, and their redshift. As a final consistency check, we run two additional minimization chains while planting 20 higher-redshift SNe up to z = 5 following our initial fit to the original Union2.1 sample. The first case includes unbiased SNe, and the second case includes SNe biased by ∼ 10% as above. The first chain results as expected, in cosmological parameters (Ω m and w) identical to those obtained The mock SNe imitate observed, magnified SNe, demagnified back to their unlensed luminosities with a magnification factor biased by the amount specified in each subfigure. This luminosity bias can be created if one neglects the cosmology assumed for the lens model (see Figs. 1-3). The solid black lines show our fit to the original Union2.1 sample, and the magenta dash-dotted lines show the fit to the entire sample including the mock SNe. The corresponding best-fit values, assuming a flat universe and a fixed equation of state parameter, are shown in the Legends, and demonstrate the overall bias created. We also specify for each subfigure some input restrictions used for generating the mock catalogs (see §3 for more details). by the fit to the original Union2.1 sample, but with errors lower by ∼ 15 − 20%, indicating, as expected, that including higher-redshift SNe improves the constraints on the cosmological parameters. In the second chain, we work on the sample containing the ∼ 10%-biased mock SNe, but now take into account this additional systematic uncertainty in the fit, increasing the errors on the planted SNe, correspondingly, to include the ∼ 10% uncertainty originating from the bias. This, in order to examine, briefly, if including magnified (i.e. possibly biased) SNe in the fit is worthwhile. We obtain that the cosmological parameters are reproduced with a > 99.99% accuracy, and the errors on them remain the same as for the original Union2.1 sample (but not smaller, despite including higher-redshift galaxies). This indicates that it is indeed worthwhile including magnified SNe in the fit, if the possible bias discussed in this work is accounted for as an additional error on their magnitude or distance modulus, so the resulting cosmological parameters will indeed remain unbiased. Since our goal here was simply to introduce the effect of the cosmological parameters assumed for constructing the mass model on the measured magnification, assess its order of magnitude, and show how it can be corrected for when fitting for the cosmological parameters, we brought one simple example using an idealized parametrization of a (circularly symmetric cluster) power-law mass profile, to rescale the magnification with cosmology. Clearly, this power-law approximation cannot describe the usually more-complex mass profile to a tee, and thus in practice, can create its own model-dependent bias (although we know from previous analyses the approximation is reasonable for the inner SL region, e.g. Broadhurst et al. 2005;Zitrin et al. 2009). Other, analytic or more flexible parameterizations, which may be better fitted per cluster, can be developed in future studies. To estimate the dependence of the magnification on the cosmological parameters more generally, these can include, for example, non-parametric (free-form mass profile) methods marginalizing over the cosmological parameters, or a Taylor-expansion of the magnification in the cosmological parameters. The rate of SNe behind clusters, as mentioned, was examined before in various works (e.g. Sullivan et al. 2000;Barbary et al. 2012;Goobar et al. 2009;Riehm et al. 2011;Postman et al. 2012;Li et al. 2012;Salzano et al. 2013;Quartin et al. 2013;Graur et al. 2013), and we gather that an order of magnitude of few Type Ia SNe within the HST's FOV are expected, per observed cluster with a typical depth of say, ∼ 27 AB spread over few years, with ∼weekly-to-monthly visits. However, as these are very crude numbers and depend exhaustively on the observational plan and lensing strength, we refer the reader to the works mentioned above for specific details. In our work here, we merely introduce and characterize the bias in question and do not attempt to assess its realistic distribution in Universe, following for example SN luminosity functions convolved with realistic mass models and a cluster mass function. We leave such estimates for future studies. Lastly, one should also comment on the weak-lensing regime, in which the magnification is typically small, approaching 1 in the outskirts of the cluster. Despite the smaller magnification, the significantly larger area covered by the weak-lensing regime (i.e. out to the virial radius and beyond) is advantageous, and large numbers of slightly-magnified SNe might be uncovered, to further reduce the statistical errors and form a useful representative sample, which could make use of similar corrections to those outlined here, albeit these are expected to be correspondingly smaller. SUMMARY A cluster mass model constructed from various sets of multiple images can be used to estimate the magnification at the position where a highly magnified Type Ia SN is seen. If the demagnified SN brightness were then to be used as part of a sample fitted to constrain the cosmological parameters, to avoid a bias originating from the cosmology assumed for the lens model, the latter should be accounted for. We showed that in principle this can be done in a simple and elegant way, by approximating the resulting mass profile with a known analytic form. More importantly, and especially since such a correction is by itself model dependent, we quantified the effect of ignoring the cosmology assumed for the lens model, on the magnification estimate of lensed SNe. We have found that a systematic error of typically a few percent, up to a few-dozen percent, per magnified SN, can be propagated onto a cosmological parameter fit, unless the cosmology assumed for the mass model is taken into account. In some specific cases, the bias can be even larger, for example if the SN is lying very near the critical curves. We then simulated how such a bias, per SN, propagates onto the cosmological parameter fit using the Union2.1 sample, when supplemented with strongly magnified SNe. The resulting bias turns out to be nonnegligible. We found that the bias on the deduced cosmological parameters is generally of the order of a few percent, if only few biased SNe are included, and increasing with the number of lensed SNe, their redshift, and the original bias from the lens model. Ultimately, we verified that the cosmological parameters are indeed accurately reproduced, if the bias on each magnified SNe is taken into account in the fit. Several SNe magnified by galaxy clusters are already in hand (e.g. Amanullah et al. 2011) or anticipated to be uncovered soon: several magnified Type Ia SNe were used in the Union2.1 compilation (Suzuki et al. 2012); expected or recently found in the CLASH Salzano et al. 2013;Patel et al. 2013;Whalen et al. 2013, see also Graur et al. 2013), and Frontier Fields programs; and many more are expected further ahead, for example with the James Webb Space Telescope (JWST; e.g. Pan & Loeb 2013). We conclude, especially given the leap over recent years in strong-lens modeling accuracy, that the effect calculated here should be readily taken into account with existing and soon to come data, to take proper advantage of magnified SNe when these are in turn used for constraining the cosmological parameters. In addition, the magnitude of the effect investigated here can be useful for related purposes, such as for estimating the additional error on the derived magnification of lensed high-z galaxies, originating from the choice of cosmological parameters used for the lens model. ACKNOWLEDGMENTS We kindly thank the anonymous reviewer of this work for most valuable comments. AZ greatly thanks Miguel Quartin, Matthias Bartelmann, Richard Ellis, Steve Rodney, Massimo Meneghetti, Or Graur, Saurabh Jha, Brandon Patel, Keiichi Umetsu and Matt Schenker, for useful discussions and comments. Support for AZ is provided by NASA through Hubble Fellowship grant #HST-HF-51334.01-A awarded by STScI. Part of this work was supported by contract research "Internationale Spitzenforschung II/2-6" of the Baden Württemberg Stiftung. Fig. 1 . 1-Bias created on the estimated luminosity of a lensed SN if the cosmology assumed for the lens model differs from the true underlying cosmology, as a function of lens redshift, for different input configurations. The different configurations are noted on each subfigure, such as the cosmologies used, the SN redshift (z SN ), and its distance from the center (θ SN ). Fig. 2 . 2-Same as Fig. 1, now showing the magnitude of the effect as a function of the magnified SN redshift, for various configurations. Fig. 3 . 3-Same as Figs. Fig. 4 . 4-Effect of the demagnified-luminosity bias discussed in this work, on the overall cosmological fit to the Union2.1 sample when supplemented with (de)lensed SNe.Figure shows different examples of mock, lensed Type Ia SNe (red error-bars), on top of the Union2.1 sample (blue error bars). 3, Ω Λ = 0.7] model, the magnifications of the second, [Ω m = 1, Ω Λ = 0] model, deviate by 17.3% on average throughout this FOV, and by a median of 1.9%. These values increase as the FOV0.2 0.4 0.6 0.8 0.992 0.994 0.996 0.998 1 1.002 Cluster redshift μ model / μ real [Ω m , Ω Λ ] model =[0.30,0.70] [Ω m , Ω Λ ] real =[0.27,0.73] z SN =1.5, θ SN =35" 0.9966 0.9970 0.9977 0.9984 0.992 0.994 0.996 0.998 1 1.002 q model / q real 0.2 0.4 0.6 0.8 1 1.1 1.2 1.3 1.4 Cluster redshift μ model / μ real [Ω m , Ω Λ ] model =[0.30,0.70] [Ω m , Ω Λ ] real =[1,0] z SN =1.5, θ SN =35" 1.0875 1.0639 1.0441 1.0270 1 1.1 1.2 1.3 1.4 q model / q real 0.2 0.4 0.6 0.8 0.992 0.994 0.996 0.998 1 Cluster redshift μ model / μ real [Ω m , Ω Λ ] model =[0.30,0.70] [Ω m , Ω Λ ] real =[0.27,0.73] z SN =2, θ SN =25" 0.9966 0.9970 0.9977 0.9984 0.992 0.994 0.996 0.998 1 q model / q real 0.2 0.4 0.6 0.8 1 1.1 1.2 1.3 1.4 Cluster redshift μ model / μ real [Ω m , Ω Λ ] model =[0.30,0.70] [Ω m , Ω Λ ] real =[1,0] z SN =2, θ SN =25" 1.0875 1.0639 1.0441 1.0270 1 1.1 1.2 1.3 1.4 q model / q real (a) 10 SNe, 10% luminosity bias per delensed SN, z < 20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2626, w=−0.9697 (b) 10 SNe, 10% luminosity bias per delensed SN, z < 3 0 0.5 1 1.5 2 2.5 3 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2420, w=−0.9291 (c) 5 SNe, 10% luminosity bias per delensed SN, z < 3 0 0.5 1 1.5 2 2.5 3 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2717, w=−0.9884 (d) 5 SNe, 20% luminosity bias per delensed SN, z < 3 0 0.5 1 1.5 2 2.5 3 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2485, w=−0.9426 (e) 10 SNe, 5% luminosity bias per delensed SN, z < 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2719, w=−0.9876 (f) 30 SNe, 10% luminosity bias per delensed SN, z < 3 0 0.5 1 1.5 2 2.5 3 32 34 36 38 40 42 44 46 48 50 Redshift Distance Modulus Union2.1 Ω m =0.2776, w=−1.0005 Mock Ω m =0.2080, w=−0.8711 http://www.stsci.edu/hst/campaigns/frontier-fields/ arXiv:1311.5224v3 [astro-ph.CO] 7 May 2014 http://supernova.lbl.gov/Union/ . 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[]
[ "The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) Mission Concept", "The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) Mission Concept" ]
[ "Regina Caputo \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Marco Ajello \nDepartment of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA\n", "Carolyn A Kierans \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Jeremy S Perkins \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Judith L Racusin \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Luca Baldini \nUniversitá di Pisa and Istituto Nazionale di Fisica Nucleare\nSezione di Pisa I56127PisaItaly\n", "Matthew G Baring \nDepartment of Physics and Astronomy\nRice University\n6100 Main Street77251-1892HoustonTexasUSA\n", "Elisabetta Bissaldi \nDipartimento Interateneo di Fisica dell'Universitá e del Politecnico di Bari\nvia Amendola 17370126BariItaly\n\nIstituto Nazionale di Fisica Nucleare\nSezione di Bari\nVia E. Orabona 470125BariItaly\n", "Eric Burns \nDepartment of Physics & Astronomy\nLouisiana State University\n70803Baton RougeLAUSA\n", "Nicholas Cannady \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nUniversity of Maryland\nBaltimore County\n21250BaltimoreMDUSA\n", "Eric Charles \nDepartment of Physics\nW. W. Hansen Experimental Physics Laboratory\nSLAC National Accelerator Laboratory\nKavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA\n", "Rui M Curado Da Silva \nLaboratório de Instrumentação e Física Experimental de Partículas\nDepartamento de Física\nUniversidade de Coimbra\nP-3004-516CoimbraPortugal\n", "Ke Fang \nDepartment of Physics\nUniversity of Wisconsin-Madison\nMadisonWisconsinUSA\n", "Henrike Fleischhack \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nCatholic University of America\n620 Michigan Ave NE20064WashingtonDCUSA\n", "Chris Fryer \nCenter for Theoretical Astrophysics\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "Yasushi Fukazawa \nDepartment of Physics\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan\n", "J Eric Grove \nSpace Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA\n", "Dieter Hartmann \nDepartment of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA\n", "Eric J Howell \nOzGrav\nUniversity of Western Australia\n6009CrawleyWestern AustraliaAustralia\n", "Manoj Jadhav \nArgonne National Laboratory\n60440LemontILUSA\n", "Christopher M Karwin \nDepartment of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA\n", "Daniel Kocevski \nNASA Marshall Space Flight Center\n35808HuntsvilleALUSA\n", "Naoko Kurahashi \nDepartment of Physics\nDrexel University\n19104PhiladelphiaPAUSA\n", "Luca Latronico \nINFN\nSezione di Torino\nVia Pietro Giuria 110125TorinoItaly\n", "Tiffany R Lewis \nNASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Richard Leys \nKarlsruhe Institute of Technology\nKIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen\n", "Amy Lien \nDepartment of Chemistry, Biochemistry, and Physics\nUniversity of Tampa\n401 W. Kennedy Blvd33606TampaFLUSA\n", "Lea Marcotulli \nDepartment of Physics\nYale University\n52 Hillhouse Avenue06511New HavenCTUSA\n", "Israel Martinez-Castellanos \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nDepartment of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA\n", "Mario Nicola Mazziotta \nIstituto Nazionale di Fisica Nucleare\nSezione di Bari\nVia E. Orabona 470125BariItaly\n", "Julie Mcenery \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Jessica Metcalfe \nArgonne National Laboratory\n60440LemontILUSA\n", "Kohta Murase \nDept. of Physics and Dept. of Astronomy and Astrophysics\nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\nUniversity ParkPennsylvaniaUSA\n\nCenter for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyotoJapan\n", "Michela Negro \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nUniversity of Maryland\nBaltimore County\n21250BaltimoreMDUSA\n", "Lucas Parker \nLos Alamos National Laboratory\n87544Los AlamosNMUSA\n", "Bernard Phlips \nSpace Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA\n", "Chanda Prescod-Weinstein \nDepartment of Physics & Astronomy\nUniversity of New Hampshire\n03824DurhamNHUSA\n", "Soebur Razzaque \nCentre for Astro-Particle Physics\nDepartment of Physics\nUniversity of Johannesburg\nPO Box 5242006Auckland ParkSouth Africa\n\nDepartment of Physics\nThe George Washington University\n20052WashingtonDCUSA\n", "Peter S Shawhan \nDepartment of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA\n", "Yong Sheng \nDepartment of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA\n", "Tom A Shutt \nSLAC National Accelerator Laboratory\n94025Menlo ParkCAUSA\n\nKavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA\n", "Daniel Shy \nNational Research Council Research Associate resident at the Naval Research Laboratory\nWashington DC20375USA\n", "Clio Sleator \nSpace Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA\n", "Amanda L Steinhebel \nNASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Nicolas Striebig \nKarlsruhe Institute of Technology\nKIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen\n", "Yusuke Suda \nDepartment of Physics\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan\n", "Donggeun Tak \nDeutsches Elektronen-Synchrotron (DESY)\nPlatanenallee 615738ZeuthenGermany\n", "Hiro Tajima \nInstitute for Space-Earth Environmental Research and Kobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\nJapan\n", "Janeth Valverde \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nUniversity of Maryland\nBaltimore County\n21250BaltimoreMDUSA\n", "Tonia M Venters \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Zorawar Wadiasingh \nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA\n\nDepartment of Astronomy\nUniversity of Maryland\n20742College ParkMarylandUSA\n", "Richard S Woolf \nSpace Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA\n", "Eric A Wulf \nSpace Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA\n", "Haocheng Zhang \nNASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n", "Andreas Zoglauer \nSpace Sciences Laboratory\nUniversity of California at Berkeley\n7 Gauss Way94720BerkeleyCAUSA\n", "\nSpace Science Division\nU. S. Naval Research Laboratory\n20375WashingtonDCUSA\n" ]
[ "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Department of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Universitá di Pisa and Istituto Nazionale di Fisica Nucleare\nSezione di Pisa I56127PisaItaly", "Department of Physics and Astronomy\nRice University\n6100 Main Street77251-1892HoustonTexasUSA", "Dipartimento Interateneo di Fisica dell'Universitá e del Politecnico di Bari\nvia Amendola 17370126BariItaly", "Istituto Nazionale di Fisica Nucleare\nSezione di Bari\nVia E. Orabona 470125BariItaly", "Department of Physics & Astronomy\nLouisiana State University\n70803Baton RougeLAUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "University of Maryland\nBaltimore County\n21250BaltimoreMDUSA", "Department of Physics\nW. W. Hansen Experimental Physics Laboratory\nSLAC National Accelerator Laboratory\nKavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA", "Laboratório de Instrumentação e Física Experimental de Partículas\nDepartamento de Física\nUniversidade de Coimbra\nP-3004-516CoimbraPortugal", "Department of Physics\nUniversity of Wisconsin-Madison\nMadisonWisconsinUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "Catholic University of America\n620 Michigan Ave NE20064WashingtonDCUSA", "Center for Theoretical Astrophysics\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Department of Physics\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan", "Space Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA", "Department of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA", "OzGrav\nUniversity of Western Australia\n6009CrawleyWestern AustraliaAustralia", "Argonne National Laboratory\n60440LemontILUSA", "Department of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA", "NASA Marshall Space Flight Center\n35808HuntsvilleALUSA", "Department of Physics\nDrexel University\n19104PhiladelphiaPAUSA", "INFN\nSezione di Torino\nVia Pietro Giuria 110125TorinoItaly", "NASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA", "Karlsruhe Institute of Technology\nKIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen", "Department of Chemistry, Biochemistry, and Physics\nUniversity of Tampa\n401 W. Kennedy Blvd33606TampaFLUSA", "Department of Physics\nYale University\n52 Hillhouse Avenue06511New HavenCTUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "Department of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA", "Istituto Nazionale di Fisica Nucleare\nSezione di Bari\nVia E. Orabona 470125BariItaly", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Argonne National Laboratory\n60440LemontILUSA", "Dept. of Physics and Dept. of Astronomy and Astrophysics\nInstitute for Gravitation and the Cosmos\nThe Pennsylvania State University\nUniversity ParkPennsylvaniaUSA", "Center for Gravitational Physics\nYukawa Institute for Theoretical Physics\nKyotoJapan", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "University of Maryland\nBaltimore County\n21250BaltimoreMDUSA", "Los Alamos National Laboratory\n87544Los AlamosNMUSA", "Space Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA", "Department of Physics & Astronomy\nUniversity of New Hampshire\n03824DurhamNHUSA", "Centre for Astro-Particle Physics\nDepartment of Physics\nUniversity of Johannesburg\nPO Box 5242006Auckland ParkSouth Africa", "Department of Physics\nThe George Washington University\n20052WashingtonDCUSA", "Department of Physics\nUniversity of Maryland\n20742College ParkMarylandUSA", "Department of Physics and Astronomy\nClemson University\n29634ClemsonSCUSA", "SLAC National Accelerator Laboratory\n94025Menlo ParkCAUSA", "Kavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCAUSA", "National Research Council Research Associate resident at the Naval Research Laboratory\nWashington DC20375USA", "Space Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA", "NASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA", "Karlsruhe Institute of Technology\nKIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen", "Department of Physics\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan", "Deutsches Elektronen-Synchrotron (DESY)\nPlatanenallee 615738ZeuthenGermany", "Institute for Space-Earth Environmental Research and Kobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\nJapan", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "University of Maryland\nBaltimore County\n21250BaltimoreMDUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology\nNASA/GSFC\n20771GreenbeltMarylandUSA", "Department of Astronomy\nUniversity of Maryland\n20742College ParkMarylandUSA", "Space Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA", "Space Science Division\nU.S. Naval Research Laboratory\n20375WashingtonDCUSA", "NASA Postdoctoral Fellow\nNASA Goddard Space Flight Center\nGreenbeltMDUSA", "Space Sciences Laboratory\nUniversity of California at Berkeley\n7 Gauss Way94720BerkeleyCAUSA", "Space Science Division\nU. S. Naval Research Laboratory\n20375WashingtonDCUSA" ]
[]
The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) is designed to identify and characterize gamma rays from extreme explosions and accelerators. The main science themes include: supermassive black holes and their connections to neutrinos and cosmic rays; binary neutron star mergers and the relativistic jets they produce; cosmic ray particle acceleration sources including Galactic supernovae; and continuous monitoring of other astrophysical events and sources over the full sky in this important energy range. AMEGO-X will probe the medium energy gamma-ray band using a single instrument with sensitivity up to an order of magnitude greater than previous telescopes in the energy range 100 keV to 1 GeV that can be only realized in space. During its three-year baseline mission, AMEGO-X will observe nearly the entire sky every two orbits, building up a sensitive all-sky map of gamma-ray sources and emission. AMEGO-X was submitted in the recent 2021 NASA MIDEX Announcement of Opportunity.
10.1117/1.jatis.8.4.044003
[ "https://export.arxiv.org/pdf/2208.04990v2.pdf" ]
253,213,376
2208.04990
5043b9f46e29537140c6cdbc7321a0a7992b2ad8
The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) Mission Concept 4 Nov 2022 Regina Caputo NASA Goddard Space Flight Center GreenbeltMDUSA Marco Ajello Department of Physics and Astronomy Clemson University 29634ClemsonSCUSA Carolyn A Kierans NASA Goddard Space Flight Center GreenbeltMDUSA Jeremy S Perkins NASA Goddard Space Flight Center GreenbeltMDUSA Judith L Racusin NASA Goddard Space Flight Center GreenbeltMDUSA Luca Baldini Universitá di Pisa and Istituto Nazionale di Fisica Nucleare Sezione di Pisa I56127PisaItaly Matthew G Baring Department of Physics and Astronomy Rice University 6100 Main Street77251-1892HoustonTexasUSA Elisabetta Bissaldi Dipartimento Interateneo di Fisica dell'Universitá e del Politecnico di Bari via Amendola 17370126BariItaly Istituto Nazionale di Fisica Nucleare Sezione di Bari Via E. Orabona 470125BariItaly Eric Burns Department of Physics & Astronomy Louisiana State University 70803Baton RougeLAUSA Nicholas Cannady NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA University of Maryland Baltimore County 21250BaltimoreMDUSA Eric Charles Department of Physics W. W. Hansen Experimental Physics Laboratory SLAC National Accelerator Laboratory Kavli Institute for Particle Astrophysics and Cosmology Stanford University 94305StanfordCAUSA Rui M Curado Da Silva Laboratório de Instrumentação e Física Experimental de Partículas Departamento de Física Universidade de Coimbra P-3004-516CoimbraPortugal Ke Fang Department of Physics University of Wisconsin-Madison MadisonWisconsinUSA Henrike Fleischhack NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA Catholic University of America 620 Michigan Ave NE20064WashingtonDCUSA Chris Fryer Center for Theoretical Astrophysics Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Yasushi Fukazawa Department of Physics Hiroshima University 1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan J Eric Grove Space Science Division U.S. Naval Research Laboratory 20375WashingtonDCUSA Dieter Hartmann Department of Physics and Astronomy Clemson University 29634ClemsonSCUSA Eric J Howell OzGrav University of Western Australia 6009CrawleyWestern AustraliaAustralia Manoj Jadhav Argonne National Laboratory 60440LemontILUSA Christopher M Karwin Department of Physics and Astronomy Clemson University 29634ClemsonSCUSA Daniel Kocevski NASA Marshall Space Flight Center 35808HuntsvilleALUSA Naoko Kurahashi Department of Physics Drexel University 19104PhiladelphiaPAUSA Luca Latronico INFN Sezione di Torino Via Pietro Giuria 110125TorinoItaly Tiffany R Lewis NASA Postdoctoral Fellow NASA Goddard Space Flight Center GreenbeltMDUSA Richard Leys Karlsruhe Institute of Technology KIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen Amy Lien Department of Chemistry, Biochemistry, and Physics University of Tampa 401 W. Kennedy Blvd33606TampaFLUSA Lea Marcotulli Department of Physics Yale University 52 Hillhouse Avenue06511New HavenCTUSA Israel Martinez-Castellanos NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA Department of Physics University of Maryland 20742College ParkMarylandUSA Mario Nicola Mazziotta Istituto Nazionale di Fisica Nucleare Sezione di Bari Via E. Orabona 470125BariItaly Julie Mcenery NASA Goddard Space Flight Center GreenbeltMDUSA Jessica Metcalfe Argonne National Laboratory 60440LemontILUSA Kohta Murase Dept. of Physics and Dept. of Astronomy and Astrophysics Institute for Gravitation and the Cosmos The Pennsylvania State University University ParkPennsylvaniaUSA Center for Gravitational Physics Yukawa Institute for Theoretical Physics KyotoJapan Michela Negro NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA University of Maryland Baltimore County 21250BaltimoreMDUSA Lucas Parker Los Alamos National Laboratory 87544Los AlamosNMUSA Bernard Phlips Space Science Division U.S. Naval Research Laboratory 20375WashingtonDCUSA Chanda Prescod-Weinstein Department of Physics & Astronomy University of New Hampshire 03824DurhamNHUSA Soebur Razzaque Centre for Astro-Particle Physics Department of Physics University of Johannesburg PO Box 5242006Auckland ParkSouth Africa Department of Physics The George Washington University 20052WashingtonDCUSA Peter S Shawhan Department of Physics University of Maryland 20742College ParkMarylandUSA Yong Sheng Department of Physics and Astronomy Clemson University 29634ClemsonSCUSA Tom A Shutt SLAC National Accelerator Laboratory 94025Menlo ParkCAUSA Kavli Institute for Particle Astrophysics and Cosmology Stanford University 94305StanfordCAUSA Daniel Shy National Research Council Research Associate resident at the Naval Research Laboratory Washington DC20375USA Clio Sleator Space Science Division U.S. Naval Research Laboratory 20375WashingtonDCUSA Amanda L Steinhebel NASA Postdoctoral Fellow NASA Goddard Space Flight Center GreenbeltMDUSA Nicolas Striebig Karlsruhe Institute of Technology KIT-ADL) -Hermann-von-Helmholtz-Platz 1D-76344Eggenstein-Leopoldshafen Yusuke Suda Department of Physics Hiroshima University 1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan Donggeun Tak Deutsches Elektronen-Synchrotron (DESY) Platanenallee 615738ZeuthenGermany Hiro Tajima Institute for Space-Earth Environmental Research and Kobayashi-Maskawa Institute for the Origin of Particles and the Universe Nagoya University Japan Janeth Valverde NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA University of Maryland Baltimore County 21250BaltimoreMDUSA Tonia M Venters NASA Goddard Space Flight Center GreenbeltMDUSA Zorawar Wadiasingh NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology NASA/GSFC 20771GreenbeltMarylandUSA Department of Astronomy University of Maryland 20742College ParkMarylandUSA Richard S Woolf Space Science Division U.S. Naval Research Laboratory 20375WashingtonDCUSA Eric A Wulf Space Science Division U.S. Naval Research Laboratory 20375WashingtonDCUSA Haocheng Zhang NASA Postdoctoral Fellow NASA Goddard Space Flight Center GreenbeltMDUSA Andreas Zoglauer Space Sciences Laboratory University of California at Berkeley 7 Gauss Way94720BerkeleyCAUSA Space Science Division U. S. Naval Research Laboratory 20375WashingtonDCUSA The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) Mission Concept 4 Nov 20221multimessengergamma-ray telescopehigh-energy astrophysicsgamma-ray mission *Corresponding author reginacaputo@nasagov The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) is designed to identify and characterize gamma rays from extreme explosions and accelerators. The main science themes include: supermassive black holes and their connections to neutrinos and cosmic rays; binary neutron star mergers and the relativistic jets they produce; cosmic ray particle acceleration sources including Galactic supernovae; and continuous monitoring of other astrophysical events and sources over the full sky in this important energy range. AMEGO-X will probe the medium energy gamma-ray band using a single instrument with sensitivity up to an order of magnitude greater than previous telescopes in the energy range 100 keV to 1 GeV that can be only realized in space. During its three-year baseline mission, AMEGO-X will observe nearly the entire sky every two orbits, building up a sensitive all-sky map of gamma-ray sources and emission. AMEGO-X was submitted in the recent 2021 NASA MIDEX Announcement of Opportunity. Introduction Multimessenger astrophysics is one of the most exciting and rapidly advancing fields of science, providing unparalleled access to extreme processes that sculpt the universe. Gamma-ray observations have been central to the advent of the multimessenger initiative and will continue to be so as the field matures. As a priority theme of the Astro2020 Decadal Survey report 1 , the science of New Messengers New Physics is poised to revolutionize our understanding of the extreme universe. Data from NASA's Fermi mission has demonstrated that many of the extreme processes that produce gravitational waves and neutrinos and accelerate cosmic rays also produce gamma rays. In other words, multimessenger sources are gamma-ray sources. These sources are, however, brightest in the under-explored MeV band. We have developed a mission and submitted a proposal to the 2021 NASA Astrophysics Medium Explorer (MIDEX) Announcement of Opportunity 2 to observe these critical energies and fully capitalize on this exciting new era of multimessenger astrophysics. The All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X) is a wide-field survey telescope designed to discover and characterize gamma-ray emission from multimessenger sources using imaging, spectroscopy, and polarization. During its three-year baseline mission, AMEGO-X will observe in the critical 100 keV -1 GeV energy band over nearly the entire sky every two orbits, building up a sensitive all-sky map of gamma-ray sources and diffuse emission. It will also access >50% (<10 MeV) and >20% (>10 MeV) of the sky instantaneously, maximizing transient detections and rapid alerts, which will be openly distributed to the astrophysics communities. AMEGO-X will deliver breakthrough discoveries in a MIDEX class mission in areas highlighted as the highest scientific priority for Explorer-scale missions in the Astro2020 Decadal Survey Report: multimessenger astrophysics and time-domain astronomy. AMEGO-X complements the recently selected COSI Small Explorer mission, 1 which has excellent energy resolution in the 0.2-5 MeV band to probe the origins of Galactic positrons and uncover sites of nucleosynthesis. AMEGO-X will provide a leap (×10-50) in continuum sensitivity, similar to Fermi/LAT at higher energies, in the long sought-after MeV gamma-ray gap. The AMEGO-X mission employs a single instrument with subsystems delivered by Argonne National Laboratory, the Naval Research Laboratory and NASA Goddard Space Flight Center and has partnered with Lockheed Martin Space for the high-heritage spacecraft. Science After a long hiatus from the first multimessenger event SN 1987A, 2 which was discovered in optical and neutrinos, a new era of multimessenger astrophysics was heralded in 2017 when the electromagnetic counterparts to sources of both neutrinos 3,4 and gravitational waves 5,6 were observed for the first time. The AMEGO-X mission has three overarching goals which tie together the extreme explosions and extreme accelerators which produce all the cosmic messengers: supermassive black holes, neutron stars and their mergers, and the remnants of Galactic supernovae. The following section will expand upon these overarching science goals. Supermassive Black Holes and their connections to Neutrinos and Cosmic Rays Active galactic nuclei (AGN), accreting supermassive black holes at the centers of galaxies, provide extreme conditions that are conducive to the particle acceleration that is inferred from observations of non-thermal X rays and gamma rays. This can take place near the black hole (core) and/or in the relativistic jets that 10% of AGN display. Those AGN with a relativistic jet aligned along our line of sight are called blazars 7 and are the most powerful persistent sources of electromagnetic radiation in the universe. Blazars are widely believed to be sites for rapid acceleration of electrons in their jets to near the speed of light because their low energy spectra are consistent with electron synchrotron emission and optical polarization measurements confirm this. 8 They are also powerful enough to accelerate protons and may be responsible for the generation of ultra-high energy cosmic rays (UHECRs). The detection of the high-energy neutrino, IC-170922A, by the IceCube Neutrino Observatory (IceCube) coincident with the brightly flaring gamma-ray blazar, TXS 0506+056, marked a milestone in multimessenger astrophysics. 3,4 It established observationally that at least some blazars accelerate protons: neutrinos are a unique signature of proton acceleration and production because they cannot be produced in purely leptonic sources, instead being produced as decay products of pions and muons in photohadronic collisions. However, the picture is far from complete, as it is not clear which among jetted or non-jetted AGN can accelerate protons. The "smoking gun" signatures of proton acceleration in AGN are either the coincident detection of high-energy PeV neutrinos and medium-energy (MeV) gamma rays in a particular AGN, and/or detection of polarized emission 9 in the MeV band. AMEGO-X, with its sensitive all-sky monitoring, will detect over 400 blazars, which is determined considering the log N -log S distributions of Swift/BAT and Fermi/LAT blazars. [10][11][12] Using the 2nd FAVA catalog, 13 it will also detect around 150 blazar flares/year, and for the brightest of these also measure the polarization fraction for more than 10 blazar flares/year, which is the sensitivity to measure polarization within a week. This will allow us to answer long-standing questions about the source of cosmic neutrinos, extragalactic cosmic rays, while also providing critical gamma-ray capabilities complementary to upgraded neutrino observatories in the era of multimessenger astrophysics. Blazars and their Flares Relativistic jets produce gamma rays through the interactions of the particles they accelerate. In leptonic models, accelerated electrons produce all observed gamma rays, principally via synchrotron and inverse Compton emission. In hadronic models, accelerated protons produce gamma rays through proton-synchrotron radiation 14 or via synchrotron radiation from secondary particles generated by proton interactions with jet photons (via e.g., photo-pion production), the latter of which also produces neutrinos. 15 For most flare data from currently operating observatories, it is not possible to determine which scenario better describes the data -both hadronic and leptonic models are consistent with observed blazar spectra. 8,16,17 Some hadronic emission dominated models predict that significant high-energy (GeV) gamma rays interact with jet photons (effectively being absorbed) and are reprocessed into the medium-energy gamma-ray band where they can finally escape the jet. By monitoring the entire medium-energy gamma-ray (100 keV-1GeV) sky every three hours, AMEGO-X will observe in real time the potential sources of IceCube neutrinos and correlate the timing between gamma-ray and neutrino flares. Figure 1 (left) shows a simulated AMEGO-X light curve for which the first and second neutrino flare of TXS 0506+056 could have been detected, since gamma rays and neutrinos are produced via the same processes. 18 While hadronic models link neutrinos with gamma rays in the AMEGO-X band, the same cannot be said for higher energies. The brightest neutrino events may not be detected by higher-energy gammaray telescopes, such as the Fermi/LAT, because the radiation fields required for efficient neutrino production make the source opaque to high-energy gamma rays [e.g., 17,19 ]. Figure 1 (right) shows that during the 2014-2015 TXS 0506+056 orphan neutrino flare, AMEGO-X may have detected a significant MeV flare if the emission was from hadronic interactions within the AGN corona. By monitoring nearly the full sky every three hours, AMEGO-X will detect ∼150 blazar flares/ year, the most promising counterparts to IceCube high-energy neutrinos which are released through real-time alerts. 20 AMEGO-X will provide an all-sky catalog of more than 400 MeV-peaked blazars for neutrino searches. 10 These are the most powerful blazars, whose maximum energy output is in the AMEGO-X band. In blazars, the number of high-energy neutrinos correlates with the total 1 keV -1 PeV flux, which is well approximated by the 100 keV -1 GeV flux, since this is the band where blazars release the most energy. 21 The number of expected IceCube-AMEGO-X coincident detections (over three years) for the baseline and threshold missions are 7.7(±2.5) and 6(±2), respectively (See Section 3), with the brightest sources expected to emit more than one high-energy neutrino within the 3 years. This has been computed from the number of expected neutrinos from each source following: 21 N ν = N ν,max (f blazar ) * f corr * 3/4(1) where N ν,max is the maximum number of expected neutrinos (in 4 years) for a given blazar with an average flux f blazar . f corr =0.0089 is a correction that takes into account realistic blazar spectra, neutrino flavors and different classes of blazars (conservative estimate 21 ). The factor 3/4 accounts for the different integration time (4 years in Ref. 21 ) compared with the expected 3-year duration of the AMEGO-X survey). Moreover, the AMEGO-X survey of the most powerful blazars will also be crucial to determine their contribution to the IceCube diffuse astrophysical neutrino flux, the origins of which remain a mystery. 22 Polarization measurements of Jetted Active Galactic Nuclei Multiwavelength spectral observations alone cannot yet distinguish if blazars accelerate only leptons or leptons and hadrons [e.g., 24,25 ]. In blazar models for which at least part of the gamma-ray emission is from proton acceleration, the polarization signature is prominent in the medium-energy gamma-ray band. 9 This is because the relativistic Compton scattering dominates the high-energy emission in leptonic models and produces a much lower fraction of polarization than the proton and cascade synchrotron emission in the hadronic model. 26 AMEGO-X polarization measurements of blazars will provide an independent constraint on proton acceleration in AGN complementary to coincident neutrino detection. 27 While IXPE 28 can detect X-ray polarization from blazar jets, this polarization probes the acceleration mechanism of the relativistic electrons, which produce the emission in the X-ray band. On the other hand, AMEGO-X will explore hadronic signatures via MeV polarimetry for blazars, unambiguously identifying proton acceleration and neutrino production in blazar jets, which are essential to multimessenger astronomy. Blazar spectral models have , AMEGO-X will be able to discover blazars whose high-energy emission is of hadronic nature. 9 (Right) AMEGO-X observations complement IXPE's because polarization at different energies probes different blazar populations, radiation processes and possibly even different acceleration mechanisms. Polarization fractions for blazars with a fully hadronic, leptonic or hybrid (mix of leptonic and hadronic) emission models in the IXPE and AMEGO-X band. shown that many BL Lacertae objects (BL Lacs), in particular intermediate-synchrotron-peaked and high-synchrotron-peaked BL Lacs, have a significant or even dominating contribution from the primary electron synchrotron component. 29 In these cases, IXPE detected polarization would indicate leptonic, not hadronic, interactions in the X-ray band. Furthermore, MeV polarimetry can uniquely probe proton synchrotron emission in blazar jets. As shown in Figure 2, AMEGO-X can clearly distinguish the model with leptonic Compton scattering and hadronic cascades (hybrid model) versus the fully hadronic models with proton synchrotron emission (hadronic model), which is not possible at lower energies. This is because these sources, mostly flat-spectrum radio quasars, are too faint at X-ray energies for IXPE to distinguish polarization degrees of a few percentage (like those, e.g., in Figure 2.) Discovering fully hadronic sources (those with proton synchrotron emission) will allow AMEGO-X to unveil the origin of ultra-high-energy cosmic rays, something not possible without observations in the MeV gamma-ray regime. AMEGO-X will measure the gamma-ray polarization fraction as low as 10% for at least 10 bright blazar flares per year (e.g., from 3C 279, PKS 1510-089) and two nearby persistent blazars (3C 273, 3C 454.3), determining whether AGN jets accelerate high-energy cosmic-ray (CR) protons for the first time at these energies. A measured polarization fraction >30% will allow AMEGO-X to independently establish blazars as extragalactic CR accelerators . 9, 26 The AGN Core IceCube has recently identified NGC 1068, a nearby AGN lacking a relativistic jet, as a potential neutrino source; 30 this provides evidence that AGN without a jet may also accelerate protons and produce neutrinos. Within the immediate vicinity of the supermassive black hole (AGN core), accretion cultivates extreme environments conducive to particle acceleration. Clouds of hot thermal electrons exist above the disk in the corona and have a thermal energy distribution. 31 However, there is recent evidence of accelerated (non-thermal) electrons that may produce MeV emission via inverse Compton scattering of disk photons. 32,33 Similarly, protons can be accelerated near the disk. 34 AMEGO-X's continuum sensitivity will allow detection of these thermal and non-thermal leptonic and/or hadronic components ( Figure 3 for NGC 1068) in off-axis jetted (e.g., Cen A, M87) and non-jetted (e.g., NGC 1068, NGC 3516, NGC 4258) AGN. 34 The solid black line shows the AMEGO-X sensitivity assuming a 3 year mission. Binary Neutron Star Mergers and their Jets In-spiraling and merging binary neutron stars (BNSs) emit gravitational waves (GWs), ripples in spacetime that can be detected by the worldwide GW network. 35 After the merger, particles are accelerated in narrowly collimated relativistic jets launched along the rotational axis. When these jets point towards Earth, we observe a bright flash of gamma rays, a short gamma-ray burst (SGRB), in the tens of keV to few MeV energy band. GW170817/GRB170817A was the first BNS merger jointly detected both with gamma rays and GWs, informing telescopes around the world to follow up the event. 5,6,36 Despite the wealth of information gained through multi-wavelength observations of this event and the associated radioactively powered isotropic kilonova emission 37 and the offaxis afterglow, fundamental questions regarding the nature of BNS mergers remain. For example, a higher-mass BNS merger, GW 190425, 38 had no associated SGRB detected by current missions, which could be due to a different merger remnant or to an off-axis jet that was pointed away from Earth. The large 90% localization region of 10,200 deg 2 was not fully covered by current instruments. Swift/BAT reported that 45.73% of the localization probability region was outside the field-of-view 39 and only 55.6% of the probability region was viewable by Fermi/GBM 40 . A population of joint observations of BNS mergers, in GW and gamma rays, will address these questions, enabling science which is impossible to perform independently or at other wavelengths. The medium-energy gamma-ray band, where SGRBs are the brightest, provides unique insight into the physics of BNS mergers making AMEGO-X ideally positioned to provide a new, comprehensive view of this population. Model Dependent Statistical Fractions and Structure of GRB jets A BNS merger can produce one of four possible remnants: a stable NS, a long-lived massive NS supported by its fast rigid-body like rotation, a short-lived massive NS supported by internal differential rotation, or prompt black hole formation. Each outcome leads to different predictions for the presence or absence of a relativistic jet, as well as the characteristic delay between the merger event and the formation and launching of the jet. 41 For events detected jointly (estimates are later in the section), the GW data provide information on NS masses, spins, and distances, but AMEGO-X observations are required to probe the nature of the merger remnant and the structure of the jet. Joint observations of ten BNS mergers can constrain the maximum NS mass to ∼1% precision, improving the current constraints by nearly an order of magnitude. 42 Typical AMEGO-X on-board SGRB localizations will have a factor of 400 smaller localization uncertainties (90% confidence regions) compared to Fermi/GBM, 43, 44 which will dramatically decrease the time needed to search for and precisely localize the SGRB afterglow or kilonova emission via X-ray, optical, and radio follow-up. This increases the fraction of events with early broadband observations including spectroscopy and an opportunity to characterize the early evolution of these objects. The AMEGO-X 10 deg 2 90% confidence region ( Figure 4, left) is well matched to the Fields of View (FOVs) of current facilities like the Gravitational wave Optical Transient Observer [GOTO 45 ] and the Zwicky Transient Facility [ZTF 46 ], and the future Vera C. Rubin Telescope. 47 Those multiwavelength observations will provide insights on the host galaxy including redshift, source dynamics and kilonova emission, complementing the AMEGO-X jet energetics and emission mechanisms. AMEGO-X's rapid SGRB localizations enable the >2000 observers (subscribers to the gamma-ray coordinates network, GCN) to quickly search and identify the potential counterparts. This population would inform not only the uniqueness of GW170817 but also the nature of the progenitors and remnant objects from compact binary mergers. Notably under-luminous compared to previously detected SGRBs, 5, 36 the energetic core of GRB170817A's relativistic jet was not completely aligned with our line-of-sight 51 (∼20-25 • offaxis). The observed emission supports the structured jet scenario, in which the jet interacts with the ejecta from the BNS merger resulting in a narrowly collimated energetic jet core with a gradual decrease in flux and Lorentz factor away from the core . 52,53 Under this structured jet scenario, the observed luminosity has a strong dependence on the viewing angle with respect to the jet axis and can be parameterized by a Gaussian function defined by the opening angle (θ core ) and luminosity of the jet (Figure 4, right panel) and only nearby SGRBs can be detected at wider viewing angles. Most BNS mergers are expected near the GW detectability horizon (or the maxiumum distance the GW network is sensitive) where the volumetric rate will be higher. This distance-viewing angle phase space ( Figure 4, right panel) provides a means to directly probe the structured jet profile. Here we note that both AMEGO-X detection and non-detections of GW-detected BNS mergers will set fundamental constraints on the structure of the jet. Of the 80 +180 −50 /yr GW-detected BNS mergers that will fall within AMEGO-X's FOV in the late 2020s (all orientations), 54 ∼40% (30 +68 −19 /yr) will have jets oriented within 30 • with a 90% CL. This assumes 80% duty cycle per GW detector (ie: 64% for both LIGO detectors) and that AMEGO-X observes 50% of the sky instantaneously for GRBs. Among those, the fraction of detected SGRBs will strongly constrain the jet structure. For example, for the narrowest and widest jet models of Figure 4 (compatible with GRB170817A) one expects an average ∼10% (3 +5 −2 /yr) and ∼25% (6 +10 −4 /yr) joint GW-SGRB detections, respectively. 48 Knowledge of the GW distances will allow AMEGO-X observations (both detections and non-detections) to place constraints on different jet structure models and their parameterization such as the parameters of the Gaussian structured jet, or more complicated models such as two-component structured jet ( Figure 4) . 48 Although not the favored scenario for GRB170817A, a cocoon shock breakout can produce SGRBs over a much wider range of viewing angles, despite carrying only a fraction of the energy of the jet. 55 In all jointly observed GW-SGRB events, AMEGO-X will be able to detect or rule out the soft (<200 keV) emission associated with shock breakout. This population of BNS mergers will constrain the jet parameters providing insights into the formation of relativistic outflows. Beyond BNS mergers, AMEGO-X will also provide observations of more than a hundred GWdetected neutron star black hole (NSBH) mergers during its mission lifetime. These observations will determine if NSBH mergers emit gamma rays. 56 Without measurements in the medium-energy gamma-ray band, a critical piece for understanding the remnant and potential jet production of the GW-detected NSBH mergers will be missing. Recent Very High-Energy (VHE) detections of GRB afterglows have accentuated the issue of whether gamma-ray afterglows violate the maximum synchrotron energy or if a synchrotron self Compton component is needed to explain GeV/TeV afterglows. [57][58][59] AMEGO-X will measure afterglow spectra between hard X-ray and GeV, where the distinction between these models is most significant. Previously we have only been able to interpolate between hard X-ray and GeV energies, 60 with the exception of a handful of NuSTAR observations. 61 Cosmic ray sources in the Galaxy Energetic charged particles (CRs, mostly protons and electrons) are ubiquitous in our galaxy. They are accelerated in a variety of environments, such as shocks and regions with strong magnetic fields. Stellar nurseries and remnants of massive stellar explosions provide such environments. After decades of observations, two main questions regarding galactic CRs remain: in what environments are protons accelerated and what is the origin of the positron excess? AMEGO-X, with its projected sensitivity, and timing accuracy, will open the possibility to discover new galactic proton accelerators and test whether pulsars are the source of the positron excess. Supernova remnants, Novae and Star-formation Regions The smoking gun to identify accelerators of CR protons is to detect the characteristic neutral piondecay (π 0 → 2γ) feature, or pion bump, produced in the interaction of protons with the interstellar material. Each photon produced has an energy of 67.5 MeV (in the π 0 rest frame) , 62 which is ideally matched to the AMEGO-X band. Proton acceleration is thought to happen in supernova remnants (SNRs), novae, and star-forming regions (SFRs). In SNRs, a strong shock, launched by the supernova, sweeps up the ambient medium and provides an ideal site for proton acceleration. 63 In novae, the material accreted on the white dwarf from the companion eventually undergoes explosive thermonuclear burning, creating a shock that can accelerate protons in the interaction with the companion's wind. 64 In SFRs, supernovae shocks and massive stars' winds provide the means to accelerate protons and low-energy cosmic rays (<1 TeV) play a fundamental role in shaping the chemical richness of the interstellar medium, determining the dynamical evolution of molecular clouds. 65,66 However, even in the best-studied case (e.g., SNR IC 443; Figure 5), the models are ambiguous, within the large observational uncertainties because of the lack of observations below 60 MeV. AMEGO-X, with its smaller PSF (factor >2) at 60 MeV than Fermi/LAT (see Sec. 4), will measure the gamma-ray energy spectra of the most promising galactic hadronic accelerators, beyond SNR IC 443, including W44, and W51C; the star-forming regions Cygnus Cocoon, Westerlund 1 and 2; the recurrent T CrB and RS Oph novae and ∼one nova/year, 64 and will determine the gamma-ray production mechanisms to confirm potential hadronic accelerators found by Fermi/LAT. Based on extrapolating the best fit spectra from the Fermi/LAT 4FGL DR2 catalog 67, 68 and the Fermi/LAT Supernova Remnant catalog, 69 AMEGO-X will detect 20-40 other SNRs. Even if these sources show no evidence of a pion bump, studying energies below 200 MeV will allow measurements of the Bremmstrahlung emission. This, together with radio observations, will allow AMEGO-X to determine the magnetic field 70 in these environments and provide insight into particle acceleration in the galaxy. Pulsars and Pulsar Wind Nebulae The flux of galactic CR positrons (electron anti-particles) in the 10-200 GeV range is well measured, yet it exceeds expectations from positrons generated by propagating CRs in interactions with interstellar gas. 72,73 These additional positrons, which are referred to as a positron excess, could be produced by the known leptonic accelerators, such as pulsars, or by new physics, such as dark-matter annihilation. 74 Pulsars are the rapidly rotating remnant cores of massive stars that have collapsed and exploded leaving behind a highly magnetized neutron star. Multiwavelength observations have shown that pulsars primarily accelerate electrons and produce electron and positron pairs via interactions in their magnetospheres. These particles, accelerated by powerful magnetic fields generated by the spinning pulsar, interact with the interstellar medium and are advected into pulsar wind nebulae (PWNe). 75 There is a known population of pulsars whose peak spectral energy distributions (SEDs) lie between 300 keV and 100 MeV. 76 Their signals are likely dominated The green and yellow lines show the Bremsstrahlung (with break) and π 0 decay models from Ref. 71 The red solid line shows the AMEGO-X 3 σ sensitivity. by pair synchrotron radiation, 77 and they may possess very different leptonic densities relative to the population of pulsars detected by Fermi/LAT in GeV gamma rays. 78 Based on extrapolating the best fit spectra from the Fermi/LAT catalog 4FGL DR2 67, 68 and the 2nd Fermi/LAT catalog of gamma-ray pulsars, 78 AMEGO-X will detect more than 15 medium-energy gamma-ray peaked pulsars (mePSRs), 76,79 observe the phase-resolved spectra of at least 5 mePSRs, and deliver 100 keV -1 MeV polarization measurements for three pulsars (B1509-58, Vela, the Crab), testing whether synchrotron is the main emission mechanism at MeV energies. AMEGO-X will constrain the location of the emission region and the pair multiplicity (i.e., number of electrons and positrons produced), thereby measuring the contribution of different pulsar populations to the CR positron excess. Pulsars provide an instantaneous snapshot of the number of e + e − pair produced, while the surrounding PWNe provide a long-term average (tens of thousands of years) of the pairs over the life-time of the pulsar. The pulsar relativistic pairs are further accelerated at a wind termination shock at the inner nebula boundary. 80 The energy of the pairs is regulated by synchrotron radiation losses in turbulent magnetic fields near the shock. This sets a natural scale of ∼150 MeV for the peak of the SEDs (independent of the field strength 81 ). There PWNe acts like a calorimeter that represents the accumulation of pairs over the history of the pulsar and provides a measure of the net pair energy output. AMEGO-X will measure the e + e − content of more than 10 PWNe, and therefore their contribution to the CR positron excess. 82 It will also search for counterparts to the extended gamma-ray halos around middle-aged pulsars found by air shower gamma-ray experiments. 83 68 into the AMEGO-X energy range. Blazars dominate the population (>85%) followed by pulsars. The Galactic diffuse in this energy range primarily includes emission from inverse Compton and Bremsstrahlung (with contribution from π 0 at higher energy). The map is convolved with a 2D Gaussian kernel to account for the angular resolution of the instrument. We have verified that the Crab flux is in agreement with measurements. 86 The keV to GeV Gamma-ray Sky With the order of magnitude increase in sensitivity in this energy band and based on extrapolations from the Fermi/LAT 10-year source catalog 67 and Swift/BAT X-ray catalog, 79 AMEGO-X will detect many additional medium-energy gamma-ray-producing sources during normal mission operations. AMEGO-X will deliver science results of significant interest for the astrophysical community and a multi-year catalog of the full medium-energy gamma-ray sky. A simulated sky map is shown in Figure 6 covering the energy range 1 − 30 MeV. The map includes emission from gamma-ray binaries (including accreting black holes in our galaxy), Galactic diffuse continuum emission, and high-redshift blazars. 10,[87][88][89][90] The Galactic diffuse emission is calculated with GALPROP 85 and the individual sources are modeled based on an extrapolation of the Fermi/LAT 4FGL-DR2 catalog. Additional sources that will be detectable by AMEGO-X (although not shown in the sky map) include long and short GRBs, magnetar bursts and giant flares, 88 the extragalactic gamma-ray background, 10 and possibly jetted tidal disruption events 89,91 and large scale bubbles. 90 AMEGO-X Gamma-Ray Telescope The Gamma-Ray Telescope (GRT) is the AMEGO-X mission's sole instrument. It is a wide-field survey instrument designed to discover and characterize gamma-ray emission from multimessenger sources using imaging, spectroscopy, and polarization. The GRT is composed of two detector Together, they characterize gamma rays from 100 keV (25 keV for transients) to 1 GeV. 92 The GRT baseline capabilities are summarized in Table 1. The MeV range covers three different photon-matter interactions that dictate three detection techniques for GRT, as shown in Figure 7. Between ∼100 keV and ∼10 MeV, photons predominantly Compton scatter. The measured position and energy of a Compton scattering interaction and subsequent absorption kinematically and geometrically constrain the initial direction of the primary gamma ray to a circle in the sky. 93 Such an event is referred to as an untracked Compton event. Compton scattering is inherently polarization sensitive, and a linearly polarized source generates a sinusoidal scattering angle distribution in the instrument. 94 If the direction of the first Compton-scattered electron is measured in the Tracker, this additional kinematic information constrains the photon direction to an arc and these tracked Compton events allow for improved background rejection. 95 High-energy gamma rays (>10 MeV) convert to an electron-positron pair, which in turn is detected through ionization tracks in the instrument. The direction of the incoming photon is determined from the positions of the interactions in the Tracker and the total energy is determined by the electromagnetic shower(s) detected in the Calorimeter. 96 At energies below the Compton regime (<100 keV), photons predominantly undergo photoelectric absorption in a single pixel in the Tracker. While these single-site events have no imaging capability, they can be used to localize transient sources using the aggregate signal. 97 For enhanced low-energy sensitivity to GRBs, AMEGO-X enables short duration (<100 s) readout of single-site events to measure emission down to the Tracker threshold of 25 keV. 97 Gamma-ray Detector The GRD is the primary GRT science subsystem. It consists of four identical Detection Towers (Figure 7), each with a Tracker and Calorimeter Module, a dedicated low-voltage power supply (LVPS), and Digital Input and Output (I/O) board. Although each Detection Tower operates independently, signals are combined in the Main Electronics Box (MEB) such that events are reconstructed using data from the full GRD. The Towers' data acquisition (DAQ) electronics and thermal management hardware are positioned along the sides of the GRD to reduce the amount of passive material within the sensitive instrument volume. Pixelated Silicon Tracker The GRD Tracker's main functionality is to measure the energy and position of gamma-ray and charged-particle interactions with high precision. Each of the four Tracker Modules consist of 40 identical stacked Tracker Segments (45×45 cm 2 ) of silicon APS detectors, separated by 1.5 cm. The Tracker Segments each contain 95 Quad Chips (Figure 8), which consist of four identical APS arrays cut out from a single silicon wafer. The AMEGO-X APS chip, AstroPix, is a 2×2 cm 2 array of 19×17 pixels measuring 1×1 mm 2 . Each pixel contains a charge-sensitive preamplifier and comparator, where the active circuitry within the pixel results <1% loss in charge-collection volume. The APSs are 0.5 mm thick and operate at full depletion. The major strength of the APS detectors is low noise, which is achieved through integration of the readout electronics within the detecting material. The AMEGO-X APS performance parameters have been determined through measurements of ATLASPix (designed for the ATLAS experiment at CERN) 98 and the first two prototype versions of AstroPix (Figure 8), in addition to simulations from the designers at Karlsruhe Institute of Technology (KIT). AMEGO-X leverages more than 10 years of development in CMOS monolithic active pixel silicon detectors from ground-based particle physics experiments. [99][100][101][102] By optimizing these APSs for space applications 98, 103 the detectors enable observations at lower photon energies, achieve an overall increase in sensitivity, and are simpler to integrate compared to previous silicon detector technologies. The GRT design uses minimal passive material and carbon fiber reinforced polymers to reduce photon attenuation and backgrounds from activation. In this regard, the Tracker is designed with high thermal conductivity K1100/Cyanate Ester (CE) for heat extraction from the APS detectors. The APS Support Frame (Figure 8) is CNC cut K1100/CE laminate bonded to M55J/CE perimeter closeouts and stand-offs for additional stiffness. Hodoscopic CsI Calorimeter The main functionality of the Calorimeter is to measure the position and energy of Comptonscattered photons and the electromagnetic showers produced from electron and positron pairs over a broad energy range. Situated directly below the Tracker subsystem, the Calorimeter is composed of four layers of thallium-doped cesium iodide (CsI:Tl) bars, hodoscopically arranged. One of the four Calorimeter Modules is shown in Figure 9. Each layer consists of 25 bars, each with a dimension of 1.5×1.5×38 cm 3 . The bars are wrapped in reflective material to pipe scintillation photons to each end, where readout occurs via an array of silicon photomultipliers (SiPMs). To achieve a large dynamic range in a single Calorimeter bar and to mitigate the effects of saturation, a mixture of small and large ONSemi SiPMs are used to cover two gain ranges, 104 as shown in Figure 9. Each end of each CsI bar is read out with a low energy SiPM array, which is a sum of eight 3×3 mm 2 SiPMs, and a high energy array, which is a sum of four 1×1 mm 2 SiPMs. Although SiPMs are sensitive to damage from the orbital radiation environment, work done at NRL has demonstrated that the damage effects can be successfully mitigated by proper instrument configuration without affecting instrument performance. [105][106][107] The position of the interaction along the bar is determined from the relative amplitude of signals on each end. The AMEGO-X Calorimeter utilizes a design based on Fermi/LAT. 96 The Calorimeter team at Naval Research Laboratory (NRL) designed, developed, assembled, tested, and currently operates the Fermi/LAT CsI Calorimeter. Furthermore, the team has built and demonstrated the performance of a prototype AMEGO-X GRT Calorimeter with SiPM readout in gamma-ray beam tests. 104, 108 Anti-coincidence Detector AMEGO-X uses a dedicated Anti-Coincidence Detector (ACD) to reduce the significant cosmicray background. The ACD comprises five plastic scintillator panels that surround the GRD to enable vetos associated with incident charged particles while being transparent to gamma rays ( Figure 10). Each ACD panel has three wavelength-shifting bars (WLS) on each of its four edges. Scintillation light entering the WLS bars is transmitted to the ends, where signals are measured by arrays of 6×6 mm 2 ONSemi SiPMs. Each WLS is read out independently to recover the energy deposited in each ACD panel. The ACD has an energy threshold of 200 keV, which is well below the minimum ionizing particle (MIP) average energy deposition of 2.5 MeV. The ACD interfaces directly with the MEB, which provides I/O and FEE power. Instrument Performance For Compton and pair telescopes, much of the performance relies on accurate event reconstruction and background rates, and therefore simulations were performed with the state-of-the-art MEGAlib analysis package, 109 which is built around Geant4. 110 A detailed mass model that reproduces the active and passive material is simulated with the detector performance to determine interactions of photons and particles and the resulting measured signals. Background Model Background radiation is one of the dominant factors that can limit the sensitivity of an MeV telescope, and therefore detailed background simulations are necessary for accurate performance predictions. The background models used in MEGAlib are based on measurements by COMPTEL, 111 INTEGRAL/SPI, 112 Fermi/LAT, 113 and NuSTAR, 114 and cosmic-ray population measurements, with an assumed AMEGO-X orbit of 575 km and an inclination of 6 • . The models and components (including prompt and delayed emission from cosmic-rays, extragalactic diffuse, and Albedo emission) are orbit averaged and are described in more detail in Ref. 115 The measured signals from charged cosmic rays is a significant background to gamma-ray observations, but almost all of these events can be vetoed by the ACD. Another background-rejection capability is through event reconstruction; only events which have a valid Compton sequence or pair-conversion tracks will be identified as "good" events for higher-level analysis. Figure 11 shows the resulting measured background spectrum after the ACD veto and event reconstruction, Fig 12: The projected angular resolution (left) and effective area (right) are determined through onaxis, mono-energetic, point source simulations. In addition to background rate, these parameters are the main contribution to the sensitivity of the instrument. separated for each event type as classified by MEGAlib. 97 The most significant background rate is measured at low energies as single-site events, which correspond to a single triggered pixel in the Tracker. This measured flux is dominated by cosmic and atmospheric photons, and without imaging capabilities to separate any source emission from background, single-site events can only be used for transient detection when the source rate is high. Most background events which have more than one trigger in the GRT, but are not vetoed by the ACD and do not leave clear straight charged-particle tracks in the Tracker, are classified as Untracked Compton events, since that is the most inclusive event category in MEGAlib. The dominant background component in the Compton regime is activation of the passive and active instrument material by cosmic rays, where the delayed emission of radiation cannot by vetoed by the ACD. In the Pair regime, the dominant background is albedo photons, 115 which also are not vetoed by the ACD. Further background rejection can be achieved through fine-tuned event selections depending on the observation, such as the total photon energy, pair opening angle, Compton scatter angle, or distance between interactions. Angular Resolution and Effective Area Monoenergetic point source simulations are performed in MEGAlib to determine the energy resolution, effective area, and angular resolution of AMEGO-X. Figure 12 shows the measured angular resolution and effective area as a function of energy for each event classification from these simulations. The angular resolution in the Compton regime is defined as the FWHM of the angular resolution measure (ARM) histogram, which is a projection of the point spread function in one dimension, as is standard for Compton telescopes. In the pair regime, the angular resolution is defined as the 68% containment radius, following the standard Fermi/LAT definition. There is no angular resolution for the single-site events since there is no imaging capabilities from these interactions. The predicted angular resolution is 4 • at 1 MeV and 2.5 • at 100 MeV. Measuring the track of a Compton-scattered electron does not improve the angular resolution in the Compton regime as the uncertainty in the scattering angle is often large. The angular resolution for Tracked Compton events are in fact slightly worse than those for Untracked Compton events due to the inherent selection of events which have larger scattering angles. Also shown in Figure 12 are the measured angular resolution of COMPTEL, Fermi/LAT, and the requirements for COSI. The angular resolution of AMEGO-X is better than Fermi/LAT since the LAT's resolution is limited by multiple scattering in the tungsten conversion foils within the tracker. The simulated effective area, shown in Fig. 12, is a measure of the detection efficiency of the telescope. It is defined as the required area of an ideal detector, i.e. 100% efficient, to detect an equivalent number of photons. It is calculated here as the number of valid events classified by MEGAlib divided by the initial number of simulated photons, and scaled by the area of the mass model's surrounding sphere. The expected effective area is 1200 cm 2 at 100 keV, 500 cm 2 at 1 MeV, and 400 cm 2 at 100 MeV. Fig. 12 also shows the effective area for a single Fermi/GBM BGO and NaI detector, as well as COMPTEL and Fermi/LAT. Continuum and Transient Sensitivity The sensitivity of AMEGO-X is determined for steady-state (i.e. continuum) sources and transient sources based on the above MEGAlib simulated performance parameters. From the measured effective area, angular resolution, and energy resolution, the continuum sensitivity can be calculated as the minimum detectable source flux: F min (E) = n 2 + n (n 2 + 4N b ) 2A eff T Obs where n is the required detection significance (3σ) per energy band, N b is the number of background counts, A eff is the effective area, and T Obs is the observation time. The 3-year AMEGO-X sensitivity is shown in Figure 13. To account for the survey-mode observation, the effective area is conservatively taken from simulations of sources at 37 • off-axis, and the observation time is estimated to be 20% of the full mission (T Obs = 3 yr × 0.2). The continuum sensitivity is calculated for Untracked Compton, Tracked Compton, and Pair events separately. When there are significant contributions from two event types, for example Pair events at 10 MeV have a similar effective area to Tracked events at this energy, the sensitivities are combined: F combined = F tracked F pair F tracked + F pair The continuum sensitivity is 3.2 × 10 −12 erg/cm 2 /s at both 1 MeV and 100 MeV. The regime where Compton and Pair events overlap, around 10 MeV, has worse sensitivity due to the limited classification capability currently implemented in MEGAlib. Recent developments in event identification and reconstruction with neural networks have shown dramatic improvements in sensitivity. 116 Additionally, the projected performance in the pair regime (>10 MeV) is expected to improve, as MEGAlib does not currently include reconstruction and shower profiling techniques developed for Fermi/LAT. 117 The transient sensitivity is determined from the measured signal-to-noise ratio from simulations of canonical GRBs. It is defined for each event type (single site and Compton): where S is the number of source counts detected (from GRB source simulations) and B is the number of background counts (from background simulations). The minimum detectable flux at which the combined SN ratios F min = SN 2 single site + SN 2 Compton ≥ 6.5 σ and SN Compton ≥4.5 σ is the transient sensitivity. The AMEGO-X transient sensitivity is 0.5 γ/cm 2 /s between 25 keV-1 MeV for 1 second. The derived GRB rates account for the dependence of the effective area with off-axis angle, and the transient sensitivity is described in Ref. 97 SN = S √ S + B AMEGO-X Mission Implementation Spacecraft The AMEGO-X flight system consists of the GRT instrument integrated onto a spacecraft bus which leverages Lockheed Martin Space's (LMS) standard subsystem architectures ( Figure 14). These include structure, mechanisms, power, attitude determination and control (ADCS), C&DH, and flight software (FSW) from cost-capped planetary missions dating back >15 years, and hardware and software from LMS spacecraft including MAVEN, Lucy, IRIS, OSIRIS-REx, and Juno. Observation plan AMEGO-X has been proposed to launch into Low Earth Orbit (LEO) in 2028 to start a 3-year baseline science campaign with the potential for an extended mission. Science campaign operations are straight-forward, consisting of scanning the sky with the GRT instrument 30 • North or South of zenith every other orbit, providing a nearly all-sky view every two orbits ( Figure 15) . The baseline AMEGO-X LEO orbit ( Figure 15) is circular with a 600 km altitude and inclination of 5 • . This inclination provides the low background radiation environment needed to achieve the required GRT sensitivity. The normalized exposure map for 5 MeV gamma rays (as an illustration) assuming low-Earth orbit (600 km) with a low inclination (5 • ). Rocking alternately ±30 • from zenith every orbit enables the GRT to uniformly observe nearly the entire sky every two orbits (∼3 hours). Over the course of 24 hours, the GRT survey covers the entire sky. Science Data Plan and Products The AMEGO-X mission performs observations of the full medium energy gamma-ray sky every three hours. The data include four different gamma-ray event classes distinguished by the energy of the incoming gamma-ray: Single-Site (photoelectric absorption, used only for fast transients), Compton (Tracked or Untracked), and Pair events. AMEGO-X data will be event-based, where each gamma-ray interaction in the instrument is analyzed separately. Images, light curves, polarization analysis, and other science products are generated on the ground as illustrated in Figure 16. For the rapid detection and localization of gamma-ray transients, AMEGO-X uses on-board Transient Alert (TA) Logic based on Fermi/GBM algorithms. 118,119 TAs are identified by the MEB CPU as a significant (6.5σ) rate increase above the background through a combination of Tracker triggers and events with triggers in Tracker and Calorimeter. Based on simulations, TAs are expected 3-5/day and the probability of a false TA detection is <1 per year. The AMEGO-X Data Plan identifies multiple levels of scientific data products that begin as raw binary files downlinked from the spacecraft and end as scientific data products, such as source spectra and light curves. Users download photon and housekeeping data from the science archive, along with instrument response functions, diffuse maps, and source lists used to perform analyses on any time, energy, and area scale. The TA data contains the spacecraft attitude information, a coarsely binned light curve, and localization from on-board event reconstruction. After straightforward Science Operations center (SOC) processing, the TA is sent to the Gamma-ray Coordinates Network (GCN) to enable multiwavelength follow-up. After the survey data is processed on the ground, a more accurate localization and full light curves will update the initial GCN alert. The GRT instrument team will provide python-based analysis tools and tutorials to make AMEGO-X data accessible to the scientific community, similar to Fermi/LAT 120 and current tools being developed for COSI. Photon and spacecraft data will be accessible through a custom photon data server (similar to Fermi/LAT). Weekly photon and spacecraft files, all catalogs, and higher-level data products will be available via the High Energy Astrophysics Science Archive Research Center (HEASARC). Summary AMEGO-X will deliver breakthrough discoveries as a MIDEX class mission addressing areas highlighted as the highest scientific priority for Explorer-scale missions in the Astro2020 Decadal Survey Report: multimessenger astrophysics and time-domain astronomy. During its three year mission, it will survey the gamma-ray sky with unprecedented sensitivity in the energy range from 100 keV to 1 GeV. The Gamma-Ray Telescope design is well understood with a combination of instrument subsystems that leverage large investments in detector technologies by the Department of Energy, now tailored for space use, while also taking advantage of the extensive flight heritage from Fermi. Its all-sky coverage enables a sensitivity to transients from milliseconds to years. AMEGO-X is complementary to Fermi and COSI with broad MeV continuum, transient and polarization capabilities. The AMEGO-X team has experts from Fermi and COSI who have built the instruments, simulation software, data pipelines and data analysis tools. AMEGO-X's spacecraft partner, Lockheed Martin Space (LMS), has a demonstrated track record of successful Explorer-class bus design and operation. The AMEGO-X science and instrument teams include members from NASA GSFC, Argonne National Laboratory, the Naval Research Laboratory, and LMS, as well as university science partners who are members of the LIGO Collaboration, the IceCube Collaboration, and the Cherenkov Telescope Array Consortium. Together the science team will ensure the maximum return from this unique and groundbreaking mission. Tables 1 The Gamma-Ray Telescope baseline capabilities. With a minimum detectable polarization of 25 % for faint sources during the 3 year survey (left), AMEGO-X will be able to discover blazars whose high-energy emission is of hadronic nature. 9 (Right) AMEGO-X observations complement IXPE's because polarization at different energies probes different blazar populations, radiation processes and possibly even different acceleration mechanisms. Polarization fractions for blazars with a fully hadronic, leptonic or hybrid (mix of leptonic and hadronic) emission models in the IXPE and AMEGO-X band. 3 Expected emission from the AGN corona of NGC 1068. The blue dashed line shows the corona thermal emission, the light blue and green dashed lines show the non-thermal leptonic and hadronic components respectively. 34 The solid black line shows the AMEGO-X sensitivity assuming a 3 year mission. 4 List of List of Figures Comparison of the detection rate vs localization accuracy for AMEGO-X, Swift/BAT, Fermi/GBM and COSI (left). The distance by AMEGO-X, versus jet observing angle for SGRBs (right). The solid, dot-dashed and dotted lines show jet structure models (from respectively Refs. [48][49][50] ) compatible with the observations of the GRB170817A. An event falling within the yellow shaded area will be detected both by AMEGO-X and the GW network. Observations or non-observations of events that lie below the blue dashed line and within the hatched area will indicate the extent of the GRB jet opening angle. Events in the orange area are detectable by AMEGO-X, but outside of the range of the GW network. 5 Fermi/LAT SED of SNR IC 443. The green and yellow lines show the Bremsstrahlung (with break) and π 0 decay models from Ref. 71 The red solid line shows the AMEGO-X 3 σ sensitivity. 6 Simulated AMEGO-X 3 yr all-sky map in the 1 − 30 MeV energy range. The map contains Galactic diffuse emission continuum emission calculated with GAL-PROP, 85 as well as individual sources extrapolated from the Fermi/LAT 4FGL-DR2 catalog 68 into the AMEGO-X energy range. Blazars dominate the population (>85%) followed by pulsars. The Galactic diffuse in this energy range primarily includes emission from inverse Compton and Bremsstrahlung (with contribution from π 0 at higher energy). The map is convolved with a 2D Gaussian kernel to account for the angular resolution of the instrument. We have verified that the Crab flux is in agreement with measurements. 86 7 (An exploded view of the Gamma-Ray Telescope on board AMEGO-X (left). The Tracker (red) and Calorimeter (purple) together characterize gamma-rays with three distinct detection techniques (right). Single-site events increase the sensitivity and low energy response (<100 keV) for transients only. Untracked and Tracked Compton events provide imaging <10 MeV, where the energy (E) and position ( r) of interactions are used to determine the initial Compton scatter angle (θ). Pair events enable imaging >10 MeV using the same detection techniques as Fermi/LAT. 8 The AMEGO-X Tracker relies on low-noise, low power CMOS monolithic APS detectors (left). Each Quad Chip consists of four identical APS arrays of 19 × 17 pixels, with the digital periphery located at the bottom of each APS array. Each Tracker Segment consists of 95 Quad Chips with a common bus bar for power distribution and wirebonds utilizing SPI readout. The Quad Chips are supported with a K1100 carbon fiber frame. The prototype AstroPix v2 detector, a 1 × 1 cm chip, has been fabricated and tested on the bench and in heavy ion beam tests (right). 9 The GRT hodoscopic arrangement of four Calorimeter layers enables an accurate measurement of gamma-ray energies through the profile of electromagnetic showers (left). The dual-gain SiPM readout at the ends of each CsI bar has been demonstrated in the lab and enables low-noise readout across a large dynamic range (right). 10 The AMEGO-X ACD is a simple, non-segmented version of the Fermi/LAT design. The ACD consists of 5 scintillation panels read out through wave-length shifting (WLS) bars with SiPM arrays on each end. The ACD surrounds the GRD to veto incident charged particles, reducing the background rate during flight. 11 The expected measured rates for each event type from the MEGAlib background models. The event classification is performed in MEGAlib, and the dominant background at lower energies is cosmic and atmospheric photons, while at higher energies in the Compton regime, the delayed emission of activated material in the instrument is the dominant source of background. The majority of background events > 1 MeV are classified as untracked Compton events because this is the most nonrestrictive event class. 12 The projected angular resolution (left) and effective area (right) are determined through on-axis, mono-energetic, point source simulations. In addition to background rate, these parameters are the main contribution to the sensitivity of the instrument. 13 The AMEGO-X continuum sensitivity using Untracked Compton, Tracked Compton, and Pair events is shown for the three year mission compared to the sensitivity of past and present missions. The effective area is conservatively taken to be for sources 37 • off-axis, and the observation time is estimated to be 20% of the full mission (T Obs = 3 yr × 0.2). 14 The Gamma-ray Telescope (yellow box) is shown on top of a base plate attached to the spacecraft bus. The pair of single-axis gimbaled solar arrays from end to end are approximately 11 m wide. This figure illustrates the deployed high-gain antenna, and one of the two low-gain antennae attached to the spacecraft bus. The other low-gain antenna is on the back side. Radiators (also gray) are located on each side of the spacecraft bus other than the side with the high-gain antenna. 15 The normalized exposure map for 5 MeV gamma rays (as an illustration) assuming low-Earth orbit (600 km) with a low inclination (5 • ). Rocking alternately ±30 • from zenith every orbit enables the GRT to uniformly observe nearly the entire sky every two orbits (∼3 hours). Over the course of 24 hours, the GRT survey covers the entire sky. 16 The two main science data products, Transient Alerts and Survey Data, use the same mode of operation and pipeline, but different telemetry paths. Transient Alerts (90% in 30 s) use Tracking and Data Relay Satellite System (TDRSS) for rapid alerts to the Gamma-ray Coordinates Network (GCN) for mulitimessenger and multiwavelength followup. All-sky Survey Data (90% in 24 hrs) is telemetered via TDRSS Ka band 2-3 times per day for on-ground processing and dissemination to the science community. Fig 1 : 1Simulated AMEGO-X SED for the 2014 flare of TXS 0506+056. The green dashed line is representative of leptonic and leptohadronic models, while the yellow dot-dashed line is the two-zone hadronic model where one of the zones is from the AGN corona from Ref. 23 The gray data points are measurements from Fermi/LAT (left). Light-curve of TXS 0506+056. The gray data points are measurements from Fermi/LAT. The yellow band show the times of the IceCube neutrino flares (right). The green data points are simulated AMEGO-X detection of the MeV emission during those flares. See 18 for more details. Fig 2 : 2With a minimum detectable polarization of 25 % for faint sources during the 3 year survey (left) Fig 3 : 3Expected emission from the AGN corona of NGC 1068. The blue dashed line shows the corona thermal emission, the light blue and green dashed lines show the non-thermal leptonic and hadronic components respectively. Fig 4 : 4Comparison of the detection rate vs localization accuracy for AMEGO-X, Swift/BAT, Fermi/GBM and COSI (left). The distance by AMEGO-X, versus jet observing angle for SGRBs (right). The solid, dot-dashed and dotted lines show jet structure models (from respectively Refs.[48][49][50] ) compatible with the observations of the GRB170817A. An event falling within the yellow shaded area will be detected both by AMEGO-X and the GW network. Observations or non-observations of events that lie below the blue dashed line and within the hatched area will indicate the extent of the GRB jet opening angle. Events in the orange area are detectable by AMEGO-X, but outside of the range of the GW network. Fig 5 : 5Fermi/LAT SED of SNR IC 443. , 84 Fig 6 : 846Simulated AMEGO-X 3 yr all-sky map in the 1 − 30 MeV energy range. The map contains Galactic diffuse emission continuum emission calculated with GALPROP, 85 as well as individual sources extrapolated from the Fermi/LAT 4FGL-DR2 catalog Fig 7 : 7(An exploded view of the Gamma-Ray Telescope on board AMEGO-X (left). The Tracker (red) and Calorimeter (purple) together characterize gamma-rays with three distinct detection techniques (right). Single-site events increase the sensitivity and low energy response (<100 keV) for transients only. Untracked and Tracked Compton events provide imaging <10 MeV, where the energy (E) and position ( r) of interactions are used to determine the initial Compton scatter angle (θ). Pair events enable imaging >10 MeV using the same detection techniques as Fermi/LAT. subsystems, the Gamma-Ray Detector (GRD) and the Anti-Coincidence Detector (ACD), which are protected by a Micro-Meteoroid Shield (MMS)(Figure 7). The GRD consists of a Tracker, with 40 layers of silicon complementary metal-oxide-semiconductor (CMOS) monolithic Active Pixel Sensors (APS), and a Calorimeter, with four layers of Cesium Iodide (CsI) scintillator bars. Fig 8 : 8The AMEGO-X Tracker relies on low-noise, low power CMOS monolithic APS detectors (left). Each Quad Chip consists of four identical APS arrays of 19 × 17 pixels, with the digital periphery located at the bottom of each APS array. Each Tracker Segment consists of 95 Quad Chips with a common bus bar for power distribution and wirebonds utilizing SPI readout. The Quad Chips are supported with a K1100 carbon fiber frame. The prototype AstroPix v2 detector, a 1 × 1 cm chip, has been fabricated and tested on the bench and in heavy ion beam tests (right). Fig 9 : 9The GRT hodoscopic arrangement of four Calorimeter layers enables an accurate measurement of gamma-ray energies through the profile of electromagnetic showers (left). The dual-gain SiPM readout at the ends of each CsI bar has been demonstrated in the lab and enables low-noise readout across a large dynamic range (right). Fig 10 : 10The AMEGO-X ACD is a simple, non-segmented version of the Fermi/LAT design. The ACD consists of 5 scintillation panels read out through wave-length shifting (WLS) bars with SiPM arrays on each end. The ACD surrounds the GRD to veto incident charged particles, reducing the background rate during flight. Fig 11 : 11The expected measured rates for each event type from the MEGAlib background models. The event classification is performed in MEGAlib, and the dominant background at lower energies is cosmic and atmospheric photons, while at higher energies in the Compton regime, the delayed emission of activated material in the instrument is the dominant source of background. The majority of background events > 1 MeV are classified as untracked Compton events because this is the most nonrestrictive event class. Fig 13 : 13The AMEGO-X continuum sensitivity using Untracked Compton, Tracked Compton, and Pair events is shown for the three year mission compared to the sensitivity of past and present missions. The effective area is conservatively taken to be for sources 37 • off-axis, and the observation time is estimated to be 20% of the full mission (T Obs = 3 yr × 0.2). Fig 14 : 14The Gamma-ray Telescope (yellow box) is shown on top of a base plate attached to the spacecraft bus. The pair of single-axis gimbaled solar arrays from end to end are approximately 11 m wide. This figure illustrates the deployed high-gain antenna, and one of the two low-gain antennae attached to the spacecraft bus. The other low-gain antenna is on the back side. Radiators (also gray) are located on each side of the spacecraft bus other than the side with the high-gain antenna. Fig 15: Fig 16 : 16The two main science data products, Transient Alerts and Survey Data, use the same mode of operation and pipeline, but different telemetry paths. Transient Alerts (90% in 30 s) use Tracking and Data Relay Satellite System (TDRSS) for rapid alerts to the Gamma-ray Coordinates Network (GCN) for mulitimessenger and multiwavelength followup. All-sky Survey Data (90% in 24 hrs) is telemetered via TDRSS Ka band 2-3 times per day for on-ground processing and dissemination to the science community. Table 1 : 1The Gamma-Ray Telescope baseline capabilities.Parameter Energy Range 25 keV -1 GeV Energy Resolution 5% FWHM at 1 MeV, 17% (68% containment half width) at 100 MeV Point Spread Function 4 • FWHM at 1 MeV, 3 • (68% containment) at 100 MeV Localization Accuracy transient: 1 • (90% CL radius), persistent: 0.6 • (90% CL radius) Effective Area 1200 cm 2 at 100 keV, 500 cm 2 at 1 MeV, 400 cm 2 at 100 MeV Field of View 2π sr (<10 MeV), 2.5 sr (>10 MeV) Simulated AMEGO-X SED for the 2014 flare of TXS 0506+056. The green dashed line is representative of leptonic and leptohadronic models, while the yellow dotdashed line is the two-zone hadronic model where one of the zones is from the AGN corona from Ref. 23 The gray data points are measurements from Fermi/LAT (left). Light-curve of TXS 0506+056. The gray data points are measurements from Fermi/LAT. The yellow band show the times of the IceCube neutrino flares (right). 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[ "A tutorial for propensity score weighting for moderation analysis: An application examining smoking disparities among sexual minority adults", "A tutorial for propensity score weighting for moderation analysis: An application examining smoking disparities among sexual minority adults" ]
[ "Beth Ann Griffin \nRAND Corporation\n1200 S Hayes St22202ArlingtonVA\n", "Megan S Schuler \nRAND Corporation\n1200 S Hayes St22202ArlingtonVA\n", "Matt Cefalu \nRAND Corporation\n1776 Main St90401Santa MonicaCA\n", "Lynsay Ayer \nRAND Corporation\n1200 S Hayes St22202ArlingtonVA\n", "Mark Godley \nChestnut Health Systems\n448 Wylie Dr61761NormalIL\n", "Noah Griefer \nInstitute for Quantitative Social Science\n02138CambridgeMA\n", "Donna L Coffman \nUniversity of South Carolina\n29208ColumbiaSC\n", "Daniel Mccaffrey \nETS\n660 Rosedale Road08541PrincetonNJUSA\n" ]
[ "RAND Corporation\n1200 S Hayes St22202ArlingtonVA", "RAND Corporation\n1200 S Hayes St22202ArlingtonVA", "RAND Corporation\n1776 Main St90401Santa MonicaCA", "RAND Corporation\n1200 S Hayes St22202ArlingtonVA", "Chestnut Health Systems\n448 Wylie Dr61761NormalIL", "Institute for Quantitative Social Science\n02138CambridgeMA", "University of South Carolina\n29208ColumbiaSC", "ETS\n660 Rosedale Road08541PrincetonNJUSA" ]
[]
Objective. To provide step-by-step guidance and STATA and R code for using propensity score (PS) weighting to estimate moderation effects.Research Design. Tutorial illustrating the key steps for estimating and testing moderation using observational data. Steps include (1) examining covariate overlap across treatment groups within levels of the moderator, (2) estimating the PS weights, (3) evaluating whether PS weights improved covariate balance, (4) estimating moderated treatment effects, and (5) assessing sensitivity of findings to unobserved confounding. Our illustrative case study uses data from 41,832 adults from the 2019 National Survey on Drug Use and Health to examine if gender moderates the association between sexual minority status (e.g., lesbian, gay, or bisexual [LGB] identity) and adult smoking prevalence.Results. For our case study, there were no noted concerns about covariate overlap and we were able to successfully estimate the PS weights within each level of the moderator. Moreover, balance criteria indicated that PS weights successfully achieved covariate balance for both moderator groups. PS weighted results indicated there was significant evidence of moderation for the case study and sensitivity analyses demonstrated that results were highly robust for one level of the moderator but not the other.Conclusions. When conducting moderation analyses, covariate imbalances across levels of the moderator can cause biased estimates. As demonstrated in this tutorial, PS weighting within each level of the moderator can improve the estimated moderation effects by minimizing bias from imbalance within the moderator subgroups.
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[ "https://export.arxiv.org/pdf/2204.03345v4.pdf" ]
248,006,167
2204.03345
c2f04bb91fb186c1881aa4ec0e321e35f864f34c
A tutorial for propensity score weighting for moderation analysis: An application examining smoking disparities among sexual minority adults Beth Ann Griffin RAND Corporation 1200 S Hayes St22202ArlingtonVA Megan S Schuler RAND Corporation 1200 S Hayes St22202ArlingtonVA Matt Cefalu RAND Corporation 1776 Main St90401Santa MonicaCA Lynsay Ayer RAND Corporation 1200 S Hayes St22202ArlingtonVA Mark Godley Chestnut Health Systems 448 Wylie Dr61761NormalIL Noah Griefer Institute for Quantitative Social Science 02138CambridgeMA Donna L Coffman University of South Carolina 29208ColumbiaSC Daniel Mccaffrey ETS 660 Rosedale Road08541PrincetonNJUSA A tutorial for propensity score weighting for moderation analysis: An application examining smoking disparities among sexual minority adults Acknowledgments: This research was financially supported through National Institutes of Health (NIH) grants (R01DA045049, PIs: Griffin/McCaffrey; P50DA046351, PI: Stein). NIH had no role in the design of the study, analysis, and interpretation of data nor in writing the manuscript. Word Count: 4,859 Number of Figures: 2 Number of Tables: 3 Number of Text Pages: 13 Conflicts of interest for all authors: Griffin -No conflicts Schuler -No conflicts Unblinded Title Page (include all contact information) and Word Count, number of figures, tables and number of text pages Cefalu -No conflicts Ayer -No conflicts Godley -No conflicts Griefer -No conflicts Coffman -No conflicts McCaffrey -No conflicts 2moderation analysesmoderated treatment effectstreatment effect heterogeneitysubgroup effectsquasi-experimental studiessurvey data 3 Objective. To provide step-by-step guidance and STATA and R code for using propensity score (PS) weighting to estimate moderation effects.Research Design. Tutorial illustrating the key steps for estimating and testing moderation using observational data. Steps include (1) examining covariate overlap across treatment groups within levels of the moderator, (2) estimating the PS weights, (3) evaluating whether PS weights improved covariate balance, (4) estimating moderated treatment effects, and (5) assessing sensitivity of findings to unobserved confounding. Our illustrative case study uses data from 41,832 adults from the 2019 National Survey on Drug Use and Health to examine if gender moderates the association between sexual minority status (e.g., lesbian, gay, or bisexual [LGB] identity) and adult smoking prevalence.Results. For our case study, there were no noted concerns about covariate overlap and we were able to successfully estimate the PS weights within each level of the moderator. Moreover, balance criteria indicated that PS weights successfully achieved covariate balance for both moderator groups. PS weighted results indicated there was significant evidence of moderation for the case study and sensitivity analyses demonstrated that results were highly robust for one level of the moderator but not the other.Conclusions. When conducting moderation analyses, covariate imbalances across levels of the moderator can cause biased estimates. As demonstrated in this tutorial, PS weighting within each level of the moderator can improve the estimated moderation effects by minimizing bias from imbalance within the moderator subgroups. Introduction Healthcare researchers are often interested in examining potential effect heterogeneityi.e., moderationwhen evaluating treatments, interventions, or health services received by individuals. 1,2 Understanding the influence that potential moderators may have on treatment effectiveness allows for greater understanding of the types of individuals who benefit the most (and least) from specific treatments and interventions, allowing more efficient targeting of resources. 3 In the context of observational data, moderation analysesalso referred to as treatment effect heterogeneity [4][5][6] and/or subgroup analyses 7,8allow researchers to explore important variation in the relationships between key exposures and health outcomes, leading to greater insights regarding population health and health disparities. In terms of estimation, moderation is traditionally assessed in regression models by testing for an interaction effect between the potential moderator and the treatment/exposure or by using stratification methods. However, these standard approaches may yield biased estimates of the moderated effects if they do not adequately yield comparable treatment/exposure groups within each level of the moderator variable. [9][10][11][12][13][14] While covariate adjustment can be useful to minize the potential bias that might arise from imbalance between the treatment and control group within each level of the moderator, relying on covariate adjustment alone can result in extrapolation of the outcome model and sensitivity to errors in the model specification. 15 It is more prudent to use causal inference approaches to ensure better comparability of the groups than covariate adjustment alone allows. [15][16][17] Specifically, this tutorial focuses on moderation analyses using propensity score (PS) weighting to statistically balance treatment/exposure groups within all levels of the moderator with respect to potential confounders in order to mitigate confounding bias. To date, only one similar tutorial type paper has been published with attention to PS matching. 14 Similar guidance is needed for PS weighting. Numerous PS weighting approaches for estimating moderated effects have been proposed; these approaches vary regarding the estimator of choice for the PS weights and/or the outcome model. Other methods in this space use more data-driven approaches (e.g., machine learning) to assess evidence of moderation in the presence of potential confounders. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] In fact, there has been an explosion of new methods in causal machine learning to estimate heterogenous treatment effects in recent years 21-33 that includes use of quasi-oracle, orthogonal statistical learning, generalized random forests, metalearners, and double (debiased) machine learning for estimation of treatment effect heterogeneity. While numerous methods for moderation analyses have been proposed, there is no easyto-use guide on how to implement these methods for health researchers that emphasizes key required assumptions and the importance of assessing the quality of the PS weights. The goal of this tutorial is to provide practitioners with step-by-step guidance and STATA and R code for conducting robust PS weighted analyses when estimating moderation effects. Accordingly, we provide step-by-step guidance regarding PS weighting for moderation that includes (1) examining covariate overlap across treatment/exposure groups within levels of the moderator, (2) estimating the PS weights, (3) evaluating whether PS weights improved covariate balance within each moderator level, (4) estimating moderated treatment effects, and (5) assessing sensitivity of findings to potential unobserved confounding. This approach is broadly applicable to observational datain particular, our case study illustrates the application of these methods to survey-weighted data. As an illustration of these methods, we present a case study that investigates whether gender moderates the association between sexual minority status (e.g., lesbian, gay, or bisexual (LGB) identity) and smoking. Numerous prior studies have demonstrated that sexual minority individuals report more frequent substance use, including smoking, compared to heterosexual individuals. [34][35][36] These disparities are often attributed to minority stress, namely the stigma, prejudice, and discrimination uniquely experienced by socially marginalized groups (e.g., LGB individuals). [37][38][39][40] Previous studies have highlighted gender differences in substance use, 41,42 including finding that disparities between sexual minority and heterosexual women are often larger in magnitude than those between sexual minority and heterosexual men. 34,43,44 In light of this, we frame gender as a potential moderator that may modify the strength of the relationship between sexual minority status and smoking in our case study. Methods Data and measures Study Population Data were from the 2019 National Survey on Drug Use and Health (NSDUH), an annual nationally-representative survey on drug use among the civilian, non-institutionalized US population ages 12 and older. 45 The sample size for the public-use NSDUH data is 56,136 for 2019 (65% response). Our sample is restricted to individuals ages 18 years and older who identified as heterosexual, lesbian/gay, or bisexual (n=41,832, including 3,618 LGB adults); respondents ages 12-17 are excluded as the NSDUH does not ask minors about sexual identity. Respondents who did not respond to the sexual identity question or answered, "don't know" (n=907, representing 2.1% of adult NSDUH respondents) are also excluded. This study was deemed exempt from review by RAND's institutional review board, as it involved de-identified survey data. Measures Primary exposure, LGB status: Sexual identity is assessed by asking: "Which one of the following do you consider yourself to be?" ("Heterosexual, that is, straight," "Lesbian or gay," "Bisexual," "Don't know"). An indicator for LGB status is defined as "1" for those who responded either "lesbian or gay" or "bisexual" and "0" for those who responded "heterosexual." We note that we conceptualize LGB status as a proxy for exposure to minority stress related to sexual minority identity. Moderator, gender: Gender is assessed on the NSDUH as male or female 1 . Outcome, past-month smoking status: Binary indicator for smoking at least one cigarette in the past 30 days. Demographic covariates included: age (categorized as: 18-25, 26-34, 35-49, 50+), race/ethnicity (non-Hispanic White; non-Hispanic Black; Hispanic; Asian; Native American/Alaskan Native; Native Hawaiian/Other Pacific Islander; non-Hispanic other race/multiracial), education level (less than high school; high school; some college/2-year college degree; 4-year college degree), employment (full-time; part-time; student; unemployed; other), and household income (less than $20,000; $20,000-$49,999; $50,000-$74,999; $75,000+). Potential outcomes, propensity scores, and moderated treatment effects We first introduce standard potential outcomes notation to define key estimands of interest for a binary treatment or exposure. Each individual has two potential outcomes: 1 , their outcome if they experienced the exposure condition (e.g., LGB identity and the associated social stressors for our case study), and 0 , their outcome if they experienced the comparison condition (e.g., heterosexual identity). 1 and 0 exist for all individuals in the population; however, we only observe one for each participant, corresponding to their reported exposure status. Let denote our binary exposure of interest. Then, = 1 × + 0 × (1 − ) denotes the i th individual's observed outcome. The treatment effect 2 for an individual is defined as 1 − 0 , representing the difference in their potential outcomes; the average treatment effect across the population (ATE) is defined as ( 1 − 0 ). In our case study, the ATE represents the average effect of minority stress due to LGB identity on current smoking rates across the entire population. Let represent a binary moderator (e.g., gender in our case study). Often, it is the case that ( 1 − 0 | = 1) ≠ ( 1 − 0 | = 0)e.g., the ATE for minority 2 We note that some in the causal inference field have argued that treatments or exposures must be "manipulable," namely capable of being changed or randomized to individuals 46,47 in order to obtain causal effects. This viewpoint excludes investigations of the causal effects of health disparities due to race, gender, or sexual identity (i.e., generally considered non-manipulable attributes). However, our case study adopts the framework articulated by Krieger, VanderWeele, and others that causal effects can be well-defined with regard to these exposures, given the social conditions/stressors associated with being in a socially marginalized group are indeed manipulable (and have changed across time) and are a primary contributor to health outcomes. 13,48 stress due to LGB status on smoking may differ for men and women. When moderation is present, it is of interest to estimate the ATE for each level of the moderator. The moderated ATE (M-ATE) is defined as ( 1 − 0 | = ). Our proposed method uses PS weighting to create comparable groups with respect to baseline potential confounders when estimating the M-ATE. In general, the PS is defined as an individual's probability of treatment, given their pretreatment characteristics denoted by the vector : ( ) = ( = 1| ). When interest lies in estimating the M-ATE, PS estimation will additionally be conditional on the moderator, 14 as follows: ( , ) = ( = 1| , = ) PS estimates, ( , ), are then used to define PS weights as follows for a binary moderator : we weight the treatment group within each level of by 1/( , ) and weight the comparison group within each level of by 1/(1 −( , )). In order to identity the ATE, and likewise the M-ATE, we must assume strong ignorability, defined as follows: i. (Y1 ,Y0) ⊥ T|X, Ze.g., no unobserved confounders ii. 0 < ( , ) < 1e.g., positivity or overlap between exposure groups within levels of the moderator These two key assumptions (overlap and no unobserved confounders) are impossible to test in practice; however, their plausibility should be explored in any study using PS weights. This tutorial details how to assess plausibility of both the overlap assumption (in Step 1) and the no unobserved confounding assumption (in Step 5), highlighting an important and often overlooked part of many applied studies. In the writing that follows, checking for overlap implies checking for the positivity assumption. We highlight that this approach can be implemented in the context of weighted data, such as data with survey sampling or attrition weights. Prior work has demonstrated that, in the context of traditional PS weighting (no moderation), it is important that the survey or attrition weights be used as survey weights in the estimation of the PS weight. 49 Then, the PS weights can be multiplied by survey or attrition weights to obtain the needed composite weights for use in the final outcome models. Key steps for estimating moderator treatment effects Step In the context of survey or attrition weighted data, comparisons prior to PS weighting should be weighted using the survey or attrition weight. Comparisons after PS weighting should use a composite weight (i.e., PS weight multiplied by the survey or attrition weight). When using TWANG, all of this is directly handled by the package. Step 4: Estimate moderated treatment effect If PS weights have successfully created comparable groups in Step 3, then moderated effects can be estimated. Specifically, M-ATE estimates will be calculated as the difference in PS weighted means between the two exposure groups (e.g., LGB and heterosexual) within each level of the moderator (e.g., females and males). This can be done by fitting a weighted regression model that includes indicators for both the treatment/exposure and the moderator, their interaction, as well as the same set of covariates used in the PS model. This model can then be used to recover the estimated treatment effects within levels of the moderators, as described below. We suggest including all covariates used in the PS model in the final outcome model since this "doubly robust" estimation procedure is the preferred approach for estimating treatment/exposure effects with PS weights. 16,65,66 As long as one model (either the covariate-adjusted outcome model or the PS model) is specified correctly, one will obtain consistent estimates for the ATE, and by extension, the M-ATE. Including the confounding covariates in the outcome model also reduces any residual covariate imbalance remaining after weighting and can improve precision of effect estimates. Since we are using PS weights in our final model, we use standard survey package commands to obtain standard errors that account for the weighting. Recommended specification of outcome model for estimating M-ATEs: ( [ ]) = 0 + 1 × + 2 × + 3 × * + × where (. ) is the appropriate link function. In this model specification, we assume the absence of interactions between covariates and for illustrative purposes. If such interactions exist, it is best to control for these additional interaction terms in the final outcome model. 67 We also assume only one moderator of interest; if interest lies in estimation of multiple M-ATEs, it is possible to repeat this procedure independently for each candidate moderator. After estimating the outcome model, we can use post hoc estimation procedures to calculate the needed M-ATE for each level of the moderator. For example, in our case study, smoking status is binary so (. ) will be the logit link for a logistic outcome model. To obtain estimates of the M-ATE, we must use the fitted logistic regression model to estimate the marginal probabilities of the outcome for each subgroup under both exposure conditions, which can then be used to compute the needed differences in the means of the potential outcomes and risk differences for males and females. 16 That is, for every person in the sample, we estimate the probability of smoking setting sexual identity to LGB and then repeat this estimation setting sexual identity to heterosexual. For both females and males, we separately average the estimated probabilities for both LGB and heterosexual sexual identity, the difference of which (within gender groups) provides our treatment effect estimate. Averages with groups should be weighted by sampling, nonresponse or attrition weights. We report the estimates and their corresponding 95% confidence intervals. In many studies, primary interest for the moderation analyses might focus on whether there is any evidence of moderation (rather than reporting of subgroup effects). This can be done using the outcome model above by testing the null hypothesis that 3 =0 and can be examined directly from the regression output by referring to the 95% confidence interval for ̂3. When the moderator has multiple levels, then a joint test for all the interactions terms between treatment/exposure and the interaction can be used. It would also be reasonable to estimate stratified outcome models using the PS weights from each level of the moderator and to report the estimated effects from those models. Here, we focus on estimation of the joint outcome model given it allows for researchers to directly test for moderation more formally than stratified models. Step 5: Check for sensitivity to unobserved confounding Given the strong ignorability assumption, it is critical to assess the potential impact that unobserved confounders might have on the study's key findings in any analysis using PS weights. There are many tools available to examine the potential impact of unobserved confounding for main effects of treatment. [68][69][70][71][72] Conveniently, most of these tools can be applied to examine sensitivity in moderation analyses as well, though with some adaptation. Here, we use a graphical tool (the Ovtool in R) that relies on using simulations to assess how sensitive our findings may be to potential unobserved confounding by a variable that is independent of the observed confounders conditional on the treatment indicator. This graphical approach tests sensitivity both in terms of how the estimated treatment effect and statistical significance (i.e., p-value) might change as a function of the strength of the relationships between the unobserved confounders and the treatment/exposure and outcome within each level of the moderator. 68,73 Illustrative Case Study Application Table 1 shows the comparison of LGB and heterosexual adults, stratified by gender, prior to PS weighting. SMDs and KS statistics indicate clear differences between heterosexual and LGB individuals for both males and females. In general, LGB adults are more likely to be unemployed, are younger, and have lower household income than heterosexual adults. The magnitude of the difference between LGB and heterosexual adults varies by gendere.g., age differences are more disparate for females than males. While there are no notable differences between female LGB and heterosexual females on race/ethnicity, heterosexual males are less likely to be Hispanic than LGB males. LGB females are less likely than heterosexual females to be college graduates, while there are no such noted differences for males. Step. histograms stratified with respect to the moderator can also be examined. Step. Estimate the PS weights We estimate the PS model separately for males and females using the TWANG package. Code for STATA and R are provided in the Appendix. The NSDUH survey weight is included via the sampw option in TWANG which uses the weights in estimating the propensity score models and the post hoc checks for balance in the weighted covariate distributions. Step.3 Evaluate whether PS weights improved covariate balance within each level of the moderator After PS weighting, we evaluate covariate balance with respect to both SMDs and KS statistics separately for both females and males (see Figure 1, Table 2). We note that balance is assessed here using the composite weight (i.e., PS weight multiplied by the survey weight), calculated by TWANG. All absolute SMDs and KS statistics fall well below the 0.10 threshold, suggesting successful balance has been achieved within each level of the moderator. Step.4 Estimating moderated treatment effects Next, we obtain M-ATE estimates using covariate-adjusted weighted logistic regression. As in Step 3, we use the composite weights in the outcome model. We observe significant evidence that LGB status is associated with higher likelihood of smoking for both males and females ( Step 5. Assess sensitivity to unobserved confounding Discussion This tutorial provides health researchers with guidance on how to use PS weighting for estimation of moderated effects. It illustrates the key steps using an application examining the potential association between minority stress due to LGB status and smoking among adults. Critically, we detail how to assess key assumptions required for robust estimationincluding attention to overlap and unobserved confounding sensitivity analyses. Similar to prior work on PS matching, we recommend that estimation of the PS weights should occur within each level of the moderator when feasible. 14 This tutorial provides guidance for performing moderation using PS weighting and ensures careful consideration of key assumptions (namely overlap and unobserved confounding) that is often overlooked in practice. We There is no simple rule of thumb for selecting among these options but comparing the balance achieved and associated loss in precision due to each option can help analysts to choose the optimal approach for a given study. In addition, we highlight that there is a growing literature on weighting methods that deal with the lack of overlap issue, including the use of overlap weighting. 75 LGB (24) #Females results = OVtool::outcome_model(ps_object = NULL, stop.method = NULL, data = data.female, weights = "psw", treatment = "lgb_flag", outcome = "cigmon", model_covariates = c("age","race","educ","income","employ"), estimand = "ATE") summary(results$mod_results) ovtool_results_twang = ov_sim(model_results=results, plot_covariates=c("age","race","educ","income","employ"), es_grid = NULL, rho_grid = seq(0, 0.40, by = 0.05), n_reps = 100, progress = TRUE) plot.ov(ovtool_results_twang, col='color', print_graphic = '3', p_contours = c(0.05)) #Males results = OVtool::outcome_model(ps_object = NULL, stop.method = NULL, data = data.male, weights = "psw", treatment = "lgb_flag", outcome = "cigmon", model_covariates = c("age","race","educ","income","employ"), estimand = "ATE") summary(results$mod_results) ovtool_results_twang = ov_sim(model_results=results, plot_covariates=c("age","race","educ","income","employ"), es_grid = seq(-0.80, 0.80, by = 0.05), rho_grid = seq(0, 0.40, by = 0.05), n_reps = 100, progress = TRUE) plot.ov(ovtool_results_twang, col='color', print_graphic = '3', p_contours = c(0.05)) STATA Code ********************************************************************* ***This is the Stata code to run the needed steps in the tutorial ***Uses the NSDUH case study which can be downloaded from our supplementary data files ********************************************************************* ********************************************************************* *** Step 1 -check for overlap concerns ********************************************************************* use "data_appendix.dta" *** Since everything is categorical -we simply need to check for empty cells *** Create tables stratified by lgb_flag and female ********************************************************************* *** Steps 2 & 3 -Estimate PS weights using TWANG within levels of the moderator *** and assess balance within levels of the moderator ********************************************************************* *** Must specify adopath where twang STATA ado files have been downloaded adopath + "/Users/folder/twang STATA/adofiles" *** subset original data to females only keep if female == 1 save data_female.dta use "data_female.dta" *** Call twang ps function *** Note that NSDUH sampling weights are controlled for by using the sampw() option ps lgb_flag i.age i.race i.educ i.income i.employ, /// ntrees(10000) stopmethod(ks.max) estimand(ATE) sampw(analwt_c) /// rcmd(/usr/local/bin/RScript) /// objpath(/Users/folder/project_results) /// plotname(/Users/folder/project_results/plot_female.pdf) *** Examine balance figures (saved as pdf file) *** Examine balance table balance, unweighted weighted *** Save twang-generated dataset with propensity score weights *** Final weight includes the survey weight times the PS weight save data_female_wgts *** Subset original data to males only use "data_appendix.dta" keep if female == 0 save data_male.dta use "data_male.dta" *** Call twang ps function ps lgb_flag i.age i.race i.educ i.income i.employ, /// ntrees(10000) stopmethod(ks.max) estimand(ATE) sampw(analwt_c) /// rcmd(/usr/local/bin/RScript) /// objpath(/Users/folder/project_results) /// plotname(/Users/folder/project_results/plot_male.pdf) *** Examine balance figures (saved as pdf file) *** Examine balance table balance, unweighted weighted *** Save twang-generated dataset with propensity score weights *** Final weight includes the survey weight times the PS weight save data_male_wgts ********************************************************************* *** Step 4 -Estimating the treatment effects ********************************************************************* *** Combine data_male_wgts.dta and data_female_wgts.dta --will run outcome model *** on combined dataset append using "data_female_wgts.dta" save "data_combined_wgts.dta" *** Outcome model: Propensity score-weighted logistic regression svyset [pweight=ksmaxate] svy: logit cigmon lgb_flag#female i.age i.race i.educ i.income i.employ margins, dydx(lgb_flag) over(female) ********************************************************************* ********************************************************************* ********************************************************************* set.seed (24) #Moderator Level 1 results = OVtool::outcome_model(ps_object = NULL, stop.method = NULL, data = data1, weights = "psw", treatment = "treatment", outcome = "outcome", model_covariates =c("categorical.confounder1","categorical.confounder2", "categorical.confounder3","continuous.confounder1","continuous.confounder2", "continuous.confounder3"), estimand = "ATE") summary(results$mod_results) ovtool_results_twang = ov_sim(model_results=results, plot_covariates =c("categorical.confounder1","categorical.confounder2", "categorical.confounder3","continuous.confounder1","continuous.confounder2", "continuous.confounder3"), es_grid = NULL, rho_grid = seq(0, 0.40, by = 0.05), n_reps = 100, progress = TRUE) plot.ov(ovtool_results_twang, col='color', print_graphic = '3', p_contours = c(0.05)) #Moderator Level 0 results = OVtool::outcome_model(ps_object = NULL, stop.method = NULL, data = data0, weights = "psw", treatment = "treatment", outcome = "outcome", model_covariates =c("categorical.confounder1","categorical.confounder2", "categorical.confounder3","continuous.confounder1","continuous.confounder2", "continuous.confounder3"), estimand = "ATE") summary(results$mod_results) ovtool_results_twang = ov_sim(model_results=results, plot_covariates =c("categorical.confounder1","categorical.confounder2", "categorical.confounder3","continuous.confounder1","continuous.confounder2", "continuous.confounder3"), es_grid = seq(-0.80, 0.80, by = 0.05), rho_grid = seq(0, 0.40, by = 0.05), n_reps = 100, progress = TRUE) plot.ov(ovtool_results_twang, col='color', print_graphic = '3', p_contours = c(0.05)) STATA Code ********************************************************************* ***This is the Stata code to run the needed steps in the tutorial ***Assumes a generic data structure with the following key variables: ***outcome, treatment, moderator, categorical and continuous covariates ********************************************************************* ********************************************************************* *** Step 1 -check for overlap concerns ********************************************************************* use "data.dta" *** For categorical confounders check for empty cells *** Create tables stratified by treatment and moderator ********************************************************************* *** Steps 2 & 3 -Estimate PS weights using TWANG within levels of the moderator *** and assess balance within levels of the moderator ********************************************************************* *** Must specify adopath where twang STATA ado files have been downloaded adopath + "/Users/folder/twang STATA/adofiles" *** subset original data to moderator level 1 only keep if moderator == 1 save data_1.dta use "data_1.dta" *** Call twang ps function *** Note that sampling weights can be controlled for by using the sampw option Finally , we conduct sensitivity analyses for unobserved confounding for the estimated M-ATE effects separately for females and males.Figure 2shows how both the estimated effect (solid contours) and p-value (dashed contours) would change as a function of an unobserved confounder whose association with our key exposure (LGB status) is expressed through an effect size or SMD (x-axis), and whose relationship with the outcome is expressed as a correlation (y-axis). The solid contours report the adjusted M-ATE estimates that would result as the relationship between the omitted variable and the outcome and exposure group indicator increases (LGB vs not).As shown for females, the estimated treatment effect gets larger as we move to the left from zero along the x-axisgoing from 0.2 to 0.4 as we move to the upper left hand corner -and decreases as we move right from zerocrossing over a null effect of 0 as we move to the upper right hand corner. The dashed contours show how statistical significance of our result is impacted and parallel what we see for the estimated treatment effect contours. Namely, there is an area of the plot around the 0 treatment effect contour that indicates that statistically significance disappears with certain magnitudes of an unobserved confounder. For females, an unobserved confounder would need a correlation with the outcome greater than 0.10 and would need to differ between LGB and heterosexual groups with an SMD of greater than 0.42 to change our finding such that it is no longer statistically significant (at 0.05 level). Additionally, the plot shows (via the black dots) the observed correlations and effect sizes observed in the data for the different categories of the five observed covariates used in the PS weights. As shown, the observed covariates are far from the region of the plot where the unobserved covariates would need to be to impact our M-ATE estimate for females. The vast majority of the observed covariates are not correlated with the outcome at greater than 0.10 and have unweighted SMDs between exposure groups of less than 0.42. Thus, relative to the observed covariates, the omitted variable would need to have a very strong relationship with both the outcome and the treatment group in order to change our findings regarding the observed association between LGB status and a higher likelihood of smoking for females. This sensitivity analysis suggests that if omitted variables are present, our findings for females would be unlikely to substantially change if these variables were controlled for, indicating that our results are robust to unobserved confounding.In contrast, the results are notably different for males. We see evidence of a highly sensitive finding that could easily move towards the null with the addition of even a weakly correlated omitted variable, suggesting the finding for males is not as sufficiently trustworthy and should be interpreted cautiously as it could be the result of an omitted variable, unlike the more robust findings for females. highlight several additional methodological considerations in the context of moderation analysis. First, there are a wealth of methods available for healthcare researchers to examine treatment effect heterogeneity and the choice of the best method for a given case study should be driven by the scientific goals of the research. While we utilize the TWANG package to estimate our PS weights, our steps generalize beyond the use of this method for estimation of the PS weights. Researchers can find an overview to the numerous causal machine learner methods in this space as well as python code 74 using the GitHub page from Microsoft Research's Automated Learning and Intelligence for Causation and Economics (ALICE) project. Additionally, we examine a binary moderator; alternative methods are needed to address continuous moderator variables. Finally, caution is necessary when using PS weighting for moderation analyses with smaller sample sizes. While PS weighting within each level of the moderator can minimize M-ATE bias arising from imbalance within moderator subgroups, it comes at the expense of more variable weights, leading to larger standard errors for M-ATE estimates. In some studies, this loss in precision may come at too great a cost and fitting a single PS model (pooling across moderation levels) may be preferrable. Figure Legends Figure Legends Figure 1 .Figure 2 .FiguresFigure 1 . 121SMDs before and after weighting for Females and Males Results from omitted variable sensitivity analyses SMDs before and after weighting for Females and Males.* Note: "ks.max.ATE" in the figure refers to the stopping rule used in TWANG and not KS statistics directly. In TWANG, a user can select from 4 stopping rules when estimating the propensity score weights: namely rules that aim to optimize (here minimize) the mean or max SMD or the mean and max KS statistics. Figure 1 1Figure 2 . 12Results moderator. If there are obvious areas where there is a lack of overlap, it is possible to still estimate moderated effects so long as the groups for which there is a lack of overlap are removed completely fromIn the absence of moderation, an essential diagnostic step is to evaluate the extent to which PS1: Check for covariate overlap across treatment groups for each level of the moderator While there is no formal way to test for overlap across treatment/exposure groups on the multivariate distribution of the control covariates, one can assess overlap in the univariate distributions of covariates. In the traditional PS context (no moderation), one should explore baseline descriptive statistics of all covariates included in the PS model, stratified by treatment group. If variables exhibit differential range across treatment groups, this suggests violation of the overlap assumption. Extending to the moderation context, covariate overlap should similarly be assessed, stratified by both the treatment and the evaluation. Removing a subsample from the data set in order to achieve overlap between the treatment groups will diminish the generalizability of study findings but is often the only option when there are obvious areas without overlap. 50 Step 2: Estimate the PS weights In the context of the M-ATE, estimation of the PS model separately within each level of the moderator generally results in better covariate balance within moderator strata which, in turn, can minimize bias. 14 However, in the case of small sample sizes, this stratified estimation approach can result in unstable weights and overly noisy estimates of the M-ATE; in these cases, alternative estimation strategies may be preferable. We address these tradeoffs more in our discussion. Since our case study data affords adequate sample size, we estimate the PS model separately for males and females. Estimation of the PS model including selection of the most relevant covariates requires significant care; model misspecification can lead to biased results. Resultantly, machine learning methods offer a key advantage over parametric models -like logistic regression -in this setting. 51 We implement PS estimation with the TWANG package 52 which uses a nonparametric machine learner, generalized boosted modeling. 53 However, we note that alternative estimation methods could be used, as there are numerous approaches to estimating high quality PS weights, including more recent causal machine learning methods. 21,22,53-57 Additionally, variable selection requires careful consideration when estimating PS model. Selection of the covariates for use in the PS model should include variables that are associated with the outcome -often regardless of their association with the treatment indicator. However, variables that are solely predictive of treatment with no association with the outcome should be avoided. 51,58 As noted above, we properly include the NSDUH survey weight in the estimation of the PS within each level of the moderator to ensure we obtain balance among the survey weighted version of the data. Step 3: Evaluate whether PS weights improved covariate balance within each level of the moderator weighting improved covariate balance across treatment groups; there has been rich methodological guidance on how best to do so for estimation of main effects. 51,59-61 Extending to the moderation context, after PS weighting, covariate balance should be evaluated across treatment groups within each level of the moderator. Often, balance is assessed by strictly comparing PS weighted mean differences for each covariate across treatment groups. However, others have argued that ensuring covariate distributions are similar across treatment groups is essential for high quality PS weights. 62,63 We recommend using both standardized mean differences (SMD) and Kolmogorov-Smirnoff (KS) statistics to assess balance. The SMD is a measure of standardized mean differences between the groups, while KS statistics are measures of distributional differences between groups. More specifically, the KS statistic is the maximum absolute difference between the two empirical cumulative distribution functions of the treated and control groups for a given covariate. For both SMD and KS, smaller is better, with values of 0 indicating no differences between treatment groups. A recommended threshold is that both SMD and KS values should not exceed 0.10 after PS weighting; absolute standardized differences of 0.10 are considered to be small effect size differences 54,62,64 and recent work has shown this same cut-off point is reasonable for KS statistics. 62 1 Check for covariate overlap across treatment groups for each level of the moderator Given the categorical nature of our covariates, overlap concerns can be assessed by checking whether there are subgroups within each covariate that have no individuals after stratifying by both LGB identity and gender (seeTable 1). Having no individuals in one specific subgroup (e.g., less than high school) would imply that the analysis cannot estimate the causal effect of the treatment for that specific subgroup. Thus, systematically checking the covariate distribution via frequency tables can be a pragmatic way to identify lack of overlap (when present) for categorical variables. As no cells with 0% are noted, there does not appear to be any concerns regarding covariate overlap for this analysis. For continuous covariates, we recommend comparing the minimum and maximum for each treatment group within each level of the moderator; Table 3 ) 3. Using the marginal probabilities from the regression results, the estimated M-ATE risk difference of current smoking for LGB vs heterosexual males is 0.05 (95% CI = 0.00, 0.10), suggesting LGB males are moderately more likely to smoke than their heterosexual counterparts. In contrast, LGB females have substantially higher risk of smoking than heterosexual females, with an estimated M-ATE risk differenceof 0.15 (95% CI = 0.08, 0.18). There is significant evidence of moderation, as determined by the significant coefficient for the interaction term in the logistic regression model -namely 0.65 (95% CI = 0.23, 1.01). ,76 These methods target the subset of the population for which data supports overlap in baseline characteristics. That is, they estimate an average treatment effect on the overlap sample, denoted ATO. Overlap weighting methods offer an ability to obtain good balance and optimal variance on the sample where overlap occurs. However, it is not always readily apparent if the ATO is of substantive interest, though many argue this is the subset where researchers have clinical equipoise. Notably, in the context of moderation analyses, we highlight that the population subset with appropriate overlap could potentially differ across moderator levels. Future work should seek to generalize overlap weighting techniques for moderation analysis; for example, identifying the population subset which has adequate overlap for all moderator levels may be of interest such that M-ATEs are directly comparable across moderator groups. We note that one of the goals of this tutorial is to provide researchers with a way to check for violations of overlap prior to estimation of the PS weights. More traditionally, it has been recommended that one assess reasonableness of the overlap Overall, estimating robust moderation effects requires thoughtful consideration of covariate imbalances across both treatment and moderator levels; as detailed in this tutorial, PS weighting is one promising technique to minimize bias arising from covariate imbalances.assumption post hoc by comparing the estimated PS distributions after estimating the weights. 51,60 Unfortunately, such a post hoc comparison alone does not allow researchers to see what might be driving the regions where there is a lack of overlap. 20. Yuan A, Chen X, Zhou Y, et al. Subgroup analysis with semiparametric models toward precision medicine. Stat Med. 2018;37(11):1830-1845. 21. Athey S, Imbens G. Recursive partitioning for heterogeneous causal effects. Proc Natl Acad Sci U S A. 2016;113(27):7353-7360. 22. Wager S, Athey S. Estimation and inference of heterogeneous treatment effects using random forests. 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Tables1Baseline characteristics of LGB and heterosexual adults prior to PS weighting, by genderFemales Males LGB Heterosexual LGB Heterosexual Age 18-25 Years Old 36.3% 11.2% a,b 23.9% 13.5% a 26-34 Years Old 28.5% 14.5% a,b 25.8% 16.1% a 35-49 Years Old 19.9% 24.2% a 23.3% 24.8% 50 or Older 15.3% 50% a,b 27.1% 45.6% a,b Race/Ethnicity NonHisp White 59.8% 63.6% 59.8% 64.2% NonHisp Black/Afr Am 14.2% 12.5% 9.7% 11.2% NonHisp Nat.Am/AK.Nat 0.6% 0.6% 0.4% 0.5% NonHisp Nat.HI/Other PIs 0.3% 0.3% 0.6% 0.4% NonHisp Asian 3.8% 5.7% 4.8% 5.4% NonHisp multiracial/other race 4.2% 1.6% a 1.9% 1.7% Hispanic 17.0% 15.7% 22.7% 16.5% a Education Less high school 12.5% 10.9% 9.7% 12.4% High school grad 23.6% 22.4% 23.7% 26.3% Some college/Assoc Dg 36.1% 32.8% 31.0% 28.7% College graduate 27.9% 33.8% a 35.6% 32.6% Income Less than $20,000 24.2% 15.8% a 17.4% 12.4% a $20,000 -$49,999 30.5% 30.0% 30.4% 26.3% $50,000 -$74,999 14.8% 15.9% 15.8% 16.2% $75,000 or More 30.5% 38.3% a 36.4% 45.2% a Employment Employed full time 46.8% 42.0% 57.3% 57.9% Employed part time 19.4% 15.4% 13.0% 10.1% Unemployed 8.4% 3.2% a 7.0% 4.2% a Student 22.2% 37.6% a,b 19.8% 26.1% a Other 3.2% 1.7% 2.9% 1.7% Notes: a = Denotes absolute SMD > 0.1; b = Denotes KS-statistic is > 0.1 All descriptive statistics account for NSDUH survey weights Table 1 Table 2 . 12Baseline characteristics of LGB and heterosexual adults after PS weighting, by genderFemales Males LGB Hetero -sexual SMD KS LGB Hetero sexual- SMD KS Age 18-25 Years Old 14.0% 13.1% 0.03 0.01 14.8% 14.0% 0.02 0.01 26-34 Years Old 16.1% 15.5% 0.02 0.01 17.2% 16.6% 0.02 0.01 35-49 Years Old 23.9% 23.9% 0.00 0.00 25.2% 24.7% 0.01 0.01 50 or Older 46.0% 47.5% -0.03 0.02 42.8% 44.8% -0.04 0.02 Race/Ethnicity NonHisp White 64.5% 63.3% 0.03 0.01 64.4% 64.0% 0.01 0.00 NonHisp Black/Afr Am 12.6% 12.6% 0.00 0.00 11.3% 11.2% 0.01 0.00 NonHisp Nat.Am/AK.Nat 0.5% 0.6% -0.01 0.00 0.3% 0.5% -0.03 0.00 NonHisp Nat.HI/OtherPIs 0.1% 0.3% -0.03 0.00 0.4% 0.5% -0.01 0.00 NonHisp Asian 4.7% 5.6% -0.04 0.01 5.0% 5.4% -0.02 0.00 NonHisp multiracial/other race 1.8% 1.8% 0.00 0.00 1.3% 1.7% -0.04 0.01 Hispanic 15.9% 15.8% 0.00 0.00 17.3% 16.8% 0.01 0.01 Education Less high school 9.4% 11.0% -0.05 0.02 12.4% 12.3% 0.00 0.00 High school grad 21.1% 22.5% -0.03 0.01 26.3% 26.2% 0.00 0.00 Some college/Assoc Dg 34.9% 33.1% 0.04 0.02 28.0% 28.8% -0.02 0.01 College graduate 34.5% 33.4% 0.02 0.01 33.3% 32.8% 0.01 0.01 Income Less than $20,000 16.7% 16.4% 0.01 0.00 13.0% 12.6% 0.01 0.00 $20,000 -$49,999 29.6% 30.1% -0.01 0.01 25.9% 26.5% -0.01 0.01 $50,000 -$74,999 16.2% 15.9% 0.01 0.00 16.4% 16.2% 0.01 0.00 $75,000 or More 37.5% 37.7% 0.00 0.00 44.7% 44.8% 0.00 0.00 Employment Employed full time 44.0% 42.4% 0.03 0.02 59.8% 57.8% 0.04 0.02 Employed part time 15.4% 15.7% -0.01 0.00 10.0% 10.2% -0.01 0.00 Unemployed 3.8% 3.6% 0.01 0.00 4.6% 4.4% 0.01 0.00 Student 34.9% 36.5% -0.03 0.02 23.9% 25.8% -0.05 0.02 Other 1.9% 1.8% 0.00 0.00 1.7% 1.8% 0.00 0.00 Notes: a = Denotes absolute SMD > 0.1; b = Denotes KS-statistic is > 0.1 All descriptive statistics account for NSDUH survey weights. Table 3 . 3Logistic regression results for the PS weighted doubly robust logistic modelRegression Coefficient Standard Error 95% Confidence Interval Table 3 3Supplementary code for both our illustrative case study as well as a generic data setAppendix A: Code to run for our illustrative case study data Please note that case study data also available as part of this paper's supplementary material R Code #This is the R code to run the needed steps in the tutorial #Uses the NSDUH case study which can be downloaded from our supplementary data files library(twang) #Subset data down to the different levels of the moderator data.female=data[data$female==1,] data.male=data[data$female==0,] ######################################### #Step 1 -check for overlap concerns ######################################### #Since everything is categorical -we simply need to check for empty cells table(data.female$lgb_flag,data.female$age) table(data.female$lgb_flag,data.female$race) table(data.female$lgb_flag,data.female$educ) table(data.female$lgb_flag,data.female$income) table(data.female$lgb_flag,data.female$employ) table(data.male$lgb_flag,data.male$age) table(data.male$lgb_flag,data.male$race) table(data.male$lgb_flag,data.male$educ) table(data.male$lgb_flag,data.male$income) table(data.male$lgb_flag,data.male$employ) #If there was a continuous covariate, check the minimum and maximum for each level of the moderator #summary(data.female$continuous.covariate) #summary(data.male$continuous.covariate) ########################################################################### #Step 2 -estimate PS weights using TWANG within levels of the moderator ########################################################################### set.seed(175659) #Note that NSDUH sampling weights are controlled for by using the sampw option ps1.f <-ps(lgb_flag ~ age+race+educ+income+employ, data = data.female, n.trees=10000,interaction.depth=2,shrinkage=0.01,perm.test.iters=0, stop.method=c("ks.max"),estimand = "ATE",verbose=FALSE,sampw = data.female$analwt_c) ps1.m <-ps(lgb_flag ~ age+race+educ+income+employ, data = data.male, n.trees=10000,interaction.depth=2,shrinkage=0.01,perm.test.iters=0, stop.method=c("ks.max"),estimand = "ATE",verbose=FALSE,sampw = data.male$analwt_c) Appendix B: Code to run for a generic data set R Code #This is the R code to run the needed steps in the tutorial #Assumes a generic data structure with the following key variables: #outcome, treatment, moderator, categorical and continuous covariates library(twang) #Subset data down to the different levels of the moderator data1=data[data$moderator==1,] data0=data[data$moderator==0,] ######################################### #Step 1 -check for overlap concerns ######################################### #For categorical confounders check for empty cells table(data1$treatment,data1$categorical.confounder1) table(data1$treatment,data1$categorical.confounder2) table(data1$treatment,data1$categorical.confounder3) table(data0$treatment,data0$ categorical.confounder1) table(data0$treatment,data0$ categorical.confounder2) table(data0$treatment,data0$ categorical.confounder3) #For continuous covariate, check the minimum and maximum for each level of the moderator summary(data1$continuous.covariate1) summary(data0$continuous.covariate1) summary(data1$continuous.covariate2) summary(data0$continuous.covariate2) summary(data1$continuous.covariate3) summary(data0$continuous.covariate3) ########################################################################### #Step 2 -estimate PS weights using TWANG within levels of the moderator ########################################################################### set.seed(175659) #Note that sampling weights can be controlled for by using the sampw option ps.1 <-ps(treatment ~categorical.confounder1+categorical.confounder2+categorical.confounder3+ continuous.confounder1+continuous.confounder2+continuous.confounder3, data = data1, n.trees=10000,interaction.depth=2,shrinkage=0.01,perm.test.iters=0, stop.method=c("ks.max"),estimand = "ATE",verbose=FALSE,sampw = NA) ps.0<-ps(treatment ~ categorical.confounder1+categorical.confounder2+categorical.confounder3+ continuous.confounder1+continuous.confounder2+continuous.confounder3, data = data0, n.trees=10000,interaction.depth=2,shrinkage=0.01,perm.test.iters=0, stop.method=c("ks.max"),estimand = "ATE",verbose=FALSE,sampw = NA ) write.table(balance1[[1]],"balance_unwted_moderator1.csv",sep=",") write.table(balance0[[1]],"balance_unwted_moderator0.csv",sep=",") write.table(balance1[[2]],"balance_wted_moderator1.csv",sep=",") write.table(balance0[[2]],"balance_wted_moderator0.csv",sep=",") ################################################################# #Step 4 -Estimating the treatment effects ################################################################## #Assign PS weights to dataset #Final weight includes the survey weight times the PS weight data1$psw <-get.weights(ps.1, stop.method="ks.max") data0$psw <-get.weights(ps.0, stop.method="ks.max") outcome.model.psw.logit <-svyglm(outcome~treatment+moderator+treatment*moderator+ categorical.confounder1+categorical.confounder2+categorical.confounder3+continuous.confounder1+ continuous.confounder2+continuous.confounder3, design=design.ps,family = quasibinomial(link = "logit")) summary(outcome.model.psw.logit)#This command uses margins to estimate our TEs within each level of the moderators summary(margins(outcome.model.psw.logit, at = list(moderator = c(0, 1)), variables = "treatment",design=design.ps))################################################################# #Step 3 -assess balance within levels of the moderator ################################################################## plot(ps.1) balance1 <-bal.table(ps.1) balance1 summary(ps.1) plot(ps.0) balance0 <-bal.table(ps.0) balance0 summary(ps.0) plot(ps.1,plots=1) plot(ps.0,plots=1) plot(ps.1,plots=2) plot(ps.0,plots=2) plot(ps.1,plots=3) plot(ps.0,plots=3) #Combine data data=rbind(data1,data0) library(margins) library(survey) design.ps <-svydesign(ids=~1, weights=~psw, data=data) ################################################################# #Step 5 -Assessing sensitivity to unobserved confounding ################################################################## library(OVtool) table ( (categorical.confounder1) (treatment moderator), nototals table (categorical.confounder2) (treatment moderator), nototals table (categorical.confounder3) (treatment moderator), nototals #For continuous covariate, check the minimum and maximum for each level of the moderator summary(data1$continuous.covariate1) summary(data0$continuous.covariate1) summary(data1$continuous.covariate2) summary(data0$continuous.covariate2) summary(data1$continuous.covariate3) summary(data0$continuous.covariate3) Transgender and non-binary gender options are not provided in the NSDUH. 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J Am Stat Assoc. 2018;113(521):390-400. /// ntrees(10000) stopmethod(ks.max) estimand(ATE) sampw(NA) /// rcmd(/usr/local/bin/RScript) /// objpath(/Users/folder/project_results) /// plotname(/Users/folder/project_results/plot_moderator1.pdf) i.categorical.confounder1 i.categorical.confounder2 i.categorical. ) stopmethod(ks.max) estimand(ATE) sampw(NA) /// rcmdps treatment i.categorical.confounder1 i.categorical.confounder2 i.categorical.confounder3 continuous.confounder1 continuous.confounder2 continuous.confounder3, /// ntrees(10000) stopmethod(ks.max) estimand(ATE) sampw(NA) /// rcmd(/usr/local/bin/RScript) /// objpath(/Users/folder/project_results) /// plotname(/Users/folder/project_results/plot_moderator1.pdf) i.categorical.confounder1 i.categorical.confounder2 i.categorical.confounder3 continuous.confounder1 continuous.confounder2 continuous.confounder3, /// ntrees(10000) stopmethod(ks.max) estimand(ATE) sampw(NA) /// rcmd(/usr/local/bin/RScript) /// objpath(/Users/folder/project_results) /// plotname(/Users/folder/project_results/plot_0.pdf) Combine data_0_wgts.dta and data_1_wgts.dta --will run outcome model *** on combined dataset append using "data_1_wgts.dta" save "data_combined_wgts. ***, *** Combine data_0_wgts.dta and data_1_wgts.dta --will run outcome model *** on combined dataset append using "data_1_wgts.dta" save "data_combined_wgts.dta" Outcome model: Propensity score-weighted logistic regression svyset [pweight=ksmaxate] svy: logit outcome treatment#moderator i.categorical.confounder1 i.categorical. ***, confounder2 i.categorical.confounder3 continuous.confounder1 continuous.confounder2 continuous.confounder3 margins, dydx(treatment) over(moderator*** Outcome model: Propensity score-weighted logistic regression svyset [pweight=ksmaxate] svy: logit outcome treatment#moderator i.categorical.confounder1 i.categorical.confounder2 i.categorical.confounder3 continuous.confounder1 continuous.confounder2 continuous.confounder3 margins, dydx(treatment) over(moderator)
[ "https://github.com/py-why/EconML." ]
[ "Dominant Eigenvalue-Eigenvector Pair Estimation via Graph Infection", "Dominant Eigenvalue-Eigenvector Pair Estimation via Graph Infection" ]
[ "Kaiyuan Yang \nDepartment of Quantitative Biomedicine\nUniversity of Zurich\n8057ZurichSwitzerland\n", "Li Xia \nDepartment of Statistics and Data Science\nNational University of Singapore\n117546Singapore\n", "Y C Tay \nDepartment of Computer Science\nNational University of Singapore\n117417Singapore\n" ]
[ "Department of Quantitative Biomedicine\nUniversity of Zurich\n8057ZurichSwitzerland", "Department of Statistics and Data Science\nNational University of Singapore\n117546Singapore", "Department of Computer Science\nNational University of Singapore\n117417Singapore" ]
[]
We present a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. The key idea in our technique lies in approximating the solution to the first-order matrix ordinary differential equation (ODE) with the Euler method. Graphs, which can be weighted, directed, and with loops, are first converted to its adjacency matrix A. Then by a naive infection model for graphs, we establish the corresponding first-order matrix ODE, through which A's dominant eigenvalue is revealed by the fastest growing term. When there are multiple dominant eigenvalues of the same magnitude, the classical power iteration method can fail. In contrast, our method can converge to the dominant eigenvalue even when same-magnitude counterparts exist, be it complex or opposite in sign. We conduct several experiments comparing the convergence between our method and power iteration. Our results show clear advantages over power iteration for tree graphs, bipartite graphs, directed graphs with periods, and Markov chains with spider-traps. To our knowledge, this is the first work that estimates dominant eigenvalue and eigenvector pair from the perspective of a dynamical system and matrix ODE. We believe our method can be adopted as an alternative to power iteration, especially for graphs.
10.48550/arxiv.2208.00982
[ "https://export.arxiv.org/pdf/2208.00982v3.pdf" ]
251,224,172
2208.00982
e5541acdc87c4b05f6992ff3c83401fc853fd423
Dominant Eigenvalue-Eigenvector Pair Estimation via Graph Infection Kaiyuan Yang Department of Quantitative Biomedicine University of Zurich 8057ZurichSwitzerland Li Xia Department of Statistics and Data Science National University of Singapore 117546Singapore Y C Tay Department of Computer Science National University of Singapore 117417Singapore Dominant Eigenvalue-Eigenvector Pair Estimation via Graph Infection Dominant Eigenvalue · Eigenvector Centrality · First-order Matrix ODE · Euler Method · Graph Infection We present a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. The key idea in our technique lies in approximating the solution to the first-order matrix ordinary differential equation (ODE) with the Euler method. Graphs, which can be weighted, directed, and with loops, are first converted to its adjacency matrix A. Then by a naive infection model for graphs, we establish the corresponding first-order matrix ODE, through which A's dominant eigenvalue is revealed by the fastest growing term. When there are multiple dominant eigenvalues of the same magnitude, the classical power iteration method can fail. In contrast, our method can converge to the dominant eigenvalue even when same-magnitude counterparts exist, be it complex or opposite in sign. We conduct several experiments comparing the convergence between our method and power iteration. Our results show clear advantages over power iteration for tree graphs, bipartite graphs, directed graphs with periods, and Markov chains with spider-traps. To our knowledge, this is the first work that estimates dominant eigenvalue and eigenvector pair from the perspective of a dynamical system and matrix ODE. We believe our method can be adopted as an alternative to power iteration, especially for graphs. Introduction Graph epidemic models seek to describe the dynamics of contagious disease transmission over networks [11,5]. The infectious disease transmits from a node to its neighbors via connecting edges over the network. Spread of the epidemic is affected by multiple factors such as the infection rate, the recovery duration, Equal Contribution. arXiv:2208.00982v3 [math.NA] 7 May 2023 and infection severity, and particularly to graphs, the network topology and the mobility of the network structure. Graph epidemic models can encode richer and more sophisticated architectures than traditional compartmental epidemic models [11,5]. For disease spread on networks, the principal eigenvalue of the network's adjacency matrix has long been shown to be an important factor on the dynamics of disease [14,8]. In fact, the well-known basic reproduction number 'R0' is itself the dominant eigenvalue of the next generation matrix [3,2]. Interestingly, despite the clear connection between the dominant eigenvalue and the network epidemics, few have attempted to approach the principal eigenvalue in the reverse manner. Here we seek to answer the question: Can we elucidate the principal eigenvalue from the progression of the infection spread over the associated network? Another motivation of ours comes from the extant issues on eigenvalue computation from the classical power iteration method. Power iteration method [9], or power method, has been widely used for computation of the principal eigenvalue. However, when multiple dominant eigenvlaues of equal modulus exist, power iteration method cannot converge [12]. Note that such failed convergence of power method is not uncommon, especially for graphs. One prominent failure is for any bipartite graph: all non-zero eigenvalues of a bipartite graph's adjacency matrix come in real number pairs that are of opposite sign. Thus if there exists a dominant eigenvalue λ for the adjacency matrix of the bipartite graph, then so is −λ. Power iteration method can also fail in Markov chains with periods. For example, there are as many eigenvalues equally spaced around the unit circle as the period of the periodic unichain [7]. Unless the Markov chain is an ergodic unichain which has only one dominant eigenvalue of 1, problematic convergence to the dominant eigenvector from power iteration should be expected. In this paper, we aim to develop an epidemic-based method to estimate the principal eigenvalue and eigenvector of a network, and compare its applications with the classical power iteration method. Our perspective of using the fastest growing term in general solution of graph infection ordinary differential equation (ODE) to 'reverse engineer' the principal eigenvalue is new. Our proposed alternative method can also overcome several limitations of power iteration method. In particular, our method works better when there are multiple dominant eigenvalues of opposite sign or are complex conjugates. Our Method: Inspired by Euler Method for Graph Infection ODE In this section we present our proposed method. See Table 1 for the notations and symbols. The key idea in our technique lies in solving the matrix ODE that describes the graph infection process. The fastest growing term of the general solution to the ODE will reveal the adjacency matrix's dominant eigenvalue. This idea comes from a naive infection model for a graph, but note that our estimation technique is numerical and does not simulate infection. Infection severity of node vi at time t, xi(t) ∈ R ≥0 x(t) Vector form of the infection severity of each node at time t I(t) Total infection of the graph at time t, I(t) = N i=1 xi(t) λ k k-th eigenvalue of A, λ k ∈ C. |λ1| ≥ |λ2| ≥ ... ≥ |λN | µ k k-th eigenvector of A, Aµ k = λ k µ k and µ k = 0 r k Algebraic multiplicity of the k-th eigenvalue of A p k Geometric multiplicity of the k-th eigenvalue of A d k Deficit of the k-th eigenvalue of A, d k = r k − p k C k Coefficient of µ k in the linear combination of eigenvectors for initial condition x(0) m Slope of the secant line with step-size ∆t for log-scale plot I(t) vs time Graph Infection and Adjacency Matrix For our setup, our graph is static, i.e., the network structure does not vary as the infection process unfolds. For a given graph G, the edges E connecting nodes V over the network do not rewire or change weights once the infection starts. The graph can be weighted, directed, and with loops. For such a graph network, the adjacency matrix A describes the connections between its nodes, in which each entry a ij represents the edge from node j to node i. Matrix A can be symmetric for undirected networks, and potentially asymmetric for directed networks. Elements of A can be zero and one for unweighted connections. Or in the case of weighted connections, A may be an arbitrary non-negative matrix. Thus our A ∈ R N ×N ≥0 , where N is the number of nodes. A basic example demonstrating the mechanism of graph infection for one node is shown in Figure 1. The rate of change of the infection severity x i (t) at node v i can be intuitively expressed as a difference equation shown in the figure. In order to describe the infection change for the graph, we turn to matrix ODE. General Solution to Graph Infection ODE The setup for our numerical method is based on the change of severity of infection for each node in the network over time. Conceptually, we can view this model as a description of infection severity of nodes, where nodes with severely infected neighbors would receive more severe transmission. See Figure 2 for a sketch of the infection severity ODE idea. Alternatively, one may perceive this model as a pest which continuously multiplies and invades neighboring nodes over a network. Note that we make the evolution of severity deterministic, and thus remove stochastic elements in the model. The severity of node i at time t, x i (t), is related to its neighboring nodes' severity. x i (t) changes with time according to this differential equation: dx i (t) dt = β N j=1 A ij x j (t)(1) Using x(t) to represent the column vector of each node's infection severity at time t, Equation 1 becomes: dx(t) dt = βAx(t)(2) Note the general solution to the above matrix ODE of Equation 2 can be divided into four cases based on the eigenvalues of A. Since the characteristic equation of A is an N th-order polynomial with real coefficients, A has exactly N eigenvalues (including repetitions if any) that are either a real number or a pair of complex conjugate numbers. Here we distinguish the four cases based on whether the eigenvalue is real or complex and on whether it is repeated. Case One: Real Distinct Eigenvalues The first case is when the adjacency matrix A has N real and distinct eigenvalues. Then we can decompose the initial condition into N linearly independent eigenvectors, x(0) = N k=1 C k µ k In this case, the matrix ODE Equation 2 has general solution: x(t) = N k=1 C k e βλ k t µ k(3) where λ 1 , λ 2 , ..., λ N are the eigenvalues of the adjacency matrix A, and µ 1 , µ 2 , ..., µ N are the corresponding eigenvectors. Case Two: Complex Distinct Eigenvalues In the second case, A still has N distinct eigenvalues, however, some of the eigenvalues are complex. Since A is real, if a complex number λ k = a + ib is an eigenvalue of our A, then its complex conjugate λ k = a − ib is also an eigenvalue, with a, b ∈ R. Let λ k = a + ib be a complex eigenvalue of A, and µ k = a + ib be the corresponding eigenvector where vectors a and b have only real entries. Since A still has N linearly independent eigenvectors, the general solution to the matrix ODE Equation 2 remains in the form of Equation 3. However, the general solution will contain the following two terms in place of the conjugate complex eigenvalues λ k and λ k and their eigenvectors: C p e βat (a cos(βbt) − b sin(βbt)) + C q e βat (a sin(βbt) + b cos(βbt)) where C p and C q are some scalars determined by the initial condition x(0). Case Three: Repeated Complete Eigenvalues When the characteristic equation of A has a multiple root, the associated eigenvalue is called a repeated eigenvalue. Suppose λ k is a real multiple-root of the characteristic equation of A, then the algebraic multiplicity of λ k , defined as r k , is the number of times λ k appears as a root. On the other hand, geometric multiplicity of λ k , defined as p k , is the number of linearly independent eigenvectors associated with λ k . Here we distinguish whether the algebraic multiplicity and geometric multiplicity are equal or not. In Case Three, the algebraic multiplicity and geometric multiplicity for the repeated eigenvalue are equal, or r k = p k . And we say the repeated eigenvalue is complete, as in without deficit. Let λ k be the repeated eigenvalue with geometric multiplicity p k , where r k = p k , and µ k , w 1 , w 2 , ..., w p k −1 are the p k linearly independent ordinary eigenvectors associated with λ k . Thus despite repeated eigenvalues in A, there are still N linearly independent eigenvectors. The general solution to the matrix ODE Equation 2 remains in the form of Equation 3, but contains the following term for the repeated complete eigenvalue λ k : C 1 e βλ k t µ k + C 2 e βλ k t w 1 + ... + C p k e βλ k t w p k −1(5) where C 1 to C p k are some scalars determined by the initial condition x(0). Case Four: Repeated Defective Eigenvalues In Case Four, there exists a repeated eigenvalue λ k for which the geometric multiplicity is smaller than the algebraic multiplicity, or r k > p k . Hence, there are fewer than N linearly independent eigenvectors for A. We call such a repeated eigenvalue λ k as defective, and d k = r k − p k is the deficit of λ k . In order to compensate for the defect gap in the multiplicity, we need to construct d k number of generalized eigenvectors for the ODE general solution. These generalized eigenvectors are associated with the defective repeated eigenvalue λ k and based on the p k linearly independent ordinary eigenvectors. Let λ k be the repeated eigenvalue with deficit of d k , and µ k , w 1 , w 2 , ..., w p k −1 are the p k linearly independent ordinary eigenvectors associated with λ k . Note that the eigenspace must have dimension of at least one. Therefore µ k is guaranteed to be available for us to construct the generalized eigenvectors. In the simplest two scenarios, we can have either one generalized eigenvector to find or we have only one ordinary eigenvector available. In the first scenario of only needing to find one generalized eigenvector, or when we have d k = 1 and r k = p k + 1, the general solution to the matrix ODE Equation 2 will have the following term for the defective repeated eigenvalue λ k : C 1 e βλ k t µ k + C 2 e βλ k t w 1 + ... + C p k e βλ k t w p k −1 + C p k +1 e βλ k t (tµ k + g) (6) where C 1 to C p k +1 are some scalars determined by the initial condition x(0), and vector g is a generalized eigenvector such that (A − λ k I)g = µ k . In the second scenario of having only one ordinary eigenvector µ k associated with λ k , or when p k = 1 and r k = d k + 1, the general solution to the matrix ODE Equation 2 will have the following term for the defective repeated eigenvalue λ k : C 1 e βλ k t µ k + C 2 e βλ k t (tµ k +g 1 ) + C 3 e βλ k t t 2 2! µ k + tg 1 +g 2 + ... ... + C d k +1 e βλ k t t d k d k ! µ k + t d k −1 (d k − 1)!g 1 + ... + t 2 2!g d k −2 + tg d k −1 +g d k(7) where C 1 to C d k +1 are some scalars determined by the initial condition x(0), and vectorsg 1 tog d k are generalized eigenvectors such that (A − λ k I)g 1 = µ k and (A − λ k I)g j+1 =g j for j ∈ {1, ..., d k }. In between the two extreme scenarios, when the deficit 1 < d k < r k − 1, the chains of generalized eigenvectors can be made up of some combinations of the p k ordinary eigenvectors associated with the defective repeated eigenvalue λ k . Although the structure of the chains of generalized eignvectors can be complicated, we will still end up with r k number of linearly independent solution vectors involving chains of generalized eigenvectors. Therefore the general solution to the matrix ODE Equation 2 can contain r k number of terms of the ordinary and generalized eigenvectors arranged in chains, resembling the terms of Equation 6 and Equation 7. Note that it is possible for A to have repeated complex conjugate pair of eigenvalues. The method involving generalized eigenvectors as discussed above works for defective complex eigenvalue as well. For more details on the multiple eigenvalue solutions to matrix ODE, please refer to Chapter 5 in [6]. However, the repeated complex eigenvalues are not a major concern for us because of the Perron-Frobenius theorem as we will discuss in the following section. Fastest Growing Term and Perron-Frobenius Theorem We first examine the graph infection ODE's general solution from the perspective of the long-term behavior of the network. According to the Perron-Frobenius theorem extended to non-negative real matrices, there exists a real nonnegative eigenvalue that is greater than or equal to the absolute values of all other eigenvalues of that matrix. So ∃λ 1 . λ 1 ∈ R ≥0 ∀λ k . k > 1 ⇒ λ 1 ≥ |λ k | In the Case One of real and distinct eigenvalues, the dominance of the principal real eigenvalue will mean the following: λ 1 > λ k ⇒ e λ1t > e λ k t Thus βλ 1 t becomes the dominant exponent term for Equation 3. Since it is exponential, exp (βλ 1 t) very quickly overwhelms all the other terms in the summation in the Case One, as the eigenvalues are real and distinct. The dynamics along the principal eigenvalue will dominate the long-term behavior in Equation 2. Therefore the fastest growing term in the general solution for Equation 2 corresponds to: x(t) ≈ C 1 e βλ1t µ 1(8) i.e., λ 1 and µ 1 are revealed through the change in x(t). Note that this dominant exponent also works for complex eigenvalues in the Case Two of having complex eigenvalues. Since for k > 1, λ k ∈ C, let λ k = a + ib, for some real and non-zero numbers a and b. Then λ 1 ≥ |λ k | = |a + ib| ⇒ λ 1 > a ⇒ e λ1t > e at Hence, the exponent with principal real eigenvalue will still dominate the longterm behavior of the general solution to the matrix ODE, even when there are complex eigenvalues with the same norm as the principal eigenvalue in Equation 4. Thus we can derive λ 1 and µ 1 using Equation 8 similar to Case One. When there are repeated eigenvalues as in Case Three and Case Four, unless the principal real eigenvalue has algebraic multiplicity greater than 1, the asymptotic behavior of the ODE's general solution is still Equation 8. Now suppose the repeated eigenvalue is the principal real eigenvalue λ 1 and r 1 > 1. In Case Three, since the algebraic multiplicity and geometric multiplicity are the same, r 1 = p 1 , the fastest growing term in the general solution to ODE is Equation 5, which is a scalar multiple of exp (βλ 1 t), and converges to a vector spanned by the p k linearly independent ordinary eigenvectors associated with λ 1 . Similarly for Case Four, the fastest growing term in the general solution to ODE is a scalar multiple of exp (βλ 1 t). However, the associated vector in Case Four is not convergent. Table 2 gives a summary of the fastest growing term to the ODE for Case One to Four. Suppose there are multiple eigenvalues of the same magnitude or Case Dominant Eigenvalues |λ k | = λ1 Fastest Growing Term 1: Real Distinct Eigenvalues −λ k = λ1. λ k ∈ R ≤0 C1e βλ 1 t µ1 2: Complex Distinct Eigenvalues |λ k | = λ1. λ k ∈ C C1e βλ 1 t µ1 3: Repeated Complete Eigenvalues λ k = λ1. λ k ∈ R ≥0 e βλ 1 t (C1µ1 + C k µ k ) 4: Repeated Defective Eigenvalues λ k = λ1. λ k ∈ R ≥0 e βλ 1 t (C1µ1 + C k (tµ1 + g)) * * For Case Four, here we only show the simplest scenario of d1 = 1, r1 = 2, and g is a generalized eigenvector such that Ag = λ1g + µ1. modulus, say λ k and λ 1 . As shown in Table 2, in all four Cases, the fastest growing term converges to the Perron-Frobenius real dominant eigenvalue λ 1 ∈ R ≥0 . In Case One and Two, despite the presence of a λ k of opposite sign or is a complex conjugate, the fastest growing term is convergent to both the λ 1 and µ 1 . As we shall see later, this marks the major advantage over the classical power iteration method. Euler Method to Approximate Matrix ODE Let I(t) be the total severity of infection for the graph at time t: I(t) = N i=1 x i (t) = 1 T · x(t) We turn to Euler method. From matrix differential Equation 2, we can estimate change of x(t) by the finite difference approximation: x(t + ∆t) ≈ x(t) + dx(t) dt ∆t = x(t) + βAx(t)∆t by Equation 2(9) Then we rewrite the finite difference for severity of graph as: I(t + ∆t) = 1 T · x(t + ∆t) ≈ 1 T · (x(t) + βAx(t)∆t) ≈ 1 T · C 1 e βλ1t µ 1 · (1 + βλ 1 ∆t) = 1 T · x(t) · (1 + βλ 1 ∆t) = I(t) · (1 + βλ 1 ∆t)(10) Note that here we substitute x(t) with the fastest growing term for Case One and Case Two from Table 2. But it should be obvious that when there are repeated and complete dominant eigenvalues, the fastest growing term of Case Three from Table 2 substituting x(t) will also give the same derivation here at Equation 10. Secant Line of Euler Method Next we make use of the finite difference of the graph infection severity with secant line. Figure 3 shows a graphical explanation of the role of secant line and Euler method in our approximation. We plot the log of total severity of the graph at time t, ln(I(t)), versus time in number of discrete time-steps, as shown in Figure 3. Take two points along the ln(I(t)) curve separated by ∆t, and draw a secant line, that secant line will have slope m defined by: m = ln (I(t + ∆t)) − ln (I(t)) ∆t = ln (1 + βλ 1 ∆t) ∆t by Equation 10 Note that the slope m of the secant line can be measured for a given time-step ∆t as shown in Figure 3. Therefore the slope m of the secant line and the true dominant eigenvalue λ 1 is related by: e m∆t = 1 + βλ 1 ∆t ∴ λ 1 = e m∆t − 1 β∆t(11) The associated dominant eigenvector µ 1 is revealed by Equation 9 along with the iterative process. Since our method depends on the long-term behavior of the dynamical system, we iteratively calculate the secant line slope m after each time step is taken. After sufficient number of time steps, we start to zoom into large t when the matrix ODE solution is dominated by the fastest growing term. This secant-line-based method thus allows us to obtain a fair estimation of the dominant eigenvalue λ 1 via measuring m for a chosen β and ∆t as elucidated in Equation 11. (Note that infection rate β is introduced to help set up the context and perspective of epidemic and infection adopted by our method. However, β can be assumed to be always 1 for simplicity, i.e. 100% of a node's infection severity will be transmitted through a connecting outward edge per unit time.) Table 3 shows the comparison between our method with the classical power iteration method. In Case One where there are N real and distinct eigenvalues, Theoretical Comparison with Power Iteration 1: Real Distinct Eigenvalues −λ k = λ1. λ k ∈ R ≤0 × × 2: Complex Distinct Eigenvalues |λ k | = λ1. λ k ∈ C × × 3: Repeated Complete Eigenvalues λ k = λ1. λ k ∈ R ≥0 × × 4: Repeated Defective Eigenvalues λ k = λ1. λ k ∈ R ≥0 N.A. N.A. N.A. N.A. power iteration is not guaranteed to converge in particular when there is a λ k such that λ 1 = −λ k . Our method on the other hand is not affected by the presence of an opposite sign dominant eigenvalue, which is the case for all bipartite graphs such as tree graphs. In Case Two with complex distinct eigenvalues, our method can still converge to the dominant eigenvalue and eigenvector pair when power iteration will fail. Note that for Case Four in Table 3, we fill the row with N.A. or not applicable. For power iteration, the method requires the matrix to be diagonalizable or not defective. Interestingly, the same constraint applies to our method as well when the dominant eigenvalue is defective. This can be observed from the derivation in Equation 10 where we factor I(t + ∆t) with I(t). When we try to plug in the fastest growing term from Table 2, the above derivation only works for Case One to Three, i.e. when the matrix is diagonalizable. Experimental Results In this section, we compare our method with power iteration method experimentally. We demonstrate our method's advantages for tree graphs, bipartite graphs, directed graphs with periods, and Markov chains with spider-traps. For simplicity, we use infection rate of β = 100%, time step-size of ∆t = 1, and initial graph infection condition of x(0) = 1. Thus the Equation 11 to estimate the dominant eigenvalue for each secant line segment will simply be λ 1 = e m − 1. Same configurations are used for both our method and power iteration. The number of iterations for our method is defined as the number of time steps used. We measure the convergence of eigenvectors by calculating the angle between them based on their dot product. Tree Graph For unweighted tree graphs, the eigenvalues of the adjacency matrix are all real since the matrix is symmetric. Figure 4 contains a tree graph with nine nodes. The true dominant eigenvalue and eigenvector pair for this tree graph can be fairly estimated by our method after ten iteration steps. However, since there always exists another eigenvalue of the same norm as λ 1 but of opposite sign, power iteration does not converge to the true dominant eigenpair, as shown in the figure's green plots. Bipartite Graph In fact, not just for tree graphs, our method works better than power iteration for the superset of tree graphs: bipartite graphs. Figure 5 gives an example of a bipartite graph. The bipartite graph with six nodes have a true dominant eigenvalue of around 2.83, which can be estimated by our method within ten iterations. The associated eigenvector is also well estimated to within 0 • angle by dot product. Compare that to the failed convergence pattern from power iteration method plotted in green. Directed Graph with Period of Two Bipartite graphs can be abstracted as directed graphs with period of two when we convert all the edges into bi-directional edges. Therefore we further experiment with the superset of bipartite graphs: directed graphs with period of two. Figure 6 shows one such graph. This directed graph has a period of two, and the true dominant eigenvalue λ 1 = √ 2. However, like bipartite graphs, there exists another eigenvalue of the same magnitude but of opposite sign λ 2 = − √ 2. Because of this, the power iteration cannot converge to the dominant eigenvalue nor the eigenvector. Our method on the other hand can give a good estimate of the dominant eigenvalue and eigenvector pair within ten iterations. Markov Chain with Spider-trap Another prominent class of examples where our method outperforms the power iteration method is for transition probability graphs. In particular, we look into Markov chains with 'spider-traps', a name coined to describe parts of the network from which a crawler cannot escape [13]. When a Markov chain contains a spidertrap, the equilibrium distribution will be determined by the spider-trap in the long term. Since there are as many eigenvalues equally spaced around the unit circle as the period of the Markov unichain [7], the period of the spider-trap will affect whether the power iteration can converge. If the period of the spider-trap is more than one, then there are multiple eigenvalues of the same magnitude as the dominant eigenvalue λ 1 = 1. Thus the power iteration method will not converge in Markov chains with periodic spider-traps. Note that the dominant eigenvector µ 1 associated with the dominant eigenvalue λ 1 = 1 is an important attribute of the network that describes the equilibrium distribution of the Markov chain. Markov Chain with Period of Three For a Markov chain with period of three, see Figure 7. There are five nodes in the Markov chain example. Nodes 1-3 constitute a spider-trap with period of three. This Markov chain has three eigenvalues of the same magnitude of 1, two of which are complex conjugates. The remaining is the true dominant eigenvalue λ 1 = 1, and the associated dominant eigenvector µ 1 is the steady-state probability vector of the Markov chain. Our previous discussion on Case Two of complex and distinct eigenvalues comes in handy here. Because of the presence of complex eigenvalues of the same norm as the dominant real eigenvalue of λ 1 = 1, power iteration method will not converge, as shown in the magenta plots in Figure 7. Our method does not have such issues and can estimate the dominant eigenvector µ 1 associated with λ 1 accurately as shown in yellow plots. Markov Chain with Period of Four There are seven nodes in the Markov chain depicted in Figure 8, out of which Nodes 1-4 constitute a spider-trap with period of four. For this Markov chain with a spider-trap of period four, there are four eigenvalues of the same magnitude. One of them is the λ 1 = 1 with the associated µ 1 steady-state vector. The other three eigenvalues are spaced equally around the unit circle with values of −1, i, −i respectively. As discussed in the Case Two scenario, the power iteration will not converge to the dominant eigenvector µ 1 . However, our method can estimate the eigenvector µ 1 as shown in the yellow vs magenta plots in Figure 8. Discussion There are a few key ideas that play important roles in our method. One being the perspective of 'reverse-engineering' the principal eigenvalue by solving a dynamical system based on graph infection epidemic model (i.e. using infection dynamics to estimate a graph's eigenvalue) . Euler method is used as a viable way to iteratively approximate the solution to the matrix ODE that describes the dynamical system. In particular, we apply the Perron-Frobenius theorem extended to non-negative matrices to derive the relationship between the nonnegative real Perron-Frobenius eigenvalue and the fastest growing term of the ODE general solution. Finally, it is interesting to see that when we plot the trends on total infection severity, the slopes of secant line segments that arise from the Euler method help us reveal the matrix's dominant eigenvalue and eigenvector pair. Relationship with NetworkX Eigenvector-centrality Implementation (Workaround on Power Iteration) NetworkX [10] is a hugely influential Python package for network analysis. In NetworkX, there is a built-in eigenvector centrality function that returns the dominant eigenvector of a graph. This eigenvector-centrality feature has been available in NetworkX from as early as 2010 based on the NetworkX official GitHub repository git log. However, until the release of NetworkX 2.0 beta 1 in August 2017, the implementation of NetworkX's eigenvector-centrality algorithm has been the classical power iteration method which cannot converge when Case One or Case Two occurs, as we have discussed. Interestingly, based on the GitHub discussions dating back to August 2015 1 from their maintenance team of the NetworkX, it was observed that addition of A by a positive multiple of the adjacency matrix, such as (A + I), could alleviate the convergence problems for power iteration for graphs. Since the NetworkX 2.0 beta 1 release in August 2017, the eigenvector-centrality has thus been modified to be using power iteration on (A + I). Here we want to highlight that NetworkX's workaround power iteration implementation can be thought of as a special case in our method. Recall our Euler method and the finite difference approximation, where we estimate the change of x(t) by x(t + ∆t) ≈ x(t) + dx(t) dt ∆t = x(t) + βAx(t)∆t by Equation 2 When we take time step size of 1 unit, ∆t = 1, infection rate to be β = 100%, the above finite difference equation becomes x(t + 1) ≈ x(t) + dx(t) dt = x(t) + Ax(t) = (A + I)x(t) In other words, NetworkX's workaround power iteration on (A + I) is in fact operating similar to a special context of our graph infection method. Note that the consideration in NetworkX's shifted power iteration implementation is to shift the graph spectrum along the positive real axis direction. Whereas our motivation stems from trying to solve an epidemic dynamical system iteratively. Therefore, it is interesting that two conceptually different starting points eventually reached solutions that take a similar mathematical form! Matrix-Free Implementation We remark on the matrix-free nature of our method. Implementation-wise, our method just sums all the node's infection severity after each time-step, I(t) = N i=1 x i (t), and computes the slope of the secant line m by dividing the difference in log of total infection severity, ln (I(t)), over ∆t. Therefore our method is matrix-free, and does not require the whole network matrix to be stored inside the memory, which is important for large graphs and can be parallelized. Please refer to section Source Code 6 for our example implementation, only requiring a few lines of code, in Python or R language. Extension to Dynamic Graphs It is possible to expand our scope to non-static graphs. For example, there is a theoretical bound on the changes of the dominant eigenvector for strongly connected graphs under perturbation [4]. Furthermore, Chen and Tong [1] proposed an algorithm that can effectively monitor how the dominant eigenvalue (computed by our method, say) can be incrementally updated if the graph's A is perturbed, which can be applicable for fast-changing large graphs. More generally on the stability of eigenvalues for graphs, Zhu and Wilson [16,15] provide some experiments on the stability of the spectrum to perturbations in the graph, which can be useful heuristics in dynamic graphs. Limitations As can be seen from the row on Case Three in Table 3, when the dominant eigenvalue λ 1 is repeated and complete, our method cannot guarantee the convergence to the associated dominant eigenvector µ 1 . But this is the same limitation faced by power iteration method for repeated complete dominant eigenvalue, as the converged vector is spanned by the p k number of eigenvectors corresponding to the geometric multiplicity. When the dominant eigenvalue is not only repeated but also defective, as shown from the row on Case Four in Table 3, due to the complicated arrangements with generalized eigenvectors, our derivation involving the fastest growing term and slope of secant line no longer holds when the dominant eigenvalue is defective. Two things to note on the Case Four though. First is that the same limitation imposed by defective dominant eigenvalue also applies to power iteration. The Jordan form of the defective dominant eigenvalue also renders power iteration ineffective. Secondly, note that our method does not require the underlying matrix to be diagonalizable. As long as the defective eigenvalue does not happen to be the dominant eigenvalue λ 1 , our method can converge to the dominant eigenvalue (Case Three), and to its dominant eigenvector (Case One and Case Two). Conclusion We have proposed a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. To our knowledge, this is the first work that estimates dominant eigen-pair from the perspective of a dynamical system and matrix ODE. Our method overcomes several limitations of the classical power iteration when the matrix has multiple dominant eigenvalues that are complex conjugates or are of opposite sign. We believe our method can be adopted as an alternative to power iteration, especially for graphs. It is our hope that this fresh perspective of 'reverse-engineering' the dominant eigenvalue and eigenvector from matrix ODE of epidemic dynamical system can not only be of some practical use, but also to leave some food for thought in the eigenvalue algorithm literature. Source Code We provide example implementations of our method in Python or R language. Source code is available at GitHub: https://github.com/FeynmanDNA/Dominant_ EigenPair_Est_Graph_Infection. Fig. 1 . 1Graph infection with infection rate of β and the difference equation of the infection severity xi(t) at node vi. Infection spreads from infected neighbouring nodes N (i) via in-degree edges. Infection severity accumulates, think of virus count in a host. Fig. 2 . 2An example network and its evolution of infection severity. Fig. 3 . 3Secant line of graph infection plot using the adjacency matrix A from the network in Figure 2, with x(0) = 1, β = 1, ∆t = 1. (Left) Plot of graph infection I(t) vs time in number of discrete time-steps #∆t. Blue dotted lines connecting the data points are the secant lines. (Middle) Plot the log of graph infection ln(I(t)) vs discrete time-steps. Yellow segment is a highlighted secant line connecting two graph infection measurements. Red line is the slope m of the yellow secant line. (Right) Zoomed-in view of the secant line. The slope m is used to estimate the dominant eigenvalue and eigenvector. Fig. 4 . 4The dominant eigenvalue for this tree graph λ1 = 4 + √ 2 ≈ 2.327. The estimated dominant eigenvalue by our method after ten iteration steps converges to 2.321 approximately. Our estimated eigenvector also converges to ground truth eigenvector to near 0 • angle. Power iteration does not converge for either eigenvalue or eigenvector because of the presence of another eigenvalue of opposite sign λ2 = −λ1. Fig. 5 . 5The true dominant eigenvalue for this bipartite graph λ1 = 2 √ 2 ≈ 2.828. The estimated dominant eigenvalue by our method after ten iteration steps converges to approximately 2.829. Our estimated eigenvector also converges to ground truth within near 0 • angle. Power iteration does not converge for either eigenvalue or eigenvector because there is an eigenvalue of the same magnitude but of opposite sign λ2 = −λ1. Fig. 6 . 6The true dominant eigenvalue for this directed graph with period of two λ1 = √ 2 ≈ 1.414. The estimated dominant eigenvalue by our method after ten iteration steps converges to around 1.414. Our estimated eigenvector also converges to the true dominant eigenvector to within near 0 • angle. Power iteration does not converge for either eigenvalue or eigenvector because there are two eigenvalues of the same magnitude λ2 = −λ1. Fig. 7 . 7Performance on Markov chain with period of three. With our method, the dot product angle θ between the estimated dominant eigenvector and the true dominant eigenvector µ1 converges to almost θ ≈ 0 • . Power iteration estimated dominant eigenvector fails to converge to true dominant eigenvector, with difference of θ ≈ 2 • , because there are two other complex conjugate eigenvalues of the same magnitude as λ1. Fig. 8 . 8Performance on Markov chain with period of four. The dot product angle θ between the estimated dominant eigenvector and the true dominant eigenvector µ1 converges to almost θ ≈ 0 • with our method. Power iteration estimated dominant eigenvector fails to converge to true dominant eigenvector, with difference of θ ≈ 7.1 • , because there are four eigenvalues of the same magnitude of 1, λ1 = 1, λ2 = −1, λ3 = i, λ4 = −i. Table 1 . 1Table of NotationsNotation Description G Graph, G = {V, E}. Graphs can be weighted, directed, and with loops V Set of vertices of the graph, V = {v1, ..., vN }, |V| = N E Set of edges of the graph N (i) Set of in-degree neighbour nodes of node vi A Adjacency matrix of the graph, A ∈ R N ×N ≥0 β Ratio of infection severity transmitted per unit time, β ∈ R>0 xi(t) Table 2 . 2Fastest Growing Term in the general solution to ODE Equation 2 Table 3 . 3Convergence comparison: Our method with the power iteration method Converge to λ1? Converge to µ1? Case Dominant Eigenvalues Ours Power Iter. Ours Power Iter. NetworkX GitHub issue #1704: https://github.com/networkx/networkx/issues/1704 AcknowledgementWe thank Olafs Vandans, Chee Wei Tan, Johannes C. Paetzold, Houjing Huang, and Bjoern Menze for helpful comments and discussions. On the eigen-functions of dynamic graphs: Fast tracking and attribution algorithms. C Chen, H Tong, Science Journal. 102Statistical Analysis and Data Mining: The ASA DataChen, C., Tong, H.: On the eigen-functions of dynamic graphs: Fast tracking and attribution algorithms. Statistical Analysis and Data Mining: The ASA Data Science Journal 10(2), 121-135 (2017) The construction of next-generation matrices for compartmental epidemic models. O Diekmann, J Heesterbeek, M G Roberts, Journal of the Royal Society Interface. 747Diekmann, O., Heesterbeek, J., Roberts, M.G.: The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface 7(47), 873-885 (2010) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. O Diekmann, J A P Heesterbeek, J A Metz, Journal of Mathematical Biology. 284Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.: On the definition and the com- putation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28(4), 365-382 (1990) Perturbations of matrices: a theorem on the perron vector and its applications to input-output models. E Dietzenbacher, Journal of Economics. Dietzenbacher, E.: Perturbations of matrices: a theorem on the perron vector and its applications to input-output models. Journal of Economics pp. 389-412 (1988) Epidemic modelling -Some notes, maths, and code. S Dobson, Independent Publishing NetworkDobson, S.: Epidemic modelling -Some notes, maths, and code. Independent Publishing Network (Jul 2020) Differential Equations and Boundary Value Problems: Computing and Modeling. C Edwards, D Penney, D Calvis, Pearson Prentice HallEdwards, C., Penney, D., Calvis, D.: Differential Equations and Boundary Value Problems: Computing and Modeling. Pearson Prentice Hall (2008), https://books. google.ch/books?id=qi6ePwAACAAJ R G Gallager, Stochastic processes: theory for applications. Cambridge University PressGallager, R.G.: Stochastic processes: theory for applications. Cambridge University Press (2013) The effect of network topology on the spread of epidemics. A Ganesh, L Massoulié, D Towsley, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies. IEEE 24th Annual Joint Conference of the IEEE Computer and Communications SocietiesIEEE2Ganesh, A., Massoulié, L., Towsley, D.: The effect of network topology on the spread of epidemics. In: Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies. vol. 2, pp. 1455-1466. IEEE (2005) Eigenvalue computation in the 20th century. G H Golub, H A Van Der Vorst, Journal of Computational and Applied Mathematics. 1231-2Golub, G.H., Van der Vorst, H.A.: Eigenvalue computation in the 20th century. Journal of Computational and Applied Mathematics 123(1-2), 35-65 (2000) Exploring network structure, dynamics, and function using networkx. A A Hagberg, D A Schult, P J Swart, Proceedings of the 7th Python in Science Conference. Varoquaux, G., Vaught, T., Millman, J.the 7th Python in Science ConferencePasadena, CA USAHagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using networkx. In: Varoquaux, G., Vaught, T., Millman, J. (eds.) Proceedings of the 7th Python in Science Conference. pp. 11 -15. Pasadena, CA USA (2008) I Kiss, J Miller, P Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models. Springer International PublishingKiss, I., Miller, J., Simon, P.: Mathematics of Epidemics on Networks: From Exact to Approximate Models. Interdisciplinary Applied Mathematics, Springer International Publishing (2017) Numerical mathematics. A Quarteroni, R Sacco, F Saleri, Springer Science & Business Media37Quarteroni, A., Sacco, R., Saleri, F.: Numerical mathematics, vol. 37. Springer Science & Business Media (2010) Link Analysis. A Rajaraman, J D Ullman, 10.1017/CBO9781139058452.006Cambridge University PressRajaraman, A., Ullman, J.D.: Link Analysis, p. 139-175. Cambridge University Press (2011). https://doi.org/10.1017/CBO9781139058452.006 Epidemic spreading in real networks: An eigenvalue viewpoint. Y Wang, D Chakrabarti, C Wang, C Faloutsos, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings. IEEEWang, Y., Chakrabarti, D., Wang, C., Faloutsos, C.: Epidemic spreading in real networks: An eigenvalue viewpoint. In: 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings. pp. 25-34. IEEE (2003) A study of graph spectra for comparing graphs and trees. R C Wilson, P Zhu, Pattern Recognition. 419Wilson, R.C., Zhu, P.: A study of graph spectra for comparing graphs and trees. Pattern Recognition 41(9), 2833-2841 (2008) Stability of the eigenvalues of graphs. P Zhu, R C Wilson, Computer Analysis of Images and Patterns: 11th International Conference, CAIP 2005. Versailles, FranceSpringerProceedings 11Zhu, P., Wilson, R.C.: Stability of the eigenvalues of graphs. In: Computer Analysis of Images and Patterns: 11th International Conference, CAIP 2005, Versailles, France, September 5-8, 2005. Proceedings 11. pp. 371-378. Springer (2005)
[ "https://github.com/FeynmanDNA/Dominant_", "https://github.com/networkx/networkx/issues/1704" ]
[ "RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field", "RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field", "RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field", "RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field" ]
[ "Nico Gößling [email protected] \nDepartment of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany\n", "Simon Doclo [email protected] \nDepartment of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany\n", "Nico Gößling [email protected] \nDepartment of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany\n", "Simon Doclo [email protected] \nDepartment of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany\n" ]
[ "Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany", "Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany", "Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany", "Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All\nUniversity of Oldenburg\nOldenburgGermany" ]
[]
Besides suppressing all undesired sound sources, an important objective of a binaural noise reduction algorithm for hearing devices is the preservation of the binaural cues, aiming at preserving the spatial perception of the acoustic scene. A well-known binaural noise reduction algorithm is the binaural minimum variance distortionless response beamformer, which can be steered using the relative transfer function (RTF) vector of the desired source, relating the acoustic transfer functions between the desired source and all microphones to a reference microphone. In this paper, we propose a computationally efficient method to estimate the RTF vector in a diffuse noise field, requiring an additional microphone that is spatially separated from the head-mounted microphones. Assuming that the spatial coherence between the noise components in the head-mounted microphone signals and the additional microphone signal is zero, we show that an unbiased estimate of the RTF vector can be obtained. Based on real-world recordings, experimental results for several reverberation times show that the proposed RTF estimator outperforms the widely used RTF estimator based on covariance whitening and a simple biased RTF estimator in terms of noise reduction and binaural cue preservation performance.
null
[ "https://export.arxiv.org/pdf/1807.04096v2.pdf" ]
125,650,222
1807.04096
5f1e7ebd315da02e64ba30ce3233d07fda5306d4
RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field 12 Jul 2018 Nico Gößling [email protected] Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All University of Oldenburg OldenburgGermany Simon Doclo [email protected] Department of Medical Physics and Acoustics and Cluster of Excellence Hearing4All University of Oldenburg OldenburgGermany RTF-Based Binaural MVDR Beamformer Exploiting an External Microphone in a Diffuse Noise Field 12 Jul 2018Web: www.sigproc.uni-oldenburg.de Besides suppressing all undesired sound sources, an important objective of a binaural noise reduction algorithm for hearing devices is the preservation of the binaural cues, aiming at preserving the spatial perception of the acoustic scene. A well-known binaural noise reduction algorithm is the binaural minimum variance distortionless response beamformer, which can be steered using the relative transfer function (RTF) vector of the desired source, relating the acoustic transfer functions between the desired source and all microphones to a reference microphone. In this paper, we propose a computationally efficient method to estimate the RTF vector in a diffuse noise field, requiring an additional microphone that is spatially separated from the head-mounted microphones. Assuming that the spatial coherence between the noise components in the head-mounted microphone signals and the additional microphone signal is zero, we show that an unbiased estimate of the RTF vector can be obtained. Based on real-world recordings, experimental results for several reverberation times show that the proposed RTF estimator outperforms the widely used RTF estimator based on covariance whitening and a simple biased RTF estimator in terms of noise reduction and binaural cue preservation performance. Introduction Noise reduction algorithms for head-mounted assistive listening devices (e.g., hearing aids, cochlear implants, hearables) are crucial to improve speech intelligibility and speech quality in noisy environments. Binaural noise reduction algorithms are able to use the spatial information captured by all microphones on both sides of the head [1,2]. Besides suppressing undesired sound sources, binaural noise reduction algorithms also aim at preserving the listener's spatial perception of the acoustic scene to assure spatial awareness, to reduce confusions due to a possible mismatch between acoustical and visual information, and to enable the listener to exploit the binaural hearing advantage [3]. As shown in [1,2,4], the binaural minimum variance distortionless response beamformer (BMVDR) beamformer is able to preserve the binaural cues, i.e., the interaural level difference (ILD) and the interaural time difference (ITD), of the desired source. The BMVDR beamformer can either be implemented using the acoustic transfer functions (ATFs) between the desired source and all microphones or using the relative transfer functions (RTFs), relating the ATFs to a reference microphone [5]. Since estimating the RTFs (unlike the ATFs) is feasible in practice, RTF estimation has become an important task in the field of multichannel speech enhancement [6][7][8][9][10][11][12][13]. Aiming at improving the performance of (binaural) noise reduction algorithms, recently the use of an external microphone in combination with the head-mounted microphones has been explored [14][15][16][17][18][19][20][21]. It has, e.g., been shown that using an external microphone is able to improve performance in terms of noise reduction [14,16,[18][19][20][21], source localisation [17] and binaural cue preservation [16,18]. In this paper, we propose a computationally efficient method to estimate the RTF vector in a diffuse noise field using the exter- Y E (ω) desired source S(ω) Y L,1 (ω) Y L,2 (ω) Y R,1 (ω) Y R,2 (ω) head-mounted microphones external microphone nal microphone. This method requires the external microphone to be located far enough from the head-mounted microphones, such that the spatial coherence between the noise components in the head-mounted microphone signals and the external microphone signal is low. Assuming this spatial coherence to be zero, we show how an unbiased RTF estimator can be derived. Using real-world recordings, we compare the proposed RTF estimator to a simple biased RTF estimator and to the widely used RTF estimator based on covariance whitening (CW) [7][8][9][10][11] for several reverberation times and signal-to-noise ratios (SNRs). The results show that the proposed RTF estimator yields a larger SNR improvement and reduced binaural cue errors compared to the existing RTF estimators. When comparing the proposed RTF estimator to an oracle RTF estimator (using the clean speech signal as external microphone signal), only a small performance difference can be observed. Configuration and Notation We consider an acoustic scenario with one desired source S(ω) and diffuse background noise (e.g., cylindrically or spherically isotropic noise) in a reverberant enclosure. Moreover, we consider a binaural configuration, consisting of a left and a right device (each containing M microphones), and an external microphone that is spatially separated from the head-mounted microphones, cf. where X L,m (ω) denotes the desired speech component, N L,m (ω) denotes the noise component and ω denotes the angular frequency. For conciseness we will omit ω in the remainder of the paper, wherever possible. The m-th microphone signal of the right hearing device Y R,m and the external microphone Y E are similarly defined by substituting R and E for L, respectively. The microphone signals of the hearing devices can be stacked in a vector, i.e., y = Y L,1 , ..., Y L,M , Y R,1 , ..., Y R,M T ∈ C 2M ,(2) with (·) T denoting the transpose of a vector. Using (1), the vector y can be written as y = x+n,(3) where the speech vector x and the noise vector n are defined similarly as in (2). Without loss of generality, we choose the first microphone on each hearing device as reference microphone, i.e., Y L = e T L y, Y R = e T R y,(4) where e L and e R are selection vectors consisting of zeros and one element equal to 1, i.e., e L (1) = 1 and e R (M +1) = 1. In the case of a single desired source, the speech vector x is equal to x = aS,(5) where the vector a ∈ C 2M contains the ATFs between the desired source S and all microphones, including reverberation, microphone characteristics and head-shadowing. The RTF vectors a L and a R of the desired source are defined by relating the ATF vector a to both reference microphones, i.e., a L = a e T L a , a R = a e T R a .(6) The speech covariance matrix R x ∈ C 2M×2M and the noise covariance matrix R n ∈ C 2M×2M are defined as R x = E {xx H } = φ x,L a L a H L = φ x,R a R a H R ,(7)R n = E {nn H },(8) where E {·} denotes the expectation operator, (·) H denotes the conjugate transpose, and φ x, L = E {|X L | 2 } and φ x,R = E {|X R | 2 } denote the power spectral density (PSD) of the desired source in the reference microphones. Assuming statistical independence between the desired speech and noise components, the microphone signal covariance matrix is equal to R y = E {yy H } = R x +R n .(9) The output signals at the left and the right hearing device are obtained by filtering and summing all microphone signals using the complex-valued filter vectors w L and w R , respectively, i.e., Z L = w H L y, Z R = w H R y.(10) Binaural MVDR Beamformer In this section, we briefly review the well-known BMVDR beamformer [2,22,23]. The BMVDR beamformer minimizes the output noise PSD while preserving the desired speech component in the reference microphones, hence preserving the binaural cues of the desired source. The constrained optimization problem for the left filter vector is given by min w L E {|w H L n| 2 } subject to w H L a L = 1.(11) The constrained optimization problem for the right filter vector is defined similarly by substituting R for L. The solutions of these optimization problems are equal to [1,2,5] w L = R −1 n a L a H L R −1 n a L , w R = R −1 n a R a H R R −1 n a R .(12) Hence, to calculate the BMVDR beamformer an estimate of the noise covariance matrix R n and the RTF vectors a L and a R of the desired source is required. Usually, the noise covariance matrix R n is either estimated by recursively updating the matrix during speech pauses or approximated by using an appropriate model, e.g., assuming a spherically isotropic noise field. Similarly, the RTF vectors a L and a R are either estimated from the microphone signals or approximated by using -simulated or measuredanechoic RTFs corresponding to the assumed position of the desired source (e.g., in front of the user). In the following sections we will consider data-dependent RTF estimation approaches to steer the BMVDR beamformer in (12). RTF Estimation Approaches In this section, we discuss different approaches to estimate the RTF vectors a L and a R of the desired source. First, we consider a biased estimator, which only requires an estimate of the microphone signal covariance matrix R y . Second, we consider the CW estimator [8,10], which requires estimates of the microphone signal covariance matrix R y and the noise covariance matrix R n . Third, we present an RTF estimator that exploits the external microphone signal Y E , assuming the spatial coherence between the noise components in the head-mounted microphone signals and the external microphone signal is zero. Biased Estimator (B) Using (6) and (7), it can be easily shown that the RTF vectors are equal to a L = R x e L e T L R x e L , a R = R x e R e T R R x e R ,(13) i.e., a column of the speech covariance matrix R x normalized with the element corresponding to the respective reference microphone. When no reliable estimate of the speech covariance matrix R x is available, a simple but biased RTF estimate can be obtained by using the (noisy) microphone signal covariance matrix R y [24] a B L = R y e L e T L R y e L , a B R = R y e R e T R R y e R .(14) The biased estimator in (14) obviously does not lead to the same solution as (13), especially for low input SNRs. Covariance Whitening (CW) A frequently used (unbiased) RTF estimator is based on covariance whitening [7][8][9][10][11]. Using a square-root decomposition (e.g., Cholesky decomposition), the noise covariance matrix R n can be written as R n = R H/2 n R 1/2 n .(15) The pre-whitened microphone signal covariance matrix is then equal to R w y = R −H/2 n R y R −1/2 n ,(16) which can be decomposed using the eigenvalue decomposition (EVD) as R w y = VΛV H ,(17) where the matrix V ∈ C 2M×2M contains the eigenvectors and the diagonal matrix Λ ∈ R 2M×2M contains the corresponding eigenvalues. Using the principal eigenvector v max , i.e., the eigenvector corresponding to the largest eigenvalue, the RTF vectors can be estimated as [11] a CW L = R 1/2 n v max e T L R 1/2 n v max , a CW R = R 1/2 n v max e T R R 1/2 n v max .(18) Due to the EVD, this estimator has a larger computational complexity than the biased estimator. Additionally, an estimate of both the microphone signal covariance matrix R y and the noise covariance matrix R n is required, although this estimate is required anyway for the BMVDR beamformer, cf. (12). Spatial Coherence (SC) Considering a spherically isotropic noise field as an example for a diffuse noise field, the magnitude-squared coherence (MSC) between the noise components in two different microphones (neglecting head-shadowing) is equal to [25] where d denotes the distance between the two microphones and c denotes the speed of sound. Figure 2 depicts the MSC for d ∈ {0.01, 0.1, 1}m and c = 343ms −1 . It can be seen that for large distances between the microphones the MSC tends to be very small, especially for high frequencies. MSC = sinc ωd c 2 ,(19) For now, let us assume that the external microphone is sufficiently far away from the head-mounted microphones, such that E {nN * E } = 0,(20) i.e., the noise components in the head-mounted microphone signals are spatially uncorrelated with the noise component in the external microphone signal. Using (20) yields E {yY * E } = E {xX * E }+E {nN * E } = E {xX * E }.(21) Using (21) and x = X L a L = X R a R , the spatial-coherence-based RTF estimator (SC) is equal to a SC L = E {yY * E } E {Y L Y * E } , a SC R = E {yY * E } E {Y R Y * E }(22) Of course, in practice the assumption made in (20) does not perfectly hold. Hence, in the experimental evaluation in Section 5 we also consider an oracle version of the estimator in (22), which uses the clean speech signal S as the external microphone signal, such that (20) perfectly holds, i.e., a SC opt L = E {yS * } E {Y L S * } , a SC opt R = E {yS * } E {Y R S * } .(23) Compared to the CW estimator, the SC estimator does not need an estimate of the noise covariance matrix R n and has a lower computational complexity, but obviously requires an external microphone to be available. Experimental Results In this section, an experimental evaluation is presented of the BMVDR beamformer in (12) using the RTF estimators discussed in Section 4. In Section 5.1 the recording setup is described, while detailed information about the implementation is provided in Section 5.2 and the results are presented in Section 5.3. Recording setup All signals were recorded in a laboratory located at the University of Oldenburg where the reverberation time can be easily changed by closing and opening absorber panels mounted to the walls and the ceiling. The room dimensions are about (7 × 6 × 2.7) m, where the reverberation time was set approximately to the three different values T 60 ∈ {250, 500, 750}ms. The reverberation times were measured using the broad band energy decay curve of measured impulse responses. At the center of the room a KEMAR head-and-torso simulator (HATS) was placed. Two behind-the-ear hearing aid dummies with two microphones each, i.e., M = 2, were placed on the ears of the HATS. The desired source was a male English speaker played back by a loudspeaker placed at about 2 m from the center of the head at the same height and at an angle of 35 • , i.e., on to the right side of the HATS (cf. Figure 1). The external microphone was placed at about 0.5 m from the desired source, leading to a distance of about 1.5 m to the HATS, which refers to, e.g., a table microphone or a smartphone that is connected to the binaural hearing device. To generate the background noise, we used four loudspeakers facing the corners of the laboratory, playing back different multi-talker recordings. Figure 3 shows the long-term magnitude-squared coherence between the recorded noise in the reference microphone of the left hearing aid and the external microphone. It can be observed that the assumption in (20) obviously does not perfectly hold, but the coherence is fairly small. The desired source and the background noise were recorded separately in order to be able to mix them together at different input SNRs ∈ {−5, 0, 5} dB. The SNR in the external microphone signal was about 9.6 dB higher than in the head-mounted microphone signals. Please note, that streaming and directly using the external microphone signal would not include any binaural cues. The complete signal had a length of 20 s with 0.5 s of noise-only at the beginning. Implementation and Performance Measures All signals were processed at a sampling rate of 16 kHz. We used the short-time Fourier transform (STFT) with frame length T = 256, corresponding to 16 ms, overlapping by R = 128 samples, e.g., for the left reference microphone signal Y L (k,l) = T −1 ∑ t=0 y L (l·R+t)w(t)e −j2πkt/T ,(24) = X L (k,l)+N L (k,l), with k the frequency bin index, l the time frame index, y L (t) the left reference microphone signal in the time-domain, w(t) a square-root Hann window of length T and j = √ −1. To distinguish between speech-plus-noise and noise-only frames we used an oracle broad band voice activity detection (VAD), based on the energy of the speech component in the right reference microphone signal. Using this VAD, the microphone signal covariance matrixR y (k, l) and the noise covariance matrix R n (k,l) were recursively estimated aŝ R y (k,l) = α yRy (k,l−1)+(1−α y )y(k,l)y H (k,l), R n (k,l) = α nRn (k,l−1)+(1−α n )y(k,l)y H (k,l), during detected speech-plus-noise frames and noise-only frames, respectively. The forgetting factors were chosen as α y = 0.8521 and α n = 0.9841, corresponding to time constants of 50 ms and 500 ms, respectively. As initialization the corresponding long-term estimates of the covariance matrices were used. The (time-varying) estimates of the covariance matrices were then used in the biased RTF estimator (B) in (14), the covariancewhitening-based RTF estimator (CW) in (18), the oracle spatial-coherence-based RTF estimator (SC opt ) in (23) and the spatial-coherence-based (SC) RTF estimator in (22). We then The performance was evaluated in terms of noise reduction and binaural cue preservation. As a measure for noise reduction performance we used the intelligibility-weighted SNR improvement (∆iSNR) [26] between the right reference microphone signal and the output of the right hearing aid. As a measure for binaural cue preservation performance we used the reliable binaural cue errors of the direct sound of the desired speech component, i.e., ∆ILD and ∆ITD, based on an auditory model [27] and averaged over frequency. Figure 4 depicts the results for all four considered RTF estimators for different reverberation times and input SNRs. As expected, B generally shows worst performance in terms of binaural cue preservation and noise reduction performance. Considering the ILD error, it can be observed for all estimators the ILD errors generally increase for increasing T 60 and decreasing input SNR. In addition it can be observed that the SC estimator consistently outperforms the CW estimator, especially for large T 60 . Moreover, almost no difference can be observed between the SC estimator and the oracle SC opt estimator, for all T 60 and input SNRs. Results Considering the ITD errors, it can be observed that for all estimators the ITD errors generally increase for increasing T 60 and decreasing input SNRs. Contrary to the ILD error, the SC estimator typically leads to larger ITD errors than the oracle SC opt estimator, especially for T 60 = 250 ms and 500 ms. Informal listening tests showed that when using SC (and SC opt ) the desired source is perceived as a point source and sounded slightly less reverberated than the input of the reference microphones. For B and CW the binaural cue error sometimes showed large variations over frequency, which may lead to strange sounding artefacts, such that some frequencies are perceived as coming from another direction and the desired source sounds slightly diffuse. Considering the iSNR improvement, it can be observed that for all estimators the SNR improvement generally decreases for increasing T 60 and decreasing input SNR. In addition, it can be observed that the SC estimator consistently outperforms the CW estimator for all T 60 and input SNRs. Moreover, almost no difference can be observed between the SC estimator and the oracle SC opt estimator. From these results it can be concluded that the SC estimator outperforms the CW estimator. Moreover, for the considered scenario, i.e., the external microphone about 0.5 m from the desired source and about 1.5 m from the head-mounted microphones, the overall performance of the (practically implementable) SC estimator is very similar to the oracle SC opt estimator, showing that the spatial coherence assumption in (20) is valid for the considered scenario. It can be expected that placing the external microphone closer to the desired source would slightly improve the performance of the SC estimator, especially in terms of binaural cue preservation. Conclusions In this paper we have shown how an external microphone signal can be exploited to estimate the RTF vectors of a desired source in a diffuse noise field. We assumed the spatial coherence between the noise components in the head-mounted microphone signals and the noise component in the external microphone signal to be zero to derive an unbiased RTF estimator. An experimental evaluation using real-world signals for several reverberation times and input SNRs showed that a better noise reduction performance and binaural cue preservation can be obtained when using the proposed RTF estimator compared to an RTF estimator based on covariance whitening and a simple biased RTF estimator. Figure 1 : 1Top-view of the considered acoustic scenario and microphone configuration (M = 2). Figure 1 . 1The m-th microphone signal of the left hearing device Y L,m (ω) can be written in the frequency-domain as Y L,m (ω) = X L,m (ω)+N L,m (ω), m ∈ {1,...,M }, (1) Figure 2 : 2Analytical inter-microphone magnitude-squared coherence in a spherically isotropic noise field. Figure 3 : 3Measured long-term magnitude-squared coherence between the recorded noise in the left reference microphone and the external microphone. Figure 4 : 4Binaural cue errors and intelligibility-weighted SNR improvement for the RTF estimators for different reverberation times (250 ms, 500 ms, 750 ms) and different input SNRs (-5 dB, 0 dB, 5 dB). computed the (time-varying) BMVDR beamformer in (12) using the estimated RTF vectors and the estimated noise covariance matrixR n (k, l). The resulting BMVDR beamformer was then applied to the head-mounted microphone signals, i.e., Z L (k,l) = w H L (k,l)y(k,l), Z R (k,l) = w H R (k,l)y(k,l). 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[]
[ "The heavy-quark potential in an anisotropic plasma", "The heavy-quark potential in an anisotropic plasma" ]
[ "Adrian Dumitru \nInstitut für Theoretische Physik\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany\n", "Yun Guo \nHelmholtz Research School\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany\n\nInstitute of Particle Physics\nHuazhong Normal University\n430079WuhanChina\n", "Michael Strickland \nInstitut für Theoretische Physik\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany\n" ]
[ "Institut für Theoretische Physik\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany", "Helmholtz Research School\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany", "Institute of Particle Physics\nHuazhong Normal University\n430079WuhanChina", "Institut für Theoretische Physik\nJohann Wolfgang Goethe Universität\nMax-von-Laue-Str. 1D-60438Frankfurt am MainGermany" ]
[]
We determine the hard-loop resummed propagator in an anisotropic QCD plasma in general covariant gauges and define a potential between heavy quarks from the Fourier transform of its static limit. We find that there is stronger attraction on distance scales on the order of the inverse Debye mass for quark pairs aligned along the direction of anisotropy than for transverse alignment.
10.1016/j.physletb.2008.02.048
[ "https://arxiv.org/pdf/0711.4722v2.pdf" ]
118,620,045
0711.4722
d1daf06fb9ee3bd72297d0a62dde341df00f25dc
The heavy-quark potential in an anisotropic plasma 6 Mar 2008 Adrian Dumitru Institut für Theoretische Physik Johann Wolfgang Goethe Universität Max-von-Laue-Str. 1D-60438Frankfurt am MainGermany Yun Guo Helmholtz Research School Johann Wolfgang Goethe Universität Max-von-Laue-Str. 1D-60438Frankfurt am MainGermany Institute of Particle Physics Huazhong Normal University 430079WuhanChina Michael Strickland Institut für Theoretische Physik Johann Wolfgang Goethe Universität Max-von-Laue-Str. 1D-60438Frankfurt am MainGermany The heavy-quark potential in an anisotropic plasma 6 Mar 2008 We determine the hard-loop resummed propagator in an anisotropic QCD plasma in general covariant gauges and define a potential between heavy quarks from the Fourier transform of its static limit. We find that there is stronger attraction on distance scales on the order of the inverse Debye mass for quark pairs aligned along the direction of anisotropy than for transverse alignment. I. INTRODUCTION Information on quarkonium spectral functions at high temperature has started to emerge from lattice-QCD simulations; we refer to ref. [1] for recent work and for links to earlier studies. This has motivated a number of attempts to understand the lattice measurements within non-relativistic potential models including finite temperature effects such as screening [2]. A detailed discussion of the properties of the heavy-quark potential in the deconfined phase of QCD is given in ref. [3], which also provides a comprehensive list of earlier work. Also, Laine et al. have recently derived a Schrödinger equation for the finite-temperature Wilson loop to leading order within "hard-thermal loop" (HTL) resummed perturbation theory by analytic continuation to real time [4]. Aside from the well-known screened Debye potential, their result includes an imaginary part due to Landau damping of low-frequency modes of the gauge field, corresponding to a finite life-time of quarkonium states. The present paper is a first attempt to consider the effects due to a local anisotropy of the plasma in momentum space on the heavy-quark potential. Such deviations from perfect isotropy are expected for a real plasma created in high-energy heavy-ion collisions, which undergoes expansion. The HTL propagator of an anisotropic plasma has been calculated in time-axial gauge in ref. [5]. We derive the result for general covariant gauges, which allows us to define a non-relativistic potential via the Fourier transform of the propagator in the static limit. II. HARD-THERMAL-LOOP SELF-ENERGY IN AN ANISOTROPIC PLASMA The retarded gauge-field self-energy in the hard-loop approximation is given by [6] Π µν (p) = g 2 d 3 k (2π) 3 v µ ∂f (k) ∂k β g νβ − v ν p β p · v + iǫ .(1) Here, v µ ≡ (1, k/|k|) is a light-like vector describing the propagation of a plasma particle in spacetime. The self-energy is symmetric, Π µν (p) = Π νµ (p), and transverse, p µ Π µν (p) = 0. In a suitable tensor basis the components of Π µν can be determined explicitly. For anisotropic systems there are more independent projectors than for the standard equilibrium case [5]. Here, we extend the tensor basis used in [5] to a four-tensor basis appropriate for use in general covariant gauges. Specifically, A µν = −g µν + p µ p ν p 2 +m µmν m 2 (2) B µν = − p 2 (m · p) 2m µmν m 2 (3) C µν =m 2 p 2 m 2 p 2 + (n · p) 2 [ñ µñν −m ·ñ m 2 (m µñν +m νñµ ) + (m ·ñ) 2 m 4m µmν ](4)D µν = p 2 m · p 2m ·ñ m 2m µmν − (ñ µmν +m µñν ) .(5) Here, m µ is the heat-bath vector, which in the local rest frame is given by m µ = (1, 0, 0, 0), and m µ = m µ − m · p p 2 p µ(6) is the part that is orthogonal to p µ . The direction of anisotropy in momentum space is determined by the vector n µ = (0, n) ,(7) where n is a three-dimensional unit vector. As before,ñ µ is the part of n µ orthogonal to p µ . The self-energy can now be written as Π µν = αA µν + βB µν + γC µν + δD µν .(8) In order to determine the four structure functions explicitly we need to specify the phase-space distribution function. We employ the following ansatz: f (p) = f iso p 2 + ξ(p · n) 2 .(9) Thus, f (p) is obtained from an isotropic distribution f iso (|p|) by removing particles with a large momentum component along n. The function f iso (|p|) should decrease monotonically with |p|, so that the square of the Debye mass defined in eq. (13) below is guaranteed to be positive; however, in the real-time approach employed here, the distribution f iso need not necessarily be thermal. The parameter ξ determines the degree of anisotropy, ξ = (1/2) p 2 ⊥ / p 2 z − 1, where p z ≡ p · n and p ⊥ ≡ p − n(p · n) denote the particle momentum along and perpendicular to the direction n of anisotropy, respectively. If f iso is a thermal ideal-gas distribution and ξ is small then ξ is also related to the shear viscosity of the plasma; for example, for one-dimensional Bjorken expansion [7] ξ = 10 T τ η s ,(10) where T is the temperature, τ is proper time, and η/s is the ratio of shear viscosity to entropy density. In an expanding system, non-vanishing viscosity implies finite momentum relaxation rate and therefore an anisotropy of the particle momenta. Since the self-energy tensor is symmetric and transverse, not all of its components are independent. We can therefore restrict our considerations to the spatial part of Π µν , Π ij (p, ξ) = m 2 D dΩ 4π v i v l + ξ(v · n)n l (1 + ξ(v · n) 2 ) 2 δ jl + v j p l p · v + iǫ ,(11) and employ the following contractions: p i Π ij p j = p 2 β , A il n l Π ij p j = (p 2 − (n · p) 2 )δ , A il n l Π ij A jk n k = p 2 − (n · p) 2 p 2 (α + γ) ,Tr Π ij = 2α + β + γ .(12) The Debye mass m D appearing in eq. (11) is given by m 2 D = − g 2 2π 2 ∞ 0 dρ ρ 2 df iso (ρ) dρ ,(13) where ρ ≡ |p|. We do not list the rather cumbersome explicit expressions for the four structure functions α, β, γ, and δ here since they have already been determined in ref. [5]. In principle, the tensor basis (2-5) could be chosen differently, such that the individual tensors have a simpler structure. For example, one could choose C µν =ñ µñν −m ·ñ 2m 2 (m µñν +m νñµ )(14)D µν = (m ·ñ) 2 m 4m µmν −ñ µñν(15) However, in the basis (2-5) the spatial components of Π µν are identical to those from ref. [5] and so we can avoid the rather tedious re-evaluation of the four structure functions. III. PROPAGATOR IN COVARIANT GAUGE IN AN ANISOTROPIC PLASMA From the above result for the gluon self-energy one can obtain the propagator i∆ µν ab . It is diagonal in color and so color indices will be suppressed. In covariant gauge, its inverse is given by ∆ −1 µν (p, ξ) = −p 2 g µν + p µ p ν − Π µν (p, ξ) − 1 λ p µ p ν = (p 2 − α)A µν + (ω 2 − β)B µν − γC µν − δD µν − 1 λ p µ p ν(16) where ω ≡ p · m and λ is the gauge parameter. Upon inversion, the propagator is written as ∆ µν (p, ξ) = α ′ A µν + β ′ B µν + γ ′ C µν + δ ′ D µν + ηp µ p ν .(17) Using (∆ −1 ) µσ ∆ σ ν = g µν it follows that the coefficient of g µν in (∆ −1 ) µσ ∆ σ ν should equal 1 while the coefficients of the other tensor structures, for example of n µ n ν , n µ p ν and p µ p ν , should vanish. Hence, we can determine the coefficients in the propagator from the following equations α ′ = 1 p 2 − α ,(18)(p 2 − α − γ)γ ′ − δ δ ′ p 2 (p 2 − (n · p) 2 ) ω 2 = γ p 2 − α ,(19)(p 2 − α − γ)δ ′ = δ β ′ p 2 ω 2 ,(20)δ p 2 − α + δ γ ′ = (ω 2 − β)δ ′ p 2 ω 2 ,(21)1 p 2 + η λ p 2 = 0 .(22) Hence, we find that in covariant gauge the propagator in an anisotropic plasma is given by ∆ µν = 1 p 2 − α [A µν − C µν ] + ∆ G (p 2 − α − γ) ω 4 p 4 B µν + (ω 2 − β)C µν + δ ω 2 p 2 D µν − λ p 4 p µ p ν ,(23) where ∆ −1 G = (p 2 − α − γ)(ω 2 − β) − δ 2 p 2 − (n · p) 2 .(24) For ξ = 0, we recover the isotropic propagator in covariant gauge ∆ µν iso = 1 p 2 − α A µν + 1 (ω 2 − β) ω 4 p 4 B µν − λ p 4 p µ p ν .(25) IV. HEAVY QUARK POTENTIAL IN AN ANISOTROPIC PLASMA We determine the real part of the heavy-quark potential in the nonrelativistic limit, at leading order, from the Fourier transform of the static gluon propagator, V (r, ξ) = −g 2 C F d 3 p (2π) 3 e ip·r ∆ 00 (ω = 0, p, ξ)(26)= −g 2 C F d 3 p (2π) 3 e ip·r p 2 + m 2 α + m 2 γ (p 2 + m 2 α + m 2 γ )(p 2 + m 2 β ) − m 4 δ .(27)m 2 α = − m 2 D 2p 2 ⊥ √ ξ   p 2 z arctan ξ − p z p 2 p 2 + ξp 2 ⊥ arctan √ ξp z p 2 + ξp 2 ⊥   ,(28)m 2 β = m 2 D ( √ ξ + (1 + ξ)arctan √ ξ)(p 2 + ξp 2 ⊥ ) + ξp z p z √ ξ + p 2 (1+ξ) √ p 2 +ξp 2 ⊥ arctan √ ξpz √ p 2 +ξp 2 ⊥ 2 √ ξ(1 + ξ)(p 2 + ξp 2 ⊥ ) ,(29)m 2 γ = − m 2 D 2    p 2 ξp 2 ⊥ + p 2 − 1 + 2p 2 z p 2 ⊥ √ ξ arctan ξ + p z p 2 (2p 2 + 3ξp 2 ⊥ ) √ ξ(ξp 2 ⊥ + p 2 ) 3 2 p 2 ⊥ arctan √ ξp z p 2 + ξp 2 ⊥    , (30) m 2 δ = − πm 2 D ξp z p ⊥ |p| 4(ξp 2 ⊥ + p 2 ) 3 2 .(31) and p 2 = p 2 ⊥ + p 2 z .(32) The above expressions apply when n = (0, 0, 1) points along the z-axis; in the general case, p z and p ⊥ get replaced by p · n and p − n(p · n), respectively. We first check some limiting cases. When ξ = 0 then m β = m D while all other mass scales in the static propagator vanish. Hence, we recover the isotropic Debye potential V (r, ξ = 0) = V iso (r) = −g 2 C F d 3 p (2π) 3 e ip·r p 2 + m 2 D = − g 2 C F 4πr e −r ,(33) wherer ≡ rm D . Consider, on the other hand, the limit r → 0 for arbitrary ξ. The phase factor in (27) is essentially constant up to momenta of order |p| ∼ 1/r and since the masses are bounded as |p| → ∞ they can be neglected. The potential then coincides with the vacuum Coulomb potential V (r → 0, ξ) = V vac (r) = −g 2 C F d 3 p (2π) 3 e ip·r p 2 = − g 2 C F 4πr .(34) The same potential emerges for extreme anisotropy since all m i → 0 as ξ → ∞: V (r, ξ = ∞) = − g 2 C F 4πr .(35) This is due to the fact that at ξ = ∞ the phase space density f (p) from eq. (9) has support only in a two-dimensional plane orthogonal to the direction n of anisotropy. As a consequence, the density of the medium vanishes in this limit. For an anisotropic distribution, the potential depends on the angle between r and n. This can be seen analytically for small but non-zero ξ. To linear order in ξ the potential can be expressed as V (r, ξ ≪ 1) = V iso (r) − g 2 C F ξm 2 D d 3 p (2π) 3 e ip·r 2 3 − (p · n) 2 /p 2 (p 2 + m 2 D ) 2 .(36) For r parallel to the direction n of anisotropy, V (r n, ξ ≪ 1) = V iso (r) 1 + ξ 2 er − 1 r 2 − 2 r − 1 −r 6 ,(37) wherer ≡ rm D , as before. This expression does not apply forr much larger than 1, which is a shortcoming of the direct Taylor expansion of V (r, ξ) in powers of ξ. However, forr ≃ 1 the FIG. 1: Heavy-quark potential at leading order as a function of distance (r ≡ rm D ) for r parallel to the direction n of anisotropy. The anisotropy parameter of the plasma is denoted by ξ. Left: the potential divided by the Debye mass and by the coupling,V ≡ V /(g 2 C F m D ). Right: potential relative to that in vacuum. coefficient of ξ is positive, (· · ·) = 0.27 forr = 1, and thus a slightly deeper potential than in an isotropic plasma emerges at distance scales r ∼ 1/m D . When r is perpendicular to n, V (r ⊥ n, ξ ≪ 1) = V iso (r) 1 + ξ 1 − er r 2 + 1 r + 1 2 +r 3 .(38) The same limitations forr apply as in eq. (37). Here, too, the coefficient of the anisotropy parameter is positive, (· · ·) = 0.115 forr = 1, but smaller than for r n. Hence, a quark-antiquark pair aligned along the direction of momentum anisotropy and separated by a distance r ∼ 1/m D is expected to attract more strongly than a pair aligned in the transverse plane. For general ξ andr, the integral in (27) has to be performed numerically. The poles of the function are integrable 1 . In Fig. 1 we show the potential in the regionr ∼ 1 for various degrees of plasma anisotropy. One observes that in general screening is reduced, i.e. that the potential at ξ > 0 is deeper and closer to the vacuum potential than for an isotropic medium. This is partly caused by the lower density of the anisotropic plasma. However, the effect is not uniform in the polar angle, as shown in Fig. 2: the angular dependence disappears more rapidly at smallr, while at larger there is stronger binding for r parallel to the direction of anisotropy. Overall, one may therefore expect that quarkonium states whose wave-functions are sensitive to the regimer ∼ 1 are bound more strongly in an anisotropic medium, in particular if the quark-antiquark pair is aligned along n. V. DISCUSSION AND OUTLOOK We have determined the HTL gluon propagator in an anisotropic (viscous) plasma in covariant gauge. Its Fourier transform at vanishing frequency defines a non-relativistic potential for static sources. We find that, generically, screening is weaker than in isotropic media and so the potential is closer to that in vacuum, in particular if the QQ pair is aligned along the direction of anisotropy. Our results are applicable when the momentum of the exchanged gluon is on the order of the Debye mass m D or higher, i.e. for distances on the order of λ D = 1/m D or less. For realistic values of the coupling, α s ≈ 0.3, λ D is approximately equal to the scale r med (T ) ≈ 0.5 (T c /T ) fm introduced in [3,8], where medium-induced effects appear. Following the discussion in ref. [3], at short distances, r < r med (T ), the potential is given by V (r) ≃ − α r + σr ,(39) where σ ≃ 1 GeV/fm is the SU(3) string tension; color factors have been absorbed into the couplings. Since r med (T ) ∼ 1/T , it follows that at sufficiently high temperature r med (T ) is smaller than α/σ and so the perturbative Coulomb contribution dominates over the linear confining potential at the length scale λ D . Roughly, this holds for T > ∼ 2T c . In this case, our result is directly relevant for quarkonium states with wavefunctions which are sensitive to the length scale λ D ≃ r med . On the other hand, for lower T the scale r med (T ) where medium-induced effects appear may grow larger than ≃ α/σ. In this regime, quarkonium states are either unaffected by the medium; namely, if the quark mass is very large and the typical momentum component in the wave function is ≫ 1/r med (T ). Conversely, states with a root-mean square radius > ∼ r med (T ) do experience medium modifications. For such states, however, it is insufficient to consider only the (screened) Coulomb-part of the potential which arises from one-gluon exchange. Rather, one should then sum the medium-dependent contributions due to one-gluon exchange and due to the string [3]. We postpone detailed numerical solutions of the Schrödinger equation in our anisotropic potential to the future. It will also be interesting to understand how the width of quarkonium states [9] which arises in HTL resummed perturbation theory due to Landau damping of modes with low frequency [4] is affected by an anisotropy of the medium. FIG. 2 : 2Comparison ofV (r n, ξ) andV (r ⊥ n, ξ). They are simple first-order poles which can be evaluated using a principal part prescription. AcknowledgmentsWe acknowledge helpful discussions with A. Mocsy and P. Petreczky. Y.G. thanks the Helmholtz foundation and the Otto Stern School at Frankfurt university for their support. M.S. is supported . A Jakovac, P Petreczky, K Petrov, A Velytsky, arXiv:hep-lat/0611017Phys. Rev. D. 7514506A. Jakovac, P. Petreczky, K. Petrov and A. Velytsky, Phys. Rev. D 75, 014506 (2007) [arXiv:hep-lat/0611017]; . G Aarts, C Allton, M B Oktay, M Peardon, J I Skullerud, arXiv:0705.2198Phys. Rev. D. 7694513hep-latG. Aarts, C. Allton, M. B. Oktay, M. Peardon and J. I. Skullerud, Phys. Rev. D 76, 094513 (2007) [arXiv:0705.2198 [hep-lat]]. . A Mocsy, P Petreczky, arXiv:hep-ph/0512156Phys. Rev. D. 7374007A. Mocsy and P. Petreczky, Phys. Rev. D 73, 074007 (2006) [arXiv:hep-ph/0512156]. . A Mocsy, P Petreczky, arXiv:0705.2559Phys. Rev. D. 7714501hep-phA. Mocsy and P. Petreczky, Phys. Rev. D 77, 014501 (2008) [arXiv:0705.2559 [hep-ph]]. . M Laine, O Philipsen, P Romatschke, M Tassler, arXiv:hep-ph/0611300JHEP. 070354M. Laine, O. Philipsen, P. Romatschke and M. Tassler, JHEP 0703, 054 (2007) [arXiv:hep-ph/0611300]; . M Laine, O Philipsen, M Tassler, arXiv:0707.2458JHEP. 070966hep-latM. Laine, O. Philipsen and M. Tassler, JHEP 0709, 066 (2007) [arXiv:0707.2458 [hep-lat]]. . P Romatschke, M Strickland, arXiv:hep-ph/0304092Phys. Rev. D. 6836004P. Romatschke and M. Strickland, Phys. Rev. D 68, 036004 (2003) [arXiv:hep-ph/0304092]. . S Mrowczynski, M H Thoma, arXiv:hep-ph/0001164Phys. Rev. D. 6236011S. Mrowczynski and M. H. Thoma, Phys. Rev. D 62, 036011 (2000) [arXiv:hep-ph/0001164]. . arXiv:hep-ph/0608270Prog. Theor. Phys. M. Asakawa, S. A. Bass and B. Müller116eq. (6-40) in M. Asakawa, S. A. Bass and B. Müller, Prog. Theor. Phys. 116, 725 (2007) [arXiv:hep-ph/0608270]. . O Kaczmarek, F Karsch, F Zantow, P Petreczky, arXiv:hep-lat/0406036Phys. Rev. D. 7059903Erratum-ibid. DO. Kaczmarek, F. Karsch, F. Zantow and P. Petreczky, Phys. Rev. D 70, 074505 (2004) [Erratum-ibid. D 72, 059903 (2005)] [arXiv:hep-lat/0406036]. . A Mocsy, P Petreczky, arXiv:0706.2183Phys. Rev. Lett. 99211602hep-phA. Mocsy and P. Petreczky, Phys. Rev. Lett. 99, 211602 (2007) [arXiv:0706.2183 [hep-ph]].
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[ "Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems", "Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems", "Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems", "Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems" ]
[ "Aishwaryaprajna † ", "Jonathan E Rowe \nThe Alan Turing Institute\nLondonUnited Kingdom\n", "\nSchool of Computer Science\nUniversity of Birmingham\nBirminghamUnited Kingdom\n", "Aishwaryaprajna † ", "Jonathan E Rowe \nThe Alan Turing Institute\nLondonUnited Kingdom\n", "\nSchool of Computer Science\nUniversity of Birmingham\nBirminghamUnited Kingdom\n" ]
[ "The Alan Turing Institute\nLondonUnited Kingdom", "School of Computer Science\nUniversity of Birmingham\nBirminghamUnited Kingdom", "The Alan Turing Institute\nLondonUnited Kingdom", "School of Computer Science\nUniversity of Birmingham\nBirminghamUnited Kingdom" ]
[]
We present an empirical study of a range of evolutionary algorithms applied to various noisy combinatorial optimisation problems. There are three sets of experiments. The first looks at several toy problems, such as ONEMAX and other linear problems. We find that UMDA and the Paired-Crossover Evolutionary Algorithm (PCEA) are the only ones able to cope robustly with noise, within a reasonable fixed time budget. In the second stage, UMDA and PCEA are then tested on more complex noisy problems: SUBSETSUM, KNAPSACK and SETCOVER. Both perform well under increasing levels of noise, with UMDA being the better of the two. In the third stage, we consider two noisy multi-objective problems (COUNTING-ONESCOUNTINGZEROS and a multi-objective formulation of SETCOVER). We compare several adaptations of UMDA for multi-objective problems with the Simple Evolutionary Multi-objective Optimiser (SEMO) and NSGA-II. We conclude that UMDA, and its variants, can be highly effective on a variety of noisy combinatorial optimisation, outperforming many other evolutionary algorithms.
10.1162/evco_a_00320
[ "https://export.arxiv.org/pdf/2110.02288v1.pdf" ]
257,257,579
2110.02288
37ecfa7d01f4a977bb38bc368ee29fcd610a59e4
Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems 5 Oct 2021 Aishwaryaprajna † Jonathan E Rowe The Alan Turing Institute LondonUnited Kingdom School of Computer Science University of Birmingham BirminghamUnited Kingdom Evolutionary Algorithms for solving Unconstrained, Constrained and Multi-objective Noisy Combinatorial Optimisation Problems 5 Oct 2021 We present an empirical study of a range of evolutionary algorithms applied to various noisy combinatorial optimisation problems. There are three sets of experiments. The first looks at several toy problems, such as ONEMAX and other linear problems. We find that UMDA and the Paired-Crossover Evolutionary Algorithm (PCEA) are the only ones able to cope robustly with noise, within a reasonable fixed time budget. In the second stage, UMDA and PCEA are then tested on more complex noisy problems: SUBSETSUM, KNAPSACK and SETCOVER. Both perform well under increasing levels of noise, with UMDA being the better of the two. In the third stage, we consider two noisy multi-objective problems (COUNTING-ONESCOUNTINGZEROS and a multi-objective formulation of SETCOVER). We compare several adaptations of UMDA for multi-objective problems with the Simple Evolutionary Multi-objective Optimiser (SEMO) and NSGA-II. We conclude that UMDA, and its variants, can be highly effective on a variety of noisy combinatorial optimisation, outperforming many other evolutionary algorithms. Introduction Realistic optimisation problems are often affected with noisy fitness measurements. The recent theoretical analyses of evolutionary algorithms (EAs) on noisy problems defined in discrete spaces are mostly focused on simple and classical benchmark problems, such as ONEMAX. Harder and more realistic combinatorial problems such as KNAPSACK or SETCOVER in presence of noisy fitness evaluations are not easily amenable to theoretical analysis. This paper discusses a thorough empirical analysis of an array of EAs on several simple and harder noisy combinatorial problems. This study attempts to identify which EAs to choose when solving realistic noisy combinatorial problems. Noise may affect fitness evaluation in a number of ways. It may be considered as prior, in which the search point is randomly tampered with, and fitness evaluation is performed on the noisy search point. Alternatively, posterior noise (which will be the focus of our study) is where a random value gets added to the fitness of a search point. With more complex combinatorial problems (e.g. ones with constraints) the noise may enter in different ways (for example, in the evaluation of those constraints). An early theoretical result by Droste (2004) examined the performance of the hill-climber (1 + 1)-EA on ONEMAX with prior noise. This was generalised to the (µ + λ)-EA by Gießen and Kötzing (2016), showing that populations can help in both prior and posterior noise. They show the (1 + 1)-EA, however, can only tolerate posterior Gaussian noise when the variance is very small (less than 1/(4 log n)). It has been recognised for a long time that the population size can affect the ability of an EA to handle noise (Goldberg et al. (1991); Rattray and Shapiro (1998)). A more recent theoretical study by Dang and Lehre (2015) shows that a low mutation rate enables a particular mutation-population algorithm to handle arbitrary posterior noise for the ONEMAX problem in polynomial time, although the bounds given are large. Similarly, the compact genetic algorithm (cGA) is shown to handle noise with (large) polynomial runtime (Friedrich et al. (2015)). A better asymptotic runtime for ONEMAX with posterior Gaussian noise is proved for the Paired Crossover Evolutionary Algorithm (PCEA) which just uses crossover, and no mutation (Prügel-Bennett et al. (2015)). Of course, it is possible to handle noise simply by re-sampling the fitness of a potential solution many times, and taking the average as an estimate of the true fitness. Suppose the noisy problem is defined by taking a deterministic fitness function and adding Gaussian noise with mean 0 and variance σ 2 . There is a general result Akimoto et al. (2015) that states if the runtime of a black box algorithm on a problem with no noise is T , then σ 2 log T samples are required at each step leading to a runtime of σ 2 T log T . In the case of ONEMAX, the most efficient algorithm Anil and Wiegand (2009) has a runtime of Θ(n/ log n). Using this algorithm with resampling, gives a runtime for noisy ONEMAX of Θ(σ 2 n). By contrast, the PCEA algorithm Prügel-Bennett et al. (2015), when σ 2 = n, has a runtime of O(n(log n) 2 ) which is already faster than the resampling algorithm. It has been suggested by Doerr and Sutton (2019) that using the median rather than mean provides a better estimate when resampling, but this is only significant when the variance is small (less than a constant). A recent study by Rowe and Aishwaryaprajna (2019) of a new Voting algorithm on ONEMAX shows a runtime of O(n log n), when the variance of the noise distribution is σ 2 = O(n) and in O(σ 2 log n) when the noise variance is greater than this. This upper bound is the best proven runtime that we are aware of to date. Some empirical results show that the use of voting in popu-lation based algorithms (UMDA, PCEA and cGA) are effective for large population sizes. In this paper, we are interested in whether any of the algorithms with polynomial theoretical runtimes for noisy ONEMAX would be capable of solving combinatorial problems with added noise in practice, when given a reasonable but fixed time budget 1 . We proceed in three stages. First we will experimentally compare a collection of algorithms on noisy ONEMAX and noisy LINEAR problems, to see which can find solutions within a reasonable amount of time (to be defined below), bearing in mind that the asymptotic bounds for some of these algorithms, while polynomial, are actually very large. Second, we will take those algorithms which pass this first test, and see how well they handle noise in three combinatorial problems: SUBSETSUM, KNAPSACK and SETCOVER. We choose these, as they have a 'packing' structure which might make them amenable to algorithms which can solve noisy ONEMAX efficiently. We generate random problem instances within the 'easy' regime (so that the algorithms can be expected to solve them when there is no noise) and then empirically study how they degrade with added Gaussian noise. In the last stage, we look at noisy multi-objective problems. Initially, we analyse the performance of a collection of multi-objective algorithms on a toy multi-objective problem COCZ without and with high levels of noise and we attempted to identify which algorithms perform better. We study the simple hill-climber algorithm SEMO, the popular NSGA-II and some other algorithms designed on the basis of our previous experimental results. We compare our algorithms on the basis of the performance indicator, hypervolume, which provides an analysis of the spread of the non-dominated solutions found, in a reasonable time budget. We then formulate the noisy constrained SETCOVER problem as a multi-objective problem and we empirically analyse the performance of the better algorithms on this. It should be noted that in our empirical results, while error bars are not always shown, the Mann-Whitney test was used on all relevant comparisons, and results are significant at the 95% level unless explicitly indicated. Notation: We use the convention that [expr] equals 1 if expr is true, and 0 otherwise. Problem Definitions -Noisy Single Objective Problems The problems studied in this paper are defined on a Boolean search space of bit strings of length n. Let N (0, σ) denote a random number drawn from a normal distribution with mean zero, and standard deviation σ, which will be freshly generated at each fitness function evaluation. Unconstrained Single-objective noisy problems The first problem is ONEMAX, whose fitness function is defined as, ONEMAX(x) = n i=1 x i When the fitness evaluation is tampered with random noise, the fitness function becomes as follows, NOISYONEMAX(x) = n i=1 x i + N (0, σ) The WEIGHTEDLINEAR problem is defined with reference to n positive weights w 1 , . . . , w n as follows, WEIGHTEDLINEAR(x) = n i=1 x i w i with corresponding noisy variant, NOISYWEIGHTEDLINEAR(x) = n i=1 x i w i + N (0, σ) In generating random problem instances, we draw the weights uniformly at random from the range 1, . . . , 100. Thus we avoid more extreme instances such as BINVAL (in which w i = 2 i−1 for each i = 1, . . . , n). The reason for this is that when the distribution of weights is highly skewed, the addition of noise is irrelevant for those bits with very high weights, yet completely overwhelms bits with weights lower than the typical noise level. Thus most algorithms will find the more significant bits, and fail on the remainder. The SUBSETSUM problem is defined with reference to n positive weights w 1 , . . . , w n and a target θ, SUBSETSUM(x) = |θ − n i=1 x i w i | In presence of noisy fitness evaluations, the fitness function can be written as follows, NOISYSUBSETSUM(x) = |θ − n i=1 x i w i | + N (0, σ) SUBSETSUM can be seen as a generalisation of the WEIGHTEDLINEAR problem (in which the target is θ = 0). In our experiments, we generate instances by choosing weights uniformly at random from 1, . . . , 100. We take the target to be two-thirds of the sum of the weights (we have run experiments for other choices of θ and found that they do not significantly affect the empirical observations). Constrained single-objective noisy problems The KNAPSACK problem is defined with respect to a set of positive weights w 1 , . . . , w n , a capacity C and positive profits p 1 , . . . , p n as follows, KNAPSACK(x) = n i=1 x i p i if n i=1 x i w i ≤ C C − n i=1 x i w i otherwise Random instances choose weights and profits uniformly from 1, . . . , 100, and the capacity is two-thirds of the sum of the weights. We consider two noisy variants of the Knapsack problem. The first version simply considers posterior additive noise as before: NOISYKNAPSACKV1(x) = KNAPSACK(x) + N (0, σ) In the second version, the presence of noise in the judgement with respect to the weights is considered, W σ (x) = n i=1 x i w i + N (0, σ) If this (noisy) weight does not exceed the capacity, we then evaluate (noisily), the profit. Otherwise we return the excess weight: NOISYKNAPSACKV2(x) = n i=1 x i p i + N (0, σ) if W σ (x) ≤ C C − W σ (x) otherwise Note that noise is added to the weight just once, when the constraint is checked, and the same value used to report the fitness value, in the case the constraint is violated. The SETCOVER problem finds a minimal covering of m elements with a collection of sets from n pre-defined subsets. A Boolean matrix a ij with n-rows and m-columns is used to define the n subsets c 1 , . . . , c n : a i,j = [i ∈ c j ] The optimal collection of the sets would have the least number of the sets needed to cover all the m elements. The SETCOVER problem has several realworld applications such as the airline crew scheduling problem. The problem can be defined as a constrained single-objective one, as well as, a single-objective problem with a penalty term. The problem can also be defined as a multi-objective problem (discussed later). The CONSTRAINEDSETCOVER problem has a constraint that checks if the solution covers each of the m elements. The optimal solution would have the least number of sets needed to cover all the m elements. It is defined as follows, CONSTRAINEDSETCOVER(x) = n j=1 x j subject to n j=1 x j a ij ≥ 1, i ∈ 1, . . . , m For comparison-based algorithms, we always prefer feasible solutions instead of infeasible solutions. Two feasible solutions are compared by their fitness values, whereas two infeasible solutions by their constraint violations. The noisy version of the problem arises if the judgements regarding the number of elements uncovered and the number of the subsets required is noisy. NOISYCONSTRAINEDSETCOVER(x) = n j=1 x j + N (0, σ) subject to n j=1 x j a ij + N (0, σ) ≥ 1, i ∈ 1, . . . , m The fitness function of SETCOVER problem can also be defined by including a penalty term such that, if elements are under-covered by the considered collection of sets, a huge penalty µ is incurred. PENALTYSETCOVER(x) = n j=1 x j + µ i max 0, 1 − n j=1 a ij x j This gives rise to a corresponding noisy variant: NOISYPENALTYSETCOVER(x) = n j=1 x j + µ i max 0, 1 − n j=1 a ij x j + N (0, σ) Algorithms Chosen For Noisy Single-objective optimisation The (1+1)-EA The (1 + 1)-EA uses a bitwise mutation operator that produces an offspring by flipping each bit of the parent string independently with probability 1/n. This can be considered as a randomised or stochastic hill-climber which considers only one point in the search space at a time and proceeds by trying to find a point which has a superior function value. In each iteration, only one function evaluation takes place. The expected runtime of the (1+1)-EA solving the nonnoisy ONEMAX is O(n log n). The runtime remains polynomial in the posterior Gaussian noise case for σ 2 < 1/(4 log n), so we do not expect this algorithm to cope with anything but the smallest noise levels (Gießen and Kötzing (2016)). Mutation-Population Algorithm It has long been recognised that populations can help an EA handle noise. The paper by Goldberg et al. (1991) developed a population sizing equation and instigated the adoption of variance-based population sizing. Rattray and Shapiro (1998) showed that in weak selection limit, effects of Gaussian noise could be overcome by an appropriate increase of the population size. More recently, a population-based, non-elitist EA was analysed by Dang & Lehre to study how it optimises the noisy ONEMAX problem with uniform, Gaussian and exponential posterior noise distributions Lehre (2015, 2016)). They considered a recently developed fitness-level theorem for non-elitist populations to estimate the expected running time for the said problems in noisy environment. In case of additive Gaussian noise N (0, σ 2 ) with mutation rate χ n = a 3σn and population size λ = bσ 2 ln n (where a and b are constants), the considered algorithm optimizes the ONEMAX problem in expected time O(σ 7 n ln(n) ln(ln (n))). Similar results were shown for uniform and exponential noise distributions. Note that this is potentially very large, when the noise is large -in excess of n 4.5 when σ = √ n, although of course this is an upper bound, and we do not know the constants. Compact Genetic Algorithm (cGA) The compact GA (cGA) is an EDA, introduced by Harik et al. (1999). cGA is able to average out the noise and optimize the noisy ONEMAX problem in expected polynomial time, when the noise variance σ 2 is bounded by some polynomial in n, as suggested in Friedrich et al. (2015). The paper introduced the concept of graceful scaling in which the runtime of an algorithm scales polynomially with noise intensity, and suggested that cGA is capable of achieving this. It is also suggested that there is no threshold point in noise intensity at which the cGA algorithm begins to perform poorly (by which they mean having super-polynomial runtime). They proved that cGA is able to find the optimum of the noisy ONEMAX problem with Gaussian noise of variance σ 2 after O(Kσ 2 √ n log Kn) steps when K = ω(σ 2 √ n log n), with probability 1 − o(1). Note that this upper bound is in excess of n 3 when σ = √ n. Population Based Incremental Learning (PBIL) The algorithm PBIL, proposed by Baluja (1994) in 1994, combines genetic algorithms and competitive learning for optimising a function. We have included this algorithm as it is in some ways similar to the cGA, so we might expect it to have similar performance. We are not aware of any theoretical analysis of this algorithm on noisy problems. The runtime of PBIL on ONEMAX is known to be O(n 3/2 log n), for suitable choice of λ (Wu et al. (2017)). Univariate Marginal Distribution Algorithm (UMDA) The Univariate Marginal Distribution Algorithm (UMDA) proposed by Mühlenbein (1997) belongs to the EDA schema. In some ways, it is therefore similar to cGA and PBIL. However, it can also be viewed as generalising the genepool crossover scheme, in which bits are shuffled across the whole population (within their respective string positions). We have included UMDA then, to see if its behaviour is more like cGA and PBIL on the one hand (which emphasise an evolving distribution over bit values), or like PCEA on the other (which emphasises crossover). The UMDA algorithm initialises a population of λ solutions, and sorts the population according to the fitness evaluation of each candidate solution. The best µ members of the population are selected to calculate the sample distribution of bit values in each position. The next population is generated from this distribution. There are two variants of UMDA, depending on whether the probabilities are constrained to stay away from the extreme values of 0 and 1, or not. It is known that if the population size is large enough (that is, Ω( √ n log n)) then this handling of probabilities at the margins is not required (Witt (2017)). Since we will work with a large population (to match the PCEA algorithm described below), we will not employ margin handling, unless otherwise stated. In our experiments we will take µ = λ/2. We are not aware of any theoretical results concerning UMDA on problems with posterior noise, but the runtime on ONEMAX is known to be O(n log n) for µ = Θ( √ n log n) -see Witt (2017). Paired-Crossover EA (PCEA) Recently, the recombination operator has been suggested to be considerably beneficial in noisy evolutionary search. Prügel-Bennett et al. (2015) considered the problem of solving ONEMAX with noise of order σ = √ n and analysed the runtime of an evolutionary algorithm consisting only of selection and uniform crossover, the Paired-Crossover EA (PCEA). They show that if the population size is c √ n log n then the required number of generations is O ( √ n log n), giving a runtime of O(cn (log n) 2 ), with the probability of failure is O(1/n c ). The proof in that paper can be generalised to the case of σ ≥ √ n, to give a runtime of O(σ 2 log n). It is not known what happens for lower levels of noise, though it is shown that in the absence of noise, PCEA solves ONEMAX in O(n(log n) 2 ). Experiments -Simple Noisy Problems Noisy ONEMAX We investigate the performance of the algorithms described above, in solving the noisy ONEMAX problem. In the literature, some theoretical proofs exist for the expected runtime of specific algorithms on solving the noisy ONEMAX problem with additive posterior Gaussian noise (Prügel-Bennett et al. (2020)). We are interested in the algorithms' performances given a reasonable but fixed runtime budget across a wide range of noise levels, from σ = 0 up to σ = √ n. To address the question of what constitutes a reasonable budget, we compared the known theoretical results of our algorithms on noisy ONEMAX. PCEA has the lowest proven upper bound on its runtime, compared to the other algorithms for which results exist. We therefore allowed each algorithm to have twice the number of fitness evaluations that PCEA requires (on average) to find the optimum, as a reasonable budget. The function evaluation budgets calculated in this way are given in Table 4.1. The population size for the PCEA is taken to be 10 √ n log n according to the theoretical proofs and empirical study by Prügel-Bennett et al. (2015). According to the proofs by Dang and Lehre (2015), the population size λ = σ 2 log n is chosen for the Mutation-Population algorithm. According to the paper by Friedrich et al. (2015), the parameter K = 7σ 2 √ n log n is considered for cGA. In presence of additive posterir noise, PBIL and UMDA have not yet been stud- 2017)). From these, we select the best µ = λ/2 individuals. In case of UMDA, the total number of generated candidates in a particular generation is chosen as 20 √ n log n, so that the effective population size is the same as for PCEA. All these parameter settings are retained for all of our experiments in simple and constrained noisy combinatorial optimisation problems. Figure 1 illustrates a comparison of all of the considered algorithms while solving the noisy ONEMAX problem for problem size n = 100. Different levels of Gaussian additive noise with mean 0 and standard deviation σ = 1 to 10 are considered in this experiment. It can be seen that PCEA and UMDA are resistant to these noise levels as they are capable of finding the global optimum within the given budget. The runtimes for these two algorithms are shown in Figure 2. However, (1 + 1)-EA, Mutation-Population algorithm, PBIL and cGA are not able to cope with even these small levels of noise within the given fixed budget of function evaluations. For these experiments, we run the algorithms until the population converges (which they will, since we do not handle probability margins). The Mann-Whitney U-test is performed on the samples of best results achieved and the runtimes of the algorithms, with the null hypothesis that they are from distributions with equal medians. For each data point, the null hypothesis is rejected at 5% significance level. Noisy WEIGHTEDLINEAR problem Maximising the WEIGHTEDLINEAR problem as defined above in Section 2 has only one global optimum, the sum of all the weights. The ONEMAX problem is a special case of the WEIGHTEDLINEAR problem when all the weights are units. However, optimising the WEIGHTEDLINEAR problem is difficult as the bits with heavier weights get optimised with a higher preference than the bits with lower weights. The plot in Figure 3 illustrates the performance comparison of all of the considered algorithms while solving the noisy WEIGHTEDLINEAR problem for the problem size n = 100. Random problem instances were studied with 100 randomly chosen weights between 1 and 100. The results for a typical problem is shown in Figure 3 with averages over 100 runs. The standard deviation of the Gaussian noise is shown as multiples of the square root of the sum of the weights. The function evaluation budget allowed to each of the algorithms are fixed at twice the average runtime of PCEA at each noise level (see Table 2). As evident from Figure 3, the curves of PCEA and UMDA are coincident, showing that they can cope with the noise well and are resistant up to these levels of noise. The runtime of UMDA and PCEA are plotted in Figure 4. However, the performance of the (1 + 1)-EA and Mutation-Population algorithm worsen with increasing noise. Even with relatively small noise levels, It is evident from the empirical results of these simple noisy problems that uniform crossover-based PCEA and UMDA can cope with noise significantly better than the other algorithms. At this point, it is interesting to note that, UMDA employs a mechanism similar to genepool crossover, where at each bit position, the offspring bit is obtained by recombination of that bit across the whole parent population. It is hypothesised that these two algorithms are therefore highly similar in operation. Experiments -Noisy Combinatorial Problems Noisy SUBSETSUM Given the success of UMDA and PCEA on the noisy toy problems, and the failure of the others to cope with even modest levels of noise, we now move to the second stage of the study considering only UMDA and PCEA. For the noisy SUBSETSUM problem, a range of problem sizes is considered with 50, 100, 150 and 200 weights, each lying between 1 and 100, and chosen uniformly at random. Corresponding to each problem size, 10 different problems are considered. The target θ is considered to be two-third of the sum of all the weights in the set. The additive Gaussian noise considered in the SUBSETSUM problem is centered at zero and is considered to have standard deviation of integral multiples of the mean of the weights, viz., 5 × mean(W ), 10 × mean(W ), 15 × mean(W ) and 20 × mean(W ). The NOISYSUBSETSUM problem being a minimisation problem, if we obtain the (non-noisy) fitness value of zero, we obtain the global optimum. Both the algorithms are able to find the global optimum for these problems and their corresponding noise levels. We therefore plot the runtime (averaged over 100 runs) to find the optimum against the standard deviation of the noise -see Figure 5. Using the Mann-Whitney U-test it is observed that UMDA has the better runtime. Noisy KNAPSACK (Version 1) For the first version of the noisy KNAPSACK problem, instances with 50, 100, 150 and 200 weights (randomly chosen between 1 and 100) with associated profits (in the same range) are considered. The maximum capacity of the knapsack is taken to be two-thirds of the sum of all the weights considered. When noise is added, neither algorithm finds the optimal solution, so we record the best solution found (as assessed by non-noisy fitness function). PCEA is run until the population converges whereas, UMDA is run for twice that time, and we report the time taken to find the best solution encountered. For each problem instance, we plot (in Figure 6) the best solution found (averaged over 100 runs) as a fraction of the best solution ever encountered for that problem instance. This enables us to make meaningful comparisons between problem instances. The best known solution for each problem instance has a scaled fitness value of 1. For each problem size, 10 different problems are considered. Figure 7 shows the time taken (on average) to locate the best found solution in each case. We can observe in Figures 6 and 7, that both the algorithms can find good, though not optimal solutions, for NOISYKNAPSACKV1 with significant levels of noise. Observations from Mann-Whitney U-test show that UMDA is slightly better than PCEA with these parameter settings. Noisy KNAPSACK (Version 2) When the measurements of the weights is uncertain, as well as the profits, this creates a more complex noise model for the KNAPSACK problem. In the first stage, the total weight of the proposed solution is compared against the capacity, and this is done with added noise. Hence it may be thought that the proposed solution is feasible when in fact it is not. If it is considered feasible, then the benefit (total profit) is calculated, again with added noise.The parameters are considered as in the previous version of the KNAPSACK problem. 10 problems each of 50, 100, 150, and 200 weights (lying between 1 and 100) with associated profits (also lying in the same range) are considered. Figure 8 depicts how the best (non-noisy) solution varies for different problem sizes. This value is scaled with respect to the best value found when there Figure 9: Comparison of runtime of UMDA (circles) and PCEA (triangles) while solving the NOISYKNAPSACKV2 is no noise. PCEA is run until the population converges while UMDA is run for twice that time, and we report the time taken to find the best solution encountered. The Mann-Whitney U-test shows that the best solution achieved and corresponding runtime of UMDA is better than PCEA in these particular parameter settings. The runtime required to find these values is shown in Figure 9, and we see that UMDA finds its best solution considerably faster than PCEA. Noisy CONSTRAINEDSETCOVER and PENALTYSETCOVER The CONSTRAINEDSETCOVER problem is solved by initially finding the feasible solutions and then minimising the number of the selected sets. This lexicographic ordering is achieved in the selection mechanism of the considered algorithms. In PCEA, the child with least uncovered elements is selected. When both of the children have the same number of uncovered elements, the child with the minimum number of sets goes to the next population. In UMDA, the sorting of the population is based on the above mentioned lexicographic ordering. We consider margin handling in UMDA for all the following experiments in single objective-optimisation. The alternative PENALTYSETCOVER problem handles the constraint within the penalty function, hence creating a single objective. For both versions of the noisy SETCOVER problem, a range of 40 problem instances (10 for each problem size) are run with 100 elements and 50, 100, 150 and 200 subsets are available to cover those elements. The problems are created by randomly generating subsets, where the probability of including any element in any subset is p. This is set so that the probability of there being cover is large: (1 − (1 − p) n ) m = 1 − δ Therefore, we take: p = 1 − (1 − (1 − δ) 1/m ) 1/n We have chosen δ = 0.001. All the algorithms are run until 50,000 function evaluations are reached. An average of 30 runs are reported. Figure 10 reports the best feasible solution found in the fixed budget of function evaluations. As evident from the figure, neither of the algorithms can handle noise well. The noisy feasibility check significantly worsens the optimum found even for small standard deviations of noise. The parameters considered for solving the PENALTYSETCOVER are chosen same as the CONSTRAINEDSETCOVER. For each problem, we plot the best feasible solution found so far in the given function evaluation budget and the runtime in Figures 11 and 12. It is interesting that both the algorithms can solve the noisy instances in a scalable manner, with UMDA typically producing better quality solutions. Noisy Combinatorial Multi-Objective Problems In this section, we empirically examine the performances of several evolutionary algorithms on noisy combinatorial multi-objective problems. Much of the previous work on multi-objective optimisation (especially in the context of noise) has concerned continuous problems (Goh et al. (2010); Shim et al. (2013); Fieldsend and Everson (2015); Falcón-Cardona and Coello (2020)). In this paper, we focus on discrete problems, but with additive (posterior) Gaussian noise. A noisy multi-objective combinatorial problem in the search space of binary strings may be defined as follows, f (x) = (f 1 (x) + N (0, σ), f 2 (x) + N (0, σ), . . . , f k (x) + N (0, σ)) where, x ∈ {0, 1} n is a candidate solution. The objectives f 1 (x), f 2 (x), . . . , f k (x) are conflicting in nature, so there does not necessarily exist an optimal solution that will minimise all the objectives simultaneously. Instead, there exists a set of non-dominating solutions known as the Pareto optimal solution set where none of the objectives may be improved without worsening at least one of the other objectives. In the context of noisy multi-objective optimisation, the goal is to find the set of Pareto optimal solutions, as defined in the absence of noise -however, the challenge is that each time a comparison is made, noise is applied. This is particularly problematic for algorithms that make use of an archive of non-dominated solutions, as it is easy for a solution to be incorrectly placed in the archive due to the noise. In order to assess how successfully we have approximated the true Pareto optimal set, we measure the spread of a set of non-dominated solutions on the basis of the frequently used hypervolume performance indicator (Zitzler and Thiele (1998)). Where we seek to minimise each objective, this is a measure of the area (or volume) of the region bounded below by a set of candidate solutions simultaneously and bounded above by a reference point r in the objective space. The reference point r is chosen to be the maximum value each objective function can attain in each corresponding dimension of the objective space, i.e., r = (max f 1 , max f 2 , . . . , max f k ). Conversely, for maximisation problems, we take the volume between the candidate set and a lower bounding reference point (in the case of non-negative objectives, it is common to take the origin as the reference point). We use hypervolume of the population as an indicator of the spread of the non-dominated solutions in each generation of the considered algorithms. In this paper, we have studied two noisy multi-objective problems. The first is based on the toy benchmark problem Counting Ones Counting Zeroes (COCZ), in which the first objective function counts the number of ones in a string, and the second objective function counts the number of ones in the first m bits and the number of zeroes in the remainder. We seek to maximise both objectives. NOISYCOCZ (x) = n i=1 x i + N (0, σ), m i=1 x i + n i=m+1 (1 − x i ) + N (0, σ) The Pareto optimal front consists of strings of the form 1 m * (n−m) . The second problem is a multi-objective version of SETCOVER problem, with the objective function and the constraint as defined in CONSTRAINED-SETCOVER as the two objective functions. These objectives are conflicting in nature. The first objective minimizes the number of sets required to cover all the m elements of the target set, and the second objective minimizes the number of uncovered elements. The noisy version of the multi-objective SETCOVER problem is defined as follows, NOISYMULTI-OBJECTIVESETCOVER(x) = n j=1 x j + N (0, σ), i [ n j=1 a ij x j = 0] + N (0, σ) Algorithms Chosen for Noisy Multi-Objective Combinatorial Problems Simple Evolutionary Multi-objective Optimiser (SEMO) SEMO (Laumanns et al. (2004)) is one of the simplest evolutionary algorithms designed for multi-objective optimisation in discrete search space. To the best of our knowledge, it has not previously been used to solve noisy problems. SEMO is a simple population-based algorithm using one-bit mutation, and a variable population size (representing the current non-dominated solutions found). The algorithm starts with adding an initial solution x ∈ {0, 1} n chosen uniformly at random to the population P . Then a solution y is chosen randomly from P and mutated with a one-bit flip to obtain y'. If y' is dominated by anything in P it is discarded. Otherwise it is added to P and all the solutions that y' dominates in P are discarded. Then a new y is chosen from P and the process is repeated. One of the great challenges SEMO will face due to noisy dominance relations is that, often good solutions will be discarded and bad solutions will be retained in P . Non-dominated Sorting Genetic Algorithm -II (NSGA-II) NSGA-II by Deb et al. (2002) sorts the population into non-dominated fronts in each generation. Based on non-dominated sorting and using a crowding Algorithm 1: SEMO Initialise solution x and add to population P . repeat Choose y from P and mutate a random bit to get y'. If y' is not dominated by any solution in P and y' ∈ P , add y' to P and discard all solutions in P that y' dominates. heuristic to break ties, the best half of individuals become the parent population of the next generation. In case of noisy function evaluations, nondominated sorting will be affected and worse solutions will appear in better non-dominated fronts. We use the same algorithm structure as defined in Deb et al. (2002) except considering noisy function evaluations during the selection process. Variants of Multi-objective Univariate Marginal Distribution Algorithm (moUMDA) From our experiments in noisy single-objective combinatorial problems, UMDA and PCEA show significantly better performance in handling noise compared to the other algorithms we tried, with UMDA generally producing better quality solutions. From these results, we hypothesise that a multi-objective version of UMDA (denoted moUMDA) may be able to handle large levels of noise in noisy combinatorial multi-objective problems if proper diversification mechanisms are employed. In order to investigate this, we have considered several versions of moUMDA in our analysis with different diversification techniques. Pelikan et al. (2005) introduced a version of UMDA to address multi-objective problems which used non-dominated sorting in the selection mechanism. They also experimented with clustering methods, to help the algorithm generate solutions across the Pareto front. We have followed this idea, and studied several versions of UMDA adapted for multi-objective problems. Where nondominated sorting and crowding are used for selection, these are implemented identically to NSGA-II. We also consider making use of an archive, and in using hypervolume as a criterion in selection: moUMDA without duplicates Uses non-dominated sorting (with crowding to break ties) for selection. Maintains diversity by disallowing duplicates when generating the population. See Algorithm 2. moUMDA with clustering Uses non-dominated sorting (with crowding to break ties) for selection. Clusters the selected population members (using either K-means or Hierarchical Agglomeration), and produces a frequency vector for each cluster. Generates next population from these, in proportion to the number of items within each cluster. See Algorithm 3. Experiments -Noisy Multi-objective Problems Following the same strategy as for single objective problems, we initially, we choose a wide range of evolutionary multi-objective algorithms to compare their performances on a toy problem: noisy COUNTINGONESCOUNTINGZE-ROES (COCZ). The algorithms considered for solving COCZ consists of SEMO, NSGA-II and several versions of multi-objective UMDA (moUMDA) as described above. Depending on their performances on this problem, we selected Algorithm 4: moUMDA with Pareto archive Initialise frequency vector p = (0.5, . . . , 0.5) Initialise empty archive P repeat Generate population of size λ from p. Use non-dominated sorting and crowding to select the best µ individuals. Add these to archive P and remove any dominated solutions. Update frequency vector p based on archive P . Algorithm 5: moUMDA with hypervolume comparison Initialise frequency vector p = (0.5, . . . , 0.5) repeat Create empty population P repeat µ times Generate two strings, x and y from p Add string with best hypervolume to P Update frequency vector p based on population P . a smaller set of the better performing algorithms for the multi-objective noisy SETCOVER problem. Some recent studies claim that multi-objective evolutionary approaches are useful in solving single objective optimisation problems (Segura et al. (2016)). For example, the multi-objective version of SETCOVER could enable us to find good solutions to the original single-objective version (by looking at solutions generated which do not violate the constraints). Here, we consider whether this approach is also helpful in the context of noise. Noisy COUNTINGONESCOUNTINGZEROES (COCZ) In this subsection, we solve a toy multi-objective problem, the noisy COCZ with n = 30, m = 15 and with additive Gaussian noise centered at zero and having standard deviations σ = 0, 1, 3, 5, 7, 9, 11, 13 and 15. We set the parameter µ = λ/2, where λ = 20 √ n log n for all the versions of moUMDA. For NS-GAII, the parent population size is set as 10 √ n log n. All the algorithms are run for 50,000 function evaluations and the mean of 30 runs are reported. The best hypervolume of the population found so far in the fixed budget of function evaluations are reported in Figure 13. The Pareto optimal front would contain 2 15 elements and the best possible hypervolume is 780. We have used the dimension-sweep algorithm and the source code by Fonseca et al. (2006) for hypervolume calculation in the experiments. The results shown in Figure 13 show that SEMO is the worst performing algorithm, even when there is no noise, and the performance degrades slightly as noise is increased. The Pareto Archive algorithm (PAmoUMDA) is the next worst. Although it does no degrade too much with added noise, it is still clearly worse than the other algorithms. The remaining algorithms have rather similar performance, but we can still distinguish different behaviours by looking at the zoomed in section of the plot in Figure 13. The version of moUMDA that uses the hypervolume comparison operator (moUMDAHCO) performs very well when there is little or no noise. However, its performance degrades considerably as the level of noise increases. The same is true for NSGAII. When the noise reaches a standard deviation of σ = 15, these two algorithms are the worst of the remaining ones. The plain moUMDA and the version forbidding duplicates in the population both have the curious property that their performance improves with the presence of low levels of noise, and then degrade at higher levels of noise. We speculate that low levels of noise allow for much more diversity in the popula-tions. At high levels of noise (σ = 15) they are the best performing algorithms, along with the two versions of moUMDA that use clustering (moUMDA-Kmeans and moUMDA-HAC). moUMDA with no duplicates is marginally the best overall at this level of noise. Noisy multi-objective SETCOVER In this section, we compare the performance of three of our multi-objective algorithms, viz., NSGA-II, moUMDA with no duplicates allowed and moUMDA employing K-means clustering, on the noisy multi-objective SETCOVER problem. We have chosen these algorithms based on their behaviours on the COCZ. These were amongst the best algorithms we tried on that problem. There being little to distinguish the two different clustering methods, we have chosen to test just one of these (K-means clustering). We have selected the "no duplicates" version of moUMDA, as this gave a small advantage over the plain moUMDA. And we have kept NSGAII as this is a standard algorithm for any multi-objective problem. Figure 14. We observe that the clustering algorithm, moUMDA-Kmeans, handles high levels of noise significantly better than other algorithms. It is evident that, the performance of NSGA-II becomes worse as the standard deviation of noise increases and the problem size increases and indeed is the worst of the three algorithms on this problem. We also consider the multi-objective formulation of noisy SETCOVER as a means to solving the standard single objective problem. To this end, we consider the quality of the best feasible solutions found by each algorithm, averaged over the 30 runs. The results are plotted in Figure 15. Again, the two versions of moUMDA perform better than NSGAII. A comparison with Figure 11 shows that this approach can indeed produce better quality results than the single objective formulation. Conclusion We have empirically studied a range of evolutionary algorithms on a set of noisy problems. The (1 + 1)-EA, as expected, fails to cope with any degree of posterior noise. Interestingly, some algorithms (the mutation-population algorithm and cGA), where there is a theoretical polynomial runtime for noisy ONEMAX, fail to be useful in practice compared to some other algorithms. PBIL performs somewhat similar to cGA. The Paired Crossover Evolution-ary algorithm handles noise well on both the simple test problems, and on the noisy combinatorial problems we have tried. Interestingly, UMDA also handles these cases well, with even a slightly better performance than PCEA. This may be due to the fact that UMDA has a strong selection method (truncation selection) than PCEA (which uses a tournament on pairs of offspring). Of course, parameter values on each could be tweaked to produce slightly different results -our key finding is that these are the only algorithms we have tried that seem remotely practical for such problems. It seems likely that UMDA's performance is more due to its relationship with crossover algorithms (such as the genepool crossover), rather than considered as an EDA (such as PBIL). We are not aware of any previously published results on noisy combinatorial multi-objective problems. We carefully selected a set of multi-objective algorithms on the basis of the performance on noisy COCZ and tested them on the noisy multi-objective SETCOVER. We observe that multi-objective UMDA with a simple diversity mechanism that allows no duplicate solutions in the population is effective at solving the noisy SETCOVER problem in both constrained and multi-objective forms. UMDA can also benefit from using a clustering approach when dealing with noisy multi-objective problems. Figure 1 : 1Comparison of the algorithms while solving the noisy ONEMAX for different Dang and Lehre (2015); Akimoto et al. (2015); Friedrich et al. (2017); Lucas et al. (2017); Qian et al. (2018); Dang-Nhu et al. (2018); Doerr and Sutton (2019); Doerr Figure 2 : 2Runtime comparison of UMDA and PCEA for noisy ONEMAX ied much. For PBIL, the population size is taken as λ = 10n (following the theoretical requirement ofWu et al. ( Figure 3 : 3Comparison of algorithms while solving noisy WEIGHTEDLINEAR Figure 4 : 4Runtime comparison of UMDA and PCEA for noisy WEIGHTEDLIN-EAR the cGA and PBIL are not able to solve the problem within twice the runtime of PCEA. Figure 5 : 5Runtime comparison of UMDA (circles) and PCEA (triangles) while solving instances of the noisy SUBSETSUM problem. Figure 6 :Figure 7 :Figure 8 : 678Solution quality of UMDA (circles) and PCEA (triangles) while solving instances of NOISYKNAPSACKV1 Comparison of runtime of UMDA (circles) and PCEA (triangles) while solving the NOISYKNAPSACKV1 Solution quality of UMDA (circles) and PCEA (triangles) while solving the NOISYKNAPSACKV2 Figure 10 :Figure 11 :Figure 12 : 101112Solution quality of UMDA (circles) and PCEA (triangles) while solving the CONSTRAINEDSETCOVER Best solution found in stipulated budget of function evaluations by UMDA (circles) and PCEA (triangles) for NOISYPENALTYSETCOVER Runtime of UMDA (circles) and PCEA (triangles) for best solution found while solving NOISYPENALTYSETCOVER moUMDA with Pareto archive Maintains an archive of non-dominated solutions and uses this to generate the frequency vector for the next population. Uses non-dominated sorting (with crowding to break ties) for selection, and updates the archive with the selected items. See Algorithm 4. moUMDA with hypervolume comparison operator Uses binary tournament selection, comparing solutions initially by Pareto dominance. If neither dominates the other, then select the one with the better hypervolume indicator value. See Algorithm 5. Algorithm 2: moUMDA without duplicates Initialise frequency vector p = (0.5, . . . , 0.5) repeat Generate population of size λ from p, disallowing duplicates. Use non-dominated sorting and crowding to select the best µ individuals. Update frequency vector p based on selected individuals. Algorithm 3: moUMDA with clustering Set k = ⌊ √ µ⌋ as the number of clusters. Initialise frequency vectors p i = (0.5, . . . , 0.5) for each i = 1 . . . k. Set q i = µ/k for each i = 1 . . . k. repeat Generate population of size 2q i from p i , for each i = 1 . . . k.Use non-dominated sorting and crowding to select the best µ individuals from all the populations. Cluster the selected individuals into k clusters. Let q i be the number of individuals in cluster i, for each i = 1 . . . k. Update frequency vectors p i based on selected individuals in each cluster. Figure 13 : 13Comparison of the hypervolume of population while solving the noisy COCZ with n = 30, m = 15 Figure 14 :Figure 15 : 1415Best hypervolume of population obtained for MULTI-OBJECTIVESETCOVER Best feasible solution found while solving the noisy MULTI-OBJECTIVESETCOVER All the algorithms are run for 50,000 function evaluations. The best hypervolume of the population obtained in the fixed function evaluation budget for each of 30 runs is shown in Table 1 : 1Function evaluation budgets allowed for noisy ONEMAX experiments with different noise levels. Table 2 : 2Function evaluation budgets allowed for noisy WEIGHTEDLINEAR ex-periments σ 1 2 3 4 5 budget 47096 46801 47704 48350 48682 σ 6 7 8 9 10 budget 49954 50876 51429 52794 53310 This paper is an extended version ofAishwaryaprajna and Rowe (2019). Noisy combinatorial optimisation by evolutionary algorithms. Rowe Aishwaryaprajna, J E , Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO '19. the Genetic and Evolutionary Computation Conference Companion, GECCO '19New York, NY, USAACMAishwaryaprajna and Rowe, J. E. (2019). Noisy combinatorial optimisation by evolu- tionary algorithms. In Proceedings of the Genetic and Evolutionary Computation Confer- ence Companion, GECCO '19, pages 139 -140, New York, NY, USA. ACM. Analysis of runtime of optimization algorithms for noisy functions over discrete codomains. Y Akimoto, S Astete-Morales, O Teytaud, Theoretical Computer Science. 605Akimoto, Y., Astete-Morales, S., and Teytaud, O. (2015). Analysis of runtime of opti- mization algorithms for noisy functions over discrete codomains. Theoretical Com- puter Science, 605:42 -50. Black-box search by elimination of fitness functions. G Anil, R P Wiegand, Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, FOGA '09. the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, FOGA '09New York, NY, USAACMAnil, G. and Wiegand, R. P. (2009). Black-box search by elimination of fitness functions. In Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, FOGA '09, pages 67-78, New York, NY, USA. ACM. Population-based incremental learning. a method for integrating genetic search based function optimization and competitive learning. S Baluja, Carnegie-Mellon Univ Pittsburgh Pa Dept Of Computer ScienceTechnical reportBaluja, S. (1994). Population-based incremental learning. a method for integrating ge- netic search based function optimization and competitive learning. Technical report, Carnegie-Mellon Univ Pittsburgh Pa Dept Of Computer Science. Runtime analysis of non-elitist populations: From classical optimisation to partial information. D Dang, P K Lehre, Algorithmica. 753Dang, D. and Lehre, P. K. (2016). Runtime analysis of non-elitist populations: From classical optimisation to partial information. Algorithmica, 75(3):428-461. Efficient optimisation of noisy fitness functions with population-based evolutionary algorithms. D.-C Dang, P K Lehre, Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII. the 2015 ACM Conference on Foundations of Genetic Algorithms XIIIACMDang, D.-C. and Lehre, P. K. (2015). Efficient optimisation of noisy fitness functions with population-based evolutionary algorithms. In Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII, pages 62-68. ACM. A new analysis method for evolutionary optimization of dynamic and noisy objective functions. R Dang-Nhu, T Dardinier, B Doerr, G Izacard, D Nogneng, Proceedings of the Genetic and Evolutionary Computation Conference. the Genetic and Evolutionary Computation ConferenceDang-Nhu, R., Dardinier, T., Doerr, B., Izacard, G., and Nogneng, D. (2018). A new anal- ysis method for evolutionary optimization of dynamic and noisy objective functions. In Proceedings of the Genetic and Evolutionary Computation Conference, pages 1467-1474. A fast and elitist multiobjective genetic algorithm: Nsga-ii. K Deb, A Pratap, S Agarwal, T Meyarivan, IEEE transactions on evolutionary computation. 62Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multi- objective genetic algorithm: Nsga-ii. IEEE transactions on evolutionary computation, 6(2):182-197. Exponential upper bounds for the runtime of randomized search heuristics. B Doerr, International Conference on Parallel Problem Solving from Nature. SpringerDoerr, B. (2020). Exponential upper bounds for the runtime of randomized search heuristics. In International Conference on Parallel Problem Solving from Nature, pages 619-633. Springer. When resampling to cope with noise, use median, not mean. B Doerr, A M Sutton, Proceedings of the Genetic and Evolutionary Computation Conference. the Genetic and Evolutionary Computation ConferenceDoerr, B. and Sutton, A. M. (2019). When resampling to cope with noise, use median, not mean. In Proceedings of the Genetic and Evolutionary Computation Conference, pages 242-248. Analysis of the (1+ 1) ea for a noisy onemax. S Droste, Genetic and Evolutionary Computation Conference. SpringerDroste, S. (2004). Analysis of the (1+ 1) ea for a noisy onemax. In Genetic and Evolutionary Computation Conference, pages 1088-1099. Springer. Indicator-based multi-objective evolutionary algorithms: A comprehensive survey. J G Falcón-Cardona, C A C Coello, ACM Computing Surveys (CSUR). 532Falcón-Cardona, J. G. and Coello, C. A. C. (2020). Indicator-based multi-objective evo- lutionary algorithms: A comprehensive survey. ACM Computing Surveys (CSUR), 53(2):1-35. The Rolling Tide Evolutionary Algorithm: A Multiobjective Optimizer for Noisy Optimization Problems. J E Fieldsend, R M Everson, IEEE Transactions on Evolutionary Computation. 191Fieldsend, J. E. and Everson, R. M. (2015). The Rolling Tide Evolutionary Algorithm: A Multiobjective Optimizer for Noisy Optimization Problems. IEEE Transactions on Evolutionary Computation, 19(1):103-117. An improved dimensionsweep algorithm for the hypervolume indicator. C M Fonseca, L Paquete, M López-Ibánez, 2006 IEEE international conference on evolutionary computation. IEEEFonseca, C. M., Paquete, L., and López-Ibánez, M. (2006). An improved dimension- sweep algorithm for the hypervolume indicator. In 2006 IEEE international conference on evolutionary computation, pages 1157-1163. IEEE. The benefit of recombination in noisy evolutionary search. T Friedrich, T Kötzing, M S Krejca, A M Sutton, Algorithms and Computation. Elbassioni, K. and Makino, K.Berlin, Heidelberg; Berlin HeidelbergSpringerFriedrich, T., Kötzing, T., Krejca, M. S., and Sutton, A. M. (2015). The benefit of re- combination in noisy evolutionary search. In Elbassioni, K. and Makino, K., editors, Algorithms and Computation, pages 140-150, Berlin, Heidelberg. Springer Berlin Hei- delberg. The compact genetic algorithm is efficient under extreme gaussian noise. T Friedrich, T Kötzing, M S Krejca, A M Sutton, IEEE Transactions on Evolutionary Computation. 213Friedrich, T., Kötzing, T., Krejca, M. S., and Sutton, A. M. (2017). The compact genetic algorithm is efficient under extreme gaussian noise. IEEE Transactions on Evolutionary Computation, 21(3):477-490. Robustness of populations in stochastic environments. C Gießen, T Kötzing, Algorithmica. 753Gießen, C. and Kötzing, T. (2016). Robustness of populations in stochastic environ- ments. Algorithmica, 75(3):462-489. An investigation on noiseinduced features in robust evolutionary multi-objective optimization. C K Goh, K C Tan, C Y Cheong, Y.-S Ong, Expert Systems with Applications. 378Goh, C. K., Tan, K. C., Cheong, C. Y., and Ong, Y.-S. (2010). An investigation on noise- induced features in robust evolutionary multi-objective optimization. Expert Systems with Applications, 37(8):5960-5980. Genetic algorithms, noise, and the sizing of populations. D E Goldberg, K Deb, J H Clark, 5161801UrbanaGoldberg, D. E., Deb, K., and Clark, J. H. (1991). Genetic algorithms, noise, and the sizing of populations. Urbana, 51:61801. The compact genetic algorithm. G R Harik, F G Lobo, D E Goldberg, IEEE transactions on evolutionary computation. 34Harik, G. R., Lobo, F. G., and Goldberg, D. E. (1999). The compact genetic algorithm. IEEE transactions on evolutionary computation, 3(4):287-297. Running time analysis of multiobjective evolutionary algorithms on pseudo-boolean functions. M Laumanns, L Thiele, E Zitzler, IEEE Transactions on Evolutionary Computation. 82Laumanns, M., Thiele, L., and Zitzler, E. (2004). Running time analysis of multiob- jective evolutionary algorithms on pseudo-boolean functions. IEEE Transactions on Evolutionary Computation, 8(2):170-182. Efficient noisy optimisation with the multi-sample and sliding window compact genetic algorithms. S M Lucas, J Liu, D Pérez-Liébana, 2017 IEEE Symposium Series on Computational Intelligence (SSCI). IEEELucas, S. M., Liu, J., and Pérez-Liébana, D. (2017). Efficient noisy optimisation with the multi-sample and sliding window compact genetic algorithms. In 2017 IEEE Sympo- sium Series on Computational Intelligence (SSCI), pages 1-8. IEEE. The equation for response to selection and its use for prediction. H Mühlenbein, Evolutionary Computation. 53Mühlenbein, H. (1997). The equation for response to selection and its use for prediction. Evolutionary Computation, 5(3):303-346. Multiobjective hboa, clustering, and scalability. M Pelikan, K Sastry, D E Goldberg, Proceedings of the 7th annual conference on Genetic and evolutionary computation. the 7th annual conference on Genetic and evolutionary computationPelikan, M., Sastry, K., and Goldberg, D. E. (2005). Multiobjective hboa, clustering, and scalability. In Proceedings of the 7th annual conference on Genetic and evolutionary computation, pages 663-670. Run-time analysis of populationbased evolutionary algorithm in noisy environments. A Prügel-Bennett, J Rowe, J Shapiro, Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII, FOGA '15. the 2015 ACM Conference on Foundations of Genetic Algorithms XIII, FOGA '15New York, NY, USAACMPrügel-Bennett, A., Rowe, J., and Shapiro, J. (2015). Run-time analysis of population- based evolutionary algorithm in noisy environments. In Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII, FOGA '15, pages 69-75, New York, NY, USA. ACM. On the effectiveness of sampling for evolutionary optimization in noisy environments. C Qian, Y Yu, K Tang, Y Jin, X Yao, Z.-H Zhou, Evolutionary computation. 262Qian, C., Yu, Y., Tang, K., Jin, Y., Yao, X., and Zhou, Z.-H. (2018). On the effective- ness of sampling for evolutionary optimization in noisy environments. Evolutionary computation, 26(2):237-267. Noisy fitness evaluation in genetic algorithms and the dynamics of learning. M Rattray, J Shapiro, Rattray, M. and Shapiro, J. (1998). Noisy fitness evaluation in genetic algorithms and the dynamics of learning. The benefits and limitations of voting mechanisms in evolutionary optimisation. J E Rowe, Aishwaryaprajna, Proceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms. the 15th ACM/SIGEVO Conference on Foundations of Genetic AlgorithmsRowe, J. E. and Aishwaryaprajna (2019). The benefits and limitations of voting mech- anisms in evolutionary optimisation. In Proceedings of the 15th ACM/SIGEVO Confer- ence on Foundations of Genetic Algorithms, pages 34-42. Using multi-objective evolutionary algorithms for single-objective constrained and unconstrained optimization. C Segura, C A C Coello, G Miranda, C León, Annals of Operations Research. 2401Segura, C., Coello, C. A. C., Miranda, G., and León, C. (2016). Using multi-objective evo- lutionary algorithms for single-objective constrained and unconstrained optimiza- tion. Annals of Operations Research, 240(1):217-250. Multi-Objective Optimization with Estimation of Distribution Algorithm in a Noisy Environment. V A Shim, K C Tan, J Y Chia, A A Mamun, Evolutionary Computation. 211Shim, V. A., Tan, K. C., Chia, J. Y., and Mamun, A. A. (2013). Multi-Objective Optimiza- tion with Estimation of Distribution Algorithm in a Noisy Environment. Evolutionary Computation, 21(1):149-177. Upper bounds on the runtime of the univariate marginal distribution algorithm on onemax. C Witt, Proceedings of the Genetic and Evolutionary Computation Conference, GECCO '17. the Genetic and Evolutionary Computation Conference, GECCO '17New York, NY, USAACMWitt, C. (2017). Upper bounds on the runtime of the univariate marginal distribution algorithm on onemax. In Proceedings of the Genetic and Evolutionary Computation Con- ference, GECCO '17, pages 1415-1422, New York, NY, USA. ACM. Stochastic runtime analysis of the cross-entropy algorithm. Z Wu, M Kolonko, R H Möhring, IEEE Transactions on Evolutionary Computation. 214Wu, Z., Kolonko, M., and Möhring, R. H. (2017). Stochastic runtime analysis of the cross-entropy algorithm. IEEE Transactions on Evolutionary Computation, 21(4):616- 628. Multiobjective optimization using evolutionary algorithms-a comparative case study. E Zitzler, L Thiele, International conference on parallel problem solving from nature. SpringerZitzler, E. and Thiele, L. (1998). Multiobjective optimization using evolutionary algo- rithms-a comparative case study. In International conference on parallel problem solving from nature, pages 292-301. Springer.
[]
[ "A GORDIAN PAIR OF LINKS", "A GORDIAN PAIR OF LINKS", "A GORDIAN PAIR OF LINKS", "A GORDIAN PAIR OF LINKS" ]
[ "Rob Kusner ", "Wöden Kusner ", "Rob Kusner ", "Wöden Kusner " ]
[]
[]
We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.A Gordian Pair: Configurations R (rotor) and W (wing) both minimize total ropelength in a common isotopy class; no isotopy between them preserves total ropelength. Coward and Hass [5], using tools from [4], gave an example of physically distinct isotopic configurations for a 2-component link: No isotopy can be performed while preserving the ropelength of each component; however length trading among components, which is more natural in the criticality theory [2, 3] for ropelength, is not allowed. Our configurations R and W are physically distinct in the broader length-trading sense appropriate for the Gordian unknot and unlink Problems [8]: Do nontrivial ropelength-critical configurations of unknots and unlinks exist? This problem arises in-and possibly obstructs-variational approaches [6] to the Smale Conjecture [7] via the space of unknots, and its generalization to spaces of unlinks[1].Definition. A pair of link configurations is Gordian if the links are isotopic, but there is no isotopy between them with thickness at least 1 which preserves total length. In fact, we prove a stronger statement, in the context of link homotopy and Gehring [2] thickness:Theorem. The configurations R and W minimize total ropelength in their common link homotopy class, but there is no link homotopy between them with Gehring thickness at least 1 while preserving the total ropelength. Because link homotopy is coarser than isotopy, and since the Gehring thickness constraint is more permissive than that for standard [4] thickness, a fortiori this is a Gordian pair.Proof of Theorem. (i) For any minimizing configuration in this link homotopy class, each component must be a particular type of stadium curve [2, 4] surrounding 1, 2 or 4 disjoint unit disks; in the last case, there is an interval moduli space of such curves C, ranging between the square (depicted above) and equilateral-rhombic configurations of 4 unit disks. (ii) Define a map Π from the space of minimizing link configurations in this link homotopy class to the space C 4 (S 1 ) of 4point configurations on the circle, taking the given link configuration to the 4 intersection points of C with the planar spanning disks for the 4 components linking C. (iii) The image of Π lies in the closed subset of C 4 (S 1 ) where each intersection point lies in one of the 4 curved arcs of C, a deformation retract of C 4 (S 1 ). (iv) The 4-configurations Π(R) and Π(W) lie in distinct path components of C 4 (S 1 )/O(2)-corresponding to dihedral orders of 4 points on a circle-so there is no path between R and W in the moduli space of Gehring-ropelength minimizers.
10.1007/s10711-023-00783-1
[ "https://export.arxiv.org/pdf/1908.05610v1.pdf" ]
199,668,949
1908.05610
a8620af1f3b571bb268b7687900b9a7956446c2e
A GORDIAN PAIR OF LINKS 13 Aug 2019 Rob Kusner Wöden Kusner A GORDIAN PAIR OF LINKS 13 Aug 20191 We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.A Gordian Pair: Configurations R (rotor) and W (wing) both minimize total ropelength in a common isotopy class; no isotopy between them preserves total ropelength. Coward and Hass [5], using tools from [4], gave an example of physically distinct isotopic configurations for a 2-component link: No isotopy can be performed while preserving the ropelength of each component; however length trading among components, which is more natural in the criticality theory [2, 3] for ropelength, is not allowed. Our configurations R and W are physically distinct in the broader length-trading sense appropriate for the Gordian unknot and unlink Problems [8]: Do nontrivial ropelength-critical configurations of unknots and unlinks exist? This problem arises in-and possibly obstructs-variational approaches [6] to the Smale Conjecture [7] via the space of unknots, and its generalization to spaces of unlinks[1].Definition. A pair of link configurations is Gordian if the links are isotopic, but there is no isotopy between them with thickness at least 1 which preserves total length. In fact, we prove a stronger statement, in the context of link homotopy and Gehring [2] thickness:Theorem. The configurations R and W minimize total ropelength in their common link homotopy class, but there is no link homotopy between them with Gehring thickness at least 1 while preserving the total ropelength. Because link homotopy is coarser than isotopy, and since the Gehring thickness constraint is more permissive than that for standard [4] thickness, a fortiori this is a Gordian pair.Proof of Theorem. (i) For any minimizing configuration in this link homotopy class, each component must be a particular type of stadium curve [2, 4] surrounding 1, 2 or 4 disjoint unit disks; in the last case, there is an interval moduli space of such curves C, ranging between the square (depicted above) and equilateral-rhombic configurations of 4 unit disks. (ii) Define a map Π from the space of minimizing link configurations in this link homotopy class to the space C 4 (S 1 ) of 4point configurations on the circle, taking the given link configuration to the 4 intersection points of C with the planar spanning disks for the 4 components linking C. (iii) The image of Π lies in the closed subset of C 4 (S 1 ) where each intersection point lies in one of the 4 curved arcs of C, a deformation retract of C 4 (S 1 ). (iv) The 4-configurations Π(R) and Π(W) lie in distinct path components of C 4 (S 1 )/O(2)-corresponding to dihedral orders of 4 points on a circle-so there is no path between R and W in the moduli space of Gehring-ropelength minimizers. Remark. In forthcoming work (in part with Greg Buck), we develop tools giving a stronger result: The total Gehring ropelength must rise by at least 2 in any isotopy (or link homotopy) between these minimizing link configurations. Configuration spaces of rings and wickets. E Tara, Allen E Brendle, Hatcher, Commentarii Mathematici Helvetici. 881Tara E Brendle and Allen E Hatcher. Configuration spaces of rings and wickets. Commentarii Mathematici Helvetici, 88(1):131-162, 2013. Criticality for the Gehring link problem. Jason Cantarella, H G Joseph, Rob Fu, Kusner, M John, Nancy C Sullivan, Wrinkle, Geometry & Topology. 104Jason Cantarella, Joseph HG Fu, Rob Kusner, John M Sullivan, and Nancy C Wrinkle. Criticality for the Gehring link problem. Geometry & Topology, 10(4):2055-2115, 2006. . Jason Cantarella, H G Joseph, Robert B Fu, John M Kusner, Sullivan, Ropelength criticality. Geometry & Topology. 184Jason Cantarella, Joseph HG Fu, Robert B Kusner, and John M Sullivan. Ropelength criticality. Geometry & Topology, 18(4):2595-2665, 2014. Jason Cantarella, Rob Kusner, John M Sullivan, On the minimum ropelength of knots and links. Inventiones mathematicae. 150Jason Cantarella, Rob Kusner, and John M Sullivan. On the minimum ropelength of knots and links. Inventiones mathematicae, 150(2):257-286, 2002. Topological and physical link theory are distinct. Alexander Coward, Joel Hass, Pacific Journal of Mathematics. 2762Alexander Coward and Joel Hass. Topological and physical link theory are distinct. Pacific Journal of Mathemat- ics, 276(2):387-400, 2015. Möbius energy of knots and unknots. Zheng-Xu Michael H Freedman, Zhenghan He, Wang, Annals of Mathematics. 1391Michael H Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Annals of Math- ematics, 139(1):1-50, 1994. A proof of the Smale conjecture, Diff(S 3 ) O(4). E Allen, Hatcher, Annals of Mathematics. 1172Allen E Hatcher. A proof of the Smale conjecture, Diff(S 3 ) O(4). Annals of Mathematics, 117(2):553-607, 1983. . Piotr Pieranski, Sylwester Przybyl, Andrzej Stasiak, physics/0103080Gordian unknots. arXiv preprintPiotr Pieranski, Sylwester Przybyl, and Andrzej Stasiak. Gordian unknots. arXiv preprint physics/0103080, 2001.
[]
[ "Real-time Bidding campaigns optimization using attribute selection", "Real-time Bidding campaigns optimization using attribute selection" ]
[ "Luis Miralles-Pechuan \nCentre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland\n", "M Atif Qureshi \nCentre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland\n", "Brian Mac Namee \nCentre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland\n" ]
[ "Centre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland", "Centre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland", "Centre for Applied Data Analytics Research (CeADAR)\nUniversity College Dublin\nDublinIreland" ]
[]
Real-Time Bidding is nowadays one of the most promising systems in the online advertising ecosystem. In the presented study, the performance of RTB campaigns is improved by optimising the parameters of the users' profiles and the publishers' websites. Most studies about optimising RTB campaigns are focused on the bidding strategy. In contrast, the objective of our research consists of optimising RTB campaigns by finding out configurations that maximise both the number of impressions and their average profitability. An online campaign configuration generally consists of a set of parameters along with their values such as {Browser = "Chrome", Country = "Germany", Age = "20-40" and Gender = "Woman"}. Throughout the investigation, a series of experiments have been conducted in order to verify that it is possible to increase the performance of RTB campaigns by means of configuration optimisation. The experiments demonstrate that, when the number of required visits by advertisers is low, it is easy to find configurations with high average profitability, but as the required number of visits increases, the average profitability tends to go down. Additionally, configuration optimisation has been combined with other interesting strategies to increase, even more, the campaigns' profitability. Along with parameter configuration the study considers the following complementary strategies to increase profitability: i) selecting multiple configurations with a small number of visits instead of a unique configuration with a large number, ii) discarding visits according to the thresholds of cost and profitability, iii) analysing a reduced space of the dataset and extrapolating the solution, and iv) increasing the search space by including solutions below the required number of visits. The developed campaign optimisation methodology could be offered by RTB platforms to advertisers to make their campaigns more profitable.
null
[ "https://arxiv.org/pdf/1910.13292v1.pdf" ]
204,950,130
1910.13292
a7c73f77a9febd5bf9c46e0d2933a8a0ed3cf909
Real-time Bidding campaigns optimization using attribute selection October 30, 2019 Luis Miralles-Pechuan Centre for Applied Data Analytics Research (CeADAR) University College Dublin DublinIreland M Atif Qureshi Centre for Applied Data Analytics Research (CeADAR) University College Dublin DublinIreland Brian Mac Namee Centre for Applied Data Analytics Research (CeADAR) University College Dublin DublinIreland Real-time Bidding campaigns optimization using attribute selection October 30, 2019 Real-Time Bidding is nowadays one of the most promising systems in the online advertising ecosystem. In the presented study, the performance of RTB campaigns is improved by optimising the parameters of the users' profiles and the publishers' websites. Most studies about optimising RTB campaigns are focused on the bidding strategy. In contrast, the objective of our research consists of optimising RTB campaigns by finding out configurations that maximise both the number of impressions and their average profitability. An online campaign configuration generally consists of a set of parameters along with their values such as {Browser = "Chrome", Country = "Germany", Age = "20-40" and Gender = "Woman"}. Throughout the investigation, a series of experiments have been conducted in order to verify that it is possible to increase the performance of RTB campaigns by means of configuration optimisation. The experiments demonstrate that, when the number of required visits by advertisers is low, it is easy to find configurations with high average profitability, but as the required number of visits increases, the average profitability tends to go down. Additionally, configuration optimisation has been combined with other interesting strategies to increase, even more, the campaigns' profitability. Along with parameter configuration the study considers the following complementary strategies to increase profitability: i) selecting multiple configurations with a small number of visits instead of a unique configuration with a large number, ii) discarding visits according to the thresholds of cost and profitability, iii) analysing a reduced space of the dataset and extrapolating the solution, and iv) increasing the search space by including solutions below the required number of visits. The developed campaign optimisation methodology could be offered by RTB platforms to advertisers to make their campaigns more profitable. Introduction The emergence of big data technologies and the facility to store a large volume of data has allowed companies to extract valuable information for their businesses [Provost andFawcett, 2013, Grochowski andHoyt, 1996]. In particular, in the online advertising business, companies collect information about users traffic and websites performance to maximise the benefits of advertisers, publishers and advertising networks (AN). Additionally, given the ease of collecting data, online advertising has become an active area of research where machine learning techniques are frequently used to predict events related to the campaign's performance such as the probability of generating a click in a given advert [Wang et al., 2017]. The first online publicity banner appeared in October 1994 and it belonged to a company called Hotwired. Ever since the online publicity sector has been growing as the number of active users on the Internet increased. The amount of people with access to the Internet makes online marketing, an attractive choice compared to traditional marketing channels. Some of the companies that started in this niche market around two decades ago such as Google and Facebook, are nowadays among the most important companies in the world [Goldfarb, 2014]. In the beginning, the ecosystem of online advertising was very simple; advertisers negotiated a price for displaying an advert directly with a publisher. For each campaign, it was required negotiation of the price which made the ecosystem unscalable as the demand of online adverts grew. Furthermore, the minimum investment required was a barrier to small businesses. In order to overcome the drawbacks of scalability, a new ecosystem called Advertising Networks (AN) was proposed [Zeff and Aronson, 1999]. The AN acts as an intermediary between advertisers and publishers. The main tasks of ANs are: buying and selling impressions, displaying adverts on the publishers' websites, and preventing fraud on the publicity ecosystem. Generally, ANs empowers advertisers with online tools to facilitate managing the advertising process and to inform them about the yield of their campaigns almost in real-time. In the same way, those tools inform publishers in detail about the performance of their websites [Trattner and Kappe, 2013]. ANs have been a popular choice for some time but, a few years ago, a new sophisticated auctionbased model called Real-time bidding (RTB) emerged. RTB offers important advantages over the AN model; it connects directly advertisers with publishers, eliminating intermediaries and their respective commissions, and it increases the cost-benefit of both sides. RTB is a real-time auction platform, where buying and selling online impressions takes place instantaneously on a per-impression basis via programmable criteria. RTB has a hierarchical bidding system where impressions are offered through several subsystems and the highest price is selected. When a bid is won by an advertiser, the advert is displayed on the publisher's website [Wang et al., 2017]. These bidding and winning processes are executed within the time it takes a user to load a page (100 milliseconds). Additionally, RTB is able to target audiences based on interests and profiles by examining the cookies generated by the users [Yuan et al., 2014b]. RTB offers more transparency to the publisher since they can see the price the advertiser is willing to pay. RTB also gives publishers a higher control over the advert they want to display by allowing them to define a list of banned advertisers. On the other hand, advertisers can buy impressions individually in a fine-grained fashion, which leads to more effective campaigns [Yuan et al., 2013]. Moreover, RTB also allows publishers to connect with multiple ANs and advertising agencies (companies that help advertisers to create and manage publicity campaigns), making it a heterogeneous ecosystem. Finally, RTB eliminates the need for commissions or fees to ANs by allowing advertisers to directly buy individual impressions from the publisher [Yuan et al., 2013]. To guaranty the adequate operation of the RTB system, it is required to collect and analyse information of the involved roles on a regular basis. Besides that, the data needs to be processed and converted into actionable information to maximize the performance of online campaigns and to mitigate online fraud [McMahan et al., 2013, Menon et al., 2011. Machine Learning (ML) techniques are a perfect approach to address those challenges and are widely applied in the domain of online advertising [Berry and Linoff, 1997]. ML models are trained using features of the users profile (browser, operating system, access time, device kind, or type of access), and of the publishers' websites (position of the banner, category, language, or history of the page) to accurately estimate relevant aspects related to the advertising ecosystem such as Click-through rate (CTR) or conversion rate [McMahan et al., 2013]. ML methods generate models that have been successfully applied to predict the CTR of an advert, to estimate the probability of generating a conversion (a purchase, a phone call, filling out a form), and to calculate the right value for a bid [McMahan et al., 2013]. In recent years, there have been many publications related to online campaigns optimisation [Zhang et al., 2017. However, some optimisation techniques are more suitable than others depending on the digital marketing format (banners, text adverts, search engine marketing...) and where advertising campaigns are launched (web searchers, websites, apps...). For example, an effective technique in search engine marketing consists of selecting multiple less frequent and therefore cheaper keywords, but that the sum of them add up a larger number than that of more popular keywords [Abhishek and Hosanagar, 2007]. This strategy, however, cannot be applied in RTB because advertisers do not bid based on keywords but on single impressions. In contrast, it is possible to apply the proposal of Thomaidou et al. [Thomaidou et al., 2012], where an expert human is replaced by a system to save costs, to most digital marketing formats. The proposed system is able to launch, monitor, and configure a search engines campaign automatically. Regardless of the underlying platform, a key aspect of any successful campaign is to display the advertised product to the right target [Sivadas et al., 1998], and RTB, as well as ANs, allows advertisers to target specific audiences which are assumed to become receptive towards the marketing of a product. The strategy to aim campaigns towards a small group with common interests is called microtargeting (CITATION). In addition to that, behavioural targeting, where the user interest is considered to display a relevant advert has been shown to be effective [Yuan et al., 2013]. However, making appropriate matchmaking is not a simple task, and even small improvements, when are applied to billions of transactions per day, it turns out into a big amount of money. Generally, there are two types of campaigns. The first type is called brand-based campaigns and is aimed at maximising the overall long-term revenue [Broussard, 2000]. Brand-based campaigns objective is focused on establishing or increasing brand reputation. The second type is called performance-based campaigns and is aimed at maximising the short-term revenue [Chen et al., 2011]. Their objective is selling a product to generate more profits. In our research, we are focused on the last one, performance-based campaigns. In this paper, we propose a methodology to optimise advertising campaigns in RTB platforms by configuring parameters for both users (to whom the campaign is directed) and web pages (where adverts are displayed). An example of suggested configuration to an advertisers could be: {Browser = "Chrome", Device OS = "Android", Traffic Category = "Organic search", Age = "25-34", Gender = "Male", Time = "10:00-11:00", Country = "UK", Position banner = "Top", and Website Category = "Fashion"}. The major contribution of the proposed work is developing a method to suggest advertisers with configurations that increase their campaign profitability. Our method is focused on optimising the attribute selection instead of optimising the bidding strategy. To ensure the stated aim, the methodology explores all the possible configurations and ranks them according to the number of visits that meet the requirements and their average profitability. The profitability of an impression is calculated according to the price and the probability that a conversion is generated from the impression. A supervised logistic regression method is applied to estimate the profitability of the advert impressions. To the best of our knowledge, this is the first work in RTB in this direction. The rest of the paper is organised as follows: Section 2 presents a review of the current stateof-the-art in RTB. In section 3 is described the methodology used to optimise campaigns using parameter selection. In section 4, experiments are conducted to evaluate the different strategies for improving optimization based on parameters. Finally, in section 5, it is presented the conclusions and some directions for future works. Background on Real-time Bidding and Online campaign optimization In this section, it is presented the state-of-the-art of RTB optimisation. First, it is introduced a general overview of online advertising campaigns, subsequently optimisation techniques applied to RTB campaigns are discussed, and finally, some research about RTB campaigns optimisation based on parameters which is the niche targeted in this contribution is described. The general overview of the online advertising campaigns In online advertising campaigns, advertisers try to acquire enough number of visits to fulfil their target at the lowest possible price, and that can be achieved using different kinds of platforms such as search engines, blogs, or social networks. There are different strategies to target the right audience such as selecting relevant keywords on a search engine or setting up relevant parameters related to users and website attributes [Evans, 2009]. Most importantly, a key aspect of a successful campaign lies in the way it is defined (i.e., target, message, adverts and channel). The series of questions while defining an online advertising campaign are the same as for traditional advertising campaign. Questions such as: Who is the right audience and their demographics? What shall be the design and message that would appeal (e.g., emotional, rational) audience in the right way? Which platform should be used to reach out to the audience effectively? and How to plan an effective campaign within the allocated budget? [Sivadas et al., 1998, Yuan et al., 2013]. Consequently, the success of the campaign is measured in terms of the impact i.e., Did campaign met the defined objectives within the budget? Online advertising is also applied in the area of predictive models such as personalised systems, user behaviour modelling, or CTR estimation [Wang et al., 2017]. Due to monetary benefits, online advertising has become a target of cybercrime and online frauds [Daswani et al., 2008], and consequently, there are many works addressing fraud prevention such as detecting bots simulating real users [Laleh and Azgomi, 2009]. In online advertising, there exist a conflict of interests between advertisers and publishers, and a trade-off between the two competing interests is needed. On one end, advertisers desire low prices for their impressions and on the other, publishers demand higher income for displaying adverts on their sites [Yuan and Wang, 2012]. As shown in figure 1, there are two platforms that define the RTB ecosystem: the demand-side platform (DSP), representing the interests of advertisers, and the supply-side platform (SSP), representing the interests of publishers. The DSP, manages advertisers and ANs campaigns efficiently, bidding directly on the auctions. On the other hand, the SSP aims at managing and optimising publishers' web spaces. The SSP distributes the information of a publisher across multiple platforms whenever a user visits a website and selects the most costeffective advert [Zhang et al., 2014a]. In RTB, the publisher can have a reserve price, that is to say, a minimum price to sell the impression. Yuan et al. [Yuan et al., 2014a] have addressed the issue of how to efficiently calculate the optimal value for the reserved price. Performance-based display advertising in RTB platforms usually implements the CPC payment model (Cost-per-click), in which the advertiser pays only when adverts generate a click or the Cost-per-acquisition (CPA), in which advertisers pay when a conversion derived from the displayed adverts takes place [Chen et al., 2011]. Thus, in order to select the best advert, the DSP implements ML models to estimate the probability of generating a conversion or a click from a given advert [Lee et al., 2012, Miralles-Pechuán et al., 2017. Real-Time bidding campaign optimization techniques RTB campaign optimization differs from search engine optimization in the way that an advertiser does not bid on a particular keyword, but instead, bids for individual impressions [Wang et al., 2017]. To optimise an RTB campaign, several factors are analysed such as the duration of the campaign, the preferences of each advertiser segment, the competitors' bids and strategies, the reserve price of the publishers, or the number of networks [Yuan et al., 2014b]. Predicting the optimal bid price for each impression is one of the most recurring challenges in RTB campaigns. When the price is too low, it is likely that it will not win the bid, on the contrary, if the price is too high, it will be paid in excess. To optimise the price paid for a campaign, an effective bidding strategy is essential. The bidding strategy determines the optimal quantity an advertiser pays for each impression. The effectiveness of the optimization strategy depends on different parameters, such as the payment method (advertisers can pay for impressions, clicks or actions), whether the bid is transparent or not, or the bidding mechanism [Yuan et al., 2014a]. There are mainly two approaches to estimate the optimal price. The first is based on game theory, where both advertisers and publishers make intelligent decisions based on others behaviour [Roth et al., 1990]. The game theory approach called Bayes-Nash equilibrium is a popular choice [Gomes and Sweeney, 2014]. The second approach is based on statistics, through the observation of the behaviour of the clicks, bids and conversions, it can be appreciated that sometimes periodic patterns are presented. That fact suggests that models that consider the time as a variable will perform better when calculating the optimal price [Yuan et al., 2013]. Ghosh et al. [Ghosh et al., 2009] introduced a different perspective that includes two scenarios for estimating the bidding price: one in which the bidding mechanism is transparent and all participants are informed of the winning price of the bid, and another one in which only the advertiser who has won the bid is informed of the winning price. Perlich et al. [Perlich et al., 2012] have examined the problem of setting the correct price according to the type of user, the type of message to be sent, and the specific moment based on supervised models and price auction theory. The process is as follows: first, the value of a particular advert is estimated, then, a bidding strategy based on the estimated price is applied. Additionally, it proposes a more aggressive bidding strategy based on a step function to set the price of bids. Traditionally, the RTB ecosystem made use of the Generalised Second-Price (GSP) auction mechanism. In GSP, the winner (the highest bid), instead of paying the amount of the bid, the advertiser pays the amount of the second-highest bid. However, the company Getintent, after analysing 338B impressions, reported that since 2017, many RTB companies decided to change their auction model to First-price auction, growing from 5.8% in December 2017 to 43.3% in March 2018 [Benes, 2019]. The main reason for this change is that First-price auctions are more transparent by definition (the winner pays exactly what they bid for), and advertisers are more confident about the price they are changed. In RTB most advertisers apply strategies to maximise the utility to select the most appropriate set of users for their campaigns by minimising the cost. Usually, it is preferable to lose a bid than to pay an excessive price for an advert. In RTB, a dynamic approach is more appropriate for estimating the amount to bid than a static approach, because the online advertising markets are very dynamic (there are permanently new advertisers, publishers, bidding strategies, and so on) [Borgs et al., 2007]. To define a dynamic strategy, the supervised method must be updated frequently [Lang et al., 2012]. Additionally, advertisers prefer their adverts to be displayed uniformly over time to reach a more diverse audience [Lee et al., 2013]. To this end, two strategies can be applied: modifying the value of the bids, or randomly selecting whether or not to participate in a bid [Xu et al., 2015]. It is worthwhile considering the approach based on the emerging machine learning branch called reinforcement learning proposed by Busoniu et al [Busoniu et al., 2008]. In this new technique, the system learns from its own experience through a reward function. The reward is positive when the system performs well and negative when the system fails [Busoniu et al., 2008]. In the long term, the system favours those actions that maximise the number of rewards over several trials [Sutton and Barto, 1998]. Reinforcement learning is a popular choice dealing with decision making that has been recently applied in many domains such as robotics, bots playing chess, and also RTB. Cai, H. et al. [Cai et al., 2017] presents a paper in which the bidding strategy is formulated as a reinforcement learning problem. In the Markov decision process, the agent represents the advertiser, the market is the environment, the parameters of the campaign (lifetime, budget ...) are the current state, and the bids of the agent are the actions. The presented model is called Reinforcement Learning to Bid (RLB) and it surpasses the performance of other state-of-art methods. Optimizing Real-Time bidding campaigns through parameter configuration Internet publicity platforms provide monitoring tools that allows advertisers to access useful information such as if the user clicked on the advert, the amount of time the user spent on the page, which users have visited the website before, and which items have been bought [Yan et al., 2009]. With the help of these tools, the performance of an online campaign can be evaluated very accurately just a few minutes after a campaign is launched. With this advantage, the campaign can be adjusted continuously to maximise the profitability by altering the configurations which impeded the profitability. One of the main benefits of online advertising is that it enables advertisers to segment their target audience in a precise way which in traditional media is much more difficult to implement [Goldfarb and Tucker, 2011]. In online publicity, an advert is shown only to the specific set of users selected by some criteria whereas, in television and radio, adverts are shown or voiced to all users regardless of their interests and characteristics. Furthermore, online publicity networks identify users' profiles by exploiting features such as browsing history, age, gender, operating system, or device type. This strategy of directing campaigns to a small group of people having common interests is useful and is known as microtargeting [Goldfarb andTucker, 2011, Miralles-Pechuán et al., 2018] Improving the overall performance of the campaign can be addressed through an appropriate configuration of parameters related to users and websites. Y. Liu et al. [Liu et al., 2012] showed that by using user features along with metadata from web pages, it is possible to estimate which users are more likely to generate a conversion. Additionally, it is also possible to create models to predict the probability of new users accessing to new web pages [Liu et al., 2012]. Our contribution approach is related to that of Y. Liu et al. [Liu et al., 2012] which aims at optimising RTB campaigns through the selection of attributes from the users and web pages. Our approach is focused on finding parameter configurations that maximise both the number of visits and the average conversion probability for those visits. Additionally, the parameter configuration technique is combined with other approaches such as multiple configurations, cost and profitabil- ity thresholds, configuration extrapolation, and increasing the search space. The combination of multiple approaches with the configuration optimization can potentially increase even more the obtained performance. RTB Campaign Optimization methodology In this section, we present the proposed methodology to optimise RTB campaigns based on attribute configuration instead of doing so by applying the most popular approach in the literature, that is, developing a better bidding strategy. Description of the RTB Campaign Optimization methodology Our approach is based on detecting configurations (sets of parameters from users and web attributes, along with their values) to suggests them to advertisers so they can display adverts more effectively. An example of parameter configuration could be: {Browser = "Chrome", Device OS = "Android", Age = "25-34", Gender = "Male", Time = "10:00-11:00", Country = "UK"}. Figure 2 shows the overall architecture of the proposed method which is composed of two modules: the Conversion rate (CVR) estimator and the Quality Score calculator (QSC). Our approach begins with the CVR estimator module that calculates the conversion probability (the probability that a conversion will be generated when an advert is displayed to a user) of each visit based on historical data. The CVR module calculates the probability of a conversion from a given impression and it is a supervised model based on logistic regression (it is briefly described in section 3.2). The second module is the QSC which scores all of the possible configurations (i.e., combinations of different attributes). The QSC module makes use of all the attributes of the impressions and optimises them to achieve the highest performance. Firstly, QSC explores all the possible configurations with a single attribute, then, it continues with two attributes, and so on and so forth until it finally reaches all the combinations with all the attributes. All configurations are ranked according to Quality Score metric defined by equation 6. To build the CVR estimator module, we used a dataset from Criteo [Diemert et al., 2017]. The dataset is ordered by time and represents 30 days. See section4.1 for more details on the dataset. We trained the supervised model with the first 6.46 M rows of the dataset and then, we calculated the conversion probability over the rest of the dataset, which we considered the testing set and has 10M rows. For evaluating the performance of our methodology, we included the prediction of the trained model as a new column of the testing set, and we stored it in the advertising campaign dataset module. The metric called Quality Score ranks how good a configuration is. The Quality Score favours configurations with high profitability as long as they have the minimum number of visits required by the advertiser. The Quality Score is calculated for all the possible configurations without repetition (discussed in detail in the following Section 3.3). Equation 1 shows the formula to calculate the number of combinations. Finally, the higher-ranked configurations scored by the QSC are offered to advertisers to increase the performance of their campaigns. Combinations without repetitions = n k=1 n! k! (n − k)!(1) Design of the Conversion Rate prediction model In this section, we discuss the design choices that were made to construct the CVR estimator model. To estimate the CVR, a database of displayed adverts is used, where each row is the displayed advert on the website of a publisher. The dataset contains attributes related to the user, the publisher's web page, the campaign identifier, the indicator of whether the conversion was generated, and the price paid for an impression. Equation 2 shows the formulation of the profitability of an impression using the CV R and the Impression P rice . The CV R is the probability that a conversion is generated from an advert 1 , and the Impression P rice is the price of the impression whose value is extracted from the dataset. According to equation 2, the profitability will be higher for those impressions that have a high probability of generating a conversion (numerator) and a low price (denominator). Impression P rof itability = CV R Impression P rice(2) We used a train/test split to build and validate the performance of the model. Figure 3 shows a snapshot of the typical dataset that the proposed model makes use of. The dataset has a total of ten columns, out of which nine are categorical (cat1, ..., cat9) which forms the predictor variables for the model e i , i = 1, ..., 9, and the final column is the P rof itability which is the target or the output variable of the model y. The output, y, represents the probability that a conversion (a purchase, a call, the filling of a form...) takes place. For instance, a predicted value of 0.225 means that the model estimates that the user has 22.5% chance of generating a conversion from the displayed advert. The model is used to expressed p, where p = P (y|e i , ∀i = 1, ..., 9). The CVR model was built using a well-known methodology for making predictions based on a Logistic Regression model [Chapelle et al., 2015, Li et al., 2011. We configured the adaptive learning rate value by following the criteria defined in [McMahan et al., 2013]. Such paper explains that the value of alpha highly depends on the nature of the dataset and the type of data. In our particular case, we used several parameters and choose the one that gave the best performance. The results and performance of the logistic regression algorithm are presented in section 4.2. To create the model, we used a simple but effective technique called hashing trick. The hashing trick is used to reduce the sparsity of the values of the dataset, not the number of features in the dataset [Li et al., 2011]. This technique allows to generate precise models and yet consuming little memory and computational resources. The hashing trick is recommended when the dataset is large, it consists of applying the modulus operation to each of the elements of the dataset e i mod D = R, where e i is an entry, D is a constant whose value should be high enough to avoid collisions, and R is the remainder of the Euclidean division (the hashed value). Collisions occur when different elements produce the same number after the application of the modulus operation. This trick works well with Logistic Regression models because it makes use of two vectors. The first vector, n, is used to store integers and it counts the number of times a value is repeated when the hash function is applied. The second vector, w, stores real numbers that represent the weight associated with each value of the dataset. The greater the value of a weight, the greater the value of the output will be [Li et al., 2011]. The original dataset can have N different values for each category, where N can be bigger than the length of the arrays n and w. The hashing trick divides the values of N by D, where D = length(w) = length(n), so the sparsity of the values will be at maximum D. This trick works very well because it reduces the amount of required space of memory from a computer to M , and despite being a simple trick, the results report that it has a very high performance in performance, time and consumes low memory [Li et al., 2011]. Initially, all of the elements of both vectors are set to zero, afterwards, the elements of n and w are updated using equations 3 and 4. n[i] = n[i] + 1 (3) w[i] = w[i] − α × (p − y) n[i] + 1(4) Once the vectors w and n have been updated, we make the prediction of the CVR for each impression using equation 5, which is the output of the model calculated as a sigmoid function. p = 1 1 + exp − N i=1 w[f i ](5) Where each variable is described as follows: • i represents the index of the arrays w and n. • w[i] represents the weight of i − th element. • n[i] represents the number of times that the value of the i appears after the application of the hashing trick to the elements of the original vector. • p ∈ [0, 1] represents if there was a conversion and it is also the estimated output value. • α represents the heuristic adaptive learning rate which is used to optimise the weights of the model. The higher the value of the learning rate, the faster it adapts but it may overshoot the gradient. By contrast, the lower the value, the more time it takes to coverage. Design of the algorithm to rank configurations of the campaign In this phase, we attempt to find optimal configurations that maximise both, the number of visits fitting the configuration and the average profitability of those visits. To meet the objective, we defined a criterion called Quality Score as shown in equation 6. The Quality Score rewards configurations that presumably have high profitability (sets of visits with a high probability of conversion at a low price) and also, ensuring a sufficient number of impressions for the advertiser campaign. To begin calculating the value of a configuration, we first estimate the average profitability. To this end, we score the profitability (using equation 2) for all the impressions that fit the configuration, and then, we calculate the average. Even though the configurations may have some unprofitable impressions, these impressions are offset by the rest of the impressions making the metric robust. Furthermore, we multiply the P rof itability Average by the minimum between the number of impressions that satisfy the configuration and the number of visits required by the advertiser, i.e., min(rows(D ), T ) as shown equation 6. The T represents a threshold, and it is used to avoid converging to configurations with an excessive number of visits. We should bear in mind that we are looking for configurations with a sufficient number to cover the number of visits required by the advertiser. Quality Score = P rof itability Average × min(rows(D ), T ) In the following lines, it is given an example of how the process of calculating the Quality Scores for a combination of attributes works. Let's imagine that we are testing the combination of columns (2, 5, 8). First, we will select the second, fifth and eighth attribute from the dataset. The result will be a subset with three selected attributes from the original dataset. From this subset, we select the unique records (i.e., rows) and calculate the Quality Scores for each record. As shown in Figure 3, the first unique value of the subset are values: 9312274, 582437 and 9312274 respectively. Then, we operate with the subset called D' (where the second, the fifth and the eighth attribute have values 9312274, 582437 and 9312274 respectively) are extracted from the original dataset. For this subset D', where D' ⊆ D, the Quality Score is calculated using equation 6. To speed up the algorithm, we created a list of all the configurations that do not have enough visits. Then, we add the following condition: if a configuration of a campaign is a subset of a previously discarded configuration, then this new configuration gets dropped as well. Such action was taken on the basis that a subset cannot be greater than the set that contains it. For example, for attributes (1, 3, 4, 7) with values (458, 47, 58, 58) respectively, we check if the list of configurations with not enough visits contains any combination without repetition of these attributes with precisely these same values. To do so, we first check in the list, combinations of one element (attribute 1 with value 458, attribute 3 with value 47, and so on). Later, we look for in the list for combinations of two elements (attribute 1 with value 458 and attribute 3 with value 47, attribute 3 with value 47, and attribute 4 with value 58, and so on), and the same with the combinations of 3 elements. This simple improvement significantly reduces the running time to Experiments In this section, experiments are carried out to evaluate the yield of the proposed methodology. First, we describe the preprocessing step applied to the dataset before running the experiments. Secondly, we explain in detail how to build the model to estimate the CVR value: we evaluated the built models using the well-known metrics and predicted the CVR of the RTB impressions. Next, we explain the algorithm used to find the optimal configuration. Finally, we discuss the five carried out experiments, and in each experiment, we demonstrate different approaches in the context of attribute selection. The proposed experiments have common steps which are presented in the algorithm 1. The steps "Calculate the Quality Score" and "Build the CVR estimation model" have been explained in detail in the previous section (see section 3). Algorithm 1 Selecting the best online campaign configuration 1: Build the CVR estimation model 2: Predict the CVR for all impressions 3: Generate all possible campaign configurations 4: for all Unique values of a combination do 5: Calculate the Quality Score 6: Store solution in the dataset 7: end for 8: Return the best solution from the dataset To carry out the experiments, we used Python 3.6.1 and Anaconda runtime (x86_64). We performed all experiments using MacBook computer having macOS High Sierra (2.3 GHz Intel Core i5, 8GB 2133 MHz, L2 Cache:256kb, L3: 4MB). Dataset description We conducted the experiments using a dataset from the Criteo company [Diemert et al., 2017]. The dataset consists of real data collected from internet traffic for 30 days. The dataset is a subset of the total visits from that period. Each row from the dataset contains information about an advert displayed on the website of a publisher. The dataset is anonymised for privacy concerns. The dataset is composed of the following fields: • Timestamp: it starts at 0 and sorts rows according to the displayed time. • UID: Unique code to identify the user. • Campaign: unique code to identify the campaign. • Conversion: it uses "1" if a conversion in the following 30 days takes place. In other cases, "0" is used. • Conversion_timestamp: the exact moment where the conversion derived from the impression took place. "-1" is used when there is no conversion. • Conversion_id: the code of the conversion derived from the impression. It enables the creation of the timeliness (in case they are required). • Attribution: binary field to indicate if the conversion was connected to Criteo ("1" means affirmative). • Click: binary field to indicate if the user clicked the impression ("1" is affirmative) • Click_pos: when there are several clicks from a user before a conversion, this value represents the position of the click in the current impression. • Click_nb: number of clicks before the conversion (the same user can click several times in adverts of the same campaign). • Cost: a transformed version of the price paid by Criteo. • Cost-per-order(CPO): the amount paid by the advertiser when the conversion is assigned to Criteo. • Time_since_last_click: indicates in seconds the amount of time elapsed from the last click. • Cat[1-9]: these nine fields represent features related to the user and the publisher. These fields are encrypted and had been hashed by applying the well-known hashing trick to reduce the dimensions [Li et al., 2011]. In addition, these values are used to create a supervised model for predicting the conversion probability. The original dataset has a total of 16,468,027 instances from different campaigns. First, we split the dataset into the training set (first 6,468,027 rows) and the testing set (last 10 million rows). After the split, we built the CVR model using the training set, and we predicted the probability of generating a conversion for the instances of the testing set. In the testing set, there are four campaign identifiers with more than 200 thousand visits (the identifiers of the campaigns are: 10341182, 15398570, 17686799, 30801593). We further divided each of the four campaigns into two parts of 100 thousand visits each. There are also nine other campaign identifiers with more than 100 thousand visits: 5061834,15184511,29427842,18975823,6686701,28351001,26852339,31772643,497593. We discarded the rows above 100 thousand so that all the datasets had the same dimensions. Finally, we got a total of 17 datasets of 100 thousand visits each (8 as a result of dividing each of the 4 campaigns with 200k into two campaigns of 100k and the other nine campaigns with 100k visits). All the experiments were carried out using the 17 datasets and figure 4 shows the total number of possible configurations in each dataset. Figure 5 shows the Average profitability of each campaign. Implementation of the Conversion Rate model As described in section 3, first we built and trained the predictive CVR model. In addition to that, we converted all features from string values to integer values and applied modulus 2 20 to calculate the hashing table. Later, we applied the logistic regression method to build the prediction conversion model with a learning rate α = 0.1 and D = 2 20 as the length of arrays n and w. Algorithm 2 summarises the implementation of the Conversion Rate model for both training and testing it. The results of the CVR model over the testing set are shown in table 1. In table 2, we can see that the Accuracy is 95.10% and the Area Under the Curve (AUC) is 82.99%, while the Logarithmic loss (a metric commonly used in this type of problem) has a value of 0.1572, which is quite accurate. Implementation of the algorithm to find the best solution To select the optimal configuration given the requirements of the advertisers, we executed algorithm 3 which in turn calls algorithm 4. The idea is to test all possible combinations of attributes and select the unique values for each combination. For example, if the attributes (1, 5) have the values (85, 58), (7714, 424), (596, 3458), (85, 58). Then, the last one will be rejected because it is the same as the first one. For each unique value, we choose from the dataset all the rows with the value 85 in attribute 1 and the value 58 in attribute 5. For that subset of the dataset, we select the average profitability and the number of rows from the dataset that match with that configuration. The idea is to explore all the possible combinations and store the ones with higher values. Description of the experiments As shown in table 3, for each of the configurations, we collect the following information: • Average profitability: indicates the average profitability of the selected impressions in the configuration campaign. • Dataset size: indicates the number of rows of the dataset. In experiment II a number of rows are removed each time a configuration is selected. Algorithm 2 Training and testing the prediction Conversion model. Require: Dataset Ensure: Prediction Conversion model 1: data ← Randomly select 12M samples visits from the original dataset 2: data hash ← Apply the hashing trick to data 3: Divide data hash into training and testing sets 6.3M training and 10M for testing 4: D ← 2 20 Length of w and n 5: α ← 0.1 6: w ← 0 0 0 · · · 0 of length D 7: n ← 0 0 0 · · · 0 of length D 8: Training the CTR model 9: for all row k ∈ training do row represents a user visit in which adverts display their adverts 10: s ← 0 11: for all f i ∈ row k do 12: s ← s + w[f i + 1] 13: end for 14: 1,2,4) • N o Rows: indicates the number of rows of the best configuration, which is the configuration with the highest value. p k ← 1/(1 + exp(s)) 15: for all f i ∈ v k do 16: w[f i ] ← w[f i ] − α(p k − y k )/( n[f i ] + 1) 17: n[f i ] ← n[f i ] + • Selected columns: indicates the selected attributes by the best configuration. • Time (Sec): required time to find the best configuration. • Values for selected columns: in addition to which columns/attributes have to be selected, we are also interested in the values that optimise those parameters. • Quality Score: as indicated previously, this metric is used to show how good a configuration is, given the number of visits required by the user. • Dataset size: the remaining size of the dataset after subtracting the selected configurations. Next, we make a small description of the carried out experiments: • Experiment I: In the first experiment, we intend to improve the campaign performance by optimising the parameter configuration. Generally, the more specific the campaigns are, the better the yield is, but, on the other hand, the number of visits that match the configuration is lower. As shown in figure 6, the original datasets have 100K samples. In this experiment, we select the best configuration from the dataset with at least 5k (5% of 100k) visits. Then, we repeated the same process increasing the limit of selected visits by the factor of 5k until reaching 50k. If advertisers require a small number of visits, the profitability will be very high, but, to perform a successful campaign it is generally required to display a sufficient number of adverts. The exciting aspect of this experiment is to visualise how profitability decreases as the number of demanded visits increases. • Experiment II: The motivation of the second experiment is to answer if it is better to optimise N campaigns with a small number of visits S, or to optimise a campaign with a big number of visits B, where the summation of the S visits of the N small campaigns is similar to that of B. In figure 7 the two approaches can be seen graphically. Optimising more campaigns has a higher computational cost, but it is possible that a better solution offsets these costs. To avoid the problem of the intersection of visits, which occurs when two or more campaigns share visits, we perform the experiments sequentially so that when the first configuration campaign is selected, all the visits that match with such configuration are removed from the dataset. The same process is repeated until all the campaigns are chosen. This experiment is performed for the following number of visits required by the advertiser: 5k, 10k, 15k, 20k, 25k and 30k. Finally, we compare the performance of selecting a campaign with these configurations with picking configurations with 1k and 2.5k visits respectively. For example, we compare a configuration that matches at least 5k visits with the summation of two configurations with at least 2.5k, and with five configurations with at least 1k visits. • Experiment III: In this experiment, we investigate the effect of finding a profitable configuration from a subset of the dataset and observe if the same configuration stills performing well over the entire dataset. If the finding is positive, it will have a substantial advantage in terms of computational costs (time and memory). Figure 8 shows the impact of configurations extracted from the subset and extrapolated over the whole dataset. In this experiment, we extrapolate from 10%, 20%, 30%, and so on until we reach 100 % of the dataset. • Experiment IV: In the following experiment, we test a strategy that combines the optimisation of parameters with an approach based on setting an economic threshold. In the proposed approach, we discard all visits that are above a particular economic value and then, we apply a parameter optimisation on the subset as indicated in 9. The threshold is calculated as the value that divides the dataset into two equivalent halves. Then, we discard the halve of the dataset below the threshold. In the experiment, we use two different kinds of thresholds, the first is based on the economic cost of the visits, and the second one on the expected profitability of the visits. We conducted the experiment with groups of visits as 5k, 10k, 15k... 50k. • Experiment V: One of the restrictions in the previous experiments was that all configurations with a number of visits lower than the required number were rejected. But, it could be the case that a campaign is offered with a number of visits slightly lower than the demanded but with average profitability that would please the advertiser. Figure 10 shows a motivating example, where the advertiser wants 15 thousand visits, and, (a) we propose a solution of 13 thousand, but, (b) with very high performance, it is likely that the advertiser may accept it as a serendipitous solution. In this experiment, we eliminate the restriction (configurations with a number of visits lower to the threshold are not discarded) which increases the search space of possible configurations to compare the performance with a scenario in which this restriction does exist. Just bear on mind that configurations with a low number of visits will still be penalised (but not rejected) as the number of visits is a factor to calculate how good a configuration is. Results and Discussion In this section, we discuss the results of the experiments of the proposed methodology. To measure the improvement of the proposed method, we have conducted five different experiments based on parameter optimisation combined with other ideas as discussed in the previous section. Experiments have been performed using 17 datasets, and the obtained results are the summation of the average profitability of all the datasets. The designed algorithm collects some parameters that indicate the performance of the campaign. The most important ones are shown in table 3. This table shows the collected information in experiment II for the first of the 17 datasets. Other variables such as Quality Score or the remaining size of the dataset can be calculated from these values. Experiment I The objective of this first experiment is to confirm that it is possible to enhance campaigns performance by means of parameter configuration. The results of the experiment are shown in table 4, which shows the average profitability of the best campaign with a number of visits greater than 4667.60 1198 (0, 1, 2, 4, 5, 8) [25259032, 7477605, 28051086, 32440044, 1973606, 29196072] 193.82 2 4139.91 1199 (0, 3, 4, 5, 7) [25259032, 29196072, 32440044, 1973606, 23998111] 189.89 3 3426.97 1200 (1, 2, 3, 4, 8) [7477605, 28051086, 23549932, 32440044, 29196072] 186.01 4 3144.99 1242 (0, 1, 2, 4, 8) [25259032, 26597095, 477175, 32440044, 29196072] 184.8 5 2821.57 1311 (0, 3, 4, 5, 8) [27093701, 29196072, 32440044, 1973606, 29196072] 189.89 6 2946.14 1165 (1, 3, 4, 5) [7477605, 23549932, 32440044, 1973606] 214.69 7 2529.13 1340 (0, 4, 7, 8) [25259032, 32440044, 29196072, 29196072] 168.6 8 2724.94 1242 (0, 1, 2, 3, 5) [25259032,7477605,28051086,29196072,1973606] Table 3: Example of the fields collected by the algorithm in experiment II selecting the best 1000 impressions and eliminating them from the dataset. or equal to the number required by the advertiser. Figure 11 shows the summary as a bar graph, in which it can be seen that the fewer visits are needed to make the campaign, the higher the yield becomes (the percentage indicates the increment over the average profitability). It can also be seen that the performance falls abruptly between the first five limits (between 1000 and 5000), but from that point, it tends to diminish slowly in a progressive way. We can conclude from the results that finding high-performance configurations using our methodology is feasible and that as the required number grows the profitability decreases. Experiment II This experiment is conducted to find out if the technique of selecting a group of configurations with a small number of visits has a higher performance than selecting a single configuration with a large number of visits, where the combination of the visits of the group has a similar number of visits to the single configuration. Figure 12 clearly shows that when the slides are of 1,000 visits, the increase in profitability is very sensitive. In addition, it can be appreciated that as the number of required visits increases, the performance when applying this technique has much better results. However, when the slides are of 2,500 visits, the improvement is very small and as the number of required visits increases, the improvement decreases. The drawback of this strategy is that it requires a higher processing cost. For example, when searching for sets of 10,000 items, the proposed strategy will search for four sets of 2,500, which requires to execute the algorithm four times instead of one. However, the improvement of the profitability confirms that it is an efficient approach. Experiment III As shown in figure 13, extrapolating is an interesting way to save campaign expenses (as discussed in the previous section). The experiments show that by extrapolating from the first 10% of the campaigns visits, it is possible to find out very profitable configurations. In particular, these configurations have an average fitness value only 8% lower than the optimal solution. This finding 8 1942.7 1942.7 1942.7 1942.7 1942.7 1942.7 Figure 11: Experiment I confirms that as the number of required visits increases, the average profitability declines. Figure 12: The technique used in Experiment II consists of selecting multiple configurations with small visits. It performs better since it is more difficult to find large groups with high performance. Figure 13: Extrapolating configurations from a small sample, as shown in Experiment III, is a very effective technique to find good configurations without having to do an excessive expense of previous. Figure 14: Experiment IV confirms that discarding visits below a certain price or profitability significantly improves the performance of the selected parameters. is interesting in order to save economic resources, as it can be assumed that a configuration that performs well today will continue to do so on the following days. Experiment IV The results of experiment IV confirm that combining the selection of parameters with the application of a threshold (either by profitability or by price) increases the efficiency of the campaigns. Figure 14 shows that until reaching the point of the 10k visits, using the price as a threshold is better than using the one based on profitability. Figure 15 shows the increments of improvement with each threshold. However, when the number of required visits is higher than 10k, it is better to use the threshold based on profitability. We also see that as the number of required visits increases, the performance of the three approaches tends to equalise. Experiment V Increasing the search space by allowing the algorithm to explore configurations with a number of visits below-the-limit allows to improve the campaign performance and decreases the economic costs of the campaign. However, it requires a higher computational cost since the search space for the solutions is larger. As it can be seen in figure 16, this technique brings a substantial improvement within the range of 10k to 25k, but from 40k the improvement becomes marginal. Figure 15: The performance of the approach based on the price is better until reaching 10k visits, from that point on it is better to use the threshold based on profitability. Figure 16: Experiment V shows that configurations in the search space below the threshold can be very cost efficient, low cost and with a number of visits close to the required number. Performance comparison with the methods Next, we compare the performance of our methodology with some of the state-of-the-art methods. Although we have not found other researches focused on increasing the performance of campaigns by means of parameter optimisation, we would like to highlight some recent publications aimed at improving the performance of RTB campaigns. Zhang et al. [Zhang et al., 2014b] affirm that there are not many publications related to RTB because until 2013 (the year in which the Chinese RTB company iPinyou decided to make public some of its campaigns) there were no databases related to RTB campaigns. This same author performed a comparative study on the performance of bidding strategies applied to RTB. The results are shown according to a key performance indicator (KPI) defined as the summation of clicks and visits. Results indicate that the algorithm called LIN (Linear-form bidding of predicted CTR) improves the performance of the bidding model below max (MCPC) by 204.13% when using 1/32 of the budget, 24% when using 1/8 of the budget, and 8.5% when using 1/2 of the budget. In a similar way, Zhang et al. [Zhang et al., 2014b] developed an algorithm to increase the number of clicks in the campaigns. In this research, it compares the performance of a new method called ORTB (Optimal real-time bidding) with LIN. The average increment of the summation of clicks for the nine campaigns using different budgets is 84.3% when the budget is 1/64, 28.61% when the budget is 1/32, 16.14% for 1/16, 9.19% for 1/8, 4.43% for 1/4, and 1.94% for 1/2 [Zhang et al., 2014a]. Another interesting publication is that of Lee et al. [Lee et al., 2013] in which an adaptive algorithm to select quality impressions is presented. The algorithm takes into account the performance of previously displayed impressions while distributes the budget evenly overtime to reach the widest possible audience. In CPM flat campaigns, the average CTR increment for seven campaigns was 123.7%. The performance of this methodology was also evaluated using ten dynamic CPM campaigns and the increment in performance with respect to conversions (CPA) and the number of clicks (CPC) was 30.9% and 19.0% respectively. We also find relevant the publication of Do et al. [Du et al., 2017] in which they improved the performance of the RTB through a Constrained Markov Decision Process (CMDP) based on a reinforcement learning framework. A distributed representation model is used to estimate the CTR value where the estimated CTR is the state, and the price of the action and the clicks are the reward. We see that the CMDP performance in terms of the number of clicks for the sum of ten campaigns is 12.6% better and the expected cost-per-click (eCPC) is 9.13% lower than in Sparse Binary which was considered as a baseline. Shioji et al. [Shioji and Arai, 2017] used neural embedding strategies like word2vec to improve the estimation of the users' response to the displayed adverts (a.k.a. CTR). The Word2vec technique is used to learn distributed representations from the internet browser history of the users. This approach is able to improve the accuracy of the CTR estimations results as follows: 4.90% for 0.3k, 3.39% for 1k, 2.87% for 10k, and 1.38% for 100k, where the first number of the tuple indicates the side of the training data and the second the AUC improvement. Conclusions and Future Work RTB platforms are becoming a very beneficial advertising model for publishers and advertisers. It is no wonder that the estimated volume of impressions managed by RTB networks will continue to increase in the coming years. It does not seem unreasonable to say that in the near future RTB can replace advertising networks platforms or at least take a significant market share. In this paper, we propose a novel methodology based on parameter configuration to find profitable campaigns for advertisers in an automatic fashion. In this sense, we think that the proposed approach is interesting because, to our knowledge, it covers a gap in RTB campaign optimisation research. The developed experiments prove that the presented methodology improves the results of RTB campaigns in a substantial way. Not only that, the combination of parameter optimisation with other approaches such as small campaign selection, setting a threshold, configuration extrapolation, or increasing the solution search space, improves, even more, the obtained results. However, these results may vary depending on several variables of a campaign such as the target, the moment when it is launched, or the behaviour of the rest of the competitors. Implementing the methodology could enable RTB platforms as well as advertising networks that manage third-party campaigns to suggest advertisers better configurations for their campaigns. Profitable campaigns will eventually boost the performance of advertisers companies, making RTB a more attractive platform, which in turn will make RTB advertisers more willing to launch more campaigns. It could seem that as the selected number of configurations increases, the gain of the average profit by campaign becomes trivial. But, here, we argue that, first, in RTB many ad networks coexist with their advertisers, therefore, when a platform decides not to bid on a particular impression, it does not imply that it is lost, but instead, it will be disputed among the rest of the advertisers. Additionally, it may be the case that some impressions may have low conversion probability for an advertiser but a high probability for another advertiser as it depends on the nature of the advert. Secondly, there are two types of campaigns: ones based on branding and others based on performance. Impressions not valuable from a performance-based perspective could be valuable for branding-based campaigns; where the goal is to increase the brand value instead of looking for profits in the short term. As future work, it could be a good idea to combine several techniques. For example, setting an economic threshold with small campaign selection and increasing the search space. The new methodology could be tested in different scenarios with other payment methods such as pay-perclick or pay-per-acquisition. Using algorithms to find a suboptimal solution but in less time could also be a good starting point. To this end, multi-objective evolutionary algorithms such as NSGA-II (Non-dominated Sorted Genetic Algorithm) could be a good solution. Figure 1 : 1Structure of the most important modules and roles in a Real-time Bidding platform. Figure 2 : 2The methodology used to find out the bests campaign configuration. Figure 3 : 3Extracting from the original dataset those visits that fit the configuration: (columns 2, 5, and 8 with values 9312274, 582437 and 9312274 respectively. find out the optimal solution. This small improvement results in completing an experiment (see experiment II in the later section) within 193.82 seconds compared to 75 hours without applying this trick. Figure 4 : 4As we can see in the graphic there are a lot of combinations in each of the 17 datasets. Figure 5 : 5Average profitability for each of the 17 advertising campaigns. Algorithm 3 3Detecting the best combination of the dataset.1: Comb ← List of all possible column combinations (1),(2),(3),...(1,2),(1,3),(1,4),...(1,2,3),( Figure 6 : 6Each configuration has two parameters: the average profitability and the number of visits matching the parameter configuration. The first picture (a) shows all the visits and the second (b) the required visits by the advertisers. Figure 7 : 7Selecting a large group as in (a) is computationally less expensive than selecting small groups as in (b), but the summation of small groups gives better results. Figure 8 : 8If it is possible to extrapolate a good solution from a slice of the campaign (a) instead of the using the whole dataset (b), then profitable configurations can be obtained at a lower computational costs. Figure 9 : 9Discarding visits over a certain price limit or below a certain profitability value as in (b) can boost the results than when the whole campaigns (a) are used. Figure 10 : 10Increasing the searching space by looking for configurations with a smaller number of visits (b) allows to find out configurations with higher Quality Score (a). Table 1 : 1Conversion probability confusion matrix.True values Positive Negative Total Predicted values Positive 9490262 14271 9504533 Negative 475589 19878 495467 Total 9965851 34149 10000000 Table 2 : 2Metrics of the Conversion probability model.Log Reg MAE MSE RMSE AUC Acc Avg Acc Sens Spec Prec F-1 0.1572 0.0807 0.0411 0.2027 0.8299 0.9510 0.5193 0.0401 0.9985 0.5821 0.0750 Algorithm 4 Calculating the best solution.,... 2: List ← { } 3: for all d i ∈ Datasets do There are 17 datasets 4: Limit ← 5,000 5: while (Limit <= 30,000) do 6: Solution ← Calculate_best_solution(d i , Limit, Comb) 7: List ← List + Solution 8: Limit ← Limit + 5,000 9: end while 10: end for Require: Data, Comb, Limit Ensure: Tuple 1: RejectSolution ← { } 2: SolList ← { } 3: for all c i ∈ Comb do Select only columns from combination of the Data 4: Data1 ← Subset(Data[,c i ]) The value of Data1 is a subset deep copy of Data 5: Data1 ← Unique (Data1) If a configuration is repeated we do not have to calculate the value again 6: for all row ∈ Data1 do 7: for i ∈ 1:Length(row) do 8: Data2 ← Subset(Data1[,Comb[i]] = row[i]) 9: if ! RejectSolution.exists(c i , row) then 10: if Rows(Data2) >= Limit then 11: P rof it Avg ← mean(Data2[,Prof]) 12: Size ← Rows(Data2) 13: Fitness ← P rof it Avg × min(Size, Limit) 14: Sol ← Tuple(Fitness, P rof it Avg , Size, c i , row) 15: SolList ← SolList + Sol 16: else 17: RejectSolution.append(Sol[1]) Add rejected solution to the list 18: end if 19: end if 20: end for 21: end for 22: end for 23: BestSol ← OrderbyFitness(SolList) Order by Fitness 24: return BestSol Table 4 : 4In Experiment I it is calculated the configurations with the highest Average Profitability with at least the number of visits required by the advertisers.Dataset 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 1 4206.8 3244.8 2849.3 1533.5 1533.5 1478.7 1478.7 1478.7 1478.7 1478.7 2 4152.6 3152.3 2746.1 2746.1 1545 1530.3 1530.3 1530.3 1530.3 1530.3 3 2083 1213.9 1041.5 1041.5 996.1 996.1 929.4 929.4 929.4 897 4 1877.5 986 856 839.4 804.9 804.9 762.8 762.8 762.8 701 5 494.4 370.5 370.5 350.8 337 337 337 331.7 331.7 331.7 6 480.4 454.9 345 344.7 319.7 319.7 319.7 319.7 319.7 319.7 7 4939.3 2570.5 1880.8 1880.8 1880.8 1880.8 1880.8 1880.8 1774.5 1774.5 8 5062.7 3971. 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[ "Thermodynamics of Fateev's models in the Presence of External Fields", "Thermodynamics of Fateev's models in the Presence of External Fields" ]
[ "Davide Controzzi \nDepartment of Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK\n", "Alexei M Tsvelik \nDepartment of Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK\n" ]
[ "Department of Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK", "Department of Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK" ]
[]
We study the Thermodynamic Bethe Ansatz equations for a one-parameter quantum field theory recently introduced by V.A.Fateev. The presence of chemical potentials produces a kink condensate that modifies the excitation spectrum. For some combinations of the chemical potentials an additional gapless mode appears. Various energy scales emerge in the problem. An effective field theory, describing the low energy excitations, is also introduced.
10.1016/s0550-3213(00)00042-0
[ "https://export.arxiv.org/pdf/hep-th/9910026v4.pdf" ]
118,949,250
hep-th/9910026
712fec8ea216a7b6cf30014ecbb08a3dddb5ed4b
Thermodynamics of Fateev's models in the Presence of External Fields arXiv:hep-th/9910026v4 27 Jan 2000 Davide Controzzi Department of Physics University of Oxford 1 Keble RoadOX1 3NPOxfordUK Alexei M Tsvelik Department of Physics University of Oxford 1 Keble RoadOX1 3NPOxfordUK Thermodynamics of Fateev's models in the Presence of External Fields arXiv:hep-th/9910026v4 27 Jan 2000(February 16, 2022)numbers: 1110-z7110-w7420De We study the Thermodynamic Bethe Ansatz equations for a one-parameter quantum field theory recently introduced by V.A.Fateev. The presence of chemical potentials produces a kink condensate that modifies the excitation spectrum. For some combinations of the chemical potentials an additional gapless mode appears. Various energy scales emerge in the problem. An effective field theory, describing the low energy excitations, is also introduced. I. INTRODUCTION Recently V.A.Fateev introduced a series of two dimensional integrable deformations of affine Toda theories possessing the property of duality 1,2 . These models exist in two incarnations -fermionic and bosonic -and the strong coupling limit of one incarnation corresponds to the weak coupling limit of the other one. Apart from being amusing examples of integrable field theories, these models may have some interesting physical applications [3][4][5] . In this paper we consider in some details one of the simplest models of that family. The Lagrangian density of the fermionic version of the theory is given by: L (f ) = s=1,2 iψ s γ µ ∂ µ ψ s − πg 2 (ψ s γ µ ψ s ) 2 + 1 2 (∂ µ φ 1 ) 2 + (∂ µ φ 2 ) 2 − (1) M 0ψ1 ψ 1 e −βφ1 − M 0ψ2 ψ 2 e βφ2 − M 2 0 2β 2 2e β(φ2−φ1) + e −2βφ1 + e 2βφ2 where g = β 2 /(4π + β 2 ) and we can identify two types of fermions ψ 1,2 and two coupled Toda chains described by the bosonic fields φ 1,2 . In the bosonic representation the phonon modes, φ 1,2 , disappear and the Lagrangian density is : L (b) = 1 2 2 s=1 ∂ µ χ s ∂ µχs 1 + (γ/2) 2 χ sχs − M 2 0 2 [χ 1χ1 + χ 2χ2 + γ 2 /2(χ 1χ1 )(χ 2χ2 )](2) where the coupling constants of the two models are connected by the duality relation: γ = 4π/β(3) As we have mentioned above, the weak coupling limit of the bosonic theory, γ << 1 (1 − g << 1), correspond to the strong coupling limit of the fermionic one, β >> 1. The bosonic form of the model is particularly interesting because the Euclidean version of the theory can be interpreted as Ginzburg-Landau free energy of two coupled superconductors. We shall discuss this application in greater details later in the text as well as in a separate publication. Models (1), (2) have U(1)×U(1) symmetry and the corresponding conserved charges are given by Q s = dxψ s γ 0 ψ s = − i 2 dx (χ s ∂ 0 χ s − χ s ∂ 0χs ) [1 + (γ/2) 2 χ sχs ](4) Thus one can introduce two chemical potentials and modify the Hamiltonian: H → H − h 1 Q 1 − h 2 Q 2(5) The U(1)×U(1) symmetry of the Hamiltonian reflects in the symmetry of the particle multiplets. In absence of chemical potentials the spectrum of the model consists of fundamental particles carrying quantum numbers of U(1) groups and their neutral bound states. The two-body scattering matrix of fundamental particles is given by a tensor product of two sine-Gordon S-matrices multiplied by a CDD-factor responsible for cancellation of double poles: S 12 (θ) = − sinh θ − i sin(π/λ) sinh θ + i sin(π/λ)Ŝ (λ; θ) ×Ŝ(λ; θ)(6) where theŜ(λ, θ) is the soliton scattering matrix of the sine-Gordon model with λ = 2 − g = 1 + γ 2 4π + γ 2(7) and is given byŜā ,b ab (λ; θ) = e iδ λ (θ) δ aā δ bb −δ ab + (1 − δ ab ) sinh(λθ) sinh[λ(θ − iπ)] +δ ab δ bā (1 − δ ab ) i sin(πλ) sinh[λ(θ − iπ)](8) with δ λ (θ) = ∞ 0 dω sin(ωθ) ω sinh[πω(λ −1 − 1)/2] cosh(πω/2) sinh(πω/2λ) To derive thermodynamic equations we start from the discrete Bethe Ansatz equations which emerge when the system is put in a box of size L with periodic boundary conditions: e −iM sinh θj L ξ(a 1 , ...a n ) = k =j S(θ j − θ k )ξ(ā 1 , ...ā n ) (9) E = M n j=1 cosh θ j(10) where ξ(a 1 , ...a n ) is the wave function of n fundamental particles in the isotopic space. The matrix on the right-hand side of Eq.(9) is related to the trace of the monodromy matrix τ : k =j S(θ j − θ k ) = Trτ (λ = θ j ; θ 1 , ...θ n )(11) Up to the scalar factor given by the product of CDD-factors the latter matrix is equal to a tensor product of monodromy matrices of the sine-Gordon models: τ (λ, θ 1 , ...θ n ) =τ (1) (λ, θ 1 , ...θ n ) ×τ (2) (λ, θ 1 , ...θ n )(12) The eigenvalues of the sine-Gordon monodromy matrix are known, hence it is straightforward to write down (9) in the diagonal form: e −iM sinh θjL = k =j S 0 (θ j − θ k ) × m+ a=1 sinh λ[(θ j − u a ) + iπ/2] sinh λ[(θ j − u a ) − iπ/2] m− b=1 sinh λ[(θ j − v b ) + iπ/2] sinh λ[(θ j − v b ) − iπ/2] (13) n j=1 sinh λ[(θ j − u a ) + iπ/2] sinh λ[(θ j − u a ) − iπ/2] = m+ b=1 sinh λ[(u b − u a ) + iπ] sinh λ[(u b − u a ) − iπ] (14) n j=1 sinh λ[(θ j − v a ) + iπ/2] sinh λ[(θ j − v a ) − iπ/2] = m− b=1 sinh λ[(v b − v a ) + iπ] sinh λ[(v b − v a ) − iπ](15) where: S 0 (θ) = e 2iδ λ (θ) n−1 a=1 sinh θ − i sin(πa/λ) sinh θ + i sin(πa/λ)(16) The numbers m ± are related to the conserved charges Q 1,2 . As we shall demonstrate later m ± = 1 2 [n − (Q 1 ± Q 2 )](17) The simplicity of this formula cannot obscure its remarkable meaning: propagating particles are superpositions of states localized on different edges of the stripe. This result is valid for other Fateev's models where the Toda array consists of more than two elastic chains. II. THERMODYNAMIC BETHE ANSATZ (TBA) EQUATIONS In order to construct the Thermodynamic Bethe Ansatz (TBA) equations from (13,14,15) we consider the case 1 − g = 1/ν, with integer ν ≥ 2, where the complex solutions of these equations (the strings) have the simplest classification. For most of the results the fact that ν is an integer is not important and one can generalize them for arbitrary g > 1/2 replacing ν by (1 − g) −1 . To save space and make our notations more transparent we shall sometimes denote the kernels in the integral equations directly in terms of their Fourier transforms. For example, the convolution of two functions g and f with the function g having Fourier transform g(ω) will be written as: +∞ −∞ dθ ′ g(θ − θ ′ )f (θ ′ ) = g * f (θ) = {g(ω)} * f(18) Thermodynamic Bethe Ansatz (TBA) equations have been introduced in 1,4 . The free energy of the system is expressed in terms of the excitation energies E n (θ): F/L = −T n M n dθ 2π cosh θ ln[1 + e −En(θ)/T ](19) In the specific case (1),(2) we have only one bound state with energy E 1 , and the fundamental particle with energy E 0 . The function E 1 satisfies the following equation : T ln[1 + e E1(θ)/T ] − G 11 * T ln[1 + e −E1(θ)/T ] = M 1 cosh θ +G 10 * T ln[1 + e −E0(θ)/T ],(20) and is directly coupled only to E 0 , which is determined by the equation: T ln(1 + e E0(θ)/T ) − K + * T ln[1 + e −E0(θ)/T ] = M cosh θ − h + + h − 2 (21) +G 10 * T ln[1 + e −E1(θ)/T ] − T s * ln[(1 + e ǫ−1(θ)/T )(1 + e ǫ1(θ)/T )] The two masses are related by: M 1 = 2M sin[π/2(2 − g)](22) and the Fourier transform of the kernel is: K + = K sym + 2a 1 * s = 1 + tanh[πω/2(ν + 1)] sinh[π(ν − 1)ω/2(ν + 1)] cosh(πω/2)(23) where K sym is related to the two-particle scattering phase : K sym (θ) = δ(θ) − 1 2πi d dθ log(S 0 (θ)),(24) and its Fourier trasform is: K sym (ω) = sinh π/2(1 − 1/λ)ω cosh(πω/λ) cosh(πω/2) sinh(πω/2λ) . The equations for ǫ n and ǫ −n are: ǫ −ν = h − ν/2 − s * T ln[1 + e ǫ −(ν−2)/T ] (26) ǫ −n = δ n,ν−1 h − ν/2 + s * T ln[1 + e −ǫn−1/T ][1 + e ǫ−n+1/T ](27)+ δ ν−2,n s * T ln[1 + e −ǫ−ν /T ] + δ n,1 s * T ln[1 + e −E0/T ] ǫ ν = h + ν/2 − s * T ln[1 + e ǫ (ν−2)/T ] (28) ǫ n = δ n,ν−1 h + ν/2 + s * T ln[1 + e ǫn−1/T ][1 + e ǫn+1/T ](29)+ δ ν−2,n s * T ln[1 + e −ǫν /T ] + δ n,1 s * T ln[1 + e −E0/T ] The Fourier transforms of the remaining kernels have the form: s(ω) = [2 cosh[π/2λ(1 − g)ω]] −1 G 11 (ω) = 4 sinh(gπω/2λ) sinh[π(1 − g)ω/λ] sinh(πω/2λ) × cosh[(λ − 1)πω/2λ] sinh[πω/2λ] cosh(πω/2) G 10 (ω) = 2 sinh(gπω/2λ) sinh[π(1 − g)ω/2λ] cosh(πω/2) where λ is defined in (7) and can be also expressed as λ = 1 + ν −1 . The subsequent analysis will show that the fields h ± are linear combinations of the chemical potentials h 1,2 h ± = h 1 ± h 2(30) (recall Eq. (17)). III. PROPERTIES OF THE GROUND STATE In this section we consider the zero temperature limit of the TBA equations. We assume that the fields h ± are strong enough to make E 0 negative in some interval [−B, B]. Then it is obvious from Eq.(20) that E 1 > 0: E 1 = M 1 cosh θ − G 01 * E (−) 0 (31) From Eqs.(27,29)we deduce that all ǫ ±n with n = ν are also positive: ǫ ±n = −a 1 * E (−) 0 − A n,ν−2 s * ǫ (−) ±ν + (n − 1)h ± (32) where A nm (ω) = 2 coth Ω sinh[(ν − max(n, m))Ω] sinh[min(n, m)Ω] sinh νΩ (33) a n (Ω) = sinh[π(ν − n)Ω] sinh νΩ , Ω = πω(1 − g) 2(2 − g) Substituting Eq.(32) into Eq.(26,28) we get the equations for ǫ ±ν : ǫ (+) ±ν + 1 − sinh[(ν − 2)Ω] sinh νΩ * ǫ (−) ±ν = h ± + sinh Ω sinh νΩ * E (−) 0(34) At last, substituting Eq.(32) into Eq.(21) we get: E (+) 0 + K sym * E (−) 0 = M cosh θ − 1 2 (h + + h − ) + sinh Ω sinh νΩ * [ǫ (−) ν + ǫ (−) −ν ].(35) Eqs.(34,35) determine the structure of the ground state. We cannot solve them in the general case and therefore we shall consider only two special cases: a) h + << h − ; b) h + ≈ h − . In the first case ǫ −ν > 0 and ǫ ν is negative inside an interval [−Q, Q], with Q >> B and Q → ∞ as h + → 0. Since ǫ (+) ν is small it is convenient to invert the kernel in Eq.(34) to get: sinh νΩ 2 sinh Ω cosh[(ν − 1)Ω] * ǫ (+) ν + ǫ (−) ν = νh + /2 + 1 2 cosh[(ν − 1)Ω] * E (−) 0(36) For |θ| >> B it is useful to approximate the right hand side of this equation by its asymptotics: 1 2 cosh[(ν − 1)Ω] * E (−) 0 ≃ 2ζ π a ǫ exp(−ζ|θ|) (37) where a ǫ = B −B dθ ′ E 0 (θ ′ )e −ζθ ′(38) and ζ = ν + 1 ν − 1 = 2 − g g (39) ε 1 Ε 1 ε n ε −ν 0 Ε θ ε ν h + FIG. 1. Schematic diagram of the ground state energies in presence of chemical potential for h+ << h− and large B. The same situation for small B will be presented in section VII Analogously one can rewrite Eq.(35) as: E (+) 0 + K * E (−) 0 = M cosh θ − h − /2 − 1 2 cosh[(ν − 1)Ω] * ǫ (+) ν(40) where K(Ω) = sinh Ω cosh[(3ν − 1)Ω] 2 sinh νΩ cosh[(ν − 1)Ω] cosh[(ν + 1)Ω](41) Since ǫ (+) ν is very small, to first approximation this reduces to: E (+) 0 + K * E (−) 0 = M cosh θ − h − /2(42) For large B we can approximate Eqs.(36) and (42) by the Wiener-Hopf (WH) ones 7 . The solutions for ε (−) 0 (θ) = E (−) 0 (θ + B) and ε (+) ν (θ) = ǫ (+) ν (θ + B) are: ε (−) 0 (ω) = h − G (−) a (ω)G (+) a (0) 1 iω + 1 − 1 iω + 0 (43) ε (+) ν (ω) = 1 G (+) (ω) h + G (−) (0) 1 −iω + 0 − a ǫ e −ζB G (−) (iζ) 1 ζ − iω (44) where a ǫ is defined in Eq.(38), and G G (+) a (ω)G (−) a (ω) = K(ω) (45) G (+) (ω)G (−) (ω) = sinh νθ 2 sinh Ω cosh[(ν − 1)Ω] ; ζ is defined by Eq.(39) and B is determined by the following relation: M e B G (+) (i) = h − G (−) (0)(46) To study the low temperature thermodynamic in Section VI it is useful to introduce also the ground state densities for E 0 and ǫ ν , σ and Σ respectively, defined by the following equations: σ (+) + K * σ (−) = M 2π cosh θ (47) Σ (−) + sinh νΩ 2 sinh Ω cosh[(ν − 1)Ω] * Σ (+) = 1 2 cosh[(ν − 1)Ω] * σ (−) ≡ Σ 0 (48) Solutions of the WH equations forσ (−) (θ) = σ (−) (θ + B) andΣ(θ) = Σ(θ + B) are: σ (−) (ω) = h − 4πG (−) (ω)G (+) (i)(iω + 1) (49) Σ(ω) = 1 G (+) (ω) h + G (−) (0)(−iω + 0) − a ǫ e −ζB G (−) (iζ)(ζ − iω)(50) In the limit b) both ǫ ±ν are positive and we have for E 0 : E (+) 0 + K sym * E (−) 0 = M cosh θ − h(51) Again using the WH method we find the following solution: ε (−) 0 = h G (−) b (ω)G (+) b (0)(iω + 1)(iω + δ)(52) with G (+) b (ω)G (−) b (ω) = K sym (ω) . Substituting this solution in Eq.(34) we obtain the following estimate for the gap in ǫ ±ν : M ν ∼ h(M/h) (2−g)/2(53) In the limit ν −1 → 0(g → 1)M ν ∼ √ M h. A schematic diagram of these results is presented in Fig.1 and Fig.2 for the case a) IV. THE LIMIT OF FREE BOSONS To check the validity of TBA equations (20-29) and establish the connection between the fields h ± and the chemical potentials h 1,2 we consider the free boson limit of the TBAs. It is realized in the case g → 1 or ν −1 → 0. In this case the kernels in Eqs.(26-29) become proportional to the delta functions s(θ) → 1 2 δ(θ) and K + → I, G 11 = G 01 → 0, such that all TBA equations become algebraic. The mass of the bound state in this limit is M 1 = 2M and the Free Energy is equal to: F/L = − T M 2π dθ cosh θ ln 1 + e −E1/T 2 1 + e −E0/T(54) The solution of TBA equations is expressed in terms of ξ = M cosh θ/T : 1 + e −E1/T = (1 − e −2ξ ) −1(55)1 + e ǫ±n/T = sinh(h ± n + a ± )/2T sinh h ± /2T 2(56) where sinh a ± /2T sinh h ± /2T = 1 + e −E0/T Substituting this into Eq. which has the following solution: 1 + e −E0/T = sinh 2 ξ (cosh ξ − cosh(h + /2T ) cosh(h − /2T )) 2 − sinh(h + /2T ) sinh(h − /2T )(59) Combining it with (55) in the Free Energy, we get F/L = − T 2π dθM cosh θ ln(F ) (60) F = (1 − e −2ξ ) −2 sinh 2 ξ (cosh ξ − cosh(h + /2T ) cosh(h − /2T )) 2 − sinh 2 (h + /2T ) sinh 2 (h − /2T ) (61) = [1 − e −ξ−(h++h−)/2T ][1 − e −ξ+(h++h−)/2T ][1 − e −ξ−(h+−h−)/2T ][1 − e −ξ+(h+−h−)/2T ] −1 The free boson limit of TBAs (20-29) does reproduce the free energy of two non interacting complex bosonic fields. This indicates that the TBA equations are correct. Another important result is the confirmation of Eqs.(17,30). V. SEMICLASSICAL ANALYSIS OF THE LIMIT OF SMALL γ Let us consider the Euclidean form of the Lagrangian (2). Rescaling the fields, χ s → (γ/2)∆ s , it becomes: L (b) = 2 γ 2 2 s=1 ∂ µ ∆ s ∂ µ∆s 1 + ∆ s∆s + M 2 ∆ 1∆1 + ∆ 2∆2 + 2(∆ 1∆1 )(∆ 2∆2 )(62) Then in the limit γ → 0 the quantum fluctuations are suppressed and one can approximate the ground state energy as minimum of the functional E(∆ s , h s ) = 2 γ 2 M 2 ∆ 1∆1 + ∆ 2∆2 + 2(∆ 1∆1 )(∆ 2∆2 ) − s=1,2 h s ∆ s∆s 1 + ∆ s∆s .(63) We will not address this problem, already studied by Fateev 1 , but we will use this approximation to to get more insight to the problem and give an intuitive picture of the low energy physics. Let us perform the following transformation in the form (62): ∆ s = sinh ρ s e iϕs(64) Under this transformation the measure transform in DρDϕ sinh(2ϕ) and the Lagrangian density in the new fields in presence of chemical potentials assume the form: L (b) = 2 γ 2 { s=1,2 [(∂ µ ρ s ) 2 + tanh 2 ρ s (∂ µ ϕ s ) 2 − h s tanh 2 ρ s ∂ 0 ϕ s ](65) +M 2 [sinh 2 ρ 1 + sinh 2 ρ 2 + 2 sinh 2 ρ 1 sinh 2 ρ 2 ]} For h 1 = h and h 2 = 0, which corresponds to the situation b in section III, we get: L (b) = 2 γ 2 {(∂ µ ρ 1 ) 2 + tanh 2 ρ 1 [(∂ µ ϕ 1 ) 2 − h∂ 0 ϕ 1 ] + (∂ µ ρ 2 ) 2 +(66) tanh 2 ρ 2 (∂ µ ϕ 2 ) 2 + M 2 (sinh 2 ρ 1 + sinh 2 ρ 2 + 2 sinh 2 ρ 1 sinh 2 ρ 2 )} The term h(∂ 0 ϕ 1 ) can be absorbed by the shift: ϕ 1 → ϕ 1 − (h/2)t(67) which generate also an additional term −h 2 /4. Then the Lagrangian density becomes: L (b) = 2 γ 2 {(∂ µ ϕ 1 ) 2 + (∂ µ ϕ 2 ) 2 + tanh 2 ρ 1 (∂ µ ϕ 1 ) 2 + tanh 2 ρ 2 (∂ µ ϕ 2 ) 2 + V b ef f (ρ 1 , ρ 2 )}(68) where: V b ef f (ρ 1 , ρ 2 ) = V 1 (ρ 1 ) + M 2 sinh 2 ρ 2 + 2M 2 sinh 2 ρ 1 sinh 2 ρ 2(69) and V 1 (ρ 1 ) = M 2 sinh 2 ρ 1 − h 2 /4 tanh 2 ρ 1(70) For h/2 > M , V b ef f (ρ 1 , ρ 2 ) has a minimum at ρ 1 = ρ 0 and ρ 2 = 0 where exp 2ρ 0 = √ 8(h/M ). In the vicinity of the minimum we have: V ef f (ρ 1 = ρ 0 + x, ρ 2 = y) ≈ hM ( √ 2y 2 + √ 8x 2 )(71) which gives the masses going like √ hM , in accordance with (53). Thus when the chemical potential exceeds the threshold the kinks condense and gapless Goldstone mode appear. To see this explicitly we write ρ 1 = ρ 0 + x and keeping only quadratic terms in x obtain the following expression for the Lagrangian: L (b) ∼ 2 γ 2 {(∂ µ x) 2 + (M h/2)x 2 + tanh 2 ρ 0 (∂ µ ϕ 1 ) 2 + (72) (∂ µ ρ 2 ) 2 + tanh 2 ρ 2 (∂ µ ϕ 2 ) 2 + sinh 2 ρ 2 h/M } Here we can identify a gapless mode ϕ 1 with velocity equal to the bare one, a gapful field x with the mass m 2 x = hM/2 and an effective integrable field theory described by the Lagrangian density: L 2 = 1 2 ( 2 γ ) 2 {M sinh 2 ρ 2 + (∂ µ ρ 2 ) 2 + tanh 2 ρ 2 (∂ µ ϕ 2 ) 2 }(73) In the case h 1 ∼ h 2 corresponding to the situation a) of Section III, the effective potential acquires the form: V a ef f (ρ 1 , ρ 2 ) = V 1 (ρ 1 ) + V 1 (ρ 2 ) + 2M 2 sinh 2 ρ 1 sinh 2 ρ 2(74) Repeating the same analysis one finds two independent gapless modes with the same velocity. Then the semiclassical approximation gives a correct qualitative description of the low energy behavior of the system. As a byproduct we obtain another confirmation of (30). VI. LOW ENERGY PHYSICS: MASSLESS MODES AND ENERGY SCALES Let us go back to the zero temperature limit described in Section III. As we have shown, in the case h 1 ≈ h 2 there are two soft modes: E 0 and ǫ ν . At |θ| < B the mode ǫ ν closely follows E 0 ; at|θ| >> B its behavior is determined by the asymptotics (37) and this mode becomes independent. The temperature scale below which the two modes decouple can be determined by the value of of ǫ ν at θ ∼ B: T sep ∼ ǫ (−) ν (B)(75) For low temperatures, T < T sep , the two modes give independent contribution to the thermodynamics. To calculate the low temperature free energy: F/L = − T 2π M dθ cosh θ ln[1 + e −E0(θ)/T ](76) we use the Eqs. (36,42) which at finite temperature have the form: T ln(1 + e E0/T ) − K * T ln(1 + e −E0/T ) = M cosh θ − h − 2 − 1 2 cosh(ν − 1)Ω * T ln(1 + e ǫν /T ) (77) sinh νΩ 2 sinh Ω cosh(1 − ν)Ω * T ln(1 + e ǫν /T ) − T ln(1 + e −ǫν /T ) = νh + 2 − a ǫ (78) where a ǫ is defined in Eq.(37). To isolate the contributions that vanish at T=0 we rewrite (77) like: E (+) 0 + K * E (−) 0 + (I − K) * T ln(1 + e −|E0|/T ) = r.h.s. of Eq.(77)(79) where I is the identity operator. Using Eq.(47) we can rewrite the free energy as: F/L = − T 2π M dθ cosh θ(E (−) 0 + T ln(1 + e −|E0|/T )) ≡ f 0 + f ν(80) where f 0 = −T dθσ(θ) ln(1 + e −|E0|/T ) ≈ −π 2 T 2 /3V c (81) f ν = −T dθ 1 2 cosh(ν − 1)Ω σ (−) (θ)T ln(1 + e ǫν /T ) ≈ −π 2 T 2 /3V s(82) The velocities of the two modes are determined by : V c = s=± | ∂E 0 (θ) ∂θ |[2πσ (s) (B)] −1 θ=sB (83) V s = s=± | ∂ǫ ν (θ) ∂θ |[2πΣ (s) (B)] −1 θ=sB(84) For large B all the quantities appearing in these equations can be calculated using the Wiener-Hopf solutions obtained in Section III via the following relations: dE 0 (±) /dθ θ=B = dε 0 (±) /dθ θ=0 = lim ω→∞ ±(iω) 2 ǫ (±) (ω) σ (±) (B) =σ (±) (0) = lim ω→∞ ±iωσ (+) (ω) and analogous relations for ǫ ν and Σ. The results are: V c = V s = 1(85) The fact that the two velocities are the same seems to be peculiar to the model in consideration and not a general property of the family of Fateev's models. In general V s depends on g and on the number of Toda chains. Thus we have shown that the presence of chemical potentials introduces additional energy scales in the problem with non trivial crossovers. At temperatures T ≪ h, ǫ −n and ǫ n (n = ν) modes decouple from the other modes. The same happens for E 1 at energies smaller then their gap and in some physical regions there is also an intermediate energy scale, as we will see in the next Section. As one lowers the temperature below T sep the gapless modes decouple from each other and the picture of two non-interacting Luttinger liquids emerges. VII. WEAK COUPLING LIMIT: NON PERTURBATIVE RESULTS Now we wish to focus our attention on the limit g → 1 which corresponds to the weak-coupling limit of the bosonic theory. Naively one can imagine to obtain perturbative results. It turns out however that, in the presence of the chemical potential, this is not the case, the reason being that the kinks condensate survives and affects the excitation spectrum. A similar effect is present in the Klein-Gordon limit of the sine-Gordon model 6 . To be specific let us consider the situation a) of Section III, all the energies ǫ n (n = ν) have gaps of the order of h − and can be omitted at temperatures T << h − . To obtain information on the excitation spectrum one has to study the fundamental equations (31) and (42). The standard relativistic dispersion law: E 1 (θ) = M 1 cosh θ, can be modified by the last term in (31) which emerge in presence of kinks. In the limit g → 1, the kernels simplify K(ω) = K 0 (ω)(1 + O ((1 − g) 2 )), and G 10 (ω) = G 0 (ω)(1 + O((1 − g) 2 )), where K 0 (ω) = 4π 2 β 2 λ ω cosh(πω(λ + 2)/2λ) cosh(πω/2) sinh(πω/λ) (86) = π λ (1 − g)ω cosh(πω(λ + 2)/2λ) cosh(πω/2) sinh(πω/λ) and G 0 = (1 − g) πω λ sinh(πω/2λ) cosh(πω/2)(87) Again we can study TBA equations with the kernel K 0 (ω) and G 0 (ω) in the limits of large and small B. We focus the attention on this second case where the effects we want to consider can be seen more clearly. To study Eq.(31) for small B we need the kernel K 0 (θ) only for small θ. This is equivalent to approximating Eq.(86) for large ω, as: K 0 (ω) ≃ 2π λ (1 − g)|ω|(88) Approximating also the rhs of Eq. (42) for small θ we can rewrite it as: K 0 * E 0 = M θ 2 /2 − H(89) where H = h − /2 − M and * denote the convolution of support (−B, B) and K 0 is the Fourier transform of (88). One can absorb the constant part of the kernel in E 0 introducing the new functionẼ 0 = 2π λ (1 − g)E 0 . Equation (42) then becomes:K 0 * Ẽ0 = M θ 2 /2 − H,(90) K 0 is the singular integral operator corresponding to the kernelK 0 (ω) = |ω| and defined: Using this form of the kernel, Eq. (90) can be reduced to the canonical form: K 0 * Ẽ 0 (θ) = − 1 π P B −B 1 (θ − θ ′ ) 2Ẽ 0 (θ ′ )dθ ′ = 1 π ∂ ∂θ P B −B 1 (θ − θ ′ )Ẽ 0 (θ ′ )dθ ′(P B −B dθ ′ 1 (θ − θ ′ ) V (θ ′ ) = g(θ)(92) where we have introduced: V (θ) = ∂Ẽ 0 (θ)/∂θ and g(θ) = π(M θ 2 /2 − H). This equation has the following general solution 8 : V (θ) = 1 π 2 B 2 − θ 2 P B −B dθ ′ g(θ ′ ) √ B 2 − θ ′2 (θ ′ − θ)(93) which gives: E 0 (θ) = − M λ 3π 2 (1 − g) (B 2 − θ 2 ) 3/2 (94) where B 2 /4 = h−/2−M M ≡ ∆. Analogously the ground state density (47) turns out to be: σ(θ) = M λ 2π 3 (1 − g) B 2 − θ 2(95) From here we find the relationship between h − and the charge: Q = h − /2 − M 2π(1 − g)(96) The Fermi energy, E F = E 0 (θ) as function of the charge if therefore: E F = 8M 3π 2 (1 − g) 1/2 2π M Q 3/2(97) We can now use the above results to solve Eq(31). Rewriting the Fourier transform of the kernel G 0 as: G 0 ∼ 2 π ∂ ∂θ (π/2(1 − g)) sinh θ sinh 2 θ + (π/2(1 − g)) 2 cosh 2 θ(98) it is possible to extract the asymptotic behavior of E 1 : E 1 = M 1 cosh θ + M/3 cos(2 sin −1 (θ/B)), θ ≪ B M 1 cosh θ + Ae −θ , θ ≫ B(99) The numerical solution is shown on Fig. 3. One can clearly see that even in the weak coupling limit the dispersion law is drastically modified at small θ. This analysis is supposed to clarify the subject of different energy scales present in the model (cfr. Fig.4). At very small doping, Q ≪ Q 1 ≡ M (g − 1) −1/3 , E F is smaller then M and there are two scales besides h: M and E F itself. The mode E 1 is completely decoupled at low temperature. As you increase the doping you reach an intermediate region, characterized by M (1 − g) −1/3 ≪ Q ≪ Q 2 ≡ M (1 − g) −1 , where E F ≃ M but still much smaller then E 1 . At very large doping we find a very interesting region. As we have seen in Sec. III at large Q (or equivalently at large B) E F ≃ Q, while E 1 (0) ≃ √ Q. Then there is a regime in which E F is bigger then the gap of the E 1 -mode, and there will be temperatures, T ≪ E F for which the E 1 -mode becomes soft. VIII. DISCUSSION We have considered in some detail the thermodynamics of one of the integrable field theories recently introduced by Fateev. Most of the results are quite general and remain valid also for other models of the family. These models are interesting for various reasons. First of all they have peculiar mathematical properties mostly related to a dual representation of the theory. In addition they probably can find application to various physical systems. Considering the model as a (1+1)-dimensional Quantum Field Theory one can interpret it as two one dimensional conductors coupled via phonon interaction. The problem of quasi one dimensional systems with electron-phonon interaction is a very important one and cannot be approached with perturbative methods, it is then very important to have exact results for some specific model. This interpretation of the model and the possible application to physical systems have been discussed in ref. 4 , for physical regimes different from the one considered here (i.e. for g < 1/2). On the other side, the Euclidean version of the model, in the bosonic representation, describes an effective Landau-Ginzburg theory of two coupled superconductors. In the specific model considered in this work, the coupling between the superconductors is quite simple. Nevertheless for other models of the Fateev's family, where coupling between the fermionic modes is achieved via a higher number of Toda chains, the resulting coupling between the two superconductors is via elastic modes and then can be interpreted as two layered superconductors separated by an insulating stratum. In relation to the high-T c (cuprate) superconductors an interesting problem is to study the dependence of the superconducting properties on the number of insulating layers between the superconducting planes. We will address this problem in a separate publication. Let us give a brief summary of our results. As shown in Sec.VII the results are non-perturbative also in the weak coupling regime and could only be obtained through exact or non-perturbative methods. This is related to the fact that the chemical potentials generate a kink condensate that survives in the weak coupling limit. As we have shown, for some combinations of the chemical potentials two gapless modes emerge and various energy scales appears, which characterizes crossovers between different regimes. Probably the most interesting region of the phase diagram is obtained at large doping where the E 1 -mode become soft at temperatures for which the E 0 -one is still frozen. It deserves further investigation. Another important result is the one contained in Eq.(17). This equation tells us that charged particles in the Fateev's model are linear combinations of particles located on the edges, which is important for qualitative understanding of the physics of the model. and G (+) (G (−) ) are analytic in the upper (lower) half plane and satisfy the conditions: FIG. 2 . 2Schematic diagram of the ground state energies in presence of chemical potential for h+ ≈ h− and large B. (21) we get cosh(h + /2T ) 1 + e −E0/T + [cosh 2 (h + /2T ) + sinh 2 (h + /2T )e −E0/T ] dotted line represents the solution in the absence of kink condensate. In the lower figure we plot the gapless mode E0. diagram describing different energy scales as a function of the charge. We can recognize three distinct regions as described in Sec. VII. . V A Fateev, Nucl. Phys. 479594V. A. Fateev, Nucl. Phys. B479, 594 (1996). . V A Fateev, Nucl. Phys. 473509V. A. Fateev, Nucl. Phys. B473, 509 (1996). . C Pepin, A M Tsvelik, Phys. Rev. Lett. 823859C. Pepin and A. M. Tsvelik, Phys. Rev. Lett. , 82, 3859 (1999). . C Pepin, A M Tsvelik, 9983180C. Pepin and A. M. Tsvelik, cond-mat 9983180. . D Controzzi, A M Tsvelik, unpublishedD. Controzzi and A. M. Tsvelik, unpublished. . J.-S Caux, A M Tsvelik, Nucl.Phys.B. 474715J.-S. Caux and A. M. Tsvelik, Nucl.Phys.B 474, 715 (1996). . V A Fateev, E Onofri, Al A Zamolodchikov, Nucl.Phys. 406521V.A.Fateev, E.Onofri, Al.A.Zamolodchikov, Nucl.Phys. B406,521 (1993). S G Mikhlin, Integral equations. OxfordPergmon pressS.G.Mikhlin, Integral equations (Pergmon press, Oxford,1964).
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[ "Zero Inflated Poisson Model with Clustered Regression Coefficients: an Application to Heterogeneity Learning of Field Goal Attempts of Professional Basketball Players", "Zero Inflated Poisson Model with Clustered Regression Coefficients: an Application to Heterogeneity Learning of Field Goal Attempts of Professional Basketball Players" ]
[ "Guanyu Hu ", "Hou-Cheng Yang ", "Yishu Xue [email protected] " ]
[]
[]
Although basketball is a dynamic process sport, with 5 plus 5 players competing on both offense and defense simultaneously, learning some static information is predominant for professional players, coaches and team mangers. In order to have a deep understanding of field goal attempts among different players, we propose a zero inflated Poisson model with clustered regression coefficients to learn the shooting habits of different players over the court and the heterogeneity among them. Specifically, the zero inflated model recovers the large proportion of the court with zero field goal attempts, and the mixture of finite mixtures model learn the heterogeneity among different players based on clustered regression coefficients and inflated probabilities. Both theoretical and empirical justification through simulation studies validate our proposed method. We apply our proposed model to the National Basketball Association (NBA), for learning players' shooting habits and heterogeneity among different players over the 2017-2018 regular season. This illustrates our model as a way of providing insights from different aspects. *
10.1002/cjs.11684
[ "https://arxiv.org/pdf/2012.06715v1.pdf" ]
229,153,320
2012.06715
fdfe2bfccb2497a47e3fcf0fb3850fb3b8e38ec8
Zero Inflated Poisson Model with Clustered Regression Coefficients: an Application to Heterogeneity Learning of Field Goal Attempts of Professional Basketball Players 12 Dec 2020 Guanyu Hu Hou-Cheng Yang Yishu Xue [email protected] Zero Inflated Poisson Model with Clustered Regression Coefficients: an Application to Heterogeneity Learning of Field Goal Attempts of Professional Basketball Players 12 Dec 20201Bayesian NonparametricMCMCMixture of Finite MixturesModel Based Clustering Although basketball is a dynamic process sport, with 5 plus 5 players competing on both offense and defense simultaneously, learning some static information is predominant for professional players, coaches and team mangers. In order to have a deep understanding of field goal attempts among different players, we propose a zero inflated Poisson model with clustered regression coefficients to learn the shooting habits of different players over the court and the heterogeneity among them. Specifically, the zero inflated model recovers the large proportion of the court with zero field goal attempts, and the mixture of finite mixtures model learn the heterogeneity among different players based on clustered regression coefficients and inflated probabilities. Both theoretical and empirical justification through simulation studies validate our proposed method. We apply our proposed model to the National Basketball Association (NBA), for learning players' shooting habits and heterogeneity among different players over the 2017-2018 regular season. This illustrates our model as a way of providing insights from different aspects. * Introduction Analyzing players' "hotspots" is an indispensable part of basketball data analytics. Identifying such hotspots, as well as which players tend to have similar hotspot locations, provides valuable information for coaches as well as for teams who are aiming at making transactions and looking for players of a specific type. One preliminary tool for representing shot locations is the shot chart, but it is rather rough as there is no clear-cut way of defining "similarity", which calls for the need for more rigorous statistical modeling. Various tools have been proposed to model point patterns. Among them, spatial point processes is a family of models that assume event locations are random, and realized from an underlying process, which has an intensity surface. Spatial point processes have a wide range of variants, the most prominent of which being the Poisson process (Geyer, 1998), the Gibbs process (Goulard et al., 1996), and the log-Gaussian Cox process (LGCP;Møller et al., 1998). They also have a wide range of applications, including ecological studies, environmental sciences (Jiao et al., 2020b;Hu et al., 2019), and sports analytics. Reich et al. (2006) developed a multinomial logit model that incorporates spatially varying coefficients, which were assumed to follow a heterogeneous Poisson process. Miller et al. (2014) discussed creating low-dimensional representation of players' shooting habits using several different spatial point processes. These works, however, focus mainly on characterizing the shooting behavior of individual players. Which players are similar to each other, however, remains un-answered by these works. Towards this end, Jiao et al. (2020a) proposed a marked point process joint modeling approach that takes into account both shot locations and outcomes. The fitted model parameters are grouped using ad hoc approaches to identify similarities among players. This, however, still does not answer the question "how many clusters can these players be grouped into". In a study of tree locations, Jiao et al. (2020b) proposed a model-based clustering approach that incorporates the Chinese restaurant process (CRP; Ferguson, 1973) to account for the latent grouped structure. The number of clusters is readily inferred from the number of unique latent cluster labels. Yin et al. (2020b) improved the model of Jiao et al. (2020b) by using Markov random fields constraint Dirichlet process for the latent cluster belongings, which effectively encourages local spatial homogeneity. Hu et al. (2020) used LGCP to obtain the underlying intensity, and then defined a similarity measure on the intensities of different players, which was later used in a hierarchical model that employed mixture of finite mixtures (MFM; Miller and Harrison, 2018) to perform clustering. Yin et al. (2020a) proposed a Bayesian nonparametric matrix clustering approach to analyze the latent heterogeneity structure of estimated intensity surfaces. Note that in all five works, the intensity function always played a certain role, which adds another layer of modeling between the shots and the grouping structure. One natural way to model the counts directly without employing the intensity surface is the Poisson regression. Zhao et al. (2020) proposed a spatial homogeneity pursuit regression model for count value data, where clustering of locations is done via imposing certain spatial contiguity constraints on MFM. Data of basketball shots, however, poses more challenges. The first challenge comes from the fact that only few shots are made by players in the region near the half court line, which means there is a large portion of the court that corresponds to no attempts. Secondly, existing approaches either only perform clustering on the spatial domain (Zhao et al., 2020;Yin et al., 2020b), or utilize the intensity surface and cluster players in terms of shooting habits . Thirdly, to demonstrate its superiority over heuristic comparison and grouping, a model-based approach should have favorable theoretical properties such as consistent estimation for the number of clusters and clustering configurations. To tackle the three challenges, we propose a Bayesian zero-inflated Poisson (ZIP) regres-sion approach to model field goal attempts of players with different shooting habits. The contribution of this paper is three-fold. First, the large proportion of the court with zero shot attempts is accommodated in the model structure by zero inflation. Next, non-negative matrix factorization is utilized to decompose the shooting habits of players into linear combinations of several basis functions, which naturally handles the homogeneity pursuit on the spatial dimension. On the dimension across players, we for the first time introduce a MFM prior in ZIP model to jointly estimate regression coefficients and zero inflated probability and their clustering information. Finally, we provide both theoretical and empirical justification through simulations for the model's performance in terms of both estimation and clustering. The rest of the paper is organized as follows. In Section 2, we introduce the motivating data from 2017-2018 regular season. In Section 3, we first review zero inflated Poisson regression, and then propose our Bayesian clustering method based on MFM. Details of Bayesian inference are presented in Section 4, including the MCMC algorithm and post MCMC inference methods. Simulation studies are conducted in Section 5. Applications of the proposed methods to NBA players data are reported in Section 6. Section 7 concludes the paper with a discussion. Motivating Data Our data consists of both made and missed field goal attempt locations from the offensive half court of games in the 2017-2018 National Basketball Association (NBA) regular season. The data is available at http://nbasavant.com/index.php, and also on GitHub (https: //github.com/ys-xue/MFM-ZIP-Basketball-Supplemental). We focus on players that have made more than 400 field goal attempts. Also, players who just started their careers in the 2017-2018 season, such as Lonzo Ball and Jayson Tatum, are not considered. A total of 191 players who meet the two criteria above are included in our analysis. We model a player's shooting location choices and outcomes as a spatial point pattern on the offensive half court, a 47 ft by 50 ft rectangle, which is the standard size for NBA. The spatial domain for the basketball court is denoted as D ∈ [0, 47] × [0, 50]. We partition the court to 1 ft × 2 ft blocks, which means that there are in total 47 × 25 = 1175 independent blocks in the basketball court. The shot charts for five selected players are visualized in values. This abundance of 0's motivates the usage of zero-inflated models for such type of data. Here we define y = (y 1 , y 2 , . . . , y n ) where y i = (y i1 , y i2 , . . . , y iJ ) for i = 1, 2, . . . , 191 and J = 1175. Each y i,j for i = 1, . . . , 191 and j = 1, . . . , J represents the total number of shots made by the i-th player in the j-th block. For selected players, we counted the number of blocks that have no shot, one shot, two shots, . . ., and more than six shots as presented in Table 1. One thing that can be noticed that Clint Capela has the most number of blocks corresponding to zero shots, which is straightforward as he is center, and barely shoots out of the painted area. LeBron James, on the other hand, has the least number of blocks with no shots, which indicates his wide shooting range. Except for Capela, the four other players have non-trivial, positive number of blocks corresponding to 1, 2, 3, 4, 5, and 6+ shots, indicating that they are comfortable shooting from a larger range. James Harden in particular, has 45 blocks with 6+ shots, indicating that he has the largest range of "hotspots". Methodology Zero Inflated Poisson Regression In this section, we briefly discuss the zero-inflated Poisson distribution (ZIP). There are some models are capable of dealing with excess zero counts (Mullahy, 1986;Lambert, 1992), and zero-inflated models are one type of them. Zero-inflated models are two-component mixture models that combine a count component and a point mass at zero with a count distribution such as Poisson, geometric or negative binomial (see, Cameron and Trivedi, 2005, for a discussion). We denote the observed count values of the i-th player as y i = (y i1 , y i2 , . . . , y i,1175 ) . Hence, the probability distribution of the ZIP random variable y ij can be written as, pr(y ij = κ) =        ρ i + (1 − ρ i ) exp{−µ ij } if κ = 0 (1 − ρ i ) µ y ij ij exp{−µ ij } y ij ! if κ > 0 ,(1) where ρ i is the probability of extra zeros and µ ij is the mean parameter of Poisson distribution. It can be seen from (1) that ZIP reduces to the standard Poisson model when ρ i = 0. Also, we know that pr(y ij = 0) > exp{−µ ij } indicates zero-inflation. The mean parameter µ ij is linked to the explanatory variables through log links as log (µ ij ) = x j β i , where x j is a vector of covariates x j = (1, x 1,j , . . . , x p,j ) and β i = (β 0i , β 1i , . . . , β pi ) are the corresponding regression coefficients including the intercept β 0i . Non-Negative Matrix Factorization for Spatial Basis To NMF is a dimensionality reduction technique that assumes a matrix Λ can be approximated by the product of two low-rank matrices representations, which can be combined to represent the whole story (Lee and Seung, 1999). Λ ≈ W B, where the matrix Λ ∈ R N ×V + is composed of N data points of length V , In our application, with data for the 359 players as a 359 × 1175 matrix, we use the R package NMF (Gaujoux and Seoighe, 2010) to obtain K = 5 bases, each of which correspond to a certain shot type, as illustrated in Figure 2. These five bases correspond to, respectively, top of key threes, long two-pointers, restricted area, wing and corner threes and perimeter shots. A player's shooting habit can be approximated by a weighted combination of these bases. Nevertheless, looking at the scales, one can see that the maximum value is over 400, and the values are highly skewed. Natural logarithm of the basis values is taken to reduce extreme values, and a normalization step is subsequently performed so that values in each basis vector have mean 0 and standard deviation 1. When such normalized basis functions are used as covariates in modeling individual player's shots, their corresponding coefficients, or weight vectors, can be regarded as a characterization for the shooting style of a player. ZIP with Clustered Regression Coefficients Consider a zero inflated Poisson regression model with varying coefficients as follows, y i ∼ ZIP exp x j β i , ρ i , i = 1, . . . , n,(2) where β i is a p+1-dimensional regression coefficients. From Gelfand et al. (2003), a Gaussian process prior can be assigned on regression coefficients to obtain varying patterns. Assuming the existence of clusters in β and ρ i , and denote the true cluster label for player i as z i , the parameters for cluster i as β z i and ρ z i , then the model in (2) can be rewritten as y i ∼ ZIP exp x j β z i , ρ z i , i = 1, . . . , n,(3) where z i ∈ {1, . . . , k}, with k being the total number of clusters. One popular way to model the joint distribution of z 1 , . . . , z k is the Chinese restaurant process (CRP; Blackwell et al., 1973), which defines a series of conditional distributions, also know as the a Pólya urn scheme, as: P (z i = c | z 1 , . . . , z i−1 ) ∝        |c|, at an existing table labeled c α, if c is a new table ,(4) where |c| is the number of elements in cluster c. Despite its favorable property as a method to simultaneously estimate the number of clusters and the clustering configuration, it has proved to produce extraneous clusters in the posterior, even when the number of sample size goes to infinity, which renders the estimation for number of clusters inconsistent (Miller and Harrison, 2018). A slowed-down version of the CRP in terms of producing new tables, mixture of finite mixtures (MFM; Miller and Harrison, 2018) is proposed to mitigate this problem: k ∼ p(·), (π 1 , . . . , π k ) | k ∼ Dir(γ, . . . , γ), z i | k, π ∼ k h=1 π h δ h , i = 1, . . . , n,(5) where p(·) is a proper probability mass function on {1, 2, . . . , } and δ h is a point-mass at h. Compared to the CRP, the introduction of new tables is slowed down by the factor V n (t + 1)/V n (t), which allows for a model-based pruning of the tiny extraneous clusters. The coefficient V n (t) is precomputed as V n (t) = +∞ n=1 k (t) (γk) (n) p(k), where k (t) = k(k − 1) . . . (k − t + 1), and (γk) (n) = γk(γk + 1) . . . (γk + n − 1). The conditional distributions of z i , i = 2, . . . , n under (5) can be defined in a Pólya urn scheme similar to CRP: P (z i = c | z 1 , . . . , z i−1 ) ∝        |c| + γ, at an existing table labeled c V n (t + 1)/V n (t)γ, if c is a new table ,(6) with t being the number of existing clusters. Adapting MFM to our model setting for clustering, the model and prior can be expressed hierarchically as: y i ∼ ZIP exp x j β z i , ρ z i , i = 1, . . . , n, β h ∼ N (0, Σ 0 ) , h = 1, . . . , k, ρ h ∼ U (0, 1) , h = 1, . . . , k, z i | k, π ∼ k h=1 π h δ h , (π 1 , . . . , π k ) | k ∼ Dir(γ, . . . , γ), k ∼ p(·),(7) where p(·) is a Poisson(ψ) distribution truncated to be positive through the rest of the paper, which has been proved by Miller and Harrison (2018) Theoretical Property In this section, we study the theoretical property of MFM-ZIP. We assume that the parameter space Θ * is the compact parameter space for all the model parameters (i.e., mixture weights, regression coefficients and zero inflated probability) given a fixed number of clusters. The mixing measure is G = k i=1 π i δ θ i , where δ is the point mass measure, and θ i = {β i , ρ i } is the collection of regression coefficients and zero inflation probability in cluster i for i = 1, . . . , k. Let K 0 , G 0 , P 0 be the true number of clusters, the true mixing measure, and the corresponding probability measure, respectively. Then the following proposition establishes the posterior consistency and contraction rate for the cluster number K and mixing measure G. The proof is based on the general results for Bayesian mixture models in Guha et al. (2020). Proposition 1 Let Π n (· | y 1 , . . . , y n ) be the posterior distribution obtained from given a random sample y 1 , . . . , y n . Assume that the parameters of interest are restricted to a compact space Θ * . Then we have Π n (K = K 0 | y 1 , . . . , y n ) → 1, and Π n (W (G, G 0 ) (log n/n) −1/2 | y 1 , . . . , y n ) → 1, almost surely under P 0 as n → ∞, where W is Wasserstein distance. Proposition 1 shows that our proposed Bayesian method is able to correctly identify the unknown number of clusters and the latent clustering structure with posterior probability tending to one as the number of observations increases. In order to prove the Proposition 1, we need to verify the conditions (P.1)-(P.4) in Guha et al. (2020) hold. Condition (P.1) is satisfied since we restrict our parameters of interest to a compact space Θ * and uniform distribution and multivariate normal distribution are first-order identifiable. Condition (P.2) also holds since we assign an non-zero continuous prior on β's and ρ's on the parameters within a bounded support. Uniform distribution and multivariate normal distribution is sufficient for Condition (P.3). Condition (P.4) holds since we choose a truncated Poisson distribution on q(·). The proof can be finished by using the results in Theorem 3.1 of Guha et al. (2020) Bayesian Inference For the hierarchical ZIP model with MFM introduced in (7), the set of parameters is denoted as Θ = {(β i , ρ i , z i , π, k) : i = 1, · · · , n}. If we choose (k − 1) ∼ Poisson(ψ) and γ = 1 in (5), the mixture weights π 1 , · · · , π k can be constructed following stick-breaking (Sethuraman, 1994) approximation: • Step 1. Generate η 1 , η 2 , · · · iid ∼ Exp(ψ), • Step 2. k = min{j : j k=1 η k ≥ 1}, • Step 3. π h = η h , for h = 1, · · · , k − 1, • Step 4. π k = 1 − k−1 h=1 π h . Prior for the hyperparameter ψ is Gamma(1, 1). With the prior distributions specified above, the posterior distribution of these parameters based on the data D = {y i , x j : i = 1, · · · , n, j = 1, · · · , p} is given by Another important task is do the posterior inference for clustering labels. We carry out posterior inference on the clustering labels based on Dahl's method (Dahl, 2006), which proceeds as follows, π(Θ | D) ∝ L(Θ | D)π(Θ) = n i=1 f (y i , x 1 , . . . , x j | β z i , ρ z i , z i )π(Θ),• Step 1. Define membership matrices A (t) = (A (t) (i, j)) i,j∈{1,...,n} = (1(z (t) i = Z (t) j )) n×n , where t = 1, . . . , T is the index for the retained MCMC draws after burn-in, and 1(·) is the indicator function. • Step 2. Calculate the element-wise mean of the membership matrices over MCMC draws A = 1 T T t=1 A (t) . • Step 3. Identify the most representative posterior A draw based on minimizing the element-wise Euclidean distance n i=1 n j=1 (A (t) (i, j) − A(i, j)) 2 among the retained t = 1, . . . , T posterior draws. The posterior estimates of cluster memberships z 1 , . . . , z n and other model parameters β's and ρ's can be also obtained using Dahl's method accordingly. Simulation Study Simulation Setup We have two scenarios in our simulation, balanced type and imbalanced type. We examine both the estimation for number of groups as well as congruence of group belongings with the true setting in terms of modulo labeling by Rand index (RI; Rand, 1971), the computation of which is facilitated by the R-package fossil (Vavrek, 2011). The RI ranges from 0 to 1 with a higher value indicating better agreement between a grouping scheme and the true setting. In particular, a value of 1 indicates perfect agreement. Simulation Results We run our algorithm with 7,000 MCMC iterations, with the first 2,000 iterations as burn-in for each replicate. The chain length has been examined to ensure convergence and stabilization. roceeding to 100 separate replicates of data, our proposed algorithm was run, and 100 RI values are obtained by comparing with the true setting. We calculate cover rate for both scenario. For each scenario, we also calculate the cover rate, which equals the percentage of replicates that our proposed algorithm accurately recovers the true number of clusters. The cover rate for each scenario are 98% and 93%, respectively. We also compare our method to K-means algorithm, high dimensional supervised classification and clustering (hdc; Bergé et al., 2012, R-package HDclassif ) and mean shift grouping. The mean shift algorithm is a steepest ascent classification algorithm, where classification is performed by fixed point iteration to a local maxima of a kernel density estimate. This method is originally from Fukunaga and Hostetler (1975), and an implementation in R can be found in the meanShiftR package (Lisic, 2018). Grouping recovery performances of all four methods are measured using the RI. As K-means and the mean shift algorithm cannot infer the number of clusters, such values need to be pre-specified, and we supply them with the number of clusters inferred by our method in each replicate. The clustering performances are compared in Table 2. In both designs, our proposed method have the highest RI, indicating its high accuracy in clustering. K-means and hdc have RI greater than 0.9, but the average performance is not as good as the proposed model. The mean shift algorithm, however, yields the worst performance. We provide parameter estimation in Table 3 Table 3. With high clustering accuracy as indicated by the RI, the estimated β for each cluster is close to its corresponding true value, which can be reflected by the small numerical values in the MAB column. The estimation performance is stable, in the sense that the MSD values are also small. The MCR under the balanced design fluctuate around its nominal value of 0.95, and under the imbalanced design, the values are overall lower due to the influence of mis-clustered players, but still remain close to or greater than 0.9. Real Data Application In this section, we apply the proposed method to the analysis of players' shot data in the 2017-2018 NBA regular season. Only the locations of shots are considered regardless of Several interesting observations can be made from Figure 3. It can be seen that each group of players has their own favorite shooting locations. Players in Group 1 make the most shots neat the hoop, which is confirmed by the regression coefficients in Table 4 Figure 3: Visualization of shooting patterns for four selected players from each group. Drummond and Dwight Howard are also centers who rarely leave the painted area. Players in Group 2 make the most shots beyond the three-point line, as they have the largest parameter estimates for the first and fourth basis function when compared to other groups. As shown in Figure 3, JJ Redick and Stephen Curry are both well-known shooters. A first look at the plots for Group 3 indicates that players in this group are able to make all types of shots, including three-pointers, perimeter shot, and also shots over the painted area. We find the players in this groups are often the leaders in their teams, and usually have the most posession. The parameter estimates also confirm the observation. Their β 0 is the largest among all groups, indicating an overall higher probability for making shots. Compared with Group 2, their shots are more evenly distributed, which can be reflected by the larger parameter estimate for the basis functions corresponding to areas within the three-point line. For Group 4, we find that most of their shots are close to the hoop and around the perimeter, and they have fewer shot around or beyond the three-point line. From the estimation result, the coefficient for the second and third basis functions are larger than other basis functions, and similar in value. In addition, their β 1 is the second smallest among all four groups while β 4 is the smallest, which again indicates their disfavor of shooting beyond the three-point line. Note that the presented analysis is based on 2017-2018 regular season, which was before Giannis Antetokoumpo increased his three-point shots in the 2019-2020 season. Discussion Based on theoretical justifications and empirical studies, our proposed methods successfully solve the three challenges raised in Section 1. Based on the results shown in Section 5, our proposed methods accurately estimates the parameters in the ZIP model and recovers the number of clusters and clustering configurations with different proportions of zero counts. Compared with several benchmark clustering methods such as K-means, high dimensional supervised classification and clustering, and mean shift grouping, our methods have higher clustering concordance without any tuning steps. In the analysis of the NBA shot charts data, four field goal attempt patterns, their corresponding zero inflation probabilities, and regression coefficients of each group are identified. The results provide valuable insights to players, coaches, and managers. The players can obtain more descriptive analysis of their current offense patterns, and hence develop customized training plans with pertinence; the coaches can organize their offense and defense strategies more efficiently for different opponents; the mangers will make better data-informed decisions on player recruiting and trading during offseason. There are several possible directions for further investigation. Spatial correlation over the court is accounted for nonparametrically by the basis functions. Considering either stationary or nonstationary model-based spatial correlation in our proposed modeling framework is a natural extension. In this paper, our posterior sampling is based on stick-breaking representations. Developments of more scalable inference algorithms (e.g., variational inference) are critical for large scale data. Finally, building a heterogeneity learning model with auxiliary information from different players, such as age and position on the court, merits future research from both methodological and applied perspectives. Figure 1 . 1The numbers of shot attempts in each of the blocks are counted. Hence, this data consists of non-negative, highly skewed sequence counts with a large proportion of zeros, as most shots are made in the range from the painted area to the three-point line, and many of the blocks between the three-point line and mid-court line have no corresponding positive Figure 1 : 1Shot charts for selected NBA players. Figure 2 : 2capture shot styles of individual players, followingJiao et al. (2020a), we construct spatial basis functions using historical data. Shot data for a total of 359 players who made over 100 shots in regular season 2016-2017 is used as input. First, kernel density estimation is employed to estimate the shooting frequency matrix λ = (λ 1 , . . . , λ 1175 ) for each individual player. Similar checking of empirical correlation between the kernel density values on blocks on the court is performed as inMiller et al. (2014), and the existence of long-range correlations in non-stationary patterns motivates the usage of a basis construction method that Visualization of basis functions obtained by NMF for K = 5. Each basis function represents the intensity function of a particular shot type.captures such long range correlation via global spatial patterns. This need motivates the usage of non-negative matrix factorization (NMF;Sra and Dhillon, 2005) in our modeling effort. ;Geng et al. (2019) to guarantee consistency for the mixing distribution and the number of groups, γ = 1, δ h denotes the Dirac measure, and Σ 0 is hyperparameter for base distribution of β's. We refer to the hierarchical model above as MFM-ZIP. where π(Θ) is the joint prior for all the parameters. Due to the unavailability of the analytical form for the posterior distribution of Θ, we employ the MCMC sampling algorithm to sample from the posterior distribution, and then obtain the posterior estimates of the unknown parameters. Computation is facilitated by the nimble(de Valpine et al., 2017) package in R (R Core Team, 2013). The implementation code is given in supplementary materials at https://ys-xue.github.io/MFM-ZIP-Basketball-Supplemental/. A total of 75 players are separated to three different groups for each type. Under the balanced design, each group contains equal number of players. Under the imbalanced design, the group sizes are 10, 35 and 30, respectively. The spatial domain is the same as for the motivating data in Section 2. We generate data {y i,j ; ∀i = 1, . . . , 75; j = 1, . . . , 1175} from ZIP model with different mean parameter and probability parameter of extra zeros. In our simulation setting, our covariates contains an intercept term and five basis function terms (see Section 3.2). Different values of coefficient β are used: , corresponding to each cluster respectively. We set the true probability parameter of extra zeros for each cluster to be (0.1, 0.3, 0.4). . For each of the three β's, the average parameter estimate denoted by β ,m ( = 1, . . . , 75; m = 1, . . . , 6) in 100 simulations is calβ ,m,r denotes the posterior estimate for the m-th coefficient of player in the r-th replicate. We use different metrics to evaluate the posterior performance. Those metrics including the mean absolute bias (MAB), the mean standard deviation (MSD), the mean of mean squared error (MMSE) and mean coverage rate (MCR) of the 95% highest posterior 1 { β ,m,r ∈95% HPD interval} , where 1 {} denotes the indicator function. The four metrics for each β under the balanced and imbalanced designs are presented in 's positions on the court (e.g., point guard, power forward, etc.). We run 15,000 MCMC iterations and the first 5,000 iterations as burn-in period. The result from the MFM-ZIP model suggests that the 191 players are to be classified into four groups. The sizes of the four groups are 29, 110, 48 and 4 respectively. We visualize the shot attempt counts made by four selected players on blocks of the court in Figure 3. The players for each group are shown in Section 3 of the supplementary materials. Table 1 : 1Number of blocks corresponding to number of shots for selected players.Observed Value Davis James Harden Walker Capela 0 836 788 853 837 1123 1 161 212 149 159 25 2 88 91 69 82 8 3 38 35 31 35 2 4 18 16 23 17 4 5 9 8 5 9 0 6+ 25 25 45 36 13 the basis matrix B ∈ R K×V + is composed of K basis vectors, and the weight matrix W ∈ R N ×K + is composed of the N non-negative weight vectors that scale and linearly combine the basis vectors to reconstruct Λ. The matrices W and B are obtained by minimizing certain divergence criteria (e.g. Kullback-Leibler divergence, or Euclidean distance), with the constraint that all matrix elements remain non-negative. With the non-negativity restriction for both the weight vectors and basis vectors, NMF eliminates redundant cases where negative bases "cancel out" positive bases. The basis left is often more sparse, and focuses on partial Table 2 : 2Comparison of clustering performance for the proposed method and three other competing approaches.Type Cover Rate RI mfm RI kmeans RI hdc RI meanshift Balanced 98% 0.9955 0.9364 0.9344 0.6757 Imbalanced 93% 0.9836 0.9581 0.9735 0.6126 density (HPD) intervals in the following ways: MAB = 1 75 75 =1 1 100 Table 3 : 3Performance of parameter estimates under the two true cluster designs.Type MAB MSD MMSE MCR Table 4 : 4Performance of parameter estimates under the real data. as their coefficients for the third and fifth basis functions are the largest. Clint Capela and DeAndre Jordan, for example, are both good at making alley-oops and slam dunks. 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[ "Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields", "Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields" ]
[ "Marzieh Eidi *[email protected] \nMax-Planck Institut for Mathematics in the Sciences\nLeipzigGermany\n", "Jürgen Jost \nMax-Planck Institut for Mathematics in the Sciences\nLeipzigGermany\n\nSanta Fe Institute\nSanta FeNew MexicoUSA\n" ]
[ "Max-Planck Institut for Mathematics in the Sciences\nLeipzigGermany", "Max-Planck Institut for Mathematics in the Sciences\nLeipzigGermany", "Santa Fe Institute\nSanta FeNew MexicoUSA" ]
[]
We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the Z 2 homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.
10.1007/s40304-022-00314-6
[ "https://arxiv.org/pdf/2105.02567v2.pdf" ]
233,864,905
2105.02567
b3fb48b5d57e01bf452a61dc1829923af09c80d7
Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields September 22, 2021 Marzieh Eidi *[email protected] Max-Planck Institut for Mathematics in the Sciences LeipzigGermany Jürgen Jost Max-Planck Institut for Mathematics in the Sciences LeipzigGermany Santa Fe Institute Santa FeNew MexicoUSA Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields September 22, 2021 We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the Z 2 homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field. Introduction One of the key ideas of modern geometry is to extract topological information about an object from a dynamical process operating on that object. For that purpose, one needs to identify the invariant sets and the dynamical relations between them. The invariant sets generate groups, and the dynamics defines boundary operators, and when one has shown that these operators square to zero, one can then define homology groups. The first such ideas may be seen in the works of Riemann, Cayley and Maxwell in the 19th century. In 1925, Morse [23] developed his famous theory where he recovered the homology of a compact Riemannian manifold M from the critical points of a smooth function f , assuming that these critical points are all non-degenerate. The dynamics in question is that of the gradient flow of f . The basic invariant sets then are precisely the critical points of f . The theory was analyzed and extended by Milnor, Thom, Smale, Bott and others. In particular, Bott [4] extended the theory to the case where the gradient of f is allowed to vanish on a collection of smooth submanifolds of M . Based on ideas from supersymmetry, Witten constructed an interpolation between de Rham and Morse homology. Floer [10] then developed the very beautiful idea that the boundary operator in Morse theory can be simply obtained from counting gradient flow lines (with appropriate orientations) between critical points of index difference one. The first systematic exposition of Floer's ideas was given in [24] (see also [20]). The main thrust of Floer's work was devoted to infinite dimensional problems around the Arnold conjecture, see [7,8,9,12], because for his theory, he only needed relative indices, and not absolute ones, so that the theory could be applied to indefinite action functionals. But also in the original finite dimensional case, Floer's theory advanced our insight considerably and motivated much subsequent work. In fact, Floer [10] had been motivated by another beautiful theory relating dynamics and topology, that of Conley [5] (for more details, see [6] and for instance the presentations in [28,19]). Conley's theory applies to arbitrary dynamical systems, not just gradient flows. Actually, Smale [26] had already extended Morse theory to an important and general class of dynamical systems on compact Riemannian manifolds, those that besides isolated critical points are also allowed to have non-degenerate closed orbits. Similar to Morse functions, the class of Morse-Smale dynamical systems is structurally stable, that is, preserves its qualitative properties under small perturbations. It turns out, however, that these systems can also be subsumed under Morse-Bott theory. In fact, in [27] Smale proved that for every gradient-like system there exists an energy function that is decreasing along the trajectories of the flow, and Meyer [22] generalized this result to the Morse-Smale dynamical systems and defined a Morse-Bott type energy function based on the flows. Such an energy function would then be constant on the periodic orbits, and they can then be treated as critical submanifolds via Morse-Bott theory. Banyaga and Hurtubise [1,2,18] then constructed a general boundary operator for Morse-Bott systems that put much of the preceding into perspective (see also the detailed literature review in [18]). There is still another important extension of Morse theory, the combinatorial Morse theory of Forman [15] on simplicial and cell complexes. Here, a function assigns a value to every simplex or cell, and certain inequalities between the values on a simplex and on its facets are required that can be seen as analogues of the non-degeneracy conditions of Morse theory in the smooth setting. As shown in [3], this theory recovers classical Morse theory by considering PL triangulations of manifolds that admit Morse functions. This theory has found various practical applications in diverse fields, such as computer graphics, networks and sensor networks analysis, homology computation, astrophysics, neuroscience, denoising, mesh compression, and topological data analysis. (For more details on smooth and discrete Morse theory and their applications see [21,25]). In [13], Forman also extended his theory to combinatorial vector fields. In the first chapter of this part, we extend Floer's theory into the direction of Conley's theory. More precisely, we shall show that one can define a boundary operator by counting suitable flow lines not only for Morse functions, but also for Morse-Smale dynamical systems, and in fact, we more generally also allow for certain types of homoclinic orbits in the dynamical system. Perhaps apart from this latter small extension, our results in the smooth setting readily follow from the existing literature. One may invoke [22] to treat it as a Morse-Bott function with the methods of [18]. Alternatively, one may locally perturb the periodic orbits into heteroclinic ones between two fixed points by a result of Franks [17] and then use [27] to treat it like a Morse function. In some sense, we are also using such a perturbation. Our observation then is that the collection of gradient flow lines between the resulting critical points has a special structure which in the end will allow us to directly read off the boundary operator from the closed (or homoclinic) orbits and the critical points. It remains to develop Conley theory in more generality from this perspective. Moreover, our approach also readily extends to the combinatorial situation of [15,13]. Again, it is known how to construct a Floer type boundary operator for a combinatorial Morse function, and an analogue of Witten's approach had already been developed in [14]. Our construction here, however, is different from that of that paper. In fact, [14] requires stronger assumptions on the function than the Morse condition, whereas our construction needs no further assumptions. What we want to advocate foremost, however, is that the beauty of Floer's idea of counting flow lines to define a boundary operator extends also to dynamical systems with periodic orbits, in both the smooth and the combinatorial setting, and that a unifying perspective can be developed. We should point out that in this part and for both of the next chapters, we only treat homology with Z 2 coefficients. Thus, we avoid having to treat the issue of orientations of flow lines. This is, however, a well established part of the theory, see [11] or also the presentations in [24,20]. The structure of this paper is as follows: In Section 2 after reviewing basic notions, we introduce the generalized Morse-Smale systems and we define a boundary operator based on these systems by which we can compute the homology groups. In section 3 we turn to the discrete settings and present such a boundary operator for general combinatorial vector fields by which we can compute the homology of finite simplicial complexes. Both sections finish with some concrete examples of computing homology groups based on our Floer type boundary operators. 2 Generalized smooth Morse-Smale vector fields Preliminaries We consider a smooth m dimensional manifold M that is closed, oriented and equipped with a Riemannian metric whose distance function we denote by d. Let X be a smooth vector field on M and φ t : M −→ M be the flow associated to X. We first recall some basic terminology. For p ∈ M , γ(p) = ∪ t φ t (p) will denote the trajectory of X through p. Then for each p ∈ M we define the limit sets of γ(p) as α(p) := ∩ s≤0 ∪ t≤s φ t (p) ω(p) := ∩ s≥0 ∪ t≥s φ t (p). Definition 2.1. If f : M → M is a diffeomorphism, then x ∈ M is called chain recurrent if for any ε > 0 there exist points x 1 = x, x 2 , ..., x n = x (n depends on ε) such that d(f (x i ), x i+1 ) < ε for 1 ≤ i ≤ n. For a flow φ t , x ∈ M is chain recurrent if for any ε > 0 there exist points x 1 = x, x 2 , ..., x n = x and real numbers t(i) ≥ 1 such that d(φ t(i) (x i ), x i+1 ) < ε for 1 ≤ i ≤ n. The set of chain recurrent points is called the chain recurrent set and will be denoted by R(X). The chain recurrent set R(X) is a closed submanifold of M that is invariant under φ t . We can think of R(X) as the points which come within ε of being periodic for every > 0. A Morse-Smale dynamical system, as introduced by Smale [26], has the fundamental property that it does not have any complicated recurrent behaviour and the α and ω limit sets of every trajectory can only be isolated critical points p or periodic orbits O. Morse-Smale dynamical systems are the simplest structurally stable types of dynamics; that is, if X is Morse-Smale and X is a sufficiently small C 1 perturbation of X then there is a homeomorphism h : M → M carrying orbits of X to orbits of X and preserving their orientation. (Such a homeomorphism is called a topological conjugacy and we say that the two vector fields or their corresponding flows are topologically conjugate). Here, we shall consider a somewhat more general case where we allow for a certain type of homoclinic rest points and their homoclinic orbits. Definition 2.2. A periodic orbit of the flow φ t on M is hyperbolic if the tangent bundle of M restricted to O, T M O , is the sum of three derivative Dφ t invariant bundles E c ⊕ E u ⊕ E s such that: 1. E c is spanned by the vector field X, tangent to the flow. There are constants C, λ > 0, such that Dφ t (v) ≥ Ce λt v for v ∈ E u , t ≥ 0 and Dφ t (v) ≤ C −1 e −λt v for v ∈ E s , t ≥ 0 where . is some Riemannian metric. A rest (also called critical) point p for a flow φ t is called hyperbolic provided that T p M = E u ⊕ E s and the above conditions are valid for v ∈ E u or E s . The stable and unstable manifolds of a hyperbolic periodic orbit O, are defined by: W s (O) = {x ∈ M | d(φ t x, φ t y) → 0 as t → ∞ for some y ∈ O} and W u (O) = {x ∈ M | d(φ t x, φ t y) → 0 as t → −∞ for some y ∈ O}. And for a rest point and a homoclinic orbit, we define the stable and unstable manifolds analogously. Also the index of a rest point or a closed orbit is defined to be the dimension of E u . We denote an arbitrary point in a homoclinic orbit H by H 0 k where k is the index of the homoclinic orbit H and the homoclinic orbit itself is denoted by H 1 k as it is homeomorphic to a circle and therefore is one-dimensional. Similarly by O 0 k we mean an arbitrary point in a periodic orbit O of index k and by O 1 k we mean the orbit itself as a one dimensional structure, homeomorphic to a circle. In the following definition we extend the definition of Morse-Smale vector fields. 3. For each β i (p), 1 ≤ i ≤ k and each β i (O), k + 1 ≤ i ≤ l the stable and unstable manifolds W s (β i ) and W u (β i ) associated with β i intersect transversally. (Here, two such submanifolds intersect transversally if for every x ∈ W u (β i ) ∩ W s (β j ) we have: T x (M ) = T x W u (β i ) T x W s (β j ).) Note that the only difference between generalised Morse-Smale flows as defined here and standard Morse-Smale flows is the possible existence of homoclinic points and orbits. In the standard case, one simply excludes the second condition. Therefore a generalised Morse-Smale flow can be perturbed to a corresponding Morse-Smale flow where all of the homoclinic orbits β i (H), n + 1 ≤ i ≤ l, are substituted by periodic orbits β i (O). For any two distinct β i and β j in the above definition we consider W (β i , β j ) = W u (β i ) ∩ W s (β j ). Then based on the transversality condition, this intersection is either empty (if there is no flow line from β i to β j or a submanifod of dimension λ β i − λ β j + dim β i where the index of β k is denoted by λ β k [2]. In the second case the flow φ t induces an R-action on [22]). W (β i , β j ) = W u (β i ) ∩ W s (β j ). Let M (β i , β j ) = (W (β i , β j ))/R) The chain complex for generalized Morse-Smale vector fields Suppose X is a generalized Morse-Smale vector field over M . To motivate our construction, we first observe that by a slight extension of a result of Franks [17], we can replace every periodic or homoclinic orbit by two non-degenerate critical points, without changing the flow outside some small neighbourhood of that orbit. Lemma 2.5. Suppose φ t is a generalized Morse-Smale flow on an orientable manifold with a periodic or homoclinic orbit of index k. Then for any neighbourhood U of that orbit there exists a new generalized Morse-Smale flow φ t whose vector field agrees with that of φ t outside U and which has rest points t and t of index k + 1 and k in U but no other chain recurrent points in U . Proof. In [17] Franks proved that for a Morse-Smale flow φ t on an orientable manifold with a closed periodic orbit O of index k and a given neighbourhood U of O, there exists a new Morse-Smale flow φ t whose vector field agrees with that of φ t outside U and which has rest points q 1 and q 2 of index k and k + 1 in U but no other chain recurrent points in U . For the generalized Morse-Smale flow we note that each homoclinic orbit is by definition obtained in a continuous local bifurcation of a periodic orbit. Therefore if we use this bifurcation in the reverse direction and substitute again any such homoclinic orbit with its corresponding periodic orbit we can use Franks' argument for replacing all the periodic and homoclinic orbits with two rest points and two heteroclinic orbits between them. Remark 2.6. In the above figure, the qualitative features of the three cases outside the gray annulus are the same. In particular, we can bifurcate two heteroclinic orbits between two critical points (in the right) to get a homoclinic orbit and a homoclinic critical point (in the middle) by bringing the two critical points closer and closer and then bifurcate the homoclinic orbit to a periodic orbit (in the left). With this lemma, we can turn our flow into one that has only non-degenerate critical points. We could then simply utilize the Floer boundary operator for that flow. In fact, that motivates our construction, but we wish to define a Floer type boundary operator directly in terms of the periodic and homoclinic orbits and the critical points. Our simple observation is that a Floer boundary operator resulting from the replacement that Franks proposed, has some additional structure that is derived from the orbits that have been perturbed away. This allows for an arrangement of the flow lines between the critical points of the perturbed flow that leads to the definition of the boundary operator in the presence of those orbits. That is, we can read off the boundary operator directly from the relations between the orbits and the critical points without appealing to that perturbation, although the perturbation helps us to see why this boundary operator squares to 0. We define the Morse-Floer complex (C * (X), ∂) of X as follows. Let C k denote the finite vector space (with coefficients in Z 2 ) generated by the following set of rest points/orbits of the vector field: p k , O 0 k , O 1 k−1 , H 0 k , H 1 k−1 . The differential ∂ k : C k (X) −→ C k−1 (X) counts the number of connected components of M (β i , β j ) (mod 2) where β i and β j are isolated rest points p k or closed orbits (either homoclinic orbits H or periodic orbits O). Here, each such orbit, carrying topology in two adjacent dimensions, corresponds to two elements in the boundary calculus. More precisely, a periodic orbit O k of index k generates an element O 1 k in dimension k + 1 and an element O 0 k in dimension k, and analogously for the homoclinics. Thus, our boundary operator is: ∂p k = α(p k , p k−1 )p k−1 + α(p k , O k−2 )O 1 k−2 + α(p k , H k−2 )H 1 k−2 ∂O 0 k = α(O k , O k−1 )O 0 k−1 + α(O k , H k−1 )H 0 k−1 + α(O k , p k−1 )p k−1 ∂O 1 k−1 = α(O k−1 , O k−2 )O 1 k−2 + α(O k−1 , H k−2 )H 1 k−2 ∂H 0 k = α(H k , H k−1 )H 0 k−1 + α(H k , O k−1 )O 0 k−1 + α(H k , p k−1 )p k−1 ∂H 1 k−1 = α(H k−1 , H k−2 )H 1 k−2 + α(H k−1 , O k−2 )O 1 k−2 . In this definition, the sums extend over all the elements on the right hand side; for instance, the first sum in the first line is over all critical points p k−1 of index k − 1. α(p k , p k−1 ), similar to the classical Morse-Floer theory (where there is no closed orbit and therefore the vector field is, up to topological conjugacy, gradient-like), is the number of flow lines from p k to p k−1 . We observe some terms do not appear; for instance, we do not have terms with coefficients of the form α(p k , O k−1 ), nor of the form α(p k , H k−1 ). This will be important below in the proof of Thm. 2.8. The reason why such a term does not show up is that if there were a flow line from some p k to some O 0 k−1 , then there would also be a flow to the corresponding O 1 k−1 which comes from the same closed orbit. But O 1 k−1 and p k are the elements of the same C k , and by the Morse-Smale condition, there are no flow lines between critical elements of the same C k . Analogously for homoclinics. Remark 2.7. Note that in defining the chain complex and the corresponding boundary operator ∂ for X, we could first replace all the homoclinic orbits with bifurcated periodic orbits and present our definitions for the simpler case where all the closed orbits are periodic. Then we would have just three generators p k , O 0 k , O 1 k−1 for C k (X) . However here we choose not to do this to emphasize that we can construct the boundary operator also for homoclinic orbits as long as our operator is defined based on the flow lines outside the tubular neighbourhoods of orbits. Theorem 2.8. ∂ 2 = 0. In classical Morse-Floer theory, one assumes that there are only isolated critical points and no closed or homoclinic orbits, and therefore all the α coefficients in the definition of ∂ except the first one (in the first row) are zero; there to prove ∂ 2 = 0 one can then use the classification theorem of one dimensional compact manifolds where the number of connected components of their boundary mod two is zero (see [20]). Here as W (β i , β i−1 ) might have dimension bigger than one, the number of connected components of the boundary of compact two dimensional manifolds might vary. For our generalized Morse-Smale flows, however, we use Lemma 2.5 to replace any orbit of index k (both periodic O k and homoclinic H k ) by a rest point of index k and one of k + 1 which are joined by two heteroclinic orbits. When replacing H k , the resulting rest point of higher index can be taken to be the point h itself, which then will be no longer homoclinic. Proof. By the above replacement, we get a vector field Y which has no periodic and homoclinic orbits and is therefore gradient-like (up to topological conjugacy [16]). This Y has all the isolated rest points of X, two isolated rest points q up k and q down k−1 instead of every periodic orbit O k−1 of index k − 1 and two isolated rest points t up k and t down k−1 instead of every homoclinic orbit H k−1 of index k − 1. We note that all the critical points in Y are isolated and for each index k they can be partitioned into five different sets p k , t up k , t down k , q up k , q down k . This partitioning is possible because orbits and isolated rest points have pairwise empty intersection. The proof will now consist of the following main steps: 1. We define C k (Y ) to be the finite vector space (with coefficients in Z 2 ) generated by p k , q up k , q down k , t up k , t down k 2. We define a boundary operator ∂ and consequently a chain complex corresponding to (Y, C * (Y ), ∂ ). 3. and then we prove there is an isomorphism (chain map) ϕ * : C * (X) −→ C * (Y ). Since ϕ is an isomorphism we get our desired equality ∂ 2 = 0 as ∂ = ϕ −1 * ∂ ϕ * and ∂ 2 = ϕ −1 * ∂ 2 ϕ * 1. We note that in C k (Y ) , q up k comes from a periodic orbit of index k − 1 and q down k comes from the replacement of a periodic orbit of index k. Similarly t up k is obtained from replacing a homoclinic orbit of index k − 1 and t down k comes from a homoclinic orbit of index k. We define ∂ k : C k (Y ) −→ C k−1 (Y ) as follows: ∂ p k = α(p k , p k−1 )p k−1 + α(p k , q up k−1 )q up k−1 + α(p k , t up k−1 )t up k−1 ∂ q down k = α(q down k , q down k−1 )q down k−1 + α(q down k , t down k−1 )t down k−1 + α(q down k , p k−1 )p k−1 ∂ q up k = α(q up k , q up k−1 )q up k−1 + α(q up k , t up k−1 )t up k−1 ∂ t down k = α(t down k , t down k−1 )t down k−1 + α(t down k , q down k−1 )q down k−1 + α(t down k , p k−1 )p k−1 ∂ t up k = α(t up k , t up k−1 )t up k−1 + α(t up k , q up k−1 )q up k−1 These sums extend over all the elements on the right hand side and α is the number of gradient flow lines (mode 2) between the corresponding critical points. We want to prove ∂ 2 = 0 over C k (Y ) by equating ∂ with the boundary operator ∂ M of Floer theory which is of the form ∂ M (s k ) = α(s k , s k−1 )s k−1 for a gradient vector field and counts the number of gradient flow lines α (mod 2) between two rest points with relative index difference one, without any partitioning on the set of isolated rest points s k of index k. In our case, we have such a partitioning and therefore more refined relationships in the definition of ∂ . And then, for all the generators of C * (Y ), ∂ = ∂ M ; we show this equality for p k , q up k , t down k as for the other cases it can be similarly proved. If we consider such a partitioning on the set of rest points of our vector field we have: ∂ M (p k ) = α(p k , p k−1 )p k−1 + α(p k , q up k−1 )q up k−1 + α(p k , q down k−1 )q down k−1 + α(p k , t up k−1 ).t up k−1 + α(p k , t down k−1 )t down Comparing this formula with that of ∂ p k we see that we have two extra terms in the latter; as we have explained after the definition of ∂, the 3th and the 5th term are not present in the former case. To have ∂ M (q up k ) = ∂ (q up k ), the three coefficients α(q up k , q down k−1 ), α(q up k , t down k−1 ), α(q up k , p k−1 ) need to be zero. The first one is zero since there are exactly two gradient flow lines (heteroclinic orbits) from q up k to q down k−1 which correspond to replacement of an orbit O k−1 . We note that for the other q down k−1 coming from other orbits α is zero by definition of ∂ over C k (X) as otherwise in X we would have flow lines between two orbits of the same index which is not possible by the Morse-Smale condition. For the same reason, the second element is also zero since there is no flow line from q up k to t down k−1 . Also the last α is zero as otherwise there would be flow lines from a periodic orbit of index k − 1 to an isolated rest point with index k − 1 in X, again violating Morse-Smale. Finally ∂ M (t down k ) = ∂ (t down k ) if we show that α(t down k , t up k−1 ) and α(t down , q up k−1 ) are zero. If not, there would be two orbits in X with index difference two which are the boundaries of a cylinder, which is not possible. Therefore over C * (Y ), ∂ M = ∂ and hence ∂ 2 = 0 by classical Morse-Floer theory. 3. We now define ϕ * : C * (X) −→ C * (Y ). For 0 ≤ k ≤ m, we put ϕ * (p k ) = p k , ϕ * (O 0 k ) = q down k , ϕ * (O 1 k−1 ) = q up k , ϕ * (H 0 k ) = t down k , ϕ * (H 1 k−1 ) = t up k . ϕ * is an isomorphism by the above construction of the rest points of Y . To prove ϕ * is a chain map from C * (X) to C * (Y ), we should have ∂ ϕ * = ϕ * ∂. Here, we show this equality for one of the generators of C k (X) and for the others it can be similarly obtained. For O 1 k−1 we have: ϕ * ∂(O 1 k−1 ) = ϕ * α(O k−1 , O k−2 ).O 1 k−2 + α(O k−1 , H k−2 ).H 1 k−2 = α(q up k , q up k−1 ).q up k−1 + α(q up k , t up k−1 ).t up k−1 = ∂ q up k = ∂ ϕ * (O 1 k−1 ) Therefore ∂ ϕ * = ϕ * ∂ and since ∂ 2 = 0 and ∂ 2 = ϕ −1 * ∂ 2 ϕ * , ∂ 2 = 0 We can then define Z 2 Morse-Floer homology of M by putting for each k, 0 ≤ k ≤ m, H k (M, Z 2 ) = ker(∂ k ) image(∂ k+1 ) . Remark 2.9. Although here we do not treat orientations, we observe from the following figure that in the above equalities ϕ * preserves the parity of α as each connected component of M (O k−1 , O k−2 ) corresponds to exactly one gradient flow line from q up k to q up k−1 (and exactly one flow line from q down k−1 to q down k−2 ). Similarly the same happens when we consider connected components of M (O k−1 , H k−2 ). Computing Homology Groups of Smooth Manifolds We shall now illustrate the simple computation of Floer homology for some smooth vector fields. 1. Let the sphere S 2 be equipped with a vector field V which has two isolated rest points of index zero at the north (N) and the south (S) pole, and one periodic orbit O of index one on the equator. Then ∂ 0 N 0 = 0 = ∂ 0 S 0 but since N 0 + S 0 is in the image of ∂ 1 therefore we have a single generator for H 0 (M, Z 2 ). 2. If we reverse the orientation of flow lines in the previous example, the isolated rest points at the north and south pole will get index two and the index of the periodic orbit becomes zero. Therefore: 3. Consider S 2 with a vector field V which has two isolated rest points, at the north pole of index zero and at the south pole of index two, one orange homoclinic orbit H of index one and one yellow periodic orbit O of index zero. C 2 = (N 2 , S 2 ) C 1 = O 1 0 C 0 = O 0 0 ∂ 2 N 2 = O 1 0 = ∂ 0 S 2 and N 2 − S 2 is the generator for H 2 (M, Z 2 ). Also ∂ 1 O 1 1 = 0 but since O 1 1 is in the image of ∂ 2 itC 2 = S 2 , H 1 1 C 1 = H 0 1 , O 1 0 C 0 = O 0 0 , N 0 ∂ 2 S 2 = O 1 0 = ∂ 2 H 1 1 and therefore S 2 − H 1 1 is the only generator for H 2 (M, Z 2 ). ∂ 1 H 0 1 = α(H 1 , O 0 ).O 0 0 + α(H 1 , N 0 ).N 0 = O 0 0 + N 0 = 0 and therefore H 0 1 does not contribute to H 1 (M, Z 2 ). On the other hand, ∂ 1 O 1 0 = 0 but since O 1 0 is in the image of ∂ 2 it does not contribute to H 1 (M, Z 2 ) and therefore H 1 (M, Z 2 ) = 0. ∂ 0 N 0 = 0 = ∂ 0 O 0 0 but since O 0 0 + N 0 is in the im- age of ∂ 1 therefore we have just one generator for H 0 (M, Z 2 ). 4. Finally, a two dimensional Torus T 2 with a vector field V with two periodic orbits O 1 and O 0 : C 2 = O 1 1 C 1 = O 0 1 , O 1 0 C 0 = O 0 0 ∂ 2 O 1 1 = 2.O 1 0 = 0 therefore O 1 1 is the generator for H 2 (M, Z 2 ). ∂ 1 O 0 1 = 2.O 0 0 = 0 so O 0 1 is a genera- tor for H 1 (M, Z). Also Combinatorial Vector Fields Preliminaries Forman introduced the notion of a combinatorial dynamical system on CW complexes [13]. He developed discrete Morse theory for the gradient vector field of a combinatorial Morse function and studied the homological properties of its dynamic [15] . For the general combinatorial vector fields where as opposed to gradient vector fields, the chain recurrent set might also include closed paths, he studied some homological properties by generalizing the combinatorial Morse inequalities. It remains, however, to construct a Floer type boundary operator for these general combinatorial vector fields. We define a Morse-Floer boundary operator for combinatorial vector fields on a finite simplicial complex. With this tool we no longer need a Morse function to compute the Betti numbers of the complex. Combinatorial vector fields can be considered as the combinatorial version of smooth Morse-Smale dynamical systems on finite dimensional manifolds; here in contrast to the smooth case we cannot have homoclinic points and homoclinic orbits as here, we cannot have a continuous bifurcation between a pair of heteroclinic orbits and a closed one, and in particular none with a homoclinic orbit in the middle. We now recall some of the main definitions that Forman introduced. Let M be a finite CW complex of dimension m, with K the set of open cells of M and K p the set of cells of dimension p. If σ and τ are two cells of M , we write σ p if dim(σ) = p, and σ < τ if σ ⊆ τ where τ is the closure of τ and we call σ a face of τ . Suppose σ p is a face of τ p+1 , B a closed ball of dimension p + 1 and h : B → M the characteristic map for τ i.e., a homeomorphism from the interior of B onto τ . Definition 3.1. σ p is a regular face of τ p+1 if • h −1 (σ) → σ is a homeomorphism. • h −1 (σ) is a closed p-ball. Otherwise we say σ is an irregular face of τ . If M is a regular CW complex (such as a simplicial or a polyhedral complex) then all its faces are regular. Definition 3.2. A combinatorial vector field on M is a map V : K → K ∪ 0 such that • For each p, V (K p ) ⊆ K p+1 ∪ 0. • For each σ p ∈ K p , either V (σ) = 0 or σ is a regular face of V (σ). • If σ ∈ Image(V ) then V (σ) = 0. • For each σ p ∈ K p {u p−1 ∈ K p−1 | V (u) = σ} ≤ 1. To present the vector field on M for any σ ∈ K where V (σ) = 0 we usually draw an arrow on M whose tail begins at σ and extend this arrow into V (σ). Thus, for each cell σ p , there are precisely 3 disjoint possibilities: • σ is the head of an arrow (σ ∈ Image(V )). • σ is the tail of an arrow (V (σ) = 0). • σ is neither the head nor the tail of any arrow (V (σ) = 0 and σ ∈ Image(V ); In the last case we call such a σ p a zero or rest point of V of index p. Cells which are not rest points occur in pairs (σ, V (σ)) with dim V (σ) = dim σ + 1. From now on and for simplicity we restrict ourselves to the special case of simplicial complexes, instead of CW complexes. As the combinatorial version of closed periodic orbits in smooth manifolds we have the next definition: Definition 3.3. Define a V -path of index p to be a sequence γ : σ 0 p , τ 0 p+1 , σ 1 p , τ 1 p+1 , ..., τ r−1 p+1 , σ r p such that for all i = 1, ..., r − 1 : V (σ i ) = τ i and σ i = σ i+1 < τ i . A closed path γ of length r is a V -path such that σ 0 p = σ r p . Also γ is called non-stationary if r > 0. Forman showed that there is an equivalence relation on the set of closed paths by considering two paths γ and γ to be equivalent if γ is the result of varying the starting point of γ . An equivalence class of closed paths of index k will be called a closed orbit of index k and denoted by O k . Definition 3.4. We call an orbit O p twisted if in its corresponding closed path γ : σ 0 p , τ 0 p+1 , σ 1 p , τ 1 p+1 , ..., τ r−1 p+1 , σ r p = σ 0 p , there is at least one vertex of σ 0 p that does not attach to itself in σ r p (for an example see remark 3.10). Otherwise we call the orbit simple. Definition 3.5. The combinatorial chain recurrent set R(V ) for a combinatorial vector field V on M is defined to be the set of simplices σ p which are either rest points of V or are contained in some non-stationary closed path γ (γ must have index either p − 1 or p). The chain recurrent set can be decomposed into a disjoint union of basic sets R(M ) = ∪ i Λ i where two simplices σ, τ ∈ R(V ) belong to the same basic set if and only if there is a closed non-trivial V -path γ which contains both σ and τ . Forman proved that if there are no non-stationary closed paths, then V is the combinatorial negative gradient vector field of a combinatorial Morse function. However when V has closed paths, then it cannot be the gradient of a function. Subsequently he defined a combinatorial "Morse-type" function on K, called a Lyapunov function, which is constant on each basic set, and has the property that, away from the chain recurrent set, V is the negative gradient of f . Remark 3.6. This Lyapunov function can be considered as the combinatorial analogue of the Morse-Bott energy function which Mayer defined for Morse-Smale dynamical systems. The chain complex of combinatorial vector fields Forman obtained Morse-type inequalities based on the basic sets of V and showed that these sets control the topology of M [13]. In this section, we present a direct way of recovering the homology of the underlying complex from the chain recurrent set of a combinatorial vector field on M from our Floer type boundary operator; our main restriction is that the chain recurrent set should not have twisted orbits. Our operator acts on chain groups generated by the basic sets and counts the number of suitable V-paths between elements of the chain recurrent set. We consider V to be a combinatorial vector field on a finite simplicial complex M where R(V ) does not include twisted orbits. We define the Morse-Floer complex of V denoted by (M, C * (V ), ∂) as follows. Let C k denote the finite vector space (with coefficients in Z 2 ) generated by the set of rest points p k and closed orbits O k−1 of the vector field: p k , O 1 k−1 , O 0 k in which by O 1 k−1 we mean the whole closed orbit O k−1 of index k − 1 and by O 0 k we mean an arbitrary simplex with dimension k in the closed orbit O k . Similar to the smooth case, each such orbit carries topology in two adjacent dimensions, namely a closed orbit O k generates an element O 1 k in C k+1 and an element O 0 k in C k . We note that here, by definition of combinatorial vector fields, we do not have any V -path between the elements of the same C k ; but in order to get a Floer type boundary operator in the same way as in the smooth setting we have to exclude three different cases in our vector field; we assume: 1. There is no V -path from a face of a critical simplex p k to a (k − 1)-dimensional simplex in an orbit O k−1 . 2. There is no V -path from a face of a k-dimensional simplex of an orbit of index k − 1 to a critical simplex of dimension k − 1. 3. There is no V -path from a face of a k-dimensional simplex of an orbit of index k − 1 to a (k − 1)-dimensional simplex of another orbit of index k − 1. This will be used in the proof of Thm. 3.7. In the smooth setting, the excluded cases cannot occur because of the Morse-Smale transversality condition. To be able to define the combinatorial Floer-type boundary operator, we have to transfer the idea of the number of connected components of the moduli spaces of flow lines to our combinatorial setting. As we saw, the number of these components (mod 2) plays a key rule in the definition of the boundary operator in the smooth setting. In the sequel, for two simplices of the same dimension q and q , by q ⊥ q , we mean that q and q are lower adjacent, i.e., they have a common face. We have V -paths between closed orbits and rest points which make different following cases: For two orbits O k−1 and O k−2 we define the set V P (O k−1 , O k−2 ) as the set of all V-paths starting from the faces of (k − 1)-and k-dimensional simplices of O k−1 and go to respectively (k − 2)-and (k − 1)-dimensional simplices of O k−2 . If V P (O k−1 , O k−2 ) is non-empty, for O 1 k−1 and O 1 k−2 , we define the higher dimensional spanned set of V-paths in V P (O k−1 , O k−2 ), denoted by SV P (O 1 k−1 , O 1 k−2 ) to be {q ∈ K k , q ∈ Image(V ) | ∃γ ∈ V P (O k−1 , O k−2 ), q ∈ γ}. On this set we can then define a relation as follows. We say q and q in SV P (O 1 k−1 , O 1 k−2 ) are related (q ∼ q ) if q and q belong respectively to two V-paths γ : α 0 k−1 , ..., q k , ..., α r k−1 and γ : • Either α 0 k−1 and β 0 k−1 coincide (and therefore γ and γ are the same) or • α 0 k−1 ⊥ β 0 k−1 or • There is a sequences of k − 1 dimensional simplices θ 0 k−1 , ...θ z k−1 , where θ 0 k−1 , ...θ z k−1 are the faces of k dimensional simplices in O k−1 such that α 0 k−1 ⊥ θ 0 k−1 , β 0 k−1 ⊥ θ z k−1 and for each i, θ i k−1 ⊥ θ i+1 k−1 and θ i k−1 is the starting simplex of some γ ∈ V P (O k−1 , O k−2 ). It is straightforward to check that ∼ is an equivalence relation on SV P (O 1 k−1 , O 1 k−2 ). On the other side for two arbitrary simplices of dimension k − 1 and k − 2 in respectively O k−1 and O k−2 we consider the following equivalence relation( ∼ ) on SV P (O 0 k−1 , O 0 k−2 ) which is defined as follows: {q ∈ K k−1 , q ∈ Image(V ) | ∃γ ∈ V P (O k−1 , O k−2 ), q ∈ γ}. We say q and q in SV P (O 0 k−1 , O 0 k−2 ) are related (q ∼ q ) if there are w and w in SV P (O 1 k−1 , O 1 k−2 ) such that q < w and q < w and w ∼ w . By definition ∼ is also an equivalence relation on SV P (O 0 k−1 , O 0 k−2 ) and the number of its equivalence classes is exactly the number of equivalence classes of ∼ over SV P (O 1 k−1 , O 1 k−2 ) . For instance consider the following triangulation of the torus which has two closed orbits of index one and index zero, respectively shown by green and red arrows. Here, SV P (O 1 k−1 , O 1 k−2 ) is the set of all the two dimensional coloured simplices and based on the above equivalence relation, this set is partitioned into two sets of yellow and pink two dimensional simplices. Also SV P (O 0 k−1 , O 0 k−2 ) is the set of all marked (with cross sign) edges which is partitioned into two sets, represented by orange and purple signs. If for two orbits, O k−1 and O k−2 , V P (O k−1 , O k−2 ) is empty and some of the faces of O k−1 (faces of both k − 1 and k-dimensional simplices) coincide with k − 2 and k − 1 dimensional simplices in O k−2 , we say O k−2 is attached to O k−1 . In the tetrahedron shown below the bottom faces of the closed red orbit of index one, is the closed orbit of index zero with purple arrows: Also we could have V-paths, V P (p k , O k−2 ), from the faces of a critical simplex p k of index k which go to the k − 1 dimensional simplices of some orbit of index k − 2 ; we define the span set of these V-paths, denoted by SV P (p k , O 1 k−2 ) to be SV P (p k , O 1 k−2 ) := {q ∈ K k , q ∈ Image(V ) | ∃γ ∈ V P (p k , O k−2 ), q ∈ γ}. As above we can define an equivalence relation on this set in which the equivalence classes are obtained based on the following relation: q ∼ q if they belong respectively to two V-paths γ : α 0 k−1 , ..., q k , ..., α r k−1 and γ : Also for V on M , for some rest point p k and some closed orbit O k−2 , the faces of p k and k − 1 dimensional simplices in O k−2 might coincide; for instance in the above tetrahedron the faces of orange 2-d rest simplex coincides with the one dimensional simplices in the closed orbit of index zero with purple arrows. We consider this again as an attachment. In the third possible case, V-paths start from the faces of k-dimensional simplices of a closed orbit of index k, O k , and go to a rest simplex of index k − 1, p k−1 . We denote the set of such V-paths by V P (O k , p k−1 ) and we consider SV P (O 0 k , p k−1 ) := {q ∈ K k , q ∈ Image(V ) | ∃γ ∈ V P (O k , p k−1 ), q ∈ γ}. In this set we call two simplices q and q equivalent if either q and q coincide or q ⊥ q or we can find a sequence of simplices in SV P (O 0 k , p k−1 ) such as θ 0 k , ...θ z k , such that q ⊥ θ 0 k , q ⊥ θ z k and for each i, θ i k ⊥ θ i+1 k . Here we have to exclude p k−1 for determining lower adjacency of k-dimensional simplices in SV P (O 0 k , p k−1 ), namely if q ∩ q = p k−1 , they belong to different classes. As an example consider the following triangulation for the torus with four orange rest simplices, one of index two, two of index one and another one of index zero, and a closed red orbit of index one. Here, the edges marked by cross signs are the edges in SV P (O 0 1 , p 0 ), which is portioned into two pink and yellow marked edges. If V P (O k , p k−1 ) is empty, but O k and p k−1 have a non-empty intersection, we have another type of attachment. For an example of this case see the top critical vertex and the red O 1 in the above tetrahedron. Finally if we substitute O k (O 0 k ) in the third case by a rest point of index k, p k , we count the number of equivalence classes of SV P (p k , p k−1 ) := {q ∈ K k , q ∈ Image(V ) | ∃γ ∈ V P (p k , p k−1 ), q ∈ γ} where V P (p k , p k−1 ) is the set of all V-paths starting from the faces of p k and go to p k−1 by passing through k-dimensional simplices, based on the following relation: We say q and q in SV P (p k , p k−1 ) are related (q ∼ q ) if there is a V-path in V P (p k , p k−1 ) which includes both q and q . Therefore the number of equivalence classes here is the number of V-paths starting from the faces of p k which go to p k−1 . The differential ∂ k : C k (V ) −→ C k−1 (V ) counts the number of the above equivalence classes mod 2, denoted by α, and for the three types of attachments we consider α = 1. That is, ∂p k = α(p k , p k−1 )p k−1 + α(p k , O 1 k−2 )O 1 k−2 ∂O 0 k = α(O 0 k , O 0 k−1 )O 0 k−1 + α(O 0 k , p k−1 )p k−1 ∂O 1 k−1 = α(O 1 k−1 , O 1 k−2 )O 1 k−2 where the sums extend over all the elements on the right hand side; for instance the second sum in the first line is over all closed orbits O 1 k−2 of index k − 2. In Forman's discrete Morse theory where there is no closed orbit (and therefore the combinatorial vector field is the negative gradient of a discrete Morse function), α(p k , p k−1 ) is the number of gradient V-paths from the faces of the rest point p k of higher dimension to the rest point of lower dimension p k−1 (in this case all the coefficients in the above formula except the first coefficient in the first line are zero). Theorem 3.7. ∂ 2 = 0. To prove this theorem similarly to Theorem 2.8 in the smooth case, we introduce a procedure to replace any closed path of index p (correspondingly its orbit of index p, O p ) with a rest point of index p and one of index p + 1 which are joined by two gradient V-paths starting from the faces of a higher dimensional rest point and going to the lower dimensional rest point. We assume V has a finite number of simple closed non-stationary paths (orbits) and rest simplices. Choose arbitrarily one of these closed paths γ of index p, γ : σ 0 p , τ 0 p+1 , σ 1 p , τ 1 p+1 , ..., τ r−1 p+1 , σ r p = σ 0 p . γ is a sequence of p and (p + 1) dimensional simplices. Take one of the p + 1 dimensional simplices τ k p+1 where k = r − 1. (Note that for non-stationary closed paths such a τ k always exists). We consider the following two sets of the simplices of γ by preserving the orders in each of the sets: σ 0 p , ...., τ k−1 p+1 , σ k p , τ k p+1 and τ k p+1 , σ k+1 p , τ k+1 p+1 , ...., σ r p = σ 0 p where the union of the elements in these sets consists of all the simplices of γ and their intersection is the starting simplex of the closed path σ 0 p and the one of higher dimension that we took τ k p+1 . We keep the arrows in the second set as they are in γ and in the first set we reverse the direction of V-path from σ 0 p to τ k p+1 . Namely instead of a pair (σ s p , τ s p+1 ) in γ (for 0 ≤ s ≤ k) we will have (σ s p , τ s−1 p+1 ) in our vector field where the two simplices σ 0 p and τ k p+1 will no longer be the tail and head of any arrow; therefore by definition both of them become rest points and there is no other rest point in γ created in this process. We note that in this procedure we just change the arrows in O and the other pairs of the vector field (outside O) are not changed. 1. All the elements of the chain recurrent set of V are rest simplices and for each index k they can be partitioned into three different sets p k , q up k , q down k . Here, in contrast to the smooth case, orbits and rest points can have non-empty intersections; in particular for different types of attachments, the pairwise intersections are non-empty. However partitioning of rest simplices is possible since the indices of the rest simplices are the same as their dimensions and after converting orbits into two rest simplices, they will belong to different chain groups (in adjacent dimensions). 2. We define ∂ : C k (V ) −→ C k−1 (V ) as follows: ∂ p k = α(p k , p k−1 ).p k−1 + α(p k , q up k−1 ).q up k−1 ∂ q down k = α(q down k , q down k−1 ).q down k−1 + α(q down k , p k−1 ).p k−1 ∂ q up k = α(q up k , q up k−1 ).q up k−1 To prove ∂ 2 = 0 over C k (V ), we want to equate ∂ with the discrete Morse-Floer boundary operator ∂ M of a combinatorial gradient vector field of the form ∂ M (s k ) = α(s k , s k−1 )s k−1 . There we count the number of gradient V-paths α (mod 2) between two rest points of relative index difference one without any such partitioning on the set of rest simplices s k of index k. In our case where we have such kind of partitioning we should show that for all the generators of C * (V ), ∂ = ∂ M . After the preceding procedure, we have: ∂ M (p k ) = α(p k , p k−1 ).p k−1 + α(p k , q up k−1 ).q up k−1 + α(p k , q down k−1 ).q down k−1 ; comparing this formula with that of ∂ p k in the above formula we see that there is one extra term in the latter; because we exclude case (1) in our vector field, the third sum is not present in the former case. As in the previous discussion to have ∂ M (q up k ) = ∂ (q up k ), the following two coefficients should be zero: α(q up k , q down k−1 ), α(q up k , p k−1 ). In the first case if q up k and q down k−1 are coming from replacement of the same orbit O k−1 , we will have exactly two V -paths from the faces of q up k to q down k−1 and it is zero mud 2. If they are not obtained from replacement of the same orbit O k−1 , α is zero as otherwise in V we would have V -paths between two orbits of the same index which either contradicts the non-existence of V -paths between elements of the same chain group or violates our exclusion (3) on the vector field. On the other hand, the second α is zero as otherwise it violates our assumption (exclusion 2) on the vector field. 3. We define ϕ * : C * (V ) −→ C * (V ) as follows. For 0 ≤ k ≤ m, ϕ * (p k ) = p k , ϕ * (O 0 k ) = q down k , ϕ * (O 1 k−1 ) = q up k ϕ * is an isomorphism by the above partitioning method for the set of rest points of V . To prove ϕ * is a chain map from C * (V ) to C * (V ) we should have ∂ ϕ * = ϕ * ∂ Here, we show the equality for O 1 k−1 and for the other generators of C k (V ) it can be similarly obtained:. ϕ * ∂(O 1 k−1 ) = ϕ * ( α(O 1 k−1 , O 1 k−2 ).O 1 k−2 = α(q up k , q up k−1 ).q up k−1 = ∂ q up k = ∂ ϕ * (O 1 k−1 ) In the second equality, ϕ * preserves the parity of α(O 1 k−1 , O 1 k−2 ) since each equivalence class of SV P (O 1 k−1 , O 1 k−2 ) corresponds to exactly one gradient V-path from q up k to q up k−1 (and one gradient V-path from q down k−1 to q down k−2 ). Therefore ∂ ϕ * = ϕ * ∂ and since ∂ 2 = 0 and ∂ 2 = ϕ −1 * ∂ 2 ϕ * , ∂ 2 = 0 Remark 3.8. For the three types of attachments in the above equality, after replacing orbits with two rest simplices and two gradient V-paths between them, α(p k , q up k−1 ), α(q up k , q up k−1 ), α(q down k , q down k−1 ) and α(q down k , p k−1 ) are also one. For instance, in the left tetrahedron below, we have two orange rest simplices, one of index two B 2 at the bottom and one of index zero at the top T 0 and two red and purple closed orbits of indices one and zero. If we convert the red orbit into two rest simplices, marked with red crosses, one of index two R 2 and the other one of index one R 1 and similarly turn the purple orbit into two rest simplices, marked with purple crosses, one of index one P 1 and the other one of index zero P 0 (shown in the right figure), we have α(B 2 , P 1 ) = 1, α(R 2 , P 1 ) = 1, α(R 1 , P 0 ) = 1 and α(R 1 , T 0 ) = 1 (Also α(R 2 , R 1 ) = 0, α(P 1 , P 0 ) = 0). We can then define the Z 2 Morse-Floer homology of M , for each k, 0 ≤ k ≤ m by H k (M, Z 2 ) = ker(∂ k ) image(∂ k+1 ) In [13] Forman proved Morse inequalities for general combinatorial vector fields based on his combinatorial Morse type Lyapunov function. There the main components are rest points and orbits in basic sets. Here we want to present these inequalities in a much shorter way based on the idea of changing every orbit of index k − 1, O k−1 to two rest points of index k and k − 1 as above. The following result can be considered as the combinatorial version of what Franks proved for smooth Morse-Smale dynamical systems [17]. Theorem 3.9. Let V be a combinatorial vector field over a finite simplicial complex M with c k rest points of index k and A k orbits of index k. Then c k − c k−1 + .... ± c 0 + A k ≥ β k − βk − 1 + ... ± β 0 , where β k = dim H k (M, Z 2 ). Proof. We create a new vector field V over M by replacing each closed orbit with two rest points as above. Since there is no closed orbit in V , based on what Forman showed in [15], V is the gradient of some combinatorial Morse function on M . On the other hand, the indices of rest points do not change when turning V to V and V has c k rest points of index k where c k = c k + A k + A k−1 . Applying the Morse inequalities for gradient vector fields to c k in V gives us the desired inequalities. Remark 3.10. If a simplicial complex is obtained by triangulation of a non-orientable manifold, we might not get the correct (Z 2 ) homology groups when the chain recurrent set of our combinatorial vector field has non-stationary closed V-paths. However for computing the Z 2 homology, we can turn each orbit into two rest simplices and two V-paths between them as above to get a combinatorial gradient vector field on the simplex and use the classical discrete Floer-Morse theory. For instance, consider a triangulation of the Klein bottle which has two closed orbits represented by red and blue arrows of index one and zero, respectively, as shown in the left diagram below. We note that the red orbit is twisted. Here by turning orbits into two rest simplices and two V-paths between them we get the correct Z 2 -homology of the Klein bottle which is the same as the Z 2 -homology of the triangulated torus as Z 2 -homology cannot distinguish between orientable and non-orientable surfaces. Remark 3.11. Orientability of a simplicial complex is not a necessary condition for defining the Floer boundary operator for general combinatorial vector fields. For instance in the following diagram, we have a non-orientable simplex as each vertex is the intersection of three different edges. Here, the chain recurrent set has an orbit of index zero with red arrows, as well as three critical edges and one critical vertex in the middle. If we use theorem 3.7 to compute the Z 2 homology of M based on this vector field we get H 0 (M, Z 2 ) = 1 and H 1 (M, Z 2 ) = 3. Computing Homology Groups of Simplicial Complexes We now present the computation of Floer homology groups for some CW complexes. 1. Consider the tetrahedron as a symmetric triangulation of the Sphere S 2 , equipped with a vector field V which has two rest simplices, one of index (dimension) zero (p 0 ) (the vertex at the top corner) and another of index two τ 2 (the simplex at the bottom), shown in orange, and a closed red orbit O 1 of index one and one of index zero O 0 in purple. 3. Consider another combinatorial vector field on the triangulated torus where V has four orange rest simplices, one of index zero (p 0 ), a vertical edge ve 1 of index one, a horizontal edge he 1 of index one and one rest simplex τ 2 of index two and also a red orbit O of index one. We have C 2 = τ 2 , O 1 1 C 1 = ve 1 , he 1 , O 0 1 C 0 = (p o ) ∂ 2 τ 2 = 1.ve 1 + 1.he 1 = 0; also ∂ 2 O 1 1 = 0 as there is no orbit of index zero here. Therefore O 1 1 is the only generator for H 2 (M, Z 2 ). ∂ 1 ve 1 = 2.p 0 = 0 = ∂ 1 he 1 but since ve 1 + he 1 is in the image of ∂ 2 , ve 1 − he 1 is one generator for H 1 (M, Z 2 ). ∂ 1 O 0 1 = 2.p 0 = 0 and therefore O 0 1 is the other generator of H 1 (M, Z 2 ). ∂ 0 p 0 = 0 and it is the generator for H 0 (M, Z 2 ). 4. Finally to compute the Floer homology groups of the depicted cube, we consider a vector field V that has two (orange and yellow) rest simplices of index two at the top τ N 2 and at the bottom τ S 2 and three different orbits, one blue orbit (bO) 0 of index zero, one green orbit (gO) 0 of index zero and a red orbit (rO) 1 of index one. Acknowledgements The first author would like to thank Cédric De Groote, Parvaneh Joharinad and Rostislav Matveev for their enlightening comments/questions, on the first draft, which helped to improve the manuscript. Definition 2 . 3 . 23We call a smooth flow φ t on M generalised Morse-Smale if : 1. The chain recurrent set of the flow consists of a finite number of hyperbolic rest points β 1 (p),... β k (p) and/or hyperbolic periodic orbits β k+1 (O),... β n (O). 2. R(X) furthermore may have a finite number of homoclinic orbits β n+1 (H),... β l (H) that can be obtained via local bifurcation from hyperbolic periodic orbits β n+1 (O), ...., β l (O). be the quotient space by this action of the flow lines from β i to β j . Remark 2. 4 . 4In the definition of standard Morse-Smale flow the condition (1) above could be replaced by (1 ): All periodic orbits and rest points of the flow are hyperbolic and there exists a Morse-Bott type energy function (as defined by Meyer 0 since there is no closed orbit of index 0 and therefore O 1 1 is the only generator for H 2 (M, Z 2 ). ∂ 1 O 0 1 = α(O 1 , N 0 ).N 0 + α(O 1 , S 0 ).S 0 = N 0 + S 0 = 0 and therefore O 0 1 does not contribute to H 1 (M, Z 2 ) and H 1 (M, Z 2 ) = 0. does not contribute to H 1 (M, Z 2 ). Finally ∂ 0 O 0 0 = 0 and therefore O 0 0 is the only generator for H 0 (M, Z 2 ). generator for H 1 (M, Z 2 ) = 0. Finally ∂ 0 O 0 0 = 0 and therefore we have one generator for H 0 (M, Z 2 ).Remark 2.10. For a computation of the homology groups of the first two examples via Morse-Bott theory (after turning the periodic orbits into critical submanifolds of a gradient flow), see[2]. β 0 k− 1 1, ..., q k , ..., β s k−1 where α 0 k−1 and β 0 k−1 are faces of k-dimensional simplices of O k−1 and α r k−1 and β s k−1 are some k − 1 dimensional simplices in O k−2 such that one of the following situations happens: β 0 k−1 , ..., q k , ..., β s k−1 such that either α 0 k−1 and β 0 k−1 coincide or α 0 k−1 ⊥ β 0 k−1 or there is a sequence of k − 1 dimensional simplices θ 0 k−1 , ...θ z k−1 , where θ 0 k−1 , ...θ z k−1 are the faces of p k such that α 0 k−1 ⊥ θ 0 k−1 , β 0 k−1 ⊥ θ k − 1 elements of O k−2 . ) is zero; if not, we would have two closed orbits O and O in V such that the faces of O are connected to O by some V-paths and their indices differ by two which is not possible. Therefore on C * (V ), ∂ M = ∂ and ∂ 2 = 0 since by Morse-Floer theory for combinatorial gradient vector fields (∂ M ) 2 = 0. this is an attachment where the faces of τ 2 and edges in O 1 0 coincide. On the other hand, ∂ 2 O 1 1 is also equal to O 1 0 (another type of attachment) and thereforeτ 2 − O 1 1 is the only generator for H 2 (M, Z 2 ). not contribute to H 1 (M, Z 2 ). Also ∂ 1 O 1 0 = 0 but since O 1 0 is in the image of ∂ 2 , it does not contribute to H 1 (M, Z 2 ) and H 1 (M, Z 2 ) = 0 Finally ∂ 0 p 0 = 0 = ∂ 0 O 0 0 , but since O 0 0 + p 0 is in the image of ∂ 1 we have just one generator for H 0 (M, Z 2 ). 2.Let T 2 at right be a triangulation of the two dimensional torus equipped with a vector field which has two closed orbits O 1 and O 0 with green and red arrows. Since this case is actually a discrete version of example 4 in the previous section, we have analogous structures for the chain complexes and boundaries; SV P (all of the generators of C k for k = 0, 1, 2 contribute to the corresponding homology groups. therefore we have one generator for H 2 (M, Z 2 ).∂ 1 (rO) 0 1 = 1.(bO) 0 0 + 1.(gO) 0 0 = 0. ∂ 1 (bO) 1 0 = 0 = ∂ 1 (gO) 1 0 ,but since both (bO) 1 0 and (gO) 1 0 are in the image of ∂ 2 , we have no generator for H 1 (M, Z 2 ). Finally ∂ 0 (bO) 0 0 = 0 = ∂ 0 (gO) 0 0 , but as (bO) 0 0 + (gO) 0 0 is in the image of ∂ 1 we have one single generator for H 0 (M, Z 2 ). Proof. of theorem 3.7. If by the help of above procedure we replace all the closed paths (orbits) by two rest points whose indices (dimensions) differ by one we get a vector field V which has no closed path (orbit) and therefore there is a discrete Morse function on M whose gradient is V . V has all the rest points of V and two rest points q up k (the simplex of higher index) and q down k−1 (the simplex of lower index) instead of every orbit O k−1 of index k − 1. Then we have the following three steps to prove the theorem:1. We consider C k (V ) to be the finite vector space (with coefficients in Z 2 ) generated byp k , q up k , q down kwhere in this set q up k comes from an orbit of index k −1 and q down k comes from the replacement of an orbit of index k.2. we define a boundary operator ∂ and consequently a chain complex corresponding to (V , C * (V ), ∂ ).3. Then we prove there is an isomorphism (chain map) ϕ * : C * (V ) −→ C * (V ) .Since ϕ * is an isomorphism we get our desired equality ∂ 2 = 0 as ∂ = ϕ −1 ∂ and ∂ 2 = ϕ −1 ∂ 2 . Lectures on Morse homology. A Banyaga, D Hurtubise, KluwerA. Banyaga and D.Hurtubise. Lectures on Morse homology. Kluwer, 2004. Morse-Bott homology. Augustin Banyaga, David E Hurtubise, Transactions of the American Mathematical Society. 3628Augustin Banyaga and David E. Hurtubise. Morse-Bott homology. Transactions of the Amer- ican Mathematical Society, 362(8):3997-4043, 2010. Discrete Morse theory is at least as perfect as Morse theory. Bruno Benedetti, arXiv:1010.0548arXiv e-printsBruno Benedetti. Discrete Morse theory is at least as perfect as Morse theory. arXiv e-prints, page arXiv:1010.0548, October 2010. Raoul Bott, Morse theory indomitable. Publications Mathématiques de l'IHÉS. 68Raoul Bott. Morse theory indomitable. Publications Mathématiques de l'IHÉS, 68:99-114, 1988. Isolated invariant sets and the Morse index. C Conley ; Providence, R I , CBMS Reg. Conf. Ser. Math. 38AMSC. Conley. Isolated invariant sets and the Morse index. CBMS Reg. Conf. Ser. Math. 38, AMS, Providence, R.I., 1978. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. C Conley, E Zehnder, Comm. Pure Appl. Math. 37C. Conley and E. Zehnder. Morse-type index theory for flows and periodic solutions for Hamil- tonian equations. Comm. Pure Appl. Math., 37:207-253, 1984. An instanton invariant for 3-manifolds. A Floer, Comm.Math.Phys. 118A. Floer. An instanton invariant for 3-manifolds. Comm.Math.Phys., 118:215-240, 1988. Morse theory for Lagrangian intersections. A Floer, J.Diff.Geom. 28A. Floer. Morse theory for Lagrangian intersections. J.Diff.Geom., 28:513-547, 1988. A relative Morse index for the symplectic action. A Floer, Comm.Pure Appl.Math. 41A. Floer. A relative Morse index for the symplectic action. Comm.Pure Appl.Math., 41:393-407, 1988. Witten's complex and infinite dimensional Morse theory. A Floer, J. Diff. Geom. 30A. Floer. Witten's complex and infinite dimensional Morse theory. J. Diff. Geom., 30:207-221, 1989. Coherent orientations for periodic orbit problems in symplectic geometry. A Floer, H Hofer, Math. Z. 212A. Floer and H. Hofer. Coherent orientations for periodic orbit problems in symplectic geometry. Math. Z., 212:13-38, 1993. The unregularized gradient flow of the symplectic action. Andreas Floer, Communications on Pure and Applied Mathematics. 416Andreas Floer. The unregularized gradient flow of the symplectic action. Communications on Pure and Applied Mathematics, 41(6):775-813, 1988. Combinatorial vector fields and dynamical systems. R Forman, Mathematische Zeitschrift. 228R. Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, 228:629-681, 1998. Witten-Morse theory for cell complexes. R Forman, Topology. 37R. Forman. Witten-Morse theory for cell complexes. Topology, 37:945-979, 1998. Morse theory for cell complexes. Robin Forman, Advances in Mathematics. 1341Robin Forman. Morse theory for cell complexes. Advances in Mathematics, 134(1):90-145, 1998. Morse-smale flows and homotopy theory. J Franks, Topology. 18J. Franks. Morse-smale flows and homotopy theory. Topology, 18:199-215, 1979. Homology and Dynamical Systems. John M Franks, American Mathematical SocietyJohn M. Franks. Homology and Dynamical Systems. American Mathematical Society, 1982. Three approaches to Morse-Bott homology. David E Hurtubise, David E. Hurtubise. Three approaches to Morse-Bott homology, 2013. Dynamical Systems. J Jost, Springer-VerlagBerlin HeidelbergJ. Jost. Dynamical Systems. Springer-Verlag Berlin Heidelberg, 2005. Riemannian geometry and geometric analysis. J Jost, Springer7th ed.J. Jost. Riemannian geometry and geometric analysis. Springer, 7th ed., 2017. Morse theory. P Kevin, Knudson, WORLD SCIENTIFICKevin P Knudson. Morse theory. WORLD SCIENTIFIC, 2015. Energy functions for Morse Smale systems. K R Meyer, American Journal of Mathematics. 904K. R. Meyer. Energy functions for Morse Smale systems. American Journal of Mathematics, 90(4):1031-1040, 1968. Relations between the critical points of a real function of n independent variables. Marston Morse, Transactions of the American Mathematical Society. 273Marston Morse. Relations between the critical points of a real function of n independent variables. Transactions of the American Mathematical Society, 27(3):345-396, 1925. M Schwarz, Morse homology. Birkhäuser. M. Schwarz. Morse homology. Birkhäuser, 1993. Discrete Morse Theory. Nicholas A Scoville, American Mathematical SocietyNicholas A. Scoville. Discrete Morse Theory. American Mathematical Society, 2019. Morse inequalities for a dynamical system. Stephen Smale, Bulletin of the American Mathematical Society. 661Stephen Smale. Morse inequalities for a dynamical system. Bulletin of the American Mathe- matical Society, 66(1):43 -49, 1960. On gradient dynamical systems. Stephen Smale, Annals of Mathematics. 741Stephen Smale. On gradient dynamical systems. Annals of Mathematics, 74(1):199-206, 1961. Shock waves and reaction-diffusion equations. J Smoller, Springer2nd ed.J. Smoller. Shock waves and reaction-diffusion equations. Springer, 2nd ed., 1994.
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[ "Analyzing Text Representations by Measuring Task Alignment", "Analyzing Text Representations by Measuring Task Alignment" ]
[ "Cesar Gonzalez-Gutierrez [email protected] \nUniversitat Politècnica de Catalunya\nBarcelonaSpain\n", "Audi Primadhanty [email protected] \nUniversitat Politècnica de Catalunya\nBarcelonaSpain\n", "Francesco Cazzaro [email protected] \nUniversitat Politècnica de Catalunya\nBarcelonaSpain\n", "Ariadna Quattoni [email protected] \nUniversitat Politècnica de Catalunya\nBarcelonaSpain\n" ]
[ "Universitat Politècnica de Catalunya\nBarcelonaSpain", "Universitat Politècnica de Catalunya\nBarcelonaSpain", "Universitat Politècnica de Catalunya\nBarcelonaSpain", "Universitat Politècnica de Catalunya\nBarcelonaSpain" ]
[]
Textual representations based on pre-trained language models are key, especially in few-shot learning scenarios. What makes a representation good for text classification? Is it due to the geometric properties of the space or because it is well aligned with the task? We hypothesize the second claim. To test it, we develop a task alignment score based on hierarchical clustering that measures alignment at different levels of granularity. Our experiments on text classification validate our hypothesis by showing that task alignment can explain the classification performance of a given representation.
10.48550/arxiv.2305.19747
[ "https://export.arxiv.org/pdf/2305.19747v1.pdf" ]
258,987,308
2305.19747
52491ba1fa4663388ed5902ceb554d60477ec6a8
Analyzing Text Representations by Measuring Task Alignment Cesar Gonzalez-Gutierrez [email protected] Universitat Politècnica de Catalunya BarcelonaSpain Audi Primadhanty [email protected] Universitat Politècnica de Catalunya BarcelonaSpain Francesco Cazzaro [email protected] Universitat Politècnica de Catalunya BarcelonaSpain Ariadna Quattoni [email protected] Universitat Politècnica de Catalunya BarcelonaSpain Analyzing Text Representations by Measuring Task Alignment Textual representations based on pre-trained language models are key, especially in few-shot learning scenarios. What makes a representation good for text classification? Is it due to the geometric properties of the space or because it is well aligned with the task? We hypothesize the second claim. To test it, we develop a task alignment score based on hierarchical clustering that measures alignment at different levels of granularity. Our experiments on text classification validate our hypothesis by showing that task alignment can explain the classification performance of a given representation. Introduction Recent advances in text classification have shown that representations based on pre-trained language models are key, especially in few-shot learning scenarios (Ein-Dor et al., 2020;Lu et al., 2019). It is natural to ask: What makes a representation good for text classification in this setting? Is the representation good due to intrinsic geometric properties of the space or because it is well aligned with the classification task? The goal of this paper is to answer this question to better understand the reason behind the performance gains obtained with pre-trained representations. Our hypothesis is that representations better aligned with class labels will yield improved performance in few-shot learning scenarios. The intuition is simple: in this setting, the limited number of labeled samples will only provide a sparse coverage of the input domain. However, if the representation space is properly aligned with the class structure, even a small sample can be representative. To illustrate this, take any classification task. Suppose we perform clustering on a given representation space that results in a few pure clusters (with all samples belonging to the same class). Then, any training set that 'hits' all the clusters can be representative. Notice that there is a trade-off between the number of clusters and their purity. A well-aligned representation is one for which we can obtain a clustering with a small number of highly pure clusters. Based on this, we propose a task alignment score based on hierarchical clustering that measures alignment at different levels of granularity: Task Hierarchical Alignment Score (THAS). To test our hypothesis that task alignment is key we conduct experiments on several text classification datasets comparing different representations. Our results show that there is a clear correlation between the THAS of a representation and its classification performance under the few-shot learning scenario, validating our hypothesis and showing that task alignment can explain performance. In contrast, our empirical study shows that intrinsic geometric properties measured by classical clustering quality metrics fail to explain representation performance in the few-shot learning scenario. Our study suggests an answer to our main question: A good efficient representation (i.e. one that enables few-shot learning) is a representation that induces a good alignment between latent input structure and class structure. Our main contributions are: 1) We develop a score based on hierarchical clustering ( §2) that measures the extent to which a representation space is aligned with a given class structure and 2) We conduct an empirical study using several textual classification datasets ( §3) that validates the hypothesis that the best representations are those with a latent input structure that is well aligned with the class structure. Task Hierarchical Alignment Score We now present the Task Hierarchical Alignment Score (THAS) designed to measure the alignment between a textual representation and the class label for a given task. The idea is quite simple, in a good representation space, points that are close to each other should have a higher probability of belonging to the same class. Therefore, we could perform clustering of the points and obtain high purity clusters, where most points belong to the same class. We assume that we are given: a dataset S = {(x i , y i )} n i=1 of n labeled data points where x ∈ X is a text fragment and y ∈ Y its corresponding class label (e.g., a sentiment classification label) and a representation function r : X → R d mapping points in X to a d-dimensional representation space R d (e.g., a sparse bag-of-words). Our goal is to compute a metric τ (S, r) that takes some labeled domain data and a representation function and computes a real value score. Fig. 1 illustrates the steps involved in computing THAS. There are three main steps: 1) hierarchical clustering, 2) computing clustering partition alignments, and 3) computing the aggregate metric. In the first step, we compute the representation of each point and build a data dendrogram using hierarchical clustering. The data dendrogram is built by merging clusters, progressively unfolding the latent structure of the input space. Traversing the tree, for each level we get a partition of the training points into k clusters. In step 2, for each partition, we measure its alignment with the class label distribution producing an alignment curve as a function of k. Finally, we report the area under this curve. Algorithm 1 summarizes the whole procedure. Implementation details and performance information can be found in A.1. Hierarchical Clustering In the first step, we will consider the input points X = {x i | (x i , y i ) ∈ S} and the representation function r to obtain a representation of all points R = {r(x i ) | x i ∈ X}. We then apply Hierarchical Clustering (HC) to the points in R obtaining a dendrogram D = HC(R) = {P k } n k=1 that defines a set of n cluster partitions. Fig. 1 (left) shows a diagram of a Algorithm 1: THAS Input: Dataset S = {(x i , y i )} n i=1 , representation function r Output: τ (S, r) 1 Get representation: R = {r(x i ) | x i ∈ X} 2 Run Hierarchical Clustering: D = HC(R) = {P k } n k=1 3 Traverse the dendrogram: foreach partition P k ⊂ D do 4 Predict scores for all points: foreach point x i ∈ X in i = 1, . . . , n where r(x i ) ∈ C ⊂ P k do 5 Label prediction scores: foreach y ′ j ∈ Y in j = 1, . . . , |Y| doŶ k,i,j = s(x i , y ′ j ) 6 Partition alignment score: a(P k ) = AUC y + (Ŷ k , Y ) 7 Final aggregate metric: τ (S, r) = 1 n n k=1 a(P k ) dendrogram. The root of this tree is the whole set and, at the leaves, each point corresponds to a singleton. At intermediate levels, top-down branching represents set splitting. For each level k = 1, . . . , n of the dendrogram there is an associated clustering partition of the input points into k clusters P k = {C j } k j=1 . That is, for any particular level we have a family of k nonempty disjoint clusters that cover the representation R = k j=1 C j , where each representation point r(x) ∈ R is assigned to one of the k clusters. Partition Alignment Score We use the gold labels Y = {y i | (x i , y i ) ∈ S} to compute an alignment score a(P k ) for each partition P k ⊂ D. We compute it in two parts. First, for every point x ∈ X and label y ′ ∈ Y we compute a label probability score by looking at the gold label distribution of the cluster C to which the point belongs in the clustering partition: s(x, y ′ ) = 1 |C| #[y ′ ∈ C](1) where #[y ′ ∈ C] is the number of samples in cluster C with gold label y ′ . Intuitively, this assigns to a point x a label probability that is proportional to the distribution of that label in the cluster C. Second, we use the label probability scores of all dataset gold labels Y to compute a partition alignment score. We choose as a single metric the area under the precision-recall curve (AUC) because it has the nice property that it applies to tasks with both balanced and unbalanced class distributions. 1 More specifically, we compute the AUC of the target (positive) class y + ∈ Y of the dataset (more details in the experimental part in §3): pointsŶ k = {s(x i , y ′ j ) | x i ∈ X, y ′ j ∈ Y}a(P k ) = AUC y + (Ŷ k , Y )(2) Final Aggregate Metric: THAS Once we have an alignment score for every level of the hierarchical dendrogram, we are ready to define our final Task Hierarchical Alignment Score (THAS). Consider the alignment scoring function a applied to the partition corresponding to the lowest level of the dendrogram. The alignment score will be a(P n ) = 1 because every cluster in this partition is a singleton and therefore #[y ′ ∈ C] will be 1 for the gold label and 0 for any other label. At the other end, for the partition corresponding to the root of the dendrogram (where all points belong to a single cluster), the alignment score a(P 1 ) is the AUC corresponding to assigning to every point x ∈ X a prediction score for each label y ′ ∈ Y equal to the relative frequency of y ′ in Y . Consider now the alignment score as a function of the size of the partition. As we increase k we will get higher scores. A good representation is one that can get a high score while using as few clusters as possible. Instead of choosing a predefined level of granularity, we propose to leverage the alignment information across all levels. To achieve this, we consider the alignment score as a function of the number of clusters and measure the area under a(P k ). 2 We are ready to define our final metric: τ (S, r) = 1 n n k=1 a(P k )(3) Experimental Setup In this section we empirically study the correlation of few-shot learning performance with 1) THAS and 2) an unsupervised clustering quality metric. We We will compare the following representations: a sparse bags-of-words (BoW); BERT embeddings (Devlin et al., 2019) using two token average pooling strategies (BERT all and BERT cls ); GloVe (Pennington et al., 2014); and fastText (Bojanowski et al., 2017;Joulin et al., 2016). For further details, please refer to A.2. Few-Shot Performance vs. THAS Since the focus of these experiments is comparing representations, we follow previous work on probing representations and use a simple model (Tenney et al., 2019;Lu et al., 2019). More precisely, we use a linear max-entropy classifier trained with l2 regularization. To simulate a few-shot learning scenario, we create small training sets by selecting N random samples, from 100 to 1000 in increments of 100. For each point N in the learning curve we create an 80%/20% 5-fold cross-validation split to find the optimal hyper-parameters. We then train a model using the full N training samples and measure its performance on the test set. We repeat the experiment with 5 random seeds and report the mean results. As the evaluation metric, we use accuracy for the balanced datasets (IMDB and Sentiment140) and F1 for the imbalanced datasets (WikiToxic and CivilComments). We generate learning curves for each dataset and representation (A.3). To study the correlation between task alignment and few-shot learning performance, it is useful to have a single score that summarizes the learning curve: We use the area under the learning curve (ALC). Representations with a larger ALC perform better in the few-shot learning scenario. 3 We observe that BERT all is consistently the best representation followed by BERT cls and GloVe performing similarly. Representations based on word embeddings are better than the sparse baseline for all datasets, except for fastText which does not exhibit a consistent improvement. To test for correlation, we also computed THAS for each representation and dataset. (The corresponding curves can be found in A.3.) Since this metric is a measure of the alignment between a label distribution and an input representation, there is a THAS score per label. 4 In the classification tasks that we consider there is always a single target class (e.g., toxicity for WikiToxic). We measure the alignment score with respect to this class. Table 1 summarizes the results showing ALC (left) and corresponding THAS (center) for all representations and datasets. Overall, BERT all is the best representation for few-shot learning followed by GloVe and BERT cls . All the representations based on pre-trained word embeddings significantly outperform the baseline sparse BoW representation. THAS predicts accurately the relative ranking between representations and the larger gap between BERT all and the rest. Fig. 2 shows a scatter plot of THAS as a function of ALC (blue dots; each point corresponds to a dataset and representation). We compute the correlation coefficients, which are displayed in Table 2. We observe a clear positive correlation between the two metrics, providing sup- porting evidence for our main hypothesis that a good representation under few-shot learning is a representation that is well aligned with the classification task. Unsupervised Clustering Quality We now look at standard metrics of cluster quality and test if they can explain few-shot learning performance. We use the Davies and Bouldin (1979) index (DBI) to measure the quality of the cluster partitions at every level of the dendrogram. This metric measures the compactness of each cluster and their separation, with better cluster partitions scoring lower. Similar to the computation of THAS described in §2, we compute DBI as a function of the number of clusters k corresponding to each level of the dendrogram. As an aggregate metric, we calculate the area under these curves to obtain a single ADBI score. (The curves are shown in A.3.) The right side of Table 1 shows the results for the same datasets and representations used for THAS. GloVe induces the best clusters according to the ADBI metric. BERT all does not produce particularly good clusters despite being the strongest few-shot representation. Fig. 2 (red crosses) and Table 2 show that there is a low correlation between the two metrics. This suggests that the geometric properties of the clusters alone can not explain few-shot performance. Related Work Representation choice has recently gained significant attention from the active learning (AL) community (Schröder and Niekler, 2020;Shnarch et al., 2022;Zhang et al., 2017). Some work has attempted to quantify what representation is best when training the initial model for AL, which is usually referred to as the cold start problem (Lu et al., 2019). The importance of word embeddings has been also studied in the context of highly imbalanced data scenarios (Sahan et al., 2021;Naseem et al., 2021;Hashimoto et al., 2016;Kholghi et al., 2016). Most research conducted by the AL community on textual representations has focused on determining which representations lead to higher performance for a given task. However, our paper aims to investigate why a certain representation performs better in the few-shot scenario. Our work, focused on examining properties of various textual representations, is closely related to recent research on evaluating the general capabilities of word embeddings. Many studies are interested in testing the behavior of such models using probing tasks that signal different linguistic skills Conneau and Kiela, 2018;Marvin and Linzen, 2018;Tenney et al., 2019;Miaschi and Dell'Orletta, 2020). Others have targeted the capacity of word embeddings to transfer linguistic content (Ravishankar et al., 2019;Conneau et al., 2020). Looking at approaches that analyze the properties of representations directly, without intermediate probes, Saphra and Lopez (2019) developed a correlation method to compare representations during consecutive pre-training stages. Analyzing the geometric properties of contextual embeddings is also an active line of work (Reif et al., 2019;Ethayarajh, 2019;Hewitt and Manning, 2019). While these previous works focus on analyzing representation properties independently, without considering a specific task, our study investigates the relationship between representations and task labels. We conduct a comparison between this relationship and the unsupervised analysis of representation properties. Our work falls in line with broader research on the relationship between task and representation. Yauney and Mimno (2021) proposed a method to measure the alignment between documents and labels in a given representation space using a data complexity measure developed in the learning-theory community. In the computer vision area, Frosst et al. (2019) introduced a loss metric and investigated the entanglement of classes in the representation space during the learning process. Zhou and Srikumar (2021) proposed a heuristic to approximate the version space of classifiers using hierarchical clustering, highlighting how representations induce the separability of class labels, thereby simplifying the classification task. In contrast, our work specifically examines the few-shot performance and emphasizes the importance of unbalanced scenarios. We find that in these more realistic situations, the choice of representation plays a critical role, paving the way for advanced strategies in active learning. Conclusion In this paper, we asked the question: What underlying property characterizes a good representation in a few-shot learning setting? We hypothesized that good representations are those in which the structure of the input space is well aligned with the label distribution. We proposed a metric to measure such alignment: THAS. To test our hypothesis, we conducted experiments on several textual classification datasets, covering different classification tasks and label distributions (i.e. both balanced and unbalanced). We compared a range of word embedding representations as well as a baseline sparse representation. Our results showed that when labeled data is scarce the best-performing representations are those where the input space is well aligned with the labels. Furthermore, we showed that the performance of a representation can not be explained by looking at classical measures of clustering quality. The main insight provided in this work could be leveraged to design new strategies in active learning. The fact that good representations induce clusters of high purity at different granularities creates opportunities for wiser exploration of the representation space in an active manner. Similar to the work of Dasgupta and Hsu (2008), we could employ the data dendrogram to guide this exploration. Limitations In this paper, we focused on analyzing the properties of textual representations in the few-shot learning scenario. Its applicability to broader annotation scenarios could be presumed but is not supported by our empirical results. Our experimental setup is based on binary classification tasks using English datasets. While our approach is general and could be easily extended to multi-class scenarios, more work would be required to extend it to other more complex structured prediction settings such as sequence tagging. We see several ways in which this work could be extended. The most obvious extension consists of trying to generalize the notion of alignment to other tasks beyond sequence classification, such as sequence tagging. In this paper, we have used THAS to understand the quality of a given textual representation. However, since THAS is a function of a labeling and a representation, it could also be used to measure the quality of a labeling (Yan and Huang, 2018), given a fixed representation. For example, this might be used in the context of hierarchical labeling, to measure which level of label granularity is better aligned with some input representation. The goal of this paper was to provide an explanation for the success of pre-trained word embeddings for text classification in the few-shot learning scenario. We believe that with our proposed methodology we have successfully achieved this goal. However, it should be clear to the reader that we do not provide a method for picking the best representation, i.e. for model selection. This is because our analysis requires access to labeled data and if labeled data is available the best way to select a model will be via cross-validation. Figure 1 : 1Three-step process for computing THAS. use four text classification datasets with both balanced and imbalanced label distributions: IMDB (IM; Maas et al., 2011), WikiToxic (WT; Wulczyn et al., 2017), Sentiment140 (S1; Maas et al., 2011) and CivilComments (CC; Borkan et al., 2019). Figure 2 : 2Few-shot performance (ALC) vs. task alignment (THAS) and clustering quality (ADBI). (µ)ALC vs r p (p-value) r s (p-value) THAS 0.98 (< 10 −12 ) 0.99 (< 10 −17 ) and the IM WT CC S1 µ IM WT CC S1 µ IM WT CC S1 µ BERT all .84 .50 .32 .79 .61 .84 .67 .27 .75 .63 2.87 3.03 3.31 3.25 3.11 GloVe .80 .48 .26 .74 .57 .80 .63 .26 .73 .60 2.62 2.12 2.01 2.47 2.31 BERT cls .80 .48 .23 .74 .56 .80 .56 .22 .74 .58 2.81 2.97 3.15 2.92 2.96 fastText .75 .41 .18 .66 .50 .77 .57 .21 .71 .56 2.78 2.13 1.93 2.47 2.33 Table 1: Learning curve performance (ALC), task alignment (THAS), and unsupervised clustering quality (ADBI) for different representations and datasets. (Rows are sorted by average ALC.)Repr. ALC THAS ADBI BoW .76 .32 .11 .59 .45 .71 .50 .20 .68 .52 3.14 3.83 4.23 3.86 3.76 Table 2 : 2Pearson correlation coefficient (r p ) and Spearman's correlation coefficient (r s ) with the corresponding p-values for ALC vs. THAS and ALC vs. ADBI, and similar analysis for mean scores across all datasets. F1 could be a valid alternative, but this metric requires the validation of decision thresholds. We could consider weighting methods that neutralize uninformative areas in the curve. In particular, we could subtract the scores originating from a random clustering. However, this contribution is solely determined by the sample size and the prior distribution. As a result, it would not have any impact when comparing representations. Alternatively, we could have picked a single point but we believe that ALC provides a more robust measure of few-shot learning performance and allows for a more concise analysis.4 We could also aggregate the scores of different classes, for example taking the average of the scores over all labels. AcknowledgementsThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 853459. The authors gratefully acknowledge the computer resources at ARTEMISA, funded by the European Union ERDF and Comunitat Valenciana as well as the technical support provided by the Instituto de Física Corpuscular, IFIC (CSIC-UV). This research is supported by a recognition 2021SGR-Cat (01266 LQMC) from AGAUR (Generalitat de Catalunya).A AppendixA.1 THAS Implementation DetailsThe data dendrogram is obtained via hierarchical agglomerative clustering. More precisely, we use a bottom-up algorithm that starts with each sample as a singleton cluster and consecutively merges clusters according to a similarity metric and merge criterion until a single cluster is formed.We applyWard's (1963)method, which uses the squared Euclidean distance between samples and then minimizes the total within-cluster variance by finding consecutive pairs of clusters with a minimal increase. The clustering algorithm produces a list of merges that represent a dendrogram and can be traversed to generate a clustering partition for each value of k. It was implemented using Scikit-learn(Pedregosa et al., 2011)and NumPy(Harris et al., 2020).Expressed as a nearest-neighbor chain algorithm, Ward's method has a time complexity of O(n 2 )(Murtagh, 1983). THAS experiments have been performed using sub-samples of size 10K and averaged over 5 seeds. Using 32 CPUs and 16GiB of RAM, each agglomerative clustering took on average 3.3 minutes. Each task alignment curve took 3 minutes on average. In contrast, DBI curves took 7.8 hours on average.A.2 Experimental DetailsDatasets.Table 3shows the statistics of the datasets used in this paper. They were extracted from HuggingFace Datasets(Lhoest et al., 2021). For WikiToxic and CivilComments, we have applied a pre-processing consisting of removing all markup code and non-alpha-numeric characters.DatasetSize PriorTask IMDB 50K 50% sentiment WikiToxic 224K 9% toxicity Sentiment1401.6M 50% sentiment CivilComments 2M 8% toxic behav. Representations. The following is a detailed description of the text representations used in our experiments:BoW: this is a standard sparse term frequency bag-of-words representation. BERT representations were extracted using the HuggingFace Transformers library(Wolf et al., 2020)implemented in PyTorch(Paszke et al., 2019).Models. The parameters for max-entropy learning curves were validated using 5-fold crossvalidation and the results averaged over subsamples from 5 seeds.Fig. 3presents the curves used to compute the main results in §3. The left column contains the learning curves used to compute the few-shot learning performance of the different datasets and representations. The center column shows task alignment scores as a function of the number of clusters. THAS is computed as the area under these curves. The pre-trained word embeddings, in particular BERT, tend to achieve the best results. In the curves, they show higher values of alignment for a small number of clusters. The relative performance of the representations in the learning curves is paralleled in the task hierarchical alignment curves. 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[ "Breast Cancer Detection and Diagnosis: A comparative study of state-of-the-arts deep learning architectures", "Breast Cancer Detection and Diagnosis: A comparative study of state-of-the-arts deep learning architectures" ]
[ "Brennon Maistry \nSchool of Mathematics, Statistics, and Computer Science\nUniversity of KwaZulu-Natal\nKing Edward Avenue3201Pietermaritzburg Campus, Pietermaritzburg\n\nKwaZulu-Natal\nSouth Af-rica\n", "Absalom E Ezugwu [email protected] \nUnit for Data Science and Computing\nNorth-West University\n11 Hoffman Street2520PotchefstroomSouth Africa\n" ]
[ "School of Mathematics, Statistics, and Computer Science\nUniversity of KwaZulu-Natal\nKing Edward Avenue3201Pietermaritzburg Campus, Pietermaritzburg", "KwaZulu-Natal\nSouth Af-rica", "Unit for Data Science and Computing\nNorth-West University\n11 Hoffman Street2520PotchefstroomSouth Africa" ]
[]
Breast cancer is a prevalent form of cancer among women, with over 1.5 million women being diagnosed each year. Unfortunately, the survival rates for breast cancer patients in certain third-world countries, like South Africa, are alarmingly low, with only 40% of diagnosed patients surviving beyond five years. The inadequate availability of resources, including qualified pathologists, delayed diagnoses, and ineffective therapy planning, contribute to this low survival rate. To address this pressing issue, medical specialists and researchers have turned to domain-specific AI approaches, specifically deep learning models, to develop end-to-end solutions that can be integrated into computer-aided diagnosis (CAD) systems. By improving the workflow of pathologists, these AI models have the potential to enhance the detection and diagnosis of breast cancer. This research focuses on evaluating the performance of various cutting-edge convolutional neural network (CNN) architectures in comparison to a relatively new model called the Vision Transformer (ViT). The objective is to determine the superiority of these models in terms of their accuracy and effectiveness. The experimental results reveal that the ViT models outperform the other selected stateof-the-art CNN architectures, achieving an impressive accuracy rate of 95.15%. This study signifies a significant advancement in the field, as it explores the utilization of data augmentation and other relevant preprocessing techniques in conjunction with deep learning models for the detection and diagnosis of breast cancer using datasets of Breast Cancer Histopathological Image Classification.
10.48550/arxiv.2305.19937
[ "https://export.arxiv.org/pdf/2305.19937v1.pdf" ]
258,987,519
2305.19937
67caa4d18c1e5d550b9602572910ad5e184f3fe4
Breast Cancer Detection and Diagnosis: A comparative study of state-of-the-arts deep learning architectures Brennon Maistry School of Mathematics, Statistics, and Computer Science University of KwaZulu-Natal King Edward Avenue3201Pietermaritzburg Campus, Pietermaritzburg KwaZulu-Natal South Af-rica Absalom E Ezugwu [email protected] Unit for Data Science and Computing North-West University 11 Hoffman Street2520PotchefstroomSouth Africa Breast Cancer Detection and Diagnosis: A comparative study of state-of-the-arts deep learning architectures Breast CancerDeep LearningConvolutional Neural Networks, Vi- sion Transformers, Inception-v3, AlexNet, ResNet-18 Breast cancer is a prevalent form of cancer among women, with over 1.5 million women being diagnosed each year. Unfortunately, the survival rates for breast cancer patients in certain third-world countries, like South Africa, are alarmingly low, with only 40% of diagnosed patients surviving beyond five years. The inadequate availability of resources, including qualified pathologists, delayed diagnoses, and ineffective therapy planning, contribute to this low survival rate. To address this pressing issue, medical specialists and researchers have turned to domain-specific AI approaches, specifically deep learning models, to develop end-to-end solutions that can be integrated into computer-aided diagnosis (CAD) systems. By improving the workflow of pathologists, these AI models have the potential to enhance the detection and diagnosis of breast cancer. This research focuses on evaluating the performance of various cutting-edge convolutional neural network (CNN) architectures in comparison to a relatively new model called the Vision Transformer (ViT). The objective is to determine the superiority of these models in terms of their accuracy and effectiveness. The experimental results reveal that the ViT models outperform the other selected stateof-the-art CNN architectures, achieving an impressive accuracy rate of 95.15%. This study signifies a significant advancement in the field, as it explores the utilization of data augmentation and other relevant preprocessing techniques in conjunction with deep learning models for the detection and diagnosis of breast cancer using datasets of Breast Cancer Histopathological Image Classification. Introduction Breast cancer encompasses all cancers found in the breast, primarily in the epithelium of the ducts or lobules of the glandular tissue [1]. It is a metastatic disease that frequently spreads to other organs, such as the lungs, liver, and spine [2]. Unfortunately, breast cancer is often incurable. In 2020, the World Health Organization (WHO) reported 2.3 million women diagnosed with breast cancer worldwide, resulting in 685,000 deaths [1]. Additionally, it is projected that in 2022, the United States will witness 43,780 deaths from breast cancer, affecting 43,250 women and 530 men [3]. With over 1.5 million women diagnosed annually across the globe [2], breast cancer remains one of the most prevalent cancers among women. The treatment of breast cancer has shown high efficacy, boasting a five-year survival rate of 90% following diagnosis. However, this favorable probability is primarily achievable in higher-income countries like the United States [1,3]. In lower-income nations such as India and South Africa, the five-year survival rates plummet dramatically to 66% and 40% respectively [1]. This rapid decline can be attributed to limited resources, high costs of medical examinations, and extended waiting times for access to pathologists' services. Breast pathology analysis and cancer treatment decisions often rely on the skill level and experience of the pathologist, making the workflow tedious and subjective [4]. Histopathology, as defined by Robertson et al. [4], involves the examination of tissue specimens after fixation in formalin, paraffin embedding, and mounting thin histologic sections onto glass slides [4]. While mammography serves as the most popular noninvasive clinical screening technique for breast cancer diagnosis, it faces challenges such as reduced sensitivity influenced by breast density and limited effectiveness in women under the age of 40 [5]. Hence, histopathological image analysis remains the gold standard for breast cancer diagnosis and serves as the medical imagery used in this research. The introduction of whole-slide digital image scanners has transformed the pathologist's workflow into a digital process, enhancing the convenience of patient diagnostic analysis. Furthermore, numerous companies now harness the power of artificial intelligence (AI) to develop machine learning solutions that optimize the pathologists' workflow. IBM Watson Health, for instance, offers the "IBM Imaging Workflow Orchestrator with Watson," a cloud-based Software-as-a-Service (SaaS) program utilizing AI to assist radiologists in analyzing patient images and extracting associated dataset features [6]. However, a limitation of machine learning methods in the medical domain is the meticulous annotation required, which remains subjective to the annotator's domain knowledge. To optimize the pathologist's workflow using Computer-Aided Diagnosis (CAD), researchers have started exploring Deep Learning (DL) to overcome challenges encountered with machine learning models. Deep Learning, a subset of machine learning, emulates the learning process of the human brain [7]. By learning from examples, deep learning can extract necessary features based on input and output target classes. The independence of dependencies required by DL models validates their significance in constructing fully automated CAD systems. Convolutional Neural Networks (CNNs) have shown remarkable performance in breast cancer binary classification and feature extraction. In a study by researchers [8], a fine-tuned Inception-v3 model was employed to classify biopsy images as either benign or malignant tumors. This approach achieved an accuracy surpassing state-of-theart models at various levels of image magnification. Another study [9] utilized a pretrained ResNet50 CNN model for feature extraction and a Logistic Regression classifier to determine the malignancy of histopathological tumor images, achieving an accuracy of 93.27%. These results demonstrate the potential of CNNs for breast cancer diagnosis. However, researchers [10] raised a significant concern regarding the limited availability of medical data, which hampers the effective training of more complex CNN models like ResNet50. Moreover, shallower CNN models, such as AlexNet, tend to overfit smaller datasets. Although the Transformer architecture has gained prominence in Natural Language Processing (NLP), in 2020, Google researchers adopted the Transformer framework for computer vision, resulting in the Vision Transformer (ViT) model [11]. Initial findings indicate that ViT surpasses the state-of-the-art CNN models of that time. This groundbreaking research necessitates a comparison between the traditional Transformer architecture and its CNN counterparts. As advocated by authors [10], the size of the dataset significantly influences the effectiveness of CNN models used. Therefore, to develop effective Computer-Aided Diagnosis (CAD) systems, it is crucial to select the most suitable CNN architectures. Consequently, a comparative study was conducted to assess the performance of different CNN architectures in medical imagery and compare them to the recently introduced ViT model. This paper aims to employ deep learning techniques to automate breast cancer detection using histopathological images. Furthermore, it seeks to compare and evaluate the performance of AlexNet, Inception-v3, and ResNet18 CNN architectures against the ViT architecture. The technical contributions of this paper are as follows: i. Proposal of enhanced deep learning models using data augmentation techniques such as rotation and flipping, and k-fold cross-validation to assess the predictive capabilities of CNN models on unseen data. ii. Comparison of the performance of AlexNet, Inception-v3, and ResNet18 CNN architectures against the ViT architecture. The rest of this paper is organized as follows: Section 2 presents a literature review, highlighting previous work in this domain and identifying their limitations, thereby justifying the importance of this study. Section 3 outlines the methods and techniques employed to address the problem at hand. Section 4 discusses the experimental results and compares the models implemented in this paper with the findings from existing literature. Finally, Section 5 concludes the paper and suggests future extensions or improvements to this study. Literature Review A considerable amount of research has been conducted on CNN architectures for breast cancer detection, which has further extended to hybrid models incorporating machine learning classifiers and ensemble learning techniques to enhance classification accuracy. On the other hand, the utilization of ViT models in the medical domain is still in its early stages and not as extensively explored as CNN architectures in this field. This section provides an overview of some key techniques and findings in the existing literature. In their work, Senan et al. [12] proposed a system for detecting breast cancer using histopathological images by combining CNN with hierarchal voting applications. Their objective was to accurately classify images as benign or malignant tumors. The researchers employed the widely used BreakHis dataset [13] and implemented data augmentation and normalization techniques to address the challenges posed by limited training data for DL models. Transfer learning was applied to a CNN based on the AlexNet architecture, where the original classification layer (designed for classifying 1000 classes) was replaced with a binary classification-capable fully connected layer. The classification of images involved both patch-level and image-level voting applications. While this system achieved favorable results, with an accuracy of 95% and an Area Under Curve (AUC) of 99.36%, further investigation can be conducted by comparing the system against a standard CNN model using the AlexNet architecture, trained and tested on the data prepared by the proposed system. This analysis would provide insights into whether the performance improvements were due to the enhanced dataset preparation, the hierarchical voting applications, or a combination of both. Additionally, cross-validation can be employed to validate the model's performance more rigorously. In the study conducted by Achuthan et al. [14], the performance of the AlexNet architecture was compared with the VGG16 and VGG19 CNN architectures. All models were pre-trained, and a 5-fold cross-validation approach was employed. The results of the comparison indicated that the AlexNet model outperformed the other architectures, emerging as the best-performing model. The researchers further evaluated the AlexNet model in conjunction with different machine learning classifiers, including Decision Tree, Random Forest, and K-Nearest Neighbors. The model that yielded the highest accuracy (87.5%) utilized the AlexNet CNN for feature extraction combined with the Random Forest classifier, surpassing the standard AlexNet CNN by 1.5%. However, a limitation of this study was the relatively small dataset obtained from Kaggle, consisting of only 4000 images. Although data augmentation techniques were applied, the limited dataset size posed challenges for more complex models like VGG16 and VGG19, while the shallower nature of the AlexNet made it less prone to overfitting. To address this limitation, the authors suggest increasing the number of folds in crossvalidation and implementing image normalization to account for the small dataset size. In the research presented by Ranjan et al. [15], the AlexNet CNN was employed in conjunction with hierarchical CNN approaches for classification purposes. The researchers demonstrated the advantages of training the entire pre-trained AlexNet network rather than solely training the final connected layer. Training the full network resulted in a 6% improvement in accuracy, reaching 83% compared to 77% when training only the fully connected network. The classification approach explored two strategies. In the first approach, the AlexNet CNN determined whether an image was classified as "Normal" or "Rest," and if classified as "Rest," another CNN determined the specific class among "Invasive," "Insitu," or "Benign." The second approach utilized the AlexNet classifier to classify an image as "Normal" or "Rest," followed by the application of majority voting for three binary CNN models. This approach yielded better results, achieving an accuracy of 95%. Similar to the previous work, the limitation identified in this study was the dataset size. The researchers utilized the BACH Challenge 2018 dataset, which contained only 400 images. Additionally, no data augmentation was employed, highlighting the need for cross-validation to further validate the system's performance and the use of some form of data augmentation to reduce generalization error. In the study conducted by Benhammou et al. [8], preliminary performance results of a pre-trained Inception-v3 CNN model were presented. However, no data augmentation or preprocessing was applied to the BreakHis dataset, making it a preliminary result. The study also explored the DeCaf approach to transfer learning and used the Inception-v3 model for binary classification, classifying the images as "malignant" or "benign." The results showed the potential of Inception-v3 as an effective model, achieving an accuracy of 90.2% on images with 40x magnification. However, CNN models that were fed preprocessed and augmented data outperformed their approach, reaching an accuracy of 96.1%. This highlights the importance of preprocessing and data augmentation for achieving good performance metrics in a model. In the work conducted by Xiang et al. [17], an Inception-v3 CNN model was finetuned and modified for binary classification on the BreakHis dataset. This study emphasized the significance of cross-validation and data augmentation. The Inception-v3 model trained without data augmentation achieved an accuracy of 92.8%, but when cross-validation and data augmentation were applied, the performance increased by 2.9%, resulting in the best accuracy of 95.7%. These findings validate the importance of data augmentation and cross-validation when training CNN models on relatively smaller datasets, such as those found in medical imagery. In a recent study by Aljuaid et al. [18], the performance of pre-trained CNNs, including ResNet18, Inception-v3, and ShuffleNet, was compared on the BreakHis dataset. The dataset was split into 65% for training and 35% for testing. Median and Gaussian filters were utilized to remove noise from the images while preserving their original features. Data augmentation was also applied to reduce overfitting. All models were trained using the more recent, smaller ImageNet dataset and then subjected to transfer learning. The evaluation of the models revealed that the ResNet18 CNN model achieved the best results, with an accuracy of 99.7% for binary classification and an accuracy of 97.81% for multiclass classification. To further improve the study, producing generalized results for each model could provide a better understanding of how the model performs with unfamiliar data. This could be achieved by applying k-fold crossvalidation and a normalization technique for the images to obtain a more generalized performance of the model. A study conducted by Thomas et al. [19] compared the performance of state-of-theart CNN models with a ViT model in binary classification on the BreakHis dataset. The researchers utilized computer vision techniques to preprocess the images, including Adaptive Histogram Equalization, Multiscale Retinex with Color Restoration, and Median Filtering. Data augmentation techniques such as horizontal flipping and resizing to 224x224 were also applied. The study employed cross-validation as part of its training strategy, and the model achieved an outstanding accuracy of 96.7%. To further enhance this research, using normalized images and specialized techniques for histopathological images could be explored. Additionally, comparing the ViT model against end-to-end CNNs for classification could provide insights into the performance advantages or disadvantages of using a ViT model. Based on the reviewed literature, all models exhibited strengths and weaknesses based on the authors' data preprocessing and augmentation approaches. This paper aims to evaluate the highlighted DL models using standard preprocessing and augmentation techniques, while also implementing k-fold cross-validation. This comprehensive strategy allows for an effective evaluation of the DL models' performance in classifying histopathological images and accurately diagnosing breast cancer. Methods and Techniques This section outlines the methods and techniques employed in this research. The process begins with data collection, followed by preprocessing and augmentation, which includes rotation and flipping of the images. The augmented dataset is subsequently divided into 10 folds to facilitate training of the deep learning (DL) models using kfold cross-validation. After training the models, they are evaluated using unseen images from the dataset. Figure 1 provides an overview of the methodology. BreakHis Dataset For this study, the widely recognized BreakHis dataset [13] was utilized. This dataset consists of a total of 7,909 images obtained from 82 patients at four different magnification levels (40x, 100x, 200x, and 400x). The dataset is categorized into two types: benign and malignant lesions. Among the images, 2,480 correspond to benign lesions, while the remaining 5,429 represent malignant lesions. All the images possess a resolution of 460x700 pixels. The BreakHis dataset was developed in collaboration with P&D laboratories and has emerged as the standard dataset for deep learning-based binary classification of breast cancer. Figure 2 displays an example image of a malignant lesion at various magnification levels, and Table 1 provides a more detailed breakdown of the dataset. Data Preprocessing and Augmentation The removal of noise is a crucial task in image processing as it contributes to the generation of higher-quality images. In the field of pathology, one persistent challenge faced by the industry is the inconsistency in slide preparation. This inconsistency arises from variations in stain manufacturers, staining procedures, and even differences in the storage time of slides, all of which have an impact on the digital representation of histopathological images. To address this issue, Macenko et al. [21] introduced an algorithm that effectively normalizes histopathological slides, enabling their utilization for quantitative analysis. Algorithm 1 presents the algorithm proposed by Macenko et al. [21]. Algorithm 1: Step for the normalization of histopathological images [21] Input In previous attempts to develop machine learning algorithms for breast cancer, the manual crafting of meticulous features was necessary. However, the advent of Convolutional Neural Network (CNN) models, with their convolutional layers, has enabled effective identification of key image features based on texture and histology structures. This autonomous feature extraction capability comes at the cost of CNN models being highly dependent on large amounts of data, which poses a challenge in many medical domains due to the limited availability of medical datasets compared to datasets like ImageNet. To overcome the data scarcity issue, researchers have turned to data augmentation techniques. Data augmentation involves generating new images by applying transformations such as flipping, rotation, and cropping to existing ones. By employing this technique, researchers can significantly enhance the performance of deep learning models. This is supported by the findings of Zuluga-Gomez et al. [22], whose study demonstrated that effective data augmentation can result in a CNN model performing equally well as a model trained on a dataset that is 50% larger. In this study, the images are first normalized using the stain normalization algorithm proposed by Macenko et al. [21]. This algorithm, specifically designed for histopathological slides/images, was chosen over traditional image processing techniques. The data augmentation technique employed follows the approach outlined by Krishevsky et al. [23]. The training images are resized using Bilinear Interpolation, based on the input dimension of the CNN models, and random crops are taken along with their horizontal flips. For testing and validation, five crops (the four corners and the center) and their horizontal flips are used. Similar computational procedures are implemented in this study, aligning with the approach of Krishevsky et al. [23]. The image transformation process is performed on the CPU using Python code, while the models are trained on the GPU. The data is transferred to the GPU only after being inserted into batches CNN Architectures For this research, the CNN models adopted include AlexNet, Inception-v3 and Res-Net18. AlexNet Model. The groundbreaking AlexNet model [23], introduced by Alex Krishevsky et al. in 2012, had a profound impact on the adoption of CNN models for computer vision tasks. One of the notable contributions of AlexNet was its ability to train on multiple GPUs, which revolutionized the field. Additionally, the model replaced traditional activation functions like the tanh function with Rectified Linear Units (ReLU), enabling faster training times compared to using tanh. The model employed GPU parallelization by distributing its kernels across multiple GPUs, with communication occurring only on specific kernels. Although this reduced training time in standard scenarios, Krishevsky et al. [23] mentioned that using GPU parallelization with cross-validation posed a challenge, requiring careful tuning of communication patterns. To address overfitting, the model incorporated overlapping of outputs within the pooling layer. The architecture of the AlexNet model consists of eight layers, with the first five layers being convolutional and the remaining three fully connected. In the original model, the final layer's output was passed through a Softmax function to generate a distribution over 1000 classes. Due to its relatively shallow structure, the AlexNet model generally outperforms more complex CNN models when working with smaller datasets. The first convolutional layer convolves an image of size 224x224x3 with 96 kernels, each having a size of 11x11x3. The output of the first layer is then filtered by the second layer using 256 kernels of size 5x5x48. The third, fourth, and fifth layers are connected, without normalization or pooling layers, and take the output of the second layer. The third layer consists of 384 kernels of size 3x3x256, the fourth layer also contains 384 kernels of size 3x3x192, and the fifth layer employs 256 kernels of the same size, 3x3x192. Each of the fully connected layers at the end of the model comprises 4096 neurons. Figure 3 provides a visual illustration of the AlexNet model. Fig. 3. Illustration of the AlexNet model [23]. Inception-v3 Model. Following the success of AlexNet in 2012, researchers dedicated their efforts to finding CNN models that could achieve even better performance in computer vision tasks. This pursuit led to the development of the VGGNet [24] and GoogLeNet [25] models in 2014. While these models delivered excellent results, they also had significant drawbacks. VGGNet, although simpler than the Inception architecture of GoogLeNet, required substantial computation time for evaluation. On the other hand, while GoogLeNet had fewer parameters (60 million) compared to VGGNet and AlexNet, its deep complexity posed challenges for scalability. To address these limitations, Szegedy et al. [26] proposed and implemented a novel method to optimize the Inception architecture. In the study conducted by Szegedy et al. [26], several improvements were made to the Inception architecture. These enhancements included factorizing larger convolutions into multiple convolutions with smaller spatial filters and employing spatial factorization through asymmetric convolutions. Additionally, a notable feature of the Inception architecture was the incorporation of an auxiliary classifier. Deep networks often encounter the vanishing gradient problem, and the auxiliary classifier was designed to mitigate this issue by facilitating the propagation of useful gradients to the lower layers of the network, thereby making them immediately beneficial. Figure 4 illustrates the improved Inception model, while Figure 5 depicts the inception module after the factorization of n×n convolutions. Figure 6 showcases the inception module with expanded filter bank outputs. An overview of the Inception-v3 architecture, as described by the authors in [26], can be found in Table 2. ResNet-18 Model. Through the works conducted by Szegedy et al. [25,26] and Simonyan et al. [24], it becomes apparent that deeper CNN models yield better results in image classification. This is attributed to the inherent ability of deeper networks to integrate low, middle, and high-level features with classifiers, thereby creating effective end-to-end models. While Szegedy et al. [26] presented a framework for scaling up networks, such as the Inception architecture that gave rise to the Inception-v3 model, the optimization problem still persists. As the network becomes deeper, the training error rate tends to increase. To address this challenge, researchers at Microsoft introduced residual learning to the architecture of CNNs, resulting in the creation of the ResNet architecture [27]. Their proposed solution involves utilizing the additional layers of a network for identity mapping, while the remaining layers form a trained shallower network. Shortcut connections, enabling residual learning, are implemented at regular intervals of convolutional layers to facilitate effective identity mapping. The ResNet architecture draws inspiration from the VGG architecture but employs fewer filters. Figure 7 provides an illustration of the ResNet-18 architecture. ViT Model The transformer architecture has emerged as the dominant architecture for Natural Language Processing (NLP) due to its self-attention mechanism. One advantage of this architecture is that it does not suffer from performance degradation as the model scales up, in contrast to CNNs. This was a key issue that motivated researchers at Microsoft to develop the ResNet architecture. However, when it comes to image recognition and classification, utilizing CNNs with self-attention poses challenges in terms of scalability to hardware accelerators. As a result, ResNets continue to be considered the gold standard for CNN-based image recognition and classification tasks. To address the challenge of self-attention in image classification, researchers at Google drew inspiration from NLP and introduced the Vision Transformer (ViT) architecture [11]. In their work, the image is divided into patches and treated similarly to word tokens in NLP. However, the researchers acknowledged that CNNs possess a stronger inductive bias. Nevertheless, by adopting the standard NLP procedure and fine-tuning a ViT model trained on a significantly larger dataset, this issue has been overcome. A typical ViT model takes embedded image patches as input, which are then processed by a Transformer encoder. The encoder consists of alternating layers of multiheaded self-attention blocks and multilayer perceptron (MLP) blocks. Each block is connected by a residual connection, and preceding each block is a Layernorm (LN) operation. Following the Transformer encoder is an MLP head for classification. By treating images as word tokens, the ViT architecture avoids introducing inductive bias specific to images. This enables scalability to handle datasets of any size, making ViT a promising approach for image classification in large-scale data scenarios. Figure 8 provides an overview of the ViT architecture as depicted by the authors in [11]. Transfer Learning As previously mentioned, traditional machine learning models required human annotation of features for effective classification. In the medical field, this approach offers no advantages to pathologists as it remains a laborious task, and the quality of features depends on the expertise of the pathologist. To address this, deep learning (DL) models, particularly CNN models, can autonomously extract features from images using their convolutional layers. However, CNNs typically demand large amounts of data, which are currently limited in the medical domain. To overcome this challenge, researchers employ transfer learning, a common practice in NLP. Transfer learning involves training a model on a large dataset and then fine-tuning it on smaller, specialized data to solve a specific problem. In computer vision tasks, the ImageNet dataset from 2012 has emerged as the primary choice for pretraining DL models. In this research, the DL models were pretrained on the ImageNet dataset and subsequently modified by replacing their classification layers with a fully connected layer that outputs two classes: benign or malignant. This approach allows leveraging the pre-existing knowledge learned from ImageNet to enhance the performance of DL models in classifying medical images. Results and Discussion In this study, an experiment was conducted using each magnification factor of the BreakHis dataset as an independent dataset. For each dataset, 80% of the data was allocated for training and validation, while the remaining 20% was kept separate for final evaluation of the model. The training strategy employed was k-fold cross-validation, with k set to 10. The pre-trained architectures were trained twice: one with full finetuning, updating all network weights, and the other by training only the final fully connected layer to assess the model's performance as a fixed feature extractor. The impact of batch sizes on the generalizability of CNNs for histopathological images was investigated in a study by Kandel et al. [28]. Their findings demonstrated that increasing the batch size does not necessarily improve the generalizability of a CNN model. Moreover, the research indicated that a batch size of 32, coupled with a learning rate of 0.0001 using the Stochastic Gradient Descent (SGD) optimizer, outperformed a model with a batch size of 256 and a learning rate of 0.001. Based on these insights, it was deemed appropriate to utilize a batch size of 16 with a learning rate of 0.0001, employing the SGD optimizer in this research. The experiments consisted of ten folds, each spanning ten epochs. The model was trained within each fold, followed by validation. Once the validation phase was completed, the model was tested on the unseen data that had been set aside. For each experiment, ten models were created, one for each fold, and the reported results represent the average performance across those models. The evaluation metrics used to obtain the results from these experiments are as follows: Accuracy: This determines the ratio of correctly classified instances over the entire number of instances: = + + + + × 100 Sensitivity: This determines the percentage of true positives and it is computed as follows: = + × 100 Precision: Measures the number of times the label of the Positive class has been incorrectly predicted as belonging to another class: = + × 100 Specificity: Measures the proportion of correctly labeled true negatives: = + × 100 F1-score: This represents the balance between Recall and Precision: Based on the results presented and depicted in Figures 9 and 10, it is evident that fine-tuning CNN models yield significantly better performance compared to models that only have the final classification layer trained. These findings also support the claims made by He et al. [27], highlighting the positive impact of deeper neural networks on performance. Specifically, the Inception-v3 model demonstrated the best results, achieving a remarkable accuracy of 91.56% on images with 40x magnification. Additionally, the results reinforce the importance of data augmentation and preprocessing. All four fine-tuned Inception-v3 models outperformed the models developed in [8], which lacked data augmentation and preprocessing. 1 − = 2 × × + × 100 Despite the impressive outcomes, the performance of the Inception-v3 model still falls short compared to the ViT model. The ViT model achieved the highest accuracy of 95.15%, surpassing the state-of-the-art system proposed by Senan et al. [12]. However, it should be noted that the ViT model did not outperform the models created by Thomas et al. [19]. One possible explanation could be the effectiveness of the data preprocessing employed by Thomas et al. in [19], which might have been superior to AlexNet Inception-v3 ResNet-18 ViT the approach used in this research. It is worth mentioning that some images were affected by being washed out due to the utilization of default values in the stain normalization method [21], potentially impacting the classification results. Conclusion and Future Works This paper focuses on investigating various deep-learning techniques for the purpose of diagnosing breast cancer using histopathological images. The evaluation was conducted on the BreakHis dataset, where the performance of AlexNet, Inception-v3, and ResNet-18 CNN models was assessed and compared to the newly introduced ViT architecture by Google. The results indicate that the Inception-v3 model, when pretrained, demonstrated the best performance for end-to-end CNN classification. However, overall, the ViT models exhibited superior performance, surpassing even the stateof-the-art CNN classification models mentioned in [12]. This study also highlights the importance of data preprocessing and augmentation in image classification using CNNs. The Inception-v3 model developed in this paper outperformed the models in [8], which were trained on unprocessed images without any data augmentation techniques applied. The significance of this finding emphasizes the need for careful preprocessing and augmentation steps to enhance the performance of CNN-based classification models. To expand upon the findings of this paper, future research could focus on creating uniform dataset sizes for experimentation purposes. Additionally, the effect of ViT performance on histopathological images could be explored by evaluating traditional image preprocessing techniques like Histogram Equalization and comparing them against stain normalization technique [21]. Furthermore, a comparison between the performance of hierarchal voting technique applied to an Inception-v3 model and a ViT model could provide valuable insights for further analysis.. Fig. 1 . 1Overview of the methodology Fig. 2 . 2Image of the malignant lesion at different magnification factors: (a) 40x, (b) 100X, (c) 200x, (d) 400x. Fig. 4 . 4Illustration of the improved Inception module[26] Fig. 5 . 5Illustration of the Inception module after factorization of × convolutions[26]. Fig. 6 . 6Illustration of the Inception module with expanded filter bank outputs[26] Fig. 7 . 7ResNet-18 Architecture[27] Fig. 8 . 8ViT Architecture[11] Fig. 10 . 10Display of the F1-Scores of fine-tuned DL models Table 1 . 1Breakdown description of the BreakHis dataset.Magnification Benign Malignant Total 40x 652 1370 1995 100x 644 1437 2081 200x 623 1390 2013 400x 588 1232 1820 Total 2480 5429 7909 Table 2 . 2Overview of the Inception-v3 Architecture -Input size is the previous layer's output.Layer Type Patch size | stride or remarks Input size Convolutional 3 x 3 | 2 299 x 299 x 3 Convolutional 3 x 3 | 1 149 x 149 x 32 Convolutional Padded 3 x 3 | 1 147 x 147 x 32 Pooling 3 x 3 | 2 147 x 147 x 64 Convolutional 3 x 3 | 1 73 x 73 x 64 Convolutional 3 x 3 | 2 71 x 71 x 80 Table 3 .Table 5 .Table 6 .Table 7 .Table 8 .Table 9 . 356789Results for breast cancer lesion diagnosis using a fine-tuned AlexNetTable 4. Results for breast cancer lesion diagnosis using an AlexNet model with only the classification layer trained. Results for breast cancer lesion diagnosis using a fine-tuned Inception-v3 Results for breast cancer lesion diagnosis using an Inception-v3 model with only the classification layer trained. Results for breast cancer lesion diagnosis using a fine-tuned ResNet-18 Results for breast cancer lesion diagnosis using a ResNet-18 model with only the classification layer trained. Results for breast cancer lesion diagnosis using a fine-tuned ViTFig. 9. Accuracies of fine-tuned DL modelsEvaluation 40x 100x 200x 400x Accuracy % 81.75 82.39 90.15 81.59 Sensitivity % 99.14 99.24 98.2 92.8 Precision % 79.86 80.21 88.54 81.3 Specificity % 41.17 44.21 73.88 62.79 F1-Score 88.41 88.69 93.06 86.38 Evaluation 40x 100x 200x 400x Accuracy % 73.4 75.17 78.86 70.44 Sensitivity % 99.86 99.03 97.70 96.75 Precision % 72.56 74 76.98 68.81 Specificity % 11.67 21.09 40.90 26.32 F1-Score 84.03 84.70 86.08 80.39 Evaluation 40x 100x 200x 400x Accuracy % 91.56 91.34 90.99 81.48 Sensitivity % 98.71 97.44 97.40 94.21 Precision % 90.22 90.84 90 79.87 Specificity % 75 77.5 78.06 60.15 F1-Score 94.27 94 93.53 86.44 Evaluation 40x 100x 200x 400x A deep learning model using data augmentation for detection of architectural distortion in whole and patches of images. Olaide N Oyelade, Absalom E Ezugwu, Biomedical Signal Processing and Control. 65102366Oyelade, Olaide N., and Absalom E. Ezugwu. 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[ "Fully guided and phase locked Ti:PPLN waveguide squeezing for applications in quantum sensing", "Fully guided and phase locked Ti:PPLN waveguide squeezing for applications in quantum sensing" ]
[ "Renato Domeneguetti \nCenter for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark\n", "Michael Stefszky \nInstitute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany\n", "Harald Herrmann \nInstitute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany\n", "Christine Silberhorn \nInstitute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany\n", "Ulrik L Andersen \nCenter for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark\n", "Jonas S Neergaard-Nielsen \nCenter for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark\n", "Tobias Gehring \nCenter for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark\n" ]
[ "Center for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark", "Institute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany", "Institute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany", "Institute of Photonic Quantum Systems (PHOQS)\nIntegrated Quantum Optics Group\nUniversity Paderborn, Warburger Str. 10033098PaderbornGermany", "Center for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark", "Center for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark", "Center for Macroscopic Quantum States bigQ\nDepartment of Physics\nTechnical University of Denmark\nFysikvej 307DK-2800Kgs.LyngbyDenmark" ]
[]
This work reports a fully guided setup for single-mode squeezing generation on integrated titanium-indiffused periodically poled nonlinear resonators. A continuous wave laser beam is delivered and the squeezed field is collected by single-mode fibers, where up to -3.17(9) dB of useful squeezing is available in fibers. To showcase the usefulness of such a fiber-coupled device, we applied the generated squeezed light in a fiber-based phase sensing experiment, showing a quantum enhancement in the signal-to-noise ratio of 0.35 dB. Moreover, our investigation of photorefraction's impact on the cavity resonance condition suggests that it causes system instabilities at high powers.
10.1364/ol.486654
[ "https://export.arxiv.org/pdf/2306.04767v1.pdf" ]
257,931,763
2306.04767
19aa389e101de0cbbe8f8efe20f69bc49c1f7afd
Fully guided and phase locked Ti:PPLN waveguide squeezing for applications in quantum sensing Renato Domeneguetti Center for Macroscopic Quantum States bigQ Department of Physics Technical University of Denmark Fysikvej 307DK-2800Kgs.LyngbyDenmark Michael Stefszky Institute of Photonic Quantum Systems (PHOQS) Integrated Quantum Optics Group University Paderborn, Warburger Str. 10033098PaderbornGermany Harald Herrmann Institute of Photonic Quantum Systems (PHOQS) Integrated Quantum Optics Group University Paderborn, Warburger Str. 10033098PaderbornGermany Christine Silberhorn Institute of Photonic Quantum Systems (PHOQS) Integrated Quantum Optics Group University Paderborn, Warburger Str. 10033098PaderbornGermany Ulrik L Andersen Center for Macroscopic Quantum States bigQ Department of Physics Technical University of Denmark Fysikvej 307DK-2800Kgs.LyngbyDenmark Jonas S Neergaard-Nielsen Center for Macroscopic Quantum States bigQ Department of Physics Technical University of Denmark Fysikvej 307DK-2800Kgs.LyngbyDenmark Tobias Gehring Center for Macroscopic Quantum States bigQ Department of Physics Technical University of Denmark Fysikvej 307DK-2800Kgs.LyngbyDenmark Fully guided and phase locked Ti:PPLN waveguide squeezing for applications in quantum sensing 10.1364/OL.486654Letter 1 This work reports a fully guided setup for single-mode squeezing generation on integrated titanium-indiffused periodically poled nonlinear resonators. A continuous wave laser beam is delivered and the squeezed field is collected by single-mode fibers, where up to -3.17(9) dB of useful squeezing is available in fibers. To showcase the usefulness of such a fiber-coupled device, we applied the generated squeezed light in a fiber-based phase sensing experiment, showing a quantum enhancement in the signal-to-noise ratio of 0.35 dB. Moreover, our investigation of photorefraction's impact on the cavity resonance condition suggests that it causes system instabilities at high powers. Squeezed states of light are an essential tool for enhancing the sensitivity of measurements of physical quantities with the help of phase or amplitude quadratures of an electromagnetic field [1]. Often interferometers are used for those measurements, and by replacing the vacuum with squeezed states in the dark port of the main beam splitter, the shot-noise limit (SNL) can be overcome and the signal-to-noise ratio (SNR) is enhanced by an amount commensurate to the squeezing level [2,3]. For example, squeezed states are successfully used to enhance the sensitivity of gravitational-wave detectors [4,5]. A squeezed light source that is simultaneously scalable, reproducible, stable, and compact is appealing since it will allow the mass fabrication of consistent and robust photonic chips. Moreover, for squeezed light generation in the telecommunication C-band, it is interesting to deliver the squeezed light in fiber, with the potential of integrating the device into an already existing optical setup or fiber network. A fully guided squeezing source was reported that combined a commercially available spontaneous parametric down-conversion (SPDC) module with other optical fiber components using simple plug-and-play assembly, but with almost no room for improvement of its quantum efficiency [6]. Various on-chip nonclassical light sources have recently been reported due to technological advances in fabricating photonic devices with low propagation loss waveguides. For example, broadband and up to 6.3 dB of squeezed light was achieved in a periodically poled lithium-niobate (PPLN) ridge waveguide in a single-pass configuration thanks to the high second-order nonlinearity of the lithium-niobate crystal, but at the cost of high pump power in the absence of a cavity. The output was collected in free space [7] and with a fiber [8]. A reconfigurable chip for the generation of single-or two-mode squeezing in PPLN working at high pump power was described in [9]. A two-module PPLN device containing cascaded second-harmonic generation (SHG) and optical-parametric amplifier (OPA) produced 1 dB of squeezing in free space [10]. Several integrated sources of single-or two-mode squeezed light using silicon photonics have also been reported [11][12][13][14]. Silicon nitride chips have a low third-order nonlinearity, offering weak light-matter interactions, which can be compensated for with high-quality-factor resonators, made possible by silicon nitride's low propagation losses and high confinement of light. Thus, high on-chip levels of squeezing were achieved, but only small levels were actually measured due to the difficulty of efficiently coupling light from chip to single-mode fibers. Here, we report broadband squeezing generation in a titanium indiffused periodically poled lithium niobate (Ti:PPLN) waveguide resonator for quantum sensing applications. In Ref. [15], squeezing was demonstrated with these geometries in a free space setup, where a total detection efficiency, measured from the output of the chip to the homodyne detector, of 72 % was achieved. In that work, to reach high homodyne detection visibility, a second waveguide was used as a mode shaper for the local oscillator (LO). We instead constructed a fully guided setup, where the squeezed light was directly coupled into a single-mode fiber and the pump field was coupled from fiber to the chip. Homodyne detection was performed using a fiber coupler with high fringe visibility. Despite the fiber components, we achieved a total detection efficiency comparable to the free space setup, with the advantage of being more compact, robust, and ready to be integrated into different fiber-coupled setups. We furthermore locked many degrees of freedom in our setup: fiber alignment, pump-to-seed phase, and the LO phase of homodyne detection. Finally, we demonstrated the applicability of our squeezer in a quantum sensing experiment, where vacuum fluctuations were replaced by phase squeezing to enhance the signal-to-noise ratio in a phase sensing protocol. Squeezed Light Source. The central part of the squeezed light source was an optical waveguide cavity in Ti:PPLN. In the cavity, squeezed light was generated through a χ (2) -nonlinear interaction and quasi-phase matching was achieved via the periodic inversion of the ferroelectric domains with a proper poling period. Optical waveguides in the 12 mm long PPLN crystal were fabricated via indiffusion of Ti-stripes into the material, a mature and well-known fabrication technology to obtain low-loss waveguides [16]. The waveguide fabrication conditions were chosen to enable single-mode guiding in the telecommunication C-band. For such optimized waveguides, the losses are about 10 dB/m, with the best ones at around 2 dB/m, mainly caused by scattering in waveguide surfaces [17]. For the nonlinear conversion process, we exploited the strongest nonlinear coefficient d 33 of lithium niobate, which requires quasi-phase-matching with poling periods around 17 µm for a type 0 process, where all the interacting waves are extraordinarily polarized. Operation at elevated temperature ∼ 140.5 • C is necessary to mitigate the photorefractive effect [18]. To reduce the pump power required for strong nonlinear conversion, dielectric mirrors were deposited on the waveguide end facets to form a cavity. The reflectivities of these mirrors were optimized on the basis of numerical simulation results for maximum squeezing and low power threshold. The input mirror comprised a SiO2/TiO2 dielectric stack with 15 layers, which yielded a high reflectivity (HR) (>99 %) in the telecom C-band. The 11-layer stack of the output mirror had a partial reflectivity (PR) of about 64 %. The transmission of both mirrors at the pump wavelength was higher than 97 %. No cavity for the pump is allowed since it can increase the photorefraction [19]. The setup for quantum characterization and operation of the Ti:PPLN squeezer is shown in Fig. 1. Light at 1550 nm from a continuous-wave (CW) fiber laser was frequency doubled to 775 nm (NKT Koheras HARMONIK). Part of the non-converted light at 1550 nm was used as a probe for phase locking the OPO to produce squeezing at a specific quadrature, and as a local oscillator for homodyne detection. Both pump and probe fields were power and polarization controlled before coupling them into the nonlinear waveguide by using variable optical attenuators and polarization controllers in single-mode (SM) fiber at respective bands. Then, both light fields were combined in one SM fiber at 1550 nm via a wavelength-division multiplexer (WDM) from Thorlabs (WD1350A). Although this WDM is designed to operate at 1310/1550 nm, it worked very well for the pump field, showing an insertion loss of only 1 dB. Also, the output of the pump field out of the SM fiber was always transversally fundamental regardless of the power, polarization, or curvature of the fiber. The coupling of both fields into the nonlinear resonator was facilitated by a lensed fiber with 4 µm spot size which according to simulations matches better with the mode-field diameter of the waveguide for the pump field. Furthermore, the lensed fiber has a finite working distance, allowing maximum coupling efficiency before touching the waveguide facet. The maximum coupling efficiency of the pump beam into the waveguide was estimated to be 70% by measuring the power at the fiber output and comparing it with the power at the output of the waveguide on the other side after discounting propagation losses at 775 nm. A lensed fiber with 6 µm spot size collected the squeezed vacuum out of the chip at the PR port of the resonator, where a maximum coupling efficiency of 88% was reached. The estimation was accomplished by comparing the measured output power of the probe with an aspherical anti-reflection coated lens, followed by measuring the power collected via lensed fiber. Because of the cavity behavior, both measurements were realized with the cavity in resonance with the probe. It is important to remark that a coupling efficiency of 90 % was achieved using a butt-coupled AR-coated fiber instead of lensed fiber. However, we preferred to work with lensed fiber because of its finite working distance, making the alignment easier when no visual access to the coupling is available. Nevertheless, pigtailing is a good option for commercial devices when no further characterization is needed. The collected squeezing was guided to homodyne detection via SM fiber. Here, 1% of the co-propagated probe was tapped off for locking the squeezing phase and to perform self-alignment of fibers. Next, it passed through a low-pass filter (LPF) from Omega Optical LLC to eliminate the pump field from the fiber. The intrinsic loss of the LPF was less than 2 % as indicated in the datasheet, but 8 % of loss was measured which can be attributed to the mating sleeve in this connection. Homodyne detection was performed using a 3 dB coupler and a custom-made homodyne photodetection circuit. Apart from the connection to the LPF we spliced all fiber connections to minimize loss. Therefore, we achieved an estimated total propagation efficiency from the output of the resonator until the input port of the beam splitter (BS) for homodyne detection of 80 %. Because both the squeezing and LO propagates in SM fiber, the visibility of the homodyne detection is very high, ∼ 99 %. Additionally, the quantum efficiency of the homodyne detection was estimated to be 93.4 ± 1.0 %. Thus, the total detection efficiency was estimated to be ∼ 74 %. Locking system. To enable locking of the system, the probe beam was phase modulated at 60 kHz which was detected after the cavity at the 1 % port of the beam splitter. A Red Pitaya (STEM 125-14) generated the error signal by demodulating the electronic signal and produced a control signal for the actuation of the phase shifter to achieve deamplification. To avoid noise contamination from the probe beam to the squeezing, the maximum power of the probe before the HR port of the resonator had to be less than 10 µW as only in this case the probe was shot noise limited at analysis frequencies above 10 MHz. For the phase of the LO, the LO beam was phase modulated at 55 kHz which was detected by the homodyne detector feeding another Red Pitaya. We furthermore implemented a self-alignment algorithm based on gradient descent to improve the stability of the alignment between the input/output fiber and the waveguide, which suffered from small but continuous piezo drifts that limit the optimum alignment to a couple of seconds. Although all locks worked well for low pump powers (< 10 mW of power inside the waveguide), they failed to maintain the experiment stability when increasing the input power, which we mainly attributed to the photorefractive process [20]. At high pump powers, small fluctuations in the coupling efficiency of light from fiber to waveguide can notably change the power in the waveguide, where both thermal effects and photorefraction alter the resonance condition, rendering the lock ineffective. In fact, if the power is high enough, the system is expected to enter a chaotic regime due to photorefraction [21]. Results. Light coupled to waveguides increases its temperature due to absorption and redistributes the impurities of the material, leading to thermo-optic and photorefractive effects, respectively, which change the effective refractive index of the waveguide. These effects are particularly stronger at shorter wavelengths. As the pump power increases, we observed a shift in the cavity resonance towards higher frequencies, as depicted in Figure 2A. The direction of the shift indicates that the photorefractive process is responsible [22]. However, for 1 mW of pump power, we observed a slight redshift which can be explained by the thermo-optic effect being slightly more substantial at this power. By ignoring the thermo-optic effect and thermal expansion, and applying a linear fit to the corresponding shifts in Fig. 2B, we estimated the photorefraction coefficient to be −3.97(12) × 10 −3 W −1 , the same order of magnitude as in [19]. Moreover, the photorefraction alters the phase-matching temper-ature for optimum parametric-down conversion (PDC) gain, as observed in Fig. 2C. At low pump power, the first two points indicate no changes in the phase-matching temperature relative to the temperature of the SHG, for which we neglected photorefraction as the powers involved were low. However, we consistently noticed an increase in the phase-matching temperature with the increase in power, which agrees with a negative change in the refraction index. After correcting the phase mismatch due to photorefraction, the simultaneous fit of (de-) amplification in Fig. 2D led to a threshold Figure 3A shows the amount of detected squeezing and anti-squeezing at 5 MHz sideband frequency produced by the Ti:PPLN squeezer after optimizing the phase-matching. The maximum amount of detected anti-squeezing and squeezing were respectively 6.97(9) and −3.17(9) dB when the pump power inside the nonlinear waveguide was 30 mW. The estimated total efficiency, including detection and escape efficiencies, is 62(4) % from the simultaneous fit of (anti-) squeezing. Based on coatings specifications and internal propagation losses (10 dB/m in a 12 mm waveguide) we estimated the escape efficiency of the resonator to be 87 %. When considering the previously mentioned propagation and detection efficiency, the total efficiency from the fit is well explained. The lowest waveguide losses observed was 2 dB/m [17], although not in the current batch. If these losses were achieved in the current waveguide, assuming all other experimental conditions were unchanged, 5 dB of squeezing could be measured. Figure 3B shows a spectrum of the squeezing and antisqueezing noise power versus sideband frequency for a pump power of 30 mW. The (anti-) squeezing levels at low frequencies are compatible with the amount described in figure B. The full width half maximum (FWHM) obtained from the fit is 200 MHz, which is compatible with the estimation based on coatings and resonator length. Our model fitted to the data includes the effect of phase noise as this improved the quality of the fit [23], leading to an overall phase noise of the order of 20 mrad. Squeezed light enhanced phase sensing. To demonstrate our fiber-coupled Ti:PPLN waveguide squeezer in a typical phase sensing experiment, we injected the squeezing into a Mach-Zehnder interferometer (MZI) as depicted in Fig. 4A. The MZI was driven by a bright beam at 1550 nm of 1 mW power and locked at mid-fringe using a phase shifter (fiber stretcher) in one of the arms. The two outputs of the MZI were detected by photodiodes whose photocurrents were subtracted. A simulated phase signal inside one arm of the interferometer was generated at 40 kHz using another phase shifter in the other arm of the interferometer. To move away from classical noise, we intensity modulated the input beam of the MZI at 4 MHz using an electrooptical modulator. The phase signal can then be observed as 40 kHz sidebands of the 4 MHz modulation [24]. We first characterized the squeezed light in this setup without the amplitude modulation and the phase signal. Figure 4B displays the noise power measured by the homodyne detector at a sideband frequency of 4.04 MHz (resolution bandwidth 20 kHz). We detected -0.5 dB of squeezing and 3.75 dB anti-squeezing. For this measurement, we used a lower pump power to improve the stability of the system, where -2 dB of squeezing was obtained right at the output of the squeezer. The reduction of squeezing is due to optical loss from the fiber components and the number of fiber connections. In particular, optimizing the MZI's visibility was challenging, and a maximum of 94% was reached by carefully balancing the polarization and power inside the MZI. We expect that by using polarization-maintaining fibers instead, the visibility could be improved. Figure 4C shows the comparison of two measurements of the 40 kHz phase signal with and without squeezing. When no squeezing was injected into the unused port of the MZI, we observed an SNR of 8.26(25) dB (dark shading). With phase squeezing the observed SNR was 8.61(24) dB (blue shading). From the curves, it is clear that when using squeezing, the background noise is reduced while maintaining the same peak amplitude, leading to a quantum enhancement of the SNR. Conclusion. In conclusion, we presented an integrated source of single-mode squeezing guided by optical fibers. Using a fiber optical homodyne detector, we observed −3.17(9) dB of usable squeezing. We locked fiber alignments and optical phase to prepare the squeezed quadrature angle in homodyne detection. The system was robust for powers below 20 mW, and squeezed light enhanced phase measurements were demonstrated, showing that our system is suitable for fiber-based sensing protocols. However, the system becomes inoperable for powers above 30 mW. It is likely that this instability arises due to the onset of catastrophic photorefraction [21]. http://dx.doi.org/10.1364/OL.486654 Introduction. Fig. 2 . 2Classical characterization of the Ti:PPLN squeezer. A) Blueshift was observed when increasing pump power. B) Estimation of the photorefraction effect. C) Increase in the phase-matching temperature due to the photorefraction. D) PDC gain after correction. Fig. 3 . 3Quantum characterization of the Ti:PPLN squeezer. A) Anti-squeezing and squeezing levels at 5 MHz sideband frequency in terms of the pump power inside the nonlinear waveguide. B) Broadband squeezing generated by the resonator when pumped at 30 mW inside the waveguide. Fig. 4 . 4Quantum sensing experiment. A) Setup for measuring phase noise at kHz range. B) Measurement of squeezing after the interferometer. Black and blue lines represent the normalized shot noise and squeezed vacuum noise, respectively. C) Quantum enhancement of the SNR when vacuum fluctuations (dark shading) or phase squeezing (blue shading) was injected into the interferometer. Relative noise power [dB]0.0 0.5 Time [s] 0.5 0.5 1.5 2.5 3.5 4.5 B 4.035 4.040 4.045 Frequency [MHz] 103 98 93 Noise power [dBm] C SNL Phase squeezing Squeezing Source SHG 4 MHz 40 kHz EOM A Laser 1550 nm Funding. We Disclosures. The authors declare no conflicts of interest. . U L Andersen, T Gehring, C Marquardt, G Leuchs, Phys. Scripta. 9153001U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, Phys. Scripta 91, 053001 (2016). . C M Caves, Phys. Rev. D. 231693C. M. Caves, Phys. Rev. D 23, 1693 (1981). . M Xiao, L.-A Wu, H J Kimble, Phys. Rev. Lett. 3M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. (1987). Volume: 59 Issue: 3. . 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[]
[ "Code Critters: A Block-Based Testing Game", "Code Critters: A Block-Based Testing Game" ]
[ "Philipp Straubinger \nUniversity of Passau\nPassauGermany\n", "Laura Caspari \nUniversity of Passau\nPassauGermany\n", "Gordon Fraser \nUniversity of Passau\nPassauGermany\n" ]
[ "University of Passau\nPassauGermany", "University of Passau\nPassauGermany", "University of Passau\nPassauGermany" ]
[]
Learning to program has become common in schools, higher education and individual learning. Although testing is an important aspect of programming, it is often neglected in education due to a perceived lack of time and knowledge, or simply because testing is considered less important or fun. To make testing more engaging, we therefore introduce Code Critters, a Tower Defense game based on testing concepts: The aim of the game is to place magic mines along the route taken by small "critters" from their home to a tower, such that the mines distinguish between critters executing correct code from those executing buggy code. Code is shown and edited using a block-based language to make the game accessible for younger learners. The mines encode test inputs as well as test oracles, thus making testing an integral and fun component of the game.
10.1109/icstw58534.2023.00077
[ "https://export.arxiv.org/pdf/2304.02246v1.pdf" ]
257,952,139
2304.02246
92b8b31436c573a38b7e6b4f719fd8112b7fea5e
Code Critters: A Block-Based Testing Game Philipp Straubinger University of Passau PassauGermany Laura Caspari University of Passau PassauGermany Gordon Fraser University of Passau PassauGermany Code Critters: A Block-Based Testing Game Index Terms-gamificationmutationblock-basedsoftware testingeducationserious game Learning to program has become common in schools, higher education and individual learning. Although testing is an important aspect of programming, it is often neglected in education due to a perceived lack of time and knowledge, or simply because testing is considered less important or fun. To make testing more engaging, we therefore introduce Code Critters, a Tower Defense game based on testing concepts: The aim of the game is to place magic mines along the route taken by small "critters" from their home to a tower, such that the mines distinguish between critters executing correct code from those executing buggy code. Code is shown and edited using a block-based language to make the game accessible for younger learners. The mines encode test inputs as well as test oracles, thus making testing an integral and fun component of the game. I. INTRODUCTION Programming has become a common aspect of education at schools [1] as well as in higher education, and it is also a sought-after skill in industry [2]. Even though software testing is an integral part of software development in practice, it is often neglected in programming education [3], despite increasing awareness in higher education and even proposals to include testing in the curricula of schools [4]- [6]. This is often caused by a lack of awareness and skills for teaching testing, and the perception shared by both learners and programmers that testing is a tedious and boring task. As long as software testing is not taught in a more engaging and accessible way, this is unlikely to change. To educate learners on testing in a more engaging way, we introduce Code Critters, a serious game based on Tower Defense, whose main purpose is not entertainment but education. Figure 1 shows the gameboard in action: Players have to rescue human-like critters (for example shown to the right in Fig. 1) from mutants (critters with zombie-like green heads further to the left), who can be distinguished by their behavior encoded in small snippets of easily accessible block-based code. The game is played by placing magic mines (e.g., at the beginning of the dirt track in Fig. 1), which are essentially test cases, on the route the critters take from their home on the left to the target tower on the right in Fig. 1. A mine represents test inputs (e.g., coordinates, terrain type) as well as test oracles represented as block-based code snippets. Different software concepts can be integrated into the gameplay by building appropriate levels and mutants based on these concepts. Despite its importance and a growing awareness in higher education, software testing remains underrepresented during programming education [7]. A promising solution to incite developers to write tests is gamification (e.g., [8]), i.e., the use of game elements such as leaderboards, points or challenges in non-game contexts [9]. Serious games take this approach a step further, as these are games explicitly made for training, education, or simulation, where players learn from embedded information about a topic without getting the feeling of learning or working [10]. Serious games are often adaptations of well-known game types like Tower Defense, while changing facets to meet the objectives. Surprisingly, only few serious games have been proposed for software testing [11]- [13]. One testing concept that has emerged as particularly suitable for gamification and serious games is mutation testing, which has, for example, been successfully integrated into the Code Defenders game [14]. Mutation testing consists of inserting artificial defects in tested code to identify and remedy weaknesses in existing tests [15]. In Code Defenders, this is gamified as attackers creating artificial defects (mutants), which defenders aim to detect by writing tests. However, a central disadvantage of Code Defenders and other attempts to gamify testing is that they require reasonably advanced programming skills, and are therefore better suited for higher education. To also engage more inexperienced learners, testing must be integrated much earlier into programming education. A common approach to make programming accessible for younger learners is to represent code using block-based languages such as Scratch [16]. Instead of typing code as text, learners assemble predefined code blocks visually by dragging and dropping them into position, quickly creating fun games and programs. Considering the success of blockbased programming [17], in this paper we explore the idea of similarly lowering the entry barrier also for software testing using a block-based programming approach. III. CODE CRITTERS Code Critters tells the story of the critters, who are people living in peace in an unknown colony in a forest. Unfortunately, a disease outbreak causes many of the critters to turn into mutants. These mutants do not behave like the other critters and are destroying the colony, forcing the healthy critters to flee into a tower across the forest. Along their walk to the tower, the player has to place magic mines, which check the behavior of the critters and only let healthy ones pass. A. Game Concept Code Critters integrates mutation testing and block-based programming around the well-known game Tower Defense. In a classic Tower Defense game, the player has to place turrets along a route that enemies take to reach a certain point on the gameboard. The more enemies are eliminated on their way to this point, the more points the player receives in the end. Unlike traditional Tower Defense games, in Code Critters there are not only enemies but also civilians who have to be protected. The behavior of critters is represented by the critter under test (CUT) as a short block-based code snippet (Fig. 2). Enemies (mutants) are mutations of the CUT (Fig. 3). To avoid the rather violent notion of killing enemies, in Code Critters the turrets are replaced with mines that represent test cases; these mines use magic to trap the mutants instead of shooting them. B. Game Mechanics The gameboard (Fig. 4) represents both the colony (the spawn point) and the tower (the destination), including the dirty trail from one to the other with the surrounding forest. The board is made of 256 tiles, represented with x and y coordinates from 1 to 16. Each tile can hold one specific texture, namely grass, dirt, water, ice, or wood, and critters can only walk on grass, dirt, and ice. In addition, mines can only be placed on tiles where critters can walk, which is also the first move the player has to make before starting the game. The behavior of critters is described with short snippets of code which are continuously executed in a loop while the critters are exploring the gameboard. A CUT is conceptually similar to an object oriented class and consists of two parts, the initialization and the code under test. The initialization behaves like a constructor and defines the initial values of the attributes of a critter like their shirt or hair color. The code under test is like a method that is called continuously in a loop while the critter walks, and receives texture and coordinates as inputs. A simple example of a healthy critter is the CUT shown in Fig. 2, which is initialized with a red shirt, and its color changes when walking on dirt tiles. Mutant critters contain one or more code mutations, such as the incorrect shirt color initialization and wrong choice of shirt color shown in Fig. 3. The goal of the game is to let only healthy critters reach and enter the tower, while mutants are held off and trapped by the mines. Mines represent test cases: A mine is placed on exactly one tile by clicking on one of the walkable tiles, which represent the test inputs with their coordinates and texture. Clicking on a tile opens a dialog (Fig. 4) in which a new test for the CUT can be created for a given input location (tile). Players can implement assertions for the mines with the same block-based language that represents the CUT or mutants. The available blocks on the left side of the dialog in Fig. 4 can be placed on the right side via drag and drop and combined into a test case. The main block of each test is the assert-block, which is divided into the property to be checked and the value it should have at this point. Properties include the color of a critter's shirt and hair as well as their size. Shirt and hair colors are enumerations, while size properties are integers that can be checked for their value, whether they are even, odd, negative, positive or prime value. It is also possible to store and check basic information in variables and to perform basic mathematical operations on numerical values. The test represented by a mine is executed during active gameplay against any critter who steps onto the mine. To trap the mutant in Fig. 3, two different mines have to be placed along the route to detect it: The first one somewhere on a grass tile which covers the mutation in the initialization, and the second mine on a dirt tile to find the mutation in the ifcondition (see Fig. 1). Both mines need to assert the expected value of the shirt color. These mines in combination with the mutants are also an abstraction layer to generate a test suite. A secondary goal of the game is to use as few mines as possible to detect all mutants, in other words, to minimize the test suite. In general, there is no limitation on the number of mines, but points are deducted if too many of them are used. After the player has placed all necessary mines on the board, the game can be started (Fig. 1). The critters now start from the colony on the left and take a random route to the tower. A running game can be paused at any time as well as reset or sped up. A valid mine lets healthy critters pass without harm, while it traps mutants if the test fails (see Section III-C). Consequently, invalid assertions may lead to false positives and trap healthy critters rather than mutants, which leads to the player losing points. At the bottom of the screen (Fig. 4), the critters and mutants are displayed, and those who reached the tower will be marked as saved while trapped ones will be greyed out. When the last free critter or mutant reaches the tower, the game ends and the scoreboard is shown (Fig. 5). It contains information about saved critters and detected mutants, as well as the number of mines used and the given time bonus. After finishing the game, the mutants can be viewed to gain insights into how the CUT was mutated. The achieved score is accumulated with points earned from prior games and displayed on a public leaderboard on the starting page of Code Critters (Fig. 6). C. Levels Code Critters provides different levels, organized into tutorial, beginner, and advanced (Fig. 6) categories. Our current proof-of-concept integrates ten levels, but Code Critters also provides a level editor (Fig. 7) to create new ones. The level editor allows the creation of a custom map with the five different texture types and the start and end points. Each level needs a CUT to be created and, based on the CUT, one or more mutants. In addition, the number of critters, the number 1 https://developers.google.com/blockly Since each level is based on one CUT, the difficulty and learning goals can be adjusted by choosing an appropriate CUT. Players need to consider the following parts of the CUT and write tests for them to trap all mutants: Levels can capture different testing concepts by adjusting those parts. For example, using a set of mutants where each mutant has one or more changed assignment statements, the concept of full statement coverage can be taught while changing the order or the content of if-else-clauses teaches about branch coverage. Changing or even removing conditions within conjunctions or disjunctions leads to the understanding of condition coverage. Not only the CUT can be mutated in various ways to increase the difficulty, but the gameboard itself can be altered and designed in a way to add more difficulty. For example, the board can be adjusted to provide more than one path the critters and mutants can take to the tower. This leads to the requirement for different mines on different tracks because of the texture and the coordinates of the tiles; some mines may be unique in one of the routes, and others have to be added in every route to ensure all mutants are caught, essentially teaching about input partitioning [18]. D. Implementation Code Critters is designed as a web application that can be deployed on any server and reached from the internet to be playable for everyone in a web browser. Like Scratch and many other block-based programming environments, we decided to build Code Critters using the Blockly 1 library for representing and editing code. Scratch adds an abstraction layer over basic programming aspects by defining different blocks that the learners can combine into a meaningful program. In Code Critters we reuse these concepts by defining the CUT with different blocks instead of source code (Fig. 2). IV. CONCLUSIONS Code Critters is a proof-of-concept implementation that demonstrates the possibility to teach testing concepts with block-based programming in a fun way. Code Critters is work in progress, and we plan to extend it with additional game elements, programming concepts, and corresponding levels in the future. It will also be important to study and evaluate Code Critters with actual learners. The source code is available at: https://github.com/se2p/code-critters and Code Critters can be tried out online at: https://code-critters.org Fig. 1 : 1The gameboard during active gameplay II. 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[Online]. . 10.1145/3105726.3106173Available: https://doi.org/10.1145/3105726.3106173 Having fun in learning formal specifications. I S W B Prasetya, C Q H D Leek, O Melkonian, J Tusscher, J Van Bergen, J M Everink, T Van Der Klis, R Meijerink, R Oosenbrug, J J Oostveen, T Van Den Pol, W M Van Zon, Proceedings of the 41st International Conference on Software Engineering: Software Engineering Education and Training, ICSE (SEET) 2019. the 41st International Conference on Software Engineering: Software Engineering Education and Training, ICSE (SEET) 2019Montreal, QC, CanadaS. Beecham and D. E.I. S. W. B. Prasetya, C. Q. H. D. Leek, O. Melkonian, J. ten Tusscher, J. van Bergen, J. M. Everink, T. van der Klis, R. Meijerink, R. Oosenbrug, J. J. Oostveen, T. van den Pol, and W. 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G Fraser, A Gambi, M Kreis, J M Rojas, 10.1145/3287324.3287471Proceedings of the 50th ACM Technical Symposium on Computer Science Education, SIGCSE 2019. E. K. Hawthorne, M. A. Pérez-Quiñones, S. Heckman, and J. Zhangthe 50th ACM Technical Symposium on Computer Science Education, SIGCSE 2019Minneapolis, MN, USAACMG. Fraser, A. Gambi, M. Kreis, and J. M. Rojas, "Gamifying a software testing course with code defenders," in Proceedings of the 50th ACM Technical Symposium on Computer Science Education, SIGCSE 2019, Minneapolis, MN, USA, February 27 -March 02, 2019, E. K. Hawthorne, M. A. Pérez-Quiñones, S. Heckman, and J. Zhang, Eds. ACM, 2019, pp. 571-577. [Online]. Available: https://doi.org/10.1145/3287324.3287471 An analysis and survey of the development of mutation testing. Y Jia, M Harman, IEEE Transactions on Software Engineering. 375Y. Jia and M. Harman, "An analysis and survey of the development of mutation testing," IEEE Transactions on Software Engineering, vol. 37, no. 5, pp. 649-678, 2011. The scratch programming language and environment. J Maloney, M Resnick, N Rusk, B Silverman, E Eastmond, ACM Transactions on Computing Education (TOCE). 104J. Maloney, M. Resnick, N. Rusk, B. Silverman, and E. Eastmond, "The scratch programming language and environment," ACM Transactions on Computing Education (TOCE), vol. 10, no. 4, pp. 1-15, 2010. Learnable programming: blocks and beyond. D Bau, J Gray, C Kelleher, J Sheldon, F A Turbak, 10.1145/3015455Commun. ACM. 606D. Bau, J. Gray, C. Kelleher, J. Sheldon, and F. A. Turbak, "Learnable programming: blocks and beyond," Commun. ACM, vol. 60, no. 6, pp. 72-80, 2017. [Online]. Available: https://doi.org/10.1145/3015455 Practical software testing. J.-F Collard, I Burnstein, J.-F. Collard and I. Burnstein, "Practical software testing," 2002.
[ "https://github.com/se2p/code-critters" ]
[ "Handheld Haptic Device with Coupled Bidirectional Input", "Handheld Haptic Device with Coupled Bidirectional Input" ]
[ "Megh Vipul Doshi ", "Michael Hagenow1 ", "Robert Radwin ", "Michael Gleicher3 ", "Bilge Mutlu ", "Michael Zinn " ]
[]
[]
Handheld kinesthetic haptic interfaces can provide greater mobility and richer tactile information as compared to traditional grounded devices. In this paper, we introduce a new handheld haptic interface which takes input using bidirectional coupled finger flexion. We present the device design motivation and design details and experimentally evaluate its performance in terms of transparency and rendering bandwidth using a handheld prototype device. In addition, we assess the device's functional performance through a user study comparing the proposed device to a commonly used grounded input device in a set of targeting and tracking tasks.
10.48550/arxiv.2305.19381
[ "https://export.arxiv.org/pdf/2305.19381v1.pdf" ]
258,987,718
2305.19381
b713d99ecca963491edfa398b2b4400055ebab15
Handheld Haptic Device with Coupled Bidirectional Input Megh Vipul Doshi Michael Hagenow1 Robert Radwin Michael Gleicher3 Bilge Mutlu Michael Zinn Handheld Haptic Device with Coupled Bidirectional Input Index Terms-Handheldhaptic devicemobilefinger flexionbidirectionalhigh performancehaptic feedback Handheld kinesthetic haptic interfaces can provide greater mobility and richer tactile information as compared to traditional grounded devices. In this paper, we introduce a new handheld haptic interface which takes input using bidirectional coupled finger flexion. We present the device design motivation and design details and experimentally evaluate its performance in terms of transparency and rendering bandwidth using a handheld prototype device. In addition, we assess the device's functional performance through a user study comparing the proposed device to a commonly used grounded input device in a set of targeting and tracking tasks. I. INTRODUCTION Kinesthetic haptic devices offer a variety of ways to interact with users, from rendering virtual environments to providing guidance and feedback during teleoperation of robots. Traditionally, high-performance haptic devices have been grounded (e.g., the devices from Force Dimension [1] or Haption [2]), meaning that they are fixed in a location and generate haptic sensations by reacting against the environment. More recently, a variety of handheld haptic devices have been proposed that provide similar kinesthetic renderings by reacting against the user's hand or arm [3]. For example, Dills et al. [4] propose a high-performance one degree-of-freedom device employing hybrid actuation. Many of the other recent handheld devices provide haptic feedback to each finger individually either through finger-mounted devices [5], [6] or gloves [7], [8]. Notably, many of the recent handheld devices are designed for rendering virtual environments in gaming or virtual-reality applications. Alternatively, we are interested in one degree-offreedom industrial applications and propose a new handheld haptic input device that is actuated using one hand through two mechanically-coupled triggers. This work was supported in part by a NASA University Leadership Initiative (ULI) There are a variety of industrial applications that could benefit from a one degree-of-freedom haptic input device. For example, an operator could precisely control a single variable during an industrial process (e.g., flow rates, temperatures, feed rates, pressure during sanding) and the device could provide haptic cues or guidance as necessary (e.g., vibration, modulation of stiffness). In many applications, it would also be desirable for the input to be differential, meaning an operator may want to adjust a particular process variable from the current set point (e.g., go faster or slower). Finally, many industrial applications would benefit from a device that is one-handed, which frees the operators' other hand for secondary tasks (e.g., controlling a second input or feeling a surface during sanding). To address these desired qualities, we propose a one degree-of-freedom bidirectional input where the differential input is provided through two-mechanically coupled triggers, as shown in Figure l . We were inspired by existing applications leveraging finger flexion, such as musical instruments (e.g., a trumpet), that allow for input in the natural direction of the finger. Specifically, we chose to investigate a mechanism leveraging adjacent finger flexion to utilize the innate structure of the human fingers, that is more adept at pulling than pushing [9]. Additionally, the mechanicallycoupled triggers allows for simple bidirectional inputs where the triggers correspond to opposite directions. In this paper, we propose a high-performance one degree-of-freedom (DOF) haptic device that is mobile and can be actuated using a single hand through two fingers. We first discuss our design requirements for realizing a usable and high-performance haptic device followed by describing our developed prototype. Through an experimental evaluation, we demonstrate that the prototype has many desirable characteristics for generating high-quality haptics, such as a high transparency and a high bandwidth. Finally, we investigate the performance of our prototype in terms of giving precise input via a user study comparing the device to a haptic knob, a form factor that is already widely used in industrial settings. II. DEVICE DESIGN AND IMPLEMENTATION A. Design Requirements Our goal was to design a handheld and mobile alternative to a 1-DOF grounded interface.We propose a solution which is to be actuated using adjacent finger flexion (index finger and middle finger), where the inputs are mechanically coupled as can be seen in Figure 1. The range of motion of the device was designed to span half that of a typical adult male finger flexion motion and be capable of rendering up to 15 N peak force [10] [11]. The device should be transparent(i.e. low friction) and capable of rendering stiffness levels comparable to other high performance kinesthetic haptic interfaces. The device requirements are summarized in the list below. • Allow for 1-DOF haptic interactions via finger flexion • Mobility with one handed operation • High stiffness and high transparency • A maximum force of 15 N at each trigger • A stroke length of 15 mm for each trigger B. Design Implementation To achieve high transparency, we incorporated as low an inertia as we could with a simple lightweight design (device weight was 383.40 grams). The device has a 3d printed central handle, with two linear triggers that are used to take input from as well as to give haptic feedback to the user. Our device design has a two finger pushing mechanism and is similar in form factor to the trumpet, a musical instrument which also uses finger flexion motion to take input, but different in that our device uses coupled finger flexion. The central handle is also a means to ground against the kinesthetic feedback from the device. The two triggers on the device are mechanically coupled (i.e., when one trigger is pushed in, the other trigger is automatically pushed out and vice versa) via the cable drive train. The linear triggers have low friction guide rails which hold them in the same axial position and a cable drive train which transmits power from the central shaft to the triggers. The cable drive uses spectra cable which allows for smaller bend radii enabling us to build a more compact design. The cable drive train has terminations on the the triggers as well as on the shaft. The terminations on the triggers can be used to re-tension the drive train in case it is affected by creep in the cable or wear and tear of the plastic parts. The central shaft is connected to an ironless core brushed DC motor via a flexible shaft coupling which corrects for any misalignment between the drivetrain shaft and the motor shaft. We use a Maxon ironless core brushed DC motor (Model No. 339156) which can provide a maximum continous torque of upto 32.3 mNm. This along with the drive train's reduction gives us a force of up to 15 N at the trigger. The motor also has a 4096 CPT encoder which is used to measure the user's input. The motor was also used to generate haptic sensations at the device triggers. We use Copley Junus JSP 90-20 amplifiers along with a TI C2000 series Piccolo 29069M Launchpad to control the system. The system runs at a 1000 Hz in an application written using simulink in an impedance control mode. We also have a button on the device which can be used to take in an additional input from the user if needed. III. SYSTEM PERFORMANCE EVALUATION To evaluate the performance of our device, we conducted a series of experiments to evaluate (1) the rendering force and stiffness, (2) the friction (i.e. transparency), and (3) the rendering bandwidth of our prototype device. The experiments and their results are described in the following sections. A. Maximum Rendering Force and Stiffness To assess the device performance, we experimentally evaluated the maximum uncoupled stable rendering stiffness [12], [13] . We perform an experiment wherein we incrementally increase the stiffness rendered by the device in discrete steps and check for stability at each step. Stability is assessed by physically perturbing the finger triggers of the device through user touch and observing whether the system stabilizes. We can qualitatively asses stability by looking at the vibrations in the system i.e. if the vibrations are increasing in amplitude, the system is deemed unstable. To reduce the bias of any single user interaction, we performed the testing with several cohorts. The average maximum stable rendering commanded stiffness was 1.06 N/mm while the maximum stable rendering commanded force was 15.9 N. B. Friction tests to assess transparency To evaluate the transparency [14], we performed an experiment to measure the friction of the drive train. Specifically, we applied a slowly increasing motor torque and identified the torque level at which the motor and drive-train initiated movement. This experiment was repeated at various motor positions to account for friction variation as a function of device configuration. The commanded motor torques varied from 0.177 to 1.029 mNm, with an average torque of 0.665 mNm. The equivalent (reflected) friction force at the finger interface, taking into account the drive-train reduction, varied from 0.05 to 0.343 N with an average frictional loss of 0.22 N. For comparison, other handheld haptic devices like the CLAW [3] and [15] report frictional losses (or a minimum transparency) of 0.5 N and 0.54 N respectively. C. Rendering Bandwidth The frequency range over which a haptic device can accurately display forces, with minimal magnitude and phase distortion, is referred to as the rendering bandwidth. A large rendering bandwidth is important for realistic user perception [16] . To achieve a large rendering bandwidth, it is important to design a stiff drive-train such that there are no structural vibration modes present within the desired bandwidth, typically up to 100 Hz for a high performance interface. To evaluate the rendering bandwidth, we measured the frequency response of the device drive-train. The frequency response was obtained by measuring the motor position output using the 4096 CPT encoder in response to an applied torque chirp signal. A virtual stiffness was overlaid on the torque chirp to maintain centering of the device. To provide a broadspectrum evaluation, we performed an experiment to measure the frequency response across both low frequency and high frequencies. In the experiment, the disturbance chirp signal was varied from 0.1 -100 Hz over a 30 second interval. The test was repeated 10 times. The results are shown in Figure 3. As seen in Figure 3, the low frequency magnitude response is approximately flat up to 4 Hz, after which it increases as it approaches the resonance created by the introduction of the virtual centering spring. The flat portion of the curve is, as expected, approximately equal to the inverse of the rendered stiffness (0.5 rad/mNm). The high frequency magnitude response shows a peak at approximately 8 Hz, which corresponds to the induced resonance the results from the introduction of the virtual centering stiffness. Importantly, there is no evidence of drive-train flexible modes below 100 Hz. We can estimate the reflected inertia of the device from the high frequency magnitude of the frequency response which is approximately equal to 7.91 x 10-4 mNm/rad/s 2 . A low value of reflected inertia infers highly transparent device. IV. USER STUDY To assess the efficacy of the proposed device, we conducted a user study. Specifically, our study aimed to assess the accuracy and perceived usability of the proposed device across a range of common one degree-of-freedom tasks. A. Study Design Our study compared our device (referred to as the handheld condition going forward) to a grounded rotary input (i.e., knob, see Figure 4) in a within-subjects design where the order of conditions (i.e., the device used for each task) was counterbalanced. We chose the knob as a baseline based on its prevalence in haptics (e.g., DC motors) and society (e.g., control panels, thermostats). Input devices like knobs are typically used to give more precise inputs as suggested by prior studies [17] and standards [18]. In this preliminary evaluation, we focused on using the inputs for two differential control applications: reaching target locations and tracking trajectories. Accordingly, both input devices were programmed to render a static haptic stiffness around a center point, similar to a joystick. B. Participants Our study involved 11 part:Ic1pants (6 male, 5 female), aged 18-23 (M = 19.27, SD = 1.60) recruited from the university campus. All participants were right handed. None of the participants identified as having extensive experience with haptics. Participants were paid $15 an hour. C. Procedure After providing informed consent, participants were briefed on the structure of the experiment. The participants then completed a targeting task for each condition followed by a tracking task for each condition. The details of the tasks are presented in the next section. After completing each condition, participants filled out the NASA TLX [19] and the System Usability Scale (SUS) questionnaire [20]. The order of the conditions was counterbalanced across participants. For each task, the participants trained with the input device before collecting test data. Following the two tasks, participants completed a brief demographics survey and completed questions on qualitative data about the devices answering questions about their comfort levels with each device for both tasks. The study procedure lasted approximately one hour. The study was administered under a protocol approved by the university Institutional Review Board (IRB). D. Apparatus The knob condition used a Maxon motor (model no. 320165) with a rotary handle attached to the shaft. An encoder attached to the motor was used to measure the user's input. Both conditions were operated as differential inputs using a haptic overlay. For both the tasks, both input devices were programmed to render a static stiffness of 7.5 mNm/rad for the knob and 7.4 mNm/mm for the handheld haptic device which centered the knob and the triggers of the device respectively. A stiffness value that was deemed comfortable for the task was chosen through testing by the authors. This stiffness is what provides the force feedback to the user through the motors in each device, the farther the user moves from the center point of each device, the greater the force. The knob was mapped to move the object on the screen left and right by rotating the knob in those directions. Whereas, the handheld device was mapped to move the object left and right by pushing each trigger in(the upper trigger to move right and lower to move left). E. Tasks & Measurements 1) Targeting Task: We assessed the targeting performance of each condition through a Fitt's law experiment. In the targeting task, shown in Figure 5, participants used the input device to move a cursor to a designated goal location. The position and size of the goal varied between tests. Participants were instructed to reach the target as fast as possible. When possible, the design of the Fitt's law experiment followed the ISO standard proposed by Mckenzie [21]. In each trial, participants pressed a key to begin and the trial was stopped once the target was reached ( defined as when the cursor was within the target for two seconds). For each condition, participants completed 30 training trials and 60 test trials. From each trial, we calculated the index of difficulty (ID) and used it to calculate the throughput (T P) of the input device. A ID= log2(W + 1) TP= ID MT (1) where A is the amplitude of movement to the target, W is the width of the target, and MT is the time to reach the target. A higher throughput corresponds to better device performance for targeting tasks. 2) Tracking Task: We assessed the tracking performance for each condition through a series of sinusoid tracking tests. As shown in Figure 6, participants used the input device to follow the desired trajectory and we tracked the absolute value of error between the reference trajectory and participant cursor at each time-step. The frequency and amplitude of the trajectory were varied during the trials. Participants were instructed to follow the trajectory as closely as possible. To enable direct comparisons of performance across the conditions, we selected two different amplitudes and two different frequencies for the sinusoidal trajectory. The two frequencies, 0.3 Hz and 1 Hz (referred to as the low frequency and high frequency), were selected within the bandwidth of human control [22]. In the tracking task, each participant performed 16 training trials (i.e., 4 trials of each combination of amplitude and frequency) before completing four test trials. The four trials were completed without any break (e.g., the second sinusoid started immediately upon the completion of the first sinusoid). The order of the effects were randomly generated and different between the training and testing trials. Each sinusoid consisted of 4 full periods of oscillation for the low frequency trajectories and 8 full periods of oscillations for the high frequency trajectories. F. Results The results for the targeting task were analyzed using a paired two-tailed t-test. We found a significant effect on throughput for the handheld device (M = 1.65, SD = 0.70) compared to the knob (M = 2.05, SD = 0.60, t(lO) = 11.48, p < 0.05, d = 0.59) with the knob having a higher throughput. The raw NASA TLX score for this task was 30.45 on average for the knob and 38.48 on average for the handheld haptic device. We perform a paired two-tailed T-test on the TLX scores and find that the haptic device(M = 38.48, SD= 16.36) has significantly higher scores than those for the knob (M = 30.45, SD= 16.36, t(lO) = 2.28, p < 0.05), . The error results of the tracking task were analyzed using a three-way repeated measures ANOVA with device (haptic device, knob), frequency (high, low), and amplitude (high, low) as factors. The results are shown in Table I. We find no significant effect of the device on the user error for this task. We do, however, find a significant effect of the The raw average error values per segment are as shown in the Table II below. The SUS score for our proposed handheld haptic device was 86.60 (M = 86.60, SD = 3.88) while that of the knob was 87.24 (M = 87.24, SD = 2.57). This falls in the excellent category in the SUS scale for both devices. We perform a two-tailed paired T-test on the SUS scores as well and find no statistically significant difference between the scores of the two devices (t(lO) : 0.90, p = 0.37). V. DISCUSSION There are several key takeaways from the user study. In the targeting task, we observed that our handheld haptic device did not perform as well as the knob in terms of throughput. Some participants appeared to struggle with the spatial mapping with the handheld haptic device (i.e., which trigger moves the cursor up or down on the screen) and would move the object in the wrong direction before correcting themselves. This effect was not observed when participants used the knob which was both more familiar to participants and grounded to the table in an orientation that was spatially consistent with the task. In the future, we are interested to explore whether the performance with the proposed device could be improved through additional training. The amount of training trials in our experiment was limited to prevent participant fatigue. In the tracking task, the knob generally had lower average errors than our handheld haptic device for all frequencies and amplitudes. However, the average error varied by less than 13 pixels between the two devices. As seen in Figure 7, much of the difference in performance can be attributed to overshoot by the handheld haptic device, which was not observed when participants used the knob. Looking closely at the high-frequency high amplitude segment of the data in Figure 7, we can see that users commonly undershoot the desired motion when using the knob as compared to our haptic device. However, we see higher overshoots with the haptic device for high-frequency high amplitude trajectories as 1200 r ---= Lo'-" w'--' F -'-' re= ue "" n"'" c ,_H "' i""h -'-' A -" m = li"" tu"'-de =---~ r -= Lo '-" w'--' F -'-' re "' ,a"" ue "' n"' cv,_L == o'-" w--'-A"-' m-"' ,o-"' Jit "" ud ==e '--~ r----' H "' i"-' h '-' F -'-' re = ue "' n"' c'--'-" Hi "'-h "--A '-"m -'-'= li "" tu'-= d-=-e ~ r----'H '-"i-"-' ah -'-' F '-'-re == •a-= u= en "' c-'-=' vLo '-" w'--' A -"' m "" ,o"'-lit '-= u"" de =---~ -True Path well. We can infer from these observations that the handheld haptic device may be more suited to tasks where reaching the peaks of the trajectory is more important than controlling the overshoot. The overshoot might also suggest that the haptic overlay was not optimal for the handheld haptic device. In the future, we are interested in exploring the impact of the overlay on user performance. In terms of usability, both the proposed device and the knob scored in the excellent category for the SUS [20] and the difference between the average usability scores for the devices was only 0.64 percent. Users who play video games frequently (at least once per week) expressed a high degree of satisfcation with the form factor of the handheld haptic device while users who had little or no experience with gaming found the device to be fatiguing. VI. CONCLUSION In this paper, we designed and evaluated a handheld haptic device that incorporates a bidirectional coupled finger flexion interface. Through experimental validation, we demonstrated that the proposed device can render high-performance haptic effects, as measured by its transparency and rendering bandwidth. Through a user study, we showed that participants rated the proposed device as highly usable and assessed the device with a series of targeting and tracking tasks. While the performance of the device was inferior to that of a haptic knob in some instances, the overall assessment was positive. The results suggest a range of modifications to be considered in future work including improvements to the ergonomic form factor, exploration of alternative haptic effects, and evaluation of these modifications on the overall device and user performance. Specifically, we are interested to test the device in simulated and realistic scenarios. Examples include industrial process control [23] and shared autonomy for collaborative robots [24]. VII. ACKNOWLEDGEMENT We would like to thank Patrick Dills for his feedback on the mechanical design, Pragathi Praveena for her feedback on study design and Yash Wani for his assistance with programming. Fig. 1 . 1Our proposed handheld haptic device with two triggers that take input via finger flexion. The triggers are mechanically coupled (i.e., when one trigger is pushed in, the other pushes out). Fig. 2 . 2Internal (a) view, zoomed internal view (b), and cross-sectional view (c) of the device showcasing the drive-train and related design elements. :Fig. 3 . 3Estimated J :::: 7 .91 * 1 O" mNm/ rad /s~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · . . . : .. ' .. ~ . : . . . . ....... . .... . ... · .... · .. . .·. -:--: -:-Frequency response of the proposed device. The only peak in the magnitude plot occurs due to the rendered stiffness. Fig. 4 . 4Knob used as the comparison input device in our study. Fig. 5 . 5Targeting task following ISO 9421-9 to calculate throughput of device. Fig. 6 . 6Task to assess device performance for tracking. amplitude (F(l , 8) = 83.85, p < 0.001) and the frequency (F(l, 8) = 15.57, p < 0.001) of the path on the user error. The average raw NASA TLX score for this task was 42.12 (M = 42.12, SD = 17.42.) for the knob and 42.42(M = 42.42, SD = 16.56) for the handheld haptic device. We perform a paired two-tailed t-test on the TLX scores and find no significant difference (t(lO) = 0.55, p = 0.58) between the scores of this task for both devices. Since the results of the ANOVA do not show any significant relation to the device used, we do not perform any post hoc analysis on this data. Fig, 7 . 7' -------------' ,_ _ _ _ _ _ _ _ _ _ _ __, ..__ _ _ _ _ _ _ _ _ _ _ __, ,_ _ _ _ _ _ _ _ _ _ _ __,1200r -----------~ r ----------~ r -----------~ r ---------Average errors and standard deviations for trajectory tracking with the handheld haptic device and the knob. grant awarded to the UW-Madison and The Boeing Company (Cooperative Agreement# 80NSSC19M0124) and by DGE -2152163 (NRT) Integrating Robots into the Future of Work Robert Radwin is with the Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison 53706, USA [email protected] Michael Gleicher, and Bilge Mutlu are with the Department of Computer Sciences, University of Wisconsin-Madison, Madison 53706, USA [gleicherlbilge]@cs.wisc.edu1 Megh Vipul Doshi, Michael Hagenow and Michael Zinn are with the Department of Mechanical Engineering, University of Wisconsin-Madison, Madison 53706, USA [megh.doshilmhagenowlmzinn]@wisc.edu 2 3 PUSH BUTTON FOR ADDITIONAL INPUT BRUSHED DC MOTOR FOR HAPTIC TABLE I THREE I-WAY ANOVA RESULTS Independent Variables F Value Num of DF Den DF PR>F Device 0.798 1.0 8.0 0.398 Frequency 19.671 1.0 8.0 < 0.05 Amplitude 62.447 1.0 8.0 < 0.001 Device x Frequency 1.182 1.0 8.0 0.309 Device x Amplitude 0.011 1.0 8.0 0.920 Frequency x Amplitude 6.056 1.0 8.0 0.039 Device x Frequency x Amplitude 0.001 1.0 8.0 0.976 TABLE II AVERAGE ERROR PER S EGMENT Trajectory Segment Device Mean Error Standard Deviation Low Frequency Low Amplitude Hand.held Haptic Device 33.12 15.52 Knob 29.19 10.76 Low Frequency High Amplitude Handheld Haptic Device 81.10 25.26 Knob 78.18 31.86 High Frequency Low Amplitude Handheld Haptic Device 51.04 13.04 Knob 38.17 14.85 High Frequency High Amplitude Hand.held Haptic Device 124.93 31.75 Knob 11 3.57 55.67 A coupling library for the force dimension haptic devices and the 20-sim modelling and simulation environment. 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[]
[ "Adverbs, Surprisingly", "Adverbs, Surprisingly" ]
[ "Dmitry Nikolaev [email protected] \nIMS\nUniversity of Stuttgart\nGermany\n", "Collin F Baker \nInternational Computer Science Institute\nBerkeleyUSA\n", "Miriam R L Petruck \nFrameNet Corresponding Author\n\n", "Sebastian Padó \nIMS\nUniversity of Stuttgart\nGermany\n" ]
[ "IMS\nUniversity of Stuttgart\nGermany", "International Computer Science Institute\nBerkeleyUSA", "FrameNet Corresponding Author\n", "IMS\nUniversity of Stuttgart\nGermany" ]
[]
This paper begins with the premise that adverbs are neglected in computational linguistics. This view derives from two analyses: a literature review and a novel adverb dataset to probe a stateof-the-art language model, thereby uncovering systematic gaps in accounts for adverb meaning. We suggest that using Frame Semantics for characterizing word meaning, as in FrameNet, provides a promising approach to adverb analysis, given its ability to describe ambiguity, semantic roles, and null instantiation.
10.48550/arxiv.2305.19650
[ "https://export.arxiv.org/pdf/2305.19650v1.pdf" ]
258,987,976
2305.19650
fd34d8536ea9f8c9cb1d9d13d1910682c78cbc89
Adverbs, Surprisingly Dmitry Nikolaev [email protected] IMS University of Stuttgart Germany Collin F Baker International Computer Science Institute BerkeleyUSA Miriam R L Petruck FrameNet Corresponding Author Sebastian Padó IMS University of Stuttgart Germany Adverbs, Surprisingly This paper begins with the premise that adverbs are neglected in computational linguistics. This view derives from two analyses: a literature review and a novel adverb dataset to probe a stateof-the-art language model, thereby uncovering systematic gaps in accounts for adverb meaning. We suggest that using Frame Semantics for characterizing word meaning, as in FrameNet, provides a promising approach to adverb analysis, given its ability to describe ambiguity, semantic roles, and null instantiation. Introduction Adverbs are the part of speech (POS) that has seen the least attention in (computational) linguistics, likely due to its challenging nature (Conlon and Evens, 1992). As Huddleston and Pullum (2002, 563) state, "the adverb is a [. . . ] residual category [. . . ] to which words are assigned if they do not satisfy the more specific criteria for nouns, verbs, adjectives, prepositions, and conjunctions." Syntactically, they modify many POSs, except nouns (eat porridge quickly, hardly noticeable), or even complete clauses (Probably, I'll come tomorrow). They are semantically varied (Thomason and Stalnaker, 1973), ranging from intensifiers/modifiers (absolutely, beautifully) to temporal and spatial specifications (yesterday, forward), to so-called speaker-oriented adverbs yielding inferences about speaker attitudes, beliefs, and evaluations. Finally, adverbs can occupy different positions in sentences, creating complex issues of scoping and ambiguity (Alexiadou, 2004;Payne et al., 2010). Consider the following sentences: 1 (1) a. Happily, they watched TV until dinner. b. They happily watched TV until dinner. c. They watched TV happily until dinner. d. They watched TV until dinner happily. 1 Huddleston and Pullum (2002, 575) While language users tend to interpret Ex. 1b-1d as describing the TV watchers' mental state, Ex. 1a is ambiguous and can also be read as a positive evaluation of the situation by the speaker. In sum, adverbs provide crucial information not just about the where and how of events, but also about attitudes and evaluations. However, relatively little research on adverbs exists in computational linguistics, although lexical factors are generally recognized as central for many NLP tasks (Berger et al., 2000). Lexical information is generally represented either in online dictionaries or by embeddings extracted from corpora (Turney and Pantel, 2010;Devlin et al., 2019;Peters et al., 2018). As a dictionary, WordNet (Miller et al., 1990) lists adverbs but only provides a relatively impoverished account, while lexicons for sentiment analysis (Benamara et al., 2007;Dragut and Fellbaum, 2014) and hedging detection (Jeon and Choe, 2009;Islam et al., 2020) only consider specific subtypes of adverbs as to how they modulate the intensity of adjectives. On the distributional side, adverbs have been considered from a derviational perspective (Lazaridou et al., 2013); yet, they are rarely scrutinized in detail. Among the standard benchmarks, only GLUE (Wang et al., 2018) and BLiMP (Warstadt et al., 2020) cover adverbs, and then only marginally. The same is true of approaches that combine dictionaries and embeddings (Faruqui et al., 2015). As a consequence, SOTA language models consistently struggle with adverb meaning, as Section 2.2 will demonstrate empirically. This paper argues that Frame Semantics (Fillmore, 1985), as realized in FrameNet (FN) (Ruppenhofer et al., 2016), provides an efficacious framework to articulate the relevant aspects of adverb meaning. Specifically, as Ex. 1 illustrates, lexical ambiguity is captured in terms of frame ambiguity. Moreover, inferences about the arguments of adverbs, typically filled by the speaker and the lexical unit that the adverb modifies, can be cap-tured and characterized via the frame elements (i.e. semantic roles) of the frame. Notably, FrameNet mechanisms will account for null-instantiated roles, allowing it to hint at unexpressed content in cases like Example 2b (v. Section 4.2 for details). (2) a. [ In such cases specifically, FrameNet considerations of frame element realization help to explain the absence of the SPEAKER semantic role in 2b. Plan of the Paper. Section 2 defines the scope of this paper (speaker-oriented adverbs) and shows the lack of accounts for adverbs in NLP through a literature review. Section 3 presents a probing dataset for speaker-oriented adverbs on the basis of which it demonstrates empirically that current large language models do not provide accounts for adverb meaning. Section 4 provides general background information on FrameNet, gives details on the framework's approach to the description of adverb meaning, and suggests its use to improve NLP models. Section 5 concludes the paper. Scope and Motivation Scope Given the variety and heterogeneity of adverbs, we restrict the empirical scope of this paper to a subclass of them -even though we believe that the conceptual points apply to adverbs generally. We focus on speaker-oriented adverbs (Ernst, 2009). This broad class of adverbs, itself comprises several subtypes brought together by their giving rise to a range of inferences about attitudes and beliefs of the speaker, such as epistemic beliefs (Ex. 3), evaluations (Ex. 1 and 4), and speech acts (Ex. 5): (3) Peter says: "Paul is certainly right". |= Peter is certain that Paul is right. (4) Peter says: "Unfortunately, Paul arrived". |= Peter is unhappy that Paul arrived. (5) Peter says: "Frankly, Paul annoys me." |= Peter voices his frank opinion. Structurally, these entailments are similar to entailments that arise from implicative verbs (Karttunen, 1971). As sources of information about how speakers assess states of affairs, they are highly relevant for tasks like opinion mining (Pang and Lee, 2008) and stance detection (Thomas et al., 2006). However, while implicative verbs have received considerable attention in the context of textual entailment (Karttunen, 2012;Lotan et al., 2013), speakeroriented adverbs have not. Treatment of Adverbs in Computational Linguistics This section summarizes work on adverbs in computational linguistics in the four most relevant areas: WordNets, applications, distributional modeling, and semantic annotation. Section 3 covers large language models separately. WordNets. Princeton WordNet (WN, version 1.3) (Miller et al., 1990) covers about 4,500 English adverbs, comprising both single words and adverbial multi-word expressions like a priori. The information recorded includes senses (although most adverbs are monosemous) and semantic relations: almost all single-word adverbs are linked to the adjectives from which they are derived,and some adverbs have antonyms. However, WN has no information on the adverbs' syntactic or semantic behavior. The approach of corresponding Word-Net resources varies substantially: GermaNet, for German, does not treat adverbs at all (Hamp and Feldweg, 1997). In contrast, plWordNet (Maziarz et al., 2016) provides a considerably richer description of adverbs, notably regarding lexical relations, but is only available for Polish. NLP applications. Apparently, sentiment and emotion analysis are the NLP applications that have paid the most attention to adverbs (Benamara et al., 2007;Dragut and Fellbaum, 2014;Chauhan et al., 2020). Hedge detection, that is, the recognition of expressions that modulate speaker confidence in their statements boasts additional work on adverbs (Jeon and Choe, 2009;Islam et al., 2020). However, these studies, are generally limited to two specific subtypes: scalar adverbs that modify sentiment strength (intensifiers/minimizers: very/hardly nice) and adverbs that modify confidence (certainly/apparently). Haider et al. (2021) also considers locative and temporal adverbs. Confidencemodifying adverbs form a subtype of the speakeroriented adverbs addressed here, but existing studies do not offer a general account of these adverbs beyond the requirements of specific tasks. Studies on structured sentiment and emotion analysis (Barnes et al., 2021;Kim and Klinger, 2018) assume a different perspective. These works concentrate on defining and modeling the relations between sentiment-and emotion-introducing expressions and their semantic arguments, such as the experiencer of the affect and its target. As the comparison with Example 2 shows, these relations are at times tied to adverb meanings. However, we are not aware of studies in this area that deal specifically with adverbs. Distributional modeling. A number of studies investigated the interplay between word embeddings and morphology, analyzing similarity by parts of speech (Cotterell and Schütze, 2015) or investigating meaning shifts corresponding to morphological derivation (Lazaridou et al., 2013;Padó et al., 2016). Typically, these studies include adverbs, and not surprisingly find that adverbs behave highly inconsistently. Semantic annotation. In principle, frameworks for the annotation of (semantic) argument structure are promising sources for information about adverb meaning, but they differ widely in the information that they offer. The PropBank (Palmer et al., 2005) annotation scheme offers a range of modifier roles (ARGM) for the annotation of modifiers, including adverbs. However, the most fitting of these roles, ARGM-ADV, is a "catch-all" category. In addition, the PropBank analysis does not treat adverbs as predicates in their own right and does not assign roles to them. Thus, fortunately, she accepted and even she accepted would receive the same analysis. In contrast, UCCA (Abend and Rappoport, 2013) explicitly splits adverbs into adverbial modifiers proper (D) and ground elements (G), where the latter expresses the speaker's attitude toward the event. However, UCCA does not make the structural relations explicit either. AMR (Banarescu et al., 2013) offers a more nuanced approach: many adverbs are mapped to their underlying predicates and endowed with complete argument structure, 2 while others are interpreted as degree, manner, or time modifiers. However, no provision exists in the representation for speakeroriented adverbs. To illustrate, the AMR annotation of thankfully, she accepted the present either treats the adverb as describing a general state of affairs (it is good that she accepted) or simply omits it. Finally, Frame Semantics (Fillmore, 1985) offers the conceptual infrastructure to improve on these treatments and avoid their limitations. Section 4 provides justification of this understanding. Case Study: Modeling Adverb Meaning as Natural Language Inference One possibility, so far not mentioned, is that the knowledge inherent in large neural language models might provide a sufficient account of the meaning of (speaker-oriented) adverbs. In that case, at least from the NLP perspective, no (new) specific treatment would be required. However, this state of affairs is not the case, as we show below. Creating Probing Datasets To operationalize "a sufficient account," we ask language models to distinguish between valid and invalid inferences along the lines of Examples 3-5. As input data, we constructed probing examples with inferences for speaker-oriented adverbs. We examined four classes of adverbs, motivated by current FrameNet frames containing adverbs (see Section 4.3 for details). These are: likelihood adverbs (e.g. undoubtedly, probably); unattributedinformation adverbs (reportedly, allegedly, supposedly); degree adverbs (at least, approximately); and obviousness adverbs (blatantly, conspicuously). We built the datasets from combinations of premises and hypotheses containing such adverbs, formulated as templates with sets of fillers for the adverbs and different participant positions. In this manner, we assessed the LM's capabilities irrespective of specific word choice. We paired each premise with two to four unambiguous hypotheses depending on the adverb class. The premise either implies or contradicts the hypothesis. Table 1 shows an example. Hypothesis 1 negates the premise and constitutes a contradiction. Hypothesis 2 is a valid inference about speaker evaluation; and Hypothesis 3 is a valid inference about the uncertainty inherent in the premise. We report studies on two datasets with different emphases. We designed the first to be naturalistic, based on existing sentences for adverbs in FrameNet. Given the limited size of this dataset, we also created a larger synthetic dataset with simpler, more varied, sentences. The Appendix lists full details on both datasets. Naturalistic Dataset. As stated, we created this dataset based on sentences in the FrameNet database containing adverbs of the four classes enumerated above. We "templatized" the sentences Premise The celebration had been postponed, ostensibly because of the Gulf War Hyp 1 The Gulf War ostensibly had no effect on the celebration (CONTRADICTION) Hyp 2 Someone said that the celebration was postponed because of the Gulf War (ENTAILMENT) Hyp 3 The Gulf War may have had no effect on the celebration (ENTAILMENT) by treating the position of the adverb as a slot that can be filled by all semantically congruent adverbs from the respective class. In sentences where the subject is a personal name, we also treated the subject position as a slot, which we filled with twenty female and male names popular in the United States. Because the low number of sentences of the each type in the FrameNet database, and most templates have only one slot, viz. the adverb, the size of this dataset is limited. See Table 3 for example counts by adverb class. Synthetic Dataset. The goal of this dataset was to test if the performance of the model is robust with regard to the replacement of the main-event description and varying syntactic complexity of the premises and hypotheses. It covers three of the four adverb classes: unattributed-information, degree, and obviousness, where the templates from the first dataset were most restricted. In these templates, subjects are always exchangeable. In addition, we also varied the description of the main action or relation described the sentence. Table 2 shows the template set for unattributedinformation adverbs. The set of adverbs for this class comprises reportedly, allegedly, supposedly, apparently, and ostensibly. Fillers of the ACTION slot include both gerund phrases (e.g. selling the house) and noun phrases (e.g. the wedding). Entailments and contradictions are produced in pairs. For entailments, we test two valid inferences triggered by the adverb. For contradictions, we test embedded clauses with and without negation. Table 5 shows the example count for each input type. Probing Setup: NLI models Arguably the best match for these types of datasets are the family of language models optimized for the task of natural-language inference (Storks et al., 2019). Concretely, we evaluated the series of NLI models released by Nie et al. (2020) SNLI or Stanford Natural Language Inference models. These models carry out a three-way classification between ENTAILMENT, CONTRADICTION, and NEUTRAL. The author fine-tuned their models on a data set created in an iterative, adversarial, human-in-the-loop fashion, designed to remedy the shortcomings of previous NLI datasets (Belinkov et al., 2019). Preliminary experiments with different available base architectures (RoBERTa, AL-BERT, BART, ELECTRA, and XLNet) showed that RoBERTa-large 3 was the best-performing variant. Thus, we used this model for evaluations. We used our probing datasets solely for evaluation, not for further fine-tuning. For analysis, we checked the labels that the model predicted with their corresponding probabilities. In several cases, we performed additional tests to verify whether the adverbs or other properties of the sentence determined the model predictions. Table 3 shows overall results of the SNLI model on the naturalistic dataset for the four adverb classes. The adverb classes are not strictly comparable because they are represented by different input sentences (as described above), which include all types of lexical and syntactic confounds. Nevertheless, our experiments showed two consistent results: (i) the model cannot correctly draw inferences based on some set of adverbs on which it fails; (ii) the presence of adverbs increases the difficulty for the model to draw correct inferences in general. What follows is a survey of the evidence for these two claims. Evaluation on a Naturalistic Dataset Overall results Failure to Understand Adverbs Degree adverbs. The model does not understand that at least as big is incompatible with smaller. While it correctly labels the pair Lantau covers nearly twice the area of Hong Kong Island -Lantau is at least as big as Hong Kong Island as EN-TAILMENT and the same premise with Lantau is much smaller than Hong Kong Island as CONTRA-DICTION, it considers that this premise also entails Hong Kong Island is at least as big as Lantau, which is also a straightforward contradiction. The quantifier-adverb combination almost every constitutes another weak point of the model. While it correctly labels the pair Almost all assignments are challenging in different ways vs. Most of the assignments are difficult, it labels Almost every assignment is a challenge in a different way vs. the same as NEUTRAL. 4 Unattributed-information adverbs. The correct analysis of these adverbs is subtle since valid inferences may be expressed in ways that differ from the premise both lexically and syntactically. Sometimes the model answers incorrectly with extremely high confidence. The example from Table 1 is a case in point. The Gulf War ostensibly had no effect on the celebration is always correctly labeled as CONTRADICTION. The Someone said... hypothesis is also correctly labelled as ENTAIL-MENT with any adverb in the premise. Strikingly, the model gives the same result when the adverb is omitted. This suggests that the model does not take the adverb in the premise into account. The experiments with Hypothesis 3 (cf. Table 1) corroborated that understanding: regardless of the combination of the adverb in the premise and the hypothesis, the model confidently marks the pair as CONTRADICTION or NEUTRAL with almost zero probability attached to the prediction of ENTAIL-MENT. This finding shows that while the model may be able to draw a positive inference from the hearsay adverb (the reported event may have happened), it is completely unaware of the possibility of the negative inference, i.e. that the reported event 4 The model answers correctly only when there is a larger lexical overlap, as in Most of the assignments are challenging. may not have taken place: 12 times out of 16, the model confidently predicts CONTRADICTION. Adverbs Complicate Inference In another analysis, we investigate the impact of the sentences' structural complexity on prediction quality. We frequently found that the model correctly inferred when the hypothesis is structurally simple or no adverb is given, but failed when the hypothesis had an embedded clause and the premise had an adverb. Table 4 shows a concrete example, which permits three observations: 1. The model is sensitive to whether the hypothesis contains an embedded clause: the confidence for the correct prediction drops from ≈1 to ≈0.8 for all verbs in the no-adverb case. 2. The presence of the adverb is not noticeable with structurally simple hypotheses: the confidence in the correct answer remains >0.9. 3. The combination of an adverb and an embedded clause can derail the model -paradoxically most so for the verb support, where the model was most confident without an adverb. Furthermore, note that an adverb can force the model to change its decision even in the presence of a strong lexical cue. Given the hypothesis The students were obviously drunk, the model correctly identifies The students abhor/forswore/renounced alcohol as CONTRADICTION. While the model labels The students abjured alcohol as ENTAIL-MENT, (perhaps) because of an incorrect analysis of the verb, when we change the hypothesis to The students were conspicuously drunk, the model confidently and correctly labels The students abjured alcohol as CONTRADICTION. Evaluation on a Synthetic Dataset The results for the application of same model on the larger synthetic dataset are shown in Table 5. They demonstrate that in general the task of drawing correct inferences from adverbs is very difficult for the model. Instead, the model tends to consistently predict the same relation (entailment / neutral / contradiction) for all sentences for an adverb class. It is able to correctly predict inference for the quantity degree class (at least two dozen people |= many people and ̸ |= nobody). However, even syntactically trivial entailments and contradictions in other classes lead to systematic failures. E.g., while the model can correctly identify the inference James said that Mary reportedly opposed the wedding |= James said that Mary may have opposed the wed- ding, it fails on the entailment of the type James is not sure that Mary opposed the wedding. Similarly, with obviousness adverbs, while the examples of the type James blatantly criticized Mary |= James disparaged Mary are easy for the model, entailments like James tried to disparage Mary leads to near-chance performance. In the domain of adverb-modulated relations, while the model seems to do well on entailments (James is at least twice as rich as Mary |= James's net worth is at least as big as Mary's), in fact it does not understand that the relation is not symmetric and therefore cannot correctly identify contradictions (Mary's net worth is at least as big as James's). Discussion Taken together, these experiments demonstrate systematic shortcomings in the ability of current large language models to account for adverb meaning, either glossing over them completely or making rather random inferences about their meaning. Arguably, this study only looked at a specific type of language model and other types of language models would fare better. However, converging evidence from the literature exists. For instance, Nikolaev and Padó (2023) analyzed sentence transformers, which might be expected to provide the most nuanced understanding of adverbs. Instead, the study found that the sentences' main participants (subjects and objects) primarily determine the semantic similarity of sentence pairs, which is largely independent of adverbs. The paper argues that this behavior arises from the structure of the training data for sentence transformers (online conversations, duplicate questions on WikiAnswers), where sentence pairs labelled as se-mantically similar often have similar sets of main participants (subjects and objects) and can vary widely in other respects. If a similar bias is at play in the NLI models in the present study, creating larger, richer training sets that involve adverbs in a systematic manner is a way forward. However, given the relative scarcity of adverbs and their complex behavior (cf. Section 1), it seems unlikely that this effect will emerge naturally by pre-training on ever larger datasets. Instead, the evidence provided here indicates that adverb data must be created intentionally. The following section outlines a proposal to do so. Describing Adverbs in FrameNet This section will provide a brief background to FrameNet (Section 4.1), show how FrameNet can be useful for the analysis of adverbs (Section 4.2), survey the data on adverbs contained in the current version of the dataset (Section 4.3), and propose concrete directions for next steps (Section 4.4). Background to FrameNet FrameNet (FN, Ruppenhofer et al. 2016) is a research and resource-development project in corpusbased computational lexicography grounded in the theory of Frame Semantics (Fillmore, 1985). At the heart of the work is the semantic frame, a script-like knowledge structure that facilitates inferencing within and across events, situations, statesof-affairs, relations, and objects. FN defines a semantic frame in terms of its frame elements (FEs), or participants (and other concepts) in the scene that the frame captures; a lexical unit (LU) is a pairing of a lemma and a frame, characterizing that LU in terms of the frame that it evokes. FN frames may include more than one POS, and FrameNet does not claim that the LUs of a frame are synonymous, merely that they are semantically similar in referring to the same situation. Additionally, FN distinguishes between core FEs and non-core FEs; the former uniquely define a frame and the later identify concepts that characterize events or situations more generally, such as time and place. To illustrate, Example 6 shows annotation for the verb BUY, defined in the Commerce_buy frame, with the FEs BUYER, SELLER, GOODS, and MONEY. 5 (6) FrameNet annotators label approximately 20 sentences for each LU in each frame; and automatic processes tabulate the results to produce valence descriptions, or semantic-syntactic combinatorial possibilities of each LU. These also include nullinstantiated core FEs, i.e. FEs that uniquely define a frame, even when not realized linguistically. Such valence descriptions provide information about meaning-form mappings that are important for natural-language understanding. FrameNet data, or semantic parsers built from them, have proven useful for tasks such as recognizing paraphrases (Ellsworth and Janin, 2007), drawing inferences (Ben Aharon et al., 2010), machine translation (Zhai et al., 2013), question answering (Khashabi et al., 2018), or paraphrasing (Wang et al., 2019). At present, the FrameNet database (Release 1.7) holds 1,224 frames, defined in terms of 10,478 frame-specific FEs, and 13,686 LUs. Of those lexical units, 61% have lexicographic annotation, i.e. annotation for one target lemma per sentence. FrameNet for the Analysis of Adverbs We now outline how the descriptive devices of FrameNet, as outlined in Section 4.1, can capture the relevant facts about adverb meaning and address the core challenges of adverb classes, ambiguity, inferences, and null instantation of roles. Frames. Since frame definitions encompass much of the meaning of each LU, many FN frames already offer fine-grained, semantically motivated descriptions of adverb classes. For example, the Emotion_directed frame captures the semantic similarity of happy, happily, happiness, sad, and sadly and offers a starting point for the description of emotion-related adverbs, by exploiting the fact that these adverbs evoke the same background knowledge as the corresponding LUs of other parts of speech (Ruppenhofer et al., 2016). When a lemma is ambiguous, each sense gets mapped to a different frame; each mapping is a separate lexical unit (LU). For instance, Example 1 in Section 1 includes the lemma happily, which is ambiguous: In Example 1a, happily is defined in the Luck frame (along with fortunately and luckily). The definition of this frame indicates that there is someone, the PROTAGONIST, for whom a particular state of affairs is surprisingly good or bad. But this sentence does not express the PRO-TAGONIST; this is a case of null instantiation or NI (see below for details). The other three sentences, Examples 1b-1d, illustrate happily in the Emotion_directed frame. This involves an emotional response of someone, the EXPERIENCER, to a stimulus, the STIMULUS FE (here, watching TV), which evokes the emotional response, specifically happiness (recoverable from the definition of the LU happily). In these examples, the EXPERIENCER is explicit, so no inference is required (although coreference resolution will be required to resolve the referent of they). Example 7 shows the annotations of the sentences in the Luck frame (Ex. 7a) and in the Emotion_directed frame (Ex. 7b): (7) Frame Elements. In FrameNet, frame elements are associated with (classes of) inferences (Chang et al., 2002). Such inferences can capture important aspects of adverb meaning, as we have shown in Section 2. The valence patterns for the two senses of happily shown above lead to different inferences via the two sets of frame elements: Emotion_directed: An EXPERIENCER [feels or experiences] a particular emotional response to a STIMULUS or about a TOPIC. While such natural language descriptions were traditionally hard to formalize, the recent advances in "prompting" language models (Shin et al., 2020) have reestablished natural language descriptions as sufficient in many conditions (cf. also our templatebased probing dataset in Section 3). Null instantiation. FrameNet annotates information about the conceptually required "core" semantic roles of a frame even if absent from the text. FN distinguishes three types of null instantiation, one licensed by a construction and the others licensed lexically. FrameNet includes approximately 55,700 NI labels in its annotations; and roughly one-quarter of these omissions are licensed constructionally, with the remaining 75% licensed lexically (Petruck, 2019). This capability of FrameNet is particularly important for adverbs, notably speaker-oriented adverbs. By definition, these adverbs welcome inferences about the speaker, who is typically not realized unless the statement is part of reported speech or thought: The father thought: "Happily they are all watching TV." Returning to Example 2 (above), 2a illustrates an instantiated SPEAKER and 2b illustrates a nullinstantiated SPEAKER, a fact that FN records in its database. No other lexical resource used extensively in computational linguistics records such information. Current Status of Adverbs in FrameNet Currently, FrameNet (Release 1.7) contains 217 adverb LUs. Of these adverbs, 158 have annotation, with a total of 2,475 annotations of adverbs on sentences in the database, yielding a mean of 16 annotations per LU. However, like many linguistic phenomena, the annotations exhibit a highly skewed (Zipfian) distribution: 41 of the 158 LUs have only one annotation while nine have more than 50 annotations each. In line with its general principles, FrameNet chose not to define one single frame to capture all speaker-oriented adverbs, instead defining each such adverb according to the specific frame it evokes. At the same time, the class of speaker-oriented adverbs is arguably recoverable from the union of a set of frames all of which support inferences about the speaker by way of describing the speaker through a certain frame element. In this way, the existing frames and their annotations provide a suitable basis for creating data for this (and future) research. Table 6 shows the four FrameNet frames used to suggest adverbs for the experiment described in Section 3 together with the adverbs listed, illustrative example sentences, and their definitions. Next Steps As the numbers show (Section 4.3), FrameNet has not attended to adverbs either. Perhaps this fact represents a principal incompatibility: the description of adverbs may not welcome using concepts that FN developed for traditional predicates with clearcut valence. Yet, we believe that including adverbs in FrameNet both follows the spirit of what Fillmore (1985) called "semantics of understanding" and is in line with FrameNet practice. Still, it will require work on two principal levels: theoretical development and practical lexicographic analysis. A PHENOMENON is portrayed in terms of the DEGREE of likelihood that it will be perceived and known, given the (usually implicit) EVIDENCE, PERCEIVER, and CIR-CUMSTANCES in which it is considered. Degree a little (bit).adv, a lot. adv, absolutely.adv, as hell.adv, far.adv, fully.adv, in part.adv, kind of.adv, so.adv, somewhat.adv, that.adv, totally.adv, very.adv, way.adv Ex. I had ABSOLUTELY nothing to say. LUs in this frame modify a GRADABLE AT-TRIBUTE and describe intensities at the extreme positions on a scale. At the theoretical level, the FrameNet approach has seen constant development over the 25 years of the project's existence. In initial verb-centered frames, nominals tended to fill FEs, with additional attributes realized as adverbs. Next, FN added deverbal nouns to frames, which largely take the same frame elements. To expand to other types of nouns, like natural kinds and artifacts, FrameNet broadened the concept of FE to encompass qualia such as substance or purpose (Pustejovsky, 1991). Layering the annotation of nouns as FEs of verbs, and modifiers of nouns as their FEs provided a richer semantic representation. Next, FrameNet included adjectives as frame-evoking elements, permitting generalizations over domains like speed or temperature. While most aspects of adverbs description are already present in FrameNet (cf. above), theoretical analysis must make precise the implications of annotating null instantiated adverbial frame elements at scale. At the practical level, the time is ripe to add many more adverbs to appropriate existing frames and to create new frames for adverbs as needed. The principles of annotating naturally occurring text and extracting valence descriptions for LUs established on the other parts of speech carry over to adverbs. The combination of valence descriptions and annotated instances constitute essential inputs to characterize inferences. Clearly, the more annotation, the better, but large-scale expert annotation is slow and resourceintensive. Using crowdsourcing, which permits parallelizing (thus, speeding up) annotation, is a possible mitigation. Fossati et al. (2013) and Feizabadi and Padó (2014) demonstrated success with crowdsourcing for frame-semantic annotation when the task is narrowed down appropriately. Substantial promise exists to extract adverb annotation automatically from comparable corpora (Roth and Frank, 2015) and paraphrasing models (Wang et al., 2019). Even for the core task of FrameNet analysis, defining frames, Ustalov et al. (2018) proposed automatic methods. Still, full automation remains hard, given concerns of quality and consistency. Conclusion Conlon and Evens (1992) stated that adverbs are under-researched in computational linguistics; this statement is still true. Adverbs have received attention only in two applications: sentiment analysis and hedging detection. The large language models used here show systematic gaps in capturing adverb meaning. The problem is not solved. We propose that Frame Semantics, as embodied in FrameNet, along with improved techniques to mitigate the annotation effort to extend FN with new frames and annotations, can capture the meaning and implicatures of adverbs. Considering frames as lexical constructions (Fillmore, 2008), this proposal fits well with recent work to combine language models and construction grammar (Weissweiler et al., 2023). Multiple ways exist for computational modeling to use such a resource, e.g., by extending the coverage of semantic role labellers to a larger range of adverbs, or by fine-tuning language models on large annotated datasets for which our probing dataset can serve as a blueprint. Limitations We only used English data in the study, so we cannot guarantee that the findings will generalize to other languages (cf. Bender 2019). The English NLI datasets are, as usual, larger than for other languages, so we should expect models targeting other languages to have worse performance. We do, however, believe that the challenges of adverbs are comparable in other languages, particularly in typologically similar languages. Ethics Statement The paper argues for a new approach to the treatment of adverbs in the development of resources and applications in NLP. We consider better understanding of language by computational models as not posing a significant societal risk in itself. The dataset used for the computational experiment in Section 3 was created based on the data contained in the publicly available FrameNet corpus and, as far as we are aware, does not contain sensitive elements. Implementation of our proposed methodology has the same risks as any data-driven approach in computational linguistics, but we assume that we cannot safeguard against its possible misuse due to its very general nature. A Details on the Naturalistic Dataset The probing dataset includes a series of template classes. Each template class corresponds to an adverb class and contains several NLI templates with slots for adverbs and, when the structure permits it, also for the subject. In testing, we used all pairs of adverbs from the relevant class to instantiate the premise and the hypothesis. When a variable for subject exists in the premise, we used the same subject in the hypotheses. A.1 Likelihood Adverbs Adverbs: undoubtedly, surely, positively, likely, certainly, definitely, totally. 2. Premise: SUBJ can ADV find bargains in Tunis. Entailment: SUBJ will ADV find good deals in Tunis. Contradiction: SUBJ will ADV discover that everything is expensive in Tunis. 3. Premise: His friend, SUBJ, is ADV a foreigner. Entailment: SUBJ ADV is from another country. Contradiction: SUBJ ADV is a native here. A.2 Unattributed-information adverbs Adverbs: reportedly, allegedly, supposedly, apparently, ostensibly. Premise: The German government ADV opposed the quotas. Entailments: The German government ADV was against the quotas; The German government may have supported the quotas. Contradiction: The German ADV supported more quotas. Premise: The celebration had been postponed, ADV because of the Gulf War. Entailments: Someone said that the celebration was postponed because of the Gulf War; The Gulf War may have had no effect on the celebration. Contradiction: The Gulf War ADV had no effect on the celebration. A.3 Degree Adverbs Adverbs: at least, at a minimum, nearly, approximately. A.4 Obviousness Adverbs Adverbs: blatantly, obviously, clearly, ostentatiously, noticeably, visibly, conspicuously. 1. Premise: Castro ADV backed the rebels. Entailments: Castro helped the rebels; Castro tried to help the rebels. Contradiction: Castro tried to stop the rebels. Premise: The students were ADV drunk. Entailment: The students were surely drinking too much. Contradiction: The students renounced alcohol. Luck: A STATE_OF_AFFAIRS is evaluated as good (or bad) [...] for a particular PROTAGO-NIST. SPEAKER The Minister] reported [ MESSAGE that the cost had exploded]. b. [ MESSAGE The cost had] reportedly [ MESSAGE exploded]. Table 1 : 1Naturalistic dataset: Probing items , the Premise SUBJ1 said that SUBJ2 ADV opposed AC-TIONHyp 1 SUBJ1 said that SUBJ2 may have opposed ACTION (ENTAILMENT) Hyp 2 SUBJ1 is not sure that SUBJ2 opposed AC- TION (ENTAILMENT) Hyp 3 SUBJ1 is sure that SUBJ2 opposed ACTION (CONTRADICTION) Hyp 4 SUBJ1 is sure that SUBJ2 did not support ACTION (CONTRADICTION) Table 2 : 2Synthetic dataset: Probing items Adverb class Error rate (%) # sentencesLikelihood 2 5,880 Unattributed information 6 90 Degree 25 35 Obviousness 23 16 Table 3 : 3Naturalistic dataset: SNLI model error rates by adverb class VerbPrediction Hypothesis obviously clearly publicly blatantly no adverbSimple 0.94 0.94 0.95 0.96 0.97 Entailment Complex 0.60 0.62 0.70 0.71 0.85 Simple 0.05 0.05 0.05 0.04 0.02 aid Neutral Complex 0.39 0.38 0.29 0.27 0.15 Simple 0.92 0.92 0.92 0.95 0.97 Entailment Complex 0.53 0.52 0.58 0.61 0.77 Simple 0.07 0.08 0.08 0.05 0.03 help Neutral Complex 0.47 0.47 0.41 0.38 0.22 Simple 0.99 0.99 0.99 0.99 0.99 Entailment Complex 0.41 0.43 0.57 0.39 0.85 Simple 0.01 0.01 0.01 0.01 0 support Neutral Complex 0.55 0.53 0.40 0.40 0.15 Table 4 : 4Prediction of NLI model given Castro ADV backed the rebels as premise and Castro VERBed the rebels or Castro tried to VERB the rebels as hypothesis (simple and complex respectively). Boldface indicates wrong model predictions; underline indicates "borderline correct" cases where an incorrect label received a probability > 40%.Semantic type Test Entailment Neutral Contradiction Error rate (%) # sentences Entailment 1 70,188 12 0 ≈ 0 70,200 Entailment 2 134 70,066 0 ≈ 100 70,200 Contradiction 1 7,940 62,260 0 100 70,200 Unattributed information Contradiction 2 567 69,633 0 100 70,200 Entailment 31,200 0 0 0 31,200 Degree (properties of people) Contradiction 12,390 3,980 14,830 52 31,200 Entailment 840 0 0 0 840 Degree (properties of objects) Contradiction 547 0 293 65 840 Entailment 38,400 0 0 0 38,400 Degree (quantities) Contradiction 0 0 38,400 0 38,400 Entailment 1 54,600 0 0 0 54,600 Entailment 2 33,217 21,383 0 39 54,600 Contradiction 1 61 0 54,539 ≈ 0 54,600 Obviousness Contradiction 2 0 1,615 52,985 3 54,600 Table 5 : 5Synthetic dataset: Model predictions (cells with correct predictions have gray background) for each template class and error rates. [Chuck BUYER ] BOUGHT [a car GOODS ] [from Jerry SELLER ] [for $2,000 MONEY ] Table 6 : 6FrameNet Frames characterizing Speaker-Oriented Adverbs Feifei Zhai, Jiajun Zhang, Yu Zhou, and Chengqing Zong. 2013. Handling ambiguities of bilingual predicate-argument structures for statistical machine translation. In Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1127-1136, Sofia, Bulgaria. Association for Computational Linguistics. Fillers for the subject slot: Barbara, Charles, David, Elizabeth, James, Jennifer, Jessica, John, Joseph, Karen, Linda, Mary, Michael, Patricia, Richard, Robert, Sarah, Susan, Thomas, William. 1. Premise: SUBJ is ADV gonna have to check it tomorrow afternoon again. Entailment: SUBJ is ADV going to have to check it again. Contradiction: SUBJ ADV won't need to check it again. 1 . 1Premise: Lantau covers ADV twice the area of Hong Kong Island. Entailment: Lantau is at least as big as Hong Kong Island. Contradiction: Hong Kong Island is at least as big as Lantau. 2. Premise: At the moment ADV 140 persons are working to curtail the fire. Entailment: Many people are fighting the fire. Contradiction: Nobody is fighting the fire. For example, AMR treats sing in sing beautifully as the first argument of beautiful-02. ynie/roberta-large-snli_mnli_fever_anli_R1_R2_R3-nli This paper uses the following typographical conventions: frame names appear in typewriter font; FE names are in SMALL CAPS; and lexical units are in BOLD CAPS. aged, incited, indoctrinated.• Action 2, infinitive: help, encourage, disparage, incite, indoctrinate.• Action 3, past: stopped, deterred, praised, calmed, deprogrammed.• Action 3, infinitive: stop, deter, praise, calm, deprogram.Premise: SUBJ1 ADV ACTION1 SUBJ2.Entailments:1. SUBJ1 ACTION2_PAST SUBJ2.2. SUBJ1 tried to ACTION2_INF SUBJ2.Contradictions:1. SUBJ1 ACTION3_PAST SUBJ2.2. 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Transactions of the Asso- ciation for Computational Linguistics, 8:377-392. Construction grammar provides unique insight into neural language models. Leonie Weissweiler, Taiqi He, Naoki Otani, David R Mortensen, Lori Levin, Hinrich Schütze, 10.48550/ARXIV.2302.02178Georgetown University Round Table Workshop on CxGs+NLP. Leonie Weissweiler, Taiqi He, Naoki Otani, David R. Mortensen, Lori Levin, and Hinrich Schütze. 2023. Construction grammar provides unique insight into neural language models. In Georgetown University Round Table Workshop on CxGs+NLP. SUBJ1 said that SUBJ2 may have opposed ACTION. SUBJ1 said that SUBJ2 may have opposed ACTION. SUBJ1 is not sure that SUBJ2 opposed AC-TION. Contradictions: 1. SUBJ1 is sure that SUBJ2 opposed ACTION. 2. SUBJ1 is sure that SUBJ2 did not support ACTIONSUBJ1 is not sure that SUBJ2 opposed AC- TION. Contradictions: 1. SUBJ1 is sure that SUBJ2 opposed ACTION. 2. SUBJ1 is sure that SUBJ2 did not support ACTION. Degree adverbs Adverbs: at least, at a minimum, nearly, approximately. B.3 Degree adverbs Adverbs: at least, at a minimum, nearly, approx- imately. Properties of people Properties: net worth, knowledge, manners, fan base, culpability. 6 Adjectives: • Adjective 1: rich, erudite, polite, popular, guilty. B.3.1 Properties of people Properties: net worth, knowledge, manners, fan base, culpability. 6 Adjectives: • Adjective 1: rich, erudite, polite, popular, guilty. Adjective 2: big, extensive, good, large, high. Premise: SUBJ1 is ADV twice as ADJ1 as SUBJ2. Entailment: SUBJ1's PROPERTY is/are at least as ADJ2 as SUBJ2's• Adjective 2: big, extensive, good, large, high. Premise: SUBJ1 is ADV twice as ADJ1 as SUBJ2. Entailment: SUBJ1's PROPERTY is/are at least as ADJ2 as SUBJ2's. Unlike in case with adverbs and subject-slot fillers, where all combinations are used, properties and adjectives in this and the next subclass are used in parallel. I.e., when the i'th adjective from the first list is used in the premise, the corresponding i'th property and adjective from the second list will be used in the hypotheses. Contradiction: SUBJ2's PROPERTY is/are at least as ADJ2 as SUBJ1'sUnlike in case with adverbs and subject-slot fillers, where all combinations are used, properties and adjectives in this and the next subclass are used in parallel. I.e., when the i'th adjective from the first list is used in the premise, the corresponding i'th property and adjective from the second list will be used in the hypotheses. Contradiction: SUBJ2's PROPERTY is/are at least as ADJ2 as SUBJ1's. Properties: age, weight, height, width, price. Adjectives: • Adjective 1: old, heavy, tall, wide, expensive. • Adjective 2: great, big, big, big, high. Premise: SUBJ1 is ADV twice as ADJ1 as SUBJ2. Entailment: The PROPERTY of SUBJ1 is at least as ADJ2 as that of SUBJ2. Contradiction: The PROPERTY of. B.3.2 Properties of objects Subjects: the truck, the house, the hotel, the ship, the wagon, the car, the tree. SUBJ2 is at least as ADJ2 as that of SUBJ1B.3.2 Properties of objects Subjects: the truck, the house, the hotel, the ship, the wagon, the car, the tree. Properties: age, weight, height, width, price. Adjectives: • Adjective 1: old, heavy, tall, wide, expensive. • Adjective 2: great, big, big, big, high. Premise: SUBJ1 is ADV twice as ADJ1 as SUBJ2. Entailment: The PROPERTY of SUBJ1 is at least as ADJ2 as that of SUBJ2. Contradiction: The PROPERTY of SUBJ2 is at least as ADJ2 as that of SUBJ1. Quantities Times: at the moment, now, these days, this month, this week. B.3.3 Quantities Times: at the moment, now, these days, this month, this week. 7 Related-person groups: friends, relatives, acquaintances, coworkers. Activities: working on this, helping with the move, coming to visit us. Premise: TIME ADV NUMBER of SUBJ's RE-LATED_PERSONS are ACTIVITY. Numbers: two dozen, thirty, fifty. 140Entailment: Many people are ACTIVITY. Contradiction: Nobody is ACTIVITYNumbers: two dozen, thirty, fifty, 140. Related-person groups: friends, relatives, ac- quaintances, coworkers. Activities: working on this, helping with the move, coming to visit us. Premise: TIME ADV NUMBER of SUBJ's RE- LATED_PERSONS are ACTIVITY. Entailment: Many people are ACTIVITY. Contradiction: Nobody is ACTIVITY. Obviousness adverbs Adverbs: blatantly, obviously, clearly, ostentatiously, noticeably, visibly, conspicuously. Actions: 8 • Action 1: backed, supported, criticized, provoked. brainwashedB.4 Obviousness adverbs Adverbs: blatantly, obviously, clearly, ostenta- tiously, noticeably, visibly, conspicuously. Actions: 8 • Action 1: backed, supported, criticized, pro- voked, brainwashed. Similarly to the two previous subclasses, times, numbers, activities, and related-person groups in this subclass are used in parallel. I.e., when the i'th time, number, related-person group. and activity are used in the premise, the corresponding i'th activity will be used in the hypothesesSimilarly to the two previous subclasses, times, numbers, activities, and related-person groups in this subclass are used in parallel. I.e., when the i'th time, number, related-person group, and activity are used in the premise, the corresponding i'th activity will be used in the hypotheses. Similarly to adjectives and properties in the case of degree adverbs above, actions of different types are used in parallel. Similarly to adjectives and properties in the case of degree adverbs above, actions of different types are used in parallel. when the i'th element from the first list is used in the premise, corresponding i'th elements from other lists will be used in the hypotheses. I E , I.e., when the i'th element from the first list is used in the premise, corresponding i'th elements from other lists will be used in the hypotheses.
[]
[ "Localized Latent Updates for Fine-Tuning Vision-Language Models", "Localized Latent Updates for Fine-Tuning Vision-Language Models" ]
[ "Moritz Ibing \nVisual Computing Institute\nRWTH Aachen University\n\n", "Isaak Lim \nVisual Computing Institute\nRWTH Aachen University\n\n", "Leif Kobbelt \nVisual Computing Institute\nRWTH Aachen University\n\n" ]
[ "Visual Computing Institute\nRWTH Aachen University\n", "Visual Computing Institute\nRWTH Aachen University\n", "Visual Computing Institute\nRWTH Aachen University\n" ]
[]
Although massive pre-trained vision-language models like CLIP show impressive generalization capabilities for many tasks, still it often remains necessary to fine-tune them for improved performance on specific datasets. When doing so, it is desirable that updating the model is fast and that the model does not lose its capabilities on data outside of the dataset, as is often the case with classical fine-tuning approaches. In this work we suggest a lightweight adapter, that only updates the models predictions close to seen datapoints. We demonstrate the effectiveness and speed of this relatively simple approach in the context of few-shot learning, where our results both on classes seen and unseen during training are comparable with or improve on the state of the art.
10.48550/arxiv.2212.06556
[ "https://export.arxiv.org/pdf/2212.06556v1.pdf" ]
254,591,708
2212.06556
ff4f21978fca01bd6bcbe7aac92bcec3cca295b7
Localized Latent Updates for Fine-Tuning Vision-Language Models Moritz Ibing Visual Computing Institute RWTH Aachen University Isaak Lim Visual Computing Institute RWTH Aachen University Leif Kobbelt Visual Computing Institute RWTH Aachen University Localized Latent Updates for Fine-Tuning Vision-Language Models Although massive pre-trained vision-language models like CLIP show impressive generalization capabilities for many tasks, still it often remains necessary to fine-tune them for improved performance on specific datasets. When doing so, it is desirable that updating the model is fast and that the model does not lose its capabilities on data outside of the dataset, as is often the case with classical fine-tuning approaches. In this work we suggest a lightweight adapter, that only updates the models predictions close to seen datapoints. We demonstrate the effectiveness and speed of this relatively simple approach in the context of few-shot learning, where our results both on classes seen and unseen during training are comparable with or improve on the state of the art. Introduction Much of the success of deep learning when it comes to vision tasks, such as classification, object detection or segmentation, is due to ever bigger models trained on increasingly large quantity of data. A popular approach to make use of immense sources of uncurated data in the form of images with textual descriptions are vision-language models [19,25]. Here both image and description are individually mapped into a joint embedding space. This embedding is optimized, so that matching pairs are close, and the distance between all other pairs large. A model trained in this fashion can be used for zeroshot classification, as the language model can deal with every conceivable class by embedding a textual description (e.g. "a picture of [CLASS]"). Even though these models can be applied to all kinds of classification tasks, their performance sometimes is suboptimal. This might be the case if used on a dataset with specific characteristics, that differ from the original training set, e.g. if the task is to recognize the action performed in an image, even though during training the model only saw generic objects. In these cases a common technique is to fine-tune the pre-trained model for the task at hand. However, updating the complete model is quite expensive (as the employed models are large). There are two solutions proposed in the literature to tackle this problem. One is prompt-learning [35][36][37] where the context around the class ("a picture of" in the last example) is optimized for a specific dataset instead of hand crafted. The other option is to use adapters [11,34], light-weight models (usually small MLPs) that modify the embedding produced by either the visual or language model (or both), thus updating the predictions without the need to update the original networks parameters. Both these approaches however still have the problem, that even though the performance is improved for the specific domain and task the fine-tuning was done for, this comes at the cost of a decrease in performance on other tasks/domains compared to the original model [12]. The goal of this work is to reap the benefits of fine-tuning on a specific task, without losing the generalization ability of the original model. Work in this direction has already been done in the form of CoCoOp [35], where promptlearning is employed, but the context is not fine-tuned on a specific dataset, but instead a suitable context is predicted from the image to be classified. Another approach using prompt-learning is ProGrad [37] where the context update is restricted in order not to lose information from the pretraining stage. Although both methods decrease the performance loss in the zero-shot setting, they still do not reach the abilities of the original model. In contrast we choose a simple method based on adapters. The idea is to only update the embedding where we actually have training data and leave it unchanged everywhere else, thus retaining the original predictions of the model, where we cannot improve on them. Furthermore, even where we have data we want to change the embedding as little as possible, to allow sensible interpolation between fine-tuned and original embedding. This approach is extremely lightweight, as we only need to tune a small amount of parameters, and back-propagating through the original model is not necessary. Still we show an improvement in performance compared to the previous state of the art. Related Work Zero-Shot Learning Zero-shot learning describes a setting, where the set of classes at training and during testing are disjoint or at least not identical [4], thus the relation between classes and images belonging to that class needs to be learned indirectly. There are numerous works of research in this area, so we instead refer to [30,32] for an overview. A common approach to tackle this task is to relate pre-trained image and class embeddings [1,2,10]. This is conceptually very close to vision-language models, another popular framework that can be applied for zero-shot learning. Vision-Language Models The term vision-language model describes networks that learn an alignment between images and text in a joint embedding space. Today these models are usually based on contrastive learning, which has been popularized for pre-training of image models [5,13,16] and aims to maximize the distance between similar and dissimilar instances in the embedding space. Currently one of the most popular vision-language models is CLIP [25], which we use in all our experiments. It uses a Transformer [28] as text encoder and a ResNet [14] or ViT [8] as image encoder. ALIGN [19] is a similar approach, whereas DeCLIP [21] tries to improve the training procedure in order reach the same performance with less data. Fine-Tuning When it comes to fine-tuning a pre-trained vision-language model, there are broadly three types of approaches in the literature. It is possible to fine-tune the entire model, but afterwards interpolate between the original and updated weights, to counteract overfitting (WiSE-FT [31]). Alternatively, not the model itself is trained, but only an adapter, that is applied onto the embedding space (CLIP-Adapter [11]). This is the approach we choose as well. Instead of learning this adapter, it can be extracted from the fine-tuning dataset (TIP-Adapter [34]). As this requires data for every class it is evaluated on, it is however not suitable for zero-shot learning. The last approach is called prompt engineering. Here the context of the text embedding is optimized for performance on the training set (CoOp [36]). This learning can be restricted in order not to decrease the loss of information from the pre-training stage (ProGrad [37]). Another option is to predict the (textual) context for each image (CoCoOp [35]). This mitigates overfitting on the train set and thus is better at retaining zero shot ability on unseen classes, but comes at the cost of an increased training time. Method Before introducing our approach in more detail, we will give a short overview on how vision-language models work in general on the example of CLIP, which is used in all our experiments. Vision-Language Models Vision-Language models consist of two networks: an image encoder f I and a text encoder f T . Their exact implementation is of no interest to us in this context. All we need to know, is that these models embed an image or a text respectively to a (normalized) feature vector of the same dimension. The cosine distance between an embedded image and text should then correspond to their similarity i.e. how well the text describes the image. During training, we are given a batch of n images X and their textual descriptions Y . We make the simplifying assumption that each text is a perfect description of the corresponding image and all other texts are completely unrelated. Thus, we want to minimize the cosine distance between embeddings of matching image/text pairs f I (x i ),f T (y i ) and maximize the distance between all other pairs within the batch f I (x i ),f T (y j ) with i = j. Another view would be to regard the cosine distance as the likelihood, that a given text y describes the corresponding image x, or vice versa. We can compute the normalized probabilities as: p(y|x) = exp(f I (x) T f T (y)/τ ) n i=1 exp(f I (x) T f T (y i )/τ )(1) where τ is a learned temperature parameter. As we assume the embeddings to be normalized, the dot product is equivalent to the cosine similarity. The probability p(x|y) only differs in the normalization. In this view it makes now sense to maximize the probability for the correct pairs, for which we can use the Cross Entropy loss. As we want to maximize the probabilities in both directions the loss is given as: L = − 1 N N i=1 (log(p(x i |y i )) + log(p(y i |x i )))/2 (2) Zero-Shot Classification This approach leads to a semantically meaningful embedding of both images and text, that can be used for downstream tasks. Alternatively, we can use it directly for zero-shot classification. For this we embed a text for each class (e.g. a picture of [CLASS]) to a feature vector f T (y k ). Then, to classify a given image, we embed it and compute the distance between its feature vector f I (x) and all class embeddings. The class probability for a class k is then given similarly as before as p(y k |x). Local Linear Updates A big benefit of vision-language models is their generalization capability. As they are usually trained on immense datasets, they tend to show great zero-shot capabilities even on unseen domains. However, they are not necessarily well tuned for small specific datasets, that were not well represented in the original training data. Although we could fine-tune the networks on this specific set, this would come at the cost of the models ability to generalize and be quite expensive. Instead we add additional functions on the output of the model g I • f I and g T • f T respectively and optimize only those. This is much cheaper, as g has much less parameters (we will omit the subscript for both f and g if both the image and textual models are meant). Furthermore, as g is applied onto the output of f we do not even have to back-propagate through the big models. We could even precompute f on the given dataset to save further computation cost. In our case we use distinct networks for g I and g T but is possible to use the same function as well (g I = g T ) as they are applied on the same domain. The training works slightly differently than before. As we now do not have unique image/text pairs, but instead images with their classes, we leave out one half of the loss, leading to the classical Cross Entropy loss: L = − 1 N N i=1 log(p(y i |x i ))(3) A similar idea was presented in CLIP Adapter [11] (Here only g I is used). This approach significantly reduces the cost of fine-tuning, but does not solve the problem of loos-ing capabilities on the previous training dataset while overfitting on the new one. Local Interpolation To lessen the overfitting WISE-FT [31] interpolates the weights between the original model and a fine-tuned version. With a similar aim in CLIP Adapter [11] the results of f and g are interpolated: α(g • f ) + (1 − α)f(4) Here α is a global parameter. It would be more sensible to localize this interpolation to the area of the feature space, where we obtained new data. Only there do we actually have information onto how to sensibly update the embeddings. As g is a global function but we only supervise it at our training samples, it is unlikely that it represents a sensible modification away from these samples. Thus, in our case α is not a global parameter, but a function, α(x, D) = β · max d∈D (exp(−γ(1 − x T d)))(5) where D is the set of datapoints we fine-tuned on and β is a global parameter similar to how α was defined previously. γ concentrates the focus of the interpolation mask and thus influences in what range our updated embedding should be applied. If we would let it go towards zero, we would have a global α parameter, similarly to CLIP Adapter. On the other hand, if we let it go towards infinity, we would only update exactly the images and classes on which we fine-tuned. Whenever an image is close to one already seen during training, we thus use our updated features, otherwise we utilize the general knowledge of the pre-trained model. Note that we do this separately for the text and image encoder, so there are separate sets D text and D image . Clustering This approach requires us to save the feature vectors of all datapoints seen during fine-tuning. This is not a problem, as long as the dataset used is indeed very small. If this however is not the case, we cluster the feature vectors to find sensible representatives. For this we use agglomerative clustering, where we start by regarding each datapoint as an individual cluster and then iteratively merge pairs based on the maximum distance between their members until we have reached the desired number of clusters. Each cluster needs a position, which is computed as the (normalized) mean of all its members. We chose this approach, as it does not make any assumptions about cluster shapes nor does it require a sensible initialization. Furthermore, we expect the number of clusters to be in a similar order of magnitude as the number of datapoints, so we do not need many merge operations. However, as we show in the ablation (Sec.4.4), clustering only has a small effect on the performance and is mainly used to bound the memory consumption. Thus, the exact clustering algorithm is not likely to make much of a difference either. Identity Regularization So far, we have restricted the region, where we change the feature space, but not the magnitude of the update, which can become arbitrarily large. It is however desirable, that the update is as small as possible, while minimizing the training loss. As we assume the original pre-trained features to already be useful, we want to stay as close to them as possible in order to retain generality. Furthermore the interpolation between f and g • f should result in sensible embeddings, which is more likely the case, if they are close to each other. In other words, g should stay as close to the identity as possible. This is easy to enforce, if we simply choose g as an affine function g = W x + b. In this case our regularization takes the form: λ(||W − I|| 2 + ||b|| 2 )(6) where I is the identity matrix and λ a weighting parameter. Of course we could choose a more complex function and regularize g to stay close to the identity at a set of sample points. However, a dense sampling of the embedding space is infeasible, and we are interested in retaining this property wherever the interpolation weight α is non-zero. Furthermore, we initialize g as the identitiy function, which is trivial for affine functions, but not for non-linear MLPs. Evaluation For our evaluation we follow CoCoOp [35], where three problem settings are investigated: 1. Generalization to new classes within a given dataset. 2. Generalization to new datasets after fine-tuning. Generalization to domain-shift. Before presenting the conclusions, we will introduce the used datasets, and explain the training procedure. Datasets Similar to CoCoOp, we follow CoOp [36] in the choice of datasets used in evaluation. To be precise we use 11 datasets that cover a wide range of tasks: Im-ageNet [7] and Caltech101 [9] for generic object classification, OxfordPets [24], StanfordCars [20], Flowers102 [23], Food101 [3] and FGVCAircraft for more specific object classification, [22], SUN397 [33], DTD [6], EuroSAT [15] and UCF101 [27] for a diverse set of tasks. Furthermore, to evaluate domain generalization we regard Ima-geNet as source and four different versions under different types of domain shift as target. The four datasets are: Ima-geNetV2 [26], ImageNet-Sketch [29], ImageNet-A [18] and ImageNet-R [17]. The set of images for few-shot training are randomly sampled for each dataset, while using the original test set for testing. For approaches that need training we average the results over three runs. Training Our implementation is based on the published code of CoOp. We use the same learning rate and number of epochs as they do. Following CoCoOp we use ViT-B/16 as the vision backbone of CLIP. As ProGrad has been evaluated on a different backbone, we retrained it for a fair comparison. Note that both CoOp and CoCoOp have a context length of 4 initialized as the prompt: "a photo of a", whereas for CLIP Adapter and us the context is class dependent and ProGrad has a context length of 16 with a class-dependent initialization. If not stated otherwise we choose the parameters of our approach as β = 0.5, γ = 20, λ = 1e3 and the number of clusters as 512. Base to New Generalization On each dataset, the classes are split equally into a set of "base" classes on which the adapter is trained and unseen "new" classes, where we only evaluate. Thus, no matter how many shots are given for the training, on the new classes we will always do zero shot inference. We show results for different numbers of shots in Figure 2 and report exact values in Table 1. Here we also give the harmonic mean between the evaluation on base and new classes for an easier comparison of the approaches regarding their respective trade-offs. As can be seen, on the 16 shot evaluation our approach outperforms all other methods on 8 out of 11 datasets, when regarding the harmonic mean. Here we have on average an improvement of almost 3 percentage points to CoCoOp (the next best method). Furthermore, (on average) our localized adapter beats all other methods regarding new classes independent of the number of shots and is first or second on the base classes. Our method is the only one that reaches the performance of CLIP when it comes to zero-shot performance on unseen classes, whereas all other methods show a drop in performance here, that usually increases with the number of shots, hinting at overfitting. Cross-Dataset Generalization In this experiment the models are fine-tuned on Ima-geNet and then evaluated on the other datasets, thus an improvement on ImageNet (compared to CLIP) is expected. Interestingly both CoCoOp and our approach show an improvement (on average) on the other datasets as well. Apparently the training samples of ImageNet are numerous and diverse enough to avoid overfitting and the data distribution of ImageNet is closer to the other regarded datasets than the original training set of CLIP. Although our method does not reach the results of CoCoOp on this evaluation we come very close. Domain Generalization In this last comparison the models are again fine-tuned on ImageNet and then evaluated on different versions with a clear domain shift. Here we can see a slight drop of performance between our method and prompt based approaches. This might be due to the fact, that prompt-based approaches only fine-tune the input of the text encoder. As the class names and thus the text encodings are not affected by domain shift, their performance generalizes better. On the other hand we directly update the text and image embedding (and CLIP Adapter only updates the image embedding), which might be problematic as here changes caused Further Analysis Ablation Here we discuss the effect different design choices in our approach have on the result. For this we regard the average performance achieved on the Base to New training setup, when using 16 shots (Table 4). As already mentioned, it barely makes any difference, whether we use clustering or not ("no cluster"), thus it is a sensible choice to limit the memory requirements. Using the global dampening parameter β does lead to an improvement, although it is rather small ("no damp"). Not restricting the interpolation to the training sam-ples ("no mask") leaves the results on the base classes unchanged but leads to overfitting and thus a reduced performance on unseen classes. On the other hand focusing the interpolation exclusively on the training samples ("Dirac mask" equivalent to a γ parameter of infinity) leads to the same performance as CLIP for new classes, but of course this way our method cannot improve on unseen classes either, as seen in Tables 2 and 3. Note that for numerical reasons, we did not actually implement a γ parameter of infinity, but clamped α to zero or one, depending on some small distance threshold. Interestingly leaving out the identity regularization ("no reg") barely has any effect on the results, whereas an initial- ization to identity seems to be more important ("rand. init"). We assume, that the training does not include enough update steps for effects to be seen. To substantiate this claim we report the actual distance to identity in Table 4 (for relevant experiments). Here we can see, that the distance between our adapter and the identity function does correlate with performance. Lastly we can see, that only using a single Linear Layer as an adapter without any of our additional improvements leads to significantly worse results, especially on the unseen classes, thus each of our improvements makes only a small difference individually, but together they significantly increase performance. Training speed A comparison of the training speed between our approach and prompt based methods depends on both the batch size and the number of classes. As we can precompute the class embeddings, the training time of our method is almost independent of their number. Prompt based approaches instead need to compute the class embeddings in every iteration. On the other hand the number of class embeddings is independent of the batch size, whereas our adapter needs to be applied to every training sample. In Figure 3 we show the timing for a single for- . Comparison of timings. Our approach (red), CoOp (green), ProGrad (cyan) and CLIP Adapter (purple). CoCoOp is marked as an orange dot, as a batch size bigger than one does not fit into memory ward and backward pass depending on the batch size. As can be seen our method and CLIP Adapter are consistently the fastest and the difference in their timings is negligible (it is barely possible to differentiate their lines). ally the overhead of the computations due to the adapter barely matter, as can be discerned from the similar slope of CoOp and the adapter based methods. The number of classes influences the distance between these parallel lines, which signifies the overhead due to the computation of their embedding. For CoCoOp we only have a single data point, as batches with more than one sample do not fit into memory. Thus, although for a batch size of one the training speed is similar to CoOp, in practice CoCoOp is much slower, as we cannot increase the batch size. ProGrad is consistently slower than other methods due to additional computations needed for gradient decomposition. Conclusion As the requirements in size, data and compute for state of the art AI models increases, it becomes more and more important to be able use available pre-trained networks for complex downstream tasks. In order to do this we need to be able to fine-tune these models in an efficient manner, preferably without loosing the generalization capability, that makes them so useful in the first place. We have introduced an extremely simple approach for this task, introducing small linear updates to the embedding space, localized to the datapoints, where we fine-tune. Our model is fast to train and needs a minimal amount of extra parameters, but still reaches state of the art results both on fine-tuned and unseen classes. In this work we always trained our adapter for optimal performance on a single dataset. A possible future research direction would be to generalize our approach to multiple distinct fine-tuning datasets. It would be possible to use dataset-dependent adapters and interpolation weights, but some further work would be needed to make this scalable. Figure 1 . 1The basic idea of our approach is to interpolate between the original networks output (blue) and the newly trained adapter (orange) to obtain the final function (green). The interpolation weight (bottom line) is based on the distance to the samples we fine-tuned on (green points). Figure 2 . 2Comparison in the intra-class generalization setting. We compare our approach (red) vs. CoOp (green), CoCoOp (yellow), ProGrad (cyan) and CLIP Adapter (purple). Zero-shot CLIP is shown as a baseline in blue. circles mark the base classes and x the unseen new classes. Figure 3 3Figure 3. Comparison of timings. Our approach (red), CoOp (green), ProGrad (cyan) and CLIP Adapter (purple). CoCoOp is marked as an orange dot, as a batch size bigger than one does not fit into memory CLIP Adapter 71.77 92.17 86.47 60.50 67.63 82.53 22.90 62.77 42.23 47.67 63.37 62.82 LLU 72.13 92.00 89.10 65.37 71.23 86.10 24.87 64.93 44.63 47.77 67.10 65.31 Table 2. Comparison for cross dataset generalization capability. All approaches are trained on ImageNet (16 shots) and then evaluated on all 11 datasets.Table 3. Comparison for domain generalization capability. All approaches are trained on the standard version of ImageNet (16 shots) and then evaluated on 4 different types of domain shift.Source Target ImageNet Caltech101 OxfordPets StanfordCars Flowers102 Food101 FGCVAircraft SUN397 DTD EuroSAT UCF101 Average CLIP 62.57 90.17 89.10 65.30 71.40 86.10 24.80 57.53 44.40 47.80 66.70 64.33 CoOp 71.51 93.70 89.14 64.51 68.71 85.30 18.47 64.15 41.92 46.39 66.55 63.88 CoCoOp 71.02 94.43 90.14 65.32 71.88 86.06 22.94 67.36 45.73 45.37 68.21 65.74 ProGrad 72.00 92.67 89.73 64.00 68.37 85.27 20.30 64.60 43.07 44.53 65.20 63.78 Source Target ImageNet ImageNetV2 ImageNet-Sketch ImageNet-A ImageNet-R CLIP 66.73 60.83 46.15 47.77 73.96 CoOp 71.51 64.20 47.99 49.71 75.21 CoCoOp 71.02 64.07 48.75 50.63 76.18 ProGrad 72.00 64.70 48.37 49.73 75.57 CLIP Adapter 71.77 63.97 46.27 47.80 72.10 LLU 72.13 64.53 47.17 48.87 74.30 Gener - GenerCLIP Default no cluster no damp. no mask Dirac mask no reg rand. init LinearTable 4. Ablation of different design decisions in our network. The details of the different experiments are explained in subsection 4.4Base 69.34 83.48 83.35 83.58 83.41 83.28 83.32 82.91 81.02 New 74.22 74.47 74.40 73.98 72.53 74.21 74.42 72.67 32.60 Mean 71.70 78.46 78.36 78.16 77.27 78.22 78.33 76.94 45.79 Regularization - 1.95e-5 - - - - 5e-4 3.6e-3 6.5e-3 Acknowledgements This work was funded by the German Research Foundation within the Gottfried Wilhelm Leibniz programme, as well as through the project "Training the Archive" in cooperation with the Ludwig Forum Aachen and the HMKV Hartware MedienKunstVerein, Dortmund. "Training the Archive" is funded by the Digital Culture Programme of the Kulturstiftung des Bundes (German Federal Cultural Foundation). Funded by the Beauf-tragte der Bundesregierung für Kultur und Medien (Federal Government Commissioner for Culture and the Media). We would like to thank Dominik Bönisch (heading the "Training the Archive" project) for helpful discussions. 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[]
[ "Performance Analysis and Optimization of Network-Assisted Full-Duplex Systems under Low-Resolution ADCs", "Performance Analysis and Optimization of Network-Assisted Full-Duplex Systems under Low-Resolution ADCs" ]
[ "Xiangning Song ", "Zhenhao Ji ", "Member, IEEEJiamin Li ", "Pengcheng Zhu ", "Member, IEEEDongming Wang ", "Fellow, IEEEXiaohu You " ]
[]
[ "IEEE SYSTEM JOURNAL" ]
Network-assisted full-duplex (NAFD) distributed massive multiple input multiple output (M-MIMO) enables the in-band full-duplex with existing half-duplex devices at the network level, which exceptionally improves spectral efficiency. This paper analyzes the impact of low-resolution analog-to-digital converters (ADCs) on NAFD distributed M-MIMO and designs an efficient bit allocation algorithm for low-resolution ADCs. The beamforming training mechanism relieves the heavy pilot overhead for channel estimation, which remarkably enhances system performance by guiding the interference cancellation and coherence detection. Furthermore, closed-form expressions for spectral and energy efficiency with low-resolution ADCs are derived. The multi-objective optimization problem (MOOP) for spectral and energy efficiency is solved by the deep Q network and the non-dominated sorting genetic algorithm II. The simulation results corroborate the theoretical derivation and verify the effectiveness of introducing low-resolution ADCs in NAFD distributed M-MIMO systems. Meanwhile, a set of Paretooptimal solutions for ADC accuracy flexibly provide guidelines for deploying in a practical NAFD distributed M-MIMO system.
10.1109/jsyst.2022.3232628
[ "https://export.arxiv.org/pdf/2212.08832v1.pdf" ]
254,853,678
2212.08832
2ddddd23f4b9680b10aa40dd34f6598a3920b94e
Performance Analysis and Optimization of Network-Assisted Full-Duplex Systems under Low-Resolution ADCs 2022 Xiangning Song Zhenhao Ji Member, IEEEJiamin Li Pengcheng Zhu Member, IEEEDongming Wang Fellow, IEEEXiaohu You Performance Analysis and Optimization of Network-Assisted Full-Duplex Systems under Low-Resolution ADCs IEEE SYSTEM JOURNAL 12022Index Terms-Low-resolution ADCsbeamforming trainingchannel estimationNAFD distributed M-MIMOMOOP Network-assisted full-duplex (NAFD) distributed massive multiple input multiple output (M-MIMO) enables the in-band full-duplex with existing half-duplex devices at the network level, which exceptionally improves spectral efficiency. This paper analyzes the impact of low-resolution analog-to-digital converters (ADCs) on NAFD distributed M-MIMO and designs an efficient bit allocation algorithm for low-resolution ADCs. The beamforming training mechanism relieves the heavy pilot overhead for channel estimation, which remarkably enhances system performance by guiding the interference cancellation and coherence detection. Furthermore, closed-form expressions for spectral and energy efficiency with low-resolution ADCs are derived. The multi-objective optimization problem (MOOP) for spectral and energy efficiency is solved by the deep Q network and the non-dominated sorting genetic algorithm II. The simulation results corroborate the theoretical derivation and verify the effectiveness of introducing low-resolution ADCs in NAFD distributed M-MIMO systems. Meanwhile, a set of Paretooptimal solutions for ADC accuracy flexibly provide guidelines for deploying in a practical NAFD distributed M-MIMO system. I. INTRODUCTION W ITH the introduction of big-data communication scenarios, the swift growth of wireless communication data puts forward urgent demand and brings severe challenges to the existing wireless systems on the channel capacity, latency, and quality of service (QoS) [1]. Huh et al. [2] demonstrated the potential gain of massive multiple input multiple output (M-MIMO) on improving spectral efficiency (SE) and acceptable performance with low-complexity receiving and beamforming schemes such as maximum ratio transmission and combining (MRT/MRC) and zero-forcing transmission and receiving (ZFT/ZFR), which provides the possibility for orders of magnitude more data transmission. For the geometric architecture, the distributed antenna system has emerged as a promising architecture for its proximity gains and enhanced macro-diversity [3] with multiple separately-located remote antenna units (RAUs). However, the high mobility of terminal devices and the heterogeneity of the network structure result in profound asymmetry flow between uplink (UL) and downlink (DL), while the sheer scale of data traffic entails an intolerable consumption of hardware resources. A. Related Works and Motivation To meet the higher requirement for flexible duplex and asymmetric data traffic, Wang et al. [4] introduced the conceptual description of network-assisted full-duplex (NAFD) system, which achieves in-band full-duplex with the half-duplex devices and promotes flexible traffic control by dynamically allocating the UL and DL RAUs, i.e., different RAUs have different operational modes, enabling DL transmission and UL reception simultaneously on the same frequency band. Compared to the co-frequency co-time full-duplex (CCFD), NAFD introduces a geographic separation between RAUs and thus weakens the self-interference of intra-RAUs. Nevertheless, the cross-link interference (CLI) via the inter-RAUs UL-to-DL channel seriously affects the system performance and needs to be eliminated. Moreover, channel reciprocity is generally adopted for the classical time division duplex to mitigate the heavy burden of DL training and eliminate the channel state information (CSI) error. In contrast, for the cell-free MIMO, few RAUs serve a given user in a relatively wide area, which immensely weakens the channel-hardening phenomenon [5]. Inspired by the beamforming training scheme, the potential benefits of DL pilot estimation become more lavish compared to the statistical CSI in the light channel-hardening scenario [6]. Further, Li. et al. [7] verified the beamforming training in NAFD distributed M-MIMO with tractable closed-form SE. However, when NAFD distributed M-MIMO employs a perfect quantization scheme, intolerable energy consumption severely limits its future deployment. To obtain the bonus of M-MIMO, analog-to-digital converters (ADCs) face new challenges in the evolution of the wireless network, where the entire system suffers from the massive processing capacity by power-hungry ADCs with high quantization bits. Low-resolution ADCs are verified as the panacea to break through those bottlenecks in cell-free MIMO system [8], which aimed to maximize the ergodic sumrate under limited ADC bit and investigated that large-scale antennas compensated for performance degradation by lowresolution quantization. Yuan et al. [9] deduced the SE of the distributed MIMO scenario with MRC receiver and proved arXiv:2212.08832v1 [cs.IT] 17 Dec 2022 that SE has an upward trend with the increase of quantization bit. Nevertheless, in NAFD distributed M-MIMO, no work still exists on whether introducing low-resolution ADCs leads to severe debilitation on system SE. For other basic criteria to evaluate the low-resolution ADCs, combinatorial optimization is proposed to maximize energy efficiency (EE). More realistically, Li et al. [10] investigated the asymptotic achievable rates with ZFR/MRC receivers in the cell-free MIMO and issued the tradeoff between SE and EE under pilot contamination. Exploring SE/EE tradeoff to obtain the feasible ADC quantization scheme in the NAFD systems is still blank in current academic research yet. B. Contribution Motivated by the above observations, this paper first considers the interference cancellation and beamforming training scheme in NAFD distributed M-MIMO with low-resolution ADCs. The main contributions of this paper are as follows: • With the quantization effect, UL and DL signal transmissions in NAFD distributed M-MIMO systems are modeled. The two-stage channel estimation is proposed, including pilot training and beamforming training. Moreover, SE&EE for the NAFD distributed M-MIMO with low-resolution ADCs are discussed. • The multi-objective optimization problem (MOOP) for maximizing SE and EE is proposed. The joint optimization is yielded by deep Q network (DQN) and nondominated sorting genetic algorithm II (NSGA-II) for flexible bit allocation schemes. • Monte Carlo simulation verifies the theoretical derivation with numerical results and investigates the influence of quantization noise, proving the desirability of lowresolution ADCs in the NAFD distributed M-MIMO. C. Organization and Notations The structure of this paper is organized as follows. Sec. II and Sec. III introduces the system model and performs channel estimation. In Sec. IV, we derive the closed-form expressions of the system SE/EE. In Sec. V, an effective ADC bit allocation algorithm is proposed. Sec. VI gives the simulation results to verify theoretical derivation and draws conclusions. Sec. VII summarizes the whole article. Notations: Boldface lower and upper case letters denote vectors and matrices. (·) H , (·) T , E[·], cov(·) indicate the conjugate transpose, transpose, expectation and covariance operators, respectively. diag(A) returns the diagonal elements of A, x ∼ CN (0, R) represents a circularly symmetric complex Gaussian vector x with zero mean and covariance R. |·| and · represent the absolute value and spectral norm. Γ(k, θ) and Nakagami(k, θ) denote the Gamma and Nakagami distribution with parameters k and θ. For the abbreviations commonly used in equations, (·) UL , (·) DL , (·) UP , (·) DP , (·) q , (·) p , (·) ω , (·) e , (·) I , (·) pre represents the UL parameter, the DL parameter, the UL pilot parameter, the DL pilot parameter, the quantized parameter, the pilot paramter, the parameter in UL/DL mode, the effective parameter, the parameter related to interference channel, and the parameter in MR/ZF scheme. conventional cell-free MIMO like [11]- [14], NAFD's RAUs are all half-duplex, but the duplex mode of each RAU can be dynamically adjusted according to the network throughput rate, reliability or latency, enabling full duplex at the network level, which results in better SE performance than CCFD systems with suitable interference cancellation, and flexible adjustment of UL and DL RAUs to cope with transmission asymmetries effectively. A. DL Signal Transmission with Low-Resolution ADCs N DL RAUs send data pre-coded by CPU in DL, and K DL users receive signal. The data received by the k-th user is ] ∈ C NDLM ×1 is the channel of the ith DL user,ĝ DL,i is the estimation of g DL,i , and g DL,n,i = λ 1/2 DL,n,i h DL,n,i ∈ C M ×1 is the channel from the n-th DL RAU to the i-th DL user, where simplified free-space model is adopted [15]. λ DL,n,i = d −αDL n,i is the large-scale fading, d n,i is the distance between the n-th RAU and the i-th user, and α DL is the path loss. h DL,n,i represents the small-scale fading modeled as Rayleigh channel [7], [16]. Interference channel between UL user j and DL user k is defined as u I,k,j = λ 1/2 I,k,j h I,k,j , and its parameters are in one-to-one correspondence with the previous definitions [7]. n DL ∼ CN (0, 1) is the DL complex additive white Gaussian noise (AWGN). The lowresolution ADCs employed on the receiving antenna quantize the analog signal. Additive quantization noise model (AQNM) [17] is adopted for the analysis of quantization effect, and quantizes Eq. (1) as r DL,k = KDL i=1 √ p DL µ k,i s i + KUL j=1 √ p UL u I,k,j x j + n DL .(r q,DL,k = ξ k r DL,k + n q,DL,k ,(2) where ξ k = 1 − ρ k and n q,DL,k is additive quantization Gaussian noise (AQGN) with covariance matrix C n q,DL,k = ξ k ρ k diag(r DL,k r H DL,k ). ρ k is the reciprocal of the signal-toquantization noise ratio as I QUANTIZATION COEFFICIENT ρ k WITH b k BITS. b 1 2 3 4 5 ρ 0.3634 0.1175 0.0345 0.0095 √ 3/2π · 4 −b B. UL Signal Transmission with Low-Resolution ADCs K UL users send the signal concurrently in UL, and N UL RAUs receive jointly as r UL = KUL i=1 √ p UL g UL,i x i + KDL j=1 √ p DL f j s j + n UL ,(3) where g UL,i = [g H UL,1,i , . . . , g H UL,NUL,i ] ∈ C NULM ×1 , and g UL,n,i = λ 1/2 UL,n,i h UL,n,i ∈ C M ×1 is the channel from the n-th UL RAU to the i-th UL user. n UL represents the UL complex AWGN. G I ∈ C NULM ×NDLM is the interference channel, and g I,i,j = λ 1/2 I,i,j h I,i,j ∈ C M ×M is the interference channel between i-th UL RAU and j-th DL RAU [7]. Denote f j = G I w j ∈ C NULM ×1 as auxiliary interference channel. Received by the UL RAU through the low-resolution ADCs, the UL quantized signal is r q,UL = Ar UL + n q,UL ,(4) where A = diag(α 1 , α 2 , . . . , α NUL ) ⊗ I M ∈ C NULM ×NULM . α n = 1 − ρ n describes the ADC resolution of the nth UL RAU. n q,UL is AQGN with C nq,UL = A(I − A)diag(r UL r H UL ). III. TWO-STAGE CHANNEL ESTIMATION SCHEME In this section, we propose a two-stage channel estimation scheme, UL pilot estimation, and beamforming training estimation. All users send pilot sequences for UL pilot estimation within the first τ 1 time slots (τ 1 ≥ K), while all RAUs send them to the CPU for channel estimation. For beamforming training within the second τ 2 time slots, all DL RAUs send pre-coded pilot, while all DL users and UL RAUs perform channel estimation, respectively. And then, the last T −τ 1 −τ 2 time slots are used for data transmission. In the NAFD M-MIMO system, interference between DL RAUs and UL RAUs is the main factor that reduces the UL SE when some RAUs are used for UL reception while others are used for DL transmission. To improve the SE, an interference cancellation mechanism based on beamforming training is proposed. Specifically, the CPU estimates the equivalent interference channel between the RAUs, reconstructs the interference signal using the estimated equivalent CSI, and finally performs interference cancellation by subtracting the reconstructed interference signal from the received signal. A. UL Pilot Estimation In the pilot estimation stage, the k-th user sends the orthogonal pilot as φ k ∈ C τ1×1 . n-th RAU receives r p,ω n = √ p UP K k=1 g ω n,k φ k + n p n ,(5) where p UP is the power of UL pilot and g ω n,k ∈ C M ×K is the channel from K user to the n-th RAU in UL or DL operating mode. n p n is AWGN, and its element obeys i.i.d. CN (0, σ 2 UP ). The quantized signal is r p,ω n = θ ω n √ p UP K k=1 g ω n,k φ k + n p n + n p,ω n ,(6) where θ ω n depends on the ADC quantization bits on the n-th RAU, and n p,ω n is AQGN. As for the k-th user, r p,ω n,k = r p,ω n φ H k = √ p UP θ ω n g ω n,k + θ ω n n p n,k + n p,ω n,k . (7) Minimum mean square error (MMSE) estimation with estimated error g ω n,k = g ω n,k −ĝ ω n,k is adopted by CPU [8] aŝ g ω n,k = E g ω n,k + cov(g ω n,k ,r p,ω n,k ) cov(r p,ω n,k ,r p,ω n,k ) r p,ω n,k − E r p,ω n,k ,(8) where g ω n,k andĝ ω n,k are both complex Gaussian random vectors with zero mean and variances as β ω,n,k = p UP θ ω n λ 2 ω,n,k p UP λ ω,n,k + σ 2 UP ,(9a)η ω,n,k = (1 − θ ω n )p UP λ ω,n,k + λ ω,n,k σ 2 UP p UP λ ω,n,k + σ 2 UP .(9b) Due to limited space, the details can be referred to [8]. B. Beamforming Training Estimation In the beamforming training stage, DL RAUs send pilot. DL users and UL RAUs estimate the equivalent channels µ k,i , f j . 1) DL User Estimation: RAUs send WPΨ, where Ψ ∈ C KDL×τ2 is orthogonal pilot sequence. Define W = [w 1 , ..., w KDL ] ∈ C NDLM ×KDL as the beamforming matrix and P = diag( √ p DP,1 , . . . , √ p DP,KDL ) as the pilot power matrix, where p DP is the power of DL pilot. The received signal of the k-th DL user is r p k = g H DL,k WPΨ + n DP ,(10) where n DP is AWGN, and its element obeys i.i.d. CN (0, σ 2 DP ). The quantized signal is r p k = ξ k PΨ + ξ k n DP +ñ DP ,(11) where P = [ √ p DP,1 µ k,1 , . . . , √ p DP,KDL µ k,KDL ]. Correlating the quantized signal with Ψ H , we have the effective signal as r p e,k =r p k Ψ H = ξ k P + ξ k n e,DP +ñ e,DP , where n e,DP = n DP Ψ H ,ñ e,DP =ñ e,DP Ψ H . In addition, we obtain the estimation of µ k,i by MMSE algorithm aŝ µ k,i = E [µ k,i ] + cov(µ k,i ,r p,i e,k ) cov(r p,i e,k ,r p,i e,k ) r p,i e,k − E r p,i e,k ,(13) wherer p,i e,k is the i-th column ofr p e,k . With Lemma 1 in Appendix I, ||ĝ DL,k || 2 ∼ Γ(k DL,k , θ DL,k ), || g DL,k || 2 ∼ Γ(k DL,k ,θ DL,k ). The details of the distribution parameters are provided in Appendix II. Proposition 1: With MRT/ZFT pre-coders (pre ∈ {MR, ZF}) on DL RAUs, MMSE estimation can be expressed asμ pre k,i =          E pre k + √ pDPχ pre kr p,i e,k − ξ kχ pre k pDPE pre k pDP χ pre k + σ 2 DP , i = k √ pDPχ pre k pDPχ pre k + σ 2 DP , i = k .(14) • For MRT pre-coding, E MR k = Γ (k DL,k + 1/2) Γ (k DL,k ) θ 1/2 DL,k ,(15a)χ MR k =χ MR k + (1 − ξ k )(E MR k ) 2 (15b) χ MR k = k DL,k θ DL,k − E MR k 2 +k DL,kθDL,k NDLM ,(15c)χ MR k = k DL,k θ DL,k +k DL,kθDL,k NDLM ,(15d) • For ZFT pre-coding, E ZF k = Γ (tDLk DL,k + 1/2) Γ (tDLk DL,k ) θ 1/2 DL,k ,(16a)χ ZF k =χ ZF k + (1 − ξ k )(E ZF k ) 2 (16b) χ ZF k = tDLk DL,k θ DL,k − E ZF k 2 +k DL,kθDL,k NDLM , (16c) χ ZF k = k DL,k θ DL,k +k DL,kθDL,k NDLM ,(16d)t DL/UL = N DL/UL M − K DL/UL + 1 N DL/UL M . (16e) Proof: Please refer to Appendix II. Remark 1: With a reduced channel-hardening effect, the statistical CSI, without a priori knowledge of the perfect CSI, leads to severe performance degradation. Beamforming training scheme is adopted to estimate the equivalent channel µ k,i instead of the channel matrix g k,i , effectively reducing the number of pilot sequences from N DL M to K DL . 2) UL RAUs Estimation: For the estimation of interference channel G I to fulfill CLI cancellation, all UL RAUs jointly receive the pilot from the DL RAUs and send it back to CPU via backhaul link as Y p = G I WPΨ + N UP = √ p DP,1 f 1 , . . . , √ p DP,KDL f KDL Ψ + N UP ,(17) where DL RAUs send WPΨ, similar as above. N UP is AWGN. Correlating the received signal with Ψ H and extracting the i-th column, the quantized signal is y p i = A √ p DP f i + An UP,i +ñ UP,i ,(18)whereñ UP,i is AWGN with covariance Cñ UP,i = A(I − A)diag(||ỹ p i || 2 ). The MMSE estimation of f i iŝ f i = E [f i ] + cov(f i ,ỹ p i ) cov(ỹ p i ,ỹ p i ) (ỹ p i − E [ỹ p i ]) .(19) Proposition 2: In the NAFD distributed M-MIMO systems, the estimations of the effective interference channel iŝ f i = √ p DP Λ p DP Λ + σ 2 UP I M NUL ỹ p i ,(20) where Λ = diag NDL n=1 λ I,1,n , . . . , NDL n=1 λ I,NUL,n ⊗ I M NDL . Decomposing it into the multiplication of equivalent largescale fading δ I,n,i and small-scale fadingq I,n,i ,f i is rewritten asf i = [δ I,1,iqI,1,i , δ I,2,iqI,2,i , . . . , δ I,NUL,iqI,NUL,i ] H ,(21) where δ I,j,i = √ p DP α j NDL n=1 λ I,j,n N DL p DP NDL n=1 λ I,j,n + N 2 DL σ 2 UP . (22) Proof: Please refer to Appendix III. IV. SE/EE ANALYSIS OF NAFD DISTRIBUTED M-MIMO A. Derivation of the DL Achievable Rate With the estimated CSI and statistical CSI of DL users, we analyze the DL SE of the NAFD distributed M-MIMO systems with low-resolution ADCs. With the DL beamforming training estimation, we rewrite the received signal of the k-th user as r q DL,k = ξ k √ p DLμk,k s k desird signal + n eff k + ξ k n DL + n q DL,k effective noise(23) where n eff k = ξ k i =k √ p DLμk,i s i + ξ k KDL i=1 √ p DLμk,i s i + ξ k KUL j=1 √ p UL u t,k,j x j , andμ k,i represents the estimation error of µ k,i . For simplicity, each term in effective noise can be approximated as equivalent noise obeying the Gaussian distribution. The DL achievable rate of the k-th user is derived as Eq. (24), where C n DL,k = σ 2 DL + C n q,DL,k /ξ 2 k . Theorem 1: Under the estimated CSI at the DL user, the closed-form expression of the DL achievable rate of k-th user is R pre DL,k = log 2 1 + ξ k p DL (E pre k ) 2 + F pre k A pre k + B pre k + C pre k + σ 2 DL ,(25)R DL,k = E   log 2   1 + √ pDLμ k,k s k 2 i =k √ pDLμ k,i si 2 + K DL i=1 √ pDLμ k,i si 2 + K UL j=1 √ pULu I,k,j xj 2 + Cn DL,k       . (24) R UL,k = E   log 2   1 + pUL v H k Aĝ UL,k 2 i =k pUL |v H k AĝUL,i| 2 + K UL i=1 pUL |v H k AgUL,i| 2 + K DL j=1 pDL v H k Afj 2 + Cn UL,k       . (33) where F pre k = p DP (χ pre k ) 2 p DP [χ pre k + (1 − ξ k )(E pre k ) 2 ] + σ 2 DP , (26a) A pre k = i =k ξ k p DL χ pre k + KUL j=1 p UL λ I,k,j ,(26b)B pre k = ξ k (1 − ξ k )p DPχ pre k (E pre k ) 2 + ξ kχ pre k σ 2 DP χ pre k + (1 − ξ k )(E pre k ) 2 + σ 2 DP /p DL , (26c) C pre k = (1 − ξ k )p DL χ pre k + (E pre k ) 2 . (26d) Proof: Please refer to Appendix IV. Theorem 2: Under the statistical CSI, the closed-form expression of the DL achievable rate of k-th user is R pre DL,k,s = log 2 1 + ξ k p DL (E pre k ) 2 A pre k,s + B pre k,s + σ 2 DL ,(27) where A pre k,s = i =k ξ k p DL χ pre k + p DLχ pre k , (28a) B pre k,s = KUL j=1 p UL λ I,k,j + (1 − ξ k )p DL (E pre k ) 2 . (28b) Proof: The only difference between Theorems 2 and 1 exists in the gap between the effective signal: E |ξ k √ p DLμk,k s k | 2 − (E [ξ k √ p DL µ k,k s k ]) 2 = ξ 2 k p DL p DP (χ pre k ) 2 p DPχ pre k + (1 − ξ k )p DP (E pre k ) 2 + σ 2 DP .(29) After inserting Eq. (29) into denominator, we have Eqs. (27) and (28) with proper approximation. Remark 2: The closed-form expressions of the DL achievable rate obtained by the DL channel estimation and static statistics are given, respectively. The beamforming training mechanism's effect can be analyzed by comparing the two cases. B. Derivation of the UL Achievable Rate Considering interference cancellation at the UL RAUs, we analyze the UL SE of NAFD distributed M-MIMO systems with low-resolution ADCs. The quantized UL signal is r UL q = A   KUL i=1 √ p UL g UL,i x i + KDL j=1 √ p DL f j s j   + n q,UL ,(30) where n q,UL = An UL + n q,UL is the equivalent quantization noise. The UL RAUs perform CLI cancellation as r IC q,UL = r UL q − x IC ,(31)where x IC = KDL j=1 √ p DL,j Af j s j ∈ C NULM ×1 is eliminable interference andf j = f j −f j is the estimation error of f j . After interference cancellation, the signal is processed by the receiver as r UL k = v H k r IC q,UL = v H k n q,UL + n UL + v H k A   KUL i=1 √ p UL g UL,i x i + KDL j=1 √ p DLfj s j   ,(32) where the receiving vector v H k for the k-th user for MRC and ZFR areĝ UL,k andĝ UL,k ĝ H UL,kĝ UL,k −1 , respectively. After decomposing r UL k into effective signal, interference and equivalent noise in Eq. (32), the UL achievable rate of the k-th user can be expressed as Eq. (33), where C n UL,k = v H k A 2 σ 2 UL + v H k C nq,UL v k . Theorem 3: With the interference cancellation of UL RAUs, the closed-form expression of the UL achievable rate of the NAFD distributed M-MIMO systems can be derived as R pre UL,k = log 2 1 + A pre k B pre k + NUL n=1 α n C pre n,k /N UL ,(34) • With MRT pre-coding and MRC receiver, A MR k = pULM N UL n=1 α 2 n β UL,n,k + σ 2 UL /NUL N UL n=1 αn, (35a) B MR k = pULM N UL n=1 αn (1 − αn) β UL,n,k ,(35b)C MR n,k = pDL ρ UL,n,i N DL m=1 λI,n,m − δ 2 UL,n,i /NDL (35c) + i =k pULβUL,n,i + K UL i=1 pULηUL,n,i, • With ZFT pre-coding and ZFR receiver, A ZF k = tULpULM N UL n=1 α 2 n β UL,n,k ,(36a)B ZF k = tULpULM N UL n=1 αn (1 − αn) β UL,n,k ,(36b)C ZF n,k = K DL j=1 pDLρ 2 UL,n,j + K UL i=1 pULηUL,n,i + σ 2 UL . (36c) Proof: Please refer to Appendix V. Remark 3: With the CLI cancellation under low-resolution ADCs, we obtain the expression for closed-form expressions of the UL SE of the NAFD distributed M-MIMO. The total energy consumption will be analyzed to further analyze the impact of low-resolution ADCs comprehensively. C. Power Consumption of NAFD Distributed M-MIMO The high-resolution ADCs with high SE pay higher hardware cost and energy consumption. Environment-friendly communication raises an urgent call for the energy consumption with low-resolution ADCs. Inspired by [18], [19], the EE of the NAFD distributed M-MIMO systems is modeled as η EE = W · KUL k=1 R UL,k + KDL k=1 R DL,k P TC + P T + P LP + P BH ,(37) where W is the transmission bandwidth. P TC is the power consumption of the transceiver link 1 as P TC = N M P RAU + ρ SYN P SYN + KP UE + M P ADC . (38) where P RAU , P SYN and P UE represent the power consumption of the RAU, local oscillator, and the user, respectively. In the distributed antenna systems, each RAU has a crystal oscillator, ρ SYN = N . In the co-located antenna systems, base station only deploys one crystal oscillator, ρ SYN = 1. P ADC is the power consumption of the low-resolution ADCs [18], as P ADC = i a 0 · M · 2 bi + a 1 , where a 0 and a 1 are constant parameters related to energy consumption of ADCs [18] and b i is the bitwidth of i-th ADC. P T is the energy consumption of the transmitting signals as P T = K UL (T − τ 1 − τ 2 ) T · ξ p UL + K DL (T − τ 1 − τ 2 ) T · ξ p DL ,(39) where ξ is the amplifier efficiency. Denote P LP as the energy consumption of the linear receiver and pre-coder [18]: P LP = T −τ ω T 2W M N K L RAU + τ ω T        W M N K(3K +1) L RAU , (a) 3W M N K L RAU , (b)(40) where (a) and (b) mean RAUs adopt MRT pre-coder and MRC receiver, and ZFT pre-coder and ZFR receiver, respectively. L RAU is the calculation efficiency of the complex operation under unit power consumption. P BH , which can be ignored in the co-located antenna systems, is the energy consumption of the backhaul link [18] between RAUs and CPU as 1 In this paper, the effect of the low-resolution ADC is mainly considered to evaluate the contribution for EE. Other minor factors are modelled uniformly in P UE (such as the power amplifier in the transmitter chain and the lownoise-amplifier in the receiver chain) as [12], [16], [19]. P BH = N P 0 + W P BT KUL k=1 R UL,k + KDL k=1 R DL,k ,(41) where P 0 and P BT are the fixed energy consumption on each backhaul link and dynamic energy consumption associated with the backhaul rate, respectively. Remark 4: We discuss the various components of energy consumption in NAFD distributed M-MIMO systems and EE with low-resolution ADCs. The energy consumption of the transceiver can be significantly reduced by a flexible selection of quantization bits, which improves the system EE. However, the reduction of quantization bits affects the SE of the system at the same time, and in the next section, we propose the corresponding allocation scheme for feasible ADC quantization. V. BIT ALLOCATION ALGORITHM FOR ADC QUANTIZATION In this section, we discuss the delicate allocation mechanism of ADC quantization for RAUs and users in NAFD distributed M-MIMO systems, which considers the tradeoff between SE and EE to significantly reduce the system energy consumption while ensuring the system SE. A. Problem Description of ADC Bit Allocation Adaptive numerical schemes with a bit allocation mechanism show the non-negligible advantage of greater flexibility, and greener power consumption than the equal-resolution schemes [10]. The objective function is designed to reach the maximum value of total SE/EE of NAFD distributed M-MIMO with low-resolution ADCs as f 1 (b) = T − τ 1 − τ 2 T KUL k=1 R UL,k + KDL k=1 R DL,k , (42a) f 2 (b) = W · KUL k=1 R UL,k + KDL k=1 R DL,k P TC (b) + P T + P LP + P BH , (42b) where b = {(b 1 , . . . , b NUL ); (b 1 , . . . , b NDL ); (b 1 , . . . , b KDL )} is the bit allocation scheme for UL RAUs, DL RAUs, and DL users. The optimization problem is summarized as max f = [f 1 (b), f 2 (b)] s.t. C1 :B U ≥ M N +KDL n=1 b n , b n = 1, 2, ... C2 :R UL,k (b) ≥ R min UL , k = 1, ..., K UL C3 :R DL,l (b) ≥ R min DL , k = 1, ..., K DL C4 :P sum (b) ≥ P sum (b UQ,opt ), (43) where P sum (b) = P TC (b) + P T (b) + P LP (b) + P BH (b). Constraint C1 expresses the limitation of the total bits of ADCs caused by the limited backhaul capacity, where B U is the upper bound of the total ADC bits. Constraints C2 and C3 are the UL and DL average QoS requirements. Constraints C4 describes the EE lower bound as the power consumption corresponding to the optimal EE with guaranteed SE when quantized uniformly (b UQ,opt ) for UL RAUs, DL RAUs, and DL users, respectively. B. Problem Solution of ADC Bit Allocation Drawn lessons from the pioneer solutions which easily fall into local minima with slow convergence and high complexity when solving non-convex discrete optimization problems like Eq. (42), DQN and NSGA-II is adopted for optimal solutions. 1) Problem Solution Based on DQN: The solution process based on reinforcement learning (RL) methods includes agent, environment, reward, and action [20], which can be abstracted as a finite Markov decision process (MDP) [21]. Define the state and action at the t-th time-step as s t b t , a t b t ,(44) where b t and b t represent the quantization bit vector and the one-bit change at the t-th time-step. The reward is defined as r t = f 1 (b)−minf 1 (b) maxf 1 (b)−minf 1 (b) + f 2 (b)−minf 2 (b) maxf 2 (b)−minf 2 (b) − r,(45) where r represents normalization constant. Concerning the Qlearning in RL, the optimization goal is maximizing the future returns R t by deciding the actions of each step in a certain way, i.e., max π E[R t |s t , a t , π(a|s)]. Hence, define the optimal action-value function as Q i (s, a) = E r + γ max a Q i (s , a )|(s, a) ,(46) where γ is the discount factor and π(a|s) is the strategy to take action a in the s state. Meanwhile, Eq. (46) obeys the Bellman equation and guides value iteration with convergence boundary i ← ∞, Q * ← Q i . Unlike the typical linear methods, [20] trains the neural network to extract advanced features with the following loss function L i (θ i ) = E s,a∼ρ |y i − Q(s, a; θ i )| 2 ,(47) where y i = E[r + γ max a Q(s , a ; θ i−1 )|s, a] represents the target of the i-th iteration, and ρ is the behaviour distribution, which adopt the -greedy strategy (with a probability of p gdy = taking a random action a rnd ) to ensure adequate exploration of state space. And then, calculate its gradient as ∇ θi L i (θ i ) = E s,a∼ρ [(y i − Q(s, a; θ i ))∇ θi Q(s, a; θ i )]. (48) Rather than the full expectation above, stochastic gradient descent (SGD) is routinely considered as a computationallyexpedient alternative. To deal with the non-stationary distribution and high-correlation of data, Q-learning obtains the experience and pools them into replay memory M, which is randomly sampled to finish the mini-batch SGD [20]. The quantization bit allocation algorithm based on DQN is summarized in Algorithm 1 with T H iterations. 2) Problem Solution Based on NSGA-II: To analyze the problem more in-depth and choose quantization schemes more flexibly, we need to find the Pareto-optimal boundary of the bit optimization problem for ADC quantization, tricky for DQN. Multi-objective evolutionary algorithms have significant advantages when simultaneously solving all objectives in one simulation. Specifically, we employ the NSGA-II proposed in [22] with considerable reliability and effectiveness. Detail steps based on DQN are summarized in Algorithm 1. Use tournament selection to select a parent population 3: Perform selection, mutation, and crossing 4: Generate offspring population 5: Calculate SE and EE of offspring population 6: Merge the parent population and offspring populations 7: Calculate SE and EE of merged population 8: Sort the merged population based on non-dominating sorting and crowding distances 9: Select a new initial population to replace the old one. 10: until The optimization conditions are satisfied or reach the maximum generation Remark 5: When traditional methods deal with such nonconvex optimization problems, infinite local optimums exist in the set of feasible domains and convergence to the global optimum typically faces tremendous challenges since it has been proven as non-deterministic polynomial-time hard (NP-hard). The computational complexity of the DQN method is O(N b T 2 H ); the complexity of NSGA-II method is O(N obj N 2 pop ), where N b and N obj = 2, represent the number of low-resolution ADCs that need to be optimized and the objective functions, respectively. N pop is the population size. After random initialization, both methods can converge quickly to the corresponding Pareto-optimal points. Algorithm 1 Bit Allocation Algorithm Based on DQN VI. NUMERICAL RESULTS AND ANALYSIS In this section, Monte-Carlo simulation is employed to verify the closed-form expression of SE in NAFD distributed M-MIMO with low-resolution ADCs, and the bit allocation algorithm is carried out for Pareto-optimal solutions with the tradeoff between the total SE and EE. A. Parameter Setting In the NAFD distributed M-MIMO system, the simulated area is limited to a circular area with a radius of 1 km. All users and RAUs are randomly distributed, where N UL = 3, N DL = 3, K UL = 2, K DL = 3, and the minimum access distance to RAUs is 30 m. Assume that the number of antennas at each RAU is M = 10, if not otherwise specified, and each RAU employs ADCs with the same resolution. The path loss exponent is set to α UL = α DL = 3.7, α I = 3. The DL and UL noise variances are σ 2 DL = σ 2 UL = 1 W. The coherence time is T = 196 symbols, the lengths of UL and DL pilot sequence are τ 1 = K and τ 2 = K DL , respectively. Hence, the number of symbols used for data transmission is T data = T − τ 1 − τ 2 , where the pre-log factor T −τ1−τ2 T impacting the overall SE is considered. Table II presents all the parameters related to power consumption same as [18]. For the training details, the DQN is trained on the TensorFlow platform v1.14.0. The RMSProp algorithm is adopted with a mini-batch size of 32, and the learning rate is 0.01, where γ = 0.9, = 0. Fig. 2 illustrates the relationship between the average DL SE and ADC bits b of each DL user with MRT or ZFT beamforming training and with statistical CSI or estimated CSI. The curves in Fig. 2 fit well, which validates the DL rate closed-form expression derived by Theorems 1 and 2 with the acceptable error caused by the Gamma approximation. By changing the values of different variables in the closed-form expressions, we can obtain good fitting results. Due to the space limitation, we provide the validation process only for the most important parameters in low-resolution studies, and others are similar. The subsequent studies are performed by closed-form expressions rather than time-consuming Monte Carlo simulations. No matter which pre-coding is used, the average SE increases as the quantization bits increases and then converges, 8 IEEE SYSTEM JOURNAL, 2022 at each RAU is M = 10, if not otherwise specified, and each RAU employs ADCs with the same resolution. The path loss exponent is set to α UL = α DL = 3.7, α I = 3. The DL and UL noise variances are σ 2 DL = σ 2 UL = 1 W. The coherence time is T = 196 symbols, the lengths of UL and DL pilot sequence are τ 1 = K and τ 2 = K DL , respectively. Hence, the number of symbols used for data transmission is T data = T − τ 1 − τ 2 , where the pre-log factor T −τ1−τ2 T impacting the overall SE is considered. Table II presents all the parameters related to power consumption same as [18]. For the training details, the DQN is trained on the TensorFlow platform v1.14.0. The RMSProp algorithm is adopted with a mini-batch size of 32, and the learning rate is 0.01, where γ = 0.9, ϵ = 0. Fig. 2 illustrates the relationship between the average DL SE and ADC bits b of each DL user with MRT or ZFT beamforming training and with statistical CSI or estimated CSI. The curves in Fig. 2 fit well, which validates the DL rate closed-form expression derived by Theorems 1 and 2 with the acceptable error caused by the Gamma approximation. By changing the values of different variables in the closed-form expressions, we can obtain good fitting results. Due to the space limitation, we provide the validation process only for the most important parameters in low-resolution studies, and others are similar. The subsequent studies are performed by closed-form expressions rather than time-consuming Monte Carlo simulations. No matter which pre-coding is used, the average SE increases as the quantization bits increases and then converges, which means that the upper bound of system performance can be achieved via a lower-resolution ADC. Meanwhile, the DL beamforming training for the estimated CSI brings more remarkable performance improvement than the statistical CSI. The DL SE of MRT and ZFT reaches the top when ADC resolution is 5-6 bits and 7-8 bits, respectively. Because ZFT pre-coding eliminates inter-user interference, it performs better than MRT regardless of whether the beamforming training estimation is performed. Moreover, when beamforming training is adopted, the CSI sensitivity of ZFT pre-coding leads to a more significant performance improvement, about 15.0%, compared to the MRT, only about 8.9%. In summary, ZFT is more sensitive to ADC resolution, which needs a higher resolution to achieve better performance. Assume that the MRC and ZFR receivers are considered. Based on Eq. (33) in Theorem 3, Fig. 3 reveals the relationship between the average UL SE and the quantization bits b of each RAU. It illustrates that the simulated result shows good agreement with the theoretical result, which verifies the accuracy of Theorem 3. The UL SE increases with the quantization bits and then converges to an upper bound, which shows that the upper bound can be reached by low-resolution ADCs, similar to the DL. Regardless of MRC or ZFR receiver, interference which means that the upper bound of system performance can be achieved via a lower-resolution ADC. Meanwhile, the DL beamforming training for the estimated CSI brings more remarkable performance improvement than the statistical CSI. The DL SE of MRT and ZFT reaches the top when ADC resolution is 5-6 bits and 7-8 bits, respectively. Because ZFT pre-coding eliminates inter-user interference, it performs better than MRT regardless of whether the beamforming training estimation is performed. Moreover, when beamforming training is adopted, the CSI sensitivity of ZFT pre-coding leads to a more significant performance improvement, about 15.0%, compared to the MRT, only about 8.9%. In summary, ZFT is more sensitive to ADC resolution, which needs a higher resolution to achieve better performance. 8 IEEE SYSTEM JOURNAL, 2022 at each RAU is M = 10, if not otherwise specified, and each RAU employs ADCs with the same resolution. The path loss exponent is set to α UL = α DL = 3.7, α I = 3. The DL and UL noise variances are σ 2 DL = σ 2 UL = 1 W. The coherence time is T = 196 symbols, the lengths of UL and DL pilot sequence are τ 1 = K and τ 2 = K DL , respectively. Hence, the number of symbols used for data transmission is T data = T − τ 1 − τ 2 , where the pre-log factor T −τ1−τ2 T impacting the overall SE is considered. Table II presents all the parameters related to power consumption same as [18]. For the training details, the DQN is trained on the TensorFlow platform v1.14.0. The RMSProp algorithm is adopted with a mini-batch size of 32, and the learning rate is 0.01, where γ = 0.9, ϵ = 0.9, M = 2000, and the solution process needs T H = 1, 000 iterations for complete convergence. Both evaluate net and target net networks in DQN consist of fully connected DNN with 2 hidden layers with ReLU activation functions, where their dimensions are 9 and 18. The dimensions of the input and output layers of both evaluate net network and target net network are equal to 9 and 18. In NSGA-II, the population size is N pop = 200, and the max number of generations is N gen = 300. Fig. 2 illustrates the relationship between the average DL SE and ADC bits b of each DL user with MRT or ZFT beamforming training and with statistical CSI or estimated CSI. The curves in Fig. 2 fit well, which validates the DL rate closed-form expression derived by Theorems 1 and 2 with the acceptable error caused by the Gamma approximation. By changing the values of different variables in the closed-form expressions, we can obtain good fitting results. Due to the space limitation, we provide the validation process only for the most important parameters in low-resolution studies, and others are similar. The subsequent studies are performed by closed-form expressions rather than time-consuming Monte Carlo simulations. No matter which pre-coding is used, the average SE increases as the quantization bits increases and then converges, which means that the upper bound of system performance can be achieved via a lower-resolution ADC. Meanwhile, the DL beamforming training for the estimated CSI brings more remarkable performance improvement than the statistical CSI. The DL SE of MRT and ZFT reaches the top when ADC resolution is 5-6 bits and 7-8 bits, respectively. Because ZFT pre-coding eliminates inter-user interference, it performs better than MRT regardless of whether the beamforming training estimation is performed. Moreover, when beamforming training is adopted, the CSI sensitivity of ZFT pre-coding leads to a more significant performance improvement, about 15.0%, compared to the MRT, only about 8.9%. In summary, ZFT is more sensitive to ADC resolution, which needs a higher resolution to achieve better performance. Assume that the MRC and ZFR receivers are considered. Based on Eq. (33) in Theorem 3, Fig. 3 reveals the relationship between the average UL SE and the quantization bits b of each RAU. It illustrates that the simulated result shows good agreement with the theoretical result, which verifies the accuracy of Theorem 3. The UL SE increases with the quantization bits and then converges to an upper bound, which shows that the upper bound can be reached by low-resolution ADCs, similar to the DL. Regardless of MRC or ZFR receiver, interference Assume that the MRC and ZFR receivers are considered. Based on Eq. (33) in Theorem 3, Fig. 3 reveals the relationship between the average UL SE and the quantization bits b of each RAU. It illustrates that the simulated result shows good agreement with the theoretical result, which verifies the accuracy of Theorem 3. The UL SE increases with the quantization bits and then converges to an upper bound, which shows that the upper bound can be reached by low-resolution ADCs, similar to the DL. Regardless of MRC or ZFR receiver, interference cancellation brings a tremendous performance improvement, proving the DL beamforming training is helpful for UL RAUs to eliminate interference between RAUs. The gap between MRC and ZFR receivers with CLI cancellation blows up since the strength of elimination determines the SE attenuation. 2) Relationship Between the Total SE and EE: Next, we explore the relationship between the total SE and EE in Eq. (42). For the MRT pre-coding and MRC receiver, Fig. 4 3) Bit Allocation Scheme: As the ADC bitwidths increase, SE increases until saturation, but EE continues to decrease under mutual constraint and balance between SE and EE. To maximize the total SE and total EE, we design the bit allocation algorithm to obtain a tradeoff scheme. The QoS requirement (further improvement of QoS reduces the number of solutions) is R min UL = R min DL = 1.5 bit/s and the total ADC bit limit is 12M N + 12K DL . For MRT/MRC and ZFT/ZFR schemes, the power constraint is set to the consumption of the allocation schemes as {(7, 7, 7); (5, 5, 5); (6, 6, 6)} and { (8,8,8); (1, 1, 1); (7, 7, 7)}, respectively. Fig. 5 shows the tradeoff between total SE and EE, where the feasible bit allocation schemes for UL RAUs, DL RAU, and DL users (negligible with only one antenna), based on DQN or selected on the Pareto-optimal boundary by NSGA-II are {(6, 7, 7); (2, 4, 3); (6, 6, 6)} or {(7, 7, 7); (5, 4, 5); (7, 7, 7)} for MRC/MRT, and {(6, 7, 6); (1, 1, 1); (7, 7, 7)} or {(6, 7, 6); (1, 1, 1); (7, 4, 7)} for ZFT/ZFR. When quantization bitwidths become relatively large, a tiny increase results in a significant leap in the power consumption and a rapid decline of EE. Note that the SE-only or EE-only method is equivalent to linear weighting, and it provides only the endpoints of the Pareto front (EE of SE-only's optimum is too small without practical meaning, and the asymptotic lines are only provided), which leads to certain inflexibility for the tradeoff of MOOP. Further, all Pareto-optimal solutions with MRT/MRC and ZFT/ZFR schemes are obtained by NSGA-II, which validates the feasibility of the optimization methods and offers flexible ADC resolution choices for practical deployment. 15 bit/J] b = 4 b = 5 b = 6 b = 7 b = 8 b = 9 VII. CONCLUSIONS This paper analyzed the NAFD distributed M-MIMO systems with low-resolution ADCs. Two-stage channel estimation, including pilot estimation and beamforming training, was adopted to complete the CLI elimination and obtain the effective CSI for signal detection and interference cancellation. With the estimated CSI, the closed-form expressions for the achievable rates of the UL and DL, as well as EE, were derived. Further, a MOOP maximizing SE and EE under per-user QoS and total bit constraints was formulated to design an efficient ADC bit allocation scheme. The simulation results verified the theoretical analysis. Low-resolution ADCs significantly improved EE in NAFD distributed M-MIMO. Interference cancellation via beamforming training enhanced the UL SE with different receivers under low-resolution ADCs, 3) Bit Allocation Scheme: As the ADC bitwidths increase, SE increases until saturation, but EE continues to decrease under mutual constraint and balance between SE and EE. To maximize the total SE and total EE, we design the bit allocation algorithm to obtain a tradeoff scheme. The QoS requirement (further improvement of QoS reduces the number of solutions) is R min UL = R min DL = 1.5 bit/s and the total ADC bit limit is 12M N + 12K DL . For MRT/MRC and ZFT/ZFR schemes, the power constraint is set to the consumption of the allocation schemes as {(7, 7, 7); (5, 5, 5); (6, 6, 6)} and { (8,8,8); (1, 1, 1); (7, 7, 7)}, respectively. Fig. 5 shows the tradeoff between total SE and EE, where the feasible bit allocation schemes for UL RAUs, DL RAU, and DL users (negligible with only one antenna), based on DQN or selected on the Pareto-optimal boundary by NSGA-II are {(6, 7, 7); (2, 4, 3); (6, 6, 6)} or {(7, 7, 7); (5, 4, 5); (7, 7, 7)} for MRC/MRT, and {(6, 7, 6); (1, 1, 1); (7, 7, 7)} or {(6, 7, 6); (1, 1, 1); (7, 4, 7)} for ZFT/ZFR. When quantization bitwidths become relatively large, a tiny increase results in a significant leap in the power consumption and a rapid decline of EE. Note that the SE-only or EE-only method is equivalent to linear weighting, and it provides only the endpoints of the Pareto front (EE of SE-only's optimum is too small without practical meaning, and the asymptotic lines are only provided), which leads to certain inflexibility for the tradeoff of MOOP. Further, all Pareto-optimal solutions with MRT/MRC and ZFT/ZFR schemes are obtained by NSGA-II, which validates the feasibility of the optimization methods and offers flexible ADC resolution choices for practical deployment. cancellation brings a tremendous performance improvement, proving the DL beamforming training is helpful for UL RAUs to eliminate interference between RAUs. The gap between MRC and ZFR receivers with CLI cancellation blows up since the strength of elimination determines the SE attenuation. 2) Relationship Between the Total SE and EE: Next, we explore the relationship between the total SE and EE in Eq. (42). For the MRT pre-coding and MRC receiver, Fig. 4 illustrates the total EE v.s. SE with different numbers of quantization bits and antennas per RAU. Different lines represent different numbers of quantization bits B ∈ {b|4 ⩽ b ⩽ 9, b ∈ N}, and different points represent different numbers of antennas M ∈ {2m|3 ⩽ m ⩽ 16, m ∈ N}. For different lines, as the ADC accuracy increases, the SE gradually increases, and the EE increases and then decreases. For different points on the same line, as the number of antennas on each RAUs increases, the SE increases, and the EE increases and then decreases. It reveals that both EE and SE increase with the growth of the antenna and bit widths when the improvement of SE first dominates, followed by EE's prevailing. 12 3) Bit Allocation Scheme: As the ADC bitwidths increase, SE increases until saturation, but EE continues to decrease under mutual constraint and balance between SE and EE. To maximize the total SE and total EE, we design the bit allocation algorithm to obtain a tradeoff scheme. The QoS requirement (further improvement of QoS reduces the number of solutions) is R min UL = R min DL = 1.5 bit/s and the total ADC bit limit is 12M N + 12K DL . For MRT/MRC and ZFT/ZFR schemes, the power constraint is set to the consumption of the allocation schemes as {(7, 7, 7); (5, 5, 5); (6, 6, 6)} and { (8,8,8); (1, 1, 1); (7, 7, 7)}, respectively. Fig. 5 shows the tradeoff between total SE and EE, where the feasible bit allocation schemes for UL RAUs, DL RAU, and DL users (negligible with only one antenna), based on DQN or selected on the Pareto-optimal boundary by NSGA-II are {(6, 7, 7); (2, 4, 3); (6, 6, 6)} or {(7, 7, 7); (5, 4, 5); (7, 7, 7)} for MRC/MRT, and {(6, 7, 6); (1, 1, 1); (7, 7, 7)} or {(6, 7, 6); (1, 1, 1); (7, 4, 7)} for ZFT/ZFR. When quantization bitwidths become relatively large, a tiny increase results in a significant leap in the power consumption and a rapid decline of EE. Note that the SE-only or EE-only method is equivalent to linear weighting, and it provides only the endpoints of the Pareto front (EE of SE-only's optimum is too small without practical meaning, and the asymptotic lines are only provided), which leads to certain inflexibility for the tradeoff of MOOP. Further, all Pareto-optimal solutions with MRT/MRC and ZFT/ZFR schemes are obtained by NSGA-II, which validates the feasibility of the optimization methods and offers flexible ADC resolution choices for practical deployment. 15 bit/J] b = 4 b = 5 b = 6 b = 7 b = 8 b = 9 VII. CONCLUSIONS This paper analyzed the NAFD distributed M-MIMO systems with low-resolution ADCs. Two-stage channel estimation, including pilot estimation and beamforming training, was adopted to complete the CLI elimination and obtain the effective CSI for signal detection and interference cancellation. With the estimated CSI, the closed-form expressions for the achievable rates of the UL and DL, as well as EE, were derived. Further, a MOOP maximizing SE and EE under per-user QoS and total bit constraints was formulated to design an efficient ADC bit allocation scheme. The simulation results verified the theoretical analysis. Low-resolution ADCs significantly improved EE in NAFD distributed M-MIMO. Interference cancellation via beamforming training enhanced the UL SE with different receivers under low-resolution ADCs, cancellation brings a tremendous performance improvement, proving the DL beamforming training is helpful for UL RAUs to eliminate interference between RAUs. The gap between MRC and ZFR receivers with CLI cancellation blows up since the strength of elimination determines the SE attenuation. 2) Relationship Between the Total SE and EE: Next, we explore the relationship between the total SE and EE in Eq. (42). For the MRT pre-coding and MRC receiver, Fig. 4 illustrates the total EE v.s. SE with different numbers of quantization bits and antennas per RAU. Different lines represent different numbers of quantization bits B ∈ {b|4 ⩽ b ⩽ 9, b ∈ N}, and different points represent different numbers of antennas M ∈ {2m|3 ⩽ m ⩽ 16, m ∈ N}. For different lines, as the ADC accuracy increases, the SE gradually increases, and the EE increases and then decreases. For different points on the same line, as the number of antennas on each RAUs increases, the SE increases, and the EE increases and then decreases. It reveals that both EE and SE increase with the growth of the antenna and bit widths when the improvement of SE first dominates, followed by EE's prevailing. 12 3) Bit Allocation Scheme: As the ADC bitwidths increase, SE increases until saturation, but EE continues to decrease under mutual constraint and balance between SE and EE. To maximize the total SE and total EE, we design the bit allocation algorithm to obtain a tradeoff scheme. The QoS requirement (further improvement of QoS reduces the number of solutions) is R min UL = R min DL = 1.5 bit/s and the total ADC bit limit is 12M N + 12K DL . For MRT/MRC and ZFT/ZFR schemes, the power constraint is set to the consumption of the allocation schemes as {(7, 7, 7); (5, 5, 5); (6, 6, 6)} and { (8,8,8); (1, 1, 1); (7, 7, 7)}, respectively. Fig. 5 shows the tradeoff between total SE and EE, where the feasible bit allocation schemes for UL RAUs, DL RAU, and DL users (negligible with only one antenna), based on DQN or selected on the Pareto-optimal boundary by NSGA-II are {(6, 7, 7); (2, 4, 3); (6, 6, 6)} or {(7, 7, 7); (5, 4, 5); (7, 7, 7)} for MRC/MRT, and {(6, 7, 6); (1, 1, 1); (7, 7, 7)} or {(6, 7, 6); (1, 1, 1); (7, 4, 7)} for ZFT/ZFR. When quantization bitwidths become relatively large, a tiny increase results in a significant leap in the power consumption and a rapid decline of EE. Note that the SE-only or EE-only method is equivalent to linear weighting, and it provides only the endpoints of the Pareto front (EE of SE-only's optimum is too small without practical meaning, and the asymptotic lines are only provided), which leads to certain inflexibility for the tradeoff of MOOP. Further, all Pareto-optimal solutions with MRT/MRC and ZFT/ZFR schemes are obtained by NSGA-II, which validates the feasibility of the optimization methods and offers flexible ADC resolution choices for practical deployment. 15 bit/J] b = 4 b = 5 b = 6 b = 7 b = 8 b = 9 VII. CONCLUSIONS This paper analyzed the NAFD distributed M-MIMO systems with low-resolution ADCs. Two-stage channel estimation, including pilot estimation and beamforming training, was adopted to complete the CLI elimination and obtain the effective CSI for signal detection and interference cancellation. With the estimated CSI, the closed-form expressions for the achievable rates of the UL and DL, as well as EE, were derived. Further, a MOOP maximizing SE and EE under per-user QoS and total bit constraints was formulated to design an efficient ADC bit allocation scheme. The simulation results verified the theoretical analysis. Low-resolution ADCs significantly improved EE in NAFD distributed M-MIMO. Interference cancellation via beamforming training enhanced the UL SE with different receivers under low-resolution ADCs, and channel estimation of DL users refined the DL SE, especially for the ZFT pre-coding and ZFR receiver. DQN and NSGA-II were selected to solve the bit allocation problem, which obtained Pareto-optimal solutions for ADC resolution providing great flexibility for system development. APPENDIX I MATHEMATICAL RESULTS Lemma 1: x ∈ C m×1 distributes as x i ∼ CN (0, σ 2 i I) [23], , θ), and its elements are independent non-identical distributions, when x is projected to an s-dimensional subspace, the distribution of projection power can be approximated as Γ(sk/m, θ) [23]. x H i x i ∼ Γ(m, σ 2 i ), (49a) i x H i x i ∼ Γ m i σ 2 i 2 i σ 4 i , i σ 4 i i σ 2 i . (49b) Lemma 2: x ∈ C m×1 satisfies x H x ∼ Γ(k APPENDIX II PROOF OF PROPOSITION 1 Proof: Consider MRT pre-coding as w i =ĝ DL,i ĝ DL,i , E [µ k,i ] = E [g DL,k w i ]. When i = k, E [µ k,i ] = E [ ĝ DL,k ]. According to the Lemmas introduced in Appendix I, we have From the relationship between Gamma and Nakagami distribution, ĝ DL,k ∼ Nakagami(k DL,k , k DL,k θ DL,k ), ĝ DL,n,k 2 ∼ Γ (M, β DL,n,k ) (50a) ĝ DL,k 2 = NDL n=1 ĝ DL,n,k 2 ∼ Γ (k DL,k , θ DL,k ) (50b)E [ ĝ DL,k ] = Γ (k DL,k + 1/2) Γ (k DL,k ) θ 1/2 DL,k .(52) When i = k,ĝ DL,k is independent ofĝ DL,i , E [µ k,i ] = 0. Combining the above two cases, E [µ k,i ] =    Γ (k DL,k + 1/2) Γ (k DL,k ) θ 1/2 DL,k , i = k 0, i = k .(53) Moreover, we have cov(µ k,i ,r p,i e,k ) = E µ k,ir p,i e,k − E [µ k,i ] E r p,i e,k = ξ k √ p DP E µ 2 k,i − E [µ k,i ] 2 ,(54) where E µ 2 k,i = E ĝ DL,k w i 2 + E g DL,k w i 2 .(55) If RAUs use MRT pre-coding, the pre-coding vector is placed in the N DL M -dimensional space. If RAUs use ZFT pre-coding, the pre-coding vector is placed in N DL M −K DL + 1 dimensional space. With the Lemmas in Appendix I,              ĝ DL,k w i 2 ∼ Γ (k DL,+ σ 2 DP +(ξ k −ξ 2 k )p DP (E MR k ) 2 when i=k ,(59) where E ñ 2 DP,i = ξ k (1 − ξ k )E ( √ pDPµ k,i + nDP,i) 2 =    ξ k (1 − ξ k ) pDP χ MR k + (E MR k ) 2 + σ 2 DP , i = k ξ k (1 − ξ k ) pDPχ MR k + σ 2 DP , i = k .(60) Finally, we obtain E r p,i e,k = ξ k √ pDPE [µ k,i ] = ξ k √ pDPE MR k , i = k 0, i = k .(61) Next is a similar proof of the ZFT. There is only a small difference between ZFT and MRT. If RAUs use ZFT pre-coding, the pre-coding vector is placed in N DL M − K DL + 1 dimensional space. When i = k, ĝ DL,k w i 2 ∼ Γ (t DL k DL,k , θ DL,k ), and E [µ k,i ] =    Γ (t DL k DL,k + 1/2) Γ (t DL k DL,k ) θ 1/2 DL,k , i = k 0, i = k .(62) Other than that, it is the same as MRT. Therefore, we omit the proof of ZFT due to the limitation of space. APPENDIX III PROOF OF PROPOSITION 2 Proof: Since G I and w i are independent, cov(f i ,ỹ p i ) = E[f i (ỹ p i ) H ] − E [f i ] E [ỹ p i ] = √ p DP AE[f i f H i ], where E [f i ] = 0, E [y p i ] = 0. g I,k is the k-th line of the interference channel G I . Next, E g I,k w i w H i g H I,j = 0, when j = k. Otherwise, E g I,k w i w H i g H I,j = 1 NDL NDL n=1 λ I,k,n . And then, cov(ỹ p i ,ỹ p i ) = E ỹ p i (ỹ p i ) H = p DP A 2 (E f i f H i +I)+A(I−A)diag p DP E f i f H i +σ 2 UP I = p DP AΛ + Aσ 2 UP .(63) Because cov(ỹ p i ,ỹ p i ) −1/2ỹ p i can be equivalent to a smallscale fading following the distribution of i.i.d. CN (0, I), then cov(f i ,ỹ p i )cov(ỹ p i ,ỹ p i ) −1/2 can be equivalent to large-scale fading. After simple derivation, Eq. (21) is derived. APPENDIX IV PROOF OF THEOREM 1 Proof: The derivation of DL rate expression is similar for MRT and ZFT pre-coding. Firstly, for Eq. (24), we use common approximation as E log 2 1 + X Y ≈ log 2 1 + E[X] E[Y ] , which is tight when the number of antennas tends to infinity, and X, Y are sums of non-negative random variables and converge to their mean due to the law of large numbers. Thus, Calculate its numerator as R DL,k ≈ log 2   1+ E ξ k √ pDLμ k,k s k 2 E φ DL,E |ξ k √ p DLμk,k s k | 2 = ξ 2 k p DL E |μ k,k | 2 = ξ 2 k p DL (E pre k ) 2 + p DP (χ pre k ) 2 p DP χ pre k + σ 2 DP .(65) Then, calculate the first term of denominator E [φ DL,k ] = ξ 2 k i =k p DL E |μ k,i | 2 = ξ 2 k i =k p DL ξ k p DP (χ pre k ) 2 p DP χ pre k + σ 2 DP .(66) The second term of denominator is obtaine as E [ψ DL,k ] = E ξ k KDL i=1 √ p DLμk,i s i 2 = ξ 2 k KDL i=1 p DL E |µ k,i | 2 − E |μ k,i | 2 .(67) When i = k, ∆E[µ] = (1 − ξ k ) p DP (χ pre k ) 2 + σ 2 DP χ pre k p DP χ pre k + σ 2 DP .(69) and the last term of denominator as Next, for the first term of denominator, the receiver vectors with MRC are placed in N UL M -dimensional space. Any vector independent of the channel vector is placed in a onedimensional space.ĝ UL,i is independent ofĝ UL,k . Hence, C n DL,k = ξ k (1 − ξ k ) E diag r DL,k r H DL,k ,(72)E φ MRC UL,k = 1 N UL NUL n=1 p UL α 2 n β UL,n,i .(75) And the other terms of denominator can be expressed as Others are similar to MRC and we omit the proof of ZFR due to the limitation of space. R UL,k ≈ log 2   1 + E pUL v H k / v k Aĝ UL,k 2 E φ UL,k + ψ UL,k + ρ UL,k + v H k / v k A 2 σ 2 UL + v H k / v k Rn q,UL v H k / v k   .(74) Fig. 1 . 1System Model of NAFD Distributed M-MIMO Systems. 9, M = 2000, and the solution process needs T H = 1, 000 iterations for complete convergence. Both evaluate net and target net networks in DQN consist of fully connected DNN with 2 hidden layers with ReLU activation functions, where their dimensions are 9 and 18. The dimensions of the input and output layers of both evaluate net network and target net network are equal to 9 and 18. In NSGA-II, the population size is N pop = 200, and the max number of generations is N gen = 300. 9, M = 2000, and the solution process needs T H = 1, 000 iterations for complete convergence. Both evaluate net and target net networks in DQN consist of fully connected DNN with 2 hidden layers with ReLU activation functions, where their dimensions are 9 and 18. The dimensions of the input and output layers of both evaluate net network and target net network are equal to 9 and 18. In NSGA-II, the population size is N pop = 200, and the max number of generations is N gen = 300. Fig. 2 . 2Comparisons of DL SE with Different Beamforming Training Mechanisms and Different CSI Acquisition Methods. Fig. 3 . 3Comparisons of UL SE with Different Receivers. Fig. 2 . 2Comparisons of DL SE with Different Beamforming Training Mechanisms and Different CSI Acquisition Methods. Fig. 2 . 2Comparisons of DL SE with Different Beamforming Training Mechanisms and Different CSI Acquisition Methods. Fig. 3 . 3Comparisons of UL SE with Different Receivers. Fig. 3 . 3Comparisons of UL SE with Different Receivers. Fig. 4 . 4Tradeoff Between the Total SE and Total EE with Different Quantization Schemes and Antenna Numbers under MRT and MRC. Fig. 4 . 4Tradeoff Between the Total SE and Total EE with Different Quantization Schemes and Antenna Numbers under MRT and MRC. Fig. 4 . 4Tradeoff Between the Total SE and Total EE with Different Quantization Schemes and Antenna Numbers under MRT and MRC. Fig. 4 . 4Tradeoff Between the Total SE and Total EE with Different Quantization Schemes and Antenna Numbers under MRT and MRC. II. SYSTEM ANALYSIS OF NAFD DISTRIBUTED M-MIMOFig. 1illustrates an NAFD M-MIMO system under lowresolution ADCs. Distributed RAUs jointly serve both UL and DL in the same time-frequency resources. UL/DL users maintain transmission simultaneously[4]. The system with low-resolution ADCs contains randomly distributed N RAUs with M half-duplex antennas and K users with one halfduplex antenna, including K UL UL users and K DL DL users. At each moment, there exist N UL RAUs for UL reception from the UL users to UL RAUs and N DL RAUs for DL transmission from DL RAUs to DL users, where N UL + N DL = N . Meanwhile, two types of interference channels, inter-user and inter-RAU between UL and DL, are considered. Compared to Uplink Signal Interference from UL-user to DL-userUL-RAU UL-RAU DL-RAU DL-RAU DL-RAU DL-user DL-user UL-user DL-user UL-user Data Data CPU RF Low-Resolution ADCs DSP Interference from DL-RAU to UL-RAU Downlink Signal 1 ) 1where p DL/UL represents the DL or UL transmit power, respectively. s i is the data signal with E[s i s H i ] = 1 and x j is the data signal transmitted by the j-th UL user with E[x k x H k ] = 1. Auxiliary channel µ k,i = g H DL,k w i is obtained by the beamforming training mechanism. The pre-coding vectors w i ∈ C NDLM ×1 for MRT and ZFT, areĝ DL,i ĝ DL,i andĝDL,i(ĝ H DL,iĝDL,i ) −1 ĝ DL,i(ĝ H DL,iĝ DL,i) −1 , respectively. g DL,i = [g H DL,1,i , . . . , g H DL,NDL,i TABLE Input:1: Initialize the state s0 and the replay memory M 2: Initialize action-value function Q with random weights Output:3: The state smax with the optimal future reward 4: for t = 1 : TH doUse -greedy behavior policy to select action:Calculate the target yi and perform SGD of Eq. (48) 11: end for Algorithm 2 Bit Allocation Algorithm Based on NSGA-II Input: Randomly generate the initial value Output: Pareto-optimal solutions; SE and EE1: repeat5: at = max a Qt(st, a; θ), p gdy = 1 − a rnd , p gdy = . 6: Execute action at and obtain the reward rt 7: Set the next stage st+1 = st {at} 8: Store {st, at, rt, st+1} to M 9: Sample random minibatch of transitions from M 10: 2: TABLE II PARAMETERS IIOF POWER CONSUMPTION[10]. Considering the parameters mentioned above, assume all DL users are equipped with the same resolution ADC, and MRT pre-coding and ZFT pre-coding are considered. Based on Eq. (25) in Theorem 1 and Eq. (27) in Theorem 2,Parameter Value UL transmission power p UL 500 mW DL transmission power p DL 500 mW UL pilot power p UP 500 mW DL pilot power p DP 1.00 W Power consumption at RAU P RAU 100 mW Power consumption at user P UE 100 mW Local oscillator power consumption P SYN 1.00 W Calculation efficiency L RAU 12.8 Gflops/W Amplifier efficiency ξ 0.4 Fixed power consumption P 0 0.825 W Dynamic power consumption P BT 0.25 W/(Gbit/s) ADC power coefficient a 0 , a 1 10 −4 , 0.02 B. Results and Discussion 1) Verification of Closed-Form Expressions: TABLE II PARAMETERS IIOF POWER CONSUMPTION [10]. Verification of Closed-Form Expressions: Considering the parameters mentioned above, assume all DL users are equipped with the same resolution ADC, and MRT pre-coding and ZFT pre-coding are considered. Based on Eq. (25) in Theorem 1 and Eq. (27) in Theorem 2,Parameter Value UL transmission power p UL 500 mW DL transmission power p DL 500 mW UL pilot power p UP 500 mW DL pilot power p DP 1.00 W Power consumption at RAU P RAU 100 mW Power consumption at user P UE 100 mW Local oscillator power consumption P SYN 1.00 W Calculation efficiency L RAU 12.8 Gflops/W Amplifier efficiency ξ 0.4 Fixed power consumption P 0 0.825 W Dynamic power consumption P BT 0.25 W/(Gbit/s) ADC power coefficient a 0 , a 1 10 −4 , 0.02 B. Results and Discussion 1) TABLE II PARAMETERS IIOF POWER CONSUMPTION [10]. Verification of Closed-Form Expressions: Considering the parameters mentioned above, assume all DL users are equipped with the same resolution ADC, and MRT pre-coding and ZFT pre-coding are considered. Based on Eq. (25) in Theorem 1 and Eq. (27) in Theorem 2,Parameter Value UL transmission power p UL 500 mW DL transmission power p DL 500 mW UL pilot power p UP 500 mW DL pilot power p DP 1.00 W Power consumption at RAU P RAU 100 mW Power consumption at user P UE 100 mW Local oscillator power consumption P SYN 1.00 W Calculation efficiency L RAU 12.8 Gflops/W Amplifier efficiency ξ 0.4 Fixed power consumption P 0 0.825 W Dynamic power consumption P BT 0.25 W/(Gbit/s) ADC power coefficient a 0 , a 1 10 −4 , 0.02 B. Results and Discussion 1) illustrates the total EE v.s. SE with different numbers of quantization bits and antennas per RAU. Different lines represent different numbers of quantization bits B ∈ {b|4 b 9, b ∈ N}, and different points represent different numbers of antennas M ∈ {2m|3 m 16, m ∈ N}. For different lines, as the ADC accuracy increases, the SE gradually increases, and the EE increases and then decreases. For different points on the same line, as the number of antennas on each RAUs increases, the SE increases, and the EE increases and then decreases. It reveals that both EE and SE increase with the growth of the antenna and bit widths when the improvement of SE first dominates, followed by EE's prevailing. X. SONG et al.: PERFORMANCE ANALYSIS AND OPTIMIZATION OF NETWORK-ASSISTED FULL-DUPLEX SYSTEMS UNDER LOW-RESOLUTION ADCS 9 cancellation brings a tremendous performance improvement, proving the DL beamforming training is helpful for UL RAUs to eliminate interference between RAUs. The gap between MRC and ZFR receivers with CLI cancellation blows up since the strength of elimination determines the SE attenuation.2) Relationship Between the Total SE and EE: Next, we explore the relationship between the total SE and EE in Eq. (42). For the MRT pre-coding and MRC receiver,Fig. 4illustrates the total EE v.s. SE with different numbers of quantization bits and antennas per RAU. Different lines represent different numbers of quantization bits B ∈ {b|4 ⩽ b ⩽ 9, b ∈ N}, and different points represent different numbers of antennas M ∈ {2m|3 ⩽ m ⩽ 16, m ∈ N}. For different lines, as the ADC accuracy increases, the SE gradually increases, and the EE increases and then decreases. For different points on the same line, as the number of antennas on each RAUs increases, the SE increases, and the EE increases and then decreases. It reveals that both EE and SE increase with the growth of the antenna and bit widths when the improvement of SE first dominates, followed by EE's prevailing.12 14 16 18 20 22 24 1 1.2 1.4 1.6 1.8 Total SE [bit/s/Hz] Total EE [×10 7 Fig. 5. SE and EE Tradeoff for Different Quantization Schemes..05 15.1 15.15 15.2 15.25 1 1.2 1.4 1.6 (15.11, 1.50) (15.19 ,1.48) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (a) MRT Pre-coding and MRC Receiver 22.6 22.8 23 23.2 23.4 1 1.5 2 2.5 ( 22.88 , 2.21) (22.87 ,2.22) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only 22.6 22.8 23 23.2 23.4 1 1.5 2 2.5 ( 22.88 , 2.21) (22.87 ,2.22) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (b) ZFT Pre-coding and ZFR Receiver Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (b) ZFT Pre-coding and ZFR Receiver Fig. 5. SE and EE Tradeoff for Different Quantization Schemes..05 15.1 15.15 15.2 15.25 1 1.2 1.4 1.6 (15.11, 1.50) (15.19 ,1.48) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (a) MRT Pre-coding and MRC Receiver 22.6 22.8 23 23.2 23.4 1 1.5 2 2.5 ( 22.88 , 2.21) (22.87 ,2.22) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only 22.6 22.8 23 23.2 2 1 1.5 2 2.5 ( 22.88 , 2.21) (22.87 ,2.22) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] SE-only's AsymptoticLine Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (a) MRT Pre-coding and MRC Receiver ( 22.88 , 2.21) (22.87 ,2.22) EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only EE-only SE-only's Asymptotic Line Total SE [bit/s/Hz] Total EE [×10 7 bit/J] Pareto-Optimal Border by NSGA-II Feasible solutions by NSGA-II Feasible solution by DQN Feasible solution by EE-only (b) ZFT Pre-coding and ZFR Receiver Fig. 5. SE and EE Tradeoff for Different Quantization Schemes.VII. CONCLUSIONSThis paper analyzed the NAFD distributed M-MIMO systems with low-resolution ADCs. Two-stage channel estimation, including pilot estimation and beamforming training, was adopted to complete the CLI elimination and obtain the effective CSI for signal detection and interference cancellation. With the estimated CSI, the closed-form expressions for the achievable rates of the UL and DL, as well as EE, were derived. Further, a MOOP maximizing SE and EE under per-user QoS and total bit constraints was formulated to design an efficient ADC bit allocation scheme. The simulation results verified the theoretical analysis. Low-resolution ADCs significantly improved EE in NAFD distributed M-MIMO. Interference cancellation via beamforming training enhanced the UL SE with different receivers under low-resolution ADCs, (b) ZFT Pre-coding and ZFR ReceiverFig. 5. SE and EE Tradeoff for Different Quantization Schemes..05 15.1 15.15 15.2 15.25 1 1.2 1.4 1.6 (15.11, 1.50) (15.19 ,1.48) EE-only 22.6 22.8 23 23.2 23.4 1 1.5 2 2.5 22.6 22.8 23 23.2 1 1.5 2 2.5 ( 22.88 , 2.21) (22.87 ,2.22) k , θ DL,k ) , i = k ĝ DL,k w i 2 ∼ Γ 1 N DL M k DL,k , θ DL,k , i = k g DL,k w i 2 ∼ Γ 1 N DL Mk DL,k ,θ DL,k , i = k. (56) Therefore, E µ 2 k,i =        k DL,k θ DL,k +k DL,kθDL,k N DL M , i = k k DL,k θ DL,k +k DL,kθDL,k N DL M , i = k . (57) Substituting Eqs. (57) and (53) into Eq. (54), we have cov(µ k,i ,r p,i e,k ) = ξ k √ p DPχ MR k , i = k ξ k √ p DP χ MR k , i = k . (58) Meanwhile, cov(r p,i e,k ,r p,i e,k ) = E[ξ 2 k p DP µ 2 k,i +ξ 2 k n 2 DP,i +ñ 2 DP,i ]−E [ξ k √ p DP µ k,i ] 2 = ξ k p DPχ MR k k +ψ DL,k +ρ DL,k + Cn DL,k  . (64) When i = k, denote ∆E[µ] = E |µ k,k | 2 − E |μ k,k | ∆E[µ] = (1 − ξ k ) p DPχ2 pre k (E pre k ) 2 +χ pre k σ 2 DP p DPχ pre k + (1−ξ k )p DP (E pre k ) 2 + σ 2 DP . Substituting Eqs. (68) and (69) into Eq. (67), we obtainE [ψ DL,k ] = ξ 2 k p DL (1 − ξ k ) p DPχWe get the third term of denominator asE[|ξ k KUL j=1√ p UL u I,k,j x j | 2 ] = ξ 2 UL λ I,k,j ,pre k (E pre k ) 2 +χ pre k σ 2 DP p DPχ pre k + (1 − ξ k )p DP (E pre k ) 2 + σ 2 DP + ξ 2 k i =k p DL (1 − ξ k ) p DP (χ pre k ) 2 + σ 2 DP χ pre k p DP χ pre k + σ 2 DP . (70) k KUL j=1 p where E diag r DL,k r H KUL j=1 p UL E |u I,k,j | 2 + σ 2 DL . Substituting Eqs. (65-72) into Eq. (24), we can obtain the result in Eqs. (25) and (26).APPENDIX V PROOF OF THEOREM 3Proof: Firstly, dividing the numerator and denominator of the SINR in Eq. (33) by v k 2 , the rate expression is approximated as Eq. (74) at the top of next page. For MRC receiver, substitute v k =ĝ UL,k into Eq. (74). We have the numerator as M Ξ n,k = M ∼ Γ Mk UL,k ,θ UL,k , (73a)DL,k = KDL i=1 p DL E |µ k,i | 2 + NUL n=1 α 2 n β UL,n,k , where Aĝ UL,k 2 = N UL n=1 αnĝ UL,n,k 2 k UL,k = Ξ 2 n,k N UL n=1 α 4 n β 2 UL,n,k ,θ UL,k = N UL n=1 α 4 n β 2 UL,n,k Ξ n,k . 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Energy-efficient downlink transmission for multicell massive DAS with pilot contamination. J Zuo, J Zhang, C Yuen, W Jiang, W Luo, IEEE Trans. Veh. Technol. 662J. Zuo, J. Zhang, C. Yuen, W. Jiang, and W. Luo, "Energy-efficient downlink transmission for multicell massive DAS with pilot contamina- tion," IEEE Trans. Veh. Technol., vol. 66, no. 2, pp. 1209-1221, 2016. Playing Atari with deep reinforcement learning. V Mnih, K Kavukcuoglu, D Silver, A Graves, I Antonoglou, D Wierstra, M Riedmiller, arXiv:1312.5602V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wier- stra, and M. Riedmiller, "Playing Atari with deep reinforcement learn- ing," arXiv:1312.5602, 2013. Reinforcement learning: An introduction. R S Sutton, A G Barto, MIT pressR. S. Sutton and A. G. Barto, Reinforcement learning: An introduction. MIT press, 2018. A fast and elitist multiobjective genetic algorithm: NSGA-II. K Deb, A Pratap, S Agarwal, T Meyarivan, IEEE Trans. Evol. Comput. 62K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182-197, 2002. Downlink spectral efficiency of distributed massive MIMO systems with linear beamforming under pilot contamination. J Li, D Wang, P Zhu, J Wang, X You, IEEE Trans. Veh. Technol. 672J. Li, D. Wang, P. Zhu, J. Wang, and X. You, "Downlink spectral effi- ciency of distributed massive MIMO systems with linear beamforming under pilot contamination," IEEE Trans. Veh. Technol., vol. 67, no. 2, pp. 1130-1145, 2017. She received the B.S. degree (summa cum laude) in information engineering from Southeast University. Her research interests include distributed massive MIMO, low-resolution ADCs, and cooperative communications. Nanjing, ChinaNational Mobile Communications Research Laboratory, Southeast UniversityShe is currently pursuing the master degree in communication and information system at theXiangning Song was born in Shandong province, China, in 1999. She received the B.S. degree (summa cum laude) in information engineering from Southeast University, Nanjing, China, in 2021. She is currently pursuing the master degree in communication and information system at the National Mobile Communications Research Laboratory, Southeast University. Her research interests include distributed massive MIMO, low-resolution ADCs, and cooperative communications.
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[ "Learning Object-Centric Neural Scattering Functions for Free-Viewpoint Relighting and Scene Composition", "Learning Object-Centric Neural Scattering Functions for Free-Viewpoint Relighting and Scene Composition" ]
[ "Hong-Xing Yu \nStanford University\n\n", "Michelle Guo \nStanford University\n\n", "Alireza Fathi \nStanford University\n\n", "Yen-Yu Chang \nStanford University\n\n", "Eric Ryan Chan \nStanford University\n\n", "Ruohan Gao \nStanford University\n\n", "Thomas Funkhouser \nStanford University\n\n", "Jiajun Wu \nStanford University\n\n" ]
[ "Stanford University\n", "Stanford University\n", "Stanford University\n", "Stanford University\n", "Stanford University\n", "Stanford University\n", "Stanford University\n", "Stanford University\n" ]
[]
Photorealistic object appearance modeling from 2D images is a constant topic in vision and graphics. While neural implicit methods (such as Neural Radiance Fields) have shown high-fidelity view synthesis results, they cannot relight the captured objects. More recent neural inverse rendering approaches have enabled object relighting, but they represent surface properties as simple BRDFs, and therefore cannot handle translucent objects. We propose Object-Centric Neural Scattering Functions (OSFs) for learning to reconstruct object appearance from only images. OSFs not only support free-viewpoint object relighting, but also can model both opaque and translucent objects. While accurately modeling subsurface light transport for translucent objects can be highly complex and even intractable for neural methods, OSFs learn to approximate the radiance transfer from a distant light to an outgoing direction at any spatial location. This approximation avoids explicitly modeling complex subsurface scattering, making learning a neural implicit model tractable. Experiments on real and synthetic data show that OSFs accurately reconstruct appearances for both opaque and translucent objects, allowing faithful free-viewpoint relighting as well as scene composition. Project website with video results: https://kovenyu.com/OSF.
10.48550/arxiv.2303.06138
[ "https://export.arxiv.org/pdf/2303.06138v3.pdf" ]
257,482,263
2303.06138
0c597d8f2e782e1d8036197d440d967b89fb015e
Learning Object-Centric Neural Scattering Functions for Free-Viewpoint Relighting and Scene Composition Hong-Xing Yu Stanford University Michelle Guo Stanford University Alireza Fathi Stanford University Yen-Yu Chang Stanford University Eric Ryan Chan Stanford University Ruohan Gao Stanford University Thomas Funkhouser Stanford University Jiajun Wu Stanford University Learning Object-Centric Neural Scattering Functions for Free-Viewpoint Relighting and Scene Composition Published in Transactions on Machine Learning Research (05/2023) 2 Google Research * Contributed equally. Reviewed on OpenReview: https: // openreview. net/ forum? id= NrfSRtTpN5 Photorealistic object appearance modeling from 2D images is a constant topic in vision and graphics. While neural implicit methods (such as Neural Radiance Fields) have shown high-fidelity view synthesis results, they cannot relight the captured objects. More recent neural inverse rendering approaches have enabled object relighting, but they represent surface properties as simple BRDFs, and therefore cannot handle translucent objects. We propose Object-Centric Neural Scattering Functions (OSFs) for learning to reconstruct object appearance from only images. OSFs not only support free-viewpoint object relighting, but also can model both opaque and translucent objects. While accurately modeling subsurface light transport for translucent objects can be highly complex and even intractable for neural methods, OSFs learn to approximate the radiance transfer from a distant light to an outgoing direction at any spatial location. This approximation avoids explicitly modeling complex subsurface scattering, making learning a neural implicit model tractable. Experiments on real and synthetic data show that OSFs accurately reconstruct appearances for both opaque and translucent objects, allowing faithful free-viewpoint relighting as well as scene composition. Project website with video results: https://kovenyu.com/OSF. Introduction Modeling the geometry and appearance of 3D objects from captured 2D images is central to many applications in computer vision, graphics, and robotics, such as shape reconstruction (Mescheder et al., 2019;Park et al., 2019;Chang et al., 2015), view synthesis (Hedman et al., 2018;Mildenhall et al., 2020;Sitzmann et al., 2019a), relighting (Zhang et al., 2021c;Bi et al., 2020b;a), and object manipulation (Simeonov et al., 2021). Traditional inverse rendering approaches (Zhou et al., 2013;Nam et al., 2018) focus on directly estimating objects' shape and material properties; they then compose them using the estimated surface properties. These explicitly reconstructed object representations often lead to visible artifacts or limited fidelity when rendering images using the recovered object assets. Recently, neural implicit methods have attracted much attention in image-based appearance modeling. They implicitly represent objects and scenes using deep neural networks (Lombardi et al., 2019;Sitzmann et al., 2019a;. Neural Radiance Fields (NeRFs) (Mildenhall et al., 2020) are one of the most representative methods due to their high-fidelity view synthesis results. NeRFs implicitly model a scene's volumetric density and radiance via coordinate-based deep neural networks, which can be learned from images via direct volume rendering techniques (Max, 1995). However, NeRFs only reconstruct the outgoing radiance fields under a fixed lighting condition. Thus, they cannot relight the captured objects or compose multiple objects into new scenes. To address this problem, a few neural inverse rendering methods (Zhang et al., 2021b;a;Boss et al., 2021a;Zhang et al., 2021c) have been proposed to jointly estimate lighting, material properties, and geometry from Figure 1: We propose a new neural object representation, Object-Centric Neural Scattering Functions (OSFs), to reconstruct object appearance from only images. OSFs can handle objects with complex materials or shapes (e.g., the translucent bunny), and support both free-viewpoint relighting and scene composition. images. The disentanglement of illumination enables relightable object appearance modeling. However, these methods all assume simple Bidirectional Reflectance Distribution Functions (BRDFs) when modeling surface properties, without taking subsurface light transport into consideration * . Therefore, they can only model the appearance of opaque objects and cannot handle translucent objects with complex material properties. We observe that subsurface scattering effects are the key to relighting translucent objects. However, accurately modeling subsurface scattering can be complex and even intractable for neural implicit representations, because each sampling operation for numerical integration requires a complete forward pass of the network. This leads to a prohibitive computational cost in both time and memory. To address this challenge, we propose Object-Centric Neural Scattering Functions (OSFs) for image-based relightable neural appearance reconstruction. An OSF learns to approximate the cumulative radiance transfer function, which models radiance transfer from a distant light to any outgoing direction at any spatial location for an object. This enables the use of a volumetric rendering formulation to learn from images, while being able to approximate both opaque and translucent object appearances. Thus, OSFs allow free-viewpoint relighting from only 2D images (see Figure 1). Modeling object-level light transport allows neural scene composition with trained OSF models, as light transport effects such as object shadows and indirect lighting can be naturally incorporated. To demonstrate this, we present a simple and computationally tractable method for scene composition. To further accelerate neural scene rendering, we introduce two improvements on OSFs scene rendering that allow magnitudes of acceleration without compromising rendering quality. Experiments on both real and synthetic objects demonstrate the effectiveness of our OSF formulation. We show that OSFs enable free-viewpoint relighting of both translucent and opaque objects by learning from only 2D images. We also show that in scene composition using the reconstructed objects, our approach significantly outperforms baseline methods. In summary, our contributions are threefold. First, we introduce the concept of the cumulative radiance transfer function that models object-centric light transport. Second, we propose the Object-Centric Neural Scattering Functions (OSFs) that learn to approximate the cumulative radiance transfer function from only 2D images. OSFs allow free-viewpoint relighting and scene composition of both opaque and translucent objects. Third, we show experimental results on both real and synthetic objects, demonstrating the effectiveness of OSFs in free-viewpoint relighting and scene composition. Related Work Neural appearance reconstruction. Traditional methods use Structure-From-Motion (Hartley & Zisserman, 2003) and bundle adjustment (Triggs et al., 1999) to reconstruct colored point clouds. More recently, a number of learning-based novel view synthesis methods have been proposed, but they require 3D geometry as inputs (Hedman et al., 2018;Thies et al., 2019;Meshry et al., 2019;Aliev et al., 2020;Martin-Brualla et al., 2018). Recently, neural volume rendering approaches have been used to reconstruct scenes from only images (Lombardi et al., 2019;Sitzmann et al., 2019a). However, the rendering resolution of these methods is limited by the time and computational complexity of the discretely sampled volumes. To address this issue, Neural Radiance Fields (NeRFs) (Mildenhall et al., 2020) directly optimizes a continuous radiance field representation using a multi-layer perceptron. This allows for synthesizing novel views of realistic images at an unprecedented level of fidelity. Various extensions of NeRF have also been proposed to improve efficiency (Liu et al., 2020;Yu et al., 2021;Garbin et al., 2021;Reiser et al., 2021), generalization (Niemeyer & Geiger, 2021;Yu et al., 2022;Yang et al., 2021), quality (Oechsle et al., 2021;Verbin et al., 2021;, compositionality (Ost et al., 2021) etc. While these neural methods produce high-quality novel views of a scene, they do not support relighting. In contrast, our approach aims at relightable appearance reconstruction. Learning-based relighting. Learning-based methods for relighting without explicit geometry have been proposed (Sun et al., 2019;Xu et al., 2018;Zhou et al., 2019), but they are not 3D-aware. Following the recent surge of neural 3D scene representations and neural differentiable rendering (Tewari et al., 2020;Mildenhall et al., 2020), neural relightable representations (Bi et al., 2020b;a;Boss et al., 2021a;Zhang et al., 2021c;Baatz et al., 2021) and neural inverse rendering methods (Zhang et al., 2021b;a;Boss et al., 2021b) have been developed to address this limitation. However, most of them assume simple BRDFs which do not consider subsurface scattering in their formulations, and thus they cannot model translucent object appearances for relighting. In contrast, our method allows the reconstruction of both opaque and translucent object appearances. Some existing methods also approximate subsurface light transport for participating media (Kallweit et al., 2017;Zheng et al., 2021). While Kallweit et al. (2017) demonstrate high-quality scattering, they require 3D groundtruth data whereas our method learns from only images. Zheng et al. (2021) focus on relevant tasks as we do, using specific designs such as spherical harmonics for scattering and visibility modeling. In contrast, our model is simple and allows adopting recent advances in accelerating neural rendering. We demonstrate interactive frame rate in rendering by our KiloOSF. Precomputed light transport. Our OSFs learn to approximate a cumulative radiance transfer function, which is based on the classical idea from the precomputed light transport methods including the precomputed radiance transfer (Sloan et al., 2002) (also see (Lehtinen, 2007) and (Ramamoorthi, 2009) for overviews). These methods map incoming basis lighting to outgoing radiance, while we additionally learn to encapsulate the object-specific light transport at the visible surface. Related ideas include the classical reflectance field (Debevec et al., 2000;Garg et al., 2006). Our method is also related to classical techniques for modeling aggregate scattering behavior (Moon et al., 2007;Lee & O'Sullivan, 2007;Meng et al., 2015;Blumer et al., 2016). These techniques consider modeling the full asset-level internal scattering, a more general form than our cumulative radiance transfer. Our cumulative radiance transfer function borrows the general idea from these classical methods and adapts to neural implicit volumetric rendering framework for relightable and compositional appearance modeling from images. This adaptation allows using the power of deep networks to approximate complex light transport. Recently a few neural methods have been proposed to model light transport of complex materials (Kuznetsov, 2021;Baatz et al., 2021), e.g., Baatz et al. (2021) represent surface texture as neural reflectance fields which is technically relevant to ours. However, they need meshes as explicit object geometry, while ours learns object representations from only images. Approach We aim for relightable neural appearance reconstruction of both opaque and translucent objects from 2D images. To achieve this, we propose Object-centric neural Scattering Functions (OSFs) that learn to approximate subsurface scattering effects in a computationally tractable way. In the following, we first introduce our OSF formulation and discuss its relations to phase functions and BSSRDFs (Sec. 3.1). Then we show how we can model translucent objects (Sec. 3.2) and opaque objects (Sec. 3.3) with OSFs. Finally, we discuss how to learn OSFs from 2D images (Sec. 3.4) and compose multiple learned OSFs (Sec. 3.6). Distant light = ( , ℎ , out ) ℎ ℎ ℎ out out = ∫ , in , out " ! , in in , in in = & & , , light , out ℎ |cos | light in ! # ℎ BSSRDFs Phase function Cumulative radiance transfer (Ours) out out Figure 2: Relations to classical graphics concepts for modeling subsurface scattering effects. The process of light transport from an unobstructed distant light to a point x inside a translucent object is complex: 1) the light (radiance) hit a point on the visible surface area; 2) the surface transmits some amount of the light into the object interior; 3) the transmitted light scatters within the object; 4) some light finally arrives at x from incoming directions ω in ; 5) the arrived light scatters to directions ω out ; and 6) this process happens at all points on the visible surface, and the final outcome is the accumulated result. Phase functions only model 4) and 5), so one has to account for the other steps by multiple integrations. BSSRDFs encapsulate steps 2) -5), so they require two integrations for step 1) and step 6). In contrast, our OSF formulation encapsulates the entire process that takes all steps into account. This abstraction of modeling the whole process as cumulative radiance transfer eliminates the need of any integration, making it computationally tractable to use in neural volume rendering. OSF Formulation Our OSFs are based on Neural Radiance Fields (NeRFs) (Mildenhall et al., 2020), which learn view-dependent emitted radiance at any point in space. NeRFs can be formulated by NN Θ : (x, ω out ) → (L out , σ) , where NN denotes a learnable deep neural network with parameters Θ, x denotes a 3D spatial location, ω out denotes an outgoing radiance direction, L out denotes the outgoing radiance, and σ denotes the volumetric density. Inspired from the principles of volume rendering (Kajiya & Von Herzen, 1984), the computational model for the radiance L(o, ω out ) arriving at the camera location o from direction ω out (a.k.a. the radiance of a camera ray r(t) = o − tω out ) with near and far bounds t n and t f is given by: L(o, ω out ) = t f tn T (t)σ(r(t))L out (r(t), ω out )dt,(1) where T (t) = exp(− t tn σ(r(s))ds) denotes the transmittance along the ray from t n to t. While a NeRF can render high-fidelity novel views of a captured object, it does not support relighting because it is based on the absorption-plus-emission model (Max, 1995), which itself is light source-agnostic. Namely, NeRFs model L out as the emitted radiance after all light transport has occurred. To address this limitation, we propose OSFs that model L out as the scattered radiance due to external light sources instead. More specifically, OSFs learn to approximate the cumulative radiance transfer from an unobstructed distant light, in addition to the volume density: NN Θ : (x, ω light , ω out ) → (ρ, σ),(2) where ρ(x, ω light , ω out ) denotes the cumulative radiance transfer function, which is defined as the ratio of two quantities. The numerator is the outgoing radiance L out (x, ω out ) in the direction of ω out at location x that is caused by an unobstructed distant light L light (ω light ) from direction ω light . The denominator is L light (ω light ). It is cumulative about all paths that L light (ω light ) scatters to x. See Figure 2 for an illustration. Given the cumulative radiance transfer function ρ, the scattered outgoing radiance L out is given by L out (x, ω out ) = S 2 ρ(x, ω light , ω out )L light (ω light )dω light .(3) We assume the object itself does not emit light. We argue that this distant light assumption in ρ is acceptable and can allow relighting with environment maps that are also distant light. Phase Functions vs. BSSRDFs. vs. Cumulative Radiance Transfer. The cumulative radiance transfer function ρ approximates a complex radiance transfer process from an unobstructed distant light to an outgoing direction at a location, including those inside the object. It is conceptually akin to a phase function that describes the scattered radiance distribution at a specific location in a scattering process. It is also related to the Bidirectional Scattering Surface Reflectance Distribution Functions (BSSRDFs) (Jensen et al., 2001). See Figure 2 for a detailed comparison of these concepts. Below we list the major distinctions against phase functions: • Generally, a phase function is local, i.e., it depends only on the material/media (and in some cases also geometry) at the location. In contrast, the cumulative radiance transfer function ρ is objectspecific and spatially varying even if the object is made from a uniform homogeneous material. This is because ρ potentially needs to account for a complex process, including the light transport for all surface points that are visible to the distant light, the subsurface scattering effects, and the radiance transfer at a specific location. Thus, this process depends on both the material and the object geometry at x, as well as the light direction. • Phase functions for some naturally occurring isotropic media only need 1 Degree of Freedom (DoF) for angular distribution, i.e., the cosine between incoming and outgoing radiance directions (see Chapter 11 in Pharr et al. (2016)). However, ρ generally needs 4 DoF for angular distribution: 2 DoF of light direction because the light direction can interact with object geometry to affect the incoming radiance distribution at any given x, and 2 DoF of outgoing direction as we want to model view-dependent effects as in NeRFs (Mildenhall et al., 2020). Compared to BSSRDFs which are surface models, ρ is used in a volumetric rendering process. OSFs do not explicitly model surface properties. Modeling Translucent Objects with OSFs Our OSF formulation can model the appearance of translucent objects in a computationally tractable way for volumetric neural implicit methods, while directly modeling the radiance transfer in a volumetric scattering process is intractable. In direct modeling, L out (x, ω out ) = S 2 p(x, ω, ω out )L in (x, ω)dω, where p denotes a phase function, L in denotes the incoming radiance at location x from the direction ω. The recursive dependency of L in over the unit sphere in the media makes the equation very difficult to solve (Novák et al., 2018;Max & Chen, 2005). From the computational perspective, to numerically compute this integral to train the neural network, one needs to account for the whole sphere (due to subsurface scattering) through dense sampling. However, for neural implicit methods, each sample requires a forward pass of the deep network, leading to a high computation cost. Moreover, training deep networks often requires saving intermediate variables for auto-differentiation (Paszke et al., 2017), which demands a massive amount of GPU memory for dense sampling. As a result, it can become computationally intractable to train the neural network in this way. In contrast, during training, we use lighting setups that match well the assumption of a single directional light source; the spherical integration in Eq. (3) thus reduces to a single evaluation. The key feature of OSFs is that they approximate the complex cumulative radiance transfer process using the expressive power of deep neural networks. Modeling Opaque Objects with OSFs Apart from modeling translucent objects, OSF formulation is also able to model opaque objects. Theoretically, the formulation may subsume surface light transport for opaque convex objects. To see this, consider a single distant light source from ω light , the light transport at a visible surface point can be described by L out (x surf , ω out ) = f (x surf , ω light , ω out )L light |n(x surf ) · ω light | ,(4) where f denotes a BRDF, x surf denotes the first surface point hit by the camera ray r, and n(x) denotes the unit normal vector at x. We have used the single distant light assumption to solve the integration. Now we consider an ideal case, where an OSF learns that σ(x) = δ(x − x surf ) for all x surf . This solves the integration in Eq. (1) and gives L(o, ω out ) = L out (x surf , ω out ) = ρ(x surf , ω light , ω out )L light .(5) We can see that if the learned OSF satisfies ρ( x surf , ω light , ω out ) = f (x surf , ω light , ω out ) |n(x surf ) · ω light | for all surface points, then it subsumes the surface light transport with the BRDF. Although a neural network cannot perfectly represent a delta function for σ(x), in our experiments, we observe that empirically OSFs learn to approximate the surface light transport for opaque objects of both convex and concave shapes. Learning OSFs from Images We aim to learn OSFs from only 2D images, allowing easy capture of real objects. As discussed in Section 3.1, we prefer a tractable training setup. Thus, we propose capturing object images using only a single distant light source, which leads to a light source radiance function L light (ω) = L 0 δ(ω − ω 0 ), where L 0 and ω 0 are obtained from the capture setup. We can define the sampling Probabilistic Density Function (PDF) for lighting as PDF light (ω) = δ(ω − ω 0 ) and this gives an analytic solution of the integration in Eq. (3): L out (x, ω out ) = E ω∼PDF light (ω) ρ(x, ω, ω out ) L light (ω) PDF light (ω) (6) = ρ(x, ω 0 , ω out )L 0 .(7) We capture images of the object-to-reconstruct under different ω 0 in a dark room. We describe our accessible real data capture setup using two iPhones in Section 4.1. Given Eq. (7), the camera ray radiance L(o, ω out ) defined in Eq. (1) is trivially differentiable with respect to OSFs. We follow the learning strategy from NeRFs (Mildenhall et al., 2020), which models the pixel color of a ray in an analog to radiance instead of the raw radiometric quantity to simplify learning. This may be interpreted as allowing the neural network to implicitly deal with the camera response that maps camera ray radiance to pixel color (also see a recent discussion in ). We also follow NeRFs to use quadrature to compute Eq. (1), and use positional encoding and hierarchical volume sampling to facilitate training (i.e., we use a coarse network for coarse sampling a ray to give a rough estimation of transmittance distribution, and then use a fine network to do additional informed sampling to further reduce variance). The loss function for learning an OSF is thus defined as L(Θ) = E r ∥L coarse (o, ω out ) − C(r)∥ 2 + ∥L fine (o, ω out ) − C(r)∥ 2 ,(8) where C(r) denotes the groundtruth pixel color (which is up to a constant of the radiance assuming a linear camera response) for ray r = o − tω out , L coarse (o, ω out ) denotes the predicted radiance by the coarse network, and L fine (o, ω out ) denotes the predicted radiance by the fine network. During testing, we only use the predicted radiance of the fine network. KiloOSF for Accelerating Rendering Neural volumetric rendering is slow. We show a complexity analysis in the following subsection. To accelerate OSFs, in the same spirit as KiloNeRF (Reiser et al., 2021), we introduce a variant of our model called KiloOSF that represents the scene with a large number of independent and small MLPs. KiloOSF represents each object with thousands of small MLPs, each responsible for a small portion of the object, making each individual MLP sufficient for high-quality rendering. Specifically, we subdivide each object into a uniform grid of resolution s = (s x , s y , s z ). Each grid cell is of 3D index i = (i x , i y , i z ). We define a mapping m from position x to index i through spatial binning as follows: m(x) = ⌊(x − b min )/(b max − b min )⌋,(9) where b min and b max are the respective minimum and maximum bounds of the axis-aligned bounding box (AABB) enclosing the object. For each grid cell, a small MLP neural network with parameters Θ(i) is used to represent the corresponding portion of the object. Then, the radiance transfer and density values at a point x can be obtained by first determining the index m(x) responsible for the grid cell that contains this point, then querying the respective small MLP: NN Θ(m(x)) : (x, ω light , ω out ) → (ρ, σ),(10) Following KiloNeRF, we use a "training with distillation" strategy. We first train an ordinary OSF model for each object and then distill the knowledge of the teacher model into the KiloOSF model. We also use empty space skipping and early ray termination to increase rendering efficiency (Reiser et al., 2021). Composing Multiple Learned OSFs We show how we use learned OSFs for visually plausible scene composition. Due to the complexity of the problem and the approximations made by OSFs, physical correctness may not be guaranteed in the following discussion. Below we present a simple and computationally tractable approximation for composing multiple learned OSFs. We leave the design of physically correct scene composition algorithms as future work. We restrict our discussion to a scene with multiple objects represented by OSFs, and we assume a single distant light source to illuminate the scene. In this case, the light distribution in Eq. (3) can potentially contain two parts: one for direct lighting, and the other for indirect lighting. We discuss them separately below, and then provide an algorithm for rendering a composed scene. Direct lighting. Direct lighting is assumed to be a distant light, so the major point to consider is solving visibility. Notice that although OSFs allow reconstructing translucent object appearances, we do not model advanced light transport effects including transmitted lights or caustics in this work, but only consider shadows caused by light visibility. While solving for hard visibility can be tricky for OSFs, because they do not have the concept of surface just like NeRFs, the transmittance T in Eq. (1) can be seen as a soft measure for visibility. Therefore, we use a shadow ray r shadow (t) = x − tω light to solve for visibility at x. We volume-render this shadow ray for all other OSFs in the scene with compositional rendering (Niemeyer & Geiger, 2021), and use the obtained transmittance T (t) = exp(− t tn σ(r shadow (s))ds) to serve as visibility: L direct = T (t f )L 0 , where we abuse t f to denote the far bound of the shadow ray and L 0 to denote the light radiance. Given L direct , we compute outgoing radiance using Eq. (7). Notice that this is a biased estimation for translucent objects, because the visibility is solved by the ray cast from a single point x, while the cumulative transfer function ρ assumes the full light visibility for the object surface ( Figure 2). Therefore, if a translucent object is partially occluded from the distant light, the outgoing radiance is underestimated for those points casting occluded shadow rays, and overestimated for those points casting unobstructed shadow rays. This also happens for concave opaque objects due to global transport. But for ideally learned OSFs of convex opaque objects (i.e., they degenerate to BRDFs) as discussed in section 3.3, the estimation is unbiased. Indirect lighting. Besides direct lighting, we also want to account for indirect lighting in evaluating Eq. (3). To do this, we use Monte Carlo integration with uniform sampling of solid angles. Specifically, we cast a ray from x to the sampled direction, and volume-render the ray using Eq. (1) to obtain the incoming radiance. Due to high computational cost, we only consider one-bounce indirect lighting. Similar to direct lighting, the estimation is biased for translucent objects, because ρ assumes distant light, while inter-reflection from objects do not meet this assumption. Nonetheless, our experiments show that our approximation can create reasonably faithful scene composition results in Sec. 4.3. Excluding non-intersecting rays to accelerating scene composition. The standard rendering procedure must be repeated per pixel to render an image. The total cost of rendering a single image with N pixel pixels and N object objects is thus where N sample denotes the number of samples per ray-object pair, N light denotes the number of light sources, and d denotes the number of light bounces we model. There are many samples for which we must query OSFs, which can result in significant rendering time. In addition to KiloOSF, we further accelerate scene composition by excluding non-intersecting rays. Specifically, we implement a sampling procedure that precludes the evaluation of rays that do not intersect with objects. Our sampling procedure assumes access to (rough) bounding box dimensions for each object. In practice, such a bounding box can be automatically computed from a trained OSF by extracting the object's bounding volume (using the predicted alpha values). As shown in Fig. 3, for each ray and each object of interest, we intersect the ray with the object's bounding box. If the ray does not intersect with the object's bounding box, then no computation is required for the ray-object pair. If the ray does intersect with the object's bounding box, we adopt the intersection points as the near and far sampling bounds for the ray-object pair. Thus the N pixel N object (number of primary rays) and N light N object (number of secondary rays) are upper bounds on the number of rays that need to be evaluated. In practice, a single ray often only intersects with at most one object in the scene, which means that the proposed rendering procedure is not significantly more expensive than the single object setting. O((N pixel N sample N object )(N light N sample N object ) d−1 ),(11) Compared with traditional volumetric path tracing methods, our OSF model does not require running path tracing within each object (by evaluating itself during secondary ray computations) to simulate intra-object light bounces, because it learns the object-level scattering function that directly predicts the effects after all light bounces (reflections) and occlusions (shadows) within an object have occurred. Therefore, OSFs are faster than alternative methods such as NRFs (Bi et al., 2020a), which relies on simulating intra-object light bounces while querying its learned BRDF model. Experiments We validate our approach by free-viewpoint relighting on both synthetic and real-world objects, including translucent and opaque ones. Furthermore, we also demonstrate that OSFs allow visually plausible scene composition by showcasing a scene with both synthetic and real objects. We leave implementation details and ablation studies in our supplementary material. Data and baselines In our experiments, we use six synthetic objects, two captured real translucent objects and five real opaque objects from the Diligent-MV dataset (Li et al., 2020). (Li et al., 2020). This dataset provides multi-view photometric images of five objects that feature different levels of shininess. For each object, there are 20 views forming a circle around the object. For each view, there are 96 calibrated light sources spatially fixed relative to the camera. We use Real image capture setup. We use an easily accessible image-capturing setup, where we take photos with two iPhone 12 in a dark room. We use the flash of a cellphone P light as a distant light source. We position P light such that it is far away from the object (the distance to the object is about 10 times compared to the object diameter). We use the other cellphone P image as a camera without using its flash. We capture 20 images from random viewpoints for a single random light direction, and we repeat this for 10 different light directions. This gives us a dataset of 200 images from different viewpoints. We split our dataset such that all 20 images of a randomly chosen light direction are held out for testing and all other images for training. We use a standard Structure from Motion (SfM) method, COLMAP (Schonberger & Frahm, 2016), to solve for camera poses for all images. To calibrate light position, we take a photo using P light for every light position, and solve the camera pose together with other captured images. Photos taken by P light are only used for light direction calibration but not for training. In order for SfM to work well in the dark room, we place highly-textured newspaper pieces surrounding the captured object to provide robust features and use a foreground segmentation network (Qin et al., 2020) to extract clean object images after SfM. Table 3: Free-viewpoint relighting on real opaque objects from Diligent-MV (Li et al., 2020). Diligent-MV dataset Synthetic images. We use 5 synthetic objects from the ObjectFolder (Gao et al., 2021) dataset, as well as a Stanford bunny. We assign translucent materials to half of them and opaque materials to the other half. For each synthetic object, we render 1,000 images, each of which is under a random light direction and a random viewpoint. We sample cameras uniformly on an upper hemisphere, and light directions uniformly on solid angles. We use 500 images for training and 500 for testing. We generate synthetic images in Blender 3.0, using the Cycles path tracer. We use the Principled BSDF (Burley, 2012) with Christensen-Burley approximation to the physically-based subsurface volume scattering (Christensen, 2015). We also evaluate OSFs on different levels of translucency and on a two-light setting in our supplementary material. Baselines. We consider the following baselines: o-NeRF (Mildenhall et al., 2020): We train a NeRF for each object and refer to it as o-NeRF. Since o-NeRF is agnostic to lighting and cannot do relighting, we show view synthesis results. PhySG (Zhang et al., 2021b): A recent representative neural inverse rendering method that jointly estimates lighting, materials, and geometry from multi-view images and masks of shiny objects. NeRD (Boss et al., 2021a): A recent representative neural relightable representation that decomposes the appearance into a neural BRDF field and environment lighting. IRON (Zhang et al., 2022): A state-of-the-art neural inverse rendering method. Note that IRON assumes collocated lighting for training images and assumes little self-shadow in training images. These assumptions do not hold for most of our data, so we only test IRON on Diligent-MV dataset which has less self-shadows. We use the following standard metrics in all our comparisons: peak signal-to-noise ratio (PSNR), structural similarity (SSIM) (Wang et al., 2003), and a perceptual metric LPIPS (Zhang et al., 2018). Free-Viewpoint Relighting OSFs take light directions as input in addition to the camera viewpoint. Thus, it inherently supports simultaneous relighting and novel view synthesis. Translucent objects. We show qualitative ( Figure 6 and Figure 4) and quantitative (Table 1 and Table 2) comparisons on both synthetic and real translucent objects. Our approach significantly outperforms the baseline methods. It successfully models the material and shape of the objects, and accurately relights them from varied viewpoints compared to the ground truth. In contrast, o-NeRF fails due to its assumption on fixed illumination. Methods that parameterize object materials by BRDFs, such as PhySG and NeRD, fail to produce subsurface scattering effects. Moreover, since they assume opaque surfaces, subsurface scattering effects in the training images seem to be explained by geometry, producing inaccurate geometry reconstruction (e.g., the bowl in Figure 6). Notably, for the real soaps, OSFs can produce correct highlights on the soap surface as well as subsurface scattering effects (e.g., notice how the shading changes non-uniformly along the fracture surface of the orange soap in Figure 4). Furthermore, only OSFs can reproduce the text on the soaps. Opaque objects. We show free-viewpoint relighting results for real objects from Diligent-MV dataset in Figure 5 (we drop NeRD as it has a convergence issue on this dataset despite our best efforts) and Reference Ours Ours (indirect only) Ours w/o indirect Ours w/o ind. or shadow Figure 9: Analysis of light transport effects including shadow and indirect lighting. Note that here we generate the ray-traced reference image with only two light bounces as we only compute two light bounces in our composition. Images are tone-mapped to sRGB space for display. synthetic opaque objects in Figure 7. We observe that OSFs produce faithful free-viewpoint relighting results, correctly synthesizing hard self-shadows (e.g., the airplane in Figure 7 and the bear in Figure 5) and highlight reflections (e.g., the buddha's belly and the cow in Figure 5). In contrast, baseline methods do not synthesize these effects well. PhySG produces some self-shadows, while it does not reconstruct geometries well on Diligent-MV objects. IRON assumes collocated lighting and it assumes little self-shadows in training images. Thus it does not reconstruct or relight the objects correctly. We show quantitative results in Table 4 and Table 3, where OSFs significantly outperform all baselines. Scene Composition Since OSFs are object-centric representations that model light transport for an object, they allow trained models to be readily composed into new scenes. To showcase neural scene composition, we use a translucent and an opaque object with randomized poses, lighting directions, and viewpoints. We show relighting and view synthesis of this example scene in Figure 8. We compare with the o-NeRF baseline. Qualitatively, while o-NeRF produces incorrect shadows and artifacts, our method generates more realistic compositions that better match the ray-traced groundtruth images. We also show quantitative results in Table 5, where we observe that OSFs significantly outperform o-NeRF. Light transport analysis. To analyze the light transport effects including shadows and indirect lighting in our composition method, we showcase a simple scene consisting of a grey cube and a red floor in Figure 9, where we show our scene composition and variants ablating shadows and indirect lighting. From Figure 9 we observe increased realism when we add shadow and indirect lighting. Specifically, we see a color bleeding effect from the red floor onto the cube by the indirect lighting. With these light transport effects, it correctly synthesizes the scene up to two light bounces (i.e., direct lighting and one-bounce indirect lighting). While here we showcase a simplistic scene and compute one bounce for indirect lighting due to computational constraints, our method may benefit from future advances in efficient neural rendering to scale to more complex scenes and more light bounces. Evaluation on KiloOSF We introduce KiloOSF for accelerating OSF rendering. Thus, in this section, we evaluate KiloOSF in terms of rendering speed and relighting performance. To this end, we test KiloOSF on all synthetic translucent objects. We show the rendering speed (frames per second) and the rendering quality in Table 6. KiloOSF achieves 60× (16.13 vs. 0.27 FPS) speed up while maintaining comparable relighting performances, suggesting the benefits brought by the KiloOSF design. Conclusion We presented a novel, learning-based object representation-Object-Centric Neural Scattering Functions (OSFs). An OSF models a cumulative radiance transfer function, allowing both free-viewpoint relighting and scene composition. Moreover, OSFs can model objects of complex shape and materials, and can relight both opaque and translucent objects. Our work offers a new promising neural graphics method for modeling real-world scenes. Limitations. The major limitation of our model comes from the assumption of unobstructed distant lighting, which leads to biased estimation in, e.g., computing inter-reflection, especially considering shiny objects. In real capture, distant light can be approximated by holding a flashlight far away from small objects, but for big objects this is difficult. We envision wider applications in future research that relaxes the assumption. Learning Object-Centric Neural Scattering Functions for Free-viewpoint Relighting and Scene Composition: Supplementary Material A Supplementary Video In the supplementary video, we first give an overview of our method, Object-centric Neural Scattering Functions (OSFs), and explain how it learns cumulative radiance transfer. Then, we show demos of freeviewpoint relighting on both opaque and translucent objects. Finally, we show how we compose multiple OSFs into a new scene configuration. B Implementation Details We use a multilayer perception (MLP) with rectified linear activations. The predicted density σ is viewinvariant, while the cumulative radiance transfer function ρ is dependent on the incoming and outgoing light directions. We use an eight-layer MLP with 256 channels to predict σ, and a four-layer MLP with 128 channels to predict ρ. Following NeRF (?), we similarly apply positional encoding to our inputs and employ a hierarchical sampling procedure to recover higher quality appearance and geometry of learned objects. For positional encoding, we use W = 10 to encode the position x and W = 4 to encode the incoming and outgoing directions (ω light , ω out ), where W is the highest frequency level. To avoid ρ from saturating in training, we adopt a scaled sigmoid (?) defined as S ′ (ρ) = δ(S(ρ) − 0.5) + 0.5 with δ = 1.2. We use a batch size of 2048 rays. For synthetic datasets, we sample N c = 64 coarse samples and N f = 128 fine samples per ray. For real world datasets, we sample N c = 64 coarse samples and N f = 64 fine samples per ray. We use the Adam optimizer (?) with a learning rate of 0.001, β 1 = 0.9, β 2 = 0.999, and ϵ = 10 −7 . C Additional Experiments C.1 Ablation study on translucency We study how well OSFs can model different levels of translucency. To evaluate this, we use the Stanford Bunny model with different translucency intensities including {0, 0.1, 0.3, 0.6, 1}, where 0 means opaque and 1 means highly translucent with strong subsurface scattering. For each of these 5 objects, we generate images with identical camera poses and light directions. We show results in Figure S1 and Table S1. From the figure, Ground truth OSF (ours) Figure S1: Qualitative examples of ablation study on translucency. we see that OSFs can model all objects well, especially translucent objects that have smoother shadows. A major reason is that the appearances of translucent objects vary smoothly w.r.t. changing view angles and lighting directions, and thus a learned neural implicit model is suitable to represent them. Opaque concave objects can have harsh self-shadows which are very high-frequency signals that are difficult for neural models to represent and interpolate (?). C.2 Free viewpoint relighting with two light sources Since OSFs learn radiance transfer functions, they support relighting with multiple distant lights due to the linearity of radiance. To demonstrate this, we evaluate OSFs on free-viewpoint relighting with two light sources, while all models are trained with only one light source. We use synthetic opaque objects. For each object, we generate a new test set with the same camera poses and light directions as the original one-light test set, but we add an additional light source. We show results in Figure S2 and in Table S2. From the figure, we observe that the visual results are faithful for both settings. From the table, we validate this via numerical metrics. We note that the widely-used environment maps are essentially collections of distant lights. Figure S2: Qualitative examples of relighting with two distant light sources. Figure 3 : 3Sampling procedure. (a) Scene with a camera, light source, and object bounding boxes. Primary rays are sent from the camera into the scene. Rays that do not intersect with objects are pruned. Of the intersecting rays, we sample points within intersecting regions. (b) Shadow rays from each sample are sent to the light source, and samples within intersecting regions are evaluated. Figure 4 :Figure 5 : 45Results on free-viewpoint relighting for real translucent objects captured by a cellphone. Free-viewpoint relighting for real opaque objects from Diligent-MV(Li et al., 2020). a front view and a back view (view 10 and view 20) as the testset, and the other 18 views as the training set. Note that the light directions in the training set are completely disjoint from the testset since the light sources move with the camera. Figure 7 :Figure 8 : 78Free-viewpoint relighting for synthetic opaque objects from ObjectFolder(Gao et al., 2021). Scene composition for a translucent bowl, an opaque airplane, and an opaque floor. Table 1 : 1Free-viewpoint relighting on synthetic translucent objects.PSNR↑ SSIM↑ LPIPS↓ o-NeRF 17.63 0.725 0.331 PhySG 20.09 0.823 0.171 NeRD 21.94 0.815 0.182 OSF (ours) 27.06 0.865 0.051 Table 2 : 2Free-viewpoint relighting on real translucent soaps.PSNR↑ SSIM↑ LPIPS↓ o-NeRF 31.36 0.944 0.038 PhySG 30.09 0.944 0.046 IRON 14.80 0.907 0.081 OSF (ours) 39.07 0.970 0.020 Table 4 : 4Free-viewpoint relighting on synthetic opaque objects.PSNR↑ SSIM↑ LPIPS↓ o-NeRF 17.21 0.678 0.244 OSF (ours) 35.57 0.927 0.031 Table 5 : 5Free-viewpoint relighting results for scene composition.FPS↑ SSIM↑ LPIPS↓ OSF 0.27 0.923 0.034 KiloOSF 16.13 0.938 0.037 Table 6 : 6Evaluation of KiloOSF on relighting synthetic opaque objects. Hong-Xing Yu * 1 , Michelle Guo * 1 , Alireza Fathi 2 , Yen-Yu Chang 1 , Eric Ryan Chan 1 , Ruohan Gao 1 , Thomas Funkhouser 2 , Jiajun Wu 1 Reviewed on OpenReview: https: // openreview. net/ forum? id= NrfSRtTpN5 Our supplementary material consists of: A. Supplementary video. B. 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