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1Department of Astronomy & Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON, M5S 3H4, Canada 2Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK 3European Southern Observatory (ESO), Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany Type Ia supernovae are generally agreed to arise from thermonuclear explosions of carbon-oxygen white dwarfs.The actual path to explosion, however, remains elusive, with numerous plausible parent systems and explosion mechanisms suggested. Observationally, type Ia supernovae have multiple subclasses, distinguished by their lightcurves and spectra.This raises the question whether these reflect that multiple mechanisms occur in nature, or instead that explosions have a large but continuous range of physical properties.We revisit the idea thatandsupernovae can be understood as part of a spectral sequence, in which changes in temperature dominate.Specifically, we find that a single ejecta structure is sufficient to provide reasonable fits of both thetype Ia supernova SN 2011fe and theSN 2005bl, provided that the luminosity and thus temperature of the ejecta are adjusted appropriately. This suggests that the outer layers of the ejecta are similar, thus providing some support of a common explosion mechanism. Our spectral sequence also helps to shed light on the conditions under which carbon can be detected in pre-maximum SN Ia spectra – we find that emission from iron can “fill in" the carbon trough in cool SN Ia. This may indicate that the outer layers of the ejecta of events in which carbon is detected are relatively metal poor compared to events where carbon is not detected.§ INTRODUCTIONWhile most type Ia supernovae (SN Ia) are remarkably uniform, it has long been known that there are subclasses whose events share distinguishing features, such as the over-luminous “91T-like” and sub-luminous “91bg-like” <cit.>.The number of subclasses has grown with the number of available spectra, with recent reviews distinguishing almost ten <cit.>.For many of the subclasses there are associated proposed evolutionary channels and physical explosion mechanisms.For few, however, there is consensus on which one is correct. Furthermore, just as supernova types II, IIb, Ib, and Ic have turned out to reflect not different physical origins but rather a sequence in the amount of stellar envelope stripped before core collapse, some of the observed Ia subtypes may reflect variation in explosion properties rather than different explosion mechanisms.<cit.> was among the first to propose that SN Ia belonging to different subclasses could originate from a spectral sequence in luminosity. This parameter sets the temperature profile of the ejecta, determining the ionisation state of the elements therein and thus shaping the observed spectral features. In this scenario, the amount of ^56Ni synthesized in the explosion would be the primary factor driving the sequence. A similar approach was employed by <cit.> to investigate in detail the strength of silicon lines.Indeed, observationally, some subclasses seem to form continuous sequences. For instance, <cit.> note that their four subclasses, which are based on the pseudo equivalent widths of the λ6355 and λ5972 Si2 absorption features, show a continuous distribution (see Figure <ref> below). Similarly, <cit.> and <cit.> show that using different techniques (velocity gradient of the λ6355 Si2 line and principle-component analysis, respectively) the known sub-groups can be recovered but that they are not strongly disjoint, with some objects classified as “transitional” types.Here, we focus specifically on theandsubclasses. Compared tosupernovae,events are fainter, decline faster, have a “titanium trough” near 4200Å and show stronger Si2 λ5972 and O1 λ7774 lines in their spectra <cit.>. Moreover, the velocity inferred from the Si2 λ6355 line evolves more rapidly and on average settles at lower values in the post-maximum spectra <cit.>. Also forandSN Ia, it is currently unclear if they share the same progenitor scenario or arise from different channels. For instance, <cit.> tried to classify explosion mechanisms based on their predicted spectra, and found that, while all mechanisms had shortcomings, equal-mass violent mergers fitSN Ia quite well, while detonations of sub-Chandrasekhar white dwarfs and delayed-detonations better representedSN Ia. From the distribution of inferred ^56Ni masses, <cit.> suggest thatandSN Ia might arise from the double detonation <cit.> and direct collision <cit.> scenarios, respectively. In contrast, e.g., <cit.> suggest a common origin on statistical grounds: whileSN Ia are found preferentially in massive galaxies, they seem to “cut into the share” of theSN Ia.We focus on the spectroscopic differences between two very well studied supernovae that are thought to be representative of their subclasses: theSN 2011fe andSN 2005bl.For both, using the technique known as abundance tomography (e.g., ), the observed spectral evolution has been used to probe different layers of the ejecta, with deeper ones exposed as the photosphere recedes.For SN 2011fe, <cit.> concluded that the progenitor likely had sub-solar metallicity, and that the density profile of the ejecta was in between the prediction from the bench-mark W7 model of <cit.> and the WDD1 delayed-detonation model of <cit.>.For SN 2005bl, <cit.> first optimized their synthetic spectra assuming a W7 density profile, and then used energy and mass scalings to improve their fits, finding best agreement when the original mass (of 1.38 M_⊙) was maintained, but the kinetic energy was reduced by 30%.Thus, beyond differences in explosion energy, the two analyses yielded somewhat different density profiles. Furthermore, also the abundances differ.For instance, for titanium, the element responsible for the 4200Å titanium trough distinctive ofsupernovae, SN 2011fe yielded a total mass >10 times higher than SN 2005bl in the outer layers (v≳ 7,800 km s^-1), whereas SN 2005bl contains more titanium in the inner layers.A known issue with tomography is the presence of strong degeneracies, i.e., multiple sets of parameters may give good agreement. Hence, in this first paper in a series in which we hope to investigate the observational constraints on the explosions with tomography, our primary goal is to determine whether, even though the density and abundance profiles found for both supernovae were different, the spectral differences can, in fact, be understood from variations in just one or a few parameters. We also report on a by-product of our investigation, which is that along only a limited range in our spectral sequences the carbon feature is visible; this may shed light on why carbon is only sporadically detected in SN Ia (∼ 30% of events with early enough spectra; e.g. ). § METHODOLOGYTo explore the physical properties that distinguishfromSN Ia, we use the spectral synthesis code TARDIS <cit.>. TARDIS is a Monte Carlo code very similar to the code used to analyse SN 2011fe and SN 2005bl <cit.>, and hence our approach is to start from the available tomography <cit.>. After verifying that we reproduce the earlier results, our goal is then to see for each object if we can make the synthetic spectra approach the observed spectra of the other object by a low-dimensional sequence in physical properties.Here, for our first attempt, we follow <cit.> and <cit.> and search for a sequence based on explosion brightness. We extend previous work by analyzing not just spectra taken near maximum light, but also before and after, and by investigating the possibility of a sequence in titanium mass fraction X(Ti).Specifically, for our simulations we used TARDIS v1.5dev2685, which calculates a synthetic spectrum for given density and abundance profiles, an observed brightness, the time since the explosion, and the position of the pseudo photosphere (i.e., below which the ejecta are optically thick).In TARDIS, the ejecta are assumed to be in homologous expansion, with parameters given in velocity space, and the modeled photon interactions are absorption and emission by lines (in the Sobolev approximation), and electron scattering (for details, see ).We adopt TARDIS settings similar to those in previous works – see appendix <ref> and <cit.> for a detailed explanation. For our “default" ejecta structure of SN 2011fe we use the data available in Table 6 and Fig. 10 of <cit.> and for SN 2005bl we use the same input parameters as <cit.> for their best model “05bl-w7e0.7”, as in their table A1). Starting with these, we then scale the luminosity of the explosion, following what was done by <cit.> to investigate the strength of the Si2 λ6355 and λ5972 lines, but focusing on the spectral sequence itself, not restraining the analysis to near maximum spectra, and varying the titanium abundance as well.For comparison purposes, all the spectra shown in this paper are normalised by the mean flux over a wavelength window of 4000–9000 Å.§ RESULTSWe compare our models with observed spectra[Reduced spectra were retrieved from the WISeREP archive <cit.> at <https://wiserep.weizmann.ac.il/>.For details of the data reduction, see <cit.> and <cit.>.] for three epochs in Figure <ref> (sets a–f). One sees that there is good agreement with the default models, showing that TARDIS allows one to reproduce the previous results, although the agreement is worse at late times, especially for SN 2005bl (see <ref>).In detail, also near and pre-maximum, our synthetic spectra are not entirely consistent with those previously published. This may reflect differences between TARDIS and the <cit.> code, small differences in the (interpretation of the) density and abundance profiles of SN 2011fe, and choice of atomic data. Nevertheless, the differences between the codes are no larger than those between the simulated and observed spectra, giving us confidence that relative changes are reliable. §.§ Luminosity scaling In Figure <ref>, we show simulated spectral sequences in which the luminosity of the SN 2011fe and SN 2005bl models are scaled down and up, respectively, towards the luminosity of the other supernova. One sees that the observed SN 2005bl spectra are quite well reproduced by some of the fainter SN 2011fe model spectra (sets a–c.)Indeed, our cooler SN 2011fe models not only approach the overall spectral shape of SN 2005bl, but also succeeds in reproducing some of the general spectroscopic features ofSN Ia (see <ref>.) For instance, the formation of the titanium trough and the stronger features due to the Si2 λ5972 and O1 λ7774 lines. This is accomplished without enhancing the abundance of these elements, but simply as an effect of the changes in ionisation states.The above corroborates the idea that luminosity is a determining factor for the spectral differences. Looking in more detail, we see that the luminosity scale factor of 0.25 for SN 2011fe that produces a good match to SN 2005bl near maximum corresponds to a brightness similar to the observed one for the latter. However, the same is not true for the pre- and post-maximum models (sets a and c,) where the best match is for a luminosity ratio of 0.33, while the observed ratio is about 0.18.In contrast, looking at the inner temperatures (T_inner; see Fig. <ref>;associated with the models, we find that near maximum, the 0.25 L_11fe model is closest in temperature to the default SN 2005bl model (T_inner=9547K vs 8920 K). More importantly, the temperature profile of the ejecta across the line-forming regions is similar between the scaled models for SN 2011fe and SN 2005bl. This is relevant because the precise inner temperature can vary significantly (± 1000K) depending on the adopted inner radius, and still produce consistent spectra. This is because the effective radius at which photons originate and the temperature at that radius are what determine the luminosity and the spectrum, and those may not be affected by the precise location of the inner boundary. This suggests the idea that rather than comparing supernovae at the same phase or brightness, one ought to compare the spectra produced when the ejecta are at a similar state, with a similar temperature profile.To also match the luminosity, one then may have to take into account the difference in rise time (∼ 17 and 19 days for SN 2005bl and SN 2011fe, respectively <cit.>), and thus correct for the timescale on which the supernova evolves (see Sect. <ref>).While it was previously known that the temperature is important (e.g. ,) the fact that a sequence of spectra produced from a common ejecta structure can produce both subclasses of SN Ia by a change in temperature lends support to the idea that their outer ejecta are quite similar, despite degeneracies in, for example, the chosen density profile.Consistent with the dominance of temperature in determining the spectral sequence is that not only do the fainter synthetic spectra based on SN 2011fe resemble SN 2005bl, the reverse is true as well: the brighter synthetic spectra based on the SN 2005bl model qualitatively resemble those ofSNe – in particular, there is no clear 4200 Å Ti trough.In more detail, however, the cooler SN 2011fe models (sets a–c) seem to exhibit a better agreement with the observed spectra of SN 2005bl than the hotter models of SN 2005bl do when compared to SN 2011fe observations (sets d–f). This suggests that the density profile found for SN 2011fe might be closer to a common density profile for SN Ia than the energy scaled W7 density profile employed for the SN 2005bl simulations.Our cooler SN 2011fe models not only approach the overall spectra shape of SN 2005bl, but also succeeds in reproducing some of the general spectroscopic features ofSN Ia (see <ref>.) For instance, the formation of the titanium trough and the stronger features due to the Si2 λ5972 and O1 λ7774 lines. This is accomplished without enhancing the abundance of these elements, but simply as an effect of the changes in ionisation states.§.§ Luminosity and Titanium Sequences In Figure <ref>, we start with the default synthetic spectrum of SN 2011fe at maximum to explore how the spectra are affected in more detail by both the luminosity and thus temperature (left) and the titanium abundance (right). As a function of luminosity, one sees that the 4200Å titanium trough becomes progressively clearer as the luminosity decreases below∼0.27 L_11fe and it gets blended with the Si2 feature near 4000Å. Other spectral features also change continuously, with, as the luminosity decreases, the Ca2 near-IR triplet feature near 8300Å becoming stronger (as fewer calcium atoms are more than singly ionized), and the Si2 feature near 6100Å first becoming stronger (similar reason) and then starting to blend with other features for L≲0.24 L_11fe.We scaled the titanium and chromium mass fractions throughout the ejecta by a factor 0–20, while imposing the condition thatX(Ti)=X(Cr), as in <cit.>. Both elements are adjusted at the expense of the most abundant element in each layer (for which this is a small change: the default model contains only ∼3× 10^-3 M_⊙ of titanium above the photosphere). We use two luminosities, 1.0 and 0.25 L_11fe,max, chosen such that we can both investigate how much titanium a typicalSN Ia could “hide” at maximum, without developing a Ti trough, and by what amount we can decrease titanium in cooler spectra, without loosing the trough.From Figure <ref>, one sees that the amount of Ti in the ejecta will strongly influence the shape of the underlying continuum and thus the color. This effect holds for both luminosities and is as expected, since titanium and chromium, like other iron-group elements, have many transitions in the blue and ultraviolet range and thus efficiently block UV/blue light <cit.>. For the spectra with L=L_11fe,max (top set), the shapes of the spectral features remain nearly unchanged, except for the Ca2 H and K feature near 3700Å. At L=0.25 L_11fe,max (bottom set), the photosphere is cooler (9500 K vs 14500 K) and the changes in color are more drastic. The Ti trough near 4200Å is conspicuous as long as X(Ti) is larger than 0.05 times the default, and it disappears when Ti (and Cr) are absent from the ejecta.For large amounts of Ti, at factors ≳5 above default, the relative depth of the trough diminishes, as the spectra become redder. Overall, like <cit.>, we conclude that the 4200Å trough primarily reflects the degree of ionization of titanium. §.§ Carbon features <cit.> report the likely detection of unburned carbon in the early spectra of SN 2005bl, due to the C2 λ6580 transition (seen next to the red shoulder of the silicon feature near 6100Å). While this feature is not restrained toSN Ia <cit.>, we note that our scaled model for SN 2011fe fails to reproduce it (see set a). Interestingly, this feature is present in our near maximum synthetic spectra (set b), but only for L=0.50 L_11fe,max. This might shed light on why carbon is only sporadically detected in the early spectra of SN Ia. In particular, the spectral series in Fig. <ref> reveals that the carbon signature is only clearly present in the luminosity range of 0.42L_11fe,max≲ L ≲ 0.98L_11fe,max.Analysing the spectral energy distribution according to each element in our spectra, we find that the presence of the carbon trough arises from a balance between iron emission “filling in” the carbon trough in the cooler spectra and not enough carbon being singly ionised in the hottest cases. This finding could explain the correlation that events in which carbon is detected tend to exhibit bluer optical/near-ultraviolet colours than their counterparts in which it is not <cit.>: since iron-group elements effectively redistribute UV light to longer wavelengths, blue continua and carbon features could both be signatures of relatively low metal content (, but see ). We discuss this further in <ref>. §.§ The Strengths of the Si2 λ6355 and λ5972 features To put the effects of changes in luminosity and temperature in a larger context, we investigated how they affect features that have been used to classify SN Ia.Specifically, we use the strong features near 6100 and 5700Å – due to Si2 λ6355 and λ5972 – which define the core-normal, shallow-silicon, cool and broad-line subclasses of <cit.>. Here, the cool subclass overlaps with the faint group defined by <cit.>, which we generically call . For each spectrum shown in the left panel of Fig. <ref>, we compute pseudo-equivalent widths (pEW) following the prescription of <cit.>. We estimate uncertainties following <cit.>, using a Monte Carlo routine to generate mock spectra with noise properties like the data (estimated from by smoothing the spectrum and computing the root mean square between the smoothed spectrum and the data in a wavelength window of 40Å).By analysing multiple TARDIS runs with identical inputs, we found the resulting uncertainties slightly underestimate the true ones by ∼20% (for which we correct).We remeasured pEWs of objects from the Berkeley Supernova Ia Program (BSNIP; ) and show them in Figure <ref> (which can be compared with Fig. 10 from ; our pEW are consistent).Overlaid are the pEW we measured from the synthetic spectra based on the SN 2011fe model at maximum, but with scaled luminosity.One sees that the spectral sequence crosses through all the four subclasses displayed.The brightest explosions are in the shallow-silicon region, reflecting that most of the silicon is more than singly ionized at the corresponding high temperatures.Towards lower brightness, the fraction of singly-ionised Si increases strongly (while only a negligible fraction is neutral) and the Si2 λ6355 feature strengthens until it saturates near a value of ∼130Å, while the Si2 λ5972 feature becomes progressively stronger <cit.>.§ MODELLING UNCERTAINTIES An important limitation of our analysis follows from the approximation of the incoming radiation field from the pseudo-photosphere as black-body radiation. This approximation becomes progressively worse starting about a week after maximum, as the ejecta evolve and the photosphere moves in too far. For comparison with previous work, we nevertheless included some simulations at such late epochs.It is generally suggested that theTARDIS mode for line treatment is used instead of themode that we employ, since the former provides a more physical approximation.Specifically, theapproximation better describes deexcitation cascades (and multiple-photon excitation, though those are less likely to be important) than does the simplermode. Nevertheless we retain theapproximation for consistency with the earlier abundance tomography models on which our study is based: to consistently use themode would require re-doing the tomography of SN 2011fe and SN 2005bl, which is beyond the scope of this work.However, while <cit.> found few substantial differences betweenandcalculations for the model they considered, our particular model is noticeably affected by this choice. In Fig. <ref>, we show synthetic spectra for the “default” model for the early spectra of SN 2011fe and SN 2005bl (set a in Fig. <ref>), computed using both theandmodes. Looking at the simulations in detail, we find that the emission and absorption of Si and Mg in particular differs substantially between the two treatments, with cooler spectra more strongly affected. Clearly, this is important for detailed fits, and we hope to investigate the origin of this difference in the future.To help give a sense of the differences, we have added an extra version of Figs. <ref>–<ref> in the appendix <ref>, in which the synthetic spectra are computed with theline interaction mode. While our main conclusions are not affected by these differences, the comparison between SN 2005bl and a “fainter” version of SN 2011fe are less strikingly similar. Of course, for these simulations, the starting models are not optimal, since the tomography was done using a different treatment. It will require tomography using themode, however, to test whether starting from proper initial conditions, the spectra would look more similar again.§ CONCLUSIONS§.§ Implications for theandsubtypesWe have shown that for the same ejecta structure, synthetic spectra can reproduce observations of both theSN Ia SN 2011fe and theSN 2005bl, with only the input luminosity adjusted such that the temperatures in the ejecta are similar to those found in earlier fits in the literature. In particular, the titanium trough and the evolution of the strength of the silicon, oxygen and calcium features are in agreement with the data. The luminosities themselves are not always consistent with the observed ones. One would likely obtain better agreement with the observed spectra at the correct luminosity if one were to take into account that SN 2005bl evolved substantially faster than SN 2011fe (Δ m_15, 05bl = 1.93 ± 0.10 mag <cit.> vs. Δ m_15, 11fe = 1.07 ± 0.06 mag <cit.>). From initial experiments, we find that there is a degeneracy between the time since explosion and the luminosity, i.e., reducing the value of time since explosion in the calculations for the scaled down SN 2011fe model, we can approach the temperature structure and thus the SN 2005bl spectrum similarly well for the correct luminosity.We have also investigated the effects of changing the titanium and chromium mass fractions, finding that the spectra are relatively insensitive to these: starting with the SN 2011fe model at maximum, abundance changes by a factor 0.05–20 influence the color but do not affect the shape of the Ti trough much. At lower luminosity, a Ti trough is a generic feature, and will exist even for 20 times lower than the nominal abundance of ∼3× 10^-3 M_⊙ above the photosphere <cit.>. Overall, we conclude thatSN Ia do not have to be Ti poor compared to theircounterparts, but simply hotter, causing titanium and chromium to be doubly ionized, thus preventing the formation of the Ti trough (as also found by .)Our results seem suggestive of a common explosion mechanism forandSN Ia, with the subclasses arising from a continuous set of parameters within a single mechanism.Of course, it does not directly distinguish between mechanisms: indeed, both have been reproduced by delayed detonation <cit.>, violent mergers <cit.> and double detonation <cit.> models.A remaining caveat is that, since the spectra of SN 2005bl can be reproduced by models with distinct ejecta profiles, we cannot truly determine whether its ejecta structure was similar to that of SN 2011fe; in the end, it is mostly an appeal to Occam's razor.Similarly, our results do not exclude potential subdivisions within each of the subclasses.For instance, <cit.> used the epoch of maximum B-V, whether one or two peaks are seen in the near-infrared, and the epoch of the first near-infrared peak relative to that in the blue, to argue that there are two distinct groups ofSN Ia, of which one exhibits a continuous set of properties withSN Ia, while the other does not.SN 2005bl was part of their sample, and was classified as belonging to the second, unconnected group, although lack of some of the photometric evidence made the classification tentative.Fortunately, the similarity can be tested: e.g., if indeedSN Ia have ejecta structure similar to that ofSN Ia, it is possible to predict how their early spectra – which have not yet been observed – will look.Furthermore, with updated simulations, it should be possible to extend the comparison to later times (e.g. ), when the inner parts of the ejecta become visible, which surely should be different:SN Ia must necessarily produce less ^56Ni.More generally, our investigation could be extended to other SN Ia subtypes, such as the overluminous 91T-like objects, as well as to objects classified as “transitional” betweenandSN Ia, such as SN 1986G <cit.> and SN 2004eo <cit.>.At the same time, Fig. <ref> shows that observed SN Ia do not follow a one-parameter family; it will be interesting to explore what other properties need to be changed to reproduce the SN Ia more distant from our models.§.§ Metal content in the outer layersOur cooler models indicate that the detection of the carbon feature near 6100Å might be hindered by emission from iron, even at pre-maximum, where it had not hitherto been considered (cf., ). This suggests that the carbon trough might preferentially be seen in low metallicity progenitors or in strongly stratified objects, where no or very little of the iron-group elements formed deeper in the ejecta are mixed up to the outer layers.It is worth noting thatSN Ia occur preferentially in massiveelliptical galaxies (e.g. ), which tend to be relatively metal rich. Therefore, it might be the case that the same mechanism that causes such explosions to produce less ^56Ni, also affects the formation of the carbon trough.Carbon was also detected in SN 2011fe <cit.>. For SN 2011fe, a sub-solar metallicity was inferred from the tomography of <cit.>, as well as from comparisons with its “twin,” SN 2011by (which also exhibits a carbon trough), with<cit.> concluding SN 2011fe had a lower luminosity, and <cit.> noting that the distance to SN 2011by might be underestimated, but that then both these SNe Ia would arise from low metallicity progenitors. In contrast to this, however, <cit.> found that, in the context of delayed detonations models, the strength of the carbon feature is more easily reproduced using solar than sub-solar abundances.Yet another example that might help to elucidate the relationship between the carbon trough and the Fe content of the ejecta is SN iPTF13asv. This overluminuous event exhibited a weak (but persistent) carbon signature, while the early spectra showed an absence of Fe features. In addition, this explosion was UV bright near maximum, its ejecta was clearly stratified and it originated from a metal-poor host galaxy <cit.>.Fortunately, this also is amenable to verification.With more detailed models, it should be possible to understand exactly where in the ejecta the presence of iron-group elements makes the carbon feature harder to detect. 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C., Aldering, G., Antilogus, P., et al. 2011, , 743, 27[Walker et al.(2012)Walker, Hachinger, Mazzali, Ellis, Sullivan, Gal Yam, & Howell]Walker2012_metallicity Walker, E. S., Hachinger, S., Mazzali, P. A., et al. 2012, , 427, 103[Yaron & Gal-Yam(2012)]Yaron2012_WISeREP Yaron, O., & Gal-Yam, A. 2012, , 124, 668§ APPENDIX A: SIMULATION PARAMETERSHere we present the set of parameters used in our simulations. We attempted to reproduce the tomography analysis of SN 2011fe, as published by <cit.>, and of SN 2005bl, published by <cit.>. Specifically, to make our assumptions as similar as possible to those used in previous work, we adopt the , ,andmodes to treat line interaction, radiative rates, excitation and ionization, respectively.The model inputs and the temperature at the inner boundary are given in Table <ref>, while the abundance stratification adopted is given in Table <ref>. For comparison purposes, we have also run the simulations adopting the more physical line interaction mode, , while keeping all other parameters unchanged. While the differences in the computed spectra are not negligible, they do not affect our main conclusions.ccccccSimulation properties. 2*Model t lg L v_inner T_innera T_innerb(d) (L_bol/L_⊙) (km s^-1) (K) (K) 5*05bl -6 8.520 8350 9765 9639 -5 8.617 8100 10 136 9857 -3 8.745 7600 10 633 104634.8 8.861 6800 8931 892012.9 8.594 3350 10072 9916 8*11fe 3.7 7.903 13 300 10 800 10 789 5.9 8.505 12 400 12 100 12 0129.0 9.041 11 300 14 500 13 97412.1 9.362 10 700 14 900 14 47016.1 9.505 9000 15 100 14 85619.1 9.544 7850 14 700 14 60422.4 9.505 6700 14 100 13 90128.3 9.362 4550 13 500 13 515 aTemperature at the inner radius as given in the literature (.) bTemperature at the inner radius obtained from the simulations used in this work. [ph] Abundance stratification. Model v (km s^-1)X(C) X(O) X(Na) X(Mg) X(Al) X(Si) X(S) X(Ca) X(Ti) X(Cr) X(Fe)_0a X(Ni)_0a 5*05bl 16 000 - 33 000 0.4184 0.5726 0.0000 0.0030 0.0000 0.0050 0.0010 0.0000 0.0000 0.0000 0.0000 0.00008400 - 16 000 0.0600 0.8600 0.0060 0.0400 0.0025 0.0200 0.0100 0.0004 0.0004 0.0003 0.0003 0.0000 8100 - 8400 0.0300 0.1300 0.0030 0.0300 0.0025 0.6800 0.1000 0.0004 0.0100 0.0070 0.0150 0.0000 7500 - 8100 0.0000 0.0100 0.0000 0.0000 0.0015 0.7000 0.1100 0.0004 0.0533 0.0367 0.0900 0.00006600 - 7500 0.0000 0.0000 0.0000 0.0000 0.0000 0.7100 0.0700 0.0004 0.0550 0.0400 0.1150 0.01003300 - 6600 0.0000 0.0000 0.0000 0.0000 0.0000 0.7700 0.0000 0.0005 0.0167 0.0167 0.0650 0.130011*11fe 19500 - 240000.98040.01200.00000.00300.00000.00400.00050.00000.00000.00000.00010.0000 16000 - 195000.02600.86050.00000.03000.00000.06000.02000.00200.00000.00000.00050.0010 13500 - 160000.03100.70300.00000.03000.00000.20000.03000.00300.00050.00050.00100.0010 12000 - 135000.0000.3510.0000.0200.0000.4400.0800.0030.0030.0030.0600.040 11000 - 120000.0080.1100.0000.0000.0000.5630.1500.0030.0050.0050.0060.150 9000 - 110000.00800.09000.00000.00000.00000.47850.15000.00300.00500.00500.00050.2600 8500 - 90000.00800.09000.00000.00000.00000.19850.07000.00300.00500.00500.00050.6200 8000 - 85000.00800.02000.00000.00000.00000.22850.07000.00300.00500.00500.00050.6600 7500 - 80000.00000.00000.00000.00000.00000.21650.07000.00300.00500.00500.00050.7000 7000 - 75000.00000.00000.00000.00000.00000.20000.06000.00300.00500.00500.00050.7265 3500 - 70000.00000.00000.00000.00000.00000.09000.03000.00010.00100.00100.15000.7279 aFe and Ni mass fractions are, for convenience, given at t = 0, before the decay rates are taken into account. | http://arxiv.org/abs/1707.08572v1 | {
"authors": [
"E. Heringer",
"M. H. van Kerkwijk",
"S. A. Sim",
"W. E. Kerzendorf"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726180000",
"title": "Spectral sequences of Type Ia supernovae. I. Connecting normal and sub-luminous SN Ia and the presence of unburned carbon"
} |
[email protected] Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, Slovakia We derive general conditions for the compatibility of channels in general probabilistic theory. We introduce formalism that allows us to easily formulate steering by channels and Bell nonlocality of channels as generalizations of the well-known concepts of steering by measurements and Bell nonlocality of measurements. The generalization does not follow the standard line of thinking stemming from the Einstein-Podolsky-Rosen paradox, but introduces steering and Bell nonlocality as entanglement-assisted incompatibility tests. We show that all of the proposed definitions are, in the special case of measurements, the same as the standard definitions, but not all of the known results for measurements generalize to channels. For example, we show that for quantum channels, steering is not a necessary condition for Bell nonlocality. We further investigate the introduced conditions and concepts in the special case of quantum theory and we provide many examples to demonstrate these concepts and their implications. Conditions for the compatibility of channels in general probabilistic theory and their connection to steering and Bell nonlocality Martin Plávala December 30, 2023 ==================================================================================================================================§ INTRODUCTIONIncompatibility of measurements is the well-known quantum phenomenon that gives rise to steering and Bell nonlocality. Historically, the idea of measurement incompatibility dates back to Bohr's principle of complementarity. Steering was first described by Schrödinger <cit.> and Bell nonlocality was first introduced by Bell <cit.>, both as a reply to the paradox of Einstein, Podolsky and Rosen <cit.>. It is known that incompatibility of measurements is necessary and in some cases sufficient for both steering and Bell nonlocality, but the operational connection between incompatibility, steering and Bell nonlocality was so far not described in general terms that would also fit channels, not only measurements.There was extensive research into properties of quantum incompatibility of measurements <cit.>, quantum incompatibility of measurements and its noise robustness, or degree of compatibility <cit.>, connection of quantum incompatibility of measurements and steering <cit.>, connection of quantum incompatibility of measurements and Bell nonlocality <cit.> and connection between steering and Bell nonlocality <cit.>, for a recent review see <cit.>. In recent years, the problems of incompatibility of measurements on channels <cit.>, compatibility of channels <cit.>, the connection of channel steering to measurement incompatibility <cit.> and incompatibility in general probabilistic theory <cit.> were all studied.The aim of this paper is to heavily generalize the recent results of <cit.>, where compatibility, steering and Bell nonlocality of measurements were formulated using convex analysis and the geometry of tensor products. In this paper, we will generalize the ideas and results of <cit.> for the case of two channels in general probabilistic theory. The generalizations are not straightforward and we will have to introduce several new operational ideas and definitions, e.g. we introduce the operational interpretation of direct products of state spaces and we define steering and Bell nonlocality as very simple entanglement-assisted incompatibility test, that boil down to the problem whether there exists a multipartite state with given marginal states.During all of our calculations we will restrict ourselves to finite-dimensional general probabilistic theory and to only the case of two channels. We will restrict to only two channel just for simplicity, as one may easily formulate many of our results for more than two channels using the same operational ideas as we will present.The paper is organized as follows: in Sec. <ref> we describe our motivation for using general probabilistic theory. We provide several references to known applications and their connections to each other. In Sec. <ref> we introduce general probabilistic theory. Note that in subsection <ref> we introduce the operational interpretations of direct products in general probabilistic theory. In Sec. <ref> we define compatibility of channels and we derive a condition for compatibility of channels. In Sec. <ref> we show that our condition for compatibility of channels yields the condition for compatibility of measurements that was presented in <cit.>. In Sec. <ref> we derive specific conditions for the compatibility of quantum channels. In Sec. <ref> we propose an idea for a test of incompatibility of channels, that will not work at first, but will eventually lead to both steering and Bell nonlocality. In Sec. <ref> we define steering by channels as one-side entanglement assisted incompatibility test and we derive some basic results. In Sec. <ref> we show that for the special case of measurements our definition of steering leads to thestandard definition of steering <cit.> in the formalism of <cit.>. In Sec. <ref> we derive the specific conditions for steering by quantum channels, we show that every pair of incompatible channels may be used for steering of maximally entangled state and that there are entangled states that are not steerable by any pair of channels, among other results. In Sec. <ref> we define Bell nonlocality of channels as a two-sided entanglement assisted incompatibility test and we derive some basic results, then in Sec. <ref> we show that, when applied to measurement, the general definition of Bell nonlocality yields the standard definition of Bell nonlocality <cit.> in the formalism of <cit.> and we also show that for measurements steering is a necessary condition for Bell nonlocality. In Sec. <ref> we derive conditions for the Bell nonlocality of quantum channels, we formulate a generalized version of the CHSH inequality, we show that for such inequality Tsirelson bound <cit.> both holds and is reached, we show an example of violation of the generalized version of CHSH inequality and we build on the example from Sec. <ref> of an entangled state not steerable by any pair of channels to show that, even though the state is not steerable by any pair of channels, it leads to Bell nonlocality, which shows that steering is not a necessary condition for Bell nonlocality for quantum channels. In Sec. <ref> we conclude the paper by presenting the many open questions and possible areas of research opened by our paper.§ MOTIVATIONS FOR USING GENERAL PROBABILISTIC THEORY There are few motivations to using general probabilistic theory. The first motivation is mathematical as general probabilistic theory is a unified framework capable of describing both classical and quantum theory, as well as other theories. In the current manuscript the mathematical motivation is (according to the personal opinion of the author) even stronger as some of the formulations of the presented ideas and some of the proofs of the presented theorems turn out to be clearer in the framework of general probabilistic theory.The second motivation comes from foundations of quantum theory as general probabilistic theory provides insight into the structure of entanglement and incompatibility.The third and most promising motivation comes from information theory. There were developed several models <cit.> that have very interesting information-theoretic properties and that can be described by general probabilistic theory, albeit sometimes it needs to be extended even more <cit.>. Apart from the well-known results on the properties of Popescu-Rohrlich boxes <cit.>, it was showed that there are theories in which one can search a N-item database in O(√(N)) queries <cit.> and that there is a general probabilistic theory that can be simulated by a probabilistic classical computer that can perform Deutsch-Jozsa and Simon's algorithm <cit.>.The aforementioned results show that studying general probabilistic theory is interesting even from practical viewpoint and that it could have potential applications in information processing.§ INTRODUCTION TO GENERAL PROBABILISTIC THEORY General probabilistic theory is a unified framework to describe the kinematics of different systems in a mathematically unified fashion. The may idea of general probabilistic theory is an operational approach to setting the axioms and then carrying forward using convex analysis. Useful bookns on convex analysis are <cit.>. The beautiful aspect of general probabilistic theory is that it is only little bit more general than dealing with the different systems on their own, but we do not have to basically rewrite the same calculations over and over again for different theories.During our calculations we will use two recurring examples, one will be finite-dimensional classical theory and other will be finite-dimensional quantum theory. The finite-dimensional classical theory is closely tied to the known results about incompatibility, steering and Bell nonlocality of measurements and we will mainly use it to verify that the definitions we will propose are, in the special case of measurements, the same as the known definitions. The quantum theory is our main concern as this is the theory we are mostly interested in. Some results, that we will only prove for quantum theory, may be generalized for general probabilistic theory, but we will limit the generality of our calculations to make them more understandable to readers that are not so far familiar with general probabilistic theory.Given that we will work with many different spaces, their duals, their tensor products and many isomorphic sets, all isomorphisms will be omitted unless explicitly stated otherwise. §.§ The state space and the effect algebra of general probabilistic theoryThere are two central notions in general probabilistic theory: the state space that describes all possible states of the system and the effect algebra that describes the measurements on the system. We will begin our construction from the state space and then define the effect algebra, but we will show how one can go the other way and start from an effect algebra and obtain state space afterwards. We will restrict ourselves to finite-dimensional spaces and always use the Euclidean topology.Let V denote a real, finite-dimensional vector space and let X ⊂ V, then by (X) we will denote the convex hull of X, by (X) we will denote the affine hull of X. We will proceed with the definition of relative interior of a set X ⊂ V.Let X ⊂ V, then the relative interior of X, denoted (X) is the interior of X when it is considered as a subset of (X).For a more throughout discussion of relative interior see <cit.>.Let K be a compact convex subset of V, then K is a state space. The points x ∈ K represent the states of some system and their convex combination is interpreted operationally, that is for x, y ∈ K, λ∈ [0, 1] ⊂ℝ the state λ x + (1-λ) y corresponds to having prepared x with probability λ and y with probability 1-λ.To define measurements we have to be able to assign probabilities to states, that is we have to have a map f: K → [0, 1] such that, to follow the operational interpretation of convex combination, we have assign the convex combination of probabilities to the convex combination of respective states. In other words for x, y ∈ K, λ∈ [0, 1] we have to havef(λ x + (1-λ) y) = λ f(x) + (1-λ) f(y),which means that f is an affine function. Such functions are called effects because they correspond to assigning probabilities of measurement outcomes to states. We will proceed with a more formal definition of effects and of effect algebra.Let A(K) denote the set of affine functions K →ℝ. A(K) is itself a real linear space, moreover it is ordered as follows: let f, g ∈ A(K), then f ≥ g if f(x) ≥ g(x) for every x ∈ K. There are two special functions 0 and 1 in A(K), such that 0(x) = 0 and 1(x) = 1 for all x ∈ K.The set A(K)^+ = { f ∈ A(K): f ≥ 0 } is the convex, closed cone of positive functions. The cone A(K)^+ is generating, that is for every f ∈ A(K) we have f_+, f_-∈ A(K)^+ such that f = f_+ - f_-, and it is pointed, that is if f ≥ 0 and -f ≥ 0, then f = 0.Although we will provide a proper definition of measurement in subsection <ref>, we will now introduce the concept of yes/no measurement, or two-outcome measurement, that will motivate the definition of the effect algebra. Our notion of measurement might seem different to the standard understanding and one may argue that what we will refer to as measurements are should be called entanglement-breaking maps, but this way of defining measurement is standard in general probabilistic theory, hence we will use it. A measurement is a procedure that assigns probabilities to possible outcomes based on the state that is measured. If we have only two outcomes and we know that the probability of the first outcome is p ∈ [0 ,1], then, by normalization, the probability of the second outcome must be 1-p. This shows that a two-outcome measurement needs to assign only probability to one outcome and the other probability follows.Since assigning probabilities to states is a function f: K → [0, 1] and due to our operational interpretation of convex combination we want such function to be affine. Traditionally the functions that assign probabilities to states are called effects and the set of all effect is called effect algebra.The set E(K) = { f ∈ A(K): 0 ≤ f ≤ 1 } is called the effect algebra.In general, one may define effect algebra in more general fashion, using the partially defined operation of addition and a unary operation ⊥, that would in our case correspond to f^⊥ = 1 - f, see <cit.> for a more throughout treatment.Let f ∈ E(K) then the two outcome measurement m_f corresponding to the effect f is the procedure that for x ∈ K assigns the probability f(x) to the first outcome and the probability 1-f(x) to the second outcome. Note that we did not mention any labels of the outcomes. Usually the outcomes are labeled yes and no, or 0 and 1, or -1 and 1, but from operational perspective this does not matter.We provide two standard examples of special cases of our definitions. [Classical theory] In classical theory, the state space K is a simplex, that is the convex hull of a set of affinely independent points x_1, …, x_n. The special property of the simplex is that every point x ∈ K can be uniquely expressed as convex combination of the points x_1, …, x_n, due to their affine independence. [Quantum theory] Letdenote a finite-dimensional complex Hilbert space, let B_h() denote the real linear space of self-adjoint operators on , for A ∈ B_h() let (A) denote the trace of the operator A and let A ≥ 0 denote that A is positive semi-definite. We say that A ≤ B if 0 ≤ B - A. Let B_h()^+ = {A ∈ B_h(): A ≥ 0 } denote the cone of positive semi-definite operators.In quantum theory the state space is given as_ = {ρ∈ B_h(): ρ≥ 0, (ρ) = 1 }which is the set of density operators on . The effect algebra E(_) is given asE(_) = { M ∈ B_h(): 0 ≤ M ≤}.The value of the effect M ∈ E(_) on the state ρ∈_ is given asM(ρ) = ( ρ M ). §.§ The structure of general probabilistic theoryThis subsection will be rather technical, but we will introduce several mathematical results that we will use later on.Let x ∈ K and consider the map x: A(K) →ℝ, that to f ∈ A(K) assigns the value f(x). This is clearly a linear functional on A(K). Moreover for x, y ∈ K, λ∈ [0, 1] we haveλ x + (1-λ) y = λx + (1-λ) yas the functions in A(K) are affine by definition.We conclude that the state space K must be affinely isomorphic to some subset of the dual of A(K). Since the aforementioned isomorphism is going to be extremely useful in later calculations we will describe it in more detail. Let A(K)^* denote the dual of A(K), that is the space of all linear functionals on A(K). For ψ∈ A(K)^* and f ∈ A(K) we will denote the value the functional ψ reaches on f as ψ, f. The dual cone to A(K)^+ is the cone A(K)^*+ = {ψ∈ A(K)^*: ψ, f ≥ 0, ∀ f ∈ A(K)^+ } that gives rise to the ordering on A(K)^* given as follows: let ψ, φ∈ A(K)^*, then ψ≥φ if and only if (ψ - φ) ∈ A(K)^*+, i.e. if ψ - φ≥ 0.It is straightforward that the state space K is isomorphic to a subset of the cone A(K)^*+, moreover it is straightforward to realize that the functionals isomorphic to K must map the function 1 ∈ A(K) to the value 1. Let _K = {ψ∈ A(K)^*+: ψ, 1= 1 }. We call _K the state space of the effect algebra E(K). It might be confusing at this point why we call _K a state space, but this will be cleared by the following. _K is affinely isomorphic to K. It is clear that the map x →x maps K to a convex subset of _K. It is easy to show the inclusion of _K in the image of K using Hahn-Banach separation theorem, see <cit.> for a proof. We will omit the isomorphism between K and _K, so for any x, y ∈ K, α∈ℝ we will write α x + y instead of αx + y∈ A(K)^*. Still, one must be careful when omitting this isomorphism, beacuse if 0 ∈ V denotes the zero vector and 0 ∈ K, then 0∈ A(K)^* is not the zero functional as by construction we have 0, 1= 1. We will do our best to avoid such possible problems by choosing appropriate notation.There are two more result we will heavily rely on:_K is a base of A(K)^*+, that is for every ψ∈ A(K)^*+, ψ≠ 0 there is a unique x ∈ K and λ∈ℝ, λ≥ 0 such that ψ = λ x. Let ψ∈ A(K)^*+, ψ≠ 0, then ψ, 1 ≠ 0 as if ψ, 1= 0 and ψ≥ 0, then ψ= 0, because 1 ∈ (A(K)^+). Let ψ' = 1/ψ, 1 ψ. It is straightforward that ψ' ∈_K.A(K)^*+ is a generating cone in A(K)^*, that is for every ψ∈ A(K) there are φ_+, φ_-∈ A(K)^*+ such that ψ = φ_+ - φ_-. The result follows from the fact that A(K)^+ is a pointed cone, see <cit.>.§.§ Tensor products of state spaces and effect algebras Tensor products are a way to describe joint systems of several other systems. There are several approaches to introducing a tensor product in general probabilistic theory. There is a category theory based approach <cit.> that is a viable way to introduce the tensor products, but we will use a simpler, operational approach. Note that the state space of the joint system will be a compact convex subset of a real, finite-dimensional vector space as it itself must be a state space of some general probabilistic theory. Also keep in mind that describing a tensor product of state spaces K_A, K_B is equivalent to describing the tensor product of the cones A(K_A)^*+, A(K_B)^*+. This is going to be useful as some things are easier to express in terms of the positive cones.Let V, W be real finite-dimensional vector spaces and let v ∈ V, w ∈ W. v ⊗ w will refer to the element of the algebraic tensor product V ⊗ W, see e.g. <cit.>. We will first describe the minimal and maximal tensor products of state spaces that set bounds on the real state space of the joint system. Note that when describing the joint state space of two state spaces or states of two systems, we will refer to them as bipartite state space or bipartite states.Let K_A, K_B denote two state spaces of Alice and Bob respectively. For every x_A ∈ K_A, x_B ∈ K_B there must be a state of the joint system describing the situation that Alice's system is in the state x_A and Bob's system is in the state x_B. We will denote such state x_A ⊗ x_B and we will call it a product state. Since the state space must be convex, the state space of the joint system must contain at least the convex hull of the product states. This leads to the definition of minimal tensor product.The minimal tensor product of state spaces K_A and K_B, denoted K_AK_B is the compact convex setK_AK_B = ( { x_A ⊗ x_B: x_A ∈ K_A, x_B ∈ K_B } ). The bipartite states y ∈ K_AK_B are also called separable states. For the positive cones we getA(K_AK_B)^*+ = ( {ψ_A ⊗ψ_B:ψ_A ∈ A(K_A)^*+, ψ_B ∈ A(K_B)^*+} ).In quantum theory, the minimal tensor product __ is the set of all separable states, that is of all states of the form ∑_i=1^n λ_i ρ_i ⊗σ_i for n ∈ℕ and ρ_i ∈_, σ_i ∈_, 0 ≤λ_i for i ∈{1, …, n}, ∑_i=1^n λ_i = 1. In a similar fashion, let f_A ∈ E(K_A), f_B ∈ E(K_B), then we can define a function f_A ⊗ f_B as the unique affine function such that for x_A ∈ K_A, x_B ∈ K_B we have(f_A ⊗ f_B) ( x_A ⊗ x_B) = f_A(x_A) f_B(x_B).This function is used in the most simple measurement on the joint system, such that Alice applies the two-outcome measurement m_f_A and Bob applies the two outcome measurement m_f_B, so f_A ⊗ f_B must be an effect on the joint state space. This leads to the definition of the maximal tensor product.The maximal tensor product of the state spaces K_A and K_B, denoted K_AK_B, is defined asK_AK_B = { ψ∈ A(K_A)^* ⊗ A(K_B)^*: ∀ f_A ∈ A(K_A)^+, ∀ f_B ∈ A(K_B)^+, ψ, f_A ⊗ f_B ≥ 0 } States in K_AK_B ∖ K_AK_B are called entangled states. Equivalent definition, in terms of the positive cones would beA(K_AK_B)^*+ = ( A(K_A)^+ A(K_B)^+ )^*+whereA(K_A)^+ A(K_B)^+ = ( { f_A ⊗ f_B:f_A ∈ A(K_A)^+, f_B ∈ A(K_B)^+ } ).As we see, the definition of tensor product of cones of positive functionals goes hand in hand with the definition of tensor product of cones of positive functions. In quantum theory, the maximal tensor product of the cones B_h()^+B_h()^+ is the cone of entanglement witnesses <cit.>, i.e. W ∈ B_h()^+B_h()^+ if for every ρ∈_, σ∈_ we have (W ρ⊗σ) ≥ 0. Note that this does not imply the positivity of W. From the constructions it is clear that the state space of the joint system has to be a subset of the maximal tensor product and it has to contain the minimal tensor product. But there is no other specification of the state space of the joint system in general, it has to be provided by the theory we are working with.We will call the joint state space of the systems described by the state spaces K_A and K_B the real tensor product of K_A and K_B and we will denote it K_AK_B. We always haveK_AK_B ⊆ K_AK_B ⊆ K_AK_B. In quantum theory, the real tensor product of the state spaces is defined as the set of density matrices on the tensor product of the Hilbert spaces, that is__ = _⊗.It is tricky to work with the tensor products in general probabilistic theory as the real tensor product is not always specified, or it may not be clear what it should be. We will always assume that every tensor product we need to be defined is defined. Moreover when working with a tensor product of more than two state spaces, say K_A, K_B, K_C we will always assume that(K_AK_B)K_C = K_A(K_BK_C)and we will simply write K_AK_BK_C. In the applications of general probabilistic theory to quantum and classical theory it will always be clear how to construct the needed tensor products and we consider this sufficient for us since we are mainly interested in the applications of our results.We will state and prove a result about classical state spaces that we will use several times later on.Let S be a simplex with the extremal points x_1, …, x_n, i.e. S = ( { x_1, …, x_n } ) and let K be any state space, then we haveSK = SK.Let S be a simplex and let x_i ∈ A(S)^*+, i ∈{ 1, …, n }, be the extreme points of S. The points x_1, …, x_n form a basis of A(S)^*. Let ψ∈ SK then we haveψ = ∑_i=1^n x_i ⊗φ_i,for some φ_i ∈ A(K)^*. Our aim is to prove that φ_i ∈ A(K)^*+ then ψ∈ SK follows by definition.Let b_1, …, b_n denote the basis of A(S) dual to the basis x_1, …, x_n of A(S)^*, i.e. we have b_i(x_j) = δ_ij, where i, j ∈{ 1, …, n } and δ_ij is the Kronecker delta. We have b_i ∈ E(S) because S is a simplex. For any f ∈ E(K) we have0 ≤ (ψ, b_i ⊗ f) = (φ_i, f)for all i ∈{1, …, n}, which implies φ_i ∈ A(K)^*+.Note that tensor product of the simplexes S_1, S_2 is also a simplex, so we haveKS_1S_2 = KS_1S_2.§.§ Direct product of state spaces and effect algebras For certain reasons we will need to use direct products together with tensor product. The idea of why they will be used is going to be clear in the end, but now we will present several of their properties that will be required later. As in the Subsec. <ref> we will work mostly with the cones of the positive functionals.Let K_A, K_B be two state spaces. Given A(K_A)^*+ and A(K_B)^*+ there are two ways to define the direct product of these cones. The first is to use the cone A(K_A)^*+× A(K_B)^*+. The second is to realize that we can construct K_A × K_B that will be a compact and convex set, i.e. a state space that gives rise to the cone A(K_B_1× K_B_2)^*+.It may seem that these cones are fairly similar, but they are not and they have different physical interpretations. Let ψ∈ A(K_A × K_B)^*+, then there are unique λ∈ℝ, x_A ∈ K_A, x_B ∈ K_B such that ψ = λ (x_A, x_B). Now let φ∈ A(K_A)^*+× A(K_B)^*+, then there are y_A ∈ K_A, y_B ∈ K_B, α_A, α_B ∈ℝ, α_A, α_B ≥ 0 such that φ = (α_A y_A, α_B y_B). In other words the normalization may be different in every component of the product. This can be rewritten asφ = (α_A y_A, α_B y_B) = (α_A + α_B) (α_Aα_A + α_B y_A, α_Bα_A + α_B y_B ) = (α_A + α_B) ( α_Aα_A + α_B (y_A, 0) + α_Bα_A + α_B ( 0, y_B ) )that shows that every element of A(K_A)^*+× A(K_B)^*+ can be uniquely expressed as a multiple of a convex combination of elements of the form (y_A, 0) and (0, y_B). The operational interpretation of such states is that we do not even know which system we are working with, but we know that with some probability p we have the first system and with probability 1-p we have the second system.The operational interpretation of A(K_A × K_B)^*+ is a bit harder to grasp. We may understand ψ∈ A(K_A × K_B)^*+ as a (multiple of) conditional state. That is, we will interpret the object (x_A, x_B) as a state that corresponds to making a choice in the past between the systems K_A and K_B and keeping track of both of the outcomes at once. The cone A(K_A × K_B)^*+ will play a central role in our results on incompatibility, steering and Bell nonlocality, because in the problem of incompatibility we wish to implement two channels at the same time and in steering and Bell nonlocality we are choosing between two incompatible channels.At last we will need to describe the set A(K_A × K_B) and its structure with respect to the sets A(K_A) and A(K_B). We will show that A(K_A × K_B) corresponds to a certain subset of A(K_A) × A(K_B) by using the following two ideas: since all of the vector spaces are finite dimensional we have that A(K_A) × A(K_B) is the dual to A(K_A)^* × A(K_B)^* and A(K_A × K_B)^* can be identified with a subset of A(K_A)^* × A(K_B)^*. Note that this identification holds only between the vector spaces and not between the corresponding state spaces. We haveA(K_B_1× K_B_2)^*+⊂ A(K_B_1)^*+× A(K_B_2)^*+.The idea of the proof is that if we have φ∈ A(K_A)^*+× A(K_B)^*+ such that φ = (α_A y_A, α_B y_B) then φ∈ A(K_A × K_B)^*+ if and only if α_A = α_B. Therefore we can identify A(K_A × K_B)^*+ with the set {ψ∈ A(K_A)^*+× A(K_B)^*+: ψ, (1, -1)= 0 }. It is easy to verify this constraint on the positive cones and since it is linear it must hold everywhere else. The above proof shows that the function (1, -1) ∈ A(K_A) × A(K_B) is equal to zero when restricted to A(K_A × K_B)^*, or in other words (1, 0) = (0, 1) when restricted to A(K_A × K_B)^*. We introduce a relation of equivalence on A(K_A) × A(K_B) as follows: for f, g ∈ A(K_A) × A(K_B) we say that f and g are equivalent and we write f ∼ g if f - g = k (1, -1) for some k ∈ℝ. Equivalently, f ∼ g if for every ψ∈ A(K_A × K_B)^* we have ψ, f= ψ, g. A(K_A × K_B) corresponds to the set of equivalence classes of A(K_A) × A(K_B) with respect to the relation of equivalence ∼.To demonstrate this, consider the constant function 1 ∈ E(K_A × K_B) and let x ∈ K_A, y ∈ K_B, then we have(x,y), (1,0)=x, 1 = 1 =(x,y), 1 , (x,y), (0,1)=y, 1 = 1 =(x,y), 1 .This is not a coincidence, because (1, 0) - (0, 1) = (1, -1) so we have (1, 0) ∼ (0, 1). §.§ Channels and measurements in general probabilistic theory It is not easy to define channels in general probabilistic theory as we would like all of the channels to be completely positive. We will use the following definition:Let K_A, K_B be state spaces, then channel Φ is a linear mapΦ: A(K_A)^* → A(K_B)^*that is positive, i.e. for every ψ∈ A(K_A)^*+ we have Φ(ψ) ∈ A(K_B)^*+ and that for ψ∈ K_A we have Φ(ψ) ∈ K_B. One may also require a channel to be completely positive, that is if K_C is some state space such that we can define K_CK_A, then we can consider the map id ⊗Φ : K_CK_A → K_CK_B and require it to be positive. In the applications of general probabilistic theory to classical and quantum theories, we always know how to create joint systems of given two systems so in the examples we will always require complete positivity of channels, but one still has to bear in mind that in the general case, complete positivity is not a well-defined concept.One can identify the channel Φ: A(K_A)^* → A(K_B)^* with an element of A(K_A) ⊗ A(K_B)^* as follows: let x ∈ K_A and f ∈ A(K_B), then the expression Φ(x), f gives rise to a linear functional on A(K_A)^* ⊗ A(K_B). This means that we have Φ∈ A(K_A) ⊗ A(K_B)^*, where we omit the isomorphism between the channel and the functional. If we also consider the positivity of the channel on the elements of the form x ⊗ f ∈ K_AE(K_B) we getΦ∈ A(K_A)^+A(K_B)^*+.This is a well known construction that may be also used to define the tensor product of linear spaces <cit.>.There is one more construction with channels that will be important in our formulation of compatibility of channels: compositions with effect. Let Φ: K_A → K_B be a channel and let f ∈ E(K_B), then they give rise to an effect (f ∘Φ) ∈ E(K_A) defined for x_A ∈ K_A asx_A, (f ∘Φ)= Φ(x_A), f .By the same idea we can define a map f ⊗ id: A(K_B)^* ⊗ A(K_C)^* → A(K_C)^* such that for x_B ∈ K_B and x_C ∈ K_C we have (f ⊗ id)(x_B ⊗ x_C) = f(x_B)x_C and we extend the map by linearity. Also given a channel Φ: K_A → K_BK_C we can compose the map f ⊗ id with the channel Φ to obtain (f ⊗ id) ∘Φ': A(K_A)^* → A(K_C)^* such that the corresponding functional on A(K_A) ⊗ A(K_C)^* is for x_A ∈ K_A and g ∈ A(K_C) given as(f ⊗ id) ∘Φ, x_A ⊗ g= Φ(x_A), f ⊗ g .Specifically we will be interested in the expressions (1 ⊗ id) ∘Φ and (id ⊗ 1) ∘Φ. If Φ is a channel then (1 ⊗ id) ∘Φ and (id ⊗ 1) ∘Φ are channels as well and they are called marginal channels of Φ.A special type of channel is a measurement.A channel m: K_A → K_B is called a measurement if K_B is a simplex.The interpretation is simple: the vertices of the simplex correspond to the possible measurement outcomes and the resulting state is a probability distribution over the measurement outcomes, i.e. an assignment of probabilities to the possible outcomes. Since we require all state spaces to be finite-dimensional this implies that we consider only finite-outcome measurements. Let K_B be a simplex with vertices ω_1, …, ω_n, then we can identify a measurement m with an element of A(K_A)^+A(K_B)^*+ of the formm = ∑_i=1^n f_i ⊗δ_ω_iwhere for i ∈{1, …, n} we have f_i ∈ E(K_A), ∑_i=1^n f_i = 1 and δ_ω_i∈ (K_B) are the functionals corresponding to the extreme points of K_B (where we have not omitted the isomorphism this time). This expression has an operational interpretation that for x ∈ K_A the measurement m assigns the probability f_i(x) to the outcome ω_i. Quantum channels are completely positive, trace preserving maps Φ: _→_. The complete positivity means that for any ρ≥ 0 we have (id ⊗Φ)(ρ) ≥ 0. We denote the set of channels Φ: _→_ as _→.Let |1, …, |n be an orthonormal base of . To every quantum channel we may assign its unique Choi matrix C(Φ) defined asC(Φ) = (Φ⊗ id) ( ∑_i, j=1^n |iijj| ) ,where we use the shorthand |i i= |i⊗ |i. Note that C(Φ) ≥ 0 and _1 (C(Φ)) =, where _1 denotes the partial trace. Also every matric C ∈ B_h(⊗) such that C ≥ 0 and _1(C) = is a Choi matrix of some channel, see <cit.>.The Choi matrix C(Φ) is isomorphic to a state 1/ C(Φ), which corresponds to the channel Φ⊗ id acting on the maximally entangled state |ψ^+ψ^+|, where|ψ^+ = 1√()∑_i=1^n |i i .§ COMPATIBILITY OF CHANNELSLet K_A, K_B_1, K_B_2 be state spaces and let Φ_1, Φ_2 be channelsΦ_1: K_A → K_B_1, Φ_2: K_A → K_B_2.We say that Φ_1, Φ_2 are compatible if and only if there exists a channelΦ : K_A → K_B_1 K_B_2such that Φ_1 and Φ_2 are the marginal channels of Φ, i.e. we haveΦ_1 = (id ⊗ 1) ∘Φ,Φ_2 = (1 ⊗ id) ∘Φ.The channel Φ is also called the joint channel of the channels Φ_1, Φ_2. The operational meaning of compatibility of channels is that if the channels Φ_1, Φ_2 are compatible, then we can apply them both to the input state at once and selecting which one we actually want the output from later. If the channels are incompatible we have to choose from which one we want the output before applying anything. For a more in-depth explanation see <cit.>. The important thing is that there is a choice from which channel we want to get the output so we can expect to see A(K_B_1× K_B_2)^*+ come up in the calculations.Consider the channel Φ : K_A → K_B_1 K_B_2. One can realize that the maps (id ⊗ 1): Φ↦ (id ⊗ 1) ∘Φ and (1 ⊗ id): Φ↦ (1 ⊗ id) ∘Φ are linear maps of channels. Moreover the Eq. (<ref>), (<ref>) both have Φ on the right hand side in the same position. We are going to exploit this to obtain simpler condition for compatibility of the channels Φ_1, Φ_2. To do so we have to introduce a new map J.Let us define a map J: A(K_A) ⊗ A(K_B_1)^* ⊗ A(K_B_2)^* → A(K_A) ⊗ A(K_B_1× K_B_2)^* given for Ξ∈ A(K_A) ⊗ A(K_B_1)^* ⊗ A(K_B_2)^* asJ(Ξ) = ( (id ⊗ 1) ∘Ξ, (1 ⊗ id) ∘Ξ).For Ξ = f ⊗ψ⊗φ we haveJ(Ξ) = f ⊗ ( φ, 1 ψ, ψ, 1 φ ). J is a linear mapping. Let Ξ_1, Ξ_2 ∈ A(K_A) ⊗ A(K_B_1⊗ K_B_2)^* and λ∈ℝ, then we haveJ(λΞ_1 + Ξ_2)= ( λ (id ⊗ 1) ∘Ξ_1 + (id ⊗ 1) ∘Ξ_2, 0 ) + ( 0, λ (1 ⊗ id) ∘Ξ_1 + (1 ⊗ id) ∘Ξ_2 ) = λ( (id ⊗ 1) ∘Ξ_1, (1 ⊗ id) ∘Ξ_1 ) + ( (id ⊗ 1) ∘Ξ_2, (1 ⊗ id) ∘Ξ_2 ) = λ J(Ξ_1) + J(Ξ_2).Assume that the channels Φ_1, Φ_2 are compatible and that Φ is their joint channel then we must haveJ(Φ) = (Φ_1, Φ_2)which is just a more compact form of the Eq. (<ref>), (<ref>). The channels Φ_1, Φ_2 are compatible if and only if there is Φ∈ A(K_A)^+A(K_B_1 K_B_2)^*+ such thatJ(Φ) = (Φ_1, Φ_2).If the channels Φ_1, Φ_2 are compatible then Eq. (<ref>) must hold for their joint channel Φ. If Eq. (<ref>) holds for some Φ∈ A(K_A)^+A(K_B_1 K_B_2)^*+, then the channels Φ_1, Φ_2 are compatible and Φ is their joint channel. The operational interpretation is that (Φ_1, Φ_2) represents a conditional channel in the same way as the states from A(K_B_1× K_B_2)^*+ represent conditional states that keep track of some choice made in the past. If the channels are compatible, then we actually do not have to make the choice of either using Φ_1 or Φ_2, but we can use their joint channel, that has the property that its marginals reproduce the outcomes of the two channels Φ_1, Φ_2. We will investigate several of the properties of the map J. For every (ξ_1, ξ_2) ∈ A(K_A) ⊗ A(K_B_1× K_B_2)^* there is a Ξ∈ A(K_A) ⊗ A(K_B_1)^* ⊗ A(K_B_2)^* such thatJ(Ξ) = (ξ_1, ξ_2).Moreover if we have(1, 1) ∘ (ξ_1, ξ_2) = 1then(1 ⊗ 1) ∘Ξ = 1.Let f_1, …, f_n be a basis of A(K_A), then we haveξ_1= ∑_i=1^n f_i ⊗ψ_iξ_2= ∑_i=1^n f_i ⊗φ_ifor some ψ_i ∈ A(K_B_1)^* and φ_i ∈ A(K_B_2)^*. Since we must have(1,0) ∘ (ξ_1, ξ_2) = (0,1) ∘ (ξ_1, ξ_2)we obtain∑_i=1^n ψ_i, 1 f_i = ∑_i=1^n φ_i, 1 f_iwhich impliesψ_i, 1= φ_i, 1= k_ifor all i ∈{1, …, n } as f_1, …, f_n is linearly independent. LetΞ = ∑_i=1^n k_i^-1 f_i ⊗ψ_i ⊗φ_ithen we haveJ(Ξ)= ∑_i=1^n k_i^-1 f_i ⊗ ( φ_i, 1_B_2ψ_i, ψ_i, 1_B_1φ_i ) = ∑_i=1^n f_i ⊗ (ψ_i, φ_i ).If we have 1 ∘ (ξ_1, ξ_2) = 1 then∑_i=1^n k_i f_i = 1and we get(1 ⊗ 1) ∘Ξ = (1 ⊗ 1) ∘ ( ∑_i=1^n k_i^-1 f_i ⊗ψ_i ⊗φ_i ) = ∑_i=1^n k_i^-1ψ_i, 1 φ_i, 1f_i = 1.We haveJ(A(K_A)^+A(K_B_1)^*+ A(K_B_2)^*+) = = A(K_A)^+A(K_1 × K_B_2)^*+.Let (ξ_1, ξ_2) ∈ A(K_A)^+A(K_B_1× K_B_2)^*+ then as in the proof of Prop. <ref> we haveξ_1= ∑_i=1^n f_i ⊗ψ_iξ_2= ∑_i=1^n f_i ⊗φ_ibut now we have f_i ≥ 0, ψ_i ≥ 0 and φ_i ≥ 0 for i ∈{ 1, …, n }. It follows by the same construction as in the proof of Prop. <ref> thatwe can construct Ξ = ∑_i=1^n k_i^-1 f_i ⊗ψ_i ⊗φ_i and we get Ξ∈ A(K_A)^+A(K_B_1)^*+ A(K_B_2)^*+.Let Ξ∈ A(K_A)^+A(K_B_1)^*+ A(K_B_2)^*+, then we have Ξ = ∑_i=1^n f_i ⊗ψ_i ⊗φ_i such that f_i ≥ 0, ψ_i ≥ 0, φ_i ≥ 0 for all i ∈{1, …, n }, moreover without lack of generality we can assume ψ_i, 1_B_1 = φ_i, 1_B_2 = 1. We haveJ(Ξ) = ∑_i=1^n f_i ⊗ (ψ_i, φ_i) ∈ A(K_A)^+A(K_B_1× K_B_2)^*+which concludes the proof. It would be very useful to know what is the image of the cone A(K_A)^+A(K_B_1 K_B_2)^*+ when mapped by J. We will denote the resulting cone Q = J(A(K_A)^+A(K_B_1 K_B_2)^*+). The cone is important due to the following: The channels Φ_1, Φ_2 are compatible if and only if(Φ_1, Φ_2) ∈ Q = J(A(K_A)^+A(K_B_1 K_B_2)^*+).Follows from Prop. <ref>.A(K_A)^+A(K_B_1× K_B_2)^*+⊂ Q. SinceA(K_A)^+A(K_B_1 K_B_2)^*+⊂ A(K_A)^+A(K_B_1 K_B_2)^*+we must haveJ(A(K_A)^+A(K_B_1 K_B_2)^*+) ⊂ Q.The result follows from Prop. <ref>.Q ⊂ A(K_A)^+A(K_B_1× K_B_2)^*+. Since we haveA(K_A)^+A(K_B_1 K_B_2)^*+⊂ A(K_A)^+A(K_B_1 K_B_2)^*+we must haveQ ⊂ J(A(K_A)^+A(K_B_1 K_B_2)^*+).Let Ξ∈ A(K_A)^+A(K_B_1 K_B_2)^*+, then for ψ∈ A(K_A)^*+ and (f_1, f_2) ∈ A(K_B_1× K_B_2)^+ we getJ(Ξ), x ⊗ f= ( (id ⊗ 1) ∘Ξ, (1 ⊗ id) ∘Ξ), x ⊗ f = Ξ(x), f_1 ⊗ 1+ Ξ(x), 1 ⊗ f_2 ≥ 0,that shows we have J(A(K_A)^+A(K_B_1 K_B_2)^*+) ⊂ A(K_A)^+A(K_B_1× K_B_2)^*+ which concludes the proof. We can also construct Q as the cone we get when we factorize the cone A(K_A)^+ A(K_B_1 K_B_2)^*+ with respect to the relation of equivalence given as follows: Ξ_1 ≈Ξ_2 if and only if J(Ξ_1) = J(Ξ_2), or equivalently if and only if Ξ_1 = Ξ_2 + Ξ, such that J(Ξ) = 0.Note that since J is a linear map, as we showed in Prop. <ref>, it is clear that Q is a convex cone. For two given channels Φ_1: K_A → K_B_1, Φ_2: K_A → K_B_1 one may write a primal linear program that would check the condition for compatibility given by Cor. <ref>. We will write such linear program for quantum channels later.§ COMPATIBILITY OF MEASUREMENTS We will apply the results of Sec. <ref> to the problem of compatibility of measurements. We will obtain the same results that were recently presented in <cit.>, that are generalization a of <cit.>.Let K_A be a state space and let S_1, S_2 be simplexes and let m_1: K_A → S_1, m_2: K_A → S_2be measurements. According to Prop. <ref> the measurements m_1, m_2 are compatible if and only if(m_1, m_2) ∈ J(A(K_A)^+A(S_1S_2)^*+).Since both S_1 and S_2 are simplexes, then we have S_1S_2 = S_1S_2 and the condition for compatibility reduces according to Prop. <ref> to(m_1, m_2) ∈ A(K_A)^+A(S_1 × S_2)^*+. Due to the simpler structure of simplexes one may get even more specific results about measurements, see <cit.>.For demonstration of the derived conditions we will reconstruct the result of <cit.> about compatibility of two-outcome measurements. According to our definition, a measurement is two-outcome if the simplex it has as a target space has two vertexes, i.e. it is a line segment. Let K be a state space, f, g ∈ E(K) and m_f: K → S, m_g: K → S be two-outcome measurements given asm_f= f ⊗δ_ω_1 + (1-f) ⊗δ_ω_2, m_g= g ⊗δ_ω_1 + (1-g) ⊗δ_ω_2.The state space given by A(S × S)^*+ is a square given as ( (δ_ω_1, δ_ω_1), (δ_ω_1, δ_ω_2), (δ_ω_2, δ_ω_1), (δ_ω_2, δ_ω_2) ), that is just affinely isomorphic to S × S. We have(m_1, m_2)= f ⊗ (δ_ω_1, 0 ) + (1-f) ⊗ (δ_ω_2, 0 ) + g ⊗ (0, δ_ω_1) + (1-g) ⊗ (0, δ_ω_2) = f ⊗ (δ_ω_1, δ_ω_2) + (1-f) ⊗ (δ_ω_2, δ_ω_2) + g ⊗ (0, δ_ω_1 - δ_ω_2),where in the second step we have used the basis (δ_ω_1, δ_ω_2), (δ_ω_2, δ_ω_2), (0, δ_ω_1 - δ_ω_2) of A(S× S)^* to express (m_1, m_2) in a more reasonable form. To have (m_1, m_2) ∈ A(K)^+A(S × S)^*+ we must have(m_1, m_2)= h_11⊗ (δ_ω_1, δ_ω_1) + h_12⊗ (δ_ω_1, δ_ω_2) + h_21⊗ (δ_ω_2, δ_ω_1) + h_22⊗ (δ_ω_2, δ_ω_2) = ( h_11 + h_12 ) ⊗ (δ_ω_1, δ_ω_2) + ( h_21 + h_22 ) ⊗ (δ_ω_2, δ_ω_2) + ( h_11 + h_21 ) ⊗ (0, δ_ω_1 - δ_ω_2),for some h_11, h_12, h_21, h_22∈ E(K). This implies the standard conditions for the compatibility of two-outcome measurements m_f, m_g:f= h_11 + h_12, 1-f= h_21 + h_22, g= h_11 + h_21,see e.g. <cit.>.§ COMPATIBILITY OF QUANTUM CHANNELS In this section we will derive results more specific to the compatibility of quantum channels. Let Φ_1: _→_, Φ_2: _→_ be quantum channels, then according to Prop. <ref> they are compatible if and only if there is a channel Φ: _→_⊗ such that for all ρ∈_ we have(Φ_1 (ρ), Φ_2(ρ) ) = ( _2( Φ (ρ) ), _1 ( Φ ( ρ ) ) ).This is equivalent to the definition of compatibility of quantum channels already stated in <cit.>. It is straightforward that Eq. (<ref>) implies that( C(Φ_1), C(Φ_2) ) = ( _2( C(Φ) ), _1 ( C(Φ) ) ),we will show that they are equivalent. This will help us to get rid of the state ρ in Eq. (<ref>).The channels Φ_1: _→_, Φ_2: _→_ are compatible if and only if there exists a channel Φ: _→_⊗ such that( C(Φ_1), C(Φ_2) ) = ( _2( C(Φ) ), _1 ( C(Φ) ) ).Let ρ∈_, then we have_2( Φ (ρ) )= _2, E ( C(Φ) ⊗⊗ρ^T ) = _E ( _2 ( C(Φ) ) ⊗ρ^T ) = _E ( C(Φ_1) ⊗ρ^T ) = Φ_1 (ρ).The same follows for Φ_2. As we already showed in Sec. <ref>, the cone Q = J( A(_)^+A( _⊗)^*+) is of interest for the compatibility of channels. In the case of quantum channels we will use Prop. <ref> to formulate similar cone in terms of Choi matrices of the channels and we will write a semi-definite program for the compatibility of quantum channels based on this approach.Denote P = { (_2 (C), _1 (C)) : C ∈_→⊗}, then according to Prop. <ref> the channels Φ_1: _→_, Φ_2: _→_ are compatible if and only if( C(Φ_1), C(Φ_2) ) ∈ P.Note that, by our definition, P is not a cone, but it generates some cone just by adding all of the operators of the form λ C, where C ∈ P and λ∈ℝ, λ≥ 0.It would be very interesting to obtain more specific results on the structure of P, but the task is not trivial. To make it simpler we will investigate the structure of the dual cone P^* given asP^* = {(A, B) ∈ B_h() × B_h():C, (A, B) ≥ 0,∀ C ∈ P }.Notice that (A, B) ∈ B_h() × B_h() is simply a block-diagonal matrix having blocks A and B. Also every C ∈ P is a block diagonal matrix, let C = (C_1, C_2), then(C_1, C_2), (A, B)= (C_1 A) + (C_2 B).Let C ∈ P, then by definition there exist a channel Φ: _→_⊗ such thatC = ( _2 (C(Φ)), _1 ( C(Φ)) ).Let (A, B) ∈ P^*, then we haveC, (A, B)= ( _2 (C(Φ)) A + _1 ( C(Φ)) B ) =( C(Φ) (à + ⊗ B ) ) ≥ 0,where à is the operator such that ( _2 (C(Φ)) A ) =( C(Φ) à ). If A = A_1 ⊗ A_2, then à = A_1 ⊗⊗ A_2. In general one can write A as a sum of factorized operators and express à in such way, because the map A ↦à is linear.The result is that à + ⊗ B must correspond to a positive function on quantum channels, hence we must have à + ⊗ B ≥ 0, see <cit.>. We have proved the following: The channels Φ_1: _→_, Φ_2: _→_ are compatible if and only if( C(Φ_1) A ) + ( C(Φ_2) B ) ≥ 0for all A, B ∈ B_h(⊗) such thatà + ⊗ B ≥ 0.This allows us to formulate the semi-definite program <cit.> for the compatibility of quantum channels as follows:Given channels Φ_1: _→_, Φ_2: _→_, the semi-definite program for the compatibility of quantum channels isinf_A, B( C(Φ_1) A ) + ( C(Φ_2) B )à + ⊗ B ≥ 0,where à is given as above.If the reached infimum is negative, then the channels are incompatible, if the reached infimum is 0 then the channels are compatible. The result follows from Prop. <ref>. One may see that the infimum is at most 0 because one may always chose A = B = 0. § PRELUDE TO STEERING AND BELL NONLOCALITY We will propose a possible test for the compatibility of channels that will not work, but it will motivate our definitions of steering and Bell nonlocality.Let K_A, K_B_1, K_B_2 be state spaces and let Φ_1: K_A → K_B_1, Φ_2: K_A → K_B_2 be channels. The channels Φ_1, Φ_2 are compatible if Eq. (<ref>) is satisfied for some channel Φ: K_A → K_B_1 K_B_2. This is the same as saying the channels Φ_1, Φ_2 are compatible if for all x ∈ K_A we have(Φ_1(x), Φ_2(x)) = ( ((id ⊗ 1) ∘Φ)(x), ((1 ⊗ id) ∘Φ)(x) ).If the channels Φ_1 and Φ_2 are compatible, then for every x ∈ K_A there must exist a state y ∈ K_B_1 K_B_2 such thatΦ_1(x)= (id ⊗ 1)(y),Φ_2(x)= (1 ⊗ id)(y).Would it be a reasonable test for the compatibility of the channels Φ_1 and Φ_2 if we considered the state x ∈ K_A fixed and we would test whether, for the fixed state x, there exists y ∈ K_B_1 K_B_2 such that Eg. (<ref>), (<ref>) are satisfied? It would not, because for a fixed x ∈ K_A one always has Φ_1 (x) ⊗Φ_2 (x) ∈ K_B_1 K_B_2 that satisfies Eg. (<ref>), (<ref>).Still, throwing away this line of thinking would not be a good choice, because going further, on may ask: if there would be another system K_C, such that K_CK_A is defined, then what if we would use the entanglement between the systems K_A and K_C to obtain a better condition for the compatibility of the channels Φ_1, Φ_2 using the very same line of thinking? As we will see, this approach leads to the notions of steering and Bell nonlocality.§ STEERING Steering is one of the puzzling phenomena we find in quantum theory but not in classical theory. It is usually described as a two party protocol, that allows one side to alter the state of the other in a way that would not be possible in classical theory by performing a measurement and announcing the outcome. Although originally discovered by Schrödinger <cit.>, steering was formalised in <cit.>. Recently there was introduced a new formalism for steering in <cit.>.So far it was always only considered that during steering one party performs a measurement. Since a measurement is a special case of a channel, one may ask whether it is possible to define steering by channels. We will use our formalism for compatibility of channels to introduce steering by channels by continuing the line of thoughts presented in Sec. <ref>. We will have to formulate steering in a little different way than it usually is formulated for measurements, but we will show that for measurements we will obtain the known results.Let K_A, K_B_1, K_B_2, K_C be finite-dimensional state spaces, such that K_CK_A is defined and letΦ_1 : K_A → K_B_1,Φ_2 : K_A → K_B_2,be channels. We can construct channelsid ⊗Φ_1: A(K_C)^*+ A(K_A)^*+→ A(K_C)^*+ A(K_B_1)^*+, id ⊗Φ_2: A(K_C)^*+ A(K_A)^*+→ A(K_C)^*+ A(K_B_2)^*+.Moreover we can construct the conditional channelid ⊗ (Φ_1, Φ_2) : A(K_C)^*+ A(K_A)^*+→→ A(K_C)^*+ A(K_B_1× K_B_2)^*+.These channels play a central role in steering and we will keep this notation throughout this section. First, we will introduce a handy name for the output state of id ⊗ (Φ_1, Φ_2). Let ψ∈ K_CK_A be a bipartite state, then we call (id ⊗ (Φ_1, Φ_2))(ψ) a bipartite conditional state. Steering may be seen as a three party protocol that tests the compatibility of channels. The parties in question will be named Alice, Bob and Charlie. Alice and Charlie share a bipartite state ψ∈ K_CK_A and Alice has the channels Φ_1 and Φ_2 at her disposal, that would send her part of the state ψ to Bob. Since Alice can choose between the channels Φ_1 and Φ_2, she will be, in our formalism, applying the conditional channel (Φ_1, Φ_2) and the resulting state will be a bipartite state from A(K_C)^*+ A(K_B_1× K_B_2)^*+. The structure of the resulting bipartite conditional state (id ⊗ (Φ_1, Φ_2) )(ψ) will not only depend on the input state ψ, but also on the compatibility of the channels Φ_1 and Φ_2. Let us assume that the channels Φ_1 and Φ_2 are compatible, then there is a channel Φ: K_A → K_B_1 K_B_2 such that (Φ_1, Φ_2) = J(Φ) and we have( id ⊗ (Φ_1, Φ_2) ) (ψ)= (id ⊗ J(Φ) ) (ψ) = (id ⊗ J') ( (id ⊗Φ) (ψ) )where J': A(K_B_1 K_B_2)^*→ A(K_B_1× K_B_2)^*, J'(ψ) = ( (id ⊗ 1)(ψ), (1 ⊗ id)(ψ)). The calculation shows that if the channels Φ_1, Φ_2 are compatible, then we must have( id ⊗ (Φ_1, Φ_2) )(ψ) ∈ (id ⊗ J') ( K_CK_B_1 K_B_2 )which does not have to hold in general if the channels are not compatible. This shows that we can define steering of a state by channels as an entanglement assisted incompatibility test.The bipartite state ψ∈ A(K_C)^*+ A(K_A)^*+ is steerable by channels Φ_1: A(K_A)^*+→ A(K_B_1)^*+, Φ_2: A(K_A)^*+→ A(K_B_2)^*+ if(id ⊗ (Φ_1, Φ_2))(ψ) ∉ (id ⊗ J') ( K_CK_B_1 K_B_2 )Now we present the standard result about the connection between compatibility of the channels and steering. The result follows from our definition immediately. The bipartite state ψ∈ A(K_C)^*+ A(K_A)^*+ is not steerable by channels Φ_1: A(K_A)^*+→ A(K_B)^*+, Φ_2: A(K_A)^*+→ A(K_B)^*+ if the channels Φ_1 and Φ_2 are compatible. If the channels Φ_1, Φ_2 are compatible, then we have (Φ_1, Φ_2) = J(Φ) for some Φ: K_A → K_B_1 K_B_2 and for every ψ∈ K_CK_A we have(id ⊗ (Φ_1, Φ_2))(ψ) ∈ (id ⊗ J') ( K_C(K_B_1 K_B_2) ). The bipartite state ψ∈ A(K_C)^*+ A(K_A)^*+ is not steerable by channels Φ_1: A(K_A)^*+→ A(K_B)^*+, Φ_2: A(K_A)^*+→ A(K_B)^*+ if ψ∈ A(K_C)^*+ A(K_A)^*+, i.e. if ψ is separable. Every separable state is by definition a convex combination of product states, i.e. of states of the form x_C ⊗ x_A, where x_A ∈ K_A, x_C ∈ K_C. Since the maps id ⊗ (Φ_1, Φ_2) and id ⊗ J' are linear it is sufficient to prove that for every product state x_C ⊗ x_A ∈ K_CK_A we have (id ⊗ (Φ_1, Φ_2))(x_C ⊗ x_A) ∈ (id ⊗ J')(K_CK_B_1 K_B_2). It follows by our construction in Sec. <ref> that product states are not steerable by any channels as one can always take x_C ⊗Φ_1(x_A) ⊗Φ_2(x_A). Remember that during steering, we fix not only the channels, but also the bipartite state, so the presented construction is valid. § STEERING BY MEASUREMENTS We will show that the definition of steering given by Def. <ref> follows the standard definition of steering <cit.> in the formalism introduced in <cit.>, when we replace measurements by channels. Let S_1, S_2 be simplexes and let m_1: K_A → S_1, m_2: K_A → S_2 be measurements, then a state ψ∈ K_CK_A is steerable by m_1, m_2 if and only if(id ⊗ (m_1, m_2))(ψ) ∉ K_C(S_1 × S_2 ).The result follows from the fact that K_CS_1S_2 = K_CS_1S_2. To obtain the standard definition of steering, one only needs to note that if ξ∈ K_C(S_1 × S_2 ), then there are x_i ∈ K_C,s_i ∈ S_1 × S_2 and 0 ≤λ_i ≤ 1 for i ∈{1, …, n} such thatξ = ∑_i=1^n λ_i x_i ⊗ s_i,where the interpretation of s_i is that it is a conditional probability, conditioned by the choice of the measurement. At this point it is straightforward to see that Eq. (<ref>) corresponds to <cit.>.§ STEERING BY QUANTUM CHANNELS Steering plays an important role in quantum theory. It has found so far applications in quantum cryptography <cit.> as an intermediate step between quantum key distribution and device-independent quantum key distribution.We will prove several results and present a simple example of steering by quantum channels. Given the standard, operational, interpretation of steering by measurements the example may seem strange, but rather expected.Let Φ_1: _→_, Φ_2: _→_ be channels and let |ψ^+ = ()^-1/2∑_i=1^ |i i be the maximally entangled vector. We will show that the maximally entangled state |ψ^+ψ^+| is steerable by the channels Φ_1, Φ_2 whenever they are incompatible.The proof is rather simple as the bipartite conditional state we obtain is (id ⊗ (Φ_1, Φ_2) )(|ψ^+ ψ^+| ). If the channels Φ_1, Φ_2 are compatible then the state |ψ^+ψ^+| is not steerable by compatible channels. Now let us assume that there is a state in ρ∈_⊗⊗ such that we have(id ⊗ (Φ_1, Φ_2)) (|ψ^+ ψ^+|) = (id ⊗ J')(ρ),i.e. that the state state |ψ^+ψ^+| is not steerable by the channels Φ_1, Φ_2. Eq. (<ref>) implies that we must have(id ⊗Φ_1) (|ψ^+ ψ^+|) = _3 ( ρ ),that, after taking trace over the second Hilbert space, gives1 = _23 ( ρ ).Now the picture becomes clear: (id ⊗Φ_1) (|ψ^+ ψ^+|) is isomorphic to the Choi matrix C(Φ_1) and Eq. (<ref>) implies that the state ρ must be isomorphic to a Choi matrix of some channel Φ. This together with Prop. <ref> means that Eq. (<ref>) holds if and only if the channels are compatible. Thus we have proved: The maximally entangled state |ψ^+ψ^+| is steerable by channels Φ_1: _→_, Φ_2: _→_ if and only if they are incompatible. We will investigate steering by unitary channels. We will see a phenomenon that is impossible to happen for steering by measurements - it is possible to steer a state when the two channels we are testing for incompatibility are two copies of the same channel. Let U, V be unitary matrices, i.e. U U^* = V V^* =, where U^* denotes the conjugate transpose matrix to U and let Φ_U, Φ_V be the corresponding unitary channels, that is for ρ∈_ we haveΦ_U ( ρ )= U ρ U^*,Φ_V ( ρ )= V ρ V^*.Note that we have Φ_ = id, i.e. the unitary channels given by an identity matrix is the identity channel. The bipartite state ρ∈_⊗ is steerable by the unitary channels Φ_U, Φ_V if and only if it is steerable by two copies of the identity channel id. The state ρ∈_⊗ is steerable by the channels Φ_U, Φ_V if and only if there is a state σ∈_⊗⊗ such that_3 ( σ )= (id ⊗Φ_U)(ρ),_2 ( σ )= (id ⊗Φ_V)(ρ).If such state σ exists, then for σ̃ = (id ⊗Φ_U^*⊗Φ_V^*)(σ) we have_3 ( σ̃ )= ρ,_2 ( σ̃ )= ρ,i.e. the state ρ is not steerable by two copies of id.The same holds other way around by almost the same construction; if the state ρ is not steerable by two copies of id then it is not steerable by any unitary channels Φ_U, Φ_V.Note that similar result would hold if only one of the channels would be unitary, but then only that one unitary channel would be replaced by the identity map id.Clearly if the state ρ would be separable, then it would not be steerable by any channel. The converse does not hold, even if the state ρ is entangled it still may not be steerable by any channels. We will provide a useful condition for the steerability of a given state ρ∈_⊗ that will help us to show that even if the state ρ is entangled, it does not have to be steerable by any pair of channels Φ_1: _→_, Φ_2: _→_.The state ρ∈_⊗ is steerable by the channels Φ_1: _→_, Φ_2: _→_ only if it is steerable by two copies of the identity channel id: _→_. Assume that the state ρ∈_⊗ is not steerable by two copies of the identity channel id: _→_, then there exists a state σ∈_⊗⊗ such that_3 ( σ )= ρ,_2 ( σ )= ρ.Let Φ_1: _→_, Φ_2: _→_ be any two channels and denoteσ̃ = (id ⊗Φ_1 ⊗Φ_2) (σ),then we have_3 ( σ̃ )= (id ⊗Φ_1) (ρ),_2 ( σ̃ )= (id ⊗Φ_2) (ρ),so the state ρ is not steerable by the channels Φ_1, Φ_2.Note that one may get other conditions for steering by replacing only one of the channels by the identity map id.One may generalize this result to the general probabilistic theory but it may be rather restrictive and not as general as one would wish. One may also use the idea of the proof of Prop. <ref> together with the result of Prop. <ref> to obtain the results on compatibility of channels that are concatenations of other channels, similar to the results obtained in <cit.>.We will present an example of an entangled state that is not steerable by any pair of channels. Let = 2 with the standard basis |0, |1 and let |W∈⊗⊗ be given as|W = 1√(3) ( |001 + |010 + |100).The projector |WW| ∈_⊗⊗ is known as W state. We haveρ_W = _2 ( |WW| ) = _3 ( |WW| ) ∈_⊗,that shows that the state ρ_W is not steerable by a pair of the identity channels id: _→_, which as a result of Prop. <ref> means that it is not steerable by any channels Φ_1: _→_, Φ_2: _→_. Moreover it is known that the state ρ_W is entangled <cit.>.Since it will be useful in later calculations we will show that the state |WW| is the only state from _⊗⊗ such that ρ_W = _2 ( |WW| ) = _3 ( |WW| ). Let |φ = 1/√(2) ( |01 + |10), then we haveρ_W = 13 |0000| + 23 |φφ|.Let σ∈_⊗⊗ denote the state such that ρ_W = _2 ( σ ) = _3 ( σ ). We have ρ_W |11 = 0 that implies (σ |1111| ⊗ ) = (σ |11| ⊗⊗ |11| ) = 0 that implies 111|σ|111 = 110| σ|110 = 101| σ|101 = 0 as σ≥ 0. We will show that this implies σ|111 = σ|110 = σ|101 = 0.Let A ∈ B_h(), A ≥ 0 and let |ψ∈. Let ‖ψ‖ = √(ψ| ψ) denote the norm given by inner product. Assume that we have ψ| A |ψ = 0, then‖√(A)ψ‖^2 = √(A)ψ | √(A)ψ = ψ| A |ψ = 0and in conclusion we have √(A)|ψ = 0 and A |ψ = √(A) ( √(A) |ψ) = 0. Finally let us denote |φ^⊥ = 1/√(2)( |01 - |10 ). We have ρ_W |φ^⊥ = 0 that implies ( σ |φ^⊥φ^⊥| ⊗) = 0 which yields σ |φ^⊥ 0= σ |φ^⊥ 1= 0. We still use the shorthand |φ^⊥ 0= |φ^⊥⊗ |0.The eight vectors |000, |001, |φ 0, |φ 1, |φ^⊥ 0, |φ^⊥ 1, |110, |111 form an orthonormal basis of ⊗⊗. We have already showed that we must haveσ |φ^⊥ 0= σ |φ^⊥ 1= σ |110 = σ | 111= 0so in general we must haveσ = a_00 |000000| + a_01 |001001| + a_φ0 |φ 0φ 0| + a_φ1 |φ1φ1| + b_1 |000001| + b̅_1 |001000| + b_2 |000φ 0| + b̅_2 |φ 0000| + b_3 |000φ 1| + b̅_3 | φ 1 000| + b_4 |001φ 0| + b̅_4 |φ 0001| + b_5 |001φ 1| + b̅_5 | φ 1001| + b_6 |φ 0 φ 1| + b̅_6 |φ 1φ 0|. Using the above expression for σ we get_2 (σ)= a_00 |0000| + a_01 |0101| + a_φ 02⊗ |00| + a_φ 12⊗ |1 1| + b_1 |0001| + b̅_1 |0100| + b_2√(2) |0010| + b̅2√(2) |1000| + b_3√(2) |0011| + b̅_3√(2) |1100| + b_4√(2) |0110| + b̅_4√(2) |1001| + b_5√(2) |0111| + b̅_5√(2) |1101| + b_62⊗ |01| + b̅_62⊗ |10|,that implies a_0 0 = a_φ 1 = 0, a_φ 0 = 2/3, a_01 = 1/3, b_1 = b_2 = b_3 = b_5 = b_6 = 0 and b_4 = √(2)/3. In conclusion we haveσ = 13 ( |001001| + 2 |φ 0φ 0| + √(2) |001φ 0| + √(2) |φ 0 001| ) =|WW|.§ BELL NONLOCALITY Bell nonlocality is, similarly to steering, a phenomenom that we do not find in classical theory, but is often used in quantum theory. Bell nonlocality <cit.> was formulated as a response to the well-known EPR paradox <cit.>. Although in the original formulation, the operational idea was different than the one we will present, we will see that Bell nonlocality may be understood as an incompatibility test, in the same way as steering.Let us assume that we have four parties: Alice, Bob, Charlie and Dan. Alice has two channels Φ_1^A: K_A → K_B_1 and Φ_2^A: K_A → K_B_2 that she can use to send a state to Bob and Charlie has two channels Φ_1^C: K_C → K_D_1 and Φ_2^C: K_C → K_D_2 that he can use to send a state to Dan. Assume that K_CK_A is defined and let ψ∈ K_CK_A be a bipartite state shared by Alice and Charlie. The idea that we use to define Bell nonlocality is very simple: if we were able to use (id ⊗ (Φ_1^A, Φ_2^A))(ψ) and ((Φ_1^C, Φ_2^C) ⊗ id)(ψ) as non-trivial incompatibility test, we may as well investigate whether ((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) provides an incompatibility test in the same manner. Let ψ∈ K_CK_A and letΦ_1^A: K_A → K_B_1,Φ_2^A: K_A → K_B_2,Φ_1^C: K_C → K_D_1,Φ_2^C: K_C → K_D_2be channels. We call the state ((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) bipartite biconditional state. Assume that the channels Φ_1^A and Φ_2^A are compatible, so that we have (Φ_1^A, Φ_2^A) = J(Φ^A) for some channel Φ^A: K_A → K_B_1 K_B_2 and also that the channels Φ_1^C and Φ_2^C are compatible, so there is a channel Φ^C: K_C → K_D_1 K_D_2 such that (Φ_1^A, Φ_2^A) = J(Φ^A). Let ψ∈ K_CK_A, then we have((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) = (J' ⊗ J') ( (Φ_C ⊗Φ_A) (ψ) ),where the maps J' are defined as before, with the exception that we denote them the same even though they map different spaces.We present a definition of Bell nonlocality using the same line of thinking as we used in Def. <ref>. For simplicity we will denoteQ_DC = (J' ⊗ J') ( K_D_1 K_D_2 K_C_1 K_C_2 ).Let ψ∈ K_CK_A be a bipartite state and let Φ_1^A: K_A → K_B_1, Φ_2^A: K_A → K_B_2, Φ_1^C: K_A → K_C_1 and Φ_2^C: K_A → K_D_2 be channels. We say that the bipartite biconditional state ((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) is Bell nonlocal if((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) ∉ Q_DC.Otherwise we call the bipartite biconditional state Bell local. The following result follows immediatelly from Def. <ref>.Let ψ∈ K_CK_A be a bipartite state and let Φ_1^A: K_A → K_B_1, Φ_2^A: K_A → K_B_2, Φ_1^C: K_A → K_C_1 and Φ_2^C: K_A → K_D_2 be channels. The bipartite biconditional state ((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) is Bell nonlocal only if the channels Φ_1^A, Φ_2^A and Φ_1^C, Φ_2^C are incompatible. We will show that entanglement plays a key role in Bell nonlocality.Let ψ∈ K_CK_A be a separable bipartite state and let Φ_1^A: K_A → K_B_1, Φ_2^A: K_A → K_B_2, Φ_1^C: K_A → K_C_1 and Φ_2^C: K_A → K_D_2 be channels. The bipartite biconditional state ((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) is Bell local. It is again sufficient to consider ψ = x_C ⊗ x_A for x_A ∈ K_A, x_C ∈ K_C due to the linearity of the maps (Φ_1^A, Φ_2^A) and (Φ_1^C, Φ_2^C). Consider the state φ∈ K_D_1 K_D_2 K_C_1 K_C_2 given asφ = Φ_1^C(x_C) ⊗Φ_2^C(x_C) ⊗Φ_1^A(x_A) ⊗Φ_2^A(x_A),then we have((Φ_1^C, Φ_2^C) ⊗ (Φ_1^A, Φ_2^A))(ψ) = (J' ⊗ J') (φ).§ BELL NONLOCALITY OF MEASUREMENTS We will again show that the Def. <ref> follows the standard definition of Bell nonlocality <cit.> in the formalism of <cit.>. Let S_1^A, S_2^A, S_1^C and S_2^C, be simplexes and let m_1^A: K_A → S_1^A, m_2^A: K_A → S_2^A, m_1^C: K_C → S_1^C, m_2^C: K_C → S_2^C be measurements. Let ψ∈ K_CK_A, then the bipartite biconditional state ((m_1^C, m_2^C) ⊗ (m_1^A, m_2^A))(ψ) is Bell nonlocal if((m_1^C, m_2^C) ⊗ (m_1^A, m_2^A))(ψ) ∉ (S_1^C × S_2^C)(S_1^A × S_2^A).By direct calculation we haveQ_CD = (J' ⊗ J') ( S_1^CS_2^CS_1^AS_2^A ) = ( S_1^C × S_2^C)( S_1^A × S_2^A).One may again use the interpretation that both S_1^C × S_2^C and S_1^A × S_2^A are spaces of conditional measurement probabilities, so if we have ψ∈ ( S_1^C × S_2^C)( S_1^A × S_2^A) then we must have 0 ≤λ_i ≤ 1, for i ∈{1, …, n}, ∑_i=1^n λ_i = 1, such thatψ = ∑_i=1^n λ_i s_i^C ⊗ s_i^Awhere in standard formulations both s_i^C ∈ S_1^C × S_2^C and s_i^A ∈ _1^A × S_2^A are represented by probabilities, i.e. by numbers, so the tensor product between them is omitted.We will provide proof of the standard and well-known result about connection of steering and Bell nonlocality of measurements. Let S_1^A, S_2^A, S_1^C and S_2^C, be simplexes and let m_1^A: K_A → S_1^A, m_2^A: K_A → S_2^A, m_1^C: K_C → S_1^C, m_2^C: K_C → S_2^C be measurements. Let ψ∈ K_CK_A. If(id ⊗ (m_1^A, m_2^A))(ψ) ∈ K_C(S_1^A × S_2^A),i.e. if the bipartite state is not steerable by measurements m_1^A, m_2^A, then((m_1^C, m_2^C) ⊗ (m_1^A, m_2^A))(ψ) ∈ (S_1^C × S_2^C)(S_1^A × S_2^A).Let(id ⊗ (m_1^A, m_2^A))(ψ) ∈ K_C(S_1^A × S_2^A),then for n ∈ℕ, i ∈{1, …, n}, there are 0 ≤λ_i ≤ 1, x_i ∈ K_C and s_i ∈ S_1^A × S_2^A, ∑_i=1^n λ_i = 1, such that we have(id ⊗ (m_1^A, m_2^A))(ψ) = ∑_i=1^n λ_i x_i ⊗ s_i.We get((m_1^C, m_2^C) ⊗ (m_1^A, m_2^A))(ψ) = ∑_i=1^n λ_i (m_1^C, m_2^C)(x_i) ⊗ s_iand since we have (m_1^C, m_2^C)(x_i) = (m_1^C(x_i), m_2^C(x_i)) ∈ S_1^C × S_2^C we have((m_1^C, m_2^C) ⊗ (m_1^A, m_2^A))(ψ) ∈ (S_1^C × S_2^C)(S_1^A × S_2^A). Note that the same result would also hold for steering by the measurements m_1^C, m_2^C.One may think that steering is somehow half of Bell nonlocality, or that it is some middle step towards Bell nonlocality as even our constructions in Sec. <ref> and <ref> would point to such a result. We will show that this is not true in general, as we will provide a counter-example using quantum channels in example <ref>.§ BELL NONLOCALITY OF QUANTUM CHANNELS Bell nonlocality of quantum measurements is a deeply studied topic in quantum theory, with several applications in various device-independent protocols <cit.>, randomness generation and randomness expansion <cit.> and others, for a recent review on Bell nonlocality see <cit.>.Bell nonlocality of quantum channels follows very similar rules to steering by quantum channels. We will derive results specific for quantum theory in the same manners as in Sec. <ref>.Let ρ∈_⊗ and let Φ_1^1: _→_, Φ_2^1: _→_, Φ_1^2: _→_, Φ_2^2: _→_ be channels. The bipartite biconditional state ((Φ_1^1, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2))(ρ) is Bell nonlocal only if the bipartite biconditional state ((id, id) ⊗ (id, id))(ρ) is Bell nonlocal. If the bipartite biconditional state ((id, id) ⊗ (id, id))(ρ) is Bell local, then there exist σ∈_⊗⊗⊗ such that_24 (σ)= ρ,_23 (σ)= ρ,_14 (σ)= ρ,_13 (σ)= ρ.Letσ̃ = (Φ_1^1 ⊗Φ_2^1 ⊗Φ_1^2 ⊗Φ_2^2) (σ)then_24 (σ̃)= (Φ_1^1 ⊗Φ_1^2 ) (ρ),_23 (σ̃)= (Φ_1^1 ⊗Φ_2^2 ) (ρ),_14 (σ̃)= (Φ_2^1 ⊗Φ_1^2 ) (ρ),_13 (σ̃)= (Φ_2^1 ⊗Φ_2^2 ) (ρ). Note that again we do not have to replace all of the channels by the identity channels id, but we may replace only some.Let ρ∈_⊗ and let Φ_1^1: _→_, Φ_2^1: _→_, Φ_1^2: _→_, Φ_2^2: _→_ be channels, moreover let Φ_1^1 = Φ_U be a unitary channel given by the unitary matrix U, then the bipartite biconditional state ((Φ_U, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2))(ρ) is Bell nonlocal if and only if the bipartite biconditional state ((id, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2))(ρ) is Bell nonlocal. Using the very same idea as before, if thebipartite biconditional state ((Φ_U, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2))(ρ) is Bell local, then there is σ∈_⊗⊗⊗ such that_24 (σ)= (Φ_U ⊗Φ_1^2 ) (ρ),_23 (σ)= (Φ_U ⊗Φ_2^2 ) (ρ),_14 (σ)= (Φ_2^1 ⊗Φ_1^2 ) (ρ),_13 (σ)= (Φ_2^1 ⊗Φ_2^2 ) (ρ).Letσ̃ = (Φ_U^*⊗ id ⊗ id ⊗ id) (σ)then we get_24 (σ̃)= (id ⊗Φ_1^2 ) (ρ),_23 (σ̃)= (id ⊗Φ_2^2 ) (ρ),_14 (σ̃)= (Φ_2^1 ⊗Φ_1^2 ) (ρ),_13 (σ̃)= (Φ_2^1 ⊗Φ_2^2 ) (ρ). One may obtain similar results if some other of the channels Φ_1^1, Φ_2^1, Φ_1^2, Φ_2^2 is unitary as well as if more or even all of them are unitary.The most iconic and most studied aspect of Bell nonlocality are the Bell inequalities. We are going to present a version of CHSH inequality for quantum channels. Assume that = 2 and let |0, |1 denote any orthonormal basis of . We will use the shorthand |00 = |0⊗ |0. Let i, j ∈{1, 2} and letE(Φ_i^1, Φ_j^2)= 00| (Φ_i^1 ⊗Φ_j^2 )(ρ) |00- 01| (Φ_i^1 ⊗Φ_j^2 )(ρ) |01- 10| (Φ_i^1 ⊗Φ_j^2 )(ρ) |10+ 11| (Φ_i^1 ⊗Φ_j^2 )(ρ) |11=( (Φ_i^1 ⊗Φ_j^2 )(ρ) A )whereA = |0000| - |0101| - |1010| + |1111|.The quantity E(Φ_i^1, Φ_j^2) is to be interpreted as the correlation between the marginals _1((Φ_i^1 ⊗Φ_j^2 )(ρ)) and _2((Φ_i^1 ⊗Φ_j^2 )(ρ)). Since we have -≤ A ≤ it is straightforward that we have -1 ≤ E(Φ_i^1, Φ_j^2) ≤ 1. Define a quantityX_ρ = E(Φ_1^1, Φ_1^2) + E(Φ_1^1, Φ_2^2) + E(Φ_2^1, Φ_1^2) - E(Φ_2^1, Φ_2^2),we will show that X_ρ corresponds to the quantity used in CHSH inequality. It is straightforward to see that -4 ≤ X_ρ≤ 4 is the algebraic bound on X_ρ.If the biconditional bipartite state ((Φ_1^1, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2) )(ρ) is Bell local, then we have -2 ≤ X_ρ≤ 2. If the biconditional bipartite state ((Φ_1^1, Φ_2^1) ⊗ (Φ_1^2, Φ_2^2) )(ρ) is Bell local then there is σ∈_⊗⊗⊗ such that_24 (σ)= (Φ_1^1 ⊗Φ_1^2 ) (ρ),_23 (σ)= (Φ_1^1 ⊗Φ_2^2 ) (ρ),_14 (σ)= (Φ_2^1 ⊗Φ_1^2 ) (ρ),_13 (σ)= (Φ_2^1 ⊗Φ_2^2 ) (ρ).This yieldsE(Φ_1^1, Φ_1^2)=( (Φ_1^1 ⊗Φ_1^2 ) (ρ) A ) = ( _24 (σ) A ) =( σ ( |00| ⊗⊗ |00| ⊗- |00| ⊗⊗ |11| ⊗- |11| ⊗⊗ |00| ⊗+ |11| ⊗⊗ |11| ⊗ ) ).In the same manner we getE(Φ_1^1, Φ_2^2) =( σ (|00| ⊗⊗⊗ |00| -|00| ⊗⊗⊗ |11| -|11| ⊗⊗⊗ |00| +|11| ⊗⊗⊗ |11| ) ), E(Φ_2^1, Φ_1^2) =( σ ( ⊗ |00| ⊗ |00| ⊗-⊗ |00| ⊗ |11| ⊗-⊗ |11| ⊗ |00| ⊗+⊗ |11| ⊗ |11| ⊗ ) )andE(Φ_2^1, Φ_2^2) =( σ (⊗ |00| ⊗⊗ |00| -⊗ |00| ⊗⊗ |11| -⊗ |11| ⊗⊗ |00| +⊗ |11| ⊗⊗ |11| ) ).Together we getX_ρ = 2 ( σ (|00000000| + |00010001| -|00100010| - |00110011| +|01000100| - |01010101| +|01100110| - |01110111| -|10001000| + |10011001| -|10101010| + |10111011| -|11001100| - |11011101| +|11101110| + |11111111| ) )that implies -2 ≤ X_ρ≤ 2. At this point one may ask whether there exists an equivalent of Tsirelson bound <cit.> for the inequality given by Prop. <ref>, or what is the maximum violation of the aforementioned inequality. We will show that the Tsirelson bound 2√(2) is both reachable and maximum violation by quantum channels.For any state ρ∈_⊗ and any four channels Φ_1^1: _→_, Φ_2^1: _→_, Φ_1^2: _→_, Φ_2^2: _→_ we haveX_ρ≤ 2 √(2).We define the adjoint channel Φ_1^1* to channel Φ_1^1 as the the linear map Φ_1^1*: B_h() → B_h() such that for all σ∈_ and E ∈ B_h(), 0 ≤ E ≤ we have( Φ_1^1 (σ) E) = ( σΦ_1^1*(E) ).Since Φ_1^1 is a channel we have 0 ≤Φ_1^1*(E) ≤ and Φ_1^1*() =. This approach of mapping effects instead of states is called the Heisenberg picture.Let i, j ∈{1, 2}, then we have( (Φ^1_i ⊗Φ^2_j)(ρ) |0000|) = ( ρΦ_i^1*(|00|) ⊗Φ_j^2*(|00|)).DenotingM_i^1= Φ_i^1*(|00|)M_j^2= Φ_j^1*(|00|)we see that we haveE(Φ_i^1, Φ_j^2)= ( ρ M_i^1 ⊗ M_j^2) - ( ρ (-M_i^1) ⊗ M_j^2) - ( ρ M_i^1 ⊗ (-M_j^2)) + ( ρ (-M_i^1) ⊗ (-M_j^2)) = E(M_i^1, M_j^2),where E(M_i^1, M_j^2) is a correlation for the two-outcome measurements given by the effects M_i^1 and M_j^2. It is well known result <cit.> that we always haveE(M_1^1, M_1^2) + E(M_1^1, M_2^2) + E(M_2^1, M_1^2) - E(M_2^1, M_2^2) ≤ 2 √(2).It is very intuitive that the Tsirelson bound, reachable by measurements, will be also reachable by channels. To prove this, let M, N ∈ B_h(), 0 ≤ M ≤, 0 ≤ N ≤ and define channels Φ_M: B_h() → B_h(), Φ_N: B_h() → B_h() such that for σ∈_ we haveΦ_M(σ)= (σ M) |00| +(σ (-M)) |11|,Φ_N(σ)= (σ N) |00| +(σ (-N)) |11|.It is easy to verify that the maps Φ_M, Φ_N are quantum channels and that they are also measurements as they map the state space _ to the simplex { |00|, |11| }. Let ρ∈_⊗, then we have( (Φ_M ⊗Φ_N) (ρ) A )= ( ρ (M ⊗ N - ( - M) ⊗ N) - ( ρ (M ⊗ ( - N) ) ) + ( ρ (( - M) ⊗ ( - N) ) ) = E(M, N).This proves that any set of correlations and any violation of CHSH inequality reachable by measurements is also reachable by quantum channels as a violation of the bound given by Prop. <ref>.To generalize the proposed inequality one may replace the projectors |00| and |11| by any pair of effects M, N ∈ B_h(), 0 ≤ M ≤, 0 ≤ N ≤ and haveA = M ⊗ N - ( - M) ⊗ N - M ⊗ ( - N) + ( - M) ⊗ ( - N). From now on we will consider a special case. Keep = 2 and let|ψ^+ψ^+| = 12 ( |0000| + |1100| + |0011| + |1111| )be the maximally entangled state, let U_1, U_2, V_1, V_2 be unitary matrices and let Φ_1^1 = Φ_U_1, Φ_2^1 = Φ_U_2, Φ_1^2 = Φ_V_1, Φ_2^2 = Φ_V_2 be unitary channels given by the respective unitary matrices. We will consider the bipartite biconditional state ((Φ_U_1, Φ_U_2) ⊗ (Φ_V_1, Φ_V_2))(|ψ^+ψ^+|) and we will show that the correlations for the given bipartite biconditional state are of a particular nice form. We have(Φ_U_i⊗Φ_V_j)(|ψ^+ψ^+|) = (id ⊗Φ_V_j U_i^T)(|ψ^+ψ^+|)where i, j ∈{1, 2} and for U^T denotes the transpose of the matrix U. For the correlation we haveE(Φ_U_i, Φ_V_j)= ( (id ⊗Φ_V_j U_i^T)(|ψ^+ψ^+|) A ) = 12 ( |0| V_j U_i^T |0|^2 + |1| V_j U_i^T |1|^2 - |0| V_j U_i^T |1|^2 - |1| V_j U_i^T |0|^2 ). We will provide an example of a violation of the bound given by Prop. <ref> by incompatible unitary channels. Let = 2 and let ϑ∈ℝ be a parameter. Let U_1, U_2, V_1, V_2 be unitary matrices given asU_1= 1√(2)[11;1 -1 ], U_2= [ 1 0; 0 1 ], V_1= 1√(1 + ϑ)[√(ϑ) 1; 1 -√(ϑ) ], V_2= 1√(1 + ϑ)[1 √(ϑ); √(ϑ) -1 ].Consider the bipartite biconditional state (Φ_U_1, Φ_U_2) ⊗ (Φ_V_1, Φ_V_2)(|ψ^+ψ^+|). Using Eq. (<ref>) we can obtain X_|ψ^+ψ^+| as a function of ϑ. The function is plotted in Fig. <ref>, where it is shown that for certain values of ϑ the bipartite biconditional state violates the bound given by Prop. <ref>.It is also easy to see that the bipartite biconditional state ((id, id) ⊗ (id, id))(|ψ^+ψ^+|) does not violate the bound given by Prop. <ref>, because all of the correlations are the same, yet according to Prop. <ref> we know that it must be a Bell nonlocal bipartite biconditional state. This shows that not all Bell nonlocal bipartite biconditional states violate the inequality given by Prop. <ref>. One may wonder whether there is or is not a connection between steering and Bell nonlocality. As we have already showed in Prop. <ref>, for measurements Bell nonlocality implies steering. We will show that for channels the same does not hold. Let = 2. Let ρ_W ∈_⊗ be given as in example <ref> as a partial trace over the state |WW|. We already know that the state ρ_W is not steerable by any pair of channels. Consider the bipartite biconditional state ((id, id) ⊗ (id, id))(ρ_W), if it is Bell local, then there must be a state σ∈_⊗⊗⊗ such that_13(σ) = _14 (σ) = _23(σ) = _24 (σ) = ρ_W.Observe that _1(σ) ∈_⊗⊗ is such that _3 (_1(σ)) = _4 (_1(σ)) = ρ_W which implies that, according to our calculations in example <ref>, we must have_1(σ) = |WW|.According to <cit.> this implies that there is a state ρ∈_ such that σ = ρ⊗ |WW|. This implies that we have _23 (σ) = ρ⊗13 (2 |00| + |11|) which is clearly a separable state. This is a contradiction as we should have had _23 (σ) = ρ_W, which is an entangled state. § CONCLUSIONS We have introduced the general definition of compatibility of channels in general probabilistic theory through the idea of conditional channels. We have also shown that a naive idea for a compatibility test leads to a simple and straightforward formulation of steering and Bell nonlocality. These formulations of steering and Bell nonlocality are overall new even when we consider only measurements instead of channels. Throughout the paper we have shown that all of our definitions and result are in correspondence with the known result for measurements and we have also provided several examples and results about the introduced concepts in quantum theory.The paper has opened several new questions and areas of research. For example, a possible area of research would be to look at the structure of conditional states and conditional channels and to try to connect them to Bayesian theory.Concerning the compatibility of channels, one may formulate different notions of degree of (in)compatibility or of robustness of compatibility in general probabilistic theory and look at their properties, in a similar way as it was already done in quantum theory <cit.>. For quantum channels one may wonder which types of channels are compatible. This would generalize the no broadcasting theorem <cit.> which states that two unitary channels can not be compatible.One may also consider our formulations of steering and Bell nonlocality as a case of the problem of finding a multipartite state with given marginals. Such problems were studied in recent years <cit.>, but not in the form that would be applicable to the problems of steering and Bell nonlocality as incompatibility tests. This opens questions whether one may characterize the structure of the cone Q_CD and of other cones of interest in quantum theory. From a geometrical viewpoint this question is closely tied to the question of existence of other Bell inequalities for channels than the one we presented. Existence and exact form of the generalized Bell inequalities is also a very interesting possible area of research.We may also consider the use of steering and Bell nonlocality of channels in the context of quantum information theory and quantum communication. Both steering and Bell nonlocality of measurement were used to formulate new quantum protocols and it is of great interest whether exploiting the steering and Bell nonlocality of channels may lead to even better or more useful applications.One may also try and clarify the lack of connection between steering and Bell nonlocality of channels. As we have showed in example <ref>, even if two channels can not steer a state, when applied to both parts of the state the resulting biconditional bipartite state may be Bell nonlocal. This may even have interesting applications in quantum theory of information as so far steering has been considered to lead to one-side device-independent protocols that were seen as a middle step between the original protocol and device-independent protocol.It may also be interesting to consider the resource theories of channel incompatibility, of steering by channels and of Bell nonlocality of channels. Several similar resource theories were already constructed, see <cit.> for a review. The author is thankful to Anna Jenčová, Michal Sedlák, Mário Ziman, Daniel Reitzner and Tom Bullock for interesting conversations on the topic of compatibility. This research was supported by grant VEGA 2/0069/16 and by the grant of the Slovak Research and Development Agency under contract APVV-16-0073. The author acknowledges that this research was done during a PhD study at Faculty of Mathematics, Physics and Informatics of the Comenius University in Bratislava. | http://arxiv.org/abs/1707.08650v2 | {
"authors": [
"Martin Plávala"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727071742",
"title": "Conditions for the compatibility of channels in general probabilistic theory and their connection to steering and Bell nonlocality"
} |
=1 =1#1#2#3#4#1 #2, #3 (#4) | http://arxiv.org/abs/1707.08812v2 | {
"authors": [
"Brandon S. DiNunno",
"Sašo Grozdanov",
"Juan F. Pedraza",
"Steve Young"
],
"categories": [
"hep-th",
"gr-qc",
"hep-ph",
"nucl-th"
],
"primary_category": "hep-th",
"published": "20170727105136",
"title": "Holographic constraints on Bjorken hydrodynamics at finite coupling"
} |
plain thmTheorem lem[thm]Lemma prop[thm]Proposition cor[thm]Corollary mproof[1][] ⊗ ℂ ℝ Re ℐ 𝒜 𝒢 ℳ 𝒩 𝒟 𝒞 𝒞 𝒟 𝕀 d diag S ℋ U d (d) ℱ ℰ [email protected] of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China [email protected] Research Institute, Allahabad, 211019, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,Mumbai 400085, [email protected] of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, [email protected] Harish-Chandra Research Institute, Allahabad, 211019, [email protected] of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China The operational characterization of quantum coherence is the corner stone in the development ofresource theory of coherence. We introduce a new coherence quantifier based on max-relative entropy. We prove thatmax-relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of max-relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states. This advantage of coherent states in subchannel discrimination can be exactly characterized by the max-relative entropy of coherence. By introducing suitable smooth max-relative entropy of coherence, we prove that the smooth max-relative entropy of coherence provides a lower bound of one-shot coherence cost, and the max-relative entropy of coherence is equivalent to the relative entropy of coherence in asymptotic limit. Similar to max-relative entropy of coherence, min-relative entropy of coherence has also been investigated. We show that the min-relative entropy of coherence provides an upper bound of one-shot coherence distillation, and in asymptotic limit the min-relative entropy of coherence is equivalent to the relative entropy of coherence. Max- relative entropy of coherence : an operational coherence measure Junde Wu December 30, 2023 ===================================================================== § INTRODUCTIONQuantumness in a single system is characterized by quantum coherence, namely, the superposition of a state in a given reference basis. The coherence of a state may quantify the capacity of a system in many quantum manipulations, ranging frommetrology <cit.> to thermodynamics <cit.> . Recently, various efforts have been made to develop a resource theory of coherence <cit.>. One of the earlier resource theories is that of quantum entanglement <cit.>, which is a basic resource for various quantum information processing protocols such as superdense coding <cit.>, remote state preparation <cit.> and quantum teleportation <cit.>. Other notable examples include the resource theories of asymmetry <cit.>, thermodynamics <cit.>, and steering <cit.>. One of the main advantages that a resource theory offers is the lucid quantitative and operational description as well as the manipulation of the relevant resources at ones disposal, thus operational characterization of quantum coherence is required in the resource theory of coherence. A resource theory is usually composed of two basic elements: free states and free operations. The set of allowed states (operations) under the given constraint is what we call the set of free states (operations). Given a fixed basis, say |i⟩^d-1_i=0 for a d-dimensional system, any quantum state which is diagonal in the reference basis is called an incoherent state and is a free state in the resource theory of coherence. The set of incoherent states is denoted by . Any quantum state can be mapped into an incoherent state by a full dephasing operation Δ, where Δ(ρ):=∑^d-1_i=0iρii. However, there is no general consensus on the set of free operations in the resource theory of coherence. We refer the following types of free operations in this work: maximally incoherent operations (MIO) <cit.>, incoherent operations (IO) <cit.>, dephasing-covariant operations (DIO) <cit.> and strictly incoherent operations (SIO) <cit.>. By maximally incoherent operation (MIO), we refer to the maximal set of quantum operations Φ which maps the incoherent states into incoherent states, i.e., Φ()⊂ <cit.>. Incoherent operations (IO) is the set of allquantum operations Φ that admit a set of Kraus operators K_i such that Φ(·)=∑_iK_i(·)K^†_i and K_i K^†_i⊂ for any i <cit.>. Dephasing-covariant operations (DIO) are the quantum operations Φ with [Δ,Φ]=0 <cit.>. Strictly incoherent operations (SIO) is the set of all quantum operations Φ admitting a set of Kraus operators K_i such that Φ(·)=∑_iK_i(·)K^†_i and Δ (K_iρ K^†_i)=K_iΔ(ρ) K^†_i for any i and any quantum state ρ. Both IO and DIO are subsets of MIO , and SIO is a subset of both IO and DIO <cit.>. However, IO and DIO are two different types of free operations and there is no inclusion relationship between them (The operational gap between them can be seen in <cit.>). Several operational coherence quantifiers have been introduced as candidate coherence measures, subjecting to physical requirements such as monotonicity under certain type of free operations in the resource theory of coherence. One canonical measure to quantify coherence is the relative entropy of coherence, which is defined as C_r(ρ)=S(Δ(ρ))-S(ρ), where S(ρ)=-ρlogρ is the von Neumann entropy <cit.>. The relative entropy of coherence plays an important role in the process of coherence distillation, in which it can be interpreted as the optimal rate to distill maximally coherent state from a given state ρ by IO in the asymptotic limit <cit.>. Besides, the l_1 norm of coherence <cit.>, which is defined as C_l_1(ρ)=∑_i≠ j|ρ_ij| with ρ_ij=iρj, has also attracted lots of discussions about its operational interpretation <cit.>. Recently, an operationally motivated coherence measure- robustness of coherence (RoC) - has been introduced, which quantifies the minimal mixing required to erase the coherence in a given quantum state <cit.>. There is growing concern about the operational characterization of quantum coherence and further investigations are needed to provide an explicit and rigorous operational interpretation of coherence. In this letter, we introduce a new coherence measure based on max-relative entropy and focus on its operational characterizations. Max- and min- relative entropies have been introduced and investigated in <cit.>. The well-known (conditional and unconditional) max- and min- entropies<cit.> can be obtained from these two quantities. It has been shown that max- and min-entropies are of operational significance in the applications ranging from data compression <cit.> to state merging <cit.> and security of key <cit.>. Besides, max- and min- relative entropies have been used to define entanglement monotone and their operational significance in the manipulation of entanglement has been provided in <cit.>. Here, we define max-relative of coherence C_max based on max-relative entropy and investigate the properties of C_max. We prove that max-relative entropy of coherence for a given state ρ is the maximum achievable overlap with maximally coherent states under DIO, IO and SIO, which gives rise to an operational interpretation of C_max and shows the equivalence among DIO, IO and SIO in an operational task. Besides, we show that max-relative entropy of coherence characterizes the role ofquantum states in an operational task: subchannel discrimination. Subchannel discrimination is an important quantum information task which distinguishes the branches of a quantum evolution for a quantum system to undergo <cit.>. It has been shown that every entangled or steerable state is a resource in some instance of subchannel discrmination problems <cit.>. Here, we prove that that every coherent state is useful in the subchannel discrimination of certain instruments, where the usefulness can be quantified by the max-relative entropy of coherence of the given quantum state. By smoothing the max-relative entropy of coherence, we introduce ϵ-smoothed max-relative entropy of coherence C^ϵ_max for any fixed ϵ>0 and show that the smooth max-relative entropy gives an lower bound ofcoherence cost inone-shot version. Moreover,we prove that for any quantum state, max-relative entropy of coherenceis equivalent to the relative entropy of coherence in asymptotic limit.Corresponding to the max-relative entropy of coherence, we also introduce the min-relative entropy of coherence C_min by min-relative entropy, which is not a proper coherence measure as it may increase on average under IO. However, it gives an upper bound for the maximum overlap between the given states and the set of incoherent states. This implies that min-relative entropy of coherencealso provides a lower bound of a well-known coherence measure, geometry of coherence <cit.>. By smoothing the min-relative entropy of coherence, we introduce ϵ-smoothed min-relative entropy of coherence C^ϵ_min for any fixed ϵ>0 and show that the smooth max-relative entropy gives an upper bound ofcoherence distillation inone-shot version. Furthermore, we show that the min-relative of coherence is also equivalent to distillation of coherence in asymptotic limit. The relationship among C_min, C_max and other coherence measures has also been investigated.§ MAIN RESULTSLetbe a d-dimensional Hilbert space and () be the set of density operators acting on . Given two operatorsρ and σ with ρ≥ 0, ρ≤ 1 and σ≥ 0,the max-relative entropy of ρ with respect to σ is defined by <cit.>,D_max(ρ||σ): = minλ:ρ≤2^λσ.We introduce a new coherence quantifier by max-relativeentropy: max-relative entropy of coherence C_max,C_max(ρ):=min_σ∈D_max(ρ||σ),whereis the set of incoherent states in ().We now show that C_max satisfies the conditions a coherence measure needs to fulfil. First, it is obvious that C_max(ρ)≥ 0. And since D_max(ρ||σ)=0 iff ρ=σ <cit.>, we have C_max(ρ)=0 if and only if ρ∈. Besides, as D_max is monotone under CPTP maps <cit.>,we have C_max(Φ(ρ))≤ C_max(ρ) for any incoherent operation Φ. Moreover, C_max is nonincreasing on average under incoherent operations, that is, for any incoherent operation Φ(·)=∑_i K_i(·)K^†_i with K_i K^†_i⊂, ∑_i p_iC_max(ρ̃_i)≤ C_max(ρ), where p_i=K_iρ K^†_i and ρ̃_i=K_iρ K^†_i/p_i, see proof in Supplemental Material <cit.>.Remark We have proven that the max-relative entropy of coherence C_max is a bona fide measure of coherence. Since D_max is not jointly convex, we may not expect that C_max has the convexity, which is a desirable (although not a fundamental) property for a coherence quantifier. However, we can prove that for ρ=∑^n_ip_iρ_i, C_max(ρ)≤max_iC_max(ρ_i). Suppose that C_max(ρ_i)=D_max(ρ_i||σ^*_i) for some σ^*_i, then from the fact thatD_max(∑_ip_iρ_i||∑_ip_iσ_i)≤max_iD_max(ρ_i||σ_i) <cit.>, we have C_max(ρ)≤ D_max(∑_ip_iρ_i||∑_ip_iσ^*_i) ≤max_iD_max(ρ_i||σ^*_i)=max_iC_max(ρ_i). Besides, although C_max is not convex, we can obtain a proper coherence measure with convexity from C_max by the approach of convex roof extension, see Supplemental Material <cit.>. In the following, we concentrate on the operational characterization of the max-relative entropy of coherence, and provide operational interpretations of C_max.Maximum overlap with maximally coherent states.—At first we show that 2^C_max is equal to the maximum overlap with the maximally coherent state that can be achieved byDIO, IO and SIO.Given a quantum state ρ∈(), we have2^C_max(ρ) = dmax_ℰ,|Ψ⟩ F(ℰ(ρ),Ψ)^2,where F(ρ,σ)=|√(ρ)√(σ)| is the fidelity between states ρ and σ <cit.>, |Ψ⟩∈ andis the set of maximally coherent states in (), ℰ belongs to either DIO or IO or SIO.(See proof in Supplemental Material <cit.>.)Here although IO, DIO and SIO are different types of free operations in resource theory of coherence <cit.>, they have the same behavior in the maximum overlap with the maximally coherent states. From the view of coherence distillation <cit.>, the maximum overlap with maximally coherent states can be regarded as the distillation of coherence from given states under IO, DIO and SIO. As fidelity can be used to define certain distance, thus C_max(ρ)can also be viewed as the distance between the set of maximally coherent state and the set of(ρ)_∈θ, where θ=DIO, IO or SIO.Besides distillation of coherence, another kind of coherence manipulation is the coherence cost <cit.>. Now we study the one-shot version of coherence cost under MIO based on smooth max-relative entropy of coherence. We define the one-shot coherence cost of a quantum state ρ under MIO asC^(1),ϵ_C,MIO(ρ):=min_ℰ∈ MIOM∈ℤlog M: F(ρ,ℰ(Ψ^M_+))^2≥1-ϵ,where |Ψ^M_+⟩=1/√(M)∑^M_i=1|i⟩, ℤ is the set of integer and ϵ>0. The ϵ-smoothed max-relative entropy of coherence of a quantum state ρ is defined by,C^ϵ_max(ρ):=min_ρ'∈ B_ϵ(ρ)C_max(ρ'),where B_ϵ(ρ):=ρ'≥0:ρ'-ρ_1≤ϵ, ρ'≤ρ. We find that the smooth max-relative entropy of coherence gives a lower bound of one-shot coherence cost. Given a quantum state ρ∈(), for any ϵ>0,C^ϵ'_max(ρ)≤C^(1),ϵ_C,MIO(ρ),where ϵ'=2√(ϵ), see proof in Supplemental Material <cit.>.Besides, in view of smooth max-relative entropy of coherence, we can obtain the equivalence between max-relative entropy of coherence and relative entropy of coherence in the asymptotic limit. Sincerelative entropy of coherence is the optimal rate to distill maximally coherent state from a given state under certain free operations in the asymptotic limit <cit.>, the smooth max-relative entropy of coherence in asymptotic limit is just the distillation of coherence. That is, given a quantum state ρ∈(), we havelim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n) =C_r(ρ).(The proof is presented in Supplemental Material <cit.>.)Maximum advantage achievable in subchannel discrimination.– Now, we investigateanother quantum information processing task: subchannel discrimination, which can also provide an operational interpretation of C_max. Subchannel discrimination is an important quantum information task which is used to identify the branch of a quantum evolution to undergo. We consider some special instance of subchannel discrimination problem to show the advantage of coherent states.A linear completely positive and trace non-increasing mapis called a subchannel. If a subchannelis trace preserving, thenis called a channel. An instrument ℑ=_a_a for a channelis a collection of subchannels _a with =∑_a_aand every instrument has its physical realization <cit.>. A dephasing covariant instrument ℑ^D for a DIOis a collection of subchannels _a_a such that =∑_a_a. Similarly, we can define incoherent instrument ℑ^I and strictly incoherent instrument ℑ^S for channel ∈ IO and ∈ SIO respectively.Given an instrument ℑ=_a_a for a quantum channel , let us consider a Positive Operator Valued Measurement (POVM) M_b_b with ∑_bM_b=𝕀. The probability of successfully discriminating the subchannels in the instrument ℑ by POVM M_b_b for input state ρ is given byp_succ(ℑ,M_b_b, ρ) =∑_a_a(ρ)M_a.The optimal probability of success in subchannel discrimination of ℑ over all POVMs is given byp_succ(ℑ,ρ)=max_M_b_b p_succ(ℑ,M_b_b, ρ).If we restrict the input states to be incoherent ones, then the optimal probability of success among all incoherent states is given byp^ICO_succ(ℑ)=max_σ∈p_succ(ℑ,σ).We have the following theorem. Given a quantum state ρ, 2^C_max(ρ) is the maximal advantage achievable by ρ compared with incoherent states in all subschannel discrimination problems of dephasing-covariant, incoherent and strictly incoherent instruments,2^C_max(ρ) = max_ℑp_succ(ℑ,ρ)/p^ICO_succ(ℑ),where ℑ is either ℑ^D or ℑ^I or ℑ^S, denoting the dephasing-covariant, incoherent and strictly incoherent instrument, respectively.The proof of Theorem <ref> is presented in Supplemental Material <cit.>. This result shows that the advantage of coherent states in certain instances of subchannel discrimination problems can be exactly captured by C_max, which provides another operational interpretation of C_max and also shows the equivalence among DIO, IO and SIO in the information processing task of subchannel discrimination.Min-relative entropy of coherence C_min(ρ).–Giventwo operatorsρ and σ with ρ≥ 0, ρ≤ 1 and σ≥ 0,max- and min- relative entropy of ρ relative to σ are defined asD_min(ρ||σ):=-logΠ_ρσwhere Π_ρ denotes the projector onto suppρ, thesupport of ρ. Corresponding to C_max(ρ) defined in (<ref>), we can similarly introduce a quantity defined by min-relative entropy,C_min(ρ):=min_σ∈D_min(ρ||σ).Since D_min(ρ||σ)=0 if supp ρ=supp σ <cit.>, we have ρ∈ ⇒ C_min(ρ)=0. However, converse direction may not be true, for example, let ρ=1/20+1/2+ with |+⟩= 1/√(2)(|1⟩+|2⟩), then ρ is coherent but C_min(ρ)=0. Besides, as D_min is monotone under CPTP maps <cit.>, we haveC_min(Φ(ρ))≤ C_min(ρ) for any Φ∈ IO. However, C_min may increase on average under IO (see Supplemental Material <cit.>). Thus, C_min is not be a proper coherence measure as C_max.Although C_min is not a good coherence quantifier, it still has some interesting properties in the manipulation of coherence. First, C_min gives upper bound of the maximum overlap with the set of incoherent states for any given quantum state ρ∈(),2^-C_min(ρ)≥max_σ∈F(ρ,σ)^2.Moreover, if ρ is pure state |ψ⟩, then above equality holds, that is,2^-C_min(ψ)=max_σ∈F(ψ,σ)^2,see proof inSupplemental Material <cit.>.Moreover, for geometry of coherence defined by C_g(ρ)=1- max_σ∈F(ρ,σ)^2 <cit.>, C_min also provides a lower bound for C_g as follows C_g(ρ)≥ 1-2^-C_min(ρ). Now let us consider again the one-shot version of distillable coherence under MIO by modifying and smoothing the min-relative entropy of coherence C_min. We define the one-shot distillable coherence of a quantum state ρ under MIO asC^(1),ϵ_D,MIO(ρ):=max_ℰ∈ MIOM∈ℤlog M: F(ℰ(ρ),Ψ^M_+)^2≥1-ϵ,where |Ψ^M_+⟩=1/√(M)∑^M_i=1|i⟩ and ϵ>0.For any ϵ>0, we define the smooth min-relative entropy of coherence of a quantum state ρ as followsC^ϵ_min(ρ):=max_ 0≤ A≤𝕀 Aρ≥ 1-ϵmin_σ∈-logAσ,where 𝕀 denotes the identity. It can be shown that C^ϵ_min is a upper bound of one-shot distillable coherence,C^(1),ϵ_D,MIO(ρ)≤ C^ϵ_min(ρ)for any ϵ>0, see proof in Supplemental Material <cit.>.The distillation of coherence in asymptotic limit can be expressed asC_D,MIO=lim_ϵ→ 0lim_n→∞1/nC^(1),ϵ_D,MIO(ρ).It has been proven that C_D,MIO(ρ)=C_r(ρ) <cit.>. Here we show that the equality in inequality (<ref>) holds in the asymptotic limit as the C_min is equivalent to C_r in the asymptotic limit. Given a quantum state ρ∈(), thenlim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n) =C_r(ρ).(The proof is presented in Supplemental Material <cit.>.)We have shown that C_min gives rise to the bounds for maximum overlap with the incoherent states and for one-shot distillable coherence. Indeed the exact expression of C_min for some special class of quantum states can be calculated. For pure state |ψ⟩=∑^d_i=1ψ_i|i⟩ with ∑^d_i=1|ψ_i|^2=1, we have C_min(ψ)=-logmax_i|ψ_i|^2. For maximally coherent state |Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, we have C_min(Ψ)=log d, which is the maximum value for C_min in d-dimensional space.Relationship between C_max and other coherence measures.– First, we investigate the relationship among C_max, C_min and C_r. Since D_min(ρ||σ)≤ S(ρ||σ)≤ D_max(ρ||σ)for any quantum states ρ and σ <cit.>, one hasC_min(ρ)≤ C_r(ρ)≤ C_max(ρ).Moreover, as mentioned before, these quantities are all equal in the asymptotic limit.Above all, C_max is equal to the logarithm of robustness of coherence, as RoC(ρ)=min_σ∈s≥0|ρ≤(1+s)σ and C_max(ρ)=min_σ∈minλ:ρ≤2^λσ <cit.>, that is,2^C_max(ρ)=1+RoC(ρ). Thus, the operational interpretations of C_max in terms of maximum overlap with maximally coherent states and subchannel discrimination, can also be viewed as the operational interpretations of robustness of coherence RoC. It is known that robustness of coherence plays an important role in a phase discrimination task, which provides an operational interpretation for robustness of coherence <cit.>. This phase discrimination taskinvestigated in <cit.> is just a special case of the subchannel discrimination in depasing-covariantinstruments. Due to the relationship between C_max and RoC, we can obtain the closed form of C_max for some special class of quantum states. As an example, let us consider a pure state |ψ⟩=∑^d_i=1ψ_i|i⟩. Then C_max(ψ)=log((∑^d_i=1|ψ_i|)^2))=2log(∑^d_i=1|ψ_i|). Thus, for maximally coherent state |Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, we have C_max(Ψ)=log d, which is the maximum value for C_max in d-dimensional space.Since RoC(ρ)≤ C_l_1(ρ) <cit.> and 1+RoC(ρ)=2^C_max(ρ), then C_max(ρ)≤log(1+C_l_1(ρ)). We have the relationship among these coherence measures,C_min(ρ)≤ C_r(ρ)≤ C_max(ρ) = log(1+RoC(ρ))≤ log(1+C_l_1(ρ)),which implies that 2^C_r(ρ)≤ 1+C_l_1(ρ) (See also <cit.>). § CONCLUSIONWe have investigated the properties of max- and min-relative entropy of coherence, especially the operational interpretation of the max-relative entropy of coherence. It has been found that the max-relative entropy of coherence characterizes the maximum overlap with the maximally coherent states under DIO, IO and SIO, as well as the maximum advantage achievable by coherent states compared with all incoherent states in subchannel discrimination problems of all dephasing-covariant, incoherent and strictly incoherent instruments, which also provides new operational interpretations of robustness of coherence and illustrates the equivalence of DIO, IO and SIO in these two operational taks. The study of C_max and C_min also makes the relationship between the operational coherence measures (e.g. C_r and C_l_1 ) more clear. These results may highlight the understanding to the operational resource theory of coherence.Besides, the relationships among smooth max- and min- relative relative entropy of coherence and one-shot coherence cost and distillation have been investigated explicitly. As both smooth max- and min- relative entropy of coherence are equal to relative entropy of coherence in the asymptotic limit and the significance of relative entropy of coherence in the distillation of coherence, further studies are desired on the one-shot coherence cost and distillation. This work is supported by the Natural Science Foundation of China (Grants No. 11171301, No. 10771191, No. 11571307 and No. 11675113) and the Doctoral Programs Foundation of the Ministry of Education of China (Grant No. J20130061).apsrev4-1§ STRONG MONOTONICITY UNDER IO FOR C_MAXWe prove this property based on the method in<cit.> and the basic facts of D_max <cit.>. Due to the definition of C_max, there exists an optimal σ_*∈ such that C_max(ρ)=D_max(ρ||σ_*). Let σ̃_i=K_iσ_* K^†_i/K_iσ_* K^†_i, then we have∑_ip_i D_max(ρ̃_i||σ̃_i)≤ ∑_i D_max(K_iρ K^†_i||K_iσ_* K^†_i)≤ ∑_i D_max(E𝕀iUραU^†𝕀i|| ×E𝕀iUσ_*αU^†𝕀i)≤ ∑_i D_max(𝕀iUραU^†𝕀i|| ×𝕀iUσ_*αU^†𝕀i)=D_max(UραU^†||Uσ_*αU^†)= D_max(ρα||σ_*α)= D_max(ρ||σ_*)= C_max(ρ),where the first inequality comes from the proof of Theorem 1 in <cit.>, the second inequality comes from the fact that there exists an extended Hilbert space _E, a pure |α⟩∈_E and a global unitary U on _E such that E𝕀iUραU^†𝕀i=K_iρ K^†_i <cit.>, the third inequality comes from the fact that D_max is monotone under partial trace <cit.>, the last inequality comes from the fact that for any set of mutually orthogonal projectors P_k, D_max(∑_kP_kρ_1 P_k||∑_kP_kρ_2 P_k)=∑_kD_max(P_kρ_1 P_k||P_kρ_2 P_k) <cit.>and the first equality comes from the fact that D_max is invariant under unitary operation and D_max(ρ_1 P||ρ_2 P)=D_max(ρ_1||ρ_2) for any projector P <cit.>. Besides, since C_max(ρ̃_i)=min_τ∈D_max(ρ̃_i||τ)≤ D_max(ρ̃_i||σ̃_i), we have ∑_i p_iC_max(ρ̃_i)≤ C_max(ρ).§ COHERENCE MEASURE INDUCED FROM C_MAXHere we introduce a proper coherence measure from C_max by the method of convex roof and prove that it satisfies all the conditions (including convexity) a coherence measure need to fulfil. We define the convex roof of C_max as followsC̃_max(ρ) =min_ρ=∑λ_iψ_i∑_iλ_iC_max(ψ_i),where the minimum is taken over all the pure state decompositions of state ρ. Due to the definition of C̃_max and the properties of C_max, the positivity and convexity of C̃_max are obvious. We only need to prove that it is nonincreasing on average under IO.Given a quantum state ρ∈(), for any incoherent operation Φ(·)=∑_μ K_μ(·)K^†_μ with K_μ K^†_μ⊂,∑_μ p_μC̃_max(ρ̃_μ)≤C̃_max(ρ),where p_μ=K_μρ K^†_μ and ρ̃_μ=K_μρ K^†_μ/p_μ. Due to the definition of C̃_max(ρ), there exists a pure state decomposition of state ρ=∑_jλ_jψ_j such that C̃_max(ρ)=∑_jλ_jC_max(ψ_j). Thenρ̃_μ = K_μρ K^†_μ/p_μ= ∑_jλ_j/p_uK_μψ_jK^†_μ= ∑_jλ_jq^(μ)_j/p_μϕ^(μ)_j,where |ϕ^(μ)_j⟩=K_μ|ψ_j⟩/√(q^(μ)_j) andq^(μ)_j=K_μψ_jK^†_μ.Thus, C̃_max(ρ̃_μ)≤∑_jλ_jq^(μ)_j/p_μC_max(ϕ^(μ)_j)and ∑_μp_μC̃_max(ρ̃_μ) ≤ ∑_j,μλ_jq^(μ)_jC_max(ϕ^(μ)_j)= ∑_jλ_j∑_μq^(μ)_jC_max(ϕ^(μ)_j)= ∑_jλ_j∑_μq^(μ)_jlog(1+C_l_1(ϕ^(μ)_j))≤ ∑_jλ_jlog(1+∑_μq^(μ)_j C_l_1(ϕ^(μ)_j))≤ ∑_jλ_jlog(1+C_l_1(ψ_j))= ∑_jλ_jC_max(ψ_j)= C̃_max(ρ),where the third line comes from the fact that for pure state ψ, C_max(ψ)=log(1+C_l_1(ψ)), the forth line comes from the concavity of logarithm and the fifth lines comes from the fact that monotonicity of C_l_1 under IO as Φ(ψ_j)=∑_μK_μψ_jK^†_μ=∑_μq^(μ)_jϕ^(μ)_j. § THE OPERATIONAL INTERPRETATION OF C_MAXTo prove the results, we need some preparation. First of all, Semidefinite programming (SDP) is a powerful tool in this work—which is a generalization of linear programming problems <cit.>. A SDP over 𝒳=ℂ^N and 𝒴=ℂ^M is a triple (Φ, C, D), where Φ is a Hermiticity-preserving map from ℒ(𝒳) (linear operators on 𝒳) to ℒ(𝒴) (linear operators on 𝒴), C∈ Herm(𝒳) (Hermitian operators over 𝒳), and D∈ Herm(𝒴) (Hermitian operators over 𝒴). There is a pair of optimization problems associated with every SDP (Φ, C, D), known as the primal and the dual problems. The standard form of an SDP (that is typically followed for general conic programming) is <cit.>Primal problem Dual problemminimize: ⟨ C,X ⟩,maximize: ⟨ D,Y ⟩,subject to: Φ(X) ≥ D,subject to: Φ^*(Y) ≥ C, X ∈ Pos(𝒳). Y ∈ Pos(𝒴).SDP forms have interesting and ubiquitous applications in quantum information theory. For example, it was recently shown by Brandao et. al <cit.> that there exists a quantum algorithm for solving SDPs that gives an unconditional square-root speedup over any existing classical method. Given a quantum state ρ∈(),min_σ≥0 Δ(σ)≥ρσ=max_τ≥0 Δ(τ)=𝕀ρτ.First, we prove thatmax_τ≥0 Δ(τ)=𝕀ρτ =max_τ≥0 Δ(τ)≤𝕀ρτ.For any positive operator τ≥0 with Δ(τ)≤𝕀, define τ'=τ+𝕀-Δ(τ)≥0, thenΔ(τ)=𝕀 and ρτ'≥ρτ. Thus we obtain the above equation.Now, we prove thatmin_σ≥0 Δ(σ)≥ρσ=max_τ≥0 Δ(τ)≤𝕀ρτ.The left side of equation (<ref>) can be expressed as the following semidefinite programming (SDP)minBσ, s.t. Λ(σ)≥ C, σ≥ 0,where B=𝕀, C=ρ and Λ=Δ. Then the dual SDP is given bymaxCτ, s.t. Λ^†(τ)≤ B, τ≥ 0.That is,maxρτ, s.t. Δ(τ)≤𝕀, τ≥ 0.Note that the dual is strictly feasible as we only need to choose σ=2λ_max(ρ) 𝕀, where λ_max(ρ) is themaximum eigenvalue of ρ. Thus, strong duality holds, and the equation (<ref>) is proved. For maximally coherent state |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩, we have the following facts,(i) For any ℰ∈ DIO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.(ii) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ DIO such that τ=dℰ^†(Ψ_+). (iii) For any ℰ∈ IO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.(iv) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ IO such that τ=dℰ^†(Ψ_+). (v) For any ℰ∈ SIO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.(vi) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ SIO such that τ=dℰ^†(Ψ_+).(i) Since ℰ is a CPTP map, ℰ^† is unital. Besides, as ℰ∈ DIO, [ℰ,Δ]=0 implies that [ℰ^†, Δ]=0. Thus Δ(τ)=dℰ^†(Δ(Ψ_+))=ℰ^†(𝕀)=𝕀. (ii) For any positive operator τ≥ 0 with Δ(τ)=𝕀, τ=d, thus τ=dτ̂ with τ̂∈() and Δ(τ̂)=1/d𝕀. Consider the spectral decomposition ofτ̂=∑^d_i=1λ_iψ_i with ∑^d_i=1λ_i=1, λ_i≥ 0 for any i∈1,.., d. Besides,for any i∈1,..., d, |ψ_i⟩ can be written as |ψ_i⟩=∑^d_j=1c^(i)_j|j⟩ with ∑^d_j=1|c^(i)_j|^2=1. Let us define K^(i)_n=∑^d_j=1c^(i)_jj for any n∈1,...,d, then K^(i)_n|Ψ_+⟩=1/√(d)|ψ_i⟩ and ∑^d_n=1K^(i)_nΨ_+K^(i)†_n=ψ_i. Let M_i,n=√(λ_i)K^(i)†_n, then∑_i,nM^†_i,nM_i,n = ∑_i,nλ_iK^(i)_nK^(i)†_n= d∑^d_i=1λ_iK^(i)_1K^(i)†_1= d∑^d_i=1λ_i∑^d_j=1|c^(i)_j|^2j= d∑^d_j=1∑^d_i=1λ_i|c^(i)_j|^2j= d∑^d_j=11/dj=𝕀, where ∑^d_i=1λ_i|c^(i)_j|^2=∑_iλ_i|ψ_ij|^2=jτ̂j=1/d. Then ℰ(·)=∑_i,nM_i,n(·) M^†_i,n is a CPTP map. Since M_i,n is diagonal, the quantum operation ℰ(·)=∑_i,nM_i,n(·) M^†_i,n is a DIO. Moreover, ℰ^†(Ψ_+)=∑_i,nM^†_i,nΨ_+M_i,n =∑_i,nλ_iK^(i)_nΨ_+K^(i)†_n=∑_iλ_iψ_i=τ̂.(iii) Ifis an incoherent operation, then there exists a set of Kraus operators K_μ such that (·)=∑_μK_μ(·) K^†_μ and K_μ K^†_μ∈. Thusdi^†(Ψ_+)i = d∑_μi K^†_μΨ_+K_μi= ∑_μ∑_m,niK^†_μmnK_μi= ∑_μ∑_miK^†_μmmK_μi= ∑_μiK^†_μK_μi =1,where the third line comes from the fact that for any K_μ, there exists at most one nonzero term in eachcolumn which implies that ⟨i|K^†_μ|m⟩⟨n|K_μ|i⟩≠ 0 only if m=n, and the forth line comes from the fact that ∑_μK^†_μK_μ=𝕀. Therefore, Δ(d^†(Ψ_+))=𝕀.(iv) This is obvious, as the DIOgiven in (ii) is also an incoherent operation.(v) This is obvious as SIO⊂ DIO.(vi) This is obvious as the DIOgiven in (ii) also belongs to SIO. Given a quantum state ρ∈(), one hasmax_ℰ∈ DIO F(ℰ(ρ),Ψ_+)^2 = max_ℰ∈ DIO |Ψ⟩∈F(ℰ(ρ),Ψ)^2.where |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩ andis the set of maximally coherent states. Due to <cit.>, every maximally coherent can be expressed as |Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, that is, |Ψ⟩=U_Ψ|Ψ_+⟩ where U_Ψ=∑^d_j=1e^iθ_jj. Obviously, [U_Ψ, Δ]=0, thus U_Ψ∈ DIO andF(ℰ(ρ),Ψ)^2= F(ℰ(ρ), U_ΨΨ_+U^†_Ψ)^2= F(U^†_Ψℰ(ρ)U_Ψ,Ψ)^2= F(ℰ'(ρ),Ψ_+)^2,where ℰ'(·)=U^†_Ψℰ(·)U_Ψ∈ DIO as ℰ, U_Ψ∈ DIO. After these preparation, we begin to prove Theorem 1. [Proof of Theorem 1] If ℰ belongs to DIO, that is, we need to prove2^C_max(ρ)=dmax_ℰ∈ DIO |Ψ⟩∈F(ℰ(ρ),Ψ)^2,whereis the set of maximally coherent states. In view of Lemma <ref>,we only need to prove2^C_max(ρ)=dmax_ℰ∈ DIO F(ℰ(ρ),Ψ_+)^2,where |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩.First of all,2^C_max(ρ) = min_σ∈minλ|ρ≤λσ= min_σ≥0σ|ρ≤Δ(σ)= min_σ≥0 Δ(σ)≥ρσ. Second,dF(ℰ(ρ),Ψ_+)^2= dℰ(ρ)Ψ_+= dρℰ^†(Ψ_+)= ρτ,where τ=dℰ^†(Ψ_+. According to Lemma <ref>, there is one to one correspondence between DIO and the set τ≥0|Δ(τ)=𝕀. Thus we have dmax_ℰ∈ DIO F(ℰ(ρ),Ψ_+)^2 =max_τ≥0 Δ(τ)=𝕀ρτ.Finally, according to Lemma <ref>, we get the desired result (<ref>). Similarly, we can prove the case where ℰ belongs to either IO or SIO based on Lemma <ref>. § SUBCHANNEL DISCRIMINATION IN DEPHASING COVARIANT INSTRUMENT[Proof of Theorem 2] First, we consider the case where instrument ℑ is dephasing-covariant instrumentℑ^D. Due to the definition of C_max(ρ), there exists an incoherent state σ such that ρ≤ 2^C_max(ρ)σ. Thus, for any dephasing-covariant instrument ℑ^D and POVM M_b_b,p_succ(ℑ^D,M_b_b,ρ)≤ 2^C_max(ρ) p_succ(ℑ^D, M_b_b, σ),which implies thatp_succ(ℑ^D,ρ)≤ 2^C_max(ρ) p^ICO_succ(ℑ^D).Next, we prove that there exists a dephasing-covariant instrument ℑ^D such that the equality in (<ref>) holds. In view of Theorem 1, there exists a DIOsuch that2^C_max(ρ)=d(ρ)Ψ_+,where |Ψ_+⟩=1/√(d)∑^d_j=1|j⟩. Let us consider the following diagonal unitariesU_k=∑^d_j=1e^ijk/d2πj,k∈1,..,d.The set U_k|Ψ_+⟩^d_k=1forms a basis of the Hilbert space and ∑^d_k=1U_kΨ_+U^†_k=𝕀. Let us define subchannels _k_k as _k(ρ)=1/dU_k(ρ) U^†_k. Then the channel =∑^d_k=1_k is a DIO. That is, the instrument ℑ^D=_k_k is a dephasing-covariant instrument.For any POVM M_k_kand any incoherent state σ, the probability of success isp_succ(ℑ^D, M_k_k,σ)= ∑_k_k(σ)M_k= 1/d(σ)∑_kU^†_kM_kU_k.Since M_k_k is a POVM, then ∑_kM_k=𝕀. As U_k_k are all diagonal unitaries , we haveΔ(∑_kU^†_kM_kU_k)= ∑_k U^†_kΔ(M_k)U_k = ∑_kΔ(M_k)= Δ(∑_k M_k)= Δ(𝕀)=𝕀.Thus,p_succ(ℑ^D, M_k_k,σ)= 1/d(σ)∑_kU^†_kM_kU_k= 1/dΔ((σ))∑_kU^†_kM_kU_k= 1/d(σ)Δ(∑_kU^†_kM_kU_k)= 1/d(σ) =1/d,where the second equality comes from the fact that (σ)∈ for any incoherent state σ,and the second last equality comes from that fact that Δ(∑_kU^†_kM_kU_k)=𝕀. That is,p^ICO_succ(ℑ^D)=1/d. Besides, taking the POVMN_k_k with N_k=U_kΨ_+U^†_k, one has _k(ρ)N_k=1/d(ρ)Ψ_+ andp_succ(ℑ^D, N_k_k,ρ)= ∑_k_k(ρ)N_k= ∑_k1/d(ρ)Ψ_+= (ρ)Ψ_+= 2^C_max(ρ)/d= 2^C_max(ρ)p^ICO_succ(ℑ^D).Thus, for this depasing-covariant instrument ℑ^D=_k_k,p_succ(ℑ^D,ρ)/p^ICO_succ(ℑ^D)≥2^C_max(ρ). Finally it is easy to see that the above proof is also true for ℑis ℑ^I or ℑ^S. Note that the phasing discrimination game studied in <cit.> is just a special case of the subchannel discrimination in the dephasing-covariant instruments. In the phasing discrimination game, the phase ϕ_k is encoded into a diagonal unitary U_ϕ_k=∑_j e^ijϕ_kj. Thus the discrimination of a collection of phase ϕ_k with a prior probability distribution p_k is equivalent to the discrimination of the set of subchannel _k_k, where _k=p_k𝐔_k and 𝐔_k(·)=U_ϕ_k(·) U^†_ϕ_k.§ C^Ε_MAX AS A LOWER BOUND OF ONE-SHOT COHERENCE COST The ϵ-smoothed max-relative entropy of coherence of a quantum state ρ is defined by,C^ϵ_max(ρ):=min_ρ'∈ B_ϵ(ρ)C_max(ρ'),where B_ϵ(ρ):=ρ'≥0:ρ'-ρ_1≤ϵ, ρ'≤ρ. ThenC^ϵ_max(ρ) = min_ρ'∈ B_ϵ(ρ)min_σ∈D_max(ρ'||σ)= min_σ∈ D^ϵ_max(ρ||σ),where D^ϵ_max(ρ||σ)is the smooth max-relative entropy <cit.> and defined asD^ϵ_max(ρ||σ)=inf_ρ'∈ B_ϵ(ρ)D_min(ρ'||σ). [Proof of Equation (5)] Suppose ℰ is MIOsuch that F(ℰ(Ψ^M_+),ρ)^2≥ 1-ϵ and C^(1),ϵ_C,MIO(ρ)=log M. Since F(ρ,σ)^2≤ 1-1/4ρ-σ^2_1 <cit.>, then ℰ(Ψ^M_+)-ρ_1≤2√(ϵ). Thus ℰ(Ψ^M_+)∈ B_ϵ'(ρ), where ϵ'=2√(ϵ). As C_maxis monotone under MIO, we have C^ϵ'_max(ρ) ≤ C_max(ℰ(Ψ^M_+))≤ C_max(Ψ^M_+)=log M= C^(1),ϵ_C,MIO(ρ).§ EQUIVALENCE BETWEEN C_MAX AND C_R IN ASYMPTOTIC CASE We introduce several lemmas first to prove the result. For any self-adjoint operator Q on a finite-dimensional Hilbert space, Q has the spectral decomposition as Q=∑_iλ_iP_i, where P_i is the orthogonal projector onto the eigenspace of Q. Then we define the positive operator Q≥ 0=∑_λ_i≥ 0P_i, andQ>0, Q≤ 0, Q<0 are defined in a similar way.Moreover, for any two operators Q_1 and Q_2, Q_1≥ Q_2 is defined as Q_1-Q_2≥ 0.<cit.> Given two quantum states ρ,σ∈(), thenD^ϵ_max(ρ||σ)≤λfor any λ∈ and ϵ=√(8ρ>2^λσρ).Note that in <cit.>, the above lemma is proved for bipartite states. However, this lemma also holds for any state. (Fannes-Audenaert Inequality <cit.>) For any two quantum states ρ and σ with ϵ=1/2ρ-σ_1, the following inequality holds:|S(ρ)-S(σ)|≤ϵlog(d-1)+H_2(ϵ),where d is the dimension of the system and H_2(ϵ) =-ϵlogϵ-(1-ϵ)log(1-ϵ) is the binary Shannon entropy. Based on these lemmas, we can prove the equivalence between C_max and C_r in asymptotic limit. [Proof of Equation (6)] First, we prove thatC_r(ρ)≤lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n). SinceC^ϵ_max(ρ^ n)= min_σ_n∈_nD^ϵ_max(ρ^ n||σ_n)= D_max(ρ_n,ϵ||σ̃_n),where ρ_n,ϵ∈ B_ϵ(ρ^ n), σ̃_n∈_n and _n is the set of incoherent states of ^ n.Due to the definition of _max, we haveρ_n,ϵ≤ 2^C^ϵ_max(ρ^ n)σ̃_n.ThenC_r(ρ_n,ϵ)≤S(ρ_n,ϵ||σ̃_n)= ρ_n,ϵlogρ_n,ϵ-ρ_n,ϵlogσ̃_n≤ ρ_n,ϵ(C^ϵ_max(ρ^ n)+logσ̃_n)-ρ_n,ϵlogσ̃_n≤C^ϵ_max(ρ^ n),where ρ_n,ϵ≤ρ^ n=1. Besides, as ρ_n,ϵ-ρ^ n≤ϵ, due to the Fannes-Audenaert Inequality (<ref>), we haveC_r(ρ_n,ϵ)≥C_r(ρ^ n)-ϵlog(d-1)-H_2(ϵ)= nC_r(ρ)-ϵlog(d-1)-H_2(ϵ).Thus,lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n) ≥ C_r(ρ). Next, we prove thatC_r(ρ)≥lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n).Consider the sequence ρ̂=ρ^ n^∞_n=1 and σ̂_I=σ^ n_I^∞_n=1, where σ_I∈ such that C_r(ρ)=S(ρ||σ_I)=min_σ∈ S(ρ||σ). DenoteD(ρ̂||σ̂) :=infγ:lim_n→∞supρ^ n≥ 2^nγσ^ n_Iρ^ n=0 .Due to the Quantum Stein's Lemma <cit.>,D(ρ̂||σ̂) =S(ρ||σ_I)=C_r(ρ). For any δ>0, let λ=D(ρ̂||σ̂)+δ=S(ρ||σ_I)=C_r(ρ)+δ. Due to the definition of the quantity D(ρ̂||σ̂), we havelim_n→∞supρ^ n≥ 2^nλσ^ n_Iρ^ n=0.Thenfor any ϵ>0, there exists an integer N_0 such that for any n≥ N_0, ρ^ n≥ 2^nλσ^ n_Iρ^ n<ϵ^2/8. According to Lemma <ref>, we haveD^ϵ_max(ρ^ n||σ^ n_I)≤ nλ =nC_r(ρ)+nδ.for n≥ N_0. Hence,C^ϵ_max(ρ^ n)≤ nC_r(ρ)+nδ.Thereforelim_ϵ→ 0lim_n→∞1/n C^ϵ_max(ρ^ n)≤ C_r(ρ)+δ.Since δ is arbitrary, lim_ϵ→ 0lim_n→∞1/n C^ϵ_max(ρ^ n)≤ C_r(ρ).§C_MIN MAY INCREASE ON AVERAGE UNDER IOFor pure state |ψ⟩=∑^d_i=1ψ_i|i⟩, we have C_min(ψ)=-logmax_i|ψ_i|^2. According to <cit.>, if C_min is nonincreasing on average under IO, it requires that C_min should be a concave function of its diagonal part for pure state. However, C_min(ψ)=-logmax_i|ψ_i|^2 is convex on the diagonal part of the pure states, hence C_min may increase on average under IO.Besides, according the definition of C_min for pure state |ψ⟩, one has2^-C_min(ψ) =max_σ∈F(ψ,σ)^2.However, this equality does not hold for any states. For any quantum state, the inequality (13) in the main context holds.[Proof of Equation (13)] There exists a σ_*∈ such that F(ρ,σ_*)^2=max_σ∈F(ρ,σ)^2. Let us consider the spectrum decomposition of the quantum state ρ, ρ=∑_iλ_iψ_i with λ_i>0 and ∑_iλ_i=1. Then the projector Π_ρ onto the support of ρcan be written as Π_ρ=∑_iψ_i and 2^-C_min(ρ)=max_σ∈Π_ρσ.Besides, there exists pure state decomposition of σ_*=∑_iμ_iϕ_i such that F(ρ,σ_*)=∑_i√(λ_iμ_i)ψ_iϕ_i <cit.>. ThusF(ρ,σ_*)^2 = (∑_i√(λ_iμ_i)ψ_iϕ_i)^2≤ (∑_iλ_i)(∑_iμ_iψ_iϕ_i)≤ ∑_iμ_iΠ_ρϕ_i= Π_ρσ_*≤2^-C_min(ρ),where the first inequality is due to the Cauchy-Schwarz inequality and the second inequality comes from the fact that ∑_iλ_i=1 and ψ_i≤Π_ρ for any i.§ EQUIVALENCE BETWEEN C_MIN AND C_R IN ASYMPTOTIC CASEFor any ϵ>0, the smooth min-relative entropy of coherence of a quantum state ρ is defined as followsC^ϵ_min(ρ):=max_ 0≤ A≤𝕀 Aρ≥ 1-ϵmin_σ∈-logAσ,where 𝕀 denotes the identity. ThenC^ϵ_min(ρ)= max_ 0≤ A≤𝕀 Aρ≥ 1-ϵmin_σ∈-logAσ= min_σ∈max_ 0≤ A≤𝕀 Aρ≥ 1-ϵ -logAσ= min_σ∈D^ϵ_min(ρ||σ),where D^ϵ_min(ρ||σ) is the smooth min-relative entropy <cit.> and defined asD^ϵ_min(ρ||σ)= sup_ 0≤ A≤𝕀 Aρ≥ 1-ϵ -logAσ. Given a quantum state ρ∈(), for any ϵ>0,C^ϵ_min(ρ)≤ C^ϵ_max(ρ)-log(1-2ϵ). SinceC^ϵ_max(ρ) = min_σ∈D^ϵ_max(ρ||σ),C^ϵ_min(ρ) = min_σ∈D^ϵ_min(ρ||σ),we only need to prove thatfor any two states ρ and σ,D^ϵ_min(ρ||σ) ≤ D^ϵ_max(ρ||σ)-log(1-2ϵ). First, there exists a ρ_ϵ∈ B_ϵ(ρ) such that D^ϵ_max(ρ||σ)=D_max(ρ_ϵ||σ)=logλ. Hence λσ-ρ_ϵ≥ 0.Second, let 0≤ A≤𝕀, Aρ≥ 1-ϵ such that D^ϵ_min(ρ||σ)=-logAσ. Since for any two positive operators A and B, AB≥ 0. Therefore (λσ-ρ_ϵ)A≥0, that is,Aρ_ϵ≤λAσ.Since ρ-ρ_ϵ_1=|ρ-ρ_ϵ|<ϵ and Aρ≥1-ϵ, one gets|Aρ_ϵ-Aρ|≤ A|ρ-ρ_ϵ|≤ |ρ-ρ_ϵ|<ϵ.Thus,Aρ_ϵ≥Aρ-ϵ≥ 1-2ϵ,which implies that1-2ϵ≤λAσ.Take logarithm on both sides of the above inequality, we have-logAσ≤logλ-log(1-2ϵ).That is,D^ϵ_min(ρ||σ) ≤ D^ϵ_max(ρ||σ)-log(1-2ϵ).The following lemma is a kind of generalization of the Quantum Stein' Lemma <cit.> for the special case of the incoherent state set , as the the set of incoherent states satisfies the requirement in <cit.>.Note that this lemma can be generalized to any quantum resource theory which satisfies some postulates <cit.> and it is called the exponential distinguishability property (EDP) (see <cit.>).Given a quantum state ρ∈(),(Direct part) For any ϵ>0, there exists a sequence of POVMs A_n, 𝕀-A_n_n such thatlim_n→∞(𝕀-A_n)ρ^ n=0,and for every integer n and incoherent state w_n∈_n with _n is the set of incoherent states on ^ n,-logA_nw_n/n+ϵ≥ C_r(ρ). (Strong converse) If there exists ϵ>0 and a sequence of POVMs A_n, 𝕀-A_n_n such that for every integer n>0 and w_n∈_n,-logA_nw_n/n-ϵ≥ C_r(ρ),thenlim_n→∞(𝕀-A_n)ρ^ n=1. Now, we are ready to prove the equivalence between C_min and C_r in asymptotic limit. [Proof of Equation (18)] First we prove thatlim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n) ≤ C_r(ρ).Since C_r(ρ)=lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n) and C^ϵ_min(ρ)≤ C^ϵ_max(ρ)-log(1-2ϵ), then we havelim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n) ≤ C_r(ρ).Now we prove thatlim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n) ≥ C_r(ρ)According to Lemma <ref>, for any ϵ>0, there exists a sequence of POVMs A_n such that for sufficient large integer n,ρ^ nA_n≥1-ϵ, and thusC^ϵ_min(ρ^ n) ≥min_σ∈-logA_nσ≥ n(C_r(ρ)-ϵ),where the last inequality comes from the direct part of Lemma <ref>. Therefore,lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n) ≥ C_r(ρ).§ C^Ε_MIN AS AN UPPER BOUND OF ONE-SHOT DISTILLABLE COHERENCEGiven a quantum state ρ∈(), then for any ℰ∈ MIO,C^ϵ_min(ℰ(ρ)) ≤ C^ϵ_min(ρ). Let 0≤ A≤𝕀 and Aℰ(ρ)≥ 1-ϵ such thatC^ϵ_min(ℰ(ρ)) =min_σ∈-logAσ =-logAσ_*.ThenC^ϵ_min(ρ)≥-logℰ^†(A)σ_*= -logAℰ(σ_*)≥ min_σ∈-logAσ= C^ϵ_min(ℰ(ρ)),where the first inequality comes from the fact that ℰ^†(A)ρ=Aℰ(ρ)≥ 1-ϵ and 0≤ℰ^†(A)≤𝕀 as 0≤ A≤𝕀 and ℰ^† is unital. [Proof of Equation (17)] Suppose that ℰ is the optimal MIO such that F(ℰ(ρ), Ψ^M_+)^2≥1-ϵ with log M=C^(1),ϵ_D,MIO(ρ). By Lemma <ref>, we haveC^ϵ_min(ρ)≥C^ϵ_min(ℰ(ρ))= max_ 0≤ A≤𝕀 Aℰ(ρ)≥ 1-ϵmin_σ∈-logAσ≥ min_σ∈-logΨ^M_+σ= log M=C^(1),ϵ_D,MIO(ρ),where the second inequality comes from the fact that 0≤Ψ^M_+≤𝕀 and Ψ^M_+ℰ(ρ)=F(ℰ(ρ),Ψ^M_+)^2≥1-ϵ. | http://arxiv.org/abs/1707.08795v2 | {
"authors": [
"Kaifeng Bu",
"Uttam Singh",
"Shao-Ming Fei",
"Arun Kumar Pati",
"Junde Wu"
],
"categories": [
"quant-ph",
"math-ph",
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],
"primary_category": "quant-ph",
"published": "20170727094131",
"title": "Max- relative entropy of coherence: an operational coherence measure"
} |
example.epsgsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore=10000 łtheoTheorem[section]lem[theo]Lemma sublem[theo]Sublemma prop[theo]Proposition conj[theo]Conjecture cor[theo]Corollary rem[theo]Remark definit[theo]Definition | http://arxiv.org/abs/1707.09041v4 | {
"authors": [
"Giorgio Patrizio",
"Andrea Spiro"
],
"categories": [
"math.CV",
"math.DG",
"32Q45, 32U35, 32G05, 32W20"
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"primary_category": "math.CV",
"published": "20170727205339",
"title": "Propagation of regularity for Monge-Ampère exhaustions and Kobayashi metrics"
} |
[ [ December 30, 2023 =====================Identification and counting of cells and mitotic figures is a standard task in diagnostic histopathology. Due to the large overall cell count on histological slides and the potential sparse prevalence of some relevant cell types or mitotic figures, retrieving annotation data for sufficient statistics is a tedious task and prone to a significant error in assessment. Automatic classification and segmentation is a classic task in digital pathology, yet it is not solved to a sufficient degree. We present a novel approach for cell and mitotic figure classification, based on a deep convolutional network with an incorporated Spatial Transformer Network. The network was trained on a novel data set with ten thousand mitotic figures, about ten times more than previous data sets. The algorithm is able to derive the cell class (mitotic tumor cells, non-mitotic tumor cells and granulocytes) and their position within an image. The mean accuracy of the algorithm in a five-fold cross-validation is 91.45 %. In our view, the approach is a promising step into the direction of a more objective and accurate, semi-automatized mitosis counting supporting the pathologist. Computer GraphicsI.5.4Picture/Pattern RecognitionApplications – Computer Vision § INTRODUCTION The assessment of cell types in histology slides is a standard task in pathology. Especially in tumor diagnostics, determining the relative amount of mitotic figures, a marker for tumor proliferation and aggressiveness, is another important task for the diagnostic pathologist <cit.>.However, evaluation of the complete slide for mitotic figures is usually too time consuming in routine diagnostics. Therefore it is suggested that only 10 high power fields (an area of assumed equal size used for statistic comparison), presumed to contain the highest density of mitoses, are subjectively chosen by the pathologist. The area of these fields is, however, not well-defined, as it depends on the optical properties of the microscope and which may vary significantly in their content of mitotic figures <cit.>. The final count thus strongly depends on the randomly but not necessarily representatively selected high power fields thus the resulting mitotic count is usually observer-dependent <cit.>. In addition, mitotic figures may be very variable in their histologic phenotype, which may also lead to inter-observer variability between pathologists. The aim of this work is to develop a more objective and accurate, automatized approach to counting of mitotic figures by assisting pathologists in the selection of fields with the highest mitotic counts and with more constant parameters of mitotic figure identification.Detection and annotation of mitotic cells in histology slides is a well-known task in images processing, and subject of several challenges in recent years <cit.><cit.>.Mitosis comprises a number of different phases in the cell cycle (prophase, metaphase, anaphase, and telophase). In each phase, the nucleus is shaped differently. This means that the variance in images showing a mitotic cell is high (see figure <ref>). On top of that, there is also atypical mitosis, adding yet another factor of variance to the picture. However, publicly available databases for mitosis detection feature a rather low number of mitotic figures (e.g. the 2014 ICPR MITOS-ATYPIA-14 dataset with 873 images,the 2012 ICPR dataset with 326 images<cit.>, or the AMIDA13 dataset with 1083 images<cit.>), especially for robust detection.Automatic detection of mitotic figures has been widely performed using the classical machine learning workflow on textural, morphological and shape features (e.g. <cit.><cit.>).Cireşan et al. were the first to employ deep learning-based approaches for mitosis detection <cit.>, yielding significant improvements over traditional approaches <cit.>. Yet, deep learning technologies suffer considerably from insufficient data amounts, as they have a large number of trainable parameters and, because of this, are likely to overfit the data. Particularly in the field of mitosis detection, we assume that detection performance could be improved if the whole variance of mitotic processes can be captured in the networks, requesting for a substantial increase in training data for such networks.§ RELATED WORK Typically, the process of object detection is parted into two sub-processes: Segmentation and classification. This setup is especially sensible for histology since the images represent a large amount of data and classification is usually the more complex process compared to segmentation. Sommer et al. used pixel-wise classification for candidate retrieval and then object shape and texture features for mitotic cell classification <cit.>. Irshad used active contour models for candidate selection and statistical and morphological features for classification <cit.>. Those hand-crafted features have significant drawbacks, however: Given the often small data sets, automatic selection of features is prone to random correlation, while using higher-dimensional classification approaches on the complete set increases overfitting <cit.>. Further, it is questionable, if those approaches can represent the variability in shape and texture of mitotic figures <cit.>.Triggered by the ground-breaking initial works of Lecun <cit.>, Convolutional Neural Networks (CNN) have spread widely in the use for various image classification tasks. CNN-based recognition algorithms have won all major image recognition challenges in recent years because of their ability to capture complex shapes and still remain sensitive to minor variations in the image. In the field of mitosis detection, CNN-based approaches have been used for classification <cit.>, feature extraction <cit.> as well ascandidate generation <cit.>. Yet, CNNs, through their inherent ability to capture complex structures, are also prone to overfitting, a problem which is usually targeted by data augmentation and regularization strategies like dropout and other mechanisms or by means of transfer learning. Another regularization strategy is to constrain the capacity of the approach <cit.> by reducing effectively the free parameters of the model. We aim to attempt this by splitting the problem into an attention task and a classification task. The general issue however, that the training data might be a non-representative sample of the classification task and thus parts of real-world data are not recognized because the data set does not generalize well, can be best targeted with a bigger training data set, as it was the base for this work. § MATERIAL For this study, digital histopathological images were acquired using Aperio ScanScope (Leica Biosystems Imaging, Inc., USA) slide scanner hardware at a magnification of 400x. Candidate patches for three different cell types (mitotic cells, eosinophilic granulocytes and normal tumor cells) were annotated by an expert with profound knowledge on cell differentiation and classified by a trained pathologist. The cells were selected from histologic images of 12 different paraffin-embedded canine mast cell tumors, stained with standard hematoxylin and eosin (H&E). In order to train a deep learning classifier with a sufficient amount of data, our emphasis was not on complete annotation of the slides but on finding enough candidates for the above-mentioned cell types within the image. More specifically, the emphasis was on finding mitotic cells. Commonly, in all major related works, the number of mitotic cells in the data set was proportional to the actual occurrence in the respective slides, as whole slides where annotated, resulting in a relatively low number of mitotic cells. On the contrary, we purposely selected a similar number of cells from each category to not assume any priors in distribution. We acknowledge that this procedure might add a certain bias in cell selection, and that our dataset might not be representative. However, this argument can also be made for the case where only a small number of mitotic figures is available. Further, because of the high inter-rater variability in mitosis expert classifications, we assume that an unbiased ground truth is hard to retrieve and a minor bias by image pre-selection can be tolerated. Finally, we do not target at finding all mitotic cells, but rather to guide the pathologist in finding a representative part of the slide and to thus reduce variability in expert grading.In the data set, we have approx. 37,800 single annotations of cells of the three different types (about 10,400 granulocytes, 10,800 mitotic figures and 16,600 normal tumor cells). The majority of cells was rated by the pathologist to be normal tumor cells, however also a significant amount of mitotic cells and eosinophil granulocytes was annotated. § METHODS Spatial Transformer Networks (STN), first described by Jaderberg et. al, provide a learnable method to focus the attention of a classification network on a specific subpart of the original image <cit.>. To achieve this, parameters of an affine transformation matrix θ are regressed by the network, alongside with the optimization of the actual classifier.Spatial Transformer Networks were originally successfully employed on a distorted MNIST data set, where translation, scale, rotation and clutter were used to increase the difficulty for the detection task. The approach has shown to be able to – without any prior knowledge about the actual transform that was applied beforehand – increase accuracy of the classification network by focusing its attention to the area where the number was present and by compensating for the deformation <cit.>. The approach can be used in a joined learning approach, where both the transform and the classification are learned end-to-end, something that could be described as a weakly supervised learning approach for the transformation. The optimization on the MNIST data set is, however, a much easier task than on real-world data. In a typical patch extracted from a histology slide, a lot of similar and valid objects may be contained in the image, and joined optimization suffers from local extrema in the gradient descend approach.In this work, we aim to use STN as a method of not only directing the attention of a classification network to a sub-area of a larger image, and thus hopefully improving classification performance, but also as a segmentation approach to derive the information about where the respective cells are located.We believe that Spatial Transformer Networks are an ideal candidate for this kind of task because they can be used to model two sources of natural variance into the machine learning process with a comparatively small overhead in complexity: Scaling and translation. Scaling is relevant in microscopy for two reasons: Firstly, the actual magnification of the microscope is dependent on the optical properties of the ocular, notably on the field number<cit.>. Secondly, cells differ in size, dependent on their function and the species they originate from. §.§ Image PreprocessingAll images were cropped around the cell center in a first processing step. In a second processing step, we introduce a random translation Δ_x, Δ_y to the origin area of the input image before cropping, so it is no longer centered around the cell, i.e. the cell can be anywhere on the image, with the restriction that the whole cell will be within the image (see figure <ref>). From the introduced translation, we can derive a new ground truth transformation vectorθ = [ [ ϑ_s 0 ϑ_x; 0 ϑ_s ϑ_y ]] where ϑ_s is the (in our case) fixed scaling vector. The scaling vector is dependent on the (manually chosen) expected cell size d_c. For our data, prior investigation has shown that all typical cells in our case are fully contained within an area of 64 px around the cell center, so with d_i = 128 px being the length of the input image, we can derive: ϑ_s = d_i/d_c = 0.5The (relative) coordinate grid for the STN is spaced from -1.0 to 1.0, with 0.0 being the center pixel. The translation elements of the ground truth transformation vector in eqn. <ref> thus become: ϑ_{x,y} = - 2Δ_{x,y}/d_i §.§ Network layout Our network consists of three main blocks, as depicted in Figure <ref>: The localizer, the classifier and the Spatial Transformer Network. The localizer is a deep convolutional network with two stacked convolutional and max-pooling layers, one inception layer and two fully connected layers. Itregresses an estimate θ of the transform matrix θ.The classifier is a rather small convolutional neural network with 7 layers, using also convolutional, max-pooling and inception layers. It outputs a vector of dimension 3, which represents the class probabilities for the three cell types depicted in Figure <ref>.Inception <cit.> blocks were introduced by Szegedy et al. in 2014, and have been since then widely used in classification tasks. They are based on the idea that visual information should be processed at different scales, and described to be particularly useful for localization <cit.>. We incorporated an inception layer, much like Szegedy, between the initial convolutional and max-pooling layers and the fully connected layers. In our case, the inception layer increased convergence andperformance in both localizer and classifier.§.§ TrainingThe network was trained with the TensorFlow framework using the Adam optimizer. Each image was augmented with an arbitrarily rotated copy of itself to increase robustness of the system. To not assume priors for the cell types, the distributions for the training were made uniform by random deletion of non-minority classes within the training set. A five-fold cross-validation was used.§.§.§ Classification networkIn order to achieve good localization and classification performance, we propose a three stage process: In a first step, centered cell images are presented to the network, and the classification-part of the network is trained for 50 epochs using an initial learning rate of 10^-3. This serves as a good initialization of the network for later use. As loss function, denoting the (one hot coded) ground-truth cell class c and the estimated class probabilities c, standard cross-entropy is used:l_cla = -∑_i=1^3 ln(c_i) · c_i§.§.§ Training of the localization networkIn the next step, the localization network is trained. For this, the images were cropped with a random offset from the original image, as described in section <ref>. Knowledge of this random offset enables to define a ground truth transformation matrix θ for optimizing the network. This is used to regress the estimated transformation vector θ with its elements θ = [ [ ϑ_1 ϑ_2 ϑ_x; ϑ_3 ϑ_4 ϑ_y ]] We want θ to be an affine transform with no skew and known scale ϑ_s. To achieve this, we first derive the scaling of the estimated transform as: ϑ_s_x = √(ϑ_1^2 + ϑ_3^2) ϑ_s_y = √(ϑ_2^2 + ϑ_4^2) Further, we want the diagonal elements ϑ_1 and ϑ_4 to be equal and the off-diagonal elements ϑ_2 and ϑ_3 equal with opposite sign, resulting in a rotation matrix with scale. These constraints compile into the loss for the localization network:l_loc = |ϑ_x - ϑ_x|^2 + | ϑ_y - ϑ_y|^2 + | ϑ_s_x - ϑ_s |^2 + | ϑ_s_y - ϑ_s |^2 + | ϑ_1 - ϑ_4|^2 + | ϑ_2 + ϑ_3|^2 The rotation angle of the transform is a degree of freedom and thus not covered by the loss. The localization part of the network is trained for 200 epochs using an initial learning rate of 10^-4. §.§.§ Final refinement of the classification networkFinally, the whole network is trained for 100 epochs, using an initial learning rate of 10^-4. This final step is calculated on the translated images that were estimated by the localization network and the STN, and it is using a combined loss: l = l_loc + κ· l_cla This loss thus incorporates knowledge about the proper class of the image, about the position of the cell within the image and about the scaling of the patch representing the cell, yet the rotation angle is not known. §.§ Baseline comparison It is hard to compare our results to other authors' works, because unlike them, we consider different cell types within the image and our data set is sparse and not fully annotated. For a baseline comparison, we took a 12-layer CNN like the one described by Cireşan et al. <cit.> for Mitosis detection, but aimed at a three class problem and with an input size of 128x128 px. This classification network was trained for 200 epochs using an initial learning rate of 10^-3. § RESULTS AND DISCUSSION There were only minor differences in the results of the individual test sets in cross-validation, which is why we concatenated the respective test vectors and calculated the following metrics on the ensemble. We achieved an accuracy of 91.8 %, with precisions reaching from 90.4 % to 93.4 % and recall reaching from 90.1 % to 92.8 %, as described in table <ref>. Compared to the baseline CNN described in section <ref>, this is a significant increase, with the added benefit of retrieving also segmentation information.Regarding misclassifications, it is noteworthy that for many false decisions the root cause of error seems to be within the scope of the localizer (see right column of figure <ref>). In the top and bottom examples depicted there, the localizer selected a different cell than the one originally annotated. Particularly for tumor cells, this is not always a definite fault, since we do not consider annotation information of the direct environment of the annotated cell. If, in a direct surrounding of a tumor cell, a granulocyte or mitotic cell is present, the localizer in fact behaves completely correct in presenting this cell to the classifier. Since we do not aim at finding or classifying all cells, this is no major drawback. In fact, we inherently prioritize classification this way: Since we crop around a known sparse event (mitotic cells or granulocyte), and give this label to our classifier, we incorporate the knowledge that sparse events are more important than others into the loss function.We think that the acquired data set provides a good fundament for further approaches in mitosis detection, where in our opinion the lack of a sufficient amount of samples may limit the methodic progress. The acquired data set is also a very interesting candidate for transfer learning. Assuming that many known CNN-approaches suffer from networks that partially do not have well defined filters due to lack of training data, in-domain transfer learning from our mitosis data to other, fully labeled data sets like the competition data sets should be beneficial. § SUMMARY In this work the potential of Spatial Transformer Networks within a convolutional neural network approach, applied to segmentation and classification tasks in digital histology images, has been shown. The presented approach focuses the attention of a classification network to a part of the original image where most likely a sparsely distributed cell type (mitosis or granulocyte) can be found. Further, we have acquired and introduced a data set of cell images from different classes of H&E stained histology images, with at least ten thousand pathologist-rated samples per class. Modeling the localization and classification process independently but with a joint training cuts down on computational complexity of the overall system.We believe that this work is an important step towards a microscope-embeddable algorithm that can help the pathologist in counting of mitotic figures by finding a representative area within a histology slide, an algorithm which could reduce inter-rater-variability and thus improve overall quality of tumor grading systems.eg-alpha-doi WCRB*14[CDW*16]Chen:2016vv Chen H., Dou Q., Wang X., Qin J., Heng P. A.: Mitosis Detection in Breast Cancer Histology Images via Deep Cascaded Networks. In Thirtieth AAAI Conference on Artificial Intelligence (2016).[CGGS13]Ciresan:2013up Cireşan D. C., Giusti A., Gambardella L. M., Schmidhuber J.: Mitosis Detection in Breast Cancer Histology Images with Deep Neural Networks. International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 16, Pt 2 (2013), 411–418.[Goo16]goodfellow2016deep Goodfellow I.: Deep Learning. The MIT Press, Cambridge, Massachusetts, 2016.[Irs13]Irshad:2013tu Irshad H.: Automated Mitosis Detection in Histopathology using Morphological and Multi-channel Statistics Features. Journal of Pathology Informatics 4, 1 (2013), 10.[JSZ*15]Jaderberg:J0WMXL0g Jaderberg M., Simonyan K., Zisserman A., et al.: Spatial transformer networks. In Advances in Neural Information Processing Systems (2015), pp. 2017–2025.[LBBH98]LeCun:1998fv LeCun Y., Bottou L., Bengio Y., Haffner P.: Gradient-based Learning Applied to Document Recognition. In Proceedings of the IEEE (November 1998), vol. 86, pp. 2278–2324.[Lea96]Leardi:1996ez Leardi R.: Genetic Algorithms in Feature Selection. In Genetic Algorithms in Molecular Modeling. Elsevier, 1996, pp. 67–86.[MMG16]Meuten:2016jh Meuten D. J., Moore F. M., George J. W.: Mitotic Count and the Field of View Area. Veterinary Pathology 53, 1 (Jan. 2016), 7–9.[RRL*13]Roux:2013kn Roux L., Racoceanu D., Loménie N., Kulikova M., Irshad H., Klossa J., Capron F., Genestie C., Le Naour G., Gurcan M. N.: Mitosis Detection in Breast Cancer Histological Images - An ICPR 2012 Contest. Journal of Pathology Informatics 4 (2013), 8.[SFHG12]Sommer:2012wy Sommer C., Fiaschi L., Hamprecht F. A., Gerlich D. W.: Learning-based Mitotic Cell Detection in Histopathological Images. In Proceedings of the 21st International Conference on Pattern Recognition (ICPR2012) (2012), IEEE, pp. 2306–2309.[SLJ*14]Szegedy:2014tb Szegedy C., Liu W., Jia Y., Sermanet P., Reed S., Anguelov D., Erhan D., Vanhoucke V., Rabinovich A.: Going Deeper with Convolutions. arXiv.org (Sept. 2014). http://arxiv.org/abs/1409.4842v1 arXiv:1409.4842v1.[VvDW*14]Veta:2014bi Veta M., van Diest P. J., Willems S. M., Wang H., Madabhushi A., Cruz-Roa A., Gonzalez F., Larsen A. B. L., Vestergaard J. S., Dahl A. B., Schmidhuber J., Giusti A., Gambardella L. M., Tek F. B., Walter T., Wang C.-W., Kondo S., Matuszewski B. J., Precioso F., Snell V., Kittler J., de Campos T. E., Khan A. M., Rajpoot N. M., Arkoumani E., Lacle M. M., Viergever M. A., Pluim J. P. W.: Assessment of Algorithms for Mitosis Detection in Breast Cancer Histopathology Images. arXiv.org, 1 (Nov. 2014), 237–248. http://arxiv.org/abs/1411.5825v1 arXiv:1411.5825v1.[WCRB*14]Wang:2014ka Wang H., Cruz-Roa A., Basavanhally A., Gilmore H., Shih N., Feldman M., Tomaszewski J., Gonzalez F., Madabhushi A.: Mitosis Detection in Breast Cancer Pathology Images by Combining Handcrafted and Convolutional Neural Network Features. Journal of Medical Imaging 1, 3 (Oct. 2014), 034003. | http://arxiv.org/abs/1707.08525v1 | {
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"published": "20170726162139",
"title": "A Guided Spatial Transformer Network for Histology Cell Differentiation"
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theoremTheorem[section] definition[theorem]Definition lemma[theorem]Lemma example[theorem]Example proposition[theorem]Proposition corollary[theorem]Corollary hypothesis[theorem]Hypothesis remark[theorem]Remark | http://arxiv.org/abs/1707.08394v1 | {
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"title": "The Classical Moment Problem and Generalized Indefinite Strings"
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AIP/123-QED]Dynamics of Relativistic Shock Waves Subject to a Strong Radiation Drag: Similarity Solutions and Numerical Simulations [email protected] and Beverly Sackler School of Physics & Astronomy, Tel Aviv University, Tel Aviv 69978, Israel We examine the effect of Compton drag on the dynamics of a relativistic shock wave.Similarity solutions describing a radiation-supported shock are obtained for certainprofiles of the external radiation intensity and the density of the unshocked ejecta, and are compared with 1D numerical simulations of a blast wave expanding into an ambientmedium containing isotropic seed radiation.Both the analytic model and the simulations indicatethat under realistic conditions the radiation drag should strongly affect the dynamics and structure of the shocked layer.In particular, our calculations show significant strengthening of the reverse shock and weakening of the forward shock over time of the order of the inverse Compton cooling time. We conclude that the effect of radiation drag on the evolution of the emitting plasma should affect the resultant light curves and, conceivably, spectra of the observed emission during strong blazar flares.[ Amir Levinson December 30, 2023 =====================§INTRODUCTION High-energy emission is a defining property of compact relativistic astrophysical sources. There is ampleevidence that the emission is produced in relativistic outflows launched byan accreting black hole or a magnetar.The interaction of these outflows with their environments results in formationof expanding cocoons and shocks, that decelerate the flow and lead to lower energy emissions over alarge range of scales, e.g., afterglow emission in GRBs and radio lobes in AGNs and micro-quasars.In certain circumstances the relativistic jets interactalso with ambient radiation fields emitted by the surrounding gas. This interaction can considerably alter the dynamicsof dissipative fluid shells at relatively small scales,and is likely to be the origin of the variable gamma-energy emission observed in certain classes of high-energy sources.Direct evidence for such interactions is found in blazars, where the properties of the ambient radiation field can be measured. This radiation is contributed primarily bythe accretion disk around the black hole,by gaseous clouds in the broad lineregion (BLR) and, at larger scales, by a dusty molecular torus <cit.>. The conventional wisdom has been thatthe high-energy emission seen in powerfulblazarsresults from inverse Compton scattering of ambient seed photons by non-thermal electrons acceleratedin dissipative regions in the jet <cit.>. In most early works the dynamics of the radiating fronts is not computed in a self-consistent manner.More recent worksincorporated realistic shock models to compute light curves of flared emission <cit.>. However, the effect of the radiation drag on the structure and dynamics of the shock has been neglected. In this paper we construct a model for the dynamics of a relativistic shock wave in the presence of intense radiation field.In section <ref> we derive the governing equations of a radiation-supported shock. In section<ref>we present self-similar solutions, obtained under several simplifyingassumptions regarding the properties of the ejecta and the intensity profile of the external radiation field,and study the dependency of the evolved structure on the magnitude of the radiation drag.In section<ref>we present results of numerical simulations of a uniform, spherical ejecta propagating through an ambient mediumcontaining isotropic seed radiation.The simulations enable analysis of the temporal evolution from theinitial stage under more realistic conditions.While the analysis is motivated by the application to blazars, mainlybecause there we have sufficient information to asses the conditions in the source, it might also be relevant toother high-energy astrophysical systems. § RADIATION-SUPPORTED SHOCK MODEL Quite generally, the collision of relativistic fluid shells leads to the formationof a double-shocked layer.Some examples include the interaction of expanding ejecta withan ambient medium and collision of fast and slow shells (internal shocks). In the absence of external forces, the dynamics of the double-shock system is dictated solely by the properties of the unshocked media.In the presence of external radiation field, the shocked layers are subject to radiation drag thatcan alter the dynamics significantly.When the drag is strong enough it should lead to a weakening of the forward shock and strengthening of the reverse shock.Inessence, the compression of the shocked ejecta (or the shocked plasma of the fast shellin case of internal shocks) is supported by the radiation pressure imposed on it. Thus, as a simple approximation one can ignore the entire region beyond the contact discontinuity, and obtain solutions describing a radiation-supported shock.This is the treatment adopted in section <ref> below. The validity of this approximation is checked in section <ref> using numerical simulations. Below we adopt the following notation: The velocities of the reverse shock and the contact surface are denoted by V_r and V_c, respectively, and the corresponding Lorentz factors by Γ_r and Γ_c.Plasma quantities are denoted bylower case letters, and subscripts u and s refer to the unshocked and shocked ejecta, respectively.§.§ Flow equationsThe dynamics of a relativistic fluid is governed by the continuity equation,∂_μ(ρ u^μ)=0,and energy-momentum equations,∂/∂ x^αT^αμ=S^μ,here expressed in terms of the energy-momentum tensorT^μν=wu^μ u^ν + g^μν p,where ρ, p and w are, respectively, the proper density, pressure and specific enthalpy,u^μ=γ(1,v_1,v_2,v_3)is the fluid 4-velocity, and g^μν is the metric tensor, here taken to be the Minikowsky metric g_μν= diag(-1,1,1,1).The source term on the right hand side of Eq. (<ref>) accounts for energy and momentum exchange between the fluid and the ambient radiation field, and is derived explicitly below. The above set of equations must be augmented by an equation of state. We adopt the relation w=ρ+γ̂/γ̂-1p≡ρ+ap,with a=4 for a relativistically hot plasma and a=5/2 for sub-relativisric temperatures.We shall restrict our analysis to spherical (or conical) flows, and adopt a spherical coordinate system (r,θ,ϕ).Equations (<ref>) and (<ref>)for the shocked fluid then reduce to:∂_t(ργ)+1/r^2∂_r(r^2ργ v)=0, ∂_t(wγ^2-p)+1/r^2∂_r(r^2 wγ^2 v)=S^0,wdlnγ/dt+dp/dt-γ^-2∂_tp=v(S^r-v S^0),here d/dt=∂_r +v∂_r denotes the convective derivative.A common approach is to seek solutions of Eqs. (<ref>)-(<ref>) inside theshocked layer, given the propertiesof the unshocked flows upstream of the forward and reverse shocks as input.The jumpconditions at the forward and reverse shocks then serve as boundary conditions for the shocked flow equations.In addition, the fluid velocity and total momentum flux mustbe continuous across the contact discontinuity surface.In the approximate treatment adopted here,whereby the reverse shock is fully supported by radiation and the layer enclosed between the contact and theforward shock can be ignored, the location of the contact is fixed by the requirement that the total energy isconserved, given formally by Eq. (<ref>) below. §.§ Jump conditions at the reverse shockFor the range of conditions envisaged here the shocked plasma is optically thin.The shock, in this case, is collisionless, i.e.,mediated by collective plasma processes (as opposed to radiation mediated shocks), and its width, roughly the skin depth, is much smaller thanthe Thomson length. Thus, the shock transition layer can be treated as a discontinuity in the flow parameters.The local jump conditions at the reverse shock can be expressed in terms of the local shock velocity V_r=dR_r/dt, where R_r(t) is theshock radius at time t, as <cit.>ρ_uγ_u(v_u-V_r)=ρ_sγ_s(v_s-V_r),w_uγ_u^2v_u(v_u-V_r)+p_u=w_sγ_s^2v_s(v_s-V_r)+p_s, w_uγ_u^2(v_u-V_r)+p_uV_r=w_sγ_s^2(v_s-V_r)+p_sV_r. We shall henceforth assume that the unshocked medium is cold and set p_u=0. The jump conditions at the forward shock can, in principle, be derived in a similar manner, but are redundant under our radiation-supported shock approximation.§.§ Compton drag termsBehind the shock electrons are heated and accelerated to relativistic energies. We denote the energy distribution of electrons there by F_e(γ_e), whereγ_e=ϵ/m_ec^2 is the dimensionless electron energy, as measured in the fluid frame.The total electronproper density is then given by n_e=∫ F_e(γ_e) dγ_e, and the average energy and second energy moment by <γ_e>=1/n_e∫γ_eF_e(γ_e) dγ_e, <γ_e^2>=1/n_e∫γ_e^2F_e(γ_e) dγ_e.If only the thermal electrons are taken into account one expectsm_e<γ_e>≃ m_pγ^'_u/2 for plasmain rough equipartition, whereγ^'_u=γ_uΓ_r(1-v_uV_r) is the Lorentz factor of the upstream flow, as measured in the shock frame, and it is assumed that γ^'_u>1.If the contribution of non-thermal electrons is substantial, then <γ_e> may be larger.Since the shocked electron plasma is relativistic we have m_ec^2n_e<γ_e^2>=w<γ_e>.The derivation of the source terms is given in Ref GL15. In terms of the energy density of theradiation intercepted by the flow, u_rad(r), and the quantities defined above they can be expressed as S^0=-8/3γ_s^3<γ_e^2>u_radσ_T n_e,and S^r=v_s S^0+S^0/3γ_s^2.From Eqs. (<ref>) and (<ref>) it is readily seen that the deceleration timeof the shocked fluid is given byt_dec=-3wγ_s^2/v_s S^0=2 γ_s t_IC^', where t^'_IC≃ 3m_ec/(4σ_T u^'_rad<γ_e>) is the inverse Compton cooling time of an electron having an energy m_ec^2<γ_e>, as measured in the fluid rest frame, andu_rad^'=4γ_s^2 u_rad/3.Substantial deceleration occurs if t_dec is much shorter than the outflow time, t_f=r/c. For convenience, we define the drag coefficient asα=3t_f/t_dec=8/3 m_e c^2γ_s<γ_e> σ_T u_rad r.As noted above, for thermal electrons γ_s<γ_e>≃ (m_p/2m_e)γ^'_uγ_s =(m_p/4m_e)γ_u.If only the contribution of thermal electrons is accounted for, thenα=8 m_p σ_T/12 m^2_e c^2γ_u u_radr.The energy loss term can now be expressed as:S^0=-αγ_s^2 w r^-1. To get an estimate of the value of α anticipated in blazars, suppose that a fraction η of the nuclear luminosity, L=10^45 L_45 erg/s, is scattered across the jet.The corresponding energy density of the radiation intercepted by the jet is roughlyu_rad(r)=η L/4π r^2=3×10^-4η L_45/r_pc^2 ergscm^-3, where r_pc is the radius in parsecs. From Eq. (<ref>) one obtains α= 0.7 γ_uηL_45 r_pc^-1.A more realistic estimate can be obtained using recent calculations ofthe ambient radiation intensity contributed by the various radiation sources surrounding a blazar jet <cit.>. It is found that for a prototypical blazar the intensity profile of radiation intercepted by the jet is flat up to a distance of about one parsec,with u_rad≃ 10^-3 erg cm^-3, followed by a gradual decline . For an internal shockproduced by colliding shells, Equation (<ref>) yields for the drag coefficient α≈ 3γ_u r_pc.Typically γ_u >10, implying significant drag on sub-parsec scales in blazars.§.§ Global energy conservationThe change over time in the total energy of the shocked ejecta (or shocked fast shell) must equal the difference between the energy injected throughthe reverse shock and the energy radiated away through the contact. The energy accumulated behind the forward shock is neglected in our approximate treatment.The time change of the total energy per steradian of the shocked ejecta is∂_t E= ∂_t ∫_R_r(t)^R_c(t)T^00r^2 dr = ∫_R_r(t)^R_c(t)∂_t(T^00r^2) dr + R_c^2 T^00_s(R_c)V_c-R_r^2 T^00_s (R_r)V_s,where R_r(t) and R_c(t) are, respectively, the radii of the reverse shock and the contact surface, andV_r =dR_r/dt, V_c=dR_c/dt are the corresponding velocities.Integrating Eq. (<ref>) from the shock, R_r(t), to the contact, R_c(t), one obtains: ∫_R_r(t)^R_c(t)∂_t(T^00r^2) dr +R_c^2 w_s(R_c)γ_s^2(R_c) v_s(R_c) -R_r^2 w_s(R_r)γ_s^2(R_r) v_s(R_r)=∫_R_r(t)^R_c(t)S^0 r^2 dr.Combining Eqs. (<ref>) and (<ref>), using the jump condition(<ref>) with p_u=0 (cold ejecta) and the relation T^00=w γ^2 -p, and recalling that the fluid velocity is continuous across the contact, viz., v_s(R_c)=V_c, yields∂_t E =R_r^2 ρ_uγ_u^2(v_u-V_r)-R_c^2p_s(R_c)V_c +∫_R_r(t)^R_c(t)S^0 r^2 dr.The first term on the right hand side accounts for the power incident through the reverse shock, the second term for pdV work at the contact and the last term for radiative losses.§ SELF-SIMILAR SOLUTIONSSimilarity solutions describing the interaction of a relativistic shell with an ambient medium, in the absenceof radiative losses,were derived in Ref NS06, and their stability was subsequently analyzed <cit.>.Such solutions can be obtained for a freely expanding ejecta characterized by a velocity profile v_u=r/t at time t after the explosion, and a proper density of the formρ_u=a_u/t^3γ_u^n,where a_u is a normalization constant, and γ_u=1/√(1-v_u^2) is the corresponding Lorentz factor (it can bereadily seen that the continuity equation is satisfied for this choice of ρ_u and v_u).Self-similarity requires that the Lorentz factors of the forward shock, reverse shock and the contact discontinuity have a similartime evolution, viz., Γ^2_f=At^-m, Γ^2_r=Bt^-m, Γ^2_c=Ct^-m, whereA,B,C and m are constants determined upon matchingthe solutions in region 1 (shocked ejecta) and region 2 (shocked ambient medium) at the contact discontinuity<cit.>. For an ambient density profile of the form ρ_i∝ r^-k, the index m is given by <cit.>m=6-2k/n+2. In cases where the shocked plasma is subject to strong radiative losses it is still possible to obtain self-similar solutions provided the energy source term scales as S^0∝γ_s^2 w r^-1 (see Eq. (<ref>) and (<ref>)), that is,α in Eq. (<ref>) is constant. We shall henceforth make this assumption even though it implies a somewhat artificial intensity profile of the external radiation field.We note, however, that as long as the deceleration length is smaller than the radius of the shock these details are unimportant. This is confirmed by numerical simulations, presented in the next section.As explained above, in the presence of a strong radiation drag the dynamics of the system is dictated by the interaction of the external radiation field with the shocked plasma enclosed between the contact and the reverse shock.To a good approximation one can then ignore the contribution of the forward shock to the overall dynamics.The evolution of the radiation-supported shock still satisfiesΓ^2_r=Bt^-m,Γ^2_c=Ct^-m, but the index m is now determined from global energy conservation, as will be shown below. Now,to order O(Γ_r^-2) the trajectoryof the reverse shock is given byR_r(t)=∫_0^t(1-1/2Γ_r^2) dt^'=t-t/2(m+1)Γ_r^2,from which we obtain for the velocity of the ejecta crossing the shock: v_u(R_r)=R_r/t=1-1/[2(m+1)Γ_r^2]. The corresponding Lorentz factor is thus given, to the same order, byγ_u^2=(m+1)Γ_r^2,and the density profile byρ_u= a_u/(m+1)^n/2 B^3/mΓ_r^(6/m-n).We adopt the similarity parameter introduced in Ref BMK76,χ=[1+2(m+1)Γ_r^2](1-r/t).With this choice the reverse shock is located at χ=1, and the contact at χ_c=Γ^2_r/Γ_c^2=B/C<1. Following Ref. NS06 we define the self-similar variables, G(χ), H(χ) and F(χ), such that at the reverse shock they satisfy the boundary conditions G(1)=H(1)=F(1)=1.Upon solving the jump conditions (<ref>)-(<ref>), the shocked fluid quantities can be expressed in terms of the self-similar variables as γ_s^2=qΓ_r^2G(χ), ρ_sγ_s=m q ρ_uγ_u/(m+1)(q-1)H(χ), p_s=mρ_u/a(q-1)+2(1-√(q/(m+1))) F(χ),w_s=ρ_s+ap_s≡ K(χ)p_s,where the parameter √(q) is the only positive root of the polynomial equationγ̂x^3+(2-γ̂)√(m+1) x^2-(2-γ̂)x-γ̂√(m+1) x=0.Upon substituting these relations into Eqs. (<ref>)-(<ref>), and using Eq. (<ref>) with a constant drag coefficient, α= const [More generally, α can be taken to be any function of the similarity coordinate χ. In the presence of strong drag the width of the shocked layer is much smaller the its radius, hence we anticipate little variations in α across it.], we obtain the following set of equations for the self-similar variables:2(1 + qGχ)∂_χln F-(1-qGχ)K∂_χln G= mn-6-(m-2/3α)K/(m+1)qG,2(1 - qGχ)∂_χln F-γ̂(1+qGχ)∂_χln G= 6-mn+(m-4)γ̂-2/3(γ̂-1) α K/m+1qG,2(1 - qGχ)∂_χln H - 2∂_χln G= -(mn-m-2)/(m+1)qG,subject to the boundary conditions G(1)=F(1)=H(1)=1.These equations reduce to those derived inRef. NS06 in the special case α=0. As can be seen, at the contact, χ=χ_c, the Lorentz factor must satisfy qG_cχ_c=1, where for short we denote G(χ_c)=G_c.This relation defines the limit of integration.The self-similar equations involve two eigenvalues; the index m and the location of the contact χ_c. Thus, two conditions are needed to find them. The first one is the relation q G_cχ_c=1mentioned above. The second one is global energy conservation, Eq. (<ref>). To order O(γ_s^-2) we have T_s^00=w_sγ_s^2-p_s≃ w_sγ_s^2,r=t and dr=-t dχ/[2(m+1)Γ_r^2],from which we obtainE(t) = ∫_R_r(t)^R_c(t)T^00r^2dr = t^n m/2 q m a_u (1-√(q/(m+1)))/2(m+1)^n/2+1B^n/2[a(q-1)+2]× ∫_χ_c^1K(χ)F(χ)G(χ) dχ,and ∂_t E=nm E/2t.To the same order one has ∫_R_r^R_cS^0r^2dr = -α E/t.Substituting these results into Eq. (<ref>) yields the constraint ∫_χ_c^1K(χ) F (χ)G(χ) dχ=(m+1)/q(α+nm/2)× [ a(q-1)+2/1-√(q/(m+1))-2F_c],where F_c=F(χ_c). §.§ ResultsFor a given choice of the drag coefficient α we guess the value of m and integrate Eqs. (<ref>)-(<ref>) from χ=1 to the point χ_c at which Eq. (<ref>) is satisfied. We then check if Eq. (<ref>) is satisfied.If not, we change the value of m and repeat the process until a solution satisfying all constraints is found.Sample profiles are shown in Fig. <ref>for n=1 and different values of α.The radiation free case (α=0) is shown for a comparison.It was computed using the full solution of the two-shock model described in Ref. NS06.As seen, the width of the shocked layer, Δχ=1-χ_c, decreases with increasing drag coefficient α, as naively expected. Moreover, larger radiative losses lead to increased non-uniformityof the Lorentz factor and pressure in the shocked layer.The divergence of the density at the contactis a basic feature of the similarity solutions <cit.> and occurs even in the absence of radiative losses (α=0), as seen in the upper panel of Fig. <ref>.Figure <ref> depicts the dependenceof the index m on α. For a comparison, the value of m obtained in the case α=0 for a blast wave propagating in a uniform density medium (k=0) is m=2 (see Eq. (<ref>) with k=0 and n=1). The 4-velocity of the unshocked fluid, as measured in the frame of the reverse shock,is given byu_u^'=γ_uΓ_r(v_u-V_r)=m/2√(m+1),and it is seen that the shock becomes substantially stronger as α increases.Note that the power dissipated behind the shock, ρ_uγ^'_uu^'_u, increases roughly linearly with the index m. At sufficiently large drag the entire power incident through the shock is radiated away, and the solution becomes independent of α, as seen in Fig. <ref>.For our choice of parameters, specifically n=1, this occurs at α 50, for which m≃ 19 and u^'_u≃2.1.We emphasize that this limit can be approached provided the Lorentz factor γ_u of the unshocked shell is sufficiently large.To be more concrete, Eq. (<ref>) implies that γ_u≃ 4.6Γ_r as α→∞, while our analysis is valid only for Γ_r>>1. The Compton drag terms given in Eqs. (<ref>) and (<ref>) assume that the intensity of ambient radiation is highlybeamed in the frame of the shocked fluid. Once γ_s decelerates to modest values, γ_s1, thedrag force is strongly reduced, ultimately becoming ineffective. § NUMERICAL SIMULATIONSNumerical simulations were performed using the PLUTO code <cit.>, that was modified to incorporate energy and momentum losses of the shocked plasma through Compton scattering.To clarify the presentation we adopted an external intensity profile of the form u_s∝ r^-1 for which α in Eq. (<ref>) is constant. Over the deceleration scale this profile is a reasonable approximation to the flat profile expected in blazars <cit.>. We start with the basic spherical blast wave problem, releasing an ejecta with a uniform Lorentz factor γ_e = 10 into a stationary ambient medium (henceforth, quantities in the unshocked ejecta are designated by subscript "e"). In our setup, both the ejecta and the ambient medium are taken to be uniform initially with a density ratio ρ_e/ρ_i = 100.The calculations are performed in "simulation units" in which a fluid element traveling with a unit velocity (the speed of light) passes a unit of length perunit of time. The initial impact occurs at a radius R_0 = 10^3 and time t = 0. Due to the collision a double-shock structure forms and a shocked layer is created. As the simulation progresses the shocked layer widens andCompton drag is then applied on the shocked fluid contained between the forward and the reverse shocks.In order to suppress artificial transients created by abrupt changes we apply the radiation drag gradually by increasing the drag coefficient over time as α(t) = α (1- e^-t/t_0).As our model assumes that the drag only affects the hot shocked plasma, the in-simulation drag is applied only in the region where the shocked fluid velocity is in the range [0.1u_e, 0.9u_e], where u_e is the 4-velocity of the unshocked ejecta.Preliminaryruns have shown that for the radiation free case the shocked layer becomes sufficiently developed by t=30 (see Fig. <ref>), which is why the time constant for applying the radiation drag was chosen to be t_0=10 (by t=30 the drag reaches its maximum). For sufficiently high values of α, the shocked layer decelerates and the reverse shock quickly becomes radiation-supported.§.§ Test case As a check, we ran a test case with α=0 for the same setup described above, and compared the results with an analytic solution obtained under the thin shell approximation, whereby the shocked layers are assumed to be uniform. Under this approximation the shocked ejecta and shocked ambient medium have the same Lorentz factor, γ_s=Γ_c. In terms of the ratiosq_e=(γ_e/Γ_r)^2 and q=(γ_s/Γ_r)^2, the jump conditions, Eqs. (<ref>) - (<ref>), yield2q_e(1-√(q/q_e))+(a(q-1)+2)(q-q_e/q+1)=0,andq/q_e=3/4γ_e^2(ρ_e/ρ_i)(q_e-1)(√(q/q_e))/q(q-1)+2,where ρ_e, q and q_e are functions of time. Solving these equations for the given initial conditions we obtain q_e=3.765, q=1.2 at t=0.As the ejecta expands, its density just upstreamof the reverse shock evolves as ρ_e(t)∝ [R_r(t)]^-2. Solving the above equations at any given time t using ρ_e(t) in Eq. (<ref>),one obtains the Lorentz factor of the contact discontinuity, Γ_c(t)=γ_e√(q(t)/q_e(t)).The contact 4-velocity, U_c(t)=√(Γ_c^2(t)-1), is shown as a dashed line in Fig. <ref>, along with the 4-velocity, pressure and densityobtained from the simulations of the radiation free case (α=0), at different times.As seen from Fig. <ref>, the jump at the shock agrees well with the analytic result.§.§ Simulation ResultsFor numerical reasons we were only able to run cases with modest values of α, however these suffice to illustrate the main trends. Below we present resultsfor α =3, 6, 10, 15. The initial conditions in all cases were set as described above. Fig <ref> displays snapshots of the plasma 4-velocity at different times for α=10, showing the evolution of the entirestructure. The x-axis gives the distance from the reverse shock, x=R-R_r(t), so that the reverse shock is locatedat x=0 at all times. The vertical red and blue lines indicate the location of the contact discontinuity x_c and the forward shock x_f, respectively.The deceleration of the shocked fluid, that leads to strengthening of the reverse shock and weakening of the forward shock is clearly seen. The radiation force gives rise to compression of the fluid near the contact which is communicatedto the reverse shock and decelerates it (Fig. <ref>).The shocked layer gradually expands as time progresses, but initially less than in the case α=0.The apparent sudden expansion of the shocked layer at late times (between t=600 and t=900) commences when the shock becomes mildly relativistic, as can be observed from Fig. <ref>. This is mainly due to the fact that the width of the shocked layer evolves with time as t/Γ_r^2.We note that at such small Lorentzfactors (Γ_r<2.5 at t>600) our choice of Compton drag terms, that assume perfect beaming of the external radiation in therest frame of the shocked fluid, overestimates the actual drag.In reality the drag is expected to be suppressed as the shocked fluid becomes mildly relativistic , so that the Lorentz factor of the reverse shock will eventually saturate once Γ_r becomes sufficiently low. Figs <ref> shows the profiles of the 4-velocity, pressure and density around the shock atthe same simulation time, t=800, for four values of αas indicated in the figure label.The radius ofthe reverse shock at this time is also indicated for convenience, and it ranges from R_r≈1776 (or (R_r-R_0)/R_0 ≃ 0.77) for α=3 to R_r≈1716 for α=15.The result of the run with no radiation drag (α=0), presented in Fig. <ref>, is plotted as a dotted-dashed line and is exhibited for a comparison.The contact location is found from the density profile, noticing that, as in the radiation-free case, the density peaks towards the contact and drops toa minimum right after it. In the resulting structure we see the transition from "material - supported" to"radiation-supported" shock; as α is increased the forward shock becomes progressively weaker and ultimately negligible, while the reverse shock strengthens.This trend justifies the neglect of the shocked ambient pressure in theself-similar solution outlined in section <ref>. Fig <ref> confirms that, as long as the shock is sufficiently relativistic, the width of the shocked layer shrinks and the shocked fluid velocity decreases with increasing radiation drag, in accord with the self-similar solution.The widening of the shocked layer for the α=10 and α=15 cases stems from the transition to the mildly relativistic regime, as explained above.The increase of the pressure with increasing α further indicates strong compression by the radiation force.The formation of the cold dense shell near the contact, seen in the lower left panel of Fig. <ref>, results from the rapid cooling of the compressed plasma. The left panel of figure <ref> exhibits the evolution of the 4-velocity of the shocked ejecta.Gradual deceleration of the shocked flow over a timescale of the order of the cooling time is observed, as expected. The evolution of the 4-velocity of the unshocked ejecta with respect to the reverse shock isexhibited in the right panel of figure <ref>, indicating a substantial increase in shock efficiency with increasing α. § SUMMARY AND CONCLUSIONS We studied the effect of radiation drag on the dynamics of shocks that form in relativistic outflows. Such situations are expected in cases where the outflow propagates through a quasi-isotropic, ambient radiation field, on scales at which the inverse Compton cooling time is significantly shorter than the outflow expansion time. The observations of the continuum emission in blazars suggest that these conditions may prevail on sub-parsec to parsec scales in those objects. For certain profiles of the external radiation intensity and the density of the unshocked ejecta we were able to find self-similar solutionsof the radiation hydrodynamics equations, describing a radiation-supported shock.We also performed 1D numerical simulations of a uniform, spherical shell interacting with an ambient medium that contains cold gas and seed radiation.For that purpose we used the PLUTO code, that we modified to incorporate energy and momentum losses of the shocked plasma through Compton scattering. In both, the analytical model and the simulation results, we find significant alteration of the shock dynamics when the ratio of dynamical time and Compton cooling time exceeds a factor of a few. Quite generally, substantial radiation drag leads to a faster deceleration, strengthening of the reverse shock and weakening of the forward shock. In the self-similar model with n=1 the 4-velocity of the upstream flow with respect to the shock increases from u^'_u≃ 0.5 for a uniform ambient density profile and no radiation (α=0), to about u^'_u≃ 2.1 in the asymptotic limit of maximum efficiency (obtained for α >50). The Lorentz factor in the asymptotic limit evolves roughly as Γ∝ t^-10.For numerical reasons the simulations were limited to modest values of α, but show similar trends. The numerical experiments enabled us to follow the evolution of the system from the onset of fluid collision, and demonstrate the gradual strengthening of the reverse shock (and weakening of the forward shock) over time. For convenience we adopted intensity profile (of the seed radiation) that scales as r^-1, which is close to the flat profile obtained for blazars from detailed calculations.Since for a sufficiently large drag the deceleration scale is much smaller than the shock radius, the details of the intensity profile has little effect on the evolution.For our choice of constant drag coefficient, the deceleration continues indefinitely. In reality the drag will be strongly suppressed once the Lorentz factor of the emitting (shocked) fluid drops to values at which our "beaming approximation" breaks down. In this regime strong cooling still ensues, but with little momentum losses.Our late time results for the cases α=10 and α=15 are therefore not reliable.More detailed calculations of the drag terms are needed to follow the evolution in the mildly relativistic regime.The effect of radiation drag on the dynamics of the shock might have important implications for the resultant emission. On the one hand, the strengthening of the reverse shock leads to enhanced efficiency of internal shocks, in particular under conditions at which the reverse shock is sub-relativistic in the absence of external radiation.On the other hand, the deceleration of the shocked fluid, that emits the observed radiation, leads to a dramatic change in the beaming factor. This temporal change in beaming factor can significantly alter the light curves, particularly for observers viewing the source off-axis.Detailed calculations of variable emission from dragged shocks is beyond the scope of this paper, but our analysis suggests that detailed emission models, at least in blazars, should account for such effects.This research was supported bya grant from the Israel Science Foundation no. 1277/13. 14 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Joshi, Marscher, and Bottcher(2014)]JMB14 author author M. Joshi, author A. P. Marscher,and author M. Bottcher,title title Seed photon fields of blazars in the internal shock scenario, @noopjournal journal Astrophys. 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Levinson, title title Pair cascades in extragalactic jets. 1: Gamma rays, @noopjournal journal Astrophys. J. volume 441,pages 79–95 (year 1995)NoStop [Ghisellini and Madau(1996)]GM96 author author G. Ghisellini and author P. Madau, title title On the origin of the gamma-ray emission in blazars, @noopjournal journal Mon. Not. Roy. Ast. Soc. volume 280, pages 67–76 (year 1996)NoStop [Joshi and Bottcher(2011)]JB11 author author M. Joshi and author M. Bottcher, title title Time-dependent radiation transfer in the internal shock model scenario for blazar jets,@noopjournal journal Astrophys. J.volume 727, pages 21 (year 2011)NoStop [Nakamura and Shigeyama(2006)]NS06 author author K. Nakamura and author T. Shigeyama, title title Self-similar solutions for the interaction of relativistic ejecta with an ambient medium, @noopjournal journal Astrophys. J. volume 645, pages 431–4325 (year 2006)NoStop [Levinson(2010)]Le10 author author A. Levinson, title title Relativistic rayleigh-taylor instability of a decelerating shell and its implications for gamma-ray bursts, @noopjournal journal GApFD volume 104, pages 85–111 (year 2010)NoStop [Golan and Levinson(2015)]GL15 author author O. Golan and author A. Levinson, title title Blazar flares from compton dragged shells, @noopjournal journal Astrophys. J. volume 809,pages 23 (year 2015)NoStop [Levinson(2009)]Le09 author author A. Levinson, title title Convective instability of a relativistic ejecta decelerated by a surrounding medium: An origin of magnetic fields in gamma-ray bursts? @noopjournal journal Astrophys. J. volume 705, pages L213–L216 (year 2009)NoStop [Blandford and McKee(1976)]BMK76 author author R. D. Blandford and author C. F. McKee, title title Fluid dynamics of relativistic blast waves, @noopjournal journal Phys. Fluids volume 19, pages 1130–1138 (year 1976)NoStop [Note1()]Note1 note More generally, α can be taken to be any function of the similarity coordinate χ. In the presence of strong drag the width of the shocked layer is much smaller the its radius, hence we anticipate little variations in α across it.Stop [Mignone et al.(2007)Mignone, Bodo, Massaglia, andet al.]PLUTO07 author author A. Mignone, author G. Bodo, author S. Massaglia,andauthor et al., title title Pluto: A numerical code for computational astrophysics,@noopjournal journal Astrophys. J. Supp. volume 170, pages 228 (year 2007)NoStop | http://arxiv.org/abs/1707.08654v1 | {
"authors": [
"Ilia Leitus",
"Amir Levinson"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726214926",
"title": "Dynamics of Relativistic Shock Waves Subject to a Strong Radiation Drag: Similarity Solutions and Numerical Simulations"
} |
Bandit Convex Optimization forScalable and Dynamic IoT ManagementTianyi Chen and Georgios B. Giannakis Work in this paper was supported by NSF 1509040, 1508993, and 1711471.T. Chen and G. B. Giannakis are with the Department of Electrical and Computer Engineering and the Digital Technology Center, University of Minnesota, Minneapolis, MN 55455 USA. Emails: {chen3827, georgios}@umn.edu December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================empty empty Robots performing manipulation tasks must operate under uncertainty about both their pose and the dynamics of the system. In order to remain robust to modeling error and shifts in payload dynamics, agents must simultaneously perform estimation and control tasks. However, the optimal estimation actions are often not the optimal actions for accomplishing the control tasks, and thus agents trade between exploration and exploitation. This work frames the problem as a Bayes-adaptive Markov decision process and solves it online using Monte Carlo tree search and an extended Kalman filter to handle Gaussian process noise and parameter uncertainty in a continuous space. MCTS selects control actions to reduce model uncertainty and reach the goal state nearly optimally. Certainty equivalent model predictive control is used as a benchmark to compare performance in simulations with varying process noise and parameter uncertainty.§ INTRODUCTION §.§ Motivation Flexibility and robustness in real-world robotic systems is essential to making them human-friendly and safe. Planning for important tasks in robotics, such as localization and manipulation, requires an accurate model of the dynamics <cit.>. However, the dynamics are often only partially known. For example, order-fulfillment robots move containers with varying loads in warehouses<cit.>, autonomous vehicles traverse variable terrain and interact with human drivers <cit.>, and nursing robotic systems use tools and interact with people <cit.>. Payload shifts, environment conditions, and human decisions can dramatically alter the system dynamics. Even for motion determined by physical laws, there are often unknown parameters like friction and inertial properties. In these cases, the robot must estimate the system dynamics to achieve its goals. Unfortunately, this estimation task often conflicts with the original goal task. The robot must balance exploration to gain knowledge of the dynamics and exploitation of its current knowledge to obtain rewards. §.§ Related Work Often this is solved with certainty equivalent control where a robot plans assuming an exactdynamics model<cit.>. This is usually the most likely or mean dynamics model. Typically, this dynamics model will update every time it receives an observation and make a new plan up to a fixed horizon that is optimal for the updated dynamics model. This approach is known as model predictive control (MPC) <cit.>. In some cases, performance can improve if a range of dynamic models are considered. This approach is known as robust MPC <cit.>. These approaches work well in many cases, however they do not encourage exploration, and are thus suboptimal for some problems <cit.>.There have been a variety of principled approaches to handle the exploration-exploitation trade-off. One body of research has referred to this challenge as the “dual control” problem <cit.>. It was shown that an optimal solution could be found using dynamic programming (DP). However, since the state, action, and belief spaces are continuous, the exact solution is generally intractable. Approximate solutions include a variety of methods such as adaptive control, sliding mode control, and stochastic optimal control <cit.>. Adaptive controllers typically first perform an estimation task to reveal unknown parameters and then perform the control task. This is suboptimal, the agent could potentially use the estimation actions to also begin performing the control task.Another popular approach is reinforcement learning (RL) <cit.>. In RL, the underlying planning problem is a discrete or continuous Markov decision process (MDP) with unknown transition probability distributions. RL agents interact with the environment to accrue rewards and may learn the transition probabilities along the way if it helps in this task. If the prior distribution of these transition probabilities is known, the policy that will collect the most reward in expectation, optimally balancing exploration and exploitation, may be calculated by solving a Bayes-adaptive Markov decision process (BAMDP) <cit.>. Unfortunately BAMDPs are, in general, computationally intractable, so approximate methods are used <cit.>. Since any BAMDP can be recast as a partially observable Markov decision process (POMDP), similar approximate solution methods are used. Both offline <cit.> and online <cit.> POMDP methods have been used for RL in discrete state spaces, and others <cit.> are easily adapted.There has been limited work on continuous spaces. Value iteration has been used for problems with Gaussian uncertainty <cit.>. Controls are locally optimized along a pre-computed trajectory as an extended Kalman filter (EKF) updates the belief over the model parameters. Exploration is encouraged by heuristically penalizing model uncertainty in the reward function. The control designer chooses how much to explore rather than the algorithm optimally calculating it. §.§ Contribution This research applies Monte Carlo tree search (MCTS) to solve a close approximation to the BAMDP for a problem with continuous state, action, and observation spaces, an arbitrary reward function, Gaussian process noise, and Gaussian uncertainty in the model parameters. An EKF updates the belief of the unknown parameters and double progressive widening (DPW) <cit.> guides the expansion of the tree in continuous spaces. <Ref> gives an introduction to the problems and methods considered, <Ref> give detailed descriptions of the problem and approach, and <Ref> shows simulations comparing MCTS with a certainty-equivalent MPC benchmark for two problems.§ BACKGROUND This section reviews techniques underlying our work: sequential decision making models, MCTS, and EKFs.§.§ MDPs, POMDPs, BAMDPs A Markov decision process is a mathematical framework for a sequential decision process in which an agent will move, typically stochastically, between states over time, accruing various rewards for entering certain states, but may affect their trajectory and rewards by taking actions at each time step. A MDP is defined by the tuple (𝒮,𝒜,T,R,γ), where: * 𝒮 is the set of states the system may reach,* 𝒜 is the set of actions the agent may take,* T(s'| s,a) is the probability of transitioning to state s' by taking action a at state s,* R(s,a) is the reward (or cost) of taking action a at state s, and* γ∈ [0,1] is the discount factor for future rewards. Solving an MDP develops a policy, π(s):𝒮→𝒜, which maps each state to an optimal action that will accrue the most rewards in expectation over some planning horizon.For infinite-horizon, discounted MDPs, the optimal policy has been shown to satisfy the Bellman equation <cit.>.For small, discrete state and action spaces, the Bellman equation may be solved explicitly with DP. However, for problems with large or continuous state and action spaces approximate solutions use approximate DP (ADP) <cit.>. Next, we discuss MCTS, the online ADP algorithm applied in this work.A partially observable MDP (POMDP) is an extension of an MDP, where the agent cannot directly observe its true state, only receiving observations which are stochastically dependent on this state. A POMDP thus contains a model of the probability of seeing a certain observation at a specific state. The agent forms a belief state, b which encodes the probability of being in each state s. The agent updates its belief at each step, depending on its previous action, the reward received, and the observation it took. Since the agent may hold any combination of beliefs about its location in the state space, the belief space ℬ is continuous, making POMDPs computationally intensive to solve.A Bayes-adaptive MDP (BAMDP) has transition probabilities that are only partially known. That is, the decision-making agent initially only knows a prior distribution of the transition probabilities. As the agent interacts with the environment, it extracts information of the transition probabilities from the history of states and actions that it has visited. A BAMDP becomes a POMDP by augmenting the state with the unknown parameters defining the transition probabilities.A POMDP is actually an MDP where the state space is the belief space of the original POMDP <cit.>, sometimes called a belief MDP. The transition dynamics of the belief MDP are defined by a Bayesian update of the belief when an action is taken and an observation received. Not all belief MDPs are POMDPs; in a POMDP, the reward function is specified only in terms of the POMDP state and action and is not a general function of the belief, so penalties cannot be explicitly assigned to uncertainty in the belief. When the belief update is computationally tractable, ADP techniques designed for MDPs may be applied to POMDPs by applying them to the corresponding belief MDP.§.§ Monte Carlo Tree Search MCTS is a sampling-based online approach for approximately solving MDPs. MCTS uses a generative model G to get a random state and reward (s',r) ∼ G(s,a). It performs a forward search through the state space, using G to draw prospective trajectories and rewards. In MCTS, a tree is created with alternating states and actions, with a roll-out performed to select a policy <cit.>. In general, MCTS involves stages: selection, expansion, rollout, and propagation <cit.>.§.§ Extended Kalman Filter In problems with linear Gaussian dynamics and observation functions, it has been shown that the optimal observer is a Kalman filter. A Kalman filter is an iterative algorithm that can exactly update Gaussian beliefs over the state, given the action taken, the observation received, and the transition and observation models <cit.>.For systems with nonlinear dynamics, an EKF may be used to approximately update the beliefs with each step <cit.>. Let x_k, a_k, and o_k be the state, action, and observation at time t=k. For a system with nonlinear-Gaussian dynamics, the transition and observation models can be expressed as x_k+1 = f(x_k,a_k) + v_ko_k = h(x_k,a_k) + w_k, where f, h are nonlinear functions and v, w are normally distributed. The Gaussian belief has mean x̂ and covariance Σ. The EKF updates the belief at each timestep by predicting a new state and covariance based on the action taken, i.e. x̂_k| k-1 = f(x̂_k-1| k-1,a_k-1)Σ_k| k-1 = F_k-1Σ_k-1 | k-1F_k-1^T + Q_k-1, where F is the linearized dynamics about the current belief. F_k-1 = . ∂ f/∂ x |_x̂_k-1 | k-1,u_k-1 A residual between the observation and predicted state is: r_k = o_k - h(x̂_k | k-1,a_k-1). The residual covariance, S_k and Kalman gain, K_k are S_k = H_k Σ_k | k-1 H_k^T + R_kK_k = Σ_k | k-1H_k^T S_k^-1 where H_k is the linearized observation function. H_k = . ∂ h/∂ x |_x̂_k | k-1,u_k-1 Finally, the belief update is completed according to x̂_k | k = x̂_k | k-1 + K_k r_kΣ_k | k = (I-K_k H_k)P_k | k-1. Due to the linearization of the dynamics and observation models, EKFs will not always converge and are generally not optimal observers. However, in practice they performwell for most systems where the true belief is unimodal.§ PROBLEM FORMULATION Let us consider a robot trying to control a system with linear-Gaussian dynamics. The system is described byx_k+1 = f(x_k,u_k;p) + v_k, where x_k, u_k are the state and control action at time t=k, p is a set of parameters that the system dynamics depend on, the process noise v_k ∼𝒩(0,Q), and f is a time-invariant function that is linear with respect to x_k and u_k, f(x_k,u_k;p) = A(p)x_k + B(p)u_k. The observation model is described by the linear equation o_k = h(x_k, u_k;p) + w_k, with observation, o_k taken at time t=k, the measurement noise w_k ∼𝒩(0,R), and h is a time-invariant linear function h(x_k,u_k;p) = C(p)x_k+D(p)u_k. While f and h are physical equations known a priori, the parameters p are not known beforehand. They can be appended to the state vector to form a state-parameter vector, s_k = [ [ x_k; p ]]. Thus, the system dynamics for s_k may be described by s_k+1 = [ [ A(p)0;0I ]] s_k + [ [ B(p);0 ]] u_k + ṽ_k, where ṽ∼𝒩(0,(Q,P)) and P is a parameter drift matrix.The observation model is described by o_k = [ [ C(p)0 ]]s_k + D(p)u_k + w_k. If we use an EKF to describe the belief about the current state in the state-parameter space, we form a belief MDP over all possible EKF states. This belief MDP is described by the tuple (𝒮,𝒜,T,R), where: * 𝒮 is the space of all possible beliefs. Since the belief maintained by the EKF is Gaussian, it can be described by the mean and covariance, b_k = (ŝ_k,Σ_k).* 𝒜 is all possible actions that the agent may take.* T(b' | b,a) is a distribution over possible EKF states after a belief update. This distribution depends on the observation model. It is difficult to represent explicitly, so it is implicitly defined by the generative model, G.* R(b,a) is a reward function for a given belief and action. It is constructed as desired for a given control task. In our work, we approximate R(b,a) = R(ŝ_k,a), a linear reward for the estimated mean state and action. The generative model for the belief MDP isb_k+1 = G(b_k, a_k)with G defined by equations (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>). The observation, o_k, used in (<ref>) is a random variable determined by (<ref>) and (<ref>); it is not the most likely observation. Solving this belief MDP gives a policy that optimally maximizes the sum of expected rewards over some planning horizon.§ APPROACHThis section discusses the use of MCTS and MPC with an EKF to control a system with unknown parameters. §.§ MCTS Our approach uses the upper confidence tree (UCT) <cit.> and DPW <cit.> extensions of MCTS. The tree is built as UCT expands action nodes maximizing an upper confidence estimate UCB(s, u) = Q̃(s,u) + c √(log N(s)/N(s,u)), where Q̃(s,u) estimates the state-action value function from rollout simulations and tree search, N(s,u) counts the times action u is taken from state s, with exploration constant, c. This balances exploration-exploitation as the tree expands.DPW defines tree growth for large or continuous state and action spaces. To avoid a shallow search, the number of children of each state-action node (s, u) is limited to k N(s,u)^α, where k and α are parameter constants tuned to control the widening of the tree. With an increase in N(s, u) the number of children also grows, widening the tree. The number of actions explored at each state is limited similarly. §.§ Model Predictive Control MPC is a technique for online calculation of a policy <cit.>. Its extensive use in control system design is due to its ability to explicitly meet state and control constraints. For our implementation of MPC, the optimization problem is constrained by the dynamics, f(x_k, u_k, p), and the maximum control effort, u_max. At each step a series of control actions to maximize a reward function are found for a fixed horizon from the current state and the first action is taken <cit.>. §.§ Basic Approach A large number of nonlinear stochastic systems can be near-optimally controlled with an EKF and MCTS or MPC. The purpose of this method is to produce and execute a control policy. Any system described by a model in the form of (<ref>) with an approximately Gaussian initial belief state can use MCTS or MPC to find a suitable control action.Taking this action propagates the true state, s, which will be partially observed. This observation and action will update the belief state with the EKF, improving our estimates about the parameters. This process is shown in Algorithm <ref>. §.§ Implementation Details Measurement noise was removed from the tests so the effects of process noise and estimated parameter uncertainty were isolated. The magnitude of control inputs was limited to u_max (see Table <ref>). Filtering the control signal is recommended for experimental validation to not damage actuators.A linear reward function was implemented with a weighted L1 norm penalty for the position, speed, and control effort R_L1(x_k,u_k) = [[ R_pos 0; 0 R_vel ]] x_k + R_u u_k . The values for R_pos, R_vel, and R_u are given in Table <ref>. For performance comparison purposes a quadratic reward function is used with a L2 norm penalty for the same terms R_L2(x_k,u_k) = x_k^T [[ R_pos 0; 0 R_vel ]] x_k + u_k^T R_u u_k. Real-world constraints were included as a minimum threshold for physical values such as mass, friction, and inertia so the dynamic models held. The MCTS rollout used a position controller to select a force proportional to the distance from the goal state. The MCTS with DPW implementation is from the POMDPs.jl package <cit.>. The optimization in the MPC controller was solved with the Convex.jl package <cit.>.§ SIMULATION AND RESULTSModels of a 1D double integrator and robot performing planar manipulation tested our MCTS approach against MPC, a certainty equivalent optimal control benchmark. §.§ 1D Double Integrator Model We first considered the control problem where an agent applies force to a point mass in one dimension with dynamics x_k+1 = [ [ 1 0; Δ t 1 ]] x_k + [ [Δ t/m; ( Δ t/m)^2 ]] f_k + v_ko_k = [ [ 1 0; 0 1 ]]x_k + w_k. Measurement noise is 0 to reduce the number of parameters.Fig. <ref> gives physical intuition to the 1D double integrator model. The point mass starts at a given position and velocity with a goal state at the origin. The state profiles for MCTS appear smoother than MPC. This is due to MPC applying large forces for short duration caused by the large penalty an L1 reward function gives small errors. §.§ Planar Manipulation (PM) ModelWe then consider a robot R pushing a box B in the plane, where the agent may apply an arbitrary force in the x- and y-directions, in addition to a torque. This problem, with its related quantities, is illustrated in Figure <ref>.We can describe the dynamics of the system about its center of mass, B_cm. These are given by ∑F⃗_B = F⃗ - μ_v v⃗ =m a⃗∑T⃗_B= T⃗ + r⃗×F⃗ = J α⃗, where μ_v is linear friction. The system in discrete-time is v_k+1 = a_kΔ t+ v_k p_k+1 = v_k Δ t + p_k, where Δ t is the discretization step. The system can be put in state-space form with the state vector x_k = [ [ p_x,k, p_y,k, θ_k, v_x,k, v_y,k,ω_k ]]^T, corresponding to the linear and angular positions and velocities of B in the global frame N. Using (<ref>) and (<ref>) and an explicit-time integration x_k+1 = f(x_k,u_k) = A x_k + B(θ) u_k + v_k, where A = [ [ 𝕀^3x3𝕀^3x3Δ t;; [-0.8em]0^3x3 𝕀^3x3 ]],B=[[ 3c 0^3x3; ;Δ t/m00;0Δ t/m0;B^3,1B^3,2Δ t/J ]],B^3,1 =Δ t/J(cos(θ)r_y + sin(θ)r_x),B^3,2 =Δ t/J(cos(θ)r_x - sin(θ)r_y),u_k = [ [ F_x,k; F_y,k; T_k ]], where m is the mass of B and J is B's moment of inertia.For a robot with noisy sensors which measure its position, velocity, and acceleration in N, the observation model isy_k = h(x_k,u_k) + w_k, where y_k = [ p_x,k,p_y,k,θ_k,v_x,k,v_y,k,ω_k,a_x,k,a_y,k,α_k ]^T.The measurement functions are given by p⃗ = r⃗_cm + r⃗_bv⃗ = v⃗_cm + ω⃗×r⃗_b α⃗ = ∑T⃗_B/J a⃗ = ∑F⃗_⃗B⃗/m+ α⃗×r⃗_b + ω⃗×(ω⃗×r⃗_b). This model uses the same stationary goal at the origin with an orientation of 0 degrees. It is given an initial position and orientation with Cartesian and angular velocities. §.§ Results The simulations performed on both models compare the rewards for MCTS and MPC for two varied parameters: process noise and initial parameter uncertainty. Fig. <ref> compares rewards of MCTS and MPC for the 1D double integrator model while varying process noise. The simulation parameters are detailed in Table <ref>. While MPC and MCTS perform similarly for small process noise, the reward accrued by MPC decreases significantly faster as process noise increases. Process noise greater than 1.0 has little impact on the reward, indicating saturation when the process noise is so large the policy cannot reach the goal region. 30 trials were run for each case with standard error of the mean indicated by the error bars. The standard deviation of the initial parameter estimate was then varied for a constant process noise with variance 3.0. This process noise level was chosen because it highlights the rapid change in reward caused by a poorer initial mass estimate. Fig. <ref> shows the MCTS reward mostly unaffected by this uncertainty, while the MPC reward decreases rapidly. This highlights the advantage of exploration-exploitation from MCTS. Searching for actions using multiple estimates of the mass and moving towards the goal allows it to get better knowledge of the system and increase reward. MPC takes the first action of an open-loop certainty-equivalent plan recalculated every step.Another interesting topic to consider is the performance of MCTS and MPC considering parameter uncertainty with different reward functions.The relative rewards of MCTS and MPC with linear (L1) and quadratic (L2) reward functions are shown in Fig. <ref>.In both cases the process noise had variance 1.0 and the initial mass estimate had variance 10.0. While MCTS performs significantly better with an L1 reward, it performs slightly worse with an L2 reward. These results indicate that, with an L2 reward in these domains, taking uncertainty into account with MCTS is not beneficial. This is due to the L2 reward function placing a small penalty on small errors, allowing MPC to perform well without considering parameter uncertainty. In another study with a quadratic state reward <cit.>, the reward function is augmented with a term penalizing parameter uncertainty. We suspect that, without this explicit stimulus to explore rather than simply exploit, planning algorithms would not need to actively gather information about the parameters to perform well in domains with quadratic reward. A chief advantage of the POMDP approach is that the solver automatically gathers information necessary to maximize the reward. Explicit penalization of uncertainty should not be necessary if the state reward function is chosen to accurately reflect theperformance requirements. It should be noted that convex reward functions were used to allow easy comparison with MPC, but there is no need for the reward function used by MCTS to be convex. Designers can use reward functions that precisely prescribe the desired behavior.The simulation for varying process noise in the PM model is displayed in Fig. <ref>. It behaves like the 1D model; MCTS has higher reward for all noise levels. As iterations per step decreases, MCTS tracks closer to MPC. This smaller reward difference than in the 1D case indicates the EKF has trouble reducing uncertainty for five estimated parameters in a non-linear and time-varying system. Thus, rewards decrease with increasing process noise for both controllers.For a constant process noise with standard deviation 0.1, again where MCTS and MPC have a large difference in reward, the standard deviation of the initial parameter estimate is varied for the PM model in Fig. <ref>. As the uncertainty of the initial parameters increases, the reward of MPC decreases at a slower rate than seen in the 1D double integrator model. This indicates difficulty stabilizing and reaching the goal for any process noise. The large standard deviations of the initial parameter estimations negatively impact MPC reward while MCTS is unaffected. For real-world tasks like warehouse robots manipulating varying loads, MCTS performs better.These results indicate MPC's open-loop planning executed in a closed-loop fashion cannot estimate parameters or system propagation as well as MCTS for even simple models with substantial noise. In terms of real-time performance, MCTS using 10 iterations per step ran with an average of 0.156 seconds per step. All simulations were run on an i7 processor. This runtime depends on implementation and hardware, but the same order of magnitude run-time indicates MCTS could be used on systems in real-time. Increasing the number of iterations per step to improve performance when computation time is available provides flexibility onboard.§ CONCLUSIONSThis paper posed the control of a robot performing a manipulation task in 1D and 2D for a payload with unknown parameters as a BAMDP. An online, sampling-based approach, MCTS, was used to approximately solve these continuous control problems with Gaussian uncertainty. Empirically, it was shown that this approach effectively balances exploration and exploitation. Even with few samples, MCTS improved parameter estimation and exploitation of the system dynamics to reach a goal state. In simulations with large process and parameter uncertainty, this approach provides policies with significantly higher reward than the commonly used MPC. MCTS algorithms may be a promising way to address estimation and control for real-world autonomous systems.§ ACKNOWLEDGEMENTS The authors thank Maxime Bouton for help with the EKF. ieeetr | http://arxiv.org/abs/1707.09055v1 | {
"authors": [
"Patrick Slade",
"Preston Culbertson",
"Zachary Sunberg",
"Mykel Kochenderfer"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170727214437",
"title": "Simultaneous active parameter estimation and control using sampling-based Bayesian reinforcement learning"
} |
arabic empty copyrightspace [ Alexey Ovchinnikov Received: date / Accepted: date ===================================Elections seem simple—aren't they just counting? But they have a unique, challenging combination of security and privacy requirements. The stakes are high; the context is adversarial; the electorate needs to be convinced that the results are correct; and the secrecy of the ballot must be ensured. And they have practical constraints: time is of the essence, and voting systems need to be affordable and maintainable, and usable by voters, election officials, and pollworkers. It is thus not surprising that voting is a rich research area spanning theory, applied cryptography, practical systems analysis, usable security, and statistics. Election integrity involves two key concepts:convincing evidence that outcomes are correct and privacy, which amounts to convincing assurance that there is no evidence about how any given person voted.These are obviously in tension. We examine how current systems walk this tightrope.§ INTRODUCTION: WHAT IS THE EVIDENCE?The Russians did three things … The third is that they tried, and they were not successful, but they still tried, to get access to voting machines and vote counting software, to play with the resultsFormer CIA Acting Director Michael Morell, Mar. 15, 2017These are baseless allegations substantiated with nothing, done on a rather amateurish, emotional levelKremlin spokesman Dmitry Peskov, Jan. 9, 2017It would take an army to hack into our voting system. Tom Hicks, EAC Commissioner, Oct. 6, 2016It is not enough for an election to produce the correct outcome. The electorate must also be convinced that the announced result reflects the will of the people. And for a rational person to be convinced requires evidence.Modern technology—computer and communications systems—isfragile and vulnerable to programming errors and undetectable manipulation. No current system that relies on electronic technology alone to capture and tally votes can provide convincing evidence that election results are accurate without endangering or sacrificing the anonymity of votes.[ Moreover, the systems that come closest are not readily usable by a typical voter. ]Paper ballots, on the other hand, have some very helpful security properties: they are readable (and countable, and re-countable) by humans; they are relatively durable; and they are tamper-evident. Votes cast on paper can be counted using electronic technology; then the accuracy of the count can be checked manually to ensure that the technology functioned adequately well. Statistical methods allow the accuracy of the count to be assessed by examining only a fraction of the ballots manually, often a very small fraction. If there is also convincing evidence that the collection of ballots has been conserved (no ballots added, lost, or modified) then this combination—voter-verifiablepaper ballots, a mechanized count, and a manual check of the accuracy of that count—can provide convincing evidence that announced electoral outcomes are correct.Conversely, absent convincing evidence that the paper trail has been conserved, a manual double-check of electronic results against the paper trail will not be convincing. If the paper trail has been conserved adequately, then a full manual tally of the ballots can correct the electronic count if the electronic count is incorrect.These considerations have led many election integrity advocates to push for a voter-verifiable paper trail (VVPAT).[ Voter-marked paper ballots or ballots marked using a ballot-marking device are preferable to VVPAT, a cash-register style printout that the voter cannot touch. ]In the 2016 presidential election, about three quarters of Americans voted usingsystems that generated voter-verifiable paper records. The aftermath of the election proved that even if 100% of voters had used such systems, it would not have sufficed to provide convincing evidence that the reported results are accurate. * No state has (or had) adequate laws or regulations to ensure that the paper trail is conserved adequately,and that provide evidence to that effect. * No state had laws or regulations that provided adequate manual scrutiny of the paperto ensure that the electronically generated results are correct; most still do not. * Many states that have a paper trail also have laws that make it hard for anyone tocheck the results using the paper trail—even candidates with war chests for litigation. Not only can other candidates fight attempts to check the results, the states themselves can fight such attempts.This treats the paper as a nuisance, rather than a safeguard. The bottom line is that the paper trail is not worth the paper it's printed on. Clearly this must change.Other techniques like software independence and end-to-end verifiability can offer far greater assurance in the accuracy of an election's outcome, but these methods have not been broadly applied. §.§ Why so hard?Several factors make it difficult to generate convincing evidence that reported results are correct.The first is the trust model.No one is trusted In any significant election, voters, election officials, and equipment and software cannot necessarily be trusted by anyone with a stake in the outcome.Voters, operators, system designers, manufacturers, and external parties are all potential adversaries. The need for evidence Because officials and equipment may not be trustworthy, elections should be evidence-based.Any observer should be able to verify the reported results based on trustworthy evidence from the voting system.Many in-person voting systems fail to provide sufficient evidence; and as we shall see Internet systems scarcely provide any at all. The secret ballot Perhaps the most distinctive element of elections is the secret ballot, a critical safeguard that defends against vote selling and voter coercion.In practical terms, voters should not be able to prove how they voted to anyone, even if they wish to do so.This restricts the types of evidence that can be produced by the voting system.Encryption alone is not sufficient, since the voters may choose to reveal their selections in response to bribery or coercion.The challenge of voting is thus to use fragile technology to produce trustworthy, convincing evidence of the correctness of the outcome while protecting voter privacy in a world where no person or machine may be trusted. The resulting voting system and its security features must also be usable by regular voters. The aim of this paper is to explain the important requirements of secure elections and the solutions already available from e-voting research, then toidentify the most important directions for research. Prior to delving into our discussion, we need to make a distinction in terminology. Pollsite voting systems are those in which voters record and cast ballots at predetermined locations, often in public areas with strict monitoring. Remote voting refers to a system where voters fill out ballots anywhere, and then send them to a central location to cast them, either physically mailing them in the case of vote-by-mail, or sending them over the Internet in the case of Internet voting.The next section defines the requirements, beginning with notions of election evidence, then considering privacy, and concluding with more general usability and security requirements. Section <ref> describes the cryptographic, statistical, and engineering tools that have been developed for designing voting systems with verifiably correct election outcomes. Section <ref> discusses the challenge of satisfying our requirements for security using the tools presented in real-worldelection systems.Section <ref> concludes with the promise and problems associated with Internet voting. § REQUIREMENTS FOR SECURE VOTING Trustworthiness before trustOnora O'Neill§.§ Trust, Verifiability, and Evidence For an election to be accepted as legitimate, the outcome should be convincing to all—and in particular to the losers—leaving no valid grounds to challenge the outcome.Whether elections are conducted by counting paper ballots by hand or using computer technology, the possibility of error or fraud necessitates assurances of the accuracy of the outcome.It is clear that a naive introduction of computers into votingintroduces the possibility of wholesale and largely undetectablefraud.If we can't detect it, how can we prevent it? §.§.§ Risk-Limiting AuditsStatistical post-election audits provide assurance that a reported outcome is correct, by examining some or all of an audit trail consisting of durable, tamper-evident, voter-verifiable records. Typically the audit trail consists of paper ballots.The outcome of an election is the set of winners. An outcome is incorrect if it differs from the set of winners output by a perfectly accurate manual tabulation of the audit trail. An audit of an election contest is a risk-limiting audit (RLA) with risk limit α if it has the following two properties: * If the reported contest outcome under audit is incorrect, the probability that the audit leads to correcting the outcome is at least 1-α.* The audit never indicates a need to alter a reported outcome that is correct. (In this context, “correct” means “what a full manual tally of the paper trail would show.” If the paper trail is unreliable, a RLA in general cannot detect that.RLAs should be preceded by “compliance audits” that check whether the audit trail itself is adequately reliable to determine who won.) Together, these two properties imply that post-RLA, either the reported set of winners is the set that a perfectly accurate hand count of the audit trail would show, or an event with probability no larger than α has occurred. (That event is that the outcome was incorrect, but the RLA did not lead to correcting the outcome.) RLAs amount to a limited form of probabilistic error correction: by relying on appropriate random sampling of the audit trail and hypothesis tests, they have a known minimum probability of correcting the outcome.They are not designed to ensure that the reported numerical tally is correct, only that the outcome is correct.The following procedure is a trivial RLA: with probability 1-α, perform a full manual tally of the audit trail. Amend the outcome to match the set of winners the full hand count shows if that set is different.The art in constructing RLAs consists of maintaining the risk limit while performing less work than a full hand count when the outcome is correct. Typically, this involves framing the audit as a sequential test of the statistical hypothesis that the outcome is incorrect. To reject that hypothesis is to conclude that the outcome is correct. RLAs have been developed for majority contests, plurality contests, and vote-for-k contests and complex social choice functions including D'Hondt and other proportional representation rules—see below.RLAs have also been devised to check more than one election contest simultaneously <cit.>.§.§.§ Software IndependenceRivest and Wack introduced a definition targeted specifically at detecting misbehavior in computer-based elections:<cit.> A voting system is software independent if an undetected change or error in its software cannot cause an undetectable change or error in an election outcome. Software independence clearly expresses that it should not be necessaryto trust software to determine election outcomes, but it does not say what procedures or types of evidence should be trusted instead.A system that is not softwareindependent cannot produce a convincing evidence trail, but neithercan a paper-based system that does not ensure that the paper trail is completeand intact, a cryptographic voting system that relies on an invalidcryptographic assumption, or a system that relies on audit proceduresbut lacks a means of assuring that those procedures are properly followed.We could likewise demand independence of many other kindsof trust assumptions: hardware, paper chain-of-custody, cryptographic setup,computational hardness, procedures, good randomness generation etc.Rivest and Wack also define a stronger form of the property that includes error recovery: <cit.> A voting system is strongly software independent if it is software independent and a detected change or error in an election outcome (due to the software) can be corrected without rerunning the election. A strongly software-independent system can recover from software errors or bugs, but that recovery in turn is generally based on some other trail of evidence.A software independent system can be viewed as a form of tamper-evident system: a material software problem leaves a detectable trace. Strongly software independent systems are resilient: not only domaterial software problems leave a trace, the overall election system can recover from a detected problem.One mechanism to provide software independence is to record votes on a paper record that provides physical evidence of voter's intent, can be inspected by the voter prior to casting the vote, and—if preserved intact—can later be manually audited to check the election outcome.Risk-limiting audits (see Section <ref>) can then achieve a pre-specified level of assurance that results are correct; machine assisted risk-limiting audits <cit.>,can help minimize the amount of labor required for legacy systems that do not provide a cast-vote record for every ballot, linked to the corresponding ballot. * How can systems handle errors in the event that elections don't verify? Can they recover? §.§.§ End-to-end verifiabilityThe concern regarding fraud and desire for transparency has motivated the security and crypto communities to develop another approach to voting system assurance: end-to-end verifiability (E2E-V). An election that is end-to-end verifiable achieves software independence together with the analagous notion of hardware independence as well as independence from actions of election personnel and vendors.Rather than attempting to verify thousands of lines of code or closely monitor all of the many processes in an election, E2E-V focuses on providing a means to detect errors or fraud in the process of voting and counting.The idea behind E2E-V is to enable voters themselves to monitor the integrity of the election; democracy for the people by the people, as it were. This is challenging because total transparency is not possible without undermining the secret ballot, hence the mechanisms to generate such evidence have to be carefully designed. (adapted from <cit.>) A voting system is end-to-end verifiable if it has the following three kinds of verifiability:* Cast as intended:Voters can independently verify that their selections are correctly recorded.* Collected as cast:Voters can independently verify that the representation of their vote is correctly collected in the tally.* Tallied as collected:Anyone can verify that every well-formed, collected vote is correctly included in the tally.If verification relies on trusting entities, software, or hardware, the voter and/or auditor should be able to choose them freely.Trusted procedures, if there are any, must be open to meaningful observation by every voter.Note that the above definition allows each voter to check that her vote is correctly collected, thus ensuring that attempts to change or delete cast votes are detected.In addition, it should also be possible to check the list of voters who cast ballots, to ensure that votes are not added to the collection (i.e., to prevent ballot-box stuffing).This is called eligibility verifiability <cit.>. =-1 §.§.§ Collection Accountability In an E2E-V election protocol, voters can check whether their votes have been properly counted, but if they discover a problem, there may not be adequate evidence to correct it. An election system that is collection-accountable provides voters with evidence of any failure to collect their votes.An election system is collection accountable if any voter who detects that her vote has not been collected has, as part of the vote-casting protocol, convincing evidence that can be presented to an independent party to demonstrate that the vote has not been collected. Another form of evidence involves providing each voter with a code representing her votes, such that knowledge of a correct code is evidence of casting a particular vote <cit.>. Yet another mechanism is a suitable paper receipt. Forensic analysis may provide evidence that this receipt was not forged by a voter <cit.>. * Can independently verifiable evidence be provided by the voting system for incorrect ballot casting? §.§.§ Dispute ResolutionWhile accountability helps secure the election process, it is not very useful if there is no way to handle disputes.If a voter claims, on the basis of accountability checks provided by a system, that something has gone wrong, there needs to be a mechanism to address this.This is known as dispute resolution:<cit.> A voting system is said to have dispute resolution if, when there is a dispute between two participants regarding honest participation, a third party can correctly resolve the dispute. An alternative to dispute resolution is dispute-freeness:<cit.> A dispute-free voting system has built-in prevention mechanisms that eliminate disputes among the active participants; any third party can check whether an active participant has cheated. * Can effective dispute resolution for all classes of possible errors exist in a given system?* Are there other reasonable definitions and mechanisms for dispute resolution?* Can a system offer complete dispute resolution capabilities in which every dispute can be adjudicated using evidence produced by the election system? §.§.§ From Verifiable to Verified Constructing a voting system that creates sufficient evidence to reveal problems is not enough on its own. That evidence must actually be used—and used appropriately—to ensure the accuracy of election outcomes.An election result may not be verified, even if it isgenerated by an end-to-end verifiable voting system. For verification of the result,we need several further conditions to be satisfied:* Enough voters and observers must be sufficiently diligent in performing the appropriate checks.* Random audits (including those initiated by voters) must be sufficiently extensive and unpredictable that changes that affect election outcomes have a high chance of being detected.* If checks fail, this must be reported to the authorities who, in turn, must take appropriate action.These issues involve complex human factors, including voters' incentives to participate in verification. Little work has been done on this aspect of the problem.An E2E-V system might give an individual voter assurance that her vote has not been tampered with if that voter performs certain checks.However, sufficiently many voters must do this in order to provide evidence that the election outcome as a whole is correct. Combining risk-limiting audits with E2E-V systems can provide a valuable layer of protection in the case that an insufficient number of voters participate in verification. Finally, another critical verification problem that has received little attention to date is how to make schemes that are recoverable in the face of errors. We do not want to have to abort and rerun an election every time a check a fails. Certain levels of detected errors can be shown to be highly unlikely if the outcome is incorrect, and hence can be tolerated.Other types and patterns of error cast doubt on the outcome and may require either full inspection or retabulation of the paper trail or, if the paper trail cannot be relied upon, a new election. Both Küsters <cit.> and Kiayias <cit.> model voter-initiated auditing <cit.> and its implications for detection of an incorrect election result.Both definitions turn uncertainty about voter initiated auditing into a bound on the probability of detecting deviations of the announced election result from the truth.* Can systems be designed so that the extent and diligence of checks performed can be measured?* Can verification checks be abstracted from voters, either by embedding them in election processes or automating them? §.§ Voter AuthenticationA significant challenge for election systems is the credentialing of voters to ensure that all eligible voters, and no one else, can cast votes.This presents numerous questions: what kinds of credentials should be used?How should they be issued?Can they be revoked or de-activated?Are credentials good for a single election or for an extended period?How difficult are they to share, transfer, steal, or forge?Can the ability to create genuine-looking forgeries help prevent coercion?These questions must be answered carefully, and until they are satisfied for remote voting, pollsite voting is the only robust way to address these questions—and even then, in-person credentialing is subject to forgery, distribution, and revocation concerns (for instance, the Dominican Republic recently held a pollsite election where voters openly sold their credentials <cit.>). In the U.S., there is concern that requiring in-person credentialing, in the form of voter ID, disenfranchises legitimate voters.* Is there a sufficiently secure way credential Internet voting?* Can a traditional PKI ensure eligibility for remote voting?* How does use of a PKI change coercion assumptions?§.§ Privacy, Receipt Freeness, and Coercion ResistanceIn most security applications, privacy and confidentiality are synonymous. In elections, however, privacy has numerous components that go well beyond typical confidentiality.Individual privacy can be compromised by “normal” electionprocesses such as a unanimous result.Voters may be coerced if they canproduce a proof of how they voted, even if they have to work to do so.Privacy for votes is a means to an end: if voters don't express their true preferences then the election may not produce the right outcome. This section gives an overview of increasingly strong definitions of what it means for voters to be free of coercion. §.§.§ Basic ConfidentialityWe will take ballot privacy to mean that the election does not leak any information about how any voter voted beyond what can be deduced from the announced results.Confidentiality is not the only privacy requirement in elections, but even simple confidentiality poses significant challenges.It is remarkable how many deployed e-voting systems have been shown to lack even the most basic confidentiality properties (e.g., <cit.>).Perhaps more discouraging to basic privacy is the fact that remote voting systems (both paper and electronic) inherently allow voters to eschew confidentiality. Because remote systems enable voters to fill out their ballots outside a controlled environment, anyone can watch over the voter's shoulder while she fills out her ballot. In an election—unlike, say, in a financial transaction—even the candidate receiving an encrypted vote should not be able to decrypt it. Instead, an encrypted (or otherwise shrouded) vote must remain confidential to keep votes from being directly visible to election authorities. Some systems, such as code voting <cit.> and the Norwegian and Swiss Internet voting schemes, defend privacy against an attacker who controls the computer used for voting; however, this relies on assumptions about the privacy and integrity of the code sheet.Some schemes, such as JCJ/Civitas <cit.>, obscure who has voted while providing a proof that only eligible votes were included in the tally.Several works <cit.> <cit.>, following Benaloh <cit.> formalize the notion of privacy as preventing an attacker from noticing when two parties swap their votes.* Can we develop more effective, verifiable forms of assurance that vote privacy is preserved? * Can we build means of privacy for remote voting through computer-based systems?§.§.§ Everlasting Privacy Moran and Naor expressed concern over what might happen to encrypted votes that can still be linked to their voter's name some decades into the future, and hence decrypted by superior technology.They define a requirement to prevent this:<cit.> A voting scheme has everlasting privacy if its privacy does not depend on assumptions of cryptographic hardness.Their solution uses perfectly hiding commitments to the votes, which are aggregated homomorphically. Instead of privacy depending upon a cryptographic hardness assumption, it is the integrity of an election that depends upon a hardness assumption; and only a real-time compromise of the assumption can have an impact.§.§.§ Systemic Privacy LossWe generally accept that without further information, a voter is more likely to have voted for a candidate who has received more votes, but additional data is commonly released which can further erode voter privacy.Even if we exclude privacy compromises, there are other privacy risks which must be managed.If voters achieve privacy by encrypting their selections, the holders of decryption keys can view their votes.If voters make their selections on devices out of their immediate control (e.g. official election equipment), then it is difficult to assure them that these devices are not retaining information that could later compromise their privacy.If voters make their selections on their own devices, then there is an even greater risk that these devices could be infected with malware that records (and perhaps even alters) their selections (see, for instance, the Estonian system <cit.>). * Are there ways to quantify systemic privacy loss?* Can elections minimize privacy loss?* Can elections provide verifiable integrity while minimizing privacy loss?§.§.§ Receipt-freenessPreventing coercion and vote-selling was considered solved with the introduction of the Australian ballot.The process of voting privately within a public environment where privacy can be monitored and enforced prevents improper influence. Recent systems have complicated this notion, however. If a voting protocol provides a receipt but is not carefully designed, the receipt can be a channel for information to the coercive adversary. Benaloh and Tuinstra <cit.> pointed out that passive privacy is insufficient for resisting coercion in elections: A voting system is receipt free if a voter is unable to prove how she voted even if she actively colludes with a coercer and deviates from the protocol in order to try to produce a proof. Traditional elections may fail receipt-freeness too.In general, if a vote consists of a long list of choices, the number of possible votes may be much larger than the number of likely voters.This is sometimes called (a failure of) the short ballot assumption <cit.>.Prior to each election, coercers assign a particular voting pattern to each voter.When the individual votes are made public, any voter who did not cast their pattern can then be found out.This is sometimes called the Italian attack, after a once prevalent practice in Sicily.It can be easily mitigated when a vote can be broken up, but is difficult to mitigate in systems like IRV in which the vote is complex but must be kept together.Mitigations are discussed in Sections <ref> and <ref>.Incoercibility has been defined and examined in the universally composable framework in the context of general multiparty computation <cit.>.These definitions sidestep the question of whether the voting function itself allows coercion (by publishing individual complex ballots, or by revealing a unanimous result for example)—they examine whether the protocol introduces additional opportunities for coercion.With some exceptions (such as <cit.>), they usually focus on a passive notion of receipt-freeness, which is not strong enough for voting. §.§.§ Coercion Resistance Schemes can be receipt-free, but not entirely resistant to coercion.Schemes like<cit.> that rely on randomization for receipt-freeness can be susceptible to forced randomization, where a coercer forces a voter to always choose the first choice on the ballot. Due to randomized candidate order, the resulting vote will be randomly distributed. If a specific group of voters are coerced in this way, it can have a disproportionate impact on the election outcome.If voting rolls are public and voting is not mandatory, this has an effect equivalent to prevent forced abstention, wherein a coercer refuses to let a voter vote. Schemes that rely on credentialing are also susceptible to coercion by forced surrender of credentials. One way to fully resist forced abstention is to obscure who voted.However, this is difficult to reconcile with the opportunity to verify that only eligible voters have voted (eligibility verifiability), though some schemes achieve both <cit.>.Moran and Naor <cit.> provide a strong definition of receipt freeness in which a voter may deviate actively from the protocol in order to convince a coercer that she obeyed.Their model accommodates forced randomization.A scheme is resistant to coercion if the voter can always pretend to have obeyed while actually voting as she likes.A voting scheme is coercion resistant if there exists a way for a coerced voter to cast her vote such that her coercer cannot distinguish whether or not she followed the coercer's instructions. Coercion resistance is defined in <cit.> to include receipt freeness and defence against forced-randomization, forced abstention and the forced surrender of credentials.More general definitions include <cit.>, which incorporates all these attacks along with Moran and Naor's notion of a coercion resistance strategy.Note that if the coercer can monitor the voter throughout the vote casting period, then resistance is futile. For in-person voting, we assume that the voter is isolated from any coercer while she is in the booth (although this is questionable in the era of mobile phones).For remote voting, we need to assume that voters will have some time when they can interact with the voting system (or the credential-granting system) unobserved. §.§.§ More Coercion ConsiderationsSome authors have tried to provide some protection against coercion without achieving full coercion resistance.Caveat coercitor <cit.> proposes the notion of coercion evidence and allows voters to cast multiple votes using the same credential. * Can we design usable, verifiable, coercion-resistant voting for a remote setting?§.§ AvailabilityDenial-of-Service (DoS) is an ever-present threat to elections which can be mitigated but never fully eliminated. A simple service outage can disenfranchise voters, and the threat of attack from foreign state-level adversaries is a pressing concern.Indeed, one of the countries that regularly uses Internet voting, Estonia, has been subject to malicious outages <cit.>. A variant of DoS specific to the context of elections is selective DoS, which presents a fundamentally different threat than general DoS. Voting populations are rarely homogeneous, and disruption of service, for instance, in urban (or rural) areas can skew results and potentially change election outcomes. If DoS cannot be entirely eliminated, can service standards be prescribed so that if an outcome falls below the standards it is vacated? Should these standards be dependent on the reported margin of victory? What, if any, recovery methods are possible?Because elections are more vulnerable to minor perturbations than most other settings, selective DoS is a concern which cannot be ignored.§.§ Usability A voting system must be usable by voters, poll-workers, election officials, observers, and so on.Voters who may not be computer literate—and sometimes not literate at all—should be able to vote with very low error rates.Although some error is regarded as inevitable, it is also critical that the interface not drive errors in a particular direction. For instance, a list of candidates that crosses a page boundary could cause the candidates on the second page to be missed. Whatever security mechanisms we add to the voting process should operate without degrading usability, otherwise the resulting system will likely be unacceptable. A full treatment of usability in voting is beyond the scope of this paper. However, we note that E2E-V systems (and I-voting systems, even when not E2E-V) add additional processes for voters and poll workers to follow.If verification processes can't be used properly by real voters, the outcome will not be properly verified.One great advantage ofstatistical audits is to shift complexity from voters to auditors. * How effectively can usability be integrated into the design process of a voting system?* How can we ensure full E2E-V, coercion resistance, etc., in a usable fashion? §.§ Local Regulatory Requirements A variety of other mechanical requirements are often imposed by legal requirements that vary among jurisdictions.For example:* Allowing voters to “write-in” vote for a candidate not listed on the ballot.* Mandating the use of paper ballots (in some states without unique identifying marks or serial numbers; in other states requiring such marks)* Mandating the use of certain social choice functions (see <ref> Complex Election Methods below).* Supporting absentee voting.* Requiring or forbidding that “ballot rotation” be used (listing the candidates in different orders in different jurisdictions).* Requiring that voting equipment be certified under government guidelines. Newer electronic and I-voting systems raise important policy challenges for real-world adoption. For example, in STAR-Vote <cit.>, there will be multiple copies of every vote record: mostly electronic records, but also paper records. There may be instances where one is damaged or destroyed and the other is all that remains. When laws speak to retention of “the ballot”, that term is no longer well-defined. Such requirements may need to be adapted to newer voting systems.§.§.§ Complex Election MethodsMany countries allow voters to select, score, or rank candidates or parties.Votes can then be tallied in a variety of complex ways <cit.>.None of the requirements for privacy, coercion-resistance, or the provision of verifiable evidence change.However, many tools that achieve these properties for traditional "first-past-the-post" elections need to be redesigned.An election method might be complex at the voting or the tallying end.For example, party-list methods such as D'Hondt andhave simple voting, in which voters select their candidate or party, but complex proportional seat allocation.Borda, Range Voting, and Approval Voting allow votes to be quite expressive but are simple to tally by addition.Condorcet's method and related functions <cit.> can be arbitrarily complex, as they can combine with any social choice function.Instant Runoff Voting (IRV) and the Single Transferable Vote (STV) are both expressive and complicated to tally.This makes for several challenges.* Which methods for cast-as-intended verification(e.g. code voting <cit.>) work for complex voting schemes?* How can we apply Risk-limiting audits to complex schemes? See Section <ref> for more detail. * How can failures of the short ballot assumption <cit.> be mitigated with complex ballots?* Can we achieve everlasting privacy for complex elections?§ HOW CAN WE SECURE VOTING?These truths are self-evident but not self-enforcing Barack ObamaThe goal of this section and the next is to provide a state-of-the-art picture of current solutions to voting problems and ongoing voting research, to motivate further work on open problems, and to define clear directions both in research and election policy.§.§ The Role of Paper and Ceremonies Following security problems with direct-recording electronic voting systems (DREs) <cit.>, many parts of the USA returned to the use of paper ballots. If secure custody of the paper ballots is assumed, paper provides durable evidence required to determine the correctness of the election outcome.For this reason, when humans vote from untrusted computers, cryptographic voting system specifications often use paper for security, included in the notions of dispute-freeness, dispute resolution, collection accountabilityand accountability <cit.> (all as defined in Section <ref>).Note that the standard approach to dispute resolution, based on non-repudiation, cannot be applied to the voting problem in the standard fashion, because the human voter does not have the ability to check digital signatures or digitally sign the vote (or other messages that may be part of the protocol) unassisted.Dispute-freeness or accountability are often achieved in a polling place through the use of cast paper ballots, and the evidence of their chain of custody (e.g., wet-ink signatures).Paper provides an interface for data entry for the voter—not simply to enter the vote, but also to enter other messages that the protocol might require—and data on unforgeable paper serves many of the purposes of digitally signed data. Thus, for example, when a voter marks a Prêt à Voter <cit.> or Scantegrity <cit.> ballot, she is providing an instruction that the voting system cannot pretend was something else. The resulting vote encryption has been physically committed to by the voting system—by the mere act of printing the ballot—before the voter “casts” her vote. Physical ceremony, such as can be witnessed while the election is ongoing, also supports verifiable cryptographic election protocols (seeSection <ref>).Such ceremonies include the verification of voter credentials, any generation of randomness if required for the choice between cast and audit, any vote-encryption-verification performed by election officials, etc.The key aspect of these ceremonies is the chance for observers to see that they are properly conducted. * Can we achieve dispute-resolution or -freeness without the use of paper and physical ceremony?§.§ Statistics and Auditing Two types of Risk Limiting Audits have been devised: ballot polling and comparison <cit.>.Both types continue to examine random samples of ballots until either there is strong statistical evidence that the outcome is correct, or until there has been a complete manual tally.“Strong statistical evidence” means that the p-value of the hypothesis that the outcome is incorrect is at most α, within tolerable risk.Both methods rely on the existence of a ballot manifest that describes how the audit trail is stored.Selecting the random sample can include a public ceremony in which observers contribute by rolling dice to seed a PRNG <cit.>.Ballot-polling audits examine random samples of individual ballots.They demand almost nothing of the voting technology other than the reported outcome. When the reported outcome is correct, the expected number of ballots a ballot-polling audit inspects is approximately quadratic in the reciprocal of the (true) margin of victory, resulting in large expected sample sizes for small margins.Comparison audits compare reported results for randomly selected subsets of ballots to manual tallies of those ballots.Comparison audits require the voting system to commit to tallies of subsets of ballots (“clusters”) corresponding to identifiable physical subsets of the audit trail.Comparison audits have two parts: confirm that the outcome computed from the commitment matches the reported outcome, and check the accuracy of randomly selected clusters by manually inspecting the corresponding subsets of the audit trail. When the reported cluster tallies are correct, the number of clusters a comparison audit inspects is approximately linear in the reciprocal of the reported margin.The efficiency of comparison audits also depends approximately linearly on the size of the clusters.Efficiency is highest for clusters consisting of individual ballots: individual cast vote records.To audit at the level of individual ballots requires the voting system to commit to the interpretation of each ballot in a way that is linked to the corresponding element of the audit trail.In addition to RLAs, auditing methods have been proposed with Bayesian <cit.> or heuristic <cit.> justifications.All post-election audits implicitly assume that the audit trail is adequately complete and accurate that a full manual count would reflect the correct contest outcome.Compliance audits are designed to determine whether there is convincing evidence that the audit trail was curated well, by checking ballot accounting, registration records, pollbooks, election procedures, physical security of the audit trail, chain of custody logs, and so on. Evidence-based elections <cit.> combine compliance audits and risk-limiting audits to determine whether the audit trail is adequately accurate, and if so, whether the reported outcome is correct.If there is not convincing evidence that the audit trail is adequately accurate and complete, there cannot be convincing evidence that the outcome is correct. §.§.§ Audits in Complex ElectionsGenerally, in traditional and complex elections, whenever an election margin is known and the infrastructure for a comparison audit is available, it is possible to conduct a rigorous risk-limiting comparison audit. This motivates many works on practical margin computation for IRV <cit.>.However, such an audit for a complex election may not be efficient,which motivates the extension of Stark's sharper discrepancy measure to D'Hondt and related schemes <cit.>.For Schulze and some related schemes, neither efficient margin computation nor any other form of RLA is known (see <cit.>); a Bayesian audit <cit.> may nonetheless be used when one is able to specify suitable priors. * Can comparison audits for complex ballots be performed without exposing voters to “Italian” attacks?* Can RLAs or other sound statistical audits be developed for systems too complex to compute margins efficiently?* Can the notion of RLAs be extended to situations where physical evidence is not available (i.e. Internet voting)?§.§ Cryptographic Tools and Designs §.§.§ Major Approaches to Voting Cryptography Typically E2E-V involves providing each voter with a protected receipt—an encrypted or encoded version of their vote—at the time the vote is cast.The voter can later use her receipt to check whether her vote is included correctly in the tabulation process.Furthermore, given the set of encrypted votes (as well as other relevant information, like the public keys), the tabulation is universally verifiable: anyone can check whether it is correct.To achieve this, most E2E-V systems rely on a public bulletin board, where the set of encrypted ballots is published in an append-only fashion. The votes can then be turned into a tally in one of two main ways. Homomorphic encryption schemes <cit.> allow the tally to be produced on encrypted votes.Verifiable shuffling transforms a list of encrypted votes into a shuffled list that can be decrypted without the input votes being linked to the (decrypted) output.There are efficient ways to prove that the input list exactly matches the output <cit.>.=-1 §.§.§ Techniques for Cast-as-Intended Verification How can a voter verify that her cast vote is the one she wanted? Code Voting, first introduced by Chaum <cit.>,gives each voter asheet of codes for each candidate.Assuming the code sheet is valid, thevoter can cast a vote on an untrusted machine by entering the code corresponding to her chosen candidate and waiting to receive the correct confirmation code.Modern interpretations of code voting include <cit.>.Code voting only provides assurance that the correct voting code reached the server, it does not of itself provide any guarantees that the code will subsequently be correctly counted. A scheme that improves on this is Pretty Good Democracy <cit.>, where knowledge of the codes is threshold shared in such a way that receipt of the correct confirmation code provides assurance that the voting code has been registered on the bulletin board by a threshold set of trustees, and hence subsequently counted. The alternative is to ask the machine to encrypt a vote directly, but verify that it does so correctly.Benaloh <cit.> developed a simple protocol to enable vote encryption on an untrusted voting machine. A voter uses a voting machine to encrypt any number of votes, and casts only one of these encrypted votes. All the other votes may be “audited” by the voter. If the encryption is audited, the voting system provides a proof that it encrypted the vote correctly, and the proof is public. The corresponding ballot cannot be cast as the correspondence between the encryption and the ballot is now public, and the vote is no longer secret. Voters take home receipts corresponding to the encryptions of their cast ballots as well as any ballots that are to be audited. They may check the presence of these on a bulletin board, and the correctness proofs of the audited encryptions using software obtained from any of several sources. However, even the most dilligent voters need only check that their receipts match the public record and that any ballots selected for audit display correct candidate selections.The correctness proofs are part of the public record that can be verified by any individual or observer that is verifying correct tallying.§.§.§ Formal models and security analyses of cast-as-intended verification protocolsIn addition to the work of Adida on assisted-human interactive proofs (AHIPs, see <cit.>), there has been some work on a rigorous understanding of one or more properties of single protocols, including the work of Moran and Naor <cit.> andKüsters et al. <cit.>.There have also been formalizations of voting protocols with human participants, such as by Moran and Naor <cit.> (for a polling protocol using tamper-evident seals on envelopes) and Kiayias et al. <cit.>.However, there is no one model that is sufficient for the rigorous understanding of the prominent protocols used/proposed for use in real elections. The absence of proofs has led to the overlooking of vulnerabilities in the protocols in the past, see <cit.>.Many systems use a combination of paper, cryptography, and auditing to achieve E2E-V in the polling place, including Markpledge <cit.>, Wombat <cit.>, Demos <cit.>, <cit.>, STAR-Vote <cit.>, and Moran and Naor's scheme <cit.>.Their properties are summarised more thoroughly in the following section. The cryptographic literature has numerous constructions of end-to-end verifiable election schemes (e.g., <cit.>).There are also detailed descriptions of what it means to verify the correctness of the output of E2E-V systems (e.g., <cit.>). Others have attempted to define alternative forms of the E2E-V properties <cit.>.There are also less technical explanations of E2E-V intended for voters and election officials <cit.>. * Can we develop a rigorous model for humans and the use of paper and ceremonies in cryptographic voting protocols?* Can we rigorously examine the combination of statistical and cryptographic methods for election verification? §.§.§ Techniques for Coercion Resistance Some simple approaches to coercion resistance have been suggested in the literature.These include allowing multiple votes with only the last counting and allowing in-person voting to override remotely cast votes (both used in Estonian, Norwegian, and Utah elections <cit.>).It is not clear that this mitigates coercion at all.Alarm codes can also be provided to voters: seemingly real but actually fake election credentials, along with the ability for voters to create their own fake credentials.Any such approach can be considered a partial solution at best, particularly given the usability challenges.One voting system, Civitas <cit.>, based on a protocol by Juels, Catalano and Jakobsson <cit.>, allows voters to vote with fake credentials to lead the coercive adversary into believing the desired vote was cast. Note that the protocol must enable universal verification of the tally from a list of votes cast with both genuine and fake credentials, proving to the verifier that only the ones with genuine credentials were tallied, without identifying which ones they were.* Can we develop cryptographic techniques that provide fully coercion resistant remote voting?§.§.§ Cryptographic Solutions in Complex ElectionsCast-as-intended verification based on creating and then challenging a vote works regardless of the scheme (e.g. Benaloh challenges).Cut-and-choose based schemes such asand Scantegrity II need to be modified to work.Both uses of end-to-end verifiable voting schemes in government elections, the Takoma Park run of Scantegrity II and the Victorian run of , used IRV (and one used STV).Verifiable IRV/STV counting that doesn't expose individual votes to the Italian attack has been considered <cit.>, but may not be efficient enough for use in large elections in practice, and was not employed in either practical implementation. =-1 * Is usable cast-as-intended verification for complex voting methods possible?§.§.§ Blockchains as a Cryptographic SolutionBlockchains provide an unexpectedly effective answer to a long-standing problem in computer science—how to form a consistent public ledger in a dynamic and fully distributed environment in which there is no leader and participants may join and leave at any time <cit.>.In fact, the blockchain process effectively selects a "random" leader at each step to move things forward, so this seems at first to be a natural fit for elections—citizens post their preferences onto a blockchain and everyone can see and agree upon the outcome of the election.However, blockchains and elections differ in significant ways. Elections typically already have central authorities to play the leadership role, an entity that administrates the election: what will be voted on, when,who is allowed to vote, etc.).This authority can also be tasked with publishing a public ledger of events.Note that (as with blockchains) there need be no special trust in a central authority as these tasks are all publicly observable.So to begin with, by simply posting something on a (digitally signed) web page, an election office can do in a single step what blockchains do with a cumbersome protocol involving huge amounts of computation.Blockchains are inherently unaccountable.Blockchain miners are individually free to include or reject any transactions they desire—this is considered a feature. To function properly in elections, a blockchain needs a mechanism to ensure all legitimate votes are included in the ledger, which leads to another problem: there's also no certainty in traditional blockchain schemes.Disputes are typically resolved with a "longest chain wins" rule.Miners may have inconsistent views of the contents of blockchains, but the incentives are structured so that the less widely held views eventually fade away—usually. This lack of certainty is not a desirable property in elections. In addition to lacking certainty and accountability, blockchains also lack anonymity. While modifications can be made to blockchain protocols to add anonymity, certainty, and accountability,balancing these modifications on top of the additional constraints of voting is difficult, and simpler solutions already exist as we discuss.In short, blockchains do not address any of the fundamental problems in elections, and their use actually makes things worse. § CURRENT SOLUTIONS I am committed to helping Ohio deliver its electoral votes to the president next year. Walden O'Dell, Diebold CEO, 2003 Below we provide a brief analysis of several real-world voting systems developed by the scientific community. These systems use the properties discussed in Sections <ref> and <ref>. We include both pollsite and remote systems. This collection is by no means exhaustive, but hopefully the abundance of verifiable, evidence-based voting systems will convince the reader thatthere are significant technological improvements that can greatly improve election security.Our analysis is graphically represented in Table 1. §.§ Pollsite SystemsThe systems below were developed specifically with the requirements from Section <ref> in mind. As such, all satisfy the end-to-end verifiability criteria from Section <ref>, and to a varying degree provide collection accountability, receipt-freeness, and coercion resistance.§.§.§ Prêt à Voter <cit.> ballots list the candidates in a pseudo-random order, and the position of the voter's mark serves as an encryption of the vote. The ballot also carries an encryption of the candidate ordering, which can be used, with the mark position, to obtain the vote.Voters can audit ballots to check that the random candidate order they are shown matches the encrypted values on their ballot. vVote In the 2014 state election the Australian state of Victoria conducted a small trial of end-to-end verifiable pollsite voting, using a system called vVote derived from <cit.>. §.§.§ ScantegrityThe Scantegrity <cit.> voter marks ballots that are very similar to optical scan ballots, with a single important difference. Each oval has printed on it, in invisible ink, a confirmation code—the encryption corresponding to this vote choice. When voters fill the oval with a special pen, the confirmation number becomes visible. The same functionality can be achieved through the use of scratch-off surfaces. Scantegrity II was used by the City of Takoma Park for its municipal elections in 2009 and 2011 <cit.>, the first secret-ballot election for public office known to use an E2E voting system within the U.S. §.§.§ VeriScan VeriScan <cit.>, like Scantegrity, uses optical scan ballots. But the ballots are ordinary – using regular ink – and are filled by voters using ordinary pens.Optical scanners used by VeriScan are augmented to hold the ballot deposited by a voter and to print a receipt consisting of an encryption of the selections made by the voter (or a hash thereof).Once the receipt has been given to the voter by the scanner, the voter can instruct the scanner to either retain the ballot or to return the ballot to the voter.A returned ballot should be automatically marked as no longer suitable for casting and effectively becomes a challenge ballot as in STAR-Vote (below).All encrypted ballots – whether cast or retained by a voter – are posted to a public web page where they can be checked against voter receipts.The cast ballots are listed only in encrypted form, but the retained ballots are listed in both encrypted and decrypted form so that voters can check the decryptions against their own copies of the ballots.§.§.§ STAR-Vote STAR-Vote <cit.> is an E2E-V, in-person voting system designed jointly with Travis County (Austin), Texas, and is scheduled for wide-spread deployment in 2018. STAR-Vote is a DRE-style touch-screen system, which prints a human-readable paper ballot which is deposited into a ballot box. The system also prints a receipt that can be taken home. These two printouts serve as evidence for audits. STAR-Vote encodes a Benaloh-style cast-or-spoil question <cit.> as the depositing of the ballot into the ballot box. Each voting machine must commit to the voter's ballot without knowing if it will be deposited and counted or spoiled and thereby challenged.STAR-Vote posts threshold encrypted cast and spoiled ballots to a web bulletin board. Voters can then check that their cast ballots were included in the tally, or that the system correctly recorded their vote by decrypting their challenged ballots. STAR-Vote is collection accountable only to the extent that paper receipts and ballot summaries are resistent to forgery.It is coercion resistant and software independent, and allows for audits of its paper records.§.§.§ PPAT While many of the above schemes provide most of the required properties laid out in Section <ref>, most do not account for everlasting privacy. However, by integrating the Perfectly Private Audit Trail (PPAT) <cit.>, many of the previously discussed systems can attain everlasting privacy. Notably, PPAT can be implemented both with mixnet schemes like Scantegrity <cit.> and Helios <cit.> as well as with homomorphic schemes like that used in STAR-Vote <cit.>. =-1§.§ Remote Systems §.§.§ RemotegrityThe Remotegrity <cit.> voting system specification provides a layer over local coded voting systems specifications to enable their use in a remote setting. It is the only known specification that enables the voter to detect and prove attempts by adversaries to change the remote vote.Voters are mailed a package containing a coded-vote ballot and a credential sheet. The sheet contains authorization codes and lock-in codes under scratch-offs, and a return code. To vote, voters scratch-off an authorization code at random and use it as a credential to enter the candidate code. The election website displays the entered information and the return code, which indicates to the voter that the vote was received.If the website displays the correct information, the voter locks it in with a random lock-in code. If not, the voter uses another computer to vote, scratching-off another authorization code. For voter-verifiability, voters may receive multiple ballots, one of which is voted on, and the others audited.The credential authority (an insider adversary) can use the credentials to vote instead of the voter. If this happens, the voter can show the unscratched-off surface to prove the existence of a problem. Remotegrity thus achieves E2E-V, collection accountability, and software independence. Since there is no secret ballot guarantee, there is no coercion resistance. Remotegrity was made available to absentee voters in the 2011 election of the City of Takoma Park, alongside in-person voting provided by Scantegrity. §.§.§ Helios Helios <cit.> is an E2E-V Internet voting system. Voters visit a web page “voting booth” to enter their selections.After voters reviewtheir ballots, each ballot is encrypted using a threshold key generated during election set up.Voters cast a ballot by entering credentials supplied for this election. Alternatively, voters can anonymously spoil their ballots to decrypt them, to show that their selections were accurately recorded. Voters can cast multiple ballots with only the last one retained, as a weak means of coercion mitigation.When the election closes, the cast votes are verifiably tallied—either using homomorphic tallying or a mixnet. Independent verifiers have been written to check the tallying and decryptions of each spoiled ballot. Confirmation that the vote is received is then emailed to the voter.Helios is used for elections by a variety of universities and professional societies including the Association for Computing Machinery and the International Association for Cryptologic Research. Helios lacks collection accountability, but is still E2E-V and software independent through its spoil function.§.§.§ Selene Selene <cit.> is a remote E2E-V system that revisits the tracker numbers of Scantegrity, but with novel cryptographic constructs to counter the drawbacks. Voters are notified of their tracker after the vote/tracker pairs have been posted to the web bulletin board, which allows coerced voters to identify an alternative tracker pointing to the coercer's required vote. Voter verification is much more transparent and intuitive, and voters are not required to check the presence of an encrypted receipt.For the same reasons as Remotegrity, Selene is software independent and provides collection accountability.* Is there a cast-as-intended method that voters can execute successfully without instructions from pollworkers?* Is it possible to make E2E-Vprotocols simpler for election officials and pollworkers to understand and administer? § INTERNET VOTING “People of Dulsford,” began Boris, “I want to assure you that asyour newly elected mayor I will not just represent the people who voted for me...”“That's good,” said Derrick, “because no-one voted for him.”“But the people who didn't vote for me as well,” said Boris. There was asmattering of half-hearted clapping from the crowd. R. A. Spratt,Nanny Piggins and the Race to PowerIn this section we present the challenges of secure Internet voting through a set of (possibly contradictory) requirements. No system has addressed the challenges sufficiently so far, and whether it is possible to do so remains an open problem.We begin by introducing prominent contemporary instances of I-voting as case studies. Then we examine the Internet voting threat model, along the way showing how these Internet systems have failed to adequately defend themselves. We look at voter authentication, verification of the correctness of a voting system's output, voter privacy and coercion resistance, protections against denial-of-service, and finally the usability and regulatory constraints faced by voting systems.One major roadblock faced exclusively by I-voting is the underlying infrastructure of the Internet. The primary security mechanism for Internet communication is Transport Layer Security (TLS), which is constantly evolving in response to vulnerabilities. For instance, the website used in the iVote systemwas vulnerable to the TLS FREAK <cit.> and LogJam <cit.> vulnerabilities.Researchers discovered this during the election period and demonstrated that they could exploit it to steal votes <cit.>. At the time, LogJam had not been publicly disclosed, highlighting the risk to I-voting from zero-day vulnerabilities. Internet voting systems must find ways to rely on properties like software independence and E2E-V before they can be considered trusted.In 2015, the U.S. Vote Foundation issued an export report on the viability of using E2E-verifiability for Internet voting <cit.>. The first two conclusions of the report were as follows. * Any public elections conducted over the Internet must be end-to-end verifiable.* No Internet voting system of any kind should be used for public elections before end-to-end verifiable in-person voting systems have been widely deployed and experience has been gained from their use. Many of the possible attacks on I-voting systems could be performed onpostal voting systems too.The main difference is the likelihood that avery small number of people could automate the manipulation of a very large number of votes, or a carefully chosen few important votes, without detection.§.§ I-voting in Government ElectionsEstonia <cit.> Estonia's I-voting deployment—the largest in the world by fraction of the electorate—was used to cast nearly a third of all votes in recent national elections <cit.>.The Estonian system uses public key cryptography to provide a digital analog of the “double envelope” ballots often used for absentee voting <cit.>. It uses a national PKI system to authenticate voters, who encrypt and digitally sign their votes via client-side software.Voters can verify[We use the term “verify” loosely in this subsection; these systems provide no guarantee that what is shown when voters “verify” their votes proves anything about the correctness of vote recording and processing.see <ref>.] that their votes were correctly received using a smartphone app, but the tallying process is only protected by procedural controls <cit.>. The voting system does not provide evidence of a correct tally, nor does it provide evidence that the vote was correctly recorded if the client is dishonest. A 2013 study showed that the Estonian system is vulnerable to vote manipulation by state-level attackers and client-side malware, and reveals significant shortcomings in officials' operational security <cit.>. iVote <cit.> The largest online voting trial by absolute number of votes occurred in 2015 in New South Wales, Australia, using a web-based system called iVote.It received 280,000 votes out of a total electorate of over 4 million.The system included a telephone-based vote verification service that allowed voters to dial in and hear their votes read back in the clear. A limited server-side auditing process was performed only by auditors selected by the electoral authority. Thus no evidence was provided that received votes were correctly included inthe tally. At election time, the electoral commission declared that,“1.7% of electors who voted using iVote also used the verification service and none of them identified any anomalies with their vote.” It emerged more than a year later that 10% of verification attempts had failed to retrieve any vote at all.This error rate, extrapolated to all 280,000 votes, would have been enough to change at least one seat.Norway <cit.> In 2011 and 2013 Norway ran trials of an I-voting system. In the 2013 trial, approximately 250,000 voters (7% of the Norwegian electorate) were able to submit ballots online <cit.>. Voters are given precomputed encrypted return codes for the various candidates they can vote for. Upon submitting a ballot, the voter receives an SMS message with the return code computed for the voter's selections. In principle, if the return codes were kept private by the election server, the voter knows the server correctly received her vote. This also means that ballots must be associated with the identity of those who cast them, enabling election officials to possibly coerce or selectively deny service to voters. The voting system did not provide publicly verifiable evidence of a correct tally.Switzerland <cit.> In Switzerland, the Federal Chancellery has produced a clear set of requirements.More stringent verifiability properties come into force as a larger fraction of the votes are carried over the Internet.Many aspects of this way of proceeding are admirable. However, the final systems are dependent on a code-verification system, and hence integrity depends on the proper and secret printing of the code sheets.If the code-printing authorities collude with compromised devices, the right verification codes can be returned when the votes are wrong.Utah <cit.> In March 2016 the Utah Republican party held its caucus, running pollsite voting in addition to an online system. Voters could register through a third-party website and have a voting credential sent to their phone via SMS or email. Any registered voter could receive a credential, but as the site was unauthenticated, anyone with a voter roll[That is, a publicly available list of registered voters, their party affiliations, home addresses, and other relevant information] could submit any registered voter's information and receive that person's credential. On the day of the election, said credentials were used to log onto the website <ivotingcenter.us> to fill out and submit ballots. The system provided voters with a receipt code that voters could check on the election website. The system does not provide evidence that the vote was correctly recorded if the client is dishonest, nor does it provide evidence of a correct tally. Election day saw many voters fail to receive their voting credentials or not be able to reach the website to vote at all, forcing as many as 13,000 of the 40,000 who attempted to register to vote online to either vote in person or not vote at all <cit.>.All of these systems place significant trust in unverifiable processes, at both client and server sides, leading to serious weaknesses in privacy and integrity.Their faults demonstrate the importance of a clear and careful trust model that makes explicit who does and does not have power over the votes of others, and reinforce the importance of providing convincing evidence of an accurate election outcome.§.§ E2E-V I-voting in Government ElectionsInternet voting presents numerous challenges that have not been adequately addressed. First among these is the coercion problem which is shared with other remote voting systems in widespread use today (such as vote-by-mail).However, I-voting exacerbates the problem by making coercion and vote-selling a simple matter of a voter providing credentials to another individual.Client malware poses another significant obstacle.While E2E-verifiability mitigates the malware risks by providing voters with alternate means to ensure that their votes have been properly recorded and counted, many voters will not avail themselves of these capabilities.We could therefore have a situation were a large-scale fraud is observed by a relatively small number of voters.While the detection of a small number of instances of malfeasance can bring a halt to an election which provides collection accountability, the required evidence can be far more fleeting and difficult to validate in an Internet setting.An election should not be overturned by a small number of complaints if there is no substantive evidence to support these complaints.Targeted denial-of-service is another serious unresolved threat to I-voting. Ordinary denial-of-service (DoS) is a common threat on the Internet, and means have been deployed to mitigate — although not eliminate — these threats. The unique aspect in elections is that while ordinary DoS can slow commerce or block access to a web site for a period, the effects of a targeted DoS attack on an election can be far more severe.Since voting paterns are far from homogeneous, an attacker can launch a targeted DoS attack against populations and regions which are likely to favor a particular candidate or position. By merely making it more difficult for people in targeted populations to vote, the result of an election can be altered.As yet, we have no effective mitigations for such attacks.Finally, as was observed in the U.S. Vote Foundation study <cit.>, we simply don't yet have much experience with large-scale deployments of E2E-verifiable election systems in the simpler and more manageable setting of in-person voting.It would be angerous to jump directly to the far more challenging setting of Internet voting with a heavy dependence on a technology that has not previously been deployed at scale.§.§ Alternatives to Internet VotingThere are numerous alternatives to Internet voting that can help enfranchise voters who can not easily access a poll site on the day of an election.Early voting is in widespread use throughout the U.S.By extending the voting window from a single day to as much as three weeks, voters who may be away or busy on the date of an election can be afforded an opportunity to vote in person, at their convenience, at a poll site with traditional safeguards. Early voting also mitigates many of the risks of traditional systems since, for example, an equipment failure ten days prior to the close of an election is far less serious than one that takes place during a single day of voting.Some U.S. jurisdictions have adopted a vote center system in which voters may vote in person outside of their home precincts.This option has been facilitated by the use of electronic poll books, and it allows voters to, for instance, vote during a lunch break from work if they will be away from their homes during voting hours.The vote center model could potentially be extended from the current model of voters away from their home precincts but still within their home counties by allowing voters to use any poll site in the state or country. It would even be possible to establish remote voting kiosks overseas in embassies, conslates, or other official sites, and roming voting kiosks could be established with as little as two poll workers and a laptop computer.Security and accountability in all of these non-local voting scenarios can be greatly enhanced by the use of E2E-verifiability.Blank-ballot electronic delivery is another option which has gained in popularity. While there are numerous risks in using the Internet for casting of ballots, the risks a far less in simply providing blank ballots to voters. Electronic delivery of blank-ballots can save half of the round-trip time that is typical in absentee voting, and traditional methods of ballot return can be used which are less susceptible to the large-scale attacks that are possible with full Internet voting. § A LOOK AHEADThere is no remedy now to a process that was so opaque that it could have been manipulated at any stageMichael Meyer-Resende and Mirjam Kunkler, on the Iranian 2009 Presidential electionVoting has always used available technology, whether pebbles dropped in an urn or marked paper put in a ballot box; it now uses computers, networks, and cryptography. The core requirement,to provide public evidence of the right result from secret ballots, hasn't changed in 2500 years.Computers can improve convenience and accessibility over plain paper and manual counting. In the polling place there are good solutions, including Risk Limiting Audits and end-to-end verifiable systems. These must be more widely deployed and their options for verifying the election result must actually be used.Many of the open problems described in this paper—usable and accessible voting systems, dispute resolution, incoercibility—come together in the challenge of a remote voting system that is verifiable and usable without supervision. The open problem of a system specification that (a) does not use any paper at all and (b) is based on a simple procedure for voters and poll workers, will motivate researchers for a long time.Perhaps a better goal is a hybrid system combining paper evidence with some auditing or cryptographic verification. Research in voting brings together knowledge in many fields—cryptography, systems security, statistics, usability and accessibility, software verification, elections, law and policy to name a few—to address a critical real-world problem.The peaceful transfer of power depends on confidence in the electoral process. That confidence should not automatically be given to any outcome that seems plausible—it must be earned by producing evidence that the election result is what the people chose. Insisting on evidence reduces the opportunities for fraud, hence bringing greater security to citizens the world over. § ACKNOWLEDGMENTS This work was supported in part by the U.S. National Science Foundation awards CNS-1345254, CNS-1409505, CNS-1518888, CNS-1409401, CNS-1314492, and 1421373, the Center for Science of Information STC (CSoI), an NSF Science and Technology Center, under grant agreement CCF-0939370, the Maryland Procurement Office under contract H98230-14-C-0127, and FNR Luxembourg under the PETRVS Mobility grant. #1 =0=1§REFERENCES -3PT PLUS 1.55PT MKBOTH [enumi] [#1] =0pt=1pt-enumiempty `=̇1000=0=0abbrv | http://arxiv.org/abs/1707.08619v2 | {
"authors": [
"Matthew Bernhard",
"Josh Benaloh",
"J. Alex Halderman",
"Ronald L. Rivest",
"Peter Y. A. Ryan",
"Philip B. Stark",
"Vanessa Teague",
"Poorvi L. Vora",
"Dan S. Wallach"
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"categories": [
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"published": "20170726193131",
"title": "Public Evidence from Secret Ballots"
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=1 | http://arxiv.org/abs/1707.08961v1 | {
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"Jeong-Hyuck Park"
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"primary_category": "hep-th",
"published": "20170727135139",
"title": "Stringy Gravity: Solving the Dark Problems at `short' distance"
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Incoherent superconductivity well above T_ c in high-T_ c cuprates]Incoherent superconductivity well above T_ c in high-T_ c cuprates – harmonising the spectroscopic and thermodynamic data ^1 Robinson Research Institute, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand ^2 MacDiarmid Institute, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand [email protected] Cuprate superconductors have long been known to exhibit an energy gap that persists high above the superconducting transition temperature (T_ c). Debate has continued now for decades as to whether it is a precursor superconducting gap or a pseudogap arising from some competing correlation. Failure to resolve this has arguably delayed explaining the origins of superconductivity in these highly complex materials. Here we effectively settle the question by calculating a variety of thermodynamic and spectroscopic properties, exploring the effect of a temperature-dependent pair-breaking term in the self-energy in the presence of pairing interactions that persist well above T_ c. We start by fitting the detailed temperature-dependence of the electronic specific heat and immediately can explain its hitherto puzzling field dependence. Taking this same combination of pairing temperature and pair-breaking scattering we are then able to simultaneously describe in detail the unusual temperature and field dependence of the superfluid density, tunneling, Raman and optical spectra, which otherwise defy explanation in terms a superconducting gap that closes conventionally at T_ c. These findings demonstrate that the gap above T_ c in the overdoped regime likely originates from incoherent superconducting correlations, and is distinct from the competing-order “pseudogap” that appears at lower doping.74.72.Gh, 74.25.Bt, 74.25.nd, 74.55.+v Keywords: cuprates, scattering, specific heat, superfluid density, Raman spectroscopy, tunneling, optical conductivity [ J G Storey^1,2 December 30, 2023 =====================§ INTRODUCTION A prominent and highly debated feature of the high-T_ c cuprates is the presence of an energy gap at or near the Fermi level which opens above the observed superconducting transition temperature. It is generally known as the “pseudogap”. Achieving a complete understanding of the pseudogap is a critical step towards the ultimate goal of uncovering the origin of high-temperature superconductivity in these materials. For example, knowing where the onset of superconductivity occurs sets limits on the strength of the pairing interaction. The community has long been divided between two distinct viewpoints<cit.>. These can be distinguished by the doping dependence of the so-called T^* line<cit.>, the temperature below which signs of a gap appear. The first viewpoint holds that the pseudogap represents precursor phase-incoherent superconductivity or “pre-pairing”. In this case of a single d-wave gap the pseudogap opens at T^* and evolves into the superconducting gap below T_ c. The underlying Fermi surface is a nodal-metal, appearing as an arc due to broadening processes<cit.>. Here the T^* line merges smoothly with the T_ c dome on the overdoped side (see figure <ref>(a)). The second viewpoint is that the pseudogap arises from some as yet unidentified competing and/or coexisting order. In this two-gap scenario the pseudogap is distinct from the superconducting gap with a different momentum dependence, likely resulting from Fermi surface reconstruction<cit.>. The T^* line in this case bisects the T_ c dome and need not be a transition temperature in the thermodynamic sense or “phase transition”, where it would instead mark a crossover region defined by the energy of a second order parameter given by E_g ≈ 2k_ BT^* (see figure <ref>(b)). Ironically, the multitude of different techniques employed to study the pseudogap has lead to much confusion over the exact form of the T^* line. However, an alternative picture is beginning to emerge that encompasses both viewpoints (see figure <ref>(c)). Small superconducting coherence lengths in the high-T_ c cuprates give rise to strong superconducting fluctuations that are clearly evident in many techniques.Thermal expansivity<cit.>, specific heat<cit.>, resistivity<cit.>, Nernst effect<cit.>, THz conductivity<cit.>, IR conductivity<cit.> and Josephson effect<cit.> measurements show that although the fluctuation regime persists as high as 150 K<cit.>, it is confined to a narrower region above T_ c and does not track the T^* line<cit.> which extends to much higher temperatures at low doping. An effective superconducting gap feature associated with these fluctuations which tails off above T_ c can be extracted from the specific heat <cit.>. And pairing gaps above T_ c have been detected by scanning tunneling microscopy in this temperature range<cit.>.Evidence for a second energy scale, which from here will be referred to specifically as the pseudogap, includes a downturn in the normal-state spin susceptibility<cit.> and specific heat<cit.>, a departure from linear resistivity<cit.>, and a large gapping of the Fermi surface at the antinodes by angle-resolved photoemission spectroscopy (ARPES)<cit.>. The opening of the pseudogap at a critical doping within the T_ c dome can be inferred from an abrupt drop in the doping dependence of several properties. These include the specific heat jump at T_ c<cit.>, condensation energy<cit.>, zero-temperature superfluid density<cit.>, the critical zinc concentration required for suppressing superconductivity<cit.>, zero-temperature self-field critical current<cit.>, and the Hall number<cit.>, most of which represent ground-stateproperties. The last signals a drop in carrier density from 1+p to p holes per Cu, and can be explained in terms of a reconstruction from a large to small Fermi surface<cit.>. At or above optimal doping the pseudogap becomes similar or smaller in magnitude than the superconducting gap and, since many techniques return data that is dominated by the larger of the two gaps, it has been historically difficult to determine which gap is being observed. In this work it will be demonstrated explicitly that in this doping range it is in fact the superconducting gap persisting above T_ c that is being observed, thereby ending the confusion over the shape of the T^* line.This work was inspired by two recent studies. The first by Reber et al.<cit.> fitted the ARPES-derived tomographic density of states using the Dynes equation<cit.> I_TDoS=Reω-iΓ_ s/√((ω-iΓ_ s)^2-Δ^2)to extract the temperature dependence of the superconducting gap Δ and the pair-breaking scattering rate Γ_ s. They found that Δ extrapolates to zero above T_ c while Γ_ s increases steeply near T_ c. They also found that T_ c occurs when Δ = 3Γ_ s. Importantly, these parameters describe the filling-in behaviour of the gapped spectra with temperature (originally found in tunneling experiments e.g. <cit.> and also inferred from specific heat and NMR<cit.>), as opposed to the closing behaviour expected if Δ was to close at T_ c in the presence of constant scattering.The second study, by Kondo et al.<cit.>, measured the temperature dependence of the spectral function around the Fermi surface using high-resolution laser ARPES. This was fitted using the phenomenological self-energy proposed by Norman et al.<cit.>Σ(k,ω)=-iΓ_single+Δ^2/ω+ξ(k)+iΓ_pairwhere ξ(k) is the energy-momentum dispersion, Γ_single is a single-particle scattering rate and Γ_pair is a pair-breaking scattering rate. The gap is well described by a d-wave BCS temperature dependence with an onset temperature T_pair above the observed T_ c. Γ_pair increases steeply near T_ c, with T_ c coinciding with the temperature where Γ_pair=Γ_single. The aim of the present work is to investigate whether other experimental properties are consistent with this phenomenology. The approach is to fit the bulk specific heat using <ref> then, using the same parameters, calculate the superfluid density, tunneling and Raman spectra, and optical conductivity. To reiterate, the focus here is the overdoped regime near T_ c where the pseudogap and subsidiary charge-density-wave order are absent<cit.>.§ RESULTS §.§ Specific HeatThe Green's function with the above self-energy (<ref>) is given byG(ξ,ω)=1/ω-ξ+iΓ_single-Δ^2/ω+ξ+iΓ_pairThe superconducting gap is given by Δ = Δ_0δ(T)cos 2θ, where Δ_0=2.14k_ BT_ p and δ(T) is the d-wave BCS temperature dependence. θ represents the angle around the Fermi surface relative to the Brillouin zone boundary and ranges from 0 to π/2. The density of states g(ω) is obtained by integrating the spectral function A(ξ,ω)=π^-1ImG(ξ,ω)g(ω) = ∫A(ξ,ω)dξ dθThe electronic specific heat coefficient γ(T)=∂ S/∂ T is calculated from the entropyS(T) = -2k_ B∫[fln f + (1-f)ln (1-f)]g(ω)dωwhere f is the Fermi distribution function. The temperature dependence of Γ_pair is extracted by using it as an adjustable parameter to fit specific heat data under the following assumptions: i) the superconducting gap opens at T_ p = 120 K, at the onset of superconducting fluctuations; and ii) A linear-in-temperature Γ_single ranging from 5 meV at 65 K to 14 meV at 135 K, similar to values reported by Kondo et al.<cit.>. A difficulty in applying this approach over the whole temperature range is that the T-dependence of the underlying normal-state specific heat γ_ n must be known. Therefore attention will be focused close to T_ c on Bi_2Sr_2CaCu_2O_8+δ data<cit.> with a doping of 0.182 holes/Cu, where γ_ n can be taken to be reasonably constant. In practice the quantity fitted is the dimensionless ratio of superconducting- to normal-state entropies S_ s(T)/S_ n(T).Fits and parameters are shown in figure <ref> for data measured at zero and 13 T applied magnetic field. Γ_pair increases steeply near T_ c in a very similar manner to the scattering rates found from the ARPES studies mentioned above. No particular relationship between Γ_pair, Γ_single and T_ c is observed, however the peak of the specific heat jump occurs when Γ_pair = Δ. In other words, once the pair-breaking becomes of the order the superconducting gap the entropy changes less rapidly with temperature, which intuitively makes sense. This appears to differ significantly with the result Δ(T_ c)=3Γ_ s(T_ c) from Reber et al.<cit.>, but note that fitting with the Dynes equation (<ref>) returns a smaller scattering rate Γ_ s equal to the average of Γ_single and Γ_pair.A puzzling feature of the cuprate specific heat jump is its non-mean-field-like evolution with magnetic field<cit.>. Rather than shifting to lower temperatures, it broadens and reduces in amplitude with little or no change in onset temperature. The fits explain this in terms of an increase in Γ_pair with field, without requiring a reduction in gap magnitude. Note that taking Δ(H)=Δ_0√(1-(H/H_c2)^2) from Ginzburg-Landau theory<cit.>, the estimated reduction in the gap at 13 T near T_ c is only 7 to 2 percent for upper critical fields in the range 50 to 100 T. Other properties will now be calculated using the parameters in figure <ref>.§.§ Superfluid DensityThe two scattering rates, Γ_pair and Γ_single are inserted into the anomalous Green's function F as followsF(ξ,ω)=Δ/(ω+ξ+iΓ_pair)(ω-ξ+iΓ_single-Δ^2/ω+ξ+iΓ_pair)The superfluid density ρ_ s is proportional to the inverse square of the penetration depth (λ) calculated from<cit.>1/λ^2(T) = 16π e^2/c^2V∑_kv_x^2∫dω^' dω^''lim_q → 0[f(ω^'')-f(ω^')/ω^''-ω^']× B(k+q,ω^')B(k,ω^'')where the anomalous spectral function B is given by the imaginary part of F. For a free-electron-like parabolic band ξ(k)=ħ^2(k_x^2+k_y^2)/2m-μ,v_x=ħ k_x/m = √(2(ξ+μ)/m)cosθ and changing variables to ξ and θ gives1/λ^2(T)∝∫(ξ+μ)cos^2θ∫[f(ω^'')-f(ω^')/ω^''-ω^']× B(ξ,θ,ω^')B(ξ,θ,ω^'')dω^' dω^''dξ dθ The T-dependence of Γ_pair causes a clear steepening of ρ_ s away from the BCS T-dependence, with the main onset being pushed down from T_ p to T_ c, see figure <ref>(a). The same result can be obtained using one scattering rate equal to the average of Γ_single and Γ_pair at each temperature. When plotted in terms of reduced temperature T/T_ c, there is a very good match with experimental data from optimally doped cuprates (figure <ref>(b)). The data, taken by different techniques, includes a YBa_2Cu_3O_7-δ (YBCO) crystal<cit.> and film<cit.> with T_ c's near 90 K, as well as a (BiPb)_2(SrLa)_2CuO_6+δ crystal<cit.> with a T_ c of 35 K. This raises the question as to whether the mooted Berezinskii-Kosterlitz-Thouless universal jump in superfluid density may not simply be attributable to the rapid increase in pair breaking scattering rate near T_ c arising from fluctuations on a pairing scale that exceeds T_ c<cit.>. Although the tail above T_ c is not evident in the selected experimental data, it is observed elsewhere in the literature<cit.>. There is a resemblance to an approximate strong-coupling T-dependence (dotted line in figure <ref>(a)), calculated from a rescaled BCS gap of magnitude Δ_0 = 2.9k_ BT_ c closing at T_ c = 94 K, in the absence of strong pair-breaking. However, as will be seen in the following sections, this interpretation of ρ_ s(T) is inconsistent with other observations. The suppression in superfluid density with field bears a qualitative similarity to field-dependent measurements on a YBCO thin film<cit.>, but because of that sample's apparent low upper critical field the calculated suppression is much smaller in magnitude. §.§ TunnelingThe current-voltage curve for a superconductor-insulator-superconductor (SIS) tunnel junction is calculated from<cit.>I(V) ∼∫g(E)g(E-eV)[f(E)-f(E-eV)]dEwhere g(E) is the density of states given by <ref>. The tunneling conductance dI/dV is plotted in figure <ref> for several temperatures around T_ c. The evolution of the spectra with temperature is very consistent with experimental observations<cit.>. These show a filling-in of the gap with temperature and a broadening and suppression of the peaks at 2Δ, with little or no shift in their positions. This is contrary to the expected shift toward zero voltage that would occur for a strong coupling gap closing at T_ c in the absence of pair-breaking scattering. A depression persists above T_ c and vanishes as T_ p is approached, where the superconducting gap closes. Remember that a pseudogap is not included in these calculations. The linearly sloping background seen in the experimental data can be reproduced by adding a linear-in-frequency term, as seen in ARPES<cit.>, to Γ_single. §.§ Raman SpectroscopyAnother property that supports the persistence of Δ above T_ c is the Raman B_1g response given by<cit.>χ^''(ω)=∑_k(γ_k^B_1g)^2∫dω^'/4π[f(ω^')-f(ω^'+ω)] ×[A(k,ω^'+ω)A(k,ω^')-B(k,ω^'+ω)B(k,ω^')]The Raman B_1g vertex γ_k^B_1g∝cos k_x-cos k_y ∼cos 2θ probes the antinodal regions of the Fermi surface where Δ(k) is largest. Changing variables from k to ξ and θ givesχ^''(ω)∝∫dξ dθcos^2(2θ)∫dω^'[f(ω^')-f(ω^'+ω)] ×[A(ξ,θ,ω^'+ω)A(ξ,θ,ω^')-B(ξ,θ,ω^'+ω)B(ξ,θ,ω^')] The superconducting Raman B_1g response function with the normal-state response at 122 K subtracted is shown in figure <ref>(a) for several temperatures around T_ c. The resemblance to experimental data, reported in <cit.>, is striking. Like the tunneling results above, the peak at 2Δ broadens and reduces in amplitude and barely shifts with temperature indicating that the gap magnitude is still large at T_ c<cit.>. Figure <ref>(b) shows the normalized area under the curves in (a) versus reduced temperature (T/T_c=94K), together with data from <cit.>. The calculations show that data plotted in this waygives little indication of a gap above T_c. §.§ Optical ConductivityThe final property considered in this work is the ab-plane optical conductivity calculated from<cit.> σ(ω)=e^2/ω∑_kv_ab^2(k)∫dω^'/π[f(ω^')-f(ω^'+ω)] ×[A(k,ω^')A(k,ω^'+ω)+B(k,ω^')B(k,ω^'+ω)]where v_ab(k)=√(v_x^2+v_y^2). Again a change of variables is made from momentum to energy and Fermi surface angle as followsσ(ω)∝1/ω∫dξ dθ(ξ+μ)∫dω^'[f(ω^')-f(ω^'+ω)] ×[A(ξ,θ,ω^')A(ξ,θ,ω^'+ω)+B(ξ,θ,ω^')B(ξ,θ,ω^'+ω)] Spectra at several temperatures around T_ c are plotted in figure <ref>. A suppression is visible below 2Δ at low temperature that fills in as temperature is increased. A gap closing at T_ c would result in the onset of this suppression shifting to lower frequency. The calculations bear a strong qualitative resemblance to the overdoped data reported by Santander-Syro et al.<cit.> § DISCUSSIONAs summarized in table <ref> only the superfluid density, and more approximately the zero-field specific heat, can be interpreted by a strong-coupling gap closing at T_ c in the absence of scattering. The non-mean-field T-dependence of all properties examined in this work is instead well described in terms of a superconducting gap that persists above T_ c, in the presence of a steep increase in scattering. This result is insensitive to the addition of linear-in-frequency terms or a cos2θ momentum dependence to Γ_pair and Γ_single. The scattering is further enhanced by magnetic field. What is the origin of the scattering and can it be suppressed to bring T_ c up to T_ p? A rapid collapse in quasiparticle scattering below T_ c, also found in microwave surface impedance measurements<cit.>, is expected when inelastic scattering arises from interactions that become gapped or suppressed below T_ c<cit.>. The spin fluctuation spectrum is a plausible candidate and has been investigated extensively<cit.>, although those calculations assumed that the superconducting gap closes at T_ c. The work presented here illustrates that the merging of the T^* line on the lightly overdoped side of the T_ c dome is not a product of the pseudogap per se, but rather the persistence of the superconducting gap into the fluctuation region between T_ c and T_ p. As doping increases, this region becomes narrower and experimental properties become more mean-field-like. Switching direction, as doping decreases the pseudogap opens, grows, and eventually exceeds the magnitude of the superconducting gap at the antinodes. When this occurs, the gap associated with T^* changes to the pseudogap. In other words, T^* is given by the larger of T_ p and E_g/2k_ B (see figure <ref>(c)). Such an interpretation makes immediate sense of the phase diagram presented by Chatterjee et al.<cit.>Supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. The author acknowledges helpful discussion with J.L. Tallon. § REFERENCES10PG3 M. R. Norman, D. Pines, and C. Kallin. The pseudogap: friend or foe of high T_c? Adv. Phys., 54:715–733, 2005.OURWORK1 J. L. Tallon and J. W. Loram. The doping dependence of T^* - what is the real high-T_c phase diagram? Physica C, 349:53–68, 2001.FERMIARCS2 A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee, S. Souma, A. Kaminski, H. M. Fretwell, S. Rosenkranz, M. Shi, T. Sato, T. 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B, 64:224522, Nov 2001.CHATTERJEE Utpal Chatterjee, Dingfei Ai, Junjing Zhao, Stephan Rosenkranz, Adam Kaminski, Helene Raffy, Zhizhong Li, Kazuo Kadowaki, Mohit Randeria, Michael R. Norman, and J. C. Campuzano. Electronic phase diagram of high-temperature copper oxide superconductors. Proceedings of the National Academy of Sciences, 108(23):9346–9349, 2011. | http://arxiv.org/abs/1707.08648v1 | {
"authors": [
"J. G. Storey"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170726211925",
"title": "Incoherent superconductivity well above $T_c$ in high-$T_c$ cuprates - harmonizing the spectroscopic and thermodynamic data"
} |
roman EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) July 26, 2017The LHCb collaboration[Authors are listed at the end of this Letter.] The first observation of the and decays is reported using a sample of proton-proton collisions collected by LHCb at a center-of-mass energy of 8, and corresponding to 2of integrated luminosity. The corresponding branching fractions are measured using as normalization the decay , where the two muons are consistent with coming from the decay of aormeson. The results are ()=(±±±) and ()=(±±±), where the uncertainties are statistical, systematic, and due to the limited knowledge of the normalization branching fraction. The dependence of the branching fraction on the dimuon mass is also investigated.Published in Phys. Rev. Lett. 119 (2017) 181805 CERN on behalf of the collaboration, licence http://creativecommons.org/licenses/by/4.0/CC-BY-4.0.plain arabicDecays of charm hadrons into final states containing dimuon pairs may proceed via the short-distance → flavor-changing neutral-current process, which in the standard model can only occur through electroweak-loop amplitudes that are highly suppressed by the Glashow-Iliopoulos-Maiani mechanism <cit.>. If dominated by these short-distance contributions, the inclusive → X branching fraction, where X represents one or more hadrons, is predicted to be 𝒪(10^-9) <cit.> and can be greatly enhanced by the presence of new particles, making these decays interesting for searches for physics beyond the standard model. However, long-distance contributions occur through tree-level amplitudes involving intermediate resonances, such as , where V represents a , or vector meson, and can increase the standard model branching fraction up to 𝒪(10^-6) <cit.>. The sensitivity to the short-distance amplitudes is greatest for dimuon masses away from resonances, though resonances populate the entire dimuon-mass spectrum due to their long tails. Additional discrimination between short- and long-distance contributions can be gained by studying angular distributions and charge-parity-conjugation asymmetries, which in scenarios beyond the standard model could be as large as 𝒪(1%) <cit.>. Decays of mesons to four-body final states(Fig. <ref>) are particularly interesting in this respect as they give access to a variety of angular distributions. These decays were searched for by the Fermilab E791 collaboration and upper limits were set on the branching fractions in the range 10^-5–10^-4 at the 90% confidence level (CL) <cit.>. More recently, a search for nonresonant decays (the inclusion of charge-conjugate decays is implied) was performed by the LHCb collaboration using 7 -collision data corresponding to 1of integrated luminosity <cit.>. An upper limit of 5.5×10^-7 at the 90% CL was set on the branching fraction due to short-distance contributions, assuming a phase-space decay.This Letter reports the first observation of and decays using data collected by the experiment in 2012 at a center-of-mass energy √(s)=8 and corresponding to an integrated luminosity of 2. The analysis is performed using mesons originating from → decays, with the meson produced directly at the primary -collision vertex (PV). The small phase space available in this decay allows for a large background rejection, which compensates for the reduction in signal yield compared to inclusively produced mesons. The signal is studied in regions of dimuon mass, , defined according to the known resonances. For decays these regions are: (low-mass) <525, (η) 525–565, (/) 565–950, (ϕ) 950–1100, and (high-mass) >1100. The same regions are considered for decays, with the exception of the ϕ and high-mass regions, which are not present because of the reduced phase space, and the / region, which extends from 565up to the kinematic limit. In the regions where a signal is observed a measurement of the branching fraction is provided, otherwise 90% and 95% CL upper limits are set; no attempt is made to distinguish between the short- and long-distance contributions in each dimuon-mass region. The branching fraction is measured using as a normalization the decay in the dimuon-mass range 675–875, where the contribution from the /→ decay is dominant. The branching fraction was recently measured to be (±) <cit.> and provides a more precise normalization than that used in the previous LHCb search <cit.>.The detector is a single-arm forward spectrometer <cit.>. It includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the -interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. Particle identification is provided by two ring-imaging Cherenkov detectors, an electromagnetic and a hadronic calorimeter, and a muon system composed of alternating layers of iron and multiwire proportional chambers. Events are selected online by a trigger that consists of a hardware stage, which is based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction <cit.>. The hardware trigger requires the presence in the event of a muon with transverse momentum, , exceeding 1.76. A first stage of the software trigger selects events with a charged particle of >1.6 and significant impact parameter, defined as the minimum distance of the particle trajectory from any PV, or alternatively with >1 if the particle has associated hits in the muon system. In a second stage of the software trigger, dedicated algorithms select candidate decays, where h is either a kaon or a pion, from combinations of four tracks, each having momentum p>3 and >0.5, that form a secondary vertex separated from any PV. Two oppositely charged particles are required to leave hits in the muon system and the scalar sum of their is required to exceed 3. The mass of the candidate, , has to be in the range 1800–1940 and its momentum must be aligned with the vector connecting the primary and secondary vertices.In the offline analysis, candidates satisfying the trigger requirements are further selected through particle-identification criteria placed on their decay products. They are then combined with a charged particle originating from the same PV and having >120, to form a → candidate. When more than one PV is reconstructed, the one with respect to which the candidate has the lowest impact-parameter significance is chosen. The vertex formed by the and mesons is constrained to coincide with the PV and the difference between the and masses, , is required to be in the range 144.5–146.5. A multivariate selection based on a boosted decision tree (BDT) <cit.> with gradient boosting <cit.> is then used to suppress background from combinations of unrelated charged particles. The features used by the BDT to discriminate signal from this combinatorial background are as follows: the momentum and transverse momentum of the pion from the decay, the smallest impact parameter of the decay products with respect to the PV, the angle between the momentum and the vector connecting the primary and secondary vertices, the quality of the secondary vertex, its separation from the PV, and its separation from any other track not forming the candidate. The BDT is trained separately forand decays, due to their different kinematic properties, using simulated <cit.> decays as signal and data candidates with between 1890 and 1940as background. To minimize biases on the background classification, the training samples are further randomly split into two disjoint subsamples. The classifier trained on one sample is applied to the other, and vice versa. Another source of background is due to the hadronic four-body decays → and →, where two pions are misidentified as muons. The misidentification occurs mainly when the pions decay in flight into a muon and an undetected neutrino. Although this process is relatively rare, the large branching fractions of the hadronic modes produce a peaking background which is partially suppressed by a multivariate muon-identification discriminant that combines the information from the Cherenkov detectors, the calorimeters and the muon chambers. Thresholds on the BDT response and on the muon-identification discriminant are optimized simultaneously by maximizing ϵ_/(5/2 + √(N_bkg)) <cit.>, where ϵ_ is the signal efficiency and N_bkg is the sum of the expected combinatorial and peaking background yields in the range 1830–1900(signal region). Candidate decays are selected using the response of the BDT trained on the signal, when they are used as normalization for the measurement of (), and that of the BDT trained on the signal, when used as normalization for (). After selection, a few percent of the events contain multiple candidates, of which only one is randomly selected if they share at least one final-state particle. To avoid potential biases on the measured quantities, candidate decays in the signal region are examined only after the analysis procedure has been finalized, with the exception of those populating the /ω and ϕ dimuon-mass regions of the sample.The and signal yields are measured with unbinned extended maximum likelihood fits to the distributions (Figs. <ref> and <ref>, respectively). The fits include three components: signal, peaking background from misidentified hadronic decays, and combinatorial background. The signal is described with a Johnson's S_U distribution <cit.> with parameters determined from simulation. To account for known differences between data and simulation, the means and widths of the signal distributions are corrected using scaling factors adjusted on the normalization channel. The mass shape of the peaking background is determined using separate data samples of → decays where the mass is calculated assigning the muon-mass hypothesis to two oppositely charged pions. The combinatorial background is described by an exponential function, which is determined from data candidates with between 150 and 160that fail the BDT selection. All shape parameters are fixed and only the yields are allowed to vary in the fits, which are performed separately in each range.The resulting signal yields are reported in Table <ref>. No fit is performed in the η region of the dimuon-mass spectrum, where only two candidates are observed. An excess of candidates with respect to the background-only hypothesis is seen with a significance above three standard deviations in all dimuon-mass ranges with the exception of the η region of both decays and the high-region of . The significances are determined from the change in likelihood from fits with and without the signal component. The signal yields, N^i_, in each range i are converted into branching fractions using wordcount^i() = N_^i ()/R^i_ϵ N_, where N_ is the yield of the normalization mode, which is determined to be(1806±48) after the selection optimized for() decays. The ratios of geometrical acceptances, and reconstruction and selection efficiencies of the signal relative to the normalization decays, , are reported in Table <ref>. They are determined using simulated events and corrected to account for known differences between data and simulation. In particular, particle-identification and hardware-trigger efficiencies are measured from control channels in data.Systematic uncertainties affect the determination of the signal and normalization yields, and of the efficiency ratio. For the determination of the yields, effects due to uncertainties on the shapes are investigated. A possible dependence on the decay mode or on the range of the scaling factors, used to account for data-simulation differences, is quantified using fits to the and data and is found to be negligible. To assess the impact of π→μν decays in flight, alternative shapes are tested for the → background by changing the muon-identification and the requirements on the misidentified pions. The largest observed variation in the ratio of to yields (1.4%) is assigned as a systematic uncertainty for both modes and all dimuon-mass ranges. Changes in the shape of the peaking background introduced by the different trigger requirements used to select the hadronic decays are negligible. The fit to the data is repeated using alternative descriptions of the combinatorial background, determined from data sidebands defined by different BDT and requirements, and results in negligible variations of the signal and normalization yields.Systematic uncertainties affecting the efficiency ratio include data-simulation differences that are not accounted for and limitations in the data-driven methods used to determine the particle-identification and trigger efficiencies. The signal decays are simulated with an incoherent sum of resonant and nonresonant dimuon and dihadron components, while the resonant structure in data is unknown. A systematic uncertainty of 3.4% on the signal efficiency is determined by varying the relative fractions of these components. A systematic uncertainty of 1.0% on the efficiency ratio is assigned due to the criteria used in simulation to match the reconstructed and generated particles. Muon- and hadron-identification efficiencies are determined from data by weighting the kinematic properties of the calibration samples to match those of the signal samples. Variations of the choice of the binning scheme used in the weighting procedure change the efficiency ratio by up to 0.8%, which is taken as systematic uncertainty. The data-driven method that evaluates the hardware-trigger efficiency ratio is validated in simulation to be unbiased within 1.3%, which is assigned as a systematic uncertainty. The efficiencies of the BDT requirement for the simulated normalization and decays are compared to those obtained from background-subtracted data. A difference in the efficiency ratio of 1.3% is observed and assigned as systematic uncertainty.Finally, the statistical uncertainty on the normalization yield introduces a relative uncertainty of 2.6% (2.7%), which is propagated to the systematic uncertainty on the () branching fractions.Table <ref> reports the measured values and upper limits on the and branching fractions in the various ranges of , where the first uncertainty accounts for the statistical component, the second for the systematic, and the third corresponds to the 10% relative uncertainty on () <cit.>. The upper limits are derived using a frequentist approach based on a likelihood-ratio ordering method that includes the effects due to the systematic uncertainties <cit.>. For the η region of , where no fit is performed, the limit is calculated assuming two signal candidates and zero background. Integrating over dimuon mass, and accounting for correlations <cit.>, the total branching fractions are measured to be() =(±±±), () =(±±±).The two results have a correlation of 0.497 and are consistent with the standard model expectations <cit.>.In summary, a study of the and decays is performed in ranges of the dimuon mass using collisions collected by the experiment at √(s)=8. Significant signal yields are observed for the first time in several dimuon-mass ranges for both decays; the corresponding branching fractions are measured and found to be consistent with the standard model expectations <cit.>. For the dimuon-mass regions where no significant signal is observed, upper limits at 90% and 95% CL are set on the branching fraction. The total branching fractions are measured to be ()=(±±±) and ()=(±±±), where the uncertainties are statistical, systematic, and due to the limited knowledge of the normalization branching fraction. These are the rarest charm-hadron decays ever observed and are expected to provide better sensitivity to short-distance flavor-changing neutral-current contributions to these decays.§ ACKNOWLEDGEMENTSWe express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy);NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland);NASU (Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple opensource software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union),Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU,Région Auvergne (France), RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust (United Kingdom). tocsectionReferences inbibliographytrue LHCb§ SUPPLEMENTAL MATERIALThe correlations between () and () in the dimuon-mass regions are reported in <ref> and <ref>, respectively. The correlation between the total branching fractions is 0.497.LHCb collaboration R. Aaij^40, B. Adeva^39, M. Adinolfi^48, Z. Ajaltouni^5, S. Akar^59, J. Albrecht^10, F. Alessio^40, M. Alexander^53, A. Alfonso Albero^38, S. Ali^43, G. Alkhazov^31, P. Alvarez Cartelle^55, A.A. Alves Jr^59, S. Amato^2, S. Amerio^23, Y. Amhis^7, L. An^3, L. Anderlini^18, G. Andreassi^41, M. Andreotti^17,g, J.E. Andrews^60, R.B. Appleby^56, F. Archilli^43, P. d'Argent^12, J. Arnau Romeu^6, A. Artamonov^37, M. Artuso^61, E. Aslanides^6, G. Auriemma^26, M. Baalouch^5, I. Babuschkin^56, S. Bachmann^12, J.J. Back^50, A. Badalov^38,m, C. Baesso^62, S. Baker^55, V. Balagura^7,b, W. Baldini^17, A. Baranov^35, R.J. Barlow^56, C. Barschel^40, S. Barsuk^7, W. Barter^56, F. Baryshnikov^32, V. Batozskaya^29, V. Battista^41, A. Bay^41, L. Beaucourt^4, J. Beddow^53, F. Bedeschi^24, I. Bediaga^1, A. Beiter^61, L.J. Bel^43, N. Beliy^63, V. Bellee^41, N. Belloli^21,i, K. Belous^37, I. Belyaev^32, E. Ben-Haim^8, G. Bencivenni^19, S. Benson^43, S. Beranek^9, A. Berezhnoy^33, R. Bernet^42, D. Berninghoff^12, E. Bertholet^8, A. Bertolin^23, C. Betancourt^42, F. Betti^15, M.-O. Bettler^40, M. van Beuzekom^43, Ia. Bezshyiko^42, S. Bifani^47, P. Billoir^8, A. Birnkraut^10, A. Bitadze^56, A. Bizzeti^18,u, M. Bjørn^57, T. Blake^50, F. Blanc^41, J. Blouw^11,†, S. Blusk^61, V. Bocci^26, T. Boettcher^58, A. Bondar^36,w, N. Bondar^31, W. Bonivento^16, I. Bordyuzhin^32, A. Borgheresi^21,i, S. Borghi^56, M. Borisyak^35, M. Borsato^39, F. Bossu^7, M. Boubdir^9, T.J.V. Bowcock^54, E. Bowen^42, C. Bozzi^17,40, S. Braun^12, T. Britton^61, J. Brodzicka^27, D. Brundu^16, E. Buchanan^48, C. Burr^56, A. Bursche^16,f, J. Buytaert^40, W. Byczynski^40, S. Cadeddu^16, H. Cai^64, R. Calabrese^17,g, R. Calladine^47, M. Calvi^21,i, M. Calvo Gomez^38,m, A. Camboni^38,m, P. Campana^19, D.H. Campora Perez^40, L. Capriotti^56, A. Carbone^15,e, G. Carboni^25,j, R. Cardinale^20,h, A. Cardini^16, P. Carniti^21,i, L. Carson^52, K. Carvalho Akiba^2, G. Casse^54, L. Cassina^21, L. Castillo Garcia^41, M. Cattaneo^40, G. Cavallero^20,40,h, R. Cenci^24,t, D. Chamont^7, M. Charles^8, Ph. Charpentier^40, G. Chatzikonstantinidis^47, M. Chefdeville^4, S. Chen^56, S.F. Cheung^57, S.-G. Chitic^40, V. Chobanova^39, M. Chrzaszcz^42,27, A. Chubykin^31, P. Ciambrone^19, X. Cid Vidal^39, G. Ciezarek^43, P.E.L. Clarke^52, M. Clemencic^40, H.V. Cliff^49, J. Closier^40, J. Cogan^6, E. Cogneras^5, V. Cogoni^16,f, L. Cojocariu^30, P. Collins^40, T. Colombo^40, A. Comerma-Montells^12, A. Contu^40, A. Cook^48, G. Coombs^40, S. Coquereau^38, G. Corti^40, M. Corvo^17,g, C.M. Costa Sobral^50, B. Couturier^40, G.A. Cowan^52, D.C. Craik^58, A. Crocombe^50, M. Cruz Torres^1, R. Currie^52, C. D'Ambrosio^40, F. Da Cunha Marinho^2, E. Dall'Occo^43, J. Dalseno^48, A. Davis^3, O. De Aguiar Francisco^54, S. De Capua^56, M. De Cian^12, J.M. De Miranda^1, L. De Paula^2, M. De Serio^14,d, P. De Simone^19, C.T. Dean^53, D. Decamp^4, L. Del Buono^8, H.-P. Dembinski^11, M. Demmer^10, A. Dendek^28, D. Derkach^35, O. Deschamps^5, F. Dettori^54, B. Dey^65, A. Di Canto^40, P. Di Nezza^19, H. Dijkstra^40, F. Dordei^40, M. Dorigo^41, A. Dosil Suárez^39, L. Douglas^53, A. Dovbnya^45, K. Dreimanis^54, L. Dufour^43, G. Dujany^8, P. Durante^40, R. Dzhelyadin^37, M. Dziewiecki^12, A. Dziurda^40, A. Dzyuba^31, S. Easo^51, M. Ebert^52, U. Egede^55, V. Egorychev^32, S. Eidelman^36,w, S. Eisenhardt^52, U. Eitschberger^10, R. Ekelhof^10, L. Eklund^53, S. Ely^61, S. Esen^12, H.M. Evans^49, T. Evans^57, A. Falabella^15, N. Farley^47, S. Farry^54, R. Fay^54, D. Fazzini^21,i, L. Federici^25, D. Ferguson^52, G. Fernandez^38, P. Fernandez Declara^40, A. Fernandez Prieto^39, F. Ferrari^15, F. Ferreira Rodrigues^2, M. Ferro-Luzzi^40, S. Filippov^34, R.A. Fini^14, M. Fiore^17,g, M. Fiorini^17,g, M. Firlej^28, C. Fitzpatrick^41, T. Fiutowski^28, F. Fleuret^7,b, K. Fohl^40, M. Fontana^16,40, F. Fontanelli^20,h, D.C. Forshaw^61, R. Forty^40, V. Franco Lima^54, M. Frank^40, C. Frei^40, J. Fu^22,q, W. Funk^40, E. Furfaro^25,j, C. Färber^40, E. Gabriel^52, A. Gallas Torreira^39, D. Galli^15,e, S. Gallorini^23, S. Gambetta^52, M. Gandelman^2, P. Gandini^57, Y. Gao^3, L.M. Garcia Martin^70, J. García Pardiñas^39, J. Garra Tico^49, L. Garrido^38, P.J. Garsed^49, D. Gascon^38, C. Gaspar^40, L. Gavardi^10, G. Gazzoni^5, D. Gerick^12, E. Gersabeck^12, M. Gersabeck^56, T. Gershon^50, Ph. Ghez^4, S. Gianì^41, V. Gibson^49, O.G. Girard^41, L. Giubega^30, K. Gizdov^52, V.V. Gligorov^8, D. Golubkov^32, A. Golutvin^55,40, A. Gomes^1,a, I.V. Gorelov^33, C. Gotti^21,i, E. Govorkova^43, J.P. Grabowski^12, R. Graciani Diaz^38, L.A. Granado Cardoso^40, E. Graugés^38, E. Graverini^42, G. Graziani^18, A. Grecu^30, R. Greim^9, P. Griffith^16, L. Grillo^21,40,i, L. Gruber^40, B.R. Gruberg Cazon^57, O. Grünberg^67, E. Gushchin^34, Yu. Guz^37, T. Gys^40, C. Göbel^62, T. Hadavizadeh^57, C. Hadjivasiliou^5, G. Haefeli^41, C. Haen^40, S.C. Haines^49, B. Hamilton^60, X. Han^12, T.H. Hancock^57, S. Hansmann-Menzemer^12, N. Harnew^57, S.T. Harnew^48, J. Harrison^56, C. Hasse^40, M. Hatch^40, J. He^63, M. Hecker^55, K. Heinicke^10, A. Heister^9, K. Hennessy^54, P. Henrard^5, L. Henry^70, E. van Herwijnen^40, M. Heß^67, A. Hicheur^2, D. Hill^57, C. Hombach^56, P.H. Hopchev^41, Z.-C. Huard^59, W. Hulsbergen^43, T. Humair^55, M. Hushchyn^35, D. Hutchcroft^54, P. Ibis^10, M. Idzik^28, P. Ilten^58, R. Jacobsson^40, J. Jalocha^57, E. Jans^43, A. Jawahery^60, F. Jiang^3, M. John^57, D. Johnson^40, C.R. Jones^49, C. Joram^40, B. Jost^40, N. Jurik^57, S. Kandybei^45, M. Karacson^40, J.M. Kariuki^48, S. Karodia^53, N. Kazeev^35, M. Kecke^12, M. Kelsey^61, M. Kenzie^49, T. Ketel^44, E. Khairullin^35, B. Khanji^12, C. Khurewathanakul^41, T. Kirn^9, S. Klaver^56, K. Klimaszewski^29, T. Klimkovich^11, S. Koliiev^46, M. Kolpin^12, I. Komarov^41, R. Kopecna^12, P. Koppenburg^43, A. Kosmyntseva^32, S. Kotriakhova^31, M. Kozeiha^5, M. Kreps^50, P. Krokovny^36,w, F. Kruse^10, W. Krzemien^29, W. Kucewicz^27,l, M. Kucharczyk^27, V. Kudryavtsev^36,w, A.K. Kuonen^41, K. Kurek^29, T. Kvaratskheliya^32,40, D. Lacarrere^40, G. Lafferty^56, A. Lai^16, G. Lanfranchi^19, C. Langenbruch^9, T. Latham^50, C. Lazzeroni^47, R. Le Gac^6, J. van Leerdam^43, A. Leflat^33,40, J. Lefrançois^7, R. Lefèvre^5, F. Lemaitre^40, E. Lemos Cid^39, O. Leroy^6, T. Lesiak^27, B. Leverington^12, P.-R. Li^63, T. Li^3, Y. Li^7, Z. Li^61, T. Likhomanenko^68, R. Lindner^40, F. Lionetto^42, V. Lisovskyi^7, X. Liu^3, D. Loh^50, A. Loi^16, I. Longstaff^53, J.H. Lopes^2, D. Lucchesi^23,o, M. Lucio Martinez^39, H. Luo^52, A. Lupato^23, E. Luppi^17,g, O. Lupton^40, A. Lusiani^24, X. Lyu^63, F. Machefert^7, F. Maciuc^30, V. Macko^41, P. Mackowiak^10, S. Maddrell-Mander^48, O. Maev^31, K. Maguire^56, D. Maisuzenko^31, M.W. Majewski^28, S. Malde^57, A. Malinin^68, T. Maltsev^36,w, G. Manca^16,f, G. Mancinelli^6, P. Manning^61, D. Marangotto^22,q, J. Maratas^5,v, J.F. Marchand^4, U. Marconi^15, C. Marin Benito^38, M. Marinangeli^41, P. Marino^41, J. Marks^12, G. Martellotti^26, M. Martin^6, M. Martinelli^41, D. Martinez Santos^39, F. Martinez Vidal^70, D. Martins Tostes^2, L.M. Massacrier^7, A. Massafferri^1, R. Matev^40, A. Mathad^50, Z. Mathe^40, C. Matteuzzi^21, A. Mauri^42, E. Maurice^7,b, B. Maurin^41, A. Mazurov^47, M. McCann^55,40, A. McNab^56, R. McNulty^13, J.V. Mead^54, B. Meadows^59, C. Meaux^6, F. Meier^10, N. Meinert^67, D. Melnychuk^29, M. Merk^43, A. Merli^22,40,q, E. Michielin^23, D.A. Milanes^66, E. Millard^50, M.-N. Minard^4, L. Minzoni^17, D.S. Mitzel^12, A. Mogini^8, J. Molina Rodriguez^1, T. Mombacher^10, I.A. Monroy^66, S. Monteil^5, M. Morandin^23, M.J. Morello^24,t, O. Morgunova^68, J. Moron^28, A.B. Morris^52, R. Mountain^61, F. Muheim^52, M. Mulder^43, D. Müller^56, J. Müller^10, K. Müller^42, V. Müller^10, P. Naik^48, T. Nakada^41, R. Nandakumar^51, A. Nandi^57, I. Nasteva^2, M. Needham^52, N. Neri^22,40, S. Neubert^12, N. Neufeld^40, M. Neuner^12, T.D. Nguyen^41, C. Nguyen-Mau^41,n, S. Nieswand^9, R. Niet^10, N. Nikitin^33, T. Nikodem^12, A. Nogay^68, D.P. O'Hanlon^50, A. Oblakowska-Mucha^28, V. Obraztsov^37, S. Ogilvy^19, R. Oldeman^16,f, C.J.G. Onderwater^71, A. Ossowska^27, J.M. Otalora Goicochea^2, P. Owen^42, A. Oyanguren^70, P.R. Pais^41, A. Palano^14,d, M. Palutan^19,40, A. Papanestis^51, M. Pappagallo^14,d, L.L. Pappalardo^17,g, W. Parker^60, C. Parkes^56, G. Passaleva^18, A. Pastore^14,d, M. Patel^55, C. Patrignani^15,e, A. Pearce^40, A. Pellegrino^43, G. Penso^26, M. Pepe Altarelli^40, S. Perazzini^40, P. Perret^5, L. Pescatore^41, K. Petridis^48, A. Petrolini^20,h, A. Petrov^68, M. Petruzzo^22,q, E. Picatoste Olloqui^38, B. Pietrzyk^4, M. Pikies^27, D. Pinci^26, A. Pistone^20,h, A. Piucci^12, V. Placinta^30, S. Playfer^52, M. Plo Casasus^39, F. Polci^8, M. Poli Lener^19, A. Poluektov^50,36, I. Polyakov^61, E. Polycarpo^2, G.J. Pomery^48, S. Ponce^40, A. Popov^37, D. Popov^11,40, S. Poslavskii^37, C. Potterat^2, E. Price^48, J. Prisciandaro^39, C. Prouve^48, V. Pugatch^46, A. Puig Navarro^42, H. Pullen^57, G. Punzi^24,p, W. Qian^50, R. Quagliani^7,48, B. Quintana^5, B. Rachwal^28, J.H. Rademacker^48, M. Rama^24, M. Ramos Pernas^39, M.S. Rangel^2, I. Raniuk^45,†, F. Ratnikov^35, G. Raven^44, M. Ravonel Salzgeber^40, M. Reboud^4, F. Redi^55, S. Reichert^10, A.C. dos Reis^1, C. Remon Alepuz^70, V. Renaudin^7, S. Ricciardi^51, S. Richards^48, M. Rihl^40, K. Rinnert^54, V. Rives Molina^38, P. Robbe^7, A. Robert^8, A.B. Rodrigues^1, E. Rodrigues^59, J.A. Rodriguez Lopez^66, P. Rodriguez Perez^56,†, A. Rogozhnikov^35, S. Roiser^40, A. Rollings^57, V. Romanovskiy^37, A. Romero Vidal^39, J.W. Ronayne^13, M. Rotondo^19, M.S. Rudolph^61, T. Ruf^40, P. Ruiz Valls^70, J. Ruiz Vidal^70, J.J. Saborido Silva^39, E. Sadykhov^32, N. Sagidova^31, B. Saitta^16,f, V. Salustino Guimaraes^1, C. Sanchez Mayordomo^70, B. Sanmartin Sedes^39, R. Santacesaria^26, C. Santamarina Rios^39, M. Santimaria^19, E. Santovetti^25,j, G. Sarpis^56, A. Sarti^26, C. Satriano^26,s, A. Satta^25, D.M. Saunders^48, D. Savrina^32,33, S. Schael^9, M. Schellenberg^10, M. Schiller^53, H. Schindler^40, M. Schlupp^10, M. Schmelling^11, T. Schmelzer^10, B. Schmidt^40, O. Schneider^41, A. Schopper^40, H.F. Schreiner^59, K. Schubert^10, M. Schubiger^41, M.-H. Schune^7, R. Schwemmer^40, B. Sciascia^19, A. Sciubba^26,k, A. Semennikov^32, A. Sergi^47, N. Serra^42, J. Serrano^6, L. Sestini^23, P. Seyfert^40, M. Shapkin^37, I. Shapoval^45, Y. Shcheglov^31, T. Shears^54, L. Shekhtman^36,w, V. Shevchenko^68, B.G. Siddi^17,40, R. Silva Coutinho^42, L. Silva de Oliveira^2, G. Simi^23,o, S. Simone^14,d, M. Sirendi^49, N. Skidmore^48, T. Skwarnicki^61, E. Smith^55, I.T. Smith^52, J. Smith^49, M. Smith^55, l. Soares Lavra^1, M.D. Sokoloff^59, F.J.P. Soler^53, B. Souza De Paula^2, B. Spaan^10, P. Spradlin^53, S. Sridharan^40, F. Stagni^40, M. Stahl^12, S. Stahl^40, P. Stefko^41, S. Stefkova^55, O. Steinkamp^42, S. Stemmle^12, O. Stenyakin^37, M. Stepanova^31, H. Stevens^10, S. Stone^61, B. Storaci^42, S. Stracka^24,p, M.E. Stramaglia^41, M. Straticiuc^30, U. Straumann^42, L. Sun^64, W. Sutcliffe^55, K. Swientek^28, V. Syropoulos^44, M. Szczekowski^29, T. Szumlak^28, M. Szymanski^63, S. T'Jampens^4, A. Tayduganov^6, T. Tekampe^10, G. Tellarini^17,g, F. Teubert^40, E. Thomas^40, J. van Tilburg^43, M.J. Tilley^55, V. Tisserand^4, M. Tobin^41, S. Tolk^49, L. Tomassetti^17,g, D. Tonelli^24, F. Toriello^61, R. Tourinho Jadallah Aoude^1, E. Tournefier^4, M. Traill^53, M.T. Tran^41, M. Tresch^42, A. Trisovic^40, A. Tsaregorodtsev^6, P. Tsopelas^43, A. Tully^49, N. Tuning^43, A. Ukleja^29, A. Usachov^7, A. Ustyuzhanin^35, U. Uwer^12, C. Vacca^16,f, A. Vagner^69, V. Vagnoni^15,40, A. Valassi^40, S. Valat^40, G. Valenti^15, R. Vazquez Gomez^19, P. Vazquez Regueiro^39, S. Vecchi^17, M. van Veghel^43, J.J. Velthuis^48, M. Veltri^18,r, G. Veneziano^57, A. Venkateswaran^61, T.A. Verlage^9, M. Vernet^5, M. Vesterinen^57, J.V. Viana Barbosa^40, B. Viaud^7, D. Vieira^63, M. Vieites Diaz^39, H. Viemann^67, X. Vilasis-Cardona^38,m, M. Vitti^49, V. Volkov^33, A. Vollhardt^42, B. Voneki^40, A. Vorobyev^31, V. Vorobyev^36,w, C. Voß^9, J.A. de Vries^43, C. Vázquez Sierra^39, R. Waldi^67, C. Wallace^50, R. Wallace^13, J. Walsh^24, J. Wang^61, D.R. Ward^49, H.M. Wark^54, N.K. Watson^47, D. Websdale^55, A. Weiden^42, M. Whitehead^40, J. Wicht^50, G. Wilkinson^57,40, M. Wilkinson^61, M. Williams^56, M.P. Williams^47, M. Williams^58, T. Williams^47, F.F. Wilson^51, J. Wimberley^60, M.A. Winn^7, J. Wishahi^10, W. Wislicki^29, M. Witek^27, G. Wormser^7, S.A. Wotton^49, K. Wraight^53, K. Wyllie^40, Y. Xie^65, Z. Xu^4, Z. Yang^3, Z. Yang^60, Y. Yao^61, H. Yin^65, J. Yu^65, X. Yuan^61, O. Yushchenko^37, K.A. Zarebski^47, M. Zavertyaev^11,c, L. Zhang^3, Y. Zhang^7, A. Zhelezov^12, Y. Zheng^63, X. Zhu^3, V. Zhukov^33, J.B. Zonneveld^52, S. Zucchelli^15.^1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil^2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil^3Center for High Energy Physics, Tsinghua University, Beijing, China^4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France^5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France^6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France^7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France^8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France^9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany^10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany^11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany^12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany^13School of Physics, University College Dublin, Dublin, Ireland^14Sezione INFN di Bari, Bari, Italy^15Sezione INFN di Bologna, Bologna, Italy^16Sezione INFN di Cagliari, Cagliari, Italy^17Universita e INFN, Ferrara, Ferrara, Italy^18Sezione INFN di Firenze, Firenze, Italy^19Laboratori Nazionali dell'INFN di Frascati, Frascati, Italy^20Sezione INFN di Genova, Genova, Italy^21Universita & INFN, Milano-Bicocca, Milano, Italy^22Sezione di Milano, Milano, Italy^23Sezione INFN di Padova, Padova, Italy^24Sezione INFN di Pisa, Pisa, Italy^25Sezione INFN di Roma Tor Vergata, Roma, Italy^26Sezione INFN di Roma La Sapienza, Roma, Italy^27Henryk Niewodniczanski Institute of Nuclear PhysicsPolish Academy of Sciences, Kraków, Poland^28AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland^29National Center for Nuclear Research (NCBJ), Warsaw, Poland^30Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania^31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia^32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia^33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia^34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia^35Yandex School of Data Analysis, Moscow, Russia^36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia^37Institute for High Energy Physics (IHEP), Protvino, Russia^38ICCUB, Universitat de Barcelona, Barcelona, Spain^39Universidad de Santiago de Compostela, Santiago de Compostela, Spain^40European Organization for Nuclear Research (CERN), Geneva, Switzerland^41Institute of Physics, Ecole PolytechniqueFédérale de Lausanne (EPFL), Lausanne, Switzerland^42Physik-Institut, Universität Zürich, Zürich, Switzerland^43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands^44Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands^45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine^46Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine^47University of Birmingham, Birmingham, United Kingdom^48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom^49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom^50Department of Physics, University of Warwick, Coventry, United Kingdom^51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom^52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom^53School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom^54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom^55Imperial College London, London, United Kingdom^56School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom^57Department of Physics, University of Oxford, Oxford, United Kingdom^58Massachusetts Institute of Technology, Cambridge, MA, United States^59University of Cincinnati, Cincinnati, OH, United States^60University of Maryland, College Park, MD, United States^61Syracuse University, Syracuse, NY, United States^62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to ^2^63University of Chinese Academy of Sciences, Beijing, China, associated to ^3^64School of Physics and Technology, Wuhan University, Wuhan, China, associated to ^3^65Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to ^3^66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to ^8^67Institut für Physik, Universität Rostock, Rostock, Germany, associated to ^12^68National Research Centre Kurchatov Institute, Moscow, Russia, associated to ^32^69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to ^32^70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain, associated to ^38^71Van Swinderen Institute, University of Groningen, Groningen, The Netherlands, associated to ^43^aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil^bLaboratoire Leprince-Ringuet, Palaiseau, France^cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia^dUniversità di Bari, Bari, Italy^eUniversità di Bologna, Bologna, Italy^fUniversità di Cagliari, Cagliari, Italy^gUniversità di Ferrara, Ferrara, Italy^hUniversità di Genova, Genova, Italy^iUniversità di Milano Bicocca, Milano, Italy^jUniversità di Roma Tor Vergata, Roma, Italy^kUniversità di Roma La Sapienza, Roma, Italy^lAGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland^mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain^nHanoi University of Science, Hanoi, Viet Nam^oUniversità di Padova, Padova, Italy^pUniversità di Pisa, Pisa, Italy^qUniversità degli Studi di Milano, Milano, Italy^rUniversità di Urbino, Urbino, Italy^sUniversità della Basilicata, Potenza, Italy^tScuola Normale Superiore, Pisa, Italy^uUniversità di Modena e Reggio Emilia, Modena, Italy^vIligan Institute of Technology (IIT), Iligan, Philippines^wNovosibirsk State University, Novosibirsk, Russia^†Deceased | http://arxiv.org/abs/1707.08377v2 | {
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"categories": [
"hep-ex"
],
"primary_category": "hep-ex",
"published": "20170726111942",
"title": "Observation of $D^0$ meson decays to $π^+π^-μ^+μ^-$ and $K^+K^-μ^+μ^-$ final states"
} |
APS/123-QEDSchool of Natural Sciences, Far Eastern Federal University, 6 Sukhanova Str., 690041 Vladivostok, Russia Institute for Automation and Control Processes FEB RAS, 5 Radio Str., 690041 Vladivostok, Russia Institute for Automation and Control Processes FEB RAS, 5 Radio Str., 690041 Vladivostok, Russia Institute for Automation and Control Processes FEB RAS, 5 Radio Str., 690041 Vladivostok, Russia School of Natural Sciences, Far Eastern Federal University, 6 Sukhanova Str., 690041 Vladivostok, Russia School of Natural Sciences, Far Eastern Federal University, 6 Sukhanova Str., 690041 Vladivostok, Russia Institute for Automation and Control Processes FEB RAS, 5 Radio Str., 690041 Vladivostok, Russia Lebedev Physical Institute, Russian Academy of Sciences, 53 Leninsky prospect, Moscow 119991, Russia ITMO University, 49 Kronverksky prospect, St. Petersburg 197101, Russia [email protected] School of Natural Sciences, Far Eastern Federal University, 6 Sukhanova Str., 690041 Vladivostok, Russia Institute for Automation and Control Processes FEB RAS, 5 Radio Str., 690041 Vladivostok, Russia Pulsed-laser dry printing of noble-metal microrings with a tunable internal porous structure, which can be revealed via an ion-beam etching post-procedure, was demonstrated. Abundance and average size of the pores inside the microrings were shown to be tuned in a wide range by varying incident pulse energy and a nitrogen doping level controlled in the process of magnetron deposition of the gold film in the appropriate gaseous environment. The fabricated porous microrings were shown to provide many-fold near-field enhancement of incident electromagnetic fields, which was confirmed by mapping of the characteristic Raman band of a nanometer-thick covering layer of Rhodamine 6G dye molecules and supporting finite-difference time-domain calculations. The proposed laser-printing/ion-beam etching approach is demonstrated to be a unique tool aimed at designing and fabricating multifunctional plasmonic structures and metasurfaces for spectroscopic bioidentification based on surface-enhanced infrared absorption, Raman scattering and photoluminescence detection schemes.Fabrication of porous microrings via laser printing and ion-beam post-etching A. Kuchmizhak=============================================================================Surface-enhanced Raman scattering (SERS) is an ultra-sensitive non-invasive spectroscopic technique based on a label-free identification of different molecules placed in the vicinity of plasmonic-active nanostructured metallic substrates <cit.>. Intensity of the characteristic Raman signal defining the specific vibrational signatures of individual molecules is usually very weak. However, this signal can be significantly increased near nanotextured surfaces or nanostructures generating localized, strongly enhanced plasmon-mediated electromagnetic fields. Since the first observation of SERS signal from single molecule <cit.>, multiple attempts were undertaken to increase the efficiency of SERS-active nanotextured substrates in terms of achieved maximal enhancement factor inside a single “hot spot” as well as number (density) of “hot spots” per individual nanostructure <cit.>.To address both issues, variety of nanotextured structures, predominantly having large surface-to-volume ratio and generating dense hot spots (tipped structures, nanostructures with inner porosity, intra-gap structures or self-assembly superstructures, etc.) were fabricated and tested as versatile SERS substrates<cit.>. Specifically, porous materials, nanostructures and nanoparticles, routinely reaching uniform SERS enhancement sufficient to overcome single-molecule detection limit independently on excitation/detection conditions, are of growing interest <cit.>. Several papers reported fabrication of such sponge-like structures, using dealloying of the two-component template via its dissolving in a corrosive environment <cit.>. Meanwhile, to design sensitive elements for advanced biosensors, along with desirable porosity, it is also important to control the overall size and shape of such porous templates as well as to arrange them into well-ordered arrays at specific point on a substrate with micrometer-scale lateral accuracy. Despite the latter issue can be resolved by using well-established but rather time-consuming non-scalable and expensive electron- or ion-beam lithography techniques <cit.>, contamination of nanotextures during wet etching procedure as well as complete dealloying of the fabricated nanostructures present issues, which are still to be resolved.Pure chemical techniques as seed-mediated growth, chemical vapor deposition or electrodeposition are suitable only for cheap inexpensive nanofabrication of disordered porous nanotextures and nanoparticles <cit.>, providing high controllability of geometric shapes and porosity, but requiring, however, additional pre-processing fabrication steps to arrange the isolated elements into their ordered arrays. Rapid laser-assisted thermal annealing and post-dealloying enable higher performance in comparison to the above mentioned approaches, requiring, on the other hand, additional steps in the processing chain, and, more importantly, often cause undesirable decrease of pore density <cit.>. In this way, high-performing, easy-to-implement, “dry" and “green" (without hazardous chemicals) technology for fabricating porous plasmonic nanostructures is still missing.In this Letter, an ion-beam assisted, liquid-free nanosecond (ns) laser printing of isolated plasmonic ring-shaped nanostructures with pronounced porosity is demonstrated. Besides the recently reported possibility to control and tune over the main geometric parameters of the laser-printed rings, such as diameter, wall thickness and height <cit.>, herein we demonstrate that an additional degree of tunability over the inner structure of the fabricated microrings can be achieved via utilization of a nitrogen-doped Au film. Variation of N_2 concentration in a buffer gas during the process of magnetron deposition of such Au film was demonstrated to pre-determine size and density of resulting nanopores, appearing as a result of boiling inside the molten rim produced upon single-shot ns-laser film ablation. The contours of the microring as well as the inner pores can be further unveiled in the process of ion-beam post-polishing, yielding in isolated rough plasmonic nanostructures with pronounced porosity. The fabricated porous microrings were shown to produce many-fold near-field enhancement of incident electromagnetic fields, which was confirmed via mapping of the characteristic Raman bands of a self-organized monolayer of the Rhodamine 6G (R6G) dye molecules and supporting finite-difference time-domain (FDTD) calculations. The proposed procedure for fabricating the isolated porous rings includes two consecutive steps schematically illustrated in Fig.1(a). First, a 100-nm thick glass- or Si-supported Au film is irradiated with single second-harmonic (532 nm), 7-ns pulses delivered by a Nd:YAG laser system (Brio, Quantel) focused with a dry objective lens of numerical aperture NA=0.6. In all our experiments, we used the same focusing conditions, while the onlyincident pulse energy was varied by means of an adjustable attenuator. In a certain range of incident pulse energies <cit.>, such single-pulse ns-laser ablation of the metal film leaves a through hole surrounded by a smooth resolidified rim typically 2.5-3 times thicker, comparing to the initial thickness of the deposited Au film. The second step involves removal of the unprocessed parts of the film using a liquid-free etching with an accelerated argon-ion beam (IM4000, Hitachi) at fixed values of acceleration voltage of 3kV, discharge current of 105 μA and gas flow of 0.1 cm^3/min. These parameter set provides relatively slow rate of 1 nm/s to avoid film melting. The ion-beam etching procedure was preliminary calibrated in terms of etching rate using as-deposited gold films of different thickness (for additional details, see Ref.<cit.>). The removal of the residual, initially deposited 100-nm thick Au film leaves the isolated microrings on the substrate surface. In our previous study, main geometric parameters of the microrings produced with the developed two-step procedure were shown to be tuned by varying the thickness of the as-deposited film and the incident pulse energy <cit.>. In the present study, we demonstrate that an additional degree of tunability of the internal structure of the produced microrings can be provided by minor variation of the chemical composition of the sputtered metal film via its controlled magnetron deposition in appropriate gaseous environments. To deposit the Au film on a glass substrate a commercial magnetron sputter was used (Quorum Technologies)<cit.>. Typically, argon is used as a discharge gas for such magnetron sputtering. Then, ns-laser printing followed by the subsequent ion-beam polishing usually results in formation of smooth microrings with a relatively small amount of defects demonstrated by their detailed SEM inspection (see Fig.1(b)).Surprisingly, detailed imaging of the structures produced with the single-shot ns-pulse ablative patterning of a Au film sputtered in a mixture of the argon and nitrogen (50/50 wt.%) reveals dense pores inside the fabricated microrings (Fig.1(c)). The nanosized (appox. 60-nm diameter) spherical-shape pores initially located inside the resolidified metal rim can be further visualized through the ion-polishing post-procedure, as indicated by the series of the AFM images (Fig.1(d)). This series presents the several consecutive polishing steps from the as-irradiated film with the smooth walls to the complete removal of the non-treated regions of the film, revealing the micro-sized contours of the rings as well as their inner nanopores. Moreover, even more pronounced porosity with significantly larger averaged pore size was found for such structures produced on the surface of the Au film deposited in the nitrogen atmosphere (Fig.2(a)). The statistical study of the distribution of the pore size inside the microrings produced at the same pulse energy indicates the clear correlation between the nitrogen concentration in the deposition chamber and the average nanopore diameter in the produced microrings (Fig.2(a)).Interestingly, detailed high-resolution SEM and AFM inspection of the untreated films sputtered in various gas environments indicates negligibly small variation of the film morphology (roughness, average grain size, etc.), comparing to the film deposited in the argon atmosphere<cit.>. Specifically, for all these tested as-deposited films this analysis indicates the appearance of the through cracks, apparently owing to low-vacuum sputtering conditions (≈10^-2 Torr) and relatively fast sputtering rate of 1 nm/s (see insets in Fig.2(c)). The remelting of such defects on the nanosecond-laser exposure timescale can potentially explain such appearance of the small amount of nanosized pores inside the rings produced in the “Ar-deposited" films (Fig.2(a), top). However, considering the similarity of the morphologies of the film deposited in the various gaseous environments, such explanation can not be used for other films, demonstrating pronounced porosity and significantly larger average diameter of pores (Fig.2(a), middle and bottom<cit.>. In this way, subsurface boiling around such nitrogen-doped cracks and N_2-reach areas of the film appears to be the key mechanism responsible for the formation of the densely-packed nanopores inside the molten rim<cit.>. As known, gold can interact with nitrogen, producing gold-nitride phase Au_xN, upon Au target bombardment with accelerated nitrogen ions <cit.>. Also, Au_xN films were fabricated, using ablative pulsed laser deposition in a N_2 containing atmosphere<cit.>. Our energy-dispersive X-ray (EDX, ThermoDry) spectroscopic analysis performed at the 5-kV e-beam acceleration voltage under its oblique (47^∘ to the sample normal) incidence shows the detectable nitrogen peak for the Au films sputtered in the nitrogen atmosphere (Fig.2(c)). Additionally, we have probed the same films by secondary-ion mass spectrometry (SIMS, Hyden Analytical EQS1000 installed in the Zeiss CrossBeam 1540 apparatus), utilizing a Ga^+ beam at 30 kV and 200 pA for sampling of 200x200 μm^2 areas of each film. For each sample, we averaged the obtained mass spectra over the depth profile. As seen, such comparative SIMS study reveals higher signals of nitrogen atoms (see Fig.2(d) and inset therein as well as Ref.<cit.>) in the films deposited in the nitrogen-containing ambients, while the carbon level for all films remains at the constant level. More generally, nitrogen-to-gold peak ratios averaged over the film depth are 0.23, 0.15, 0.14 for the films deposited in the nitrogen, nitrogen-argon mixture and argon ambients. Despite both methods indicate the increased amount of the nitrogen inside the metal films sputtered in the appropriate atmosphere (Fig.2(c,d)), the X-ray photoelectron spectroscopy should be undertaken during this ongoing research to prove the chemical bounding of nitrogen to gold.Additionally, since the boiling of the nitrogen-rich areas of the Au film appears to be a temperature-driven process, the size of the nanopores also should demonstrate similar tendency. This is clearly illustrated by a series of SEM images of the microrings printed at the increasing pulse energy on the N_2-doped Au film (Fig. 2(e)), where the evident increase of the nanopore diameter can be identified. Specifically, in the certain range of the incident pulse energies E, the molten rim becomes hydrodynamically unstable undergoing the periodical modulation of its height (“crowning"<cit.>) governed by the Rayleigh-Plateau hydrodynamic instability. The printed microrings, with their well-controlled micrometer-scale diameter and pronounced nanotexture governed by the unveiled multiple circumferentially-spaced pores, are expected to provide strong near-field enhancement of incident electromagnetic waves in visible and IR spectra ranges, making them promising for chemo- and biosensing applications. In this respect, the porous rings arranged into the ordered array can be considered as multifunctional substrates, providing enhancement of both photoluminescence and SERS signals from adsorbed analyte molecules commonly pumped in the optical range, while giving also possibility to utilize near-to-mid infrared wavelengths to excite main dipolar localized plasmon resonances of the microrings.Without loss of generality, in this Letter we restrict our studies of biosensing performance for the laser-printed porous microrings by measuring their Raman response from a self-organized monolayer of R6G molecules and mapping surface intensity distribution of their specific main bands. A commercial Raman microscope (Alpha, WiTec), utilizing a 532-nm CW semiconductor laser source focused by a 0.9-NA dry lens (100x, Carl Zeiss) and a grating-type spectrometer (600 lines/mm) with a electrically cooled CCD camera, was used to measure SERS performance of the fabricated microrings. An ethanol solution of R6G with a concentration of 10^-5M was drop-casted on the sample surface and after its complete drying was rinsed in a distilled water producing a monolayer. All Raman spectra were measured at the 1-mW excitation laser power and the accumulation time of 0.5 s per point. Similar parameters were used for mapping of the surface intensity distribution of the main R6G Raman bands near the several smooth and porous microrings within the 2x2 μm^2 square sample area with 0.2-μm sampling step. The build-in software (WiTec Project) was used to obtain the 2D maps related to each spectrally narrow Raman band.Typical Raman spectra acquired from the R6G layer on both smooth and porous microrings, having almost the same outer diameter and the wall thickness, reveal all main Raman bands characteristic for R6G molecules, while these “fingerprints" can not be identified on the glass substrate even under 10 times increased accumulation time (Fig.3(a)). This measurement indicates strong SERS performance of the fabricated structures. The maximal Raman intensity averaged over all main spectral bands marked in Fig.3(a) for smooth microrings reaches 300± 60 counts(s^-1mW^-1), while increases to 1550± 170 counts(s^-1mW^-1) for porous microrings, yielding in the average SERS enhancement factor of ≈ 10^5 when weighted as signal per molecule of R6G solution <cit.>. Also, it points out a clear plasmonic contribution to the SERS enhancement, rather than the concentration effect. Mapping the surface distribution of the Raman intensity at the 1361-cm^-1 band of the R6G molecules, covering the smooth microring, reveals the position of the local electromagnetic “hot spots”, which were found to depend on the polarization direction of the exciting laser source (orange arrows in Fig.3(b-e)).According to our supporting FDTD calculations (for details see <cit.>), such 532-nm, normal-incident linearly-polarized laser irradiation provides low-efficiency excitation of the smooth microring walls with quite moderate 5-fold enhancement of the squared electric field amplitude (Fig.3(d)), while the corresponding hot spots, following the linear polarization direction, are distributed non-uniformly along the ring circumference, yielding in similar distribution of SERS signal (Fig.3(b)). On the contrary, according to our FDTD calculation performed for the porous microring, nanosized surface features produce multiple strongly enhanced E-fields, homogeneously distributed along the microring surface independently on the polarization direction (Fig.3(e)) and generally correlating with the acquired SERS map (Fig.3(c)).In conclusion, in this work we have fabricated porous microrings from thin gold films deposited in nitrogen and nitrogen-argon mixed atmospheres, by their ns-laser ablation and the following ion-beam etching post-procedure, and studied their SERS performance. The proposed approach allows us to fabricate both smooth and porous microrings with variable geometric parameters as well as tunable inner structure (nanopore size and density). 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"authors": [
"S. Syubaev",
"A. Nepomnyashchiy",
"E. Mitsai",
"E. Pustovalov",
"O. Vitrik",
"S. Kudryashov",
"A. Kuchmizhak"
],
"categories": [
"physics.app-ph",
"physics.optics"
],
"primary_category": "physics.app-ph",
"published": "20170726030639",
"title": "Fabrication of porous microrings via laser printing and ion-beam post-etching"
} |
empty 1 Mathematical and Algorithmic Sciences Lab, Paris Research Center, Huawei Technologies France2 Communications and Electronics Department, Telecom ParisTech, Paris, 75013, France*[email protected] multiplexed (PDM) transmission based on the nonlinear Fourier transform (NFT) is proposed for opticalfiber communication. The NFT algorithms are generalized from the scalar nonlinear Schrödinger equation for one polarization to the Manakov system for two polarizations. The transmission performance of the PDMnonlinear frequency-division multiplexing (NFDM)and PDM orthogonal frequency-divisionmultiplexing (OFDM) are determined. It is shown that the transmission performance in terms of Q-factor is approximately the same in PDM-NFDM and single polarization NFDM at twice the data rate and that the polarization-mode dispersion does not seriously degrade system performance. Compared with PDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized to multi-mode fibers in the strong coupling regime, paving the way for the application of the NFT to address the nonlinear effects in space-division multiplexing. (060.2330) Fiber optics communications,(060.4230) Multiplexing, (060.4370) Nonlinear optics, fibersosajnl § INTRODUCTION Nonlinear frequency-division multiplexing (NFDM) is an elegant method to address the nonlinear effectsin optical fiber communication. The scheme can be viewed as a generalization of communication using fiber solitons <cit.> and goes back to the original idea of eigenvalue communication <cit.>.In this approach, information is encoded in the nonlinear spectrum of the signal, definedby means of the nonlinear Fourier transform (NFT), <cit.>. The evolution of the nonlinear spectral components in fiber is governed by simple independent equations. As a result, the combined effects of the dispersion and nonlinearity can be compensated in the digital domain by the application of an all-pass-like filter. Furthermore, interference-free communication can be achieved in network environments.The nonlinear spectrum consists of a continuous part and, in the case of the anomalousdispersion fiber, also a discrete part (solitonic component). In principle all degrees-of-freedom can be modulated. Prior workhas mostly focused on either continuous spectrum modulation <cit.>, or discrete spectrum modulation <cit.>. Transmission based on NFT has been experimentally demonstratedfor the continuous spectrum <cit.>, discrete spectrum <cit.>, as well asthe full spectrum <cit.>. Modern coherent optical fiber systems are based on polarization-division multiplexed (PDM) transmission to improvethe achievable rates. However, research on data transmission using the NFT is limited tothe scalar nonlinear Schrödinger equation, which does not take into account polarizationeffects (with the exception of <cit.>). In this paper we overcome this limitation by generalizing the NFDM to the Manakov system, proposingPDM-NFDM. We develop a stable and accurate algorithm to compute the forward and inverse NFT of a two-dimensional signal. It is shown that the PDM-NFDM based on the continuous spectrum modulation is feasible, and that the data ratecan be approximately doubled compared to the single polarization NFDM. Compared to the PDM-OFDM, the PDM-NFDM exhibits a peak Q-factor gain of 6.4 dB, in a system with 25 spans of 80 km standard single-mode fiber and 16 QAM.In this paper, we set the discrete spectrum to zero and modulate only the continuous spectrum. Even though not all available degrees of freedom are modulated, the achievable rates reach remarkably close to the upper bound <cit.>.The differential group delay and randomly varying birefringence give rise to temporal pulse broadening due topolarization-mode dispersion (PMD). The PMD might be compensated in conventional systems, and may even be beneficial in reducing the nonlinear distortions <cit.>. On the other hand, PMD changes the nonlinear interaction between signals in the two polarizations. The impact of PMD on NFDM is not fully investigated yet <cit.>. In this paper, we show that linear polarization effects can be equalized in PDM-NFDM at the receiver using standard techniques, and that the PMD does not seriously degrade performance.§ CHANNEL MODEL WITH TWO POLARIZATIONS Light propagation in two polarizations in optical fiber is modeled by thecoupled nonlinear Schrödinger equation (CNLSE) <cit.>. The fiberbirefringence usually varies rapidly and randomly along the fiber in practical systems(on a scale of 0.3 to 100 meters). Under this assumption, the averaging of the nonlinearity in the CNLSE leads to the Manakov-PMD equation <cit.>:∂A/∂ Z =-jΔβ_0/2σ_ZA-Δβ_1/2σ_Z ∂A/∂ T -α/2A + jβ_2/2∂^2 A/∂ T^2- jγ8/9A^2A.Here A≡A(Z,T) is the 2× 1 Jones vector containing the complex envelopes A_1 and A_2 of the two polarization components, Z denotes the distance along the fiber, T represents time, σ_Z is a 2× 2 Pauli matrix (dependingon the state of polarization at Z), andΔβ_0, Δβ_1, α, β_2 and γ are constant numbers. The term ∂A/∂ T is responsible for the PMD <cit.>, while the second line represents loss, chromatic dispersion and Kerr nonlinearity. The factor 8/9 in front of the nonlinear term stems from polarization averaging. The NFT applies to the integrable equations. However, the Manakov-PMD system (<ref>)apparently is not integrable in the presence of loss or PMD. The loss may be approximately compensatedusing ideal distributed Raman amplification, leading to a lossless modelwith distributed additive white Gaussian noise (AWGN). Discrete amplification with short spans can also be modeled similarly, but with a modified nonlinearity coefficient (depending on loss); see <cit.>.Ignoring the PMD and, for the moment, noise, (<ref>) is reduced to ∂A/∂ Z =jβ_2/2∂^2 A/∂ T^2- jγ8/9A^2A. It is convenient to normalize (<ref>). Let Z_0=1 andT_0=√((β_2Z_0)/2),A_0=√(2/(8/9γ Z_0)).Introducing the normalized variables z=Z/Z_0, t=T/T_0 and q=A/A_0, (<ref>) is simplified to the Manakov equation j∂q/∂ z = ∂^2 q/∂ t^2- 2s q^2q,where s=1 in the normal dispersion (defocusing) regime and s=-1 in the anomalous dispersion (focusing or solitonic) regime. It is shown in <cit.> that (<ref>) is integrable. In what follows, we develop thecorresponding NFT.§ NFT OF THE TWO-DIMENSIONAL SIGNALS In this section, basics of the NFT theory of the two-dimensional signals, as well as numerical algorithms tocompute the forward and inverse NFT, are briefly presented <cit.>.§.§ Brief Review of the TheoryEquation (<ref>) can be represented by a Lax pair L̂ and M̂. This means that,operators L̂ and M̂ can be found such that (<ref>) is in one-to-one correspondence with the Lax equation ∂L̂/∂ z = [M̂,L̂]. The Lax pair for the Manakov equationwas found by Manakov in 1974 <cit.>. The operator L̂ is:L̂ = j[∂/∂ t -q_1 -q_2; sq_1^* -∂/∂ t0; sq_2^*0 -∂/∂ t ]. To simplify the presentation, consider the focusing regime with s=-1. The eigenvalue problem L̂ v=λ vcan be solved for the Jost function v (eigenvector) assuming that thesignals vanish at t=±∞. This gives rise to six boundary conditions for v, denoted by j_±:j_±^(0)→ e^(0)exp(-jλ t),j_±^(i)→ e^(i)exp(jλ t),i=1,2, as t→±∞,where e^(k) are unit vectors, i.e., e^(k)_l=δ_kl, k,l=0,1,2. Each of the boundary conditions(<ref>) is bounded when λ∈^+ or λ∈^-. The eigenvalueproblem (<ref>) under the boundary conditions (<ref>) can be solved,obtaining six Jost functions{j^(i)_±(t,λ)}_i=0,1,2for all t. It can be shown that {j^(i)_+(t,ł)}_i=0,1,2 and {j^(i)_-(t,ł)}_i=0,1,2 each form anorthonormal basis for the solution space of (<ref>). Thus we can expand j^(0)_-(t,λ) in the basis of {j^(i)_+(t,ł)}_i=0,1,2:j_-^(0)(t,ł) = a(ł) j_+^(0)(t,ł)+b_1(ł) j_+^(1)(t,ł)+b_2(ł) j_+^(2)(t,ł),where a(ł), b_1(ł) and b_2(ł) are called nonlinear Fourier coefficients. It can be shownthat in the focusing regime the inner product of two Jost functions corresponding to the same eigenvalue does not depend on time. This implies that a(λ) and b_i(λ) do not depend on time (as the notation in (<ref>) suggests).The NFT of the q=[q_1,q_2] is now defined asNFT(q)(ł) =q̂_i = b_i(ł)/a(ł),λ∈R,q̃_i = b_i(ł_j)/a'(ł_j),ł_j ∈C^+,i=1,2,where ł_j, j=1,2,⋯, N, are the solutions of a(ł_j)=0 in ł_j∈C^+.An important property of NFT(q(t,z))(ł) is that it evolves in distance according to the all-pass-like filterH(λ) = e^4jsλ^2 L. Remark. Note that if q_1 or q_2 is set to zero in the Manakov equation and the associated L̂ operator (<ref>), the equation and the operator are reduced, respectively, to the scalar NLSE and the corresponding L̂ operator <cit.>.Also, the theory in this section can straightforwardly be generalized to include any number ofsignals q_i, i=1,2,..., for instance, LP modes propagating in a multi-mode fiber in the strong coupling regime.Remark Since the Jost vectors are orthonormal at all times, (<ref>) implies the unimodularity condition|a(λ)|^2+|b_1(λ)|^2+|b_2(λ)|^2 = 1,which will be used in the algorithm.§.§ Forward NFT Algorithm We develop the NFT algorithms for the continuous spectrum that is considered in this paper. The forward and inverse NFT algorithms are, respectively, based on the Ablowitz-Ladik and discrete layer peeling (DLP) methods. These algorithms generalize the corresponding ones in <cit.>.We begin by rewriting the eigenvalue problem L̂v=λ v in the form ∂ v/∂ t = Pv, where P = [ -jλq_1(t)q_2(t); -q_1^*(t)jλ 0; -q_2^*(t) 0jλ ].Here and in the remainder of this section we suppress the dependence on the coordinate z. We discretize the time interval [T_0,T_1] according to t[k] = T_0 + kΔ T, where Δ T= (T_1-T_0)/N such that t[0]=T_0 and t[N]=T_1. We set q_i[k] = q_i(T_0+kΔ T) and similarly for the vector v. The Ablowitz-Ladik method is a discretization of ∂ v/∂ t = Pv as follows v[k+1,ł] = c_k[ z^1/2Q_1[k]Q_2[k]; -Q_1[k]^*z^-1/2 0; -Q_2[k]^* 0z^-1/2 ]v[k,ł],where Q_i[k]=q_i[k]Δ T, z:= e^-2jλΔ T and k=0,1,⋯, N-1.With this discretization v becomes a periodic function of ł with period π/Δ T. We introduced a normalization factor c_k=1/√( P[k]) in (<ref>), whereP[k]=1+Q_2[k]^2+Q_2[k]^2, to improve numerical stability. The iterative equation (<ref>) is initialized with v[0,ł]=j_-^(0)(T_0,ł)=e^(0)z^T_0/2Δ T.After N iterations, the nonlinear Fourier coefficients are obtained as projections onto the Jost-solutions j_+^(i)(T_1,λ),a[ł]= z^-N/2-T_0/2Δ Tv_0[N,ł],b_i[ł]= z^N/2+T_0/2Δ Tv_i[N,ł], i=1,2.The continuous component in the (q)(λ) is then computed based on (<ref>).The algorithm can be efficiently implemented in the frequency domain.Let us write (<ref>) asV[k+1,ł] = c_k[1 Q_1[k]z^-1 Q_2[k]z^-1;-Q_1^*[k] z^-10;-Q_2^*[k]0 z^-1 ]V[k,ł],where V[k,ł] (A[k,ł],B_1[k,ł],B_2[k,ł])^T, and A[k,λ]= a[k,λ]B_i[k,λ]=z^-N-T_0/Δ T+1/2b_i[k,ł], i=1,2.We discretize λ on the interval [Λ_0,Λ_1] with Λ_1=-Λ_0=π/(2Δ T) such that λ[k] = Λ_0+kΔΛ. Let tilde '∼' denote the action of the discrete Fourier transform DFT with respect to ł[k], e.g., Ã[.,l]=DFT(A[.,ł[k]]), where l=0,1,,⋯,N-1 is the discrete frequency. Note thatDFT{z^-1B_i[.,ł[k]]}[l]={B̃_̃ĩ}[l], wheredenotes circular right shift of the array by one element.Equation (<ref>) in the frequency domain isÃ[k+1,l]= c_k(Ã[k,l]+Q_1[k][B̃_̃1̃[k]][l] +Q_2[k][B̃_̃2̃[k]][l]), B̃_̃1̃[k+1,l]= c_k( -Q_1[k]^*Ã[k,l]+[B̃_̃1̃[k]][l]), B̃_̃2̃[k+1,l]= c_k(-Q_2[k]^*Ã[k,l]+[B̃_̃2̃[k]][l]).The initial condition is given by Ã[0] = DFT[a[k]][0] and B̃_̃ĩ[0] = 0. At k=N-1, a and b_i arefound by recovering V[N,λ] through an inverse DFT and using (<ref>). §.§ Inverse NFT Algorithm The inverse NFT algorithm consists of two steps. First, we compute v[N,λ] from the continuous spectra q̂_1,2(λ) and invert the forward iterations (<ref>). Second, at each iteration,we compute Q[k] from v[k,λ].Substituting b_i(ł)=q̂_i(ł) a(ł) in the unimodularity condition (<ref>), we can compute |a(ł)| |a(ł)| = 1/√(1+|q̂_1(ł)|^2+|q̂_2(ł)|^2).In the absence of a discrete spectrum, from (<ref>), a(ł)≠ 0 for all ł. We have (log(a)) = log(|a|) and (log(a)) = ∠ a. Since a(ł) and therefore log(a) are analytic functions of ł we can recover the phase of a as ∠a = ℋ(log(|a|)), where ℋ denotes the Hilbert transform. The inverse iterations are obtained by inverting the matrix in (<ref>) and droppingterms of order Q^2_i∼Δ T^2 (to yield the same accuracy as for the forward iterations):v[k,ł] = c_k[ z^-1/2-Q_1[k]-Q_2[k]; Q_1^*[k]z^1/20; Q_2^*[k]0z^1/2 ]v[k+1,ł]. In practice, as in the forward NFT, it is better to invert the frequency domain iterations (<ref>). The signal Q[k] can then be obtained from V[k+1,l] as follows. Recall that[B̃_i[k]][l]=B̃_i[k,l-1]. Using the initial conditions for the forward iterations Ã[0,l]=δ_0,l and B̃_i[0,l]=0, it is straightforward to showthat Ã[k,l]=0 and B̃_i[k,l]=0 for l ≥ k>0. In particular, B̃_i[k,N-1]=0 for k<N. For the first element of B̃_i[k], from (<ref>) and [B̃_̃1̃[k]][-1] = [B̃_̃1̃[k]][N-1]=0, we obtainÃ[k+1,0]= c_kÃ[k,0], B̃_̃1̃[k+1,0]= -c_kQ_1[k]^*Ã[k,0], B̃_̃2̃[k+1,0]= -c_kQ_2[k]^*Ã[k,0].These equations can be solved for Q_i^*[k] = -B̃_i[k+1,0]/Ã[k+1,0]. §.§ Testing the NFT AlgorithmsThe forward and inverse NFT numerical algorithms are tested as follows.Figure <ref> compares a signal containing two polarization components in the nonlinear Fourier domain with its reconstruction after successive INFT and NFT operations. We take two displaced Gaussians with standard deviation σ=√(2) as input signals for the two polarization components q̂_i, i=1,2. We set time windows to T=64 and take 2048 samples in time and nonlinear Fourier domain. In the discrete layer peeling method, the signal is periodic in nonlinear Fourier domain with period π/Δ T≈ 100. Deviations occur as the amplitude of the signal is increased. The accuracy of the NFT (INFT) can beenhanced by increasing the sampling rate in time (nonlinear frequency), as this reduces errorsdue to to the piecewise-constant approximation of the signal. We find that we roughly need twice thenumber of samples to achieve the same accuracy as in the single-polarization case.§ PDM-NFDM TRANSMITTER AND RECEIVERIn this section, we describe the transmitter (TX) and receiver (RX) digital signal processing (DSP) in PDM-NFDM and PDM-OFDM. A schematic diagram of the polarization-division multiplexed NFDM and OFDM transmission systems is shown in Fig. <ref>.The TX DSP produces digital signals for the in-phase and quadrature components of both polarization components, which are fed into the IQ-Modulator. The modulated signals for the two polarization components are combined in the polarization beam combiner before they enter the transmission line consisting of multiple fiber spans. We consider the practically relevant case of lumped amplification using Erbium-doped fiber amplifiers (EDFAs). After propagation through the fiber the two polarization components are separated in the polarization beam splitter and fed into the polarization-diversity intradyne coherent receiver, which provides the input to the RX DSP.We briefly describe the TX and RX DSP in OFDM and NFDM. We first map the incoming bitstosignals taken from a QAM constellation. We then oversample the discrete-timesignal in the time domain (by introducing zeros in the frequency domain outside the support of the signal) and thenadd guard intervals in the time domain. Increasing the guard interval increases the accuracy of the INFT. Unless stated otherwise, we do not increase the guard intervals during the computation of the INFT as the amplitude is increased, so there is a penalty due to inaccuracies in the algorithm. In NFDM, these steps are performed in what is called the U-domain; see <cit.>. The signal is subsequently mapped from the U-domain to nonlinear Fourier domain through the transformationU_i (λ) = √(-log (1-q̂_i(λ)^2))e^j∠q̂_i(λ), i=1,2.This step has no analogue in OFDM. Note that contrary to the single-polarization case here the energy of the signal in the timedomain is only approximately proportional to the energy in the U-domain. At this point, we perform the inverse NFT in the case of NFDM and an inverse FFT in case of OFDM. Then we obtain the unnormalizedsignal by introducing units using (<ref>). Finally we combine the OFDM or NFDM bursts to obtain the signal to be transmitted. The output of the DSP are the in-phase and quadrature components of the signals in the two polarization components. All steps in the DSP are performed independently for the two polarizations, except for the INFT, which requires joint processing.At the RX we invert the steps of the TX DSP. The signal processing begins with burst separation and signal normalization using (<ref>), followed by the forward NFT (FFT) in case of NFDM (OFDM). In NFDM, we subsequently equalize the channel using (<ref>), which isa single-tap phase compensation. Similarly, in OFDM weeither compensate the dispersion with the phase exp(-jβ_2ω^2 N_spanL/2), or perform digitalbackpropagation (DBP) for a fixed number of steps per span. We then remove the guard intervals in time, down-sample the signal in the time domain, and obtain the outputsymbols.The bit error rate (BER) is calculated using a minimum distance decoder for symbols.In Fig. <ref> we compare the time domain signals of NFDM and OFDM at the TX for one of the polarization components. We use 112 subcarriers over a burst duration of T_0=2ns in both cases. These parameters are the same as in Ref. <cit.> and correspond to a burst data rate of 224GBit/s per polarization at a Baudrate of 56GBaud. In NFDM the shape of the signal in the time domain changes as the amplitude |q̂_1,2|is increased in the nonlinear Fourier domain. At a signal power of 0.5 dBm, one can see a significant amount of broadening of the NFDM signal in the time domain. This effect makes it difficult to control the time duration of signals in NFDM. Pulse broadening in the time domain due to the chromatic dispersion can be estimated by <cit.>Δ T = 2π |β_2| LB,where B is the signal bandwidth. The guard time intervals for minimizing the interaction among bursts can be approximated using (<ref>). For the parameters in Table <ref>, we obtain Δ T∼ 15ns, to which we add a 20% marginto obtain a total symbol duration of T=T_0+T_guard=(2+18)ns=20ns. This corresponds to an effective data rate of 44.8Gbit/s. The right panel of Fig. <ref> shows the time domain signals in NFDM and OFDM in one polarization at the RX after propagation over 2000 km governed by the integrable model (<ref>) (we consider transmission over the realistic fiber model (<ref>) below in Sec. <ref>). We observe that the amount of the temporal broadening of theNFDM and OFDM signals is about the same. In our simulations we do not employ pre-compensation at the TX, which may be addedto further increase the data rate <cit.>.§ SIMULATION RESULTS In this section, we consider the system shown in Fig. <ref> and compare the PDM-OFDM andPDM-NFDM via simulations, taking into account loss and PMD. First, we consider the model withloss and periodic amplification,setting thePMD to zero (namely, neglecting the terms in the first line of the Eq. <ref>). Next, the PMD effects are studiedin Sec. <ref>. The system parameters used are summarized in Table <ref>.§.§ Effect of loss and lumped amplification In the lumped amplification scheme, the signal is periodically amplified after each span. The channel is described by the model (<ref>) including the attenuation term, which is manifestly not integrable. The effect of lumped amplification on NFT transmission has been studied previously <cit.>.We illustrate the effect of the attenuation by comparing the BER as a function of the OSNR in Fig. <ref> for four models: 1) the back-to-back (B2B) configuration; 2) lossless model (<ref>) (there is propagation but without loss);3) lossy model with lumped amplification;and 4) a transformed-lossless model that is introduced below. In producing Fig. <ref>, weartificially introduced AWGN at the receiver, to exclude the effects of thesignal-noise interaction from the comparison. We fix the power P=-3.1dBm to keep the accuracy of the NFT algorithms the same and vary the OSNR by changing the noise power.We see that the BER in B2B and lossless models are approximately the same.Thisresult verifies that the implementation of the NFT is correct.It also shows that the effect of burst interaction due to the finite guard interval is negligible.The BER of the lossy model is significantly higher than the BER in the B2B and lossless models at high OSNRs.The BER of the lossy model seems to flatten out as the OSNR is increased. Fiber loss can be taken into account in the NFT as follows. The attenuation term in (<ref>) can beeliminated by a change of variable A(z,t)=A_0(z,t)e^-α/2z <cit.>. This transformsthe non-integrable equation (<ref>) to the integrable equation (<ref>) (thus with zero loss)and a modified nonlinearity parameter γ_eff(L) = 1/L∫_0^L γ e^-α zdz = γ(1-e^-α L)/(α L).We refer to the resulting model as the transformed-lossless model.Note that the power of the signal A_0(z,t) in the transformed-lossless model is higher than the power of the A(z,t) in the original lossy model. That is because γ_eff<γ, which yields higher amplitudes according to (<ref>).Figure <ref> shows that the proposed scheme for loss cancellation is indeed very effective. Using γ_eff(L) in the inverse and forward NFTs, the BER of the lossy and transformed-lossless modelsare almost identical. §.§ Transmission performance PDM-NFDM In order to assess the transmission performance of the PDM-NFDM transmission, we performedsystem simulations as described in Sec. <ref>. For now we focus on deterministic impairments and neglect PMD. We simulate transmission of 112 subcarrier NFDM and OFDM pulses based on a 16QAM constellationover N_span=25 spans of 80km of standard single-mode fiber. We assume ideal flat-gain amplifiers with a noise figure of N_F=6.2dB. The system parameters are summarized in Table <ref>. They are comparable with those of Ref. <cit.> to facilitate a comparison with the single-polarization case.We first compare single-polarization NFT transmission based on the NLSE with polarization-multiplexed transmission based on the Manakov equation. Figure <ref> shows that in the linear regime the two curves are offset by 3dB because polarization multiplexing doubles the transmission power without increasing the signal to noise ratio. We note that in this calculation we increased the guard band and thereby the sampling rate in frequency domain by a factor of two in the computation of the PDM-NFT (the guard band in transmission is the same) compared to the single-polarization case. This is to avoid penalties due to inaccuracies in the NFT, which here amount to approximately 1dB at peak power if the same guard bands are used. Without this penalty, the Q-factor at peak power is roughly the same in both cases. This implies that data rates can approximately be doubled using polarization multiplexing. We show in Sec. <ref> that this conclusion still holds in presence of polarization effects.In Fig. <ref> we compare the Q-factor as a function of launch power in PDM-OFDM and PDM-NFDM transmission. The optimum launch power in OFDM is significantly smaller than in NFDM. OFDM transmission is limited by nonlinearity, while NFDM is mainly limited by nonlinear signal-noise interaction.The Q-factor at optimum launch power is significantly higher in NFDM than in OFDM. We observe an overall gain of 6.4dB in Q-factor. Figure <ref> shows the corresponding received constellations at the respective optimal launch power.In Fig. <ref> we further report results obtained by digital backpropagation after OFDM transmission. For a fair comparison with NFDM, we use the same sampling rate in the DBP and apply it without prior downsampling. OFDM with DBP achieves similar performance to NFDM for around 10 DBP steps per span and exceeds it for 16 steps per span. We note that here we considered a single-user scenario. The full potential of NFDM is leveraged in a network scenario, where DBP is less efficient. Our results are in good qualitative agreement with those reported in Ref. <cit.> for the single-polarization case, while achieving twice the data rate. §.§ Effect of PMD§.§.§ Simulation of PMDThe fiber model (<ref>) describes light propagation in linearly birefringent fibers <cit.>. Here Δβ_1 is the difference in propagation constants for the two polarizationstates which are induced by fiber imperfections or stress. In real fibers, this so-called modal birefringence varies randomly, resulting in PMD.We simulate PMD with the coarse-step method, in which continuous variations of the birefringence are approximated by a large number of short fiber sections in which the birefringence is kept constant. PMD is thus emulated in a distributed fashion <cit.>. We use fixed-length sections of length 1km, larger than typical fiber correlation lengths. At the beginning of each section, the polarization is randomly rotated to a new point on the Poincaré sphere. We apply a uniform random phase in the new reference frame. The latter accounts for the fact that in reality the birefringence will vary in the sections where it is assumed constant, which will lead to a random phase relationship between the two polarization components <cit.>. The differential group delay (DGD) of each section is selected randomly from a Gaussian distribution. This way artifacts in the wavelength domain caused by a fixed delay for all sections are avoided <cit.>.Within each scattering section, the equation (<ref>) without PMD terms is solved using standard split-step Fourier integration.To speed up the simulations, we employ a CUDA/C++ based implementation with a MEX interface to Matlab.The resulting DGD of the fiber is Maxwell distributed <cit.>,p_Δ t(Δ t) = 32/π^2Δ t^2/Δ t^3exp(-4Δ t^2/πΔ t^2).For the Maxwell distribution, the mean is related to the root-mean-square (RMS) delay by Δ t√(3π/8)=√(Δ t^2). The average DGD varies with the square root of the fiber length <cit.>,Δ t∼√(Δ t^2) = D_PMD√(L),where D_PMD is the PMD parameter. Typical PMD values for fibers used in telecommunications range from 0.05 in modern fibers to 0.5 ps/√(km).§.§.§ PMD impact on NFDM In this section, we consider polarization effects on the NFT transmission.Since the nonlinear term in the Manakov equation is invariant under polarization rotations, the equalization can be done in in the nonlinear Fourier domain (cf. Fig. <ref>). We use a simple training sequence based equalization algorithm to compensate linear polarization effects. The samples of the first NFDM symbol are used as the training sequence. The filter taps of the equalizer are determined using least-squares estimation. To obtain the Q-factor we average the BER over 120 random realizations of PMD for each data point.Due to its statistical nature, the effects of PMD are often quantified in terms of outage probabilities. In practice, the system is designed to tolerate a certain amount of PMD, in this case by fixing the number of taps in the equalizer. When the DGD exceeds this margin, the system is said to be in outage.We first determine the number of taps to achieve a given outage probability. Fig. <ref> shows the Q-factor as a function of the number of taps for different values of the PMD parameter. For D_PMD=0 we only have random polarization rotations and no DGD. Hence there is no interaction between nonlinearity and PMD and we use this case as the reference. The upturn for a small number of taps is due to the fact that a couple of taps are required to fully compensate the polarization rotations in presence of noise. For finite PMD, the curves for different PMD values converge to the same result within error bars. We can estimate the required number of taps from the Maxwell distribution. In order for the system to fully compensate PMD in 98.7% of the cases (1.3% outage probability), the equalization must cover a time interval equal to the mean plus 3 standard deviations of the DGD distribution. For example, for D_PMD = 0.1ps/√(km) this interval corresponds to 12.8ps. Using the sampling frequency we find that we approximately need 12 taps to converge, consistent with the figure.We note that by construction our equalizer also corrects potential phase rotations and therefore at least part of the nonlinear effects. In order to separate effects due to interaction of nonlinearity and PMD, we performed a calculation where we reversed the polarization rotations and phases exactly (instead of using the equalizer), by keeping track of the randomly generated angles and DGD values in the fiber simulation. We find a small phase rotation consistent with but smaller than the one reported in <cit.>. In our case it remains negligible at peak power. In Fig. <ref> we compare the Q-factor as a function of launch power with and without PMD effects. Here we use the number of taps determined as described above. We observe a penalty of roughly 0.3dB at peak power for D_PMD=0.2ps/√(km) relative to the case without PMD (D_PMD=0ps/√(km)). The penalty is therefore not serious for typical fibers used in telecommunication. Compared to the case without PMD effects and random birefringence (labeled “no birefringence"), we find a penalty of roughly 1.2dB at peak power for the case of zero PMD due to the equalization.§ CONCLUSIONS In this paper, we have proposed polarization-division multiplexing based on the nonlinear Fourier transform. NFT algorithms are developed based on the Manakov equations. Our simulations demonstrate feasibility of polarization multiplexed NFDM transmission over standard single-mode fiber. The results show that data rates can approximately be doubled in polarization-multiplexed transmission compared to the single-polarization case. This is an important step to achieve data rates that can exceed those of conventional linear technology.Numerical simulations of polarization multiplexed transmission over a realistic fiber model including randomly varying birefringence and polarization mode dispersion have shown that penalties due to PMD in real fibers do not seriously impact system performance. Our fiber simulations are based on the Manakov equations. As a next step, which is beyond the scope of this paper, the results should be verified experimentally.The equation governing light propagation of N modes in multi-mode fibers in the strong coupling regime can be written in the form <cit.>∂A/∂ Z =jβ̅_2/2∂^2 A/∂ T^2- jγκA^2A,where A=(A_1,…,A_N)^T and β̅_2 denotes the average group velocity dispersion. The NFT, as well as the algorithms presented here, generalize to this equation in a straightforward manner. Our approach therefore paves the way to combining the nonlinear Fourier transform with space-division multiplexing.§ ACKNOWLEDGEMENTSJ-W.G. would like to thank Wasyhun A. Gemechu for helpful discussions. H.H. acknowledges useful discussions with Djalal Bendimerad, Majid Safari, Sergei K. Turitsyn and Huijian Zhang. | http://arxiv.org/abs/1707.08589v1 | {
"authors": [
"Jan-Willem Goossens",
"Mansoor I. Yousefi",
"Yves Jaouën",
"Hartmut Hafermann"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170726180201",
"title": "Polarization-Division Multiplexing Based on the Nonlinear Fourier Transform"
} |
^1 Departamento de Fisica, Universidade Federal de Sergipe 49100-000, Sao Cristovao, Brazil.^2 Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil^3 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA^4 Complexity Science Hub Vienna, Josefstadter Strasse 39, 1080 Vienna, Austria We study the one-dimensional transverse-field spin-1/2 Ising ferromagnet at its critical point. We consider an L-sized subsystem of a N-sized ring, and trace over the states of (N-L) spins, with N→∞. The full N-system is in a pure state, but the L-system is in a statistical mixture. As well known, for L >>1, the Boltzmann-Gibbs-von Neumann entropy violates thermodynamical extensivity, namely S_BG(L) ∝log L, whereas the nonadditive entropy S_q is extensive for q=q_c=√(37)-6, namely S_q_c(L) ∝ L. When this problem is expressed in terms of independent fermions, we show that the usual thermostatistical sums emerging within Fermi-Dirac statistics can,for L>>1, be indistinctively taken up to L terms or up to log L terms. This is interpreted as a compact occupancy of phase-space of the L-system, hence standard BG quantities with an effective length V ≡log L are appropriate and are explicitly calculated. In other words, the calculations are to be done in a phase-space whose effective dimension is 2^log L instead of 2^L. The whole scenario is strongly reminiscent of a usual phase transition of a spin-1/2 d-dimensional system, where the phase-space dimension is 2^L^d in the disordered phase, and effectively 2^L^d/2 in the ordered one. 05.70.Jk, 05.30.-dEntanglement can preserve the compact nature of the phase-space occupancy A. M. C. Souza^1,2, P. Rapčan^2, and C. Tsallis^2,3,4 December 30, 2023 =========================================================================The Boltzmann-Gibbs (BG) theory refers to ensembles, which constitute pillars of statistical mechanics <cit.>. The microcanonical ensemble, for example, is associated with the set of points in the phase-space in which one can choose a given total energy. In this case, it is assumed a priori the ergodic hypothesis, in which the trajectories of the particles cover the hypersurface of the phase-space corresponding to that energy in a time scale sufficient to carry out the measurements. In many cases, it is not necessary for the system to cover the entire phase-space associated with the ensemble in question, but only a finite part of it, for instance half of it. A typical example is usual phase transitions. Below a certain critical temperature, the system has a spontaneous symmetry breaking and effectively occupies only half the phase-space. However, we may still assume the ergodic hypothesis in this half, and thus remain within the BG theory.A more complex situation occurs in disordered glass-like systems <cit.>, in which the particles cover a small volume of the phase-space corresponding to a vanishing Lebesgue measure <cit.>. In this case, we can think in two situations: (i) the particles have trajectories that cover a compact subspace of the total phase-space, or, (ii) the particles have trajectories that do not cover the total phase-space, at all relevant time scales, and it is not possible to identify a compact subspace. In studies of conservative nonlinear dynamical systems, some examples of this latter situation has been found <cit.>. In this case, ergodicity might be broken in such a complex manner that the use of BG theory may be not legitimate. Weak chaotic regimes have been found and the q-statistical generalization<cit.> of the BG theory has emerged as an appropriate description.Here, we analyze the nature of the phase-space occupancy as function of the entanglement. We focus on the one-dimensional transverse-field spin-1/2 Ising ferromagnet at its zero-temperature critical point <cit.>. We consider an L-sized subsystem of a N-sized ring, and trace over the states of (N-L) spins, with N→∞. The full N-system is in a pure state, but the L-system is in a statistical mixture. We show that the quantum entanglement becomes responsible for trapping part of the particles into not physically attainable energy states. In this sense, particles do not cover the total volume of the phase-space. However, they do not destroy the nature of the phase-space ocuppancy, covering a compact subspace of the total phase-space. Therefore, the BG theory can continue to be legitimately used. In this case, we can recover extensivity for physical quantities such as the entropy, which can linearly grow with the system size, by redefining the size of the system.Entanglement stands out as a key-feature in the mechanics underlying quantum phase transitions <cit.>. A few years ago, Vidal et al <cit.> proposed an entanglement measure for pure states based on the von Neumann entropy of the reduced density matrix accounting for a subset of the total system. This can be done by simply tracing out the undesired external degrees of freedom. Entanglement undergoes a prominent increase in the vicinity of a critical point <cit.>, at which the von Neumann entropy acquires finite values while we approach the quantum phase transition point, where it diverges. Fortunately, conformal field theory allows for an analytical description of the physical properties right at the transition point <cit.>. It can be shown <cit.> that entanglement increases with the size of the subsystem L. Its divergence can be further associated to a given universality class provided by conformal field theory <cit.>.We stress that, when computing entanglement via the entropy, there is no need to impose conditions that usually follow from their definition in order to establish a connection with thermodynamics. However, here the entanglement entropy naturally assumes the role of thermodynamic entropy as well, by allowing for extensive thermodynamic variables corresponding to an effective volume of the system.The one-dimensional transverse-field spin-1/2 Ising ferromagnet with N sites is described by the following Hamiltonian <cit.>Ĥ= - ∑_i=0^N-1( σ_i^xσ_i+1^x + λσ_i^z),where σ_i^α is the αth Pauli matrix at site i and λ denotes the magnetic field along the z direction. The Hamiltonian (<ref>) can be diagonalized by a Jordan-Wigner transformation, which maps the spin chain onto a spinless fermionic system, followed up by a Bogoliubov linear transformation <cit.>. The Hamiltonian then assumes the diagonal formĤ= - ∑_k( ω_kâ_k^†â_k + λω_k),where â_k are operators that obey the usual fermionic anticommutation relations,k=-N/2,-N/2+1,...,N/2-1 and ω_k= √( [sin(2π k/N)]^2 + [cos(2π k/N) - λ]^2).The ground-state properties of this model strongly depend on λ. A zero-temperature quantum phase transition occurs when λ = 1. The ground-state behavior is further revealed by the interplay between entanglement and the ground-state structure <cit.>. Right at the critical point λ = 1 the spins are mostly entangled and, in this case, it is possible to define a proper entanglement witness which brings about signatures of a quantum phase transition.One of the most commonly-used entanglement measures for such a task is the so-called entanglement entropy <cit.>. Given a pure state, it quantifies how much a given subsystem, which can be properly described by a reduced density matrix, is entangled with the remaining part. For a spin chain with N sites, we obtain the state describing a given block of L spins ρ_L by tracing out the subsystem of length (N-L) of the overall density matrix ρ_N. We have taken the thermodynamic limit (N →∞).The von Neumann entanglement entropy reads <cit.>S_vN(L,λ)= - Tr [ ρ_Llog ( ρ_L)],the Rényi entropy <cit.> readsS_α^R(L,λ) = 1 /1-αlog [Tr (ρ_L)^α] ,and the q-entropy <cit.> readsS_q(L,λ) = 1- Tr (ρ_L)^q/q-1 .Note that many other entanglement measures can be defined <cit.>. Regardless of the choice though, all the relevant information is contained in the reduced density matrix ρ_L.Let us first discuss the entanglement properties when we are away from the critical point, that is λ≠1 (recall that L →∞). Using the mapping between the quantum d=1 model and the classical d=2 model, it is possible to express ρ_L as a product of density matrices account for an infinite number of uncorrelated spinless free fermions <cit.>. The energy levels of the fermions areϵ_λ (n) = {[(2n+1)ϵ_λ, for λ < 1 ,;2nϵ_λ, for λ > 1 , ].with n=0,1,2,... andϵ_λ = πI(√(1-y^2))/I(y),whereI(y) = ∫_0^1dx/√((1-x^2)(1-y^2x^2))is the complete elliptic integral of the first kind and y=min[λ,λ^-1]. Therefore, ρ_L →∞= ⊗_nρ̃_n, whereρ̃_n = 1/ 1+e^ -ϵ_λ (n)( [ 1 0; 0 e^-ϵ_λ(n) ]). Once we have obtained the reduced density matrix, we can calculate the von Neumann entropy for, say, λ > 1, usingS_vN(∞,λ) = ∑_n=0^∞[ log(1+e^-2nϵ_λ) + 2nϵ_λ/1+e^-2nϵ_λ].In the vicinity of the critical point (λ→ 1), we have that ϵ_λ→ 0 and the sum above can be approximated by the integralS_vN(∞,λ) ≃∫_0^∞ dx [ log(1+e^-2xϵ_λ) + 2xϵ_λ/1+e^-2xϵ_λ] S_vN(∞,λ) ≃π^2/12ϵ_λ→∞.We can unveil the behavior of the Renyi and of the q-entropy by a similar procedure <cit.>.At the critical point, a similar analysis can be carried out. By considering now an L-sizedsubsystem, its reduced density matrix ρ_L is obtained from the following matrix <cit.>Γ_L = ( [ Π_0 Π_1 ⋯ Π_L-1;Π_-1 Π_0 ⋮; ⋮ ⋱ ⋮; Π_1-L ⋯ ⋯ Π_0; ]),whereΠ_l = ( [ 0 -4/π (2l+1); -4/π (2l-1) 0 ]).An orthogonal matrix transforms Γ_L into a block-diagonal matrix corresponding to purely imaginary eigenvalues ± iν_n (n=0,..,L-1). In Fig. <ref>(a), we show the imaginary part of the eigenvalues of Γ_L obtained through straightforward numerical diagonalization for various block sizes. The analytical outcome for ν_n reads ν_n=tanh [(2n+1)ϵ_L /2], where ϵ_L will be obtained later on. The 2^L eigenvalues of ρ_L are given byμ_x_1x_2...x_L = ∏_n1+(-1)^x_nν_n/2where x_n=0,1 ∀ n.Analogously to our previous discussion for the λ≠1 case, at the critical point the model can also be mapped onto a system featuring spinless free fermions and thus ρ_L= ⊗_nρ̂_n, where ρ̂_n has a similar form to that of Eq. (<ref>), with ϵ_λ being replaced by ϵ_L. For large L, we can write the energy spectrum asϵ_L (n) = (2n+1)ϵ_L n=0,1,...,L-1.Similar spectrum was obtained by Peschel <cit.> working with a matrix which commutes with Γ_L. We obtained ϵ_L theoretically, observing that ϵ_L→ 0 for L →∞, so that, analogously to Eqs. (<ref>), (<ref>), and (<ref>), we can writeS_vN(L,1) ≃π^2/12ϵ_L.Using that <cit.>S_vN(L,1) ≃1/6log (L),we obtainϵ_L = π^2/2log (L).Our analytical results for the eigenvalues of Γ_L are represented in Fig. <ref>(a), which are in excellent agreement with the numerical outcomes. Our numerical proof was made for finite L, and in these cases, for high energies, we obtained a nonlinear spectrum in n. This nonlinearity decreases with increasing L and strongly suggests a linear spectrum in n for infinite L. Let us stress that the region of high energies are physically irrelevant as we shall see below.We can think of the problem not solely for the purpose of entanglement analysis, but also regarding the spin block as a physical system of interest by itself. Thus, it becomes relevant to discuss its thermodynamic properties, which is carried out in what follows. We have a system of L free fermions whose Hamiltonian reads asĤ_L = E_L ∑_n=0^L-1 (2n+1) ĉ^†_nĉ_n,where ĉ^†_n (ĉ_n) are the creation (annihilation) fermionic operators at site n for a one-dimensional lattice. E_L = ϵ_L ϵ_0, where both ϵ_0 and Ĥ_L have the dimension of energy.The Hamiltonian (<ref>) represents tightly-bounded electrons in a uniform electric field <cit.>. This model has been extensively studied, both on theoretical and experimental grounds (see, e.g., <cit.>). Our case, however, embodies the limit of localized atomic electrons, where nearest-neighbor hopping is neglected. In this extreme limit, the equidistant energy levels are identified as Stark ladders <cit.>. The concept of Stark ladder was put forward by Wannier <cit.> and confirmed experimentally in several setups, for instance, in GaAs-GaAlAs superlattices subjected to electric fields <cit.> and in an elastic-rod apparatus <cit.>.The thermodynamic properties of the free fermions at temperature T are determined from the partition function of the canonical ensembleZ(L) = Tr[ e^-βĤ_L] = ∏_n=0^L-1 (1+e^-β (2n+1)E_L),where β=1/(k_BT). We obtain the Helmholtz free energyF (L) = - 1/βln [Z(L)]= - 1/β∑_n=0^L-1ln [1+e^-β (2n+1)E_L]and the internal energyU (L) = - ∂/∂βln [Z(L)]= ∑_n=0^L-1 (2n+1)E_L /1+e^β (2n+1)E_L.As in Eq. (<ref>), the above sums can be approximated by integrals and we obtainF (L) ≃ - π^2/24 E_Lβ^2 = - 1/ 12β^2ϵ_0log(L)andU (L) ≃π^2/24 E_Lβ^2 = 1/12β^2ϵ_0log(L).Consequently, it becomes straightforward to obtain the entropy, which readsS (L) = 1/T [U(L) - F(L)] ≃k_B/6βϵ_0log(L).One can recover the entanglement entropy by assuming βϵ_0=1 and thus S_vN(L,1) ≃k_B/6log(L). We can also write the Rényi <cit.> and q-statistics entropies <cit.> asS_α^R(L,1) ≃(α+1)/12αlog(L)andS_q(L,1) ≃ L^(1/q-q)1/12 - 1 /1-q,respectively.Note that the q-entropy can be used by satisfying the requirement of extensivity, i.e., (1/q-q)1/12=1, henceq_c=√(37)-6 ≃ 0.08 <cit.>. In this case,S_q_c(L,1) ≃ L,and the desired thermodynamic extensivity is recovered.We now make a crucial observation: this system has an effective number of unattainable physical energy states<cit.>, characterized by density matrices followingρ̂_n≃( [ 1 0; 0 0 ]).Another way to put this is by thinking that we have a set of free fermions frozen in the ground state that depends on the manner through which entanglement was established in the original problem. Therefore, since only a part of the free fermions becomes thermodynamically accessible, this is the very subset on which we build up our analysis. For the number of accessible free fermion states versus the block size L, for L > 300, we obtain numericallyL̃ = 1.227 log L + 2.88(r^2=0.999999).Those coefficients depend on the chosen numerical precision, which we set to 10^-6 for real numbers. However, it definitely has no influence on the qualitative features we point out next.Fig. <ref>(b) shows ν_n [also featured in Fig. <ref>(a)] now as a function of n/(2V), whereV≡ 1.227 log L. We can observe a data collapse at which ν_n is independent of L.Further numerical analysis yields the conclusion that the entire physical behavior of the subsystem composed by L free fermions can be completely evaluated by considering only the first V particles. This allows us to writeĤ_V = E_L ∑_n=0^V (2n+1) ĉ^†_nĉ_n.Using this expression, we confirm that the results of Eqs. (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) are precisely the same.The thermodynamic properties are extracted from the free energyF (T,V) = - k_B^2T^2/12 ϵ_0 V,such thatS_BG (T,V) = - ( ∂ F/∂ T)_V = k_B^2T/6 ϵ_0 VandU (T,V) = F+TS = k_B^2T^2/12 ϵ_0 Vare extensive thermodynamic quantities. For completeness, we can also define the intensive quantityP (T,V) = - ( ∂ F/∂ V)_T = k_B^2T^2/12 ϵ_0,so that we can write U=PV. All the above expressions are consistent with standard thermodynamics.The entanglement behavior of the system mandates that only a given part of energy states is thermodynamically relevant. As a consequence, the standard BG quantities are associated with an effective length V ≡log L, and the phase-space has an effective dimension 2^log L instead of 2^L. The effective number of microstates grows with L as a power-law, in variance with the exponential growth corresponding to standard nonentangled systems.The above analysis suggests a scenario where the physical systems are essentially grouped into three classes, in terms of their phase-space occupancy, ergodicity and Lebesgue measure, namely (i) ergodicity occurs in the entirephase-space or in a compact subspace whose Lebesgue measure remains different from zero in the thermodynamic limit; (ii) ergodicity occurs only in a compact subspace whose Lebesgue measure vanishes in the thermodynamic limit; and (iii) ergodicity does not occur, the trajectories covering a noncompact subspace whose Lebesgue measure vanishes in the thermodynamic limit (typically an hierarchical structure like a multifractal). For each class, there is an appropriate statistical mechanics. Typical examples of the first class are physical systems with or without usual phase transitions. The BG theory perfectly describes this class and the von Neumann/Boltzmann entropy is an extensive thermodynamic quantity. For systems that fall in the second class, we exhibit in the present workhow the BG theory can still be used. Here, we can find a particular value of q such that the q-entropy satisfies the requirement of extensivity within the total volume, while the von Neumann/Boltzmann entropy is an extensive thermodynamic quantity within an appropriate effective volume. Some of the systems exhibiting the area-law <cit.> for the entropy might also belong to this class. For the third class, we do not expect the use of the BG theory to be legitimate. This is the case for say systems with long-range interactions, for which theories such as q-statistics have been satisfactorily applied <cit.>.Acknowledgements We thank G. M. A. Almeida, L. J. L. Cirto and E. M. F. Curado for useful remarks, and Y. N. Fernández, K. Hallberg and A. Gendiar for very fruitful discussions at early stages of this effort. We also acknowledge partial financial support from the John Templeton Foundation-USA (Grant nº53060) and from the Brazilian agencies CNPq and CAPES.40 book L. E. Reichl, A Modern Course in Statistical Physics (J. Wiley and Sons, New York,3rd edition, 2009). palm R. G. Palmer, Broken ergodicity, Adv. Phys. 31, 669 (1982). lebe H. Kestelman, Modern Theories of Integration (Dover, New York, 2nd edition, 1960). tirn U. Tirnakli and E. P. Borges, The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics, Scientific Reports 6, 23644 (2016). tsal1 C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988). pfeu P. Pfeuty, The One-Dimensional Ising Model with a Transverse Field, Annals of Phys. 57, 79 (1970). sach S. Sachdev, Quantum Phase Transitions (Cambridge Universuty Press, Cambridge, 1999). amic L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008). its A. R. Its, B.-Q. Jin, and V. E. Korepin, Entanglement in the XY spin chain, J. Phys. A 38, 2975 (2005). lato J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quantum Inf. Comput. 4, 48 (2004). cala P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.: Theory Exp. 2004, P06002 (2004). isla R. Islam, R. Ma, P. M. Preiss,M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature 528, 77 (2015). vida G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in Quantum Critical Phenomena, Phys. Rev. Lett. 90, 227902 (2003). holz C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 443 (1994). fran F. Franchini, A. R. Its, and V. E. Korepin, Renyi entropy of the XY spin chain, J. Phys. A 41, 025302 (2008). hast M. B. Hastings, I. Gonzales, A. B. Kallin, and R. G. Melko,Measuring Renyi entanglement entropy in quantum Monte Carlo simulations Phys. Rev. Lett. 104, 157201 (2010). horo R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). caru F. Caruso, and C. Tsallis, Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics, Phys. Rev. E 78, 021102 (2008). jinB.-Q. Jin and V. E. Korepin, Quantum spin chain, Toeplitz determinants and Fisher-Hartwig conjecture, J. Stat. Phys. 116, 79 (2004). pesc I. Peschel, On the reduced density matrix for a chain of free electrons. Journal of Statistical Mechanics: Theory and Experiment, 2004, P06004 (2004). fuku H. Fukuyama, R. A. Bari, and H. C. Fogedby, Tightly bound electrons in a uniform electric field, Phys. Rev. B 8, 5579 (1973). sait M. Saitoh, Electronic states in a finite linear crystal in an electric field, J. Phy. C: Solid State Phys. 6, 3255 (1973). mend E. E. Mendez, F. Agullo-Rueda, and J. M. Hong, Stark localization in GaAs-GaAlAs Superlattices under an Electric Field, Phys. Rev. Lett. 60, 2426 (1988). guti L. Gutierrez, A. Diaz-de-Anda, J. Flores, R. A. Mendez-Sanchez, G. Monsivais, and A. Morales, Wannier-Stark ladders in one-dimensional elastic systems, Phys. Rev. Lett. 97, 114301 (2006). kane E. O. Kane, Zener tunneling in semiconductors, J. Phys. Chem. Solids 12, 181 (1959). wann G.H. Wannier, Wave functions and effective Hamiltonian for Bloch electrons in an electric field, Phys. Rev. 117, 432 (1960). herd C. M. Herdman, P.-N. Roy, R. G. Melko, and A. Del Maestro, Entanglement area law in superfluid ^4He, Nature Phys. 13, 556 (2017). cala2 P. Calabrese, M. Mintchev and E. Vicari, Entanglement entropies in free fermion gases for arbitrary dimension, Eur. Phys. Lett. 97, 20009 (2012). tsa_fp C. Tsallis, On the foundations of statistical mechanics, Eur. Phys. J. Special Topics 226, 1433 (2017). tsal2 C. Tsallis, Nonadditive entropy and nonextensive statistical mechanics - an overview after 20 years, Braz. J. Phys. 39, 337 (2009). tsal3 C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, 2009). | http://arxiv.org/abs/1707.08527v1 | {
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IMB UMR5584, CNRS, Univ. Bourgogne Franche-Comt, F-21000 Dijon, [email protected] Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, [email protected] of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan [email protected] work was partially funded by Grant-in-Aid for JSPS Fellows Number 15F15751and Grant-in-Aid for Scientific Research of JSPS No. 15K04805. The research was done during visits of the first and second authors at Saitama University, and during visits of the third author at the Institut de Mathmatiques de Bourgogne. The authors thank these institutions for their generous supports and the excellent working conditions offered.[2000]14R20; 14R25; 14R05; 14L30, 14D06Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X→ X//𝔾_a restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper 𝔾_a-actions, whose algebraic quotient morphisms π:X→ X//𝔾_a are surjective with only isolated degenerate fibers, all isomorphic to the affine plane 𝔸^2 when equipped with their reduced structures.Equivariant extensions of 𝔾_a-torsors over punctured surfaces Takashi Kishimoto =============================================================§ INTRODUCTION Algebraic actions of the complex additive group 𝔾_a=𝔾_a,ℂ on normal complex affine surfaces S are essentially fully understood: the ring of invariants 𝒪(S)^𝔾_a,ℂ is a finitely generated algebra whose spectrum is a smooth affine curve C=S//𝔾_a, and the inclusion 𝒪(S)^𝔾_a⊂𝒪(S) defines a surjective morphism π:S→ C whose general fibers coincide with general orbits of the action, hence are isomorphic to the affine line 𝔸^1 on which 𝔾_a acts by translations. The degenerate fibers of such 𝔸^1-fibrations are known to consist of finite disjoint unions of smooth affine curves isomorphic to 𝔸^1 when equipped with their reduced structure. A complete description of isomorphism classes of germs of invariant open neighborhoods of irreducible components of such fibers was established by Fieseler <cit.>. In contrast, very little is known so far about the structure of 𝔾_a-actions on complex normal affine threefolds. For such a threefold X, the ring of invariants 𝒪(X)^𝔾_a is again finitely generated <cit.> and the morphism π:X→ S induced by the inclusion 𝒪(X)^𝔾_a⊂𝒪(X) is an 𝔸^1-fibration over a normal affine surface S. But in general, π is neither surjective nor equidimensional. Furthermore, it can have degenerate fibers over closed subsets of pure codimension 1 as well as of codimension 2. All of these possible degeneration are illustrated by the following example:The restriction of the projection pr_x,y to the smooth threefold X={x^2(x-1)v+yu^2-x=0} in 𝔸^4 is an 𝔸^1-fibration π:X→𝔸^2 which coincides with the algebraic quotient morphism of the 𝔾_a-action on X associated to the locally nilpotent derivation ∂=x^2(x-1)∂_u-2yu∂_v of its coordinate ring. The restriction of π over the principal open subset x^2(x-1)≠0 of 𝔸^2 is a trivial principal 𝔾_a-bundle, but the fibers of π over the points (1,0) and (0,0) are respectively empty and isomorphic to 𝔸^2. Furthermore, for every y_0≠0, the inverse images under π of the points (0,y_0) and (1,y_0) are respectively isomorphic to 𝔸^1 but with multiplicity 2, and to the disjoint union of two reduced copies of 𝔸^1.Partial results concerning the structure of one-dimensional degenerate fibers of 𝔾_a-quotient 𝔸^1-fibrations were obtained by Gurjar-Masuda-Miyanishi <cit.>. In the present article, as a step towards the understanding of the structure of two-dimensional degenerate fibers, we consider a particular type of non equidimensional surjective 𝔾_a-quotient 𝔸^1-fibrations π:X→ S which have the property that they restrict to 𝔾_a-torsors[sometimes also referred to as Zariski locally trivial principal 𝔾_a-bundles]over the complement of a finite set of smooth points in S. These are simpler than the general case illustrated in the previous example since they do not admit additional degeneration of their fibers over curves in S passing through the given points. The local and global study of some classes of such fibrations was initiated by the second author <cit.>. He constructed in particular many examples of 𝔾_a-quotient 𝔸^1-fibrations on smooth affine threefolds X with image 𝔸^2 whose restrictions over the complement of the origin are isomorphic to the geometric quotient SL_2→SL_2/𝔾_a of SL_2 by the action of unitary upper triangular matrices. One of the simplest examples of this type is the smooth threefold X_0⊂𝔸_x,y,p,q,r^5 defined by the equationsX_0: xr-yq =0yp-x(q-1) =0pr-q(q-1) =0and equipped with the 𝔾_a-action associated to the locally nilpotent ℂ[x,y]-derivation x^2∂_p+xy∂_q+y^2∂_r of its coordinate ring. The equivariant open embedding SL_2={xv-yu=1}↪ X_0 is given by (x,y,u,v)↦(x,y,xu,xv,yv). The 𝔾_a-quotient morphism coincides with the surjective 𝔸^1-fibration π_0:pr_x,y:X_0→𝔸^2. Its restriction over 𝔸^2∖{(0,0)} is isomorphic to the quotient morphism SL_2→SL_2/𝔾_a, while its fiber over (0,0) is the smooth quadric {pr-q(q-1)=0}⊂𝔸_p,q,r^3, isomorphic to the quotient SL_2/𝔾_m of SL_2 by the action of its diagonal torus (see Example <ref>). A noteworthy property of this example is that the 𝔾_a-quotient morphism π:X_0→𝔸^2 factors through a locally trivial 𝔸^1-bundle ρ:X_0→𝔸̃^2 over the the blow-up τ:𝔸̃^2→𝔸^2 of the origin. It is a general fact that every irreducible component of a degenerate fiber of pure codimension one of a 𝔾_a-quotient 𝔸^1-fibration π:X→ S on a smooth affine threefold is an 𝔸^1-uniruled affine surface (see Proposition <ref>). We do not know whether every 𝔸^1-uniruled surface can be realized as an irreducible component of the degenerate fiber of a 𝔾_a-extension. But besides the smooth affine quadric SL_2/𝔾_m appearing in the previous example, the following one confirms that the affine plane 𝔸^2 can also be realized (see also Examples <ref> and <ref> for other types of surfaces that can be realized): Let X_1⊂𝔸_x,y,z_1z_2,w^5 be the smooth affine threefold defined by the equations X_1: xw-y(yz_1+1) =0xz_2-z_1(yz_1+1) =0z_1w-yz_2=0,equipped with the 𝔾_a-action associated to the locally nilpotent ℂ[x,y]-derivation x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w of its coordinate ring. The morphism SL_2↪ X_1 given by (x,y,u,v)↦(x,y,u,uv,yv) isequivariant open embedding. The 𝔾_a-quotient morphism coincides with the surjective 𝔸^1-fibration π_1=pr_x,y:X_1→𝔸^2, whose fiber over the origin is the affine plane 𝔸^2=Spec(ℂ[z_2,w]) and whose restriction over 𝔸^2∖{(0,0)} is again isomorphic to the quotient morphism SL_2→SL_2/𝔾_a. A special additional feature is that the 𝔾_a-action on X_1 extending that on SL_2 is not only fixed point free but actually proper: its geometric quotient X_1/𝔾_a is separated. One can indeed check that X_1/𝔾_a is isomorphic to the complement 𝔸̃^2∖{o_1} of a point o_1 supported on the exceptional divisor E of the blow-up 𝔸̃^2 of 𝔸^2 at the origin (see Example <ref>).Relaxing the hypothesis that the 𝔸^1-fibration π:X→ S arises as the quotient of a 𝔾_a-action on an affine threefold X to consider the broader problem of describing the geometry of degeneration of 𝔸^1-fibrations over irreducible closed subsets of pure codimension two of their base, we are led to the following more general notion:Let (S,o) be a pair consisting of a normal separated 2-dimensional scheme S essentially of finite type over a field k of characteristic zero and of a closed point o contained in the smooth locus of S. A 𝔾_a-extension of a 𝔾_a-torsor ρ:P→ S∖{o} is a 𝔾_a-equivariant open embedding j:P↪ X into an integral scheme X equipped with a surjective morphism π:X→ S of finite type and a 𝔾_a,S-action, such that the commutative diagramP [r,hook,"j"] [d,"ρ"'] X [d,"π"] S∖{o}[r,hook] Sis cartesian. The examples X_0 and X_1 above provide motivation to study the following natural classes of 𝔾_a-extensions π:X→ S of a 𝔾_a-torsor ρ:P→ S∖{o}, which are arguably the simplest possible types of 𝔾_a-extensions from the viewpoints of their global geometry and of the properties of their 𝔾_a-actions:- (Type I) Extensions for which π factors through a locally trivial 𝔸^1-bundle over the blow-up τ:S̃→ S of the point o, the fiber π^-1(o) being then the total space of a locally trivial 𝔸^1-bundle over the exceptional divisor of τ. - (Type II) Extensions for which π^-1(o)_red is isomorphic to the affine plane 𝔸_κ^2 over the residue field κ of S at o, X is smooth along π^-1(o) and the 𝔾_a,S-action on X is proper.The first main result of this article, Proposition <ref> and Theorem <ref>, is a complete description of 𝔾_a-extensions of Type I together with an effective characterization of which among them have the additional property that the morphism π:X→ S is affine. Our second main result, Theorem <ref>, consists of a classification of 𝔾_a-extensions of Type II, under the additional assumption that the morphism π:X→ S is quasi-projective. More precisely, given a 𝔾_a-torsor ρ:P→ S∖{o} and a 𝔾_a-extension π:X→ S with proper 𝔾_a,S-action and reduced fiber π^-1(o)_red isomorphic to 𝔸_κ^2, we establish that the possible geometric quotients S'=X/𝔾_a belong to a very special class of surfaces isomorphic to open subsets of blow-ups of S with centers over o which we fully describe in<ref>. We show conversely that every such surface is indeed the geometric quotient of a 𝔾_a-extension of ρ:P→ S∖{o} with the desired properties. In a second step, we tackle the question of existence of 𝔾_a-extensions π:X→ S of Type II for which the structure morphism π is not only quasi-projective but affine. Our method to produce extensions with this property is inspired by the observation that the threefolds X_0 and X_1 above are not only birational to each other due to the property that they both contain SL_2 as open subset, but in fact that the birational morphism η:X_1→ X_0,(x,y,z_1,z_2,w)↦(x,y,p,q,r)=(x,y,xz_1,yz_1+1,w)expresses X_1 as a 𝔾_a-equivariant affine modification of X_0 in the sense of Kaliman and Zaidenberg <cit.>. This suggests that extensions of Type II for which X is affine over S could be obtained as equivariant affine modification in a suitable generalized sense from extensions of Type I with the same property. Using this technique, we are able to show in Theorem <ref> that for each possible geometric quotient S' above, there exist 𝔾_a-extensions π:X→ S of ρ:P→ S∖{o} with geometric quotient X/𝔾_a=S' such that π is an affine morphism. As an application towards the initial question of the structure 𝔾_a-quotient 𝔸^1-fibrations on affine threefolds, we in particular derive from this construction the existence of uncountably many pairwise non isomorphic smooth affine threefolds X endowed with proper 𝔾_a-actions, containing SL_2 as an invariant open subset with complement 𝔸^2, whose geometric quotients are smooth quasi-projective surfaces which are not quasi-affine, and whose algebraic quotients are all isomorphic to 𝔸^2.The scheme of the article is the following. The first section begins with a review of general properties of 𝔾_a-extensions. We then set up the basic tools which will be used through all the article: locally trivial 𝔸^1-bundles with additive group actions and equivariant affine birational morphisms between these. In section two, we study 𝔾_a-extensions of Type I. The last section is devoted to the classification of quasi-projective 𝔾_a-extensions of Type II. § PRELIMINARIESIn the rest of the article, the term surface refers to a normal separated 2-dimensional scheme essentially of finite type over a field k of characteristic zero. A punctured surface S_*=S∖{o} is the complement of a closed point o contained in the smooth locus of a surface S. We denote by κ the residue field of S at o. We do not require that the residue field κ of S at o is an algebraic extension of k. For instance, S can very well be the spectrum of the local ring 𝒪_X,Z of an arbitrary smooth k-variety X at an irreducible closed subvariety Z of codimension two in X and o its unique closed point, in which case the residue field κ is isomorphic to the field of rational functions on Z.In this section, we first review basic geometric properties of equivariant extensions of 𝔾_a-torsors over punctured surfaces. We then collect various technical results on additive group actions on affine-linear bundles of rank one and their behavior under equivariant affine modifications.§.§ Equivariant extensions of 𝔾_a-torsorsA 𝔾_a-torsor over punctured surface S_*=S∖{o} is an S_*-scheme ρ:P→ S_* equipped with a 𝔾_a-action μ:𝔾_a,S_*×_S_*P→ P for which there exists a Zariski open cover f:Y→ S_* of S_* such that P×_S_*Y is equivariantly isomorphic to 𝔾_a,Y acting on itself by translations. In the present article, we primarily focus on 𝔾_a-torsors ρ:P→ S_* whose restrictions P×_S_*U→ U∖{o} over every Zariski open neighborhood U of o in S are nontrivial. Since in this case the total space of P is affine over S (see e.g. <cit.> whose proof carries over verbatim to our more general situation), it follows that for every 𝔾_a-extension j:P↪ X the fiber π^-1(o)⊂ X of the surjective morphism π:X→ S has pure codimension one in X. Two important families of examples of non trivial normal 𝔾_a-extensions j:SL_2→ X of the 𝔾_a-torsor ρ:SL_2→SL_2/𝔾_a≃𝔸^2∖{(0,0)}, where 𝔾_a acts on SL_2 via left multiplication by upper triangular unipotent matrices, were constructed in <cit.>. Various other extensions were obtained from these by performing suitable equivariant affine modifications. One can observe that for all these extensions, the fiber π^-1({(0,0)}) is an 𝔸^1-ruled surface, a property which is a consequence of the following more general fact:Let ρ:P→ S_* be a non trivial 𝔾_a-torsor over the punctured spectrum S∖{o} of a regular local ring of dimension 2 over an algebraically closed field k and with residue field κ(o)=k, and let π:X→ S be a 𝔾_a-extension of P. If X is smooth along π^-1(o), then every irreducible component F of π^-1(o)_red is a uniruled surface. Furthermore, if X is affine then F is 𝔸^1-uniruled, hence 𝔸^1-ruled when it is normal. Since π^-1(o) has pure codimension one in X and X is smooth along π^-1(o), every irreducible component of π^-1(o) is a 𝔾_a-invariant Cartier divisor on X. The complement X' in X of all but one irreducible component of π^-1(o) is thus again a 𝔾_a-extension of P, and we may therefore assume without loss of generality that F=π^-1(o)_red is irreducible. Let x∈ F be a closed point in the regular locus of F. Since F and X are smooth at x and X is connected, there exists a curve C⊂ X, smooth at x and intersecting F transversally at x. The image π(C) of C is a curve on S passing through o, and the closure B of π^-1(π(C)∩ S_*) in X is a surface containing C. Since ρ:P→ S_* is a 𝔾_a-torsor, the restriction of π to B∩ P is a trivial 𝔾_a-torsor over the affine curve π(C). So π|_B:B→π(C) is an 𝔸^1-fibration. Let ν:C̃→π(C) be the normalization of π(C). Then π|_B lifts to an 𝔸^1-fibration θ:B̃→C̃ on the normalization B̃ of B. The fiber of θ over every point in ν^-1(o) is a union of rational curves. Since the normalization morphism μ:B̃→ B is surjective, one of the irreducible components of ν^-1(o) is mapped by μ onto a rational curve in F passing through x. This shows that for every smooth closed point x of F, there exists a non constant rational map h:ℙ^1 F such that x∈ h(ℙ^1). Thus F is uniruled. If X is in addition affine, then B and B̃ are affine surfaces, and the fibers of the 𝔸^1-fibration θ:B̃→C̃ consist of disjoint union of curves isomorphic to 𝔸^1 when equipped with their reduced structure. This implies that F is not only uniruled but actually 𝔸^1-uniruled.Let X be the smooth affine threefold in 𝔸^2×𝔸^4=Spec(k[x,y][c,d,e,f]) defined by the equations xd-y(c+1) =0xc^2-y^2e =0yf-c(c+1) =0xf^2-(c+1)^2e =0de-cf =0equipped with the 𝔾_a-action induced by the locally nilpotent k[x,y]-derivation xy∂_c+y^2∂_d+x(2c+1)∂_f+(2x^2f-2xye)∂_eof its coordinate ring. The morphism j:SL_2={xv-yu=1}→ X defined by (x,y,u,v)↦(x,y,yu,yv,xu^2,xuv) is an open embedding of SL_2 in X as the complement of the fiber over o=(0,0) of the projection π=pr_x,y:X→𝔸^2. So j:SL_2→ X is an affine 𝔾_a-extension of the 𝔾_a-torsor ρ:SL_2→SL_2/𝔾_a=𝔸^2∖{o}, for which π^-1(o) consists of the disjoint union of two copies D_1={x=y=c=0}≃Spec(k[d,f]) and D_2={x=y=c+1=0}≃Spec(k[d,e]) of 𝔸^2. Note that the induced 𝔾_a-action on each of these is the trivial one. Let X be the affine 𝔾_a-extension constructed in the previous example and let C⊂ D_1 be any smooth affine curve. Let τ:X̃→ X be the blow-up of X along C, let i:X'↪X̃ be the open immersion of the complement of the proper transform of D_1∪ D_2 in X̃ and let π'=π∘τ∘ i:X'→𝔸^2. Since C and D_1∪ D_2 are 𝔾_a-invariant, the 𝔾_a-action on X lifts to a 𝔾_a-action on X̃ which restricts in turn to X'. By construction, π' is surjective, with fiber π'^-1(o) isomorphic to C×𝔸^1 and τ∘ i:X'→ X restricts to an equivariant isomorphism between X'∖π'^-1(o) and X∖π^-1(o)≃SL_2. So π':X'→𝔸^2 is a 𝔾_a-extension of the 𝔾_a-torsor ρ:SL_2→SL_2/𝔾_a=𝔸^2∖{o}.§.§ Recollection on affine-linear bundles Affine-linear bundles of rank one over a scheme are natural generalization of 𝔾_a-torsors. To fix the notation, we briefly recall their basic definitions and properties By a line bundle on a scheme S, we mean the relative spectrum p:M=Spec(Sym^·ℳ^∨)→ S of the symmetric algebra of the dual of an invertible sheaf of 𝒪_S-module ℳ. Such a line bundle M can be viewed as a locally constant group scheme over S for the group law m:M×_SM→ M whose co-morphism m^♯:Sym^·ℳ^∨→Sym^·ℳ^∨⊗Sym^·ℳ^∨≃Sym^·(ℳ^∨⊕ℳ^∨)is induced by the diagonal homomorphism ℳ^∨→ℳ^∨⊕ℳ^∨. An M-torsor is then an S-scheme θ:W→ S equipped with an action μ:M×_SW→ W which is Zariski locally over S isomorphic to M acting on itself by translations. This is the case precisely when there exists a Zariski open cover f:Y→ S and an 𝒪_Y-algebra isomorphism ψ:f^*𝒜→Sym^·f^*ℳ^∨ such that over Y'=Y×_SY the automorphism p_1^*ψ∘p_2^*ψ^-1:Sym^·ℳ_Y'^∨→Sym^·ℳ_Y'^∨ of the symmetric algebra of ℳ_Y'^∨=p_2^*f^*ℳ^∨=p_1^*f^*ℳ^∨ is affine-linear, i.e. induced by an 𝒪_Y'-module homomorphism ℳ_Y'^∨→Sym^·ℳ_Y'^∨ of the form β⊕id:ℳ_Y'^∨→𝒪_Y'⊕ℳ_Y'^∨↪⊕_n≥0(ℳ_Y'^∨)^⊗ n=Sym^·ℳ_Y'^∨for some β∈Hom_Y'(ℳ_Y'^∨,𝒪_Y')≃ H^0(Y',ℳ_Y') which is a Čech 1-cocyle with values in ℳ for the Zariski open cover f:Y→ S. Standard arguments show that the isomorphism class of θ:W→ S depends only on the class of β in the Čech cohomology group Ȟ^1(S,ℳ), and one eventually gets a one-to-one correspondence between isomorphism classes of M-torsors over S and elements of the cohomology group H^1(S,M)=H^1(S,ℳ)≃Ȟ^1(S,ℳ) with the zero element corresponding to the trivial torsor p:M→ S. It is classical that every locally trivial 𝔸^1-bundle θ:W→ S over a reduced scheme S can be equipped with the additional structure of a torsor under a uniquely determined line bundle M on S. The existence of this additional structure will be frequently used in the sequel, and we now quickly review its construction (see also e.g. <cit.>). Letting 𝒜=θ_*𝒪_W, there exists by definition a Zariski open cover f:Y→ S and a quasi-coherent 𝒪_Y-algebra isomorphism φ:f^*𝒜→𝒪_Y[u]. Over Y'=Y×_SY equipped with the two projections p_1 and p_2 to Y, the 𝒪_Y'-algebra isomorphism Φ=p_1^*φ∘p_2^*φ^-1 has the form Φ:𝒪_Y'[u]→𝒪_Y'[u], u↦ au+bfor some a∈Γ(Y',𝒪_Y'^*) and b∈Γ(Y',𝒪_Y') whose pull back over Y”=Y×_SY×_SY by the three projections p_12,p_23,p_13:Y”→ Y' satisfy the cocycle relations p_13^*a=p_23^*a·p_12^*a and p_13^*b=p_23^*a·p_12^*b+p_23^*b in Γ(Y”,𝒪_Y”^*) and Γ(Y”,𝒪_Y”) respectively. The first one says that a is a Čech 1-cocycle with values in 𝒪_S^* for the cover f:Y→ S, which thus determines, via the isomorphism H^1(S,𝒪_S^*)≃Pic(S), a unique invertible sheaf ℳ on S together with an 𝒪_Y-module isomorphism α:f^*ℳ^∨→𝒪_Y such that p_1^*α∘p_2^*α^-1:𝒪_Y'→𝒪_Y' is the multiplication by a. The second one can be equivalently reinterpreted as the fact that β=p_2^*(^tα)(b)∈Γ(Y',ℳ_Y') is a Čech 1-cocycle with values in ℳ for the Zariski open cover f:Y→ S. Letting Sym^·(α):Sym^·f^*ℳ^∨→𝒪_Y[u] be the graded 𝒪_Y-algebra isomorphism induced by α, the isomorphism ψ=Sym^·(α^-1)∘φ:f^*𝒜→Sym^·f^*ℳ^∨ has the property that p_1^*ψ∘p_2^*ψ^-1 is affine-linear, induced by the homomorphism β⊕id:ℳ_Y'^∨→𝒪_Y'⊕ℳ_Y'^∨. So θ:W→ S is a torsor under the line bundle M=Spec(Sym^·ℳ^∨), with isomorphism class in H^1(S,M) equal to the cohomology class of the cocyle β. Summing up, we obtain;Let θ:W→ S be a locally trivial 𝔸^1-bundle. Then there exists a unique pair (M,g) consisting of a line bundle M on S and a class g∈ H^1(S,M) such that θ:W→ S is an M-torsor with isomorphism class g.§.§ Additive group actions on affine-linear bundles of rank oneGiven a locally trivial 𝔸^1-bundle θ:W→ S, which we view as an M-torsor for a line bundle M=Spec(Sym^·ℳ^∨)→ S on S, with corresponding action μ:M×_SW→ W, every nonzero group scheme homomorphism ξ:𝔾_a,S→ M induces a nontrivial 𝔾_a,S-action ν=μ∘(ξ×id):𝔾_a,S×_SW→ W on W. A nonzero group scheme homomorphism ξ:𝔾_a,S=Spec(𝒪_S[t])→ M=Spec(Sym^·ℳ^∨) is uniquely determined by a nonzero 𝒪_S-module homomorphism ℳ^∨→𝒪_S, equivalently by a nonzero global section s∈Γ(S,ℳ). The following proposition asserts conversely that every nontrivial 𝔾_a,S-action on an M-torsor θ:W→ S uniquely arises from such a section. (<cit.>) Let θ:W→ S be a torsor under the action μ:M×_SW→ W of a line bundle M=Spec(Sym^·ℳ^∨)→ S on S and let ν:𝔾_a,S×_SW→ W be a non trivial 𝔾_a,S-action on W. Then there exists a non zero global section s∈Γ(S,ℳ) such that ν=μ∘(ξ×id) where ξ:𝔾_a,S→ M is the group scheme homomorphism induced by s. Let 𝒜=θ_*𝒪_W and let f:Y→ S be a Zariski open cover such that there exists an 𝒪_Y-algebra isomorphism φ:f^*𝒜→𝒪_Y[u], and let Φ=p_1^*φ∘p_2^*φ^-1:𝒪_Y'[u]→𝒪_Y'[u], u↦ au+bbe as in (<ref>) above. Since θ:W→ S is an M-torsor, φ also determines an 𝒪_Y-module isomorphism α:f^*ℳ^∨→𝒪_Y such that p_1^*α∘p_2^*α^-1:𝒪_Y'→𝒪_Y' is the multiplication by a. The 𝔾_a,S-action ν on W pulls back to a 𝔾_a,Y-action ν×id on W×_S̃Y. The co-mophism η:𝒪_Y[u]→𝒪_Y[u]⊗𝒪_Y[t] of the nontrivial 𝔾_a,Y-action φ∘(ν×id)∘(id×φ^-1) on Spec(𝒪_Y[u]) has the form u↦ u⊗1+1⊗γ t for some nonzero γ∈Γ(Y,𝒪_Y). Letting ℐ=γ·𝒪_Y be the ideal sheaf generated by γ, η factors as η=(id⊗ j)∘η̃:𝒪_Y[u]→𝒪_Y[u]⊗Sym^·ℐ→𝒪_Y[u]⊗𝒪_Y[t]where η̃ is the co-morphism of an action of the line bundle Spec(Sym^·ℐ)→ Y on 𝔸_S^1×_SY≃ W×_SY and j:Sym^·ℐ→𝒪_Y[t] is the homomorphism induced by the inclusion ℐ⊂𝒪_Y. Pulling back to Y', we find that p_2^*γ=a·p_1^*γ, which implies that ^tα(γ)∈Γ(Y,f^*ℳ) is the pull-back f^*s to Y of a nonzero global section s∈Γ(S,ℳ). Letting D=div_0(s) be the divisors of zeros of s, we have ℳ^∨≃𝒪_S(-D)⊂𝒪_S and f^*ℳ^∨≃𝒪_Y(-f^*D)⊂𝒪_Y is equal to the ideal ℐ=γ·𝒪_Y. The global section f^*s viewed as a homomorphism f^*ℳ^∨→𝒪_Y coincides via these isomorphisms with the inclusion γ·𝒪_Y↪𝒪_Y. We can thus rewrite η in the form η=(id⊗Sym^·f^*s)∘η̃:𝒪_Y[u]→𝒪_Y[u]⊗Sym^·f^*ℳ^∨→𝒪_Y[u]⊗𝒪_Y[t].By construction η̃=(φ⊗id)∘ f^*μ^♯∘φ^-1 where f^*μ^♯ is the pull-back of the co-morphism μ^♯:𝒜→𝒜⊗Sym^·ℳ^∨ of the action μ:M×_SW→ W of M on W. It follows that the pull-back f^*ν^♯ of the co-morphism of the action ν:𝔾_a,S× W→ W factors as f^*ν^♯=(id⊗Sym^·f^*s)∘ f^*μ^♯=f^*𝒜→ f^*𝒜⊗Sym^·f^*ℳ^∨→ f^*𝒜⊗𝒪_Y[t]This in turn implies that ν^♯ factors as (id⊗Sym^·s)∘μ^♯:𝒜→𝒜⊗Sym^·ℳ^∨→𝒜⊗𝒪_Y[t] as desired.In the setting of Proposition <ref>, letting U⊂ S be the complement of the zero locus of s, the morphism ξ restricts to an isomorphism of group schemes ξ|_U:𝔾_a,U→ M|_U for which W|_U equipped with the 𝔾_a,U-action ν|_U:𝔾_a,U×_UW|_U→ W|_U is a 𝔾_a,U-torsor. This isomorphism class in H^1(U,𝒪_U) of this 𝔾_a,U-torsor coincides with the image of the isomorphism class g∈ H^1(S,ℳ) of W by the composition of the restriction homomorphism res:H^1(S,ℳ)→ H^1(U,ℳ|_U) with the inverse of the isomorphism H^1(U,𝒪_U)→ H^1(U,ℳ|_U) induced by s|_U.§.§ 𝔾_a-equivariant affine modifications of affine-linear bundles of rank one Recall <cit.> that given an integral scheme X with sheaf of rational functions 𝒦_X, an effective Cartier divisor D on X and a closed subscheme Z⊂ X whose ideal sheaf ℐ⊂𝒪_X contains 𝒪_X(-D), the affine modification of X with center (ℐ,D) is the affine X-scheme σ:X'=Spec(𝒪_X[ℐ/D])→ X where 𝒪_X[ℐ/D] denotes the quotient of the Rees algebra 𝒪_X[(ℐ⊗𝒪_X(D))]=⊕_n≥0(ℐ⊗𝒪_X(D))^nt^n⊂𝒦_X[t]of the fractional ideal ℐ⊗𝒪_X(D)⊂𝒦_X by the ideal generated by 1-t. In the case where X=Spec(A) is affine, D=div(f) is principal and Z is defined by an ideal I⊂ A containing f then X̃ is isomorphic to the affine modification X'=Spec(A[I/f]) of X with center (I,f) in the sense of <cit.>. Now let S be an integral scheme and let θ:W→ S be a locally trivial 𝔸^1-bundle. Let C⊂ S be an integral Cartier divisor, let D=θ^-1(C) be its inverse image in W and let Z⊂ D be a non empty integral closed subscheme of D on which θ restricts to an open embedding θ|_Z:Z↪ C. Equivalently, Z is the closure in D of the image α(U) of a rational section α:C→ D of the locally trivial 𝔸^1-bundle θ|_D:D→ C defined over a non empty open subset U of C. The complement F of θ|_Z(Z) in C is a closed subset of C hence of S. Letting i:S∖ F↪ S be the natural open embedding, we have the following result: Let σ:W'→ W be the affine modification of W with center (ℐ_Z,D). Then the composition θ∘σ:W'→ S factors through a locally trivial 𝔸^1-bundle θ':W'→ S∖ F in such a way that we have a cartesian diagramW' [d]_θ '[r]^σW [d]^θS∖ F [r]^i S.The question being local with respect to a Zariski open cover of S over which θ:W→ S becomes trivial, we can assume without loss of generality that S=Spec(A), W=Spec(A[x]), C=div(f) for some non zero element f∈ A. The integral closed subscheme Z⊂ D is then defined by an ideal I of the form (f,g) where g(x)∈ A[x] is an element whose image in (A/f)[x] is a polynomial of degree one in t. So g(x)=a_0+a_1x+x^2fR(x) where a_0∈ A, a_1∈ A has non zero residue class in A/f and R(x)∈ A[x]. The condition that θ|_Z:Z→ C is an open embedding implies further that the residue classes a_0 and a_1 of a_0 and a_1 in A/f generate the unit ideal. The complement F of the image of θ|_Z(Z) in C is then equal to the closed subscheme of C with defining ideal (a_1)⊂ A/f, hence to the closed subscheme of S with defining ideal (f,a_1)⊂ A. The algebra A[t][I/f] is isomorphic to A[x][u]/(g-fu)=A[x][u-x^2R(x)]/(a_0+a_1x-f(u-t^2R(x))≃ A[x][v]/(a_0+a_1x-fv).One deduces from this presentation that the morphism θ∘σ:W'=Spec(A[I/f])→Spec(A) corresponding to the inclusion A→ A[I/f] factors through a locally trivial 𝔸^1-bundle θ':W'→ S∖ F over the complement of F. Namely, since a_0 and a_1 generate the unit ideal in A/f, it follows that a_1 and f generate the unit ideal in A[x][u]/(g-fu). So W' is covered by the two principal affine open subsets W'_a_1 ≃Spec(A_a_1[x][v]/(a_0+a_1x-fv))≃Spec(A_a_1[v])≃ S_a_1×𝔸^1W'_f ≃Spec(A_f[x][v]/(a_0+a_1x-fv))≃Spec(A_f[x])≃ S_f×𝔸^1on which θ' restricts to the projection onto the first factor. With the notation above, θ:W→ S and θ':W'→ S∖ F are torsors under the action of line bundles M=Spec(Sym^·ℳ^∨) and M'=Spec(Sym^·ℳ'^∨) for certain uniquely determined invertible sheaves ℳ and ℳ' on S and S∖ F respectively.(<cit.>) Let σ:W'→ W be the affine modification of W with center (ℐ_Z,D) as is Lemma <ref>. Then ℳ'=ℳ⊗_𝒪_S𝒪_S(-C)|_S∖ F and the cartesian diagram of Lemma <ref> is equivariant for the group scheme homomorphism ξ:M'→ M induced by the homomorphism ℳ⊗_𝒪_S𝒪_S(-C)→ℳ obtained by tensoring the inclusion 𝒪_S(-C)↪𝒪_S by ℳ. Since M and M' are uniquely determined, the question is again local with respect to a Zariski open cover of S over which θ:W→ S, hence M, becomes trivial. We can thus assume as in the proof of Lemma <ref> that S=Spec(A), W=Spec(A[x]), that C=div(f) for some non zero element f∈ A and that Z⊂ D is defined by the ideal (f,g) for some g=a_0+a_1x+fx^2R(x)∈ A[x]. Furthermore, the action of M≃𝔾_a,S=Spec(A[t]) on W≃ S×𝔸^1 is the one by translations x↦ x+t on the second factor. Let N=Spec(Sym^·𝒪_S(C))≃Spec(Sym^·f^-1A) where f^-1A denotes the free sub-A-module of the field of fractions Frac(A) of A generated by f^-1. As in the proof of Proposition <ref>, the inclusion 𝒪_S(-C)=f·𝒪_S↪𝒪_S induces a group-scheme homomorphism ξ:N→ M whose co-morphism ξ^♯ concides with the inclusion A[t]⊂Sym^·f^-1A=A[(f^-1t)]. The co-morphism of the corresponding action of N on W is given by A[x]→ A[x]⊗ A[f^-1t], x↦ x⊗1+1⊗ t=x⊗1+f⊗ f^-1t.This action lifts on W'≃Spec(A[x][v]/(a_0+a_1x-fv)) to an action ν:N×_SW'→ W' whose co-morphismA[x][v]/(a_0+a_1x-fv)→ A[x][v]/(a_0+a_1x-fv)⊗ A[f^-1t]is given by x↦ x⊗1+1⊗ t and v↦ v⊗1+a_1⊗ f^-1t. By construction, the principal open subsets W'_a_1≃Spec(A_a_1[v])≃Spec(A_a_1[v/a_1]) and W'_f≃Spec(A_f[x])≃Spec(A_f[x/f]) of W' equipped with the induced actions of N|_S_a_1 and N|_S_f respectively are equivariantly isomorphic to N|_S_a_1 and N|_S_f acting on themselves by translations. So θ':W'→ S∖ F is an N|_S∖ F-torsor, showing that ℳ'=ℳ⊗_𝒪_S𝒪_S(-C)|_S∖ F as desired.§ EXTENSIONS OF 𝔾_A-TORSORS OF TYPE I: LOCALLY TRIVIAL BUNDLES OVER THE BLOW-UP OF A POINT Given a surface S and a locally trivial 𝔸^1-bundle θ:W→S̃ over the blow-up τ:S̃→ S of a closed point o in the smooth locus of S, the restriction of W over the complement S̃∖ E of the exceptional divisor E of τ is a locally trivial 𝔸^1-bundle τ∘θ:W|_S̃∖ E→S̃∖ E≃→S∖{o}. This observation combined with the following re-interpretation of an example constructed in <cit.> suggests that locally trivial 𝔸^1-bundles over the blow-up of closed point o in the smooth locus of a surface S form a natural class of schemes in which to search for nontrivial 𝔾_a-extension of 𝔾_a-bundles over punctured surfaces.Let o=V(x,y) be a global scheme-theoretic complete intersection closed point in the smooth locus of a surface S. Let ρ:P→ S∖{o} and π_0:X_0→ S be the affine S-schemes with defining sheaves of ideals (xv-yu-1) and (xr-yq,yp-x(q-1),pr-q(q-1)) in 𝒪_S[u,v] and 𝒪_S[p,q,r] respectively. The morphism of S-schemes j_0:P→ X_0 defined by (x,y,u,v)↦(x,y,xu,xv,yv) is an open embedding, equivariant for the 𝔾_a,S-actions on P and X_0 associated with the locally nilpotent 𝒪_S-derivations x∂_u+y∂_v and x^2∂_p+xy∂_q+y^2∂_r of ρ_*𝒪_P and (π_0)_*𝒪_X_0 respectively. It is straightforward to check that ρ:P→ S∖{o} is a 𝔾_a,S_*-torsor and that π_0:X_0→ S is a 𝔾_a-extension of P whose fiber over o is isomorphic to the smooth affine quadric {pr-q(q-1)=0}⊂𝔸_κ^3. Viewing the blow-up S̃ of o as the closed subscheme of S×_kProj(k[u_0,u_1]) with equation xu_1-yu_0=0, the morphism of S-schemes θ:X_0→S̃ defined by (x,y,p,q,r)↦((x,y),[x:y])=((x,y),[q:r])=((x,y),[p:q-1])is a locally trivial 𝔸^1-bundle, actually a torsor under the line bundle corresponding to the invertible sheaf 𝒪_S̃(-2E), where E≃ℙ_κ^1 denotes the exceptional divisor of the blow-up. Given a surface S and a closed point o in the smooth locus of S, with residue field κ, we denote by τ:S̃→ S be the blow-up of o, with exceptional divisor E≃ℙ_κ^1. We identify S̃∖ E and S_*=S∖{o} by the isomorphism induced by τ. For every ℓ∈ℤ, we denote by M(ℓ)=Spec(Sym^·𝒪_S̃(-ℓ E)) the line bundle on S̃ corresponding to the invertible sheaf 𝒪_S̃(ℓ E).The aim of this section is to give a classification of all possible 𝔾_a-equivariant extensions of Type I of a given 𝔾_a-torsor ρ:P→ S_*, that is 𝔾_a-extensions π:W→ S that factor through locally trivial 𝔸^1-bundles θ:W→S̃.§.§ Existence of 𝔾_a-extensions of Type I By virtue of Propositions <ref> and <ref>, there exists a one-to-one correspondence between 𝔾_a-equivariant extensions of a 𝔾_a-torsor ρ:P→ S_* that factor through a locally trivial 𝔸^1-bundle θ:W→S̃ and pairs (M,ξ) consisting of an M-torsor θ:W→S̃ for some line bundle M on S̃ and a group scheme homomorphism ξ:𝔾_a,S̃→ M restricting to an isomorphism over S̃∖ E, such that W equipped with the 𝔾_a,S̃-action deduced by composition with ξ restricts on S_*=S̃∖ E to a 𝔾_a,S_*-torsor θ|_S_*:W|_S_*→ S_* isomorphic to ρ:P→ S_*. The condition that ξ:𝔾_a,S̃→ M restricts to an isomorphism outside E implies that M≃ M(ℓ) for some ℓ, which is necessarily non negative, and that ξ is induced by the canonical global section of 𝒪_S̃(ℓ E) with divisor ℓ E.Let ρ:P→ S_* be a 𝔾_a,S_*-torsor. Then there exists an integer ℓ_0≥0 depending on P only such that for every ℓ≥ℓ_0, P admits a 𝔾_a-extension to a uniquely determined M(ℓ)-torsor θ_ℓ:W(P,ℓ)→S̃ equipped with the 𝔾_a,S̃-action induced by the canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)) with divisor ℓ E. The invertible sheaves 𝒪_S̃(nE), n≥0, form an inductive system of sub-𝒪_S̃-modules of the sheaf 𝒦_S̃ of rational function on S̃, where for each n, the injective transition homomorphism j_n,n+1:𝒪_S̃(nE)↪𝒪_S̃((n+1)E) is obtained by tensoring the canonical section 𝒪_S̃→𝒪_S̃(E) with divisor E with 𝒪_S̃(nE). Let i:S_*=S̃∖ E↪S̃ be the open inclusion. Since E is a Cartier divisor, it follows from <cit.> that i_*𝒪_S_*≃colim_n≥0𝒪_S̃(nE). Furthermore, since E≃ℙ_κ^1 is the exceptional divisor of τ:S̃→ S, we have 𝒪_S̃(E)|_E≃𝒪_ℙ_κ^1(-1), and the long exact sequence of cohomology for the short exact sequence0→𝒪_S̃(nE)→𝒪_S̃((n+1)E)→𝒪_S̃((n+1)E)|_E→0, n≥0,combined with the vanishing of H^0(ℙ_κ^1,𝒪_ℙ_κ^1(-n-1)) for every n≥0 implies that the transition homomorphisms H^1(j_n,n+1):H^1(S̃,𝒪_S̃(nE))→ H^1(S̃,𝒪_S̃((n+1)E)), n≥0,are all injective. By assumption, S whence S̃ is noetherian, and i:S_*→S̃ is an affine morphism as E is a Cartier divisor on S̃. We thus deduce from <cit.> and <cit.> that the canonical homomorphismψ:colim_n≥0H^1(S̃,𝒪_S̃(nE))→ H^1(S_*,𝒪_S_*)obtained as the composition of the canonical homomorphisms colim_n≥0H^1(S̃,𝒪_S̃(nE))→ H^1(S̃,colim_n≥0𝒪_S̃(nE))=H^1(S̃,i_*𝒪_S_*)and H^1(S̃,i_*𝒪_S_*)→ H^1(S_*,𝒪_S_*) is an isomorphism. Let g∈ H^1(S_*,𝒪_S_*) be the isomorphism class of the 𝔾_a,S_*-torsor ρ:P→ S_*. If g=0, then since ψ is an isomorphism, we have ψ^-1(g)=0 and, since the homomorphisms H^1(j_n,n+1) are injective, it follows that ψ^-1(g) is represented by the zero sequence (0)_n∈ H^1(S̃,𝒪_S̃(nE)), n≥0 . Consequently, the only 𝔾_a-extensions of P are the line bundles W(P,ℓ)=M(ℓ), ℓ≥0, each equipped with the 𝔾_a,S̃-action induced by its canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)). Otherwise, if g≠0, then h=ψ^-1(g)≠0, and since the homomorphisms H^1(j_n,n+1), n≥0 are injective, it follows that there exists a unique minimal integer ℓ_0 such that h is represented by the sequence h_n=H^1(j_n-1,n)∘⋯∘ H^1(j_ℓ_0,ℓ_0+1)(h_ℓ_0)∈ H^1(S̃,𝒪_S̃(nE)), n≥ℓ_0for some non zero h_ℓ_0∈ H^1(S̃,𝒪_S̃(ℓ_0E)). It then follows from Proposition <ref> that for every ℓ≥ℓ_0, the M(ℓ)-torsor θ_ℓ:W(P,ℓ)→S̃ with isomorphism class h_ℓ equipped with the 𝔾_a,S̃-action induced by the canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)) is a 𝔾_a-extension of P. Conversely, for every 𝔾_a-extension of P into an M(ℓ)-torsor θ:W→S̃ equipped with the 𝔾_a,S̃-action induced by the canonical global section s_ℓ∈Γ(S̃,𝒪_S̃(ℓ E)), it follows from Proposition <ref> again that the image of the isomorphism class h_ℓ∈ H^1(S̃,𝒪_S̃(ℓ E)) of W in H^1(S̃∖ E,𝒪_S̃(ℓ E)|_S̃∖ E)≃ H^1(S_*,𝒪_S_*) is equal to g. Letting h∈colim_n≥0H^1(S̃,𝒪_S̃(nE)) be the element represented by the sequence h_n=(H^1(j_n-1,n∘⋯∘ j_ℓ,ℓ+1)(h_ℓ))_n≥ℓ∈ H^1(S̃,𝒪_S̃(nE)), n≥ℓwe have ψ(h)=g and since ψ is an isomorphism, we conclude that W≃ W(P,ℓ). §.§ 𝔾_a-extensions with affine total spacesThe extensions θ:W→S̃ we get from Proposition <ref> are not necessarily affine over S. In this subsection we establish a criterion for affineness which we then use to characterize all extensions θ:W→S̃ of a 𝔾_a-torsor ρ:P→ S_* whose total spaces W are affine over S. Let S=Spec(A) be an affine surface and let o=V(x,y) be a global scheme-theoretic complete intersection point in the smooth locus of S . Let τ:S̃→ S be the blow-up of o with exceptional divisor E and let θ:W→S̃ be an M(ℓ)-torsor for some ℓ≥0. Then the following hold:a) H^1(W,𝒪_W)=0. b) If H^1(W,θ^*𝒪_S̃(ℓ E))=0 for some ℓ≥2 then W is an affine scheme. Since o is a scheme-theoretic complete intersection, we can identify S̃ with the closed subvariety of S×_kℙ_k^1=S×_kProj(k[t_0,t_1]) defined by the equation xt_1-yt_0=0. The restriction p:S̃→ℙ_k^1 of the projection to the second factor is an affine morphism. More precisely, letting U_0=ℙ_k^1∖{[1:0]}≃Spec(k[z]) and U_∞=ℙ_k^1∖{[0:1]}≃Spec(k[z']) be the standard affine open cover of ℙ_k^1, we have p^-1(U_0)≃Spec(A[z]/(x-yz) and p^-1(U_∞)≃Spec(A[z']/(y-xz')). The exceptional divisor E≃ℙ_κ^1 of τ:S̃→ S is a flat quasi-section of p with local equations y=0 and x=0 in the affine charts p^-1(U_0) and p^-1(U_∞) respectively. Every M(ℓ)-torsor θ:W→S̃ for some ℓ≥0 is isomorphic to the scheme obtained by gluing W_0=p^-1(U_0)×Spec(k[u]) with W_∞=p^-1(U_∞)×Spec(k[u']) over U_0∩ U_∞ by an isomorphism induced by a k-algebra isomorphism of the form A[(z')^±1]/(y-xz')[u']∋(z',u')↦(z^-1,z^ℓu+p)∈ A[z^±1]/(x-yz)[u]for some p∈ A[z^±1]/(x-yz). Since H^1(W,𝒪_W)≃Ȟ^1(W,𝒪_W)≃Ȟ^1({W_0,W_∞},𝒪_W), it is enough in order to prove a) to check that every Čech 1-cocycle g with value in 𝒪_W for the covering of W by the affine open subsets W_0 and W_∞ is a coboundary. Viewing g as an element g=g(z^±1,u)∈ A[z^±1]/(x-yz)[u], it is enough to show that every monomial g_s=hz^ru^s where h∈ A, r∈ℤ and s∈ℤ_≥0 is a coboundary, which is the case if and only if there exist a(z,u)∈ A[z]/(f-gz)[u] and b(z',u')∈ A[z']/(y-xz')[u'] such that g=b(z^-1,z^ℓu+p)-b(z,u). If r≥0 then g∈ A[z]/(x-yz)[u] is a coboundary. We thus assume from now on that r<0. Suppose that s>0. Then we can write u^s=z^-ℓ s(z^ℓu+p)^s-R(u) where R∈ A[z^±1]/(x-yz)[u] is polynomial whose degree in u is strictly less than s. Then since r<0, hz^ru^s=hz^r-ℓ s(z^ℓu+p)^s-hz^rR(u) =b(z^-1,z^ℓu+p)-hz^rR(u)where b(z',u')=h(z')^-r+ℓ s(u')^s∈ A[z']/(y-xz')[u']. So g_s is a coboundary if and only if -hz^rR(u) is. By induction, we only need to check that every monomial g_0=hz^r∈ A[z^±1]/(x-yz)[u] of degree 0 in u is a coboundary. But such a cocycle is simply the pull-back to W of a Čech 1-cocycle h_0 with value in 𝒪_S̃ for the covering of S̃ by the affine open subsets p^-1(U_0) and p^-1(U_∞). Since the canonical homomorphism H^1(S,𝒪_S)=H^1(S,τ_*𝒪_S̃)→ H^1(S̃,𝒪_S̃)≃Ȟ^1({p^-1(U_0),p^-1(U_∞)},𝒪_S̃)is an isomorphism and H^1(S,𝒪_S)=0 as S is affine, we conclude that h_0 is a coboundary, hence that g_0 is a coboundary too. This proves a). Now suppose that H^1(W,θ^*𝒪_S̃(ℓ E))=0 for some ℓ≥2. Let η:V→ℙ_k^1 be a non trivial 𝒪_ℙ_k^1(-ℓ)-torsor and consider the fiber product W×_p∘θ,ℙ_k^1,ηV:@C-3exW×_p∘θ,ℙ^1_k,η V [dl] [dr] W [dr]_p∘θ V [dl]^ηℙ^1_kBy virtue of <cit.>, V is an affine surface. Since p∘θ:W→ℙ_k^1 is an affine morphism, so is pr_V:W×_ℙ_k^1V→ V and hence, W×_ℙ_k^1V is an affine scheme. On the other hand, since p^*𝒪_ℙ_k^1(-1)≃𝒪_S̃(E), the projection pr_W:W×_ℙ_k^1V→ W is a θ^*M(ℓ)-torsor, hence is isomorphic to the trivial one q:θ^*M(ℓ)→ W by hypothesis. So W is isomorphic to the zero section of θ^*M(ℓ), which is a closed subscheme of the affine scheme W×_ℙ_k^1V, hence an affine scheme. We are now ready to prove the following characterization:A 𝔾_a,S_*-torsor ρ:P→ S_* admits a 𝔾_a-extension to a locally trivial 𝔸^1-bundle whose total space is affine over S if and only if for every Zariski open neighborhood U of o, P×_S_*U→ U_*=U∖{o} is a non trivial 𝔾_a,U_*-torsor. When it exists, the corresponding locally trivial 𝔸^1-bundle θ:W→S̃ is unique and is an M(ℓ_0)-torsor for some ℓ_0≥2, whose restriction to E≃ℙ_κ^1 is a non trivial 𝒪_ℙ_κ^1(-ℓ_0)-torsor. The scheme W is affine over S if and only if its restriction W|_E over E⊂S̃ is a nontrivial torsor. Indeed, if W|_E is a trivial torsor then it is a line bundle over E≃ℙ_κ^1. Its zero section is then a proper curve contained in the fiber of π=τ∘θ:W→ S, which prevents π from being an affine morphism. Conversely, if W|_E is nontrivial, then it is a torsor under a uniquely determined line bundle 𝒪_ℙ_κ^1(-m) for some m≥2 necessarily. Since by construction π restricts over S_* to ρ:P→ S_* which is an affine morphism, π is affine if and only if there exists an open neighborhood U of o in S such that π^-1(U) is affine. Replacing S by a suitable affine open neighborhood of o, we can therefore assume without loss of generality that S=Spec(A) is affine and that o is a scheme-theoretic complete intersection o=V(x,y) for some elements x,y∈ A. By virtue of <cit.> every nontrivial 𝒪_ℙ_κ^1(-m)-torsor, m≥2 , has affine total space. The Cartier divisor D=W|_E in W is thus an affine surface, and so H^1(D,𝒪_W((n+1)D)|_D)=0 for every n∈ℤ. By a) in Lemma <ref>, H^1(W,𝒪_W)=0, and we deduce successively from the long exact sequence of cohomology for the short exact sequence0→𝒪_W(nD)→𝒪_W((n+1)D)→𝒪_W((n+1)D)|_D→0in the case n=0 and then n=1 that H^1(W,𝒪_W(D))=H^1(W,𝒪_W(2D))=0. Since 𝒪_W(2D)≃θ^*𝒪_S̃(2E), we conclude from b) in the same lemma that W is affine. The condition that P×_S_*U→ U_* is nontrivial for every open neighborhood U of o is necessary for the existence of an extension θ:W→S̃ of P for which W|_E is a nontrivial torsor. Indeed, if there exists a Zariski open neighborhood U of o such that the restriction of P over U_* is the trivial 𝔾_a,U_*-torsor, then the image in H^1(U_*,𝒪_U_*) of the isomorphism class g of P is zero and so, arguing as in the proof of Proposition <ref>, every 𝔾_a-extension θ:W→S̃ restricts on τ^-1(U) to the trivial M(ℓ)|_τ^-1(U)-torsor M(ℓ)|_τ^-1(U)→τ^-1(U), hence to a trivial torsor on E⊂τ^-1(U). Now suppose that ρ:P→ S_* is a 𝔾_a,S_*-torsor with isomorphism class g∈ H^1(S_*,𝒪_S_*) such that P×_S_*U→ U_* is non trivial for every open neighborhood U of o. The inverse image h=ψ^-1(g)∈colim_n≥0H^1(S̃,𝒪_S̃(nE)) of g by the isomorphism (<ref>) is represented by a sequence of nonzero elements h_n∈ H^1(S̃,𝒪_S̃(nE)) as in (<ref>) above. By the long exact sequence of cohomology of the short exact sequence (<ref>), the image h_n of h_n in H^1(E,𝒪_S̃(nE)|_E)≃ H^1(ℙ^1,𝒪_ℙ_κ^1(-n)) is nonzero if and only if h_n is not in the image of the injective homomorphism H^1(j_n,n-1). Since h_n coincides with the isomorphism class of the restriction W_n|_E of an M(n)-torsor θ_n:W_n→S̃ with isomorphism class h_n, we conclude that there exists a unique ℓ_0≥2 such that the restriction to E of an M(ℓ_0)-torsor θ_ℓ_0:W_ℓ_0→S̃ with isomorphism class h_ℓ_0∈ H^1(S̃,𝒪_S̃(ℓ_0E)) is a nontrivial 𝒪_ℙ_κ^1(-ℓ_0)-torsor. §.§ Examples In this subsection, we consider 𝔾_a-torsors of the punctured affine plane. So S=𝔸^2=Spec(k[x,y]), o=(0,0) and 𝔸_*^2=𝔸^2∖{o}. We let τ:𝔸̃^2→𝔸^2 be the blow-up of o, with exceptional divisor E≃ℙ^1 and we let i:𝔸_*^2↪𝔸̃^2 be the immersion of 𝔸_*^2 as the open subset 𝔸̃^2∖ E. We further identify 𝔸̃^2 with the total space f:𝔸̃^2→ℙ^1 of the line bundle 𝒪_ℙ^1(-1) in such a way that E corresponds to the zero section of this line bundle.§.§.§ A simple case: homogeneous 𝔾_a-torsorsFollowing <cit.>, we say that a non trivial 𝔾_a,𝔸_*^2-torsor ρ:P→𝔸_*^2 is homogeneous if it admits a lift of the 𝔾_m-action λ·(x,y)=(λ x,λ y) on 𝔸_*^2 which is locally linear on the fibers of ρ. By <cit.>, this is the case if and only if the isomorphism class g of P in H^1(𝔸_*^2,𝒪_𝔸_*^2) can be represented on the open covering of 𝔸_*^2 by the principal open subsets 𝔸_x^2 and 𝔸_y^2 by a Čech 1-cocycle of the form x^-my^-np(x,y) where m,n≥0 and p(x,y)∈ k[x,y] is a homogeneous polynomial of degree r≤ m+n-2. Equivalently, P is isomorphic the 𝔾_a,𝔸_*^2-torsorρ=pr_x,y:P_m,n,p={ x^mv-y^nu=p(x,y)}∖{x=y=0}→𝔸_*^2,which admits an obvious lift λ·(x,y,u,v)=(λ x,λ y,λ^m-du,λ^n-dv), where d=m+n-r, of the 𝔾_m-action on 𝔸_*^2. Let q:𝔸_*^2→𝔸_*^2/𝔾_m=ℙ^1 be the quotient morphism of the aforementioned 𝔾_m-action on 𝔸_*^2. Then it follows from <cit.> that the inverse image by the canonical isomorphism ⊕_k∈ℤH^1(ℙ^1,𝒪_ℙ(k))≃ H^1(ℙ^1,q_*𝒪_𝔸_*^2)→ H^1(𝔸_*^2,𝒪_𝔸_*^2)of the isomorphism class g of such an homogeneous torsor is an element h of H^1(ℙ^1,𝒪_ℙ(-d)). Furthermore, the 𝔾_m-equivariant morphism ρ:P→𝔸_*^2 descends to a locally trivial 𝔸^1-bundle ρ:P/𝔾_m→ℙ^1=𝔸_*^2/𝔾_m which is an 𝒪_ℙ^1(-d)-torsor with isomorphism class h∈ H^1(ℙ^1,𝒪_ℙ(-d)).Since f^*𝒪_ℙ^1(-d)≃𝒪_𝔸̃^2(dE), the fiber product W(P,d)=𝔸̃^2×_ℙ^1P/𝔾_m is equipped via the restriction of the first projection with the structure of an M(d)-torsor θ:W(P,d)→𝔸̃^2 with isomorphism class f^*h∈ H^1(𝔸̃^2,𝒪_𝔸̃^2(dE)). On the other other hand, W(P,d) is a line bundle over P/𝔾_m via the second projection, hence is an affine threefold as P/𝔾_m is affine. By construction, we have a commutative diagram@C+2ex@R-1exW(P,d) @->'[d]^-θ[dd] [dr] P [dd]_ρ[ur]^j[rr]P/𝔾_m [dd]_ρ𝔸̃^2 [dr]^f 𝔸^2_*[ur]^i[rr]^qℙ^1 in which each square is cartesian. In other words, W(P,d) is obtained from the 𝔾_m-torsor P→ P/𝔾_m by “adding the zero section”. The open embedding j:P↪ W(P,d) is equivariant for the 𝔾_a-action on W(P,d) induced by the canonical global section of 𝒪_𝔸̃^2(dE) with divisor dE (see Proposition <ref>). By Theorem <ref> , θ:W(P,d)→𝔸̃^2 is the unique 𝔾_a-extension of ρ:P→𝔸_*^2 with affine total space. In the simplest case d=2, the unique homogeneous 𝔾_a,𝔸_*^2-torsor is the geometric quotient SL_2→SL_2/𝔾_a of the group SL_2 by the action of its subgroup of upper triangular unipotent matrices equipped with the diagonal 𝔾_m-action, and we recover Example <ref>. §.§.§ General case Here, given an arbitrary non trivial 𝔾_a-torsor ρ:P→𝔸_*^2, we describe a procedure to explicitly determine the unique 𝔾_a-extension θ:W→𝔸̃^2 of P with affine total space W from a Čech 1-cocycle x^-my^-np(x,y), where m,n≥0 and p(x,y)∈ k[x,y] is a non zero polynomial of degree r≤ m+n-2, representing the isomorphism class g∈ H^1(𝔸_*^2,𝒪_𝔸_*^2) of P on the open covering of 𝔸_*^2 by the principal open subsets 𝔸_x^2 and 𝔸_y^2. Write p(x,y)=p_d+p_d+1+⋯+p_r where the p_i∈ k[x,y] are the homogeneous components of p, and p_d≠0. In the decomposition H^1(𝔸_*^2,𝒪_𝔸_*^2)≃ H^1(ℙ^1,q_*𝒪_𝔸_*^2)≃⊕_s∈ℤH^1(ℙ^1,𝒪_ℙ^1(s))a non zero homogeneous component x^-my^-np_i of x^-my^-np(x,y) corresponds to a non zero element of H^1(ℙ^1,𝒪_ℙ^1(-m-n+i)). On the other hand, since for every ℓ∈ℤ, 𝒪_𝔸̃^2(ℓ E)=f^*𝒪_ℙ^1(-ℓ) and f:𝔸̃^2→ℙ^1 is the total space of the line bundle 𝒪_ℙ^1(-1), it follows from the projection formula that H^1(𝔸̃^2,𝒪_𝔸̃^2(ℓ E))≃ H^1(ℙ^1,f_*𝒪_𝔸̃^2⊗𝒪_ℙ^1(-ℓ))≃⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ)).The image of x^-my^-np(x,y) in ⊕_s∈ℤH^1(ℙ^1,𝒪_ℙ^1(s)) belongs to ⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ)) if and only if ℓ≥ℓ_0=m+n-d≥2. Given such an ℓ, the image (h_t)_t≥0∈⊕_t≥0H^1(ℙ^1,𝒪_ℙ^1(t-ℓ)) of x^-my^-np(x,y) then defines a unique M(ℓ)-torsor θ_ℓ:W(P,ℓ)→𝔸̃^2 whose restriction over the complement of E is isomorphic to ρ:P→𝔸_*^2 when equipped with the action 𝔾_a-action induced by the canonical section of 𝒪_𝔸̃^2(ℓ E) with divisor ℓ E. On the other hand, the restriction of W|_E→ E over E is an 𝒪_ℙ^1(-ℓ)-torsor with isomorphism class h_0∈ H^1(ℙ^1,𝒪_ℙ^1(-ℓ)). By definition, h_0 is non zero if and only if ℓ=ℓ_0, and we conclude from Theorem <ref> that θ_ℓ_0:W(P,ℓ_0)→𝔸̃^2 is the unique 𝔾_a-extension of ρ:P→𝔸̃^2 with affine total space. § QUASI-PROJECTIVE 𝔾_A-EXTENSIONS OF TYPE IIIn this section we consider the following subclass of extensions of Type II of a 𝔾_a-torsor over a punctured surface. A 𝔾_a-extension π:X→ S of a 𝔾_a-torsor ρ:P→ S_* over a punctured surface S_*=S∖{o} is said to be a quasi-projective extension of Type II if it satisfies the following propertiesi) X is quasi-projective over S and the 𝔾_a,S-action on X is proper, ii) X is smooth along π^-1(o) and π^-1(o)_red≃𝔸_κ^2. Let o=V(x,y) be a global scheme-theoretic complete intersection closed point in the smooth locus of a surface S and let ρ:P→ S∖{o} be the 𝔾_a-torsor with defining sheaf of ideals (xv-yu-1)⊂𝒪_S[u,v] as in Example <ref>. Let π_1:X_1→ S be the affine S-scheme with defining sheaf of ideals (xw-y(yz_1+1),xz_2-z_1(yz_1+1),z_1w-yz_2)⊂𝒪_S[z_1,z_2,w]. The morphism of S-schemes j_1:P→ X_1 defined by (x,y,u,v)↦(x,y,u,uv,yu) is an open embedding, equivariant for the 𝔾_a,S-action on X_1 associated with the locally nilpotent 𝒪_S-derivation x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w of π_*𝒪_X_1. The fiber π_1^-1(o) is isomorphic to 𝔸_κ^2=Spec(κ[z_2,w]) on which the 𝔾_a,S-action restricts to 𝔾_a,κ-action by translations associated to the derivation ∂_z_2 of κ[z_2,w]. It is straightforward to check that X_1 is smooth along π_1^-1(o). We claim that the geometric quotient of the 𝔾_a,S-action on X_1 is isomorphic to the complement of a κ-rational point o_1 in the blow-up τ:S̃→ S of o. Such a surface being in particular separated, the 𝔾_a,S-action on X_1 is proper, implying that j_1:P↪ X_1 is a quasi-projective extension of P of Type II.Indeed, let us identify S̃ with the closed subvariety of S×_kProj(k[u_0,u_1]) with equation xu_1-yu_0=0 in such a way that τ coincides with the restriction of the first projection. The morphism f:X_1→S̃ defined by(x,y,z,u,v)↦((x,y),[x:y])=((x,y),[yz_1+1:w])is 𝔾_a-invariant and maps π_1^-1(o) dominantly onto the exceptional divisor E≃pr_S^-1(o)≃Proj(κ[u_0,u_1]) of τ. The induced morphismf|_π^-1(o):π^-1(o)=Spec(κ[z_2,w])→ E,(z_2,w)↦[1:w]factors as the composition of the geometric quotient π_1^-1(o)→π_1^-1(o)/𝔾_a,κ≃Spec(κ[w]) with the open immersion π_1^-1(o)/𝔾_a,κ↪ E of π_1^-1(o)/𝔾_a,κ as the complement of the κ-rational point o_1=((0,0),[0:1])∈ E. On the other hand, the composition τ∘ f∘ j_1:P≃⟶X_1∖π_1^-1(o)→S̃∖ E≃⟶S∖{o}coincides with the geometric quotient morphism ρ:P→ S∖{o}. So f:X_1→S̃ factors through a surjective morphism q:X_1→S̃∖{o_1} whose fibers all consist of precisely one 𝔾_a-orbit. Since q is a smooth morphism, q is a 𝔾_a-torsor which implies that X_1/𝔾_a≃S̃∖{o_1}.The scheme of the classification of quasi-projective extensions of Type II of a given 𝔾_a-torsor ρ:P→ S_* which we give below is as follows: we first construct in <ref> families of such extensions, in the form of 𝔾_a-torsors q:X→ S' over quasi-projective S-schemes τ:S'→ S such that τ^-1(o)_red is isomorphic to 𝔸_κ^1, S' is smooth along τ^-1(o), and τ:S'∖τ^-1(o)→ S_* is an isomorphism. We then show in<ref> that for quasi-projective 𝔾_a-extension π:X→ S of Type II of a given 𝔾_a-torsor ρ:P→ S_*, the structure morphism π:X→ S factors through a 𝔾_a-torsor q:X→ S' over one of these S-schemes S'. In the last subsection, we focus on the special case where π:X→ S has the stronger property of being an affine morphism.§.§ A family of 𝔾_a-extensions over quasi-projective S-schemes Let again (S,o) be a pair consisting of a surface and a closed point o contained in the smooth locus of S, with residue field κ. We let τ_1:S_1→ S be the blow-up of o, with exceptional divisor E_1≃ℙ_κ^1. Then for every n≥2, we let τ_n,1:S_n=S_n(o_1,…,o_n-1)→S_1 be the scheme obtained from S_1 by performing the following sequence of blow-ups of κ-rational points: a) The first step τ_21:S_2(o_1)→S_1 is the blow-up of a κ-rational point o_1∈E_1 with exceptional divisor E_2≃ℙ_κ^1, b) Then for every 2≤ i≤ n-2, we let τ_i+1,i:S_i+1(o_1,…,o_i)→S_k(o_1,…,o_i-1) be the blow-up of a κ-rational point o_i∈E_i, with exceptional divisor E_i+1≃ℙ_κ^1. c) Finally, we let τ_n,n-1:S_n(o_1,…,o_n-1)→S_n-1(o_1,…,o_n-2) be the blow-up of a κ-rational point o_n-1∈E_n-1 which is a smooth point of the reduced total transform of E_1 by τ_1∘⋯∘τ_n-1,n-2. We let E_n≃ℙ_κ^1 be the exceptional divisor of τ_n,n-1 and we let τ_n,1=τ_2,1∘⋯∘τ_n,n-1:S_n(o_1,…,o_n-1)→S_1.The inverse image of o in S_n(o_1,…,o_n-1) by τ_1∘τ_n,1 is a tree of κ-rational curves in which E_n intersects the reduced proper transform of E_1∪⋯∪E_n-1 in S_n(o_1,…,o_n-1) transversally in a unique κ-rational point.For every κ-rational point o_1∈E_1, we let S_1(o_1)=S_1∖{o_1}, E_1=E_1∩ S_1≃𝔸_κ^1 and we let τ_1:S_1(o_1)→ S be the restriction of τ_1. For n≥2, we let S_n(o_1,…,o_n-1)=S_n(o_1,…,o_n-1)∖E_1∪⋯∪E_n-1 and E_n=S_n(o_1,…,o_n-1)∩E_n≃𝔸_κ^1. We denote by τ_n,1:S_n(o_1,…,o_n-1)→S_1 the birational morphism induced by τ_n,1 and we let τ_n=τ_1∘τ_n,1:S_n(o_1,…,o_n-1)→ S.The following lemma summarizes some basic properties of the so-constructed S-schemes:For every n≥1, the following hold for S_n=S_n(o_1,…,o_n-1):a) τ_n:S_n→ S is quasi-projective and restricts to an isomorphism over S_* while τ_n^-1(o)_red=E_n, b) S_n is smooth along τ_n^-1(o)c) τ_n^*:Γ(S,𝒪_S)→Γ(S_n,𝒪_S_n) is an isomorphism.Moreover for n≥2, the morphism τ_n,1:S_n→S_1 is affine. The first three properties are straightforward consequences of the construction. For the last one, let D=E_1+∑_i=2^n-1a_iE_i where a_i is a sequence of positive rational numbers and let m≥1 be so that mD is a Cartier divisor on S_n. Then a direct computation shows that the restriction of 𝒪_S_n(mD) to τ_n,1^-1(o_1)_red=⋃_i=2^nE_i is an ample invertible sheaf provided that the sequence (a_i)_i=2,…,n-1 decreases rapidly enough with respect to the distance of E_i to E_1 in the dual graph of E_1∪⋯∪E_n-1. Since τ_n,1 restricts to an isomorphism over S_1∖{o_1}, it follows from <cit.> that 𝒪_S_n(mD) is τ_n,1-ample on S_n. Since by definition τ_n,1 is the restriction of the projective morphism τ_n,1:S_n→S_1 to S_n=S_n∖E_1∪⋯∪E_n-1=S_n∖Supp(D), we conclude that τ_n,1 is an affine morphism.By construction, τ_1^-1(o)=E_1 in S_1(o_1), but for n≥2, we have τ_n^-1(o)=mE_n for some integer m≥1 which depends on the sequence of κ-rational points o_1,…,o_n-1 blown-up to construct S_n(o_1,…,o_n-1). For instance, it is straightforward to check that m=1 if and only if for every i≥1, o_i∈E_i is a smooth point of the reduced total transform of E_1 in S_i(o_1,…,o_i-1).The structure morphism of a 𝔾_a-torsor being affine, hence quasi-projective, the total space of any 𝔾_a-torsor q:X→ S_n over an S-scheme τ_n:S_n=S_n(o_1,…,o_n)→ S is a quasi-projective S-scheme π=τ_n∘ q:X→ S equipped with a proper 𝔾_a,S-action. Furthermore π^-1(o)_red=q^-1(E_n)≃ E_n×𝔸_κ^1≃𝔸_κ^2 and X is smooth along π^-1(o) as S_n is smooth along E_n. On the other hand, π:X→ S is by construction a 𝔾_a-extension of its restriction ρ:P→ S_n∖ E_n≃ S_* over S_n∖ E_n, hence is a quasi-projective 𝔾_a-extension of P of Type II. The following proposition shows conversely that every 𝔾_a-torsor ρ:P→ S_* admits a quasi-projective 𝔾_a-extension of Type IIinto a 𝔾_a-torsor q:X→ S_n. Let ρ:P→ S_* be a 𝔾_a-torsor. Then for every n≥1 and every S-scheme τ_n:S_n(o_1,…,o_n-1)→ S as in Notation <ref> there exist a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1) and an equivariant open embedding j:P↪ X such that in the following diagramP [hook,r,"j"] [d,"ρ"'] X [d,"q"] S_n(o_1,… , o_n-1)∖ E_n [hook,r] [d,"τ_n"',"≀"] S_n(o_1,… , o_n-1) [d,"τ_n"] S_* [hook,r] S all squares are cartesian. In particular, j:P↪ X is a quasi-projective 𝔾_a-extension of P of Type II.Letting S_n=S_n(o_1,…,o_n), we have to prove that every 𝔾_a-torsor ρ:P→ S_n∖ E_n≃ S_* is the restriction of a 𝔾_a-torsor q:X→ S_n, or equivalently that the restriction homomorphism H^1(S_n,𝒪_S_n)→ H^1(S_n∖ E_n,𝒪_S_n∖ E_n) is surjective. It is enough to show that there exists a Zariski open neighborhood U of E_n in S_n and a 𝔾_a-torsor q:Y→ U such that Y|_U∖ E_n≃ P|_U∖ E_n. Indeed, if so then a 𝔾_a-torsor q:X→ S_n with the desired property is obtained by gluing P and Y over U∖ E_n by the isomorphism Y|_U∖ E_n≃ P|_U∖ E_n. In particular, we can replace S_n by the inverse image by τ_n:S_n→ S of any Zariski open neighborhood of o in S. We can thus assume from the very beginning that S=Spec(A) is affine and that o=V(f,g) is a scheme-theoretic intersection for some f,g∈ A. Up to replacing f and g by other generators of the maximal ideal of o in A, we can assume that the proper transform L_1 in τ_1:S_1→ S of the curve L=V(f)⊂ S intersects E_1 in o_1. We denote by M_1⊂S_1 the proper transform of the curve M=V(g)⊂ S. We first treat the case n=1. The open subset U_1=S_1∖ L of S_1 is then affine and contained in S_1. Furthermore U_1∖ E_1=S_1∖τ_1^-1(L)≃ S∖ L is also affine. The Mayer-Vietoris long exact sequence of cohomology of 𝒪_S_1 for the open covering of S_1 by S_1∖ E_1 and U_1 then reads 0→H^0(S_1,𝒪_S_1)→ H^0(U_1,𝒪_S_1)⊕ H^0(S_1∖ E_1,𝒪_S_1)→ H^0(U_1∖ E_1,𝒪_S_n)→⋯ ⋯→H^1(S_1,𝒪_S_1)→ H^1(U_1,𝒪_U_1)⊕ H^1(S_1∖ E_1,𝒪_S_1)→ H^1(U_1∖ E_1,𝒪_U_1)→⋯.Since U_1∖ E_1 is affine, H^1(U_1∖ E_1,𝒪_U_1)=0 and so, the homomorphism H^1(S_1,𝒪_S_1)→ H^1(S_1∖ E_1,𝒪_S_1) is surjective as desired. In the case where n≥2, the open subset V_1=S_1∖ M_1 of S_1 is affine and it contains o_1 since M_1 intersects E_1 in a point distinct from o_1. Since τ_n,1:S_n→S_1 is an affine morphism by Lemma <ref>, U_n=τ_n,1^-1(U_1) is an affine open neighborhood of E_n in S_n. By construction, S_n is then covered by the two open subset U_n and S_n∖ E_n which intersect along the affine open subset U_n∩ S_n∖ E_n=U_n∖ E_n=τ_n,1^-1(S_1∖τ_1^-1(M)) of S_n. The conclusion then follows from the Mayer-Vietoris long exact sequence of cohomology of 𝒪_S_n for the open covering of S_n by S_n∖ E_n and U_n. §.§ ClassificationThe following theorem shows that every quasi-projective 𝔾_a-extension of Type II of a given 𝔾_a-torsor ρ:P→ S_* is isomorphic to one of the schemes q:X→ S_n constructed in<ref>.Let ρ:P→ S_* be a 𝔾_a-torsor and letP [r,hook,"j"] [d,"ρ"'] X [d,"π"] S_* [r,hook] Sbe a quasi-projective 𝔾_a-extension of P of Type II. Then there exists an integer n≥1 and a scheme τ_n:S_n(o_1,…,o_n-1)→ S such that X is a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1)≃ X/𝔾_a and ρ:P→ S_* coincides with the restriction of q to S_n(o_1,…,o_n-1)∖ E_n≃ S_*. Since the 𝔾_a,S-action on X is proper, the geometric quotient X/𝔾_a,S exists in the form of a separated algebraic S-space δ:X/𝔾_a,S→ S. Furthermore, since by definition of an extension π^-1(S_*)≃ P, we have π^-1(S_*)/𝔾_a,S≃ P/𝔾_a,S≃ S_* and so δ restricts to an isomorphism over S_*. On the other hand, π^-1(o)≃𝔸_κ^2 is equipped with the induced proper 𝔾_a,κ-action, whose geometric quotient 𝔸_κ^2/𝔾_a,κ is isomorphic to 𝔸_κ^1. It follows from the universal property of geometric quotient that δ^-1(o)=𝔸_κ^2/𝔾_a,κ=𝔸_κ^1. Since X is smooth in a neighborhood of π^-1(o), X/𝔾_a,S is smooth in neighborhood of δ^-1(o). In particular, π^-1(o) and δ^-1(o) are Cartier divisors on X and X/𝔾_a,S respectively. Let τ_1:S_1→ S be the blow-up of o. Then by the universal property of blow-ups <cit.>, the morphisms π:X→ S and δ:X/𝔾_a,S→ S lift to morphisms π_1:X→S_1 and δ_1:X/𝔾_a,S→S_1 respectively, and we have a commutative diagramX [r,"π_1"] [d]S_1 [d,"τ_1"]X/𝔾_a [r,"δ"] [ur,"δ_1"] S Furthermore, since δ:X/𝔾_a,S→ S and τ_1:S_1→ S are separated, it follows that δ_1:X/𝔾_a,S→S_1 is separated. By construction, the image of π^-1(o)_red/𝔾_a,κ by δ_1 is contained in E_1. If δ_1 is not constant on π^-1(o)_red/𝔾_a,κ then δ_1 is a separated quasi-finite birational morphism. Since S_1 is normal, δ_1 is thus an open immersion by virtue of Zariski Main Theorem for algebraic spaces <cit.>. Since π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1, the only possibility is that S_1∖δ_1(X/𝔾_a,S) consists of a unique κ-rational point o_1∈E_1 and δ_1:X/𝔾_a,S→ S_1(o_1)=S_1∖{o_1} is an isomorphism. So π_1:X→ S_1(o_1) is 𝔾_a-torsor whose restriction to S_1(o_1)∖ E_1≃ S_* coincides with ρ:P→ S_*. Otherwise, if δ_1 is constant on π^-1(o)_red/𝔾_a,κ, then its image consists of a unique κ-rational point o_1∈E_1. The same argument as above implies that π_1:X→S_1 and δ_1:X/𝔾_a,S→S_1 lift to a 𝔾_a,S-invariant morphism π_2:X→S_2(o_1) and a separated morphism δ_2:X/𝔾_a,S→S_2(o_1) to the blow-up τ_2,1:S_2(o_1)→S_1 of S_1 at o_1, with exceptional divisor E_2. If the restriction of δ_2 to π^-1(o)_red/𝔾_a,κ is not constant then δ_2 is an open immersion and the image of π^-1(o)_red/𝔾_a,κ is an open subset of E_2 isomorphic to 𝔸_κ^1. The only possibility is that δ_2(π^-1(o)/𝔾_a,κ)=E_2∖E_1. Indeed, otherwise S_2∖δ_2(X/𝔾_a,S) would consist of the disjoint union of a point in E_2∖(E_1∩E_2) and of the curve E_1∖(E_1∩E_2) which is not closed in S_2, in contradiction to the fact that δ_2 is an open immersion. Summing up, δ_2:X/𝔾_a,S→ S_2(o_1)=S_2(o_1)∖E_1 is an isomorphism mapping π^-1(o)_red/𝔾_a,κ isomorphically onto E_2. So π_2:X→ S_2(o_1) is 𝔾_a-torsor whose restriction to S_2(o_1)∖ E_2≃ S_* coincides with ρ:P→ S_*. Otherwise, if δ_2 is constant on π^-1(o)_red/𝔾_a,κ, then δ_2(π^-1(o)/𝔾_a,κ) is a κ-rational point o_2∈E_2, and there exists a unique minimal sequence of blow-ups τ_k+1,k:S_k+1(o_1,…,o_k)→S_k(o_1,…,o_k-1), k=2,…,m-1 of successive κ-rational points o_k∈E_k⊂S_k(o_1,…,o_k-1), with exceptional divisor E_k+1⊂S_k+1(o_1,…,o_k) such that π_2:X→S_2(o_1) and δ_2:X/𝔾_a,S→S_2(o_1) lift respectively to a 𝔾_a,S-invariant morphism π_m:X→S_m(o_1,…,o_m-1) and a separated morphism δ_m:X/𝔾_a,S→S_m(o_1,…,o_m-1) with the property that the restriction of δ_m to π^-1(o)_red/𝔾_a,κ is non constant. By Zariski Main Theorem <cit.> again, we conclude that δ_m is an open immersion, mapping π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1 isomorphically onto an open subset of E_m≃ℙ_κ^1. As in the previous case, the image of π^-1(o)_red/𝔾_a,κ in E_m must be equal to the complement of the intersection of E_m with the proper transform of E_1∪⋯∪E_m-1 in S_m(o_1,…,o_m-1) since otherwise S_m(o_1,…,o_m-1)∖δ_m(X/𝔾_a,S) would not be closed in S_m(o_1,…,o_m-1). Since π^-1(o)_red/𝔾_a,κ≃𝔸_κ^1, it follows that E_m intersects the proper transform of E_1∪⋯∪E_m-1 in a unique κ-rational point, implying in turn that o_m-1∈E_m-1 is a smooth κ-rational point of the reduced total transform E_1∪⋯∪E_m-1 of E_1 in S_m-1(o_1,…,o_m-2). Summing up, δ_m:X/𝔾_a,S→S_m(o_1,…,o_m-1)∖E_1∪⋯∪E_m-1is an isomorphism with an S-scheme of the form S_m(o_1,…,o_m-1) as constructed in <ref>, mapping π^-1(o)_red/𝔾_a,κ isomorphically onto E_m=S_m(o_1,…,o_m-1)∩E_m. It follows in turn that π_m:X→ S_m(o_1,…,o_m-1) is a 𝔾_a-torsor whose restriction to S_m(o_1,…,o_m-1)∖ E_m≃ S_* coincides with ρ:P→ S_*. This completes the proof. §.§ Affine 𝔾_a-extensions of Type II In this subsection, given a 𝔾_a-torsor ρ:P→ S_* we consider the existence of quasi-projective 𝔾_a-extensions of Type IIP [r,hook,"j"] [d,"ρ"'] X [d,"π"] S_* [r,hook] Swith the additional for which X is affine over S. As in the case of extension to 𝔸^1-bundles over the blow-up of o treated in<ref>, a necessary condition for the existence of such extensions is that the restriction of P over every open neighborhood of the closed point o in S is nontrivial. Indeed, if there exists an affine open neighborhood U of o over which P is trivial, then P≃ U∖{o}×𝔸_k^1 is strictly quasi-affine, hence cannot be the complement of a Cartier divisor π^-1(o) is any affine U-scheme X|_U. The next theorem shows that this condition is actually sufficient:Let ρ:P→ S_* be a 𝔾_a-torsor such that for every open neighborhood U of o in S, the restriction P×_S_*U→ U∖{o} is non trivial. Then for every n≥1 and every S-scheme τ_n:S_n(o_1,…,o_n-1)→ S as in Notation <ref> there exists a quasi-projective 𝔾_a-extension of P of Type II into the total space of a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1) for which π=τ_n∘ q:X→ S is an affine morphism.The following example illustrates the strategy of the proof given below, which consists in constructing such affine extensions π:X→ S by performing a well-chosen equivariant affine modification of extensions of ρ:P→ S_* into locally trivial 𝔸^1-bundles θ:W(P)→S̃ over the blow-up τ:S̃→ S of the point o.Let again X_0 and X_1 be the 𝔾_a-extensions of ρ:P={xv-yu=1}→ S∖{o} considered in Example <ref> and <ref>. Recall that X_0 and X_1 are the affine S-schemes in 𝔸_S^3 defined respectively by the equations X_0: xr-yq =0yp-x(q-1) =0pr-q(q-1) =0 and X_1: xw-y(yz_1+1) =0xz_2-z_1(yz_1+1) =0z_1w-yz_2=0equipped with the 𝔾_a,S-actions associated with the locally nilpotent 𝒪_S-derivations ∂_0=x^2∂_p+xy∂_q+y^2∂_r and ∂_1=x∂_z_1+(2yz_1+1)∂_z_2+y^2∂_w respectively. The morphism π_0:X_0→ S factors through the structure morphism θ:X_0→S̃ of a torsor under a line bundle on the blow-up τ:S̃→ S of the origin, with the property that the restriction of X_0 to exceptional divisor E=ℙ_κ^1 of τ is a nontrivial torsor under the total space of the line bundle 𝒪_ℙ_κ^1(-2). The 𝔾_a,S-action on X_0 restricts to the trivial one on X_0|_E=π_0^-1(o). More precisely, ∂_0 is a global section of the sheaf 𝒯_X_0⊗𝒪_X_0(-2X_0|_E) of vector fields on X_0 that vanish at order 2 along X_0|_E. One way to obtain from X_0 a 𝔾_a-extension π:X→ S of ρ:P→ S∖{o} with fiber π^-1(o)_red isomorphic to 𝔸_κ^2 and a fixed point free action is thus to perform an equivariant affine modification which simultaneously replaces X_0|_E by a copy of 𝔸_κ^2 and decreases the “fixed point order of ∂_0 along X_0|_E”, typically a modification with divisor D equal to X_0|_E and whose center Z⊂ X_0|_E is supported by a curve isomorphic to 𝔸_κ^1 which is mapped isomorphically onto its image by the restriction of θ. The birational S-morphismη:X_1→ X_0,(x,y,z_1,z_2,w)↦(x,y,xz_1,yz_1+1,w)is equivariant for the 𝔾_a,S-actions on X_0 and X_1 and corresponds to an equivariant affine modification of this type: it restricts to an isomorphism outside the fibers of π_0 and π_1 over o, and it contracts π_1^-1(o)=Spec(κ[z_2,w]) onto the curve {p=q-1=0}⊂π_0^-1(o)={pr-q(q-1)=0}. This curve is isomorphic to 𝔸_κ^1=Spec(κ[r]) and it is mapped by the restriction θ|_π_0^-1(o):π_0^-1(o)≃{pr-q(q-1)=0}→ E=ℙ_κ^1, (p,q,r)↦[p:q-1]=[q:r]of θ isomorphically onto the complement of the κ-rational point [0:1]∈ℙ_κ^1. By virtue of Theorem <ref>, there exists a unique integer ℓ_0≥2 such that ρ:P→ S_* is the restriction of a torsor θ_1:W_1→S_1 under the line bundle M_1(ℓ_0)=Spec(Sym^·𝒪_S_1(-ℓ_0E_1))→S_1 whose total space W_1 is affine over S_1. We now treat the case of S_1(o_1) and S_n(o_1,…,o_n-1), n≥2 separately.Given a κ-rational point o_1∈E_1, the restriction of W_1 over E_1=E_1∖{o_1}≃𝔸_κ^1 is the trivial 𝔸^1-bundle E_1×𝔸_κ^1. Since on the other hand the restriction θ_1|_E_1:W_1|_E_1→E_1 is a non trivial 𝒪_ℙ^1(-ℓ_0)-torsor (see Theorem <ref>), it follows that for every section s:E_1→ W_1|_E_1 the image Z_1 of E_1 in W_1|_E_1 is a closed curve isomorphic to E_1. Indeed, otherwise if Z_1 is not closed in W_1|_E_1 then its closure Z_1 would be a section of θ_1|_E_1 in contradiction with the fact that θ_1|_E_1:W_1|_E_1→E_1 is a non trivial 𝒪_ℙ^1(-ℓ_0)-torsor. Let D_1=θ_1^-1(E_1) and let σ_1:W_1'→ W_1 be the affine modification of W_1 with center (ℐ_Z_1,D_1) . By virtue of Lemmas <ref> and <ref>, θ_1∘σ_1:W_1'→S_1 factors through a torsor θ_1':W_1'→S_1∖{o_1}=S_1(o_1) under the line bundle M_1'(ℓ_0-1)=Spec(Sym^·𝒪_S_1(o_1)((-ℓ_0+1)E_1))→ S_1(o_1).Now since E_1≃𝔸_κ^1 is affine, the restriction of θ_1' over E_1⊂ S_1(o_1) is the trivial M_1'(ℓ_0-1)|_E_1-torsor. Letting D_2=θ_1'^-1(E_1) and Z_2⊂ D_2 be any section of θ'_1|_D_2:D_2→ E_1, the affine modification σ_2:W_2'→ W_1' with center (ℐ_Z_2,D_2) is then an M_1'(ℓ_0-2)-torsor θ'_2:W_2'→ S_1(o_1). Iterating this construction ℓ_0-1 times, we reach a 𝔾_a,S_1(o_1)-torsor q=θ_ℓ_0+1':X=W'_ℓ_0+1→ S_1(o_1). Since σ_1:W_1'→ W_1 and each σ_i:W_i'→ W_i-1', i≥2, restricts to an isomorphism over the complement of E_1, the restriction of q:X→ S_1(o_1) over S_1(o_1)∖ E_1≃ S_* is isomorphic to ρ:P→ S_*. Furthermore, since the morphisms σ_i, i=1,…,ℓ_0+1 are affine and τ_1∘θ_1:W_1→ S is an affine morphism, it follows that τ_1∘ q=τ_1∘θ_1∘σ_1∘⋯σ_ℓ_0+1:X→ Sis an affine morphism. So q:X→ S_1(o_1) is a 𝔾_a-extension of ρ:P→ S_* with the desired property. Now suppose that n≥2. It follows from the construction of the morphism τ_n,1:S_n=S_n(o_1,…,o_n-1)→S_1 given in subsection <ref> that τ_n,1^*𝒪_S_1(ℓ_0E_1)≃𝒪_S_n(mE_n) for some m≥2. The fiber product W_n=W_1×_S_1S_n is thus a torsor θ_n:W_n→ S_n under the line bundle M_n(m)=Spec(Sym^·𝒪_S_n(-mE_n))→ S_nwhose restriction to S_n∖ E_n≃ S_* is isomorphic to ρ:P→ S_*. Furthermore, since τ_n,1 is an affine morphism by virtue of Lemma <ref>, so is the projection pr_W_1:W_n→ W_1. Since τ_1∘θ_1:W_1→ S is an affine morphism, we conclude that τ_n∘θ_n=τ_1∘τ_n,1∘θ_n=τ_1∘θ∘pr_W_1:W_n→ S is an affine morphism as well. Since E_n≃𝔸_κ^1, the restriction of θ_n over E_n is the trivial M_n(m)|_E_n-torsor. The desired 𝔾_a,S_n-torsor q:X→ S_n extending ρ:P→ S_* is then obtained from θ_n:W_n→ S_n by performing a sequence of m successive affine modifications similar to those applied in the previous case. In the case where S is affine, the total spaces X of the varieties q:X→ S_n(o_1,…,o_n-1) of Theorem <ref> are all affine. To our knowledge, these are the first instances of smooth affine threefolds equipped with proper 𝔾_a-actions whose geometric quotients are smooth quasi-projective surfaces which are not quasi-affine.We do not know in general if under the conditions of Theorem <ref> every quasi-projective 𝔾_a-extensions of P of Type II into the total space of a 𝔾_a-torsor q:X→ S_n(o_1,…,o_n-1) has the property that π=τ_n∘ q:X→ S is an affine morphism. In particular, we ask the following:Is the total space X of a quasi-projective 𝔾_a-extension π:X→𝔸^2 of ρ=pr_x,y:SL_2={xv-yu=1}→𝔸_*^2 of Type II always an affine variety ?§.§ Examples In the next paragraphs, we construct two countable families of quasi-projective 𝔾_a-extensions of the 𝔾_a-torsor SL_2→SL_2/𝔾_a≃𝔸^2∖{(0,0)} of Type II with affine total spaces. As a consequence of <cit.>, for any nontrivial 𝔾_a-torsor ρ:P→ S_* over a local punctured surface S_*, these provide, by suitable base changes, families of examples of 𝔾_a-extensions of P whose total spaces are all affine over S. §.§.§A family of 𝔾_a-extensions of SL_2 of “Type II-A”Let S=𝔸^2=Spec(k[x,y_0]) and let X_n⊂𝔸_S^n+2=Spec(k[x,y_0][z_1,z_2,y_1,…,y_n]), n≥1, be the smooth threefold defined by the system of equationsy_iy_j-y_ky_ℓ=0 i,j,k,ℓ=0,…,n, i+j=k+ℓz_2y_i-z_1y_i+1=0 i=0,…,n-1xy_i+1-y_i(y_0z_1+1) =0 i=0,…,n-1xz_2-z_1(y_0z_1+1) =0.The threefold X_n can be endowed with a fixed point free 𝔾_a,S-action induced by the locally nilpotent k[x,y_0]-derivation x∂_z_1+(2y_0z_1+1)∂_z_2+∑_i=1^niy_0y_i-1∂_y_iof its coordinate ring. The scheme-theoretic fiber over o={(0,0)} of the 𝔾_a-invariant morphism π_n=pr_x,y,:X_n→ S is isomorphic 𝔸^2=Spec(k[z_2,y_n]), on which the induced 𝔾_a-action is a translation induced by the derivation ∂_z_2 of k[z_2,y_n]. On the other hand, the morphism j:SL_2={ xv-y_0u=1}→ X_n defined by (x,y,u,v)↦(x,u,uv,y,yv,yv^2,…,yv^n)is an equivariant open embedding of SL_2 equipped with the 𝔾_a-action induced by the locally nilpotent derivation x∂_u+y_0∂_v of its coordinate ring into X_n with image equal to π^-1(𝔸^2∖{o}). So j:SL_2↪ X_n is a quasi-projective 𝔾_a-extension of SL_2 into the affine variety X_n, with π_n^-1(o)≃𝔸_k^2. The restrictions of the projection 𝔸_S^n+3→𝔸_S^n+2 onto the first n+2 variables induce a sequence of 𝔾_a-equivariant birational morphisms σ_n+1,n:X_n+1→ X_n. The threefolds X_n thus form a countable tower of 𝔾_a-equivariant affine modifications of X_1. It follows from Example <ref> that X_1 is a quasi-projective extension of SL_2 of Type II with geometric quotient isomorphic to a quasi-projective surface of the form S_1(o_1). More generally, we have the following result. For every n≥2, the morphism j:SL_2↪ X_n is a quasi-projective 𝔾_a-extension of Type II. The geometric quotient X_n/𝔾_a is isomorphic to a quasi-projective surface S_n=S_n(o_1,…,o_n) as in<ref> for which S_n(o_1,…,o_n-1)∖ S_n consists of a chain of n-1 smooth rational curves with self-intersection -2, i.e. the exceptional set of the minimal resolution of a surface singularity of type A_n-1. To see this, we consider the following sequence of blow-ups: the first one τ_1:S_1→ U_0=𝔸^2 is the blow-up of the origin, with exceptional divisor E_1, and we let U_1≃𝔸^2=Spec(k[x,w_1]) be the affine chart of S_1 on which τ_1:S_1→𝔸^2 is given by (x,w_1)↦(x,xw_1). Then we let τ_2,1:S_2(o_1)→S_1 be the blow-up of the point o_1=(0,0)∈ U_1⊂S_1 with exceptional divisor E_2, and we let U_2≃𝔸^2=Spec(k[x,w_2]) be the affine chart of S_2(o_1) on which the restriction of τ_2,1:S_2(o_1)→S_1 coincides with the morphism U_2→ U_1, (x,w_2)↦(x,xw_2). For every 2<m≤ n, we define by induction the blow-up τ_m,m-1:S_m(o_1,…,o_m-1)→S_m-1(o_1,…,o_m-2)of the point o_m-1=(0,0)∈ U_m-1⊂S_m-1(o_1,…,o_m-2) with exceptional divisor E_m and we let U_m≃𝔸^2=Spec(k[x,w_m]) be the affine chart of S_m(o_1,…,o_m-1) on which the restriction of τ_m,m-1 coincides with the morphism U_m→ U_m-1, (x,w_m)↦(x,xw_m). By construction, we have a commutative diagram@C8exS_n(o_1,…,o_n-1) [r]^τ_n,n-1 S_n-1(o_1,…,o_n-2) [r]^-τ_n-1,n-2 ⋯ [r]^τ_2,1 S_1[r]^τ_1 𝔸^2 U_n [u] [r] U_n-1[u] [r]⋯ [r] U_1 [u] [r]𝔸^2=U_0. @=[u] The total transform of E_1 in S_n(o_1,…,o_n-1) is a chain E_1∪E_2∪⋯∪E_n-1∪E_n is a chain formed of n-1 curves with self-intersection -2 and the curve E_n which has self-intersection -1. The morphism π:X_n→ S lifts to a morphism π_1:X_n→S_1 defined by (x,z_1,z_2,y_0,y_1,…,y_n)↦((x,y_0),[x:y_0])=((x,y),[y_0z_1+1:y_1]).This morphism contracts π^-1(o) onto the point o_1=((0,0),[1:0]) of the exceptional divisor E_1 of τ_1. The induced rational map π_1:X_n U_1 is given by (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_1/y_0z_1+1)and it contracts π^-1(o) onto the origin o_1=(0,0). So π_1 lifts to a morphism π_2:X_n→S_2(o_1), and with our choice of charts, the induced rational map π_2:X_n U_2 is given by (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_2/(y_0z_1+1)^2)If n=2 then the image of π^-1(o)=Spec(k[z_2,y_2]) by π_2 is equal to E_2∩ U_2 and π_2^-1(E_2∩ U_2) is equivariantly isomorphic to (E_2∩ U_2)×Spec(k[z_2]) on which 𝔾_a acts by translations on the second factor. So π_2:X_n→S_2(o_1) factors through a 𝔾_a-bundle q_2:X_2→ S_2(o_1)=S_2(o_1)∖E_1 and X_2/𝔾_a≃ S_2(o_1). Otherwise, if n>2 then π_2 contracts π^-1(o) onto the point o_2=(0,0)∈E_2∩ U_2⊂S_2(o_1). So π_2:X_n→S_2(o_1) lifts to a morphism π_3:X_n→S_3(o_1,o_2). With our choice of charts, for each 2<m<n, the induced rational map π_m:X_n U_m is given by (x,z_1,z_2,y_0,y_1,…,y_n)↦(x,y_m/(y_0z_1+1)^m)hence contracts π^-1(o) onto the point o_m=(0,0)∈ U_m⊂S_m(o_1,…,o_m-1). It thus lifts to a morphism π_m:X_n→S_m(o_1,…,o_m-1). At the last step, the image of π^-1(o)=Spec(k[z_2,y_n]) by the rational map π_n:X_n U_n induced by π_n:X_n→S_n(o_1,…,o_n-1) is equal to E_n∩ U_n, and we conclude as above that π_n:X_n→S_n(o_1,…,o_n-1) factors through a 𝔾_a-bundle q_n:X_n→ S_n(o_1,…,o_n-1)=S_n(o_1,…,o_n-1)∖(E_1∪⋯∪E_n-1),hence that X_n/𝔾_a is isomorphic to the quasi-projective surface S_n(o_1,…,o_n-1).§.§.§ A family of 𝔾_a-extensions of SL_2 of “Type II-D” To conclude this section, we present as an illustration of the proof of Theorem <ref> another countable family of quasi-projective 𝔾_a-extensions of SL_2 of Type II withaffine total spaces. Let again τ_1:S_1→ S=𝔸^2 be the blow-up of the origin o={(0,0)} in 𝔸^2=Spec(k[x,y]) with exceptional divisor E_1≃ℙ^1, identified with closed subvariety of 𝔸^2×ℙ_[w_0:w_1]^1 with equation xw_1-yw_0=0 in such a way that τ coincides with the restriction of the first projection. The second projection identifies S_1 with the total space p:S_1→ℙ^1 of the invertible sheaf 𝒪_ℙ^1(-1). We fix trivializations p^-1(U_∞)=Spec(k[z_∞][u_∞]) and p^-1(U_0)=Spec(k[z_0][u_0]) over the open subsets U_∞=ℙ^1∖{[0:1]}=Spec(k[z_∞]) and U_0=ℙ^1∖{[1:0]}=Spec(k[z_0]) in such a way that the gluing of p^-1(U_∞) and p^-1(U_0) over U_0∩ U_∞ is given by the isomorphism (z_0,u_0)↦(z_∞,u_∞)=(z_0^-1,z_0u_0).For every n≥1, we let S_2n+3,0=Spec(k[z_0,u_0^±1]),S_2n+3,∞=Spec(k[z_∞,u_∞,v_∞]/(u_∞^nv_∞-z_∞^2-u_∞)),and we let S_2n+3 be the surface obtained by gluing S_2n+3,0 and S_2n+3,∞ along the open subsets S_2n+3,0∖{z_0=0} and S_2n+3,∞∖{z_∞=u_∞=0} by the isomorphism (z_0,u_0)↦(z_∞,u_∞,v_∞)=(z_0^-1,z_0u_0,(z_0u_0)^-n(z_0^-2+z_0u_0)).The canonical open immersion S_2n+3,0↪ p^-1(U_0) and the projection pr_z_∞,u_∞:S_2n+3,∞→ p^-1(U_∞) glue to a global birational affine morphism τ_2n+3,1:S_2n+3→S_1 restricting to an isomorphism S_2n+3∖{z_∞=u_∞=0}→S_1∖E_1 where we identified the closed subset E_2n+3={z_∞=u_∞=0}≃Spec(k[v_∞]) of S_2n+3,∞ with its image in S_2n+3. We leave to the reader to check that with the notation of<ref>, S_2n+3=S_2n+3(o_1,…,o_2n+2) for a surface τ_2n+3,1:S_2n+3,1(o_1,…,o_2n+2)→S_1 obtained by first blowing-up the point o_1=(0,0)∈ p^-1(U_∞) with exceptional divisor E_2, then the point o_2=E_1∩E_2 with exceptional divisor E_3, then a point o_3∈E_3∖(E_1∪E_2) with exceptional divisor E_4 and then a sequence of point o_i∈E_i∖E_i-1 with exceptional divisor E_i+1, i=5,…,2n+2 in such a way that the total transform of E_1 in S_2n+3,1 is a tree depicted in Figure <ref>. Letting τ_2n+3=τ_1∘τ_2n+3,1:S_2n+3→𝔸^2, we have τ_2n+3^-1(o)_red=E_2n+3≃𝔸^1 and τ_2n+3^*(o)=2E_2n+3. Now we let q:X_2n+3→ S_2n+3 be the 𝔾_a-bundle defined as the gluing of the trivial 𝔾_a-bundles X_2n+3,0=S_2n+3,0×Spec(k[t_0]) and X_2n+3,∞=S_2n+3,∞×Spec(k[t_∞]) over S_2n+3,0 and S_2n+3,∞ respectively along the open subsets X_2n+3,0∖{z_0=0} and X_2n+3,∞∖{z_∞=u_∞=0} by the 𝔾_a-equivariant isomorphism (z_0,u_0,t_0)↦(z_∞,u_∞,v_∞,t_∞)=(z_0^-1,z_0u_0,(z_0u_0)^-n(z_0^-2+z_0u_0),t_0+z_0^-1u_0^-2).Let π_2n+3=τ_1∘τ_2n+3,1∘ q:X_2n+3→𝔸^2. For every n≥1, the variety X_2n+3 is affine and there exists a 𝔾_a-equivariant open embedding j:SL_2↪ X_2n+3 which makes π_2n+3:X_2n+3→𝔸^2 a quasi-projective 𝔾_a-extension of SL_2 of Type II, with fiber π_2n+3^-1(o) isomorphic to 𝔸^2 of multiplicity two, and geometric quotient X_2n+3/𝔾_a≃ S_2n+3.Let j_1:SL_2↪ W=W(SL_2,2) be the 𝔾_a-extension of SL_2 into a locally trivial 𝔸^1-bundle θ:W→S_1 with affine total space constructed in Example <ref>. Recall that the image of j_1 coincides with the restriction of θ to S_1∖E_1=𝔸^2∖{o}. With our choice of coordinates, the open subsets W_0=θ^-1(q^-1(U_0)) and W_∞=θ^-1(q^-1(U_∞)) of W are respectively isomorphic to p^-1(U_0)×Spec(k[w_0]) and p^-1(U_∞)×Spec(k[w_∞]) glued over U_0∩ U_∞ by the isomorphism (z_0,u_0,w_0)↦(z_∞,u_∞,w_∞)=(z_0^-1,z_0u_0,z_0^2w_0+z_0).The 𝔾_a-action on W_0 and W_∞ are given respectively by α·(z_0,u_0,w_0)=(z_0,u_0,w_0+α u_0^2) and α·(z_∞,u_∞,w_∞)=(z_∞,u_∞,w_∞+α u_∞^2). Let W'=W×_S_1S_2n+3, equipped with the natural lift of the 𝔾_a-action on W. Since τ_2n+3,1:S_2n+3→S_1 restricts to an isomorphism over S_1∖E_1, the composition j'=τ_2n+3,1^-1∘ j_1:SL_2→ W' is a 𝔾_a-equivariant open embedding. Furthermore, since W is affine and τ_2n+3,1 is an affine morphism, it follows that W' is affine. By construction, W' is covered by the two open subsets W'_0=W×_p^-1(U_0)S_2n+3,0 ≃ S_2n+3,0×Spec(k[w_0])W'_∞=W×_p^-1(U_∞)S_2n+3,∞ ≃ S_2n+3,∞×Spec(k[w_∞]).The local 𝔾_a-equivariant morphisms β_0:X_2n+3,0=S_2n+3,0×Spec(k[t_0])→ W_0' β_∞:X_2n+3,∞=S_2n+3,∞×Spec(k[t_∞])→ W_∞' of schemes over S_2n+1,0 and S_2n+3,∞ respectively defined by t_0↦ w_0=u_0^2t_0 and t_∞↦ w_∞=u_∞^2t_∞ glue to a global 𝔾_a-equivariant birational affine morphism β:X_2n+3→ W', restricting to an isomorphism over S_2n+3∖ E_2n+3≃𝔸^2∖{o}. Summing up, X_2n+3 is affine over W' hence affine, and the composition β^-1∘ j':SL_2↪ X_2n+3 is a 𝔾_a-equivariant open embedding which realizes π:X_2n+3→𝔸^2 as a 𝔾_a-extension of SL_2 of Type II with affine total space. By construction, π_2n+3^-1(o)=q^-1(2E_2n+3) is isomorphic to 𝔸^2, with multiplicity two, while the geometric quotient X_2n+3/𝔾_a is isomorphic to S_2n+3.For every n≥1, the birational morphism S_2(n+1)+3,∞→ S_2n+3,∞, (z_∞,u_∞,v_∞)↦(z_∞,u_∞,u_∞v_∞) extends to a birational morphism S_2(n+1)+3→ S_2n+3 which lifts in turn in a unique way to a 𝔾_a-equivariant birational morphism γ_n+1,n:X_2(n+1)+3→ X_2n+3. So in a similar way as for the family constructed in<ref>, the family of threefolds X_2n+3, n≥1, form a tower of 𝔾_a-equivariant affine modifications of the initial one X_5.amsplain 99 DuPhD04 A. Dubouloz, Sur une classe de schémas avec actions de fibrés en droites, Ph.D Thesis, Université Joseph-Fourier-Grenoble I, <https://tel.archives-ouvertes.fr/tel-00007733/>, 2004.DuTG05 A. Dubouloz Danielewski-Fieseler surfaces, Transformation Groups vol. 10, no. 2, (2005), 139-162.Du05 A. Dubouloz, Quelques remarques sur la notion de modification affine, math.AG/0503142, (2005).DF14 A. Dubouloz and D. R. Finston,On exotic affine 3-spheres, J. Algebraic Geom. 23 (2014), no. 3, 445-469.Du15 A. Dubouloz, Complements of hyperplane sub-bundles in projective spaces bundles over ℙ^1, Math. Ann. 361 (2015), no 1-2, 259-273.EGA1 A. Grothendieck, Éléments de Géométrie Algébrique, I, Publ.Math. IHES, 4, 1960.EGA3 A. Grothendieck, Éléments de Géométrie Algébrique, III, Publ.Math. IHES, 11 and 17, 1961 and 1963.Fie94 K-H. Fieseler, On complex affine surfaces with ℂ_+-action, Comment. Math. Helv. 69 (1994), no. 1, 5-27.GMM12 R.V. Gurjar, K. Masuda and M. Miyanishi, 𝔸^1-fibrations on affine threefolds,Journal of Pure and Applied Algebra Volume 216, Issue 2 (2012), 296-313.He15 I. Hedén, Affine extensions of principal additive bundles over a punctured surface, Transform. Groups 21 (2),(2016), 427-449.KaZa99 S. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4 (1999), no. 1, 53-95.Ke80 G. Kempf, Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves, Rocky Mountain J. Maths. Volume 10, Number 3, 1980, 637-646.Na59 M. Nagata, Lectures on the Fourteenth Problem of Hilbert, Lecture Notes, Tata Institute, Bombay, 31, (1959). StackP The Stacks Project Authors, Stacks Project, <http://stacks.math.columbia.edu>, 2017. | http://arxiv.org/abs/1707.08768v1 | {
"authors": [
"Adrien Dubouloz",
"Isac Hedén",
"Takashi Kishimoto"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170727080154",
"title": "Equivariant extensions of Ga-torsors over punctured surfaces"
} |
firstpage–lastpage [ Jorge Drumond Silva December 30, 2023 =======================We present 1.3- and/or 3-mm continuum images and 3-mm spectral scans, obtained using NOEMA and ALMA, of 21 distant, dusty, star-forming galaxies (DSFGs). Our sample is a subset of the galaxies selected by <cit.> on the basis of their extremely red far-infrared (far-IR) colours and low Herschel flux densities; most are thus expected to be unlensed, extraordinarily luminous starbursts at z 4, modulo the considerable cross-section to gravitational lensing implied by their redshift. We observed 17 of these galaxies with NOEMA and four with ALMA, scanning through the 3-mm atmospheric window.We have obtained secure redshifts for seven galaxies via detection of multiple CO lines, one of them a lensed system at z=6.027 (two others are also found to be lensed); a single emission line was detected in another four galaxies, one of which has been shown elsewhere to lie at z=4.002.Where we find no spectroscopic redshifts, the galaxies are generally less luminous by 0.3–0.4 dex, which goes some way to explaining our failure to detect line emission. We show that this sample contains amongst the most luminous known star-forming galaxies.Due to their extreme star-formation activity, these galaxies will consume their molecular gas in 100 Myr, despite their high molecular gas masses, and are therefore plausible progenitors of the massive, `red-and-dead' elliptical galaxies at z ≈ 3.galaxies: high-redshift – galaxies: starburst – galaxies: ISM– ISM: molecules§ INTRODUCTION It has been known since the 1970s and 1980s that a large fraction of the energy produced by vigorously star-forming galaxies in the nearby Universe is radiated by cool dust which mingles with their reservoirs of molecular gas <cit.>.A decade on, the existence of a more distant population of dusty galaxies was inferred by <cit.> from the detection of the cosmic far-IR background using FIRAS aboard the Cosmic Background Explorer, individual examples of which were quickly detected by <cit.> in the submillimeter (submm) waveband.If their initial stellar mass function (IMF) is normal, these galaxies form stars at tremendous rates, sometimes >1000M_⊙yr^-1 <cit.>. Deeper submm observations in cosmological deep fields <cit.> confirmed the abundance of these so-called submm galaxies (SMGs), sometimes known now as dusty, star-forming galaxies <cit.>.In the decades since then, the SPIRE camera <cit.> aboard Herschel <cit.> and the SCUBA-2 camera <cit.> on the James Clerk Maxwell Telescope (JCMT) have together detected orders of magnitude more of these DSFGs. Conventional optical and near-IR spectroscopic observations confirmed that DSFGs are considerably more abundant (≈ 1,000×) at high redshift than in the local Universe, with a redshift distribution for those selected at 850 μm that peaks at z∼1–3 <cit.>.Those selected at >1 mm by the South Pole Telescope <cit.> are more distant while those selected at the far-IR wavelengths imaged by Herschel are typically at z<2.In the local Universe, massive early-type galaxies have old stellar populations, >2 Gyr, and are therefore red in optical color – so-called `red-and-dead' galaxies.They have little gas or dust, and star-formation activity has ceased <cit.>. The majority of these galaxies experienced an intense phase of star formation around 5–10 Gyr ago <cit.>, and current observational evidence suggests that DSFGs at z≈ 2 are their likely progenitors.It is also well established that there exists a population of massive elliptical galaxies at z∼ 2–3.It has been claimed that most of these are high-redshift analogs of local, massive red-and-dead galaxies <cit.>.The existence of these galaxies at z∼2–3 suggests intense star-formation episodes must occur at even higher redshifts, perhaps implying that DSFGs are common at z 4 <cit.>.Only a small number of DSFGs were known at z 4 until recently, most of them gravitationally lensed <cit.>.To address this issue, <cit.> recently exploited the widest available far-IR imaging survey, H-ATLAS <cit.>, to create a sample of the faintest, reddest dusty galaxies, further improving their photometric redshifts via ground-based photometry from SCUBA-2 <cit.> and LABOCA <cit.>.The galaxies thus selected are expected to be largely unlensed[Despite expecting a low lensing fraction,and others have shown that strongly lensed galaxies are common at z>4 due to the increase with redshift of the optical depth to lensing and the magnification bias; <cit.> present high-resolution ALMA imaging of this sample, showing that the fraction of lensed galaxies is indeed relatively high.], luminous and very distant.Their vigorous star-formation activity thus tallies with the star-formation history required to build up the large mass of stars found in spheroidal galaxies at z≈ 2.To confirm that the ultrared DSFGs selected bydo lie at z 4, which will strengthen their links with red-and-dead galaxies at z∼2-3, requires robust spectroscopic confirmation of their photometric redshifts.This is non-trivial when working in the traditional optical and near-IR regime, verging on impossible with current telescopes and instrumentation.Following the success of <cit.>, who scanned the 3-mm atmospheric window to determine the redshift of one of the brightest, reddest, lensed galaxies to emerge from H-ATLAS <cit.>, we have therefore obtained 3-mm spectral scans of 21 ultrared DSFGs from thesample, as well as interferometric 0.85- and 1.3-mm imaging to better pinpoint their positions.Our primary objective here is to determine robust spectroscopic redshifts for these DSFGs, via the detection of multiple molecular and/or atomic emission lines.Using these to fine-tune the far-IR/submm photometric techniques employed bythen allows us to more reliably determine the space density of DSFGs at z 4.In addition, we use our improved measurements of IR luminosity and our CO line luminosities to estimate physical properties, such as SFR and molecular gas mass.Finally, we compare these derived properties with those of other DSFGs at low and high redshifts, subject as usual to the considerable uncertainties imposed by α_ CO and the assumed IMF.Where applicable, we assume a flat Universe with (Ω_m,Ω_Λ,h_0) = (0.3,0.7,0.7).In this cosmology, an arcsecond corresponds to 7.1 kpc at z=4.§ SAMPLE SELECTION Our targets – see Table <ref> – were chosen from the faint, `ultrared' galaxy sample of <cit.>, taking those best suited to the latitudes of the telescopes we employ, with photometric redshifts consistent with z4. Here, we briefly summarize the selection method used, referring readers to <cit.> for more details.The sample was selected from the SPIRE images used to construct H-ATLAS Data Release 1 <cit.>, employing an optimal extraction kernel to minimize the effects of source confusion, which is especially pernicious at 500 μm. The reddest galaxies were isolated based on their SPIRE colors, such that S_500/S_250≥1.5 and S_500/S_350≥0.85, where S_250 is the flux density measured at 250 μm (see Fig. <ref>). The galaxies thus selected have a median S_500∼ 50 mJy, such that the majority are not expected to be lensed gravitationally <cit.>.The reddest of these SPIRE-selected galaxies were then imaged with SCUBA-2 <cit.> on the 15-m JCMT and/or with LABOCA <cit.> on the 12-m Atacama Pathfinder Telescope (APEX) so that better photometric redshifts could be determined. These data are also utilized here, in <ref>, to aid us in spatially localizing any line emission.Of the 109 objects thus targeted by , 17 galaxies were selected for further observations with the Institute Radioastronomie Millimetrique's (IRAM's) Northern Extended Millimeter Array (NOEMA) and four galaxies for further observations with the Atacama Large Millimeter Array (ALMA), based on their accessibility to those telescopes and their high photometric redshifts. The SPIRE flux densities and photometric redshifts determined by <cit.> are listed in Table <ref>.§ OBSERVATIONS §.§ NOEMA 3-mm spectral scans Our observations with NOEMA[http://iram-institute.org/EN/noema-project.php] were conducted as two programs (Program IDs: W05A, X0C6; Co-PIs: R. J. Ivison, M. Krips). Table <ref> lists the galaxies observed.Both projects acquired data using five or six antennas in NOEMA's most compact (D) configuration. W05A was carried out between 2012 June and 2013 April, and 14 targets were observed. X0C6 took place between 2013 November and 2014 June, where four targets were observed. One target, G09-83808, was observed during both periods.We employed multiple receiver tunings together with the WideX correlator – which provides 3.6 GHz of instantaneous dual-polarization bandwidth – to cover the 80–101.6-GHz part of the 3-mm atmospheric window, in which we expect to find at least one ^12CO transition for galaxies at z>3.6 – see, for example, Figure 2 of <cit.>, where for 3.6<z<7.5 we always expect ^12CO(4–3), ^12CO(5–4) and/or ^12CO(6–5) in our frequency search range, with other lines such as C i(1–0) and H_2O(211–202) also present for some redshifts.Different approaches were used during the two projects to maximize the probability of detecting multiple emission lines from each target, necessary to yield an unambiguous redshift <cit.>. In W05A, once a single emission line was detected during an initial sweep of the 3-mm atmospheric window, the remaining 3-mm tunings were skipped and we instead tuned to a higher frequency, outside the 3-mm band, to search for a higher CO transition, having used the initial line and/or continuum detection to quantize the possibilities as well as to improve the photometric redshift estimate. In X0C6, spectra covering the 3-mm atmospheric window were obtained for all targets (∼21 GHz in total), then targets with emission lines were observed again to search for emission lines at higher frequencies, in the 2-mm band.Some of the targets therefore have less than 21 GHz of coverage (e.g. NGP-190387, NGP-126191 and G09-59393); others have coverage larger than 21 GHz (NGP-284357, NGP-246114 and G09-81106). Average on-source time per tuning was 20 min for W05A, and 120 min for X0C6, of which 15 and 90 min remained after flagging, respectively.Calibration of the data was carried out by using the gildas package[http://www.iram.fr/IRAMFR/GILDAS]. The typical resulting r.m.s. noise levels were 2.7 and 1.1 mJy beam^-1 in 100- channels for data taken in W05A and X0C6. Calibrated visibilities were converted into FITS format for export, then into MS format to be imaged by casa <cit.>. The average synthesized beam size was 5fwhm during both runs, with considerable diversity in beam shape due to the relatively short tracks.To study any 3-mm continuum emission from our targets, we integrated our data over all observed frequencies, imaging with the clean task in casa, with a map size of 1^'× 1^', sufficient to cover the 3-mm primary beam. §.§ ALMA 3-mm spectral scans Four ultrared galaxies were observed using ALMA (see Table <ref>), with five separate tunings to cover the 3-mm window (Program ID: 2013.1.00499.S; PI: A. Conley). Data were acquired during 2014 July 02–03 and August 28, with typically 8.6–9.7 min spent on-source for each tuning, in addition to 20 min of calibration – pointing, phase, flux density (Neptune) and bandpass.Data were calibrated using the ALMA pipeline, with only minor flagging required. Calibrated data were imaged using clean within casa, using the natural weighting scheme to maximize sensitivity.The resulting r.m.s. noise levels ranged between 0.73 and 0.80 mJy beam^-1 in channels binned to 100 . Because the observations were carried out on several different dates, with different antenna configurations, at frequencies ranging from 84 to 115 GHz, the resulting synthesized beamsizes varied between 0.6 and 1.2” fwhm.As with our NOEMA data, 3-mm continuum images were created using all the available data, with a map size of 1^'× 1^'. §.§ NOEMA 1.3-mm continuum observationsWe have also carried out 1.3-mm observations of ten galaxies lacking continuum detections, and hence accurate positions, in our earlier 3-mm work. Table <ref> lists those targets observed during 2015 December (Program ID: W15ET; PI: M. Krips), again using the most compact NOEMA configuration, with six antennas. The typical resulting synthesized beam size was ∼ 1.5” fwhm. Calibration was accomplished following the standard procedures, using gildas, with little need for significant flagging. The average time spent on-source was 25 min, yielding typical noise level of 0.47 mJy beam^-1.We also use data from an earlier programme which observed another five of our targets – G09-81106, G09-83808, NGP-101333, NGP-126191 and NGP-246114 – taken during 2013 in the compact 6C configuration, with a typical resulting synthesized beam size of 1.0”× 1.3” fwhm, the major axis at a position angle of 25^∘ (Program ID: W0BD; Co-PIs: F. Bertoldi, I. Perez-Fournon).§ RESULTSIf detecting faint line emission from distant galaxies is challenging, doing so in the absence of an accurate position is considerably more so.For this reason, our first step is to explore the 3-mm continuum images described in <ref>, hoping that thermal dust emission from our luminous, dusty starbursts will betray the precise position of our targets. §.§ Continuum emissionTo determine the significance of any continuum emission, we measured the r.m.s. noise level of the maps, and then created the signal-to-noise ratio (SNR) images shown in Fig. <ref>. All four sources observed at 3 mm with ALMA are clearly detected in continuum, at >8σ significance.For the objects observed at 3 mm with NOEMA, the sensitivity is much reduced compared to ALMA, so we begin by overlaying the 3-mm continuum images with contours from the deep SCUBA-2 850-μm imaging of <cit.>, where the unsmoothed fwhm of the SCUBA-2 images is around 13”, and the r.m.s. pointing accuracy of the JCMT for a single visit to a target is ∼ 2–3”.We then searched for faint 3-mm continuum sources coincident with SCUBA-2 850-μm emission, finding eight plausible sites.We discount the faint 3-mm emission seen towards G09-59393, favoring the 1.3-mm position a few arcsec to the east, which is considerably more significant.The most dubious of the others is NGP-113609, although the close proximity of the 3-mm peak to the SCUBA-2 850-μm emission lends extra confidence.NGP-126191 displays ≥4-σ emission; again, the near-coincidence with 850-μm and/or 1.3-mm emission gives additional confidence.For the five remaining sources, 3-mm continuum emission was detected at >5σ.Of the targets observed in continuum at 1.3 mm using NOEMA, we were able to measure positions and flux densities for 13 of 15.The flux densities and coordinates of all these continuum detections are quoted in Tables <ref> and <ref>, respectively, corrected for primary beam attenuation, including the small number of tentative examples (which are marked as such).The contribution from emission lines to the continuum flux density is negligible, as we shall see in what follows.It is worth noting here that none of the ultrared galaxies observed in 1.3- or 3-mm continuum are revealed as doubles, as would be expected in the simulations of <cit.>, though it remains possible that some or all of the ∼ 20% of targets that remain undetected in continuum have been pushed below our interfeometric detection threshold by multiplicity. §.§ Searching for emission lines To determine reliable, unambiguous redshifts for a DSFG, we must detect two or more emission lines.Ideally we must extract their spectra at known positions, typically betrayed by interferometric continuum detection in the cases of DSFGs, thereby maximising the significance of any line detections. If we extract spectra blindly, we must correct our statistics for the number of independent sightlines explored.Here, our known positions come from the 1.3- and 3-mm continuum imaging with NOEMA and ALMA, as described in <ref>; for the 17 sources with reliable coordinates (Table <ref>), we extracted spectra at the precise positions of the corresponding continuum detections. In the three cases where we have no continuum detection at either 1.3 or 3 mm, tagged as such in Table <ref>, we searched blindly for emission lines in data cubes that had not been corrected for the primary beam response.We convolved these data cubes along their frequency axis with box-car kernels of width 3, 4 and 7 channels, corresponding to velocity widths of ≈ 200–500 km s^-1, typical for DSFG emission lines <cit.>.For each convolved cube we created a SNR cube, then searched for peaks above 5σ, where the significance of detections at this stage have not been corrected for the number of independent sightlines we have explored.We also performed the same blind line-search procedure on continuum-detected sources to look for any additional line emission. Only known lines were recovered.As a result of these emission-line searches, we detected multiple (two or more) emission lines from seven of our targets, one of these following the detection of three lines by <cit.>, as well as single emission lines from four targets, where more lines have been detected subsequently in one case <cit.>.We thus report the first eight robust, accurate, unambiguous redshifts for faint, largely unlensed and thus intrinsically very luminous starbursts.For all the detected emission lines, we have fitted single-component Gaussians, measuring the frequency of the line center, and its fwhm.Continuum emission was subtracted with the UVCONTSUB task in casa, using all available channels except those close to emission lines.The flux of each emission line has been measured with the casa IMFIT task, from the zeroth moment map (created by integrating along the frequency axis across the emission line).There are no significant discrepencies between these values and those found from the Gaussian fits.The measured properties of the emission lines are summarized in Table <ref>. §.§ Unambiguous redshifts via detection of multiple emission lines We detect multiple emission lines towards seven of our targets, such that the redshifts of these sources and species/transitions of the emission lines are confirmed unambiguously.The properties of those sources are discussed in <ref>.For NGP-190387, two emission lines are detected at 85.10 and at 106.23 GHz (see Fig. <ref>), CO(4–3) and CO(5–4) at z=4.420 (z=4.418 and 4.425, respectively, for the two lines). NGP-190387 lies close to a group of three faint (K_ AB≈ 21–22) galaxies, likely at z1, revealed by NIRI on the 8-m Gemini North telescope (Fig. <ref>), which amplify the DSFG gravitationally by a factor we cannot constrain meaningfully at the present time.Towards G09-81106 we have detected two emission lines, CO(4–3) and CO(5–4), at 83.36 and 104.19 GHz (Fig. <ref>), both at z=4.531.There is no suggestion of gravitational lensing for G09-81106, either via the presence of unusually bright near-IR galaxies in the field, or via its submm morphology as seen in high-resolution ALMA continuum imaging <cit.>.Towards G09-83808 we have detected two faint emission lines, at 82.02 and 98.39 GHz (Fig. <ref>), corresponding to CO(5–4) and CO(6–5) at z=6.026±0.001 and 6.028±0.001, respectively, so an average of 6.027±0.001.These lines were also noted by <cit.> in a spectrum obtained using the Large Millimeter Telescope.G09-83808 is near-coincident with a foreground galaxy, seen clearly in near-IR imaging from the VIKING survey <cit.>, indicative of gravitationally lensing. This foreground galaxy has a spectroscopic redshift of 0.778, obtained using X-shooter on the 8-m Very Large Telescope (see Fig. <ref>).A lens model based on the morphology determined by high-resolution ALMA continuum imaging predicts a gravitional amplification of 8.2± 0.3 <cit.>.For NGP-284357, we find at least two emission lines, at 97.75 and 136.9 GHz (Fig. <ref>). If these are CO(5–4) and CO(7–6), the redshifts are 4.895 and 4.892, respectively, so an average of z=4.894.At this redshift, the fine-structure lines of neutral carbon are expected at 83.54 and 137.41 GHz, respectively, and we see strong hints of corresponding emission – a discrete feature where C i(1–0) is expected, and C i(2–1) appears to be broadening the CO(7–6) line.In the case of NGP-246114, two emission lines are detected at 95.08 and 142.71 GHz (Fig. <ref>), which must be CO(4–3) and CO(6–5) at z=3.849 and 3.845, respectively, so an average of z=3.847.SGP-196076 has been studied in detail by <cit.>, who referred to the galaxy as SGP-38326, an H-ATLAS nomenclature pre-dating <cit.>; for further details we refer the readers to that paper.Summarising the main results obtained from our 3-mm spectral scans of SGP-196076: we have detected the CO(5–4) and CO(4–3) transitions at 84.97 and 106.19 GHz, so at z=4.425 (Fig. <ref>). C i is also seen, at low significance. Both the continuum emission, from dust, and the CO(5–4) line emission, indicate clearly that SGP-196076 comprises multiple (≥ 3) components, with the star formation in each one presumably triggered by their close proximity – a ongoing merger or strong interaction.<cit.> explored the velocity field of the two largest components, via their CO and [C ii] emission, finding ordered disk-like rotation.SGP-261206 displays emission lines at 87.95 and 109.01 GHz (Fig. <ref>), CO(4–3) and CO(5–4) at z=4.242± 0.001, around 1.7σ below its photometric redshift. C i is also seen, at low significance. Very dust-obscured, distant galaxies should not be coincident with near-IR sources at the depth of our available imaging, unless those near-IR sources are gravitationally lensing the dusty galaxy.However, the K-band image[Gravitational lensing is found likely for three of the galaxies in this sample, as revealed by K-band imaging – see Fig. <ref>; the rest are devoid of close near-IR counterparts, though the depth of the available near-IR imaging does not exclude the possibility of distant (z 1) lenses.] of SGP-261206 shown in Fig. <ref>, from VIKING <cit.>, contains a clear K-band counterpart, coincident with the dust emission. This suggests that SGP-261206 is gravitationally lensed by the foreground galaxy detected in the near-IR image, a hint confirmed by high-resolution ALMA imaging <cit.>.§.§.§ CO line ratios Our spectra allow us to determine line luminosity ratios for those galaxies for which multiple lines were detected, typically anchored to ^12CO J=5–4.In Table <ref> we list the CO line luminosity ratios (i.e.L^'_CO(i - i-1)/L^'_CO(j - j-1)) which we find are consistent with the average values found for SMGs by <cit.>.§.§.§ Rest-frame stacking For the eight spectra for which we have accurate, unambiguous redshifts, we can shift the data to the corresponding rest-frame frequencies and stack them to search for features fainter than the relatively bright ^12CO lines, following <cit.> and <cit.>.The resulting stacked spectrum is shown in Fig. <ref> where we find the expected ^12CO ladder between J=4–3 and J=7–6, the latter broadened by C i(2–1), as well as weak C i(1–0) line emission.Absorption due to the collisionally excited H_2O 1_1,0–1_0,1 ground transition[Due to its very high critical density, this line is very difficult to excite in emission, but it can be seen relatively easily in absorption, where there is strong background continuum.In the cold ISM, water is normally frozen out, forming icy mantles on dust grains; detecting this transition in absorption suggests water is gaseous, perhaps because of turbulence or shock heating.] may be seen, at low (≈2.5σ) significance.§.§.§ Detection of single emission linesTowards four of our galaxies, single emission lines were detected, insufficient to determine the redshift of the source unambiguously, as the species and/or transition of the emission line is unknown. However, combining the redshift constraint available by virtue of far-IR/submm color, often only a handful of strong emission lines become plausible candidates.Towards NGP-126191 we detected a clear emission line at 85.77 GHz (Fig. <ref>), with a fwhm of 570± 180 km s^-1. As outlined earlier, this line emission is ≈ 3” from weak 3-mm continuum emission, which may be spurious, or may be from a companion, or the dust emission may be slightly displaced from the line emission – a relatively common finding amongst DSFGs <cit.>.With a far-IR/submm photometric redshift of 4.9, the most likely identification for this emission line is ^12CO(4–3) at z=4.38; however, ^12CO(5-4) would then be expected at 107.1 GHz, with a similar significance given the typical spectral-line energy distributions of DSFGs, and such a line is not detected (Fig. <ref>).^12CO(3–2) and ^12CO(5–4) are the other most likely possibilities, at z=3.03 and z=5.71.Towards NGP-111912 we detected a weak emission line at 95.15 GHz (Fig. <ref>), with a fwhm of 440± 200 km s^-1. The line emission is coincident spatially with 1.3-mm continuum emission (Fig. <ref>).With a photometric redshift estimate of 3.28^+0.36_-0.26, the emission line may be ^12CO(4–3) at z=3.84, in which case we would not expect any other lines in our current frequency coverage, consistent with our data.SGP-32338 is a similar case: we detected an emission line at 101.07 GHz, with a fwhm of 630± 80 km s^-1.The line emission is again coincident with its 3-mm continuum emission (Fig. <ref>).The photometric redshift, 4.51^+0.47_-0.39, makes ^12CO(5–4) at z=4.70 the most likely candidate emission line.Because the line lies close to the center of the spectral coverage, we would not then expect to detect any other lines, despite the high sensitivity and the wide frequency range available.Follow-up observations are required to determine unambiguous redshifts for these three galaxies.In SGP-354388, dubbed the `Great Red Hope' because it is amongst the reddest galaxies seen by Herschel, our ALMA spectrum reveals a line at 98.34 GHz, coincident with 3-mm continuum emission.Extensive further follow-up observations of SGP-354388, reported by <cit.>, confirm that the line at 98.34 GHz is, in fact, the C i(1–0) transition at z=4.002±0.001, a rare ≈2σ deviation from the photometric redshift which can be attributed at least partially to dusty galaxies surrounding SGP-354388, at the same redshift, which contaminate the flux densities measured at ≥500 μm by SPIRE and LABOCA <cit.>.§.§.§ Galaxies where no emission lines are detectedIn our remaining spectral scans, regardless of whether or not we have secure positions via continuum detections, we have found no compelling evidence of line emission (Fig. <ref>). Note that the mean [median] log_10 far-IR luminosity of this subsample, 13.2 [13.1], for an average [median] photometric redshift of 3.79 [3.70], which is 0.3–0.4 dex below that of the sample in which line emission has been detected.For a fwhm line width of 500and typical brightness temperature and /CO ratios (see later, <ref>), this equates to a peak line flux density in CO(4–3) of 1.6 [1.4] mJy, comparable to the r.m.s noise levels in our spectral scans, which goes some considerable way towards explaining why we detected no line emission for this sub-sample. §.§ Spectral energy distributionsSince we have added continuum flux density measurements at 1.3 and/or 3 mm for many of our targets, as well as some unambiguous spectroscopic redshifts, it is worth repeating the SED fits performed by <cit.>.We have constructed the SEDs of our targets, utilizing data from SPIRE at 250, 350 and 500 μm, from SCUBA-2 at 850 μm <cit.>, from NOEMA at 1.3 mm, and from NOEMA and/or ALMA at 3 mm – see Table <ref>.For details of the SED fits for SGP-32386, we refer readers to <cit.>.Like , we employ SED templates representative of high-redshift DSFGs: the average SEDs from <cit.>, <cit.> and <cit.>, and the observed SEDs of individual targets – the Cosmic Eyelash <cit.>, HFLS 3 <cit.>, G15.141 <cit.> and Arp 220 <cit.>.We have restricted our SED work to the sources with unambiguous redshift determinations, such that we need shift only the flux density scale of the templates to fit the observed SEDs. We adopted the lowest χ^2 values, calculated from the difference between the templates and the observed flux densities, with inverse weighting of the flux density uncertainties.These best-fitting SEDs are plotted in Fig. <ref>, with the corresponding IR luminosities (L_ 8-1000 μ m) and SFRs listed in Table <ref>, the latter calculated using the calibration of <cit.>, <cit.> and <cit.>, with a Salpeter IMF <cit.>.On the basis of high-resolution ALMA continuum and line observations, <cit.> found that SGP-196076 at z=4.425 comprises at least three components, their on-going merger driving large masses of turbulent gas to form stars, as is ubiquitous amongst objects with such high intrinsic IR luminosties <cit.>.§.§.§ Modified blackbody fits To better quantify the thermal dust emission we have performed SED fits using modified blackbody (MBB) spectra, again by minimizing χ^2.We adopted an optically thin model with single dust temperature (i.e.S_ν(T_ d)∝ (ν/ν_ c)^β B_ν(T_ d)), where ν_c is the frequency at which the optical depth is unity, B_ν(T_ d) is Planck function at frequency, ν, and dust temperature, T_ d, and β is the dust emissivity index.We fixed ν_ c to 1.5 THz <cit.> and adopted κ_850 μ m = 0.15m^2 kg^-1 <cit.>.Dust emissivity, β, being poorly constrained by our data, was fixed to values of 1.5, 2.0 or 2.5 <cit.>.The assumption of a single dust temperature means that we are measuring the emission-weighted mean dust temperature and dust mass of all the dust components in the galaxy.Another advantage of this approach is that it allows us to compare directly with other high-redshift DSFGs, which are usually described in terms of single-temperature MBB fits (see Table <ref>).Finally, the modest sampling of our SEDs, especially at the short wavelengths required to constrain hot dust components, prevents meaningful multi-temperature MBB fitting.The best MBB fits are plotted in Fig. <ref>.Minimum χ^2 were obtained with β=1.5 for NGP-190387, β=2.0 for NGP-246114 and NGP-284357.G09-81106 proved difficult to reconcile with a value of β below 2.5 – probably the result of using a single temperature MBB <cit.>.Changing β results in an increase/decrease of the dust temperature by ∼ 2–3 k, and the dust mass by ∼ 0.1 dex.The resulting dust temperatures are in the range ∼31–36 k, and dust masses, ∼ 0.2–4.1×10^9 M_⊙. For their high IR luminosities, the dust temperatures of our galaxies are relatively low <cit.>.Individual values of T_ d and M_ d are listed in Table <ref>. §.§ Molecular gas masses Although low-J transitions of CO, or C i, are much preferred when tracing the remaining reservoirs of molecular hydrogen <cit.>, we can estimate those gas masses from higher-J CO lines by assuming the average CO line ratio for SMGs, as measured by <cit.> and <cit.>, and tabulated by the latter: L^'_CO(7-6)/L^'_CO(1-0) = 0.18, L^'_CO(6-5)/L^'_CO(1-0) = 0.21, L^'_CO(5-4)/L^'_CO(1-0) = 0.32 and L^'_CO(4-3)/L^'_CO(1-0) = 0.41, bearing in mind that there can be large variations.We find that L_ IR-L^'_ CO as derived from different – usually neighbouring – transitions are consistent.We have taken the average L^'_CO(1-0) from the available high-J CO transitions, then calculated the molecular gas masses using α_ CO=0.8M_⊙ (K km s^-1 pc^2)^-1, the value often assumed for high-redshift starbursts and local ULIRGs since the work of <cit.>. Estimates of gas-to-dust mass ratios then lie in the range 50–140, consistent with those of local galaxies. Our M_ H_2 estimates are listed in Table <ref>. Since luminous CO(5–4) and CO(4–3) emission can be generated by the presence of a massive molecular gas reservoir, or by a much smaller amount of highly excited molecular gas, follow-up observations of low-J CO transitions are required to better determine M_ H_2, modulo the effects of cosmic rays laid out by <cit.>. §.§ L_ IR - L^'_ CO correlationThe relationship between star formation and total molecular gas content is often shown via a plot of the key observables, L_ IR versus L^'_ CO, and can reveal if and how star-formation efficiency (SFE) changes with the amount of molecular gas available for star formation.We have constructed a plot of L_ IR - L^'_ CO using our z>4 IR-luminous galaxies, other high-redshift unlensed DSFGs <cit.> and local U/LIRGS <cit.> — see Fig. <ref>.A linear fit to all the data has a slope, 1.15±0.02 <cit.>.Caution is required here, however, since most of the high-redshift targets, ours included, are detected in mid-J CO transitions. Using only CO(1–0) observations, with self-consistent determinations of IR luminosity, <cit.> reported a slope significantly below unity, showing that adopting mid-J CO transitions for high-redshift galaxies and CO(1–0) transitions for low-redshift galaxies may artificially steepen the slope.Differential amplification is likely also an issue for the lensed galaxies in Fig. <ref>, where the amplifications derived for the dust, the CO(1–0) and/or high-J CO lines likely differ significantly. Finally, we note that several studies have suggested that our adopted value of α_ CO is too low, including <cit.> and <cit.>; indeed, if we were to apply the formalism of <cit.>, who use optically thin long-wavelength dust emission to probe the mass of molecular gas, we arrive at a value ∼ 3.5× higher, the equivalent of α_ CO≈ 3M_⊙ (K km s^-1 pc^2)^-1. §.§ Depletion timescale The average gas-depletion timescale of our galaxies, t_ depl, is around 50 Myr, modulo the possibility of considerably higher gas masses noted in <ref>.Taken at face value, this t_ depl is consistent with the idea that our targets are rapidly building a significant mass of stars, which may be picked up in a later phase at z∼2–3 as massive `red-and-dead' galaxies by near-IR imaging surveys <cit.>.§ CONCLUSIONS We report spectral scans of Herschel-selected ultra-red galaxies with photometric redshifts estimated to lie at ≳4. For each of 21 galaxies we have covered Δν≈ 20 GHz using ALMA and NOEMA in the 3-mm waveband.We have determined the redshifts of seven galaxies unambiguously, in the range z=3.85–6.03, detecting multiple emission lines, usually CO rotational transitions. One of these redshifts was determined independently by <cit.>.For an additional four galaxies, single emission lines are detected, one of which has been shown by <cit.> to lie at z=4.002. Candidate redshifts are suggested, based on their photometric redshifts. Follow-up observations are required to measure their redshifts unambiguously, except in that one case.Since the comparison of photometric and spectroscopic redshifts for this sample by <cit.>, two new spectroscopic redshifts have been determined, one below and one above the respective photometric redshifts.Although the offsets for these two galaxies are larger than the expected uncertainties in z_ phot, the overall scatter in (z_ phot - z_ spec) / (1 + z_ spec) is still consistent with (actually, slightly better than) that of the training set. In the worst case, the offset can be understood in terms of contamination of flux densities measured at ≥500 μm by a cluster of dusty galaxies <cit.>.Our sample of redshift-confirmed galaxies contains extraordinarily IR-luminous starbursts, with an average SFR of ≈ 2900 M_⊙ yr^-1.They are also among the most massive known, in terms of molecular gas mass, and dust mass, with M_ H_2≈ 1.8×10^11M_⊙ on average, and M_ d∼ 0.9–4.1×10^9 M_⊙.Lurking amongst our IR-luminous galaxies we find three lensed systems. These would otherwise have been hailed as the most luminous known starbursts.It is notable that the vast majority of the brightest systems selected by Herschel have been revealed as either lensed galaxies, groups/clusters of starburst galaxies, starbursts with buried AGN, or some combination of the three <cit.>, which suggests strongly that there exists a limit to the luminosity of individual starbursting galaxies.Combining local U/LIRGs, other high-redshifts DSFGs and our new redshift-confirmed galaxies, the resulting L_ IR - L^'_ CO correlation has slope close to unity, 1.14±0.02, suggesting slightly higher star-formation efficiency in the most IR-bright galaxies. The gas-depletion timescale of our galaxies, around 50 Myr, is consistent with the idea that our targets may be picked up in a later phase at z∼2–3 as massive `red-and-dead' galaxies by near-IR imaging surveys. § ACKNOWLEDGEMENTS RJI, IO, VA, LD, SM, JMS and ZYZ acknowledge support from the European Research Council in the form of the Advanced Investigator Programme, 321302, COSMICISM.JMS also acknowledges financial support through an EACOA fellowship.DR acknowledges support from the National Science Foundation under grant number AST-1614213. HD acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2014 Ramón y Cajal program, MINECO RYC-2014-15686. We thank the referee, Francoise Combes, for her rapid and constructive feedback. This work was based on observations carried out with the IRAM Interferometer, NOEMA, supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00499.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile.The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 090.A-0891(A).Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil).mnras . | http://arxiv.org/abs/1707.08967v1 | {
"authors": [
"Y. Fudamoto",
"R. J. Ivison",
"I. Oteo",
"M. Krips",
"Z. Y. Zhang",
"A. Weiss",
"H. Dannerbauer",
"A. Omont",
"S. C. Chapman",
"L. Christensen",
"V. Arumugam",
"F. Bertoldi",
"M. Bremer",
"D. L. Clements",
"L. Dunne",
"S. A. Eales",
"J. Greenslade",
"S. Maddox",
"P. Martinez-Navajas",
"M. Michalowski",
"I. Pérez-Fournon",
"D. Riechers",
"J. M. Simpson",
"B. Stalder",
"E. Valiante",
"P. van der Werf"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727180006",
"title": "The most distant, luminous, dusty star-forming galaxies: redshifts from NOEMA and ALMA spectral scans"
} |
^1Division of Geological and Planetary Sciences, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA ^2Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109-1107, USA ^3Department of Geophysical Sciences, The University of Chicago, 5734 South Ellis Ave., Chicago, IL 60637, USA ^4European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748, Garching, Germany ^5School of Space Research, Kyung Hee University, 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 17104, Korea The Earth and other rocky bodies in the inner solar system contain significantly less carbon than the primordial materials that seeded their formation. These carbon-poor objects include the parent bodies of primitive meteorites, suggesting that at least one process responsible for solid-phase carbon depletion was active prior to the early stages of planet formation. Potential mechanisms include the erosion of carbonaceous materials by photons or atomic oxygen in the surface layers of the protoplanetary disk. Under photochemically generated favorable conditions, these reactions can deplete the near-surface abundance of carbon grains and polycyclic aromatic hydrocarbons by several orders of magnitude on short timescales relative to the lifetime of the disk out to radii of ∼20–100+ au from the central star depending on the form of refractory carbon present. Due to the reliance of destruction mechanisms on a high influx of photons, the extent of refractory carbon depletion is quite sensitive to the disk's internal radiation field. Dust transport within the disk is required to affect the composition of the midplane. In our current model of a passive, constant-α disk, where α = 0.01, carbon grains can be turbulently lofted into the destructive surface layers and depleted out to radii of ∼3–10 au for 0.1-1 grains. Smaller grains can be cleared out of the planet-forming region completely. Destruction may be more effective in an actively accreting disk or when considering individual grain trajectories in non-idealized disks. § INTRODUCTIONThe earliest stages of planet formation begin within a primordial disk of gas and submicron-sized dust grains surrounding a young star, formed due to the conservation of angular momentum through the collapse of a dense core in a molecular cloud. This material inherited from the interstellar medium (ISM) experiences a range of physical conditions and chemical environments. The dust grains within the disk concentrate toward the midplane due to the gravitational pull from the central star over time, increasing their likelihood of interaction. Dust grain aggregates are thought to grow through collisions, forming bodies that are orders of magnitude larger in size than typical interstellar dust as part of the planet formation process <cit.>. Provided that the original grains are of approximately interstellar composition, the amount of carbon available at the onset of planet formation can be roughly estimated. The carbon-to-silicon abundance ratio of the ISM is similar to that of the solar photosphere, C/Si∼10 (, hereinafter B15; ).About 60% of carbon in the ISM is present in an unidentified solid, and thus potentially refractory, form <cit.>. Suggested components include graphite, amorphous carbon or hydrocarbon grains with a combination of aliphatic-rich and aromatic-rich components <cit.>, large organics, and/or polycyclic aromatic hydrocarbons (PAHs). Once in the protoplanetary disk, carbon retained in the gas phase is subject to accretion by the central star or gas-giant planets or else to dissipation, ultimately being largely cleared from the disk. Any remaining volatile carbon species would likely be found in the form of ices in only the cold outer regions of the disk. In contrast, one would expect more-refractory carbon sources with vaporization temperatures of 425–626 K <cit.> to be readily available and incorporated into solid planetary bodies, including planetesimals, throughout the entire disk except for the very innermost radii <cit.> and the directly irradiated surface. However, observations of rocky bodies in our solar system reveal a significant depletion in carbon compared to the progenitorial interstellar dust (see Table <ref>; ; ).The C/Si ratio of the bulk silicate Earth (BSE; the entire Earth, including the oceans and atmosphere, excluding the core) is about four orders of magnitude lower than that of the interstellar dust that seeded its formation . This depletion of carbon could be the result of various events throughout Earth's formation and early stages, including devolatilization during processing of accreted material or impact events and sequestration of carbon via core formation (seeand references therein). However, carbonaceous chondrites, some of the most primitive materials known in our solar system, also exhibit carbon depletions <cit.> relative to the interstellar value. This suggests that a significant amount of the refractory carbon, which would have otherwise been incorporated into these solid bodies at a level of ∼6× that of silicon, was destroyed early in the solar system's history and prior to the formation of rocky planets and differentiated planetesimals. Not all solid bodies in our solar system are carbon poor. The dust of comet Halley has a C/Si ratio similar to that of the ISM (; ) and anhydrous interplanetary dust particles (IDPs), argued to be of cometary origin <cit.>, also have high carbon content. The increased carbon retention by these outer-solar-system bodies cannot be explained by volatile ices alone; at least 50% of the carbon in comets is thought to be in refractory form . Unlike Halley, the Sun-grazing comets resemble carbonaceous chondrites in terms of carbon abundance. These differences suggest that the mechanism responsible for refractory carbon destruction did not operate uniformly throughout the solar system and may have been less active in some comet-forming regions. Evidence for carbon-deficient bodies is also found beyond our solar system. Spectra of polluted white dwarf atmospheres provide an opportunity to view the chemical composition of extrasolar planestimals. The strong gravitational forces of white dwarfs are expected to deplete heavier elements from their photospheres on relatively short timescales. Therefore, heavy elements observed in the atmospheres of white dwarfs are believed to have originated in rocky bodies that are destroyed near the tidal radius and accreted by the star <cit.>. Elemental measurements reveal these bodies to be carbon-poor with C/Fe ratios similar to those measured in chondritic meteorites in our own solar system <cit.>.Explaining the C/Si record of solid bodies in the solar system requires a mechanism that (1) selectively removes refractory carbon from the condensed phase while leaving silicates intact, (2) operates prior to the formation of the parent bodies of meteorites, (3) accounts for a range in carbon content among different solar-system bodies, and (4) may be a common process that is not unique to the specific conditions of our solar system. Once the disk enters the passive state, meaning that it is no longer accreting material from the interstellar cloud and is heated mainly by irradiation from the central star, vaporization of refractory carbon will be ineffective throughout most of the disk because the dust temperatures are too low <cit.>. Alternative destruction mechanisms include those driven by energetic radiation. For example, photochemically activated oxidation of carbonaceous material via chemical reactions with O-bearing species, including OH and free atomic O, which can erode the surface of carbon grains releasing carbon into the gas phase <cit.>. <cit.> calculated that combustion through reactions with atomic O, which is abundant in the photochemically active surface layers of the disk, can efficiently destroy carbon grains <10 in size and given sufficient vertical transport in the disk, potentially explain the observed carbon deficiencies of rocky bodies in our solar system. Directphotochemical destruction of refractory carbon sources, resulting in the ejection of small hydrocarbons, represents an additional destruction mechanism. <cit.> experimentally* Note.—See <cit.> and sources therein[!b]lc Bulk C/Si in Solar-system Bodies Source C/Si ISM (dust) 6 Earth (BSE) 0.001 Meteorites (CI, CM) 0.4–0.7 Meteorites (CO, CV) 0.07–0.08 Comets (Halley, dust) 6 Comets (Halley, gas + dust) 8 Comets (Sun-grazing) 0.08–0.2 Interplanetary Dust Particles (IDPs) 2 investigated the rate of vacuum ultraviolet (VUV) photolysis for interstellar hydrogenated amorphous carbon (HAC) analogs; whereas the ability of large aromatic carbon species to survive photodissociation has been evaluated through modeling by <cit.>. <cit.> and <cit.> also explored the potential for PAH destruction via more energetic extreme ultraviolet (EUV) and X-ray photons. Here we expand upon the work of <cit.>, exploring the possibility that refractory carbon sources in the disk can be depleted before grains grow to sufficient sizes and while most of the solid mass remains in 0.1–1 baseline seed particles. We use a full chemical disk model to explore the oxidation and photochemical destruction of two sources of refractory carbon: carbon grains and PAHs. With this model, we attempt to estimate the efficacy of these mechanisms throughout the disk and determine the radial distance to which carbon can be depleted for comparison with our own solar system. Section 2 describes the disk model used in our analysis, while Section 3 provides the results of the model and subsequent analysis of the capabilities of the proposed destruction mechanisms. A discussion of the results is presented in Section 4 followed by a final summary in Section 5. § MODELHere we model a static, passive protoplanetary disk around a T-Tauri star, a young, low-mass star thought to be similar to our early Sun. Chemistry within the disk is predicted by solving a set of ordinary differential equations describing the chemical reaction rates of the system. To the standard chemical model, we add a source of refractory carbon, either carbon grains or PAHs, and appropriate destruction mechanisms in order to track the abundance of these sources throughout the disk. §.§ Physical ConditionsTo provide the conditions representative of a disk surrounding a typical T-Tauri star, we use a physical disk model similar to that of <cit.>. The model is two-dimensional and azimuthally symmetric. To approximate the dust distribution of an early disk, prior to substantial grain growth and sequestration of mass in the midplane, we include a single population of small dust (radii ≤1–10) that follows the distribution of the gas. The dust composition is 80% astronomical silicates <cit.> and 20% graphite. Although ice coatings on the dust may affect the resulting opacity <cit.>, the model assumes the general approximation of bare grain surfaces. Given our static model, these grains are considered only for calculating the physical conditions of the disk and are not directly included in the chemical reactions. This dust distribution therefore remains unchanged as we destroy carbon-bearing grains. The gas and dust share a fixed radial surface density profile described by a power law with an exponential cutoff, Σ_gas(R) = Σ_c (R/R_c)^-1exp[-(R/R_c)] where Σ_c =3.1 g cm^-2 and R_c = 135 au from <cit.>, and are related by assuming a vertically integrated gas-to-dust mass ratio of 100.The total mass of the disk gas is 0.039 M_⊙.The UV field, including the propagation of Ly α photons, and the X-ray radiation field within the disk were generated using a Monte Carlo radiative transfer code as described by <cit.>. The input UV spectrum is taken to be that observed for TW Hya <cit.>.The modeled X-ray spectrum has a total luminosity of 10^29.5 erg s^-1 between 1–20 keV, corresponding to that of a typical T-Tauri star. The gas temperature is estimated based on the computed UV field and gas density distributions using the thermochemical models of <cit.>, as described in <cit.>. Figure <ref> shows the final density, temperature, and radiation distributions provided to the chemical model.§.§ Chemical NetworkThe disk chemistry model combines the higher-temperature (≲800) gas-phase network of <cit.> with the gas-grain network of <cit.>. In total, over 600 chemical species and nearly 7000 reactions are specified, including gas-phase reactions among ions, neutrals, and electrons; photodissociation; X-ray and cosmic-ray ionization; gas-grain interactions (freeze-out, sublimation, photodesorption, and cosmic-ray-induced desorption); and formation of OH, H_2O, and H_2 on grains. The model is run for radii from 0.25 to 100 au from the central star, divided into a grid of 42 logarithmically spaced radii each containing 45 linearly spaced vertical zones. With the exception of H_2 and CO self-shielding, which are considered throughout the vertical column, chemistry within each grid space is modeled independently. The disk model is static, with no mixing between grid spaces. To this chemical network, we added sources of refractory carbon as described in the following sections. §.§ Sources of Refractory Carbon§.§.§ Carbon GrainsThe first source of refractory carbon included in our model is solid grains. Different grain sizes are considered, with radii of 0.01–10. Sizes are larger than the average interstellar values to account for growth within the disk. Large, ∼1–10, silicate grains have been observed in emission at 10 and 18 originating from the irradiated layers. Modeling the grains as having compact, spherical structures provides an upper limit on the destruction timescale. However, during growth due to collisions the aggregate structures can become open. Laboratory experiments using dust analogs find that initial growth through collisions resulting in the sticking and freezing together of individual particles leads to the formation of fractal structures <cit.>. In addition, interplanetary dust particles collected from Earth's atmosphere are described as typically being porous aggregates composed of ∼10^6 grains <cit.>. To account for porosity in our modeled carbon grains, volume-enlargement factors from the collisional evolution simulations of <cit.> are used. The resulting porosities are assumed to be 0% for the 0.01 and 0.1 grains, 88% for the 1 grains, and 96% for the 10 grains. The porosity (ϕ) represents the percentage of the grain volume that is unoccupied. Compared to a compact configuration of atoms, a porous grain of the same size will only contain (100-ϕ)% of the amount of carbon. Oxidation of these carbon grains is modeled based on the approach of <cit.>. An oxygen atom that collides with a carbon grain removes a single C atom to form gaseous CO. Oxidation by OH is also possible <cit.>, but O is more abundant in the photoactive surface layers of the disk.The reaction occurs at a rate of R_ox = n_cgr n_oσ v_o Y, where n_cgr is the number density of carbon grains, n_o is the number density of oxygen atoms, σ is the cross-section of a carbon grain, v_o is the thermal velocity of an oxygen atom, and Y is the yield of the reaction (, Y = 170exp(-4430/T_gas) if T_gas > 440 and Y = 2.30exp(-2580/T_gas) if T_gas≤ 440). The abundance of carbon grains is determined by taking the initial abundance of refractory carbon and dividing it by the number of carbon atoms per grain, equal to [ρ_cgr4/3π r^3 (100%-ϕ)]/m_c where ρ_cgr is taken to be the density of graphite, 2.24 g cm^-3; r is the grain radius; ϕ is the porosity; and m_c is the mass of a carbon atom. Photolysis rates for carbon grains of amorphous structure are taken from <cit.>, who measured the production rates of hydrogen and small hydrocarbons released from a plasma-produced HAC surface irradiated by VUV photons. Methane was the C-bearing product of highest yield and is the sole product considered here. The rate is R_UV = n_cgrσ Y_CH_4 (Φ^FUV_ISRF/1.69) F_UV, where n_cgr and σ are the same as above,Y_CH_4 is the photo-production yield of CH_4 per incoming photon (∼8×10^-4), Φ^FUV_ISRF is the far ultraviolet (FUV) flux of the interstellar radiation field divided by 1.69 to convert from the Draine to Habing values, and F_UV is the FUV flux relative to the interstellar value from the disk model. Here the term “photolysis" is used to refer to the photon-induced release of small hydrocarbons from the surface of a grain containing ∼10^6 or more C atoms, distinct from the photodissociation of the PAHs, where the structure of a ≲100 C atom molecule is broken resulting in the loss of small hydrocarbons.Small grains are also subject to destruction by higher energy, X-ray photons through heating to the point of sublimation or grain charging increasing electrostatic stress to the point of shattering. The conditions under which X-ray radiation will lead to grain destruction have been estimated by <cit.> for emission from γ-ray bursts. X-ray emission from T-Tauri stars is dominated by 1–2 keV photons. The energy deposited by these photons will be insufficient to heat 0.01–10 grains above the sublimation temperature of refractory carbon throughout most of the disk <cit.>. Shattering due to the buildup of electrostatic charge relies on successive interactions with X-ray photons. However, in the protoplanetary disk, grains will cycle between the surface and the midplane. Whereas X-ray ionization rates may dominate at the disk surface, in denser gas toward the midplane recombination with free electrons will be a competitive process potentially preventing the buildup of charge on grains. Cosmic rays and high-energy X-rays, if present in these dense layers, will be key sources of ionization in the inner disk resulting in the production of free electrons <cit.>. The susceptibility of silicate grains to shattering is slightly greater than that of carbonaceous grains <cit.>. Therefore, this mechanism would not selectively destroy carbonaceous grains and efficient reformation of silicate grains would be required to explain the observed composition of solar-system bodies. Further modeling is required to understand the effectiveness of X-rays in reducing refractory carbon abundances.§.§.§ PAHsPAHs represent the second source of refractory carbon included in our model. Refractory carbon is introduced initially as large, 50 C atom PAHs representative of interstellar species. For comparison of destruction rates versus PAH size, additional models start with 20 C atom PAHs to demonstrate the breakdown of smaller PAHs after some initial destruction in the disk. As time progresses in the model, reactions with O, H, and OH dismantle the initial PAHs removing fragments containing ∼2 C atoms per reaction. Rates involving small PAHs from pyrene (16 C atoms) down to a single aromatic ring are from <cit.>. Where necessary, reverse-reaction rate coefficients are calculated using the thermodynamic data they provided.[We derive the reverse-reaction rate coefficient from k_r=k_f/K, where the forward reaction rate coefficient provided is k_f = A T^nexp(-E/RT), and A, n, and the activation energy E are given by <cit.>, R is the gas constant, and T is the gas temperature. The equilibrium constant is given by K =exp(- Δ G/RT) =exp(- Δ H/RT + Δ S/R) where Δ G, Δ H and Δ S are the change in the standard Gibbs free energy, enthalpy, and entropy of the reaction.] For larger PAHs, rates are taken as those of the analogous reaction (O, H, or OH) with benzene as is the case for pyrene, phenanthrene, and naphthalene in <cit.>. Given that the specific structure of PAHs in astrophysical environments is largely unknown, the molecular structure of the large PAHs was not strictly considered aside from ensuring conservation of mass within the reaction network.<cit.> provide UV photodissociation rates resulting in the loss of two C atoms for PAHs of different sizes based on theoretical calculations. These rates incorporate the modulated UV field, specifically the FUV field, at each grid location in the disk. C_2H is successively removed to break down the larger PAHs and reaction products from the photodissociation of PAHs smaller than pyrene were selected to mimic the oxidation reaction pathways where appropriate. The rate of photodissociation by X-ray photons follows <cit.> using the cross-section for PAHs from <cit.>, N_C × C^abs where N_C is the number of carbon atoms per PAH and C^abs≃ 10^-17× (20/E)^2.2 cm^2 is the cross-section per carbon atom for photon energies (E) greater than 20 eV. The energy-dependent cross section is multiplied by the X-ray flux distribution at E integrated over E = 1-20 keV. X-ray photons carry enough energy such that a single interaction can dismantle an entire 50 C PAH. In this case, we assumed the products of the reaction to be small hydrocarbons in our reaction network containing four or fewer C atoms. However, the photodestruction of large PAHs via X-rays has not been characterized through laboratory experiments and the dispersion of energy from X-ray photons among different molecular processes is not well known. Given the uncertainty in how to define an X-ray photodissociation rate for large PAHs <cit.>, we also tested models where X-ray absorption resulted in the removal of C_2H (analogous to the network used for UV photodissociation of PAHs above) for reaction yields of 0.5 and 1.0.The breakdown of PAHs with more than 60 C atoms may result in the production of fullerenes, such as C_60, that are highly stable and resistant to destruction. <cit.> suggest that ≲1% of the instellar PAH abudance may be in the form of C_60. PAHs in the surface layers will likely have predominately neutral charge. Abundant PAH cations are not produced by the radiation field of T-Tauri stars <cit.> and PAH anions will only be present in denser regions of the disk toward the midplane. The inclusion of PAH formation reactions is not important since we found the formation of benzene (as a proxy) to be inefficient in comparison to the rate of PAH destruction under the conditions in our disk model. §.§.§ Model SetupAt the onset of the model, carbon is divided between the refractory and volatile phases. Several iterations of the model were run with different initial forms for the refractory carbon. This phase is represented by either carbon grains of a single size and porosity (six different cases: R = 0.01, 0.1, 1, or 10,and for the larger two sizes: porous or non-porous structures) or PAHs (two cases: initial PAHs of 50 or 20 C atoms in size). For the carbon grain scenarios, the total abundance of carbon <cit.> is divided equally between the grains and the volatile phase. In the case of the PAHs, an abundance of 1.5 × 10^-5 C relative to H, approximately the abundance observed in the ISM <cit.>, is distributed among the 50 or 20 C molecules. Volatile species are set to 1 × 10^-4 relative to H. Nearly all the volatile carbon is initially stored in CO (with minor amounts in CN, C, HCN, C^+, HCO^+, H_2CO, and C_2H). All chemical species are spread uniformly in abundance relative to H over the entire grid of the disk at the beginning of the model run. Radial and vertical mixing within the disk are not considered here.§ RESULTSA chemical model of an irradiated (passive) protoplanetary disk is used to identify the regions where destruction of refractory carbon sources occurs. Further analysis of the timescales of grain transport and the distribution of grains within the disk allow for the estimation of the radial extent of refractory carbon depletion due to the mechanisms described here. §.§ Regions of Active Refractory Carbon Destruction The model described in Section 2 is used to identify the locations in the disk where oxidation and photochemical destruction could cause significant depletion of refractory carbon sources. Figures <ref> and <ref> show the abundance of refractory carbon remaining after running the model for 10^6 years, on the order of the lifetime of a protoplanetary disk. Each panel illustrates a vertical cross-section of the disk above the midplane with contours representing the level of depletion of a particular refractory carbon source (with an abundance relative to H of 10^-4 representing zero depletion for the carbon grains and 1.5×10^-5 likewise for the PAHs). Here the PAH abundance is considered to be the total abundance of all aromatic components larger than benzene in our model. As demonstrated in these panels, oxidation and photochemical destruction are only operative near the disk surface. Whereas oxidation is restricted to within a radius that depends fairly weakly on the size and nature of the refractory carbon sources, photochemical destruction can be unlimited in the radial direction out to at least 100 au.§.§.§ Photochemical Destruction vs. OxidationIn a passive disk, irradiation from the central star and the ISM produces environmental conditions—temperatures of at least a few 100 and high atomic oxygen abundances—in the surface layers that allow oxidation to proceed at an appreciable rate. The exponential dependence of the chemical kinetics on the gas temperature is the main limiting factor toward the disk midplane. Additionally, the dominant carriers of oxygen vary within the vertical column at each radius. Atomic O is most abundant above the cool middle layers that are protected from stellar photons, where H_2O ice and other molecular forms dominate, and below the surface of the disk where at radii ≲10 au ionized O^+ begins to form. Radially, at a certain distance from the star the sufficiently dense layers of the disk no longer reach the required temperatures and atomic oxygen abundances to activate oxidation and the reaction shuts off. This radial cutoff occurs at ∼20–65 au depending on the refractory carbon source.Photolysis occurs over a larger region of the disk due to the lack of temperature dependence in the reaction rate. The layer to which UV photons reach extends below the hot region where oxidation can occur and to larger radii in the disk. However, the reaction is limited by the modulation of the UV field by disk material, leaving the midplane of the disk shielded. UV photolysis of HAC grains occurs down to interstellar UV levels implying the need for reformation of carbonaceous grains in the ISM (e.g., ), if they are indeed of HAC composition as predicted by current interstellar grain models <cit.>. UV photodissociation is only efficient for small PAHs, occurring within the disk lifetime for sizes ≲24 C atoms and FUV photons as shown by <cit.>. Significant depletion, by four orders of magnitude or greater in total refractory C abundance, occurs deeper in the disk for UV photodissociation on these small PAHs than for oxidation or X-ray photodissociation. However, the vertical cutoff for photodissociation of PAHs by X-rays is not as sharply defined as that in the UV. Minor amounts of depletion, less than 0.01% of the total refractory C abundance, continue deeper into the disk approaching the midplane.The timescales for UV photochemical destruction are faster than those of oxidation for all carbon grain sizes and for small PAHs (Figures <ref> and <ref>). In the case of the PAHs, UV photodissociation will accelerate the depletion of small PAHs following the destruction of 50 C PAHs via oxidation or X-ray photodissociation. In comparison to X-ray photodissociation, oxidation of PAHs occurs over a smaller region of the disk, but at a faster rate.§.§.§ Carbon Grain and PAH SizesThe extent of refractory carbon depletion has some dependence on the size and structure of the carbon sources. Fig. <ref> shows the abundance of the different carbon grains over time at radii of 1 and 10 au in the surface layers where rapid oxidation and photolysis are occurring. Larger and less porous grains are destroyed more slowly because they have a smaller [cross-section]/[occupied volume] ratio reducing the frequency of their interaction with O atoms and UV photons. For the smaller or more porous grains, faster destruction rates facilitate depletion over a larger range of vertical heights and, in the case of oxidation, radii. The cutoff in disk radius for oxidation of different grain sizes decreases from ∼65 au to ∼45 au for grain radii from 0.01 to 10 (porous) in panels (a), (c), and (e) of Fig. <ref> and down to ∼20 au for 10 compact grains. There is no such radial cutoff within 100 au for photolysis until grain sizes of 10 are reached, where depletion by a factor of 100 occurs beyond 100 au but depletion by 10^4 only occurs out to 80 au. In addition, the vertical extent of depletion is slightly less for larger grains at large radii. At distances of 10s of au, depletion occurs down to ∼4 scale heights above the midplane for 0.01 grains vs. just above 5 scale heights for porous 10 grains.Within 10^6 years, oxidation depletes abundances of the initial 20 or 50 C PAHs by several orders of magnitude at radii out to 100 au in the surface layers of the disk. However, this is only the first step. Breakdown of the subsequent PAH products depends on the reaction rates of each of the following steps in the network. PAH depletion occurs rapidly in the inner few au for all species. Further out in the disk where conditions become less ideal for oxidation, the results will be more dependent on the chemical pathways included in the model. For the network selected here, in 10^6 years, all PAH species are cleared from the surface layers as shown in Fig. <ref> for radii ≲30 au regardless of initial size. However, where oxidation operates, the total abundance of C in PAHs is lower when the PAHs start out in 50 C molecules as can be observed by comparing the surface layers of the disk in panels (a) and (b) of Fig. <ref>. Since the rate-limiting steps in the breakdown process are the destruction of the smaller, 2- or 3-ring-containing species, the refractory carbon piles up in these small PAHs. In breaking down the larger 50 C PAHs, a larger portion of the total carbon has been removed in the form of small fragments before reaching these bottleneck species. Starting with larger PAHs does slow down the rate of carbon depletion by oxidation, as shown in Fig. <ref>.UV photodissociation here only affects small PAHs. Large PAHs (greater than 24 carbon atoms, ) cannot be photodissociated by FUV radiation within the disk lifetime and therefore can only be broken down by oxidation or more energetic photons. The inclusion of EUV photons would result in the break down of 50 C PAHs in addition to faster depletion of 20 C PAHs in the uppermost layers of the disk, about 1–2 scale heights above the current vertical cutoff for FUV photodissociation <cit.>. X-ray photodissociation rates are slower than those for FUV photons. In the case of the complete dissociation of a PAH molecule by a single X-ray photon, destruction occurs faster (Fig. <ref>) for the larger PAHs due to the dependence of the absorption cross-section on the number of carbon atoms. PAH destruction rates are decreased by an order of magnitude when considering the successive removal of C_2H compared to complete dissociation of the PAH molecule (Fig. <ref>), which also limits the radial and vertical extent of PAH depletion (Fig. <ref>).§.§.§ Chemical Modeling SummaryFigures <ref> and <ref> demonstrate that under favorable conditions present in the surface layers of the disk, depletion of refractory carbon sources occurs very rapidly compared to the disk lifetime. In terms of the chemistry, the amount of refractory carbon destroyed in these regions can exceed that required to explain the disparity among carbonaceous chondrites and the Earth relative to the interstellar dust (1–2 and ∼4 orders of magnitude, respectively). As shown in Fig. <ref> and <ref>, these carbon-deficient regions in the surface layers reach out beyond the terrestrial-planet- and asteroid-forming regimes of the disk.However, material in the midplane remains largely unaffected by oxidation and photochemical destruction in the static chemical model. §.§ Radial Extent of Refractory Carbon Depletion The conditions required for oxidation and photochemical destruction become less attainable (higher up in the disk) at larger distances from the central star suggesting that there may be a radial cutoff beyond which the mechanism is ineffective and refractory carbon will remain in the condensed phase at abundances similar to the interstellar value. In contrast to snow lines, this boundary, akin to the “soot line" described by <cit.> for the abundance of PAHs, would mark the location of an irreversible transition. Once carbon in the inner disk enters the gas phase it will likely remain volatile given the limited mechanisms available to return it to a more-refractory state. The high abundance of refractory carbon in the ISM will cause a potentially drastic change in the carbon chemistry from one side of the transition region, where nearly all the cosmically available carbon would be in volatile form and—in the warm inner disk—largely removed from the planetesimal formation process, to the other, where about half of this carbon would exist in a more-refractory phase and be available to be incorporated into forming planetesimals. Therefore, estimating the radial extent of carbon depletion mechanisms such as oxidation and photochemical destruction may allow us to relate the chemistry of the disk to the carbon content of solar-system bodies. As shown in Section 3, active oxidation and photochemical destruction of refractory carbon is limited to the photochemically active surface layers. However, the majority of the disk material is concentrated in the midplane and it is in this region that planet formation occurs. As suggested by <cit.>, mixing within the disk could introduce dust from these deeper layers into the destructive regions causing depletion throughout the vertical column. Turbulent motion is thought to counteract dust grain settling, continually stirring up disk material and maintaining the presence of grains in layers high above the midplane <cit.>. Fully assessing the extent to which destruction in the surface layers could affect material in the midplane requires a treatment of dust evolution and transport including processes such as growth, settling, fragmentation, and radial drift. The extent will further depend upon the underlying mechanism driving disk accretion and causing the turbulent motions throughout the disk. The dominant radial flow in the disk midplane is inward, but there is also evidence for outward transport in the solar nebula (e.g. crystalline silicates found in the comet Wild 2 by the NASA Stardust mission, ) that could carry carbon-depleted materials from close to the Sun to larger radii. Given these complexities, here we start by only considering the limit where vertical motion is more rapid than this radial transport and where the turbulence in the disk is characterized by a constant value of the Shakura & Sunyaev α parameter ranging from 10^-4 to 10^-2 <cit.>.The vertical height to which grains can be suspended due to turbulent motion is estimated by comparing the timescale of grain stirring to that of grain settling throughout the disk as described by <cit.> and used by <cit.>. The stirring timescale (t_stir =Sc(z) z^2/[α c_s H]) describes the amount of time required for grains to diffuse to a height z above the midplane, effectively describing the time required for components within a vertical column between the midplane and z to become well-mixed. The Schmidt number, Sc(z), is assumed to equal one such that the small particles considered here are perfectly coupled to the turbulence. This number can greatly exceed one in the low-density, upper layers of the disk making the stirring timescale used here a lower limit. In addition to the height above the midplane, t_stir depends on the level of turbulence parameterized by α (a more turbulent disk has shorter stirring timescales) and the temperature structure of the disk, which affects the sound speed c_s(z) and disk scale height H = c_s(0) / Ω_K, where Ω_K is the Keplerian rotation rate at radius R. The settling timescale (t_sett = [4 σ c_s(z) ρ_gas(z)]/[3 m Ω_K^2]) is shorter for larger and more compact grains due to its dependence on the cross-section/mass ratio [σ / m = 3 / (4 ρ_cgr (100%-ϕ) r)] of the grains but is also affected by the disk temperature and density [ρ_gas(z)] structure. Grains can be lofted up to a maximum height, where t_stir = 100 × t_sett, above which the downward gravitational force sufficiently overcomes the turbulent motion causing grains to be completely settled out. Figure <ref> shows the maximum height reached by grains of sizes 0.01–10 within the disk lifetime given the conditions in our physical disk model. Due to their longer settling times, smaller and more porous grains can reach heights closer to the disk surface. Excluding the larger (1–10) compact grains, all other grains can be lofted into the photo-destructive layer shown in Fig. <ref> out to radii of nearly 100 or more au for sufficiently turbulent disks (α = 10^-3–10^-2).This distance is only ∼10–100+ au for compact 1 grains and 0–2 au for compact 10 grains for the same range in α. At low α, t_stir approaches the disk lifetime of several million years and therefore restricts the upward mobility of grains of all sizes. For α = 10^-4, only grains smaller or more porous than compact 1 grains can be lofted into the photo-destructive layers and within a limited distance from the star of ∼15 au.PAHs in the gas phase would remain coupled to the gas, similar to the small carbon grains, and are not subject to settling. However, given that hydrocarbons have a sublimation temperature of ∼400 , in dense regions of the disk where the chance of interaction with grains is high, PAHs will likely freeze out and remain on grain surfaces. We would expect these PAHs to attach mainly to the smallest grains because they represent the largest surface area. Therefore, in the limit where all of the PAHs are frozen out, the results would be similar to those of the 0.01 grains. PAHs that evade interaction with grains would be destroyed at the rates presented in Section 3.1 at the disk surface where conditions are appropriate for oxidation, but would remain largely unaffected in deeper layers of the disk. Even if grains are able to reach the heights where destruction occurs within the disk, the amount of time grains spend at those heights is equally important to ensure their destruction. Where the stirring timescale is short compared to the disk lifetime, grains will cycle through the vertical column multiple times. In lower α disks with longer stirring timescales, grains tend to experience longer excursions through a smaller number of different environments. Alternatively, in higher α disks with shorter stirring timescales, grains rapidly change environments spending brief intervals of time in each one <cit.>. The average time spent in a particular region over multiple stirring cycles ends up being only weakly dependent on α, however, and related primarily to the density structure of the disk. Here the fraction of the disk lifetime that an average grain spends above a height z is taken to be approximately the ratio of the integrated gas density above height z to the total gas density of the vertical column above the midplane. Figure <ref> compares the time spent at heights above the midplane for a disk lifetime on the order of 10^6 years to the time required to deplete the 0.1 grains by one order of magnitude. Although carbon grain destruction occurs rapidly, grains would only be exposed to destructive conditions several scale heights above the midplane for ∼1–10 years for oxidation or ∼100–1000 years for photolysis. Where the depletion timescale, represented by the filled contours in Fig. <ref>, is longer than the exposure timescale and therefore a filled contour crosses over solid contour lines corresponding to comparable or shorter amounts of time, depletion of refractory carbon in 0.1 grains can occur within the disk lifetime. For UV photolysis in panel (a), the crossover occurs around 10 au, where the depletion and exposure times are on the order of 10 years. In comparison, this crossover occurs at ∼0.5 au for oxidation. These numbers represent upper estimates because they assume that the particles spend all of their time behaving aerodynamically as small monomers.This may be true in very turbulent cases, where collisions become destructive and prohibit growth or if grains are part of very porous aggregates.Growth to large or more compact aggregates may limit the time spent at these higher altitudes <cit.>.In the exposure time, refractory carbon abundances can be depleted to the levels observed in solar-system bodies out to maximum radii shown in Figure <ref>. For 0.01 grains, the maximum is about 3–5 au for depletion levels similar to the carbonaceous chondrites via oxidation. However, UV photolysis will clear out these grains throughout the planet-forming region. Larger, from compact 0.1 to porous 1, grains are depleted out to less than 1 au via oxidation but 3–10 au via photolysis. Once carbon has been incorporated into grains of 10 or larger (or compact 1 grains), it will likely survive these processes and remain in refractory form with the exception of porous grains in very turbulent disks within 1 au.For a constant-α disk, depletion requires sufficient turbulence to allow for lofting of the grains into the surface layers within the disk lifetime. Values of α∼10^-2 are needed to deplete carbon from 1 grains out to even a few au. For lower turbulence disks, depletion is limited to submicron grain sizes out to a few au (α=10^-3) or 0.01 grains out to ∼10 au (α=10^-4).The sizes of the depleted regions in the surface layers of the disk from Fig. <ref> represent the upper limit for the radial extent of carbon depletion assuming perfectly efficient transfer of material from the midplane to the surface and sufficient exposure times to destruction mechanisms. The actual extent of depletion in the midplane will almost certainly be less and highly dependent on the transport processes, grain evolution, and the temperature and density structure within the disk. In the simplest case of a constant-α model with our physical disk structure, depletion of submicron to micron-sized carbon grains can occur within ≲3–10 au in turbulent disks and average interstellar-sized carbon grains can be cleared out of the planet-forming region.§ DISCUSSIONThe analysis of the potential refractory carbon depletion due to oxidation and photochemical destruction depends on several unconstrained factors. The extent to which grains can be destroyed depends on their growth rate. Smaller grains are destroyed faster and can be lofted higher allowing them to reach the surface layers more easily. Larger grains are more likely to survive destruction. Therefore, the effectiveness of this destruction mechanism depends on the timescale and efficiency of grain growth. When growth is accompanied by collisional erosion and fragmentation, maintaining a constant supply of small grains in the disk, photochemical destruction and combustion remain effective. However, the formation of sufficiently large bodies that sweep up small dust grains prevents the small grains from traveling efficiently from the midplane to the surface <cit.> and therefore hinders the destruction of refractory carbon. The locations where the tested destruction mechanisms occur at an appreciable rate depend on the radiation field and temperature structure (for oxidation) within the disk. In order to have sufficient UV photons for photochemical destruction of refractory carbon in disks around low-mass stars where the UV field is dominated by accretion luminosity, the star must be accreting. The current analysis is based on an approximately solar-mass star. Increasing the stellar mass and therefore its luminosity would result in a warmer disk and potentially extend the distances out to which destruction can occur. However, changing the stellar mass would alter the input stellar irradiation field as well, which is a topic to be explored in future work. The assumed disk properties are also important. Younger disks may be more massive, preventing radiation from penetrating as deeply into the disk affecting both the temperature structure and atomic O abundances, causing the photoactive and oxidative regions to be higher above the midplane away from the majority of the disk material. An active disk undergoing accretion and potentially subject to large bursts of episodic accretion similar to those of FU Orionis objects will have a vastly different temperature profile than the passive disk modeled here. Warming dense regions of the disk may increase the effectiveness of oxidation throughout the disk. Furthermore, this analysis depends on the composition, optical properties, and distribution of materials within the disk. Dust grains may not have pure carbon surfaces and in cooler regions of the disk may be coated in volatile ice. This would reduce the efficiency of destruction mechanisms tested here. The change in opacity due to icy mantles could also alter the modeled temperature distribution in the disk. However, evidence suggests that ice coatings may not be a concern for this analysis. Observed water emission due to UV desorption of water ice from grains in the outer disk is relatively weak. This may be the result of differential settling where larger, ice-coated grains typically reside below the UV-irradiated layers and the small grains above have bare surfaces <cit.>. Additional settling of dust grains, approximated by increasing the fraction of the dust mass in larger grains and restricting the large grain population to lower scale heights, has little effect on the location of the high temperature (>100) gas in our model.Mass transport within the disk is required in order for this mechanism to reproduce the observed carbon deficit in solar-system bodies relative to interstellar dust. In the constant-α model, sufficiently turbulent disks are able to loft grains above the midplane. However, increasing the amount of time grains spend exposed to destructive conditions requires some asymmetry in their vertical motion that would cause them to spend additional time in the upper layers of the disk. Consideration of different angular momentum transport processes may be important for this analysis. For example, a disk model including wind-driven accretion, where <cit.> found that at 1 au the accretion flow occurs in a thin layer offset from the midplane by multiple scale heights, may alter the patterns of dust migration relative to the destructive regions. To further explore grain motion, a numerical (i.e. non-ideal) turbulent disk model could be employed, coupling dust evolution to the physical and chemical state of the disk and allowing timescales and lofting heights to be determined by averaging over the trajectories and lifetimes of individual grains in the simulation <cit.>. Ultimately, such models could be combined with dust coagulation and planetesimal formation scenarios to provide a quantitative assessment of the refractory carbon distribution in the disk prior to and through the formation of planetesimals. § CONCLUSIONWe ran a chemistry model for an irradiated, passive disk, including destruction mechanisms for solid carbon grains and PAHs – two potential sources of refractory carbon that could have been inherited from the ISM and present in the protoplanetary disk. Oxidation and photochemical destruction rapidly deplete refractory carbon but are limited to the photochemically active surface layers of the disk. Oxidation and photolysis of large grains are further limited within a particular radial distance depending on the size and structure of the refractory carbon source. The maximum radial distance to which refractory carbon can be oxidized at the surface is ∼20–65 au after 10^6 years for the 0.01–10 grains and ∼30 au for PAHs whereas photochemical destruction can extend out to 100+ au.Motion of grains within the disk is required to deplete refractory carbon at the midplane to the levels observed in solar-system bodies. This motion is difficult to model analytically. Approximations for the timescales of the average motion of dust grains are used to constrain the extent to which refractory carbon can become depleted at the midplane. For our model of a passive, constant-α disk with high turbulence, carbon grains smaller or more porous than compact 10 grains (and PAHs frozen to their surfaces) can be lofted into the destructive regions within 10–100+ au from the central star but their depletion is limited by the amount of time they are exposed to the destructive conditions. UV photolysis has proved to be an important mechanism in depleting refractory carbon. The fast reaction rates allow for destruction of grains in the surface layers of the disk even if they spend very little time there and the lack of temperature dependence extends the destructive region to any surface layers with sufficient UV radiation.Early on in planet formation, cm-sized and larger “pebbles" are built up through the interaction of smaller grains starting with interstellar sizes and compositions. These initial small grains will be subjected to physical and chemistry processes in the protoplanetary disk. While these grains remain small, photolysis (and to a lesser extent oxidation) can selectively erode the refractory carbon component of the population releasing it into the volatile phase in the inner portions of the disk. Photolysis can destroy the0.01 carbon grains throughout the planet-forming region of the disk assuming sufficient turbulence. However, the 0.1–1 grains may be a more significant source of material for planetesimals. If the primordial grain size distribution is similar to that of the ISM, small grains will represent most of the surface area but the bulk of the mass will be in the 0.1–1 grains. Once grains reach 10 in size they will be largely unaffected by oxidation and photochemical destruction unless they are broken down and rebuilt from smaller grains. Therefore, in order to be effective, destruction of refractory carbon grains will need to occur early in the lifetime of the disk prior to significant grain growth and the building of planetesimals.Based on our model, submicron to micron-sized carbon grain abundances can be depleted down to the levels of the carbonaceous chondrites and planetesimals sampled in the atmospheres of polluted white dwarfs out to a few to 10 au in sufficiently turbulent disks. Interstellar-sized grains can be cleared out of the planet-forming region of the disk in such turbulent disks and up to ∼10 au even in disks with lower turbulence. However, this analysis depends on several unconstrained parameters in the disk including the temperature and density structure, the amount of turbulence present, and the nature of the carbon sources. Further exploration of refractory carbon depletion in protoplanetary disks may therefore require consideration of alternative disk structures, dust transport, and/or accretion mechanisms. Ultimately, estimating the efficacy and radial cutoff of refractory carbon destruction mechanisms within the protoplanetary disk, including oxidation and photochemical destruction, may provide an explanation for the carbon content of planetary bodies in our solar system and how it relates to their place of origin.The authors thank the anonymous reviewer whose comments and suggestions improved this work. This material is based upon work supported by the National Science Foundation, via the Graduate Research Fellowship Program under Grant No. DGE-1144469 and the Astronomy and Astrophysics Research Grants Program under Grant No. AST-1514918.Regions of the disk where active depletion of refractory carbon occurs are determined by the physical conditions present. Figure <ref> aids in the direct comparison of the abundance of refractory carbon after 10^6 years for models shown inFigures <ref> and <ref> to the gas temperature and radiation fields from Figure <ref> for select radii. | http://arxiv.org/abs/1707.08982v1 | {
"authors": [
"Dana E. Anderson",
"Edwin A. Bergin",
"Geoffrey A. Blake",
"Fred J. Ciesla",
"Ruud Visser",
"Jeong-Eun Lee"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170727180731",
"title": "Destruction of Refractory Carbon in Protoplanetary Disks"
} |
Notes on the Kitaev ⇔ Tight-binding correspondenceextracted from a BSc project in Physics presented at Universidade de Aveiro Rui Carlos Andrade MartinsDepartamento de Física, Universidade de Aveiro July 14, 2017 =============================================================================================================================== In this project, we study the properties of a non-trivial topological system which exhibits localized edge states. In our study, we address the Kitaev chain, a one-dimensional chain of atoms deposited on top of a p-wave superconductor that induces superconductivity in the chain by proximity effect. We establish a correspondence between the Kitaev chain and a tight-binding lattice with a particular geometry for a particular case of the system parameters. This correspondence allows one to find the exact energy levels and eigenstates of the t=|Δ| Kitaev chain when μ=0 for an arbitrary chain size.roman arabicCHAPTER: INTRODUCTIONIn 1997, Alexei Y. Kitaev proposed the concept of topological quantum computation <cit.>. This theoretical model employs the properties of non-abelian anyons to construct the logic gates necessary for the computer to function. It also provides a way to avoid decoherence at the physical level since the encoding of information using a topological quantum computer is done in a non-local fashion. In mathematics, very briefly, the concept of topological equivalence classifies surfaces in ℝ^n by the number of holes in them <cit.>. Two objects are topologically equivalent if one can be transformed into the other by a homeomorphism <cit.>. For example, a spherical surface is topologically equivalent to the surface of an ellipsoid since one shape can be deformed into the other continuously. Topological invariants are quantities or objects preserved during an homeomorphism. In physics, there exist quantities which are directly connected to the topology of the system, and thus only change if it undergoes a transition between topologically different phases, meaning that these quantities remain unaltered by adiabatic changes in the parameters. In the systems we are going to study, a smooth deformation of the physical parameters which leave the band gap finite in the band structure do not change the topological phase of the system. In other words, the topological phase shifts whenever the gap closes. Knowing this, we can identify topological transitions by looking at the band structure of such systems. Certain topologically invariant quantities, such as the Zak's phase <cit.>, allow to distinguish between topologically trivial and non-trivial phases of the system. A physical signature of non-trivial topological phases is given by the presence of localized states at the edges of a system with open-boundaries, which are topologically protected against perturbations. The Majorana states that we will study in this work are an example of such topological states.In this project we will study one of the simplest model in which Majorana fermions, which are quasiparticles that obey non-abelian statistics, appear. This system consists of a one-dimensional tight-binding chain deposited on top of a effectively spinless p-wave superconductor, and was introduced originally by Alexei Y. Kitaev <cit.>. As shown by Fu and Kane <cit.> such a system might occur for the surface states of a topological insulator when brought intotunneling contact with an ordinary s-wave superconductor . In order to lay the foundations for our analysis, we will first introduce the tight-binding approximation and study simple 1D tight-biding systems in Chapter 2, finding their electronic band structure. We will consider a tight-binding Hamiltonian for a chain of atoms with open boundary conditions and for a quantum ring, which is a chain of atoms with periodic boundary conditions, both with and without magnetic flux. In Chapter 3 we will introduce the Kitaev chain and study some of its properties, such as its topological characterization and the existence of Majorana end states. Still in the same chapter we establish a correspondence between the Kitaev chain system and a specific tight-binding problem, for a special case of the parameters. We will study this correspondence for the particular case of the three site Kitaev chain, providing the groundwork required to establish the correspondence for a chain with a larger number of sites. In the last section of this chapter, we provide a set of rules that allows one to construct the tight-binding lattices and relate the solutions of both problems. Finally, we present the conclusions on Chapter 4. CHAPTER: THE TIGHT-BINDING APPROXIMATIONIn this Chapter we will introduce the tight-binding approximation and study 1D tight-binding models with both open and periodic boundary conditions and also in the presence of flux in the latter case. This Chapter is based on references <cit.> and <cit.>. § THE METHODThe tight-binding model is an approach to the calculation of the electronic band structure of a lattice that relies on the fact that electrons are either tightly bound to the atoms or that these are well separated. Very briefly, the method consists of expressing the eigenstates of a lattice in terms of the localized atomic orbitals of the atoms.Let us start by defining localized atomic orbitals ϕ_n(r⃗-R⃗_j), where R⃗_j is the lattice vector correspondent to site j, r⃗ is the position vector. These are the eigenstates of the atomic Hamiltonian H_a(R⃗_j), which is the Hamiltonian of a non-interacting atom located at site j. We haveH_a(R⃗_j)ϕ_n(r⃗-R⃗_j)=ϵ_nϕ_n(r⃗-R⃗_j) ,where ϵ_n is the energy and n stands for the set of all necessary quantum numbers to fully describe an orbital of the atom. In the context of our work, we will only consider one type ofatomic orbital to be important to model the transport properties of the system, so we will drop the index n. Now, we assume that these atomic orbitals quickly decay so that they overlap only slightly, i.e, they are almost orthogonal, and so, they constitute a complete set of orthogonal functions. One can, therefore, describe these functions in terms of the Wannier functions, Φ(R⃗_j), of the lattice .Bloch's theorem states the eigenfunctions of a system with a periodic repeating potential are Bloch's functions, which are plane waves with the same periodicity as the lattice. These functions can be expanded on the basis of the Wannier functions as follows |ψ_k⃗⟩=∑_je^ik⃗·R⃗_⃗j⃗|Φ(R⃗_⃗j⃗)⟩where k⃗ is a wave momentum and a vector from the reciprocal lattice contained in the first Brillouin zone. Eigenfunctions corresponding to different k⃗ vectors are orthogonal. As we said before, we can approximate the atomic orbitals by this set of Wannier functions (such that the atomic orbital of site jwill be approximated by the Wannier function corresponding to site j) since they constitute a complete set of orthogonal functions, |ψ_k⃗⟩=∑_je^ik⃗·R⃗_⃗j⃗|ϕ(r⃗-R⃗_j)⟩.In the context of our work we will only focus on one-dimensional systems. With this in mind, let us writethe solution of the one-dimensional problem in second quantization formalism using the result above, c_k^†|0⟩=∑_j e^ikjc_j^†|0⟩where c_k^† is the creation operator of an extended electronic state, a Bloch function, with a well defined momentum k, c_j^† is the creation operator of a localized electron at site j, a Wannier state, and |0⟩ is the vacuum state of the system. Notice that we dropped the vector notation for k given that we are considering 1D systems. Also, we assumed the lattice parameter a to be 1, so that k⃗·R⃗_⃗j⃗=kja=kj.Having stated all this, we are ready to write the tight-binding Hamiltonian of a lattice in second quantization. One can write the lattice Hamiltonian as H=∑_R_j H_W (R_j)+ΔH,where H_W (R_j) is a Hamiltonian whose eigenfunctions are Wannier functions centered on R_j and ΔH is a correction term associated with the overlap between atomic orbitals, necessary to reproduce the full Hamiltonian. As we said before, we assume that only one orbital of the atom is present, and therefore, only one Wannier state is important to describe the transport properties of the system. With this in mind, we define -t_i,j = ⟨0|c_iΔHc_j^†|0⟩=∫ dr ϕ(r⃗-R⃗_i)^*ΔHϕ(r⃗-R⃗_j), ϵ_j = ⟨0|c_j∑_j^' H_W (R_j^')c_j^†|0⟩ =∫ dr ϕ(r⃗-R⃗_j)^*∑_j^' H_W (R_j^')ϕ(r⃗-R⃗_j) = ∫ dr ϕ(r⃗-R⃗_j)^*δ_j j^'H_W (R_j^')ϕ(r⃗-R⃗_j),where we took into account that the atomic orbital of site j was approximated by thethe Wannier state of site j, so that ϕ(r⃗-R⃗_j)→ c_j^†|0⟩. Now we can write the Hamiltonian in second quantization formalism as follows, H=∑_i,j(δ_ijϵ_j-t_i,j)c_i^† c_j,where ϵ_j is the energy associated with an electron occupying the orbital of the atomat site j and t_i,j is the hopping parameter, which in turn is the kinetic energy parameter associated with the probability of transition of an electron from the atom at site j to the atom at site i.Finally, we will assume the system to be isotropic, as we have been implying already, since the atomic orbital functions are similar and only differ in where they are centered, and we have ϵ_j=ϵ. Moreover, we only consider hopping between nearest-neighboring sites, so we have { t_i,j=t, |i-j|=1,t_i,j=0, otherwise. . This way, the Hamiltonian becomes H=ϵ∑_jc_j^† c_j-t∑_j(c_j^† c_j+1+c^†_j+1c_j). § QUANTUM RING - CHAIN WITH PERIODIC BOUNDARY CONDITIONS Now we focus on studying a quantum ring, which is a structure composed by N atoms in a periodic arrangement.§.§ Quantum ring without flux We consider a system described by the following Hamiltonian H=-t∑_j=1^N(c_j^† c_j+1+c^†_j+1c_j), where we assume periodic boundary conditions, such that site 1 is equivalent to site N+1, and that the on-site energy ϵ is 0. Since the Hamiltonian has translational invariance it is diagonalized by Bloch's states, therefore we perform a Fourier transform to go into k-space, c_j^†=1√(N)∑_ke^-ikjc_k^† , c_j=1√(N)∑_ke^ikjc_k,where, due to periodic boundary conditions, k=2π n/N, which results from the equalityc_1^†=1√(N)∑_ke^-ikc_k^†=1√(N)∑_ke^-ik(N+1)c_k^†=c_N+1^†,where { n=-N/2,-N/2+1,...,N/2-1,when N is evenn=-N-1/2,-N-1/2+1,...,N-1/2,when N is odd.so that k is contained in the first Brillouin zone ]π/a,π/a]. By using the orthogonality relation ∑_je^i(k-k^')j=δ_kk^'N ,we arrive at the following diagonalized Hamiltonian H=-2t∑_kcos(k)c_k^† c_k,and the band structure of the quantum ring. In Fig. <ref>, we plot the energy levels of a quantum ring with ten sites and in Fig. <ref> and <ref>, we plot the amplitude distribution of the ground and highest energy eigenstates respectively. §.§ Quantum ring with flux Now we examine the ring while it is under a perpendicular magnetic field. Let us consider the magnetic field pierces the ring in a region where the amplitude of the wavefunction is almost null, so no forces are exerted on the electrons. Even so, due to the Aharonov-Bohm effect <cit.>, the wave function of the electron when the ring is under a magnetic field is altered by a phase compared to the wavefunction of the ring in the absence of a magnetic field. The phase shift of the wavefunction when the electron travels along a path L is called the Peierls phase, and it is given by exp[ iq/ħ∫_LA⃗· dl⃗]. where q is the electron charge, A⃗ is the vector potential and ħ the reduced Planck's constant. We can relate the Wannier state of an electron when the ring is under a magnetic field, |j⟩_B, to the Wannier state in the absence of field |j⟩_0 by |j⟩_B=exp[ iq/ħ∫_j-1^jA⃗· dl⃗]|j⟩_0where the path of integration will be the straight line connecting nearest-neighbors. Since the system must be gauge invariant, we consider the gauge where the Peierls phase is the same on every path L between nearest-neighbors, in order to preserve the translational symmetry, so that ∫_j-1^jA⃗· dl⃗=1/N∫_1^NA⃗· dl⃗=ϕ/N,where ϕ is the total magnetic flux piercing the ring. Taking into account this choice of gauge, the Hamiltonian becomes H=-t∑_j=1^N e^iϕ^'/Nc_j^† c_j+1+e^-iϕ^'/Nc^†_j+1c_j, where ϕ^'=2πϕ/ϕ_0 is the reduced flux and ϕ_0=h/q is the flux quantum. Considering this choice of gauge, the Hamiltonian has translational invariance and so Bloch waves are its eigenstates. The method to find the solution follows the same procedure of the case without flux, and we get H=-2t∑_kcos(k+ϕ^'/N)c_k^† c_k. In Fig. <ref>, we plot the energy as a function of the reduced magnetic flux ϕ^', for an open chain with N=10 and hopping parameter t=-1.§ CHAIN WITH OPEN BOUNDARY CONDITIONS Now we study an N-site tight-binding chain with open boundaries, meaning that the amplitude of the wavefunction must be zero beyond the ends of the chain. We consider the following Hamiltonian H=-t∑_j=1^N-1(c_j^† c_j+1+c^†_j+1c_j). The wavefunctions in the bulk of the chain should be similar to the case of the quantum ring (at least, if we consider a big enough open chain). However, due to the boundary conditions, only the wavefunctions that obey the following conditions are allowed {E_k^'ψ_1=-tψ_2 E_k^'ψ_N=-tψ_N-1. where ψ_j is the amplitude of the eigenfunction at site j and E_k^' its energy. To solve this problem, we add two virtual sites to the chain, site j=0 and site j=N+1 and impose that the amplitude of the wavefunction at these sites must be zero. We assume that the eigenfunctions are linear combinations of Bloch waves of the same energy and opposite momenta of the form|k,open⟩=1/√(2)(|k⟩-|-k⟩)=C(∑_jsin(kj)c^†_j)|∅⟩where |k,open⟩ is the eigenstate of the system, |k⟩ is a Bloch wave with momentum k, |∅⟩ is the vacuum and C is some normalization constant. These linear combinations originate standing waves that allow both conditions to be satisfied. On the one hand, they allow the amplitude of the wavefunctions to vanish at the border sites and, on the other hand, they are still eigenstates of the bulk Hamiltonian, since they combine Bloch waves with the same energy.The only thing left to find the solution is to determine the normalization constant C, that results from the fact that only certain k are allowed as a solution. We should distinguish between the case where the solutions are standing waves with odd parity and with even parity with relation to the center of the chain. Let us start by taking into account the odd parity solutions (see Fig. <ref>). In this case, the problem corresponds to one of a tight-binding ring with N+1 sites, with boundary conditions |0⟩=|N+1⟩ and the added condition that the amplitude of the wavefunction is zero at site j=0. We obtain the quantization for |k,open⟩ by applying the boundary conditionsin(k(N+1))=2π m , m=0,...,Nthat results in k=2π/N+1m. However, we should recall that every k has a direct correspondence to a k contained in the first Brillouin zone, meaning that they represent the same wavefunction. By taking a look at all k contained in the first Brillouin zone it is clear that only half of the solutions are present, since every pair k and -k correspond to the same wavefunction solution, since |k,open⟩=-|-k,open⟩. Besides, k=0 is not a solution. So, until now, we have obtained the set of unique solutions k=2π/N+1m^', where m^'=1,..,N/2 (we have excluded k=0 and assumed N even) and the wave functions are |k,open⟩=√(2/N+1)(∑_jsin(kj)c^†_j)|∅⟩Now, we take into account the even parity solutions (see Fig. <ref>). In this case, the standing wave will not repeat itself after it reaches the boundary, but would repeat itself were it to travel 2(N+1) sites. In this sense, the problem now corresponds to one of a tight-binding ring of 2(N+1) sites, with boundary conditions |0⟩=|2(N+1)⟩ and with the added condition that the amplitude of the wavefunction at site N+1 is zero, which will naturally be obeyed by any stationary wave of the form |k,open⟩. We find the quantization by applying the boundary condition sin(k(2(N+1)))=2π m , n=0,...,2N+1that results in k=2π/2(N+1)n. However, same as above, half of the solutions are repeated, so we take those out, and k becomes k=2π/2(N+1)m^', where m^'=1,...,N. Moreover, we have already took into consideration some of these k values in the odd parity situation, so we take those out, which corresponds to taking out the even m^', so we get m^'=1,3,...,N-1. The wavefunctions for the even parity standing waves will be |k,open⟩=√(2/N+1)(∑_jsin(kj)c^†_j)|∅⟩.In Fig. <ref> we plot the energy levels for a open chain with ten sites in units of t and in Fig. <ref> we plot the amplitude distribution of the ground state for the same chain.CHAPTER: KITAEV CHAIN"Implementing a full-scale quantum computer is a major challenge to modern physics and engineering" <cit.>. One of the main problems is that it is very difficult to isolate quantum states from unwanted perturbations, and so the information being manipulated is subject to error. Although this problem is also recurrent in classical computation, error-correcting codes are enough to keep the information coherent. Quantum decoherence is the process of loosing information from a system to the environment and, to perform quantum computation, one must be able to preserve coherence of the states being manipulated. Several quantum error-correcting algorithms can be used to cover the effects of quantum decoherence. However, the problem might be solved at the physical level, by constructing decoherence protected degrees of freedom. Motivated by this, in this chapter we will study the Kitaev chain. The system is a simple toy model that exhibits unpaired Majorana fermions. It consists of a one-dimensional tight-binding chain, a quantum wire, that lies on the surface of a p-wave type superconductor, which induces a non standard superconducting term in the Hamiltonian, that couples only electrons with the same spin, so the problem effectively becomes one of spinless fermions, meaning that the site occupation can only be 0 or 1. § MAJORANA FERMIONS Derivation We start by considering the following Hamiltonian that describes an N site tight-binding chain of effectively spinless fermions with an induced superconducting pairing term, H = -t∑^N-1_j=1(c^†_j+1c_j+c^†_jc_j+1)+∑^N-1_j=1(Δ c_jc_j+1+Δ^* c^†_j+1c^†_j)-μ∑^N_j=1c^†_jc_j,where t is the hopping parameter, μ is the chemical potential and Δ is the p-wave superconducting gap. For the special case t=Δ and μ=0, the Hamiltonian simplifies to H=-t∑^N-1_j=1( c^†_jc_j+1+c^†_j+1c_j-c^†_j+1c^†_j-c_jc_j+1).Kitaev proposed a method to diagonalize the Hamiltonian by defining the following Majorana operators <cit.> γ_j,1=c_j+c^†_j ,γ_j,2=i(c^†_j-c_j).Each fermionic operator corresponds to a combination of Majorana operators, c_j=1/2(γ_j,1+iγ_j,2), c_j^†=1/2(γ_j,1-iγ_j,2) .To an unpaired Majorana operator, i.e., one decoupled from the system, one calls a Majorana fermion. Majorana operators correspond to particles which are their own anti-particle (γ^†≡γ) and obey the following anti-commutation relation, {γ_i,α,γ_j,β}=2δ_ijδ_αβ.Using (<ref>) and the usual fermionic anti-commutation relations {c_i,c^†_j}=δ_ij, {c_i,c_j}={c^†_i,c^†_j}=0,one arrives at the following Hamiltonian, written in the basis of Majorana operators, H=it∑^N-1_i=1γ_i,2γ_i+1,1.Now we perform another change in basis by defining new fermionic operators c̃_j that are a linear combinations of Majorana operators from nearest-neighboring sites, as shown at the bottom of Fig. <ref> , c̃_j=1/2(γ_j+1,1+iγ_j,2) ,c̃_j^†=1/2(γ_j+1,1-iγ_j,2).The following inverse relations hold γ_j+1,1=c̃^†_j+c̃_j ,γ_j,2=i(c̃^†_j-c̃_j).In the basis of these new fermionic operators, the Hamiltonian becomes diagonalized, H=it∑^N-1_j=1γ_j,2γ_j+1,1=-2t∑^N-1_j=1(c̃^†_jc̃_j-1/2).These new fermionic operators represent quasiparticles of the system with energy -2t. We notice that we only need (N-1) quasiparticle fermionic operators to write the Hamiltonian in this new basis, so one more state must exist. This state can be constructed from the unpaired Majorana operators γ_N,2 and γ_1,1, which are missing from the Hamiltonian in (<ref>) and, since they are decoupled from the system, they combine to create a Majorana state of zero energy. In the quasiparticle fermionic basis, these Majorana operators can be written as c̃_M=1/2(γ_N,2+iγ_1,1), c̃^†_M=1/2(γ_N,2-iγ_1,1).The Majorana state givenby c_M^† is a highly non-local state with finite amplitude only at the edges of the chain. Due to its non-local nature, this state is immune to perturbations.We may now look at the problem in a different way, where we have N-1 sites, which can be occupied by quasiparticles, each adding -2t energy when occupied and another extra site, which can be occupied by the fermion defined in equation (<ref>), with no energetic cost. To fully characterize the state of the system, we must specify the action of every quasiparticle creation operator, including c̃^†_M, on the vacuum of the system. Since the operators in equation (<ref>) do not show up in the Hamiltonian, hence do not affect the energy of the state on which they act, every energy should always be, at least, doubly degenerate.The above argument has only been made for a very special case. However, one can show that Majorana end states still exist as long as the condition |μ|<2t is verified <cit.>. In the general case, Majorana end states are not completely localized at the edges of the chain and exhibit a decaying tail to the bulk.Computational Results We will now, very briefly, show some numerical results obtained using Matlab for the Kitaev chain. We will show a plot of the energy levels and energy dispersion relation with t for the N=3, N=4 and N=5 Kitaev chain and draw simple conclusions from them. The simulations are made for Δ=1 and μ=0. By looking at the energy levels for the three chains of Fig. <ref>, Fig. <ref>, and Fig. <ref>, we can distinguish between three regimes where we know the exact solution of the problem. One regime, already analyzed in this section, is where t=Δ and μ=0. We notice that, in this regime, for any N, the energy degeneracy reaches its maximum, being equal to the number of sites of the chain. This is to be expected if we look at the Hamiltonian in equation (<ref>) since by adding one c̃_j^† quasiparticle the energy of the system decreases by 2t and there only exists N-1 quasiparticle operators for a chain with N sites, so one can only decrease the energy N-1 times. Besides, in this regime, as will be studied from section <ref> on, the system as a correspondence between two independent and identical tight-binding lattices with 2^N-1 sites. The other regime occurs for t=0. Here we can conclude that the system will correspond to a tight-binding problem constituted of two independent lattices, by the same reasoning as in section <ref>. The lattices will have the shape of Fig. <ref> for N=3. The correspondent tight-binding lattice would be the same were we to analyze the regime Δ→∞. Finally, the last regime occurs when t→∞ and in this case the correspondent tight-binding lattice will have the form of Fig. <ref> for N=3. The correspondent tight-binding lattice would be the same were we looking at the regime Δ=0. In Fig. <ref>, we plot the energy spectrum for N=4 as a function of t/Δ, where the different regimes are indicated by the vertical lines, and in Fig. <ref>, we plot the energy levels in units of t for the case t=20Δ. We verify that the energy levels in units of t when t→∞ tend to the energy levels in units of Δ when t=0 (compare Fig. <ref> with Fig. <ref>), which is expected because the lattices in Fig. <ref> and Fig. <ref> should have the same energy levels.For every regime, it is straightforward to find the eigenfunctions and eigenvalues of the Kitaev chain because of the correspondence with simple tight-binding systems. In the following sections, we analyze the Kitaev chain and study the first regime (t=|Δ|), providing the rules of construction for the correspondent tight-binding lattices for any N.§ BAND STRUCTURE OF THE KITAEV CHAINIn this section, we will calculate the band structure of the bulk of the Kitaev chain. We start by considering the full Kitaev Hamiltonian with periodic boundary conditions, H =∑^N-1_j=1( -tc_j^† c_j+1+Δ c_j+1^†c_j^† +H.c.)-μ∑_j=1^Nc_j^† c_j.Using equations (<ref>) and the orthogonality relation (<ref>), we can write the Hamiltonian in k-space, H=∑_k (-te^ikc^†_k c_k+Δ e^-ikc^†_k c^†_-k+H.c)-μ∑_k c^†_k c_k.We can avoid double counting and further simplify the result by restricting the sum to k>0, ∑_k-te^ik c^†_k c_k=∑_k>0 -te^ikc^†_k c_k-te^-ikc^†_-k c_-k-tc^†_k=0 c_k=0= ∑_k>0-te^ikc^†_k c_k-te^-ik(1-c_-kc^†_-k)= ∑_k>0 -te^ikc^†_k c_k+te^-ikc_-kc^†_-k,∑_k Δ e^-ik c^†_k c_-k^†=∑_k>0Δ e^-ikc^†_k c_-k^†+Δ e^ikc^†_-k c_k^†+Δ c^†_k=0 c^†_k=0= ∑_k>0 Δ e^-ikc^†_k c_-k^†+Δ e^ik(1-c^†_k c_-k^†)=∑_k>0 -2iΔsin(k)c^†_k c_-k^†,where we dropped the surface term at k=0 and any other constant terms. Analogously, we restrict the sum in the other terms of the Hamiltonian and we get H=∑_k>0[ c_k^†c_-k ][ -2tcos(k)-μ-2iΔsin(k); 2iΔsin(k)2tcos(k)+μ ][c_k; c_-k^† ].The only thing left to find the band structure is to diagonalize the Hamiltonian above. In the diagonal basis we get H=∑_k>0[ a_k^† b_k^† ][ -√((2tcos(k)+μ)^2+4Δ^2sin^2(k)) 0; 0√((2tcos(k)+μ)^2+4Δ^2sin^2(k)) ][ a_k; b_k ],where the operators a_k^† and b_k^† are linear combinations of c_k^† and c_-k given by a_k^†=1/√(2)(c_k^†+e^iϕ_1(k)c_-k) , b_k^†=1/√(2)(c_k^†+e^iϕ_2(k)c_-k),where ϕ_1(k) and ϕ_2(k) are phases with an elaborate dependence on k. In Fig. <ref> and Fig. <ref> we plot the band structure in units of t for the case μ=2t and μ=-2t respectively.In the regime where t=|Δ| we can calculate Zak's phase, which is a topological invariant, that will allow us to characterize if the system is in a topological trivial or non-trivial phase. Zak's phase is defined byΓ_β=i∫_-π^π dk⟨β(k)|d/dkβ(k)|⟩where |β(k)⟩=β_k^†|0⟩, with β=a,b. We get the following values {Γ_a=Γ_b=0,for |μ|>2t, Γ_a=Γ_b=π,for |μ|<2t..The π shift between these two μ regimes is indicative of a transition from a topological trivial to a non-trivial phase.§ KITAEV ⇔ TIGHT-BINDING CORRESPONDENCE FOR A THREE-SITE CHAIN A property of the system we have been studying is that it has a correspondence to a periodic tight-binding one-dimensional lattice. In order to establish this correspondence, we will consider a Kitaev chain of only three sites. We will then generalize for any number of sites N of the Kitaev chain.We consider the Hamiltonian of the Kitaev chain for N=3 and μ=0, H = -t∑^2_j=1c^†_j+1c_j+H.c.+Δ∑^2_j=1c_jc_j+1+H.c.The set of states B={|000⟩,|110⟩,|101⟩,|011⟩,|100⟩,|010⟩,|001⟩,|111⟩} constitutes an orthogonal basis in which we can express the Hamiltonian[We used the notation where, for example, the state |111⟩ is c_1^† c_2^† c_3^†|000⟩, where |000⟩ is the vacuum.]. In matrix form the Hamiltonian is given by H=[0 -Δ0 -Δ0000; -Δ^*0 -t00000;0 -t0 -t0000; -Δ^*0 -t00000;00000 -t0 -Δ;0000 -t0 -t0;00000 -t0 -Δ;0000 -Δ^*0 -Δ^*0 ].Notice that only states with the same parity in the number of particles are coupled to each other. In this sense, there are two independent subspaces in the total Hilbert space, one consisting of states with an odd number of particles and another with an even number of particles, which we will call "odd" and "even" subspaces, respectively. Furthermore, if we take a closer look at the Hamiltonian, we can see that it is block diagonal with an identical periodic tight-binding Hamiltonian on each block. Due to these properties, we can establish a correspondence between the three-site Kitaev chain and two independent tight-binding rings with four sites each. The Hamiltonian of the system becomes H =H_even+H_odd,H_even= -Δ|1⟩⟨2|-t|2⟩⟨3|-t|3⟩⟨4|-Δ^*|4⟩⟨1|+H.c.,H_odd= -t|5⟩⟨6|-t|6⟩⟨7|-Δ|7⟩⟨8|-Δ^*|8⟩⟨5|+H.c.We have renamed the vectors in basis B to simplify the writing and so, hereafter, B={|1⟩,|2⟩,|3⟩,|4⟩,|5⟩,|6⟩,|7⟩,|8⟩}. A pictorial representation of the system can be seen in Fig. <ref>.Having made the correspondence between both problems, we now find out the specific the correspondences between the states of the tight-binding chain and the states of the Kitaev chain. We will study only the case where t=|Δ|, so Δ=te^iϕ, where ϕ is the superconducting phase. Let us start by finding the eigenstates of the tight-binding problem.Eigenstates of the tight-binding rings The eigenstates of an Hamiltonian with translational invariance are obtained using Bloch's theorem, as it was already discussed in Chapter 2. The Hamiltonian of the tight-binding rings will have translational invariance if we perform the following gauge transformations e^iϕ|1⟩→|1̃⟩ , e^-iϕ|8⟩→|8̃⟩.The new basis becomes B̃={|1̃⟩,|2⟩,|3⟩,|4⟩,|5⟩,|6⟩,|7⟩,|8̃⟩} and we define B̃_even={|1̃⟩,|2⟩,|3⟩,|4⟩} and B̃_odd={|5⟩,|6⟩,|7⟩,|8̃⟩}. Using Bloch's theorem, the eigenstates of ring "even" and ring "odd" are, respectively, |k_even⟩=1/2∑_j_1e^ik_evenj_1|j_1⟩ ,|k_odd⟩=1/2∑_j_2e^ik_oddj_2|j_2⟩,where j_1 and j_2 run through all the states in basis B̃_even and B̃_odd, respectively, and k_even and k_odd are contained in the first Brillouin zone.Eigenstates of the Kitaev chain Now let us find the states of the Kitaev chain and express them in basis B. As it was discussed above, the state of the Kitaev chain is obtained by specifying the action of the quasiparticle creation operators in the vacuum of the system.Knowing this, let us start by finding the vacuum state written in basis B and build the other states of the system from that. Hereafter, we will drop the phase in Δ, so ϕ=0, because it does not make a difference in the solutions of the problem. In fact, we do not need to worry about it because it can be absorbed in the definition of the Majorana operators,γ_j,1=e^-iϕ/2c^†_j+e^iϕ/2c_j, γ_j,2=ie^-iϕ/2c^†_j-ie^iϕ/2c_j, yielding the same solution as in <ref>. Finding the Vacuum State The vacuum state |∅_vac⟩ obeys the following set of equations c̃_1|∅_vac⟩=0, c̃_2|∅_vac⟩=0, ... c̃_N-1|∅_vac⟩=0, c̃_M|∅_vac⟩=0.To find the solution written in terms of the vectors of basis B, we start by expressing the c̃_j operators in terms of the original fermionic operators c_j and c_j^†: c̃_j=1/2(γ_j+1,1+iγ_j,2)=1/2(c_j+1^†+c_j+1-c_j^†+c_j); c̃_M=i/2(c_N^†-c_N+c_1^†+c_1).Since basis B is a complete set of orthonormal states, we are able to express the vacuum as |∅_vac⟩=∑_j ⟨j|∅_vac||%s⟩⟩j,where j runs through all the basis vectors and ⟨j|∅_vac|$⟩ are coefficients. In matrix form and written in baseB, the vacuum is |∅_vac⟩=[ a; b; c; d; e; f; g; h ],⟨∅_vac|∅_vac|=⟩1where the letters are coefficients. Because we know the action of the original fermionic operators on basisBwe can solve the following equations for the case of the Kitaev chain of 3 sites c̃_1|∅_vac⟩=1/2(c_2^†+c_2-c_1^†+c_1)(∑_j ⟨∅_vac|j||%s⟩⟩j)=0, c̃_2|∅_vac⟩=1/2(c_3^†+c_3-c_2^†+c_2)(∑_j ⟨∅_vac|j||%s⟩⟩j)=0, c̃_M|∅_vac⟩=1/2(c_3^†-c_3+c_1^†+c_1)(∑_j ⟨∅_vac|j||%s⟩⟩j)=0.The solution for this set of equations is[We chose a descending order of operation of the fermionic operators, so that, for example, we act with c_2^† before we act with c_1^†] { a=-bc=-de=-fg=-h . ,{ a=-db=-ce=-hf=-g . ,{ a=-cb=-de=gf=h . .The only possible normalized solution, apart from a global phase factor, is given by |∅_vac⟩=1/2[0;0;0;0; +1; -1; +1; -1 ],Eigenstates of the system Since we are now working on the eigenbasis of the Hamiltonian, we can construct the eigenstates of the system by acting in all possible configurations with the creation operators on the vacuum state. The eigenstates will be, |∅_vac⟩, c̃_M^†|∅_vac⟩, c̃_1^†|∅_vac⟩, c̃_M^†c̃_1^†|∅_vac⟩... c̃_M^†c̃_2^†c̃_1^†|∅_vac⟩.Written in matrix form, the states are, |∅_vac⟩=1/2[0;0;0;0; +1; -1; +1; -1 ],c̃_M^†|∅_vac⟩=i/2[ -1; +1; -1; +1;0;0;0;0 ],c̃_1^†|∅_vac⟩=1/2[ -1; -1; +1; +1;0;0;0;0 ],c̃_1^†c̃_M^†|∅_vac⟩=i/2[0;0;0;0; -1; -1; +1; +1 ] c̃_2^†|∅_vac⟩=1/2[ , +1; -1; -1; +1;0;0;0;0 ],c̃_2^†c̃_M^†|∅_vac⟩=i/2[0;0;0;0; +1; -1; -1; +1 ],c̃_1^†c̃_2^†|∅_vac⟩=1/2[0;0;0;0; +1; +1; +1; +1 ],c̃_1^†c̃_2^†c̃_M^†|∅_vac⟩=i/2[ -1; -1; -1; -1;0;0;0;0 ] Notice that every energy is at least doubly degenerate, as expected, so any linear combination of eigenstates with the same energy will also be an eigenstate. Knowing this, we can explicitly establish the correspondence between the states above and the states of the tight binding rings constructed for the "even" and "odd" subspaces. State-site correspondence The states of the tight binding rings are the following[Notice that we dropped the phase factor in Δ, so |1⟩=|1̃⟩], |k_even=0⟩ = 1/2(|1⟩+|2⟩+|3⟩+|4⟩),|k_even=π/2⟩=1/2(i|1⟩-|2⟩-i|3⟩+|4⟩),|k_even=π⟩ = 1/2(-|1⟩+|2⟩-|3⟩+|4⟩),|k_even=-π/2⟩=1/2(-i|1⟩-|2⟩+i|3⟩+|4⟩),|k_odd=0⟩ = 1/2(|5⟩+|6⟩+|7⟩+|8⟩),|k_odd=π/2⟩=1/2(i|5⟩-|6⟩-i|7⟩+|8⟩),|k_odd=π⟩ = 1/2(-|5⟩+|6⟩-|7⟩+|8⟩),|k_odd=-π/2⟩=1/2(-i|5⟩-|6⟩+i|7⟩+|8⟩).Just by looking at the states above one can directly establish the following correspondences, -i|k_even=0⟩ → c̃_1^†c̃_2^†c̃_M^†|∅_vac⟩,|k_odd=0⟩→c̃_1^†c̃_2^†|∅_vac⟩,-i|k_even=π⟩ → c̃_M^†|∅_vac⟩,-|k_odd=π⟩→|∅_vac⟩,and, since any linear combination of eigenstateswith the same energy are also eigenstates, one possible case of the remaining correspondences are given by 1+i/2 |k_even=π/2⟩ +1-i/2|k_even=-π/2⟩→c̃_1^†|∅_vac⟩,1-i/2 |k_even=π/2⟩ +1+i/2|k_even=-π/2⟩→c̃_2^†|∅_vac⟩,-1+i/2 |k_odd=π/2⟩ +1+i/2|k_odd=-π/2⟩→c̃_1^†c̃_̃M̃^†|∅_vac⟩,1+i/2 |k_odd=π/2⟩ +-1+i/2|k_odd=-π/2⟩→c̃_2^†c̃_̃M̃^†|∅_vac⟩.In general, the eigenstates of the Kitaev chain correspond to linear combinations of the eigenstates of the tight-binding ring with the same energy.§ GENERALIZATION OF THE KITAEV ⇔ TIGHT-BINDING CORRESPONDENCE In this section we will generalize the concept of establishing a correspondence between the Kitaev chain problem and a tight-binding problem for any number of sitesN. This will be done by providing a rule of construction of the tight-binding lattices and a rule of attributing the states of the original basis of the Kitaev chain to the sites of the tight binding lattices.The general tight-binding problem that corresponds to anNsite Kitaev chain consists of two independent and identical lattices that have the form of2^N-1side regular polygons. The sites of one of these lattices are associated only with states of even parity and the sites of the other are associated only with states of odd parity and, since they are identical and share the same rules of construction, we will hereafter be describing only the "even" lattice, keeping in mind that an identical lattice describes the "odd" subspace. The lattices are always constituted by four unit cells of2^N-3sites each.One interesting property of these lattices, and also one that will allow us to describe them for anyN, is that the unit cell of the lattice correspondent to theN+1sites Kitaev problem is obtained by cutting in half the lattice of theNsites Kitaev problem, as can be checked by looking at Fig. <ref>. We distinguish between three types of connections in these lattices. The first type connects nearest-neighboring sites in the same unit cell, the second connects sites in the same unit cell that are not nearest-neighbors and the third connects different unit cells, each of these connections being represented in a different color in Fig. <ref>.Before we present the generalized Hamiltonian that describes the tight-binding lattices, let us first provide the rule that allows one to attribute the states of the Kitaev chain to the sites of these lattices. This rule allows us to discover the state-site correspondences for theN+1case knowing them for theNcase. We start by considering one of the lattices forN=3with the state-site correspondences already made in section <ref>, as shown in the left square of Fig. <ref>. If we now add one zero to the end of the label of every state (see right square of Fig. <ref>) the state-site correspondences obtained are still valid, only now they are linking states of theN=4Kitaev chain. When we add one site to the Kitaev chain each state of the basis couples with one more state, so there is still one connection missing for each site. However, there is only one possible state that can couple to each of the states of the right square in Fig. <ref>, since two out of three connections are already established, and these missing states will couple with each other in a similar fashion to the ones shown in the figure. So, we can imagine that we duplicate the square and then pull it outwards , so that each state couples with the one directly above it, as exemplified in Fig. <ref>. The label of the duplicated statesfollows the rule The state(...x0),links to(...y1){ifx=0 y=1 ifx=1 y=0 . .This is exemplified in Fig.<ref>. The only thing left to obtain theN=4tight-binding lattice of Fig. <ref> is to project this shape into a two dimensional plane. We do so by following a simple rule, which consists of taking one ofthe longest paths that starts in the first site of any unit cell and ends in the same site along the 3-dimensional shape created by pulling the square outwards and, by each site we pass through we attribute it in clockwise order to the vertexes of a2^N-1side polygon, which is precisely the shape of the tight-binding lattice of theN+1Kitaev chain. Let us follow this rule for the case ofN=4.We take the site correspondent to the vacuum state(0000)[Here, the notation(0000) stands for |0000⟩.], as the site where we start taking the path from. Now if we repeat the pattern up-clockwise turn-down-clockwise turn, along the shape at the left of Fig. <ref> and attribute each site we pass through to the polygon in clockwise order we get the state-site correspondences forN=4, as shown at the right of Fig. <ref>. To find the shape of theN=5tight-binding chain, show in Fig. <ref>, one has to follow the same pattern, that is, duplicate the octagon and pull it outwards, labeling the states by the rule (<ref>) and then take the longest path starting at the first site of any unit cell along the duplicated octagon and, when projected into a plane, we find theN=5shape of the tight-binding lattice. The process is repeated to find the shapes for higherN.Given this repeating pattern, it is possible to present a generalized Hamiltonian of the tight-binding lattices for any number of sitesNof the Kitaev chain, H=-t∑_l=1^4( ∑_i=1^N_uc-1(c^†_l,ic_l,i+1)+∑_i=1^N_uc(c^†_l,ic_l+1,N_uc+1-i)+ ∑_j=1^N_d-2∑_u=1^1/2d_j-1∑_s=0^(d_N_d+1-j)-1(c^†_l,u+sd_jc_l,d_j(1+s)+1-u) +H.c.),withN_ucthe number of sites in the unit cell,d_jthe jth divisor ofN_uc,N_dis the number of divisors[The first divisor of N_uc is N_uc itself and 1 is the last divisor. For example, for N_uc=4 one has d_1=4, d_2=2, d_3=1, so N_d=3] andc^†_l,iis the creation operator associated to siteiof unit celll. This expression is only valid forN>4but the Hamiltonian's forN≤4are easy to find. The first sum in (<ref>) runs through all the unit cells. The first term in the parenthesis expresses the hopping between nearest-neighboring sites in the same unit cell whereas the second term in the parenthesis expresses the hopping between sites of different adjacent unit cells. Lastly, the third term in the parenthesis expresses the hopping between sites in the same unit cell that are not nearest-neighboring sites.§ BAND STRUCTURE OF THE TIGHT-BINDING CHAINFinally, in this section, we calculate the band structure of the tight-binding lattice by expressing the Hamiltonian ink-space for any number of sitesNof the Kitaev chain. Let us start by considering the Hamiltonian of theN=4tight-binding chain, H=-t∑_j=1^4(c_1,j^† c_2,j+1+c_1,j^† c_2,j+c_2,j^† c_1,j+1+H.c.),wherec^†_i,jis the creation operator associated to siteiof unit cellj. Using Bloch's theorem, we have c_1,j^†=1/√(N)∑_k e^-ikjc^†_1,k, c_2,j^†=1/√(N)∑_k e^-ikjc^†_2,k.Substituting these relations in the Hamiltonian of (<ref>) we get, H=-t∑_k (2cosk+1)c^†_1,kc_2,k+(2cosk+1)c^†_2,kc_1,k.Written in matrix representation the Hamiltonian becomes, H=-t∑_k>0[ c^†_1,k c^†_2,k ][ 0 α_k; α_k 0 ][ c_1,k; c_2,k ],whereα_k=1+2coskandk=2πn/4, wherenis the same as in (<ref>). Analogously, we can find the Hamiltonian for anyN. For instance, the results forN=5andN=6are H=-t[ 0 1 0 α_k; 1 0 α_k 0; 0 α_k 0 1; α_k 0 1 0; ] ,for N=5,H=-t[ 0 1 0 1 0 0 0 α_k; 1 0 1 0 0 0 α_k 0; 0 1 0 1 0 α_k 0 0; 1 0 1 0 α_k 0 0 0; 0 0 0 α_k 0 1 0 1; 0 0 α_k 0 1 0 1 0; 0 α_k 0 0 0 1 0 1; α_k 0 0 0 1 0 1 0; ] ,for N=6.Notice that the matrix representation of the Hamiltonian is following a pattern. The elements of the anti-diagonal are alwaysα_kand the Hamiltonian forN+1copies the shape of the Hamiltonian forNin the diagonal blocks (substitutingα_kfor 1). This repeating pattern allows us to write a generalized expression for the Hamiltonian ink-space for any givenN, H=-t( α_k(∏_⊗^N-3σ_x)+∑_i=1^N-4((∏_⊗^iσ_x)⊗(∏_⊗^N-3-iσ_0)))whereσ_0is the2×2identity matrix,σ_xis the first Pauli matrix and the notation∏_⊗^Nσ_xstands forσ_x⊗σ_x...⊗σ_x N times. Regardless of theNconsidered, the correspondent tight-binding lattices will always be periodic with four unit cells (see Fig. <ref>). Therefore, the allowedk-states in the band structure for everyNare alwaysk=0,±π/2,π. In Fig. <ref>, we plot the energy spectrum of one of the tight-binding lattices correspondent to theN=5Kitaev chain. The energies at the allowedk-states are consistent with those found in Fig. <ref> fort=Δ, as expected.CHAPTER: CONCLUSIONIn this project we studied the Kitaev chain, a simple toy model that exhibits unpaired Majorana fermions. We started by taking Kitaev's approach to find the exact solution of the problem for the caset=|Δ|, where we found the presence of non-local zero energy states. We then studied the energy spectrum of the bulk of the chain, drawing conclusions on the different topological phases of the system. We performed numerical calculations to explore the energy levels of the chain for different parameterst,Δandμ, and we concluded that there are three regimes where we know the exact solution of the problem. We further explored the regimet=|Δ|, making a correspondence between the Kitaev chain forN=3and a tight-binding problem consisting of two identical and independent rings with four sites each. Building on that, we generalized the correspondence between the Kitaev chain and a tight-binding problem for an arbitrary number of sites, providing the rules of construction of the tight-binding lattices. Finally, we studied the energy spectrum of the tight-binding lattices and presented a general expression for the matrix representation of the Hamiltonian ink-space.The correspondence of the Kitaev chain with two tight-binding rings will allow, in principle, to understand more easily the modifications of the behavior of the Kitaev chain due to possible perturbations. In particular, it would be interesting to find a perturbation of the Kitaev chain that maps onto a magnetic flux in the tight-binding rings. unsrt | http://arxiv.org/abs/1707.08930v1 | {
"authors": [
"Rui Carlos Andrade Martins",
"Anselmo Miguel Magalhães Marques",
"Ricardo Assis Guimarães Dias"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727165119",
"title": "Notes on the Kitaev \\leftrightarrow Tight-binding correspondence"
} |
[1]#1 | http://arxiv.org/abs/1707.08456v1 | {
"authors": [
"Marek Mozrzymas",
"Michał Studziński",
"Sergii Strelchuk",
"Michał Horodecki"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170726141528",
"title": "Optimal Port-based Teleportation"
} |
Some Extensions of the Crouzeix-Palencia Result Trevor Caldwell Anne Greenbaum Kenan Li University of Washington, Applied Mathematics Dept., Box 353925, Seattle, WA 98195.This work was supported in part by NSF grant DMS-1210886.December 30, 2023 =========================================================================================================================================================================================================== In [The Numerical Range is a (1 + √(2))-Spectral Set, SIAM J. Matrix Anal. Appl. 38 (2017),pp. 649-655], Crouzeix and Palencia show that the closure of the numerical range of a square matrix or linear operator A is a (1 + √(2))-spectral set for A; that is, for any function f analytic in the interior of the numerical range W(A) and continuous on its boundary, the inequality f(A) ≤ (1 + √(2) )f _W(A) holds, where the norm on the left is the operator 2-norm and f _W(A) on the right denotes the supremum of | f(z) | over z ∈ W(A).In this paper, we show how the arguments in their paper can be extended to show that other regions in the complex plane that do not necessarily contain W(A) are K-spectral sets for a value of K that may be close to 1 + √(2).We also find some special cases in which the constant (1 + √(2)) for W(A) can be replaced by 2, which is the value conjectured by Crouzeix. § INTRODUCTIONLet A be a square matrix or a bounded linear operator on a Hilbert space. In <cit.>, Crouzeix and Palencia establish that closure of the numerical range of A,W(A) = {⟨ Aq,q ⟩ : ⟨ q,q ⟩ = 1 } ,is a (1 + √(2))-spectral set for A; that is, for any function f analytic in the interior of W(A) and continuous on its boundary, f(A) ≤ ( 1 + √(2) )f _W(A) ,where the norm on the left is the operator 2-norm and f _W(A) on the right denotes the supremum of | f(z) | over z ∈ W(A).Crouzeix's conjecture is that f(A) ≤ 2f _W(A) <cit.>.In this paper we show how the arguments in <cit.> can be extended to show thatother regions Ω in the complex plane that do not necessarily contain W(A) are K-spectral sets for a value of K that may be close to 1 + √(2). We present numerical results that show what these values K are for various regions that contain the spectrum of A but not necessarily all of W(A).In particular weconsider disks of various sizes containing the spectrum of A.In part of the paper, we limit the discussion to n by n matrices.In this case,for any proper open subset Ω of ℂ containing the spectrum of A, there is a function f that attainssup_f̂∈ H ( Ω )f̂ (A)/ f̂_Ω, where H ( Ω ) denotes the set of analytic functions in Ω.Furthermore, if Ω is simply connected, the form of f is known <cit.>:f(z) = B ∘φ ( z) ,where φ is any bijective conformal mapping from Ω to the unit disk D and B is a Blaschke product of degree at most n-1,B(z) = e^i θ∏_j=1^n-1z - α_j/1 - α̅_j z , | α_j | ≤ 1 .[Note that we have allowed | α_j | = 1 in this definition so that the degree of B(z) can be less than n-1, since factors with | α_j | = 1 are just unit scalars.] We show that for this “optimal” B, if B ( φ (A) )> 1 and if y is a right singular vector of B( φ (A) ) corresponding to the largest singular value, then⟨ By,y ⟩ = 0.Using this result, we are able to replace the bound 1 + √(2) in <cit.> by 2 in some special cases. In particular, we give a new proof of the Okubo and Ando result <cit.> that if W(A) ⊂ D then D is a 2-spectral set for A.The paper is organized as follows.In Section 2, we prove the basic theorems extendingresults in <cit.> to other regions Ω containing the spectrum of A but notnecessarily all of W(A).The proofs of these theorems involve minor modifications of the arguments in <cit.>.We then limit the discussion to n by n matrices. In Section 3, we use results from Section 2 to show that if one replaces W(A) by an appropriate region Ω a distance about ϵ inside W(A), then this region is a K-spectral set for K = 1 + √(2) + O( ϵ ). Section 4 contains numerical studies of the values of K derived in Section 2, when Ω is a disk containing the spectrum of A. In Section 5 we derive a property of the optimal Blaschke product (<ref>).This is used in Section 6 to describe cases in which the bound 1 + √(2) can be replaced by 2 and in which the bound in Theorem <ref> can be replaced by 2 + δ from Lemma <ref>. Section 7 describes a case in which the bound of Lemma <ref> is sharp, and in Section 8 we mention remaining open problems.§ EXTENSIONS OF THE CROUZEIX-PALENCIA ARGUMENTSLet Ω be a region with smooth boundary containing the spectrum of A in its interior. Any function f ∈ A ( Ω ) :=H( Ω ) ∩ C( ( Ω )),is defined via the Cauchy integral formula asf(A) = 1/2 π i∫_∂Ω ( σ I - A )^-1 f( σ ) d σ .If we parameterize ∂Ω by arc length s running from 0 to L, then this can be written asf(A) = ∫_0^L σ' (s)/2 π i R( σ (s) , A ) f( σ (s)) ds ,where R( σ , A ) is the resolvent, ( σ I - A )^-1.A key idea in <cit.> was to look atg(A) = ∫_0^L σ' (s)/2 π i R( σ (s) , A ) f( σ (s) ) ds .Note that since f( σ (s) ) is not analytic, one cannot apply the Cauchy integral formula in its simplest form to the integral in (<ref>). Crouzeix and Palencia analyzed the operatorS := f(A) + g(A )^* = ∫_0^L μ ( σ (s) , A ) f( σ (s) ) ds ,where the Hermitian operator μ ( σ (s) , A ) isμ ( σ (s) , A ) = σ' (s)/2 π i R( σ (s), A ) + [σ' (s)/2 π i R( σ (s), A ) ]^* .They argued that if Ω is a convex region that contains W(A), then μ ( σ , A ) is positive semidefinite for σ∈∂Ω.We are interested in regions Ω that do not necessarily contain W(A), so we will defineM( σ , A ) := μ ( σ , A ) - λ_min ( μ ( σ , A ) ) I ,where λ_min ( μ ( σ , A )) is the infimum of the spectrum ofμ ( σ , A ) at the point σ on ∂Ω.Thus, by definition,M( σ , A ) is positive semidefinite on ∂Ω.Using the same method of proof as in <cit.>, we establish the following: Let Ω be a region with smooth boundary containing the spectrum of A in its interior. For f ∈ A ( Ω ) with f _Ω = 1, letS = f(A) + g(A )^* + γ I , γ := - ∫_0^L λ_min( μ ( σ (s), A ) )f( σ(s) ) ds .Then S ≤ 2 + δ , whereδ =- ∫_0^L λ_min ( μ ( σ (s),A) ) ds .Let u and v be any two unit vectors and, for convenience, write M(s) for M( σ (s) , A ) and λ_min (s) for λ_min ( μ ( σ (s),A ) ) in (<ref>).Then| ⟨ Sv,u ⟩ | =| ∫_0^L ⟨ M(s) v , u ⟩f( σ (s)) ds | ≤ ∫_0^L | ⟨ M(s) v , u ⟩ | ds ≤ ∫_0^L ⟨ M(s) u , u ⟩^1/2·⟨ M(s) v , v ⟩^1/2 ds ≤ ( ∫_0^L ⟨ M(s) u, u ⟩ ds )^1/2( ∫_0^L ⟨ M(s) v, v ⟩ ds )^1/2 =⟨( ∫_0^L M(s) ds ) u , u ⟩^1/2⟨( ∫_0^L M(s) ds ) v , v ⟩^1/2 =⟨( 2 - ∫_0^L λ_min ( s ) ds ) u ,u ⟩^1/2⟨( 2 - ∫_0^L λ_min ( s) ds ) v ,v ⟩^1/2 = 2 + δ .Remark 1:Note that δ in (<ref>) can be positive or negative (but it cannot be less than -2).It is 0 if Ω = W(A), positive if Ω is a proper subset of (W(A)), and negativeif Ω is convex and (W(A)) is a proper subset of Ω. This follows from the fact shown in <cit.> that if τ lies on the tangent line to W(A)at a point σ∈∂ W(A), the infimum of the spectrum of the Hermitian part of ( σ' (s) / ( π i ) ) R( τ , A ) is 0, while on the side of this line that does not contain W(A) it is positive and on the side that does contain W(A) it is negative.Remark 2:The region Ω in Lemma <ref> need not be simply connected.For example, it could consist of a union of disks or other smooth regions, each of which encloses a part of the spectrum.The remainder of <cit.> is aimed at relating f(A) + g(A )^* to f(A). We note that in many numerical experiments in which A is an n by n matrix and f is the function with f _W(A) = 1 that maximizes f̂ (A) over all f̂∈ A (W(A)) with f̂_W(A) = 1 (or, at least, our best attempt at computing this function via numerical optimization of roots ofa Blaschke product),we have always found that f(A) ≤ f(A) + g(A )^*.If this could be proved, then it would establish Crouzeix's conjecture.Since we do not know a proof, however, more work is needed to bound f(A) in terms of f(A) + g(A )^*.We now assume that Ω is a bounded convex domain with smooth boundary. It is shown in <cit.> that if g(z) is defined in Ω byg(z) = 1/2 π i∫_∂Ωf( σ )/σ - z d σ ,then g ∈ A ( Ω ) (when g is extended continuously to ∂Ω), g(A) satisfies (<ref>), and g _Ω≤ f _Ω .It is further shown that g( ∂Ω ) := { g( σ ) : σ∈∂Ω} is contained in the convex hull of f( ∂Ω ) := {f( σ ) : σ∈∂Ω}.For any bounded set Ω containing the spectrum of A in its interior, with the spectrum of A bounded away from ∂Ω, there is a minimal constant c_Ω (A) (which, for convenience, we denote as simply c_Ω) such that for all f ∈ A ( Ω ),f(A) ≤ c_Ω f _Ω .One such constant can be derived from the Cauchy integral formula:f(A) ≤1/2 π( ∫_∂Ω ( σ I - A )^-1 | d σ | )f _Ω ,but this usually is not optimal.It was shown in <cit.> that even if the spectrum of A is not bounded away from ∂Ω, if Ω⊃ W(A), then such aconstant exists and is finite. The following theorem uses Lemma <ref> and (<ref>) to obtain a different upper bound on c_Ω.[The authors thank Felix Schwenninger and an anonymous referee for an improvement to the bound obtained in the original version of this theorem.] Let Ω be a convex domain with smooth boundary containing the spectrum of A in its interior.Thenc_Ω≤ 1 + δ/2 + √(2 + δ + δ^2 / 4 + γ̂) ,where δ = - ∫_0^L λ_min ( μ ( σ (s) , A )) ds , γ̂ = ∫_0^L | λ_min ( μ ( σ (s) , A )) | ds .Let f ∈ A ( Ω ) satisfy f _Ω = 1.From (<ref>), we can writef(A )^* = S^* - (g(A) + γ̅ I ) .Multiply by f(A )^* f(A) on the left and by f(A) on the right to obtain[ f(A )^* f(A) ]^2 = f(A )^* f(A) S^* f(A) - f(A )^* f(A) ( g(A) + γ̅ I) f(A) .Now take norms on each side and use the fact that the norm of any function of A is less than or equal to c_Ω times the supremum of that function on Ω to find:f(A) ^4≤c_Ω^3S^* + c_Ω h(A), h(z) := f(z)(g(z) + γ̅ ) f(z) , ≤c_Ω^3 (2 + δ ) + c_Ω^2 ( 1 + γ̂ ) ,Since this holds for all f ∈ A ( Ω ) with f _Ω = 1, it follows thatc_Ω^4 ≤ c_Ω^3 ( 2 + δ ) + c_Ω^2 ( 1 + γ̂ ),and solving the quadratic inequality c_Ω^2 - (2 + δ ) c_Ω - (1 + γ̂ ) ≤ 0 for c_Ω gives the desired result (<ref>). § A PERTURBATION RESULTHow does λ_min ( μ ( σ , A ) ) vary as σ moves inside or outside ∂ W(A)?Is a region just slightly inside W(A) a K-spectral set for A for some K that is just slightly greater than 1 + √(2)?We now assume that A is an n by n matrix and that no eigenvalue of A lies on ∂ W(A). Consider a curve consisting of points σ̃ ( s̃ ), where s̃ runs from 0 to L̃ < L and σ̃' ( s̃ ) = σ' (s), for σ (s) on ∂ W(A) close to σ̃( s̃ ). For example, we might take a “center” point of W(A) as the origin and for θ∈ [0 , 2 π ) write σ ( θ ) = r( θ ) e^i θ, σ̃ ( θ ) =(1 - ϵ ) r( θ ) e^i θ, where r( θ ) is the distance from the center point to the point on ∂ W(A) with argument θ, and ϵ > 0 is small. Then . d σ̃/d s̃|_s̃ =. d σ/ds|_s if s̃ = (1 - ϵ ) s.From the Wielandt-Hoffman theorem <cit.>, | λ_min ( μ ( σ , A )) - λ_min ( μ ( σ̃ , A ) ) | ≤μ ( σ , A ) - μ ( σ̃ , A ),and the same inequality holds for the difference between every pair of ordered eigenvalues of μ ( σ , A ) and μ ( σ̃ , A ). The matrix on the right is the Hermitian part ofσ' (s)/π i[ R( σ , A ) - R( σ̃ , A ) ] ,and from the first resolvent identity this is equal to the Hermitian part of( σ̃ - σ ) [ σ' (s)/π i R( σ , A ) R( σ̃ , A ) ] .Therefore since, from <cit.>, λ_min ( μ ( σ , A ) ) = 0, it follows from(<ref>) thatλ_min ( μ ( σ̃ , A )) ≥- | σ̃ - σ | ·[ σ' (s)/2 π i R( σ , A ) R( σ̃ , A ) ] + [ σ' (s)/2 π i R( σ , A ) R( σ̃ , A ) ]^* .As ϵ→ 0, this shows that the smallest eigenvalue of μ ( σ̃ , A ) is greater than or equal to -C ϵ for a positive constant C, and δ andγ̂ in (<ref>) are therefore O( ϵ ), so that c_Ω≤ 1 + √(2) + O( ϵ ).§ NUMERICAL STUDIESTo further study the behavior of λ_min ( μ ( σ , A ) ) for σ inside or outside W(A), one can make a plot of λ_min vs. arc length s or angleθ (s), for σ (s) on various curves.[Note that the value of λ_min depends not only on the location of σ but also on the curve on which σ isconsidered to lie.]Instead of taking curves on which σ' matches the derivative at some associated point on ∂ W(A), we will now take σ to lie on a circle. We first determine the center c and radius r of the smallest circle enclosing the spectrum of A and then determine values of λ_min( μ ( σ , A ) ) at points σ on circles about c of radius R > r.On such circles, σ = c + R e^i θ, 0 ≤θ < 2 π, and s = R θ so thatσ (s) = c + R e^i s/R , σ' (s) = i e^i s/R . Figure <ref> shows the eigenvalues (x's) and numerical range (solid curve) of a random complex upper triangular matrix A of dimension n=12.It also shows thecircles on which we computed λ_min ( μ ( σ , A )) (dashed circles). Figure <ref> shows the values of λ_min ( μ ( σ (s) , A )) plotted vs. arc length s on each of these circles, where the bottom curve correspondsto the innermost circle and the curves move up as the circles become larger.From the figure,it can be seen that λ_min decreases rapidly as σ moves inside W(A) towards the spectrum, but it grows very slowly as σ moves outside W(A).Figures <ref> and <ref> show the same results for a 3 by 3 perturbed Jordan block:A = [ [ 0 1 0; 0 0 1; 0.1 0 0 ]] . Again one can see that λ_min ( μ ( σ , A )) decreases rapidly as σ moves toward the spectrum but grows slowly as it moves outside W(A). Table <ref> shows the values of δ and γ̂ in (<ref>) and the upper bound on c_Ω in (<ref>) (labeled K_δ) for each of the disks (starting with the smallest) in both problems.For comparison, we also include the upper bound on c_Ω based on the Cauchy integral formula and the resolvent norm in (<ref>) (labeled K_). In all cases, K_δ < K_. § OPTIMAL BLASCHKE PRODUCTSFrom here on we always assume that A is an n by n matrix.Then if Ω is any simply connected proper open subset of ℂ containing thespectrum of A, then there is a function f such that f _Ω = 1 and f (A)= c_Ω.This function f is of the form B ∘φ, where φ is any bijective conformal mapping from Ω to the unit disk D, and B is a Blaschke product of the form (<ref>).While we do not know an analytic formula for the roots α_j, j=1, … , n-1, of this optimal Blaschke product, the following theorem describes one property of the optimal B: Let Ψ be an n by n matrix whose spectrum is inside the unit disk Dand let B be a Blaschke product of degree at most n-1 that maximizesB̂ ( Ψ ) over all Blaschke products B̂ of degree at most n-1.Assume that B( Ψ )> 1.Then, if v_1 is a right singular vector of B( Ψ ) corresponding to the largest singular value σ_1, then⟨ B( Ψ ) v_1 , v_1 ⟩ = 0 . [The authors thank M. Crouzeix for the nice proof of this theorem, after we had observed the result in numerical experiments.] Let M = B( Ψ ) where B is the Blaschke product of the form (<ref>) for which B ( Ψ ) is maximal.Then no matrix of the form( M - α I ) ( I - α̅ M )^-1 , | α | < 1 ,can have larger norm than M since this is also a Blaschke product in Ψ. Let v_1 be a unit right singular vector of M corresponding to the largest singular value σ_1, and define w = (I - α̅ M ) v_1.Then(M - α I ) v_1=(M - α I ) ( I - α̅ M )^-1 w ≤σ_1w= σ_1(I - α̅ M ) v_1.Squaring both sides, this becomes⟨ M v_1 , M v_1 ⟩ - 2 ( α̅⟨ M v_1 , v_1 ⟩ ) + | α |^2 ≤σ_1^2 [ 1 - 2 ( α̅⟨ M v_1 , v_1 ⟩ ) + | α |^2 ⟨ M v_1 , M v_1 ⟩ ],and since ⟨ M v_1 , M v_1 ⟩ = σ_1^2,2 ( σ_1^2 - 1)( α̅⟨ M v_1 , v_1 ⟩ ) ≤ | α |^2 ( σ_1^4 - 1 ) .With the assumption that σ_1 > 1, dividing by σ_1^2 - 1 this becomes2( α̅⟨ M v_1 , v_1 ⟩ ) ≤ | α |^2 ( σ_1^2 + 1 ) .Choosing α so that α̅⟨ M v_1 , v_1 ⟩ = | α | | ⟨ M v_1 , v_1 ⟩ |, we have2 | α | | ⟨ M v_1 , v_1 ⟩ | ≤ | α |^2 ( σ_1^2 + 1 ) ,and letting | α | → 0, this implies that | ⟨ M v_1 , v_1 ⟩ | = 0.§ SOME CASES IN WHICH 1 + √(2) CAN BE REPLACED BY 2 It was noted after Lemma <ref> that in all numerical experiments that we have performed – determining f = B ∘φ by first finding a conformal mapping φ from W(A) to D and then using an optimization code with many differentinitial guesses to try to find the roots α_1 , … , α_n-1 in (<ref>)that maximize f(A)=B( φ (A) ), and then finding the matrix g(A) in (<ref>) that corresponds to this f – it has always been the case that f(A) ≤ f(A) + g(A )^*.We now describe some cases in whichthat can be proved and also cases in which it can be shown that for other regionsΩ, f(A) ≤ S in Lemma <ref>. §.§ Ω a Disk If Ω is a closed disk with center c, the formula (<ref>) for g(z) can beevaluated very simply:For z inside Ω, g(z) ≡f(c)<cit.>. Thus g(A) = f(c) I.If f = B ∘φ is afunction that maximizes f(A) over all functions with f _Ω = 1, if f(A)> 1, and if v_1 is a unit right singular vector of f(A) corresponding to the largestsingular value and u_1 = f(A) v_1 /f(A) v_1is the corresponding unit left singular vector, thenu_1^* ( f(A) + g(A )^* + γ I ) v_1 = u_1^* f(A) v_1 =f(A),since, by Theorem <ref>, u_1^* v_1 = 0.It follows that f(A) is less than or equal to max{ 1, S } in (<ref>). This provides a new proof of the statement:,since in this case δ in (<ref>) is less than or equal to 0.[A priori, the assumption that the spectrum of A is contained in the interior of W(A) is needed, but this can be easily avoided.]Note also that c_Ω < 2 if W(A) ≠Ω since in this case δ is negative. The proof of (<ref>) in <cit.> was based on a paper ofAndo <cit.>, and a similar construction had been carried out by Okubo andAndo <cit.> in a more general setting.Our proof is simpler, but the previous proofs showed, in addition, that Ω is a complete 2-spectral set for A.Even if Ω does not contain all of W(A), the estimatef(A) ≤max{ 1, 2 + δ} holds. This provides a better upper bound on c_Ω in the experiments of Section 4.For comparison, Table <ref> lists this upper bound on c_Ω and also the largest value returned by our optimization code, which we believe to be the true value of c_Ω but, at least, it is a lower bound.In some cases, these are quite close.§.§.§ A Different Bound on c_W(A)Since we know that a disk containing the spectrum of a matrix Ψ is amax{ 1, 2 + δ}-spectral set for Ψ, we can use this to obtain (numerically) a different bound on c_W(A).Let φ be a bijective conformal mapping from W(A) to the unit disk D.Then D is a K-spectral set for φ (A), where K = max{ 1, 2 + δ_φ (A)}, and δ_φ (A) = - ∫_0^2 πλ_min ( μ ( σ (s) , φ (A) ) ) ds ,where σ (s) = e^is.It follows that W(A) is a K-spectral set for A, with the same value of K, since for any f ∈ A (W(A)),f(A)=f ∘φ^-1 ( φ (A) ) ≤Kf ∘φ^-1_ D = Kf _W(A) . Using this result, one can determine numerically better bounds on c_W(A) for the problems considered in Section 4.For the 12 by 12 random upper triangular matrix, δ_φ (A) = 0.0113, so W(A) is a 2.0113 spectral set for A. For the 3 by 3 perturbed Jordan block, δ_φ (A) = -0.0013, so W(A) is a 1.9987-spectral set for A.It is an open question whether such bounds can be determined theoretically, and the numerical result for the first problem suggests that this will not be a way to prove Crouzeix's conjecture. §.§ Matrices for which Crouzeix's Conjecture has been ProvedBesides matrices whose numerical range is a circular disk, Crouzeix's conjecture has beenproved to hold for a number of other classes of matrices – e.g., 2 by 2 matrices <cit.>, matrices of the forma I + DP or a I + PD, where a ∈ℂ, D is a diagonal matrix, and P is a permutation matrix <cit.>; 3 by 3 tridiagonal matrices with elliptic numerical range centered at an eigenvalue <cit.>; etc.In all of these cases, while the conformal mapping φ may be a complicated function, φ (A) is just a linear function of A: φ (A) = α A + β I.From our experiments, it appears that in all of these cases the function g corresponding to the optimal f has the formg(A )^* = c_0 I + c_1 ( f(A )^* )^-1 .Then (assuming f(A)> 1),| u_1^* ( f(A) + g(A )^* + γ I ) v_1 | = |f(A)+ c_1/ f(A) | .If ( c_1 ) ≥ | c_1 |^2 / ( 2f(A) ^2 ), then this is greater than orequal to f(A), hence f(A) ≤ S.§ A CASE IN WHICH THE BOUND IS SHARPIt was shown in the previous section that if Ω is a disk containing the spectrum of A in its interior, then max_f ∈ A ( Ω ) f(A)/f _Ω≤max{ 1, 2 + δ}, where δ is defined in (<ref>). It turns out that if A is a 3 by 3 Jordan block, then this bound is sharp for all disks of radius less than or equal to 1 centered at the eigenvalue of A,which, for convenience, we will take to be 0. Let A be a 3 by 3 Jordan block with eigenvalue 0 and let Ω be any disk about the origin with radius r ≤ 1.Thenmax_f ∈ A( Ω ) f(A) / f _Ω = 2 + δ ,where δ is defined in (<ref>).The function f that achieves the maximum in (<ref>) is of the form B ∘φ, where B is a Blaschke product and φ (z) = z/r mapsΩ to the unit disk.Since φ (A) is a scalar multiple (1/r) of a Jordan block, it is easy to see that the optimal B is B(z) = z^2, assuming r ≤ 1.(For r=1, one could take B(z) = z or B(z) = z^2 since the norm of a 3 by 3 Jordan block with eigenvalue 0 and its square are both equal to 1.) Thus, the left-hand side of (<ref>) is 1/ r^2.Next we show that the right-hand side of (<ref>) is equal to 1/ r^2. Since σ (s) = r e^i s/r on ∂Ω, we can writeσ' (s)/2 π i R ( σ (s), A ) = e^i s/r/2 π( r e^is/r I - A )^-1 = 1/2 π r[ [1e^-is/r/r e^-2is/r/r^2;01e^-is/r/r;001 ]] ,andμ ( σ (s) , A ) = 1/2 π r[ [2e^-is/r/r e^-2is/r/r^2; e^is/r/r2e^-is/r/r;e^2is/r/r^2 e^is/r/r2 ]] .It is easy to check that for r ≤ 1, the smallest eigenvalue of this matrix isλ_min ( μ ( σ (s) , A ) ) = 2 r^2 -1/2 π r^3 ,independent of s.Since the length of ∂Ω is 2 π r,δ = -(2 - 1/ r^2 ) and 2 + δ = 1/ r^2. § SUMMARY AND OPEN PROBLEMSWe have shown how the arguments in <cit.> can be modified to give information about regions that contain the spectrum of A but not necessarily all of W(A).Perhaps the most interesting regions are disks about the spectrum, which we have shown to be K-spectral sets for K = max{ 1, 2 + δ}, thereby providing a new proof that if W(A) ⊂ D, then D is a 2-spectral set for A <cit.>.We derived one property of optimal Blaschke products; i.e., Blaschke products thatmaximize B( Ψ ) where Ψ is a given matrix whose spectrum is in D. Specifically, we showed that if B( Ψ )> 1, then the left and right singular vectors of B( Ψ ) corresponding to the largest singular value must be orthogonal to each other.An interesting open problem is to determine other properties of the optimal B.Perhaps the most interesting question raised is whether it is true that if f = B ∘φ is the optimal f then f(A) ≤ f(A) + g(A )^*.A proof of this would establish Crouzeix's conjecture, and a counterexample might lead to a new line of attack.Acknowledgments:The authors thank Michel Crouzeix, Michael Overton, and Abbas Salemi for many helpful discussions and suggestions.We especially thank Michel Crouzeix for providing the proof of Theorem <ref> and Abbas Salemi for pointing out that δ in Lemma <ref> could be allowed to take on negative values.We also thank two anonymous referees for very helpful comments, especially for an improvement of theresult in Theorem <ref>.99Ando T. Ando, Structure of operators with numerical radius 1,Acta Sci. Math. 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Lindström, Finite Blaschke product interpolation on the closed unit disc, J. Math. Anal. Appl. 273 (2002), pp. 417-427.Henrici P. Henrici, Applied and Computational Complex Analysis, Vol. III, Wiley, 1986.HW1953 A. Hoffman and H. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), pp. 37-39.OkuboAndo K. Okubo and T. Ando, Constants related to operators of class C_p, Manuscripta Math 16 (1975), pp. 385-394.Remmert R. Remmert, Theory of Complex Functions, Springer, 1991. | http://arxiv.org/abs/1707.08603v2 | {
"authors": [
"Trevor Caldwell",
"Anne Greenbaum",
"Kenan Li"
],
"categories": [
"math.NA",
"15A60, 65F35"
],
"primary_category": "math.NA",
"published": "20170726183617",
"title": "Some Extensions of the Crouzeix-Palencia Result"
} |
ApJ MNRAS ApJS New Astronomy Review P^3MMpch^-1MpchMpc^-1 Mpc^-1h^-1kpc h^-1kpc kms^-1 M_⊙h^-1 M_⊙h^-2 M_⊙R_ virv_ nv_ t#1𝐫_#1#1𝐬_#1θ_ spin1.16pt>-7.0pt 3.06pt∼1.16pt<-7.0pt 3.06pt∼ > ∼ < ∼∝∼ .5ex.5ex.5ex.5ex.5exM_r-5loghM_ sphM_ haloM_ subt_ crv_ virr_ virR_ virα_ infallELUCID: Galaxy Quenching and its Relation to EnvironmentWang H.Y. et al.1Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, Anhui 230026, China; [email protected] of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China3Department of Astronomy, University of Massachusetts, Amherst MA 01003-9305, USA4Astronomy Department and Center for Astrophysics, Tsinghua University, Beijing 10084, China5Department of Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China6IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China7Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA8Purple Mountain Observatory, the Partner Group of MPI für Astronomie,2 West beijing Road, Nanjing 210008, China9School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275, China10Departamento de Física Teórica, Módulo 15, Facultad de Ciencias, Universidad Autónoma de Madrid, E-28049 Madrid, Spain11Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China We examine the quenched fraction of central and satellite galaxies as a function of galaxy stellar mass, halo mass, and the matter density of their large scale environment. Matter densities are inferred from our ELUCID simulation, a constrained simulation of local Universe sampled by SDSS, while halo masses and central/satellite classification are taken from the galaxy group catalog of Yang et al. The quenched fraction for the total population increases systematically with the three quantities. We find that the `environmental quenching efficiency', which quantifies the quenched fraction as function of halo mass, is independent of stellar mass. And this independence is the origin of the stellar mass-independence of density-based quenching efficiency, found in previous studies. Considering centrals and satellites separately, we find that the two populations follow similar correlations of quenching efficiency with halo mass and stellar mass, suggesting that they have experienced similar quenching processes in their host halo. We demonstrate that satellite quenching alone cannot account for the environmental quenching efficiency of the total galaxy population and the difference between the two populations found previously mainly arises from the fact that centrals and satellites of the same stellar mass reside, on average, in halos of different mass. After removing these halo-mass and stellar-mass effects, there remains a weak, but significant, residual dependence on environmental density, which is eliminated when halo assembly bias is taken into account. Our results therefore indicate that halo mass is the prime environmental parameter that regulates the quenching of both centrals and satellites. § INTRODUCTION In the low-redshift universe, galaxies are observed to exhibit bimodal distributions in their colors and specific star formation rates. According to these bimodal distributions, galaxies can be divided into two distinctive sequences, a red sequence of galaxies quenched in star formation,and a blue sequence of star-forming galaxies (e.g. Strateva et al. 2001; Blanton et al. 2003; Baldry et al. 2004; Brinchmann et al. 2004). The bimodal distribution is correlated with galaxy mass, with the red sequence dominated by massive galaxies, while the blue population by lower-mass ones. Furthermore, it is also known that the red, quiescent galaxies tend to reside in high density regions, while the blue star-forming galaxies display an opposite trend with environments (e.g. Oemler 1974; Dressler 1980; Hogg et al. 2003; Kauffmann et al. 2004; Baldry et al. 2006; Peng et al. 2010; Zheng et al. 2017). All these suggest that the mechanisms responsible for quenching star formation in a galaxy must be related to galaxy mass as well as to the environment.In the current cold dark matter (CDM) cosmogony, galaxies are assumed to form and evolve within dark matter halos (e.g. White & Rees 1978; Mo, van den Bosch & White 2010). Lower-mass halos on average form earlier and subsequently merge to form more massive ones. In this process, the galaxy that forms on the main branch of a halo merging tree is expected to be the dominant galaxy residing near the center of the halo (the central galaxy),while other galaxies that form in the sub-branch progenitor halos, are expected to orbit the central as satellite galaxies. These two populations of galaxies are expected to have experienced different processes that quench their star formation: while supernovae and active galactic nuclei (AGNs) feedbacks, and shock heating of cold accretion flow may affect both central and satellite galaxies, processes like ram pressure and tidal stripping are believed to operate only on satellite galaxies (e.g. White & Frenk 1991; Kang et al. 2005; Dekel & Birnboim 2006; Bower et al. 2006; Croton et al. 2006; Kereš et al. 2009; Lu et al. 2011; Guo et al. 2011; Vogelsberger et al.2014; Schaye et al. 2015). These models have successfully reproduced many global properties of observed galaxies. However, there are still significant discrepancies between model predictions incorporating these processes and observational data in terms of the fraction of the quenched population (see e.g. Hirschmann et al. 2014; Vogelsberger et al. 2014; Henriques et al. 2016), indicating that our understanding of the quenching processes is still incomplete. Observing correlations between galaxy properties and different aspects of their environment could help to distinguish different galaxy formation models. A variety of quantities have been used to describe the environment around a galaxy (Haas, Schaye & Jeeson-Daniel 2012). These environmental parameters are usually designed for different purposes, and an optimal decision has to be made for a specific question (Leclercq et al. 2016). The traditional environmental parameter is the (projected) number density of galaxies, which is often adopted in observational studies of galaxy properties on various scales (e.g. Dressler 1980;Hogg et al. 2003; Kauffmann et al. 2004; Hirschmann et al. 2014). It can be directly obtained from the galaxy redshift survey without any additional assumption. The other widely used parameter is the host halo mass (e.g. Weinmann et al. 2006; Wetzel, Tinker & Conroy 2012; Woo et al. 2013), which is closely linked to galaxy formation in the CDM paradigm. Indeed, the halo occupation distribution (HOD) models (e.g. Jing et al. 1998; Peacock & Smith 2000; Zheng et al. 2005; Zu & Mandelbaum 2016), conditional luminosity function (CLF) models(e.g., Yang et al. 2003;van den Bosch et al. 2007), abundance matching models (e.g., Mo et al. 1999; Kravtsovet al. 2004; Vale & Ostriker 2006; Behroozi, Conroy, & Wechsler 2010; Hearin & Watson 2013) and halo-based empirical models (e.g., Yang et al. 2013; Lu et al. 2014, 2015; Moster, Naab & White 2017) have all used halo masses to link galaxies to dark matter halos.Using halo mass inferred from galaxy group catalog (Yang et al. 2007) as environmental parameter, Weinmann et al. (2006) found that the quenched fraction of satellite galaxies is much lower than that in model predictions and increases strongly with host halo mass (see also Liu et al. 2010; Wetzel et al. 2012). This has motivated later semi-analytic galaxy formation models (SAMs) to employ an incremental stripping of hot gas associated with satellites through ram-pressure and tidal stripping (e.g. Kang & van den Bosch 2008; Font et al. 2008; Weinmann et al. 2010; Guo et al. 2011; Henriques et al. 2015). Moreover, halo mass is also found to have significant impact on the quenching of star formation in centrals of given galaxy masses (Weinmann et al. 2006; Woo et al. 2013; 2015; Bluck et al. 2014;2016). Here, AGN feedback is thought to be the major quenching mechanism, and its strength is likely to depend on halo mass (e.g. Croton et al. 2006; Henriques et al. 2016), qualitatively consistent with the observation results.In addition to environmental effects that are confined within halos, there are also observational indications that the environmental effects may operate on scales beyond their boundaries. For example, at a fixed halo mass, the clustering of galaxy groups is found to depend on the star formation rate and color of the central galaxies (Yang, Mo, & van den Bosch 2006; Lacerna, Padilla, & Stasyszyn 2014). Similarly, Kauffmann et al. (2013) found that the star formation rates of central galaxies are correlated with that of their neighbours on scales up to several Mpcs, far beyond their halo virial radii (see also Berti et al. 2017, and references therein). These results suggest that large scale environments may also affect the star formation of galaxies embedded in them. However, it is unclear whether this is due to a causal connection between star formation and large-scale environments, or is produced by a correlation induced by some intermediate connections. For example, such large-scale effects may be produced by the dependence of star formation on halo assembly history (see e.g. Hearin, Watson, & van den Bosch 2015; Lim et al. 2016; Tinker et al. 2016; Zentner et al. 2016), combined with halo assembly bias that links halo formation with large-scale structure (e.g. Gao, Springel, & White 2005; Wechsler et al. 2006; Jing, Suto & Mo 2007), or may be produced by the preheating of the intergalactic gas owing to the formation of large scale structure (e.g. Mo et al. 2005; Kauffmann et al. 2013).Using galaxy number density as environmental parameter, Baldry et al. (2006) found that the quenched fraction depends both on galaxy stellar mass and environmental density, and the dependence can be well described by a simple functional form. Peng et al. (2010) studied the environmental quenching efficiency, which is defined as the probability for a galaxy to be quenched in high-density regions relative to that in low-density regions, where environmental effects are expected to be weak. Remarkably, the efficiency defined in this way is found to be almost independent of stellar mass. Subsequently, Peng et al. (2012) suggested that the independence may be explained if environmental quenching is assumed to be important only for satellite galaxies, and if both the quenching efficiency of satellite galaxies and the satellite fractionare independent of galaxy mass. However, these assumptions are not supported by the results obtained for centrals from galaxy groups, which clearly show that environmental quenching of central galaxies is also important (e.g. Weinmann et al. 2006; Woo et al. 2013; 2015), and by the results for satellites, which show that both the quenching efficiency (Knobel et al. 2015) and satellite fraction (Mandelbaum et al. 2006; Cooray 2006; Tinker et al. 2007; van den Bosch et al. 2007) depend on galaxy mass. The difference between centrals and satellites also attracted particular attention (e.g. van den Bosch et al. 2008; Skibba 2009; Wetzel et al. 2012;2013; Peng et al. 2012; Hirschmann et al. 2014; Knobel et al. 2015; Spindler & Wake 2017). It has been found that the quenched fraction of the central population is lower than that of satellites of the same mass. This difference has been used to quantify the efficiency of various satellite-specific quenching processes, such as strangulation, tidal stripping and ram-pressure stripping (van den Bosch et al. 2008). However, Knobel et al. (2015) found that centrals and satellites of the same mass respond to their environments in a similar way, as long as centrals have massive satellites. Moreover, Hirschmann et al. (2014) studied the failures of current galaxy formation models in matching observational data and suggested that centrals and satellites should be treated not as differently in their response to environments as previously assumed.Clearly, more investigations are required in order to understand these contradictory results in the literature. It is essential to identify and characterize the contribution to the quenching of star formation of the relevant parameters, such as galaxy stellar mass, halo properties and large-scale density field. In particular, it is important to see whether the independence of environmental quenching efficiency on galaxy mass can be reproduced if only halo masses and halo assembly bias are taken into account, and what roles centrals and satellites play in establishing the galaxy-mass independence of the quenching efficiency found earlier, and whether the star formation quenching in centrals and satellites is dominated by different processes.In this paper, the fourth of a series, we use the environmental information provided by the ELUCID project and galaxy groups selected from the Sloan Digital Sky Survey (SDSS; York et al. 2000) to investigate the quenching of galaxies in different environments. The ELUCID project (Wang et al. 2014; 2016; Tweed et al. 2017) aims to reconstruct the initial conditions responsible for the formation of the structures in the observed low-redshift universe, and to recover the mass distribution in the local universe by constrained simulations. The constrained simulations give the full information about the dynamical state and formation history of the large scale structure within which the observed galaxies reside. This provides an unique opportunity to systematically investigate the quenched population of galaxies of different masses in different environments.Our paper is organized as follows. In Section <ref>, we describe the galaxy sample, group catalog and environmental quantities used for our analysis. Section <ref> shows how the quenched galaxy population depends on galaxy mass, halo mass and environmental density for the total population, as well as separately for the central and satellite populations. In Section <ref>, we investigate the galaxy stellar mass independence of the environmental quenching efficiency using the matter density field as inferred from our ELUCID simulation, with a special focus on the role of central and satellite galaxies. In Section <ref>, we investigate the quenching efficiencies using halo mass as environmental parameter. In Section <ref> we discuss the implications of our results by comparing the data to three simple models that incorporate the dependence on galaxy stellar mass, halo mass, environmental density, and halo assembly history. In Section <ref> we discuss whether central galaxies are special in their quenching properties in comparison with satellites. Finally, we summarize ourresults and discuss their implications in Section <ref>. § GALAXY SAMPLE AND ENVIRONMENTAL QUANTITIES§.§ The galaxy sample The galaxy sample used here is extracted from the New York University Value-added Galaxy Catalog (NYU-VAGC, Blanton et al. 2005) of the SDSS DR7 (Abazajian et al. 2009). We select all galaxies in the main galaxy sample, with r-band apparent magnitudes ≤17.72, and with redshift completeness C≥0.7, and within the reconstruction region of ELUCID simulation (see below and Wang et al. 2016 for the details). The first two selection criteria ensure that most of the selected galaxies are contained in the Yang et al. (2007) group catalog (with extension to DR7), and the third ensures that we have reliable estimates of our galaxies' environmental densities. Among these galaxies, 1,707 are members of groups that have only a fraction of 0<f_ edge≤ 0.6 of their virial volumes contained within the survey boundary. These galaxies are removed from our sample (see Yang et al. 2007). A total of 4,233 galaxies that do not have star formation rate estimates are also discarded. Our final sample contains 317,791 galaxies.Stellar masses for these galaxies, indicated by m (with unit ), are computed using the relations between stellar mass-to-light ratio and (g - r) color as given in Bell et al. (2003), adopting a Kroupa (2001) initial mass function (IMF). We refer to Yang et al. (2007) for details. Star formation rates (SFR) for these galaxies are taken from the the MPA-JHU DR7 release website[ http://www.mpa-garching.mpg.de/SDSS/DR7/], which are estimated by using an updated version of the method presented in Brinchmann et al. (2004) and calibrated to the Kroupa IMF. For given galaxy stellar mass, the distribution of SFR is known to be bimodal (e.g. Brinchmann et al. 2004), with a high SFR mode corresponding to the star forming population and a low SFR mode corresponding to a quenched population. In this paper, we adopt the division line proposed by Woo et al. (2013) to separate the two populations:log SFR =0.64logm-1.28logh-7.22 ,where the reduced Hubble constant h (Hubble constant in units of 100km s^-1Mpc^-1) is used to transfer the unit of the stellar mass fromused here toused in Woo et al. (2013).In our analyses, each galaxy is assigned a weight w=1/(V_ maxC) to take into account the Malmquist bias and redshift (spectroscopic) incompleteness, with the latter taken from the NYU-VAGC. Since the geometry of our reconstruction region is not regular, we calculate V_ max in the following way. For each galaxy, we first obtain its Petrosian photometry in ugriz bands and its redshift. We then use these data as input to the K-correction utilities (v4_2) of Blanton & Roweis (2007) to estimate z_ min and z_ max, the minimum and maximum redshifts, between which the galaxy can be observed with the r-band limit of 17.72 mag. Finally, we measure V_ max as the volume of the reconstruction region that is between z_ min and z_ max. §.§ Host halos of galaxies Galaxy groups and clusters (hereafter referred to together as galaxy groups), when properly selected from a galaxy sample, can be used to represent the host dark halos of galaxies. In this paper we make use of the galaxy groups identified by Yang et al. (2012) from the SDSS DR7 to represent halos in which galaxies reside. This group catalog was constructed with the halo-based group finder developed by Yang et al. (2005), which assigns new galaxies into groups based on the size (virial radius) and velocity dispersion of the host dark halo represented by the current members assigned to a tentative group. Iterations are performed until the identification of member galaxies as well as the estimation of halo mass converge. The halo masses in the catalog are estimated via the ranking of two mass proxies: the total luminosity or total stellar mass of all members brighter than M_r=-19.5+5log(h) in the r-band. For our analysis, we adopt the halo masses, M (with unit ), estimated using the total stellar mass. Following commonpractice, we define the central galaxy of a group to be the most massive member, and all other members are referred to as satellites. The reconstruction region is restricted to the redshift range 0.01<z<0.12, where groups with log(M) 12 are complete.§.§ Environmental density In Wang et al. (2016; hereafter paper III), we presented a series of methods to reconstruct the initial density field that is responsible for the galaxy distribution in the local Universe, and we used a high-resolution N-body simulation to evolve the initial conditions to the present-day. These simulation results can be used to obtain reliable estimates for the environmental densities within which the observed SDSS galaxies reside.The reconstruction is restricted to the Northern Galactic Cap of the SDSS DR7 region and to the redshift range 0.01≤ z≤0.12. In order to avoid problems near the survey boundary, where the reconstruction is less reliable, we exclude galaxies and groups whose distances to the SDSS survey boundary are smaller than 5. For each galaxy, we correct for its redshift-space distortions (see Paper III, Wang et al. 2012 and Shi et al. 2016, for details) and estimate its real-space location. The environmental density for a galaxy is determined by computing the matter density, smoothed with a Gaussian kernel of 4, at its real-space location in our constrained simulation. Tests based on mock galaxy catalogs demonstrate that the uncertainties in this density estimate are typically 0.10 dex. In what follows, we quantify the matter density for each galaxy using the quantityΔ =ρ/ρ̅ ,where ρ is the smoothed mass density at the location of the galaxy, and ρ̅ is the mean density of the universe.In the literature, one environmental indicator commonly used is the local number density (or over-density) of galaxies. Our tests show that the galaxy densities are positively correlated with the mass densities, although the scatter is rather large (see also Baldry et al. 2006). Because of the redshift distortion effect and the complicated correlation with underlying mass density, galaxy density hampers a meaningful interpretation of its correlation between galaxy properties (see Weinberg et al. 2006 and Woo et al. 2013). Therefore, we consider the matter densities, Δ, used here to be a superior, and more physical, quantity to characterize the large-scale environment of a galaxy. § THE QUENCHED POPULATIONSIn this section we examine how quenching of star formation depends on the intrinsic and environmental properties of galaxies. For any subset of galaxies with a given set of properties {g}, we calculate the average quenched fraction asF_ q({g})=∑_i∈{g}q_iw_i/∑_i∈{g}w_i ,where w_i is the weight for a galaxy i, and q_i is set to 1 for a quenched galaxy, otherwise zero.In our following analysis, we will consider the total population, and the central and satellite populations separately. The quenched fractions of the sub-populations are denoted by F_ q, c (centrals) and F_ q, s (satellites), respectively. §.§ Marginalized dependence on stellar mass, halo mass and density Figure <ref> shows, for the whole population and for centrals and satellites separately,the quenched fraction of galaxies as a function of galaxy stellar mass, halo mass and environmental mass density. The quenched fractions as a function of one of the three parameters are obtained by marginalizing over the other two parameters. Note that we only consider galaxies with log(m)≥9.0 and results are shown only for these galaxies in the left two panels. In this and the following figures, error bars are all evaluated from 1,000 bootstrap re-samplings and only data points that contain at least 10 galaxies are shown.It is clear that the quenched fraction increases with increasing stellar mass, halo mass and environmental density, consistent with previous studies (e.g. Brinchmann et al. 2004; Weinmann et al. 2006; Baldry et al. 2006).These correlations are often considered as observational evidence for quenching processes, such as mass quenching, halo quenching and environmental quenching. We then examine centrals and satellites separately. At given stellar mass or density, satellites tend to be more often quenched than centrals. However, the trend is reversed if halo mass is used instead as the control parameter. It is well known that more massive galaxies tend to live in more massive halos and more massive halos tend to reside in higher density regions, and so it is not obvious whether the density dependence reflects a causal connection or is induced by the correlation between halo mass and density. In order to disentangle the different effects, we need to examine the joint distribution of the quenched population with respect to the parameters in question (see below).§.§ Dependence on stellar mass and environmental density We first disentangle the dependencies of the quenched fraction on stellar mass and environmental density. The left panel of Figure <ref> shows the quenched fraction of the total population, F_ q(m,Δ), as a function of the environmental density for galaxies in various stellar mass bins. Consistent with the marginalized result, there is significant dependence on Δ, even for galaxies in a given narrow m bin. The dependence is seen to be stronger at low m and weaker at the lower Δ end. At a given Δ, the quenched fraction increases with m, and the increase is more significant for galaxies of lower masses.We then investigate the quenched fractions separately for the central and satellite populations and present the results in the middle and right panels of Figure <ref>. For central galaxies, the Δ-dependence is rather weak for all stellar mass bins in comparison to the result shown in the right panel of Figure <ref>. This suggests that the marginalized Δ-dependence for centrals is primarily due to the fact that more massive centrals tend to reside in higher density region. In contrast, for satellites, the Δ-dependence is strong in most of the stellar mass bins. In particular, the marginalized Δ-dependence is very similar to that for the lowest stellar mass galaxies, indicating that the marginalized result for satellites is dominated by low-mass galaxies over the whole density range. At given Δ, the m-dependence is significant for both centrals and satellites. The m-dependence for centrals is similar to that for satellites at low Δ, but is stronger at high Δ, suggesting that the difference shown in the marginalized m-dependence between the two populations is mainly caused by the difference in the high density region. These results are in qualitative agreement with the results previously obtained by Peng et al. (2010; 2012) and Knobel et al. (2015). §.§ Dependence on stellar mass and halo mass Figure <ref> shows the quenched fraction as a function of halo massfor galaxies in different stellar mass bins. At a given m, the quenched fraction increases with M, and the increase is steeper for lower mass galaxies. These trends are consistent with those obtained before (e.g. Weinmann et al. 2006;Wetzel et al. 2012). Most intriguingly, the behaviors of central and satellite galaxies are almost indistinguishable, although it is important to stress that the halo mass range covered by central galaxies is rather limited, in particular for galaxies with low masses. This suggests that the difference between centrals and satellites found in the marginalized dependencies on m and M are mainly caused by the different m and M ranges covered by the two populations. For example, satellites usually reside in more massive halos than centrals of the same stellar mass, this combined with the M-dependence of the quenched fraction can explain why the quenched fraction for satellites is higher than that for centrals of the same stellar mass (left panel of Figure <ref>). Similarly, since by construction centrals are more massive than satellites in halos of a given mass, centrals are expected to be more often quenched than satellites because of the m-dependence (middle panel of Figure <ref>).For the stellar mass bin 10.6≤log(m)≤11, the quenched fraction of centrals appears to be somewhat higher than for satellites in the same halo mass bin. However, upon closer inspection, we find that this is mainly an artifact of the finite bin-sizes used, combined with the fact that at given halo mass, the satellite galaxies are strongly biased towards log(m)∼10.6, while centrals are biased towards log(m)∼11. To demonstrate this, we split the galaxies in this mass bin into two sub-samples at log(m)=10.8. The results for these two sub-samples are shown in the bottom left panel of Figure <ref>. As one can see, this significantly reduces the differences between the centrals and satellites, and we therefore conclude that there is no significant difference in the halo mass dependence of the quenched fraction between centrals and satellites.We note again that numerous previous studies (e.g., van den Bosch et al. 2008; Weinmann et al. 2009; Pasquali et al. 2010; Wetzel et al. 2012; Peng et al. 2012; Knobel et al. 2013; Bluck et al. 2014; Fossati et al. 2017) have shown that, at a given stellar mass, satellites are more often quenched than centrals and suggested that some satellite-specific environmental processes have played important roles in quenching satellite galaxies. However, our results portray a different picture; centrals and satellites follow the same correlationbetween quenched fraction and host halo mass, suggesting that they experienced similar environment-dependent quenching, and the only reason that satellite and centrals appear different at fixed stellar mass is that they sample different ranges in host halo mass. Further investigations on this are presented in Sections <ref> and<ref>.§.§ Dependence on halo mass and environmental densityIt is known that more massive halos tend to locate in higher density regions, an effect usually referred to as halo bias (e.g. Mo & White 1996). Therefore the density dependence shown in Figure <ref> and the halo mass dependence shown in Figure <ref> may be connected. To examine this, we split galaxies further into four sub-samples based on Δ and calculate the quenched fraction as a function of M. The results are shown in Figure <ref>. Overall, the M-dependence of the quenched fraction is similar for different sub-samples of Δ, suggesting that halo mass may be the dominating factor in determining the quenched fraction.However, for a given M, there is a weak but systematic trendof increasing quenched fraction with increasing Δ. This suggests that factors other than halo mass affect galaxy quenching, and is broadly consistent with some previous findings that galaxy groups of a given halo mass have different clustering properties depending on the color or star formation of their member galaxies (e.g. Yang et al. 2006). All of this is most likely related to assembly bias (Gao et al. 2005), the fact that halo bias depends not only on halo mass, but also on other halo properties such as assembly history, if the quenching processes depend on these additional halo properties. Unfortunately, the observational sample is still too small, and the uncertainties in the density dependence shown in Figure <ref> are still too large, to draw any quantitative conclusions. We will revisit this issue in Section <ref>. § DENSITY-BASED ENVIRONMENTAL QUENCHING EFFICIENCIES A useful parameter to quantify the efficiency of galaxy quenching is the relative environmental quenching efficiency (hereafter `quenching efficiency' for brevity) ε(m, Δ | Δ_0) ≡F_ q(m,Δ)-F_ q(m,Δ_0) 1-F_ q(m,Δ_0) .(Peng et al. 2010), which specifies the probability for a star-forming galaxy of mass m to be quenched when it transits from an environment characterized by some zero-point matter density, Δ_0, to a region with Δ. We choose Δ_0 to correspond to the lowest-density environment probed by our data, as the environmental effects are expected to be minimal in these void-like environments. As shown in Figure <ref>, the quenched fraction of galaxies, in most of the stellar mass bins, has the lowest value at logΔ≤ 0, where the Δ-dependence of the quenched fraction is also weak. We thus choose galaxies with logΔ≤0 to define the zero point. §.§ A dearth of stellar mass dependence for the total population The upper-left panel of Figure <ref> shows the quenching efficiency ε(m, Δ | Δ_0) of the total population as a function of Δ. As one can see, the efficiency is close to zero at Δ∼Δ_0, by definition, and increases rapidly with Δ. Remarkably,the efficiency is almost independent of m over the range 9≤log(m)≤11. To see this more clearly, we show ε(m, Δ | Δ_0) as a function of m for various Δ bins in Figure <ref> as black squares. The quenching efficiency is almost a constant in the entire mass range,except for the highest stellar mass bin. This is in good agreement with Peng et al. (2010), although they used galaxy number density instead of the more physical matter density used here. The efficiency for the highest stellar mass bin is higher than for the other stellar mass bins, but the uncertainties are large. Note that there are only very few massive galaxies in low density regions with Δ < 1, which can produce a large statistical uncertainty in the denominator of Eq. (<ref>). The discrepancy for the highest m bin is, therefore, not conclusive. We will come back to this issue in Section <ref>.Baldry et al. (2006) found that the quenched fraction can be fitted by a very simple equation:F_ q(m,Δ)=1-exp[-(Δ/Δ_⋆)^b] exp[-(m/m_⋆)^d] .Inserting it into Equation (<ref>), we obtainε(m, Δ | Δ_0)=1-exp[-(Δ/Δ_⋆)^b +(Δ_0/Δ_⋆)^b] .We use this equation to fit the ε data points, with logΔ_0=-0.2, which is the median value for galaxies with logΔ_0 ≤ 0. The best fit gives Δ_⋆=11.5±0.26 and b=0.975±0.035 (see Figure <ref> for the fitting curve). We then use Equation (<ref>) to fit the quenched fraction for the total population with Δ_⋆=11.5 and b=0.975. The best fit values for the other two parameters are m_⋆=5.29±0.04×10^10 and d=0.749±0.006, and the results are shown in the left panel of Figure <ref>. These fitting results are presented here as a convenient way to represent the data. §.§ Central versus satellite populations For central galaxies, we define a quenching efficiency similar to Equation (<ref>):ε_ c(m, Δ | Δ_0) = F_ q,c(m,Δ)-F_ q,c(m,Δ_0) 1-F_ q,c(m,Δ_0) ,where F_ q,c(m,Δ) is the quenched fraction of central galaxies. Here again we use galaxies at logΔ_0≤ 0 to calculate F_ q,c(m,Δ_0). So defined, this efficiency characterizes the probability for a star forming central to quench if it were to move to a higher density environment, Δ, while remaining a central. As is evident from the upper-middle panel of Figure <ref>, ε_ c has a much weaker dependence on Δ than the corresponding quenching efficiency of the total population. Nevertheless, even for centrals there is a significant tendency for ε_ c to increase with Δ. Moreover, there is also a trend for ε_ c to increase with stellar mass (see also Figure <ref>), in particular at high Δ. As for the total population, the most massive bin has very few galaxies with logΔ≤0, so that the corresponding ε_ c carries large uncertainties.Similarly, we can define a quenching efficiency for satellite galaxies. Unfortunately, the total number of satellites at logΔ≤ 0 is small, and so the derived efficiency will have large uncertainties. However, F_ q,s(m,Δ_0) is close to F_ q,c(m,Δ_0) for most of stellar mass bins where both can be measured reliably (Figure <ref>). We thus define an alternative efficiency for satellites asε'_ s(m, Δ | Δ_0) = F_ q,s(m,Δ)-F_ q,c(m,Δ_0) 1-F_ q,c(m,Δ_0) .The results are shown in the upper-right panel of Figure <ref>. As one can see, ε'_ s(m, Δ | Δ_0) increases rapidly with Δ, but its dependence on m is weak.Finally we consider a satellite-specific quenching efficiency, which is defined asε_ s(m, Δ) = F_ q,s(m,Δ)-F_ q,c(m,Δ)/1-F_ q,c(m,Δ) ,(see van den Bosch et al. 2008). Here, the central galaxies are used as the control sample (zero point) in (m, Δ) space, against which the quenching of satellites is measured. The lower-left panel of Figure <ref> shows ε_ s as a function of Δ for satellite galaxies of different masses. As one can see, the quenching efficiency increases quite rapidly with Δ in most stellar mass bins. For a given Δ,ε_ s decreases with m, and the decrease is larger for more massive galaxies.As pointed out by Wetzel et al. (2013), ε_ s defined in this way actually measures a combined effect of the satellite-specific quenching processes and the evolution of central galaxies. So it is not straightforward to use it to interpret the satellite quenching efficiency. However, it does not change our conclusion in the subsequent subsection that there exists an unexpected connection between centrals and satellites. §.§ A conspiracy between centrals and satellites? Peng et al. (2010) argued that the stellar mass independence of ε(m, Δ | Δ_0) can be fully understood in terms of the quenching of satellite galaxies. Assuming that the environmental effect on centrals is negligible, they wrote the environmental quenching efficiency as,ε'(m, Δ | Δ_0)=f_ s(m, Δ) ε_ s(m, Δ) ,where, f_ s(m, Δ) is the satellite fraction, and ε_ s(m, Δ) is the satellite-specific quenching efficiency of Equation. (<ref>). Figure <ref> shows the satellite fraction as a function of Δ for galaxies in different m bins. In order to explain the m-independence of the quenching efficiency, Peng et al. (2010; 2012) suggested that both f_ s(m, Δ) and ε_ s(m, Δ) are independent of m. This is clearly inconsistent with our data shown in the lower-left panel of Figure <ref> and Figure <ref>. Since the hypothesis made by Peng et al. has far-reaching implications (i.e., the the environment dependence of the quenching efficiency is entirely due to the quenching of satellites), it is important to address this discrepancy in some detail. In fact, there are a number of factors that play a role. First of all, Peng et al. used the over-density of galaxies as their environment indicator, rather than the more physical matter density used here. In addition, they used colors to split their population into star-forming and quenched, whereas we use actual star formation rates. Since dust extinction can make a star forming galaxy appear red, and thus `quenched' based on color, using actual star formation rates yields more accurate estimates of the true quenched fraction. Secondly, Peng et al., assumed F_ q,c(m,Δ) to be independent of Δ, even though this is not supported by their own data. Since centrals on average reside in lower Δ environments than satellites of the same stellar mass (e.g. Knobel et al. 2015), the average of F_ q,c over Δ is biased towards low-density regions. As a consequence, adopting the average of F_ q,c can lead to an overestimation of ε_ s, in particularly at the high-Δ end. For low-mass galaxies, this bias is negligible, because F_ q,c is on average much smaller than both F_ q,s and unity. This is consistent with the weak m-dependence of ε_ s we find for these galaxies. For massive galaxies, on the other hand, the value of F_ q,c is higher and the denominator in Equation (<ref>) smaller. This results in a significant bias that weakens the m-dependence of ε_ s at the high mass end. Indeed, when taking into account the dependence of F_ q,c on galaxy number density, Knobel et al. (2015) also found that ε_ s decreases with m at a given galaxy number density (see their figure 3). Finally, as is evident from Figure <ref>, the satellite fraction depends strongly on m. In fact, it is well known that the satellite fraction increases with decreasing stellar mass, which has been demonstrated using galaxy group catalogs (e.g., van den Bosch et al. 2008), subhalo abundance matching (e.g., Wetzel et al. 2013), galaxy-galaxy lensing (e.g., Mandelbaum et al. 2006), galaxy clustering (e.g., Cooray 2006; Tinker et al. 2007; van den Bosch et al. 2007) and combinations thereof (e.g., Cacciato et al. 2013).In the lower-middle panel of Figure <ref>, we show ε' as a function of Δ. For low-mass galaxies with log(m)≤ 10.2, ε' is almost independent of m, in agreement with the hypothesis of Peng et al. (2010; 2012). However, for more massive galaxies, ε' clearly depends on m. Thus, the m-independence of ε of the total population shown in Section <ref> is not due to the m-independence of f_ s and ε_ s, and the quenching efficiency of the total galaxy population cannot be explained by the quenching of satellites alone.As shown in Figure <ref>, central galaxies do exhibit some weak but non-trivial Δ-dependence in all the mass bins considered, and at log(m)≥ 10.6 the dependence is actually comparable to or slightly stronger than that for satellites. For centrals, the Δ-dependence is stronger for more massive galaxies, a trend opposite to that seen for satellites. To understand the importance of quenching for centrals in the total population, we express the quenching efficiency for the total population in terms of ε_ c and ε'. At very low density, logΔ≤ 0, environmental quenching is very weak and f_ s(m, Δ) is quite small. Thus, to good approximation F_ q(m,Δ_0)≈ F_ q,c(m,Δ_0), and we can rewrite Equation (<ref>) asε (m, Δ| Δ_0) ≃F_ q(m,Δ)-F_q,c(m,Δ_0)/1-F_ q,c(m,Δ_0)= ε_ c(m, Δ | Δ_0)+[1-ε_ c(m, Δ | Δ_0)]ε'(m, Δ | Δ_0) ,where the second equation is obtained by inserting Equations (<ref>) and (<ref>)into the right-hand-side of the first line. Note that this reduces to Equation (<ref>) in the limit ε_ c→ 0. Using Equation (<ref>) to compute the quenching efficiencies yields the results that are in excellent agreement with those shown in Figure <ref> obtained using the original definition [Equation. (<ref>)]. Since the difference in all m bins is less than 0.03, we do not show the results. The good agreement justifies our approximation that F_ q(m,Δ_0)≈ F_ q,c(m,Δ_0).Note once more that ε(m, Δ| Δ_0) reveals only a very weak dependence on m, except for the most massive galaxies, for which the statistics is extremely poor (see Section <ref>). It seems to conflict with the fact that ε is the combination of ε_ c and ε', which both strongly depend on m. To understand this apparent discrepancy, we show ε, ε_ c and ε' as a function of m for various Δ bins in Figure <ref>. Apparently, the opposite trends in the m-dependence of ε_ c and ε' counterbalance each other so as to yield a ε that depends only weakly on m.Our results clearly demonstrate that the environmental effect on centrals has to be taken into account in order to reproduce the m-independence of the quenching efficiency seen for the total population. This is particularly important for massive centrals. Peng et al. (2012) also found a significant environmental dependence for central galaxies. However, they suspected that it is caused by the misidentification of satellites as centrals. We indeed find some signals for such misidentification in our results. For example, some abnormal behavior of the lowest mass galaxies in high-density bins can be seen in ε_ s, as well in ε_ c. More recently, Hirschmann et al. (2014) investigated the misidentification problem using mock galaxy catalogs constructed from a semi-analytic model of galaxy formation, and found that the contamination in centrals is less than 10% for most galaxy masses and environmental densities. The average contamination is less than 6.5%, roughly independent of m (see also Lange et al. 2017). Since the difference between F_ q,c and F_ q,s rapidly decreases with increasing m (Figure <ref>), the impact of the contamination is expected to decrease with m. In contrast, the observed Δ-dependence is more important for more massive galaxies. Based on these results, we are confident that central-satellite contamination does not significantly impact our conclusion that satellite quenching alone cannot account for them-independence of the quenched efficiency, at least for massive galaxies.An interesting question is why the increase of ε_ c with increasing m apparently compensates the decrease of ε' with increasing m. It might reflect some deeper connection between the quenching processes for centrals and satellites. As shown above, the quenched fractions of centrals and satellites of the same stellar mass correlate with the environment as characterized by halo mass in the same way. This suggests that centrals and satellites may experience similar quenching processes which are ultimately related to the host halo mass. In Section <ref> we will construct simple models to investigate this issue. § HALO-BASED QUENCHING EFFICIENCIES As discussed earlier, halos play a crucial role in shaping galaxy properties. Many quenching processes are thought to correlate with halo mass and stellar mass (see Mo et al. 2010). In order to examine this, we define two new quenching efficiencies in the same vein as the environmental quenching efficiency of Equation (<ref>). These are the halo-based environmental quenching efficiency, defined asε(m, M | M_0) ≡F_ q(m,M)-F_ q(m,M_0) 1-F_ q(m,M_0) ,and the stellar mass quenching efficiency, defined as (see also Peng et al. 2010)ε_ m(m, M | m_0) ≡F_ q(m,M)-F_ q(m_0,M) 1-F_ q(m_0,M) .Here M_0 and m_0 are the halo mass and stellar mass `zero-points' against which the dependencies on M and m are compared. Note that ε(m, M | M_0) and ε_ m(m, M | m_0) characterize the dependence on halo mass and stellar mass, respectively, of the combined effect of all quenching processes, including `environmental processes' (such as ram-pressure/tidal stripping and strangulation) and `internal processes' (such as quenching induced by AGN and supernova feedback). §.§ Quenching efficiencies for the total population Ideally, one would like to adopt the lowest halo mass as `zero-point' environment to calculate the environmental efficiency. However, because no massive galaxy (log(m)>10.6) resides in halos of log(M)<12, choosing the lowest halo mass bin is inappropriate. We therefore adopt 13≤log(M)<13.5 to define the quenching zero-point, at which the estimates of the quenched fraction are robust for all stellar mass bins (see Figure <ref>). The corresponding efficiency, obtained from Equation (<ref>), is shown as a function of halo mass in Figure <ref>. Here again the environmental quenching efficiency is almost independent of stellar mass, although it increases strongly with halo mass. It is easy to see that, if the efficiency is independent of stellar mass, this independence holds regardless of the value of M_0 used. This result clearly shows that the environmental dependence of the quenched fraction can be well separated from the dependence on stellar mass, independent of whether the environmental parameter is the large scale matter density or the mass of the host halo in which the galaxies reside.Motivated by Equation (<ref>) and that the dependence of F_ q on halo mass can be well described by a power law function (Figure <ref>), we propose to use a simple formula to describe the quenched fraction:F_ q(m,M)=1-exp[-(m/m_⋆)^d](alog M-ac) .It results in an environmental quenching efficiency given byε(m, M | M_0)=1/c-log M_0logM/M_0 .We first use Equation (<ref>) with log(M_0)=13.25 to fit the data points shown in Figure <ref>. The best fit gives c=16.37±0.06 and is shown as the grey line in Figure <ref>. We then use Equation (<ref>) to fit the quenched fraction for the total population shown in Figure <ref>, with c fixed to 16.37. The best fit values for the other three parameters are a=-0.2±0.004, m_⋆=6.5±0.2×10^10 and d=0.61±0.02.The results for ε_ m(m, M | m_0) are presented in Figure <ref>. Here we choose 9.8≤logm_0< 10.2 so that we have robust estimation of F_ q(m_0,M) for all the halo-mass bins. As one can see, the ε_ m(m, M | m_0) - m relation is quite independent of halo mass in all the halo mass bins except the most massive halo bin. Since the quenched fraction can be well fitted by Equation (<ref>), ε_ m(m, M | m_0) can be described byε_ m(m, M | m_0) ≡ 1-exp[-(m/m_⋆)^d +(m_0/m_⋆)^d] .The prediction of this equation with m_⋆=6.5×10^10, d=0.61 and m_0=10^10 is plotted as the grey line in Fig. <ref>. §.§ Quenching efficiencies for centrals and satellites As shown in Sections <ref> and <ref>, the density-based quenching efficiencies for central and satellite populations depend on stellar mass in an opposite way, and they counterbalance each other to produce a m-independent efficiency for the total population. It is thus also interesting to examine the quenching efficiencies for centrals and satellites separately by using halo mass, instead of the density, as the environmental parameter. However, we will not repeat the same analyses as in Section <ref> and <ref>, for the following two reasons. First, the stellar masses of centrals are strongly correlated with the halo masses of their host groups, so that it is difficult to find a single halo mass bin to define the environmental zero point for central galaxies of different masses. Second, for satellites with given (m,M), it is difficult to select a large number of centrals of the same (m,M) to form a control sample to calculate the satellite-specific quenching efficiency. Because of these we adopt a different approach, as described below.To start with, we look at the satellite fraction as a function of halo mass in different stellar mass bins, as shown in Figure <ref>. Unlike the smooth relation between the satellite fraction with Δ shown in Figure <ref>, f_ s(m, M) as a function of M resembles roughly a step function, particularly for low-massgalaxies. The fraction is close to zero at M<M_ tr and about one at M>M_ tr, where M_ tr is the mass scale at the transition of the step function and increases with increasing stellar mass. Therefore, centrals and satellites dominate the galaxy population in different regions in the (m,M) plane and the two populations are comparable in number only in a narrow region in the (m,M) plane. This property of the satellite fraction can be used to understand the quenching efficiency of centrals and satellites over a large range in m and M.As shown in Figure <ref>, ε(m, M | M_0) for galaxies of a given m follows the same correlation with M below and above M_ tr(m) (marked by the vertical lines in the figure), where galaxies are dominated by centrals and satellites, respectively. This indicates that the quenched fractions in both the central and satellite populations depend on the host halo mass in a similar way not only in the M range where the two populations overlap, but also over the whole M range. Similar analysis can also be performed on the basis of the mass quenching efficiency ε_ m. For example, the four stellar mass bins in 9.0<log(m)<10.6 are all dominated by centrals in halos with log(M)<12, but by satellites at log(M)>13. However, the mass quenching efficiency follows the same trend with stellar mass in different halo mass bins, no matter whether the galaxies in the halo mass bin are dominated by centrals or by satellites. All these suggest that, whatever the quenching processes are, they tend to produce a quenching efficiency that depends on halo and stellar masses in a similar way for both centrals and satellites. The results also suggest that the similarity in quenching efficiency between centrals and satellites exists not only in the region of the (m,M) plane where the two populations overlap, but also over the whole range of (m,M) covered by the sample. § HALO MASS AND ASSEMBLY DRIVE ENVIRONMENTAL QUENCHING As we have shown above, the quenched fraction of galaxies depends on both halo mass and environmental density. It is therefore important to examine whether it is the halo mass or the environmental density that plays the dominating role. To address this question, we construct two simple models, in which the quenched fraction is assumed to be determined by the stellar mass combined with one of the two environmental quantities. We also try to understand whether assembly histories of dark matter halos affect the galaxy properties in a third model. These models will also help us to understand the apparent discrepancy between the results based on environmental density and halo mass. Namely why is the density-based quenching efficiency independent of the stellar mass for the total population but stellar-mass dependent when centrals and satellites are analyzed separately (Section <ref>), while centrals and satellites follow very similar trends in the halo-based quenching efficiency (Section <ref>)?In the first model (hereafter Model A), the environmental density, Δ, is assumed to be the primary driver of the environmental dependence of quenching. For each real galaxy in our SDSS sample, we construct a corresponding model galaxy, which has exactly the same m, M, Δ, and the same identification as either a central or a satellite. We assign a given model galaxy i a quenching probability, q_i=F_ q(m, Δ), according to its position in the (m, Δ) space (Figure <ref>). Note that centrals are treated differently from satellites, because the observed F_ q,c(m, Δ) is very different from F_ q,s(m, Δ). We then use Equation (<ref>) to estimate the average quenched fraction for any given subset of the model galaxies. Figure <ref> shows the predicted quenched fraction as a function of halo mass, M, for galaxies in different m bins.As one can see, Model A predicts a positive dependence of the quenched fraction on M, in rough agreement with the observation. This dependence is expected from the fact that dark matter halos are tracers of the matter density field. Note, however, that there is a marked difference between the model prediction and the observational data. First of all, the model predicts no significant dependence of F_ q,c on M, contrary to the data which reveals a clear trend of increasing F_ q,c with increasing M (cf., Figure <ref>). Second, the predicted M-dependence for satellite galaxies is also weaker than observed.This discrepancy is particularly large for low mass galaxies in low mass halos, where the model overpredicts F_ q,s by as much as 0.2. Finally, the model predicts significant differences between the quenched fractions of centrals and satellites of similar m and M, while such differences are absent in the observational data (cf., Figure <ref>).In the second model (hereafter Model B), halo mass instead of environmental density is assumed to be the primary driver of the observed environmental dependence of quenching. To test this hypothesis, we construct a model galaxy sample by assigning each real galaxy in our SDSS sample a quenching probability q_i = F_ q(m, M) based on its position in the (m, M) plane (the black solid lines in Figure <ref>). Since the dependence of F_ q on M and m is very similar between centrals and satellites, we do not distinguish between them when assigning q_i. Hence, in this model the probability for a galaxy to be quenched is solely determined by the galaxy's stellar mass and host halo mass, with centrals and satellite being treated in exactly the same way.The quenched fractions predicted by Model B as a function of Δ and m [computed using Equation (<ref>)] are shown in Figure <ref>. The results closely resemble those for the real galaxies. To better illustrate the quality of the model, Figure <ref> plots the differences between the model prediction and the data, δ F_ q(m,Δ). Note that the differences are fairly small, typically less than ∼ 0.05 for the entire galaxy sample. For centrals the discrepancies are slightly larger, while for satellites the differences are comparable to the observational uncertainties. Hence, Model B provides a fairly accurate description of the data.Taken together, the results from Models A and B strongly suggest that the mass of the host halo is a far more important environmental parameter for regulating quenching than is the matter density.Before moving on, we try to understand some of the general trends predicted by Model B. The resulting dependence of F_ q on Δ is very different for centrals and satellites, although the two populations are assumed to have exactly the same dependence of F_ q on m and M. This arises because centrals and satellites of a given m cover very different ranges in M. Centrals with 9≤log (m)≤ 10.2 usually reside in halos with log (M) ∼ 12. The halo bias at this mass scale is close to unity with little dependence on M (see Sheth, Mo & Tormen 2001). Hence, the mean halo mass in the low-density regions is not very different from that in the high-density regions, which explains why F_ q,c is almost independent of Δ for galaxies in this m range. As the mass of the central galaxy increases, so does the mass of its host halo, which pushes it into the regime where halo bias is larger than unity and has a strong dependence on halo mass. Consequently, halos in high density regions are, on average, more massive than those in low density regions, even if they contain centrals of the same stellar mass. This explains why the dependence of F_ q,c on Δ becomes stronger for centrals with higher stellar masses, as shown in Figure <ref>. However, for a given m, the distribution in M is quite narrow (see Yang et al. 2009), so the Δ-dependence predicted by Model B remains weak. For satellites, the situation is very different. At the low m end, satellites reside in halos that cover a very wide range in M, which gives rise to a very strong dependence on Δ. As m increases, the dispersion in M decreases, which weakens the dependence of F_ q,s on Δ. As evident from Figure <ref>, Model B slightly overestimates F_ q,c at the low-Δ end while underestimating it at the high-Δ end, a trend that is evident for every stellar mass bin. It suggests that the quenching of (central) galaxies depends not only on halo mass and stellar mass, but also on some other halo properties that are correlated with the environmental density. One such halo property is halo assembly history, which is correlated with the environmental density (known as assembly bias). In order to see if the discrepancy can be explained by halo assembly bias, we need to know, for each individual group, the formation redshift (hereafter z_ f) that characterizes its assembly history. Our ELUCID simulation is a constrained simulation in the SDSS DR7 region and can reliably reproduce most of massive groups (see paper III), and here we make use of the information it provides to estimate the formation redshifts for individual groups. To do this, for each group, we search all halos in the simulation that have mass differences less than 0.3 dex with, and distance less than 5 to, the group in question.Most of the groups (∼97.4%) have at least one halo companion defined in this way, and we assign the formation redshift of the nearest halo to the group. The formation redshift is defined as the highest redshift at which half of the final halo mass has assembled into progenitors more massive than 10^11.5 (see Neistein, van den Bosch & Dekel 2006; Li et al. 2008). The choice of this mass limit is motivated by the fact that it corresponds to the halo mass at which the star formation efficiency is the highest at different redshifts (see Lim et al. 2017). For groups with log(M)<12.0 that do not have accurate halo mass estimates in the group catalog, we only search for halo companions with 11.7<log(M)<12.0, where the lower mass limit (11.7) is adopted so that all halos have reliable estimates of halo formation redshifts in the ELUCID simulation. We have also used other definitions of halo formation redshift, such as the redshift at which the main progenitor reaches half of the present-day halo mass. The results are very similar.A third model, Model C, is then constructed on top of Model B. In Model C, the quenching probability of a model galaxy depends not only on m and M, but also on the formation redshift of its halo. The simplest way to link z_ f to quenching probability is to assume that a galaxy is quenched when z_ f>z_ th, where z_ th is a formation redshift threshold. In reality, however,galaxies with given z_ f, m and M, must have some dispersion in their star formation rate. To mimic this, we introduce a dispersion in z_ f for each system before applying the criterion z_ f>z_ th to select the quenched fraction. In practice, for a galaxy i with z_ f=z^i_ f, we use a Monte Carlo method to generate 500 mock galaxies, with their formation redshifts (z^i_ f,m) randomly drawn from a Gaussian distribution with the mean value equal to z^i_ f and a width σ_ z. Then, for a given z_ th, the quenching probability of the model galaxy, q_i, is set to be the fraction of the mock galaxies that have z^i_ f,m> z_ th among the 500 mock galaxies. In order to introduce the dependence on m and M, the threshold z_ th is required to be a function of the two quantities, and is determined by the criterion that the dependence on m and M for the model galaxies is exactly the same as that for real galaxies. Note that in our model the formation redshift dependence is only considered for central galaxies; satellites are treated in exactly the same way as in Model B.When σ_ z is set to be 0, the dependence of F_ q,c(m, Δ) on Δ for the model galaxies is found to be much stronger than that for real galaxies. We have experimented a series of values for σ_ z, and found that the model matches the observation the best when σ_ z∼ 0.8. Figure <ref> shows the quenched fraction as a function of m and Δ obtained from this best model. A significant increasing trend of the quenched fraction with Δ is now produced for central galaxies in most of m bins, as is seen in the observational data. The lower two panels of Figure <ref> show the difference between the observational quenched fraction and the predictions of Model C for the total population and centrals, respectively. In contrast to Model B, the dependence of δ F_ q(m,Δ) on Δ almost completely disappear in Model C.We then compute the environmental quenching efficiency, defined in Equation (<ref>) with the zero point estimated from data at logΔ≤ 0, for Model B and C. The results are presented in the left panels of Figure <ref>. The quenching efficiency predicted by Model B exhibits a very weak dependence on m. This is not surprising since the halo-based environmental efficiency, ε_ c(m, M | M_0), is independent of stellar mass (Figure <ref>) and since halo mass is the only driver of the environmental quenching in Model B. There is a small deviation between the model prediction and the observation for most of the stellar mass bins. Such deviation is absent in Model C, suggesting that it has the same origin as that shown in Figure <ref>.We also show the predictions of Model B and C for ε_ c(m, Δ | Δ_0), defined by Equation (<ref>), and for ε'(m, Δ | Δ_0), defined by Equation (<ref>), in the middle and right panels of Figure <ref>. In Model B, one can see a clear trend that, at a fixed Δ, ε_ c increases, while ε' decreases, with m. As discussed above, the halo mass distribution becomes broader for centrals but narrower for satellites as stellar mass increases, which explains the opposite trends in the quenched efficiency as a function of stellar mass for centrals and satellites. After taking into account the assembly bias effect (Model C), the dependencies on stellar mass for both centrals and satellites are enhanced and the predictions resemble more closely the observational results.Moreover, the predicted efficiency for the most massive galaxies is now closer to the other mass bins than the observational result, giving further support to the hypothesis that the discrepancy found for the most massive galaxies in the observational data is mainly caused by the uncertainty in the estimation of F_ q(m,Δ_0). The uncertainty is largely eliminated in the model predictions, because the value of q_i assigned to galaxies in the low-density regions is the average over a larger sample of galaxies. § THE QUENCHING OF CENTRALS IS NOT SPECIAL As shown in Section <ref> and <ref>, we find that centrals and satellites follow the same correlation of quenched fraction with stellar mass and host halo mass. This result appears to be in conflict with those obtained by some earlier studies (e.g. van den Bosch et al. 2008; Wetzel et al. 2012; Peng et al. 2012), where quenching is found to depend strongly on whether a galaxy is a central or a satellite. However, as discussed above, host halo mass ranges covered by centrals and satellites of the same stellar mass are very different. Thus, if halo mass is not used as a control parameter when comparing the two populations, as is the case in most of the previous studies, then one is comparing centrals in low mass halos with satellites in massive ones and neglecting the strong dependence on halo mass. We believe that this is the origin of the discrepancy and that there is no conflict between our results and those obtained in these earlier studies.It is possible that the similarity between centrals and satellites we find in the data is not real, but caused by errors in the group finder used to identify groups and central galaxies in them. As one can see from Figure <ref>, the satellite fraction, f_ s(m,M), resembles roughly a step function and is close to zero at M<M_ trand about one at M>M_ tr (M_ tr is the mass scale at the transition of the step function). Thus, even if the mis-identification fraction is small in the whole population (see Hirschmann et al. 2014), the galaxies that are identified as satellites at M<M_ tr may be significantly contaminated by centrals. In this case, the similarity between centrals and `satellites' of a given stellar mass in their quenched fraction may be produced by the false identification between centrals and satellites, rather than a real similarity.However, it is difficult to explain the m-independence of the environmental quenching efficiency and the M-independence of the stellar mass quenching efficiency (Section <ref>) by central/satellite mis-identifications alone. If centrals and satellites had quenching properties that depend on the halo and stellar masses in significantly different ways, the environmental quenching efficiency, ε(m, M | M_0), would be expected to correlate with the halo mass in different ways depending on whether M<M_ tr or M>M_ tr, as the galaxy populations in the two mass ranges are dominated by centrals and satellites, respectively (see Fig. <ref>). Moreover, since M_ tr increases with increasing stellar mass, ε(m, M | M_0) would also be expected to vary with stellar mass. Similarly, it would also lead to M-dependence in the mass quenching efficiency, ε_ m(m, M | m_0). Both are inconsistent with the observational results. We thus conclude that the similar quenching properties of centrals and satellites are not produced by mis-identifications between the two populations, and are real within the statistical uncertainties of the data.It should be pointed out that this interpretation is based on the premise that the inferred halo mass is sufficiently accurate. Recently, Campbell et al (2015) used a mock catalog including galaxy color to check the color-dependent statistics inferred from group catalogs. They compared the marginalized dependence of the quenched (red) fraction on halo mass obtained from their mock group catalog with that obtained directly from the simulation used to construct the mock catalog, and found that group finders tend to reduce the difference in the quenched fraction between centrals and satellites (see their figure 13). This systematic error is produced by the combined effect of group membership determination, central/satellite designation, and halo mass assignments. If the real difference between centrals and satellites is small, it may be washed out by this error. However, we want to point out that if this systematic error shown in the mock group catalog in Campbell et al. indeed affects the SDSS group catalog in a similar way, one would expect to see that the marginalized dependence of the quenched fraction on halo mass is similar between centrals and satellites for the SDSS group catalog. It is apparently inconsistent with what we found (the middle panel of Figure <ref>). This suggests that the error caused by group finder depends on the galaxy formation model that is used to construct the mock catalog. Unfortunately this also means that the impact of the inaccuracy of the group finder on the results obtained here from the SDSS group catalog is unclear.One way to bypass the uncertainty in the central versus satellite identification is to study the dependence of galaxy quenching on the locations of galaxies in their host halos. Instead of looking at centrals versus satellites, we can look at galaxies at different locations in a halo. By definition, centrals belong to the innermost population. Previous studies (e.g. Wetzel et al. 2012; Woo et al. 2013; Bluck et al. 2016) found that the quenched fraction increases with decreasing distance to the halo center (halo-centric radius). In contrast, the similarities between centrals and satellites found in this paper seem to suggest that quenching is quite independent of the location within halos. To find the cause of this discrepancy, we divide the total galaxy population into two: the inner and outer sub-populations, according to whether or not their projected distances to the halo centers (R_ p) are smaller or larger than half of the virial radii (R_ vir). Here the center of a halo is the luminosity-weighted position of its member galaxies. Figure <ref> shows the quenched fraction as a function of stellar mass for the two sub-populations in the six halo mass bins used above. To compare with the central galaxies in the corresponding halo mass bin, the quenched fraction is scaled with the quenched fraction of centrals and the stellar mass is also scaled with the median of the stellar mass of central galaxies (m̃_ c). For comparison, the satellite fractions in the two sub-populations are also shown in the figure.For small galaxies, the quenched fraction clearly increases with the decrease of halo-centric radius, consistent with previous studies, while for massive galaxies the dependence on halo-centric radius is very weak. The characteristic stellar mass above which the R_p dependence becomes weak is log m≃log(m̃_ c)-0.7 (indicated by the vertical lines) and almost independent of halo mass. This clearly demonstrates that the quenching probability is independent of its location in the halo, as long as a galaxy has a stellar mass larger than about one fifth of the mass of its central galaxy. For galaxies with log m∼logm̃_ c, the scaled quenched fraction is close to unity for both the inner and outer populations. At this stellar mass, the satellite fraction is close to zero for halos with log(M)<12.5 and increases to >60% for log(M)>14.5. It indicates that, when galaxies have stellar masses comparable to those of their centrals, the quenched fraction is the same as that of their centrals, no matter where they are located and whether they are identified as centrals or not. Alternative separations of inner and outer regions at 0.3R_ vir and 0.4R_ vir have no significant impact on this conclusion. These demonstrate again that centrals are not special as far as their quenching properties are concerned. Our results are also not in conflict with previous finding, as a strong dependence on halo-centric radius is present but only for galaxies that are much less massive than the centrals in their corresponding halos. § SUMMARY AND DISCUSSION In this paper, we present a detailed investigation about the environmental dependence of quenching of star formation using a large sample of galaxies constructed from the SDSS. We adopt two quantities to describe the different aspects of galaxy environments: the environmental mass densities, smoothed on a scale of 4 (half width of Guanssian kernel) at the positions of individual galaxies, and the masses of the host halos within which galaxies reside. The mass densities are obtained from the ELUCID simulation, a constrained N-body simulation in the SDSS volume,while the halo masses are based on a galaxy group catalog constructed with a halo-based group finder. Our main findings are summarized as follows. *The quenched fraction of galaxies increases systematically with galaxy stellar mass, environmental density and host halo mass. When analyzed separately, centrals and satellites show very different density-dependence: while the dependence is strong for satellites, it is weak or even absent for central galaxies. The environmental effect is stronger for centrals with higher stellar masses, while satellites show the opposite trend. In contrast, the dependence of the quenched fraction on halo mass is almost the same for both centrals and satellites, although the two populations cover different ranges of halo masses. *For the total galaxy population, the quenching efficiency, defined as the quenched fraction of galaxies in a given environment relative to a zero-point population, is found to be almost independent of galaxy stellar mass over a wide mass range, 9≤log (m)≤11.4), no matter which environmental parameter (mass density or halo mass) is adopted. This suggests that the strong stellar mass-dependence of the quenched fraction is predominantly produced by such a dependence of the zero point. *When central and satellite galaxies are analyzed separately, the density-based quenching efficiency is found to increase systematically with stellar mass for centrals but to decrease for satellites, and the stellar mass dependence is stronger for galaxies of higher stellar masses. The opposite trends seen in centrals and satellites compensate each other so as to make an almost stellar mass-independent quenching efficiency for the total population described above. The results thus do not support the hypothesis proposed in previous studies that the mass independence of the quenching efficiency is only due to satellite quenching. *Centrals and satellites are found to follow the same trends in both the halo mass and stellar mass based quenching efficiencies, contrary to the efficiency defined by the environmental density. It indicates that quenching does not depend strongly on the location within halo, at least for galaxies with stellar masses comparable to those of the centrals. Further investigation of the dependence of the quenched fraction on the distance to halo center shows that the distance dependence is only important for galaxies with stellar masses that are lower than one fifth of the masses of their centrals but insignificant for more massive galaxies. *A model, in which the quenching probability of a galaxy is assumed to be determined by the galaxy stellar mass combined with the host halo mass (Model B), can well reproduce the observed dependence of the quenched fraction on environmental density for the total population, as well as separately for centrals and satellites. In contrast,a model in which the quenching probability is assumed to be determined by galaxy mass and environmental density (Model A) predicts too weak dependence on halo mass and different halo-mass dependence between centrals and satellites, in conflict with observation. These suggest that halo properties are the driver of the environmental quenching seen in the observational data. *Model B is found to slightly overestimate the quenched fraction in the low density end and to underestimate it at the high density end for central galaxies. It suggests that galaxy quenching depends not only on halo mass but also on some other halo properties that are correlated with the environmental density. A model (Model C), which takes into account halo assembly bias, can explain the discrepancy between Model B and the observational results. *The environmental quenching efficiency based on the mass density predicted by both Model B and Model C (where the halo-mass is the primary driver of the environmental effect) is found to be independent of stellar mass, consistent with observational results.This strongly suggests that the stellar mass independence of the density-based efficiency originates from the stellar mass independence of the halo-based efficiency. *The difference in star formation quenching between centrals and satellites found in this paper and in numerous previous studies are mainly due to the difference in the host halo mass ranges covered by the two populations, and not produced by the difference in the correlation of quenching probability with halo mass between the two populations. Many mechanisms of quenching star formation in galaxies have been proposed in the literature, such as virial shock heating to accretion flows, the feedback from active galactic nuclei and supernova, and the stripping of hot and cold gas associated with galaxies (see e.g. Gabor et al. 2010 for a more comprehensive discussion). The strengths of these quenching mechanisms depend on stellar mass and halo mass in different ways. Furthermore, because of the special positions assumed for central galaxies in their halos, some of the processes may have different impact between centrals and satellites. In what follows we discuss how our findings can be used to constrain these different quenching mechanisms.AGN feedback has been proposed to quenchcooling flows in massive halos and suppress star formation in massive galaxies (e.g. Croton et al. 2006; Bower et al. 2006; Cui, Borgani, & Murante 2014). The strength of AGN feedback is usually assumed to be more efficient for more massive galaxies that contain more massive black holes in more massive halos. AGN feedback may thus be able to produce the positive dependency of the quenched fraction and quenching efficiency on galaxy stellar mass and halo mass found in this paper.Moreover, AGN feedback may affect both centrals and satellites in a similar way, again consistent with our observation. Numerous observations have provided evidence for AGN feedback through radio jet and massive outflows driven by radiation pressure (e.g. Best et al. 2007; Wang et al. 2011b; Fabian et al. 2012). However, the details how AGN feedback is coupled with the gas and regulates the star formation in galaxies are still uncertain.Hydrodynamical simulations (e.g. Kereš et al. 2005; 2009) have revealed that galaxies acquire their baryonic mass primarily through cold gas flows alongfilamentary structures around halos, and such cold accretion can be heated and suppressed by virial shocks in the host halo (see also Dekel & Birnboim 2006). This process can result in a decrease in the cold accretion as a function of halo mass, as radiative cooling is less effective in more massive halos (Ocvirk, Pichon & Teyssier 2008; Kereš et al. 2009). If the star formation in galaxies is fuelled mainly by the cold accretion, an increasing trend of the quenched fraction and the quenching efficiency with halo mass is expected. Moreover, galaxies in simulations are found to continue to acquire cold gas after becoming satellites, and the cold accretion is also affected by the shock in the host halos (Kereš et al. 2009), which is consistent with our finding that the quenching properties of centrals and satellites are similar. Unfortunately, it is unclear how the suppression of cold accretion depends on the stellar mass of a galaxy in a halo, and so it is unclear if this mechanism can accommodate the dependence of quenching on stellar mass seen in the observation.The stripping of gas from galaxies by ram-pressure in hot halo is expected to be more efficient for galaxies of lower masses living in higher mass halos(see Henriques et al. 2016 for discussion). Therefore this mechanism predicts that quenching should be important only in massive halos with hot halo gas, and that the quenched fraction and quenching efficiency should increase with decreasing stellar mass.These predictions seem to be at odds with the positive dependence of the quenched fraction(quenching efficiency) on stellar mass found in this paper, and with the fact no characteristic halo mass is seen in the quenching fraction - halo mass relation.In addition,ram pressurestripping is expected to be important only for satellite galaxies that are orbiting in halos, but unimportant for centrals that sit close to the bottoms of the halo gravitational potential wells. If such stripping process dominates quenching of star formation,then centrals and satellites are expected to be affected differently.This seems contrary to the results that the quenching efficiencies of centrals and satellites are correlated with halo mass and galaxy mass in a similar way. Our results thus suggest that ram pressure stripping alone cannot be the dominant quenching processes for the whole population. Similar argument may also be made for tidal stripping. However, since the effect of the tidal stripping is determined by the local mass density relative to the mass density of the galaxy to be stripped, this effect may also be important in relatively low-mass halos, which may be in better agreement with observation.The dependence of quenching on halo-centric radius for low-mass galaxies suggests that the significance of the underlying physical processes depends on the locations of galaxies in their host halos. The satellite-specific processes, such as ram pressure stripping and tidal stripping,are expected to be more efficient near the halo center,and so may be able to produce the dependence on halo-centric radius observed in the data. However, as discussed above, if these processes dominate the quenching of satellites and do not operate on centrals,then why do central galaxies have quenching properties similar to satellite galaxies of the same stellar masses?It may be that centrals are not special, and these satellite-specific processes also operate on centrals.This is in fact consistent with the fact that centrals are not at rest within the halo potential wells, and not even located at the halo center (e.g. Skibba et al. 2011). Indeed, one of the main conclusions that can be drawn from our results is that the central in a halo is not special, as far as its star formation quenching is concerned. Peng et al. (2010) found that the environmental quenching and mass quenching can be well separated from each other for galaxies at z∼ 1. It suggests that such a conclusion also holds for these galaxies. However, as mentioned above, this similarity between centrals and satellites is not expected in some galaxy formation models, where centrals and satellites are assumed to be affected by different quenching processes.The processes discussed above, which are all confined within halos, can not account for the residual dependence on environmental density after removing the halo-mass effects. We thus include halo assembly history as an additional parameter, which is known to be influenced by large scale environment (e.g. Gao et al. 2005). Local tidal field is thought to play a key role in shaping the halo assembly bias (Wang et al. 2007;2011a; Hahn et al. 2009; Shi et al. 2015; Paranjape et al. 2017; Borzyszkowski et al. 2017). For example, for a small halo in a high density region, the material around the halo is accelerated by the local tidal field so that the halo growth is significantly suppressed. These processes (on the scale much larger than the galactic scale) are unlikely to directly influence the star formation in galaxies. Therefore, a correlation between the star formation and the halo assembly history has to be introduced in order to explain the residual dependence. It is worthwhile to note that it is unclear whether these large scale processes affect baryonic gas and dark matter in the same way. If it is not, additional effect on star formation should be taken into account.The discussions given above provide some qualitative assessments about some of the quenching processes that have proposed in the literature, in connection to the observational results we find in this paper. To constrain theoretical models in a quantitative way, however, detailed modeling of the various quenching processes, as well as thorough analyses of all potentially important observational selection effects are needed. In a forthcoming paper,we will use mock catalogs constructed from hydrodynamic simulations and semi-analytic models of galaxy formation to compare galaxy formation models with the observational results obtained here (E. Wang et al. in preparation).§ ACKNOWLEDGMENTS We thank an anonymous referee for a useful report. This work is supported by the 973 Program (2015CB857002), NSFC (11522324, 11733004, 11421303, 11233005, 11621303),and the Fundamental Research Funds for the Central Universities. H.J.M. would like to acknowledge the support of NSF AST-1517528 and NSFC-11673015. S.H.C and Y.Y. are supported by the Fund for Fostering Talents in Basic Science of the National Natural Science Foundation of China NO.J1310021. FvdB is supported by the Klaus Tschira Foundation and by the US National Science Foundation through grant AST 1516962. 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"authors": [
"Huiyuan Wang",
"H. J. Mo",
"Sihan Chen",
"Yang Yang",
"Xiaohu Yang",
"Enci Wang",
"Frank C. van den Bosch",
"Yipeng Jing",
"Xi Kang",
"Weipeng Lin",
"S. H. Lim",
"Shuiyao Huang",
"Yi Lu",
"Shijie Li",
"Weiguang Cui",
"Youcai Zhang",
"Dylan Tweed",
"Chengliang Wei",
"Guoliang Li",
"Feng Shi"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727190531",
"title": "ELUCID IV: Galaxy Quenching and its Relation to Halo Mass, Environment, and Assembly Bias"
} |
Polarization-Division Multiplexing Based on the Nonlinear Fourier Transform Jan-Willem Goossens,1,2,* Mansoor I. Yousefi,2 Yves Jaouën2 and Hartmut Hafermann1 Received ***date***; Accepted ***date*** ======================================================================================§ INTRODUCTIONRecent years have seen exciting progress in understanding the connection between entanglement and geometry <cit.>. However, in the context of the AdS/CFT correspondence, our ability to decipher the bulk geometry (or bulk physics, more generally) from information in the boundary CFT remains very incomplete. The challenges are most pronounced if one considers physics behind the horizon of a black hole. Consider for example the eternal AdS black hole, which is dual to the thermofield double (TFD) state <cit.> |TFD(t_L,t_R)⟩=1/√(Z_β)∑_ie^-βE_i/2 e^-iE_i(t_L+t_R) |i⟩_L |i⟩_R . This describes an entangled state of the two copies of the CFT associated with the asymptotic boundaries (see figure <ref>), which are joined by a wormhole, an Einstein-Rosen bridge (ERB), in the bulk <cit.>. The AdS/CFT correspondence demands that the interior region have an equivalent description in terms of the boundary field theory. But now, in addition to the usual difficulties involved in probing behind the horizon, we have another conundrum: the boundary field theory reaches thermal equilibrium very quickly, on the order of the thermalization time 1/T, while the ERB continues to grow on much longer timescales <cit.>. Therefore, there must be some quantity in the field theory that corresponds to this fine-grained information – which is evidently not captured by entanglement entropy <cit.> – that continues to evolve long after thermal equilibrium is reached. These considerations led Susskind to introduce holographic complexity as the boundary entity whose growth corresponds to the evolution of the ERB <cit.>. In particular, with his collaborators, he developed two new gravitational observables, both of which successfully probe the late-time growth of the ERB. The first of these is referred to as the complexity=volume (CV) conjecture, which posits that the complexity of the boundary state is proportional to the volume of a maximal codimension-one bulk surface ℬ that extends to the AdS boundary, and asymptotes to the time slice Σ on which the boundary state is defined <cit.>: 𝒞_V(Σ) = max_Σ=∂ℬ [𝒱(ℬ)/G_N ℓ] ,where ℓ is some length scale associated with the bulk geometry, the AdS radius or the radius of the black hole. For example, in the eternal AdS black hole, this bulk surface connects the time slices denoted t_L and t_R on the left and right boundaries through the ERB; see the left panel in figure <ref>. The second proposal is the complexity=action (CA) conjecture. This identifies the complexity of the boundary state with the gravitational action evaluated on a bulk region known as the Wheeler-DeWitt (WDW) patch <cit.>: 𝒞_A(Σ)=I_WDW/π ħ .One can think of the WDW patch as the causal development of the spacelike surface ℬ picked out by the CV construction. The right panel in figure <ref> illustrates the WDW patch for the example of the eternal AdS black hole, where the CFT state is again evaluated on the t_L and t_R slices of the left and right boundaries, respectively. Both proposals have their merits, as well as certain shortcomings. In any case, they bring to our attention two new classes of interesting gravitational observables which should certainly be studied in further detail. In fact, various aspects of the proposals and these new observables have been examined in a number of recent papers, <cit.>. And while both the CV and CA conjectures appear to provide viable candidates for holographic complexity, this research program is still at a very preliminary stage. In particular, one would like to establish a concrete translation of the new observables in the bulk to a specific quantity in the boundary theory, as was recently found for holographic entanglement entropy <cit.>. However, a stumbling block to this endeavor is finding the answer to an even simpler question: what does “complexity” mean in the boundary CFT?This question is the focus of the present paper. Specifically, our objective is to provide the first steps towards defining circuit complexity in quantum field theory (QFT).[We also refer the reader to ref. <cit.> for a recent complementary investigation in this direction.] A precise understanding of this quantity will not only shed light on the CV and CA proposals, but is also an interesting question deserving of study in its own right. For example, it may also provide new insights into quantum algorithms for the simulation of quantum field theories<cit.>, or more generally into Hamiltonian complexity <cit.>, or the efficient description of many-body wave functions <cit.>.In computer science, the notion of computational complexity refers to the minimum number of operations necessary to implement a given task <cit.>. In the present context, the task of interest will be the preparation of a state in the QFT, and we will define the complexity in terms of a quantum circuit model. That is, we will begin with a simple reference state |ψ_R⟩, and construct a unitary transformation U that produces the desired target state |ψ_T⟩ via |ψ_T⟩=U |ψ_R⟩ . The unitary U will be constructed from a particular set of simple elementary or universal gates, which can be applied sequentially to the state. When working with such discrete operations, we should also introduce a toleranceso that even if we cannot achieve the precise equality above, we may still judge the transformation to be successful when the two states are sufficiently close to one another according to some distance measure, |||ψ_T⟩-U |ψ_R⟩ ||^2≤ .Of course, there will not be a unique circuit which implements the desired transformation circuitDef: generally there will exist infinitely many sequences of gates which produce the same target state. However, the complexity of the state |ψ_T⟩ may be defined as the minimum number of gates required to produce the transformation (<ref>), the complexity is the number of elementary gates in the optimal or shortest circuit. The challenge then is to identify this optimal circuit from amongst the infinite number of possibilities.Our work takes inspiration from the geometric approach of Nielsen and collaborators <cit.>,[See <cit.> for another application of Nielsen's ideas in holography. We also refer the interested reader to ref. <cit.>, which introduces an interesting connection between quantum algorithms and geodesics on the Fubini-Study metric.] which itself was developed using ideas from the theory of optimal quantumcontrol, <cit.>. In Nielsen's case, the question of interest was to find the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation U (without the use of ancilla qubits). Neilsen approaches this question as the Hamiltonian control problem of finding a time-dependent Hamiltonian H(t) that synthesizes the desired U,U=P exp[∫_0^1dt H(t) ]where H(t)=∑_I Y^I(t) M_I , where the Hamiltonian is expanded in terms of generalized Pauli matrices, denoted here as M_I,[Our notation diverges from that of Neilsen, in order to increase the similarity of these equations with our notation in the main text. In particular, note that we have absorbed a factor of -i in M_I so that these are now anti-Hermitian operators.] and the P indicates a time ordering such that the Hamiltonian at earlier times is applied to the state first, the circuit is built from right to left. In <cit.>, the control functions Y^I form a 4^n-1-dimensional vector space, and can be seen as specifying the tangent vector to a trajectory in the space of unitaries, U(t)=P exp[∫_0^t t̃ H(t̃) ] .In this general space, the paths of interest satisfy the boundary conditions U(t=0)=1 and U(t=1)=U. Neilsen's idea is then to define a cost for the various possible paths D(U(t))=∫_0^1t FU(t),U̇(t) , and to identify the optimal circuit or path by minimizing this functional. In general, the cost function F(U,v) is some local functional of the position U in the space of unitaries and a vector v in the tangent space at this point. Neilsen further argues that for the present problem, a physically reasonable cost function must satisfy a number of desirable features:0.8ex 1.Continuity: F should be continuous, F∈ C^0. 0.8ex 2.Positivity: F(U,v)≥0 with equality if and only if v=0.0.8ex 3.Positive homogeneity: F(U,λ v)= λF(U,v) for any positive real number λ.0.8ex 4.Triangle inequality: F(U,v+v')≤ F(U,v)+F(U,v') for all tangent vectors v and v'.0.8exthese four properties come very close to defining a class of geometries known as Finsler manifolds. In particular, if we replace the first condition above with 0.8ex1'.Smoothness: F should be smooth, F∈ C^∞, 0.8exthen eq. costco defines length functional for a Finsler manifold, a particular class of differential manifolds equipped with a quasimetric structure in which the length of any curve is measured by a length functional of the form costco, with a Finsler metric F satisfying the four properties enumerated above, see <cit.>. While the familiar notion of Riemannian manifolds would fall within this definition, Finsler geometry provides a generalization to a broader class of manifolds where the norm on the tangent space is not (generally) induced by a metric tensor. Hence Neilsen has identified the problem of finding an optimal circuit with the problem of finding extremal curves, geodesics, in a Finsler geometry, and the complexity is then identified with the length of the geodesic.[For future reference, when referring to general paths or circuits, we will use “size,” “length,” “cost,” and “depth” interchangeably; however, “complexity” will be reserved for the length of the optimal path or circuit.] Of course, this still leaves open the question of the precise form of the cost function, and various possibilities are examined in <cit.>:[The functions F_1 and F_p are not technically Finsler metrics, since both fail to meet the smoothness requirement. However, as explained in <cit.>, they can be approximated arbitrarily well by metrics which are Finsler. This subtlety will not be important for our analysis.]F_1(U,Y)= ∑_I |Y^I| ,F_p(U,Y) = ∑_I p_I |Y^I| ,F_2(U,Y)= √(∑_I Y^I^2) , F_q(U,Y) = √(∑_I q_IY^I^2) .In the two measures on the right, p_I and q_I are penalty factors which can be chosen to favour certain directions in the circuit space over others, to give a higher cost to certain classes of gates. We do not include such factors in most of our analysis, but we return to this issue in section <ref>. Of course, the F_2 measure yields a standard Riemannian geometry — and in fact, it will be the focus of much of our discussion. The preceding exposition of Nielsen's approach is of course very incomplete, and the interested reader is referred to <cit.> for more details. The key feature of this approach is that it enables one to bring the full power of differential geometry to bear on the problem of constructing the optimal quantum circuit, and this provides an objective manner in which to measure the complexity as the length of extremal paths in the geometry. However, at many points our approach will necessarily differ from that of Nielsen since we are studying a different problem, namely complexity in a quantum field theory. The primary purpose of the above presentation was to provide motivation for our geometrical analysis, but we should add that the details of Finsler geometry will not play any role in the following. Rather, a simpler physics-oriented perspective is to view the problem of finding the optimal circuit as a trajectory in the space of all possible circuits, as a classical mechanics problem for the motion of particle governed by the usual Lagrangian in eq. costco.This paper is organized as follows: we begin in section <ref> by examining complexity for a simple free scalar field theory. Following the preceding discussion, this requires identifying a simple reference state, introducing a set of elementary gates, and also identifying a family of interesting target states. However, the first step will be to regulate the theory by placing it on a lattice, which reduces the scalar field theory to a family of coupled harmonic oscillators. Hence, as a warm up problem, we consider the case of a single pair of harmonic oscillators. Then, having built up some intuition, we shall geometrize the problem in section <ref>. The main ideas from Nielsen's approach are implemented here: we represent the circuit as a path-ordered exponential analogous to eq. row, show that our space of circuits forms a representation of , and construct the appropriate (Euclidean) metric. With this in hand, we proceed to find the geodesics, and identify the complexity of the ground stateas the geodesic length of the global minimum. In section <ref>, we return to the field theory problem by generalizing these results to a lattice of coupled oscillators. Given the complexity for the (regulated) field theory, we then ask how our results compare to holographic complexity, and we find some surprising similarities. In section <ref>, we conduct a preliminary exploration of the effects of introducing penalty factors for nonlocal gates. Finally, we close in section <ref> with a brief discussion of our results and directions for future work. Various technical details have been relegated to several appendices: we construct some explicit example circuits using the elementary gates given in section <ref> in appendix <ref>, elaborate on some geometrical details in appendix <ref>, derive the normal-mode frequencies for a one-dimensional lattice in appendix <ref>, find a closed-form approximation to the circuit complexity for the d-dimensional lattice in appendix <ref>, and compute an approximation to the optimal circuit in the presence of penalty factors in appendix <ref>.§ COMPLEXITY FOR HARMONIC OSCILLATORSAs a first step towards understanding circuit complexity in QFT, we will consider for simplicity a free scalar field in d spacetime dimensions. However, having identified this particular QFT, we must first regulate the theory by placing it on a lattice,[Our experience with holographic complexity suggests that we will not be able to sensibly define complexity in a QFT without a UV regulator in place <cit.>.] which reduces the system to an infinite family of harmonic oscillators. This in turn suggests the much simpler warm-up problem of two coupled harmonic oscillators. As it turns out, this simple model retains enough of the structure of the original problem that we will be able to learn several important lessons, which we can then carry over to the problem of circuit complexity in our scalar field theory. As in the general case, to study complexity in the two oscillator problem, we must identify a target state, a reference state, and a suitable family of elementary gates.We begin with the Hamiltonian of a free scalar field in d spacetime dimensions, H=1/2∫^d-1x[π(x)^2+∇⃗ϕ(x)^2+m^2ϕ(x)^2] . As mentioned above, our first step is to regulate the theory by placing it on a (square) lattice with lattice spacingδ, in which case the Hamiltonian becomes: H=1/2∑_{p()^2/δ^d-1+δ^d-1[1/δ^2∑_iϕ()-ϕ(-x̂_i)^2+m^2ϕ()^2]} , where x̂_i are unit vectors pointing along the spatial directions of the lattice. The resulting theory is essentially a quantum mechanical problem with an infinite family of coupled (one-dimensional) harmonic oscillators. We can make this description manifest by redefining X()=δ^d/2ϕ(), P()=p()/δ^d/2, M=1/δ, ω=m and Ω=1/δ, whereupon the Hamiltonian eq:Hlattice takes the familiar form H=∑_{P()^2/2M+1/2 M [ω^2 X()^2+Ω^2∑_iX()-X(-x̂_i)^2]} . Hence the frequency of the individual masses is given by ω=m, and the inter-mass coupling is given by Ω=1/δ.Now, the above suggests that we begin with an even simpler warm-up problem, namely, the case of two coupled harmonic oscillators: H=1/2[ p_1^2+p_2^2+ω^2x_1^2+x_2^2+Ω^2x_1-x_2^2] ,where x_1,x_2 label their spatial positions, and we have set M_1=M_2=1 for simplicity. Of course, to solve this system, one simply rewrites the Hamiltonian in terms of the normal modes, H=1/2p̃_+^2+ω̃_+^2x̃_+^2+p̃_-^2+ω̃_-^2x̃_-^2 ,where[When working in the normal-mode basis, we denote variables (positions, frequencies), with a tilde to clearly distinguish from the physical basis. The utility of this convention will become apparent later. ] x̃_±≡1/√(2)x_1±x_2 ,ω̃_+^2=ω^2 ,ω̃_-^2=ω^2+2Ω^2 .This recasts the problem as that of two decoupled simple harmonic oscillators, and hence it is now straightforward to solve for the eigenstates and eigen-energies of the Hamiltonian. For example, we can write the ground-state wave function as the product of the ground-state wave functions for the two individual oscillators: ψ_0(x̃_+,x̃_-)=ψ_0+(x̃_+)ψ_0-(x̃_-) =ω̃_+ω̃_-^1/4/√(π) exp[-1/2ω̃_+x̃_+^2+ω̃_-x̃_-^2] , where the normalization has been chosen such that ∫ d^2x |ψ_0|^2=1. We may also express this wave function in terms of the physical positions of the two masses: ψ_0(x_1,x_2)=ω_1ω_2-β^2^1/4/√(π)exp[-ω_1/2x_1^2-ω_2/2x_2^2-βx_1x_2] , where ω_1=ω_2=1/2ω̃_++ω̃_- ,β≡1/2ω̃_+-ω̃_-<0 . We note in passing that our notation for the wave function in eq. eq:targetPhys is slightly more general than necessary; however, these Gaussian wave functions constitute an interesting family of target states for the present exercise.[For example, Gaussian states play an important role in quantum optics, and much of our analysis is closely related to ideas developed in the quantum information literature for this purpose, <cit.>.]The next step is to identify a simple reference state. Motivated by discussions of holographic complexity <cit.>, as well as cMERA <cit.>, we choose a reference state where the two masses are unentangled, namely a factorized Gaussian state, ψ_R(x_1,x_2)=√(ω_0/π) exp[-ω_0/2x_1^2+x_2^2] . For the time being, we will simply leave ω_0 as a free parameter which characterizes our reference state. We shall examine specific choices of this frequency in section <ref>. Having chosen our reference and target states, it remains to identify a simple set of unitary gates with which to construct the desired unitary U, which implements ψ_T=U ψ_R. The natural operators appearing in the quantum mechanics problem of the two coupled oscillators are the positions x_1, x_2 and the momenta p_1=-i∂_1, p_2=-i∂_2, which satisfy the canonical commutation relations [x_a,p_b]=i δ_ab.We can use these operators to build an interesting set of elementary gates for our problem:H=e^ix_0p_0 , J_a=e^ix_0p_a ,K_a=e^ix_ap_0 ,Q_ab =e^ix_ap_b(with ab) , Q_aa=e^i/2x_ap_a+p_ax_a=e^/2 e^ix_ap_a , where x_0 and p_0 are c-number constants. A key point is that we have introduced an infinitesimal parameter ≪1 into the exponent of each one of these operators. This ensures that the action of any one of these gates only produces a small change on the wave function. The action of each of these gates can be understood with the following general examples: H ψ(x_1,x_2) =e^ip_0x_0ψ(x_1,x_2)(global) phase changeJ_1 ψ(x_1,x_2) =ψ(x_1+x_0,x_2)shift x_1 by constant x_0K_1 ψ(x_1,x_2) =e^ip_0x_1ψ(x_1,x_2)shift p_1 by constant p_0Q_21 ψ(x_1,x_2) =ψ(x_1+x_2,x_2)shift x_1 by x_2 (entangling gate)Q_11 ψ(x_1,x_2) =e^/2ψe^x_1,x_2 scale x_1 →e^x_1 (scaling gate)When working with position-space wave functions, the momentum shift produced by K_1 (or K_2) amounts to introducing a small plane-wave component in the wave function, as illustrated in (<ref>). We refer to Q_11 and Q_22 as scaling gates, for the obvious reason that these operators scale the corresponding coordinate by a small amount. Note that they also introduce an overall normalization factor, which ensures that the norm of the wave function is preserved. The operators Q_21 and Q_12 mix the positions of the two masses, thereby increasing (or decreasing) the entanglement between the two oscillators; hence we refer to these as the entangling gates. The scaling and entangling gates will play a key role in the circuits we construct below.Of course, one could extend the ensemble of gates introduced in eq. eq:gates with operators like exp[i p_0/x_0x_1x_2] orexp[i x_0/p_0p_1^2] .Furthermore, one could also introduce gates with even higher powers of x's and p's in the exponent. However, we know that the collection of gates in eq. eq:gates is sufficient to implement the unitary transformation from the specified reference state eq:refPhys to the desired target state eq:targetPhys. Hence for simplicity, we shall work within this subset of all possible unitary gates.A circuit then consists of a sequence of these gates, whose action on ψ_R produces the desired state ψ_T. For example, consider the following circuit: ψ_T=Uψ_R≡Q_22^α_3 Q_21^α_2 Q_11^α_1 ψ_R . Here, Q_11 acts first, and by acting with the appropriate number of times α_1, we will increase the reference frequency ω_0 appearing in front of x_1^2 in eq. eq:refPhys to the desired frequency ω_1 appearing in eq. eq:targetPhys. Similarly, the number of times that the Q_21 and Q_22 gates are required to appear in the circuit, namely α_2 and α_3, are uniquely fixed by the desired ω_2 and β in the target state. The details of the corresponding calculations are given in appendix <ref>, and the final result isα_1=1/2logω_1/ω_0 , α_2=1/√(ω_0/ω_1)β/√(ω_1ω_2-β^2) , α_3=1/2logω_1ω_2-β^2/ω_0 ω_1 .We then define the circuit depth as the total number of gates in the circuit. In the above example, we have simply 𝒟(U) = |α_1|+|α_2|+|α_3|= 1/[1/2 logω_1ω_2-β^2/ω_0^2+√(ω_0/ω_1)|β|/√(ω_1ω_2-β^2)] .Note the use of the absolute values in the first line. At a pragmatic level, this is required because α_2 is negative in this particular example, β<0. But this means that we are giving an equal complexity cost for the inverse gates Q_ij^-1 as for the original gates Q_ij, we count the appearance of Q_ij^-1 as one gate in a circuit. We refer to the result in eq. leach as the circuit depth of the particular circuit U given in eq. eq:eg1. But we must distinguish this from the complexity of the target state ψ_T, which is the minimum number of gates required to produce the desired transformation. In other words, the complexity is the circuit depth of the optimal circuit. At present, we have no reason to believe that the simple circuit proposed in eq. eq:eg1 is the optimal circuit, and in fact, our calculations below will show that it is not.We can describe the general form of the result in eq. leach as being an overall factor of 1/ϵ, and a coefficient determined by the various physical parameters characterizing the target and reference states. More generally, the circuit depth might be given by an expansion in ϵ, beginning with a 1/ϵ term followed by a finite term and then potentially terms involving positive powers of ϵ. However, since ϵ≪1, determining the complexity essentially requires finding the circuit which minimizes the coefficient of the leading 1/ϵ term. For further discussion and additional examples, the interested reader may turn to appendix <ref>. In the next section, we apply Neilsen's approach of geometrizing the circuit complexity to find the optimal circuit. Before leaving present example however, for comparison to later results it is convenient to express the circuit depth in eq. leach in terms of the normal-mode frequencies using eq. eq:omega12pm. This substitution yields 𝒟_1=1/[ 1/2log_+/ω_0+1/2log_-/ω_0+_--_+/√(2_+_-)√(ω_0/_++_-) ] .Recall that _->_+ from eq. qm3 (and implicitly, we are assuming ω̃_±>ω_0). § GEOMETRIZING COMPLEXITY In the introduction, we discussed Neilsen's approach <cit.> of geometrizing the problem of finding the optimal circuit. We now wish to apply this geometric approach to the problem of finding the optimal preparation of the ground-state of two coupled harmonic oscillators. Our first step is to represent the circuit U as a path-ordered exponential, U=𝒫 exp∫_0^1s Y^I(s) _I ,ψ_Tx_1,x_2=Uψ_Rx_1,x_2 . This structure replaces the representation of the circuits as products of the discrete gates in eq. eq:gates. The connection with these gates comes about since we choose the operators _I appearing in the exponential to be precisely those appearing in the scaling and entangling gates introduced previously; that is, we writeQ_ab = exp[ _ab] with_ab=(i x_a p_b+1/2 δ_ab) .Our notation in eq. eq:pathPsi is that the sum over I runs over the pairs ab, I∈{11,12,21,22}. Hence in the path-ordered exponential, we can think of s as parametrizing a (continuous) product of gates, and the functions Y^I(s) as indicating whether the I'th type of gate is turned on or off in this sequence (analogous to the control functions in Nielsen's time-dependent Hamiltonian (<ref>)). In the integral appearing in the exponent, the differential s plays a role analogous to that of the infinitesimal parameter . Finally, the path-ordering symbol indicates that we build the circuit from right to left, the operators at smaller values of s act on the wave function before those at larger values of s. Furthermore, with this framework, we consider a particular circuit as being constructed by following a particular trajectory, specified by Y^I(s), through the space of unitary circuits. Hence we begin with U(s=0)=1, and have the family of unitaries U(s)=𝒫 exp∫_0^ss̃ Y^I(s̃) _I . Eq. eq:pathPsi then specifies the final unitary at the end-point s=1, which corresponds to the desired circuit that generates the target state, U_fin=U(s=1) with ψ_T=U_finψ_R. From this perspective, Y^I(s) specifies the velocity vector tangent to this trajectory, in a manner in which we will make precise below. In more geometric language which may be familiar from general relativity, we would say that the Y^I(s) are the components of the velocity in a particular frame basis, rather than in a coordinate basis.As in the example in section <ref> above, the circuit depth is determined by counting the total number of gates appearing in the full sequence comprising the circuit, eq. leach. For our path-ordered exponential eq:pathPsi, the analogous expression becomes[Actually this expression cost1 is the continuum limit of the cost function 𝒟(U)=∑ϵ |α_i|. Including the extra factor of ϵ in the sum eliminates the 1/ϵ factor, so the circuit depth remains finite in the limit ϵ→0.] 𝒟(U)= ∫_0^1s∑_I|Y^I(s)|= ∫_0^1s[|Y^11(s)|+|Y^12(s)|+|Y^21(s)|+|Y^22(s)|] .This cost function corresponds to the F_1 metric in the notation of <cit.> — see eq. (<ref>). Our goal of finding the optimal circuit then amounts to finding the functions Y^I(s) which yield the desired unitary U_fin while minimizing this cost function. However, having also identified Y^I(s) as the velocity along the trajectories U(s), we can use our physical intuition to think of this as a classical mechanics problem where we aim to find the extremal trajectory given a particular set of boundary conditions and the somewhat unusual Lagrangian in eq. cost1.A mentioned in the introduction, we can also make other choices for the cost function, and the analysis will go through essentially unchanged. Hence in order to develop the present problem most easily, we shall consider the F_2 or F_q metric in eq. (<ref>). That is, we replace eq. cost1 with 𝒟(U)=∫_0^1s√(G_IJ Y^I(s) Y^J(s)) . This expression should be familiar as the action of a particle moving in a curved space, and hence the optimal path corresponds to a geodesic in the corresponding (Riemannian) geometry. As we mentioned above, Y^I(s) are the components of the velocity in a particular frame, for which the metric G_IJ then defines the inner product. In our examples, G_IJ is taken to be a purely constant (and usually diagonal) matrix. We will begin by studying the simple Euclidean metric G_IJ=δ_IJ, which corresponds to the F_2 metric above. With this choice, motion in every direction in the space of unitaries is assigned the same cost, the cost of each type of gate is the same. However, our notation is sufficiently general to allow for the assignment of penalty factors for particular gates, as in the F_q metric. We shall return to this possibility in section <ref>.To proceed further, we must find a prescription to explicitly identify the functions Y^I(s). Given eq. path2, it is straightforward to show that Y^I(s) _I=_sU(s) U^-1(s) . However, this expression is not particularly useful as it stands. In Neilsen's construction <cit.>, one works with unitary matrices acting on qubits, rather than operators acting on wave functions. Hence the components of the velocity analagous to eq. v12 can be isolated by simply tracing over the corresponding matrix generators. This procedure does not immediately lend itself to eq. v12, so in order to make progress, we shall re-express our problem in terms of matrices. Recall that we reduced the problem to evaluating the complexity of the ground state (<ref>) of two coupled harmonic oscillators, starting from a factorized Gaussian reference state (<ref>). That is, we begin and end with a Gaussian wave function; furthermore, it is straightforward to show that the scaling and entangling operators preserve the general Gaussian form of the wave function, all of the intermediate wave functions take a form analogous to eq. (<ref>). Therefore, since we're only working with Gaussian states, we may think of the space of states as the space of (positive) quadratic forms. In other words, the states under consideration are all of the form ψ≃exp[-1/2x_a A_ab x_b] ,and thus we may think of the relevant space of states as the three-dimensional space of 2×2 positive symmetric matrices A, with A_ab=A_ba, detA>0, and A_11,A_22>0.[These positivity constraints ensure that both eigenvalues of A_ab are positive.] In particular, the reference and target states become, respectively, A_R=ω_01 ,A_T=(ω_1 ββ ω_2) ,where ω_1, ω_2 and β are given by eq. eq:omega12pm.We now translate the scaling and entangling gates to this matrix representation. That is, we build a representation of these operators as 2×2 matrices which act on the symmetric matrices A. In particular, one finds that the gate matrices act asA' = Q_abA Q_ab^T ,where Q_ab = exp[ M_ab] with[M_ab]_cd = δ_ac δ_bd .In this notation, [M_ab]_cd is a 2×2 matrix, where c and d denote row and column indices, respectively.[A quick way to construct these matrices is to consider the action of _ab on the column vector x_1,x_2^T, and then build the matrix M^T_ab which yields the same result. One can verify that the commutators of the M_ab match those of the _ab. Note that, while the action of the Q_ab in eq. operate leaves the wave functions properly normalized at each step, we lose track of this normalization when working with the A_ab.] Explicitly, we shall denote the basis of generators M_I asM_11 =[ 1 0; 0 0 ] , M_12=[ 0 1; 0 0 ] , M_21 =[ 0 0; 1 0 ] , M_22=[ 0 0; 0 1 ] .With this new matrix formulation of our problem, we readily observe that the action of the gates Q_ij – or more generally, circuits constructed from Q_ij – on the vector (x_1,x_2)^T produces a vector whose elements are linear combinations of x_1 and x_2. Furthermore, since the gates are invertible, this is precisely the definition of the group of transformations .[Note that one can also see the emergence of this group by observing that the algebra of the original operator generators _ab in eq. operate close to form the algebra gl(2,ℝ).] Thus our circuits form a representation of , the U(s) are trajectories in the space oftransformations. Now, in this matrix formulation, the path-ordered exponentials in eq. (<ref>) are replaced by U(s)=𝒫 exp∫_0^ss̃ Y^I(s̃) M_I ,with A_T=U(s=1) A_R U^T(s=1) , where M_I are the generators given in eq. Msimple. The advantage of this formulation is that eq. v12 becomes Y^I(s) M_I=_sU(s) U^-1(s) Y^I(s)=_sU(s) U^-1(s)M^T_I . That is, we now have a simple expression which yields the components of the velocity vector Y^I(s). Before we can utilize this expression however, we must explicitly construct a parametrization of thetransformations. We proceed with this task in the next subsection, but first let us make a few comments. Our task will be to find the shortest geodesic in some right-invariant metric onthat connects the initial and final states, A_R and A_T, as in eq. pathA. We emphasize shortest geodesic because in fact, we will find that there is a continuous family of geodesics connecting the desired states. This non-uniqueness arises because our space of circuits is four-dimensional (since dim=4) whereas our space of states is only three-dimensional (since the 2×2 matrices A_ij are symmetric). As a result of this mismatch, we should expect to find a one-parameter family of geodesics U(s) which yield the desired transformation A_T=U(s=1) A_R U^T(s=1). However, as we have explained, the complexity is defined as the cost of the minimal or optimal circuit that obtains the specified target state. Hence this one-parameter family of solutions is merely the set of all possible circuits within this class. To find the optimal circuit, we simply need to find the geodesic within this family with the shortest length cost2.Since our ultimate aim will be to return to free scalar field theory, we note in passing that the notation introduced in the last two subsections generalizes very easily from two coupled oscillators to N coupled oscillators. We would then build a right-invariant metric on GL(N,ℝ). Furthermore, note that the dimension of the space of circuits becomes N^2, while the dimension of the space of Gaussian states or quadratic forms is only N(N+1)/2. Hence the non-uniqueness involved in finding the most efficient circuit U(s) which produces the desired transformation grows quickly. We shall discuss the extension to a lattice of oscillators in section <ref>. §.§ Geodesics on circuit space To proceed with constructing the desired geodesics, we must choose an explicit parametrization of a general element U∈= ℝ×SL(2,ℝ). Let us first consider U∈SL(2,ℝ), which can be written as U=[ x_0-x_3 x_2-x_1; x_2+x_1 x_0+x_3 ] , withx_0^2+x_1^2-x_2^2-x_3^2=1 . We recognize the constraint imposing detU=1 as the embedding of (Lorentzian) AdS_3 in ℝ^2,2. Indeed, the appearance of AdS_3 could have been anticipated since the latter is the universal cover of SL(2,ℝ). Our familiarity with this embedding then motivates the following choice of coordinates: x_0=cosτcoshρ ,x_1=sinτcoshρ ,x_2=cosθsinhρ ,x_3=sinθsinhρ ,where τ, ρ and θ are the usual time, radius, and angle, respectively, of global coordinates on AdS_3. We can easily extend this parametrization to U∈= ℝ×SL(2,ℝ) by introducing an additional coordinate to parameterize the determinant of U, U=[ x_0-x_3 x_2-x_1; x_2+x_1 x_0+x_3 ] , withx_0^2+x_1^2-x_2^2-x_3^2=e^2y . Hence we extend eq. slr to x_0=e^ycosτcoshρ ,x_1=e^ysinτcoshρ ,x_2=e^ycosθsinhρ ,x_3=e^ysinθsinhρ ,where, as before, τ, ρ, θ are coordinates on the SL(2,ℝ) subgroup, and y parametrizes the ℝ fibre. With these coordinates, we can express a general U∈ as U =e^y [cosτcoshρ-sinθsinhρ -sinτcoshρ+cosθsinhρ;sinτcoshρ+cosθsinhρcosτcoshρ+sinθsinhρ ] . We are now equipped to construct the geometry implicit in the cost function cost2, where the velocity components are given by eq. (<ref>). As mentioned above, we begin by choosing G_IJ=δ_IJ, which assigns an equal cost or weight to every gate. This choice then defines the following right-invariant metric: s^2 =δ_IJ U U^-1M^T_I U U^-1 M^T_J =2y^2+2ρ^2+2cosh(2ρ)cosh^2ρ τ^2+2cosh(2ρ)sinh^2ρ θ^2-2sinh^22ρ τθ .For later use, it is also convenient to express this in the form s^2=2y^2+2ρ^2+2x^2+2cosh(4ρ) z^2-4cosh(2ρ)x z ,where we have defined the pseudo-lightcone coordinates x≡1/2(θ+τ) ,z≡1/2(θ-τ) . Note that our metric metric1 is Euclidean, as is appropriate for defining a cost function, and so does not contain the (Lorentzian) AdS_3 geometry noted above. Indeed, a Lorentzian signature would not be suitable for the problem at hand, since certain directions would then carry negative or zero cost. We discuss the relation between our geometry and that of AdS_3 in appendix <ref>.With the geometry in hand, we now wish to find the geodesics, and thereby the optimal circuit. Inspecting the metric metric1, we can see three obvious Killing vectors: _y, _τ, _θ. However, the metric is right-invariant by construction, meaning eq. metric1 remains unchanged if we right-multiply U(s) by a constanttransformation. Therefore there must be one Killing vector for each generator of , namely, four.[We thank Lucas Hackl for discussions on this point.] In fact, it turns out that choosing G_IJ=δ_IJ results in an extra “accidental” symmetry, and so the metric above has a total of five Killing vectors. These Killing vectors (k̂_I)^i_i are explicitly constructed in appendix <ref>, and are given in eqs. (<ref>) and (<ref>).Of course, the existence of five Killing vectors implies an equal number of conserved momenta, c_I≡ (k̂_I)^i g_ij ẋ^j, which we will use to solve for the geodesics. Given the Killing vectors in eqs. (<ref>) and (<ref>), it is straightforward to evaluate the corresponding conserved quantities: c_1 =2 ẏ ,c_2 =2 sin(θ-τ)ρ̇+cos(θ-τ) [sinh(4ρ)-sinh(2ρ)θ̇-sinh(4ρ)+sinh(2ρ)τ̇] ,c_3 =2cos(θ-τ)ρ̇-sin(θ-τ) [sinh(4 ρ)-sinh(2 ρ)θ̇-sinh(4 ρ)+sinh(2 ρ)τ̇] ,c_4 =cosh4ρ-cosh2ρθ̇-cosh4ρ+cosh2ρτ̇ ,c_5 =(1-cosh2ρ) θ̇+(1+cosh2ρ) τ̇ ,where the dot denotes differentiation with respect to some affine parameter s along the geodesic. We are free to choose this parameter such that the normalization of the tangent vector is constrained to be constant, g_ijẋ^iẋ^j =2ẏ^2+2ρ̇^2+2cosh(2ρ)sinh^2ρ θ̇^2+cosh^2ρ τ̇^2-2sinh^2(2ρ) θ̇ τ̇≡k^2 . In a GR calculation, we would typically choose the normalization (for a spatial geodesic) to be +1, but this choice would leave the final value of s at the end of the circuit undetermined. However, recall that our notation for the path-ordered exponentials above is such that the circuits run over 0≤ s≤1, eq:pathPsi. Hence we shall scale the affine parameter s to lie in this range. The normalization constant k then gives the length of the geodesic, the depth of the corresponding circuit, since from eq. cost2 we have D(U)=∫_0^1s√(g_ij ẋ^i ẋ^j)≡k .The minimum value of k is then the depth of the optimal circuit, and by extension, the complexity of the target state ψ_T.Next, we must establish the boundary conditions for our geodesics. The geodesics (and paths in the circuit geometry in general) are described by (s)={τ(s), ρ(s), θ(s), y(s)}. Now, our initial condition is that U=1 at s=0, and by comparing with the parametrization in eq. (<ref>), we find that all coordinates except θ are initially zero, (s=0)={0, 0, θ_0, 0} . Note that the fact that θ=θ_0 is undetermined is not surprising since this is an angular coordinate, but the geodesic starts at the origin ρ=0. Hence the freedom to specify θ_0 is the freedom that the geodesic leave the origin in any direction. In part, this freedom reflects the fact that we do not expect the boundary conditions to uniquely fix the geodesic, but to instead give rise to a one-parameter family thereof—see the discussion at the end of the previous subsection. Now, the end-point of the geodesic is determined by A_T=U(s=1) A_R U^T(s=1), as in eq. pathA, where the quadratic forms for the reference and target states are given in eq. eq:stateMatrix. Substituting the initial state A_R=ω_0 1 and the explicit representation of the unitaries (<ref>) into this relation, we have A_T=ω_0 UU^T= ω_0e^2y_1[ cosh(2ρ_1)-sin(θ_1+τ_1)sinh(2ρ_1)cos(θ_1+τ_1)sinh(2ρ_1);cos(θ_1+τ_1)sinh(2ρ_1) cosh(2ρ_1)+sin(θ_1+τ_1)sinh(2ρ_1) ] ,where the subscript 1 denotes the value of the coordinate at s=1, y_1=y(s=1). Comparing the entries of the matrix on the right-hand side with those of A_T in eq. eq:stateMatrix, we arrive at the following boundary conditions for the end of the geodesic: ω_1/ω_0 =e^2y_1[cosh(2ρ_1)-sin(θ_1+τ_1) sinh(2ρ_1)] ,ω_2/ω_0 =e^2y_1[cosh(2ρ_1)+sin(θ_1+τ_1) sinh(2ρ_1)] ,β/ω_0 =e^2y_1cos(θ_1+τ_1) sinh(2ρ_1) .Implicitly, these constraints allow us to identify the final coordinates (s=1) for the geodesics corresponding to circuits which produce the desired transformation. Explicitly, we may solve this system to obtain e^2y_1=√(ω_1ω_2-β^2)/ω_0 ,cosh(2ρ_1)=ω_1+ω_2/2√(ω_1ω_2-β^2) ,tan(θ_1+τ_1)=ω_2-ω_1/2β .However, there is an obvious ambiguity here since θ_1 and τ_1 appear only in the combination θ_1+τ_1. Since only this linear combination is fixed by eq. fini, we have a one-parameter family of final boundary conditions—the linear combination θ_1-τ_1 remains unspecified. Naïvely, this might lead one to suspect a two-parameter family of allowed solutions, since the initial conditions left θ_0 unfixed as well. But this is not the case: rather, the geodesic equations of motion relate the freedom in the boundary conditions at s=0 and s=1, and the freedom in the initial and final conditions combine to yield the one-parameter family of geodesics anticipated above. This situation is illustrated in figure <ref>, which shows a one-parameter family of solutions beginning at the origin and ending on the spiral given by θ+τ=θ_1+τ_1 and radius ρ=ρ_1. To determine the complexity of the final state A_T, we must find the minimum length geodesic within this family, and thereby the optimal circuit.Having specified the boundary conditions, we proceed to solve for the geodesicsby examining the conserved momenta (<ref>). The first of these gives the simplest constraint: c_1=2 ẏ. Integrating with respect to the affine parameter s then yields: y(s) = c_1 s/2+y_0. In this case the undetermined coefficients are easily fixed by the boundary conditions, y(s=1)=y_1 and y(s=0)=0, hence: c_1=2 y_1andy_0=0y(s) =y_1 s .Next, we consider c_4 and c_5. These two constraints may be solved to obtain τ̇=c_5+c_4-c_5/4cosh^2ρ ,θ̇=c_5+c_4+c_5/4sinh^2ρ .We then observe that θ̇ diverges at the origin ρ=0 unless c_4=-c_5, which we must therefore impose in order to be compatible with the initial conditions. Implicitly, we are setting the angular momentum, the conserved momentum associated with the Killing vector _θ, to zero, which is characteristic of geodesics passing through the (radial) origin ρ=0. With this condition, the θ equation can be trivially integrated to yield θ=c_5 s+θ_0, where we have already imposed θ(s=0)=θ_0. Imposing the final boundary condition then yieldsc_5=Δθ≡θ_1-θ_0 θ(s)=Δθ s+θ_0 .Furthermore, the above allows us to simplify the τ̇ equation to τ̇=Δθ1-1/2cosh^2ρ . Now, combining our expressions for θ̇ and τ̇ with the constraints c_2 and c_3 in eq. eq:conserved, we find a relatively simple equation for ρ̇: ρ̇^2=c_2^2+c_3^2/4-Δθ^2/4 tanh^2ρ . In principle, we should now solve for the general solutions of eqs. tauDot4 and rhoDot4 subject to the boundary conditions in eqs. eq:coordinit and fini. While it is possible to carry out this exercise, the final solutions are not particularly illuminating.[The general solution for eq. rhoDot4 is given by sinhρ=c/√(c^2-Δθ^2) sinhs/2√(c^2-Δθ^2) ,where c^2=c_2^2+c_3^2 is fixed by substituting the boundary condition ρ=ρ_1 at s=1 into this equation. Furthermore, given this result, it is possible to integrate eq. tauDot4 to obtain τ(s); one finds τ=Δθ s-tan^-1Δθ/√(c^2-Δθ^2)tanhs/2√(c^2-Δθ^2) . Substituting τ=τ_1 at s=1 into this expression fixes τ_1 in terms of Δθ and c^2. Combining this result with the boundary condition for θ_1+τ_1 in eq. fini, we can then determine θ_1. In turn, θ_0 is now fixed since we know Δθ and θ_1. ] Instead, let us point out the particularly simple solution that arises for Δθ=0. In this case the expressions for τ̇ and ρ̇ reduce to τ̇=0 τ=0 ,ρ̇=1/2√(c_2^2+c_3^2) ρ=ρ_1 s ,which combine with y=y_1 s and θ=θ_0 from eqs. yrun and gonzo to describe a simple “straight-line” geodesic. Substituting this solution into eq. Umatrix, we can write the corresponding circuit as U_0(s) =e^y_1s[ coshρ_1s- sinθ_0 sinhρ_1s cosθ_0 sinhρ_1s; cosθ_0 sinhρ_1scoshρ_1s+sinθ_0 sinhρ_1s ] =exp[[ 1 0; 0 1 ] y_1 s+[ -sinθ_0cosθ_0;cosθ_0sinθ_0 ] ρ_1 s] .Note that an explicit path-ordering is not needed in the second expression since it is simply the exponential of a fixed matrix.[For this simple case, it is straightforward to identify the exponential form in the second line of eq. (<ref>) given the expression appearing in the first. In general however, one would apply eq. eq:metricRightInv to identify the components of Y^I(s) and then substitute these into eq. (<ref>).] The circuit depth of U_0, the length of the geodesic, is given by eqs. eq:geodesic and costa4, which for this simple solution yields D(U_0)=√(2(y_1^2+ρ_1^2)) .Again, in principle, we should determine all of the other geodesics satisfying the appropriate boundary conditions, and compare their respective circuit depths to D(U_0) in order to determine the minimum. However, we shall instead provide a more indirect but less technically challenging proof that this simple straight-line solution is in fact the shortest possible geodesic, and hence that it describes the optimal circuit.To prove that the straight-line solution above is the geodesic whose length is the (global) minimum, we recall from eq. eq:geodesic that the length of any geodesic is given by the normalization constant k. Now into this expression, we substitute our general solutions for y(s) and θ(s) from eqs. yrun and gonzo, respectively, as well as the expression for τ̇ from eq. tauDot4, whereupon we find k^2 =2y_1^2+2ρ̇^2+1-1/2cosh^2ρ Δθ^2 . This equation holds point-by-point along any geodesic satisfying the appropriate boundary conditions, but what we would like to argue (without explicitly solving for ρ(s)) is that k^2 is minimized by choosing Δθ=0.To begin, consider motion in the ρ-direction along any of our geodesics. The average velocity is given by ∫_0^1sρ̇=ρ_1 .Additionally, we have 0≤∫_0^1s(ρ̇-ρ_1)^2=∫_0^1s ρ̇^2 -ρ_1^2,and hence we may conclude that ∫_0^1 s ρ̇^2≥ρ_1^2, and that this inequality is only saturated when ρ̇=ρ_1 along the entire geodesic. Now, examining the coefficient of Δθ^2 in eqn. eq:geodesic3, we have 1/2≤1-1/2cosh^2ρ≤1 ,where the lower inequality is only saturated at ρ=0, and the upper inequality is saturated at ρ→∞.[In accordance with its interpretation as a radial coordinate, we do not consider negative values of ρ.] Given that all of our geodesics must start at ρ=0 and end at ρ=ρ_1, upon averaging over any of these geodesics, we find 1/2<∫_0^1 s1-1/2cosh^2ρ < 1 .Finally, let us average eq. eq:geodesic3 over any of our geodesics: k^2 =2y_1^2+2∫_0^1s ρ̇^2 + Δθ^2 ∫_0^1 s1-1/2cosh^2ρ ≥2y_1^2+2ρ_1^2 +Δθ^2/2 ≥2(y_1^2+ρ_1^2) .Comparing this to eq. solver4b, we have established the inequality k≥ D(U_0). Furthermore, our argument has established that this inequality can only be saturated with ρ̇(s)=ρ_1 (ρ=ρ_1 s) and Δθ=0 (θ(s)=θ_0, which implies τ(s)=0 via eq. tauDot4). We have therefore proved that the simple straight-line geodesic indeed constitutes the global minimum for our cost function costa4, and hence that eq. solver4b is in fact the complexity of the Gaussian wave function in this framework: (A_T)=√(2y_1^2+ρ_1^2) . As an exercise, we can compare the above result for D(U_0) in eq. solver4b, which was evaluated using eq. costa4, with the result found by evaluating eq. cost2. In this case, we must identify the components Y^I(s), which is easily done by examining the exponential expression in eq. solver4a:[Recall that ourgenerators are given in eq. Msimple.] Y^11 = y_1-ρ_1 sinθ_1 ,Y^22=y_1+ρ_1 sinθ_1 , Y^12=Y^21=ρ_1 cosθ_1 .Since these components are all constant, the integral over s in eq. cost2 is trivial, and the circuit depth (with G_IJ=δ_IJ) reduces to D(U_0) = √((Y^11)^2 +(Y^12)^2+(Y^21)^2+(Y^22)^2)= √( y_1-ρ_1sinθ_1^2+2ρ_1cosθ_1^2 +y_1+ρ_1sinθ_1^2) =√(2y_1^2+ρ_1^2) , in agreement with eq. (<ref>). §.§ Normal-mode subspace To properly interpret the complexity, we must re-express our result solver4b in terms of the physical parameters of the two coupled oscillators qm1, as well as the frequency ω_0 in the reference state eq:omega12pm. However, one finds that the complexity is most elegantly described in terms of the normal-mode frequencies _+ and _- given in eqs. qm2 and qm3. Using eq. eq:omega12pm, the final boundary conditions fini simplify to y_1=1/4 log_+_-/ω_0^2 ,ρ_1=1/4 log_-/_+ ,θ_1+τ_1=π .Substituting these expressions for y_1 and ρ_1 into eq. solver4b then yields the complexity of the ground state, (A_T) =D(U_0)= 1/2√(log^2ω̃_+/ω_0+log^2ω̃_-/ω_0) . At this point, let us also note that the boundary condition θ_1+τ_1=π (along with Δθ=0 and τ(s)=0) implies that the initial angle is θ_0=π. This straight-line geodesic is illustrated by the green line in figure <ref>. The corresponding circuit solver4a simplifies to U_0(s)=e^y_1s[coshρ_1s -sinhρ_1s; -sinhρ_1scoshρ_1s ] =exp[[y_1 -ρ_1; -ρ_1y_1 ] s ] ,with y_1 and ρ_1 given by eq. fin2.The simple and elegant form eq:CpmPre of the complexity in terms of the normal-mode frequencies suggests that we should investigate the optimal circuit solver4a in terms of the normal modes. The relationship between the physical positions of the masses and the normal-mode coordinates was given in eq. qm3, but we can understand this change of coordinates in terms of a simple rotation. In particular, we can perform the coordinate transformation via the orthogonal rotation matrix R,[Our transformation matrix certainly satisfies R R^T=R^T R=1. However, with the conventions adopted above, we note that detR=-1 and as a result, we actually have that as a numerical matrix R is symmetric, as shown with the eq. eq:rot. However, we still distinguish R and R^T in the following because R provides a mapping from the physical positions to the normal coordinates, while R^-1=R^T provides the inverse mapping. In other words, the columns of R are labeled 1,2 while the rows are labeled +,– and vice versa for R^T. ]R=1/√(2)[11;1 -1 ][ x̃_+; x̃_- ]=R[ x_1; x_2 ] . Introducing the short-hand notation x=x_1, x_2^T and x̃=x̃_+,x̃_-^T, the transformation (<ref>) may be concisely written x̃ =Rx, and the inverse transformation becomes x =R^T x̃. Of course, we can also use this transformation to re-express the target Gaussian wave function in terms of the normal-mode coordinates, ψ_T∼exp[-1/2 x^TA_Tx]=exp[-1/2x̃^T RA_TR^T x̃]Ã_T=R A_T R^T , where Ã_T denotes the quadratic form describing the ground state in the normal-mode space. Explicitly performing this rotation, one finds Ã_T=[ ω̃_+0;0 ω̃_- ] .That is, the target state becomes a factorized Gaussian in the normal-mode basis, eq. qm2. Of course, this decoupling was the essential point of introducing the normal-mode coordinates in the first place. Furthermore, if we apply this transformation to the reference state in eq. eq:stateMatrix, we see that it retains its simple form, Ã_R = RA_RR^T =ω_01 .That is, the reference state remains a factorized Gaussian when written in terms of the normal modes. Now, given the action of the gates and circuits on the quadratic forms, eq. pathA, we can transform our minimal circuit solver5 to act in the normal-mode space: Ũ_0(s) ≡R U_0(s) R^Twhere Ã_T=Ũ_0(s=1)Ã_R Ũ^T_0(s=1) . This transformation effects a remarkable simplification of the circuit solver5 to Ũ_0(s) =exp[[ y_1-ρ_1 0; 0 y_1+ρ_1 ] s] =exp[[ 1/2 log_+/ω_0 0; 0 1/2 log_-/ω_0 ] s]=[ _+/ω_0^s/20;0 _-/ω_0^s/2 ],where in the second line we have used eq. fin2.The important lesson learned here is as follows: from the perspective of the normal modes, both the target state and the reference state are factorized Gaussians, as shown in eqs. Atilde and Atilde2. The optimal circuit Ũ_0(s) then simply acts in a diagonal fashion to “amplify” each of the diagonal entries in the corresponding quadratic forms, taking ω_0 to _± in a simple linear manner. It is rather intuitive that this should be the optimal way to prepare Ã_T from Ã_R, since if any off-diagonal entries (entanglement) were introduced along the circuit, they would simply have to be removed by the time the trajectory reaches its end-point. This feature of the optimal circuit will greatly simplify our considerations of a lattice of coupled oscillators in the next section.Before turning to this generalization however, we wish to emphasize that the original circuit solver4a is performing the same operation of amplifying the normal modes—this is simply a matter of re-expressing U_0 in an alternative basis of generators. To properly clarify this, we need to introduce some additional notation. In the above, we adopted a tilde to denote various quantities in the normal-modes basis.[At this point, we wish to alert the reader to a subtle distinction that arises in our notation here: as established in footnote <ref>, we have introduced tilde's to distinguish quantities related to the normal modes from similar quantities in the position basis. Beginning with eq. Atilde, a state, circuit, or generator carrying a tilde acts in the normal-mode space, on wave functions written in terms of normal modes. However, this should be distinguished from the instances described here, where we place the tilde's on the indices. These tilded indices indicate that a normal-mode “basis” may still appear on objects acting in the oscillator position space. For example, above eq. bark, M_Ĩ indicates certain linear combinations of the standard generators Msimple, which still act on wave functions written in terms of x_1,x_2, but in a way that scales or entangles the normal modes.] We also introduced the index notation I={11,22,12,21} to label the components of the velocity Y^I(s) and the generators M_I. Here we would like to combine these two conventions to introduce a new index label Ĩ={++,+-,-+,–} to denote the same objects with components acting in the normal-mode basis. Thus the natural basis of generators M̃_Ĩ with which to construct the circuits acting on the states described in the normal-mode basis areM̃_++=[ 1 0; 0 0 ] , M̃_+-=[ 0 1; 0 0 ] , M̃_-+=[ 0 0; 1 0 ] ,M̃_–=[ 0 0; 0 1 ] .As numerical matrices, these M̃_Ĩ are of course identical to the M_I given in eq. Msimple, but the two sets of generators act in different spaces. Via the transformation eq:rot, we can also transform these generators to act on the states in the original position basis, M_Ĩ=R^T M̃_Ĩ R: M_++= 1/2[ 1 1; 1 1 ]= 1/2(M_11+M_22+M_12+M_21 ) ,M_+-= 1/2[1 -1;1 -1 ] = 1/2(M_11-M_22-M_12+M_21 ) , M_-+= 1/2[11; -1 -1 ]= 1/2(M_11-M_22+M_12-M_21) , M_–= 1/2[1 -1; -11 ] = 1/2(M_11+M_22-M_12-M_21) . The action of these generators can be read off from the indices, M_++ scales the x_+ coordinate or amplifies the corresponding normal mode. Of course, we could also transform the original generators M_I in eq. Msimple with M̃_I=RM_IR^T to construct the corresponding normal-mode basis. For example, M̃_11 would still scale the x_1 coordinate but would act on states in the normal-mode basis, it acts on Gaussian wave functions written in terms of x̃_±.With this new notation in hand, we would like to express our optimal circuit U_0 in terms of the generators M_Ĩ. It is easily shown, either by examining eq. solver5 directly or by transforming the expression in eq. solver5a with U_0(s)=R^T Ũ_0(s) R, that the optimal circuit can be expressed as U_0(s)=exp[M_++(y_1-ρ_1)+M_–(y_1+ρ_1) s] ,where M_±± are the linear combinations of the original generators given in eq. bark. In this form, we again recognize that the optimal circuit is simply amplifying the two normal modes, without introducing (and then having to remove) any entanglement between x_±.We can also observe that this simple circuit only involves two commuting generators, M_++ and M_–. Since the generators commute, it is straightforward to show that the geometry of corresponding normal-mode subspace is flat. That is, if we consider general circuits of the form U(y,ρ)=exp[M_++(y-ρ)+M_–(y+ρ)] ,then the corresponding metric becomes[This conclusion is slightly premature, since we have not shown that the metric metric1 is invariant under the change of basis from the original generators Msimple to those in eq. bark, but we shall prove this below in eq. metric1a. Note that we have also used that the new basis of generators still satisfies M_Ĩ M_J̃^T=δ_ĨJ̃.] s_n-m^2 =δ_ĨJ̃ U U^-1M^T_Ĩ U U^-1 M^T_J̃ =(y-ρ)^2+(y+ρ)^2=2y^2+2ρ^2 .Hence we recognize the normal-mode subspace as precisely the θ,τ=π,0 plane in our extended geometry metric2.[Implictly, we may allow ρ to run over positive and negative values in eq. swer4. Hence this subspace also includes θ,τ=0,0.] This perspective also makes clear why the optimal geodesic remains in the normal-mode subspace. Examining the full metric metric2, it is clear that motion in the θ and τ directions only extends the length of the trajectory. Thus since the start and end points both lie in this plane, there is no advantage to be gained by moving out of the normal-mode subspace. This argument also relies on the fact that g_yy and g_ρρ in the full metric metric2 are constants, independent of θ and τ, which precludes the existence of “short-cuts” to be found by moving off the normal-mode subspace (we return to this point in section <ref>). This is another important feature that extends to the case of a lattice of coupled oscillators in the next section.To close this section, we wish to introduce some additional technology which will prove useful in those that follow. Thus far, we have two particularly useful sets of generators for our gates and circuits, namely, M_I and M_Ĩ given in eqs. Msimple and Msimple2, respectively. While these generators all act on states and circuits in the physical basis, M_I acts to scale or entangle the physical positions x_1,2, while M_Ĩ scales or entangles the normal-mode coordinates x_±. The transformation between the two bases is given in eq. bark, but we would like to build an explicit transformation matrix R̂: M_Ĩ = R_Ĩ J M_JwhereR_Ĩ J =1/2[1111;1 -11 -1;11 -1 -1;1 -1 -11 ]=R_ka⊗R_ℓb .Note that in the final equality, R is the rotation matrix in eq. eq:rot, and we are identifying the indices as follows: Ĩ=(kℓ) with k,ℓ∈{+,-}, and J=(ab) with a,b∈{1,2}.[ Recall that as defined in eq. eq:rot, R is the matrix which transforms the `1,2' indices of the oscillator position basis to the `+,–' indices of the normal-mode basis—see footnote <ref>.] This identification is really the origin of the interesting tensor product structure R=R⊗ R. The expression in eq. bark2 indicates that the first (second) R is rotating the first (second) component of the pairs which comprise the Ĩ and J indices on the two generators. Given this expression, we immediately see thatR is also an orthogonal rotation matrix. Hence we can easily invert the transformation between the basis generators via M_I=(R^T)_IJ̃M_J̃=R_J̃ I M_J̃. Similarly, this transformation acts on the velocity components as Y^I=Y^J̃R_J̃ I. These transformations will prove useful in examining the complexity with cost functions written in different bases. For example, in the present context, we can see that the cost function remains unchanged if we express it directly in the normal-mode basis. We can also transform the metric metric1 as follows: s^2 =δ_IJ U U^-1M^T_I U U^-1 M^T_J =R_Ĩ IR_J̃ J δ_IJ U U^-1M^T_Ĩ U U^-1 M^T_J̃=δ_Ĩ J̃ U U^-1M^T_Ĩ U U^-1 M^T_J̃ ,where we have used the fact that R is an orthogonal matrix. In going from the second to third line, we have used the identity R_Ĩ I δ_IJ (R^T)_JJ̃=δ_ĨJ̃. Note that the invariance of the metric under this change of basis was already used in evaluating the metric on the normal-mode subspace in eq. metric5x. We extend this discussion of changing between the position and normal-mode bases to the case of a linear lattice of N oscillators in appendix <ref>.§ A LATTICE OF OSCILLATORS In this section, we wish to return to the original problem of a free scalar field regulated by a lattice, eq:Hlattice. That is, we will consider evaluating the complexity of the ground state of a lattice of coupled oscillators hqm. Drawing on our experience with the two coupled oscillators, this becomes a straightforward calculation. In particular, as we saw above, both the ground state and the reference state are described by factorized Gaussians in the normal-mode space. And in this space, the optimal circuit simply amplifies each of the diagonal entries in the corresponding quadratic forms in a linear manner. To simplify the technicalities in the following discussion, we will explicitly consider the case of a one-dimensional lattice, and discuss more general dimensions in the next subsection. Hence, we begin with N oscillators on a one-dimensional circular lattice, H=1/2∑_a=0^N-1[p_a^2+ω^2x_a^2+Ω^2x_a-x_a+1^2] ,with periodic boundary conditions x_a+N=x_a.[Note that for convenience, we have labeled the first oscillator with a=0, rather than a=1, the sum in eq. qm88 runs over a∈{0,1,⋯,N-1}.] As in the two oscillator problem, we have set the masses M_a=1 for simplicity but we should think of the frequencies as being related to the field theory parameters by ω=m and Ω=1/δ, as in eq. hqm. The Hamiltonian qm88 then corresponds to the lattice version of a (one-dimensional) free scalar field on a circle of length L=N δ. Of course, to solve the above system, one simply rewrites the Hamiltonian in terms of the normal modes, H=1/2∑_k=0^N-1[ |p̃_k|^2+_k^2|x̃_k|^2 ] ,where the transformation to the normal-mode basis is achieved by a (discrete) Fourier transform, x̃_k≡1/√(N)∑_a=0^N-1exp-2πi k/N ax_a . where k∈{0,…,N-1}, and we note that x̃_k^†=x̃_N-k.[We can see this result as a combination of two simpler identities: x̃_k^†=x̃_-k, which follows from the complex conjugation of eq. eq:Fourier, and x̃_k=x̃_k+N, which follows from the periodicity of the lattice. Note that our convention for the range of k was chosen to match the range of the position labels a, rather than shifting the range of k to run over positive and negative values, k∈{-⌈ N/2⌉+1,-⌈ N/2⌉+2,,⋯, ⌊ N/2⌋}, which is a more typical convention. Furthermore, for future reference, note that we can define u⃗_k≡ [u_k]_a=exp -2π i k a/N as the orthogonal basis of an N-dimensional vector space,satisfying the normalization condition u⃗^†_ k·u⃗_k'=∑_a=1^N[u^†_ k]_a [u_k']_a=∑_a=0^N-1exp-2πi(k-k')/N a=N δ_k,k' . Hence we use the usual definition for the normal-mode momenta p̃_k≡1/√(N)∑_a=0^N-1exp2πi k/Nap_a . Note the change in the sign in the exponential in comparison to eq. eq:Fourier. This definition then produces the standard commutation relations: [x̃_k,p̃_k']=iδ_kk' and [x̃_k,x̃_k']=0=[p̃_k,p̃_k']. ]The normal-mode frequencies _k are defined in terms of the physical frequencies ω and Ω in the Hamiltonian qm88 as follows: _k^2=ω^2+4Ω^2 sin^2πk/N , (see appendix <ref>). As desired, eq. qm288 reduces the problem to N decoupled harmonic oscillators, which enables us to easily write the ground-state wave function as ψ_0(x̃_0,x̃_1, x̃_2,⋯) =∏_k=0^N-1 (_k /π)^1/4 exp[-1/2 _k |x̃_k|^2] . As before, this ground state will be the target state in our complexity computations. While eq. targetk will suffice to describe the ground state, in principle, one would also like to express the wave function in terms of the original variables x_a in the position basis. This transformation is facilitated using notation introduced in section <ref>. In particular, following eq. eq:rot, we write the Fourier transformation eq:Fourier between the position and normal-mode bases as x̃ = R_ Nx, with R_N≡1/√(N)[ 1 1 1 … 1; 1 μ μ^2 … μ^N-1; 1 μ^2 μ^4 …μ^2(N-1); ⋮ ⋮ ⋮ ⋱ ⋮; 1 μ^N-1μ^2(N-1) … μ^(N-1)^2; ] ,where μ≡exp-2π i/N.[As discussed in footnote <ref> for the matrix R in eq. eq:rot, we distinguishfrom ^T even though the numerical matrix in eq. logan is symmetric. Note that if we write out the transformation to show the indices, we have x̃_k = [R_ N]_kax_a. That is, the row index ofhas values in the momenta k while the column index has values in the lattice position a.In passing, we also observe that eq. logan reduces to eq. (<ref>) for the special case N=2, for which we have μ=exp-iπ=-1.] Sinceis a unitary matrix, ^†=1, the inverse transformation is given by x =^† x̃. Now let us adopt the notation of section <ref> (and in particular, of eq. gauss) to write the target state targetk as ψ_T(x̃_k)=∏_k=0^N-1_k/π^1/4 exp[-1/2 x̃^†Ã_Tx̃]withÃ_T=diag_0,…,_N-1 .Using the rotation logan, we can write this target state in terms of the physical coordinates,[The relation _k=_N-k ensures that A_T is real. ] ψ_T(x_a)=∏_k=0^N-1ω_k/π^1/4 exp[-1/2x^T A_T x ]withA_T=^†Ã_T .We are now prepared to extend our complexity calculations to this lattice of coupled oscillators. We have already identified the target state as the ground state targetk. In analogy with eq. eq:refPhys, the reference state will be a factorized Gaussian state, ψ_R(x_a)=ω_0/π^N/4exp[-1/2x^TA_R x ]withA_R=ω_0 1 .where the individual oscillators are completely unentangled.[Recall our tilde notation to distinguish the normal-mode space from the physical space. In particular, the reference frequency ω_0 is independent of the normal-mode frequency with k=0, ω_0≠_0!]An important feature of our reference state is that it is invariant under translations on the lattice, the Gaussian of each oscillator has the same width ω_0. As a result, it remains a factorized Gaussian when expressed in terms of the normal-mode coordinates: ψ_R(x̃_k)=ω_0/π^N/4exp[-1/2x̃^†Ã_R x̃ ]withÃ_R=A_R ^†=ω_0 1 .Lastly, we need to consider the elementary gates with which we will build the circuit U that implements the desired transformation ψ_T=U ψ_R. With the notation introduced in eq. eq:gates, the set of gates (particularly the entangling and scaling gates) is easily enlarged for the present problem by simply extending the range of the indices: a,b∈{1,2} ⟶ a,b∈{0,1,2,⋯,N-1}. These discrete gates are then easily extended to the path-ordered exponentials introduced in eqs. eq:pathPsi and path2, U(s)=𝒫 exp[∫_0^ss̃ Y^I(s̃) _I], where the index I runs over the N^2 values corresponding to pairs (ab), and the operators _I take the same form as in eq. operate. In discussing the target and reference states with the notation of eq. gauss, we also anticipated mapping these exponentials to the matrix formulation introduced in eqs. matrix0, matrix and pathA for Gaussian states. In fact, the generators have precisely the form given in eq. matrix, where again the indices run over the range a,b,c,d∈{0,1,2,⋯,N-1}. That is, we now have N^2 generators which are N× N matrices. This extends thegroup found in section <ref> to the group GL(N,ℝ) in the present problem.Following the analysis in section <ref>, we use the analogous F_2 cost function, i.e.,[In the position basis, there is no need for the complex conjugations appearing in eqs. cost5 or metric5 since all of the relevant quantities are real. However, we are including them here in anticipation that later on, we will transform these formulae to the normal-mode space. Thesetransformations are accomplished within eq. logan, which is a complex unitary matrix.Hence using, M^† rather than M^T allows us to use precisely the same expressions without change.Of course, as defined in eq. eq:Fourier, the normal modes are generally complex, but as we commented above, they also satisfy the “reality condition” x̃_k^†=x̃_N-k—which ensures that we have not doubled the number of degrees of freedom.]𝒟(U)=∫_0^1s√(δ_IJ Y^I(s)Y^J(s)^*) , whereY^I(s)=_sU(s) U^-1(s)M^†_I .Hence the optimal circuit will correspond to a geodesic in the GL(N,ℝ) geometry given by a right-invariant metric, analogous to eq. metric1. To simplify the discussion of the metric here (and in the next section), we introduce the following notation: s^2=δ_IJ Y^IY^J^* withY^I=U U^-1M^†_I .However, extending the detailed calculations above to the full N^2-dimensional geometry would be very involved. In particular, the next step would require finding the analog of eq. Umatrix, a convenient parametrization of a general group element U∈GL(N,ℝ), which would naturally involve N^2 coordinates. Thus at this point, we rely on the lessons learned from the case of two coupled oscillators in the previous section. In particular, there we found that since both the ground state and the reference state are described by factorized Gaussians in the normal-mode basis, the optimal circuit simply acts to amplify each of the diagonal entries in the corresponding quadratic forms in a simple linear manner. We have already noted by way of eqs. raffle and refk that the former statement about factorized Gaussians also applies in our lattice problem. Hence it is natural that the most efficient circuit simply amplifies the Gaussian width for each of the normal-mode coordinates, ω_0→_k. In particular, the circuit does not introduce any entanglement between the normal modes at any stage, since this entanglement would have to be removed before arriving at the final target state raffle. Via eq. rotateU, let us write the optimal circuit acting in the normal-mode basis; we have U_0(s) = ^† Ũ_0(s)whereÃ_T=Ũ_0(s=1)Ã_R Ũ^†_0(s=1) , and thus the straight-line circuit Ũ_0(s) becomes Ũ_0(s)=exp[M̃_0 s] with M̃_0=diag(1/2log_0/ω_0,1/2log_1/ω_0, ⋯, 1/2log_N-1/ω_0 ) .This circuit certainly accomplishes the desired transformation with Ũ_0(s=1) = exp[M̃_0], but the intuition from the previous analysis of two coupled oscillators suggests that it is also the optimal circuit.We can add to this intuitive picture as follows: in the discussion around eqs. swer4 and metric5, we identified the normal-mode subspace as consisting of those circuits U which only involve the scaling generators for the normal modes. Consequently, it is straightforward to show that the geometry of the normal-mode subspace is flat since these generators all commute with one another. In the present case, the normal-mode subspace becomes a N-dimensional subspace of U∈GL(N,ℝ) with the form U= Ũ^†, where[In general, the coordinates ỹ_k are complex but satisfy the normal-mode “reality condition” ỹ_k^†=ỹ_N-k.] Ũ_n-m=exp[M̃_n-m] withM̃_n-m=diag(ỹ_0, ỹ_1, ⋯,ỹ_N-1 ) .Substituting this expression into eq. metric5, one finds the following flat Cartesian metric induced on this subspace: s^2_n-m=|ỹ_0|^2+|ỹ_1|^2+⋯+|ỹ_N-1|^2 .Therefore any geodesic within the normal-mode subspace will simply take the form of a straight line. It is then straightforward to show that if we confine the circuit to this normal-mode subspace metric6, the optimal circuit is described by the simple circuit in eq. nmU, which we write as U_0(s) = ^† Ũ_0(s)= exp[^†M̃_0 s] ,where Ũ_0(s) and M̃_0 are defined in eq. raffle2. There are actually some subtleties in the preceding argument which make the conclusion somewhat premature. The first is that eq. metric5 which defines the metric is written in the position basis, whereas eq. metric6 was implicitly calculated for an expression raffle4 written in the normal-mode basis. That is, in eq. raffle4, we worked with M̃_n-m=Ỹ^ĨM̃_Ĩ with a particular choice of Ỹ^Ĩ.[Again, the tilde on the index Ĩ indicates that it runs over pairs of momentum labels (kℓ), while the tilde on M indicates that these generators act on Gaussian wave functions written with the normal-mode coordinates x̃_k.] However, we show in appendix <ref> that this was nonetheless a valid approach since the metric takes precisely the same form when written in terms of the normal-mode space. This requires extending the discussion around eq. bark2 describing the change of bases for the case of two coupled oscillators to the analogous transformation for our linear lattice of N oscillators. Secondly, to properly establish that the optimal circuit follows a straight line in the normal-mode subspace, as in eq. straight9, we must show that no shorter path can be found by making an excursion outside this subspace. To begin, we note that implicitly we assumed in eq. raffle4 that all of the other coordinates in thegeometry could be set to zero. Recall that in themetric metric1, the metric on the normal-mode subspace was completely independent of the other coordinates, we had s^2_n-m=2 y^2 +2ρ^2 irrespective of the values of θ and τ. In particular, recall that the optimal circuit was a straight line in this subspace with θ=π and τ=0.We would like to establish a similar result for the present N^2-dimensional geometry. For simplicity, we will work in the normal-mode space. We proceed by expressing general circuits Ũ using the Iwasawa (or KAN) decomposition of ; see for example <cit.>. This states that any Ũ∈ can be uniquely written as the product of three matrices, Ũ = KAN, where K is an orthogonal matrix, A is a diagonal matrix with positive entries,[We denote this diagonal matrix with the traditional A, but it should not be confused with the quadratic forms specifying the Gaussian states, eq. gauss. Similarly, K here should not be confused with the gates producing a momentum shift in eq. eq:gates, nor should N be confused with the total number of oscillators. We trust that these distinctions will be clear from context.] and N is an upper triangular matrix with every diagonal element equal to 1. Clearly, we are interested in the A component as this describes the normal-mode subspace, as in eq. raffle4.As a warm up exercise, let us consider translating Ũ_n-m by some fixed angles and shifts. In particular, we write Ũ=K_0 Ũ_n-m N_0 where only the ỹ_k in Ũ_n-m vary (eq. raffle4) and ask what is the metric on the corresponding subspace. Since N_0 acts on the right, and the metric is right-invariant by construction, it has no effect on the geometry. Our experience in changing bases in appendix <ref> allows us the eliminate the K_0 rotation as well: following eq. tran1, we write the differentials Ỹ^Ĩ=(Ũ Ũ M̃^†_Ĩ) as Ỹ^Ĩ= Ũ_n-mŨ_n-m^-1 [K_0^T M̃_Ĩ K_0]^† . Now in the last factor, K_0 acts by a similarity transformation on the generators which effectively produces a change of basis. However, using the special form of the generators lot1, it is straightforward to show – following a series of steps analogous to those given in eqs. lot2 or lot2a – that K_0^T M̃_Ĩ K_0 = [K_0]_ĨJ̃ M̃_J̃whereK_0=K_0⊗K_0 .It follows that K_0 is an orthogonal matrix since K_0 is orthogonal, and hence this rotation of the generator basis leaves the metric unchanged. Therefore we find that the induced metric on this subspace is s^2_n-m=δ_ĨJ̃ Ỹ^Ĩ_n-m ( Ỹ^J̃_n-m)^*whereỸ^Ĩ_n-m=Ũ_n-mŨ_n-m^-1M̃_Ĩ^† ,which again yields the simple answer given in eq. metric6.This result establishes that there are indeed no short-cuts to be found by running the circuit through the angle and shift directions. That is, the circuit must run from ỹ_k=0 to ỹ_k=1/2log(ω̃_k/ω_0) as in eq. raffle2. Eq. tran6 further establishes that there will be a fixed distance or cost associated with this displacement, irrespective of the orientation of the normal-mode subspace in the full geometry, that is, irrespective of the angles and shifts chosen in K_0 and N_0. Since the full geometry is Euclidean, moving in these “orientation directions” will only add to the distance. Thus the best strategy is to fix the shifts and angles at the beginning of the circuit (to zero, as required by U(s=0)=1) and then move only in the normal-mode directions.This argument is still not quite sufficient to establish that the simple straight-line circuit is a geodesic in the full N^2-dimensional geometry. In particular, non-vanishing off-diagonal terms in the metric which mix ỹ_k with the other coordinates would force the geodesic to move away from the normal-mode subspace in the additional angle and shift directions. But since evaluating the full metric would require a rather lengthy and involved calculation, we instead consider small deviations of the circuits around the subspace specified by Ũ=K_0 Ũ_n-m N_0, we extend our initial ansatz to allow small excursions in the K and N directions, Ũ=K_0 exp[ M̃^rot_Ĩθ_Ĩ] Ũ_n-mexp[M^shift_Ĩη_Ĩ] N_0 ,where θ_Ĩ, η_Ĩ≪1. Here, the (small) change in K only involves the (antisymmetric) rotation generators [M^rot_kℓ]_pq=δ_kpδ_ℓq- δ_ℓpδ_kqwithk<ℓ , while the (small) change in N only involves the shift generators[M^shift_kℓ]_pq=δ_kpδ_ℓqwithk<ℓ . The rotation generators are, of course, a linear combination of the original generators given in eq. lot1, and hence are not orthogonal to the shift generators in the sense that M^rot_kℓ [M^shift_pq]^†=δ_kp δ_ℓq .Of course, all of these generators are orthogonal to the diagonal generators appearing in Ũ_n-m, which will become the key point momentarily.With our extended circuits extend0, we now evaluate the differentials Ỹ^Ĩ=( Ũ Ũ M̃^†_Ĩ) on the normal-mode subspace, at θ_Ĩ=0=η_Ĩ, Ỹ^Ĩ=[K_0]_ĨJ̃ [Ỹ^J̃_n-m +M̃^rot_K̃ dθ_K̃ M̃_J̃^†+ Ũ_n-m M^shift_ĨŨ_n-m^-1 dη_Ĩ M̃_J̃^†], where K_0 is the orthogonal matrix given in eq. tran5 and Ỹ^Ĩ_n-m are the differentials along the normal-mode directions identified in eq. tran6. As before, the rotation of the differentials by K_0 can be ignored since this transformation leaves δ_IJ in the metric unchanged. Next we observe that the only non-vanishing components of Ỹ^J̃_n-m are along the diagonal directions, J̃=(kk). It is then easy to show that the other two differentials are orthogonal to these. Given the explicit form of the rotation generators in eq. rotgen, it is clear that the second term only contributes in the off-diagonal directions, J̃=(kℓ) with kℓ. Similarly, one can show that the same is true of the third term via eqs. raffle4 and shiftgen, Ũ_n-m M^shift_kℓ Ũ_n-m^-1=e^ỹ_k-ỹ_ℓM^shift_kℓwithk<ℓ .The key point then is that Ỹ^J̃_n-m are orthogonal to the other two differentials in eq. tran7. In fact, it is straightforward to show that the full metric on the normal-mode subspace (θ_Ĩ=0=η_Ĩ) becomes s^2_n-m=|ỹ_0|^2+|ỹ_1|^2+⋯+|ỹ_N-1|^2 +∑_k<ℓ[(dθ_kℓ)^2 + |dθ_kℓ+e^ỹ_k-ỹ_ℓ dη_kℓ|^2 ] .Hence there are no off-diagonal terms in the metric, which would drive the geodesic away from the normal-mode subspace.[In general, we are asking that there are no source terms in the linearized equations for the θ_Ĩ and η_Ĩ. In turn, this means that we are asking that there are no linear terms in the cost function cost5 when expanding about the straight-line trajectories raffle2. Here we have explicitly shown that no such linear terms arise as off-diagonal terms in the metric, involving the differentials of θ_Ĩ and η_Ĩ. In principle, we should also verify that the metric components g_ỹ_kỹ_k are not varied at linear order in the perturbations. However, our previous analysis shows that the metric on the normal mode subspace is completely independent of the coordinates parametrizing the K and N transformations, which ensures that this class of potential terms linear in θ_Ĩ and η_Ĩ vanishes. Therefore the above discussion is sufficient to ensure that the geodesic equations in the full GL(N,ℝ) geometry have no source terms which would push the straight-line trajectories away from the normal-mode subspace.] We may therefore conclude that the optimal circuit indeed takes the form of the simple straight-line circuit in eq. raffle2. Thus, for the cost function (<ref>), the complexity for our lattice of oscillators is obtained by simply summing up the circuit elements in the normal-mode basis.Using eqs. (<ref>) and (<ref>), we find =1/2√(∑_k=0^N-1( logω̃_k/ω_0)^2) , where the normal-mode frequencies are given in eq. eq:eigenfreq. Recall that in our lattice regularization eq:Hlattice, we had ω=m and Ω=1/δ, and so we can express the complexity complexityN in terms of the field theory parameters via _k^2=m^2+4/δ^2 sin^2πk/N . Furthermore, we can replace N=L/δ where L is the total length of the one-dimensional lattice of oscillators. Of course, ω_0 remains the (as yet unspecified) frequency which specifies the Gaussian reference state refX.The entire discussion in this section is easily extended (albeit with a somewhat tedious extension of the notation) to the evaluation of the complexity of a (d-1)-dimensional spatial lattice of N^d-1 oscillators, and the final result is =1/2√(∑_{k_i}=0^N-1( logω̃_k⃗/ω_0)^2) , where k_i are the components of the momentum vector k⃗=(k_1,k_2,⋯,k_d-1), and the normal-mode frequencies are given by _k⃗^2=m^2+4/δ^2 ∑_i=1^d-1sin^2πk_i/N . The linear size of each spatial direction here is L=Nδ, and so the total (spatial) volume of the system is V=L^d-1=N^d-1δ^d-1. Hence the total number of oscillators can be expressed as N^d-1=V/δ^d-1 ,which will prove useful below. §.§ Comparison with holography Eq. complexityNd gives our result for the complexity of the ground state of a free scalar field in d spacetime dimensions. We would now like to compare this result with the analogous results arising from the proposals for holographic complexity discussed in the introduction. Of course, we must note that we are trying to compare complexities for disparate QFTs, a free theory with a single degree of freedom in the present case versus a strongly coupled theory with a large number of degrees of freedom in holography. Hence there is no a priori reason to expect that the results should agree in the two cases. Nevertheless, we will find that with certain choices, our QFT calculations share a number of qualitative features with holographic complexity. We can interpret these similarities as providing guidance towards understanding the cost function that underlies the holographic complexity conjectures.Examining eq. complexityNd, we see that the expression under the square root essentially involves an integration over the spatial momenta. Our experience with QFT thus suggests that the result will be dominated by the UV modes, by modes withω̃_k⃗∼ 1/δ. Hence as an approximation which allows us to identify the leading contribution to the complexity, we may replace all of the ω̃_k⃗ with 1/δ in eq. complexityNd to obtain[See appendix <ref> for more accurate estimates of the large N behaviour of the complexity complexityNd.] ≈N^d-1/2/2 log1/δ ω_0∼V/δ^d-1^1/2 ,where we have used eq. vold to re-express the leading power of N in terms of V/δ^d-1. The leading UV divergence in holographic complexity for both the CA and CV proposals was studied in some detail in <cit.>. Hence we can compare our QFT result croot with the analogous results for holographic complexity; denoting the latter collectively as _holo, these were found to take the form_holo∼V/δ^d-1 . Thus we see that the leading terms in the QFT and holographic complexities differ by the power of 1/2 appearing in eq. croot. However, the origin of this square root is clear: it is simply the overall square root appearing in eq. complexityNd, which in turn arises from our use of the F_2 cost function in eq. cost5. Now, there is nothing wrong with the result in eq. croot per se, but it does suggest that if our QFT complexity is to emulate the leading behaviour found in holographic complexity, then we should make an alternative choice for the cost function.[An alternative approach <cit.> would be to simply assign each gate the cost N^d-1/2. However, it seems this may be problematic if, we wish to compare complexities for different UV cut-offs. ] For example, the cost function defined by the F_1 measure in eq. eq:Fmetrics, which involves the first power of a single sum over the modes, would produce the desired behaviour. More generally, a natural family of cost functions which would reproduce the divergence in eq. Cholo is 𝒟_κ=∫_0^1s∑|Y^Ĩ(s)|^κ , where the natural choice would be that κ is a positive integer, but any positive real value (with κ≥1) will suffice for most of the following discussion. Note that we have defined the cost function here with the sum running over the normal-mode basis—we return to this point below in section <ref>.Here, we should note that only κ=1 (equivalently, the F_1 cost function), satisfies the condition of positive homogeneity as described in the introduction; that is, for general κ>1, doubling the amplitude of Y^Ĩ does not double the cost. This issue can be described as saying that only the case κ=1 yields a reparametrization-invariant cost function, that replacing s→ŝ(s) leaves the cost unchanged. However, we may proceed with the physics intuition that we can think of 𝒟_κ as different kinds of actions describing the motion of a particle in the space of circuits.Now it is relatively straightforward to show that in fact the straight-line circuit in eq. raffle2 minimizes all of these cost functions. This circuit only acts with scaling gates on the various normal modes. It is clear that if the circuit were to make an excursion away from the normal-mode subspace, if the path also moved in the entangling directions, this would only turn on new components of the velocity Y^Ĩ and thereby increase the cost of the circuit. To establish that the straight-line path is favoured by the general κ cost function, let us consider a more general trajectory in the normal-mode subspace, Ũ_1(s)=exp[M̃_1(s)], where M̃_1=diag(f_0(s)/2log_0/ω_0 ,f_1(s)/2log_1/ω_0, ⋯, f_N-1(s)/2log_N-1/ω_0 ) ,and each of the f_k(s) is an arbitrary function satisfying f_k(s=0)=0andf_k(s=1)=1 .With this ansatz, the cost function Dalpha evaluates to 𝒟_κ(Ũ_1)=1/2^κ∑_k |log_k/ω_0|^κ×∫_0^1s |∂_s f_k(s)|^κ .However, with reasoning along the lines of that in eqs. fun1 and fun2, one can argue that ∫_0^1 s |∂_s f_k(s)|^κ≥1 (for κ≥1), and that the inequality is saturated if and only if ∂_s f_k(s)=1, f_k(s)=s.[An exception to this result arises for κ=1. In this case, the bound is saturated by any functions f_k(s) satisfying ∂_s f_k(s)≥0 everywhere. We return to this point in the discussion section <ref>.] That is, the general κ cost function Dalpha is minimized by the straight-line circuit Ũ_0(s) = exp[M̃_0 s]. Furthermore, working with the UV approximation ω̃_k⃗= 1/δ, the leading contribution to the complexity then becomes ≈V/δ^d-1| log1/ω_0 δ|^κ .An interesting feature of this result is that in limit δ→0, this contribution appears to diverge faster than the power law 1/δ^d-1 in the first factor.This last observation, however, depends on the choice of ω_0 which defines our reference state refX, which we have hitherto left unspecified. Considering this choice, there seem to be a number of reasonable options. First, ω_0 could be associated with some ultraviolet frequency at the lattice scale. For example, ω_0=e^-σ/δ where e^-σ provides a numerical scale that ensures ω_0> ω̃_k⃗ for all k⃗. In this case, the leading contribution in eq. calpha reduces to ≈σ^κ V/δ^d-1 .With this choice, the extra logarithmic factors in the δ→0 divergence have been eliminated and we are only left with the 1/δ^d-1 factor. However, this choice also entails the interesting feature that the (subleading) infrared contributions to the complexity will involve the UV cut-off scale. That is, the full sum over momenta in eq. Dalpha includes summing over the infrared modes, modes with ω̃_k⃗∼ m. These infrared contributions will take the form _IR≈-log^κ(mδ) .An alternative choice would be to associate ω_0 with some infrared scale, ω_0≪1/δ. One might choose this scale to be a physical scale in the problem, such as the mass m or the volume V, but this would tie the reference state to the properties of the QFT.[Furthermore, if we choose ω_0∼ V^-1/(d-1), the complexity becomes superextensive.] This appears problematic if we wish to compare the complexities of states in different theories, with different masses—see below. Hence it seems that we are instead led to choose some arbitrary IR scale to define the reference frequency, which then becomes a part of our definition of the complexity of QFT states. In this sense, the appearance of ω_0 here is not very different from the appearance of the arbitrary numerical factor σ in the complexity with the previous UV choice. Of course, if ω_0 is a fixed IR frequency, the additional logarithmic factor in the complexity calpha survives and contributes to the leading divergence in the limit δ→ 0.With the above in mind, we find surprising similarities when we compare our result with the CA proposal for holographic complexity. The leading divergence appearing in the latter ccaction takes the form <cit.> _A∼V/δ^d-1 log(L_AdS/α δ) ,where δ is the short-distance cut-off scale in the boundary CFT, L_AdS is the AdS curvature scale of the bulk spacetime, and α is an arbitrary (dimensionless) coefficient which fixes the normalization of the null normals on the boundary of the WDW patch. Since _A is a quantity which is to be defined in the boundary CFT, it should not depend on the bulk AdS scale. However, we can eliminate this factor with the freedom in choosing α, we set α = ω_0 L_AdS where ω_0 is some arbitrary frequency. In this case,eq. leader reduces to _A∼V/δ^d-1 log(1/ω_0 δ) . While this choice eliminates the AdS scale, the holographic complexity still depends on the choice ofω_0, just as in our QFT result calpha. Furthermore, all of the issues discussed above with respect to this choice also appear in the case of the CA conjecture. In particular, we emphasize that with the choice ω_0=e^-σ/δ, the UV cut-off appears in infrared contributions to the holographic complexity arising from joint terms deep in the bulk <cit.>. Whereas in <cit.> this ambiguity and the associated issues seemed problematic for holographic complexity, here can view them as a natural feature of complexity for QFTs.We can go further in comparing our QFT results with holography. In particular, in order for the leading divergence in eq. calpha to match the holographic result leader2 more closely, we should choose κ=1 in eq. Dalpha, the F_1 cost function. Of course, this reasoning only applies when the reference frequency is chosen in the IR and the logarithmic factor modifies the form of the leading divergence. In the case where ω_0 is set by the cut-off scale, the leading divergence is a simple power law and the exponent κ only modifies the overall numerical pre-factor, which we have not specified here. However, κ also appears in the IR contribution in eq. calpha3. Again the analogous contributions in holography would be linear in log(δ) because of the form of the corresponding boundary terms in the gravitational action <cit.>. Hence this reasoning again favours the choice κ=1. That said, given the aforementioned disparity between the field theories that we are comparing, it is not clear how much weight to give this observation. One can also look at the form of subleading corrections to the leading divergence. As discussed in appendix <ref>,the first subleading correction comes from the mass and has the form V m^2/δ^d-3. This form could be anticipated by simple dimensional analysis, and analogous results can be found in holographic complexity as well <cit.>. Specifically, if the boundary CFT is perturbed by a relevant operator of dimension Δ, the corresponding coupling λ will have dimension d-Δ, and the first subleading correction to the holographic complexity then takes the form V λ^2/δ^2Δ-(d+1). For the CV conjecture, these calculations follow in close parallel with the analogous calculations of corrections to holographic entanglement entropy induced by relevant operators <cit.>. However, we expect that analogous results will appear for the CA conjecture. Such corrections to holographic complexity were considered in <cit.> as arising from placing the boundary theory in a curved spacetime or from evaluating the state of a curved time slice. It may be interesting to understand how to extend our present QFT calculations of complexity to incorporate such situations. § PENALTY FACTORS In eq. cost2, the cost function was written with a general metric G_IJ, which allows us the freedom to include penalty factors to weight certain directions or classes of gates more heavily than others. This is particularly relevant for the lattice of oscillators representing the regulated scalar field. In the previous section, our circuits implicitly included entangling gates Q_ab, which coupled points on the lattice which were arbitrarily far apart. However, if we want complexity to be a physical attribute of a QFT, then we would expect it to reflect the notion of locality. That is, gates which couple far-separated points should be more expensive – incur a higher cost in the geometric distance function – than those which couple nearest neighbours. To gain some experience with this idea, we return to the problem of two coupled oscillators and we introduce a penalty factor weighting the entangling gates, which act on two oscillators (or sites), more heavily than the scaling gates, which act on a single oscillator (or site). Specifically, we may penalize the “off-diagonal” directions by choosing G_IJ=diag(1,^2,^2,1) ,with >1.As a result, our original metric metric1 is replaced by the following more complicated metric: s^2 =G_IJU(s) U^-1(s)M^T_I U(s) U^-1(s)M^T_J =2 y^2+2[^2-(^2-1)sin^2(θ+τ)]ρ^2+(^2-1)sin2(θ+τ)sinh(2ρ) ρ (τ-θ) +2[^2cosh(2ρ)-(^2-1)cos^2(θ+τ) sinh^2ρ]cosh^2ρ τ^2 +2[^2cosh(2ρ)-(^2-1)cos^2(θ+τ) cosh^2ρ]sinh^2ρ θ^2-[2^2-(^2-1)cos^2(θ+τ)]sinh^2(2ρ) τ θ . Of course, this geometry reduces to eq. (<ref>) upon setting =1. The metric has a slightly simpler expression in terms of the pseudo-lightcone coordinates (<ref>), where θ=x+z, τ=x-z: s^2 = 2y^2+2[^2-^2-1sin^22x]ρ^2-2 (^2-1)sin(4 x)sinh(2 ρ) ρz + 2 ^2x^2+2[^2cosh(4ρ)-(^2-1)cos^2(2x)sinh^22ρ]z^2-4 ^2cosh(2 ρ) xz , which reduces to eq. metric2 when =1. Although these coordinates somewhat obscure our physical intuition for the geometry, they are computationally much simpler. Therefore we will work with the metric in the form penalty2 for most of the following. As in the unpenalized case, this metric enjoys the four Killing vectors (k̂_I)^i given in eq. (<ref>).[Again, these arise from the right-invariance of the expression in the first line of eq. penalty1. The fifth “accidental” Killing vector ∂_x in eq. Killed5 no longer gives rise to a symmetry for the penalized metric, as is clear from eq. penalty2.] For the metric (<ref>), the associated conserved quantities ĉ_I=(k̂_I)^ig_ijẋ^i areĉ_1≡2ẏ , ĉ_2≡ -ẋ[2 ^2 sinh(2 ρ)cos(2 z)] +ż [cos(2 z) sinh(4 ρ)(2^2-(^2-1) cos^2 (2 x)) -(^2-1)sin(2 z) sinh(2 ρ) sin(4 x) ] + ρ̇[2sin(2 z) (^2-(^2-1) sin^2 (2 x))-(^2-1) cos(2 z)cosh(2 ρ) sin(4 x) ] , ĉ_3≡ẋ[2 ^2 sinh(2 ρ)sin(2 z)] - ż [ sin(2 z) sinh(4 ρ)(2^2-(^2-1) cos^2 (2 x))+(^2-1) cos(2 z) sinh(2 ρ) sin(4 x) ] + ρ̇[2cos(2 z) (^2-(^2-1) sin^2 (2 x))+(^2-1) sin(2 z) cosh(2 ρ) sin(4 x) ] , ĉ_4≡ -2 ^2 cosh(2 ρ)ẋ +2 (^2 cosh(4 ρ)-(^2-1) cos^2 (2 x)sinh^2(2 ρ) ) ż- (^2-1)sinh(2 ρ) sin(4 x) ρ̇ . One can check that these quantities indeed reduce to those given in eq. (<ref>) when =1. Solving the first equation for y is trivial, and we simply recover eq. (<ref>), y=y_1 s. The next three equations may be solved for ρ̇, ẋ, and ż: ρ̇=1/4 ^2{(^2-1) cosh(2 ρ) sin(4 x) (ĉ_2 cos(2 z)-ĉ_3 sin(2 z)) +2(^2-^2-1cos^2 (2 x)) (ĉ_2 sin(2 z)+ĉ_3 cos(2 z))-(^2-1) ĉ_4 sinh(2 ρ) sin(4 x)} ẋ= 1/4 ^2{1/sinh(2 ρ)[2cosh(4ρ)+^2-1cos^2(2x)cosh^22ρĉ_2cos(2z)-ĉ_3sin(2 z) +^2-1sin(4x)cosh2ρĉ_2sin(2z)+ĉ_3cos(2z)] -2(1+(^2-1) cos^2 (2 x))cosh2ρĉ_4} ż= 1/4 ^2{1/sinh(2ρ)[2(1+(^2-1) cos^2 (2 x))cosh2ρ(ĉ_2 cos(2 z)-ĉ_3 sin(2 z)) +(^2-1) sin(4 x) (ĉ_2 sin(2 z)+ĉ_3 cos(2 z))] -2(1+(^2-1) cos^2 (2 x))ĉ_4}Now recall that in the unpenalized case, the expression for θ̇ in eq. trun diverged at the origin ρ(s=0)=0 unless the conserved quantities were properly tuned. This divergence is simply the usual angular momentum barrier at the origin, and the tuning amounts to setting the angular momentum to zero. The same issue arises here, as reflected in the fact that both ẋ and ż have the same pole structure as ρ→0, in this limit, θ̇=ẋ+ż diverges but τ̇=ẋ-ż does not. Taking the limit ρ→0, and setting x_0=z_0 (since τ(s=0)=0) in eq. revel1, one finds that this divergence is avoided by choosing ĉ_2=ĉ_3/^2tan2z_0 .Substituting eq. lov2 back into the expressions for derivatives thus renders them well-behaved at the origin, as required by the initial condition ρ(s=0)=0, but since their forms are not appreciably simpler we shall not write them out here.Now with the metric penalty2, the normalization of the tangent vector k^2=g_ij ẋ^iẋ^j becomes k^2=2 y_1^2+2[^2-^2-1sin^22x]ρ̇^2-2 (^2-1)sin(4 x)sinh(2 ρ) ρ̇ ż + 2 ^2ẋ^2 +2[^2cosh(4ρ)-(^2-1)cos^2(2x)sinh^22ρ]ż^2-4 ^2cosh(2 ρ) ẋ ż .In principle, one can substitute the expressions in eq. revel1 for ρ̇, ẋ, and żinto this expression to obtain an explicit formula for the geodesic length, with no derivatives. Unfortunately, the resulting expression appears quite intractable, and a general solution remains beyond our reach. However, we are ultimately only interested in the optimal trajectory. Given our experience with the unpenalized metric, one might reasonably conjecture that the global minimum is again obtained with the simple straight-line circuit solver5, and as we now verify, this trajectory remains a geodesic in the penalized geometry penalty2. Recall that the first constraint in eq. revel0 yielded the desired behaviour for y, y(s)=y_1 s, as in eq. yrun. The straight-line solution also had τ and θ fixed with τ(s)=0 and θ(s)=π. This then implies that x and z are fixed with x(s)=π/2 and z(s)=π/2. Combining the latter with eq. lov2 then yields ĉ_2=0. Substituting these values of x and z into the last two expressions in eq. revel1, we obtain ẋ=-ĉ_4/2 cosh(2ρ) andż=-ĉ_4/2 .Hence consistency with the condition ẋ=ż=0 demands that we set ĉ_4=0. Finally, the ρ̇ equation yields ρ̇=-ĉ_3/2^2ĉ_3=-2^2 ρ_1 ,and we arrive at the desired solution: ρ(s)=ρ_1 s. Having shown that this simple trajectory remains a geodesic in the penalized geometry penalty2, we substitute this geodesic into eq. (<ref>) to obtain k^2=2y_1^2+^2ρ_1^2≡k_0^2 , where we have introduced the label k_0 to denote the geodesic length of the straight-line circuit to avoid confusion with other lengths considered below. Note that eq. (<ref>) is the natural generalization of eq. (<ref>) to the case with >1. However, it turns out that this is not the minimum geodesic for the penalized metric: shorter trajectories can be found. In particular, examining the geometry penalty2 more closely, we see thatg_ρρ=2[^2-^2-1sin^22x]depends on the x coordinate. This contrasts with the unpenalized metric metric1 (or the lattice metric metric6a) where we found that the geometry of the normal-mode subspace (the metric for the y and ρ directions) was independent of the other coordinates. The latter property was essential to showing that the straight-line circuit was indeed the optimal trajectory. Examining eq. rats6, it is clear that there should be short-cuts for the motion along ρ if we move away from the normal-mode subspace, away from x=π/2.[As an amusing observation, let us add that it is also clear that there are no such short-cuts if <1. That is, if we weight the scaling gates more heavily than the entangling gates, then the straight-line circuit solver5 will remain the optimal circuit.] For example, we might consider the following simple path consisting of two segments: a)0≤s̅≤1 :y=0 , ρ=ρ_1 s̅ , x=π/4 , z=π/4 ; b)1≤s̅≤2 :y=y_1(s̅-1) , ρ=ρ_1 , x=π/4 s̅, z=π/4 , where s̅ provides some arbitrary parametrization of the path. This segmented path is not a geodesic, but does connect the initial point at the origin to the desired end-point at y=y_1, ρ=ρ_1 and x=π/2. The first segment moves only in the ρ direction at the optimal value of x, and then the second segment moves uniformly in both the x and y directions to arrive at the required end-point. The total length of this path is k_s=√(2)ρ_1+√(2π/4^2^2+2y_1^2) , where we use the subscript s to denote “segmented”, in contrast with k_0 above. Of course, the relation between k_0 and k_s now depends on the details of the various parameters y_1, ρ_1, and . It is naturalthat all three of these coefficients are large, in which case one generally finds k_0>k_s. However, to simplify the analysis and illustrate this result, let us consider the regime where the penalty factor is the largest constant, ≫ρ_1,y_1 and ρ_1,y_1≫1. Then we may approximate the two lengths with k_s ≃ π/2√(2) +√(2)ρ_1+ 2√(2)/π y_1^2/^2 +⋯ , k_0 ≃ √(2)ρ_1+ y_1^2/√(2)ρ_1 +⋯ ,k_0/ k_s≃4/π ρ_1≫1 . Thus in the penalized geometry penalty2, the length of the segmented path is much shorter than the straight-line geodesic in this regime. Again, the segmented path is not a geodesic and so cannot describe the optimal path, but we shall find that it gives a remarkably good approximation to the optimal geodesic. We would like to find the optimal geodesic but as noted below eq. eq:kPenFull, obtaining the general solution seems out of reach. However we can make progress with a simplifying assumption: if we examine the penalized metric penalty2, we see that as the radius ρ increases, the most rapidly growing component of the metric is g_zz∼^2 e^4ρ (for generic x). This suggests that motion in the z direction will quickly be suppressed as the geodesics move out from the origin. Therefore we simplify our problem by considering trajectories confined to a constant-z submanifold, for which the relevant metric is given by s^2=2y^2+2[^2-^2-1sin^22x]ρ^2+2 ^2x^2 . We will return to justify our assumption of no (or little) motion in the z direction below. Obviously, eq. (<ref>) is a much simpler geometry, and the analysis of the geodesics becomes much more tractable. We leave the details of solving for the resulting geodesic to appendix <ref> and only refer to certain key results in the comparison below. Our expectation is that the new geodesic is the optimal trajectory, at least for large , but we must add that we have not provided an irrefutable proof of this result. Furthermore, using the results of appendix <ref>, we also explicitly show below that the segmented path segments provides a good approximation of this optimal geodesic in the regime where the penalty factor is large.Our analysis in appendix <ref> suggests we use the following quantity to more conveniently compare the lengths of the various paths: k̅^2≡k^2/2-y_1^2 .For the straight-line geodesic, we have simply k̅_0= ρ_1 ,while for the segmented path, we have k̅_s = π/4[1+8ρ_1/π2ρ_1/π+√(1+4y_1/π^2)]^1/2≈ π/4+ρ_1+8/π^2 y_1^2/^2 ρ_1+… ,where as above, the expansion in the second line assumes ≫ρ_1,y_1. Now for the optimal geodesic, we have from eq. eq:kbarCoords k̅ ≈ /2tan^-1√(^2-1)+ρ_1≈ π/4+ρ_1-1/2-1/12^2+1/^4 .Comparing these results, we see that in this largeregime, k̅ for the optimal geodesic is much smaller than k̅_0 for the straight-line geodesic, and extremely close to k̅ for the segmented path. Furthermore, we note that both k̅_0 and k̅ are completely independent of y_1, while it only appears in k̅_s at ordery_1^2/^2.In figure <ref>, we plot k̅, k̅_0, and k̅_s as functions offor fixed values of the variable(see eq. eq:kEps), as well as for various values of y_1 in the case of k̅_s. Recall the definition newer which shows that these quantities are giving us direct information about the length of the corresponding paths. Hence one clearly sees in the figure that the new optimal geodesic is shorter than the straight-line circuit for all values of . In the right panel, we also see that the length of the segmented path quickly approaches the length of the optimal geodesic for large values of the penalty factor, in agreement with eqs. kbars and kbaropt. In fact, if we take the difference of these two equations in the largelimit, we find k̅_s-k̅≃1/2 . -2ex Given that in this limit, both k̅ and k̅_s are diverging, it is impressive to find the simple (1) difference shown above. Figure <ref> examines this difference in more detail numerically. Of course, the above results support the conjecture that the new geodesic represents the optimal geodesic and hence yields the shortest possible distance between the origin and the end-point. Since the segmented path segments is not itself a geodesic, it must have a longer length. However, the impressive agreement in eq. eq:overhead seems to indicate that this path is coming very close to the optimal geodesic. We can confirm this very clearly by examining x(s) and ρ(s) numerically. As shown in figure <ref>, the geodesic essentially has two phases: the first in which ρ increases uniformly with fixed x=π/4 and the second in which ρ̇→0 and x increases uniformly from π/4 to π/2. As shown in the figure, these two distinct phases are separated by an abrupt but smooth transition, which becomes particularly obvious for larger . We might note that the growth in y is uniform throughout the entire span 0≤ s≤ 1. However, for large , the transition occurs for small s (see further comments below) and so y is growing primarily in the second phase where x increases. Hence we can see very explicitly that the behaviour of the optimal geodesic is indeed very similar to that of the segmented path segments for large . Let us examine the behaviour of the transition point in more detail. For computational purposes, we define this as the value of s=s_trans at which the two (normalized) curves for ẋ and ρ̇ cross in figure <ref>, ẋ(s)/ẋ|_max =ρ̇(s)/ρ̇|_max at s=s_trans. In the limit ≪1, this point can be well-approximated by[This expression is obtained by equating the normalized quantities ẋ/ẋ|_max and ρ̇/ρ̇|_max (from eqs. (<ref>) and (<ref>), respectively) to solve for the critical point x_crit at which the curves in the right plot in figure <ref> cross. Upon substituting this into (<ref>) for s(x), and using (<ref>) for k̅ and (<ref>) for c̅_2, one obtains an expression that depends only onand , s,, in which we then take ≪1.] s_trans≈1-Π1-^2, h(), 1-/Π1-^2, 1- , where h()≡^-1[^2-1+√(5^2-4)/2^3--√(5^2-4)]^1/2 . We plot s_trans() in figure <ref>. In conjunction with eq. (<ref>), one sees that a large penalty factor strongly suppresses the duration of the first phase, and thus the circuit spends most of its “time” – in terms of some fixed total affine parameter – on the second phase. Additionally, one sees that the switchover point appears to go to zero as →∞. We can verify this by first approximating h() for largeas h()≈^-1[2/√(5)-1]^1/2 , and then expanding eq. (<ref>) in the limit →∞, →0: s_trans()≈1/[1/πlog1/+^0]+1/^2 , where ^0≈1.11502. Comparing this expression with eq. (<ref>) for large , we see that the leading-order term in s_trans() above may be written as s()≈4 ρ_1/π . This result has an intuitive explanation: ρ_1 sets the radial distance that must be covered in the first phase (which is large in the →0 limit while the “angular” change in x remains fixed), while as explained above, a large penalty factorcompels the circuit to complete this motion as quickly as possible. To close this discussion of the penalized geometry, let us reiterate that we have argued that the geodesic confined to the constant-z subspace eq:metricSimple is the optimal geodesic, and hence that its length gives the complexity of the state. That is, when we penalize the entangling gates with eq. pen99, the complexity of the ground state becomes =√(2)[π/4 +ρ_1-1/2+2 y_1^2/π+1/^2] .in the regime ≫ρ_1,y_1. §.§ New optimal circuit Given the optimal geodesic for the penalized geometry, we would now like to examine the properties of the corresponding circuit. For simplicity, we rely on the fact that for ≫1 the optimal geodesic is well approximated by the segmented path described in eq. segments and explicitly build the circuit for the latter path. First however, let us rewrite eq. segments in terms of θ=x+z and τ=x-z: a)0≤s̅≤1 :y=0 , ρ=ρ_1 s̅ , θ=π/2 , τ=0 ; b)1≤s̅≤2 :y=y_1(s̅-1) , ρ=ρ_1 , θ=π/4(s̅+1), τ=π/4(s̅-1) . Recall that s̅ is some arbitrary parameter along the path. Furthermore, given this form, it is interesting to plot this segmented path in the (ρ, θ, τ) space in order to visually compare it to the straight-line geodesic—see figure <ref>. This is essentially a comparison of the optimal geodesic in the original geometry metric1 to that in the new penalized geometry penalty1. One feature that the figure emphasizes is that these two geodesics end at different points along the allowed (blue) spiral at θ+τ=π, ρ=ρ_1.Using the expression for a general element ofin eq. (<ref>), we write the segmented circuit U_s(s̅) as U_s(s̅)= U_a(s̅)=[ e^-ρ_1 s̅ 0; 0 e^ρ_1s̅ ] for 0≤s̅≤1 , U_b(s̅)=e^y_1(s̅-1)[ cosπ/4(s̅-1)e^-ρ_1 -sinπ/4(s̅-1)e^ρ_1; sinπ/4(s̅-1)e^-ρ_1cosπ/4(s̅-1)e^ρ_1 ] for 1≤s̅≤2 ,Note that U_a(s̅=1)=U_b(s̅=1), as required by continuity along the path. Furthermore, observe that the circuit along the second segment can be re-expressed as U_b(s̅)=e^y_1(s̅-1)R̅(s̅) U_a(s̅=1) ,where we have defined the rotation matrix R̅(s̅)≡[cosπ/4(s̅-1) -sinπ/4(s̅-1);sinπ/4(s̅-1)cosπ/4(s̅-1) ] . The interpretation of eq. (<ref>) is that upon completing the first segment with U_a(s̅=1), the circuit performs a rotation (as well as multiplying by the exponential involving y_1) along the second segment until we reach the desired target state. The additional evolution along this second segment is therefore captured entirely by e^y_1(s̅-1)R̅(s̅). In passing, we also note that at the end-point, s̅=2, the rotation matrix reduces to R̅(s̅=2)=1/√(2)[1 -1;11 ] ,which is closely related but distinct from the rotation matrix R defined previously in eq. (<ref>). At its end-point, this new circuit becomes U_s(s̅=2)=e^y_1/√(2)[ e^-ρ_1 -e^ρ_1; e^-ρ_1e^ρ_1 ] ,which we might compare to the end-point of the straight-line circuit solver5, U_0(s=1)=e^y_1[coshρ_1 -sinhρ_1; -sinhρ_1coshρ_1 ] .These are clearly different, in accordance with our comment about the geodesics ending at different points infigure <ref>. However, it is straightforward to show that the transformations implemented in the segmented and straight-line circuits are related by the rotation in eq. dead, U_s(s̅=2)=U_0(s=1) R̅(s̅=2) .Both of these transformations act on the reference state to produce the target state (see eq. eq:stateMatrix) as A_T =U A_R U^T. Since the reference state is proportional to the identity, the additional rotation in eq. eq:U2phys leaves this state invariant, A_R=R̅(s̅=2)A_R R̅^T(s̅=2), and so both U_s(s̅=2) and U_0(s=1) will produce the same target state, as required.It is useful to re-express the new circuit eq:segU in the normal-mode space using eq. rotateU, Ũ(s)=R U(s) R^T, which yields Ũ_s(s̅)= Ũ_a(s̅)=[coshρ_1 s̅ -sinhρ_1 s̅; -sinhρ_1 s̅coshρ_1 s̅ ] for 0≤s̅≤1 , Ũ_b(s̅)=e^y_1(s̅-1)[cosπ/4(s̅-1)sinπ/4(s̅-1); -sinπ/4(s̅-1)cosπ/4(s̅-1) ] Ũ_a(s̅=1) for 1≤s̅≤2 ,where we have expressed Ũ_b(s̅) in a form analogous to eq. eq:seg2. The key observation to note here is that the optimal circuit involves off-diagonal components when expressed in the normal-mode space. That is, in terms of the normal modes, the new circuit is utilizing entangling gates, gates which entangle (or disentangle) the normal-mode coordinates. Since we know that the reference state Atilde and the target state Atilde2 are unentangled in this space, it must be that along the first segment U_a(s̅), the circuit is introducing entanglement in the state, but then this entanglement is removed along the second segment U_b(s̅) on the second segment. We can see this explicitly by examining the state à along the trajectory. In particular, along the first segment (for 0≤s̅≤1), we find Ã(s̅)= ω_0 Ũ_a(s̅) Ũ^T_a(s̅)=ω_0 [cosh2ρ_1 s̅ -sinh2ρ_1 s̅; -sinh2ρ_1 s̅cosh2ρ_1 s̅ ].Here we see the entanglement (the off-diagonal terms) begins at zero and steadily grows to a maximum ats̅=1 at the end of the first segment. Subsequently, along the second segment (for 1≤s̅≤2), we find Ã(s̅) = ω_0 Ũ_b(s̅) Ũ^T_b(s̅)= ω_0 e^2y_1(s̅-1)[ cosh2ρ_1 -sinh2ρ_1sinπ/2(s̅-1)-sinh2ρ_1cosπ/2(s̅-1);-sinh2ρ_1cosπ/2(s̅-1) cosh2ρ_1 +sinh2ρ_1sinπ/2(s̅-1) ]. Here, the entanglement shrinks steadily back to zero as s̅ runs over this second interval. Recall that y_1 and ρ_1 are given in terms of the normal-mode frequencies in eq. fin2.To reiterate our key observation, with eq. pen99 penalty factors are introduced to increase the cost of the entangling gates in the position space. As a result (for large ), the optimal geodesic is deformed to be close to the segmented path described in eqs. segments and segmentsX. However, in the normal-mode space, the new geodesic is driven off of the normal-mode subspace. That is, even though the initial and final states are unentangled when written in terms of the normal modes, the optimal circuit still introduces entanglement (among the normal modes) at intermediate steps along the trajectory. One gains some insight into this behaviour by transforming the penalized metric pen99 to the normal-mode basis using the orthogonal matrixdefined in eq. bark2. The new metric then becomes G_ĨJ̃ = _Ĩ IG_I J ^T_JJ̃=1/2[ 1+^200 1-^2;0 1+^2 1-^20;0 1-^2 1+^20; 1-^200 1+^2 ].Here we see that in the normal-mode basis, there is no extra cost attributed to the entangling gates relative to the scaling gates. There are also a number of curious negative entries in the off-diagonal components, but this does not fundamentally distinguish the scaling and entangling gates.§ DISCUSSION In this paper, we took the first steps towards defining circuit complexity in quantum field theory.The key idea, due to Nielsen <cit.>, was to endow the space of circuits with an appropriate geometry which allows one to translate the task of finding the optimal circuit into the task of finding the minimum geodesic (with appropriate boundary conditions). We implemented this approach for a simple free scalar field theory. The first step however was to introduce a UV regulator by placing the theory on a lattice, which reducedthe scalar field theory to a family of coupled harmonic oscillators. In this context, we were able to construct a interesting set of elementary gates eq:gates, in particular, scaling and entangling gates. We also chose our reference state to be a factorized Gaussian state refX, whose simplicity lies in the fact that there is no entanglement between different points on the lattice. For the purposes of this preliminary study, we chose the target state to be the ground state targetX of the system, which is also a Gaussian state. To gain some intuition for the problem, we began by studying the simple case of a pair of harmonic oscillators. The fact that both the reference and target states were Gaussian allowed for the simplification that we could translate from an operator language to a matrix language. It was then straightforward to show that with the F_2 cost function, the desired geometry was given by a right-invariant metric metric1 on . The optimal geodesic was then a simple straight line, which only moved through a flat two-dimensional subspace of the full, more complicated geometry. Translating this geodesic to the optimal circuit, the latter had a particularly simple interpretation in the normal-mode basis, where it only consisted of scaling gates amplifying the individual normal modes. This was a reflection of the fact that the ground state also takes the form of a factorized Gaussian when written in terms of the normal modes. These results for the two coupled oscillators were then extended to the full problem of a lattice of coupled oscillators with relative ease. In particular, we were able to show that the optimal circuit was given by the analogous straight-line geodesic moving in the normal-mode subspace, without constructing the full right-invariant metric on the N^2-dimensional geometry of . §.§ Comparison with holography:As discussed in the introduction, a primary motivation for this paper came from recent efforts to understand “holographic complexity,” and so it was interesting in section <ref> to compare our results to those obtained from the holographic proposals. Here we must reiterate the caveat that this comparison involves two very different QFTs, namely a free theory with a single degree of freedom in our scalar field model versus a strongly coupled theory with a large number of degrees of freedom in holography. Hence there is no a priori reason to expect that the results should agree in the two cases. Nevertheless, we found that if the cost function is chosen appropriately, the scalar field complexity exhibits remarkable similarities with holographic complexity. Our tentative interpretation of this concordance is that it provides insight into the implicit cost functions that underly the holographic complexity conjectures.In particular, the leading divergences Cholo in both the CV and CA proposals are extensive, they are proportional to the volume of the time slice on which the boundary state is evaluated, as shown in <cit.>. While the F_2 cost function gave a result proportional to V^1/2 in the scalar field theory, it is straightforward to construct a family of cost functions in eq. Dalpha, all of which yield an extensive complexity for the scalar field theory.[Again, we remind the reader that one can continue to work with the F_2 measure if the cost of the individual gates is set proportional to V/δ^d-1^1/2—see footnote <ref>.] With the new cost functions Dalpha, the leading contribution also contained a logarithmic factor, which was ambiguous in that it depended on the choice of the frequency ω_0 specifying the reference state refX. However, this precisely matched an ambiguity in the holographic complexity <cit.> found for the CA construction ccaction. In the latter case, the logarithmic factor came from joint terms <cit.> in the gravitational action, and the ambiguity arose from the freedom to choose the normalization of the null normals on the boundary of the WDW patch.Whereas this ambiguity had originally been seen as problematic for the CA conjecture, our scalar field calculation indicates that it is a perfectly natural feature associated with the freedom in the choice of the reference state that we can anticipate in any definition of complexity for a QFT. It might then seem mysterious that no such ambiguity arises for the CV conjecture volver. However, as explained in section <ref>, the additional logarithmic factor in the leading term calpha becomes a simple numerical coefficient if we choose ω_0= e^-σ/δ, and so such a choice may indeed be an integral part of the microscopic rules implicit in the CV construction. Unfortunately, this does not explain the absence of infrared terms of the form given in eq. calpha3 which might be expected with this choice. While it would be premature to conclude that the CV conjecture is incorrect, we might note that there is an alternative proposal in the literature suggesting that the volume of a maximal time slice in the bulk should be dual to the information metric rather than the complexity in the boundary theory <cit.>.As further noted in section <ref>, our scalar field complexity emulates the CA proposal ccaction most closely if we choose the F_1 cost function, κ=1 in eq. Dalpha. Given the aforementioned disparity in the two field theories in question, it is unclear how much weight to give this observation. However, we might add that the F_1 measure is a natural choice since it adheres most closely to the original definition of complexity, which involved simply counting the number of gates in the optimal circuit. Furthermore, let us add that the F_1 cost function will also feature again in the discussion of cMERA networks below.Of course, it would be interesting if a more precise connection can be found between the holographic and QFT calculations with regards to the ambiguity in the reference state discussed above. At present, it is actually not clear how the reference state enters in the holographic calculations at all, but perhaps one can draw upon the proposal for a state-surface conjecture in <cit.>. Undoubtedly, making this connection concrete would bring us closer to an explicit translation for the complexity between the bulk and boundary.§.§ Ambiguities and other miscellaneous complaints: In the introduction, our “definition” of complexity was rather imprecise, as it left open the choice of the reference state |ψ_R⟩, the choice of the set of elementary gates which would be used to construct U, and the choice of the tolerance (and measure) in eq. scotch. Clearly, even though it is easy to set out interesting questions for complexity (what is the complexity of a particular state in a particular QFT?), the precise value of the complexity will depend on the details of all of these choices.[Of course, the tolerance does not play a role in the geometric framework adopted here because the gates are no longer discrete—see discussion around eq. success.] The ambiguity in our reference state, the choice ω_0, was already seen to modify the complexity in an interesting way in the preceding discussion. Here one might recall early discussions of entanglement entropy from a hep-th perspective: the explicit dependence of the leading contributions on the UV cut-off was certainly seen as problematic (or at least, it was by one of the present authors). However, with some experience, we learned to find universal information in the entanglement entropy and to apply it as a useful diagnostic of QFTs in various ways, <cit.>. In fact given our experience with entanglement entropy, the non-universality of the leading contributions to the complexity (because of the power-law dependence on δ, as in eqs. croot and calpha), was assumed to be self-evident in the present discussion and not even commented upon. Analogously, we would advocate that complexity is again a new quantity with what initially seems to be unusual and perhaps undesirable features, but that we must develop our experience to learn how complexity can inform us about interesting physics and universal properties of QFTs and holography. Hence rather than regarding the ambiguities discussed above as a problem per se, they should be collectively considered as a new feature which we must learn to accommodate in working with complexity.In the context of the present calculations, clearly evaluating the complexity for a single state, the ground state, will not be particularly informative. Instead we might compare the complexities of different states, and while extending our calculations to the complexity of excited states would allow such a comparison, it is beyond the scope of the present paper. However, we can certainly compare the complexities of the ground states of different scalar theories, in particular theories with different masses. Here our experience with holographic complexity <cit.> suggests that there may be interesting information that could be extracted from the finite or logarithmic contributions. For example, for even spacetime dimensions (even d), there will be an interesting contribution that is intrinsically independent of the cut-off, although it may depend on the reference frequency ω_0. This explicitly appears in eq. eq:Cd1, where we are examining the case d=2 and κ=1. In this instance, we may isolate this constant in[Of course, we are inspired to formulate this quantity by the constructions using entanglement entropy to examine RG flows of three-dimensional QFTs <cit.>.] (L ∂_L-1)=-a_0 .where L is the linear (spatial) size of the system. This result is independent of both the short-distance cut-off scale and the reference frequency. Hence it would be interesting to better understand the meaning of the coefficient of a_0, and whether it carries some universal information about the underlying theories. Of course, it is straightforward to extend this simple example to higher dimensions. It would also be interesting to compare these results to similar calculations for holographic complexity where the boundary CFT is deformed by a relevant operator—see discussion at the end of section <ref>.1.5exLet us also remark here that the reference state refX is an unusual state from the textbook perspective of QFT. Recall thatthis state was chosen since it has no entanglement between different points (on the lattice). Such a factorized Gaussian is precisely the kind of reference state that appears in the cMERA construction <cit.>—see the discussion below. However, such an unentangled state is a very unusual statein standard QFT, which for example would typically have a divergent energy density. Of course, the vacuum energy density of the ground state is also divergent, and so to make this statement meaningful, we may evaluate the difference in the energies of |ψ_R⟩ and the ground state |ψ_T⟩: ⟨ψ_R| H |ψ_R⟩-⟨ψ_T| H |ψ_T⟩ = 1/4 N^d-1 ω_0+1/4ω_0 ∑ω_k⃗^2-1/2 ∑ω_k⃗ ≈V/δ^d[ω_0δ/4+1/ω_0δ-1] . We therefore see that that generically, if ω_0δ≪ 1 or ωδ≫1 (which were advocated to be the natural choices in section <ref>), the “renormalized” energy density of |ψ_R⟩ diverges as δ→0.[Note that with some fine-tuning of ω_0, we could arrange the difference of energy densities to be finite.] Hence this would not be a state that would be considered to be part of the standard Hilbert space that one builds with particle excitations on top of the vacuum. However, one should simply regard this as another unusual feature of complexity. As we have seen both here with our QFT calculations as well as in holographic complexity, the complexity can only be sensibly defined with a finite value of the regulator, in which case the reference state is certainly a sensible state within the associated Hilbert space. 1.5exWhile introducing a UV regulator was an essential step in sensibly defining the complexity in the scalar field theory, let us add that this does not regulate the size of the Hilbert space in the present case.[We thank Edward Witten for making this observation and raising the following question.] With the lattice regulator, the scalar field theory is reduced to N^d-1 normal-mode oscillators, but the Hilbert space of each of these oscillators is infinite! It is an interesting question whether or not an additional regulator should be introduced to render the total number of states finite as well. Otherwise it would seem that even within the UV regulated theory, there will be states of infinite complexity. §.§ Penalty factors and locality: In section <ref>, we experimented with the introduction of penalty factors in the case of two coupled oscillators. In particular, we gave a higher cost to the entangling gates than the scaling gates with pen99. This certainly resulted in a different optimal circuit, but ultimately the circuit could not avoid incorporating the entangling gates, and so the complexity increased to (), as shown in eq. loft. Perhaps the most interesting lesson to be learned from these calculations is that the introduction of penalty factors (in the position-space cost function) tends to drive the optimal circuit away from the normal-mode subspace, the restriction to which played a central role in the previous analysis of section <ref>.However, in our simple experiment in section <ref>, the optimal circuit was still required to introduce entanglement using the entangling gates, and therefore our calculations did not really address the motivation discussed at the beginning of that section. Namely, we expected that penalty factors could be used to introduce a notion of locality in the complexity of the scalar field theory. In particular, our calculations in section <ref> included entangling gates Q_ab, which coupled points on the lattice that were arbitrarily far apart, all with equal cost. It seems natural that using gates which couple far-separated points should incur a higher cost than using those which couple nearest neighbors. To gain some insight into this problem,let us return for a moment to the discrete gates in eq. (<ref>) and consider a one-dimensional lattice of N coupled oscillators. We will show that our set of entangling gates is over-complete in the sense that any of these gates can be constructed from nearest-neighbour entangling gates.For example, it is a straightforward calculation (using the Baker-Campbell-Hausdorff formula) to show that Q_13=Q_12^-1Q_23^-1Q_12Q_23^1/ . In other words, the next-to-nearest neighbour entangling gate Q_13 is equivalent to 4/ nearest-neighbour entangling gates. A simple generalization of the above result is Q_14=Q_13^-1 Q_34^-1 Q_13 Q_34^1/ , which implies that the next-to-next-to-nearest neighbour entangling gates are equivalent to 8/^2+2/ nearest-neighbour gates (8/^2 from the use of the Q_13's and another 2/ from the Q_34's). These calculations can be easily generalized to show thatnonlocal gates Q_a,a+1+n or Q_a+1+n,a, which entangle oscillators that are separated by n intermediate sites, can be constructed by the use of c(n)=2/1+c(n-1) nearest-neighbor entangling gates, where c(0)≡ 1. Thus to leading order, the “cost” of these nonlocal gates in terms of nearest-neighbour gates grows like c(n)∼1/^n.Following Neilsen's approach <cit.>, we would not eliminate these nonlocal gates from the elementary gate set, but would instead modify the geometry by introducing (heavy) penalty factors to discourage the geodesics from moving along the corresponding directions. The structure of eq. tsoc suggests increasing the penalty factors as a power law to match the growth of the nonlocality, the directions corresponding to I=(a,a+1+n) and (a+1+n,a) would be assigned a penalty factor ^2n. Note that this does not penalize the nearest-neighbour gates at all, in contrast to our experiment in section <ref>. Of course, for a periodic chain of oscillators, the maximum penalty factor would be ^N-2 and ^N-3 for even and odd N, respectively.We can gain further insight by translating eq. eq:Q13 into a macroscopic circuit described by a path-ordered exponential pathA. In particular, consider the following path: Y^23(s) =α [ 1-Θ(s-1) -Θ(s-2) +Θ(s-3)] , Y^12(s) =α [ Θ(s-1)-Θ(s-2) +Θ(s-3) -Θ(s-4) ] ,where 0≤ s≤4, Θ(x) is the Heaviside theta-function, and the Y^I are implicitly zero for all other values of I. That is, we turn on the M_23 generator with amplitude α for the interval 0≤ s≤ 1;M_12 is then turned on with amplitude α for 1≤ s≤ 2; next, M_23 is turned on with amplitude -α for 2≤ s≤ 3; andfinally M_12 is turned on with amplitude -α for 3≤ s≤ 4. Note that the precise parametrization of this path is not important. The circuitU_1(s)=𝒫 exp∫_0^xs̃ Y^I(s̃) M_Ithen yields U_1(s=4)=exp[α^2 M_13], following the same calculation that yields eq. eq:Q13. Hence we could accomplish the same transformation with U_2(s)=𝒫 exp∫_0^xs̃ Y^13(s̃) M_13 whereY^13(s)=α^2/4 for 0≤s≤4 .Now let us compare the costs of these two circuits using the F_q measure eq:Fmetrics, where the nearest neighbour gates are assigned cost 1 while the next-to-nearest neighbour gates are assigned cost . The cost functions are then easily evaluated to be D(U_1)=∫_0^4s √(δ_IJY^I(s)Y^J(s))=4α , D(U_2) = 4|Y^13|= α^2 . Hence with an appropriate penalty factor, we can suppress the use of the nonlocal gates in favour of the nearest neighbour gates.While it would be interesting to examine the effect of the above scheme of penalty factors in more detail, we leave this for future work. §.§ cMERA: The AdS/MERA correspondence was the first proposal for a novel connection between holography and tensor networks <cit.>. This proposal suggests that the MERA (Multiscale Entanglement Renormalization Ansatz)tensor network <cit.> provides a discrete representation of a time slice of (three-dimensional) AdS space. As illustrated in figure <ref>, the MERA network consists of unitary operators which, starting from the simple product state |0>⊗…⊗|0>, generate the ground state in d=2 critical systems.In other words, the MERA network can be thought of as a quantum circuit. The AdS/MERA correspondence was certainly a source of motivation/inspiration for the early discussions of holographic complexity, in particular, of the CV conjecture <cit.>. Furthermore, in these discussions, it was implicitly considered the optimal circuit for the preparation of the CFT ground state.There has been some progress towards developing a continuum version of MERA, however, these constructions are limited to describing very simple QFTs <cit.>—see also <cit.>. In particular, one example is the cMERA description of the ground state of a free scalar field. That is, there is a cMERA circuit which more-or-less performs precisely the transformation for which the circuits studied herein were constructed. Hence, our original expectation was that our analysis would find that the optimal circuit was something like a cMERA network. However, we instead found the straight-line circuit described in section <ref>. The key difference between the two circuits is that the cMERA circuit is organized to systematically introduce entanglement scale-by-scale, to order the amplification of the normal modes according to their wavelength <cit.>. However, by almost all of the measures considered in section <ref>, including the F_2 cost function and the κ cost functions in eq. Dalpha, the straight-line circuit is the optimal circuit. The one exception to this rule is the F_1 (or κ=1) cost function. This last describes an unusual geometry,[Let us add here that the F_1 measure also exhibits some unusual properties under a change of basis, as discussed in appendix <ref>.] which is sometimes called the “Manhattan metric.” The key feature of this geometry is that the length is the same for all paths as long as they do not back-track at any point, and hence the straight-line circuit and the cMERA circuit have identical costs for this measure. This question certainly deserves further study. It appears that there are two possible approaches: the first would be to study more exotic cost functions in order to identify those which favour the cMERA circuit. This may be useful since given the AdS/MERA duality, it may provide better insight into the properties of the cost function that appears in holographic complexity. We also mention that this is likely not a straightforward approach since we found that introducing penalty functions (in position space) seems to drive the optimal circuit out of the normal-mode subspace, whereas the cMERA circuit is confined to this particular slice of the full circuit geometry by construction. A second option might be to introduce new physics in the selection of the “optimal” circuit. That is, while the straight-line and cMERA circuits have equivalent costs according to the F_1 cost function, there may be additional physics considerations, some relation to renormalization group flows, which lead holography to favour a cMERA-like circuit. §.§ Future directions: This paper provides only a preliminary investigation towards understanding circuit complexity in quantum field theory. We already mentioned a number of future directions that we expect will be fruitful. Some examples include extending the present calculations to evaluate the complexity of excited states, producing a more concrete connection between the ambiguities arising in our QFT calculations and those in holographic calculations of the complexity, and studying in detail the effect of penalty factors on the complexity and the structure of the optimal circuit for a lattice of oscillators. Other obvious extensions of the present work would include evaluating the complexity in fermionic theories or in interacting QFTs. In closing, we would like to draw a comparison with entanglement entropy in QFT. Entanglement entropy has a simple textbook definition: first one must construct the reduced density matrix ρ_A of the particular subsystem under study, and then one evaluates the von Neumann entropy of this density matrix as S_EE=-∑λ_ilogλ_i, where λ_i are the eigenvalues of ρ_A. However, much of the progress in understanding the properties and role of entanglement entropy came from the replica trick, introduced by Calabrese and Cardy <cit.>. The latter applies familiar tools (path integrals) in a novel setting (the replicated background geometry) to evaluate the entanglement entropy. Returning to complexity, our present approach is to apply a more-or-less standard textbook definition to evaluating the complexity of states in a QFT, which is a useful preliminary step to gain an understanding of the properties of this new quantity. However, we would really like to develop a new approach, analagous to those developed for entanglement entropy above, which again uses familiar QFT techniques in a presumably novel setting to evaluate some quantity like the complexity. In other words, we are asking what is the new calculation of complexity which is the analog of Calabrese and Cardy's replica trick for entanglement entropy. Indeed, it may be that the first steps in this direction have already been taken in <cit.>—see also <cit.>.§ ACKNOWLEDGMENTSIt is a pleasure to thank Micha Berkooz, Eugenio Bianchi, Shira Chapman, Adrián Franco-Rubio, Lucas Hackl, Markus Hauru, Michal Heller, Qi Hu, Steve Jordan, Hugo Marrochio, John Preskill, Djordje Radicevic, Grant Salton, Joan Simon, Guillaume Verdon-Akzam, Guifré Vidal, Edward Witten, and Beni Yoshida for helpful conversations.Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research & Innovation. RCM is also supported by an NSERC Discovery grant, as well as research funding from the Canadian Institute for Advanced Research and from the Simons Foundation through the “It from Qubit" Collaboration. RJ is supported by the Δ-ITP consortium and the Foundation for Fundamental Research on Matter (FOM), both of which are parts of the Netherlands Organization for Scientific Research (NWO) funded by the Dutch Ministry of Education, Culture, and Science (OCW). RJ also gratefully acknowledges the support of the Perimeter Institute Visiting Graduate Fellows program. Finally, RCM would also like to thank the organizers of the It-From-Qubit “Complexity and Black Holes” workshop at Stanford University, the “Tensor Networks for Quantum Field Theories II” workshop at Perimeter Institute and the “Strings 2017” conference in Tel Aviv for the opportunity to present this work.1.5ex While this paper was in preparation, we were informed of <cit.>, which seems to have significant overlap with the present work. § EXAMPLE CIRCUITSIn this section, we analyze the circuit depth of a few discrete circuits, asintroduced in section <ref>. In particular, let us consider the example given in eq. (<ref>), ψ_T=U_1 ψ_R= Q_22^α_3 Q_21^α_2 Q_11^α_1 ψ_R ,where the target wave function is given in eq. eq:targetPhys and the reference wave function, in eq. eq:refPhys. The question then is to determine the exponents α_i in this equation, the number of times each type of gate is applied in the circuit.Intuitively, U_1 is a string of gates running from the right to the left. The first gate applied is Q_11, which rescales the coefficient of x_1^2 in the exponent of the Gaussian wave function. Next, by applying Q_21, the two oscillators become entangled. Finally, the application of Q_22 rescales x_2 to ensure that x_2^2 appears with the correct coefficient. Hence let us begin the quantitative analysis by considering: Q_11^α_1 ψ_0(x_1,x_2) =e^α_1/2ψe^α_1x_1,x_2=√(ω_0/π)e^α_1/2exp[-ω_1/2x_1^2-ω_0/2x_2^2] , where ω_1≡e^2α_1ω_0α_1=1/2logω_1/ω_0 .Next, applying the Q_21 gates yields Q_21^α_2 Q_11^α_1 ψ_0(x_1,x_2)=√(ω_0/π)e^α_1/2e^ix_2p_1exp[-ω_1/2x_1^2-ω_0/2x_2^2] =√(ω_0/π)e^α_1/2exp[-ω_1/2x_1+α_2 x_2^2-ω_0/2x_2^2] =√(ω_0/π)e^α_1/2exp[-ω_1/2x_1^2-1/2ω_0+^2α_2^2ω_1x_2^2-α_2ω_1 x_1x_2] . Note that the x_1x_2 cross-term will be rescaled in the next step, so we cannot fix any of the coefficients quite yet. Finally, we rescale x_2 with the Q_22 gates: Q_22^α_3 Q_21^α_2 Q_11^α_1 ψ_0(x_1,x_2) =√(ω_0/π)e^(α_1+α_3)/2exp[-ω_1/2x_1^2-ω_2/2x_2^2-βx_1x_2] , where α_2 and α_3 are determined by matching the second and third coefficients in the exponent, ω_2=ω_0+^2α_2^2ω_1e^2α_3 ,β≡α_2ω_1e^α_3 . Solving the above constraints then yields α_2=1/√(ω_0/ω_1)β/√(ω_1ω_2-β^2) ,α_3=1/2logω_1ω_2-β^2/ω_0 ω_1 .As a consistency check, note that with these identifications, the normalization factor of the final wave function becomes √(ω_0/π)e^(α_1+α_3)/2=ω_1ω_2-β^2^1/4/√(π) , which correctly preserves the unit norm. Of course, this was expected since, as discussed in the main text, the entangling and scaling gates, Q_ij and Q_ii, preserve the norm when acting on Gaussian wave functions.Hence the total number of gates in the circuit U_1 in eq. exam1 is given by𝒟(U_1)=|α_1|+|α_2|+|α_3| =1/2logω_1ω_2-β^2/ω_0^2+1/√(ω_0/ω_1) |β|/√(ω_1ω_2-β^2) , where we have assumed here that ω_1>ω_0 and ω_1ω_2-β^2>ω_0^2.[See further comments at the end of this appendix.] As in the main text, we refer to 𝒟(U_1) as the circuit depth, rather than the complexity, since while it counts the total number of gates in the circuit, we have no reason to expect that U_1 is the optimal circuit. Recall that we introduced absolute values in eq. eq:Deg1Phys in order to give an equal complexity cost for the inverse gates Q_ij^-1 as for the original gates Q_ij, we count the appearance of Q_ij^-1 as one gate in a circuit. At a pragmatic level, this is required because α_2 is negative in our example, β=(ω_+-ω_-)/2<0. Note that in evaluating the exponents α_i in eqs. exp1 and exp2, we are implicitly treating them as real numbers. If we insisted on having integer exponents, then we would would need to round these results up or down to the nearest integer. In this case, we would define a measure of success for our transformation and choose the integer exponents to maximize this measure. For example, we could consider the overlap |∫d^2x ψ_T^† Q_22^α_3 Q_21^α_2 Q_11^α_1 ψ_R|^2 =1-χ ,and choose the precise integer values of α_i to minimize χ. Of course, using real exponents α_i is very much in line with describing the circuits in terms of path-ordered exponentials eq:pathPsi. This discussion is related to the choice of a tolerancein eq. scotch, rather than minimizing χ, one might demand that χ≤.Now let us briefly present a few other examples of simple circuits to further familiarize the reader with the concepts discussed here. First, let us consider applying the entangling gate before either of the scaling gates: ψ_T=U_2ψ_R= Q_22^α̃_3 Q_11^α̃_1Q_21^α̃_2 ψ_R . Note that for comparison purposes, our numbering of the exponents is such that they are associated with the same gates as appear in eq. exam1. The calculation proceeds essentially as above; in the end, we must match the coefficientsω_1=ω_0e^2α̃_1 ,ω_2=1+^2α̃_2^2ω_0 e^2α̃_3 ,β=α̃_2ω_0 e^(α̃_1+α̃_3) . Solving for the exponents α_i then yields α̃_1=1/2logω_1/ω_0 ,α̃_2=1/β/√(ω_1ω_2-β^2) ,α̃_3=1/2logω_1ω_2-β^2/ω_0ω_1 ,and hence the circuit depth becomes 𝒟(U_2)=∑|α̃_i|= 1/2logω_1ω_2-β^2/ω_0^2+1/|β|/√(ω_1ω_2-β^2) . Comparing the results in eqs. exp1 and exp2 with those in eq. exp3, we see that the exponents for the scaling gates are identical, α_1=α̃_1 and α_3=α̃_3, and only the exponent for the entangling gate has changed. Hence the circuit depth is almost identical to (<ref>), except that the second term lacks the factor √(ω_0/ω_1). If we assume ω_1>ω_0 as before, this implies that the present circuit will be slightly longer, 𝒟(U_2)>𝒟(U_1).As a third simple example, let us consider instead applying the entangling gate after both of the scaling gates: ψ_T=U_3ψ_R= Q_21^α̂_2 Q_22^α̂_3Q_11^α̂_1ψ_R .Again we skip over the details of the calculation; we find that we must match the coefficientsω_1=ω_0e^2α̂_1 ,ω_2=e^2α̂_3+^2α̂_2^2 e^2α̂_1ω_0 , β=α̂_2ω_0 e^2α̂_1 . Solving for the exponents α_i then yields α̂_1=1/2logω_1/ω_0 ,α̂_2=1/β/ω_1 , α̂_3=1/2logω_1ω_2-β^2/ω_0ω_1 ,and hence the circuit depth becomes 𝒟(U_3)=∑|α̃_i|= 1/2logω_1ω_2-β^2/ω_0^2+1/|β|/ω_1 . Again, comparing with the exponents in eqs. exp1 and exp2 or in eq. exp3, we see that only the exponent for the entangling gate has changed. Hence the circuit depth here is similar to those for the previous two circuits, and whether the present circuit is longer or shorter depends on the values of the parameters ω_0, ω_1, ω_2 and βLet us consider one more general example. Another interesting circuit would be ψ_T=U_4ψ_R= Q_22^α̅_3 Q_21^α̅_2(Q_21^-1 Q_11)^α̅_1ψ_R .Note that (Q_21^-1 Q_11)^nψ(x_1,x_2) =e^n /2ψe^nx_1-e^1-e^n/1-e^ x_2,x_2 , the derivation of which is as follows: first, consider Q_11ψ=e^/2ψe^x_1,x_2 Q_21^-1Q_11ψ=e^/2ψe^x_1-e^x_2,x_2 . Then acting with this combination twice yields Q_21^-1Q_11^2ψ =e^2/2Q_21^-1ψe^2 x_1-e^ x_2,x_2=e^2/2ψe^2x_1-x_2-e^ x_2,x_2 =e^2/2ψe^2 x_1-e^e^+1x_2,x_2 . And a third time: Q_21^-1Q_11^3ψ=e^3/2Q_21^-1ψe^3 x_1-e^e^+1x_2,x_2=e^3/2ψe^3 x_1-e^e^2+e^+1x_2,x_2 . Now the pattern is clear, and we deduce Q_21^-1Q_11^nψ=e^n/2ψe^n x_1-e^∑_k=0^n-1 e^k x_2,x_2 . Since ∑_k=0^n-1e^k=1-e^n/1-e^ , this becomes (<ref>), as claimed.Now, acting with the circuit U_4 and matching coefficients as before, we find α̅_1 =1/2logω_1/ω_0 ,α̅_2 =1/√(ω_0/ω_1)β/√(ω_1ω_2-β^2)+ 1/e^--11-√(ω_0/ω_1) ,α̅_3 =1/2logω_1ω_2-β^2/ω_0ω_1 . Expanding α̅_2 near ≈0, we have α̅_2 =1/√(ω_0/ω_1)β/√(ω_1ω_2-β^2)-1/+1/2+1-√(ω_0/ω_1)+ =1/[√(ω_0/ω_1)1+β/√(ω_1ω_2-β^2)-1]-1/21-√(ω_0/ω_1)+() . We therefore find that the circuit depth for U_4 is 𝒟U_4=1/2logω_1ω_2-β^2/ω_0^2+|1/[√(ω_0/ω_1)1+β/√(ω_1ω_2-β^2)-1] -1/21-√(ω_0/ω_1)+()| . Here, as above, we assume ω_0<ω_1.In general, we can describe the form of the circuit depth as being an overall factor of 1/ϵ followed by a coefficient determined by the various physical parameters characterizing the target state and the reference state. More generally, the circuit depth might be given by an expansion in ϵ, beginning with a 1/ϵ term followed by a finite term and then potentially terms involving positive powers of ϵ. However, since ϵ≪1,determining the complexity essentially requires finding the circuit which minimizes the coefficient of the leading 1/ϵ term. For comparison to the results of the geometric approach in the main text, it is useful to express the present results in terms of the normal-mode frequencies via eq. eq:omega12pm. If we focus our attention on the first circuit U_1 in eq. exam1, the exponents given in eqs. exp1 and exp2 become α_1=1/2log_++_-/2ω_0 , α_2=-1/√(ω_0/_++_-)_–_+/√(2_+_-) ,α_3=1/2log2 _+_-/ω_0 (_++_-) .As was alluded to above, to proceed further we must decide on the value of the reference frequency ω_0 relative to the normal-mode frequencies. Given the discussion in section <ref>, there are two natural hierarchies to consider: (i)_+<_-<ω_0 or (ii) ω_0<_+<_-.[Implicitly, we chose the second hierarchy above in presenting our results in eqs. eq:Deg1Phys, horse0, and horse.] Of course, the ordering of the normal-mode frequencies is fixed and we are really only choosing ω_0 here. In particular, in the first (second) hierarchy, ω_0 is a UV (IR) frequencylarger (smaller) than any physical frequency in the coupled oscillator problem. Note that in the first case, all three exponents are negative, while in the second case, α_1,α_3>0 and α_2<0. Evaluating 𝒟(U_1)=|α_1|+|α_2|+|α_3| in these two cases yields: 𝒟(U_1)=1/√(ω_0/2 _++ ω_0/2 _-) _--_+/_++_-+1/2 ×{ logω_0^2/_+_-forω_0>_+,_- ,log_+_-/ω_0^2 forω_0<_+,_- ..Recall that _->_+ from eq. qm3. We may now compare this result with those derived using the geometric approach of section <ref>. In particular, if we recall the F_1 measure given in eq. (<ref>), the complexity would be given by C=1/2|log_+/ω_0|+1/2|log_-/ω_0|={ 1/2logω_0^2/_+_-forω_0>_+,_- ,1/2log_+_-/ω_0^2 forω_0<_+,_- ..Furthermore, recall that we should compare this with the coefficient of the 1/ϵ factor in the discrete calculations (see footnote <ref>). Hence we see the second contribution in eq. jjj precisely matches the complexity above. However, there is an additional positive term in 𝒟(U_1), and therefore we see that – at least by the F_1 measure – U_1 is not the optimal circuit. We can also describe this circuit as a trajectory in the language of the path-ordered exponentials eq:pathPsi. In this case, U_1 as given in eq. exam1 becomes 0≤s≤|α_1|/𝒟(U_1):Y^11= 𝒟(U_1) , Y^22=Y^12=Y^21=0 , |α_1|/𝒟(U_1)≤s≤|α_1|+|α_2|/𝒟(U_1) :Y^11= Y^22= Y^12=0 , Y^21=𝒟(U_1) , |α_1|+|α_2|/𝒟(U_1)≤s≤1: Y^11= 0 , Y^22=𝒟(U_1) , Y^12=Y^21=0 . This form makes clear that the circuit consists of three separate “straight” segments, and so U_1 does not correspond to a geodesic path or an optimal circuit.§ KILLING VECTORS AND MORE GEOMETRY Inspecting the metric in eq. metric1, we can see three obvious Killing coordinates: y, τ, θ. When the penalty factors were introduced in section <ref>, we found that this is reduced to two Killing coordinates, y and z=(θ-τ)/2, in the geometry described by eqs. penalty1 or penalty2. However, by construction, all of these metrics are right-invariant, and hence the corresponding geometries must have one Killing vector for each generator Msimple, namely, four.[We thank Lucas Hackl for discussions on this point.] Furthermore, as we will see below, the structure of these Killing vectors will be completely independent of the particular choice of G_IJ appearing in the metric (assuming it is a constant matrix), but rather is determined by the structure of eq. eq:metricRightInv.One way to think of a Killing vector k^i is as providing a coordinate transformation x^i →x^i+ k^iwhich leaves the geometry or line element invariant. (Noteis just an infinitesimal parameter.) For example, eq. metric1 is certainly invariant under δτ = and so we write the corresponding Killing vector as k^i∂_i = ∂_τ or k^i =δ^i_τ. So let us identify the coordinate transformations which generally leave eq. eq:metricRightInv invariant. For a general coordinate shift in eq. Umatrix, we have δU = ∂_i U δx^i.As long as G_IJ is a constant matrix, all of the coordinate dependence is hidden in the one-forms U(s) U^-1(s)M^T_I, eq. penalty1. However, it is clear that these expressions are invariant if we right-multiply U by a globaltransformation. Hence let us make the infinitesimal transformation: U→ U exp[^I M_I], where the ^I are (infinitesimal) constants. To leading order in these parameters, this reduces to δU = U M_I ^I .Equating eqs. shift1 and shift2, we have U M_I^I=_iUδx^i^I=U^-1_i U M_I^T δx^i ,where we have assumed that we are working with the orthogonal basis of generators satisfying M_I M_J^T=δ_IJ, eq. Msimple. We now observe that, since the argument of the trace contains two free indices, we may view this object as a 4×4 matrix, which we can then invert to obtain δx^i=[U^-1_i U M_I^T]^-1^I =k_I^i^I . Thus we obtain four independent Killing vectors k_I= k_I^i∂_i.Given our basis of generators in eq. Msimple and our parametrization of the circuit space in eq. (<ref>), we can easily compute k_I^i.We can then identify the Killing vectors by simply reading off this matrix row-by-row: k_1= 1/2_y-1/2sin(2z)_ρ-cos(2 z)/2sinh(2 ρ)_x-cosh(2ρ)/2sinh(2 ρ)cos(2 z) _z ,k_2= 1/2 cos(2 z)_ρ-sin(2 z)/2sinh(2 ρ) _x+1/2 (1-cosh(2ρ)/sinh(2ρ)sin(2z) )_z ,k_3= 1/2 cos(2 z)_ρ-sin(2 z)/2sinh(2 ρ)_x-1/2 (1+cosh(2ρ)/sinh(2ρ)sin(2z) )_z ,k_4= 1/2_y+1/2sin(2z)_ρ+cos(2 z)/2sinh(2 ρ)_x+cosh(2ρ)/2sinh(2 ρ)cos(2 z) _z ,where we are using the pseudo-lightcone coordinates of eq. (<ref>), with θ=x+z, τ=x-z. One can explicitly verify that these indeed satisfy the Killing equations,0=∇_i k_I_j+∇_jk_I_i =g_jℓ∇_i+g_iℓ∇_jk_I^ℓ , for either of the metrics in eqs. metric2 or penalty2. However, it is clear that eq. Killed does not organize the Killing vectors in the simplest way, so we define: k̂_1≡k_1+k_4 = _y ,k̂_2≡ -k_1+k_4=sin(2z)_ρ+cos(2 z)/sinh(2 ρ)_x+cosh(2ρ)/sinh(2 ρ)cos(2 z) _z ,k̂_3≡k_2+k_3 = cos(2 z)_ρ-sin(2 z)/sinh(2 ρ)_x-cosh(2ρ)/sinh(2ρ)sin(2z) _z ,k̂_4≡k_2-k_3 =_z . However, a simple inspection of the first metric (<ref>) reveals that _x is also an independent Killing vector, hence: k̂_5≡_x . This is an accidental symmetry that emerges with the choice G_IJ=δ_IJ.[We thank Lucas Hackl for discussions on the Killing symmetries.] However, as noted above, the four Killing vectors in eq. Killed2 apply for any (constant) choice of G_IJ.Of course, the existence of the above Killing vectors implies that there are an equal number of conserved momenta or charges which distinguish the geodesics, c_I≡ (k̂_I)^i g_ij ẋ^j. We make use of these momenta in solving for the optimal circuits in sections <ref> and <ref>. §.§ AdS_3 geometry:In section <ref>, we noted the appearance of a three-dimensional anti-de Sitter geometry in discussing the parametrization of U∈= ℝ×SL(2,ℝ).Of course, the appearance of AdS_3 is natural since it is the universal cover of the SL(2,ℝ) subgroup. Here we would like to show how the AdS_3 geometry can be realized using the formalism introduced in section <ref>. In particular, we consider the geometry that results from the choiceG_IJ=[ 1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0 ] =η_IJ .We have designated G_IJ=η_IJ because the I=3={12} and I=4={21} directions are null and hence the metric has a Minkowski signature. With this choice, eq. metric1 is replaced with s^2 =η_IJ U(s) U^-1(s)M^T_I U(s) U^-1(s)M^T_J =2y^2+2ρ^2-2cosh^2ρ τ^2+2sinh^2ρ θ^2 .Hence we have produced precisely the AdS_3×ℝ geometry anticipated in section <ref>. Eq. adds describes the natural group invariant metric for , the left- and right-invariant metric, whereas the Euclidean metric metric1 is a less symmetric metric with only right-invariance. However, the Lorentzian signature is undesirable for the problem of circuit complexity, since pieces of the circuit that correspond to null-geodesics have zero length, zero cost. This would allow the construction of arbitrarily low-complexity circuits simply by deforming the circuit along the null directions.[Of course, moving in a timelike direction also yields a negative cost.] §.§ Alternate basis of generators:The basis of matrix generators in eq. Msimple is natural in the sense that it straightforwardly extends from the problem of complexity in the case of two coupled harmonic oscillators to the case of N coupled oscillators. However, this is not the most convenient basis for certain calculations in section <ref>. Hence for the interested reader, we describe here an alternate basis of generators which simplifies some of the calculations. In particular, consider the Pauli-like basis: M_1 =1/√(2)[ 1 0; 0 1 ]=1/√(2) 1 ,M_2=1/√(2)[10;0 -1 ]=1/√(2) σ_3 ,M_3 =1/√(2)[ 0 1; 1 0 ]=1/√(2) σ_1 ,M_4=1/√(2)[01; -10 ]=-i/√(2) σ_2 .The normalization of the generators is still given by (M_I M_J^T)=δ_IJ. In fact, the new generators are easily related to the original generators in eq. Msimple by an orthogonal transformation: M_I = R_I^JM_J with R_I^J∈ O(2)× O(2)∈ O(4). In this new basis, the M_2,3,4 generators naturally form the sl(2,ℝ) subalgebra, with [_2,_3]=√(2) _4 ,[_2,_4]=√(2) _3 ,[_3,_4]=-√(2) _2 , while M_1 describes the remaining fibre over ℝ in thegroup.With this new basis, the Killing vectors which emerge from the right-invariance of the metric naturally appear in the form given in eq. Killed2.One can easily show that working with these new generators, the metrics appearing in eqs. metric1 and penalty1 are unchanged, the corresponding G_IJ are left unchanged by the rotation R_I^J introduced above. Additionally, the AdS_3 geometry in eq. adds now results from the choice G_IJ=η_IJ=diag(1,1,1,-1).§ NORMAL-MODE FREQUENCIES _KThe derivation of the normal-mode frequencies in eq. (<ref>) – or eq. freakd for a lattice of coupled oscillators – is straightforward, and can be found in a number of different sources, any elementary condensed matter textbook. For completeness, we briefly review the result (<ref>) for the periodic one-dimensional lattice discussed in section <ref>. Essentially, we need only apply the inverse Fourier transformx_a≡1/√(N)∑_k=0^N-1exp2πi k/N ax̃_k ,to re-express the Hamiltonian qm88 in terms of the normal modes, qm288. In particular, we focus on the potential V=1/2∑_a=0^N-1[ω^2x_a^2+Ω^2x_a-x_a+1^2] . Considering the second term involving the coupling between the oscillators, we findΩ^2∑_a=0^N-1 x_a-x_a+1^2=Ω^2∑_a=0^N-1[1/√(N)∑_k=0^N-1exp2π ik a/Nx̃_k1-exp2π ik/N]^2 =Ω^2/N∑_a,k,k'exp2π i(k+k')a/Nx̃_kx̃_k'1-exp2π ik/N1-exp2π ik'/N =Ω^2∑_kx̃_kx̃_-k1-exp2π ik/N1-exp-2π ik/N=2Ω^2∑_kx̃_kx̃_-k1-cos2π k/N =4Ω^2∑_k|x̃_k|^2 sin^2π k/N ,where in going to the third line we applied the normalization condition (<ref>), and in the last step we used x̃_kx̃_-k=x̃_kx̃^†_k. Here all sums run from 0 to N-1. The Fourier transform of the first term in the potential (<ref>) is trivial, and thus we find V=1/2∑_k=0^N-1[ ω^2+4Ω^2 sin^2πk/N]|x̃_k|^2 =1/2 ∑_k=0^N-1_k^2 |x̃_k|^2 . Hence we have identified the desired normal-mode frequencies, _k^2=ω^2+4Ω^2 sin^2πk/N ,(<ref>). If instead we were examining a d-dimensional free scalar field, the lattice would be extended to d-1 (spatial) dimensions, whereupon the corresponding normal-mode frequencies become _k⃗^2=ω^2+4Ω^2 ∑_i=1^d-1sin^2πk_i/N ,where k_i are the components of the momentum vector k⃗=(k_1,k_2,⋯,k_d-1). Implicitly, we have assumed here that the lattice is square with periodic boundary conditions in each direction.§ CHANGE OF BASISIn this appendix, we would like to extend the discussion around eq. bark2 describing the change of bases for the case of two coupled oscillators to the analogous transformation for a lattice of oscillators. In particular, we will focus on the case of a one-dimensional lattice of N oscillators, although it is straightforward to extend the discussion to a lattice extending in d-1 (spatial) dimensions.This transformation is particularly relevant in section <ref>, where we presented a tentative argument that the metric on the normal-mode subspace is flat, s^2_n-m=|ỹ_0|^2+|ỹ_1|^2+⋯+|ỹ_N-1|^2 ,(<ref>). However, we also noted that, at the time, this conclusion was somewhat premature, since implicitly we applied eq. metric5, which defines the metric in the position basis, to a calculation with the diagonal circuit raffle4 written in the normal-mode basis. That is, in eq. metric5, the indices I, J run over pairs of position labels (ab), and implicitly the generators act on Gaussian wave functions written in terms of coordinates x_a. In contrast, in eq. raffle4, we would write M̃_n-m=Ỹ^ĨM̃_Ĩ where the tilde on the index Ĩ indicates that it runs over pairs of momentum labels (kℓ), and the tilde on M indicates that these generators act on Gaussian wave functions written in terms of the normal coordinates x̃_k. In other words, in eq. raffle4, where we are restricting our attention to the normal-mode subspace, we are considering the diagonal generators Ỹ^kℓ=δ^kℓ ỹ_k. Hence to show that the result metric6xx is correct, we must take care to translate between the two bases of generators discussed above. As in eqs. nmU and straight9, we can transform from generators acting in the normal-mode basis to those in the position basis via[Note that this transformation removes the tilde from M but not from the index. For example, the new generator M_kk acts on Gaussian wave functions written in terms of the oscillator position coordinates x_a, but still has the effect of scaling the kth normal mode x̃_k.]M_Ĩ = ^†M̃_Ĩ .Implicitly, the normal-mode generators M̃_Ĩ have the same form as that given in eq. matrix, namely [M̃_kℓ]_pq=δ_kpδ_ℓq ,where we have denoted Ĩ=(kℓ) with momentum labels k, ℓ. Similarly p, q are the row and column indices, respectively, of the N× N matrix, which also take values as momentum labels (since the generator acts in the normal-mode space). Now let us combine these two equations to write[Note that the complex conjugation appears on the first factor in =^*⊗ because our convention is that written in terms of the normal modes, the Gaussian wave functions involve both x̃_k and x̃_k^†, the appearance of |x̃_k|^2 in eq. targetk.] [M_kℓ]_ab=[^†]_ap [M̃_kℓ]_pq []_qb =[^†]_a k []_ℓb=[^†]_c k []_ℓd [M_cd]_abM_Ĩ = []_ĨJM_J with=^*⊗ .In going from the second to third line, we used eq. matrix and identified Ĩ=(kℓ) and J=(cd). This equation generalizes eq. bark2 fromin the previous section to the case ofstudied here. Furthermore, given the properties of , one can easily see that the matrixis a unitary matrix. Hence we can invert the transformation in eq. lot2 to write M_I =[^†]_IJ̃M_J̃. Similarly, we can invert the transformation in eq. lot0, transform from generators acting in the position basis to the normal-mode basis via M̃_ I= M_I ^† ,and combine this expression with eq. matrix, [ M_ab]_cd =δ_acδ_bd, to write [M̃_ab]_kℓ=[]_kc [M_ab]_cd [^†]_dℓ =[]_ ka [^†]_bℓ=[]_pa [^†]_bq[M̃_pq]_kℓM̃_I = [^†]_IJ̃M̃_J̃ with^†=^T⊗^† .In going from the second to third line, we used eq. lot1 and identified I=(ab) and J̃=(pq). As before, we can easily invert the transformation in eq. lot2a to write M̃_Ĩ =[]_Ĩ J M̃_J. As our notation indicates,is precisely the unitary matrix appearing in eq. lot2, and hence it also plays a role in transforming the generators acting in the normal-mode space. Hence by using the special structure of the generators in eqs. matrix and lot1, we have re-organized thetransformation acting on the matrix indices in eqs. lot0 and lot0a to a transformation acting on the generator labels in eqs. lot2 and lot2a, respectively.With these tools in hand, let us consider re-expressing the cost function cost5 or the metric metric5 in terms of the normal-mode basis, using eq. lot0. Here, we show the calculation for the metric; the transformation of the cost function follows in a similar manner. Beginning with the differential Y^I= U U^-1M^†_I defined in eq. metric5, we transform the circuit to the normal-mode space via U=^† Ũ, which yieldsY^I=ŨŨ^-1 M^†_I^†=ŨŨ^-1 M̃^†_I =[]_ĨI Ỹ^Ĩwhere Ỹ^Ĩ=Ũ Ũ^-1 M̃^†_Ĩ, and we have employed eqs. lot0a and lot2a in the second and third equalities. Hence the metric metric5 transforms as s^2=[]_ĨI δ_IJ[^†]_J J̃Ỹ^Ĩ(Ỹ^J̃)^* =δ_Ĩ J̃Ỹ^Ĩ(Ỹ^J̃)^*.Note that here we are using the fact thatis a unitary matrix. Thus we have found that the metric takes precisely the same form whether expressed in terms of the oscillator position space or the normal-mode space.[The fact that this transformation preserves the cost function essentially follows from the Plancherel theorem, which states that the Fourier transform preserves the L^2 norm. We thank Adrián Franco-Rubio for a discussion on this point.] Of course, the same is true of the cost function cost5, it can also be written as 𝒟(U)=∫_0^1s√(δ_ĨJ̃ Y^Ĩ(s) ( Y^J̃(s))^*) , whereY^Ĩ(s)=_sŨ(s) Ũ^-1(s)M̃^†_Ĩ.Note that this transformation is slightly different than that expressed in eq. metric1a for the metric for two coupled oscillators. In the latter case, we are considering the metric to still be in the position basis but evaluated with a different basis of generators. The same invariance holds here for a lattice of oscillators, as can be seen by applying eq. lot2 directly to the metric metric5 to produce s^2=δ_Ĩ J̃ Y^Ĩ(Y^J̃)^* , whereY^Ĩ=U U^-1M^†_Ĩ .Of course, the same change of basis could also be performed with eq. lot2a when working in the normal-mode space. §.§ General cost functionsIn eq. Dalpha, the κ cost functions were defined with a sum over the components of the velocity Y^Ĩ in the normal-mode basis. Here we would like to apply the techniques developed above to examine the differences that arise from using the original oscillator position basis. That is, we could equally well define cost functions with 𝒟_κ=∫_0^1s∑|Y^I(s)|^κ . In the discussion of the F_2 cost function in the previous section, we found that this change of basis had no effect on the complexity; but here we will find that, in fact, the complexity is not basis independent. As a simple example, let us consider the case of two coupled oscillators for which the optimal circuit U_0(s) appears in eq. solver5, for which the velocity components in the position basis become Y^11=Y^22=y_1 ,Y^12=Y^21=-ρ_1 .These two factors are written in terms of the normal-mode frequencies in eq. fin2, but we can re-express these results as y_1=1/4(log_-/ω_0+log_+/ω_0) ,ρ_1=1/4(log_-/ω_0-log_+/ω_0) .Recall that _->_+, but in the following, we also assume that _±>ω_0, which ensures that both y_1 and ρ_1 are positive quantities. Now we evaluate the cost of U_0 using eq. DalphaP for a few values of κ,[In the case that ω_0>_±, one should replace _±/ω_0→ω_0/_∓ in these formulae. Note this substitution only really changes the results for odd κ.] 𝒟_κ(U_0) =2y_1^κ+2ρ_1^κ= log_-/ω_0 for κ=1 ,1/4(log^2_-/ω_0+log^2_+/ω_0) for κ=2 ,1/16(log^3_-/ω_0+3log_-/ω_0 log^2_+/ω_0) for κ=3 ,1/64(log^4_-/ω_0+log^4_+/ω_0+6log^2_-/ω_0 log^2_+/ω_0) for κ=4 . Hence we see that it is only for κ=2 that we reproduce the cost found using eq. Dalpha in the normal-mode basis, 𝒟_κ(U_0)≃log^κ(_-/ω_0)+ log^κ(_+/ω_0).These differences in the cost can be understood using the approach developed to implement a change of basis for a lattice of oscillators in section <ref>. In particular, transforming from the position basis to the normal-mode basis can be described in terms of the unitary matrixdefined in eq. lot2. Given the definition of the velocity components in eq. cost5, we then have Y^Ĩ=[^*]_Ĩ J Y^J ,or, inverting this expression, Y^I=[^T]_IJ̃ Y^J̃. Furthermore, recall that the quadratic construction δ_IJ Y^I (Y^J)^*= δ_ĨJ̃ Y^Ĩ (Y^J̃)^* is invariant under this change of basis. Therefore the cost evaluated with the F_2 or κ=2 cost functions are invariant as well. However, this discussion also makes clear that if we include penalty factors, then these quadratic cost functions are no longer invariant. That is, the penalty factors introduce a more general metric G_IJ, which transforms nontrivially under the change of basis, G_ĨJ̃= []_J̃ J G_IJ [^†]_IĨ ,where we assumed symmetry of the metric G_IJ=G_JI.This also suggests how we should treat the more general κ cost functions. We should generalize eq. DalphaP to allow for general penalty factors by writing 𝒟_κ=∫_0^1s∑_I_1,I_2,⋯,I_κ G_I_1I_2⋯I_κ|Y^I_1(s)||Y^I_2(s)|⋯|Y^I_κ(s)| , where G_I_1I_2⋯ I_κ is a symmetric tensor with κ indices.In eq. Dalpha, we are implicitly considering simple “penalty” tensors of the form G_I_1I_2⋯I_κ = δ_I_1I_2 δ_I_2I_3⋯δ_I_κ-1 I_κ forκ≥2 , G_I = 1forκ=1 .In general, it is clear that the unitary transformation will not leave these penalty tensors (or more general choices) invariant. This simply reflects the fact that in choosing different gates, we are treating different gates as fundamental and that in general, we expect the results for the complexity to depend on the choice of the elementary gate set.Of course, this does not mean that the complexity must be evaluated in one particular basis. However, if the cost function is fixed with a certain choice of basis, then changing the basis requires that we properly transform the cost function to the new basis. To gain a better understanding of this situation, let us investigate the case of κ=1 in more detail. In addition to the simplicity of this case, recall that this was also the cost function favoured in the comparison to holographic complexity in section <ref>.Let us begin with the case of N=2, in which case the transformation matrix =_2 takes the simple form given in eq. bark2. For κ=1, the penalty tensor tens2 becomes the four component vector G_I=(1,1,1,1) ,which is actually an eigenvector of . Hence if we transform as in eq. chan1, we find the rather surprising result that G_Ĩ = _Ĩ JG_J= (2,0,0,0) .That is, expressing our κ=1 cost function DalphaP in terms of the normal-mode basis, we are only penalizing the scaling gateassociated with x̃_+! The other (normal-mode) gates can be inserted in the circuit at zero cost. However, we must add that the transformation in eq. tens3 is slightly naive since it assumes that the absolute values in the cost function DalphaP play no role, we are assuming that all Y^Ĩ≥0 (or all Y^Ĩ≤0). However, one finds that, depending on the signs of the various velocity components, only one of the normal-mode gates is penalized at a time. For example, with Y^Ĩ≥0 for Ĩ=++,– and Y^Ĩ≤0 for Ĩ=+-,-+,[This is the case in eq. veloc for the optimal circuit with _±>ω_0. Hence in eq. trials, the κ=1 cost function only depends on _-.] one finds G_Ĩ = (0,0,0,2), only the scaling gate associated with x̃_- is penalized. Similar results arise if we begin with the κ cost functions Dalpha expressed in terms of the normal-mode basis and examine their structure in the position basis. In this case for κ=1, the original and transformed penalty tensors becomeG_Ĩ=(1,1,1,1)⟶G_I = ^T_IJ̃G_J̃= (2,0,0,0) .Hence we have the rather curious result that this cost function is only penalizing the scaling gate associated with x_1, the position of the first oscillator. Of course, we must again remind the reader that eq. tens5 assumes that the absolute values in the cost function Dalpha play no role. This assumption is more natural in this case, as with a natural choice of ω_0 we find that all Y^Ĩ≥0 (or all Y^Ĩ≤0) for the optimal circuit, all of the scaling components have a definite sign and all components in the entangling directions vanish. Furthermore, using eq. Dalpha, we might note that the cost of our straight-line circuit is simply 𝒟_κ=1(U_0) =ỹ_++ỹ_-=1/2log_+/ω_0 + 1/2log_-/ω_0 ,again assuming _±>ω_0. Here we emphasize that since only two of the velocity components were non-vanishing, namely Y^++ and Y^–, we would arrive at the same cost for a family of penalty tensors of the form G_Ĩ=(1,_1^2,_2^2,1) .In this case, transforming to the position basis as in eq. tens5 yields G_I = ^T_IJ̃G_J̃=1/2 (2+_1^2+_2^2, _2^2-_1^2, _1^2-_2^2, 2-_1^2-_2^2) .At first sight, this result seems to yield a more reasonable penalty tensor relative to eq. tens5. However, upon closer examination, we see that G_12=-G_21, and hence one of these penalty factors will be negative. That is, the cost of the circuit will be reduced by including more of one type of the entangling gates in the normal-mode basis! The only resolution of this unsatisfactory situation is to set the two penalty factors equal, _1=_2=, whereupon eq. tens7 becomes G_I = (1+^2, 0, 0, 1-^2) ,which still requires that ^2≤1 in order that G_22≥0.The above results are somewhat unsatisfying, in that a perfectly reasonable penalty tensor in one basis yields an undesirable or even inconsistent (in the case of negative penalty factors) cost function in another basis. We return to this point in the discussion in section <ref>; however, we should say that some of these issues arise because we focused on the simple case of κ=1. For example, if we consider instead the κ=2 cost function DalphaP with our penalized metric pen99, then transforming to the normal-mode basis yields eq. pen99a. While the resulting metric has negative entries, we know that this in itself is not worrisome. Rather, one must examine the eigenvalues of the new metric, and since these have not been changed by the transformation, all remain positive. To close this section, let us comment on extending this discussion to a lattice of oscillators.In particular, we observe that the essential features of the complexity noted in section <ref> using the κ cost functions Dalpha constructed in the normal-mode basis remain unchanged when working with eq. DalphaP in the position basis. For a (d-1)-dimensional spatial lattice of oscillators, the κ=1 penalty tensor in eq. tens2 becomes G_Ĩ=(N^d-1,0,0,⋯,0)in terms of the normal modes. Hence as in eq. tens4, only the scaling gate of lowest normal mode is penalized, but the cost of that single gate has been increased to N^d-1, the total number of oscillators in the lattice.[Note that in eq. tens4, the penalty associated to the M_++ was increased to 2, the number of oscillators.] The cost for the straight-line circuit then becomes D_κ=1(U_0)=1/2 N^d-1 |logm/ω_0|=V/2 δ^d-1 |logm/ω_0| .Hence the cost is still proportional to V/δ^d-1, as desired to emulate the holographic complexity. This factor is again multiplied by a logarithmic factor whose argument depends on the reference frequency ω_0. However, since only the lowest eigenfrequency ω̃_k⃗=0=m appears in the (single) logarithmic factor, the cut-off scale can only appear in this result through the reference frequency ω_0. Hence δ appears if ω_0 is chosen as a UV frequency, ω_0=e^-σ/δ, but it does not appear if the reference frequency is chosen as an IR frequency. We observe that this is the opposite of the situation discussed in section <ref>.§ APPROXIMATING THE COMPLEXITY In section <ref>, we compared our result complexityNd for the complexity of the ground state of a (d-1)-dimensional spatial lattice of N^d-1 oscillators to the analogous results for holographic complexity. We could easily identify the leading contribution in the limit of large N and a small UV cut-off distance, mδ≪1. In particular, this led us to consider the generalized family of κ cost functions given in eq. Dalpha, which yields =1/2^κ∑_{k_i}=0^N-1|logω̃_k⃗/ω_0|^κ .To identify the leading contribution to either eq. complexityNd or nothing, we made the crude approximation of replacing _k⃗∼1/δ for all momenta. In the following, we would like to avoid this approximation and examine the complexity nothing in more detail. Our result (<ref>) is still an approximation, but it allows us to consider the subleading contributions to eq. calpha. In particular, we will determine the leading corrections involving the mass. First we substitute the normal-mode frequencies freakd into the above expression for general κ to find =1/4^κ∑_{k_i}=0^N-1|logm^2/ω_0^2+2/ω_0δ^2 ∑_j=1^d-1sin^2πk_j/N|^κ Now, it is certainly true that the second term in the argument of the logarithm dominates for most of the terms in the sum over all momenta. But we would like to be more careful in retaining the leading corrections arising from the mass term. To simplify the analysis below, we will assume that the reference frequency is an IR frequency with ω_0<m.As a first step, we should isolate the infrared contributions which come from terms in the sum where the mass actually dominates or is comparable to the momentum term in the argument. For large N, we can take the usual continuum limit for these contributions with p_i=2π k_i/(Nδ). The IR contribution in eq. atall then becomes _IR=V/4^κ∫_0^Λ_IR ^d-1p/(2π)^d-1 [logm^2+p^2/ω_0^2]^κ .Note that we have dropped the absolute value symbol here since we are assuming that ω_0<m. The cut-off Λ_IR in this integral is an IR scale which delineates the boundary of the IR contributions to the momentum sum in eq. atall. Implicitly, we are also letting k_i and p_i range over positive and negative values so that all of the IR contributions come in the vicinity of k⃗=0—see footnote <ref>. Choosing this cut-off to be Λ_IR∝ m, the IR contribution takes the general form _IR=V m^d-1 ∑_a=0^κc_a [logm/ω_0]^a ,where the numerical coefficients c_a are independent of m and ω_0, but will depend on the spacetime dimension d. The leading contribution then takes the form _IR≃ c_κ V m^d-1[log m/ω_0]^κ.Having isolated the IR contribution, we return to the UV contributions to eq. atall. In these remaining terms, we can consider m^2/ω_0^2 to be a small correction to the argument of the logarithm, and so we perform a Taylor series expansion and keep only the first correction in mδ: _UV ≃ 1/4^κ∑_{k_i}>IR log[2/ω_0δ^2 ∑_j=1^d-1sin^2πk_j/N]^κ× {1+κ (mδ)^2/4(∑_j=1^d-1sin^2πk_j/Nlog[2/ω_0δ^2 ∑_j=1^d-1sin^2πk_j/N])^-1} , where the notation ∑_{k_i}>IR indicates that the summation begins at the IR cut-off, |k⃗|≥Λ_IR Nδ/(2π). The leading term will produce the expected leading contribution in eq. calpha with some overall numerical coefficient which depends on κ and d. Of course, there will also be a subleading dependence on our IR cut-off Λ_IR∝ m.To proceed further, we focus on the case κ=1, which simplifies the calculations slightly and was also the case that we found best emulated the holographic complexity. The following analysis is essentially unchanged for larger values of κ. Substituting κ=1 into eq. (<ref>) yields _UV≃1/4∑_{k_i}>IR{log[2/ω_0δ^2 ∑_j=1^d-1sin^2πk_j/N] +(mδ)^2/4[∑_j=1^d-1sin^2πk_j/N]^-1} .We now examine the two sums separately. First, we break the leading sum into two: log4/ω_0^2δ^2∑_j=1^d-1sin^2πk_j/N=2log2/ω_0δ+log∑_j=1^d-1sin^2πk_j/N . Since the first term is independent of k⃗, the corresponding sum over the UV modes yields a factor of N^d-1 (up to corrections proportional to Λ_IR^d-1). Summing over the second termis more complicated, but numerical fits for a range of N and d suggest that the sum takes the form 1/4∑_{k_i}>IRlog∑_j=1^d-1sin^2πk_j/N=a_d-1N^d-1+a_d-3N^d-3+⋯+a_0 , where the a_i are fixed numerical coefficients. Note that the constant term a_0 appears for both odd and even d, and the former may also have a logarithmic correction (log N). Turning now to the second sum in eq. (<ref>), we again carried out numerical fits to find 1/4∑_{k_i}>IR[∑_j=1^d-1sin^2πk_j/N]^-1≃b_d-1N^d-1+b_d-3N^d-3+⋯+b_0 . Collecting results, our approximation to the total complexity for κ=1 (and assuming ω_0<m) is therefore≃ V m^d-1 (c_IR 1 logm/ω_0+ c_IR 0 )+N^d-1/2log2/ω_0δ+a_d-1N^d-1+a_d-3N^d-3+⋯+a_0+mδ^2b_d-1N^d-1+b_d-3N^d-3+⋯+b_0 .To make a comparison with holographic complexity, as in section <ref>, we substitute N^d-1=V/δ^d-1, and introduce L=V^1/(d-1) as the linear size of the lattice. The complexity exit then becomes≃1/2V/δ^d-1log2/ω_0δ +V/δ^d-1a_d-1+a_d-3δ^2/L^2+⋯+m^2V/δ^d-3b_d-1+b_d-3δ^2/L^2+⋯+V m^d-1 (c_1 logm/ω_0+ c_0 ) .Hence we find the expected leading term, which corresponds to the result in eq. calpha with κ=1. We also find a subleading term proportional to V/δ^d-1, as is found in holographic complexity in the CA proposal <cit.>. Additionally, we would highlight the correction proportional to m^2V/δ^d-3, for which analogous results can again be found in holographic calculations—see section <ref> for further comments. It is interesting that we also see corrections, of the form V/(L^2δ^d-3). Of course, this term is far more suppressed that the previous one, but it also involves a fractional power of the volume, V^d-3/d-1. Such fractional powers would never arise in holographic complexity.To make the above formulae more concrete, consider the case of a one-dimensional lattice (d=2), in which case eq. exit2 reduces to =1/2L/δlog2/ω_0δ +a_1L/δ+a_0+L m (c_1 logm/ω_0+ c_0 ) ,where we have replaced V=L to emphasize that the volume is only a linear length here.§ OPTIMAL GEODESIC FOR PENALIZED GEOMETRY We would like to find the optimal geodesic in the penalized geometry penalty2, but as commented below eq. eq:kPenFull, finding the general solution for geodesics satisfying the desired boundary conditions seems out of reach. Recall that we were able to show that the simple straight-line geodesic describing the optimal circuit solver5 in the unpenalized geometry remains a geodesic in our new penalized geometry. However, it was also easy to show that the segmented path described by eq. segments was shorter than this geodesic when the penalty factor was large, ≫ρ_1,y_1; eq. compar2. To make progress towards finding the optimal geodesic in the new geometry, we make a simplifying assumption. To begin, we examine the penalized metric penalty2 and observe that as the radius ρ increases, the fastest growing component of the metric is g_zz∼^2 e^4ρ (for generic x, but this component still grows as g_zz∼ e^4ρ for x=π/2). This suggests that motion in the z direction will quickly be suppressed as the geodesics move out from the origin. Therefore, we simplify our problem by considering motion on a constant-z submanifold: s^2=2y^2+2[^2-^2-1sin^22x]ρ^2+2 ^2x^2 . The particular value of z in question will be fixed below by the boundary condition that z=x at s=0. We will return to justify our assumption of no (or little) motion in the z direction at the end of this appendix. Working with this simpler geometry (<ref>), the analysis of the geodesics becomes much more tractable. First, we observe that both _y and _ρ are now Killing vectors, for which the associated conserved quantities are 2 c̅_1≡2ẏ ,2ac̅_2≡2[ ^2-^2-1sin^22 x]ρ̇ , where the factors of 2 andwere chosen to simplify expressions below, and we have used the notation c̅_i to avoid confusion with the ĉ_i in eq. revel0. As before, the first constraint gives the usual solution yrun for y, y(s)=y_1 s with c̅_1=y_1. The second constraint yields ρ̇= c̅_2/^2-^2-1sin^22 x .The normalization of the tangent vector then becomes k^2 =2ẏ^2+2[ ^2-^2-1sin^22 x]ρ̇^2+2^2ẋ^2 =2y_1^2+2^2 c̅_2^2/^2-^2-1sin^22 x+2^2ẋ^2 . It is possible to integrate this equation to find s(x). To simplify the subsequent equations, we shall define k̅ via 2k̅^2≡k^2-2y_1^2 .Isolating ẋ in eq. eq:ktemp then yields s/x=√(2)a[ 2k̅^2-2^2c̅_2^2/^2-^2-1sin^22 x]^-1/2 .Upon integrating and choosing the constant of integration such that s(x_1=π/2)=1, the result can be simplified to s(x)=1-^2/2√(k̅^2-c̅_2^2) Π-(^2-1);-f(x) | ^2-1c̅_2^2/k̅^2-c̅_2^2 , where f(x)≡1/sin^-1(2x)√(^2+tan^2(2x)) , and Π is the incomplete elliptic integral of the third kind, which we write here as Π(-n;-z|m)=-∫_0^zt/1+nsin^2t√(1-msin^2t) . We now combine the expressions for ρ̇ and 1/ẋ in eqs. revel2 and revel3 to findρ/x=ρ̇/ẋ=√(2)^2 c̅_2/^2-^2-1sin^22 x[k̅^2-2^2c̅_2^2/^2-^2-1sin^22 x]^-1/2 .This expression can likewise be integrated to obtain ρ(x)=ρ_1-ic̅_2/2√(k̅^2-c̅_2^2) Fi g(x)| 1-^2-1c̅_2^2/k̅^2-c̅_2^2 , where g(x)≡1/sinh^-1(2x) ,and F is the incomplete elliptic integral of the first kind, F(z=ix|m)=i ∫_0^xτ/√(1+msinh^2τ) ,x∈ℝ . Note that in eq. eq:rX, we have fixed the integration constant via the boundary condition ρ(x_1=π/2)=ρ_1. Now, unfortunately (<ref>) cannot be inverted to find an analytical expression for x(s), which we could then use to obtain ρ(s) via eq. (<ref>). However, we can still study the behaviour of these geodesics numerically. Before doing so, it remains to relate the parameters c̅_2 and k (or k̅) to the boundary values ρ_1 and x_0 (as well as x_1=π/2). Let us first examine the parameter range for which we obtain a real result. It turns out that F in eq. (<ref>) is always complex; hence in order for ρ(x) to be real, the coefficient must also be imaginary, which requires k̅^2>c̅_2^2k^2>2y_1^2+c̅_2^2 . Turning now to s(x) in eq. (<ref>), the elliptic integral Π in this case is always real, and the coefficient will also be real in precisely the same regime (<ref>). Therefore this is the only restriction on our parameters required to ensure a real result. We have fixed the integration constants in both s(x) and ρ(x) via the boundary conditions at the end-point of the geodesic, namely s(x=π/2)=1 and ρ(x=π/2)=ρ_1. For the optimal geodesic, we further choose the boundary condition x(s=0)=π/4, which minimizes the cost of motion in the ρ direction, eq. rats6.[Alternatively, we could choose x(s=0)=3π/4, but the resulting trajectory is simply of a copy of the present geodesic rotated 180^o around the (θ,τ)=(π,0) axis—see figure <ref>.] However, we must be careful in evaluating eqs. eq:sX and eq:rX at this value of x; in particular, we must consider the limits lim_x→π/4^+s(x) =1-^2/2√(k̅^2-c̅_2^2) Π-(^2-1)| ^2-1c̅_2^2/k̅^2-c̅_2^2 ,lim_x→π/4^+ρ(x) = ρ_1- c̅_2/2√(k̅^2-c̅_2^2) K^2-1c̅_2^2/k̅^2-c̅_2^2 , where F and K are the complete elliptic integrals of the first and third kind, respectively, defined via K(z)=Fπ/2 | z ,Π(n | m)=Πn;π/2 | m . The parameters c̅_2 and k̅ must be chosen so that both these limits vanish, since initially we must have s=0 and ρ_0=0. In principle, we have two equations and two unknowns, but in practice the elliptic integrals are intractable. Fortunately, for our purposes a general solution is not required: we seek only a valid case to compare with the length(<ref>) of the simple straight-line geodesic.To that end, observe that the elliptic integral K is of order 1 almost everywhere, except when the argument approaches 1 in where it diverges, lim_w→1K(w)=∞. Since we want ρ to be large, let us choose ^2-1c̅_2^2/k̅^2-c̅_2^2=1-k̅^2=c̅_2^2 ^2-/1- , where 0<≪1. Note that this is within the reality domain (<ref>) since >1. The boundary condition that the limits (<ref>) should vanish then allows us to solve for ρ_1 and c̅_2; one finds: ρ_1=/2√(1-/^2-1) K1- ,c̅_2=^2/2√(1-/^2-1) Π1-^2 | 1- . One can then make ρ_1 arbitrarily large by taking →0; note that c̅_2 becomes arbitrarily large in the same limit. In fact, the divergence in both cases is logarithmic: ρ_1 =/√(^2-1)1/4log1/+log2+() ,c̅_2 =1/4√(^2-1)log1/+1/2tan^-1√(^2-1)+log2/√(^2-1)+() ,where higher-order terms vanish in the limit →0. We may now numerically compare the length of this geodesic to the proposed minimum (<ref>) associated with the straight-line circuit. Substituting c̅_2 and ρ_1 from eq. (<ref>) into k̅ given eq. (<ref>) and the analogous quantity k̅_0 from eq. (<ref>), we find k̅=^2/2√(^2-/^2-1) Π1-^2 | 1-andk̅_0=^2/2√(1-/^2-1) K1- , where 2k̅_0^2≡ k_0^2-2y_1^2. Of course, while these expressions are well-suited to numerics, we would also like to express k̅ in terms of the coordinates ρ_1, y_1, so as to compare with (<ref>) on more physical footing. We can obtain an approximation of this form by first replacing c̅_2 in (<ref>) by its expression in (<ref>), and then expanding for →0: k̅=/√(^2-1)1/4log1/+log2+/2tan^-1√(^2-1)+() ,where as above the () terms vanish as →0, and we shall drop them henceforth. Comparing this expression to eq. (<ref>), we observe that we can equivalently write this as k̅≃/2tan^-1√(^2-1)+ρ_1≃ π/4+ρ_1-1/2-1/12^2+1/^4 , where in the second approximation we have performed an expansion in the limit →∞.Additionally, it will be interesting to compare these geodesics against the segmented trajectory described in eq. segments. The length of this path is given by eq. eq:ks, and so as in eq. newerX, we define 2k̅_s^2 =k_s^2 -2 y_1^2,k̅_s =1/4[π^2^2+8ρ_12ρ_1+√(π^2^2+ (4y_1)^2)]^1/2 =/4[π^2+41-/^2-1^1/2K1- 1-/^2-1^1/2 K1-+π1+16y_1^2/π^2^2 ^1/2]^1/2 ,where in the second line we have replaced ρ_1 using eq. (<ref>). Note that unlike k̅ and k̅_0 in eq. (<ref>), the parameter y_1 still appears in this expression—although this contribution is suppressed for ≫ y_1. Again however, the second line above is more suited to numerics than physical inspection; to compare with (<ref>), we shall expand with ≫ρ_1,y_1 (as well as ρ_1,y_1≫1, and assuming ρ_1 and y_1 are roughly the same order of magnitude). Hence: k̅_s=π/4[1+8ρ_1/π2ρ_1/π+√(1+4y_1/π^2)]^1/2 ≃π/4+ρ_1+8/π^2y_1^2/^2 ρ_1+… . We mentioned above that the segmented path constitutes a remarkably good approximation to the geodesic. Comparing k̅ in (<ref>) and k̅_s in (<ref>), one can see evidence for this claim in that the leading-order behaviours are precisely the same; deviations arise only in the subleading terms, which are increasingly negligible for large values of . We discuss this point further in the main text—see eq. eq:overhead. We also explicitly confirm that the two paths are very close to one another in the largeregime by examining x(s) and ρ(s) numerically, as shown in figure <ref>.In closing this appendix, we remind the reader that in order to make progress, we confined our attention to motion in the constant-z subspace given by the simpler metric (<ref>). Hence for completeness, we should go back and examine whether or not this was a reasonable assumption.In particular, we wish to argue that, at least in the limit ≫1, the particular class of geodesics with x_0=π/4 and x_1=π/2 obtained for the constant-z subspace are a good approximation to the corresponding geodesics in the full geometry (<ref>). Intuitively, we motivated this restriction by the observation that movement in the z-direction is relatively costly. We can quantify this by considering the behaviour of ż given in eq. (<ref>). Recall that τ_0=0, and hence z_0=x_0=π/4. Then the finiteness condition (<ref>) requires that we set ĉ_3=0.[Note that ĉ_2 is still free, since we can rewrite eq. (<ref>) as ĉ_3=^2ĉ_2(2z_0)|_z_0=π/4=0.] Along the initial segment, where x=π/4, the derivatives (<ref>) then reduce to ẋ|_x=π/4=-ĉ_4cosh2ρ/^2 ,ż|_x=π/4=-ĉ_4/2^2 ,ρ̇|_x=π/4=ĉ_2/2 . Therefore, in the largelimit under consideration, motion in both the x- and z-directions is highly suppressed, while only motion along ρ is inexpensive. Along the second segment, where we rotate around to x=π/2, both ẋ and ż pick up terms which are (1) in , but ż is still exponentially suppressed in ρ_1 relative to ẋ. (Meanwhile ρ̇ decreases sharply to 0 on this segment in the limit ≫1.) Thus geodesics in the full spacetime (<ref>) can indeed be approximated by those in the constant-z subspace (<ref>), at least in the limits that we are considering. ytphys | http://arxiv.org/abs/1707.08570v2 | {
"authors": [
"Ro Jefferson",
"Robert C. Myers"
],
"categories": [
"hep-th",
"quant-ph"
],
"primary_category": "hep-th",
"published": "20170726180000",
"title": "Circuit complexity in quantum field theory"
} |
Faculty of Computer Science, "Alexandru Ioan Cuza" University, Iaşi, Romania {acf, mionita}@info.uaic.roIn this paper a new method for checking the subsumption relation for the optimal-size sorting network problem is described. The new approach is based on creating a bipartite graph and modelling the subsumption test as the problem of enumerating all perfect matchings in this graph.Experiments showed significant improvements over the previous approaches when considering the number of subsumption checks and the time needed to find optimal-size sorting networks. We were able to generate all the complete sets of filters for comparator networks with 9 channels,confirming that the 25-comparators sorting network is optimal.The running time was reduced more than 10 times, compared to the state-of-the-art result described in <cit.>.Comparator networks. Optimal-size sorting networks. Subsumption. § INTRODUCTION Sorting networks are a special class of sorting algorithms with an active research area since the 1950's<cit.>, <cit.>, <cit.>. A sorting network is a comparison network which for every input sequence produces a monotonically increasing output.Since the sequence of comparators does not depend on the input, the network represents an oblivious sorting algorithm.Such networks are suitable in parallel implementations of sorting, being applied in graphics processing units <cit.>and multiprocessor computers <cit.>. Over time, the research was focused on finding the optimal sorting networks relative to their size or depth. When the size is considered, the network must have a minimal number of comparators, while for the second objective a minimal number of layers is required. In <cit.> a construction method for sorting network of size O(n log n) and depth O(log n) is given. This algorithm has good results in theory but it is inefficient in practice because of the large constants hidden in the big-O notation. On the other side, the simple algorithm from <cit.> which constructs networks of depth O(log^2 n) has good results for practical values of n.Because optimal sorting networks for small number of inputs can be used to construct efficient larger networks the research in the area focused in the last years on finding such small networks. Optimal-size and optimal-depth networks are known for n ≤ 8 <cit.>.In <cit.> the optimal-depth sorting networks were provided for n=9 and n=10.The results were extended for 11 ≤ n ≤ 16 in <cit.>. The approaches use search with pruning based on symmetries on the first layers. The last results for parallel sorting networks are for 17 to 20 inputs and are given in <cit.>, <cit.>. On the other side, the paper <cit.> proved the optimality in size for the case n=9 and n=10. The proof is based on exploiting symmetries in sorting networks and on encoding the problem as a satisfiability problem.The use of powerful modern SAT solvers to generate optimal sorting networks is also investigated in <cit.>. Other recent results can be found in <cit.>, where a revised technique to generate, modulo symmetry, the set of saturated two-layer comparator networks is given. Finding the minimum number of comparators for n>10 is still an open problem.In this paper, we consider the optimal-size sorting networks problem. Heuristic approaches were also considered in literature, for example approaches based on evolutionary algorithms <cit.>that are able to discover new minimal networks for up to 22 inputs, but these methods cannot prove their optimality. One of the most important and expensive operation used in <cit.> is the subsumption testing. This paper presents a new better approach to implement this operation based on matchings in bipartite graphs. The results show that thenew approach makes the problem more tractable by scaling it to larger inputs. The paper is organized as follows. Section 2 describes the basic concepts needed to define the optimal-size sorting-network problem and a new model of the subsumption problem. Section 3 presents the problem of finding the minimal-size sorting network. Section 4 discusses the subsumption problem while Section 5 the subsumption testing. Section 6 presents the new way of subsumption testing by enumerating all perfect matchings. Section 7 describes the experiments made to evaluate the approach and presents the results. § BASIC CONCEPTSA comparator network C_n,k with n channels (also called wires) and size k is a sequence of comparatorsc_1=(i_1,j_1);…;c_k=(i_k; j_k) where each comparator c_t specifies a pair of channels 1 ≤ i_t < j_t ≤ n. We simply denote by C_n a comparator network with n channels, whenever the size of the network is not significant in a certain context.Graphically, a comparator network may be represented as a Knuth diagram <cit.>.A channel is depicted as a horizontal line and a comparator as a vertical segment connecting two channels.An input to a comparator network C_n may be any sequence of n objects taken from a totally ordered set, for instance elements in ℤ^n. Let x=(x_1,…,x_n) be an input sequence. Each value x_i is assigned to the channel i and it will "traverse" the comparator network from left to right. Whenever the values on two channels reach a comparator c=(i,j) the following happens: if they are not in ascending order the comparator permutes the values (x_i,x_j), otherwise the values will pass through the comparator unmodified. Therefore, the output of a comparator network is always a permutation of the input.If x is an input sequence, we denote by C(x) the output sequence of the network C.A comparator network is called a sorting network if its output is sorted ascending for every possible input.The zero-one principle <cit.> states that if a comparator network C_n sorts correctly all 2^n sequences of zero and one,then it is a sorting network. Hence, without loss of generality, from now on we consider only comparator networks with binary input sequences. In order to increase readability, whenever we represent a binary sequence we only write its bits; so 1010 is actually the sequence (1,0,1,0).The output set of a comparator network is outputs(C)={C(x) | ∀x∈{0,1}^n}. Let x be a binary input sequence of length n.We make the following notations:zeros(x)={1 ≤ i ≤ n | x_i = 0} andones(x)={1 ≤ i ≤ n | x_i = 1}. The output set of a comparator network C_n can be partitioned into n+1 clusters,each cluster containing sequences in outputs(C) having the same number of ones. We denote by cluster(C,p) the cluster containing all sequences having p ones: cluster(C,p) = {x∈ outputs(C)||ones(x)| = p }.Consider the following simple network C=(1,2);(3,4). The output clusters of C are: cluster(C,0)={0000},cluster(C,1)={0001,0100},cluster(C,2)={0011,0101,1100},cluster(C,3)={0111,1101}, cluster(C,4)={1111}.The following proposition states some simple observations regarding the output set and its clusters.Let C be a comparator network having n channels. (a) C is the empty network ⇔ |outputs(C)|=2^n.(b) C is a sorting network ⇔ |outputs(C)|=n+1 (each cluster contains exactly one element). (c) |cluster(C,p)| ≤np, 1 ≤ p ≤ n-1.(d) |cluster(C,0)|=|cluster(C,n)|=1.We extend the zeros and ones notations to output clusters in the following manner. Let C be a comparator network.For all 0 ≤ p ≤ n we denote zeros(C,p) = ⋃{zeros(x) | x∈ cluster(C,p)} and ones(C,p) = ⋃{ones(x) | x∈ cluster(C,p)}. These sets contain all the positions between 1 and n for which there is at least one sequence in the cluster having a zero, respectively an one,set at that position. Considering the clusters from the previous example, we have: zeros(C,0) = zeros(C,1) = zeros(C,2)={1,2,3,4}, zeros(C,3)={1,3}, zeros(C,4)=∅, ones(C,0) = ∅, ones(C,1) = {2,4}, ones(C,2)=ones(C,3)=ones(C,4)={1,2,3,4}.We introduce the following equivalent representation of the zeros and ones sets, as a sequence of length n,where n is the number of channels of the network, and elements taken from the set {0, 1}. Let Γ be a cluster: * zeros(Γ) = (γ_1,…,γ_n), where γ_i=0 if i ∈ zeros(Γ), otherwiseγ_i=1,* ones(Γ) = (γ'_1,…,γ'_n), where γ'_i=1 if i ∈ ones(Γ), otherwiseγ'_i=0.In order to increase readability, we will depict 1 values in zeros, respectively 0 values in oneswith the symbol *. Considering again the previous example, we have: zeros(C,3)=(0*0*) and ones(C,1) = (*1*1).If C is a comparator network on n channels and 1 ≤ i < j ≤ n we denote by C;(i,j) the concatenation of C and (i,j), i.e. the network that has all the comparators of C and in addition a new comparator connecting channels i and j. The concatenation of two networks C and C' having the same number of channels is denoted by C;C' andit is defined as the sequence of all comparators in C and C', first the ones in C and then the ones in C'. In this context, C represents a prefix of the network C;C'. Obviously, size(C;C')=size(C) + size(C').Let π be a permutation on {1,…,n}. Applying π on a comparator network C=(i_1,j_1);…;(i_k,j_k) will producethe generalized network π(C)=(π(i_1),π(j_1));…;(π(i_k),π(j_k)).It is called generalized because it may contain comparators (i,j) with i>j,which does not conform to the actual definition of a standard comparator network. An important result in the context of analyzing sorting networks (exercise 5.3.4.16 in <cit.>) states thata generalized sorting network can always be untangled such that the result is a standard sorting network of the same size. The untangling algorithm is described in the previously mentioned exercise. Two networks C_a and C_b are called equivalent if there is a permutation π such that untangling π(C_b) results in C_a.Applying a permutation π on a binary sequence x=(x_1,…,x_n) will permute the corresponding values: π(x) = (x_π(1),…,x_π(n)). Applying π on a set of sequences S (either a cluster or the whole output set) will permute the values of all the sequences in the set: π(S) = {π(x) | ∀x∈ S }. For example, consider the permutation π=(4,3,2,1) and the set of sequences S={0011,0101,1100}. Then, π(S)={1100,1010,0011}§ OPTIMAL-SIZE SORTING NETWORKS The optimal size problem regarding sorting networks is:"Given a positive integer n, what is the minimum number of comparators s_n needed to create a sorting network on n channels?".Since even the problem of verifying whether a comparator network is a sorting network is known to be Co-𝒩𝒫 complete <cit.>, we cannot expect to design an algorithm that will easily answer the optimal size problem. On the contrary.In order to prove that s_n ≤ k, for some k, it is enough to find a sorting network of size k.On the other hand, to show that s_n > k one should prove that no network on n channels having at most k comparators is a sorting network.Let R^n_k denote the set of all comparator networks having n channels and k comparators. The naive approach to identify the sorting networks is by generating the whole set R^n_k,starting with the empty network and adding all possible comparators.The algorithm to generate R^n_k In order to find a sorting network on n channels of size k,one could iterate through the set R^n_k and inspect the output set of each network. According to proposition <ref> (b), if the size of the output is n+1 then we have found a sorting network. If no sorting network is found, we have established that s_n > k.Unfortunately, the size of R^n_k grows rapidly since |R^n_k|= (n(n-1)/2)^kand constructing the whole set R^n_k is impracticable even for small values of n and k.We are actually interested in creating a set of networks N^n_k that does not include all possible networks but contains only "relevant" elements.A complete set of filters <cit.> is a set N^n_k of comparator networks on n channels and of size k,satisfying the following properties: (a) If s_n = k then N^n_k contains at least one sorting network of size k.(b) If k < s_n = k' then ∃ C^opt_n,k' an optimal-size sorting networkand ∃ C_n,k∈ N^n_k such that C is a prefix of C^opt. Since the existence of N^n_k is guaranteed by the fact that R^n_k is actually a complete set of filters, we are interested in creating such a set that is small enough (can be computed in a "reasonable" amount of time).§ SUBSUMPTION In order to create a complete set of filters in <cit.> it is introduced the relation of subsumption. Let C_a and C_b be comparator networks on n channels.If there exists a permutation π on {1,…,n} such that π(outputs(C_a)) ⊆ outputs(C_b) we say that C_a subsumes C_b, and we write C_a ≼ C_b (or C_a ≤_π C_b to indicate the permutation). For example, consider the networks C_a=(0,1);(1,2);(0,3) and C_b=(0,1);(0,2);(1,3). Their output sets are: ouputs(C_a)={{0000},{0001,0010},{0011,0110},{0111,1011},{1111}}, ouputs(C_b)={{0000},{0001,0010},{0011,0101},{0111,1011},{1111}}. It is easy to verify that π=(0,1,3,2) has the property that C_a ≤_π C_b.Let C_a and C_b be comparator networks on n channels, having |outputs(C_a)|=|outputs(C_b)|. Then, C_a ≼ C_b ⇔ C_b ≼ C_a. Assume that C_a ≤_π C_b ⇒π(outputs(C_a)) ⊆ outputs(C_b) and since |outputs(C_a)|=|outputs(C_b)| ⇒π(outputs(C_a)) = outputs(C_b).That means that π is actually mapping each sequence in outputs(C_a) to a distinct sequence in outputs(C_b). The inverse permutation π^-1 is also a mapping, this time from outputs(C_b) to outputs(C_a),implying that π^-1(outputs(C_b)) = outputs(C_a) ⇒ C_b ≤_π^-1 C_a. The following result is the key to creating a complete set of filters:Let C_a and C_b be comparator networks on n channels, both having the same size, and C_a ≼ C_b. Then, if there exists a sorting network C_b;C of size k, there also exists a sorting network C_a;C' of size k.The proof of the lemma is presented in <cit.> (Lemma 2) and <cit.> (Lemma 7).The previous lemma "suggests" that when creating the set of networks R^n_k using the naive approach,and having the goal of creating actually a complete set of filters, we should not add two networks in this set if one of them subsumes the other.The algorithm to generate N^n_k A comparator c is redundant relative to the network C if adding it at the end of C does not modify the output set: outputs(C;c) = outputs(C). Testing if a comparator c=(i,j) is redundant relative to a network C can be easily implemented by inspecting the values x_i and x_jin all the sequences x∈ outputs(C). If x_i ≤ x_j for all the sequences then c is redundant.The key aspect in implementing the algorithm above is the test for subsumption. § SUBSUMPTION TESTING Let C_a and C_b be comparator networks on n channels. According to definition <ref>, in order to check if C_a subsumes C_bwe must find a permutation π on {1,…,n} such that π(outputs(C_a)) ⊆ outputs(C_b). If no such permutation exists then C_a does not subsume C_b.In order to avoid iterating through all n! permutations,in <cit.> several results are presented that identify situations when subsumption testing can be implemented efficiently.We enumerate them as the tests ST_1 to ST_4.(ST_1) Check the total size of the output If |outputs(C_a)| > |outputs(C_b)| then C_a cannot subsume C_b.(ST_2) Check the size of corresponding clusters (Lemma 4 in <cit.>) If there exists 0≤ p ≤ n such that |cluster(C_a,p)| > |cluster(C_b, p)| then C_a cannot subsume C_b.When applying a permutation π on a sequence in outputs(C_a), the number of bits set to 1 remains the same, only their positions change. So, if π(outputs(C_a))⊆ outputs(C_b) then ∀ 0≤ p ≤ n π(cluster(C_a),p) ⊆ cluster(C_b,p),which implies that |cluster(C_a)| = |π(cluster(C_a),p)| ≤ |cluster(C_b,p)|for all 0≤ p ≤ n. (ST_3) Check the ones and zeros (Lemma 5 in <cit.>)Recall that zeros and ones represent the sets of positions that are set to 0, respectively to 1. If there exists 0≤ p ≤ n such that |zeros(C_a,p)| > |zeros(C_b,p)| or |ones(C_a,p)| > |ones(C_b,p)| then C_a cannot subsume C_b. For example, consider the networks C_a=(0,1);(2,3);(1,3);(0,4);(0,2) and C_b=(0,1);(2,3);(0,2);(2,4);(0,2). cluster(C_a, 2)={0011,00110,01010}, cluster(C_b, 2)={00011,01001,01010}, ones(C_a, 2)={2,3,4,5}, ones(C_b, 2)={2,4,5}, therefore C_a ⋠C_b. (ST_4) Check all permutations (Lemma 6 in <cit.>) The final optimization presented in <cit.> states thatif there exists a permutation π such that π(outputs(C_a)) ⊆ outputs(C_b) then ∀ 0≤ p ≤ n zeros(π(C_a,p)) ⊆ zeros(C_b,p) and ones(π(C_a, p)) ⊆ ones(C_b, p).So, before checking the inclusion for the whole output sets, we should check the inclusion for the zeros and ones sets,which is computationally cheaper.The tests (ST_1) to (ST_3) are very easy to check and are highly effective in reducing the search space. However, if none of them can be applied, we have to enumerate the whole set of n! permutations, verify (ST_4) and eventually the definition of subsumption, for each one of them. In <cit.> the authors focused on n=9 which means verifying 362,880 permutations for each subsumption test. They were successful in creating all sets of complete filters N^9_k for k=1,…,25 and actually proved that s_9=25. Using a powerful computer and running a parallel implementation of the algorithm on 288 threads,the time necessary for creating these sets was measured in days (more than five days only for N^9_14).Moving from 9! to 10!=3,628,800 or 11!=39,916,800 does not seem feasible. We have to take in consideration also the size of the complete filter sets, for example |N^9_14|=914,444.We present a new approach for testing subsumption, which greatly reduces the number of permutations which must be taken into consideration. Instead of enumerating all permutations we will enumerate all perfect matchings in a bipartite graph created for the networks C_a and C_b being tested.§ ENUMERATING PERFECT MATCHINGS Let C_a and C_b be comparator networks on n channels. The subsumption graph G(C_a,C_b) is defined as the bipartite graph (A, B; E(G))with vertex set V(G)=A ∪ B,where A=B={1,…,n} and the edge set E(G) defined as follows. Any edge e∈ E(G) is a 2-set e={i,j} with i∈ A and j∈ B (also written as e=ij) having the properties: * i ∈ zeros(C_a,p) ⇒ j ∈ zeros(C_b,p), ∀ 0 ≤ p ≤ n;* i ∈ ones(C_a,p)⇒ j ∈ ones(C_b,p),∀ 0 ≤ p ≤ n. So, the edges of the subsumption graph G represent a relationship between positions in the two output sets of C_a and C_b.An edge ij signifies that the position i (regarding the sequences in outputs(C_a)) and the position j (regarding C_b)are "compatible", meaning that a permutation π with the property π(outputs(C_a)) ⊆ outputs(C_b) might have the mapping i to j as a part of it.As an example, consider the following zeros and ones sequences,corresponding to C_a=(0,1);(2,3);(1,3);(1,4) and C_b=(0,1);(2,3);(0,3);(1,4). zeros(C_a)={00000,00000,000-0,000–,000–,—–},zeros(C_b)={00000,00000,00000,000–,000–,—–},ones(C_a)={—–,—11,1-111,11111,11111,11111},ones(C_b)={—–,—11,-1111,11111,11111,11111}.The subsumption graph G(C_a,C_b) is pictured below: A matching M in the graphG is a set of independent edges (no two edges in the matching share a common node). If ij ∈ M we say that i and j are saturated.A perfect matching is a matching that saturates all vertices of the graph. Let C_a and C_b be comparator networks on n channels.If C_a ≤_π C_b then π represents a perfect matching in the subsumption graph G(C_a,C_b).Suppose that C_a ≤_π C_b, π(i)=j and ij ∉E(G).That means that ∃ 0≤ p ≤ n such that i ∈ zeros(C_a,p) ∧ j ∉zeros(C_b,p) or i ∈ ones(C_a,p) ∧ j ∉ones(C_b,p). We will asumme the first case. Let x a sequence in cluster(C_a,p) such that x(i)=0.Since π(outputs(C_a)) ⊆ outputs(C_b) ⇒π(x) ∈ cluster(C_b, p). But π(i)=j, therefore in cluster(C_b, p) there is the sequence π(x) having the bit at position j equal to 0, contradiction. The previous lemma leads to the following result: Let C_a and C_b be comparator networks on n channels. Then C_a subsumes C_b if and only if there exists a perfect matching π in the subsumption graph G(C_a,C_b). The graph in figure <ref> has only four perfect matchings: (2,1,3,4,5), (3,1,2,4,5), (2,1,3,5,4), (3,1,2,5,4). So, when testing subsumption, instead of verifying 5!=120 permutations it is enough to verify only 4 of them.If two clusters are of the same size, then we can strengthen the previous result even more. If there is a permutation π such that π(cluster(C_a, p)) = cluster(C_b, p) then π^-1(cluster(C_b, p) = cluster(C_a, p). Using the same reasoning, when creating the subsumption graph C(G_a,C_b) we add the following two condition when defining an edge ij: * j ∈ zeros(C_b,p) ⇒ i ∈ zeros(C_a,p), ∀ 0 ≤ p ≤ n such that |cluster(C_a,p)|=|cluster(C_b,p)|,* j ∈ ones(C_b,p) ⇒ i ∈ ones(C_a,p), ∀ 0 ≤ p ≤ n such that |cluster(C_a,p)|=|cluster(C_b,p)|. In order to enumerate all perfect matchings in a bipartite graph, we have implemented the algorithm described in <cit.>. The algorithm starts with finding a perfect matching in the subsumption graph G(C_a,C_b).Taking into consideration the small size of the bipartite graph, we have chosen the Ford-Fulkerson algorithm which is very simple and does not require elaborate data structures. Its time complexity is O(n|E(G)|). If no perfect matching exists, then we have established that C_a does not subsume C_b. Otherwise, the algorithm presented in <cit.> identifies all other perfect matchings, taking only O(n) time per matching. § EXPERIMENTAL RESULTS We implemented both variants of subsumption testing:* (1) enumerating all permutations and checking the inclusions described by (ST_4) before verifying the actual definition of subsumption;* (2) verifying only the permutations that are actually perfect matchings in the subsumption graph, according to Corollary <ref>. We made some simple experiments on a regular computer (Intel i7-4700HQ @2.40GHz), using 8 concurrent threads. The programming platform was Java SE Development Kit 8.Several suggestive results are presented in the table below: t]|l|r|r|r|r|r|r|r| (n,k) |N^n_k| total sub perm_1 time_1 perm_2 time_2 (7,9) 678 1,223,426 5,144 26,505,101 2.88 33,120 0.07 (7,10) 510 878,995 5,728 25,363,033 2.82 24,362 0.06 (8,7) 648 980,765 2,939 105,863,506 13.67 49,142 0.14 (8,8) 2088 9,117,107 9,381 738,053,686 94.50 283,614 0.49 (8,9) 5703 24,511,628 29,104 4,974,612,498 650.22 1,303,340 1.96 The columns of the table have the following significations: * (n,k) - n is the number of channels, k is the number of comparators;* |N^n_k| - the size of the complete set of filters generated for the given n and k;* total - the total number of subsumption checks;* sub - the number of subsumptions that were identified;* perm_1 - how many permutations were checked, using the variant (1);* time_1 - the total time, measured in seconds, using the variant (1);* perm_2 - how many permutations were checked, using the variant (2);* time_2 - the total time, measured in seconds, using the variant (2); As we can see from this results, using the variant (2) the number of permutations that were verified in order to establish subsumption is greatly reduced. Despite the fact that it is necessary to create the subsumption graph and to iterate through its set of perfect matchings,this leads to a much shorter time needed for the overall generation of the complete set of filters.This new approach enabled us to reproduce the state-of-the-art result concerning optimal-size sorting networks, described in <cit.>. Using an Intel Xeon E5-2670 @ 2.60GHz computer, with a total of 32 cores, we generated all the complete set of filters for n=9. The results are presented in the table below.t]|l|r|r|r|r|r|r|r|r|r| k 1 2 3 4 5 6 7 8 |N^9_k| 1 3 7 20 59 208 807 3415 time(s) 0 0 0 0 0 0 0 0 k 9 10 11 12 13 14 15 16 |N^9_k| 14343 55991 188730 490322 854638 914444607164 274212 time(s) 4 48 769 6688 25186 40896 24161 5511 k 17 18 19 20 21 22 23 24 25 |N^9_k| 94085 25786 5699 1107 250 73 27 8 1 time(s) 610 36 2 0 0 0 0 0 0 In <cit.> the necessary time required to compute |N^9_14| using the generate-and-prune approachwas estimated at more than 5 days of computation on 288 threads.Their tests were performed on a cluster with a total of 144 Intel E8400 cores clocked at 3 GHz. In our experiments, the same set was created in only 11 hours, which is actually a significant improvement. § ACKNOWLEDGMENTS We would like to thank Michael Codish for introducing us to this research topic and Cornelius Croitoru for his valuable comments.Furthermore, we thank Mihai Rotaru for providing us with the computational resources to run our experiments. § CONCLUSIONS In this paper we have extended the work in <cit.>, further investigating the relation of subsumption. In order to determine the minimal number of comparators needed to sort any input of a given length, a systematic BFS-like algorithm generates incrementally complete sets of filters, that is sets of comparator networks that have the potential to prefix an optimal-size sorting network. To make this approach feasible it is essential to avoid adding into these sets networks that subsume one another. Testing the subsumption is an expensive operation, invoked a huge number of times during the execution of the algorithm. We described a new approach to implement this test, based on enumerating perfect matchings in a bipartite graph, called the subsumption graph. Computer experiments have shown significant improvements, greatly reducing the number of invocations and the overall running time. The results show that, using appropriate hardware, it might be possible to approach in this manner the optimal-size problemfor sorting networks with more than 10 channels.plain | http://arxiv.org/abs/1707.08725v1 | {
"authors": [
"Cristian Frasinaru",
"Madalina Raschip"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170727070521",
"title": "An Improved Subsumption Testing Algorithm for the Optimal-Size Sorting Network Problem"
} |
National Institute of Laser, Plasma and Radiation Physics, PO Box MG 36, RO-077125 Măgurele, Bucharest, Romania Faculty of Physics, University of Bucharest, PO Box MG 11, RO-077125 Măgurele, Bucharest, RomaniaThe stochastic advection of low energy deuterium ions is studied in a three dimensional realistic turbulence model in conditions relevant for current tokamak fusion experiments. The diffusion coefficients are calculated starting from the test particles trajectories in the framework of the semi-analitical statistical model called the Decorrelation Trajectory Method. We show that trajectory trapping determined by the space correlation of the velocity field is the main cause for anomalous diffusion and we obtain the transport regimes corresponding to different values of the parameters of the turbulence model and a trapping condition in the limit of frozen turbulence. Depending on the value of the parameters of the model, the interaction between turbulence and magnetic drifts leads to both an increase (with transport in the poloidal direction increasing up to an order of magnitude) and a decrease of the particle transport.Effects of magnetic drifts on ion transport in turbulent tokamak plasmas A. Croitoru December 30, 2023 ======================================================================== § INTRODUCTION The efficiency of magnetic fusion experiments is conditioned by the rate of transport of thermal energy and particles from the hot core of the plasma to the colder edge. The effective transport coeficients obtained in tokamaks by means of particle and power balance studies have a much larger value than what would be expected solely from collisional neoclassical transport processes <cit.>. This anomalous transport is attributed in large part to the presence of turbulent processes, such as drift waves instabilities excited by radial gradients in density and in temperature. The field is very active and there exist a vast literature dedicated to the study of the effect of various turbulence scenarios as for exemple the Ion Temperature Gradient (ITG) mode and the Trapped Electron Mode (TEM) instabilities on transport, with recent gyrokinetic simulations including also the coupling ITG-TEM <cit.> or the dissipative trapped electron mode (DTEM) that causes electrostatic turbulence in the pedestal <cit.>.While the behaviour of energetic ions has reached some satisfactory level of understanding, with studies offering scallings of transport with microturbulence [<cit.>, in both 2D <cit.> and 3D turbulence <cit.> low energy ions have received little to no attention despite studies suggesting that the diffusivity of lower energy test particles is similar to that of fast particles <cit.>. In the present paper we study the collisionless turbulent advection of low energy deuterium ions in a three-dimensional tokamak geometry and in a realistic turbulence model corresponding to the general characteristics of ITG or TEM, using a semi-analitical test particle approachdeveloped by M.Vlad et al. <cit.> called The Decorrelation Trajectory method (DTM). We show that the interaction between the neoclassical transport processes and the turbulent processes in the presence of trapping (for high values of the Kubo number) is highly nonlinear and can lead to either an increase or to a decrease of the diffusion coefficients depending on the parameters of the turbulence model. The spectrum of potential fluctuations is modeled as in <cit.> to be in agreement with the results of the numerical simulations <cit.> and includes the drift of the potential with the effective diamagnetic velocity and parallel decorrelations. Depending on the value of its velocity pitch-angle θ a particle in a three dimensional nonuniform magnetic field is either passing or trapped. We show that the value of the radial diffusion coefficient is maximum for θ=0.The paper is organized as follows. In Sec. <ref>we describe the turbulence model. Sec. <ref> offers a brief description of the DTM applied to particles moving in a 3D turbulent electromagnetic field. The results of the interaction between the drift of the turbulent potential and the magnetic drift are summerized in Sec. <ref> and the conclusions in Sec. <ref>. § TURBULENCE MODEL When the distance covered by the particles in a time length of the order of the decorrelation time is much smaller than the gradient scale length of the nonuniform quantities (such as temperature or density) we can apply a test particle approach <cit.>.Unlike in self-consistent models, the turbulence model in the test particle method is considered given a priori as a function of some turbulence parameters and independent of the distribution function of the particles. This allows us to obtain the transport coefficients as functions of the Fourier transform of the spectrum of the electrostatic potential (Eulerian correlation), leading to different transport regimes corresponding to different ranges of the parameters of the turbulence model.The electrostatic potential of a plasma in a drift type turbulence has an irregular structure of hills and wells continuosly changing in time <cit.>. This is mainly due tothe presence of density gradients which drive diamagnetic currents that restore the equilibrium. If there exists a small disturbance in the particles' pressure gradient, the response of the diamagnetic currents is also perturbed and it is propagated in a direction perpendicular to the magnetic field by ions that move with the ion polarization drift velocity <cit.>. This then drives a perturbed parallel current of the electrons so that the total current is divergence-free. If the parallel electron motion is not adiabatic and dissipates because of the interaction with the background plasma, there will be a time delay between the electron density perturbation and the plasma potential perturbation leading to a growing amplitude of the disturbance. The resulting nonzero time average product of the density and the 𝐄×𝐁 velocity causes a net transport of the plasma with an effective diamagnetic velocity wich can be either in the direction of the electron or of the ion diamagnetic velocity.The growing drift wave disturbance can interact nonlinearly with disturbances at other wavenumbers leading to perturbations at other scales, either at smaller wavenumbers due to kinetic energy perturbations or at larger wavenumbers due to density perturbations. The large scale perturbations, as for example the zonal flow, decorrelates and fragmentates the initial structure of the potential dramatically modifiyng the trasport. It follows that the spectrum of the fluctuating potential must contain two separated peaks <cit.>, one corresponding to the large scale oscillations and the other to the small scale ones. This aspect is confirmed in numerical simulations such as <cit.> that show a spectrum with two symmetrical maxima at k_y=± k_y^0 and k_x=0 and with a vanishing amplitude for k_y=0. Since ion dynamics in the poloidal direction is responsible for the potential fluctuation, the spectrum at saturation will have a similar form for both ITG and TEM instabilities, with the only difference being the typical values of the poloidal wave numbers (k_yρ_s∼ 1 for TEM and k_yρ_s ∼ 0.1-0.5 for ITG) <cit.>. We can thus approximate this spectrum by the analytical expression <cit.>S(k_1,k_2,z,t)∝ A_ϕ ^2[ exp( -(k_2-k_2^0)^2/2λ _2^2) -exp( -(k_2+k_2^0)^2/2λ _2^2) ] ×k_2/k_2^0exp( -k_1^2/2λ _1^2) exp( -|z|/λ _z- |t|/τ _c),where we also included the decorrelation of the potential due to its variation in time and to the existence of a finite parallel decorrelation length λ_z.The normalized Eulerian correlation (EC) of the potential (see figure Fig. <ref>) is obtained as the Fourier transform of S(k_1,k_2,z,t) : E(𝐱,z,t) =A_ϕ^2exp(-x^2/2λ_1^2)·d/dy[exp(-y'^2/2λ_2^2)sin(k_2^0y')/k_2^0]×exp(-v_∥ t/λ_z-t/τ_c)The potential ϕ(x,y',z,t)=ϕ(x,y-V_dt,z,t)is modeled as a stationary and homogeneous Gaussian stochastic function. The potential drift with the diamagnetic velocity V_d in the poloidal direction, specific to the drift type turbulence, is included in the argument y'=y-V_dt. The turbulence parameters are therefore the amplitude of the potential fluctuations A_ϕ , the correlation lengths along each direction λ _i, i=x (radial), i=y (poloidal), i=z (parallel), the correlation time τ _c the dominant wave number k_2^0 and the average diamagnetic velocity V_d.The normalized 3D equations of motion in a nonuniform magnetic field B⃗=B_0e^-x/L_be⃗_z≈ B_0(1-x/L_b) in the guiding center approximation in the frame of the moving potential are dx_i/dt =-K_*ϵ_ij∂_jϕ(1+x*δ b)+δ_iyV_d + δ_iyδ b/ρ_*(v_∥^2+v_⊥^2/2) , dz/dt =v_∥,δ b=v_⊥/(ω_c*L_b) is the inverse of the gradient scale-length of the magnetic field L_b normalized to the Larmor radius of the deuterium ions ρ=v_⊥/ω_c, ω_c = qB_0/M_D is the cyclotron frequency and ρ_*=ρ/a, where a is the small radius of the tokamak.The last term of the perpendicular velocity is the neoclassical gradient and curvature drift which is modelled as an average velocity with a Maxwellian distribution. The following units were used: the Larmor radius ρ for theperpendicular displacements, the small radius of the tokamak a for the parallel displacements, the thermal velocity v_th for the neoclassical velocity, the thermal velocity times the normalized Larmor radius v_*=ρ_* v_th for the diamagnetic velocity and the time needed for a particle movind radially with the thermal velocity to escape to the wall of the tokamak τ_0=a/v_th for time. The dimensionless parameterK_*=A_ϕ/B_0ρ1/v_*=eA_ϕ/Ta/ρ=eA_ϕ/T1/ρ_*is a measure of the amplitude of the turbulence.§ STATISTICAL METHOD The decorrelation trajectory method was developed by M.Vlad et al. <cit.> for the study of diffusion in incompresible stochastic velocity fieldsdx_i/dt=K v_i(𝐱,t)beyond the quasilinear regime, i.e at high values of the Kubo number K, where trajectory trapping or eddying determines anomalous statistics such as non-Gaussian distributions and memory effects expressed as long time Lagrangian correlations and also an increased degree of coherence <cit.>.Defined as the ratio between the decorrelation time of the trajectories from the potential, τ_c and the time of flight of the particles, τ_fl=λ_c/V , where λ_c is the space scale of the potential and V is the amplitude of the fluctuating velocity field, the Kubo number is a measure of the trapping of the particles in the fluctuating potential as well as a measure of the amplitude of turbulence. As emphasized in <cit.> for K→∞ the particles are trapped indefinetly in the structure of the electrostatic potential, and partially for K>1. Perturbations to the Hamiltonian structure of the system introduce additional characterstic decorrelation times, such as the parallel decorrelation time τ_z=λ_z/v_z. As shown by Taylor <cit.> the running diffusion coefficients are calculated as time integrals of the Lagrangian autocorrelation function of the velocity D_ii(t)=∫_0^t dτ L_ii(τ),where L_ii(τ)=⟨ v_i(𝐱_1,t_1)v_i(τ;𝐱_1,t_1) ⟩, with v_i(τ;𝐱_1,t_1) being the velocity of the particle at time τ calculted along the trajectory starting at 𝐱_1, t_1. The main idea of the decorrelation trajectory method is to obtain the diffusion coefficients by projecting the Langevin equation (<ref>) in subensembles S of realisations of the stochastic field, determined by fixed values of the stochastic potential and velocity in the origin of the trajectories 𝐱=0, t=0: (S):ϕ^0=ϕ(0,0) v^0=v(0,0).The 𝐄×𝐁 velocity is still Gaussian in a subensemble, but it's subensemble average ⟨𝐯(𝐱(t),t)⟩^S is usually nonzero. This allows for the definition of an average trajectory in a subensemble byd/dt⟨𝐱(t)⟩^S=⟨𝐯(𝐱(t),t)⟩^S. Since the trajectories in a subensemble are overdetermined by the initial conditions (ϕ(0,0) and its derivatives), they are very similar.We can thus replace the trajectories in a subensemble with a single trajectory, called the decorrelation trajectory, obtained as a time integral of the subensemble average of the Eulerian velocity calculated along this average trajectory d/dt𝐗^S=⟨𝐯(𝐗^S(t),t)⟩^S. The average velocity is obtained from the derivatives of the potential average in a subensemble ⟨ v_i(𝐱(t),t)⟩^S=-ϵ_ij∂_j⟨ϕ(𝐱(t),t)⟩^S,which, if we perform a change of basis from (∂_x ϕ^0(𝐱(t),t), ∂_y ϕ^0(𝐱(t),t) )to (| ∇ϕ^0(𝐱(t),t)|, β^0), where β^0 is the angle between ∂_y ϕ^0(𝐱(t),t) and the gradient of the potential and integrate over the modulus of the gradient, can be written as a function of the derivatives of the Eulerian autocorrelation of the potential: ⟨ϕ(𝐱(t),t)⟩^S'=ϕ^0E(𝐱,z,t)/E(0,0,0)+√(8/π)cosβ^0∂_y E(𝐱,z,t)/V_1-√(8/π)sinβ^0∂_x E(𝐱,z,t)/V_2,with V_1=∂_yyE(0,0,0) and V_2=∂_xxE(0,0,0), where the new subensemble S' is now defined by the values of ϕ^0 and β in the origin.Thus, the diffusion coefficient is D_ii(t)=V_i/2π√(π/2)∫_-∞^∞ dϕ^0P(ϕ^0)∫_0^2π dβ^0cosβ^0sinβ^0X_i^S',where ϕ^0 is Gaussian P(ϕ^0)=exp(-(ϕ^0)^2/2). For system (<ref>) we modeled the neoclassical velocity as an average velocity with a Maxwellian distribution functionP(v_∥,v_⊥)=(M_D/2π k_BT)^3/2exp(-(v_∥^2+v_⊥^2)/2). The decorrelation trajectories for subensemble S' are thereforedX^S'_i/dt=-K_*ϵ_ij∂_j⟨ϕ⟩^S'(1+X^S'*δ b)+δ_iyV_d + δ_iyδ b/ρ_*(v_∥^2+v_⊥^2/2),where⟨ϕ(𝐱(t),t)⟩^S' =ϕ^0E(𝐱,z,t)/E(0,0,0)+√(8/π)cosβ^0∂_y E(𝐱,z,t)/V_1-√(8/π)sinβ^0∂_x E(𝐱,z,t)/V_2+x*V_d+x*δ b/ρ_*(v_∥^2+v_⊥^2/2),and V_1=∂_yyE(0,0,0)=(k_2^0)^2+3/λ_y^2 and V_2=∂_xxE(0,0,0)=1/λ_x^2.The dimensionless parameter K_* is very similar to the Kubo number since it can be written as K_*=τ_0 / (ρ / V), where V is the amplitude of the 𝐄×𝐁 velocity fluctuations V=(A_ϕ/B_0)/ρ in the chosen normalization (see section <ref>). Defining the pitch angle θ as the angle between the modulus of the neoclassical magnetic drift and the projection of the magnetic drift on the x axis, such that v_∥=v_neo^0cosθ and v_⊥=v_neo^0sinθ, the running diffusion coefficient as function of the picth angle will beD_ii(t;θ)=V_i/2π√(π/2)∫_0^∞ dv_neo^0 P(v_neo^0) ∫_-∞^∞ dϕ^0P(ϕ^0)∫_0^2π dβ^0cosβ^0sinβ^0X_i^S', with P(v_neo^0)=exp(-(v_neo^0)^2/2)). The running diffusion coefficients (<ref>) are obtained numerically using approximately 40000 test particles. System (<ref>) is solved using an adaptive Runge Kutta method, the Cash-Karp method, implemented in the C++ boost library. The adaptive stepsize controls the error of the method and ensures stability. The trajectories are then interpolated at certain times using a spline interpolation technique and the integrals are calculated using the trapezoidal rule.In figure Fig. <ref> we plotted a decorrelation trajectory of a test particle starting at x=y=0 for increasing values of the diamagnetic velocity, in an constant magnetic field see Fig. <ref>. The drift of the potential with V_d is equivalent to the existence of an average potential xV_d. This modifies the contour lines of the total potential. Since the magnetic drift is modelled as an average drift, it has the same effect as the diamagnetic velocity leading to an average potential x*δ b/ρ_*(v_∥^2+v_⊥^2/2). Their combined effect is to modify the contour lines of the total potential creating islands of closed contour lines with a decreasing size as the size of the drifts increases. At large t, the particles will move in the direction of this combined drift velocity, in this case, in the y direction.§ RESULTS The statistical model contains 9 physical parametersK_*, λ_i, τ_c, k_2^0, V_d, δ_b and ρ_* (or equivalent the temperature of the ions). These determine several characteristic times whose ordering determine the transport regimes. The decorrelation produced by the drift of the potential with the effective diamagnetic velocity induces a normalized diamagnetic time τ_d=λ_y/V_d, the parallel motion of the particles induces a parallel decorrelation timeτ_∥=λ_z/v_∥≈λ_z/v_th, and the magnetic drift induces τ_b=λ_yρ_*/δ_b/(v_∥^2+v_⊥^2/2). The time of flight is τ_fl=λ_x/V_x+λ_y/V_y where V_x and V_y are the amplitudes of the x and y velocities respectively. In the case of a static potential with no diamagnetic drift V_d=0 and in the limit of frozen turbulence τ_c→∞ the decorrelation is provided by the magnetic drift and by the parallel motion of the particles. The characteristic time for the parallel motion is equal to the ratio between the parallel decorrelation length and the parallel velocity which is of the order of the thermal velocity. The normalized parallel decorrelation time will thus be equal to the normalized parallel decorrelation length λ_z, in units of the small radius of the tokamak, τ_∥/τ_0=λ_z . In figure Fig. <ref>(b) we plotted the running normalized diffusion coefficients for different values of the parallel decorrelation length. Trapping effects become visibile for λ_z>1. This offers a trapping condition τ_z>τ_0, since for smaller decorrelation times, particles moving radially can escape to the tokamak walls, without exploring the correlated region. As shown in <cit.> the inhomogeneity of the magnetic field does not modify the shape of the running diffusion coefficient (see Fig. <ref>(a)), its only effect being the cause of a direct transport. We thus choose a value of the normalized inverse gradient scale length of the magnetic field such that the ratio δ b/ρ_*=a/L_b≈ a/R≈ 0.3 is relevant major to tokamak devices <cit.>. This requires that deuterium ions at temperature T=1keV will have δ b=0.00041. In figure Fig. <ref>, we plotted the running diffusion coefficients for different values of the diamagnetic velocity. Since, as seen also in Fig. <ref>, at large times the particles move in the direction of the combined magnetic and diamagnetic drift, the transport in the poloidal direction increases dramatically. The antiparallel orientation of the diamagnetic velocity relative to the magnetic drift just decreases the total combined drift. In Fig. <ref>(b) the finite τ_c decorrelates the particles from the contour lines of the potential, the transport becoming diffusive.The contribution of the magnetic drift depends on the values of the pitch angle, see Fig. <ref>. In the radial direction the diffusion coefficient decreases with the value of the pitch angle as expected, since for θ>0 more particles are trapped in the magnetic field.§ CONCLUSIONS We studied the transport of low energy deuterium ions in a realistic model of tokamak microturbulence in the framework of the Decorelation Trajectory Method. The main decorrelation mechanisms are provided by the parallel motion of the ions, the diamagnetic velocity specific to the drift type turbulence and the neoclassical magnetic curvature and gradient drift due to the inhomogeneity of the magnetic field. These mechanisms induce several characteristic times whose ordering determine the transport regimes. In the limit of frozen turbulence we obtained that a neccessary condition for trapping is for the parallel decorrelation time to be greater or equal to the time needed for an ion moving radially to escape to the tokamak walls τ_z>τ_0. The magnetic drift and the drift of the potential with the diamagnetic velocity have similar effects on the trapping of the particles since they are equivalent with the existence of an additional potential x*V_d+x*δ b/ρ_*(v_∥^2+v_⊥^2/2) which modifies the contour lines of the total potential creating islands of closed lines that decrease in size as the drifts increases. The contribution of the magnetic drift depends on the value of the pitch angle and has a maximum in the radial direction at θ=0. This work was supported by the Romanian Ministry of National Education under the contracts 1EU-4 WPJET1-RO_C and PN 16 47 01 04. The contract 1EU-4 WPJET1-RO_C is included in the Programme of Complementary Research in Fusion. The views presented here do not necessarily represent those of the European Commission. The author would also like to thank also M. Vlad and F. Spineanu for support and assistance in the development of this research project.plain | http://arxiv.org/abs/1707.08797v1 | {
"authors": [
"A. Croitoru"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170727094427",
"title": "Effects of magnetic drifts on ion transport in turbulent tokamak plasmas"
} |
Composite boson description of a low density gas of excitonsA. E. GolomedovYandex,ulitsa Lva Tolstogo 16,Moscow, Russia119021 Yu. E. LozovikInstitute of spectroscopy RAS, Troitsk, Moscow, Russia108840 G. E. Astrakharchik and J. BoronatDepartament de Física, Universitat Politècnica de Catalunya Barcelona, SpainE-08034Composite boson description of a low density gas of excitonsA. E. Golomedov Yu. E. LozovikG. E. Astrakharchik J. Boronat July 26, 2017 ========================================================================= Ground state properties of a fermionic Coulomb gas are calculated using the fixed-node diffusion Monte Carlo method. The validity of the composite boson description is tested for different densities. We extract the exciton-exciton s-wave scattering length by solving the four-body problem in a harmonic trap and mapping the energy to that of two trapped bosons. The equation of state is consistent with the Bogoliubov theory for composite bosons interacting with the obtained s-wave scattering length. The perturbative expansion at low density has contributions physically coming from (a) exciton binding energy, (b) mean-field Gross-Pitaevskii interaction between excitons, (c) quantum depletion of the excitonic condensate (Lee-Huang-Yang terms for composite bosons). In addition, for low densities we find a good agreement with the Bogoliubov bosonic theory for the condensate fraction of excitons. The equation of state in the opposite limit of large density is found to be well described by the perturbative theory including (a) mixture of two ideal Fermi gases (b) exchange energy. We find that for low densities both energetic and coherent properties are correctly described by the picture of composite bosons (excitons). § INTRODUCTION The achievement of Bose-Einstein condensation (BEC) in confined alkali gases at nanokelvin temperatures has reinforced the interest in the search for other systems showing this extreme quantum behavior. In this line, the progress achieved in recent years towards the observation of a BEC state in Coulomb systems based on electrons and holes in semiconductors is of particular interest. This new candidate for a Bose condensate and superfluid state will show its macroscopic quantum behavior at much larger temperatures than BEC states in ultracold gases due to the much lower mass of the electron with respect to alkali atoms. This feature, and its expected sufficiently large lifetime, makes the study of BEC in Coulomb systems extremely interesting.Thinking on a BEC state, where the constituents are electrons and holes, leads immediately to the idea of formation of composite bosons where one electron and one hole, both of Fermi statistics, bind together. This composite particle is termed exciton and is on the basis of the search for a BEC state in electronic matter. Direct excitons are the ones in which electron and hole are not physically separated by any external potential, whereas indirect ones are carried out by physically separating electron and holes in two different layers with almost zero transition probability between them. Indirect excitons are the most studied ones and constitute the most probable scenario for observing their Bose-Einstein condensation with the advantage of a substantially larger lifetime with respect to the direct ones. In addition, spatially separated and coupled electrons and holes after Bose condensation can form nondissipating electric currents in separated layers due to superfluidity of condensed excitons<cit.>.The case of direct excitons has been less studied, probably in part due to the experimental difficulty of making the system stable for a finite lifetime. However, a gas of excitons is a clean and very interesting system from the theoretical side. It is particularly interesting to study its properties in the limit of low densities in which a description of the system in terms of composite bosons might be appropriate. Considering a gas of polarized electrons (treated as spin up particles) and polarized holes (spin down particles), the ground state at low density will be constituted by a gas of excitons where one electron and one hole couple and form a boson with integer spin. Then, these composite bosons will behave as bosons with a mass equal to the sum of the masses of electron and hole and the s-wave scattering length between excitons will be the dominant parameter of their effective interaction. In some sense, this is formally equivalent to the formation of molecules in dilute two-component Fermi gases with positive scattering lengths, i.e., beyond the unitary limit.While the excitonic description is very simple and tempting, due to the possibility of using well-established techniques (Gross-Pitaevskii equation<cit.>, Bogoliubov theory<cit.>, etc), in the last years there was a strong criticism of the very idea of such possibility. One of the strongest opponents to such description comes notably due to Monique Combescot who by introducing Shiva diagrams and performing calculations<cit.> argued that the composite-boson description of an exciton intrinsically misses a relevant part. That is, for some properties an elementary boson differs in a fundamental way from two Coulomb fermions due to the composite nature which prohibits<cit.> to describe the interaction between excitons by some effective potential even in the extremely low density limit, and eventually to make use of the usual many-body theories. Also in a classical work<cit.> by Keldysh and Kozlov dating back to 1968, it was shown that the commutation relation for the exciton creation â_ k and annihilation â^†_ k operators do not obey usual Bose commutation rule as, instead, [â_ k,â_ k'] = δ_ k, k' + O(na_0^3), where n is the particle density and a_0 the Bohr radius. As a result there, the contribution due non-bosonic commutation relation is of the order of na_0^3 and consequently it induces corrections to the energy already in the lowest order of the density, that is exactly on the same level as coming from the mean-field theory for Bose particles.It is interesting to check if the bosonic or fermionic nature of excitons manifests itself in the energetic and coherent properties of the gas. If the composite-boson description is possible, the equation of state can be expanded in powers of the gas parameter na^3, where a is the s-wave scattering length coming from the four-body problem (note that a effectively includes exchange effects). As a function of density, the mean-field contribution to the energy per particle should scale as ∝ n and beyond-mean field one as ∝ n^3/2. In addition, the equation of state might contain terms proportional to the Fermi momentum k_F = (3π^2 n)^1/3∝ n^1/3 or the Fermi energy ∝ n^2/3. Thus, by calculating the expansion of the equation of state in an ab initio microscopic simulation of a Coulomb fermionic system we should be able to see which description holds. Furthermore, we can check if the exciton-exciton interaction can be described in terms of some effective potential, namely a short-range potential with an effective s-wave scattering length a_ee. To do so we can first solve the four-body problem and extract a_ee and afterwards compare the energy in the many-body system.It can be argued, that Quantum Monte Carlo methods are extremely well suited for studying the equilibrium properties of electron and Coulomb systems. Fixed-node diffusion Monte Carlo calculations of jellium surfaces were performed by Acioli and Ceperley<cit.>. A relativistic electron gas was studied by VMC and DMC methods by Kenny et al.<cit.> The electron-hole plasma was recently studied by variational Monte Carlo<cit.> and diffusion Monte Carlo<cit.> approaches. Two-dimensional electron gas in strong magnetic fields was investigated in Ref. <cit.> by means of the variational Monte Carlo method. Finite-temperature properties can be accessed using path integral Monte Carlo method. The high-temperature phase diagram of a hydrogen plasma was obtained in Ref. <cit.>. The biexciton wave function was obtained using a quantum Monte Carlo calculation in Ref. <cit.>.In the present paper, we analyze a gas of excitons at low densities trying to verify if their description as composite bosons is compatible with the low density expansion for the energy and condensate fraction of a dilute universal Bose gas. To this end, we have performed quantum Monte Carlo simulations of the Fermi electron-hole gas using accurate trial wave functions and the fixed-node approximation to control the sign. To make the comparison feasible we have also calculated the scattering length between excitons which shows agreement with previous estimations. At low densities, the effective description of energy and condensate fraction of pairs is fully compatible with the universal law for dilute bosons without any significant contribution of purely Fermi contributions.The rest of the paper is organized as follows. In Section <ref>, we briefly describe the quantum Monte Carlo method used in the present study. Section <ref> comprises the analysis of the four-body problem in harmonic confinement used to determine the exciton-exciton s-wave scattering length. Results of the many-body problem and their effective description as composite bosons are reported in Sec. <ref>. Finally, we draw the conclusions of the work in Sec. <ref>.§ QUANTUM MONTE CARLO METHOD In the present work the electron-hole system is microscopically described using the diffusion Monte Carlo (DMC) method. DMC is nowadays a standard tool for describing quantum many-body systems that solves, in a stochastic way, the imaginary-time Schrödinger equation of the system (for a general reference on the DMC method, see for example <cit.>). For particles obeying Bose-Einstein statistics, DMC solves exactly the problem for the ground state within some statistical variance. When the system under study is of Fermi type we need to introduce an approximation to account for the non-positiveness of the wave function. This approximation, known as fixed node (FN), restricts the random walks within the nodal pockets defined by a trial wave function used as importance sampling technique during the imaginary-time evolution. Further details on the FN-DMC method can be found elsewhere.Our system is composed by a mixture of N_e electrons with mass m_e and N_h holes with mass m_h. All the electrons (holes) have the same spin up (down). The Hamiltonian of the system isH=-ħ^2/2 m_e∑_i=1^N_e∇_i^2 -ħ^2/2 m_h∑_i^'=1^N_h∇_i^'^2 +∑_i<j^N_ee^2/r_ij +∑_i^'<j^'^N_he^2/r_i^' j^' -∑_i,i^'=1^N_e,N_he^2/r_ii^' ,where i,j,… and i^',j^',… label electron and hole coordinates, respectively. In our study, we have considered equal masses m_e=m_h≡ m and used distances measured in units of the Bohr radius a_0=ħ^2/(m e^2) and energies in Hartrees, 1Ha=e^2/a_0. Therefore, in these units the Hamiltonian becomesH=-1/2∑_i=1^N_e∇_i^2 -1/2∑_i^'=1^N_h∇_i^'^2 +∑_i<j^N_e1/r_ij +∑_i^'<j^'^N_h1/r_i^' j^' -∑_i,i^'=1^N_e,N_h1/r_ii^' . The convergence of DMC method can be significantly improved by a proper choice of the trial wave function used for the importance sampling. As we are interested in the excitonic phase at low densities, our model for the wave function in the superfluid phase isΨ(R) =A (ϕ(r_1 1^')ϕ(r_2 2^') …ϕ(r_N_e N_h)),with A the antisymmetrizer operator of all the pair orbitals ϕ(r_i i'). This function is taken from the ground-state solution of the two-body problem, ϕ(r_i i^')= exp[-r_i i^'/(2 a_0)], corresponding to the electron-hole bound state with energy E_b=-ħ^2/(4 m a_0^2). It is worth noticing that a similar approach<cit.> was used in the study of the unitary limit of a two-component Fermi gas and proved its accuracy in reproducing the experimental data.In order to take into account the long-range behavior of the Coulomb interaction, we used standard Ewald summation to reduce size effects. Other possible bias coming from the use of a finite time step and number of walkers were optimized to reduce their effect to the level of the typical statistical noise.§ FOUR BODY PROBLEM. EXCITON-EXCITON SCATTERING LENGTHIf the description of excitons in terms of composite bosons is possible, the size of each composite boson is of the order of the Bohr radius a_0 which becomes small in the limit of dilute density, n a_0^3→ 0. The induced-dipole interaction between electron-hole pairs is of a Van der Waals type with 1/r^6 decay at large distances and can be treated as a short-range potential in that limit. This suggests that in the limit of dilute density, the exciton-exciton interaction potential can be described by a single parameter, the s-wave scattering length a_ ee. In this section we extract its value from the four-body problem. A textbook procedure<cit.> of finding the s-wave scattering length involves finding the low-energy asymptotic of the phase shift in the scattering problem. Alternatively, one might solve the few-body problem in a harmonic oscillator trapping and map the energy to that of a two-boson problem and take the limit of the vanishing strength of the trap<cit.>We calculate the energy of the 1e+1h and 2e+2h systems confined in a harmonic trap of different frequencies. The Hamiltonian in this case is the sum of the original Hamiltonian H, Eq. (<ref>), and the confining term, that isH_c = H + ∑_i=1^N_e1/2r_i^2 + ∑_i^'=1^N_h1/2r_i^'^2,where we consider equal masses m_e=m_h=m and use harmonic oscillator (HO) dimensionless units, that is HO length a_ ho=√(ħ/(m ω)) for distances and HO level spacing E_0=ħω for the energies. To improve the sampling, the trial wave function (<ref>) is multiplied by one-body terms which are the solution of non-interacting particles under the harmonic confinement,Ψ_c(R) = ∏_i=1^N_e e^-α r_i^2∏_i^'=1^N_h e^-α r_i^'^2 Ψ(R). The two-body problem, 1e-1h, can be solved exactly using a numerical grid method and also using the DMC method. We have verified that both results match exactly. For the four-body case, 2e-2h, we deal only with the DMC method. The energies for the two and four-body problems can be split in the following formE_2 = E_b + E_ CME_4 = 2 E_b + E_ int + E_ CM ,with E_b the binding energy of 1e-1h, E_ CM=3/2 the center-of-mass energy, and E_ int the energy associated to the exciton-exciton interaction. We are mainly interested in the last one,E_ int = (E_4-2 E_2) + 3/2 ,because from it we can extract the s-wave scattering length.The 2e-2h system in a harmonic trap can be thought as forming two dimers (excitons) obeying Bose statistics. These composite bosons interact with some short-range potential, which can be approximated as a regularized contact pseudopotential,V(r) = 4 π a_ eeδ(𝐫) ∂/∂ r (r·) .Within this approximation, one ends up with a problem of two bosons in a harmonic trap described by effective HamiltonianH_2^b = -1/2∇_1,2^2 +1/2 r_1,2^2 + 4 π a_ eeδ(𝐫_12) ∂/∂ r_12 (r_12·) ,with m_b being the mass of composite particle (m_b = m_e+m_h = 2 m in the case of equal masses). The eigenstates of Hamiltonian (<ref>) can be found analytically, see Ref <cit.>), and the s-wave scattering length a_ ee can be found as a solution of the following equation, a_ ee = 1/√(2) Γ(-E_ int/2+1/4)/Γ(-E_ int/2+3/4) ,with E_ int the energy associated to the exciton-exciton interaction (<ref>).Results for the scattering length a_ ee obtained through the combination of DMC results for the energy E_ int and the formula for a_ ee (<ref>) are reported in Fig. <ref> for different values of a_0/a_ ho. As one can see, the dependence ofa_ ee on the strength of the confinement is rather shallow, approaching a value a_ ee≃ 3 a_0 when a_0/a_ ho→ 0. This result is in nice agreement with previous estimations by Shumway and Ceperley based on finite-temperature calculations performed using the path integral Monte Carlo method<cit.> and DMC phase-shift calculations<cit.>.We will use this value for making a comparison with the Bogoliubov theory for the many-body problem.§ ELECTRON-HOLE GAS Using the formalism introduced in Sec. <ref> we have calculated the properties of a bulk electron-hole gas, mainly for very low values of the gas parameter na_0^3, with n=(N_e+N_h)/V the total density. We consider an unpolarized gas, N_e = N_h, of equal mass particles. As we are interested in the description of the excitonic phase we use as a trial wave function a determinant composed by electron-hole orbitals (see Sec. <ref>).In Fig. <ref>, we plot the energy of the electron-hole gas per particle E/N as a function of the gas parameter na_0^3. At very low densities, na_0^3 ≲ 10^-4, the energy per particle tends to half the binding energy of an electron-hole pair, -|ε_b|/2=-0.125 Ha. When the density increases the energy also increases due to the repulsive interaction between excitons. Within the picture of composite bosons, the mean-field Gross-Pitaevskii energy<cit.> of a weakly interacting composite Bose gas can be written as(E/N_ex)_MF = 1/2 g_ex n_ex ,where number of excitons is twice smaller than the number of charges, N_ex = N/2, and corresponding concentration is twice smaller, n_ex = n/2, exciton has a twice larger mass, m_ex = 2 m, the coupling constant between excitons is g_ex = 4πħ^2 a_ ee/ m_ex with a_ ee being the exciton-exciton s-wave scattering length. Equation <ref> can be written in atomic Hartee units as( E/N)_MF = π/4a_ee^2n a_ee^3 .In Fig. <ref>, we plot the mean-field energy (<ref>) shifted to be half the binding energy of the pair -|ϵ_b|/2 and compare it with the results of FN-DMC calculations. Our results match the mean-field energy with a_ee=3 a_0 at very low densities, na_0^3 ≲ 10^-4 but, when the gas parameter increases more, the FN-DMC energies increase faster than the mean-field law. Adding the Lee-Huang-Yang (LHY) correction<cit.> to the mean-field term (<ref>),( E/N)_LHY = π/4 a_ee^2n a_ee^3 [ 1 + 128/15 √(π)√(n_ex a_ee^3)]we can estimate the beyond-mean-field first correction. In Fig. <ref>, we plot LHY energy (<ref>) to be compared with the DMC data. As one can see, the LHY law reproduces our data up to densities na_0^3 ∼ 3· 10^-3 which approach the end of the universal regime, where the energy of a Bose gas is completely described solely in terms of the gas parameter. The LHY term arises from quantum fluctuations of the bosons that drop out of the condensate and in a single component LHY correction is accurate up to na^3 ≲ 10^-3<cit.>, where a is the boson-boson s-wave scattering length. It is interesting to note that the energetic behavior of the Coulomb electron-hole gas at low-density is fully described by the picture of composite bosons. These results corroborate the picture of an exciton as being considered effectively as a composite boson.When the density increases even more the energies depart from the low-density universal expansion (<ref>). At high density one expects that the system evolves to a mixture of two ideal Fermi gases with energy<cit.>E^(0)/N= 3/10ħ^2k_F^2/m ,with the Fermi wave number k_F = (3π^2 n)^1/3 or, the energy per exciton as a function of Wigner-Seitz radius r_s = [3/(4 π n)]^1/3 / a_0 in Hartree atomic units,E^(0)/N_ex= 2.21/r_s^2 .In Fig. <ref>, we plot the energy (<ref>) as a function of the gas parameter na_0^3. As one can see this energy is clearly out of our results. However, if one incorporates the exchange energy derived as a first-order perturbation theory on top of the free Fermi gas<cit.>,E^(1)/N_ex= 2.21/r_s^2 - 0.916/r_s .our results approach well to Eq. (<ref>).If the description of excitons as composite bosons, interacting with an effective s-wave scattering length a_ee, iscorrect at low densities then we have to observe a finite fraction of condensate pairs. We found that the excitonic picture of composite bosons provides a good energetic description and it is important to verify up to which level the excitonic description is valid in terms of the coherence in the correlation functions. To this end, we have calculated the two-body density matrixρ_2( r_1^', r_2^', r_1, r_2)= ⟨ψ_↑^†( r_1^') ψ_↓^†( r_2^')ψ_↑( r_1) ψ_↓( r_2)⟩ .For an unpolarized gas with N_e=N_↑=N/2 and N_h=N_↓=N/2, if ρ_2 has an eigenvalue of the order of the total number of particles N, the ρ_2 can be decomposed as,ρ_2( r_1^', r_2^', r_1, r_2)= α N/2 φ^∗( r_1^', r_2^') φ( r_1, r_2) + ρ_2^' ,ρ_2^' containing only eigenvalues of order one. The parameter α≤ 1 in Eq. (<ref>) is interpreted as the condensate fraction of pairs (excitons), in a similar way as the condensate fraction of single atoms is derived from the one-body density matrix.The spectral decomposition (<ref>) yields for homogeneous systems the following asymptoticbehavior of ρ_2ρ_2( r_1^', r_2^', r_1, r_2) →α N/2 φ^∗(| r_1^'- r_2^'|) φ(| r_1- r_2|),if | r_1- r_1^'|, | r_2- r_2^'|→∞. The wave function φ is proportional to the order parameter ⟨ψ_↑( r_1)ψ_↓( r_2)⟩=√(α N/2)φ(| r_1- r_2|), whose appearance characterizes the superfluid state of composite bosons.In Fig. <ref>, we plot the condensate fraction of excitons as a function of the gas parameter. At very low densities practically all the pairs are in the condensate, N_0/N → 1 and this value decreases monotonically towards zero with the density. The DMC estimation of the condensate fraction becomes difficult at large densities, which translates into a larger statistical noise, as can be appreciated in the figure. When the gas parameter is low enough one expects to recover the Bogoliubov law,N_0/N= 1 - 8/3 √(π)√(n_ex a_ee^3) .We compare this low density universal behavior (<ref>) with the DMC data in Fig. <ref>. As we can see, the agreement is excellent corroborating that the composite-boson picture with a_ee is fully consistent. It is interesting to note that the the universal behavior in a single component Bose gas breaks down at a similar value of the gas parameter, na^3 ∼ 10^-2<cit.>.§ CONCLUSIONS The consideration of excitons as composite bosons has been controversial for many years. Our DMC calculations have tried to contribute to this discussion using a microscopic approach, with the only restriction of the fixed-node approximation to overcome the sign problem. Working first with a four-body problem we have obtained the s-wave scattering length of the exciton-exciton interaction. The value obtained is in good agreement with previous estimations obtained in finite-temperature path integral Monte Carlo calculations. In the second part of the present study, we have calculated the properties of a homogeneous electron-hole system, focusing on the energy and the excitonic condensate fraction. Both the energy and condensate fraction agrees perfectly at low densities with the universal relations in terms of the gas parameter. Using the scattering length, obtained from the four-body problem, we reproduce the DMC data at low densities with good accuracy. In particular, we observe the relevance of the Lee-Huang-Yang term, beyond the mean field one, in describing correctly the energy. It is important to note that exchange terms appearing due to composite structure of excitons, as build up from excitons, is taken into account in the way we solve the four-body problem. As a result, the exchange effects are readily incorporated in the effective exciton-exciton s-wave scattering length a_ ee=3a_0. Only after the universal regime breaks down, the energies depart from the composite-boson picture and approach the regime of a Fermi gas with Coulomb interaction. The equation of state in the high-density regime agrees with the description in terms of the energy of two ideal Fermi gases corrected by the exchange energy arising due to Coulomb interactions. With respect to the condensate fraction of excitons, we have verified by means of a calculation of the two-body density matrix that the condensate fraction of pairs matches the Bogoliubov prediction of a Bose gas of particles interacting with an scattering length a_ee for low values of the gas parameter.Altogether, our results allow to conclude that the disputed interpretation of excitons as composite bosons is actually consistent in terms of energy and coherence with our results once the effective s-wave scattering length is extracted from the four-body problem using energy mapping to a two-boson problem.Pierbiagio Pieri and Alexander Fetter are acknowledged for useful discussions about the expansion of the equation of state for a weakly-interacting Fermi gas. 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"authors": [
"A. E. Golomedov",
"Yu. E. Lozovik",
"G. E. Astrakharchik",
"J. Boronat"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170726161619",
"title": "Composite boson description of a low density gas of excitons"
} |
Topological Data Analysis of Clostridioides difficile Infection and Fecal Microbiota Transplantation Topological Data Analysis of CDI and FMT P Petrov ST Rush Z Zhai CH Lee PT Kim G Heo Pavel Petrov, MSc Consultant at MNP, 21 West Hastings Street Vancouver , BC. V6E 0C3 Canada. Stephen Rush, PhDDepartment of Medical Sciences, University of Örebro, Örebro, 702 81, Sweden. [email protected] Zhichun Zhai,PhD Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1 Canada. [email protected] Christine Lee, MD FRCPC. Department of Microbiology, Royal Jubilee Hosptial, Victoria, BC, V8R 1J8, Canada. [email protected] Peter Kim, PhDDepartment of Mathematics & Statistics, University of Guelph, Guelph, Ontario, N1G 2W1 [email protected] Giseon Heo, PhDSchool of Dentistry; Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 1C9Canada. [email protected] * Pavel PetrovStephen T. RushZhichun ZhaiChristine H. Lee Peter T. Kimand Giseon Heo================================================================================================== Computational topologists recentlydeveloped a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity.Indeed, persistent homology studies the evolution of topological features in terms of a single index, and is able to capture higher order features beyond the usual clustering techniques. There are three descriptive statistics of persistent homology, namely barcode, persistence diagram and more recently, persistence landscape. Persistence landscapeis useful for statistical inference as it belongs to a spaceof p-integrablefunctions, a separable Banach space. We apply tools inboth computational topology and statistics toDNA sequences taken from Clostridioides difficile infected patients treated with an experimental fecal microbiota transplantation. Our statistical and topological data analysis are able todetectinterestingpatternsamong patients and donors. It also provides visualization of DNA sequences in the form of clusters and loops.Key words and phrases: Persistent homology; Persistence landscape, Discriminant analysis; Clostridioides difficile; Fecal Microbiota Transplantation; DNA sequences; 16S rRNA gene. § INTRODUCTIONTopological data analysis has become a formidable technique for analyzing high-dimensional data, especially when the purpose is for classification and discrimination.Methodological advancement has been rampant along withapplications to medical or scientific data, see for example <cit.>, <cit.> and more recently <cit.>. In this paper we propose making use of computational topological techniques to analyze gut microbiome data at the basic DNA sequence level based on data collected from sequencing the 16S rRNA gene.We are particularly interested in seeing changes in patient gut microbiome for a certain hypervirulent infectious disease following a radical experimental procedure that is gaining widespread attention and usage in medicine. Clostridioides (formerly Clostridium) difficile (C. difficile) infection (CDI) is the most frequent cause of healthcare-associatedinfections and its rates are growing in the community <cit.>. One of the major risk factors for developing CDI is through antibiotics. Thehealthy and diverse bacteria which reside within the colon are the major defense against the growth of C. difficile. Antibiotics kill these bacteria and allow C. difficile to multiply, producetoxins and cause disease. The current standard of care for this infection are the antibiotics:metronidazole, vancomycin and more recently, fidaxomicin.The efficacy of these antibiotics is limitedas vancomycin and metronidazole also suppress the growth of anaerobic bacteria such as Bacteriodesfragilis group which protect against proliferation of C. difficile.The efficacy of the recent narrower spectrum fidaxomicin is still under investigation although the initial data shows promise, <cit.>. The persistent disruption of healthycolonic flora may in part explain the reason for recurrences following a course of treatment with theseantibiotics. As an alternative to antibiotictherapy for CDI, in particular for recurrent and refractory diseases, is to infuse healthy gut bacteria directly into thecolon of infected patients to combat C. difficile by a procedure known as fecal microbiota transplantation (FMT). FMT is a process in which a healthydonor'sstool is infused into an affected patient.This can be performed using a colonoscope, nasogastric tube, enema, or more recently, in capsulized pill form, <cit.>.FMT serves to reconstitute the altered colonic flora, in contrast to treatment with antibiotic(s),which can further disrupt the establishment of key microbes essential in preventing recurrent CDI. Theliterature reveals a cumulative clinical success rate of over 90% in confirmed recurrent CDI cases <cit.>.There has been a growing interest into the microbiome of CDI patients <cit.>especially thoseinvolved with FMTs <cit.>.In case of the latter, there aredifferences in: the route of administration with all forms covered; different donor selection criteria, some used family members, some used a pool of donors; different sample sizes, although all studies had small sample sizes; and different sequencing procedures and equipment.Despite these differences, there seems to be two fundamentalpoints of agreement across all studies.The first is that CDI patients have low diversity in their microbiome, and that after receiving an FMT(s), their diversity was increased. The second fundamental agreement is CDI patients who were treated with FMT undergo changes in their microbiome thatat least initially have similarities to that of their donors. This paper provides further reinforcing evidence to support these two fundamental points in the framework of FMT delivered by enema, with a small exclusive donor pool.The novelty comes from using computational topological techniques to demonstrate this.We now summarize the paper.In Section <ref> we explain the details of the clinical data. In Section <ref> we provide topological preliminaries where we go over the Vietoris-Rips complex, and three topological descriptors: barcodes, persistence diagrams, and persistence landscapes. In the following Section <ref>, we apply these techniques and demonstrate the added value of the topological approach at the basic DNA sequence level. We complete our article in Section <ref> with a summary ofthe key findings. § CDI FMT AND 16S RRNA DATAAn earlier report provided details with regard to donor screening and the protocol used <cit.>. Including the earlier study, FMT was administered to 94 patients and several patients required multiple FMTs to achieve resolution <cit.>. From this patient pool we selected 19 patients, not necessarily randomly, for sequencing their 16S rRNA gene prior to treatment, pre-FMT, followed by a post treatment, post-FMT. Consequently, the data that is about to be presented should not be interpreted with respect to the efficacy of FMT, as the purpose is to better understand the microbial changes that came about as a result of an FMT(s).The gender of the patients was 63% female.The average age (to the time of their first FMT) was 77.11 with a standard deviation of 9.54 years, range 49 to 92 years.In-hospital patients accounted for 53% while the total peripheral white blood count (× 10^9/L) had a mean and standard deviation of 14.6 and 10.87, respectively.Three patients had temperature greater than 38^o C, and 11 patients experienced abdominal pain.Approximately half (53%) of the patients were on proton-pump inhibitors. Eight patients were refractory to treatment with metronidazole and four patients were refractory to vancomycin. These four patients were also refractory to metronidazole.No patients were on fidaxomicin as it had not been available to the public at that time. The patient characteristics are provided in Table <ref>, below. Predisposing conditions that may have resulted in CDI were:cellulitis, extreme fatigue, respiratory tract infections, septicemia, surgery and open wounds,and urinary tract infections.Some conditions were unknown.The majority of patients received the following antibiotics prior to contracting CDI: amoxicillin, azithromycin, cefazolin, cefprozil, cephalexin, ciprofloxacin, clindamycin, clarithromycin, cloxacillin, levofloxacin, moxifloxacin and nitrofurantoin. Some patients claimed no prior antibiotics used.In an interesting paper, the affects ofciprofloxacin was studied <cit.> using 16S rRNA deep sequencing.The patients all had varying pre-FMT regimens to treat their CDI. All had undergone multiple rounds of traditional antibiotic therapy with metronidazole and/or vancomycin before being administered an FMT. Seventeen received at least one course of metronidazole monotherapy, 18 received vancomycin monotherapy, 6 received vancomycin taper, and 3 received concomitant metronidazole and vancomycin therapy. Patients generally received two courses of metronidazole followed by multiple courses of vancomycin before receiving FMT(s).The number of days every patient received each therapy are summarized by their means and standard deviations and reported below in Table <ref>. Each patient had two stool samples sequenced, one representing a pre-FMT sequence, and one representing a post-FMT sequence. We attempted to sequence each patient prior to them receiving any FMT which for the most part occurred except for one patient who failed their first FMT and their stool sample sequenced was taken right after that event but prior to their next FMT. This patient had multiple FMT failures, hence we took that stool sample as the pre-FMT sequence.We also took a stool samplefollowing each patient's pre-FMT sample with at least one FMT in between.For the most part, the latter occurred following their last FMT which was the case if the patient resolved their CDI, but if they did not, then the post-FMT sequence was not necessarily their last.We would also like to make clear that 4+-FMT means that a patient had at least four treatments but could have had more.We indicate such as `4+' to be consistent with our clinical paper, <cit.>.The breakdown of the data is presented in Table <ref> with the above qualifications.All patients received a single treatment, 1-FMT and 9 of them clinically resolved their CDI.All who failed the first treatment went on to receive a second treatment.Of the remaining patients who received 2-FMT, 2 resolved, while the remaining went onto to receive a third treatment.There were 2 successes, and 1 failure, meaning they did not go on to receive additional treatments. Of the remaining patients who went on to 4+-FMT, 1 resolved, 1 resolved with antibiotics used in between treatments, and there were 3 failures. There were several patients who received antibiotics in between FMTs as described in our previous report <cit.>. In addition four healthy volunteers served as donors and were screened for transmissible pathogens as was outlinedin an earlier report <cit.>. The donors took no antibiotics for 6 months prior to stool donation.Seven donor samplestaken at various times were sequenced. All C. difficile infections were confirmed by in-hospital real-time polymerase chain reaction (PCR) testing for the toxin B gene. This study sequenced the forward V3-V5 region of the 16S rRNA gene from 19 CDI patients who were treated with FMT(s).A pre-FMT, a corresponding post-FMT, and 7 samples from four donors, corresponding altogether to 45 fecal samples were sequenced. All sequencing was performed on the 454 Life Sciences, GS Junior Titanium Series. The Qiagen Stool Extraction Kit(Omega BIO-TEK, Norcross, Georgia) was used to extract the DNA from the fecal samples following the `stool DNA protocol for pathogen detection'. Subsequent DNA amplification was done using PCR forward and reverse primers. One round of DNA amplicon purification was performed using the QIAquick PCR Purification Kit (Qiagen, Valencia, CA) followed by two rounds of purification using Agencourt AMPure XP beads (Beckman Coulter Inc., Mississauga, ON).We examined 19 pairs of pre-FMT and post-FMT patients, as well as donors, selected from the 94 CDI patients treated by the first author over the period 2008-12, <cit.>.The selection was not random and was chosen to reflect as wide a variability as possible.Furthermore, we deliberately chose 4 FMT failures to try and better understand what FMT can and cannot do.It was perhaps through the failures that we learned the most.The failed FMT cases were ultimately resolved with antibiotics even though some patients were refractory to metronidazle and/or vancomycin beforehand. Indeed FMT acted as a `gut primer' for antibiotics to fulfill it's role in clearing recurrent and refractory CDI.Although this phenomenon of failed FMT patients resolved with antibiotics afterwards has been observed in our work, <cit.>, as well as others, see for example <cit.>, as far as we are aware this is the first paper that has pursued this from a microbiome point of view. The data comes from the clinical work of the fourth author who is a practising physician. Some of this data was also examined in <cit.> and <cit.>.Table <ref> below, provides some descriptive statistics about the total and unique number of DNA sequences found in the 45 samples.Note that it is impossible to ensure that we have an approximately equal number of sequences in each sample <cit.>. Consider the DNA sequences λ and μ. Let x be the number of point dissimilarities between DNA bases in λ and μ, and let x_λ and x_μ be the number of DNA bases in λ and μ, respectively. Let x_O be the number of places where one sequence contains a DNA base and the other an O, a gap. Let y_λ and y_μ be the number of gaps in λ and μ, respectively. Let z be the length of the alignment sequences; we can assume they are equal otherwise we choose the smaller.The various dissimilarity metrics are the following.The one-gap distance d_O is defined by d_O(λ,μ)=x+min{y_λ,y_μ}/min{x_λ+y_λ,x_μ+y_μ}.It treats a string of O's flanked by any DNA bases in one sequence and the corresponding region in the other as one mismatch.The no-gap distance d_N is defined by d_N(λ,μ)=x/min{x_λ,x_μ}.It ignores gaps in a sequence and the corresponding region in the other sequence.The each-gap distance d_E is defined by d_E(λ,μ)=x+x_O/z.It treats every DNA base-placeholder pair as a mismatch.These metrics produce true mathematical distance matrices, which are symmetric and contain zeros on the diagonal. For this and most studies, the metric of interest is the one-gap metric, <cit.>.As an example, suppose there are two DNA sequences with the following base pair orientation:and. Here there are two mismatches and one gap. The distance is calculated as the number of mismatches divided by length of the shorter sequence. The length of the shorter sequence is 10 base pairs, since the gap is considered a single position. Hence distance is 0.3, <cit.>. We can use the other definitions of distance, but the definition provided is the one most commonly used. In addition, the results do not vary significantly.Most of the DNA sequences found in the samples appear several times, and hence the distance between them will be zero as they are identical. For this reason, the unique DNA sequences are taken.Table <ref> shows the minimum number of unique sequences is 147. DNA sequencing does not provide exact results, hence as the number of sequences in a sample increases, some mutations inevitably occur and these are recorded as unique sequences <cit.>. In other words, as the total number of sequences increases, the number of unique sequences is also inflated. For this reason, it is necessary to subsample from the number of unique sequences. The smallest number of unique sequences is 147, hence a weighted subsample of size 147 is taken from the number of unique sequences for our research.The pairwise distance between the 147 sequences is calculated in each of the 45 samples using the one-gap metric. Thus the data used for the primary analysis is 147 × 147 distance matrices for each of the 45 patients and donors.§ SOME TOPOLOGICAL PRELIMINARIES Persistent homology is a branch of computational topology that has been popularized by <cit.> and<cit.>. Several researchers have shown that persistenthomologyworks well on detecting topological and geometricalfeatures in high dimensional data, see <cit.> and<cit.>,for example.Let us suppose that we have points on a manifold whose dimension is not necessarily known.Topology studies the connectivity of these points. Each pointis replaced by a disk (ball) withradiuscentered at each point. If the disks overlap, connectthosecenter points with edges.As the radius increases, more points will be connected and so the number of connected components will decrease. This is analogous to clustering analysis in statistics. The connected components (clusters) are considered as 0-degree topological features. The number of connected componentsis denoted as the 0-th Betti number, β_0. Connecting the points with edges will also create simplices (convex hull of a geometrically independent set of points) andproduce topological features in higher dimensions. The low dimensional simplices are well known; a vertex (0-degree simplex), an edge (1-degree simplex), a triangle (2-degree simplex), and a tetrahedron (3-degree simplex). A loopis also called a 1-degreetopological featureand a void is a 2-degree topological feature. The k-th Betti number β_k counts the number of k-degree topological features (k-dimensional `holes').Persistent homology studies the history of topological features as the parameter ε varies. It recordsthe time at which a topological feature appears and disappears. The birth, death, andsurvival time of features are recorded as a barcode <cit.>. A true feature in the datalives over a long time while noise is short lived. We illustrate a fundamental idea of persistent homologyin Figure <ref> with 40 randomly selected points from a doubleannulus.The loops in themiddle of each annulusare prominent1-degree featuresand theirintervals in the barcode shows their persistence.Clusterscome in different shapes, see a few synthetic examples in Figure <ref>. Clusters arehomogeneous subgroups where the meaning of homogeneity is dependent on the types of similarity measure.In the double annulus, Figure <ref>, if we consider geodesic distance between points,those points are connected to form simplices (edges and triangles).All the edges and triangles together form a band which is homotopy equivalent to a circle (1-degree topological feature).The authors in <cit.> applied support vector clustering to concentric rings and were able to detect three rings as clusters. In Figure <ref>, we generate similar concentric rings as in <cit.>.Persistent homology analysis shows three persistent loops which correspond to three clusters in <cit.>. Consider another example,where points are randomly sampled fromthe number 8 with noise, see Figure <ref>.What are the clusters in this data?If one thinks of `blobs' as clusters, there are several patches made up with a few points. However,each loop, 1-degree topological feature,is represented as the set of points which are close in the sense of the shortest path distance. To study the connectivityof a space,the space need notberepresentedas a point cloud.All we need to know is how closethe points are in a space where they live. Thus point clouds or matrices of (dis-)similarity measurements among other data forms are input data for analysis of persistent homology. A good reference for algebraictopology and persistent homology is<cit.>.§.§ Vietoris-Rips complex and topological descriptors We will further explain Figure <ref> and then introduce three topological descriptors, barcode, persistence diagram <cit.> and persistence landscape <cit.>. A collection of point cloud data in a metric space is converted toa combinatorial graph whose edges are determined by closeness between the points.While agraph capturesconnectivity and clustering of data, itignores higher dimensional features. Thisidea of graph can be extended to a simplicial complex, which is a collection of simplices.Suppose there is a finite set of points {v_i}_1^ninand (dis-)similarity measure is denoted as d(v_i, v_j). A k-simplex isa set of all points x∈ such that x=∑_i=1^k, a_iv_i, where ∑ a_i=1, a_i ≥ 0. It is easy to picture low dimensional simplices; a 0-simplex a vertex, a 1-simplex an edge joining two vertices,a 2-simplex a triangle, a 3-simplex a tetrahedron. Vietoris-Rips complex _ is a set of simplices whose vertices have pairwisedistance within d(v_i, v_j) ≤. Algebraic topology adds group structure onto the complex,H_k(_), called k-th homology group.For coefficients in a field, H_k(_) is a vector space, whose basis consist of linearly independent k-dimensional cycles that are notboundaries.The k-th Betti number β_k, is the rank of H_k(_) andcounts the number of k-dimensional holes of a simplcial complex. At each fixed , homology group H_k(_)can be calculated, butcomputational topologists think of persistence.The simplical complexes grow asincreases, that is,𝒱_⊆𝒱_*, for ≤^*, this inclusion induces a linear map,H_k(_) → H_k(_^*). This allows us to examine the filtration of homology.The evolution of the simplicial complexes overincreasing values ofcan be completelytracked using barcodes orpersistence diagrams. Barcode is themultiset of intervals (', ”), where 'and ” indicate birth and death time of atopological feature.Alternatively,the birth and death times can be represented by a point (',”) in ℝ^2. The collection of these points in ℝ^2 is a persistence diagram (see Figure <ref>). Vietoris-Rips complex and its evolution are demonstrated with random points selected from a double annulus, see Figure <ref> above. In a simple summary, the point cloud data are transformed to barcodes orpersistence diagrams in eachdimension. Is it possible to calculate means and variances of barcodes or persistence diagrams? It is well known thatthe Fréchet mean of barcode (persistence diagram) is not unique. Many researchershave advanced in developingtheoriesin this research area <cit.>. Bubenik <cit.> introduceda third topological descriptor, persistence landscape. Given an interval (b, d), with b≤ d, define a function leading to an isosceles triangle,f_(b,d): →,f_(b, d)=min (t-b, d-t)_+,where u_+=max(u,0).The persistence landscapecorresponds to a multiset of intervals {(b_i, d_i) : b_i ≤ d_i} and to a set of functions,{λ(k, t): ×→} where : λ (k, t) is thek-th largest value of {f_(b_i, d_i )}. See an illustration in Figure <ref>. §.§Statistical Inference with Persistence Landscape On ×, the product of the counting measure onand the Lebesgue measure onare used. The persistence landscape,λ(k,t): ×→,is bounded and nonzero on a bounded domain. Hence persistence landscape belongs to ^p(×), with ametric induced by p-integrable functions and henceis a separable Banach space <cit.>. Bubenik also showed that when p ≥ 2, with finite first and secondmoments, persistence landscape satisfies a Strong Law of Large Numbers (SLLN) and a Central Limit Theorem (CLT). Suppose Λ_1, …, Λ_n are the random variables correspondingto persistence landscapes. The vector space structure of ^p(×) inducesthe meanlandscape as the pointwise mean, λ̅(k,t)=1/n∑_i=1^nλ_i(k, t).On p-integrable space,^p(×), where p ≥ 2and under the assumption offinite first and second moments,for any continuous linear functional f, the random variable f(λ(k,t)) also satisfiesSLLN and CLT <cit.>. There are many choices of f, but the integration of λ might be a natural choice, f(λ(k,t))=∑_k∫_λ(k, t) dt. It has a good interpretation; thevaluesof fare an enclosed totalarea of all curvesλ(k,t). Choice of f might depend ondata, see other choices in <cit.>.With the integration as a functional choice,we are ready to set hypotheses. Let Y_1=f(λ_1(k,t)) and Y_2=f(λ_2(k,t)) be random variables for groups 1 and 2. We let μ_1 and μ_2 be corresponding population means. The hypothesis of interest isH_0: μ_1-μ_2=0 H_a: μ_1- μ_2 ≠ 0. We also consider the similarity measure between persistence landscapesas well as between persistence diagrams. The measure between persistence landscape is defined as the p-norm of difference. Suppose there are two samples that have landscapes denoted as λ_1(k, t) and λ_2(k, t). Then the L_p-norm is defined in (<ref>). This compares pairwise the area under the contours between λ_1(k,t) and λ_2(k,t) ||λ_1-λ_2||_p=(∑_k∫_ℝ|λ_1(k,t)-λ_2(k,t)|^p)^1/p.For our data, we will calculate the persistence landscapedistance using both L_1 and L_2 norm in(<ref>). Wasserstein distance (see(<cit.>, for example) is a popularmeasure of dissimilaritybetween persistence diagrams. All the results based on persistance landscape (p=1, 2)and Wasserstein distance are similar and so all our statistics and presentations arebased on L_2 norm inthe following sections. § TOPOLOGICAL DATA ANALYSIS The computation of the Vietoris-Rips complex, which is computationally intensive, iscarried out using thepackage in R <cit.> on the Westgrid computer network.All 45 samples had intervals in barcode in degrees zero and one. However in degree two, only 12 of the pre-FMT samples and 15 of the post-FMT samples have intervals in barcode. Persistence diagrammakes visual pairwise comparisons easier than barcode.Figure <ref> shows the corresponding persistence diagrams of patient 10 before and after the FMTtreatment. At first glance there does not appear to be anything interesting but there are some general trends. We observe that the degree 0 persistencediagram for the pre-FMT samples have components that die a lot sooner than those in the post-FMT samples. Similarly, the birth and death times of these loops are shorter for the pre-FMT samples than for the post-FMT samples. These observationsmay indicate that there could be a true difference in the topological structure in degrees zero and one between the two groups. On the other hand,many of the intervals in degree oneare very short, thus what we observe may be noise rather than signal. Unlike barcode or persistence diagrams, we can calculate means and variances of persistence landscapes. Figure <ref> below, shows the average persistence landscapes of pre-FMT and post-FMT samples in degree 0 and 1.The trend in average persistence landscapes is the same as in persistence diagrams or barcodes, that is, the post-FMT samples have slightly longer intervals than the pre-FMT in degree 0 and 1.Quadratic discriminant analysis (QDA)was performed onthe β_0- and β_1-Isomap embedded coordinates. Three groups are well separated on the plane of both degree 0 and 1.It is interesting to see that the donors are grouped on one side and pre-FMT patientson the other side, andpost-FMT patients between the two.This is in complete agreement with what clinicians believe is happening and it is frequently reported that patients gut microbiome take on the characteristics of the donor microbiome following an FMT, see <cit.>. It would have been very interesting to compare the matched donor and the patient after FMT, but this information was not recorded during this study, <cit.>. The classification in ℝ^2 spacein degree 0in Figure <ref> shows the patients 7, 16 and 19 post-FMT, become much like donors, particularlydonors 2 and 3. Patients9 and 15 post-FMTdid not seem improved as they are similar to the cases pre-FMT. The classification in ℝ^2 spacein degree 1in <ref>show a few post-FMT patients(7, 10, 15) are close to the donors after FMT while most post-FMT patients remain similar topre-FMT.Following <cit.>, a clinical trial comparing the efficacy of frozen versus fresh FMT has been completed, see <cit.>.Here stool samples were collected at pre-FMT, followed by day-10, week-5, week-13 following a patients last FMT along with the exact donor stool sample pairing.Sequencing of this data is currently underway.We recall that our `raw' data was the dissimilarity matrix between 147uniques DNA sequences per subject, so a total of forty five 147× 147 matrices. For each matrix, we construct a Vietoris-Rips complex, then calculatepersistence landscapes which enables us to perform statistical inference.Hypothesistest (<ref>) were carried out to comparing pre-FMT and post-FMT samples.The p-values of the paired t-tests are 0.0064, 0.0083 and 0.2591 in degree 0, 1, and 2, respectively. Since all the analysis above is based on 147 sequences randomly chosen fromthose patients whose number ofof DNA sequences are bigger than 147, we repeated analysis 10 times withindependent 147 DNAsamples. The test statisticsand p-values for 10 runs are similar showing consistency of result regardless of which147 DNA sequences were applied. Hypothesis testsshow significant differenceof topological features in degree 0 and 1 between patients before and after FMT treatment. For degree 0, this implies thatthe number ofclustersand their persistence on DNA sequences in pre-FMT samplesare different fromthose in post-FMT samples. For degree 1, this implies thatthenumberof loops (cycles) and their persistence in DNA sequences in pre-FMT samplesare different fromthose post-FMT samples. WepresentDNA sequencesas pointscloud in ^3 and observepatterns of clustering and loops in DNA sequences inthe following section.§.§ Clusters and loops inDNA sequences Applying dimensional reduction methods; Isomap and multidimensional scaling (MDS), to dissimilarity measure between DNA sequences, we project the DNA sequences to ^3 and obtain embedded coordinates. Scree plots appear to indicate embedding dimension of the DNA sequences is 3. Theresidual variancein MDS was much higher than for Isomap, hence figures based on Isomapare presented below. Figure <ref> shows the Isomap embedded coordinates for donor 3, post-FMT patient 7 and pre-FMT/post-FMT patient 19.From the Figure <ref>, it can be seen that donor 3 and post-FMT patient 7 have a similar spread among the sequences and there are 2-3 distinct clusters.We have alsonoticed that donor 2 and post-FMT patient 18 also have similar structures.The donors generally have a wider spread, which would indicate more diverse DNA sequences and hencea healthier gut microbiome. For the number of of clusters, there are higher number of clusters in post-FMT samples,howeversome clusters contain only one DNA sequenceandare more spread out. For example, there are about 5 groups inpre-FMT patient 19; many clusters with fewer DNA sequences in each cluster in post-FMT patient 19. Ignoring the singleton clusters, we observe twolarge clusters in post-FMT patient 19.This information wasshownin Figure <ref> as well as onbarcode and persistence diagram.The two clusters in post-FMT patient 19 can be seen in the barcode diagram as the two points that have the highest `death' time. The spread out clusters in pre-FMT 19 are shown as the bars that have the earlier birth and earlier death times (figures are not shown here). This trend is visible in other samples and this might explain the small p-value for testing the difference in area under the persistence landscapes which was calculated in Section <ref>.§ CONCLUSIONS We illustrated how topological data analysis can be applied to similarity measures data of DNA sequences. DNA sequence data was analyzed using three summary statistics of persistent homology; namely, barcodes, persistence diagrams and persistence landscapes. The main objective was to see if there are any differences in the topological feature of DNA sequences in the gut microbiome of CDI patients before and after FMT treatment. From visual inspection of barcodes and persistence diagrams it was seen that the components in dimensions zero and one died sooner in the pre-FMT samples than in the post-FMT samples. Persistence landscapes were able to present this difference more formally, showing that there is a difference in the average area under the persistence landscapes for pre-FMT and post samples. Alternative interpretation of this was that there was a difference in the number and size of clusters in dimension zero, and the number and sizeof loops in dimension one. The post-FMTsamples had more clusters than the postsamples, whereas there was no visually obvious difference in the size of the loops, but the loops were bigger for the post samples.We performed discriminant analysis on β_0, β_1 -Isomap embedded coordinates. The classification on two dimensional space for both degree 0 and 1show good separation among three groups. For degree 0, the first twodiscriminant functionseparatespre-FMTpatients and donors, and post-FMT patients sitbetween the two. For degree 1, the first twodiscriminant functionseparatespre-FMT and post-FMT patients and donors are betweenthe two. The major drawbacks of this analysis is that information about individual sequences is lost. This project looked at the topological structure created by the sequences but no details were provided about the individual sequences. As the micro-biological technology and methods improve it may be interesting to incorporate this information.Also of interest would be meaningful identification of bacterial species. Studies of the 16S rRNA gene only measure presence of the species, but do not say anything about their functionality.For the latter one has to turn to the metabolome which is currently under investigation with the patients in <cit.>.We are grateful to all of the donors, families and patientswho took part in this study.We also appreciate the clinical and research staffs at St Joseph's Healthcare Hamilton where the clinical work had been performed.The corresponding author would also like to thank the participants of the SAMSI Working Group “NonlinearLow-dimensional Structures in High-dimensions for Biological Data" which was part of the 2013-14 SAMSI LDHD Program.Much of the discussions were centred on the work presented.Wealso thank Violeta Kovacev-Nikolic for her help with matlab code and Figure 1;Professor Patrick Schloss for his help using mothur; and Yi Zhou for his help with Figures 2 and 3. Computations in this research were largely enabled by resources provided by WestGrid and Compute Canada.We would like to acknowledge funding support provided by:CANSII CRT;CIHR 413548-2012; McIntyre Memorial Fund; Michael Smith Foundation; NSERC DG 293180, 46204; NSF DMS-1127914; and, PSI Foundation Health Research Grant 2013, 2017.The study and permission protocol was approved by the Hamilton Integrated Research Ethics Board #12-3683, the University of Guelph Research Ethics Board 12AU013 and the University of Alberta,Health Ethics approval Pro00047221.abbrvnat | http://arxiv.org/abs/1707.08774v2 | {
"authors": [
"Pavel Petrov",
"Stephen T Rush",
"Zhichun Zhai",
"Christine H Lee",
"Peter T Kim",
"Giseon Heo"
],
"categories": [
"q-bio.QM",
"stat.AP",
"62-07"
],
"primary_category": "q-bio.QM",
"published": "20170727082815",
"title": "Topological Data Analysis of Clostridioides difficile Infection and Fecal Microbiota Transplantation"
} |
Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure Irene Brito e-mail: [email protected], Filipe C. Mena e-mail: [email protected],^1Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal. December 30, 2023 ===================================================================================================================================================================== We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial spacelike hypersurface with a timelike boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.Keywords: Einstein equations; General Relativity; Initial boundary value problems; Self-gravitating systems; Spacetime matching, Elasticity § INTRODUCTION The initial value problem for the Einstein equations for perfect fluids, with suitable equations of state, is well understood in domains where the matter density is positive <cit.>. However, in physical models of isolated bodies in astrophysics one faces problems where the matter density has compact support and there are matter-vacuum interfaces. From the mathematical point of view, these physical situations can be treated as initial boundary value problems for partial differential equations (PDEs). These cases arise frequently in studies of numerical relativity (see e.g. <cit.>) and it is, therefore, important to have analytical results complementing the numerical frameworks.Rendall <cit.> proved the existence of local (in time) solutions of an initial value problem for perfect fluid spacetimes with vacuum interfaces. The fluids had polytropic equations of state and vanishing matter density at the interface. Initial boundary value problems (IBVP), where the matter density vanishes at the interface, were also studied byChoquet-Bruhat and Friedrich for charged dust matter in <cit.>, where not only existence but also uniqueness of solutions have been proved. In both cases, the field equations were written in hyperbolic form in wave coordinates and no spacetime symmetries were required.Problems where the matter density does not vanish at the interface bring a non-trivial discontinuity along the boundary. For the Einstein-fluid equations, Kind and Ehlers <cit.> use an equation of state for which the pressure vanishes for a positive value of the mass density. They proved that, for a given spherically symmetric perfect fluid distribution on a compact region of a spacelike hypersurface, and for a given boundary pressure, there exists locally in time a unique spacetime that can be matched to a Schwarzschild exterior if and only if the boundary pressure vanishes. In turn, the only existing result along those lines without special symmetry assumptions is due to Andersson, Oliynyk and Schmidt <cit.> for elastic bodies also having a jump discontinuity in the matter across a vacuum boundary. Under those circunstances, they prove local existence and uniqueness of solutions of the IBVP for the Einstein-elastic fluid system under some technical assumptions on the elasticity tensors. As in all cases above, in <cit.>, the boundary is characterised by the vanishing of the normal components of the stress-energy tensor although, in that case, conditions on the continuity of the time derivatives of the metric are also imposed. These compatibility conditions arise naturally from the matching conditions across the matter-vacuum boundary and have to be imposed on the allowed initial data.The present paper generalises the Elhers-Kind approach to fluids with tangential pressure in spherical symmetry. This includes some cases of interest, such as particular cases of elastic matter. Unlike <cit.>, we cannot ensure that, in general, the origin of the coordinates remains regular locally during the evolution. However, this can be ensured in some physically interesting cases.The plan of the paper is as follows: In Section 2, we setup our IBVP specifying the initial and boundary data. In Section 3, we obtain a first order symmetric hyperbolic (FOSH) system of PDEs and write our main result, which states existence and uniqueness of smooth solutions to the IBVP in a neighbourhood of the boundary. Section 4 contains an application of our results to elastic fluids with vanishing radial pressure and a regular centre.We use units such that c=8π=1, greek indices α, β, ..=0,1,2,3 and latin indices a,b,..=1,2,3.§ THE INITIAL BOUNDARY VALUE PROBLEM Consider a spherically symmetric spacetime (M,g) with a boundary S and containing a fluid source. This gives rise to a fluid 4-velocity u and we define a time coordinate T such that u is normal to the surfaces of constant T. We also introduce a comoving radial coordinate R.The general metric for spherically symmetric spacetimes can be written, in comoving spherical coordinates, as <cit.>g=-e^2Φ(T,R)dT^2+e^2Λ(T,R)dR^2+r^2(T,R)dΩ^2,where dΩ^2=dθ^2 +sin^2θ dϕ^2, and the components of the 4-velocity are written asu^μ=(e^-Φ,0,0,0).There is freedom in scaling the T and R coordinates which we fix by imposingΦ(T,R_0)=0, r(0,R)=R,where R_0 will correspond to the boundary of the matter.For fluids with no heat flux, the components of the energy-momentum tensor T_μν, in the above coordinates, can be written asT_TT=ρ e^2Φ, T_RR=p_1 e^2Λ, T_θθ=p_2 r^2, T_ϕϕ=p_2 r^2 sin^2 θ,where ρ is the fluid energy density, p_1 the radial pressure and p_2 the tangential pressure. The equation of state for p_1 and the energy conditions are such thatp_1=p_1(ρ) ∈ C^∞ ρ > 0 ρ+p_1 > 0s_1^2(ρ):=d p_1(ρ)/dρ ≥ 0.The equation of state for p_2 is such thatp_2=p_2(ρ)∈ C^∞and we use the notations_2(ρ):=d p_2(ρ)/dρ. We note that although we do not assume that s_1^2 necessarily remains positive when p_1=0, as in <cit.>, the system of PDEs that we derive for the general case becomes singular for s_1^2=0, so we will have to treat this case separately. §.§ Einstein and matter equations The conservation of the energy-momentum tensor, ∇_ν T^μν=0, impliesrρ̇+ rΛ̇ (ρ+p_1)+2ṙ(ρ+p_2)=0,for μ=T,rΦ' (ρ +p_1)+rp_1'+2 r' (p_1 -p_2)=0,for μ=R,where the prime and dot indicate derivatives with respect to R and T, respectively.The Einstein equations G_μν=T_μν lead to(μν)=(TT):ρ = 1/r^2[1-r'^2 e^-2Λ +ṙ^2 e^-2Φ+2rṙΛ̇e^-2Φ-2r(r”-r'Λ')e^-2Λ](μν)=(RR):p_1 =- 1/r^2[1-r'^2 e^-2Λ +ṙ^2 e^-2Φ-2r r' Φ' e^-2Λ+2r(r̈-ṙΦ̇)e^-2Φ](μν)=(RT):ṙΦ'+r' Λ̇ -ṙ'=0(μν)=(θθ)=(ϕϕ):p_2=e^-2Φ/r[ṙ(Φ̇-Λ̇)-r̈]+ e^-2Φ[Λ̇(Φ̇-Λ̇)-Λ̈]+e^-2Λ/r[r'(Φ'-Λ')+r”]+ e^-2Λ[Φ' (Φ'-Λ')+Φ”], while the contracted Bianchi identities are identically satisfied. We note that (<ref>) can be obtained from (<ref>)-(<ref>).Integrating (<ref>), with (<ref>), one gets Φ(T,R)=-∫_ρ_0^ρs_1^2(ρ̅) /ρ̅+p_1(ρ̅) dρ̅- 2 ∫_R_0^R(p_1-p_2)/ρ+p_1r'/rdR.As an example, in the case of linear equations of state p_1=γ_1 ρ and p_2=γ_2 ρ, we simply getΦ(T,R)=-γ_1/1+γ_1ln(ρ/ρ_0)-γ_1-γ_2/1+γ_1ln(r/r_0)^2, with r_0=r(T,R_0),which will happen for a particular case of elastic matter that we will consider in Section 4.Defining the radial velocity asv:=e^-Φṙand the mean density of the matter within a ball of coordinate radius R asμ:=3/r^3∫_0^Rρ r^2 r' dR̅,one obtains from (<ref>)r'^2 e^-2Λ=1+v^2 -1/3μ r^2,where we also used the condition r(T,0)=0, for regularity of the metric at the center. Then, (<ref>), (<ref>) and (<ref>), together with the evolution equations for v and μ, giveṙ =v e^Φ, v̇ =[-r/2(μ/3+p_1)-r' p_1'/ρ +p_1e^-2Λ- 2 p_1 - p_2/ρ+p_1r'^2/re^-2Λ]e^Φ, ρ̇ =[-ρ(v'/r'+2v/r)-(p_1 v'/r'+2p_2 v/r)]e^Φ, Λ̇ =v'/r' e^Φ, μ̇ =-3v/r(μ +p_1) e^Φ. To summarize, the Einstein equations resulted in the system of evolution equations (<ref>)-(<ref>) for the five variables r,v,ρ,Λ,μ together with constraints (<ref>) and (<ref>). We note that although we could close the system without (<ref>) and (<ref>), those equations will be crucial to obtain a symmetric hyperbolic form. In Section 3, we shall apply suitable changes of variables in order to write our evolution system as FOSH system. §.§ Initial data and boundary data In spherical symmetry, the free initial data (at T=0 and R∈ [0,R_0]) is expected to be[We use a "tilde" for boundary data defined on (T, R=R_0) and a "hat" for initial data defined on (T=0, R).] ρ̂( R):=ρ(0,R) and v̂( R):=v(0,R), satisfying v̂(0)=0, and constrained by1/R∫_0^Rρ̂(R̅)R̅^2 dR̅≤ 1 + v̂(R)^2,as a consequence of (<ref>), (<ref>) and (<ref>).The initial data for the remaining variables r, μ and Λ can be obtained from (<ref>), (<ref>) and (<ref>), respectively.Note that the intrinsic metric and extrinsic curvature (i.e. the first and second fundamental forms) of the initial hypersurfaceh_0= e^2Λ̂dR^2+R^2 dΩ^2, K_0=v̂' e^2Λ̂dR^2 + R v̂dΩ^2are fully known once ρ̂( R) and v̂( R) are known.At the boundary of the fluid, we must specify the two smooth boundary functions p̃_1(T):=p_1(T,R_0) and w̃ (T):=v(T,R_0)/r(T,R_0) (or μ̃(T):=μ(T,R_0), via (<ref>)) which should satisfy the corner conditions p̃_1 (0)=p̂_1(R_0) and w̃ (0)=ŵ(R_0) (or μ̃(0)=μ̂(R_0)). In terms of the initial data set {ρ̂(R), v̂(R)}, we note thatfrom (<ref>) we get (μ̇/(μ+p_1))(0,R_0)=-3v̂(R_0)/R_0, at the corner.The fact that we need w̃(T) at the boundary is reminiscent of the compatibility conditions of <cit.> arising from the matching conditions, since w̃(T) is related to the time derivative of the metric at the boundary. In fact, in spherical symmetry, the matching conditions (see the appendix) imply the continuity of the areal radius (here r) through the boundary. Therefore, ṙ(T,R_0) also has to be continuous and, for Φ(T,R_0)=0, this gives w̃(T), for the interior spacetime from the data of the exterior.In what follows, we will prove existence and uniqueness of solutions to the evolution equations (<ref>)-(<ref>), on a neighbourhood of the boundary, for the variables r,v,ρ, Λ,μ, subject to the specified initial and boundary data. We will also show that, since the initial data obeys the constraints (<ref>) and (<ref>), the solutions will also satisfy the constraints. When the fluid boundary corresponds to characteristics, then the corner data will locally determine the boundary evolution and this will be the case when s_1^2=p_1=0, as we will show in Section 3.2.§ EXISTENCE AND UNIQUENESS RESULTS ON A NEIGHBOURHOOD OF THE BOUNDARY We treat separately the cases s^2_1> 0 and s^2_1=0. §.§ Case s_1^2>0 In this case, the boundary is non-characteristic and we will be able to use the theorem of Kind-Elhers <cit.> provided we write our evolution system as a FOSH system and give the appropriate data. We thus recall the theorem (whose proof uses results of Courant and Lax <cit.>): [Kind-Elhers] Consider the systemẊ+A(U_i)Y'=F(X,Y,U_i,R)Ẏ+A(U_i)X'=G(X,Y,U_i,R)U̇_j = H_j (X,Y,U_i,R),i, j=1,...,p,where F,G,H_j and A are C^k+1 functions, for R> 0, and A is always positive. Let C^k+1 initial values X̂, Ŷ,Û_̂î on [R_1 ,R_0 ], R_1>0, and the C^k+1 boundary value Ỹ(T) be given. Assume that Ỹ(0),Ỹ̇(0),...,Ỹ^(k+1)(0) equal the values of Y,Ẏ,...,Y^(k+1) at (0,R_0), which are obtained from (<ref>) and the initial data. Then, the system (<ref>) has a unique C^k solution on a compact trapezoidal domain 𝒯, for small enough times.In order to apply this theorem we will need to use new variables and write the evolution system (<ref>)-(<ref>) in quasi-linear symmetric hyperbolic form.We thus use the Kind-Elhers variables𝒬 =ln(R/r), ℒ =∫_ρ_0^ρs_1(ρ)/ρ+p_1(ρ)dρ, ω =r',w = v/r,X =e^-Λℒ',Y =v'/ω+2v/r.Our system will then have the 8 variablesX, Y and U_i={ Q,L, ω, w, μ, Λ}.Before proceeding, note that, from (<ref>), one obtainsω̇=(v'+vΦ')e^Φ.Now, since (<ref>) is invertible, we can consider ρ as a known function of L. Moreover, for given equations of state, we can also consider p_1, p_2, s_1^ 2 and s_2 as known functions of ℒ. Regarding Φ, it is not clear from (<ref>) that it can be written as a smooth function of the new variables, so we need a further assumption:Φ, as obtained from (<ref>), is a known smooth function of the variables {X,Y,U_i,R}. Fulfilling Assumptions 1, 2 and 3 depends on the type of matter and equations of state under consideration. For example, for the linear equations of state of Remark 1 we get:ρ ( L) = ρ_0 e^(1+γ_1) L/√(γ_1) Φ( L,Q, R) =-√(γ_1) L-2(γ_1-γ_2)/1+γ_1 [lnR/r_0+ Q],which for ρ_0>0, r_0>0 and γ_1>0 satisfy the assumptions. As another example, there are cases where the coordinate system can be chosen to be synchronous and comoving so that Φ≡ 0 and Assumption 3 becomes trivial. Then, taking into account the Assumptions 1 and 2, and after a long calculation, our evolution system in terms of the new variables becomes: 𝒬̇ =-w e^Φ, ℒ̇ =-s_1 Y e^Φ-2s_1 we^Φp_2 -p_1/ρ + p_1, Λ̇ = (Y-2w) e^Φ, ẇ =-e^Φ[s_1 ω e^𝒬-ΛX/R+w^2+1/2(μ/3+p_1 )+ 2 ω^2/R^2 e^2(𝒬-Λ)p_1 -p_2/ρ +p_1], ω̇ =e^Φ[ω(Y-2w)-s_1 wXe^Λ-QR- 2 ω/Re^𝒬 p_1 -p_2/ρ+p_1], μ̇ =-3we^Φ(μ+p_1), Ẋ+s_1 e^Φ-Λ Y' =e^Φ[XY(s_1^2-ds_1/dℒ-1)+2wX] -2wXe^Φp_1-p_2/ρ+p_1+e^Φ-Λp_1-p_2/ρ+p_1[2wXe^Λds_1/dℒ+2 s_1 ω/Re^𝒬(2Y-3 w)-4s_1^2 w Xe^Λ..- 4 s_1 ω/Rwe^𝒬p_1 -p_2/ρ +p_1]+2 s_1 w X e^Φ(s_1 -s_2/s_1), Ẏ+s_1 e^Φ-Λ X' =e^Φ[X^2(s_1^2-ds_1/dℒ)-2s_1 ω e^Q-ΛX/R-(Y-2w)^2-2w^2-ρ+3p_1/2]+2X/R e^Φ-Λ+𝒬(s_2/s_1-s_1)+ e^Φp_1 -p_2/ρ +p_1[(6s_1 +2/s_1)ωX/Re^𝒬-Λ-3μ+p_1..+2ρ- 2ω^2/R^2e^2(𝒬-Λ)(1-2p_1 -p_2/ρ +p_1)+ 2 Yw(2 w-1)], which has the symmetric hyperbolic form (<ref>), under Assumption 3. Note that the system reduces to the one of <cit.> for p_1≡ p_2.In our system, the equation (<ref>) was obtained from (<ref>) using (<ref>), while (<ref>) came from (<ref>) together with (<ref>). The evolution equation for Λ was derived from (<ref>), and the evolution equation for w from (<ref>) together with (<ref>). Equation (<ref>) came from (<ref>). The evolution equations for μ, X and Y were obtained from (<ref>), (<ref>) and (<ref>), respectively.The constraints are given by the equations (<ref>), (<ref>), (<ref>) and by the derivatives of (<ref>) and (<ref>) with respect to R. They can be expressed in the following way C_1 := ℒ'-e^ΛX=0,C_2 :=R𝒬'+ω e^Q-1=0,C_3 := Re^-Qw'-ω(Y-3w)=0,C_4 := Re^-Qμ'+3ω (μ-ρ)=0,C_5 :=e^-2Λ(ω'-Λω)+Re^-Q[1/6(3ρ-μ)-w(Y-2w)]=0, and, as in <cit.>, it can be shown that for a given C^1 solution {X,Y,U_i} of (<ref>)-(<ref>), the quantities C_1,..,C_5 satisfy a linear system of the formĊ_k=∑_l=1^5A_klC_l,where A_kl are continuous functions of X,Y,U_i. We then conclude that the constraints C_k =0 are satisfied for all T if they are satisfied at T=0.From the initial data ρ̂(R) and v̂(R), using (<ref>), one obtainsℒ(0,R)=ℒ(ρ̂(R)), w(0,R)=v̂(R)/R,𝒬(0,R)=0,and the initial data for μ,Λ,ω,X,Y are specified by (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), respectively. We impose that the quantities μ,Λ,ω,X,Y: (i) satisfy the contraints initially, i.e. C_k (0,R)=0, and (ii) are smooth functions in a region [R_1,R_0], for some R_1>0.To apply Theorem <ref>, we also need the necessary boundary function which we get from (<ref>), considering Φ(T,R_0)=0, as:Ỹ(T)=Y(T,R_0)=-[ℒ̇/s_1(ℒ)](T,R_0) -2w(T,R_0)[p_2 (ℒ)-p_1 (ℒ)/ρ(ℒ)-p_1 (ℒ)](T,R_0),which, using (<ref>), can be rewritten asỸ(T)=-[ℒ̇/s_1(ℒ) +2/3(μ̇/μ+p_1(ℒ))p_2(ℒ)-p_1(ℒ) /ρ(ℒ)-p_1(ℒ)](T,R_0). This function will be known given the two smooth boundary functions p̃_1(T) and w̃ (T) (or μ̃(T)) which should satisfy the corner conditions, as described in Section 2.2.We are now in the position of applying Kind-Elhers' theorem to (<ref>)-(<ref>), given the above initial and boundary data, and this proves existence and uniqueness of solutions to the initial boundary value problem in a neighbourhood of the boundary. In detail, we get the following result:Consider a fluid matter field satisfying Assumptions 1, 2 and 3 with s_1^2>0. Then, the system (<ref>)-(<ref>) is a FOSH system of the form (<ref>) for the variables X,Y and U_j={μ,Q, ω, w, Λ, L}. Suppose that the initial data for those variables is smooth on [R_1,R_0], for some R_1>0, and that the constraints C_1,..,C_5 are satisfied at T=0. Suppose a spherically symmetric distribution of such matter is given together with the smooth boundary functions p̃_1(T) and w̃(T) (or μ̃(T)). Suppose that the given initial and boundary data for Y and their time derivatives satisfy the corner conditions at (0,R_0). Then, there exists locally in time a unique smooth solution to the Einstein equations in T. A well known case is that of a Kottler spacetime exterior (<ref>) which can be attached to the fluid if and only if p̃_1(T)=-Λ, where Λ∈. A way to see this using our framework is as follows: The conditions of the continuity of the first and second fundamental forms across the boundary (see the appendix) imply the continuity of the normal component of the pressure p̃_1(T)=-Λ (this is also a well known consequence of the so-called Israel conditions). Due to the Einstein equations and the continuity of the areal radius, this condition turns out to be equivalent to the continuity of the mass through the boundary μ̃(T)r̃^3(T)= 6m, where m is the Kottler mass.§.§ Case p_1=s_1^2=0 While the constraints equations C_2,..,C_5 remain the same in the case p_1=s_1^2=0, the quasi-linear system (<ref>)-(<ref>) is modified since X≡ L≡ 0. In that case, we get a semi-linear symmetric hyperbolic PDE system of the formU̇_j = H_j (U_i,R), i,j=1,..,6,for the variables U_j={Y,ω, w,Q, μ, Λ}. An important aspect of this system, compared to the previous one, is that the quantity s_2 does not appear in H_j. This means that we do not need to use Assumptions 1 and 2 in order to close the system, as before. Instead, we make the following assumption:Both ρ>0 and p_2 are smooth known functions ofthe variables {Y,ω, w,Q, μ, Λ, R}.In this case, the characteristics of the system (<ref>) are the vertical lines of constant R and the trapezoidal region T of Figure 1 is now a rectangle. In particular, the boundary is now characteristic. IBVP with characteristic boundaries were investigated, in more generality, by Chen <cit.> and Secchi <cit.> for quasi-linear systems.In our case, the integration along the characteristics gives, at each point (T,R), simplyU_j(T, R) = U_j (0,R)+∫_0^T H_j (U_i, R) dT̅.which, using the methods of Courant and Lax <cit.>, gives a smooth local (in time) solution to the system (<ref>), once smooth initial data Û_j (R) is prescribed. In our case, provided suitable data from an exterior spacetime at the corner (0,R_0), one can integrate from that point to get uniquely the boundary functions Ũ_j(T):=U_j(T,R_0). In this sense, the spacetime boundary is completely fixed by the initial corner conditions.As mentioned in Section 2.2, from the initial data {ρ̂, v̂} we can get h_0 and K_0 from the expressions (<ref>). The corner conditions are the matching conditions evaluated at (0,R_0), and these give the components of h_0 and K_0 at R_0.In particular, this provides the corner data Λ̂_0:=Λ̂(R_0), v̂_0:= v̂(R_0) and v̂'_0:=v̂'(R_0).In turn, this gives the remaining corner data for the systemω̂(R_0)=v̂_0/R_0, Ŷ(R_0)= v̂'_0+2v̂_0/R_0, Q̂(R_0)=0, ŵ(R_0)=1, μ̂(R_0)= -3/R_0^2 (e^-2Λ̂_0-1-v̂^2_0).We summarize this discussion as: Consider a matter field with p_1=s_1^2=0 and satisfying Assumptions 3 and 4. Then, the system (<ref>)-(<ref>) reduces to the form (<ref>) which corresponds to a semi-linear FOSH system for the variables U_j={Y,ω, w,Q, μ, Λ}. Consider the smooth initial data {ρ̂, v̂} for the variables U_j on [R_1,R_0], for some R_1>0, and suppose that the matching conditions to an exterior spacetime are satisfied at (0,R_0). Suppose that the constraints C_2,..,C_5 are satisfied at T=0. Then, there exists a unique smooth solution to the Einstein equations in the rectangle [R_1,R_0]× T_0 for small enough T_0>0. § AN APPLICATION TO SELF-GRAVITATING ELASTIC BODIES In this section, we consider a simple application of the previous result to elastic matter. In this case, the matter does not necessarily satisfy (<ref>) or (<ref>). So, before proceeding, we recall some basic facts about elastic matter adapting the presentation to the particular setting we shall consider in the context of spherical symmetry.The material space X for elastic matter is a three-dimensional manifold with Riemannian metric γ. Points in X correspond to particles of the material and the material metric γ measures the distance between particles in the relaxed (or unstrained) state of the material. Coordinates in X are denoted by y^A, A=1,2,3. The spacetime configuration of the material is described by the map ψ: M ⟶ X, where M denotes the spacetime with metric g. The differential map ψ_∗: T_pM⟶ T_ψ(p)X, which is also called relativistic deformation gradient, is represented by a rank 3 matrix with entries (y^A_μ)=∂ y^A/∂ x^μ, where x^μ are coordinates in M. In turn, the matrix kernel is generated by the 4-velocity vector u satisfying y^A_μu^μ=0.The push-forward of the contravariant spacetime metric from M to X is defined byG^AB= ψ_∗g^μν=g^μνy^A_μy^B_ν,which is symmetric positive definite and, therefore, a Riemannian metric on X. This metric contains information about the state of strain of the material, which can be described by comparing this metric with structures in X, e.g. the material metric. The material is said to be unstrained at an event p∈ M if G_AB=γ_AB at p.The dynamical equations for the material can be derived from the Lagrangian density L=-√(-detg)ρ, whereρ=ϵ h is the rest frame energy per unit volume. The density of the matter, measured in the material rest frame, is given byϵ=ϵ̃(y^a) √(detG^AB), where ϵ̃(y^a) is an arbitrary positive function.The equation of state is defined by the function h=h(y^A, G^AB), which describes the dependence of the energy on the state of strain and specifies the material.The stress-energy tensor for elastic matter can be written as <cit.>T^μ_ ν=ϵ(-h/g_00δ_ 0^μg_0 ν+2∂ h/∂ G^ABδ_ ν^Ag^μ B). In spherical symmetry, the push-forward of the metric g with line-element (<ref>) givesG^AB=ηδ^A_ 1δ^B_ 1+βδ^A_ 2δ^B_ 2 +β̃δ^A_ 3δ^B_ 3,where η=e^-2Λ,β=1/r^2 and β̃=β/sin^2θ. In the unstrained state, one has Λ=0 and r=R, so that η=1 and β=1/R^2.The matter density of the material is, in our case, given byϵ=ϵ_0(R)β√(η),assuming that ϵ̃(y^A)=ϵ_0(R)sinθ, where ϵ_0 is an arbitrary positive function.The energy-momentum tensor can then be expressed byT^μ_ ν=diag(-ρ,p_1 ,p_2 ,p_2),where ρ=ϵ h, p_1 = 2ϵη∂ h/∂η, p_2 = -1/2ϵ r ∂ h/∂ r and h=h(R,η,r). Using (<ref>), we getp_1 = 2η/h∂ h/∂ηρ, p_2 = -1/2r/h∂ h/∂ rρ.This T^μ_ ν for elastic matter falls in the class given by (<ref>) although, in general, we will not have p_1=p_1(ρ) as in Assumption 1, so Theorem 2 does not apply. We will now investigate a particular elastic fluid for which p_1=s_1^ 2=0 and, in that case, we may use Theorem 3. §.§ Magli's ansatz Magli <cit.> found a class of non-static spherically symmetric solutions of the Einstein equations corresponding to anisotropic elastic spheres. These models have vanishing radial stresses and generalize the Lemaître-Tolman-Bondi dust models of gravitational collapse, by including tangential stresses.Assuming that the equation of state for the elastic matter, prescribed by the function h, does not depend on the eigenvalue η in the material space, i.e. h=h(R,r), then it is possible to write the spherically symmetric metric as <cit.>ds^2=-e^2Φdt^2 +r'^2 h^2/1+fdR^2+r^2dΩ^2,whereΦ (T,R)=-∫1/h∂ h/∂ rr' dR+c(T),and c(T) is an arbitrary function, reflecting the invariance with respect to time rescaling, which can be chosen such that Φ(T,R_0)=0, for some R_0>0. One can then obtain a first integral to the Einstein equations ase^-2Φṙ^2=-1+2F/r+1+f/h^2,where F(R)>0 and f(R)>-1 are arbitrary functions which we assume to be smooth on [0,R_0]. With the assumption h=h(R,r), and using the first integral, we also getρ (T,R) =F'/4π r^2 r',p_1 (T,R) = 0, p_2 (T,R) =-r/2h∂ h/∂ rρ :=ℋρ,where ℋ=-r/2h∂ h/∂ r is called adiabatic index (see <cit.>, cf. (<ref>)).A physically interesting example of a spacetime with h=h(R,r) is the non-static Einstein cluster in spherical symmetry, which describes a gravitational system of particles sustained only by tangential stresses <cit.>.To make contact with our formalism, in this case, we get a semi-linear system of the form (<ref>), for the variables U_j={Y,𝒬,Λ, w, ω, μ} with smooth functionsρ ( Q, ω, R) = F'(R) e^ Q/4πω Rp_2( Q, ω, R)= ℋ( Q, R) ρ ( Q, ω, R),which clearly satisfy Assumption 4, and smoothinitial data on [R_1,R_0], R_1>0, asŶ =Ω̂'+2Ω̂/R,whereΩ̂=√(1+f/ĥ^2-(1-2F/R)), 𝒬̂ =0, Λ̂ =1/2ln(ĥ^2/1+f), ŵ = 1/R√(1+f/ĥ^2-(1-2F/R)), ω̂ =1, μ̂ =3(F(R)-F(0))/4π R^3.The free initial data here is given by a smooth function ĥ> 0, i.e. the initial equation of state, and smooth functions F and f which come from the initial density ρ̂ and radial velocity v̂ profiles, respectively, as in Section 2.2. For smooth ĥ, F, f on some interval [R_1,R_0], R_1>0, we are under the conditions of Theorem 3 which ensures existence and uniqueness of solutions in a neighbourhood of the boundary.In this case, we can also obtain similar results in a neighbourhood of the origin R=0, at least for some choices of initial data. This has to be done separately since our evolution system (<ref>) is singular at the center.Consider an open region D including the center, as in Figure <ref>, by considering the original variables of Magli. In that case, the center is regular for r(T,0)=0, smooth functions f and F such that <cit.>f(0)=h^2(0,0)-1 and F(R)=R^3φ(R), with φ(0) is finite,and equations of state satisfying the minimal stability requirement, namely that h has a minimum at r=R <cit.>. Physically, this means that the centre is unstrained and has zero radial velocity as lim_R→ 0ṙ=0.In <cit.>, Magli shows that there are open sets of C^∞ data ĥ, F and f such that the regularity conditions at the centre are fulfilled. From the PDE point of view, this implies that for such data there exists a unique C^∞ solution to (<ref>), for small enough time in a neighbourhood D of the centre. Uniqueness of the spherically symmetric initial data gives the uniqueness of solutions in the region D∩ T.We have then proved: Consider a spacetime with metric (<ref>) and containing elastic matter satisfying (<ref>). Consider smooth initial data F,f,ĥ on [0,R_0] satisfying the (corner) matching conditions to an exterior spacetime at (0,R_0) and the regularity conditions at the center. Then, there exists a unique smooth solution to the Einstein equations on [0, R_0]× T_0, for small enough T_0.A suitable exterior to such elastic spacetimes is given by the Schwarzschild solution, which can be matched initially at R_0. We omit the details of the matching conditions here since they were given in <cit.>.As a final remark, we note that the analysis of this particular IBVP can be taken much further if one has explicit solutions for r. These solutions can be obtained by assuming h=h(r)>0. In this case, equations (<ref>) and (<ref>) decouple and Φ'=-1/hd h/drr', yielding Φ (T,R)=-ln h+ln h(r(T,R_0)), which satisfies Φ(T,R_0)=0. Consider a linear stress-strain relation p_2 = k ρ, where the adiabatic index k satisfies -1≤ k≤ 1 in order to comply with the weak energy condition <cit.>. This choice of stress-strain is equivalent to set h(r)=r^-2k. In that case, one has ṙ^2=2F r^4k-1+(1+f)r^8k-r^4k which is now integrable. Remarkably, for some values of k, such as k=-1/4 and k=1/4, one can obtain explicit solutions which satisfy r(0,R)=R and r(T,0)=0 and have a regular origin for certain choices of initial data <cit.>. This allows e.g. to solve the matching conditions and obtain the boundary hypersurface <cit.>, as well as to study of global properties of the matched spacetime such as the formation of horizons and spacetime singularities <cit.>. Funding: This work was supported by CMAT, Univ. Minho, through FEDER Funds COMPETE and FCT Projects Est-OE/MAT/UI0013/2014 and PTDC/MAT-ANA/1275/2014.Acknowledgments: We thank Robert Beig and Marc Mars for useful comments. We thank the referees for the comments and suggestions. § APPENDIX: MATCHING CONDITIONS IN SPHERICAL SYMMETRY In this appendix, for completeness, we recall some basic facts about spacetime matching theory and write the matching conditions in spherical symmetry adapted to our context. These conditions are known but are of importance here since they provide compatibility conditions for our IBVP problem.Let (M^±,g^±) be spacetimes with non-null boundaries S^±. Matching them requires an identification of the boundaries, i.e. a pair of embeddings Ω^±:S ⟶ M^± with Ω^±(S) = S^±, where S is an abstract copy of any of the boundaries. Let ξ^i be a coordinate system on S. Tangent vectors to S^± are obtained by f^±α_i = ∂Ω_±^α/∂ξ^i though we shall work with orthonormal combinations e^±α_i of the f^±α_i. There are alsounique (up to orientation) unit normal vectors n_±^α to the boundaries. We choose them so that if n_+^α points into M^+ then n_-^α points out of M^- or viceversa. The first and second fundamental forms on S^± are simplyq_ij^±= e^±α_i e^±β_j g_αβ|__S^±, K_ij^±=-n^±_α e^±β_i∇^±_β e^±α_j.The matching conditions (in the absence of shells), between two spacetimes (M^±,g^±) across a non-null hypersurface S, are the equality of the first and second fundamental forms (see <cit.>): q_ij^+=q_ij^-, K_ij^+=K_ij^-. We shall now specify the matching conditions for the case where the interiorhas the anisotropic fluid metric (<ref>) and the exterioris the Kottler spacetime given by metric:g^+ = -( 1 - 2m/ϱ-Λ/3ϱ^2) dt^2 + dρ^2/1 - 2m/ϱ-Λ/3ϱ^2+ϱ^2 dΩ^2.The boundary S^+ can be parametrized as Ω^+={ t = t_0 (λ), ϱ = ϱ_0 (λ) }, where two dimensions were omitted, and we can assume ṫ_0 >0 without loss of generality. Considering our choice Φ(T,R_0)=0 for the interior spacetime, we can take the following parametrization Ω^-={ T = λ, R = R_0}, so that Ṫ=1 and Ṙ=0. Then, the equality of the first fundamental forms on S gives1 =( 1 - 2m/ϱ_0-Λ/3ϱ_0^2 ) ṫ_0^2 - ϱ̇_0^2/1 - 2m/ϱ_0-Λ/3ϱ_0^2 , r(λ,R_0) =ϱ_0(λ)and the equality of the second fundamental forms, on S, impliesΦ' e^2Φ -Λ = (-ṫ_0 ϱ̈_0 + ẗ_0ϱ̇_0+ 3m ϱ̇_0^2 ṫ_0/ϱ_0(ϱ_0 - 2m-Λ/3ϱ_0^3) - m/ϱ_0^2 ( 1- 2m/ϱ_0-Λ/3ϱ_0^2 ) ṫ_0^3),rr'e^-Λ =-ṫ_0(ϱ_0-2 m -Λ/3ϱ_0^3).Then, the matching conditions give expressions for ϱ_0 (λ), t_0(λ) and imply the continuity of the mass μ̃r̃^3=6m through the boundary, as is well known. By further substituting the Einstein equations, those conditions also imply that p̃_1=-Λ. In fact, from p̃_1=-Λ we also recover μ̃r̃^3=6m as mentioned at the end of Section 3.1.Generalising this procedure, consider now the matching where the interior is still the anisotropic fluid (<ref>) and the exterior is the spacetime given by a general spherically symmetric metric:g^+ = - e^2μ(t,ϱ) dt^2 + e^2ν(t,ϱ)dϱ^2 +r̅^2(t,ϱ) dΩ^2.The boundaries S^± are parametrized as before. Then, the equality of the first fundamental forms, for Φ(T,R_0)=0, imply-1 =-ṫ_0^2e^2μ+ϱ̇_0^2 e^2ν, r(λ,R_0) =r̅(t_0(λ),ϱ_0(λ))and the equality of the second fundamental forms gives-Φ' e^-Λ =e^μ+ν (ρ̇ṫ_0^2 μ̇+2ρ̇_0^2ṫ_0μ'+ρ̇_0^3ν̇e^2ν-2μ-ṫ_0^3μ'e^2μ-2ν-2ṫ_0^2ρ̇_0ν̇-ρ̇_0^2ṫ_0ν'),rr'e^-Λ =e^μ+νr̅( ṫ_0e^-2νr̅'+ρ̇_0e^-2μṙ̅̇).The above equations provide ρ_0(λ), t_0(λ) and p_1^-(λ)=p_1^+(λ), at the boundary. Another way to get this last relation is to use the Israel conditions n^μ_- T^-_μν=n^μ_+ T^+_μν at S (see e.g. <cit.>). With this boundary data, and knowing the exterior spacetime, one gets r̃(λ):=r(λ,R_0) from (<ref>) and, therefore, w̃(λ) as mentioned in Section 3.1. 99 Andersson Andersson, L., Oliynyk, T. A. and Schmidt, B. G., Dynamical compact elastic bodies in general relativity, Arch. Rat. Mech. Anal., 220 (2016) 849-887. Bondi Bondi, H., Spherically symmetric models in general relativity, Mon. Not. R. Astron. Soc., 107 (1947) 410. Chen Chen, S., Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary, Front. Math. China, 2 (2007) 87-102; Translated from Chinese Annals of Mathematics Ser. A, 3 (1982) 223-232. Choquet Choquet-Bruhat Y., General Relativity and the Einstein Equations (Oxford University Press, 2009) CBF Choquet-Bruhat Y. and Friedrich H., Motion of isolated bodies, Class. Quantum Grav., 23 (2006) 5941. CH Courant, R. and Hilbert D., Methods of Mathematical Physics, vol. 2, Partial Differential Equations, (Interscience, New York, 1967), ch V, sec 7. Datta Datta, B. K., Non-static spherically symmetric clusters of particles in general relativity, Gen. Rel. Gravit., 1 (1970) 19. KE Kind, S. and Ehlers, J., Initial-boundary value problem for the Einstein equations, Class. Quantum Grav., 10 (1993) 2123. MagliI Magli, G., Gravitational collapse with non-vanishing tangential stresses: a generalisation of the Tolman-Bondi model, Class. Quantum Grav., 14 (1997) 1937. MagliII Magli, G., Gravitational collapse with non-vanishing tangential stresses II: A laboratory for cosmic censorship experiments, Class. Quantum Grav., 15 (1998) 3215. Mars-Seno Mars, M. and Senovilla, J. M. M., Geometry of General Hypersurfaces in Spacetime: Junction Conditions, Class. Quant. Grav., 10 (1993) 1865-1897. R Rendall, A. D., The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys., 33 (1991) . Sarbach Sarbach, O. and Tiglio M.,Continuum and discrete initial-boundary-value problems and Einstein's field equations, Living Rev. Relativity, 15 (2012) 9. Secchi Secchi, P., Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 134 (1996) 155. SW Singh, T. P. and Witten, L., Spherical gravitational collapse with tangential pressure, Class. Quantum Grav., 14 (1997) 3489. Kramer Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E., Exact solutions of Einstein's Field Equations (Cambridge University Press, 2nd edition, 2006). | http://arxiv.org/abs/1707.08593v1 | {
"authors": [
"Irene Brito",
"Filipe C. Mena"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170726180813",
"title": "Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure"
} |
[cor1]Corresponding author add1]Manuel Zingl [add1]Institute of Theoretical and Computational Physics, Graz University of Technology, NAWI Graz, Petersgasse 16, 8010 Graz, Austria add1]Martin Nuss add1]Daniel Bauernfeind add1]Markus Aichhorncor1 [email protected] Recently solvers for the Anderson impurity model (AIM) working directly on the real-frequency axis have gained much interest. A simple and yet frequently used impurity solver is exact diagonalization (ED), which is based on a discretization of the AIM bath degrees of freedom. Usually, the bath parameters cannot be obtained directly on the real-frequency axis, but have to be determined by a fit procedure on the Matsubara axis. In this work we present an approach where the bath degrees of freedom are first discretized directly on the real-frequency axis using a large numberof bath sites (≈ 50). Then, the bath is optimized by unitary transformations such that it separates into two parts that are weakly coupled. One part contains the impurity site and its interacting Green's functions can be determined with ED. The other (larger) part is a non-interacting system containing all the remaining bath sites. Finally, the Green's function of the full AIM is calculated via coupling these two parts with cluster perturbation theory.Anderson impurity model exact diagonalization cluster perturbation theory bath optimization § INTRODUCTIONThe single-orbital Anderson impurity model (AIM) <cit.> can be represented exactly by an interacting site coupled to a bath of infinitely many non-interacting sites. In approaches based on exact diagonalization (ED), the number of sites in the interacting system is restricted, and thus the bath needs to be truncated <cit.>. This is a delicate step, because no unique procedure exists. Different ways are used, e.g., fits on the Matsubara axis or continuous fraction expansions <cit.>. Various methods improving on ED have been presented in recent years, e.g., the variational exact diagonalization <cit.>, the distributional exact diagonalization <cit.> and methods based on a restriction of the basis states <cit.>. Another way of going beyond ED is the use of cluster perturbation theory (CPT) <cit.>, i.e. the more advanced variational cluster approximation (VCA) <cit.>, as a solver for the AIM <cit.>. From now on, we assume a single-orbital AIM coupled to a finite but large bath of L-1 non-interacting sites. The basic idea of using CPT as an impurity solver is to separate the L-site AIM into a cluster of size L_C, which includes the impurity site and L_C-1 bath sites, and a non-interacting system consisting of the remaining bath sites. In general, the non-interacting Green's function is specified by the Hamiltonian H^0, that is a matrix in orbital space of size L × L. For illustration purposes (see the sketch in Fig. <ref>), we denote the upper left L_C× L_C block in H^0 as the interacting cluster, subsequently H^0_C. The remaining, lower (L-L_C)× (L-L_C) block describes the remainder of the bath, subsequently H^0_R. Additionally, there are two off-diagonalblocks T connecting H^0_C and H^0_R.The onsite Hubbard interaction H_U = U n_I,↑ n_I,↓, where I labels the impurity site, is now added to the cluster Hamiltonian, H_C=H_C^0+H_U. There are no interactions in the bath degrees of freedom, hence H_R^0 remains unchanged.In CPT both Hamiltonians (H_C and H^0_R) are solved exactly for their single-particle Green's functions G_C(ω) and G_R(ω). To obtain G_C(ω) we use the Lanczos procedure at zero temperature <cit.>.Note that G_R(ω) = G^0_R(ω), as the remainder of the bath is a non-interacting system. Subsequently, the two systems are joined to yield the single-particle Green's function of the full system G(ω) via the CPT relation <cit.> G^-1(ω)= ([ G_C(ω)0;0 G_R(ω);])^-1-Vwhere V is a L× L coupling matrix consisting only of the T blocks. Eq. <ref> is exact in the case of a non-interacting system (U=0). In the interacting case, the CPT relation is no longer exact, but a result of perturbation theory in V. CPT approximates the self-energy of the full system by the self-energy of the interacting cluster.In general, the non-interacting bath can always be transformed to a tridiagonal representation via a Lanczos tridiagonalization, yielding a chain representation of the AIM. This representation straight forwardly allows to define the separation of the interacting cluster and the remainder of the bath. However, the situation is not so clear in other representations. Consider for example the case of a star geometry, where all bath sites couple directly to the impurity site.Incorporating just a random set of these star sites into the interacting cluster will lead to a poor discretization of the bath, and hence a poor self-energy. Any unitary transformation on the non-interacting bath degrees of freedom leaves the physics of the interacting AIM invariant. However, such a transformation will influence the self-energy of the interacting cluster significantly, since it changes the cluster Hamiltonian H_C. Additionally, such transformations will also alter the off-diagonal block T, rendering the resulting perturbation in some cases larger than in others. There exist an infinite number of representations which all describe the non-interacting bath exactly and which are related via unitary transformations. However, the CPT method itself suggests which baths might be the best: Those which minimize the off-diagonal perturbative elements in T. The key idea of this work is to use unitary transformations to find those bath representations with minimal couplings between the cluster and the remainder of the bath.In the following, we outline a way to construct CPT-favorable bath representations in Sec. <ref>, and present results for a L=64 AIM with a semi-circular particle-hole symmetric bath in Sec. <ref>.§ METHODThe general form of the non-interacting Hamiltonian for an L-site AIM isH_0= ϵ_I c^†_I c_I^ + ∑_i=1^L-1(t_iI c^†_i c_I^ +h.c.) +∑_i, j = 1^L-1 t_ij c^†_i c_j^ ,where the impurity is denoted by the index I and the L-1 bath sites by i and j. We omit the spin indices.To obtain H_0 for an L-site system one can use a star representation, where each bath site couples only to the impurity site. Then, the parameters of H_0 can be determined by a discretization of the non-interacting bath DOS into equally spaced intervals. Each interval is represented by a delta peak, where the energy positions of the delta peaks correspond to the on-site energies and the hopping parameters are obtained from the spectral weights in the intervals. Of course, the higher the number of bath sites the better the result of this discretization.Under a unitary transformation R, performed in the bath only, withc_i^ =∑_α R_iα d_α^ and c_i^† = ∑_α d_α^† R^*_α i, where R R^† =, the transformed Hamiltonian readsH'_0= ϵ_I c^†_I c_I^ + ∑_α=1^L-1(h_α I d_α^† c_I^ +h.c.) + ∑_α, β = 1^L-1 h_αβd_α^† d_β^ .The parameters of the Hamiltonian transform likeh_α I = ∑_i R^*_α i t_i I andh_αβ =∑_i,j R^*_α i t_ij R_jβ. Such a transformation leaves the impurity state I and consequently ϵ_I invariant.We define an energy of a certain bath representation via the 2-norm of the off-diagonal blocks TE = 1/N_T∑_i,j|T_ij|^2where the number of elements in T is N_T = L_C· (L-L_C). Transformations on the bath degrees of freedom included in the interacting cluster do not influence the resulting self-energy. The same is true for transformations performed only in the remainder of the bath. This imposes a constraint on the energy E, namely, it has to be invariant with respect to such transformations, which is indeed fulfilled by the 2-norm. The aim is now to find an optimal bath representation for CPT by minimizing the energy E. Since the configurationspace of T_ij is high dimensional, we usea Monte Carlo procedure. Initially, we perform global updates in all dimensions with random rotation matrices to obtain a randomized starting representation of H_0. Then, we move through the space of possible H_0 by proposing random local updates R. In general, any unitary update would be allowed,but here we restrict ourselves to two-dimensional rotation matrices for thelocal updates R = ([ 1 0 … 0 0 … 0 … 0; 0 1 … 0 0 … 0 … 0; … … … … … … … … 0; 0 0 …cos(ϕ) 0 … -sin(ϕ) … 0; 0 0 … 0 1 … 0 … 0; … … … … … … … … 0; 0 0 …sin(ϕ) 0 …cos(ϕ) … 0; … … … … … … … … …; 0 0 … 0 0 … 0 … 1; ]) A local update matrix R(i,j,ϕ) is drawn by choosing two random integers i,j ∈ [1,L-1] representing the plane of rotation and one rotation angle ϕ∈ [0,2π[. A new representation with energy E' is accepted with probability p = min(1, e^-γ(E'-E)). We use simulated annealing to obtain low-energy CPT bath representations by increasing the parameter γ. Although bath rotations leave the particle-hole symmetry invariant on the L-site H_0, they destroy it on the L_C-site cluster. Therefore,as shown in Fig. <ref>, we split the bath sites into an equalamount of positive (blue elements) and negative energy (red elements) sites and one zero mode (green 0). Updates are performed simultaneously on the positive and negative modes which leaves the whole bath, the bath in the cluster as well as the remaining bath particle-hole invariant. To avoid a Kramers-degenerate ground state, clusters with an even number of sites L_C are chosen. This implies that one bath site (the zero mode) is exactly located at zero energy. Zero mode updates cannot be achieved by two-dimensional rotations without breaking the particle-hole symmetry of the cluster, but would rather require special unitary transformations involving at least three bath sites. For the proof of concept presented here, we refrain from updatingthe zero mode, i.e. the green elements in Fig. <ref> do not change. Hence, the zero mode coupling is determined by the initial discretization of the system. Although this restricts the space of trial bath representations, we leave the zero mode updates for future works.Next to the energy E (Eq. <ref>), which reflects the magnitude of the perturbation, we evaluate the influence of the CPT truncation by comparing the non-interacting single-particle impurity Green's function of the full system G^I_0(ω) to the one considering only the sites in the cluster G^I_0,C(ω). The resulting quantity χ^2_C=∫ dω|G^I_0(ω+iη)-G^I_0,C(ω+iη)|^2reflects the ability of the L_C cluster sites to represent the bath degrees of freedom. A-priori a positive correlation of E and χ^2_C is not ensured but expected. We emphazise that χ^2_C is not used in the algorithm, but only serves as a measurement for the quality of the bath optimization. A numerical broadening of η =0.02 is used to evaluate Eq. <ref>.To asses the quality of the optimization scheme we also perform plain ED calculations for a truncated L=10 system. The Hamiltonian of this 10-sites system (with the Green's function G^I_0,ED(iω_n)) is obtained by fitting G^I_0(iω) on the Matsubara axis with the cost function χ^2_ED=∑_n W_n |G^I_0(iω_n)^-1-G^I_0,ED(iω_n)^-1|^2We employ the simplex search method by Lagarias et al. <cit.> and impose particle-hole symmetry to reduce the number of fit parameters. A Matsubara grid with 1024 points at a fictitious temperature corresponding to β =100^-1 is used, but the ED solution itself is obtained at zero temperature.The cost function Eq. <ref> is a heuristic choice and can also take various other forms, e.g. with a different definition of the distance or a different weight W_n (we set W_n = 1/ω_n) <cit.> .This leads to an ad-hoc determination of the bath parameters and introduces some ambiguity to the solution of the AIM. For a given number of bath sites also the discretization on the real-frequency axisis not uniquely defined, e.g. we could use unequally spaced energy intervals.In contrast to ED, where usually only a small number of bath sites is fitted on the Matsubara axis via the minimization of Eq. <ref>, the discretization on the real-frequency axis can be easilyperformed for a large number of sites. Indubitably, the ambiguities in determining the AIM parameters are less severe for larger system sizes.§ RESULTS Here, we discuss a single interacting impurity in a particle-hole symmetric semi-circular bath with a half-bandwidth of 1. For the bath optimization we use a discretized system with a total number of L=64 sites and an interacting cluster of L_C=10. The interacting cluster includes the impurity site, one zero mode, and four additional positive and negative bath sites each. Fig. <ref> shows that smaller perturbative elements E correlate positively with smaller χ^2_C, indicating a better representation of the non-interacting bath by the sites contained in the interacting cluster. We compare intermediate bath representations (black crosses) to the random initial representation (orange star), a star representation by choosing ten sites at random to enter the interacting cluster (cyan triangle), and the chain representation cut after 10 sites (green circle). Note that the chain representation hosts only one perturbative matrix element, which is however large. In all cases one finds a higher χ^2_C with respect to the final result of the optimization (blue cross).Of course, due to the optimization the number of elements in T grows, but their magnitude becomes tiny compared to the chain or the star representation. In Fig. <ref> the self-energy Σ(iω_n) of the optimized system on the Matsubara axis is shown for different values of U. We compare it to results obtained at an inverse temperature of β=100^-1 with the TRIQS/CTHYB solver <cit.>, which is based on continuous-time quantum Monte Carlo in the hybridization expansion (CTHYB) <cit.>. Additionally, we show the self-energies calculated with plain ED for a L=10 system. As the CPT self-energy of the full AIM is given by the self-energy of the 10-sitecluster, we can only expect it to be compatible with CTHYB on a similar level as the ED self-energies are. For U<=4 the EDself-energies are even slightly closer to the CTHYB result (see Fig. <ref>). Nevertheless, the important message conveyed by Figs. <ref>and <ref> is that the minimization of the off-diagonal block T is a proper procedure to obtain a good representation of thefull AIM system by the 10-site cluster.The actual advantage of CPT is revealed on the level of the spectral function A(ω), which results from coupling the interacting clusterto the remainder of the non-interacting bath. In Fig. <ref> we show the CPT spectral function (red) of the optimized system (L=64) for interaction values U=0, 1, 2,and 3. We compare our results to spectral functions obtained with the density matrix renormalization group (DMRG) <cit.> and real-time evolution (as inRef. <cit.>) using matrix product states (MPS), colored in black.This approach is able to provide an excellent spectral resolution on the whole real-frequency axis <cit.>. We use the star representation of the L=64 AIM for the MPS calculation with a truncated weight of 10^-10. Additionally, we show the results of the 10-site ED system (gray). In contrast to ED, which exhibits strong finite size effects, the CPT optimization scheme provides smoother spectral functions which are in much better agreement with the MPS results. This difference is particularly pronounced for the smallerU values, which is a consequence of CPT becoming exact for U→ 0. Up to the influence of the discretization, CPT reproduces the non-interacting spectral function for U=0(see top grap in Fig. <ref>). Of course, for higher values of U the energy resolution is limited by the size of the exactly solved system (L_C), which can be improved on by including more sites in the interacting cluster. § CONCLUSIONIn this work we introduced a bath optimization scheme for the Anderson impurity model. Using unitary transformations in the bath degrees of freedom, we minimize the coupling between a small cluster containing the interacting impurity site and the remaining sites of the bath. These transformations leave the impurity DOS of the non-interacting bath invariant. In general, the proposed scheme can be useful for all CPT-based methods when parts of the entire system are non-interacting, but it does in principle also provide a guideline to construct finite-size representations of hybridization functions as needed, e.g., in the framework of dynamical mean-field theory. For a large enough number of bath sites, the initial AIM can be obtained directly on the real-frequency axis, and thus a fit on the Matsubara axis can be avoided. In this work we have presented a proof of concept, but anticipate to explore the bath optimization scheme for systems without particle-hole symmetry and multi-orbital impurity models.§ ACKNOWLEDGMENTSThe authors acknowledge financial support from the Austrian Science Fund (FWF) through SFB ViCoM F41 (P04 and P03), project P26220, and throughthe START program Y746, as well as from NAWI-Graz. The CTHYB results were calculated using the TRIQS library <cit.> and the TRIQS/CTHYB solver <cit.>.The MPS results were obtained using the ITensor library <cit.>. § REFERENCESelsarticle-num | http://arxiv.org/abs/1707.08841v2 | {
"authors": [
"Manuel Zingl",
"Martin Nuss",
"Daniel Bauernfeind",
"Markus Aichhorn"
],
"categories": [
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"primary_category": "cond-mat.str-el",
"published": "20170727124859",
"title": "A real-frequency solver for the Anderson impurity model based on bath optimization and cluster perturbation theory"
} |
=1 | http://arxiv.org/abs/1707.08980v1 | {
"authors": [
"Matti Heikinheimo",
"Kristjan Kannike",
"Florian Lyonnet",
"Martti Raidal",
"Kimmo Tuominen",
"Hardi Veermäe"
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"categories": [
"hep-ph",
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"primary_category": "hep-ph",
"published": "20170727180246",
"title": "Vacuum Stability and Perturbativity of SU(3) Scalars"
} |
apalike#1 1 1Gaussian Processes for Individualized Continuous Treatment Rule Estimation Pavel Shvechikovand Evgeniy Riabenko The reported study was funded by RFBR, according to the research project No. 16-07-01192 А. Department of Computer Science, Higher School of Economics December 30, 2023 ===================================================================================================================================================================================================== 0 Gaussian Processes for Individualized Continuous Treatment Rule EstimationIndividualized treatment rule (ITR) recommends treatment on the basis of individual patient characteristics and the previous history of applied treatments and their outcomes. Despite the fact there are many ways to estimate ITR with binary treatment, algorithms for continuous treatment have only just started to emerge. We propose a novel approach to continuous ITR estimation based on explicit modelling of uncertainty in the subject’s outcome as well as direct estimation of the mean outcome using gaussian process regression. Our method incorporates two intuitively appealing properties – it is more inclined to give a treatment with the outcome of higher expected value and lower variance. Experiments show that this direct incorporation of the uncertainty into ITR estimation process allows to select better treatment than standard indirect approach that just models the average. Compared to the competitors (including OWL), the proposed method shows improved performance in terms of value function maximization, has better interpretability, and could be easier generalized to multiple interdependent continuous treatments setting.Keywords: Dose Finding, Gaussian Process Regression, Personalized Medicine 1.45 § INTRODUCTION Many clinician studies conclude that reactions to a fixed treatment could vary significantly among individuals in a population <cit.>. Such heterogeneity of treatment effects <cit.> raises a question about the applicability of this "one size fits all" approach to clinical practice and leads naturally to personalized medicine paradigm – the idea of healthcare improvement as a result of tailoring treatments on the basis of patient individual characteristics.§.§ Treatment rulesOne of the crucial concepts in personalized medicine research is individualized treatment rule (ITR) (aka individualized treatment regime) which is usually defined as a mapping from measured patient characteristics to recommended treatment <cit.>. An obvious goal of ITR construction is a maximization of patients' response to treatment averaged over all individuals in a population. This goal is sometimes referred to as expected reward in a population or value function. ITR estimation is a procedure which uses data from either observational or randomized clinician studies to fulfill this goal. Personalized medicine often comes in one of the two forms <cit.>: either identification of the subgroups of patients who benefit from a given treatment, or identification of the optimal treatment for a specific patient. The latter form (which we also adhere to in this work) could be further subdivided by the approach to ITR estimation: 0em * Indirect approach usually consists of a two-stage procedure. First, modeling of either conditional mean outcomes or contrasts between mean outcomes takes place. Second, the ITR is derived as the rule assigning the treatments that maximize the model of the conditional mean outcome. * Direct approach approximates expected reward in a population with its sample estimate as a function of ITR. Maximization of this expected reward estimate with regard to parameters of ITR yields the desired treatment rule.Specific ITR estimation methods differ a lot depending on whether the treatment is discrete or continuous. In the latter case ITR is also called individualized dose rule (IDR) and is of a particular practical importance because of its usage in therapies balancing drug dose between insufficient (not causing any effect) and excessive (causing severe adverse effects). One example of such dose administering therapies is thrombosis prevention using a drug called warfarin, which is shown to be more effective when prescribed on the basis of environmental and genetic patient factors <cit.>. Another illustrative example is a treatment of inflammatory bowel disease with thiopurines <cit.>.These examples motivate a strong need in developing reliable and efficient methods for ITR estimation. There is a substantial amount of literature devoted to ITR estimation dealing with discrete (mostly binary) treatment using an indirect approach <cit.> as well as a direct one <cit.>. However, despite its importance, as far as we know, there are very few methods dealing with IDR estimation. IDR estimation poses additional challenges because the set of admissible treatments has a cardinality of the continuum. There are infinitely many treatments to choose from, only a few of which might be actually present in the data. These difficulties yield trivial generalization of discrete methods to continuous treatment cases either impossible or intractable. One of the recent works devoted to IDR estimation <cit.> follows a direct approach with the focus on policy interpretability, which is achieved through the use of decision trees with splits guided by complicated purity measure. This split criterion is based on the usual value function from discrete ITR with additional smoothing. The proposed smoothing is analogous to introducing a stochastic policy with probability mass concentrated around the deterministic policy assignments observed in the training data. An interesting peculiarity of the work is that only treatments observed in the training data are considered as a split candidates. Another recent approach to IDR estimation <cit.> is devoted to developing an extension of O-learning algorithm <cit.>. It resembles in spirit the work of <cit.> because of chosen direct approach to IDR estimation and the presence of smoothing in value function estimate. However, unlike in <cit.>, the smoothing is used only as an intermediate step required for optimization purposes, so that the annealing of the kernel bandwidth to zero gives precisely the same value function as is used for discrete ITR estimation. Specifically, <cit.> introduced a regression based approach inspired by the <cit.> and SVM; the proposed algorithm comprises global optimization task tackled by DC procedure which is known to depend on initialization and could guarantee only local convergence. All of the aforementioned algorithms follow the direct approach to IDR estimation consisting of maximizing mean conditional reward over individuals in a population (aka value function), but do not deal with variance of the patients' potential outcome, which might be crucial for a real world application. Consider a simple example when patient is told to choose between two mutually exclusive options:0em * Take a drug A with 0.99 chance of recovery (reward = 100) and 0.01 probability to acquire physical disability (reward = -1000). * Take a drug B with 0.85 chance of recovery (reward = 100) and 0.15 probability not to experience any effect (reward = 0). Mean reward in the first case is 89 while in the second one it is 85, suggesting that rational patient should always choose the first option. However, it is known that people are not rational and tend to averse losses <cit.>. This common behavior was also shown to be true in a health domain <cit.>. Therefore, it seems very natural to explicitly incorporate reasoning about mean and variance of a possible outcome into an ITR. Nevertheless, such reasoning is impossible when using direct approach.§.§ Indirect methodsOur work aims to develop an indirect approach to ITR estimation. Indirect methods are usually criticized for potential overfitting during the first stage (fitting regression model to rewards) and for possible model misspecification.In defense of indirect methods, we argue that the concerns about misspecification could be reduced to a great extent through the use of a flexible model that could fit the data well <cit.>. However, the more powerful model is, the more it is susceptible to overfitting. To cope with the overfitting problem on the level of model design, we have chosen to follow Bayesian paradigm, which allow to do natural regularization through the averaging over model parameters and is known to be free of overfitting problem <cit.>. Nevertheless, any Bayesian model becomes prone to overfitting as soon as optimization with regards to hyperparameters is involved (so-called type II maximum likelihood procedure). Careful treatment of overfitting requires either very small number of hyperparameters or fully Bayesian inference with Markov chain Monte Carlo methods <cit.>.Another argument in favor of indirect methods is their capacity to estimate the uncertainty of the expected outcome. Such estimation is impossible when following direct approach but could be invaluable in practice, providing the means for balancing exploration and exploitation. General task of ITR estimation is similar to the classic reinforcement learning (RL) setting <cit.>: patient characteristics, their treatments and responses are direct analogs of state, action and reward concepts from RL literature. Moreover, direct and indirect approaches to ITR estimation in RL community are called value-based and policy-based approaches. Recently, it was shown <cit.> that these two approaches share much more in common than it was believed before. To some extent this finding reconciles two different ITR estimation approaches and gives rise to an unexplored idea of their combination. The contributions of this work are summarized as follows.0em * We propose a new indirect approach to IDR estimation that benefits from separation of uncertainty due to finite sample estimation and irreducible variance of patient outcomes. * We show how to reduce the hard task of continuous inversion to simple univariate global optimization problem which is well-known in active learning and Bayesian optimization literature. * We evaluate the proposed method on various simulation studies, confirming its superior performance for randomized dose-finding clinical trials. § PROPOSED METHODWe assume that train data comes from randomized dose trial and every training record consists of three major components:0em * patient covariates ∈̧ℝ^p; * treatment doze a ∈ [0,1], selected at random within safe bounded dose range, which w.l.o.g. is assumed to be equal to [0, 1] interval; * patient outcome r ∈ [r_lo, r_hi], r_lo, r_hi∈ℝ. Note that we do not restrict the sign of outcomes but do require them to be bounded. We also introduce a shorthand notation _i = [_̧i, a_i] for a concatenation of covariate vector and a treatment value.As was mentioned previously, indirect approach to IDR estimation requires building a model of the outcome value given patient covariates and treatment doze.One of the models meeting all criteria of being sufficiently flexible, robust to overfitting due to Bayesian nature, and able to make probabilistic inference is Gaussian process (GP) regression model <cit.>, which is the basis of this work.§.§ Gaussian process framework We proceed with minimal introduction to GP regression. For more thorough discussion one could refer to <cit.>. Gaussian process is a collection of random variables any finite number of which have joint Gaussian distribution. A GP is completely specified with its mean and covariance functions which we will denote as m() and k(, ') respectively. For clarity and without loss of generality <cit.> we will consider Gaussian process f with zero mean m() = 0 and denote f() ∼𝒢𝒫(0, k(, ')). We assume that we were given a training dataset {(_i , y_i ) | i = 1,..., n} that consists of pairs of object description _i and noisy observation of function realization, i.e. y_i = f(_i) + ε, where the noise ε∼𝒩(0,σ_n^2 ).Having specified a joint Gaussian prior over the test output f_* and noisy train outputs y, the inference is carried out with conditioning on observed train outputs. This procedure allows to compute the mean and the variance of f_* posterior distribution given the test point _* using the following equations:f_*∼𝒩(f̅_*, f_*)f̅_*= k_*(K + σ^2_n I)ıy f_* = k(_*, _*) - k_*(K + σ^2_n I)ık_*,where k_* = (k(_*, _1), ..., k(_*, _n)) and covariance matrix K has entries K_ij = k(_i, _j). To reduce clutter we will subsequently denote (K + σ^2_n I)ı as Λ. One of the main constitutes of GP flexibility is covariance function. The choice of covariance function for particular application can be guided by Bayesian model selection <cit.>, however, for illustration purposes we have chosen probably the most commonly used squared exponential function with separate length-scales. We also extend this covariance function with the noise (aka nugget) term:k(_p, _q)= σ_f^2 exp{ - ∑_i=1^p (x_pi - x_qi)^22θ_i} + σ_n^2 δ_pq,where θ_i is called length-scale, x_pi is a p-th feature of i-th object and δ_pq is the Kronecker delta function. Note that all of the subsequent derivations could be replicated with other popular choices of covariance function (i.e. the Matérn class) and are not unique to squared exponential covariance function. §.§ Method formulation Motivation of the proposed method is as follows.To solve the prescription problem for a new patient with characteristics _̧*, doctor, given the history of previously seen patients {(_̧i, a_i, r_i)}_i=1^n, have to simultaneously solve two conflicting tasks:0em * Doctor has to be confident about potential dose effect and this confidence should be based on previous cases. The more similar are patient characteristics _̧* to the ones from previous practice _̧i, the more doctor is sure that giving the same a_i to patient with _̧* will result in r_i. * Doctor has to prescirbe the best treatment possible. The bigger is r_i, the more doctor wants a_* to be similar to a_i. These two intuitive observations coupled with the notion of loss aversion in health domain <cit.> motivate prescription of the dose which maximizes probabilistic lower surface for the potential outcome. That is, in terms of mean and variance of posterior distribution over f_*a_* = _a f̅_* - s √(f_* ), where s is a scaling coefficient controlling the penalty for the uncertainty in the outcome. We argue that this simple heuristic, which hereinafter will be referred to as LCSL (lower confidence surface learning): 0em * is a strong IDR competitive with the previously proposed approaches; * provides the means to account for loss aversion and uncertainty; * provides the means to perform adaptive trial design; * allows to explicitly address an exploration-exploitation tradeoff. §.§ Modelling correlation with exponential power family Formula (<ref>) does not gives us a method to perform maximization over treatment domain.So we need to design a constructive way to carry out such an inversion.Lets rewrite each of the terms in (<ref>) separately assuming that covariance function is squared exponential with separate length-scales.For the sake of brevity we omit length-scales in following derivationsk(_p, _q)= σ_f^2 exp{ -||_p - _q||^2_22}= σ_f^2 exp{ -||_̧p - _̧q||^2_22}exp{ -||a_p - a_q||^2_22}Lets denoteK(_̧*) = σ_f^2 ·diag( exp{ -||_̧* - _̧1||^2_2/2}, … , exp{ -||_̧* - _̧n||^2_2/2})Then one could write f̅_* in a form of a linear combination of exponential terms where α_i depends only on the train data and known covariates _̧* of test object:f̅_*= [ exp{ -||a - a_1||^22}, … , exp{ -||a - a_n||^22} ]K(_̧*)Λy= ∑_i α_i exp{ -||a - a_i||^22},where α_i = [K(_̧*)Λy]_i.The same transformation could be done for f_*:f_* = k(_*, _*) - k_*Λk_* = k (_*, _*) - ∑_i,jγ_ijexp{ -||a - a_i||^2 + ||a - a_j||^2/2}Here γ_ij = σ_f^4 Λ_ijexp{ -||_̧* - _̧i||^2 + ||_̧* - _̧j||^2/2} and ∀x_* k( _*, _*) = σ_f^2, so neither β_ij nor k( x_*, x_*) depends on unknown a.To sum up, our optimization task could be simplistically formulated as followsmax_a ∑_i α_i exp{ -(a - a_i)^2/2} - s √( k( _*, _*) - ∑_i,jγ_ijexp{ -(a - a_i)^2 + (a - a_j)^2/2})§.§ Global optimization As is easily seen, the objective (<ref>) is a sum of unnormalized exponential functions with coefficients α_i, γ_ij of arbitrary sign. Thus, to allow for explicit maximization of this nonconvex objective, one need to employ global optimization methods. Even though the task (<ref>) is a very hard one, it is surely not a new one – problems of this type are inevitable in Bayesian optimization <cit.>, arising during maximization of the so-called acquisition function. There are many methods to solve global optimization problem of the form (<ref>), i.e. DIRECT <cit.>, which is derivative free and deterministic. In this work we adhere to a simpler yet efficient two-step approach to global optimization that is often implemented in Bayesian optimization frameworks <cit.>. First, seeds are generated – either uniformly at random, in a form of fixed step grid or as a Sobol' sequence <cit.>. Second, function is evaluated at each seed and (optionally) local optimization method is started with each seed as an initial point. Frequent choice of such local optimization algorithm is L-BFGS-B <cit.>.§.§ Interpretability of the model Flexibility of any model usually comes with the price of reduced interpretability which is often of particular interest for practical applications.However, as is evident from formula (<ref>), mean predictions of GP regression are additive with respect to covariance terms k_*.This means that, given test object _*, one could argue which objects from the train dataset influence mean prediction the most based on the components of covariance vector k_* and coefficients [Λ y]_i. The same reasoning holds for variance interpretation with the only difference of quadratic dependence replacing the linear one. Additionally, the learned values of length-scales could serve as indicators of feature importances, giving rise to the so-called automatic relevance determination (ARD) effect.These simple model interpretations coupled with the recent work devoted to the tradeoff between accuracy and interpretability <cit.> along with possible interpretability increase due to additive GP modelling <cit.> allow to get even closer to interpretability level of linear and tree-based models. §.§ Applicability to observational dose trials The requirement of training data being obtained from randomized study is by no means necessary. Under usual assumptions in casual inference literature <cit.> the proposed method could be seen as a model of counterfactual mean outcome. These assumptions are as follows: 0em * Conditional exchangeability (aka strong ignorability) – potential outcomes R^a are conditionally independent of A given covariates C, i.e. ∀ a, R^a ⊥ AC. * Positivity – there is nonzero probability of assigning any of the treatment levels to any of possible objects, i.e. ∀ Cp(C) > 0, ∀ a p(aC) > 0. * Consistency – each observed outcome is precisely the one that would have been observed under given treatment level, i.e. R = R^A.In case these assumptions hold true, structural model R^aC = c of counterfactual outcome R^a is equal to conditional mean model RC = c, A = a. This imply that the proposed method falls under umbrella of outcome regression causal inference models.§ NUMERICAL RESULTS All experiments, unless stated otherwise, were conducted as follows. Train and test sample generation was repeated 50 times for each combination of scenario, train sample size, algorithm and variance penalty factor. Test sample size was fixed at the value of 1000. Rewards in train dataset were scaled to fit into [0,1]. For each train dataset we fit a Gaussian process regression that was allowed to internally restart optimization of marginal likelihood 10 times. The model maximizing marginal likelihood across these 10 restarts is chosen for subsequent evaluation on test data. Approximate model performance is obtained by plugging predicted optimal dose for test objects into the true underlying state-action value function Q(, A). Method quality is reported in terms of 𝒱̂(f), which is defined as approximate model performance averaged over all data generations. We have empirically observed that the choice of a particular global optimization method out of those described in section <ref> does not change performance much. Thus, we report all of the results using the simplest one – fixed step grid evaluation with number of grid point fixed at 50 for all of the simulation scenarios and settings. In this section we refer to our method as LCSL.X, where X specifies the value of variance penalty (<ref>) by the equation s = Φ^-1(X / 100), where Φ is normal CDF. All our experiments are built on top of Gaussian process framework <cit.>.§.§ Simulation study In this section we first present results of two simulation experiments which show that for simple scenarios our method works remarkably well.We denotefor patient covariates matrix, i.e. = [_̧1, ..., _̧n], where _̧i is a column vector of i-th patient characteristics; A for a column vector of treatments, i.e. A = [a_1, ..., a_n]; C_i for an i-th column ofmatrix.Our first scenario simulates optimal treatment dependence on single covariate in a form of parabola (figure <ref>, left):R ∼ 𝒩( Q_1(, A), 0.01),A ∼ 𝒰(0,1), _̧i ∼ 𝒰(0,1) Q_1(, A) = - 100 · (f^opt_1() - A)^2 f^opt_1() = 4 · (C_1 - 0.5)^2 Our second scenario is a more complex one – it simulates optimal treatment assignment as a highly nonlinear function with abrupt change in the middle of covariate space and local periodicity (figure <ref>, right) :R ∼ 𝒩( Q_2s(, A), 0.01),A ∼ 𝒰(0,1), _̧i ∼ 𝒰(0,1) Q_2(, A) = - 100 · (f^opt_2() - A)^2 f^runge(x) = cos(3 π (4x - 0.3)) / (1 + 25 (4x - 0.55) ^2) f^step(x) = 0.1 x · (10 + sin(20x) + sin(50x) ) - 1.3 f^opt_2() = (f^runge(C_1) + f^step(C_1) ·(C_1 > 0.5)) / 1.5 - 0.7 Because true f^opt() is nonlinear in both scenarios, we don't include linear OWL (L-OWL) from <cit.> into comparison. Kernel OWL (K-OWL) results are obtained by reusing the code from the supplementary material of <cit.>. Simulation results (table <ref>, <ref>) show that the proposed method works extremely well in both scenarios, showing unimprovable results for the first scenario. We proceed with results of LCSL on several more complex and higher dimensional simulations first introduced in <cit.>. In particular, we evaluate and compare our method on three scenarios, first two of which simulate randomized trial and the last one simulates observational trial.Scenario 3, randomized trial: R ∼ 𝒩( Q_3(, A), 1),A ∼ 𝒰(0,2), _̧i ∼ 𝒰(-1,1)^30Q_3(, A) = 8 + 4 C_1 - 2C_2 - 2C_3 - 25 · (f^opt_3() - A)^2 f^opt_3() = 1 + 0.5 C_1 + 0.5 C_2Scenario 4, randomized trial: R ∼ 𝒩(Q_4(, A), 1),A ∼ 𝒰(0,2), _̧i ∼ 𝒰(-1, 1)^10Q_4(, A) = 8 + 4 cos(2π C_2) - 2 C_4 - 8 C^3_5 - 15 · |f^opt_4()-A| f^opt_4() = 0.6 ·(|C_1| ≥ 0.5) + C^2_4 + 0.5 log(|C_7|+1)Scenario 5, observational trial: R ∼𝒩(Q_5(, A), 1),A ∼Trunc𝒩( f^opt_5(), 0, 2, 0.5), _̧i ∼ 𝒰(-1,1)^10Q_5(, A) ≡ Q_4(, A)f^opt_5() ≡ f^opt_4() Results of randomized study simulation (tables <ref>, <ref>) allow to conclude that our method outperforms every other competitor (often by a large margin) in a randomized trial setting. Such superiority originates from the randomization of treatment assignment, which increases the accuracy of global response surface model and, as a result, improves quality of treatment recommendations.However, for simulation of observational study (table <ref>) we observe that out method yields superior results only when given enough data.Small samples do not allow to properly model response surface because all of the points are already concentrated in near optimal regions, giving insufficient information about treatment dependence on covariates. This effect could be mitigated to some extent by large variance penalty (figure 2, last column). LCSL performance dependence on the variance penalty coefficient, denoted by s in objective (<ref>), needs additional elaboration. We have varied s on standard normal quantile grid (from 0.5 to 0.99 quantile with a 0.01 step) and observed a sustainable relationship between penalty value and method performance (figure <ref>). The first interesting effect, equally present in all scenarios, is that strong variance penalty always improves upon mean prediction given very small train data (positive trend on every plot in the first row of figure <ref>). The second observation is that increasing train sample size in randomized studies reduces the optimal variance penalty (first two columns in figure <ref>). This observation could be explained by increase of model accuracy and, as a consequence, diminished need in variance penalty. Note that for more complex setup (scenario 2) optimal penalty shrinks slower than for a simpler one (scenario 1). However, we also report a consistent small improvement in average 𝒱̂(f) irrespective of train data size for small values of penalty (figure <ref>). This effect can be viewed as a regularization, preventing the model of too assertive extrapolation. The last noteworthy remark concerns observational study (last column of figure <ref>). In such setting we observe that large variance penalty increases performance irrespective of train sample size.That is because data generation policy (A ∼𝒩( f^opt_4() , 0.5)) is by design close to optimal. Thus, in observational studies the value of variance penalty should be treated as a proxy of prior knowledge about the degree of data generation policy optimality.We conclude that different setups need to treat variance penalty differently. Randomized studies benefit most when treating variance penalty as a regularization. In contrast, observational studies should treat variance penalty as a means for explicit reasoning about degree of data generation policy optimality. § CONCLUSIONS We have presented an indirect approach to IDR estimation which is based on Gaussian process regression. In spite of obvious difficulties of indirect IDR estimation, we have shown that it is possible to reduce such an intractable problem to a univariate global optimization setting which is ubiquitous in Bayesian optimization and active learning.We have discussed the advantages and disadvantages of the proposed method including explicit probability reasoning about potential outcome and computational challenges arising as a consequence of using flexible nonparametric model. We have also reported various simulation results which not only show that our method is highly competitive with other algorithms of ITR estimation, but uncover unexpected variance penalty interpretations. § FUTURE WORKAs was previously noted, ITR estimation shares many similarities with policy search problem arising in reinforcement learning (RL). This suggests researches dealing with ITR and DTR (dynamic treatment regime) estimation could get inspiration from already existing methods in reinforcement learning world. For example, it would be very interesting to see an adaptation of reward weighted regression <cit.> to direct DTR estimation. Also, an extension of the proposed LCSL method to multi-stage treatment assignments would be of a great interest. Such extension could probably inherit some of the useful ideas from data-efficient RL algorithm called PILCO <cit.>.Diffusion of statistical science and reinforcement learning may bring benefits to all parties and thus is highly desirable. | http://arxiv.org/abs/1707.08405v2 | {
"authors": [
"Pavel Shvechikov",
"Evgeniy Riabenko"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170726121743",
"title": "Gaussian Processes for Individualized Continuous Treatment Rule Estimation"
} |
Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The NetherlandsInstituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The NetherlandsInstituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands Departments of Physics and Integrative Biology, University of California, Berkeley, CA 94720, USA Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands Topological states can be used to control the mechanical properties of a material along an edge or around a localized defect. The surface rigidity of elastic networks is characterized by a bulk topological invariant called the polarization; materials with a well-defined uniform polarization display a dramatic range of edge softnesses depending on the orientation of the polarization relative to the terminating surface.However, in all three-dimensional mechanical metamaterials proposed to date, the topological edge modes are mixed with bulk soft modes and so-called Weyl loops. Here, we report the design of a gapped 3D topological metamaterial with a uniform polarization that displays a corresponding asymmetry between the number of soft modes on opposing surfaces and, in addition, no bulk soft modes. We then use this construction to localize topological soft modes in interior regions of the material by including defect structures—dislocation loops—that are unique to three dimensions. We derive a general formula that relates the difference in the number of soft modes and states of self-stress localized along the dislocation loop to the handedness of thevector triad formed by the lattice polarization, Burgers vector, and dislocation-line direction. Our findings suggest a novel strategy for pre-programming failure and softness localized along lines in 3D, while avoiding extended periodic failure modes associated with Weyl loops. Localizing softness and stress along loops in three-dimensional topological metamaterials Vincenzo Vitelli December 30, 2023 ========================================================================================= § INTRODUCTIONMechanical metamaterials can control softness viaa balance between the number of degrees of freedom of their components or nodes and the number of constraints due to connections or links <cit.>. This balance, first noted by Maxwell <cit.> and later explored by Calladine <cit.>, is termed isostaticity. In isostatic materials, softness can manifest itself via large-scale deformations, for example as uniform Guest-Hutchinson modes <cit.> or via periodic soft deformations, corresponding to so-called Weyl modes <cit.>. Uniform softness can be exploited to create extraordinary mechanical response <cit.>, such as materials with a negative Poisson's ratio <cit.>. Alternatively, localized softness has been programmed into isostatic materials in one and two dimensions via a topological invariant called the polarization <cit.> that controls mechanical response and stress localization <cit.> at an edge [including the edge of a disordered sample <cit.>], an interface, or bound to a moving soliton <cit.>. These mechanical <cit.> examples of topological metamaterials <cit.> exhibit a general feature of topological matter <cit.>: a correspondence between integer invariants in the bulk and response at a boundary. Large-scale and localized deformations are deeply intertwined, as can be seen in demonstrations in which topological edge softness is created or destroyed by applying large uniform strains <cit.>. A combination of topological polarization and localized defects can be used to program in softness or failure at a specified region in the material <cit.>. Although there are a number of examples of isostatic periodic structures in one, two, and three dimensions, the three-dimensional case is unique because all prior realizations of three-dimensional isostatic lattices include large-scale periodic deformations along continuous lines in momentum space <cit.>. These Weyl lines define families of periodic soft modes in the material bulk and contain a number of modes, which scales with the linear size of the structure. As Ref. <cit.> explores, Weyl lines can be useful to create a metamaterial surface with anisotropic elasticity, but in order to create a material whose top surface is much softer than the bottom, it proves necessary to collapse two Weyl lines on top of each other.An alternative would be to find a metamaterial without Weyl lines. However, these Weyl lines are generic and have a topological character which ensures that they cannot be annihilated locally—a single Weyl loop can only be destroyed by shrinking it to a point. This presents a challenge in three-dimensional isostatic metamaterial design:to achieve a gapped topological material analogous to those in two dimensions <cit.>, in which softness can be controlled and localized. In this work, we design gapped topological materials by exploring the parameter space of the generalized stacked kagome lattice.The mechanical response of this gapped structure is characterized by a nonzero topological polarization oriented along the z-axis.This topological polarization P can be exploitedto localize soft modes in the material bulk by introducing topological defects within the lattice structure called dislocation loops. These dislocations are characterized by a topological invariant called the Burgers vector b. Along the dislocation, we show that the topological charge characterizing the softness or rigidity of the lattice (with unit cell volume V_cell) depends on the orientation ℓ̂ of the dislocation line and is given by P·( b×ℓ̂)/V_cell per unit length.§ GAPPING THE STACKED KAGOME LATTICEWe examine the mechanics of metamaterial structures by using a lattice model for the displacements of nodes and strains of the links. Within this set-up, softness corresponds to displacements with small strains, and rigidity to strains that induce only small displacements. We place a particle (or ball) at each vertex and connect the neighboring particles by linear springs. Such models capture the small-strain response of realistic structures that are either 3D-printed from soft polymers <cit.> or assembled from construction sets <cit.> or laser-cut components <cit.>. The mechanics of this ball-spring model are captured via the linearized equation of motionẌ = - D X, where X≡(x_1,…,x_N) is a d × N-dimensional vector containing the displacements of all N particles in d-dimensions relative to their equilibrium positions. For a given lattice geometry, we calculate the dynamical matrix D, which relates the forces exerted by springs to displacements of the particles. For simplicity, we work in units in which all particle masses and spring constants are one. In the linear regime, we can find the dynamical matrix by first relating the N_B-dimensional vector of spring extensions S≡(s_1,…,s_N_B) to the displacements via S=RX. The rigidity matrix R contains dN× N_B entries determined by the equilibrium positions of the particles and the connectivity of the lattice. In combination with Hooke's law, the matrix R lets us calculate the dynamical matrix D via the relationD = R^𝖳R <cit.>. Although the rigidity theory in terms of the matrix R offers an equivalent description to the dynamical matrix, it contains additional physical insight.The null spaces (kernels) of the rigidity matrix and its transpose give us information about lattice mechanics. From the relation S=RX, we note that R contains soft modes, i.e., collective displacements that (to lowest order) do not stretch or compress any of the springs. On the other hand, the matrix R^𝖳 relates forces on particles Ẍ to spring strains via Ẍ = - R^𝖳S. From this relation, one notes that R^𝖳 contains combinations of spring tensions that do not give rise to particle forces. These configurations are dubbed states of self-stress because they define load-bearing states, which put the lattice under a static tension <cit.>. Localized states of self-stress can concentrate uniformly-applied external loads, which can lead to selective buckling in pre-programmed regions <cit.>.To calculate soft modes (states of self-stress), we need to numerically solve the configuration-dependent equation RX = 0 (R^𝖳S = 0).However, a mathematical result called the rank-nullity theorem lets us compute the difference between the number of zero modes N_0 ≡null R and the number of states of self-stress N_B ≡null R^𝖳 <cit.>. This difference, the rigidity charge ν≡ N_0 - N_B, is given by the dimensionality of the R matrix: ν = dN - N_B, and can only change if balls or springs are either added or removed. We focus on the special case of isostatic, or Maxwell, lattices defined by ν = 0 when the system is considered under periodic boundary conditions. These lattices are marginally rigid and exhibit a symmetry between zero modes and states of self-stress.To design metamaterials based on simple, repeated patterns, we focus on periodic structures. Periodicity allows us to explicitly calculate zero modes and states of self-stress in a large sample. We begin with a highly symmetric, “undeformed” lattice and explore its configuration space by changing the positions (but not the connectivity) of the nodes. The specific geometry that we consider is illustrated in Fig. 1: this stacked kagome lattice has a coordination number z≡ 2N_B/N = 2d= 6 and ν = 0: the lattice is isostatic. This lattice is based on the two-triangle unit cell shown in Fig. <ref>A, which corresponds to two unit cells of the kagome lattice stacked on top of each other. We deform the lattice via the orientations of the triangles, which are governed by the three angles (ϕ,θ,ψ) of rotation around the (x,y,z)-axes, respectively.Within this structure, we fix the connectivity and explore a range for each of the three angles from -π/2 to π/2. In Fig. <ref>B, we show the deformed unit cell for a particular structure in this region, which corresponds to the choice (ϕ,θ,ψ) = (π/3,0,π/3). This unit cell builds the periodic geometry in Fig. <ref>C. We proceed to quantitatively demonstrate that this metamaterial is gapped. We calculate the spectrum of normal modes by considering plane-wave solutions of the form x_α,ℓ=e^iℓ·kx_α, where the index α refers to a particle within one unit cell and the lattice index ℓ enumerates different unit cells within a lattice. The three-component wavevector k is periodic in each component with -π≤ k_x, k_y,k_z≤π. This collection of points forms the first Brillouin zone of the lattice.The Bloch representation of the rigidity matrix in this plane-wave basis is R_αβ(k)=R_αβe^i(ℓ_α-ℓ_β)·k. A zero mode at wave-number k is a vector of unit-cell displacements x_α that solves R_βα(k)x_α=0. Thus, within this setting, zero modes correspond to the zeros of the complex function R(k). By examining this function, we find the zero modes for different configurations of the lattice. We numerically evaluate zero modes for the stacked kagome lattice (see Supporting Information [SI] for details) and for most values of the angles (ϕ,θ,ψ) find collections of zero modes along compact loops in k-space [Fig. <ref>A]. These Weyl loops, shown in Fig. <ref>B–C,are analogous to one-dimensional nodal lines <cit.> and experimentally observed zero-dimensional Weyl points <cit.> in three-dimensional electronic semimetals and photonic crystals. In isostatic lattices, the number of Weyl loops is always even, because the materials' time-reversal symmetry maintains k→-k reflection symmetry in the Brillouin zone—each Weyl loop comes in a pair with its reflected partner <cit.>.Furthermore, these loops attach to the origin of the Brillouin zone along soft directions, i.e., the lines tangent to the Weyl loops at the origin. We count the number of loops by looking at soft directions in the neighborhood of the origin and, in Fig. <ref>A, plot this count in a slice of parameter space. In this phase diagram, we note regions with up to six different loops.Strikingly, the middle of the diagram displays a region in which no Weyl loops exist [Fig. <ref>A,D].We conclude that although Weyl loops are generic, the stacked kagome lattice also exhibits gapped configurations which contain no Weyl loops.In lattices with Weyl loops, the number of soft modes scales as the linear size of the system, whereas gapped lattices have only the three uniform Guest-Hutchinson modes <cit.>.We are not aware of any other realizations of a three-dimensional isostatic lattice which is gapped. In the next section, we address the implications of the existence of a gap for the topological characterization of mechanical networks. § TOPOLOGICAL RIGIDITY IN THREE DIMENSIONSFor the gapped isostatic lattice, an integer topological invariant called the polarization can be computed from the bulk phonon spectrum <cit.>. This winding-number invariant is only well defined in the gapped case: in Weyl lattices, the closure of the gap prevents a consistent definition. That is, when Weyl loops are present, the polarization changes depending on the choice of contour in the Brillouin zone. The mechanical consequences of polarization are apparent by looking at the spectrum of the material's soft surface waves. The polarization controls which surfaces have more soft modes: this is the mechanical version of the bulk-boundary correspondence principle.For the gapped polarized lattice, a polarization vector P can be computed via P=∑_i m_i a_i - d_0, where a_i are the three lattice vectorsand, in our case, the unit-cell dipole d_0 = -3a_3/2. This dipole is computed via the expression d_0 ≡ - ∑_br_b, where r_b are the positions of the bond centers relative to the center of mass of the unit cell <cit.>. The three coefficients m_i are winding numbers computed by integrating the phase change of the complex function R( k) across a straight-line contour crossing the Brillouin zone:m_i ≡ - 1/2π i∮_C_id k_i d/d k_iln R( k),where C_i is a closed contour in the k̂_i direction <cit.>. This integral is well defined for contours along which R( k) ≠ 0. The integers m_i depend smoothly on the choice of contour and are constant. In gapped lattices, the determinant is non-zero everywhere except for the origin k=0. As a consequence, the winding numbers are independent of the chosen contour and the polarization is a topological invariant. On the other hand, Weyl loops partition the space of straight-line contours into lines going through the inside (outside) of the loop. Because R( k)=0 along the loops, contours on either side of the loop can have different winding numbers m_i. Note that the combination of any two such contours taken in opposite directions can be smoothly deformed (without intersecting the Weyl line) into a small circle enclosing the Weyl line, as shown in Fig. <ref>C. The winding number m_W around this small circle is an invariant and equal to the difference between m_i for C_i on two sides of the loop. This topological protection guarantees that a single Weyl loop cannot be destroyed, which explains why Weyl loops only vanish by shrinking to the origin within the phase diagram in Fig. <ref>A. In summary, Weyl lines are protected by a winding number, and gapped lattices can have a well-defined topological polarization.Bulk-boundary correspondence states that the topological invariants computed in the bulk can have significant effects on the mechanics of a sample with boundaries. We demonstrate this correspondence by computing the topological invariants and the spectrum of soft edge states in the stacked kagome lattice.In the bulk of the Weyl lattice corresponding to (ϕ,θ,ψ)=(5π/12,0,π/4) (c.f., Fig. <ref>A,C), the winding number around the loop is m_W=- 1. The m_3 for contours inside and outside of the loop differ by one. To see the boundary counter-part of the correspondence, we compute the spectrum of soft edge states for this lattice with a stress-free surface parallel to the xy-plane (see Fig. <ref>C). In Fig. <ref>A–B, we show the (signed) inverse penetration depth κ for soft surface modes in the two-dimensional Brillouin zone for wavevector (k_x,k_y). These plane-wave solutions of R = 0 correspond to soft modes of the form e^iℓ·(k+iκ)x_α. The middle mode of Fig. <ref>A (see zoom in Fig. <ref>B) changes sides: for κ > 0 (red region), the soft mode is attached to the bottom surface and decays upwards into the bulk, whereas for κ < 0 (blue region), the mode is localized on the top surface.For the line corresponding to the projection of the Weyl loop onto the two-dimensional Brillouin zone, the penetration depth is infinite and κ = 0, because the Weyl lines are soft modes in the bulk. The difference in m_3 between inside and outside of the Weyl loops isthe bulk invariant m_W = - 1, which corresponds to the difference between the number of soft surface modes across the projected Weyl loop. This connection follows from Cauchy's argument principle for Eq. (<ref>), which states that the third winding numbers m_3 count, up to a constant, the number of modes with zero energy (R = 0) at a boundary. These observations confirm bulk-boundary correspondence. A similar correspondence exists in the gapped lattice. There, the invariant polarization pointing along the z-axis is given by P =a_3/2, which we computed for parameters (ϕ,θ,ψ)=(π/3,0,π/3).To understand the effect of P on the boundary, note in analogy with electromagnetism that the rigidity charge ν^S in region S is related to the flux of polarization P through the region's boundary ∂ S. In a nearly-uniform, gapped lattice <cit.>ν^S=∮_∂ SdA/V_celln̂· P,where n̂ is the boundary's inward normal and V_cell=( a_1, a_2, a_3) is the volume of a unit cell.The difference between the rigidity charges at the top versus the bottom is determined by P: for the stacked kagome, we expect one more band of soft modes along the top surface relative to the bottom.The total number of soft surface modes is three as a result of three bonds being cut per unit cell <cit.>. We plot the inverse penetration depth κ for this extra band in Fig. <ref>C–D within the two-dimensional Brillouin zone.Unlike the Weyl lattice, the whole middle band in Fig. <ref>C is blue: the top surface has more soft modes and is, therefore, softer.Equation (<ref>) shows that topological polarization acts analogously to an electric polarization. Inside a homogeneous polarized material, the charge is zero. However, if homogeneity is broken by the presence of boundary or defects, charge can accumulate at these spots. figure-1§ LOCAL RIGIDITY AND SOFTNESS AT DISLOCATIONSFor many applications such as cushioning <cit.>, programmed assembly <cit.>, or controlled failure <cit.>, materials need to be designed with build-in softness or rigidity. The previous section illustrates how a topological invariant can be programmed into a material to design softness at boundaries. We now proceed to show how a flux of this topological polarization can be harnessed to create soft regions in the bulk of the material. Equation (<ref>) suggests that a nonzero polarization flux can be achieved by considering an inhomogeneous material. We choose dislocations to provide this inhomogeneity.The natural defects in three-dimensional crystals are line dislocations: displacements of unit cells along straight lines which terminate at material boundaries. In many crystalline solids, such defects control mechanical deformations and plasticity. As these lines are topologically protected, the only way to confine a line dislocation to a finite region is to form a closed dislocation loop (Fig. <ref>A–B).The topological invariant characterizing a dislocation is the Burgers vector b, which expresses the effect of a dislocation on the surrounding lattice. If a particle at point x is displaced to x+ u( x) according to displacement u( x), then the Burgers vector is b=∮_C d u, where C is any closed contour surrounding the dislocation line.Line dislocations come in two primary types called edge and screw dislocations, which are distinguished by the orientation of the Burgers vector b relative to the dislocation line direction ℓ̂ (see Fig. <ref>A). For edge dislocations, these vectors are orthogonal: the displacement u pushes unit cells apart in order to insert a half-plane of unit cells that extends from the dislocation line in the direction ℓ̂× b. In this way, edge dislocations are three-dimensional generalizations of two-dimensional point dislocations.By contrast, for screw dislocations, the b and ℓ̂ vectors are parallel: the displacement u pushes neighboring cells apart along the loop. In this way, screw dislocations give rise to an inherently three-dimensional spiral structure. Along a dislocation loop, the Burgers vector b is constant, but the line direction ℓ̂ changes. As Fig. <ref>A shows, a dislocation loop can contain both edges and screws.The interplay between line properties (ℓ̂,b) and the topological polarization P of the lattice leads to a rigidity charge that can be expressed in a simple formula (derived in the SI) for the charge line density ρ^L localized at the dislocation line: ρ^L = 1/V_cell P·( b×ℓ̂),whereV_cell is the unit cell volume of the uniform lattice. Equivalently, we could write ρ^L = V_cell^-1 ( P, b,ℓ̂). Consequently, for any configuration in which these three vectors are not linearly independent, the charge density is zero. In particular, screw dislocations do not carry charge—their Burgers vector points along the line direction. For edge dislocations, the charge is negative (positive) if P points along (against) the half-plane direction ℓ̂× b: intuitively, these rows carry extra polarization out or in. Note that for any dislocation loop, the net charge ∮_L ρ^L is zero.However, dislocations loops do separate rigidity charges in space, resulting in a topological charge dipole, as shown in Fig. <ref>A.This dipole is oriented within the b-P plane and can be quantified by a dipole moment, which captures the amount of charge separated and the distance of separation.The dipole's b (P) component is the area of the dislocation loop after it is projected onto the plane formed by the two vectors b (P) and b×P (see SI for derivation). For the loop composed of edge and screw dislocations shown in Fig. <ref>A, all of the charges are localized along the edge dislocations and the dipole moment lies along b. A positive charge density corresponds to soft modes localized along part of the dislocation loop. We investigate this localized softness within a polarized lattice using the configuration shown schematically in Fig. <ref>A and plotted for a small sample in Fig. <ref>B. In Fig. <ref>C, we demonstrate that for the softest (i.e., lowest-frequency non-translational) modes of this lattice, the unit cells with the largest displacements are localized along the near side of the loop, in agreement with Eq. (<ref>). This can be contrasted with the lowest-energy modes of a sample without a dislocation: in Fig. <ref>D, we show that the dislocated lattice has soft modes at lower frequencies. Whereas the lowest modes in the non-dislocated sample are the largest-wavelength acoustic phonons that fit within the periodic box, the lowest modes of the dislocated lattice are a combination of these acoustic phonons and the many localized modes such as the ones shown in Fig. <ref>C. § CONCLUSIONS AND OUTLOOKWe demonstrated that dislocation loops in polarized three-dimensional lattices can localize soft modes at one part of the loop. Similarly, the opposite part of the loop localizes states of self-stress. Although the total topological charge along the loop is zero, the loop acts as a topological charge dipole. The loop can be used to localize topologically protected soft modes in one region of the material, inside the material bulk. Furthermore, because the lattice we have designed is gapped, these localized soft modes do not compete with the many extended, periodic soft modes characteristic of Weyl loops.Let us conclude by discussing applications for both three-dimensional gapped lattices and for localized softness. Three-dimensional topological materials may be useful in the design of devices for cushioning, including safety devices such as helmets, in which a hard side necessary to resist stress exists in combination with a soft side necessary to cushion the body <cit.>. Localized softness can be used to pre-program large displacements and to isolate the rest of the material from strain. In active mechanical metamaterials <cit.>, for example in self-folding origami <cit.>, these mechanisms could be actuated using motors to strain the material in a well-controlled way <cit.>. Furthermore, localized states of self-stress can be used to control material failure via either buckling or fracture, which then isolates the rest of the material from failure even under a large external load.§ ACKNOWLEDGMENTSWe thank Paul Baireuther and Bryan G. Chen for fruitful discussions and T. C. Lubensky for a critical reading of the manuscript.We gratefully acknowledge funding from FOM, NWO, and Delta Institute for theoretical physics.seccntformat#1 c@#1@̧sectionthe#1§ SUPPORTING INFORMATION FOR “LOCALIZING SOFTNESS AND STRESS ALONG LOOPS IN THREE-DIMENSIONAL TOPOLOGICAL METAMATERIALS”§ GENERALIZED STACKED KAGOME LATTICE The generalized stacked kagome lattice is built by repeating a given unit cell along the following lattice vectors[a_1=a(1,0,0) a_2=a(1/2,√(3)/2,0)a_3=a(0,0,1) ],where a defines the lattice spacing. In the undeformed kagome lattice the equilibrium positions of the particles in the unit cell is given by the triangular prism[r(1)=-14 a_3r(2)=12 a_1-14 a_3r(3)=12 a_2-14 a_3;r(4)=+14 a_3r(5)=12 a_1+14 a_3 r(6)=12 a_2+14 a_3, ]where particles s=1,2,3 (s=4,5,6) constitute the bottom (top) triangle of the prism.Within the unit cell we connect the particle pairs (2,3), (3,1), (1,2), (5,6), (6,4), (4,5), (1,4), (2,5), (3,6) by springs. Horizontally between unit cells we connect pairs (3,2), (1,3), (2,1), (6,5), (4,6), (5,4) according to the nearest neighbor principle. Finally we connect (4,1), (5,2) and (6,3) between the unit cells in the positive vertical direction.We let a deformation act on the unit cell by rotating the triangles around three orthogonal axes attached to the triangles. The first two axes are coplanar to the triangle, with the first one bisecting one of its angles. The third axis is perpendicular to the triangle. The three axes meet in the center of the triangle. We then define our deformation by rotating the top triangle around the three axis by angles ϕ, θ, and ψ respectively, while rotating the bottom triangle by -ϕ, -θ, and -ψ. Note that by taking our rotation axes relative to the triangles, our rotation group is commutative and we do not need to worry about the order of rotations. Explicitly, the equilibrium positions of the particles in the generalized stacked kagome can be given as r(1) =a/2√(3)[s_ψ c_θ-√(3)c_ψ c_ϕ-c_ψ s_ϕ s_θ; -c_ψ c_θ-√(3) s_ψ c_ϕ- s_ψ s_ϕ s_θ; - c_ϕ s_θ+ √(3)s_ϕ+ √(3) ]+ar/2[ c_α; s_α; 0 ]r(2) =a/2√(3)[s_ψ c_θ+√(3)c_ψ c_ϕ-c_ψ s_ϕ s_θ; -c_ψ c_θ+ √(3)s_ψ c_ϕ- s_ϕ s_ϕ s_θ; - c_ϕ s_θ- √(3)s_ϕ+ √(3) ]+ar/2[ c_α; s_α; 0 ]r(3) =a/√(3)[ - s_ψ c_θ+c_ψ s_ϕ s_θ;c_ψ c_θ+ s_ψ s_ϕ s_θ;c_ϕ s_θ + √(3)/2 ]+ar/2[ c_α; s_α; 0 ]r(4) =a/2√(3)[- s_ψ c_θ-√(3)c_ψ c_ϕ-c_ψ s_ϕ s_θ; -c_ψ c_θ+√(3) s_ψ c_ϕ+ s_ψ s_ϕ s_θ; c_ϕ s_θ- √(3)s_ϕ- √(3) ]-ar/2[ c_α; s_α; 0 ]r(5) =a/2√(3)[- s_ψ c_θ+√(3)c_ψ c_ϕ-c_ψ s_ϕ s_θ; -c_ψ c_θ- √(3)s_ψ c_ϕ+ s_ϕ s_ϕ s_θ; c_ϕ s_θ+ √(3)s_ϕ- √(3) ]-ar/2[ c_α; s_α; 0 ]r(6) =a/√(3)[s_ψ c_θ+c_ψ s_ϕ s_θ; c_ψ c_θ- s_ψ s_ϕ s_θ; - c_ϕ s_θ - √(3)/2 ]-ar/2[ c_α; s_α; 0 ], where s_·=sin(·) and c_·=cos(·). Following the above order of bond enumeration, the compatibility matrix takes the form:R^†( k)=[ [b̂^𝖳_1;b̂^𝖳_2;b̂^𝖳_3;b̂^𝖳_4;b̂^𝖳_5;b̂^𝖳_6;b̂^𝖳_7;b̂^𝖳_8;b̂^𝖳_9; b̂^𝖳_10; b̂^𝖳_11; b̂^𝖳_12; b̂^𝖳_13; b̂^𝖳_14; b̂^𝖳_15; b̂^𝖳_16; b̂^𝖳_17; b̂^𝖳_18 ] [0 -11 b̂^𝖳_1;10 -1 b̂^𝖳_1O; -110 b̂^𝖳_1; b̂^𝖳_10 -11; b̂^𝖳_1O10 -1; b̂^𝖳_1 -110; b̂^𝖳_1; b̂^𝖳_1 -II; b̂^𝖳_1;0 e^-i k_1 -1 b̂^𝖳_1; -10 e^-i k_2 b̂^𝖳_1O; e^-i (k_1-k_2) -10 b̂^𝖳_1; b̂^𝖳_10 e^-i k_1 -1; b̂^𝖳_1O -10 e^-i k_2; b̂^𝖳_1e^-i(k_1-k_2) -10; b̂^𝖳_1; b̂^𝖳_1e^-i k_3I -I; b̂^𝖳_1;] ] where the initial spring direction b_i is defined as the normalization of r(s')- r(s), where (s,s') is the i-th spring. In this notation, each number in the matrix grid represents a multiple of the triple on the far left in the same row. This turns the matrix into an 18×18 square matrix. As the initial bond directions depend smoothly on the initial particle positions, each element of Q depends smoothly on the deformation parameters ϕ, θ, and ψ. § SOFT DIRECTIONS Weyl lines are formed by collections of points satisfying R( k)=0. This equation is often quite hard to solve. However, with observational evidence suggesting that the Weyl loops generally connect to the origin, we can formally expand the determinant around k≡|| k||=0 asR( k) =∑_n∈ℤ^3c_ ne^-i n· k =i k^3𝒫_3(k̂)+k^4𝒫_4(k̂)+𝒪(k^5),for finitely many nonzero coefficients c_ n∈ℝ, where 𝒫_m is an m-th order homogeneous polynomial with real coefficients in three variables over the unit sphere ||k̂||=1. The vanishing constant, linear, and quadratic polynomials reflect the triple multiplicity of the zero at k=0.A Weyl line can only attach to the origin at a direction k̂ where both 𝒫_3(k̂) and 𝒫_4(k̂) vanish.Suppose, for instance, that 𝒫_4(k̂)0 and note that the error term is dominated by M k^5 for all k<δ, where M and δ are fixed positive numbers. As a consequence, the relation| R( k)|> k^4(|𝒫_4(k̂)|-M k)holds for all k<δ. In particular, this means | R( k)|>0 for all k<min(δ,1/M|𝒫_4(k̂)|). We call these common zeroes of 𝒫_3(k̂) and 𝒫_4(k̂) over the unit sphere the soft directions of R. As any Weyl loop connects to the origin along two directions, we find that a configuration with 2n soft directions can accommodate no more than n pairs of Weyl loops.In the main text we show the θ=0 slice of soft-mode phase space, containing a region where the lattice does not exhibit any soft directions. This absence of soft directions is rare. Sampling of the 0<ϕ,θ,ψ<π/2 phase space with step size π/60 we obtain the following data for the prevalence of different numbers of soft directions: § RIGIDITY CHARGE BOUND TO LINE DISLOCATIONS The dislocation acts on the polarization and the unit cell volume through the lattice vectors a_i. Letting the displacement function u act on both the initial and terminal point of the vector a_i, we obtain the lattice vector field in the dislocated lattice:ã_i( x) =[ x+ a_i+ u( x+ a_i)]-[ x+ u( x)] = a_i+( a_i·∇) u( x),where we assume that higher order differentials of u are negligible. The polarization changes asP̃( x) =∑_i m_iã_i = P+( P·∇) u( x).The unit cell volume changes asṼ_cell( x) =( ã_1,ã_2,ã_3) =V_cell(1+∇ u( x)).We start from Eq. (1) in the main text. In a canonical basis with coordinates (x^1,x^2,x^3), a surface element is given by dx^ℓ∧ dx^k=ϵ^iℓ kdS_i, where d S is positively oriented. In this notation, the change in rigidity charge in the region S from the ν^S=0 periodic case is given byν̃^S-ν^S=∫_∂ S(Ṽ_cell^-1 P̃-V_cell^-1P)·(-d S) =V_cell^-1∫_∂ S P_i∂_ju_j-P_j∂_ju_i dS_i =V_cell^-1∫_∂ SP_i ϵ_ijk∂_ℓ u_j ϵ^mℓ kdS_m =V_cell^-1∫_∂ S( P× d u)_k∧ dx^k. Consider a straight dislocation line segment L in direction ℓ̂ and let S be the cylinder with axis L and radius ℛ. Without loss of generality, we choose canonical coordinates such that L={tℓ̂: 0<t<T} with ℓ̂ pointing along the third basis vector. Furthermore, we take Tℓ̂ to be an integer combination of lattice vectors a_i. If the displacement function respects the periodicity of the lattice, the top (t=T) and bottom (t=0) surfaces of S will cancel by orientation. It remains to calculate Eq. (<ref>) along the vertical boundary of S. First consider the k=1 term:∫_∂ S( P× d u)_1∧ dx^1=∫_0^2π( P×∫_L_θd u)_1 dℛcosθ=0,where ∂ S=⋃_θ L_θ is a decomposition of the cylindrical boundary in vertical lines at different angle θ. The last equality holds since ∫_L_θd u is independent of θ as can be shown by contour integration∫_L_θd u-∫_L_θ'd u=∫_θ^θ'd u|^t=T-d u|^t=0=0.The k=2 term vanishes similarly. Hence we find that the rigidity charge is given entirely by the term corresponding to the line direction. Let ∂ S=⋃_t C_t be the decomposition of the cylindrical boundary in horizontal circles at different height t. Recalling the definition of the Burgers vector, we then obtainν̃^S =V_cell^-1∫_0^T( P×∫_C_td u)_3dt=1/V_cell( P× b)· Tℓ̂.The rigidity charge density along the line can then be given by ρ_L≡ν_L/T=V_cell^-1( P, b,ℓ̂), where the three vectors P, b and ℓ̂ form the columns of a 3×3 matrix.§ DISLOCATION LOOP DIPOLE MOMENT Let r(s) be an arc length parametrization of the loop L. 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(2016) Topological Mechanics of Origami and Kirigami. Physical Review Letters 116(13):135501. | http://arxiv.org/abs/1707.08928v2 | {
"authors": [
"Guido Baardink",
"Anton Souslov",
"Jayson Paulose",
"Vincenzo Vitelli"
],
"categories": [
"cond-mat.soft",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.soft",
"published": "20170727164621",
"title": "Localizing softness and stress along loops in three-dimensional topological metamaterials"
} |
FIN QUI snr snrČebyšev 3pt3pt #1#2#3#4#5 remarkRemarkexampleExamplelemmaLemmadefinitionDefinition Line codes generated by finite Coxeter groupsEzio Biglieri, Life Fellow, IEEE, Emanuele Viterbo, Fellow, IEEE The work of E. B. was supported by Project TEC2015-66228-P, while that of E. V.was supported by ARC under Grant Discovery Project No. DP160101077. Preliminary versions of this manuscript were presented in <cit.>.Ezio Biglieri is with the Electrical Engineering Department, UCLA, and with Departament TIC, Universitat Pompeu Fabra, Barcelona, Spain (email:[email protected]). Emanuele Viterbo is with the Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, Australia (email:[email protected]).December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= Using an algebraic approach based on the theory of Coxeter groups, we design, and describe the performance of, a class of line codes for parallel transmission of b bits over b+1 wires that admit especially simple encoding and decoding algorithms. A number of designs are exhibited, some of them being novel or improving on previously obtained codes.Line coding, group codes for the Gaussian channels, permutation modulation, Coxeter groups.§ INTRODUCTION AND MOTIVATION OF THE WORKIn this paper we describe the design of vector line codes allowing an especially simple maximum-likelihood (ML) detection procedure. This consists of a linear transformation of the vector received at the output of an additive white Gaussian noise (AWGN) channel, followed by a binary slicer. The design is based on the selection of a subset of a permutation modulation (PM) codebook being the direct product of binary antipodal signaling schemes, and hence having a geometrical representation in the form of a multidimensionalorthotope (or hyper-rectangle). The encoder can also be implemented as a linear transformation of the source (binary) vector.Transmission on parallel wireline links (as those used to interconnect integrated circuits, or a television set to a set-top box) is affected by disturbances placing a number of constraints on the design of the signaling scheme. The key problem here is the design of line codes allowing the transmission of b bits over w ⩾ b wires and using a codebook W subject to some constraints to be detailed later. The general scheme is shown in Fig. <ref>.h!0.9figure1bis General scheme of vector coding.figure1bis Here, b binary information symbols ± 1 are input in parallel to the (w,b) line encoder, which is a one-to-one map from {± 1 }^b to the codebook W ⊆ℝ^w. This encoder outputs a code vector w with w real components, which is added to white Gaussian noise to obtain vector y. Vector y is processed by a detector whose output is an estimate b of the information vector.Fig. <ref> illustrates the two basic circuits for wired binary communications: (a) unipolar signaling and (b) differential signaling. | http://arxiv.org/abs/1707.08932v1 | {
"authors": [
"Ezio Biglieri",
"Emanuele Viterbo"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170727165337",
"title": "Line codes generated by finite Coxeter groups"
} |
Electron spin resonance for the detection of long-range spin nematic order Tsutomu Momoi December 30, 2023 ========================================================================== English.In this work we analyze the performances of two of the most used word embeddings algorithms, skip-gram and continuous bag of words on Italian language. These algorithms have many hyper-parameter that have to be carefully tuned in order to obtain accurate word representation in vectorial space. We provide an extensive analysis and an evaluation, showing what are the best configuration of parameters for specific analogy tasks. Italiano. In questo lavoro analizziamo le performances di due tra i più usati algoritmi di word embedding: skip-gram e continuous bag of words. Questi algoritmi hanno diversi iperparametri che devono essere impostati accuratamente per ottenere delle rappresentazioni accurate delle parole all'interno di spazi vettoriali. Presentiamo un'analisi accurata e una valutazione dei due algoritmi mostrando quali sono le configurazioni migliori di parametri su specifiche applicazioni.§ INTRODUCTIONThe distributional hypothesis of language, set forth by firth1935technique and harris1954distributional, states that the meaning of a word can be inferred from the contexts in which it is used. Using the co-occurrence of words in a large corpus, we can observe for example that the contexts in which client is used are very similar to those in which customer occur, while less similar to those in which waitress or retailer occur. A wide range of algorithms have been developed to exploit these properties. Recently, one of the most widely used method in many natural language processing (NLP) tasks is word embeddings <cit.>. It is based on neural network techniques and has demonstratedto capture semantic and syntactic properties of words taking as input raw texts without other sources of information. It represents each word as a vector such that words that appear in similar contexts are represented with similar vectors <cit.>. The dimensions of the word are not easily interpretable and, with respect to explicit representation, they do not correspond to specific concepts.In mikolov2013efficient, the authors propose two different models that seek to maximize, respectively, the probability of a word given its context (Continuous bag-of-word model), and the probability of the surrounding words (before and after the current word) given the current word (Skip-gram model). In this work we seek to further explore the relationships by generating word embedding for over 40 different parameterizations of the continuous bag-of-words (CBOW) and the skip-gram (SG) architectures, since as shown in levy2015improving the choice of the hyper-parameters heavily affect the construction of the embedding spaces.Specifically our contributions include: * Word embedding.The analysis of how different hyper-parameters can achieve different accuracy levels in relation recovery tasks <cit.>. * Morpho-syntactic and semantic analysis. Word embeddings have demonstrated to capture semantic and syntactic properties, we compare two different objectives to recover relational similarities for semantic and morph-syntactical tasks. * Qualitative analysis. We investigate problematic cases.§ RELATED WORKSThe interest that word embedding models have achieved in the NLP international community has recently been confirmed by the increasing number of studies that are adopting these algorithms in languages different from English. One of the first example is the Polyglot project that produced word embedding for 117 languages <cit.>. They demonstrated the utility of word embedding, achieving, in a part of speech tagging task, performances competitive with the state-of-the art methods in English. attardi2014adapting have done the first attempt to introduce word embedding in Italian obtaining similar results. They have shown that, using word embedding, they obtained one of the best accuracy levels in a named entity recognition task.However, these optimistic results are not confirmed by more recent studies. Indeed the performance of word embedding are not directly comparable in the accuracy test to those obtained in the English language. For example, attardi2014dependency combining the word embeddings in a dependency parser have not observed improvements over a baseline system not using such features. berardi2015word found a 47% accuracy on the Italian versus 60% accuracy on the English. The results may be a sign of a higher complexity of Italian with respect to English as we will see section <ref>.Similarly, recent work that trained word embeddings on tweets have highlighted some criticalities. One of these aspects is how the morphology of a word is opaque to word embeddings. Indeed, the relatedness of the meaning of a lemma's different word forms, its different string representations, is not systematically encoded. This means that in morphologically rich languages with long-tailed frequency distributions, even some word embedding representations for word forms of common lemmata may become very poor <cit.>.For this reason, some recent contribution on Italian tweets have tried to capture these aspects. tamburini2016bilstm trained SG on a set of 200 million tweets. He proposed a PoS-tagging system integrating neural representation models and a morphological analyzer, exhibiting a very good accuracy. Similarly, stemle2016bot proposes a system that uses word embeddings and augment the WE representations with character-level representations of the word beginnings and endings.We have observed that in these studies the authors used either the most common set-up of parameters gathered from the literature <cit.> or an arbitrary number<cit.>. Despite the relevance given to these parameters in the literature <cit.> we have not seen studies that analyze the different strategies behind the possible parametrization. In the next section, we propose a model to deepen these aspects.§ ITALIAN WORD EMBEDDINGSPrevious results on the word analogy tasks have been reported using vectors obtained with proprietary corpora <cit.>. To make the experiments reproducible, we trained our models on a dump of the Italian Wikipedia (dated 2017.05.01), from which we used only the body text of each articles. The obtained texts have been lowercased and filtered according to the corresponding parameter of each model. The corpus consists of 994.949 sentences that result in 470.400.914 tokens.The hyper-parameters used to construct the different embeddings for the SG and the CBOW models are: the size of the vectors (dim), the window size of the words contexts (w), the minimum number word occurrences (m) and the number of negative samples (n). The values that these hyper-parameters can take are shown in Table <ref>. § EVALUATIONThe obtained embedding[The trained vectors with the best performances are available at http://roccotripodi.com/ita-we] spaces are evaluated on an word analogy task, using a enriched version of the Google word analogy test <cit.>, translated in Italian by <cit.>. It contains 19.791 questions and covers 19 relations types. 6 of them are semantic and 13 morphosyntactic (see Table <ref>). The proportions of these two types of question is balanced as shown in Table <ref>. To recover these relations two different methods are used: 3CosAdd (Eq. <ref>) <cit.> and 3CosMul (Eq. <ref>) <cit.> to compute vectors analogies: 3CosAdd_b^*∈ Vcos(b^*, b - a + a^*)3CosMul_b^*∈ Vcos(b^*,b)cos(b^*,a^*)/cos(b^*,a)+ϵ These two measures try to capture different relations between word vectors. The idea behind these measures is to use the cosine similarity to recover the vector of the hidden word (b^*) that has to be the most similar vector given two positive and one negative word. In this way, it is possible to model relations such as queen is to king what woman is to man. In this case, the word queen (b^*) is represented by a vector that has to be similar to king (b) and woman (a^*) and different to man (a). The two analogy measures slightly differ in how they weight each aspect of the similarity relation. 3CosAdd allows one sufficiently large term to dominate the expression <cit.>, 3CosMul achieves a better balance amplifying the small differences between terms and reducing the larger ones <cit.>. As explained in levy2014linguistic, we expect 3CosMul to over-perform 3CosAdd in evaluating both the syntactic and the semantic tasks as it tries to normalize the strength of the relationships that the hidden term has both with the attractor terms and with the repellers term.§.§ Experimental resultsThe results of our evaluation are presented in Figure <ref>. The main trend that it is possible to notice is that accuracy increases as the number of dimensions of the embedded vectors increases. This indicates that Italian language benefits of a rich representation that can account for its rich morphology. Another important trend that emerges is the fact that the parameters have the same effect on both algorithms and that they perform very differently on all the tasks. CBOW has very low accuracy compared to SG. We can also see that the dim hyper-parameter is not correlated with the dimension of the vocabulary (model complexity) as one should expect. In fact, with increasing values of dim the accuracy increases whatever is the value of m. This hyper-parameter heavily affects the vocabulary length (see Table <ref>). However the dim hyper-parameter seems to be correlated only with the accuracy in the semantic tasks while the performances on the morpho-syntactic tasks seems not to have a big bust increasing the dimensionality.With respect to the size of the context (w) used to create the words representations we do not observe a clear difference between the 18 pairs both in the SG and in the CBOW. On the contrary a clear trend can be observed varying the n hyper-parameter, with n=1the accuracy was significantly lower than the one we obtained with n=5 or n=10. Increasing the number of negative samples constantly increases the accuracy. These results support also the claim put forward by <cit.> that the 3CosMul method is more suited to recover analogy relations. In fact, we can see that on average the right bars of the plots are higher than the left. §.§ Error analysisIf we restrict the error analysis to the most macroscopic differences in figure <ref> we can compare three different parametrizations: SG-200 w5-m5-n1, SG-500 w5-m5-n1, SG-500 w5-m5-n10. In this way we can analyze the results obtained changing the number of dimensions of the vectors and the role played by n. In Table <ref> the total number of errors and the number of different words that have not been recovered by each parametrization are presented. From this table we can see that most of the errors are done one a relatively small set of words. This phenomenon can be studied analyzing themost problematic cases. In Table <ref> we can see the list of the most common errors ranked by frequency for each method. As we can see from these lists the errors are done on the same words and this because they are the most common in the dataset (e.g.: in the dataset there are 217 queries that require Florida as answer compared to the 55 of Italia). However if we compare thefrequency of these errors in the analogy test within the three parametrisation we can observe an improvement of approximately 15% in accuracy withSG-500 w5-m5-n10. Indeed, despite many errors are not recovered for any of the parametrisation, we can observe that approximately 21% of the errors are recovered under certain parametrizations (Table<ref>).To further investigate these improvements related to the aforementioned parametrisation we focused on one of the most frequent errors in the analogy test, the word California.As we can see from the list of the analogy test solved (Table <ref>) different parametrizations are helpful to solve different types of analogies. For example an increase in the dimensionality increases the accuracy, but mainly in analogy test with words that have a representation in the training data related to a wider set of contexts(Houston:Texas; Chicago:Illinois). The best parametrisation is obtained increasing the negative sampling. As we can see from the examples provided, the analogies are resolved thanks to a contextual similarity between the two pairs (Huntsville:Alabama; Oakland:California). In these cases the negative sampling could help to filter out from each representation those words that are not expected to be relevant for the words embeddings.Similar types of improvement are noticed on analogy tests that contain a challenging word predire (predict). The results of this analysis are presented in Table <ref> where it is possible to see that an higher dimensionality improves the accuracy of analogical tests containing open domain verbs (e.g.: descrivere, vedere). Similarly to the previous case, an higher dimensionality allows for fine grained partitions improving the correct associations between terms. However, also in in this case, the best parametrizations are obtained increasing the negative sampling or both the parameters. As we can see here both the present participle and the past tense pairs are correctly solved. These example provide a preliminary evidence of how negative sampling, filtering out non informative words from the relevant context of each word, is able to build representation by opposition that are beneficial both for semantic and syntactic associations.Examples of words that almost always are not recovered correctly are presented in Table <ref>. A selected list of words problematic for all parametrizations is shown in Table <ref>. It contains plurals, feminine, currencies, superlatives and ambiguous words. The low performances on these cases can be explained by the poor coverage of these categories in the training data. In particular, it would be interesting to study the case of feminine and to analyze if it is due to a gender bias in the Italian Wikipedia, as a preliminary analysis of the most frequent errors that persist in all the parametrization seems to suggest.The words that have been benefited by the increase of n are: 4ghana pakistan irlandese migliorano scrivendo slovenia giocando serbia implementano ucraino zimbabwe namibia suonano maltese portoghese contessa messicano giordania the errors that have been introduced increasing this parameter are related to the words in Table <ref>.It is interesting to notice that given an error in an analogy test, it is possible to find the correct answer in the top five most similar words to the query. Precisely we observed this phenomenon in 26 % of the cases for SG-200-w5-m5-n10, in 27 % of the cases for SG-500-w5-m5-n1 and in 25 % for SG-500-w5-m5-n1. Furthermore, approximately in 50 % of these cases the correct answer is the second most similar. Most of the recovery errors are due to vocabulary issues. In fact, many words of the test set have no correspondence in the developed embedding spaces. This is due to the low frequency of many words that are not in the training corpus or that have been removed from the vocabulary because of their (low) frequency. For this reason we kept the m hyper-parameter very low (e.g., 1 and 5), in counter-tendency with recent works that use larger corpora and then remove infrequent words setting m with highvalues (e.g., 50 or 100). In fact, with increasing value of m the number of not given answers increases rapidly. It passes from 300 (m=1) to 893 (m=5). Some of the words that are not present in the vocabulary with m=1 include plural verbs (1st person), that probably are not used by a typical Wikipedia editor and remote past verbs (1st person), a tense that in recent years is disappearing from written and spoken Italian. Some of these verbs are:3giochiamo affiliamo rallentiamo zappiamo implementai rallentai mescolai nuotaiIn berardi2015word the number of not given answer is 1.220. The accuracy of their embeddings, obtained using a larger corpus and using the hyper-parameters that perform well on English language, is always lower than those obtained with our setting, in both the morphosyntactic and the semantic tasks. This confirms that the regularization of the parameters is crucial for good representation of the embeddings, since the berardi2015word's model has been trained on a much larger corpus and for this should outperform ours. Furthermore, this model seems to have some tokenization problem. § CONCLUSIONSWe have tested two word representation methods: SG and CBOW training them only on a dump of the Italian Wikipedia. We compared the results of the two models using 12 combinations of hyper-parameters.We have adopted a simple word analogy test to evaluate the generated word embeddings. The results have shown that increasing the number of dimensions and the number of negative examples improve the performance of both the models.These types of improvement seems to be beneficial only for the semantic relationships. On the contrary the syntactical relationship are negatively affected by the low frequency of many of its terms. This should be related to the morphological complexity of Italian. In the future it would be helpful to represent the spatial relationship regarding specific syntactical domain in order to evaluate the contribution ofhyper-parametrization to syntactical relationship accuracy. Moreover future work will include the testing of these word embedding parametrizations in practical applications (e.g. analysis of patents'descriptions and books' corpora).§.§.§ AcknowledgmentsPart of this work has been conducted during a collaboration of the first author with DocFlow Italia. All the experiments in this paper have been conducted on the SCSCF multiprocessor cluster system at University Ca' Foscari of Venice.acl | http://arxiv.org/abs/1707.08783v4 | {
"authors": [
"Rocco Tripodi",
"Stefano Li Pira"
],
"categories": [
"cs.CL",
"cs.AI"
],
"primary_category": "cs.CL",
"published": "20170727085629",
"title": "Analysis of Italian Word Embeddings"
} |
Gaussian Processes for Individualized Continuous Treatment Rule Estimation Pavel Shvechikovand Evgeniy Riabenko The reported study was funded by RFBR, according to the research project No. 16-07-01192 А. Department of Computer Science, Higher School of Economics December 30, 2023 =====================================================================================================================================================================================================We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider the problem with a combination of clamped, simply supported and free boundary conditions subject to both distributed and concentrated (point and line) loads. Extensive numerical computations are presented to verify the functionality of the estimators. Kirchhoff plate, C^1 elements, a posteriori estimates 65N30 § INTRODUCTIONThe purpose of this paper is to perform an a posteriori error analysis of conforming finite element methods for the classical Kirchhoff plate bending model. So far this has not been done in full generality as it comes to the boundary conditions. Most papers deal only with clamped or simply supported boundaries, see <cit.> for conforming C^1 elements,<cit.> for the mixed Ciarlet–Raviart method (<cit.>), and <cit.> for discontinuous Galerkin (dG) methods. The few papers that do address more general boundary conditions, in particular free, are <cit.> in which the nonconforming Morley element is analysed, <cit.> where a new mixed method is introduced and analysed, and <cit.> where a continuous/discontinuous Galerkin method is considered. One should also note that the Ciarlet–Raviart method cannot even be defined for general boundary conditions. Free boundary conditions could be treated using dG methods following an analysis similar to the one presented here. In this study, we will derive a posteriori estimates using conforming methods and allowing for a combination of clamped, simply supported and free boundaries. In addition, we will investigate the effect of concentrated point and line loads, which are not only admissible in our H^2-conformingsetting but of great engineering interest, on our a posteriori bounds in numerical experiments. We note that finite element approximation of elliptic problems with loads acting on lower-dimensional manifolds has been considered by optimal control theory, see <cit.> and all the references therein.The outline of the paper is the following. In Section 2, we recall the Kirchhoff-Love plate model by presenting its variational formulation and the corresponding boundary value problem. We perform this in detail for the following reasons. First, as noted above, general boundary conditions are rarely considered in the numerical analysis literature.Second, the free boundary conditions consist of a vanishing normal moment and a vanishing Kirchhoff shear force.These arise from the variational formulation via successive integrations by parts. It turns out that the same steps are needed in the a posteriori analysis in order to obtain a sharp estimate, i.e. both reliable and efficient.In the following two sections, we present the classical conforming finite element methods and derive new a posteriori error estimates. In the last section, we present the results of our numerical experiments computed with the triangular Argyris element. We consider the point, line and square load cases with simply supported boundary conditions in a square domain as well as solve the problem in an L-shaped domain with uniform loadingusing different combinations of boundary conditions.§ THE KIRCHHOFF PLATE MODEL The dual kinematic and force variables in the model are the curvature and the moment tensors. Given the deflection u of the midsurface of the plate, the curvature is defined through Κ(u) = -(∇ u),with the infinitesimal strain operator defined by(v) = 1/2*∇v+∇v^T,where (∇v)_ij = ∂ v_i/∂ x_j. The dual force variable, the moment tensor ,is related to Κ through theconstitutive relation(u) = d^3/12Κ(u),where d denotes the plate thickness and where we have assumed an isotropic linearly elastic material, i.e.A = E/1 + ν*A+ ν/1 - ν(tr A) I, ∀A∈ℝ^2 × 2.Here E and ν are the Young's modulus and Poisson ratio, respectively.The shear force is denoted by =(u). The moment equilibrium equation reads as (u) = (u),whereis the vector-valued divergence operator applied to tensors. The transverse shear equilibrium equation is-(u) = l,withl denoting the transverse loading. Using the constitutive relationship (<ref>), a straightforward elimination yields the well-known Kirchhoff–Love plate equation:(u) := D Δ^2 u = l,where the so-called bending stiffness D is defined asD = E d^3/12(1-ν^2). Let Ω⊂ℝ^2 be apolygonal domain that describes the midsurface of the plate.The plate is considered to be clamped on ⊂∂Ω, simply supported on ⊂∂Ω and free on ⊂∂Ω as depicted in Fig. <ref>. The loading is assumed to consist of a distributed load f∈ L^2(Ω), a load g∈ L^2(S) along the line S⊂Ω, and a point load F at an interiorpoint x_0 ∈Ω.Next, we willturn to the boundary conditions, which are best understood from the variational formulation. (Historically, this was also how they were first discovered by Kirchhoff, cf. <cit.>.) The elastic energy of the plate as a function of the deflection v is 1/2 a(v,v), with the bilinear form a defined bya(w,v) = ∫_Ω(w) : Κ(v)dx = ∫_Ωd^3/12(∇ w) : (∇ v) dx,and the potential energy due to the loading isl(v)= ∫_Ω f vdx + ∫_S g vd s + F v(x_0).Defining the space of kinematically admissible deflectionsV={v ∈ H^2(Ω) :v|_∪=0,∇ v ·n|_=0}, minimization of the total energy u=_v∈ V{1/2 a(v,v)-l(v) }leads to the following problem formulation.[Variational formulation] Find u ∈ V such thata(u,v) = l(v) ∀ v ∈ V.To derive thecorresponding boundary value problem, we recall the following integration by parts formula, valid in any domain ⊂Ω ∫_ (w) : Κ(v)dx = ∫_(w) ·∇ vdx - ∫_∂(w)n·∇ v ds = ∫_(w) vdx + ∫_∂(w) ·n v ds - ∫_∂(w)n·∇ v ds ,At the boundary ∂ R, the correct physical quantities are the components in the normal n and tangential s directions.Therefore, we write ∇ v=∂ v/∂ nn + ∂ v/∂ ssanddefine the normal shear force and the normal and twisting moments as(w)=(w)·n, (w) =n·(w) n, (w) =(w)=s·(w) n . With this notation, we can write ∫_∂(w) ·n v ds - ∫_∂(w)n·∇ v ds =∫_∂(w)vds - ∫_∂ ((w)∂ v/∂ n+(w)∂ v/∂ s)ds,and thus rewrite the integration by parts formula (<ref>) as∫_ (w) : Κ(v)dx = ∫_(w) vdx + ∫_∂(w)vds- ∫_∂ ((w)∂ v/∂ n+(w)∂ v/∂ s)ds. The key observation for deriving the correct boundary conditions is that, at any boundary point, a value of v specifies also ∂ v /∂ s. Defining theKirchhoff shear force (cf. <cit.>)(w)= (w) +∂(w)/∂ san integration by parts on a smooth partof ∂ R yields ∫_(w)vds- ∫_ (w)∂ v/∂ sds= ∫_(w) ds -|_a^b(w) v,where a and b are the endpoints of . We are now in the position to state the boundary value problem for the Kirchhoff plate model. Assuming a smooth solution u in (<ref>), we havea(u,v)= ∫ _Ω𝒜(u)vdx+∫_∂Ω(u)vds -∫_∂Ω ((u)∂ v/∂ n+(u)∂ v/∂ s)ds.With the combination of clamped, simply supported and free boundaryconditions at ∂Ω = ∪∪,we have for any v∈ V,∫_∂Ω(u)vds - ∫_∂Ω ((u)∂ v/∂ n+(u)∂ v/∂ s)ds = ∫_(u)vds- ∫_ (u)∂ v/∂ sds- ∫_∪ (u)∂ v/∂ nds.In the final step, we integrate by parts at the free part of the boundary. To this end, let =∪_i=1^m+1^i, with ^i smooth. Integrating by parts over ^i yields∫_^i (u)vds- ∫_^i (u)∂ v/∂ sds= ∫_^i (u) vds -|_c_i-1^ c_i(u) vwhere c_0 and c_m+1 are the end points ofand c_i, i=1,…, m, its successive interior corners.Combining equations (<ref>)–(<ref>), and noting that v(c_0)=v(c_m+1)=0, gives finally a(u,v)=∫ _Ω 𝒜(u)vdx - ∫_∪ (u)∂ v/∂ nds + ∑_i=1^m+1∫_^i (u) vds - ∑_i=1^m{( (u)|_c_i+ -(u)|_c_i-} v(c_i), where M_ns(u)|_c_i ± = lim_ϵ→ 0 +M_ns(u)|_c_i + ϵ(c_i± 1 - c_i) Choosingv∈ V in such a way that three of the four terms in (<ref>) vanish and the test function in the fourthterm remains arbitrary and repeating this for each term,we arrive at the following boundary value problem: * In the domain we have the distributional differential equation𝒜(u) = l Ω,where l is the distribution defined by (<ref>).* On the clamped part we have the conditionsu=0 ∂ u/∂ n=0 .* On the simply supported part it holdsu=0(u)=0. * On the free partit holds (u)=0 (u)=0^i ,i=1,…, m. * At the interior corners on the free part, we have the matching condition on the twisting moments(u)|_c_i+ =(u)|_c_i- c_i, i=1,…,m.§ THE FINITE ELEMENT METHOD AND THE A POSTERIORI ERROR ANALYSIS The finite element method is defined on a meshconsisting of shape regular triangles. We assume that the point load is applied on a node of the mesh. Further, we assume that the triangulation is such that the applied line load is on element edges. We denote the edges in the mesh by and divide them into the following parts: the edges in the interior ^i, the edges on the curve of the line load ^S ⊂^i, and the edges on the free and simply supported boundary, ^f and ^s, respectively. The conforming finite element space is denoted by V_h. Different choices for V_h are presented in Section <ref>. Note that we often write a ≲ b (or a ≳ b ) when a ≤ Cb (ora ≥ Cb) for some positive constant C independent of the finite element mesh.[The finite element method]Find u_h ∈ V_h such thata(u_h,v_h) = l(v_h) ∀ v_h ∈ V_h.Let K and K' be two adjoining triangles with normals n and n', respectively, andwith the common edgeE=K∩ K'. On E we define the following jumps (v)|_E = (v)- M_n'n'(v)and(v)|_E = (v) +V_n'(v).In the analysis, we will need the Girault–Scott <cit.> interpolation operator Π_h : V → V_h for whichthe following estimateholds ∑_K ∈ h_K^-4w-Π_h w_0,K^2 +∑_E ∈ h_E^-1∇(w-Π_h w)_0,E^2+∑_E ∈ h_E^-3w-Π_h w_0,E^2 ≲w_2^2 andΠ_h w_2 ≲w_2.Note that the Girault–Scottinterpolant uses point values at the vertices of the mesh. We use this property in the proof of Theorem <ref>to derive a proper upper bound for the error in terms of theedgeresiduals. Next, we formulate an a posteriori estimate for Problem <ref>.The local error indicators are the following: * The residual on each elementh_K^2 (u_h) -f _0,K ,K∈.* Theresidual of the normal moment jump along interior edgesh_E^1/2(u_h) _0,E ,E∈^i . * The residual of the jump in the effective shear force along interior edgesh_E^3/2(u_h)- g _0,E,E∈^S,h_E^3/2(u_h)_0,E, E∈^i∖^S.* The normal momenton edges at the free and simply supported boundariesh_E^1/2(u_h)_0,E,E∈^f∪^s. * Theeffective shear force along edges at the free boundaryh_E^3/2(u_h) _0,E, E∈^f.The global error estimator is then defined throughη^2= ∑_K∈h_K^4 (u_h) -f _0,K^2 + ∑_ E∈^S h_E^3 (u_h) -g_0,E^2+ ∑_E∈^i∖^S h_E^3 (u_h)_0,E^2+∑_ E∈^i h_E (u_h) _0,E^2+ ∑_ E∈^f h_E^3 (u_h) _0,E^2+∑_ E∈^f∪^s h_E (u_h)_0,E^2.The following estimate holdsu-u_h_2 ≲η.Let w = u-u_h and w := Π_h w ∈ V_h be its interpolant. In view of the well-known coercivity of the bilinear form a and Galerkin orthogonality, we haveu-u_h_2^2 ≲ a(u-u_h,w)= a(u-u_h,w-w)= l(w-w)-a(u_h,w-w) . Sincex_0 is a mesh node andthe interpolant uses nodal values, we haveF (w(x_0)-w(x_0))=0, and hencel(w-w)=(f, w-w)+⟨ g, w-w⟩_S .From integration by parts over the element edges, using the fact that the interpolant uses values at the nodes, it then follows that u-u_h_2^2 ≲(f, w-w)+⟨ g, w-w⟩_S-a(u_h,w-w)= (f, w-w)+⟨ g, w-w⟩_S-∑_K ∈{((u_h),w-w)_K + ⟨(u_h) ,w-w⟩_∂ K -⟨(u_h),∂∂ s (w-w)⟩_∂ K - ⟨(u_h),∂∂ n (w-w)⟩_∂ K}= (f, w-w)+⟨ g, w-w⟩_S-∑_K ∈{((u_h),w-w)_K + ⟨(u_h) ,w-w⟩_∂ K - ⟨(u_h),∂∂ n (w-w)⟩_∂ K} .Regrouping and recalling definitions (<ref>) and (<ref>),yieldsu-u_h_2^2 ≲ ∑_K ∈(f-(u_h),w-w)_K -∑_E∈^S⟨(u_h)- g ,w-w⟩_E -∑_E∈^i ∖^S⟨(u_h),w-w⟩_E - ∑_E∈^i⟨(u_h), ∂∂ n_E (w-w)⟩_E -∑_E∈^f⟨(u_h) ,w-w⟩_E- ∑_ E∈^f∪^s⟨(u_h), ∂∂ n_E (w-w)⟩_E.The asserted a posteriori estimate now follows by applying the Cauchy–Schwarz inequality and the interpolation estimate (<ref>).Instead of the jump terms in the estimatorη, we could consider the normal and twisting moment jumps h_E^1/2(u_h)_0,E,h_E^1/2(u_h) _0,E, and the normal shear force jumps h_E^3/2(u_h)_0,E, h_E^3/2(u_h)- g _0,E. In this case we cannot, however, prove the efficiency, i.e. the lower bounds.Next, we will consider the question of efficiency. Let f_h∈ V_h be the interpolant of f and define_K(f) = h_K^2‖ f-f_h‖_0,K.Similarly, for a polynomial approximationg_h of g on E ⊂ S we define_E(g) = h_E^3/2‖ g-g_h‖_0,E. In the following theorem,ω_E stands for the union of elements sharing an edge E. In its proof, we will adopt some of the techniques used in <cit.>.For all v_h ∈ V_h it holdsh_K^2 (v_h) - f_0,K ≲u-v_h_2,K + _K(f),K∈, h_E^1/2(v_h) _0,E ≲u-v_h_2,ω_E + ∑_K ⊂ω_E_K(f),E∈^i, h_E^3/2(v_h) _0,E ≲u-v_h_2,ω_E + ∑_K ⊂ω_E_K(f),E∈^i∖^S, h_E^3/2(v_h)-g_0,E ≲u-v_h_2,ω_E + ∑_K ⊂ω_E_K(f)+_E(g),E∈^S,h_E^1/2(v_h) _0,E ≲u-v_h_2,ω_E + ∑_K ⊂ω_E_K(f),E∈^f∪^s, h_E^3/2(v_h) _0,E ≲u-v_h_2,ω_E+ ∑_K ⊂ω_E_K(f),E∈^f.Denote by b_K ∈ P_6(K) the sixth order bubble that, together with its first-order derivatives, vanishes on ∂ K, i.e. let b_K = (λ_1,Kλ_2,Kλ_3,K)^2, where λ_j,K are the barycentric coordinates for K. Then we defineγ_K = b_K h_K^4((v_h)-f_h) in Kandγ_K=0 in Ω∖ K,forv_h ∈ V_h. The problem statement givesa_K(u,γ_K) = (f,γ_K)_K,where a_K(u,γ_K) = ∫_K (u) : Κ(γ_K)dx. We haveh_K^4 (v_h)-f_h_0,K^2≲ h_K^4 √(b_K)((v_h)-f_h) _0,K^2 = ((v_h)-f_h,γ_K)_K = ((v_h),γ_K)_K - (f,γ_K)_K + (f-f_h,γ_K)_K= a_K(v_h-u,γ_K)+(f-f_h,γ_K)_K.The local bound (<ref>) now follows from applying the continuity of a, the Cauchy–Schwarz inequality and inverse estimates. Next, consider inequality (<ref>). SupposeE=K_1 ∩ K_2 forthe triangles K_1 and K_2; thusω_E=K_1∪ K_2. Let λ_E ∈ P_1(ω_E) be the linear polynomial satisfyingλ_E|_E=0,and∂λ_E/∂ n_E =1,and let p_1 be the polynomial that satisfies p_1 |_ E=M_nn(v_h) |_E and ∂ p_1/∂ n_E|_E= 0. Moreover, let p_2 ∈ P_8(ω_E) be the eight-order bubble that takes value one at the midpoint of the edge E and, together with its first-order derivatives, vanishes on ∂ω_E. Definew = λ_E p_1 p_2. Since∂ w/∂ n_E|_E = ∂λ_E/∂ n_E M_nn(v_h)p_2= M_nn(v_h)p_2,scaling yields the equivalenceM_nn(v_h) _0,E^2≈∂ w/∂ n_E_0,E^2 ≈√( p_2)M_nn(v_h) _0,E^2= ⟨ M_nn(v_h),∂ w∂ n_E⟩_E. Furthermore, since∂ w/∂ s|_E = 0,w|_E ∪∂ω_E=0 and∇ w|_∂ω_E =0,the integration by parts formula(<ref>) yields ⟨ M_nn(v_h),∂ w∂ n_E⟩_E =-∫_ω_E(v_h):Κ(w) dx + ((v_h),w)_ω_E.Extending w by zero to Ω∖ω_E,we obtain from the problem statement (<ref>)∫_ω_E(u):Κ(w) dx -(f,w)_ω_E=0.Hence, usingthe Cauchy–Schwarz inequality, we get from (<ref>)⟨ M_nn (v_h), ∂ w∂ n_E⟩_E=∫_ω_E(u-v_h):Κ(w) dx+((v_h)-f,w)_ω_E≲u-v_h_2,ω_E |w|_2,ω_E + (v_h)-f_0,ω_Ew_0,ω_E.By scaling, one easily shows that|w|_2,ω_E≲h_E^-1/2∂ w/∂ n_E_0,Eandw_0,ω_E≲h_E^3/2∂ w/∂ n_E_0,E.The estimate (<ref>) then follows from (<ref>),(<ref>), (<ref>), (<ref>), and the already proved bound (<ref>).Since (<ref>) follows from (<ref>) with g=0, we prove the latter.Due to the regularity condition imposed on the mesh there exists for each edge E a symmetric pair of smaller triangles (K^'_1,K^'_2) that satisfy ω^'_E=K_1^'∪ K_2^'⊂ω_E, see Fig. <ref>.Let w^' = p_2^' ((v_h) -g_h) where p_2^' is the eight-order bubble that takes value one at the midpoint of E and, together its first-order derivatives, vanishes on ∂ω_E^'. By the norm equivalence, we first have (v_h) -g_h_0,E^2 ≈ w^'_0,E^2 ≲⟨(v_h)-g_h, w^'⟩_E .Next, we write⟨(v_h)-g_h, w^'⟩_E =⟨(v_h)-g, w^'⟩_E +⟨ g -g_h, w^'⟩_E. Due tosymmetry, ∂ w^'/∂ n|_E = 0, and hence (<ref>) and (<ref>) give⟨(v_h)-g, w^'⟩_E = ⟨(v_h), w^'⟩_E-⟨ g, w^'⟩_E =∫_ω_E^'(v_h):Κ(w^' ) dx-((v_h),w^' )_ω_E^'-⟨ g, w^'⟩_E. Extending w^' by zero to Ω∖ω^'_E, the variational form (<ref>) implies that ∫_ω_E^'(u):Κ(w^') dx -(f,w^')_ω_E^'-⟨ g,w^'⟩_E=0.Hence,⟨(v_h)-g, w^'⟩_E=∫_ω_E^'(v_h-u):Κ(w^') dx +(f-(v_h) ,w^')_ω_E^'and the Cauchy–Schwarz inequality, scaling estimates and (<ref>) give⟨(v_h)-g, w^'⟩_E≲ h_E^-3/2( u-v_h_2,ω_E^' +h_K^2 (v_h)-f_0,ω_E^')w^'_0,E≲ h_E^-3/2( u-v_h_2,ω_E + ∑_K ∈ω_E_K(f) ) w^'_0,E. The asserted estimate then follows from (<ref>), (<ref>) and (<ref>). The estimates(<ref>),(<ref>), are proved similarly to the bounds(<ref>) and(<ref>), respectively.The above estimates providethe following global bound.It holdsη≲u-u_h_2 + (f)+ (g),where (f) = √(∑_K ∈_K(f)^2 )(g) = √(∑_E∈^S_E(g)^2).§ THE CHOICE OF V_HLet us briefly discuss some possible choices of conforming finite elements for the plate bending problem. Each choice consists of a polynomial space 𝒫 and of a set of N degrees of freedom defined through a functional ℒ : C^∞→ℝ. We denote by x^k, k∈{1,2,3}, the vertices of the triangle and by e^k, k∈{1,2,3}, the midpoints of the edges, i.e.e^1 = 1/2(x^1+x^2), e^2 = 1/2(x^2+x^3), e^3 = 1/2(x^1+x^3). The simplest H^2-conforming triangular finite element that is locally H^4(K) in each K is the Bell triangle.𝒫 = { p ∈ P_5(K) : ∂ p∂ n∈ P_3(E) ∀ E ⊂ K } ℒ(w)=w(x^k),for 1≤ k ≤ 3, ∂ w/∂ x_i(x^k),for 1 ≤ k ≤ 3 and 1 ≤ i ≤ 2, ∂^2 w/∂ x_i ∂ x_j(x^k),for 1 ≤ k ≤ 3 and 1 ≤ i, j ≤ 2.Even though the polynomial space associated with the Bell triangle is not the whole P_5(K) it is still larger than P_4(K). This can in some cases complicate the implementation. Moreover, the asymptotic interpolation estimates for P_5(K) are not obtained. This can be compensated by adding three degrees of freedom at the midpoints of the edges of the triangle and increasing accordingly the size of the polynomial space.𝒫 = P_5(K),ℒ(w)=w(x^k),for 1≤ k ≤ 3, ∂ w/∂ x_i(x^k),for 1 ≤ k ≤ 3 and 1 ≤ i ≤ 2, ∂^2 w/∂ x_i ∂ x_j(x^k),for 1 ≤ k ≤ 3 and 1 ≤ i, j ≤ 2, ∂ w/∂ n(e^k),for 1 ≤ k ≤ 3.The Argyris triangle can be further generalized to higher-order polynomial spaces, cf. P. Šolín <cit.>. Triangular macroelements such as the Hsieh–Clough–Tocher triangle are not locally H^4(K) and therefore additional jump terms are present inside the elements.Various conforming quadrilateral elements have been proposed in the literature for the plate bending problem cf. Ciarlet <cit.>. The proofs of thelower boundthat we presented do not directly apply to quadrilateral elements,but thetechniques can be adapted to them as well.§ NUMERICAL RESULTS In our examples, we will use the fifth degree Argyris triangle. On a uniform mesh for a solution u∈ H^r (Ω), with r≥ 2,we thus have the error estimate <cit.> ‖ u-u_h‖_2 ≲ h^s | u |_r,with s=min{r-2,4}. Since the mesh length is related to the number of degrees of freedom N byh∼ N^-1/2 on a uniform mesh,we can also write‖ u-u_h‖_2 ≲ N^-s/2| u |_r.If the solution is smooth, say r≥ 6, we thus have the estimates ‖ u-u_h‖_2 ≲ h^4‖ u-u_h‖_2 ≲ N^-2.In fact, therate N^-2 isoptimal also on a general mesh since, except for a polynomial solution, it holds <cit.>‖ u-u_h‖_2 ≳ N^-2.In the adaptivecomputations we use the following strategy for marking the elements that will be refined <cit.>. Given a partition , error indicators η_K, K ∈ and a threshold θ∈ (0,1), mark K for refinement if η_K ≥θmax_K^'∈η_K^'. The parameter θ has an effect on the portion of elements that are marked, i.e. for θ = 0 all elements are marked and for θ = 1 only the element with the largest error indicator value is marked. We simply take θ = 0.5 which has proven to be a feasible choice in most cases.The set of marked elements are refined usingTriangle <cit.>, version 1.6, by requiring additional vertices at the edge midpoints of the marked elements and by allowing the mesh generator to improve mesh quality through extra vertices. The default minimum interior angle constraint of 20 degrees is used. The regularity of the solution depends on the regularity of the load and the corner singularities, cf. <cit.>. Below we consider two sets of problems, one where the regularity is mainly restricted by the load, and another one where the load is uniform and the corner singularities dominate. §.§ Square plate, Navier solution A classical series solution to the Kirchhoff plate bending problem, the Navier solution <cit.>, in the special case of a unit square with simply supported boundaries and the loadingf(x) =f_0,if x∈ [12-c, 12+c] × [12-d, 12+d],0,otherwise,readsu(x,y) = 16 f_0/D π^6∑_m=1^∞∑_n=1^∞sinmπ/2sinnπ/2sinm π csinnπ d/mn(m^2+n^2)^2sinm π xsinn π y.In the limit c ⟶ 0 and 2 c f_0 ⟶ g_0 we get the line load solutionu(x,y) = 8 g_0/D π^5∑_m=1^∞∑_n=1^∞sinmπ/2sinnπ/2sinnπ d/n(m^2+n^2)^2sinm π xsinn π y,and in the limit c, d ⟶ 0 and 4 c d f_0 ⟶ F_0 we obtain the point load solutionu(x,y) = 4 F_0/D π^4∑_m=1^∞∑_n=1^∞sinmπ/2sinnπ/2/(m^2+n^2)^2sinm π xsinn π y.From the series we can infer that the solution is inH^3-ϵ(Ω), H^7/2-ϵ(Ω) and H^9/2-ϵ(Ω), for any ϵ>0, for the point load, line load and the square load, respectively. in the three cases. On a uniform mesh, one should thus observe the convergence rates N^-0.5,N^-0.75, and N^-1.25.An unfortunate property of the series solutions is that the partial sums converge very slowly. This makes computing the difference between the finite element solution and the series solution in H^2(Ω) and L^2(Ω)-norms a challenging task since the finite element solution quickly ends up being more accurate than any reasonable partial sum. In fact, the "exact" series solution is practically useless, for example,for computing the shear force which is an important design parameter. The H^2(Ω)-norm is equivalent to the energy norm,v= √(a(v,v)),with which the erroris straightforward to compute. In view of theGalerkin orthogonality and symmetry, one obtainsu-u_h^2 = a(u-u_h,u)= l(u-u_h) ,i.e. the error is given by u-u_h= √( l(u-u_h)).This is especially useful for the point load for which u-u_h= √( F_0(u(12,12)-u_h(12,12))). Evaluating the series solution at the point of maximum deflection gives <cit.> u(12,12)= 4 F_0/D π^4∑_m=1^∞∑_n=1^∞(sinmπ/2sinnπ/2)^2/(m^2+n^2)^2= 4 F_0/D π^4∑_m=1^∞(sinmπ/2)^2 ∑_n=1^∞(sinnπ/2)^2/(m^2+n^2)^2= F_0/2 D π^3∑_m=1^∞(sinmπ/2)^2(sinhmπ-mπ)/m^3(1+coshmπ). We first consider a point load with F_0=1, d=1, E=1 and ν=0.3, and compare the true error with the estimator η. In this case, we have the approximate maximum displacement u(12,12) ≈ 0.1266812,computed by evaluating and summing the first 10 million terms of the series (<ref>).Starting with an initial mesh shown in Fig. <ref>, we repeatedly mark and refine the mesh to obtain a sequence of meshes, see Fig. <ref> where the values of the elementwise error estimators are depicted for four consecutive meshes. Note that the estimator and the adapted marking strategy initially refine heavily in the neighborhood of the point load as one might expect based on the regularity of the solution in the vicinity of the point load.In addition to the adaptive strategy, we solve the problem using a uniform mesh family where we repeatedly split each triangle into four subtriangles starting from the initial mesh of Fig. <ref>. The energy norm error and η versus the number of degrees of freedom N are plotted in Fig. <ref>.The results show that the adaptive meshing strategy improvessignificantlythe rate of convergence in the energy norm. In Fig. <ref>,we have also plotted, for reference, the slopes corresponding to the expected convergence rate O(N^-0.5) for uniform refinement and the optimal convergence rate for P_5 elements, O(N^-2).In Fig. <ref> is is further revealed that the energy norm error and the estimator η follow similar trends. This is exactly what one would expect given that the estimator is an upper and lower bound for the true error modulo an unknown constant. This is better seen by drawing the normalized ratio η over u-u_h, see Fig. <ref>.Since the estimator correctly follows the true error and an accurate computation of norms like u-u_h_2 is expensive, the rest of the experiments document only the values of η and N for the purpose of giving idea ofthe convergence rates.We continue with the line load case taking g_0=1 and d=1/3, and using the same material parameter values as before.The initial and final meshes are shown in Fig. <ref>. The estimator can be seen to primarly focus on the end points of the line load. The values of η and N are visualized in Fig. <ref> together with the expected and the optimal rates of convergence. Again the adaptive strategy improves the convergence of the total error in comparison to the uniform refinement strategy. The local error estimators and the adaptive process are presented in Fig. <ref>.We finish this subsection by solving the square load case with f_0=1, c=d=1/3 and the same material parameters as before. The initial and the final meshes are shown in Fig. <ref>. The convergence rates are visualized in Fig. <ref> and the local error estimators in Fig. <ref>. An improvement in the convergence rate is again visible in the results. ndofs nelems energynorm eta 70 8 0.0334469831818 1.03051270004 206 32 0.0169100336631 0.493682375884 694 128 0.00843662159162 0.247183724801 2534 512 0.00421783761299 0.123606218404ndofs nelems energynorm eta 70 8 0.0334469831818 1.03051270004 206 32 0.02365310132 0.82181176783 278 48 0.0119938888183 0.416323779618 350 64 0.00610693490316 0.212543389914 422 80 0.0032646626221 0.114554845207 494 96 0.00199970066259 0.0715111566944 566 112 0.00139625006813 0.0557785320735ndofs nelems eta 219 38 0.112263251802 774 152 0.0392580419907 2910 608 0.0138990740805 11286 2432 0.00491278489704ndofs nelems eta 219 38 0.0874824228262 318 60 0.0172422350444 732 152 0.00563138145735 1158 242 0.00175456884437 1712 362 0.000633039818195 2270 486 0.00041008970146 3394 728 0.000183071024098ndofs nelems eta 208 34 0.0380160176302 716 136 0.0080971107166 2650 544 0.00102523606838 10190 2176 0.000137845398212ndofs nelems eta 208 34 0.0380160176302 407 72 0.00957105089795 452 82 0.00538652794625 653 122 0.00227568555901 957 188 0.00119173613498 1453 292 0.0004456239461 1478 296 0.000394849345563 1907 389 0.000227748630462 2867 593 9.98047955912e-05 3074 639 7.70829141819e-05 §.§ L-shaped domain Next we solve the Kirchhoff plate problem in L-shaped domain with uniform loading f=1 and the following three sets of boundary conditions: * Simply supported on all boundaries.* Clamped on all boundaries.* Free on the edges sharing the re-entrant corner and simply supported along the rest of the boundary.Due to the presence of a re-entrant corner, the solutions belong to H^2.33(Ω), H^2.54(Ω) and H^2.64(Ω) in the cases 1, 2 and 3, respectively(see <cit.>). As before, we use fifth-order Argyris elements to demonstrate the effectiveness of the adaptive solution strategy. The initial and the final meshes are shown in Fig. <ref>. The resulting total error estimators and unknown counts are visualized in Fig. <ref>. ndofs nelems eta 593 111 2.02577144144 2179 444 1.46810976277 8348 1776 1.12381491681 32674 7104 0.871033873118ndofs nelems eta 593 111 2.02577144144 638 121 1.86546429104 727 140 1.59759130521 824 160 1.23885967393 921 180 0.969188164204 1018 200 0.762907587751 1115 220 0.603440209985 1212 240 0.479294415861ndofs nelems eta 593 111 0.924193320451 2179 444 0.595889771428 8348 1776 0.40426945804 32674 7104 0.275514833081ndofs nelems eta 593 111 0.924193320451 638 121 0.725209767782 727 140 0.579349194803 824 160 0.392423628718 921 180 0.269295860608 1018 200 0.187398321262 1115 220 0.133243366253 1212 240 0.0982385833818ndofs nelems eta 593 111 0.672667391309 2179 444 0.388381670884 8348 1776 0.229683630386 32674 7104 0.137832309494ndofs nelems eta 593 111 0.672667391309 690 131 0.391628712819 778 149 0.233378180614 866 167 0.143258369597 954 185 0.0917709133812 1042 203 0.0633457066741 1156 226 0.0454579296491 1304 255 0.032930857058§ ACKNOWLEDGEMENTSThe authors thank the two anonymous referees and Prof. A. Ernfor comments that improved the final version of the paper.siamplain | http://arxiv.org/abs/1707.08396v2 | {
"authors": [
"Tom Gustafsson",
"Rolf Stenberg",
"Juha Videman"
],
"categories": [
"math.NA",
"65N30"
],
"primary_category": "math.NA",
"published": "20170726115835",
"title": "A posteriori estimates for conforming Kirchhoff plate elements"
} |
[email protected] [email protected] School of Engineering & Applied Science, Ahmedabad University, Ahmedabad-380009, India An efficient Singular Value Decomposition (SVD) algorithm is an important tool for distributed and streaming computation in big data problems. It is observed that update of singular vectors of a rank-1 perturbed matrix is similar to a Cauchy matrix-vector product. With this observation, in this paper, we present an efficient method for updating Singular Value Decomposition of rank-1 perturbed matrix in O(n^2log(1/ϵ)) time. The method usesFast Multipole Method (FMM) for updating singular vectors in O(nlog (1/ϵ)) time, where ϵ is the precision of computation. Updating SVD Rank-1 perturbation Cauchy matrixFast Multipole Method§ INTRODUCTION SVD is a matrix decomposition technique having wide range of applications, such as image/video compression, real time recommendation system, text mining (Latent Semantic Indexing (LSI)), signal processing, pattern recognition, etc. ComputationofSVDis rather simpler in case of centralized system where entire data matrix is available at a single location. It is more complex when the matrix is distributed over a network of devices. Further, increase incomplexity is attributed to the real-time arrival of new data. Processing of data over distributed stream-oriented systems require efficient algorithms that generate results and update them in real-time. In streaming environment the data is continuously updated and thus the output of any operation performed over the data must be updated accordingly. In this paper, a special case of SVD updating algorithm is presented where the updates to the existing data are of rank-1. Organization of the paper is as follows: Section <ref> motivates the problem of updating SVD for a rank-1 perturbation and presents a characterization to look at the problem from matrix-vector product point of view. An existing algorithm for computing fast matrix-vector product using interpolation is discussed in Section <ref>. Section <ref> introducesFast Multipole Method (FMM). In Section <ref> we present an improved algorithm based on FMM for rank-1 SVD update that runs in O(n^2 log1/ϵ) time, where real ϵ >0 is a desired accuracy parameter. Experimental results of the presented algorithms are given in Section <ref>. For completeness we have explained in details matrix factorization (<ref>), solution to Sylvester Matrix (<ref>), FAST algorithm for Cauchy matrix vector product (<ref>) and Fast Multipole Method (<ref>) at the end of this paper.§ RELATED WORKIn a series of work Gu and others <cit.> present work on rank-one approximate SVD update. Apart from the low-rank SVD update, focus of their work is to discuss numerical computations and related accuracies in significant details. This leads to accurate computation of singular values, singular vectors and Cauchy matrix-vector product. Our work differs from this work as follows:we use matrix factorizationthat is explicitly based on solution of Sylvester equation <cit.> to reach Eq. (<ref>) that computes updated singular vectors (see Section <ref>). Subsequently, we reach an equation, Eq. (<ref>) (similar to equation (3.3) in <cit.>). Further, we show that with this new matrix decomposition we reach the computational complexity O(n^2 log1/ϵ) for updating rank-1 SVD. § SVD OF RANK-ONE PERTURBED MATRICES Let SVD of a m × n matrix A = UΣ V^⊤.[We consider the model and characterization as in <cit.>. A detailed factorization of this matrix is given in <ref>.]Where, U ∈ℝ^m× m, Σ∈ℝ^m× n and V ∈ℝ^n× n, where, without loss of generality, we assume, m ≤ n. Let there be a rank-one update ab^⊤ to matrix A given by, Â=A + ab^⊤ and let ÛΣ̂V̂^⊤ denote the new(updated) SVD, where a ∈ℝ^m, b ∈ℝ^n, Thus, ÂÂ^⊤ = ÛΣ̂Σ̂^⊤Û^⊤. An algorithm for updating SVD of a rank-1 perturbed matrix is given in Bunch and Nielsen <cit.>. The algorithm updates singular values using characteristic polynomial and computes the updated singular vectors explicitly using the updated singular values. From (<ref>), ÂÂ^⊤= (U Σ V^⊤+ab^⊤ )(U Σ V^⊤+ ab^⊤ )^⊤ = U Σ V^⊤ V Σ^⊤U^⊤+ U Σ V^⊤ b_b̃a^⊤+ ab^⊤ V Σ^⊤ U^⊤_b̃^⊤ + ab^⊤ b_βa^⊤ ÂÂ^⊤= UΣΣ^⊤ U^⊤+ b̃a^⊤+ ab̃^⊤+ β aa^⊤ Where, b̃ = U Σ V^⊤ b,b̃^⊤= b^⊤ V Σ^⊤ U^⊤ and β = b^⊤ b. From (<ref>) it is clear that the problem of rank-1 update (<ref>) is modified to problem of three rank-1 updates (that is further converted to two rank-1 updates in (<ref>)). From (<ref>) and (<ref>) we get, ÛΣ̂Σ̂^⊤_D̂Û^⊤= UΣΣ^⊤_DU^⊤+ b̃a^⊤+ ab̃^⊤+ β aa^⊤ ÛD̂Û^⊤=UDU^⊤+ ρ_1a_1a_1^⊤_ŨD̃Ũ^⊤ + ρ_2b_1b_1^⊤ . Refer Appendix <ref> Eq. (<ref>) for more details. Similar computation for right singular vectors is required. The following computation is to be done for each rank-1 update,i.e., the procedure below will repeat four times, two times each for updating left and right singular vectors. From (<ref>) we have, ŨD̃Ũ^⊤= UDU^⊤+ ρ_1a_1a_1^⊤ = U(D + ρ_1a̅a̅^⊤ )_BU^⊤ , where a̅ = U^⊤ a_1. From (<ref>) we have, B := D + ρ_1a̅a̅^⊤ B =C̃D̃C̃^⊤ (Schur-decomposition). From (<ref>) and (<ref>) we get, ŨD̃Ũ^⊤ =U(C̃_ŨD̃C̃^⊤ )U^⊤_Ũ^⊤. After adding the rank-1 perturbation to UDU^⊤ in(<ref>), the updated singular vector matrix is given by matrix-matrix product Ũ = U C̃. Stange <cit.>, extending the work of <cit.>, presents an efficient way of updating SVD by exploring the underlying structure of the matrix-matrix computations of (<ref>). §.§ Updating Singular Values An approach for computing singular values is through eigenvalues. Given a matrix Ŝ = S + ρ uu^⊤, its eigenvalues d̃ can be computed in O(n^2) numerical operations by solving characteristic polynomial of (S + ρ uu^⊤ )x = d̃x, where S= diag(d_i). Golub <cit.> has shown that the above characteristic polynomial has following form. w(d̃) = 1 + ρ∑_i=1^nu_i^2/d_i - d̃. Note that in the equation above d̃ is an unknown and thus, though the polynomial function is structurally similar to Eq. (<ref>), we can not use FMM for solving it. Recall, in order to compute singular values D̂ of updated matrix  we need to update D twice as there are two symmetric rank-1 updates, i.e., forŨD̃Ũ^⊤= UDU^⊤+ ρ_1a_1a_1^⊤ we will update B = D + ρ_1a̅a̅^⊤ Eq. (<ref>) and similarly for ÛD̂Û^⊤= ŨD̃Ũ^⊤+ ρ_2b_1b_1^⊤ we will update B_1 = D̃ + ρ_2b̅b̅^⊤. At times, while computing eigen-system, some of the eigenvalues and eigenvectors are known (This happens when there is some prior knowledge available about the eigenvalues or when the eigenvalues are approximated by using methods such as power iteration.). In such cases efficient SVD update method should focus on updating unknown eigen values and eigen vectors. Matrix Deflation is the process of eliminating known eigenvalue from the matrix. Bunch, Nielsen and Sorensen <cit.> extended Golub's work by bringing the notion of deflation. They presented a computationally efficient method for computing singular values by deflating the system for cases: (1) when some values of a̅ and b̅ are zero, (2) a̅ = 1 and b̅ = 1 and (3) B and B_1 has eigenvalues with repetition (multiplicity more than one). After deflation, the singular values of the updated matrix D̂can be obtained by Eq. (<ref>). §.§ Updating Singular Vectors In order to update singular vectors the matrix-matrix product of (<ref>) is required. A naive method for matrix multiplication has complexity O(n^3). We exploit matrix factorization in Stange <cit.> that shows the structure of matrix C̃ to be Cauchy. From (<ref>) and (<ref>) we get, C̃D̃C̃^⊤= D + ρ_1a̅a̅^⊤ DC̃ - C̃D̃= - ρ_1a̅a̅^⊤C̃.Equation (<ref>) is Sylvester equation with solution,(I_n ⊗ D + (-D̃)^⊤⊗ I_n)C̃ =(- ρ_1a̅a̅^⊤C̃). Simplifying L.H.S. of (<ref>) for C̃ we get,C̃ = [ [ a̅_1 ; ⋱; a̅_n ]] [ [ (μ_1 - λ_1)^-1… (μ_n - λ_1)^-1;⋮ ⋮; (μ_1 - λ_n)^-1… (μ_n - λ_n)^-1 ]] [ [ a̅^⊤ c_1 ; ⋱; a̅^⊤ c_n ]]. Where, C̃ = [[ c_1 … c_n ]] and from R.H.S. of (<ref>), c_i ∈ℝ^n arecolumns of the form, c_i = ρ_1 [[ a̅_1/μ_i - λ_1;⋮; a̅_n/μ_i - λ_n ]] a̅^⊤ c_i.Simplifying (<ref>) further we get, c_i = ρ_1 [[ a̅_1/μ_i - λ_1;⋮; a̅_n/μ_i - λ_n ]] [[ a̅_1 a̅_2 ]] [[ c_i1; c_i2 ]]c_i = -ρ_1 (a̅_1c_i1 +a̅_2c_i2) [[ a̅_1/λ_1 - μ_i;⋮; a̅_n/λ_n - μ_i ]].Thus, denoting α̂ as -ρ_1 (a̅_1c_i1 +a̅_2c_i2) we get,c_i = α̂[[ a̅_1/λ_1 - μ_i…a̅_n/λ_n - μ_i ]]^⊤ .Placing (<ref>) in (<ref>) and we have,α̂[[ a̅_1/λ_1 - μ_i;⋮; a̅_n/λ_n - μ_i ]] = α̂ρ_1 [[ a̅_1/μ_i - λ_1;⋮; a̅_n/μ_i - λ_n ]] ∑_k = 1^na̅_k^2/λ_k - μ_i.Therefore for each element of c_i,α̂a̅_j/λ_j - μ_i= α̂ρ_1a̅_j/μ_i - λ_j ∑_k = 1^na̅_k^2/λ_k - μ_i 0= α̂(1 + ρ_1 ∑_k = 1^na̅_k^2/λ_k - μ_i)_=0Equation inside the bracket of (<ref>) is the characteristic equation (<ref>) used for finding updated eigenvalues μ_i of D̃ by using eigenvalues of D i.e. λ. As μ_i are the zeros of this equation. Placing λ_k and μ_i will make the term in the bracket (<ref>) zero. Due to independence of the choice of scalar α̂, any value of α̂ can be used to scale the matrix C̅. In order to make the final matrix orthogonal, each column of C̅ is scaled by inverse of Euclidean norm of the respective column (<ref>). From Eq. (<ref>) matrix notation for C̃ can be written as C̃ = [ [ a̅_1 ; ⋱; a̅_n ]] [[ 1/λ_1 - μ_1⋯1/λ_1-μ_n;⋮⋮; 1/λ_n - μ_1⋯1/λ_n - μ_n ]]^C_C̅ = [[ c_1 … c_n ]][ [ |c_1|; ⋱; |c_n| ]]^-1.Where, c_i is the Euclidean norm of c_i. From (<ref>) it is evident that the C matrix is similar to Cauchy matrix and C̃ is the scaled version of Cauchy matrix C. In order to update singular vectors we need to calculate matrix-matrix product as given in(<ref>). From (<ref>) and (<ref>) we get,UC̃= U[ [ a̅_1 ; ⋱; a̅_n ]] [[ 1/λ_1 - μ_1⋯1/λ_1-μ_n;⋮⋮; 1/λ_n - μ_1⋯1/λ_n - μ_n ]][ [ |c_1|; ⋱; |c_n| ]]^-1 UC̃=[[ û_1 … û_n ]] C [ [ |c_1|; ⋱; |c_n| ]]^-1.Where, û_i = u_ia̅_iandi = 1, … ,n.UC̃= U_1C [ [ |c_1|; ⋱; |c_n| ]]^-1UC̃= U_2 [ [ |c_1|; ⋱; |c_n| ]]^-1Where, U_1 =[[ û_1 … û_n ]] and U_2 = U_1 C . §.§.§ Trummer's Problem: Cauchy Matrix-Vector Product In (<ref>) there are n vectors in U which are multiplied with Cauchy matrix C. The problem of multiplying a Cauchy matrix with a vector is called Trummer's problem. As there are n vectors in (<ref>), matrix-matrix product in it can be represented as n matrix-vector products, i.e., it is same as solving Trummer's problem n times. Section <ref> describes an algorithm given in <cit.> which efficiently computes such matrix-vector product in O(nlog^2(n)) time. § FAST: METHOD BASED ON POLYNOMIAL INTERPOLATION AND FFT Consider the the matrix - (Cauchy) matrix product U_2 = U_1C of Section <ref>. This product can be written as U_2 =[[ û_1 … û_n ]] C U_2 =[[ u_11⋯ u_1n;⋮⋮⋮; u_n1⋯ u_nn ]][[ 1/λ_1 - μ_1⋯1/λ_1-μ_n; ⋮⋮⋮; 1/λ_n - μ_1⋯1/λ_n - μ_n ]]. The dot product of each row vector of U_1 and the coulmn vector of the Cauchy matrix can be represented in terms of a function of eigenvalues μ_i,i = 1,…,n, f(μ_1) =[[ u_11⋯ u_1n ]][[ 1/λ_1 - μ_1; ⋮; 1/λ_n - μ_1 ]] =u_11/λ_1 -μ_1+…+u_1n/λ_n -μ_1 f(μ_1)=∑_j=1^nu_1j/λ_j - μ_1. Hence in general, f(x) =∑_j=1^nu_j/λ_j - x. Equation (<ref>) can be shown as the ratio of two polynomials f(x) =h(x)/g(x), where, g(x) = ∏_j=1^n (λ_j - x). h(x)=g(x)∑_j=1^nu_j/λ_j - x h(x)= ∏_j=1^n (λ_j - x)∑_j=1^nu_j/λ_j - x The FAST algorithm<cit.> represents the a matrix-vector product as (<ref>) and (<ref>). It then finds solutions to this problem by the use of interpolation. The FAST algorithm of Gerasoulis <cit.> has been reproduce in <ref>. § MATRIX-VECTOR PRODUCT USING FMM The FAST algorithmin <cit.> has complexity O (nlog^2n). It computes the function f(x) using FFT and interpolation. We observe that these two methods are two major procedures that contributes to the overall complexity of FAST algorithm. To reduce this complexity we present an algorithm that uses FMM for finding Cauchy matrix-vector product that updates the SVD of rank-1 perturbed matrix with time complexityO(n^2log(1/ϵ)). Recall from Section <ref>, update of singular vectors require matrix-(Cauchy) matrix product U_2 = U_1C, This product is represented as the function below. f(μ_i)=∑_j=1^n-u_j/μ_i - λ_j. We use FMM to compute this function. §.§ Fast Multipole Method (FMM) An algorithm for computing potential and force of n particles in a system is given in Greengard and Rokhlin <cit.>. This algorithm enables fast computation of forces in an n-body problem by computing interactions among points in terms of far-field and local expansions. It starts with clustering particles in groups such that the inter-cluster and intra-cluster distance between points make them well-separated. Forces between the clusters are computed using Multipole expansion. Dutt et al. <cit.> describes the idea of FMM for particles in one dimension and presents an algorithm for evaluating sums of the form, f(x) = ∑_k=1^Nα_k ·ϕ(x - x_k). Where, {α_1,…,α_N} is a set of complex numbers and ϕ(x) can be a function that is singular at x=0 and smooth every where. Based on the choice of function, (<ref>) can be used to evaluate sum of different forms for example: f(x) = ∑_k=1^Nα_k/x-x_k where, ϕ(x) = 1/x. Dutt et al. <cit.> presents an algorithm for computingsummation (<ref>) using FMM that runs in O(nlog(1/ϵ)). §.§ Summation using FMM Consider well separated points {x_1,x_2,…, x_N} and {y_1,…,y_M} in ℝ such that for points x_0,y_0 ∈ℝ and r>0, r ∈ℝ.|x_i - x_0| < r∀ i = 1,…,N|y_i - y_0| < r∀ i = 1,…,Mand|x_0 - y_0| > 4rFor a function defined as f:ℝ→ℂ such that f(x) = ∑_k=1^Nα_k/x-x_kwhere, {α_1 , …, α_k} is a set of complex numbers. Given f(x), the task is to find f(y_1),…,f(y_M).§ RANK-ONE SVD UPDATE In this section, we present Algorithm <ref> that uses FMM and updates Singular Value Decomposition in O(n^2log(1/ϵ)) time.§.§ Algorithm: Update SVD of rank-1 modified matrix using FMMINPUT A ∈ℝ^m × n = UΣ V^T, a∈ℝ^mand b∈ℝ^n OUTPUT U_n, Σ_n, and V_n STEP 1 Compute b̃ = UΣ V^Tb, ã = VΣ^TU^Ta, β = b^Tb, α = a^Ta, D_u = ΣΣ^T and D_v = Σ^TΣ. STEP 2 Compute the Schur decomposition of [[β 1; 10 ]] = Q_u [[ ρ_10;0 ρ_2 ]]Q_u^T and [[ ab̃ ]]Q_u = [[ a_1b_1 ]].STEP 3 Compute the Schur decomposition of [[α 1; 10 ]] = Q_v [[ ρ_10;0 ρ_2 ]]Q_v^T and [[ bã ]]Q_v = [[ a_2b_2 ]]. STEP 4 Compute updated left singular vector Ũ and D̃ by calling the procedure RankOneUpdate(U,a_1,D,ρ_1) - Algorithm <ref>. STEP 5 Compute left singular vector of Â, i.e., U_nand D_n by calling the procedure RankOneUpdate(Ũ,b_1D̃,ρ_2). STEP 6 Compute updated right singular vector Ṽ and D̃_̃ṽ by calling the procedure RankOneUpdate(V,a_2,D_v,ρ_3). STEP 7 Compute right singular vector of  i.e. V_n and D_vn by calling the procedure RankOneUpdate(Ṽ,b_2,D̃_̃ṽ,ρ_4). STEP 8 Find singular values by computing square root of the updated eigenvalues Σ_n = √(D_n).Note that the Schur decomposition of Steps 2 and 3 are computed over constant size matrices and thus the decomposition will take constant time.§.§ Algorithm: RankOneUpdate INPUT U,a_1,D,ρ_1OUTPUT Ũ, D̃ STEP 1 Compute a̅ = U^T a_1.STEP 2 Compute μ as zeros of the equation w(μ) = 1 + ρ_1 ∑_i=1^na̅_i^2/λ_i - μ. STEP 3 Compute C = [[ 1/λ_1 - μ_1⋯1/λ_1-μ_n;⋮⋮; 1/λ_n - μ_1⋯1/λ_n - μ_n ]]. STEP 4 Compute U_1 = U [[ a̅_1 ;⋱; a̅_n ]]STEP 5Compute C̅ = [[ a̅_1 ;⋱; a̅_n ]] [[ 1/λ_1 - μ_1⋯1/λ_1-μ_n;⋮⋮; 1/λ_n - μ_1⋯1/λ_n - μ_n ]] STEP 6 Compute U_2= U_1C as n matrix-vector product. Where each row-column dot product is represented as a function f(μ_i)=∑_j=1^n-u_j/μ_i - λ_j .for each μ_i.U_2 = FMM(λ, μ, -U_1) - (<ref>). STEP 7 Form Ũ by dividing each column of U_2 by norm of respective column of C̅. §.§ Complexity for Rank-One SVD update theoremTheorem Given a matrix A ∈ℝ^m × n such that m ≤ n and precision of computation parameter ϵ>0, the complexity of computing SVD of a rank-1 update of A with Algorithm <ref> is O(n^2log (1/ϵ)). Follows from Algorithm <ref> and Table <ref>. § NUMERICAL RESULTSAll the computations for the algorithm were performed on MATLAB over a machine with Intel i5, quad-core, 1.7 GHz, 64-bit processor with 8 GB RAM. Matrices used in these experiments are square and generated randomly with values ranging from [1,9]. The sample size varies from 2 × 2 to 35 × 35.For computing the matrix vector product we use FMM Algorithm (instead of FAST Algorithm) with machine precision ϵ=5^-10.In order to update singular vectors we need to perform two rank-1 updates (<ref>). For each such rank-1 update we use FMM.ÛD̂Û^T =UDU^T + ρ_1a_1a_1^T_ŨD̃Ũ^T + ρ_2b_1b_1^T.Figure <ref> shows the time taken by FAST Algorithm and FMM Algorithm to compute the first rank-1 update.§.§ Choice of ϵIn earlier computations we fixed the machine precision parameter ϵ for FMM based on the order of the Chebyshev polynomials, i.e., ϵ = 5^-p, where p is the order of Chebyshev polynomial. As we increase the order of polynomials - we expect reduction in the error and increase in updated vectors accuracy. This increase in accuracy comes with the cost of higher computational time. To show this we fix the input matrix dimension to 25 × 25 and generate values of these matrices randomly in the range [0,1]. Figure <ref> shows the error between updated singular vectors generated by Algorithm <ref> and exact computation of singular vector updates.The results in figure justifies our choice of fixed machine precision p = 20.Error is computed using the equation (<ref>) <cit.>, where  is the perturbed matrix, ÛΣ̂V̂^T is the approximation computed using FMM-SVDU and max σ̂ is the directly computed maximum eigenvalue of Â.Error= max| - ÛΣ̂V̂^T/max σ̂| Table <ref> shows the accuracy of rank-1 SVD update using FMM- SVDU for varying sample size. Figure <ref> shows plot of the accuracy of FMM-SVDU with varying sample size. § CONCLUSION In this paper we considered the problem of updating Singular Value Decomposition of a rank-1 perturbed matrix. We presented an efficient algorithm for updating SVD of rank-1 perturbed matrix that uses the Fast Multipole method for improving Cauchy matrix- vector product computational efficiency. An interesting and natural extension of this work is to consider updates of rank-k. § REFERENCES 10 url<#>1urlprefixURL href#1#2#2 #1#1gu93 M. Gu, Studies in numerical linear algebra, Ph.D. thesis, Yale University (1993).ge93 M. Gu, S. C. Eisenstat, A stable and fast algorithm for updating the singular value decomposition, Tech. Rep. YALEU/DCS/RR-966, Dept. of Computer Science, Yale University (1993).ge94 M. Gu, S. C. Eisenstat, A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem, SIAM journal on Matrix Analysis and Applications 15 (4) (1994) 1266–1276.ge95 M. Gu, S. C. Eisenstat, Downdating the singular value decomposition, SIAM Journal on Matrix Analysis and Applications 16 (3) (1995) 793–810.stange08 P. Stange, On the efficient update of the singular value decomposition, PAMM 8 (1) (2008) 10827–10828.bn78 J. R. Bunch, C. P. Nielsen, Updating the singular value decomposition, Numerische Mathematik 31 (2) (1978) 111–129.golub73 G. H. Golub, Some modified matrix eigenvalue problems, Siam Review 15 (2) (1973) 318–334.bnc78 J. R. Bunch, C. P. Nielsen, D. C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numerische Mathematik 31 (1) (1978) 31–48.gerasoulis88 A. Gerasoulis, A fast algorithm for the multiplication of generalized Hilbert matrices with vectors, Mathematics of Computation 50 (181) (1988) 179–188.gr87 L. Greengard, V. Rokhlin, A fast algorithm for particle simulations, Journal of computational physics 73 (2) (1987) 325–348.dgr96 A. Dutt, M. Gu, V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM Journal on Numerical Analysis 33 (5) (1996) 1689–1711.§ MATRIX FACTORIZATIONWe consider the matrix factorization in <cit.>. Let SVD of a m × n matrix A = UΣ V^⊤. Where, U ∈ℝ^m× m, Σ∈ℝ^m× n and V ∈ℝ^n× n. Here it is assumed that m ≤ n. Let there be a rank-one update ab^⊤ to matrix A Eq. (<ref>) and let ÛΣ̂V̂^⊤ denote the new(updated) SVD, where a ∈ℝ^m, b ∈ℝ^n. Â= A + ab^⊤ Â=ÛΣ̂V̂^⊤ In order to find SVD of this updated matrix  we need to compute ÂÂ^⊤ and Â^⊤ because left singular vector Û of  is orthonormal eigenvector of ÂÂ^⊤ and right singular vector V̂ of  is orthonormal eigenvector of Â^⊤Â. ÂÂ^⊤= (ÛΣ̂V̂^⊤ )(ÛΣ̂V̂^⊤ )^⊤ =ÛΣ̂V̂^⊤V̂Σ̂^⊤Û^⊤ ÂÂ^⊤=ÛΣ̂Σ̂^⊤Û^⊤ Where, V̂^⊤V̂ = I ÂÂ^⊤= (U Σ V^⊤+ab^⊤ )(U Σ V^⊤+ ab^⊤ )^⊤ = (U Σ V^⊤+ab^⊤ )(V Σ^⊤U^⊤+ ba^⊤ ) = U Σ V^⊤ V Σ^⊤U^⊤+ U Σ V^⊤ b_b̃a^⊤+ ab^⊤ V Σ^⊤ U^⊤_b̃^⊤ + ab^⊤ b_βa^⊤ ÂÂ^⊤= UΣΣ^⊤ U^⊤+ b̃a^⊤+ ab̃^⊤+ β aa^⊤ Where, V^⊤ V = I Where, b̃ = U Σ V^⊤ b,b̃^⊤= b^⊤ V Σ^⊤ U^⊤ and β = b^⊤ b. From Eq. (<ref>) it is clear that the problem of rank-1 update Eq. (<ref>) is modified to problem of three rank-1 updates that is further converted to two rank-1 updates in Eq. (<ref>). Equating Eq. (<ref>) and Eq. (<ref>) we get, ÛΣ̂Σ̂^⊤_D̂Û^⊤= UΣΣ^⊤_DU^⊤+ b̃a^⊤+ ab̃^⊤+ β aa^⊤ ÛD̂Û^⊤= UDU^⊤ + b̃a^⊤+ ab̃^⊤+ β aa^⊤ = UDU^⊤ + β aa^⊤+ b̃a^⊤+ ab̃^⊤ = UDU^⊤ + (β a + b̃)a^⊤+ ab̃^⊤ = UDU^⊤ + [[ β a + b̃ a ]][[a^⊤; b̃^⊤ ]] ÛD̂Û^⊤= UDU^⊤+ [[ a b̃ ]] [[β 1; 10 ]] [[a^⊤; b̃^⊤ ]] ÛD̂Û^⊤= UDU^⊤+ [[ a b̃ ]] Q [[ ρ_10;0 ρ_2 ]]Q^⊤[[a^⊤; b̃^⊤ ]] Where, [[β 1; 10 ]]= Q [[ ρ_10;0 ρ_2 ]]Q^⊤ (Schur-decomposition) ÛD̂Û^⊤= UDU^⊤+ [[ a b̃ ]] Q_[[ a_1b_1 ]][[ ρ_10;0 ρ_2 ]]Q^⊤[[a^⊤; b̃^⊤ ]]_[[ a_1b_1 ]]^⊤ ÛD̂Û^⊤= UDU^⊤+ [[ a_1b_1 ]] [[ ρ_10;0 ρ_2 ]][[ a_1^⊤; b_1^⊤ ]]^⊤ ÛD̂Û^⊤=UDU^⊤+ ρ_1a_1a_1^⊤_ŨD̃Ũ^⊤ + ρ_2b_1b_1^⊤ . Similarly for computing right singular vectors do the following. Â^⊤Â= (U Σ V^⊤+ab^⊤ )^⊤ (U Σ V^⊤+ ab^⊤ )= (V Σ^⊤U^⊤+ ba^⊤ )(U Σ V^⊤+ab^⊤ ) Â^⊤Â= V Σ^⊤U^⊤ U Σ V^⊤+ V Σ^⊤U^⊤ a_ãb^⊤+ ba^⊤ U Σ V^⊤_ã^⊤+ ba^⊤ a_αb^⊤ (V̂Σ̂^⊤Û^⊤ )(ÛΣ̂V̂^⊤ )= VΣ^⊤Σ V^⊤+ ãb^⊤+ bã^⊤+ α bb^⊤ Where, U^⊤ U = I . V̂Σ̂^⊤Σ̂V̂^⊤=VΣ^⊤Σ V^⊤+ ρ_3a_2a_2^⊤+ ρ_4b_2b_2^⊤ The following computation is to be done for each rank-1 update in Eq. (<ref>) and Eq. (<ref>) i.e. the below procedure will repeat four times, two times each for updating left and right singular vectors. From Eq. (<ref>) we have, ŨD̃Ũ^⊤= UDU^⊤+ ρ_1a_1a_1^⊤ = UDU^⊤+ ρ_1(UU^⊤ a_1)(UU^⊤ a_1)^⊤ = UDU^⊤+ ρ_1(Ua̅)(Ua̅)^⊤ Where, a̅ = U^⊤ a_1 = UDU^⊤+ ρ_1(Ua̅a̅^⊤ U^⊤ ) ŨD̃Ũ^⊤= U(D + ρ_1a̅a̅^⊤ )_BU^⊤ . From Eq. (<ref>) we have, B := D + ρ_1a̅a̅^⊤ B =C̃D̃C̃^⊤ (Schur-decomposition). Placing Eq. (<ref>) in Eq. (<ref>) we get, ŨD̃Ũ^⊤ =U(C̃_ŨD̃C̃^⊤ )U^⊤_Ũ^⊤. After the rank-1 update to UDU^⊤ Eq. (<ref>) the updated singular vector matrix is given by matrix-matrix product Ũ = U C̃.§ SOLUTION TO SYLVESTER EQUATIONIn Section <ref> we discussed method for updating singular vectors. For the same we derived solutions to (<ref>) using Sylvester equation. In this section we present details of how this solution can be obtained. For simplicity consider the case where the dimension of all the matrices (C, D and D̃) is 2 × 2. L.H.S = (I_n ⊗ D + (-D̃)^⊤⊗ I_n)C̃ = {([[ 1 0; 0 1 ]] ⊗[[ λ_1 0; 0 λ_2 ]])- ([[ μ_1 0; 0 μ_2 ]]^⊤⊗[[ 1 0; 0 1 ]])}[[ c_11 c_12; c_21 c_22 ]]= {[[ λ_1 0 0 0; 0 λ_2 0 0; 0 0 λ_1 0; 0 0 0 λ_2 ]] - [[ μ_1 0 0 0; 0 μ_1 0 0; 0 0 μ_2 0; 0 0 0 μ_2 ]]}[[ c_11; c_21; c_12; c_22 ]]= [[ λ_1 - μ_1 0 0 0; 0 λ_2 - μ_1 0 0; 0 0 λ_1 - μ_2 0; 0 0 0 λ_2 - μ_2 ]] [[ c_11; c_21; c_12; c_22 ]]R.H.S =(- ρ_1a̅a̅^⊤C̃) = -ρ_1([[ a̅_1; a̅_2;]][[ a̅_1 a̅_2 ]][[ c_11 c_12; c_21 c_22 ]]) = -ρ_1([[ a̅_1; a̅_2;]][[ a̅_1c_11 + a̅_2 c_21 a̅_1c_12 + a̅_2 c_22 ]]) = -ρ_1([[ a̅_1; a̅_2;]][[[ a̅_1 a̅_2 ]][[ c_11; c_21 ]] [[ a̅_1 a̅_2 ]][[ c_12; c_22 ]]]) = -ρ_1([[ a̅_1; a̅_2;]][[ a̅^⊤ c_1 a̅^⊤ c_2 ]])Where, C̃= [[ c_1 c_2 ]] and c_1 = [[ c_11; c_21 ]] c_2 = [[ c_12; c_22 ]] = -ρ_1([[ a̅_1a̅^⊤ c_1 a̅_1a̅^⊤ c_2; a̅_2a̅^⊤ c_1 a̅_2a̅^⊤ c_2 ]]) = -ρ_1[[ a̅_1a̅^⊤ c_1; a̅_2a̅^⊤ c_1; a̅_1a̅^⊤ c_2; a̅_2a̅^⊤ c_2 ]] Equating L.H.S and R.H.S we get, [[ λ_1 - μ_1 0 0 0; 0 λ_2 - μ_1 0 0; 0 0 λ_1 - μ_2 0; 0 0 0 λ_2 - μ_2 ]] [[ c_11; c_21; c_12; c_22 ]] = -ρ_1[[ a̅_1a̅^⊤ c_1; a̅_2a̅^⊤ c_1; a̅_1a̅^⊤ c_2; a̅_2a̅^⊤ c_2 ]].Therefore,[[ c_11; c_21; c_12; c_22 ]] = -ρ_1[[ λ_1 - μ_1 0 0 0; 0 λ_2 - μ_1 0 0; 0 0 λ_1 - μ_2 0; 0 0 0 λ_2 - μ_2 ]]^-1[[ a̅_1a̅^⊤ c_1; a̅_2a̅^⊤ c_1; a̅_1a̅^⊤ c_2; a̅_2a̅^⊤ c_2 ]] [[ c_11; c_21; c_12; c_22 ]] = -ρ_1[[ 1/λ_1 - μ_1 0 0 0; 0 1/λ_2 - μ_1 0 0; 0 0 1/λ_1 - μ_2 0; 0 0 0 1/λ_2 - μ_2 ]][[ a̅_1a̅^⊤ c_1; a̅_2a̅^⊤ c_1; a̅_1a̅^⊤ c_2; a̅_2a̅^⊤ c_2 ]] [[ c_11; c_21; c_12; c_22 ]] = -ρ_1[[ a̅_1a̅^⊤ c_1/λ_1 - μ_1; a̅_2a̅^⊤ c_1/λ_2 - μ_1; a̅_1a̅^⊤ c_2/λ_1 - μ_2; a̅_2a̅^⊤ c_2/λ_2 - μ_2 ]] [[ c_11; c_21; c_12; c_22 ]] = ρ_1[[ a̅_1a̅^⊤ c_1/μ_1 - λ_1; a̅_2a̅^⊤ c_1/μ_1 - λ_2; a̅_1a̅^⊤ c_2/μ_2 - λ_1; a̅_2a̅^⊤ c_2/μ_2 - λ_2 ]] Where, [[ c_11; c_21 ]] = ρ_1[[ a̅_1a̅^⊤ c_1/μ_1 - λ_1; a̅_2a̅^⊤ c_1/μ_1 - λ_2 ]] and [[ c_12; c_22 ]]= ρ_1[[ a̅_1a̅^⊤ c_2/μ_2 - λ_1; a̅_2a̅^⊤ c_2/μ_2 - λ_2 ]] i.e.c_1 = ρ_1[[ a̅_1/μ_1 - λ_1; a̅_2/μ_1 - λ_2 ]]a̅^⊤ c_1 andc_2 = ρ_1[[ a̅_1/μ_2 - λ_1; a̅_2/μ_2 - λ_2 ]]a̅^⊤ c_2In general c_i = ρ_1[[ a̅_1/μ_i - λ_1;⋮; a̅_2/μ_i - λ_n ]]a̅^⊤ c_1 By placing c_1 and c_2 in C̃ = [[ c_1 c_2 ]] we get C̃ as below. C̃= [[ c_1 c_2 ]] C̃= [[ a̅_1a̅^⊤ c_1/μ_1 - λ_1 a̅_1a̅^⊤ c_2/μ_2 - λ_1; a̅_1a̅^⊤ c_1/μ_1 - λ_2 a̅_1a̅^⊤ c_2/μ_2 - λ_2 ]] C̃= [[ a̅_10;0 a̅_2 ]][[ a̅^⊤ c_1/μ_1 - λ_1 a̅^⊤ c_2/μ_2 - λ_1; a̅^⊤ c_1/μ_1 - λ_2 a̅^⊤ c_2/μ_2 - λ_2 ]] C̃= [[ a̅_10;0 a̅_2 ]][[ 1/μ_1 - λ_1 1/μ_2 - λ_1; 1/μ_1 - λ_2 1/μ_2 - λ_2 ]][[ a̅^⊤ c_10;0 a̅^⊤ c_2 ]] § FAST ALGORITHM In this section we present FAST Algorithm <cit.> that computes functions of the form (<ref>) using polynomial interpolation in time O(n log^2 n). * Compute the coefficients of g(x) in its power form, by using FFT polynomial multiplication, in O(n(log n)^2) time. Decription: Uses FFT for speedy multiplication which reduces complexity of multiplication to O(nlog(n)) from O(n^2). Input: Function g(x) and eigenvalues of D̃ i.e. [μ_1,…,μ_n] Output: Coefficients of g(x) i.e. [a_0,a_1,…,a_n] Complexity: O(nlog^2n) * Compute the coefficients of g'(x) in O(n) time. Description: Differentiate the function g(x) and then compute its coefficients. Input: Function g(x) Output: Coefficients of g'(x) i.e. [b_0,b_1,…,b_n] Complexity: O(n) * Evaluate g(λ_i), g'(λ_i) and g(μ_i). Description: For the functions g(x) and g'(x) evaluate their values at λ_i and μ_i Input: Function g(x) and g'(x), eigenvalues [λ_1,λ_2,…,λ_n] of D and [μ_1,μ_2,…,μ_n] of D̃ Output: g(λ_i), g'(λ_i) and g(μ_i) Complexity: O(nlog^2n) * Compute h_j = u_jg'(λ_j). Description: For each eigenvalue λ_j compute its function value h(λ_j) i.e. find the points (λ_j,u_jg'(λ_j)) Input: [λ_1,λ_2,…,λ_n] Output: h_j Complexity: O(n) * Find interpolation polynomial h(x) for the points (λ_j,h_j). Description: Given the function values and input i.e. points (λ_j,h_j) find interpolation polynomial for n points. Input: Points (λ_j,h_j) Output: h(x) Complexity: O(nlog^2n) * Compute v_i = h_(μ_i)/g(μ_i). Description: Compute the ratio v_i = h_(μ_i)/g(μ_i) for each μ_i where, v_i is value of f(μ_i) at each μ_i. Input: Function h(x) and g(x) and [μ_1,μ_2,…,μ_n] Output:f(μ_i) = v_i Complexity: O(n) § FAST MULTIPOLE METHOD §.§ Interpolation and Chebyshev Nodes Chebyshev nodes are the roots of the Chebyshev polynomials and they are used as points for interpolation. These nodes lie in the range [-1,1]. A polynomial of degree less than or equal to n-1 can fit over n Chebyshev nodes. For Chebyshev nodes the approximating polynomial is computed using Lagrange interpolation. Expansions are used to quantify interactions among points. Expansions are only computed for points which are well-separated from each other. §.§ FMM Algorithm STEP 1Description: Decide the size of Chebyshev expansion i.e., p = -log_5(ϵ) = log_5 (1/ϵ) where ϵ > 1 is the precision of computation or machine accuracy parameter. Input: ϵ Output: p Complexity: O(1) STEP 2Description: Set s as the number of points in the cell of finest level (s ≈ 2p) and level of finest division nlevs = log_2(N/s) where, N is the number of points. Input: N and p Output: s and nlevs Complexity: O(1) STEP 3Description: Consider p Chebyshev nodes defined over an interval [-1,1] of the form below. t_i = cos ( 2i - 1/p·π/2) Where, i = 1,…,p. STEP 4Description: Consider Chebyshev polynomials of the form u_j(t) = ∏_k=1 k≠ j^pt - t_k/t_j - t_k. Where, j = 1,…,p. STEP 5 Description: Calculate the far-field expansion Φ_l,i at i^th subinterval of level l. For i^th interval of level l (nlevs) far-field expansion due to points in interval [x_0 - r, x_0 + r] about center of the interval x_0 is defined by a vector of size p as Φ_nlevs,i = ∑_k=1^Nα_k ·t_i/3r - t_i(x_k -x_0). Input: l = nlevs, α_k, x_k, x_0, r, t_i for i = 1,…,2^nlevs, Output: Φ_nlevs,i Complexity: O(Np) STEP 6[Bottom-up approach]Description: Compute the far-field expansion of individual subintervals in terms of far-field expansion of their children. These are represented by p × p matrix defined as below. M_L(i,j) = u_j(t_i/2+t_i) M_R(i,j) = u_j(t_i/2-t_i) Where, i = 1,…,p and j = 1,…,p. {u_1,…,u_p} and {t_1,…,t_p} are as defined in Eq. (<ref>) and Eq. (<ref>) respectively. Far-field expansion for i^th subinterval of level l due to far-field expansion of its children is computed by shifting children's far-field expansion by M_L or M_R and adding those shifts as below.Φ_l,i = M_L ·Φ_l+1,2i-l + M_R ·Φ_l+1,2i Input: M_L, M_R and Φ Output: Φ_l,i Complexity: O(2Np^2/s) STEP 7 Description: Calculate the local expansion Ψ_l,i at i^th subinterval of level l. For i^th interval of level l local expansion due to points outside the interval [y_0 - r, y_0 + r] about center of the interval y_0 is defined by a vector of size p as Ψ_i = ∑_k=1^Nα_k ·1/rt_i - (x_k - x_0). Input: t_i, α_k, r, x_k,andx_0 Output: Ψ_i Complexity: O(Np) STEP 8[Top-down approach]Description: Compute the local expansion of individual subintervals in terms of local expansion of their parents. These are represented by p × p matrix defined as below. S_L(i,j) = u_j(t_i - 1/2) S_R(i,j) = u_j(t_i + 1/2) Where, i = 1,…,p and j = 1,…,p. {u_1,…,u_p} and {t_1,…,t_p} are as defined in Eq. (<ref>) and Eq. (<ref>) respectively. Compute local expansions using far-field expansion using p × p matrix defined as below. T_1(i,j) = u_j(3/t_i - 6) T_2(i,j) = u_j(3/t_i - 4) T_3(i,j) = u_j(3/t_i + 4) T_4(i,j) = u_j(3/t_i + 6) Local expansion for each subinterval at finer level is computed using local expansion of their parents. For this, first the local expansion of parent interval are shifted by S_L or S_R and then the result is added with interactions of subinterval with other well separated subintervals (which were not considered at the parent level). Ψ_l+1,2i-1 =S_L ·Ψ_l,i + T_1 ·Φ_l+1,2i-3 + T_3 ·Φ_l+1,2i+1 + T_4 ·Φ_l+1,2i+2 Ψ_l+1,2i =S_R ·Ψ_l,i + T_1 ·Φ_l+1,2i-3 + T_2 ·Φ_l+1,2i-2 + T_4 ·Φ_l+1,2i+2 Input: S_L, S_R, T_1, T_2, T_3, T_4, Φ and Ψ Output: Ψ_l+1,2i-1 and Ψ_l+1,2i Complexity: O(8Np^2/s) STEP 9 Description: Evaluate local expansion Ψ_nlevs,i at some of {y_j} (which falls in subinterval i of level nlevs) to obtain a p size vector. Complexity: O(Np) STEP 10 Description: Add all the remaining interactions which are not covered by expansions. Compute interactions of each { y_k} in subinterval i of level nlevs with all {x_j} in subinterval i-1,i,i+1. Add all these interactions with the respective local expansions. Complexity: O(3Ns) 1 Total Complexity of FMM O(1+1+Np+(22Np^2/s)+ Np + (82Np^2/s)+Np+3Ns) = O(2+3Np+3Ns+(10Np^2/s)) = O(2+3Np+6Np+(10Np^2/2p)) = O(2+9Np+(5Np)) = O(2+14Np) = O(Np) = O(Nlog(1/ϵ)) , Where p = log(1/ϵ) | http://arxiv.org/abs/1707.08369v1 | {
"authors": [
"Ratnik Gandhi",
"Amoli Rajgor"
],
"categories": [
"cs.LG",
"math.NA"
],
"primary_category": "cs.LG",
"published": "20170726105122",
"title": "Updating Singular Value Decomposition for Rank One Matrix Perturbation"
} |
plain thmTheorem cor[thm]Corollary[ |⟩⟩⟨⟨| College of Basic Sciences and Humanities, G.B. Pant University Of Agriculture and Technology, Pantnagar, Uttarakhand - 263153, IndiaHarish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, India Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, IndiaS. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, IndiaHarish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, IndiaWe establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics. The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices.The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that non-ergodicity plays an important role in preserving information of quantum states undergoing lossy evolution. Wealso consider a model of a central qubitinteracting with a fermionic thermal bath and derive its reduced dynamics, to subsequently investigate the information loss and non-ergodicity in such dynamics. We comment on the “minimal” situations that saturate the uncertainty relations.Universal quantum uncertainty relations between non-ergodicity and loss of information Ujjwal Sen December 30, 2023 =======================================================================================§ INTRODUCTION In practical situations, it is arguably impossible to completely isolate a quantum system from its surroundings and it is subjected to information loss due to dissipation and decoherence. In modelling open quantum systems, the simpler approach is to consider the environment to be memoryless, i.e. Markovian <cit.>. The system-environment relation is however more often than not non-Markovian, and there are possibilities of information backflow into the system, which can be considered as a resource in information theoretic tasks <cit.>. The systems showing such properties are usually associated with various structured environments without the consideration of weak system-environment coupling and the Born- Markov approximation <cit.>.In a Markovian evolution, this information flow is one-way and quickly leads to an unwanted total loss of coherence and other quantum characteristics. Using structured environments, it may be possible to reduce information loss of the associated quantum system. On the other hand, an important statistical mechanical attribute of asystem interacting with an environment, with the later being in a thermal state, is the ergodicity of the system.A physical process is considered to be ergodic, if the statistical properties of the process can be realized from a long-time averaged realization. In the study of the realization of a thermal relaxation process, ergodicity plays a very important role <cit.>.Italso has important applications in quantum control <cit.>, quantum communication <cit.>, and beyond <cit.>. Here we intend to capture the notion of “non-ergodicity" from the perspective of quantum channels, i.e. considering only the reduced dynamics of a quantum system interacting with an environment. In the framework of open quantum systems, a rigorous study on ergodic quantum channels can be found in <cit.>. Ergodic quantum channels are channels having a unique fixed point in the space of density matrices <cit.>. Non-ergodicity of a dynamical process can then be quantified as the amount of deviation from a ergodic process in open system dynamics. In this work, we find a connection between information loss of a general open quantum system and non-ergodicity therein. We propose ameasure of information loss in a quantum system, based on distinguishabilityof quantum states, which in turn is based on distance measures on the space of density operators <cit.>. We quantify the non-ergodicity of the dynamics based on the distance between the time-averaged state after sufficiently long processing time and the corresponding thermal equilibrium state. Within this paradigm, we derive an uncertaintyrelation between informaton loss and the amount of non-ergodicity for an arbitrary quantum system interacting according to an arbitrary quantum Hamiltonian with an arbitraryenvironment. The derivation is not for a particular distance measure, but for all such which satisfies a set of intuitively satisfactory axioms. In the illustrations, we mainly focus on the trace distance, and to a certain extent, also on the relative entropy. We find that ourrelationsare compatible with Markovian ergodic dynamics, where the system loses all the information.Finally, we have considered a particular structured environment model, where a central qubit interacts with a collection of mutually non-interacting spins in thermal states at an arbitrary temperature. A spin-bath model of this type, which has been considered previously in the literature <cit.>, shows ahighly non-Markovian nature. Here we have derived the reduced dynamics ofa particular spin-bath model without the weak coupling and Born-Markov approximations. Subsequently, we investigate the information loss and non-ergodicity, and find the status of the uncertainty for this system. The organization of the paper is as follows. In Section II, we present the definitions for loss of information and non-ergodicity. We derive the uncertainty relations between information loss and non-ergodicity in Section III. In Section IV, we consider the central spin model, derive the reduced dynamics of the central qubit, and analyze the corresponding information loss and non-ergodicity. We conclude in Section V.§ DEFINITIONS: MEASURES FOR LOSS OF INFORMATIONANDNON-ERGODICITY Before proceeding to the main results, let usdefine the two primary quantities under present investigation, i.e. loss of information and a measure of non-ergodicity, based on distance measures.§.§ Loss of informationWe quantify the loss of information in quantum systems due to environmental interaction, in terms of distinguishability measuresfor quantum states. The loss of information, denoted by I_Δ(t), at any instant oftime, can be quantified by the maximal difference between the initial distinguishability between a pair of states, ρ_1(0), ρ_2(0), and that for the corresponding time evolved states ρ_1(t)=Φ(ρ_1(0)), ρ_2(t)=Φ(ρ_2(0)) at time t, where Φ denotes the open quantum evolution of the initial states. Mathematically, it is given byI_Δ(t)= max_ρ_1(0), ρ_2(0)(D(ρ_1(0),ρ_2(0))-D(ρ_1(t),ρ_2(t)) ),where the distance measure D(ρ,σ) must satisfy the following conditions: P1.D(ρ,σ)≥ 0 ∀ ρ, σ.P2.D(ρ,ρ)=0 ∀ ρandD(ρ,σ)=0 ⟺ ρ=σ, ∀ ρ,σ.P3.D(Φ(ρ),Φ(σ))≤ D(ρ,σ) ∀ ρ,σand ∀ completely positive trace preserving maps Φ(·), on the space of density operators, ℬ(ℋ), on the Hilbert space ℋ. The class of distance measures satisfying these conditions, includetrace distance, Bures distance, Hellinger distance <cit.>. Though the von Neumann relative entropy and Jensen-Shannon divergence also satisfy the aforementioned conditions, they are not generally considered as geometric distances, since they certain other metric properties. But also note here that the square root of Jensen-Shannon divergence does satisfy metric properties <cit.> and can be considered as a valid distance measure. It is also important to mention that all the aforementioned valid distance measures are bounded. Loss of information for time-averaged states: To draw the connection with non-ergodicity, discussed below, we now define the long time-averaged state asρ̅= lim_τ→∞1/τ∫_0^τρ(t)dt .The information loss for the time-averaged state, which we call “ average loss of information”, can then be defined asI̅_Δ= max_ρ_1(0), ρ_2(0)(D(ρ_1(0),ρ_2(0))-D(ρ̅_1,ρ̅_2) ),which is lower and upper bounded by 0and 1 respectively. From Eq. (<ref>) we can infer that, when the open system dynamics has a unique steady state or fixed point, independent of initial state, the entire information D(ρ_1(0),ρ_2(0)) is lost for arbitrary inputs ρ_1(0),ρ_2(0). Later in the paper, we draw a connection between average loss of information and non-ergodicity of the underlying dynamics, with the later being defined in the succeeding subsection.§.§ Non-ergodicityErgodicity plays an important role instatistical mechanics, to describe the realization of relaxation of the system to the thermal equilibrium. The ergodic hypothesis states that if a system evolves over a long period of time, the long time-averaged state of the system is equal to its thermal state corresponding to the temperature of the environment with which the system is interacting. Ergodicity can also be defined in terms of observables. For any observable f, if its long time average, ⟨f||_⟩T is equal to its ensemble average, ⟨f||_⟩en, the dynamics is considered to be ergodic for the observables. Here the time and ensemble averages of the observable are respectively defined as⟨ ̅|f|=⟩lim_τ→∞1/τ∫_0^τ[fρ(t)] = [fρ̅] ; ⟨f||_⟩en = [fρ_th]. Ergodicity further assumes the equality of ⟨ ̅|f|$⟩ and⟨f||_⟩en, independent of the initial state of the evolution.Therefore, non-ergodicity of the dynamics for the observable can be quantified by the difference between the time average and ensemble average, i.e. by|⟨ ̅| f|-⟩⟨f||_⟩en|= |[f (ρ̅-ρ_th)]|. Based on these understandings of ergodicity of a dynamics, we define a measure of non-ergodicity as the distance between the long time-averaged state (ρ̅) of the system and its corresponding thermal state (ρ_th), and so is given by 𝒩_ϵ(ρ̅) =D(ρ̅ , ρ_th).Here weimposetwo further conditions on the allowed distance measures:P4. The measure must be symmetric, i.e D(ρ,σ)=D(σ, ρ), ∀ρ,σ.P5. The measure must satisfy the triangle inequality, given by D(ρ,σ) ≤ D(ρ,κ)+D(κ,σ), ∀ ρ, σ,κ.The conditions P1-P5 are satisfied by the geometric distance measures like trace distance, Bures distance, and Hellingerdistance. Note the von Neumann relative entropy neither satisfies the symmetry property nor the triangle inequality and hence we cannot use it directly for our investigation. However, we will later show the possibility of overcoming such “shortcomings” of the relativeentropy distance. Interestingly, it has been shown<cit.> that Jenson-Shannon divergence satisfies the symmetry property and for its square root, the triangle inequality holds. Therefore, the square root of Jensen-Shannon divergence can also be taken as a proper distance measure for our investigation. Note that the measure of non-ergodicity, given in (<ref>), depends on the initial state. Hence, toobtain a measure ofnon-ergodicity which is state-independent, we introduce 𝒩_ϵ^M =max_ρ(0)𝒩_ϵ(ρ̅),where maximization is performed over all initial states (ρ(0)). § CONNECTING INFORMATION LOSS WITH NON-ERGODICITYWith the definitions given in the preceding section, we now establish a connection between loss of information and non-ergodicity. For the distance measures, which satisfy P1-P5, we obtainD(ρ̅_1,ρ̅_2) ≤𝒩_ϵ(ρ̅_1)+𝒩_ϵ(ρ̅_2). Using Eq. (<ref>), we therefore have the inequality I̅_Δ≥max_ρ_1(0), ρ_2(0)(D(ρ_1(0),ρ_2(0))-(𝒩_ϵ(ρ̅_1)+𝒩_ϵ(ρ̅_2))).It draws a direct connection between non-ergodicity and loss of information in open system dynamics.Using the state-independent measure of non-ergodicity (Eq. (<ref>)), we can arrive at an uncertainty relation between information loss and a measure of non-ergodicity, given byI̅_Δ + 2𝒩_ϵ^M ≥max_ρ_1(0), ρ_2(0)(D(ρ_1(0),ρ_2(0))).The above relation is valid for any distance measure which satisfies the conditions P1-P5, and for any quantum system, interacting with an arbitrary environment.In this paper, we will mainly workon the uncertainty relation based on the distance measure given byD^T(ρ,σ)=1/2|ρ-σ|for pairs of statesρandσ. The importance of quantum relative entropy <cit.> as a “distance-type” measure, notwithstanding its inability in satisfying symmetry and other relations, from the perspective of quantum thermodynamics is unquestionable, and hence obtaining uncertainity relation in terms ofquantum relative entropy can be interesting. Towards this aim, we usea relation between relative entropy and trace distance <cit.>, given byS(ρ||σ)≡[ρ(logρ-logσ)]≥ 2(D^T(ρ,σ))^2. The above inequality helps us to overcome the drawbacks of relative entropy for not satisfying P4 and P5. Let us first rewrite(<ref>) in terms of trace distance asI̅_Δ^T≥max_ρ_1(0), ρ_2(0)(D^T(ρ_1(0),ρ_2(0))-(𝒩_ϵ^T(ρ̅_1)+𝒩_ϵ^T(ρ̅_2))).Using inequalities (<ref>) and(<ref>), we arrive atI̅_Δ^T ≥max_ρ_1(0), ρ_2(0)(D^T(ρ_1(0),ρ_2(0))-(√(𝒩_ϵ^Rel(ρ̅_1)/2)+√(𝒩_ϵ^Rel(ρ̅_2)/2))),where𝒩_ϵ^Rel(ρ̅_i)=S(ρ̅_i||ρ_th)denotes the measure of non-ergodicity for the time-averaged stateρ̅_iin terms of relative entropy.As before, we can define a state-independentmeasure of non-ergodicity as𝒩_ϵ^M(Rel)=max_ρ(0)S(ρ̅||ρ_th), The above defintion and the inequality (<ref>) leads to another uncertainty relationI̅_Δ^T + √(2𝒩_ϵ^M(Rel)) ≥max_ρ_1(0), ρ_2(0)(D^T(ρ_1(0),ρ_2(0))),in terms of trace and relative entropy distances. But it is to be noted that there is a certain limitation in this relation, because of the fact that the relative entropy is not a bounded function. When ρ⊈ ρ_th, the relative entropy diverges. One such example is obtained for the zero temperature bath, whereρ_th=|0|⟨%s|⟩0|is pure. In that case, the relation (<ref>) becomes trivial. But in that case, we can find state- dependent uncertainty relations by defining state- dependent information loss asI̅_Δ(ρ̅_1,ρ̅_2) =(D(ρ_1(0),ρ_2(0))-D(ρ̅_1,ρ̅_2)).This will lead us to the state-dependent uncertainty relationI̅_Δ^T(ρ̅_1,ρ̅_2)+∑_i=1,2√(𝒩_ϵ^Rel(ρ̅_i)/2)≥ D^T(ρ_1(0),ρ_2(0)).But other than these extreme cases, the relation (<ref>) works perfectly.Note that the distinguishability measures like trace distance, Bures distance and Jensen-Shanon divergence, mentioned earlier, not only satisfies all the conditions P1-P5, but they are also bounded. But in the cases of some unbounded distance measure, to avoid the triviality of the uncertainty relation (<ref>), we can use the state-dependent uncertainty relationI̅_Δ(ρ̅_1,ρ̅_2)+ ∑_i=1,2𝒩_ϵ(ρ̅_i)≥ D(ρ_1(0),ρ_2(0)).§.§ QubitsUpto now, we have considered an arbitrary density matrix of arbitrary dimension. Let us now restrict to the case of a two- level system (TLS) as a simple example to further understand the connection between non-ergodicity and information loss. For a TLS, the pair of states maximizing the trace distance is located on the antipodes of the Bloch sphere i.e., the pair ofstates consists of pure and mutually orthogonal states <cit.>.Therefore in the case of trace distance, the uncertainty relation (<ref>), for a qubit, reads as I̅_Δ^T + 2𝒩_ϵ^M(T)≥ 1.Similarly, the uncertainty relation given in (<ref>) reduces to I̅_Δ^T + √(2𝒩_ϵ^M(Rel)) ≥ 1. Let us now consider a simple Markovian model, where a qubit is weakly coupled with a thermal bosonic environment. In absence of any external driving Hamiltonian, the qubit eventually thermally equilibrates with the environment. Under Born-Markov approximation, the master equation for this model is given by[ ρ̇(t̃)=i/ħ[ρ(t̃),H_0]+γ(n+1)(σ_-ρ(t)σ_+-1/2{σ_+σ_-,ρ(t̃)}); +γ n(σ_+ρ(t̃)σ_--1/2{σ_-σ_+,ρ(t̃)}), ] whereH_0=ħΩ_0|1|⟨%s|⟩1|is the Hamiltonian of the system,γis a constant parameter andn=1/(exp(ħΩ_0/KT̃_m)-1)is the Planck number. Hereσ_+andσ_-are respectively the raising and lowering operators of the TLS, with|1⟩being the excited state of the same. The solution of the Markovian master equation in (<ref>) is given by ρ(t̃)=ρ_11(t̃)|1|⟨%s|⟩1|+ρ_22(t̃)|0|⟨%s|⟩0|+ρ_12(t̃)|1|⟨%s|⟩0|+ρ_21(t̃)|0|⟨%s|⟩1|,with[ ρ_11(t̃)=ρ_11(0)e^-γ(2n+1)t̃+n/2n+1(1-e^-γ(2n+1)t̃),; ρ_22(t̃)=1-ρ_11(t̃),;ρ_12(t̃)=ρ_12(0)exp(-γ(2n+1)t̃/2-2iΩ_0 t̃). ]One can find from the solution given above that the long time-averaged state for this evolution is independent of initial states and equal to thethermal state corresponding to the temperature of the bathT̃_m, which can be expressed asp|0|⟨%s|⟩0|+(1-p)|1|⟨%s|⟩1|, withp=1/(1+exp(-ħΩ_0/KT̃_m)).Hence the dynamics is ergodic andwe find that the information lossI̅_Δ^T=1; i.e. the system loses all its information. It is also noteworthy that Markovianity of a quantum evolution does not mean it will be ergodic. An example of such Markovian non-ergodic evolution is the dephasing channel expressed by the master equationρ̇= iΩ_0[σ_z, ρ]+γ_d( σ_zρσ_z-ρ)Here the Lindblad operator is in the same basis as the system Hamiltonianσ_z. A system interacting with a bosonic environment can lead to such an evolution <cit.>. The solution of this equation is given by[ρ_11(t̃)=ρ_11(0),;ρ_22(t̃)=ρ_22(0),; ρ_12(t̃)=ρ_12(0)e^-2(iΩ+γ_d)t. ] We realize from Eq. (<ref>) that under this particular evolution, the system will decohere, but the digonal elements of the density matrix will remain invariant, leading to infinitely many fixed points for the dynamics. So this particular evolution will certainly be non-ergodic, since there exists infinitely many fixed points and the time averaged state will depend on the initial state of the system. This gives a definite example which proves that Markovianity does not imply ergodicity of the dynamics. § NON-ERGODICITY AND INFORMATION BACK-FLOW IN A CENTRAL SPIN MODELIn this section, we consider a specific non-Markovian model and study the status of uncertainty relation derived in Sec III. The system here consists ofa single qubit, interacting withNnumber of non-interacting spins. The total Hamiltonian of the system, governing the dynamics, is given by H̃=H̃_S+H̃_B+H̃_I,where the system HamiltonianH̃_S, bath HamiltonianH̃_B, and interaction HamiltonianH̃_Iare respectively given by[ H̃_S=ħ gω_0σ_z,;H̃_B=ħ gω/N∑_i=1^Nσ_z^i,; H̃_I=ħ gα/√(N)∑_i=1^N (σ_xσ_x^i+σ_yσ_y^i+σ_zσ_z^i). ] Hereσ_k,k=x,y,zare the Pauli spin matrices, the superscript `i' represents the ith spin of the bath,gis a constant factor with the dimension of frequency,ω_0andωare the dimensionless parameters characterizing the energy level differences of the system and the bath respectively andαdenotes the coupling constant of the system-bath interaction. By using the total angular momentum operatorsJ_k=∑_i=1^Nσ^i_k, and the Holstein-Primakoff transformation, given byJ_+=√(N)b^†(1-b^†b/2N)^1/2 , J_-=√(N)(1-b^†b/2N)^1/2b,the bath and interaction Hamiltonians can now be rewritten as[ H̃_B =-ħ gω(1-b^†b/N),; H̃_I=2ħ gα[σ_+(1-b^†b/2N)^1/2b+σ_-b^†(1-b^†b/2N)^1/2]; -ħ gα√(N)σ_z(1-b^†b/N). ] We consider the initial (uncorrelated) system-bath state asρ_S(0)⊗ρ_B(0).Let us take the initial system qubit asρ_S(0)=ρ_11(0)|1|⟨%s|⟩1|+ρ_22(0)|0|⟨%s|⟩0|+ρ_12(0)|1|⟨%s|⟩0|+ρ_21(0)|0|⟨%s|⟩1|and the initial bath state to be a thermal stateρ_B(0)=exp(-H̃_B/KT̃)in an arbitrary temperatureT̃with K being the Boltzmann constant. The reduced dynamics of the system state can then be calculated by tracing out the bath degrees of freedom and is given byρ_S(t)=_B[exp(-iHt)ρ_S(0)⊗ρ_B(0)exp(iHt)]. WhereH=H̃/ħ g, t=gt̃, T=KT̃/ħ g,are dimensionless, specifying Hamiltonian, time and temperature respectively. After solving the global Schrödinger evolution, the reduced dynamics can be exactly obtained <cit.> asρ_S(t)=(ρ_11(t)ρ_12(t) ρ_21(t)ρ_22(t) ),where [ ρ_11(t)=ρ_11(0)(1-Θ_1(t))+ρ_22(0)Θ_2(t),; ρ_12(t)=ρ_12(0)Δ(t), ] with[ Θ_1(t)=∑_n=0^N (n+1)α^2(1-n/2N)(sin(η t/2)/η/2)^2e^-ω/T(n/N-1)/Z,;; Θ_2(t)=∑_n=0^N nα^2(1-(n-1)/2N)(sin(η' t/2)/η'/2)^2e^-ω/T(n/N-1)/Z,;; Δ(t)=∑_n=0^N e^-i(Λ -Λ')t/2(cos(η t/2)-iθ/ηsin (η t/2)); ×(cos(η' t/2)+iθ'/η'sin (η' t/2))e^-ω/T(n/N-1)/Z,;; Z=∑_n=0^Ne^-ω/T(n/N-1),;;η=2√((ω_0-ω/2N-α√(N)(1-2n+1/2N))^2+4α^2(n+1)(1-n/2N)),;;η'=2√((ω_0-ω/2N-α√(N)(1-2n-1/2N))^2+4α^2 n(1-(n-1)/2N)),;; θ=2(ω_0-ω/2N+α√(N)(1-2n+1/2N)),;; θ'=-2(ω_0-ω/2N-α√(N)(1-2n-1/2N)),;Λ=-2ω(1-2n+1/2N)-α/√(N),; Λ'=-2ω(1-2n-1/2N)-α/√(N). ]The time-averaged state for this system can then be calculated as[ ρ̅_11=ρ_11(0)(1-Θ̅_1)+ρ_22(0)Θ̅_2,; ρ̅_12=ρ_12(0)Δ̅, ] with[Θ̅_1=∑_n=0^N 2(n+1)α^2(1-n/2N)(1/η^2)e^-ω/T(n/N-1)/Z,; ; Θ̅_2=∑_n=0^N 2nα^2(1-(n-1)/2N)(1/η'^2)e^-ω/T(n/N-1)/Z,; ;Δ̅=0. ]Note that in general the coherence of the time-averaged state will vanish asΔ̅=0. But there are specific resonance conditions under which there can be non-zero coherence present in the time-averaged state <cit.>. But in this work, we will not consider such situations. Before investigating the uncertainty relation in terms of trace distance given in (<ref>), we explore the behavior of loss of information at instantaneoustime with different parameters involved in this dynamics. For such study, let usrestrict ourselves to the set of pure initial qubits over which the optimization involved in (<ref>) is performed. In particular, we take the initial pair of orthogonal pure states to becosθ/2|1|+⟩sinθ/2e^-iϕ|0|$⟩ and sinθ/2|1|-⟩cosθ/2e^-iϕ|0|$⟩, with0≤θ≤π, 0≤ϕ< 2π. The instantaneous and average information losses in this case are given respectively by[ I_Δ^T(t)=Θ_1(t)+Θ_2(t), I̅_Δ^(T)=Θ̅_1+Θ̅_2. ]In Figs. <ref>,<ref>,<ref>, the instantaneous loss of information is depictedwith time for different valuesof the number of bath-spins (N), temperature (T̃) and system-bath interaction strength(α)respectively, by keeping other parameters fixed. From the figures, we deduce the following:Observation 1: The instantaneous loss of information shows oscillatory behavior whose amplitude decreases with time. Observation 2: The increase of number of spins of the bath, in temperature, as well as in the interaction strength can be seen as increaseof influence of bath on the system. Hence, expectantly in all cases, the loss of information increaseswith increase of the above system parameters.Let us now check the uncertainty relation given in (<ref>) for the qubit case, taking the same initial pair of pure orthogonal states andthe thermal state at arbitrary temperatureρ_th=p_1|0|⟨%s|⟩0|+(1-p_1)|1|⟨%s|⟩1|, wherep_1=1/2(1+tanh(ħ gω_0/KT̃)). After performing the maximization, we find I̅_Δ^T+2𝒩_ϵ^M(T) =Θ̅_1+Θ̅_2+2|p_1-Θ̅_1|.We now examine theconditions for which the uncertainty relation (<ref>) saturates. Note that for the ergodic situations, i.e. if the steady state is unique and is equal to the thermal state, the information loss is equals to unity, leading to a trivialequality in (<ref>). KeepingNfixed to 1000 and fixing the temperature to different values, we investigate the values ofI̅_Δ^T+2N_ϵ^M(T)for increasing interacting strength. We observe that the sum goes close to unity for a strong interaction strength as depicted in Fig. <ref> for high temperature. In Figs. <ref> and <ref>, we analyze the sumI_Δ^T+2𝒩_ϵ^M(T)as the number of bath spins are ramped up from 100 to 1000. Scrutinizing these figures, we can safely conclude that withthe increase in number of spins of the bath, or in the bath temperature, or in the system-bath interaction strength, the sumI_Δ^(T)+2𝒩_ϵ^M(T)goes very close to unity in this qubit case, provided the optimization involved is restricted to pure qubits. However, numerical evidence strongly suggests that for this non-Markovian model, given in Eq. (<ref>), there will be no non-trivial situation when theuncertanity relation in Eq. (<ref>) saturates to unity, provided the maximization is carried out over pure state.We find that the saturation value ofΘ̅_1andΘ̅_2, whenα⟶∞,are respectively given by [ Θ̅_1^sat=1/2∑_n=0^n=N1/4+N (1-(2n+1)/2N)/(n+1)(1-n/2N)^2e^-ħω/KT(n/N-1)/Z,; Θ̅_2^sat=1/2∑_n=0^n=N1/4+N (1-(2n-1)/2N)^2/n(1-(n-1)/2N)e^-ħω/KT(n/N-1)/Z. ] If the interaction Hamiltonian given in Eq. (<ref>), is considered in absence of thez-zinteraction, the saturated values ofΘ̅_1andΘ̅_2in the limitN→∞, α→∞will beΘ̅_1^sat=Θ̅_2^sat=1/8. In the infinite temperature limit, we havep_1=1/2, which leads to the equality in (<ref>). It is interesting that in the mentioned limit, the non-ergodicity measure is finite and equals to 3/8. Therefore, we find a non-ergodic situation, where the equality of the uncertainty relation holds.When the equality of the mentioned relations hold for a non-ergodic evolution, these relations imply that when the non-ergodicity of the dynamics increases, the information loss in the system decreases. Nonergodic dynamics are, in general, good for information processing asthey have less chance of leakage ofinformation compared to ergodic dynamics. In particular, for non-ergodic evolution for which the uncertainty relations discussed in this paper are equalities, the loss of information can be quantified by and attributed to the nonergodicity in the evolution. It is also important to mention that spin bath models do not always indicate a shows non-ergodic dynamics. In a recent work <cit.>, such dynamics hase been considered with the Born-Markov approximation region to find the effective reduced dynamics. It is shown in the mentioned work that there are situations where a unique fixed point (stationary state) can exist for the evolution, and hence in those situations, the dynamics is ergodic <cit.>. § CONCLUSIONIn open quantum dynamics, the information exchange between the system and bath plays an important role, while the time-evolved state's correspondencewith the Gibb's ensemble conspire to imply the ergodic nature of the system. In this article, we establish a relation between loss of information and a measure of non-ergodicity. Both the definitions are given in terms of distinguishability, which can be measured by a suitably chosen distance measure. We have shown that the informationloss and the quantifier of non-ergodicity followan uncertainty relation, valid for a broad class of distinguishability measures, which includes trace distance, Bures distance, Hilbert-Schmidt distance, Hellinger distance, and square root of Jensen-Shannon divergence. We have further considered trace distance between a pair of quantum states as a specific distinguishability measure and connected the corresponding information loss with non-ergodicity, which is now definedin terms of relative entropy between the time-averaged state and the thermal state, maximized over all possible initial states. We have shown that in a Markovian model, the uncertainty relation saturates and shows a complete information loss. We also considered a structured environment model of a central quantum spin interacting, according to Heisenberg interaction, with a collection of mutually non-interacting quantum spin-half particles, leading to non-Markovian dynamics. In this case, we observed that with the increase of temperature, number of spins in the bath, and the system-bath interaction strength, there is increase in information loss at instantaneous time.In this scenario, we found that the uncertainty relation shows a nonmonotonic behavior with the increase of temperature for small values of interaction strength, provided the optimization is performed over pure qubits. Moreover, we found that although the uncertainty relation in this model goes close to the saturation value, it fails to saturate exactly. Interestinglyhowever, we found that in absence ofz-zsystem-bath interaction and in the limit of large bath size, high bath temperature, and strong system bath interaction, uncertainty relation between information loss and non-ergodicity, based on trace distance measure, is saturated, providing a non-ergodic situation that saturates the uncertainty. The uncertainty relations have been obtained by using the usual notion of the ergodicity where we require to have the unique fixed point, of the dynamics, to be thermal. We note that the entire analysis goes through for a more general definition, where a single fixed point is sufficient to imply ergodicity. apsrev4-1Therefore, the average loss of information can now be expressed asI̅_Δ^T = max_|ψ_1|,⟩|ψ_2|⟩(1-D^T(|ψ_1|⟨%s|⟩ψ_1| , |ψ_2|⟨%s|⟩ψ_2|) ), and the inequality (<ref>) takes the simplerform asI̅_Δ^T ≥max_|ψ_1|,⟩|ψ_2|⟩(1-(𝒩_ϵ(|ψ_1|⟨%s|⟩ψ_1|)+𝒩_ϵ(|ψ_2|⟨%s|⟩ψ_2|)).As a particular case, we consider the initial states to be the energy eigenstates{|0|,⟩|1|}⟩. Then the average state and also the thermal state are convex combination of the energy eigenstates. If now the thermal state lies also between the statesρ̅_1andρ̅_2, then the inequality reduces to an equality relation asI̅_Δ^T +(𝒩_ϵ(|1|⟨%s|⟩1|)+𝒩_ϵ(|0|⟨%s|⟩0|)=1. Again choosing the initial states to be the energy eigenstates, in Fig. <ref> we have plotted the instantaneous information loss (I_Δ^T(t)) and the trace distance between the instantaneous state and the thermal state (D^T(ρ(t),ρ_th)), for which also the inequality (<ref>) is valid. It shows that as the information loss of the system increases, the distance between the instantaneous state and the thermal state decreases and vice versa, validating the uncertainty relation between the mentioned quantities. In the same figure we have also plotted the average information loss (I̅_Δ^T) and the total non-ergodicity (𝒩_ϵ(ρ̅_̅1̅)+𝒩_ϵ(ρ̅_̅2̅)). For the specific parameter values chosen in Fig. (<ref>), we have checked that the equality in Eq. (<ref>) also holds in this case. So for this particular case, there exists an exact complementarity between the information loss and total non-ergodicity, which validates our findings in the previous section. ] | http://arxiv.org/abs/1707.08963v2 | {
"authors": [
"Natasha Awasthi",
"Samyadeb Bhattacharya",
"Aditi Sen De",
"Ujjwal Sen"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727163016",
"title": "Universal quantum uncertainty relations between non-ergodicity and loss of information"
} |
Maximum entropy based non-negative optoacoustictomographic image reconstruction Jaya Prakash^†, Subhamoy Mandal^†, Member, IEEE, Daniel Razansky,Member, IEEE,and Vasilis Ntziachristos, Senior Member, IEEE This article has been accepted for publication in IEEE Transactions on Biomedical Engineering. DOI: 10.1109/TBME.2019.2892842 ^†J.P. and S.M. contributed equally to this work. J.P. acknowledges support from the Alexander von Humboldt Postdoctoral Fellowship Program. S.M. acknowledges support from DAAD PhD Scholarship Award (A/11/75907) and IEEE Richard E. Merwin Scholarship. D.R. acknowledges funding support from the European Research Council (ERC-2015-CoG-682379), US National Institutes of Health (R21-EY026382-01), Human Frontier Science Program (RGY0070/2016) and Deutsche Forschungsgemeinschaft (RA1848/5-1). V.N. acknowledges funding support from European Research Council (694968, ERC-PREMSOT). J.P., S.M., D.R., and V.N. are with the Institute of Biological and Medical Imaging, Helmholtz Zentrum Munich, Ingolstaedter Landstr. 1, D-85764 Neuherberg, Germany, and also with the Chair for Biological Imaging, Technical University Munich, Ismaningerstr 22, D-81675, Munich, Germany. (e-mail: [email protected]). December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= Objective: Optoacoustic (photoacoustic) tomography is aimed at reconstructing maps of the initial pressure rise induced by the absorption of light pulses in tissue. In practice, due to inaccurate assumptions in the forward model, noise and other experimental factors, the images are often afflicted by artifacts, occasionally manifested as negative values. The aim of the work is to develop an inversion method which reduces the occurrence of negative values and improves the quantitative performance of optoacoustic imaging.Methods: We present a novel method for optoacoustic tomography based on an entropy maximization algorithm, which uses logarithmic regularization for attaining non-negative reconstructions. The reconstruction image quality is further improved using structural prior based fluence correction.Results: We report the performance achieved by the entropy maximization scheme on numerical simulation, experimental phantoms and in-vivo samples.Conclusion: The proposed algorithm demonstrates superior reconstruction performance by delivering non-negative pixel values with no visible distortion of anatomical structures. Significance: Our method can enable quantitative optoacoustic imaging, and has the potential to improve pre-clinical and translational imaging applications. Optical parameters, photoacoustic tomography, inverse problems, image reconstruction, regularization theory. § INTRODUCTION Optoacoustic (OA) imaging detects broadband ultrasound (pressure) waves generated within tissue in response to external illumination with light of transient energy, due to light absorption by tissue elements and thermo-elastic expansion. Using forward models that describe sound propagation in tissue, ultrasound measurements from multiple positions surrounding the object imaged are mathematically reconstructed to resolve the spatial distribution of the initial pressure rise. The reconstructed pressure rise is proportional to the product H=μ_a ϕ, whereby μ_a is the optical absorption coefficient and ϕ is the light fluence <cit.>. The value H has only positive values in biological tissues since both absorption and light fluence are positive. However, the appearance of negative values is common in OA images due to different factors, such as the use of inaccurate forward models, inversion schemes, numerical errors, limited view detection geometry, transducer impulse response, unknown or unpredictable experimental effects or noise in the imaging system. The presence of negative values in the reconstruction does not have physical relevance. Importantly, when spectral techniques are employed, such as Multispectral Optoacoustic Tomography (MSOT) <cit.>, the presence of negative values make spectral quantification problematic. It is therefore important to treat the appearance of negative values in the OA tomography problem. Model based reconstruction has been suggested as an alternative to back-projection algorithms to improve the accuracy of OA imaging, further incorporating transducer and laser characteristics into the inversion procedure <cit.>. In principle, accurate inversion can reduce the image artifacts, but errors persist due to different experimental challenges including limited-angle signal collection, limited bandwidth detection, noise and other uncertainties, leading to incomplete data problems and results in the presence of erroneous negative values <cit.>. Consequently, methods to directly treat the problem of negative values have been considered <cit.>. Ding et. al. <cit.> compared the utility of different minimization procedures using non-negative constraints, including steepest descent, conjugate gradient, and quasi-newton based inversion. Typical non-negative constraint schemes truncate the negative values within each step of the gradient iteration, forcing a result containing only positive or zero values. This practice however may bias the solution and generate inaccuracies in the reconstruction.An alternative approach to address the problem of negative values is to use image content for image correction. Image features such as the total energy (smoothness), contrast, total variation of an image can be generally employed as prior information to direct the inversion towards pre-determined outcomes, usually based on the assumptions about the nature of the image. For example, ℓ_2- or ℓ_1-norm minimization of the total variation of an image minimizes the edges of the reconstructed image. Using this notion, negative artifacts can then be eliminated by applying an explicit non-negativity constraint along with ℓ_2-norm minimization <cit.>. Another image metric that has been considered for eliminating negative values is the entropy of an image <cit.>. Entropy is the measure of randomness in an image. Randomness of the image implies that information from each subpixel is assumed to be independent of each other and can statistically take any value irrespective of its neighboring subpixel. This becomes very useful in limited data situations; wherein the principle of maximum entropy tries to eliminate all uncertainties within each subpixel (among the different possible solutions) by imposing independent statistical structure on each pixel. Maximization of entropy (i.e. maximizing the term -xlog(x); whereby x is the vectorized image) is equal to minimizing the term xlog(x) and is a method considered inPositron Emission Tomography (PET) and multi-modal imaging <cit.> or astronomical imaging <cit.>.In this work, we examine the use of entropy as a prior in OA image inversion, in the context of nonlinear conjugate gradient minimization <cit.>. We hypothesize that the use of an entropy-based prior, which implements an implicit non-negativity constraint, can improve the accuracy of OA inversions over externally imposed non-negativity constraints. To prove this hypothesis, we first theoretically compare a conventional ℓ_2-norm minimization problem using a smoothness constraint to an entropy maximization problem. We show that images reconstructed by entropy maximization cannot take negative values. The reconstructed OA images were further improved by correcting for the fluence, the fluence was estimated using finite volume method after segmenting the imaging domain (phantom or mouse). Thereafter, we compare the performance of inversion (after fluence correction) using entropy maximization and conventional inversion with externally applied non-negativity constraint using numerical simulation, experimental phantoms and small animal imaging. We discuss the performance differences observed and the advantages and limitations of using entropy maximization.§ MATERIALS AND METHODS§.§ TheoreticalbackgroundThe propagation of the acoustic pressure wave generated due to the short-pulsed light absorption is governed by the following inhomogenous wave equation <cit.>,∂^2 p(r,t)/∂ t^2 - c^2 ρ∇.(1/ρ∇ p(r,t))= Γ∂ H(r,t)/∂ t,where the instantaneous light power absorption density in W/m^3 is indicated by H and Γ represents the medium-dependent dimensionless Grueneisen parameter. In Eq. <ref>, the tissue density is represented by ρ while c indicates the speed of sound (SoS). For our experiments, a uniform SoS of 1520 m/sec was heuristically estimated using image autofocusing method <cit.>. The initial pressure rise at position r and time t is given as p(r,t). The solution for the wave equation can then be obtained using a Green's function by assuming H(r,t)=H_r(r) δ(t), which results in <cit.>,p(r,t) = Γ/4 π c∂/∂ t∫_R=ctH_r(r')/R dr',where R=ct represents the radius of the integration circle over a line element given as dr'. The above solution is subsequently discretized into the following matrix equation <cit.>,b=Ax,where b is the boundary pressure measurements, A is the interpolated model matrix and x is the unknown image to be reconstructed, representing the initial pressure rise distribution. The above formulation represents the forward model, i.e. given the initial pressure rise one can estimate the pressure at the boundary locations detected by the transducers. Thus, the acoustic inverse problem involves reconstructing the initial pressure rise given the boundary pressure data. In the ℓ_2-norm formulation, the inverse problem is solved by minimizing a function given as, Ω_ℓ_2 = arg min_x (||Ax-b||_2^2 + λ|| Lx||_2^2),where λ is the regularization parameter. The term || Ax-b||_2^2 is called the residual term. The term || Lx||_2^2 is a ℓ_2-norm of the second order total-variation of the image x and L indicates the Laplacian operator. The value of the regularization parameter affects the resolution characteristics of the reconstructed image; higher the value of regularization the smoother the reconstructed image.§.§ Entropy Maximization and Non-negative constraintAn alternative method to the minimization problem of Eq. <ref> (optoacoustic reconstruction), is maximization of the entropy of the image. To elaborate on this point, a statistical approach is considered, wherein we assume that the image to be reconstructed follows a Gaussian distribution with estimated mean and standard deviation values. The dimension of the image to be reconstructed is N × N, i.e. a vector of size NN (=N^2). Next, we assume that each pixel j in this image will be formed by a group of subpixels indicated by m_j (>=1) and M=∑_j=1^NN m_j. With these assumptions, let us consider the following experiment: wherein K particles are distributed over all subpixels and let K_i be the number of particles that fall in pixel i. Then the number of combinations to place K particles in NN pixels such that K_j particles are present in pixel j is given as,C(K̃) = K!/Π_j=1^NN K_j!,Further we have m_j^(K_j) ways to put K_j particles into m_j subpixels. Hence, the total number of combinations to create the particle distribution V(K̃) is given as,V(K̃) = C(K̃). Π_j=1^NN m_j^(K_j),The total number of particles in the distribution is given as M^K. Now making the assumption that each particle is equally likely i.e. uniform distribution. We get the probability of distribution of K̃ as,p(K̃) = V(K̃)/M^K,Now using Stirling approximation i.e. K! ≈ K^K e^(-K), we can write,log(p(K̃)) = -K ∑_j=1^NN z_j log(z_j/m̂_j),where z_j=K_j/K and m̂_̂ĵ = m_j/M. The average value inside a pixel x_j will now be proportional to z_j i.e. x_j=S.z_j and m̃_̃j̃ = S.m̂_̂ĵ such that,∑_j=1^NN x_j = ∑_j=1^NNm̂_̂ĵ = S,with x_j ≥ 0, m̂_̂ĵ>0. Now let the prior distribution of the image vector be considered as p_A (x), which is given as,log(p_A (x)) = -K/S∑_j=1^NN x_j log(x_j/m_j),which follows the relative entropy definition and is always non-negative (not defined for negative values). Our next assumption is that the error vector or the noise is normally distributed with zero mean and standard deviation σ given as,p(r_i) = c.e^-r_i^2/2σ^2,which can be rewritten as,p(y|x) = c.e^|| A x - b||_2^2/2σ^2,Rewriting the overall expression using Bayes rule we get,log(p(x|y)) = -K/S∑_j=1^NN x_j log(x_j/m_j) - 1/2σ^2||A x - b||_2^2,Neglecting the terms independent of x. We can pose this as an entropy maximization problem which is non-linear convex maximization problem, and this can be solved by minimizing the function,Ω_maxent = arg min_x (||Ax-b||_2^2 + λ∑_i=1^NN x_i log(x_i/m_i)),where -xlog(x/m) indicates the relative entropy function of image x, typically m is assumed to be an arbitrary constant <cit.>. In this work m is assumed to be 1. Detailed mathematical analysis on the use of Eq. <ref> for applying an implicit non-negativity constraint, stability, and convergence of entropy maximization is given in <cit.>. Herein we study how positive values are retained with entropy maximization scheme. In ℓ_2-norm minimization (Eq. <ref>), the gradient update equation at iteration i is given as,x_i = x_i-1 - ( A^T ( A x_i-1 - b)) - λ L^TL x_i-1,The above update equation is obtained by taking the derivative of the objective function in Eq. <ref>. Note that in the above equation all the quantities will always be in real space i.e. (A, x_i-1, x_i, b∈IR), and can take any values due to the absence of any natural non-negativity barrier. Therefore, the ℓ_2-norm based minimization can generate negative values (which can be in IR) during the image reconstruction procedure. In case of entropy maximization (Eq. <ref>), the gradient updated equation at iteration i is given as,x_i = x_i-1 - ( A^T ( A x_i-1 - b)) - λ (1 + log(x_i-1/m_j-1)),The derivation pertaining to applying implicit positivity constraint using entropy maximization is discussed in the Appendix-I.Choice of regularization plays a key role in reconstructed image quality by defining over-smoothed or under-smoothed approximations in case of ℓ_2-norm based reconstruction. In terms of distance measure, ℓ_2-norm constraint can be considered as Euclidean distance between the prior and the expected image, i.e. || Lx||_2 = < Lx, Lx> ⟺ < Lx, Lx_pr><cit.>, therefore higher regularization will weigh the ℓ_2-norm constraint more and thus resulting in a smoother solution. Similarly entropy maximization can be related to Kullback-Leiber distance, as cross entropy between prior and the expected image, i.e. ∑ x log(x) = ∑ x log(x/x_pr), therefore higher regularization will push the subpixels (i.e. x_pr) in pixel i of image vector x to uniform distribution <cit.>. Thus, low regularization in the entropy maximization scheme will result in minimizing the residual (i.e. noisy reconstruction), whereas choosing higher regularization will result in the initial pressure rise being close to a smooth distribution having intrinsically positive values. The operating range of the regularization parameter in the entropy maximization framework can be found using the L-curve type method, cross-validation based scheme <cit.>. §.§ Choice of regularization parameter - L-curve methodTypically, the regularization parameter (λ) is chosen automatically using the L-curve method <cit.>.The L-curve method is a popular method for automatically choosing the regularization parameter for a linear inverse problem and this scheme was earlier used in diffuse optical tomography and OA tomography. In the L-curve method, a graph is plotted between the residual (||Ax-b||_2^2) and the reconstruction (||x_λ||_2^2) as function of regularization parameter (λ). This essentially means that the reconstructed solution (x_λ) is a function of regularization (λ). In an ideal case this curve will be of L-shape. The corner point of this L-shape represents the least distance from the origin, indicating an ideal balance between residual and expected solution. For the case of entropy maximization the solution norm will be replaced by entropy term i.e. (Σx_λ log(x_λ)). In this work, we use L-curve type approach to automatically estimate the regularization parameter in both the L2-norm and entropy maximization schemes.§.§ ℓ_2-norm with smoothness and non-negativity constraintMinimizing the function in Eq. <ref> was performed using a conjugate gradient method (equivalent to iterative least squares QR (LSQR) method), which has a closed form solution as <cit.>,x ≈ x_ℓ_2-lsqr = V_k (B_k^T B_k + λS_k^T S_k)^-1β_0 B_k^T e_1,where B_k, S_k, V_k, β_0, and e_1 can be obtained in the Lanczos diagonalization procedure with [ A; λ L ] and [ b; 0 ]. Here k indicates the number of iterations during the joint bidiagonalization procedure. In the ℓ_2-norm formulation with non-negativity constraint, the following minimization is solved, Ω_ℓ_2-NN = arg min_x (||Ax-b||_2^2 + λ|| Lx||_2^2) s.t. x>0,The above minimization is solved using the LSQR solver and then the obtained solution containing negative values are thresholded to 0, as negative values do not have any physical relevance (as optical absorption coefficient in biological tissue is not negative). Eq. <ref> is used to obtain the solution and then the negative values in the solution are thresholded. The regularization parameter was chosen using L-curve method (explained in Sec. II-C)<cit.>. §.§ Implementation Steps for Entropy MaximizationEq. <ref> is minimized using a non-linear conjugate gradient type method and the step-length for the conjugate gradient method is computed using a line search <cit.>. Minimization of the objective function in Eq. <ref> with conjugate gradient requires computing the derivative and then move in independent perpendicular gradient direction. The derivative used in the conjugate gradient scheme for the objective function in Eq. <ref> is computed as,∇Ω_maxent = 2 A^T (A x -b) + λ (1 + log(x_i-1/m_i-1)),The minimization is presented in more details in the Algorithm-1 section. The regularization parameter was chosen using an L-curve method (as a tradeoff between negative of entropy and residual).§.§ Fluence CorrectionThe image reconstructed in Eq. <ref> (LSQR) and with Algorithm-1 (Entropy Maximization) represents the absorbed energy distribution H_r (r) in tissue, which depends on the fluence distribution and the optical absorption coefficientμ_a (r) i.e <cit.>,x = p_0 (r) = H_r (r) = μ_a (r) Φ (r),where p_0 (r) is the initial pressure rise distribution and Φ(r) indicates the local light fluence density in mJ/cm^2. To extract the absorption coefficient map, it is therefore critical to estimate the fluence in the medium imaged. Different schemes have been developed for estimating the fluence distribution and quantitatively recover optical absorption coefficient maps, including model-based inversion schemes integrated with fluence compensation <cit.>, wavelet frameworks <cit.>, finite-element implementation of the delta-Eddington approximation to the radiative transfer equation <cit.>, diffusion equation based regularized Newton method <cit.>, or approximations with base spectra <cit.>. Herein we assumed for demonstration purposes a light propagation model based on the diffusion equation, further assuming that scattering dominates over absorption <cit.>, which is a valid approximation formost biological tissues and NIR measurements, i.e.,-∇.[D(r).∇Φ(r)] + μ_a(r) Φ(r) = S_0 (r),where D(r)=1/(3(μ_a+μ_s^')) is the diffusion coefficient and μ_s^'(r) indicates the reduced scattering coefficient at position r. S_0 (r) indicates the light source at the boundary of the imaging domain. Eq. <ref> is used for fluence estimation, and the diffusion equation is solved using the finite volume method (FVM). Optical properties were based on the known phantom specifications or estimates of absorption and scattering coefficients of tissue from the literature <cit.>. Then, we obtained absorption coefficient maps by normalizing the images with the corresponding calculated fluence distribution <cit.>. Since OA measurements of phantoms were performed in a water bath, we also employed the Beer-Lambert Law (OD = -log(I/I_0) = -μ_a d) to model photon propagation in water. The relative distances in phantom and water were assigned after segmentation of the OA images. The entire workflow of segmentation and fluence correction is integrated with the proposed non-negative entropy maximization algorithm to render improved image quality.§.§ Imaging instrumentation and protocol(s):Experimental data was acquired using the multispectral optoacoustic tomography (MSOT) scanner <cit.> (MSOT256-TF, iThera Medical GmbH, Munich, Germany). The boundary pressure readouts (time-series) were collected at 2,030 discrete time points at 40 Mega samples per second using a 256-element cylindrically focused transducer, resulting in the number of measurements (M) being 2030x256=519,680. The utilized piezocomposite transducer had a central frequency of 5 MHz with a radius of curvature of about 40 mm and an angular coverage of 270^∘. Uniform illumination was achieved with a ring type of light delivery using laser fiber bundles. Numerical simulations were performed with the same configuration as MSOT256-TF system with a realistic breast phantom having spatially varying absorption coefficient (in cm^-1) as shown in Fig. 1(a). Next, we segmented the boundary of the breast region in Fig. 1(a) and estimated the fluence distribution (shown in Fig. 1(b)) by solving the hybrid model (Sec. II.F) with the absorption coefficient and reduced scattering coefficient set to 0.2 cm^-1 and 12 cm^-1 respectively. The initial pressure rise (in kPa) was then estimated by multiplying the fluence distribution (Fig. 1(b)) with the spatially varying optical absorption (Fig. 1(a)), the initial pressure rise distribution (after scaling with acoustic parameters) is shown in Fig. 1(c). Note that we assumed point detector and did not model transducer characteristics in the simulations. The numerical breast phantom was created by using contrast enhanced magnetic resonance imaging <cit.>. Eq. 3 was used to model the acoustic propagation (on a 512×512 grid) and the pressure signals were collected at specific detector locations. The model matrix in Eq. 3 was built using interpolated model matrix method as explained inRef. <cit.>. To avoid inverse crime, the simulated data was generated on an imaging grid of size 512x512, while the reconstruction was performed on a grid of size 256x256. The simulated data was added with additive white Gaussian noise, to result in a SNR of 32 dB in the simulated data.To verify the quantitative reconstruction capabilities of the proposed entropy maximization scheme, a star shaped (irregular) phantom was created. The phantom constituted of a tissue mimicking (7% by volume of Intralipid and pre-computed volume of diluted India ink added) agar core having the optical density of 0.25. Two tubular absorbers made up of India-ink with the absorption coefficient values of 2.5 OD (calibrations done with Ocean Optics USB 4000) were inserted in the phantom. The absorbers were placed at two different depths within the phantom (one at the center and the other at the edge of the imaging domain) to test the sensitivity of the proposed scheme in reconstructing the absorbers at different imaging distances from the sensing arrays. Under normal operating conditions, the fluence at the center of the imaging domain is significantly lower as compared to the boundary of the object imaged, owing to the optical attenuation of the incident irradiation. Hence, performing fluence correction becomes indispensable to assign appropriated intensity to the absorber at the center of the imaging domain. The proposed methods were further validated on in-vivo mouse abdomen and brain datasets drawn from a standardized in-vivo murine whole body imaging database (10 mice/30 anatomical datasets) previously developed in Ref. <cit.>. The selected images were obtained at a laser wavelength of 760 nm and 800 nm, and the water (coupling medium) temperature was maintained at 34^∘ C for all experiments. Non-negativity based entropy maximization scheme was further validated using spectral measurements. Spectral measurements were acquired from a tumor bearing nude BALB-C mice with the laser wavelengths running from 680 nm to 900 nm at steps of 20 nm. All animal experiments were conducted under supervision of trained technician in accordance with institutional guidelines, and with approval from the Government of Upper Bavaria.§.§ Figure of meritTo develop an objective approach to evaluate imaging performance of different reconstruction methods, we used line plots on the reconstructed image (from phantom and tissue measurements). We also performed quantification using sharpness metric, defined as,SM = ∑dI^2/dx^2 + dI^2/dy^2/n,The sharpness metric indicates the edges in the reconstructed image (I): the higher the value of SM, the sharper the reconstructed image. This figure of metric was used for evaluating the proposed method, as the non-negative constraint tend to introduce zeros in the reconstructed image. The number of non-negative values is also reported for comparing the different reconstruction methods.Note that the number of negative pixels were calculated from the phantom or mice region (excluding the water region).Further root mean square error (RMSE) and peak signal to noise ratio (PSNR) was used to evaluate the performance of different reconstruction methods with numerical simulation. RMSE is given as,RMSE = √(∑_o (x_o^recon - x_o^true)^2/NN),is computed for comparing the performance of different algorithm. Here x_o^true is the o^th pixel of ground truth and x_o^recon is the o^th pixel of reconstructed image. PSNR is defined as,PSNR = 20 × log(max(x^true)/RMSE), The calculated sharpness metrices (for phantom and in vivo small animal images), and the RMSE/ PSNR values (for simulations) are given in section <ref>. § RESULTSFig. 1(c) shows the initial pressure distribution with the realistic numerical breast phantom used to evaluate the performance of different reconstruction methods. The reconstructed initial pressure rise distribution using the ℓ_2-norm based reconstruction is shown in Fig. 1(d). The solution pertaining to ℓ_2-norm based reconstruction (along with non-negative constraint) is indicated in Fig. 1(e). The reconstructed optoacoustic image using the entropy maximization approach is represented in Fig. 1(f). The reconstructions containing negative values are indicated with a red colormap, hence the negative pixels in Fig 1(d) are shown in red color. From the numerical simulations, it is apparent that the ℓ_2-norm based reconstruction produces negative values by just adding noise to the data and incorporating fluence effects, however these negative values do not appear after thresholding and using entropy maximization scheme as indicated by red arrows in Figs 1(e) and 1(f). Furthermore ℓ_2-norm with thresholding results in a nosier reconstruction with limited structures compared to entropy maximization scheme as shown with red arrows in Figs 1(e) and 1(f). The PSNR values for ℓ_2-norm, ℓ_2-norm with thresholding and entropy maximization reconstruction are 29.9736 dB, 30.2616 dB and 30.3529 dB respectively. The RMSE values for ℓ_2-norm, ℓ_2-norm with thresholding and entropy maximization reconstructions are 0.0453, 0.0451, and 0.0450 respectively. The number of reconstructed negative pixels with ℓ_2-norm reconstruction with numerical breast phantom is 4370. Note that the simulation studies did not model many experimental parameters like impulse response of the transducer, physical dimension of the transducer, pitch of the detector, artifacts arising due to reflections, and these parameters are known to influence the OA measurements in experimental scenarios. Further, we proceeded to study the performance of the proposed entropy maximization scheme with phantom and in-vivo datasets. Fig. 2 shows reconstructions of the star phantom, which reveal the efficacy of the proposed method vis-a-vis traditional ℓ_2-norm based reconstruction in generating positive values for both the initial pressure rise and absorption coefficient distribution. The reconstructed initial pressure rise and absorption coefficient distribution using the ℓ_2-norm based reconstruction is shown in Figs 2(a) and 2(d) respectively. The reconstructed initial pressure rise and absorption coefficient distribution using the ℓ_2-norm based reconstruction (with non-negative constraint) is indicated in Figs 2(b) and 2(e) respectively. The reconstructed initial pressure rise and absorption coefficient distribution using the entropy maximization based approach is represented in Figs 2(c) and 2(f) respectively. The reconstructions containing negative values are indicated with a red colormap, hence the negative pixels in Figs 2(a) and 2(d) are shown in red color. The proposed entropy maximization method (Fig. 2(f)) can provide accurate image representation with the ability to reconstruct the absorber (having OD of 2.5) at the center and the edge of the imaging domain along with reconstructing a star shaped background (having OD of 0.25). The negative values obtained using LSQR inversion is shown as red color in Fig. 2(a) and Fig. 2(d). The non-negative based ℓ_2-norm reconstruction is able to generate reconstruction results with positive values, but is not able to correctly reconstruct the internal volume of the star (tissue mimicking agar with 0.25 OD) phantom which is accurately reconstructed using entropy maximization. Fig. 2(g) shows the photograph of the phantom used from front-view (FV) and top-view (TV). Fig. 2(h) indicates the line plot along the vertical red dashed line shown in Fig. 2(b). Fig. 2(i) indicates the line plot along the horizontal blue dashed line shown in Fig. 2(b). The sharpness metric and the number of non-negative values are shown in Table-<ref>. The quantitative metric indicate that the proposed method can provide accurate image representation. Fig. 2(f) and the line plots in Figs 2(h) and 2(i) demonstrate that the maximum entropy based scheme can deliver better contrast while maintaining the background intensity than the standard ℓ_2-norm based reconstructions.The fluence correction was performed by using segmented (boundary) priors obtained automatically using deformable active contour models <cit.>. The results were corroborated with additional phantom (Agar block with 5% intralipid) scans which included India ink insertions of 3 different ODs in tissue relevant concentrations - 0.15, 0.30 and 0.45 OD at 800nm measured using a spectrometer (VIS-NIR; Ocean Optics). The results demonstrate that the signal intensities change proportionately with the changing OD of the insertions, and the values are in agreement with other commonly used inversion algorithm (i.e Tikhonov). The reported signal intensities were obtained by taking the mean of the different ROI's indicated in Table-I of the supplementary. Additionally, the proposed reconstruction scheme recovered higher (absolute) signal intensities while reducing negative values in reconstructed image (see supplementary Table I).Empirically selecting the regularization biases the reconstruction results.Therefore an L-curve method was used to automatically choose the regularization parameter for Tikhonov method <cit.> and entropy maximization based scheme. Previous works have used L-curve approach for automatically choosingthe regularization parameter in entropy maximization framework for estimating distance distributions of magnetic spin-pairs <cit.>. Fig. 3 indicates the L-curve criterion used to choose the regularization parameter (details regarding L-curve approach is given in Sec. II-C) as applied to star phantom OA data presented in Fig. 2. Similar approach was used for automatically selecting the regularization parameter with numerical simulations and in-vivo data. Other methods like cross-validation can also be used for automatically choosing the regularization parameter in Tikhonov and entropy based framework <cit.>. Further, we studied the effect of regularization parameter choice on reconstruction image quality. Fig. S1 in supplementary shows maximum entropy reconstruction at different regularization parameter values. It can be seen that at high regularization values, the solution leads to uniform distribution, however maximum entropy scheme seems to have a large operating range from 1 to 10,000.The maximum entropy based scheme depends on the initial guess used in the non-linear conjugate gradient scheme. The maximum entropy constraint involves a non-linear logarithmic term, and the logarithm of a negative value is not defined, therefore having a large positive value at the initial guess will always generates positive reconstruction distributions and thus plays an important role in intrinsically obtaining non-negative reconstruction. The same is elaborated in the Appendix-I. The reconstruction results corresponding to a backprojection-type initial guess (A^T b containing negative values; A^T indicates transpose of system matrix) is indicated in Fig. 4(a), the image shows the real part of the solution. The reconstruction results corresponding to the initial guess (||b||_2/|| A||_1× ones(NN,1)) is indicated in Fig. 4(b). Fig. 4(a) clearly indicates that the negative values in the entropy maximization reconstructions arises because of initial guess used in the non-linear conjugate gradient scheme i.e. (||b||_2/|| A||_1× ones(NN,1)) gives non-negative results while A^T b results in negative values.Hence, in all the reconstructions the initial guess was chosen to be (||b||_2/|| A||_1× ones(NN,1)) and the regularization parameter was chosen using the L-curve method. Note that reconstructions in Fig. 4 involve performing additional fluence correction. The colormap in the case of mouse images are normalized to maximum and minimum values and the negative values are indicated in red color.Non-negative reconstruction generated with entropy maximization approach was further improved using fluence correction method. Fig. 5(a) shows the performance of segmentation approach in delineating the interface/boundary between the mice body (at the abdominal region) and water. The segmented boundary is used as a source term (after attenuation compensation using Beer-Lambert law in water) for modeling light propagation by solving the diffusion equation. Indeed, this boundary can be a good approximation for source term, as fiber bundle in the MSOT machine are arranged to provide uniform illumination on the sample. The fluence profile obtained after solving diffusion equation is shown in Fig. 5(b), the fluence was estimated with optical properties obtained from the literature <cit.>. Fig. 5(c) represents the initial pressure rise distribution reconstructed with entropy maximization approach. Fig. 5(d) shows the absorption coefficient distribution after normalizing the initial pressure distribution (Fig. 5(c)) with the estimated fluence profile (Fig. 5(b)). It can be clearly seen that signals from deeper regions on the mice gets highlighted more, similar approach was used for other regions of the mice.The reconstruction results (corresponding to absorption coefficient distribution) pertaining to the mouse head and mouse abdominal regions using the standard and proposed method are shown in Fig. 6. The reconstruction results corresponding to ℓ_2-norm based scheme (solved using LSQR method) for the mouse head and abdominal region is indicated in Figs 6(a) and 6(e) respectively, and the corresponding results for ℓ_2-norm based non-negative scheme (solved using LSQR method with thresholding) are given by Figs 6(b) and 6(f) respectively. The reconstruction results using the entropy maximization approach (Algorithm-1 with the integrated hybrid fluence correction) for the same anatomical regions is shown in Fig. 6(c) and Fig. 6(g) respectively. The experimental phantom and in-vivo reconstructions were performed on a 200x200 pixel imaging domain which corresponds to a physical field of view of 20mm x 20mm. The optical properties used for fluence estimation was assumed to be homogenous inside the tissue and taken from literature<cit.>. Figs 6(d) and 6(h) indicate the Fourier domain representation of the reconstructed images (i.e. Fig. 6(f) and 6(g)) using L2-norm with thresholding and entropy maximization schemes respectively. We could clearly see that entropy maximization scheme (Fig. 6(h)) has more low frequency content when compared to L2-norm with thersholding (Fig. 6(d)). Fig. 6(i) indicates the line plot along the red dashed line shown in Fig. 6(b) and Fig. 6(j) shows the line plot along the red dashed line indicated in Fig. 6(f). The sharpness metric and the number of non-negative values for these reconstructions are indicated in Table-<ref>. These metrics show that the proposed method can provide accurate image reconstruction with lesser negative values and increased sharpness. Negative values should not arise during standard OA data acquisition, hence the lesser the number of negative pixels more accurate is the reconstructions. However in some scenarios the presence of negative values might indicate accurate reconstruction like temperature dependent studies <cit.>. However, we are working with standard OA acquisition, and thus more positive values indicate accurate reconstruction. Again, the colormap is normalized to maximum and minimum values, while indicating the negative values in red color. Finally, we performed a study to check if entropy maximization scheme was able to accurately recover the spectral information. Fig. <ref>(a) shows the reconstruction results pertaining to a tumor bearing mice using L2-norm based scheme with thresholding at 680 nm wavelength. Fig. <ref>(b) shows the recovered mean spectral information using entropy maximization and L2-norm based reconstruction for the red square region shown inFig. <ref>(a).Fig. <ref>(b) indicates that at wavelengths below 700 nm, we have appearance of negative values using L2-norm based reconstruction. Moreover, in some parts of the image, like the one shown using orange arrow in Fig. <ref>(a), the entire recovered spectra turned out to be negative using L2-norm based reconstruction (however maximum entropy scheme was able to recover positive spectral profile). Fig. <ref>(c) shows reconstructed mean spectra information using entropy maximization and L2-norm based reconstruction from the green square region indicated in Fig. <ref>(a). As can be seen from Figs <ref>(b) and <ref>(c), the spectral recovery of maximum entropy scheme is similar to that of L2-norm based reconstruction, however the appearance of negative values in L2-norm based reconstruction will hinder unmixing results in terms of absolute quantification. § DISCUSSION AND CONCLUSION The reconstruction results for the numerical simulations, phantom and in-vivo mouse scans indicate that the proposed entropy maximization scheme renders strictly positive image values that are also close to the a-priori known absorption values in the phantom. Employing a segmented image prior can effectively reduce the aberrations in image contrast by suitably mapping the light propagation pathway in two optically diverse domains (background and tissue), and enhance the performance of (optical) fluence correction methods<cit.>, as demonstrated in Figs 2(f) and 6(g).Moreover, when a global SoS is attribute to the entire imaging domain, small SoS variation causes aberration at the edge of the surfaces of the imaged object <cit.>, the same two compartment model can be used to remove SoS mismatch. The figure of merits (Table-<ref>), magnitude of Fourier spectrum from the reconstructed images, and the line plots indicate entropy maximization approach provides superior results in comparison with non-negativity constrained reconstructions. Importantly the proposed approach offers an opportunity for exploring a family of differential type non-negative regularization methods (like entropy scheme). The entropy maximization scheme performed better with experimental data (Figs 2 and 6) compared to numerical simulation (Fig. 1). This is because experimental OA measurements are heavily influenced by experimental factors like laser pulse width, transducer impulse response, pitch and size of the transducer, making the reconstruction problem with experimental OA measurements more challenging. From Figs 2 and 6, it can be observed that the presence of negative pixels is higher in water region and in the center of imaging domain, where the absorption/the fluence is low resulting in lower SNR in time-series OA measurements. Similarly, introduction of noise and fluence effects in simulation studies (Fig. 1) results in large number of negative values in regions where the initial pressure rise is close to 0 and also generating spurious negative values inside the numerical breast phantom.In recent studies, lot of emphasis has been placed on using ℓ_1-norm based minimizations for performing OA tomographic image reconstruction in different frameworks <cit.>. We have performed ℓ_1-norm based reconstruction as explained in <cit.> and the results pertaining to non-negativity constraint in the ℓ_1-norm minimization is shown in Fig. S2. Fig. S2 also shows the performance comparison of ℓ_1-norm minimization with entropy maximization and Tikhonov reconstruction with printed phantom data. We observe that applying a ℓ_1-norm constraint does not afflict the appearance of negative values and the reconstruction performance is similar to ℓ_2-norm based scheme in terms of reducing negative values. This also demonstrates the superiority of using entropy maximization to generate physically relevant OA reconstructions devoid of negative values. We have not taken up further comparisons with ℓ_1-norm based approach, as our goal was to demonstrate the utility of entropy maximization approach to overcome appearance of pixels with negative values.Entropy maximization scheme was evaluated with biological datasets acquired from 270^∘ detection angle wherein the acquired dataset consists of highly independent (incoherent) data. While recent developments involve building systems with handheld probes (90^∘ three-dimensional acquisition, or 145^∘ two-dimensional acquisition) with different data-collection geometry. Performing accurate reconstructions with these clinical handheld systems tend to be difficult due to acquisition of limited independent data. Evaluating the performance of the entropy scheme with the limited independent data scenarios can enable utility of OA imaging in different clinical scenarios<cit.>. The proposed method preserves the structural integrity (numerical breast phantom and star phantom) and the anatomical structures (mouse data), and was successful in correcting the effects of variations in optical fluence. As part of future work, we aim to integrate the entropy maximization with more accurate light propagation modeling (such as Monte Carlo based schemes) to obtain better representation of the absorption coefficient with the reconstruction process accelerated by means of graphics processing units <cit.>. In this work, we demonstrated a non-negative image reconstruction method with improved image quality using fluence correction step at single acquisition wavelength. Translating the same to multi-wavelength scenario for estimation of quantitative tissue parameters is a fairly complex problem, since the optical properties used for fluence estimation varies nonlinearly with wavelength and is not known beforehand. Combining these problems will lead to generation of infinite possible ways to obtain accurate spatio-spectral representation, and such spectral analysis methods are beyond the scope of the current study. In this work, we have shown that entropy maximization is able to accurately recover the spectral information compared to L2-norm based reconstruction (see Fig. 7). However, the ability to resolve intrinsic chromophores like oxyhemoglobin, deoxyhemoglobin, fat, and water by acquiring data at multiple wavelengths is a key benefit of multispectral OA imaging. The unmixing of chromophores is achieved by a solving system of linear equations (direct or non-negatively constrained), or by non-linear unmixing using an integrated fluence correction. All of these approaches use thresholding of negative values, making them suboptimal and error prone. On the other hand, entropy maximization canpurge out the inaccuracies occurring from truncated pixel information, potentially improving the performance of unmixing and image analysis algorithms. Therefore, the future work will involve comparing the different combination of reconstruction (acoustic inverse problem) and unmixing with different solvers like LSQR, non-negative LSQR and entropy maximization to bring out value among these schemes.§ CONCLUSION The proposed maximum entropy based OA image reconstruction scheme demonstrates superior reconstruction performance with no visible distortion of anatomical structures associated with delivering of non-negative pixel values. Entropy maximization reconstruction thus tends to be physically relevant and more accurate in resolving the structures (as demonstrated with numerical simulation, experimental phantoms and in-vivo case) in an imaged sample. The developed methodology has the potential to emerge as a suitable data processing tool for OA imaging, and specifically benefiting pre-clinical biomedical <cit.> and translational imaging <cit.>. § IMPLICIT NON-NEGATIVITY USING ENTROPY MAXIMIZATION The objective function in the entropy maximization scheme is given as,Ω = | |Ax - b||_2^2 + λx^T log( x/m)The gradient of the above equation can be written as,∂Ω/∂ x =A^T(Ax - b) + λ( 1 + log( x/m)) = 0Now, we can consider the above minimization problem as minimizing two models in the subspace, one is based on residual i.e. Res = | | Ax - b||_2^2 and the other being relative entropy i.e. Ent = ∑ xlog( x/m). Here the regularization parameter defines the proportion of residual and entropy term in this minimization problem. As in any optimization, the solution is always found using the search directions (these search directions are defined by the gradients). The update equation at i^th gradient iteration will turn out to be,x_i = x_i - 1 - α( ∂Ω/∂ x)_x_i - 1where α is the step length estimated using line search method and is always non-negative. As x_i - 1→ 0, ∇ Ent →- ∞, the gradient update will be pushed to a very low value using entropy constraint. Also note that as, x_i - 1→ 0, ∇ Ent will reach - ∞ faster, and the ∇ Res →- A^Tb; importantly ∇ Res cannot reach ∞ as fast as ∇ Ent → - ∞ to nullify the effect of entropy term, therefore the overall gradient will be negative i.e. ( ∂Ω/∂ x)_x_i - 1→- ve. In any gradient descent method, we traverse in the direction perpendicular to the gradient, therefore the solution will be pushed away from zero to have high positive value, i.e. as x_i - 1→ 0, x_i→+ ve.Hence, using the entropy constraint will enable the solution to move away from zero and leading to positive real numbers. Since, a natural barrier is created by including the entropy constraint into the optimization framework, this barrier will not allow the solution to take negative values and consequently positive OA reconstructions are generated. In order to converge to positive OA reconstructions, we need to start with a large positive initial guess i.e. when x_0→ IR^ + then ∇ Res →A^TAx_0 and ∇ Ent → IR^ +. 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"authors": [
"Jaya Prakash",
"Subhamoy Mandal",
"Daniel Razansky",
"Vasilis Ntziachristos"
],
"categories": [
"physics.med-ph",
"cs.CV",
"eess.IV",
"physics.optics"
],
"primary_category": "physics.med-ph",
"published": "20170726115006",
"title": "Maximum entropy based non-negative optoacoustic tomographic image reconstruction"
} |
]Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint Bangti Jin]Bangti Jin Department of Computer Science, University College London, Gower Street, London, WC1E 2BT, UK. [email protected] Buyang Li]Buyang Li Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong buyang.li@ polyu.edu.hkZhi Zhou]Zhi Zhou Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong zhizhou@ polyu.edu.hk [ [ December 30, 2023 =====================In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order α∈(0,1) in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size h and time stepsize τ, we establish the following order of convergence for the numerical solutions of the optimal control problem: O(τ^min(1/2+α-ϵ,1)+h^2) in the discrete L^2(0,T;L^2(Ω)) norm and O(τ^α-ϵ+ℓ_h^2h^2) in the discrete L^∞(0,T;L^2(Ω)) norm, with any small ϵ>0 and ℓ_h=ln(2+1/h). The analysis relies essentially on the maximal L^p-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results. optimal control, time-fractional diffusion, L1 scheme, convolution quadrature, pointwise-in-time error estimate, maximal regularity.§ INTRODUCTIONLet Ω⊂ℝ^d (d=1,2,3) be a convex polyhedral domain with a boundary ∂Ω. Consider the distributed optimal control problemmin_q ∈ J(u,q)=12u - u_d_L^2(0,T;L^2)^2 + γ2 q _L^2(0,T;L^2)^2,subject to the following fractional-order partial differential equationu-Δ u= f+q, 0<t≤ T, u(0)=0,where T>0 is a fixed final time, γ>0 a fixed penalty parameter, Δ:H^1_0(Ω)∩ H^2(Ω)→ L^2(Ω) the Dirichlet Laplacian, f:(0,T)→ L^2(Ω) a given source term, and u_d:(0,T)→ L^2(Ω) the target function. The admissible setfor the control q is defined by={ q∈ L^2(0,T;L^2): a≤ q ≤ b a.e. in Ω×(0,T)},with a,b∈ℝ and a< b. The notation u in (<ref>) denotes the left-sided Riemann-Liouville fractional derivative in time t of order α∈(0,1), defined by <cit.>u(t)= 1/Γ(1-α)/ṭ∫_0^t(t-s)^-α u(s)ṣ.Since u(0)=0, the Riemann-Liouville derivative u(t) coincideswith the usual Caputo derivative <cit.>.Further, when α=1, _0∂_t^α u(t) coincides with the first-order derivative u'(t), and thus the model (<ref>) recovers the standard parabolic problem.The fractional derivative u in the model (<ref>) is motivated by an ever-growing list of practical applications related to subdiffusion processes, in which the mean square displacement grows sublinearly with time t, as opposed to linear growth for normal diffusion. The list includes thermal diffusion in fractal media, protein transport in plasma membrane and column experiments etc <cit.>. The numerical analysis of the model (<ref>) has received much attention. However, the design and analysis of numerical methods for related optimal control problems only started to attract attention <cit.>. The controllability of (<ref>) was discussed in <cit.> and <cit.>. Ye and Xu <cit.> proposed space-time spectral type methods for optimal control problems under a subdiffusion constraint, and derived error estimates by assuming sufficiently smooth state and control variables. Antil et al <cit.> studied an optimal control problem with space- and time-fractional models, and showed the convergence of the discrete approximations via a compactness argument. However, no error estimate for the optimal control was given for the time-fractional case. Zhou and Gong <cit.> proved the well-posedness of problem (<ref>)–(<ref>) and derived L^2(0,T;L^2(Ω)) error estimates for the spatially semidiscrete finite element method, and described a time discretization method without error estimate. To the best of our knowledge, there is no error estimate for time discretizations of (<ref>)–(<ref>). It is the main goal of this work to fill this gap.This work is devoted to the error analysis of both time and space discretizations of (<ref>)-(<ref>). The model (<ref>) is discretized by the continuous piecewise linear Galerkin FEM in space and the L1 approximation <cit.> or backward Euler convolution quadrature <cit.> in time, and the control q by a variational type discretization in <cit.>. The analysis relies crucially on ℓ^p(L^2(Ω)) error estimates for the fully discrete finite element solutions of the direct problem with nonsmooth source term. Such results are still unavailable in the existing literature. We shall derive such estimates in Theorems <ref> and <ref>, and use them to derive an O(τ^min(1/2+α-ϵ,1)+h^2) error estimate in the discrete L^2(0,T;L^2(Ω)) norm for the numerical solutions of problem (<ref>)–(<ref>), where h and τ denote the mesh size and time stepsize, respectively, and ϵ>0 is small, cf. Theorems <ref> and <ref>. The O(τ^min(1/2+α-ϵ,1)) rate contrasts with the O(τ) rate for the parabolic counterpart (see, e.g., <cit.>). The lower rate for α≤1/2 is due to the limited smoothing property of problem (<ref>), cf. Theorem <ref>. This also constitutes the main technical challenge in the analysis. Based on the error estimate in the discrete L^2(0,T;L^2 (Ω)) norm, we further derive a pointwise-in-time error estimate O(τ^α-ϵ+ℓ_h^2h^2) (with ℓ_h=log(2+1/h), cf. Theorems <ref> and <ref>). Our analysis relies essentially on the maximal L^p-regularity of fractional evolution equations and its discrete analogue <cit.>. Numerical experiments are provided to support the theoretical analysis.The rest of the paper is organized as follows. In Section <ref>, we discuss the solution regularity and numerical approximation for problem (<ref>). In Section <ref>, we prove error bounds on fully discrete approximations to problem (<ref>)–(<ref>). Finally in Section <ref>, we provide numerical experiments to support the theoretical results. Throughout, the notation c denotes a generic constant which may differ at each occurrence, but it is always independent of the mesh size h and time stepsize τ.§ REGULARITY THEORY AND NUMERICAL APPROXIMATION OF THE DIRECT PROBLEMIn this section, we recall preliminaries and present analysis for the direct problemu-Δ u= g,0<t≤ T,u(0)=0 ,and its adjoint problemz-Δ z= η,0≤ t<T,z(T)=0,where the fractional derivative z is defined in (<ref>) below. In the case α∈(0,1/2], the initial condition should be understood properly: for a rough sourceterm g, the temporal trace may not exist and the initial condition should be interpreted in a weak sense <cit.>. Thus we refrain from the case of nonzero initial condition, and leave it to a future work. §.§ Sobolev spaces of functions vanishing at t=0We shall use extensively Bochner-Sobolev spaces W^s,p(0,T;L^2(Ω)). For any s≥ 0 and 1≤ p< ∞, we denote by W^s,p(0,T;L^2(Ω)) the space of functions v:(0,T)→ L^2(Ω), with the norm defined by interpolation. Equivalently, the space is equipped with the quotient normv_W^s,p(0,T;L^2(Ω)):= inf_vv_W^s,p(ℝ;L^2(Ω)) ,where the infimum is taken over all possible extensions v that extend v from (0,T) to ℝ. For any 0<s< 1, one can define Sobolev–Slobodeckiǐ seminorm |·|_W^s,p(0,T;L^2(Ω)) by| v|_W^s,p(0,T;L^2(Ω))^p := ∫_0^T∫_0^T v(t)-v(ξ)_L^2(Ω)^p/|t-ξ|^1+ps ṭξ̣,and the full norm ·_W^s,p(0,T;L^2(Ω)) byv_W^s,p(0,T;L^2(Ω))^p = v_L^p(0,T;L^2(Ω))^p+|v|_W^s,p (0,T;L^2(Ω))^p .For s>1, one can define similar seminorms and norms. LetC^∞_L(0,T;L^2(Ω)):={v=w|_(0,T): w∈ C^∞(ℝ;L^2(Ω)): (w)⊂ [0,∞)},and denote by W_L^s,p(0,T;L^2(Ω)) the closure of C^∞_L(0,T;L^2(Ω)) in W^s,p(0,T;L^2(Ω)), and by W^s,p_R(0,T;L^2(Ω)) the closure of C^∞_R(0,T;L^2(Ω)) in W^s,p(0,T;L^2(Ω)), withC^∞_R(0,T;L^2(Ω)):={v=w|_(0,T): w∈ C^∞(ℝ;L^2(Ω)): (w)⊂ (-∞,T]}.By Sobolev embedding, for v∈ W_L^s,p(0,T;L^2(Ω)), there holds v^(j)(0)=0 for j=0,…,[s]-1 (with [s] being the integral part of s>0), and also v^(j)=0 if (s-[s])p>1. For v∈ W_L^s,p(0,T;L^2(Ω)), the zero extension of v to the left belongs to W^s,p(-∞,T;L^2(Ω)), and W_L^s,p(0,T;L^2(Ω))=W^s,p(0,T;L^2(Ω)), if s<1/p. We abbreviate W^s,2_L(0,T;L^2(Ω)) as H^s_L(0,T;L^2(Ω)), and likewise H^s_R(0,T;L^2(Ω)) for W^s,2_R(0,T;L^2(Ω)).Similar to the left-sided fractional derivative u in (<ref>), the right-sided Riemann-Liouville fractional derivative v(t) in (<ref>) is defined byv(t):= - 1/Γ(1-)/ṭ∫_t^T(s-t)^-v(s) ṣ .Let any p∈(1,∞) and p'∈(1,∞) be conjugate to each other, i.e., 1/p+1/p'=1. Since for u∈ W^α,p_L(0,T;L^2(Ω)),v∈ W^α,p'_R(0,T;L^2(Ω)), we have _0∂_t^α u∈ L^p(0,T;L^2(Ω)), _t∂_T^α v∈ L^p'(0,T;L^2(Ω)). Thus, there holds <cit.>:∫_0^T (_0∂_t^α u(t)) v(t) ṭ = ∫_0^T u(t) (_t∂_T^α v(t)) ṭ , ∀ u ∈ W^α,p_L(0,T;L^2(Ω)),v ∈W^α,p'_R(0,T;L^2(Ω)).§.§ Regularity of the direct problemThe next maximal L^p-regularity holds <cit.>, and an analogous result holds for(<ref>). If u_0=0 and f∈ L^p(0,T;L^2(Ω)) with 1<p<∞, then (<ref>) has a unique solution u∈ L^p(0,T; H_0^1(Ω)∩ H^2(Ω)) such that u∈ L^p(0,T;L^2(Ω)) such thatu_L^p(0,T; H^2(Ω)) + u_L^p(0,T;L^2(Ω))≤ cf_L^p(0,T;L^2(Ω)),where the constant c is independent of f and T. Now we give a regularity result. For g∈W^s,p( 0,T;L^2(Ω )),s∈ [0,1/p) and p∈(1,∞), problem (<ref>) has a unique solution u∈ W^α+s,p( 0,T;L^2(Ω )) ∩ W^s,p(0,T;H_0^1(Ω) ∩ H^2(Ω)), which satisfiesu_W^α+s,p (0,T;L^2(Ω)) + u_W^s,p(0,T;H^2(Ω))≤ cg_W^s,p (0,T;L^2(Ω)).Similarly, for η∈ W^s,p(0,T;L^2(Ω )),s∈ [0,1/p) and p∈(1,∞), problem (<ref>) has a unique solution z∈ W^α+s,p(0,T;L^2(Ω )) ∩ W^s,p(0,T;H_0^1(Ω) ∩ H^2(Ω)), which satisfiesz_W^α+s,p (0,T;L^2(Ω)) + z_W^s,p(0,T;H^2(Ω))≤ cη_W^s,p (0,T;L^2(Ω)).For g∈ W^s,p(0,T;L^2(Ω)),s∈ [0,1/p) and p∈(1,∞), extending g to be zero on Ω×[(-∞,0)∪(T,∞)] yields g∈W^s,p(ℝ;L^2(Ω)) andg_W^s,p (ℝ;L^2(Ω))≤ cg_W^s,p (0,T;L^2(Ω)) .For the zero extension to the left, we have the identity _0∂_t^α g(t) = _-∞∂_t^α g(t) for t∈[0,T], and there holds the relation _-∞∂_t^α g = (iξ)^αg(ξ) <cit.>, where denotes taking Fourier transform int, and g the Fourier transform of g. Then, with ^∨ being the inverse Fourier transform in ξ, u= [((iξ)^α-Δ)^-1g(ξ)]^∨ is a solution of (<ref>) and(1+|ξ|^2)^α+s/2u(ξ)= (1+|ξ|^2)^α/2((iξ)^α-Δ)^-1(1+|ξ|^2)^s/2g(ξ) . Let Δ be the Dirichlet Laplacian, with domain D(Δ) = H_0^1(Ω)∩ H^2(Ω). Then the self-adjoint operator is invertible from L^2(Ω) to D(Δ), and generates a bounded analytic semigroup <cit.>. Thus the operator(1+|ξ|^2)^α/2((iξ)^α-Δ)^-1is bounded from L^2(Ω) to D(Δ) in a small neighborhood 𝒩 of ξ=0. Further, in 𝒩, the operatorξ/ξ̣(1+|ξ|^2)^α/2((iξ)^α-Δ)^-1 = α |ξ|^2/1+|ξ|^2 (1+|ξ|^2)^α/2((iξ)^α-Δ)^-1+α (1+|ξ|^2)^α/2((iξ)^α-Δ)^-1(iξ)^α((iξ)^α-Δ)^-1is also bounded. If ξ is away from 0, then(1+|ξ|^2)^α/2((iξ)^α-Δ)^-1= (iξ)^-α(1+|ξ|^2)^α/2(iξ)^α((iξ)^α-Δ)^-1 ,and in the latter case the inequalities|(iξ)^-α(1+|ξ|^2)^α/2|≤ c (iξ)^α((iξ)^α-Δ)^-1≤ cimply the boundedness of (<ref>) and (<ref>).Since boundedness of operators is equivalent to R-boundedness of operators in L^2(Ω) (see <cit.> for the concept of R-boundedness), the boundedness of (<ref>) and (<ref>) implies that (<ref>) is an operator-valued Fourier multiplier <cit.>, and thusu_W^α+s,p(ℝ;L^2(Ω)) ≤[(1+|ξ|^2)^α+s/2u(ξ)]^∨_L^p(ℝ;L^2(Ω))=[(1+|ξ|^2)^α/2((iξ)^α-Δ)^-1(1+|ξ|^2)^s/2g(ξ)]^∨_L^p(ℝ;L^2(Ω))≤ c[(1+|ξ|^2)^s/2g(ξ)]^∨_L^p(ℝ;L^2(Ω))≤ cg_W^s,p(ℝ;L^2(Ω)) .This and(<ref>) implies the desired bound on u_W^α+s,p(0,T;L^2(Ω)). The estimateu_W^s,p(0,T;H^2(Ω))≤ cg_W^s,p(0,T;L^2(Ω))follows similarly by replacing (1+|ξ|^2)^α/2((iξ)^α-Δ)^-1 with Δ((iξ)^α-Δ)^-1 in the proof.Below we only use the cases “p=2, s=min(1/2-ϵ,α-ϵ)” and “p>max(1/α,1/(1-α)), s=1/p-ϵ” of Theorem <ref>. Both cases satisfy the conditions of Theorem <ref>. A similar assertion holds for the more general case g∈ W_L^s,p(0,T;L^2(Ω)), s>0 and 1<p<∞:u_W^α+s,p (0,T;L^2(Ω)) + u_W^s,p(0,T;H^2(Ω))≤ cg_W_L^s,p (0,T;L^2(Ω)).In fact, for g∈ W_L^s,p(0,T;L^2(Ω)), the zero extension of g to t≤ 0 belongs to W^s,p(-∞,T;L^2(Ω)), which can further be boundedly extended to a function in W^s,p(ℝ;L^2(Ω)). Then the argument in Theorem <ref> gives the desired assertion. This also indicates a certain compatibility condition for regularity pickup.§.§ Numerical schemeNow we describe numerical treatment of the forward problem (<ref>), which forms the basis for the fully discrete scheme of problem (<ref>)–(<ref>) in Section <ref>.We denote by 𝒯_h a shape-regular and quasi-uniform triangulation of the domain Ω into d-dimensional simplexes, and letX_h= {v_h∈ H_0^1(Ω): v_h|_K , ∀K ∈𝒯_h}be the finite element space consisting of continuous piecewise linear functions. The L^2(Ω)-orthogonal projection P_h:L^2(Ω)→ X_h is defined by (P_h ,χ_h)=(,χ_h), for all ∈ L^2(Ω), χ_h∈ X_h, where (·,·) denotes the L^2 inner product. Then the spatially semidiscrete Galerkin FEM forproblem (<ref>) is to find u_h(t)∈ X_h such that u_h(0)=0 and( u_h(t),χ_h) + (∇ u_h(t),∇χ_h) = (g(t),χ_h),∀χ_h∈ X_h, ∀t∈(0,T],By introducing the discrete Laplacian Δ_h: X_h→ X_h, defined by -(Δ_h_h,χ_h)=(∇_h,∇χ_h), for all _h, χ_h∈ X_h, problem (<ref>) can be written asu_h(t) -Δ_h u_h(t) =P_hg(t), ∀t∈(0,T] ,u_h(0)=0.Similar to Theorem <ref>,for s<1/p there holdsu_h_W^α+s,p(0,T;L^2(Ω))+ Δ_h u_h_W^s,p(0,T;L^2(Ω))≤ c g_W^s,p(0,T;L^2(Ω)) ,where the constant c is independent of h (following the proof of Theorem <ref>). Lemma <ref> and Remark <ref> remain valid for the semidiscrete solution u_h, e.g.,Δ_hu_h_L^p(0,T;L^2(Ω)) + u_h_L^p(0,T;L^2(Ω))≤ cf_L^p(0,T;L^2(Ω)).These assertions will be used extensively below without explicitly referencing.To discretize (<ref>) in time, we uniformly partition [0,T] with grid points t_n=nτ, n,=0,1,2,…,N and a time stepsize τ=T/N≤1, and approximate φ(t_n) by (with φ^j=φ(t_j)):φ^n= τ^-α∑_j=0^nβ_n-jφ^j,where β_j are suitable weights. We consider two methods: L1 scheme <cit.> and backward Euler convolution quadrature (BE-CQ) <cit.>, for which β_j are respectively given byβ_0=1, β_j=(j+1)^1-α-2j^1-α+(j-1)^1-α,j=1,2,…,N,β_0 = 1, β_j = -β_j-1(α-j+1)/j,j = 1,2,…,N.Both schemes extend the classical backward Euler scheme to the fractional case. Then we discretize problem (<ref>) by: with g_h^n=P_hg(t_n), find U_h^n∈ X_h such thatU_h^n -Δ_hU_h^n= g_h^n, n=1,2,…,N,U_h^0 = 0.By <cit.> and <cit.>, we have the following error bound. For g∈ W^1,p(0,T;L^2(Ω)), 1≤ p≤∞, let u_h and U_h^n be the semidiscrete solution andfully discrete solution, respectively, in (<ref>) and (<ref>). Then there holdsU_h^n - u_h(t_n) _L^2≤ cτ t_n^α-1g(0)_L^2(Ω)+cτ∫_0^t_n (t_n+1-s)^-1g'(s) _L^2ṣ. Lemma <ref> slightly refines the estimates in <cit.>, but can be proved in the same way using the following estimates in the proof of <cit.>:t_n^α-1≤ c (t+τ)^α-1 and∫_t^t_n s^α-1ṣ≤ c (t+τ)^α-1 ,For any Banach space X, we define(U_h^n)_n=0^N_ℓ^p(X):= { (∑_n=0^NτU_h^n_X^p)^1/p1≤ p<∞,max_0≤ n≤ NU_h^n_Xp=∞ . .Then the maximal ℓ^p-regularity estimate holds for (<ref>) <cit.>.The solutions (U_h^n)_n=1^N of (<ref>) satisfy the following estimate:( U_h^n)_n=1^N_ℓ^p(L^2(Ω)) + (Δ_h U_h^n)_n=1^N_ℓ^p(L^2(Ω))≤ c_p(g_h^n)_n=1^N_ℓ^p(L^2(Ω)), ∀1<p<∞. §.§ Error estimatesNow we present ℓ^p(L^2) error estimates for g∈ W^s,p(0,T;L^2(Ω)), 0≤ s≤ 1, 1≤ p≤∞. Error analysis for such g is unavailable in the literature. First, we give an interpolation error estimate. This result seems standard, but we are unable to find a proof, and thus include a proof in Appendix <ref>. For v∈ W^s,p(0,T;L^2(Ω)), 1<p<∞ and s∈(1/p,1], let v̅^n = τ^-1∫_t_n-1^t_n v(t) ṭ, there holds(v(t_n) - v̅^n)_n=1^N_ℓ^p(L^2)≤ c τ^s v _W^s,p(0,T;L^2(Ω)) . Our first result is an error estimate for g∈ W_L^s,p (0,T;L^2) (i.e., compatible source). Since g may not be smooth enough in time for pointwise evaluation, we define the averages g̅_h^n=τ^-1∫_t_n-1^t_nP_h g(s)ṣ, and consider a variant of the scheme (<ref>) for problem (<ref>): find U_h^n ∈ X_h such thatU_h^n -Δ_h U_h^n = g̅_h^n , n = 1,…,N, U_h^0=0.For g∈ W^s,p_L(0,T; L^2), 1< p<∞ and s∈[0,1], let u_h and U_h^n be the solutions of problems (<ref>) and (<ref>), respectively, and u̅_h^n:=τ^-1∫_t_n-1^t_nu_h(s)ṣ. Then there holds(U_h^n - u̅_h^n)_n=1^N_ℓ^p(L^2)≤ cτ^ s g _W^s,p_L(0,T;L^2).By Hölder's inequality and (<ref>) (with s=0), we haveτ∑_n=1^Nu̅_h^n_L^2^p = τ∑_n=1^N τ^-1∫_t_n-1^t_nu_h(s)ṣ_L^2^p ≤τ^1-p∑_n=1^N(∫_t_n-1^t_nu_h(s)_L^2(Ω)ṣ)^p≤∫_0^Tu_h(s)_L^2^pṣ≤ cg_L^p(0,T;L^2)^p .Similarly, by applying Lemma <ref> to (<ref>) and the L^2(Ω) stability of P_h, we have(U_h^n)_n=1^N_ℓ^p(L^2(Ω))≤ c (g̅_h^n)_n=1^N_ℓ^p(L^2(Ω))≤ cg _L^p(0,T;L^2) .This and the triangle inequality show the assertion for s=0.Next we consider g∈ W_L^1,p(0,T;L^2), and resort to (<ref>). Since g(0)=0, by Lemma <ref>,U_h^n - u_h(t_n) _L^2≤ cτ∫_0^t_n (t_n+τ-s)^-1 g'(s) _L^2ṣ.This directly implies( U_h^n -u_h(t_n))_n=1^N_ℓ^∞(L^2) ≤c τ g _W_L^1,∞(0,T;L^2).Further, let ψ(s)=τ∑_n=1^N(t_n+τ-s)^α-1χ_[0,t_n], where χ_S denotes the characteristic function of a set S. Then clearly, we havesup_s∈ [0,T]ψ(s)=ψ(T)=τ∑_n=1^N(nτ)^α-1≤∫_0^Ts^α-1ṣ = α^-1T^α≤ c_T .Therefore,(U_h^n -u_h(t_n))_n=1^N_ℓ^1(L^2) ≤ cτ^2 ∑_n=1^N ∫_0^t_n (t_n+τ-s)^-1 g'(s) _L^2 ṣ= cτ∫_0^Tψ(s)g'(s)_L^2(Ω)ṣ≤ c_Tτ g _W^1,1(0,T;L^2).Then (<ref>), (<ref>) and Riesz-Thorin interpolation theorem <cit.>, yield for any 1<p<∞(U_h^n -u_h(t_n))_n=1^N_ℓ^p(L^2)≤c τ g _W_L^1,p(0,T;L^2).Since U_h^n-U_h^n satisfies the discrete scheme, cf. (<ref>) and (<ref>), Lemmas <ref> and <ref> imply(U_h^n - U_h^n)_n=1^N_ℓ^p(L^2(Ω))≤c (g̅_h^n - P_hg(t_n))_n=1^N_ℓ^p(L^2(Ω))≤ cτg_W^1,p(0,T;L^2(Ω)).Further, by Lemma <ref> and Remark <ref>, we have(u_h(t_n) - u̅_h^n)_n=1^N_ℓ^p(L^2)≤ cτu_h_W^1,p(0,T;L^2)≤ cτg_W_L^1,p(0,T;L^2) .The last three estimates show the assertion for s=1. The case 0<s<1 follows by interpolation. In Theorem <ref>, we compare the numerical solution U_h^n to (<ref>) with the time-averaged solution u̅_h^n, instead of u_h(t_n). This is due to possible insufficient temporal regularity of u_h: it is unclear how to define u_h(t_n) for t_n∈(0,T] for g∈ W_L^s,p(0,T; L^2) with s+α<1/p. For s∈(1/p,1], W_L^s,p(0,T; L^2) ↪ C([0,T] ;L^2) and so Theorem <ref> requires the condition g(0)=0. Such a compatibility condition at t=0 is not necessarily satisfied by (<ref>)–(<ref>). Hence, we state an error estimate below for a smooth but incompatible source g∈ W^s,p(0,T; L^2). For g∈ W^s,p(0,T; L^2) with p∈(1,∞) and s∈(1/p,1), let u_h and U_h^n be the solutions of (<ref>) and (<ref>), respectively. Then there holds(U_h^n -u_h(t_n))_n=1^N_ℓ^p(L^2)≤ c τ^min(1/p+α,s) g _W^s,p(0,T; L^2) .Moreover, if p>1/α is so large that α∈(0,1/p'), then(U_h^n -u_h(t_n))_n=1^N_ℓ^∞(L^2)≤ c τ^α g _W^1/p+α,p(0,T; L^2).For g(x,t)≡ g(x), which belongs to W^1,p(0,T;L^2(Ω)), by Lemma <ref> we have(U_h^n - u_h(t_n))_n=1^N _ℓ^p(L^2)^p =τ∑_n=1^N U_h^n - u_h(t_n)_L^2^p ≤c τ^p+1g _L^2^p ∑_n=1^Nt_n^p(α-1)≤ c τ^pα+1g _L^2^p+ c τ^pg _L^2^p ∫_τ^Tt^p(α-1) ṭ.Now for s<1, there holds∫_τ^Tt^p(α-1) ṭ≤{ c τ^pα+1 α∈ (0,1/p') c τ^p α∈ (1/p' ,1)c τ^p (1+ln (T/τ))α = 1/p'.≤{ c τ^p(1/p+α) α∈ (0,1/p') c τ^ps α∈ (1/p' ,1)c τ^ps α = 1/p'.which together with the preceding estimate implies(U_h^n - u_h(t_n))_n=1^N _ℓ^p(L^2(Ω))≤ cτ^min(1/p+α,s) g _W^s,p(0,T; L^2(Ω)) .For g∈ W^s,p(0,T; L^2), by Sobolev embedding, g(0) exists, and in the splitting g(t)=g(0) + (g(t)-g(0)), there holds g-g(0)_W_L^s,p(0,T;L^2(Ω))≤ cg_W^s,p(0,T;L^2(Ω)). Let v_h be the semidiscrete solution for the source g(t)-g(0), and v̅_h^n=∫_t_n-1^t_nv_h(t)ṭ. Since g(t)-g(0)∈ W_L^s,p(0,T; L^2), by Theorem <ref>, the corresponding fully discrete solution V_h^n by (<ref>) satisfies(V_h^n- v̅_h^n)_n=1^N_ℓ^p(L^2)≤ cτ^sg-g(0)_W_L^s,p(0,T; L^2).Further, by Lemma <ref> and (<ref>), we have(v_h(t_n)-v̅_h^n)_n=1^N_ℓ^p(L^2)≤ c τ^sv_h_W^s,p(0,T;L^2(Ω)) ≤ cτ^sg-g(0)_W_L^s,p(0,T; L^2).Similarly, for the fully discrete solution V_h^n for the source g(t)-g(0) by (<ref>), from Lemmas <ref> and <ref>, we deduce(V_h^n-V_h^n)_n=1^N_ℓ^p(L^2(Ω)) ≤ c (P_hg(t_n)-g̅_h^n)_n=1^N_ℓ^p(L^2(Ω))≤ cτ^sg_W^s,p(0,T;L^2(Ω)).These estimates together with the triangle inequality give (<ref>).Finally, (<ref>) follows by the inverse inequality in time and (<ref>) with s=1/p+α, i.e.,(U_h^n -u_h(t_n))_n=1^N_ℓ^∞(L^2) ≤ c τ^-1/p (U_h^n -u_h(t_n))_n=1^N_ℓ^p(L^2)≤ cτ^αg_W^1/p+α,p(0,T; L^2) .This completes the proof of the theorem.The estimate (<ref>) is not sharp since, according to the proof, the restriction s<1 is only needed for α = 1/p'. Nonetheless, it is sufficient for the error analysis in Section <ref>.§ THE OPTIMAL CONTROL PROBLEM AND ITS NUMERICAL APPROXIMATIONIn this section, we develop a numerical scheme for problem (<ref>)–(<ref>), and derive error bounds for the spatial and temporal discretizations. §.§ The continuous problemThe first-order optimality condition of(<ref>)-(<ref>) was given in <cit.>.Problem (<ref>)-(<ref>) admits a unique solution q∈. There exist a state u∈ L^2(0,T; D(Δ))∩ H_L^(0,T; L^2) and an adjoint z∈ L^2(0,T; D(Δ))∩ H_R^(0,T; L^2) such that (u,z,q) satisfiesu-Δ u = f+q,0<t≤ T, u(0)=0 , _t∂_T^α z-Δ z = u-u_d, 0≤ t<T, z(T)=0, (γ q + z, v-q)_L^2(0,T;L^2(Ω))≥ 0,∀ v∈.where (·,·)_L^2(0,T;L^2(Ω)) denotes the L^2(0,T;L^2(Ω)) inner product. Let P_ be the nonlinear pointwise projection operator defined byP_ (q) = max(a,min(q,b)).It is bounded on W^s,p(0,T; L^2(Ω)) for 0≤ s≤ 1 and 1≤ p≤∞ P_u_W^s,p(0,T;L^2(Ω))≤ cu_W^s,p(0,T; L^2(Ω)).This estimate holds trivially for s=0 and s=1 (see <cit.>), and the case 0<s<1 follows by interpolation. Then (<ref>) is equivalent to the complementarity condition q= P_(-γ^-1 z). Now we give higher regularity of the triple (u,z,q).For any s∈(0,1/2), let u_d∈ H^s(0,T;L^2(Ω)) and f∈ H^s(0,T;L^2(Ω)). Then the solution (u,z,q) of problem (<ref>)–(<ref>) satisfies the following estimateq_H^min(1,α+s) (0,T;L^2(Ω)) +u_H^α+s(0,T;L^2(Ω))+ z_H^α+s(0,T;L^2(Ω))≤ c. Let r=min(1,α+s). By (<ref>) and (<ref>), we haveq_H^r (0,T;L^2(Ω))≤ cz_H^r (0,T;L^2(Ω))≤ cz _H^α+s (0,T;L^2(Ω)).Applying Theorem <ref> to (<ref>) yieldsz_H^α+s (0,T;L^2(Ω)) ≤ c(u_H^s(0,T;L^2(Ω))+u_d_H^s(0,T;L^2(Ω))) ≤ c u_H^s (0,T;L^2(Ω))+c.Similarly, applying Theorem <ref> to (<ref>) givesu_H^α+s (0,T;L^2(Ω)) ≤ c(f_H^s (0,T;L^2(Ω))+q_H^s(0,T;L^2(Ω)))≤ c+cq_H^s(0,T;L^2(Ω)) .The last three estimates together implyq_H^r (0,T;L^2(Ω)) ≤ c+cq_H^s(0,T;L^2(Ω))≤ c+c_ϵ'q_L^2(0,T;L^2(Ω))+ϵ'q_H^r (0,T;L^2(Ω)) ,where the last step is due to the interpolation inequality <cit.>q_H^s(0,T;L^2(Ω))≤ c_ϵ'q_L^2(0,T;L^2(Ω))+ϵ'q_H^r (0,T;L^2(Ω)).By choosing a small ϵ'>0 and the pointwise boundedness of q, cf. (<ref>), we obtainq_H^r (0,T;L^2(Ω))≤c+c_ϵ'q_L^2(0,T;L^2(Ω))≤ c.This shows the bound on q.(<ref>) and (<ref>) give the bound on u, and that of z follows similarly. Next, we give an improved stability estimate on q. Let p>1/α be sufficiently large so that α∈(0,1/p'), u_d∈ W^α,p(0,T;L^2(Ω)) and f∈ L^p(0,T;L^2(Ω)). Then the optimal control q satisfies:q_W^1/p+α-ϵ,p(0,T;L^2(Ω))≤ c,where the constant c depends on u_d_W^α,p(0,T;L^2(Ω)) and f_L^p(0,T;L^2(Ω)). The condition α∈(0,1/p') implies r:=1/p+α-ϵ<1. Thus (<ref>) and Theorem <ref> (with s=r-α) implyq_W^r,p(0,T;L^2(Ω))≤c z_W^r,p(0,T;L^2(Ω))≤cu-u_d_W^r-α,p(0,T;L^2(Ω)) ,Since p>1/α, r-α=1/p-ϵ<α and thus Theorem <ref> (with s=0) and (<ref>) giveu-u_d_W^r-α,p(0,T;L^2(Ω))≤cu-u_d_W^α,p(0,T;L^2(Ω))≤cf+q_L^p(0,T;L^2(Ω))+c≤ c,The last two estimates together imply the desired result.§.§ Spatially semidiscrete schemeNow we give a spatially semidiscrete scheme for problem (<ref>)–(<ref>): find q_h∈ such thatmin_q_h∈ J(u_h, q_h) = 12 u_h - u_d_L^2(0,T;L^2)^2 + γ2 q_h _L^2(0,T;L^2)^2,subject to the semidiscrete problemu_h-Δ_h u_h= P_h(f+q_h),0<t≤ T, u_h(0)=0.Similar to Theorem <ref>, problem (<ref>)-(<ref>) admits a unique solution q_h∈. The first-order optimality system reads:u_h-Δ_h u_h = P_h(f+q_h), 0<t≤ T, u_h(0)=0, _t∂_T^α z_h-Δ_h z_h = P_h(u_h-u_d), 0≤ t<T, z_h(T)=0,(γ q_h + z_h, v-q_h)_L^2(0,T;L^2(Ω))≥ 0, ∀ v∈.The variational inequality (<ref>) is equivalent toq_h=P_(-γ^-1 z_h). For the approximation (<ref>)–(<ref>), see <cit.> for an error estimate.For f, u_d∈ L^2(0,T;L^2(Ω)), let (u,z,q) and (u_h,z_h,q_h) be the solutions of problems (<ref>)–(<ref>) and (<ref>)–(<ref>), respectively. Then there holdu- u_h_L^2(0,T;L^2(Ω)) +z-z_h_L^2(0,T;L^2(Ω)) +q-q_h_L^2(0,T;L^2(Ω)) ≤ ch^2, ∇( u- u_h) _L^2(0,T;L^2(Ω)) + ∇( z-z_h)_L^2(0,T;L^2(Ω)) ≤ ch.Next, we present the regularity of the semidiscrete solution (u_h,z_h,q_h). The proof is similar to the continuous case in Lemmas <ref> and <ref> and hence omitted.Let s∈(0,1/2), u_d∈ H^s(0,T;L^2(Ω)) and f∈ H^s(0,T;L^2(Ω)). Then the solution (u_h,z_h,q_h) of problem (<ref>)–(<ref>) satisfies the following estimate:q_h _H^min(1,α+s) (0,T;L^2(Ω)) +u_h_H^α+s(0,T;L^2(Ω))+ z_h_H^α+s(0,T;L^2(Ω))≤ c.Further, for u_d∈ W^α,p(0,T;L^2(Ω)), f∈ L^p(0,T;L^2(Ω)), with p>1/α and α∈(0,1/p'), there holdsq_h_W^1/p+α-ϵ,p(0,T;L^2(Ω))≤ c.Last, we derive a pointwise-in-time error estimate.For f, u_d∈ H^1(0,T;L^2(Ω)), let (u,z,q) and (u_h,z_h,q_h) be the solutions of problems (<ref>)–(<ref>) and (<ref>)–(<ref>), respectively. Then there holdsu- u_h_L^∞(0,T;L^2(Ω)) +z-z_h_L^∞(0,T;L^2(Ω)) +q-q_h_L^∞(0,T;L^2(Ω))≤ c ℓ_h^2 h^2with ℓ_h=log(2+1/h), where the constant c depends on f_H^1(0,T;L^2(Ω)) and u_d_H^1(0,T;L^2(Ω)). We employ the splitting u - u_h = (u - u_h(q)) + (u_h(q) - u_h):=ϱ+ϑ, where u_h(q)∈ X_h solves∂_t^α u_h(q) -Δ_h u_h(q) = P_h(f + q), 0<t≤ T,with u_h(q)(0)=0.Then u_h(q) is the semidiscrete solution of (<ref>) with g=f+q, and ϱ is the FEM error for the direct problem. By <cit.> and Lemma <ref>, we haveϱ_L^∞(0,T;L^2)≤ c ℓ_h^2h^2f+q_L^∞(0,T;L^2(Ω))≤ cℓ_h^2h^2.Since ϑ satisfies ∂_t^αϑ -Δ_h ϑ= P_h (q-q_h), for 0<t≤ T with ϑ(0)=0, (<ref>), L^2(Ω)-stability of P_h, the conditions (<ref>) and (<ref>), and the pointwise contractivity of P_ imply_0∂_t^αϑ_L^p(0,T;L^2(Ω))≤ P_h (q -q_h)_L^p(0,T;L^2(Ω)) ≤c q - q_h_L^p(0,T;L^2(Ω))≤ cz-z_h _L^p(0,T;L^2(Ω)).Next, it follows from (<ref>), (<ref>) and the identity P_hΔ=Δ_hR_h (with R_h:H^1(Ω)→ X_h being Ritz projection) that w_h:=P_hz-z_h satisfies w_h(T)=0_t∂_T^α w_h-Δ_h w_h= P_hu-u_h -Δ_h(P_hz-R_hz), 0≤ t<T,and thus_t∂_T^αΔ_h^-1w_h-Δ_h Δ_h^-1w_h = Δ_h^-1(P_hu-u_h) - (P_hz-R_hz) .The maximal L^p regularity (<ref>) and triangle inequality implyw_h_L^p(0,T;L^2(Ω)) ≤ cΔ_h^-1(P_hu-u_h) - (P_hz-R_hz)_L^p(0,T;L^2(Ω))≤ cP_hu-u_h_L^p(0,T;L^2(Ω))+cP_hz-R_hz_L^p(0,T;L^2(Ω)).The L^2(Ω)-stability of P_h and triangle inequality yieldP_hu-u_h_L^p(0,T;L^2(Ω))≤ cu-u_h_L^p(0,T;L^2(Ω))≤ c(ϑ_L^p(0,T;L^2(Ω))+ϱ_L^p(0,T;L^2(Ω))),and Theorem <ref> (with s=0) and lemma <ref> giveP_hz-R_hz_L^p(0,T;L^2(Ω))≤ cz_L^p(0,T;H^2(Ω)) h^2 ≤ cu-u_d_L^p(0,T;L^2(Ω)) h^2≤ ch^2. The last three estimates and (<ref>) yieldw_h_L^p(0,T;L^2(Ω))≤ cϑ_L^p(0,T;L^2(Ω))+cℓ_h^2h^2.Thus repeating the preceding argument yieldsz-z_h_L^p(0,T;L^2(Ω)) ≤z-P_hz_L^p(0,T;L^2(Ω))+w_h_L^p(0,T;L^2(Ω))≤ cϑ_L^p(0,T;L^2(Ω))+cℓ_h^2h^2.Substituting it into (<ref>) and by Sobolev embedding W^α,p (0,T;L^2(Ω))↪ L^p_α(0,T;L^2(Ω)), with the critical exponent p_α=p/(1-pα) if pα<1, and p_α=∞ if pα>1:ϑ_L^p_α(0,T;L^2(Ω))≤ c_0∂_t^αϑ_L^p(0,T;L^2(Ω))≤ cϑ_L^p(0,T;L^2(Ω))+cℓ_h^2h^2 .A finite number of repeated applications of this inequality yieldsϑ_L^∞(0,T;L^2(Ω))≤ c ϑ_L^2(0,T;L^2(Ω))+ cℓ_h^2 h^2≤ c ℓ_h^2 h^2,where we have used the fact that, by maximal L^p regularity (<ref>) and Theorem <ref>,ϑ_L^2(0,T;L^2(Ω))=u_h(q)-u_h_L^2(0,T;L^2(Ω))≤ cq-q_h_L^2(0,T;L^2(Ω))≤ ch^2.This gives the desired bound on u-u_h_L^∞(0,T;L^2). The bounds on z - z_h_L^∞(0,T;L^2) andq-q_h_L^∞(0,T;L^2) follow similarly by the contraction property of P_U_ad.§.§ Fully discrete schemeNow we turn to the fully discrete approximation of (<ref>)–(<ref>), with L1 scheme or BE-CQ time stepping. First, we define a discrete admissible setU_ad^τ = {Q⃗_h=(Q_h^n-1)_n=1^N: a≤ Q^n-1≤ b, n=1,2,...,N },and consider the following fully discrete problem:min_Q⃗_h ∈ U_ad^ττ/2∑_n=1^N ( U_h^n - u_d^n_L^2^2 + γ Q_h^n-1_L^2^2 ),subject to the fully discrete problemU_h^n-Δ_h U_h^n = f_h^n+P_h Q_h^n-1, n=1,2,...,N, with U_h^0 =0,with u_d^n=u_d(t_n) and f_h^n=P_hf(t_n). Let ∂̅_τ^αφ^n be the L1/BE-CQ approximation of _t∂_T^αφ (t_n):∂̅_τ^αφ^N-n =τ^-α∑_j=0^nβ_n-jφ^N-j .Then the fully discrete problem is to find (U_h^n, Z_h^n, Q_h^n) such thatU_h^n-Δ_h U_h^n= f_h^n+P_h Q_h^n-1, n=1,2,...,N, withU_h^0 =0 ,∂̅_τ^α Z_h^n-1-Δ_h Z_h^n-1 = U_h^n-P_h u_d^n , n=1,2,...,N,withZ_h^N =0 ,(γ Q_h^n-1+ Z_h^n-1, v-Q_h^n-1 )≥ 0, ∀v∈ L^2 a≤ v≤ b.Similar to (<ref>), (<ref>) can be rewritten asQ_h^n-1=P_U_ad(-γ^-1Z_h^n-1), n=1,2,…,N . To simplify the notation, we define a discrete L^2(0,T;L^2(Ω)) inner product [·,·]_τ by[ v, w]_τ = τ∑_n=1^N (v_n,w_n) ∀v=(v_n)_n=1^N,w=(w_n)_n=1^N∈ L^2(Ω)^N ,and denote by ·_τ the induced norm. Let v=( v^n_h)_n=1^N ∈ L^2(Ω)^N and ∂̅_τ^α w=(∂̅_τ^α w^n-1_h)_n=1^N∈ L^2(Ω)^N. Then the discrete integration by parts formula holds <cit.>[ v,w]_τ =[ v,∂̅_τ^α w]_τ∀v , w∈ L^2(Ω)^N.Thus, ∂̅_τ^α is the adjoint towith respect to [·,·]_τ. LetU_h=(U_h^n)_n=1^N,Z_h=(Z_h^n-1)_n=1^N,Q_h=(Q_h^n-1)_n=1^N , u_h=(u_h(t_n))_n=1^N,z_h=(z_h(t_n-1))_n=1^N,q_h=(q_h(t_n-1))_n=1^N . Next we introduce two auxiliary problems. Let 𝐔_h(q_h)=(U_h^n(q_h)) )_n=1^N∈ X_h^N solveU_h^n(q_h)-Δ_hU_h^n(q_h)= f_h^n + q_h(t_n-1) , n=1,…,N,U_h^0(q_h)=0 .By Lemma <ref>, the pointwise evaluation q_h(t_n) does make sense, and thus problem (<ref>) is well defined. For any v_h=(v_h^n)_n=1^N, let Z_h( v_h)= (Z_h^n-1( v_h))_n=1^N∈ X_h^N solve∂̅_τ^α Z_h^n-1( v_h) -Δ_h Z_h^n-1( v_h) = v_h^n - P_h u_d^n , n=1,2,…,N,withZ_h^N( v_h) =0 . The rest of this part is devoted to error analysis. First, we bound q_h- Q_h_τ.For 𝐐_h, 𝐪_h, Z_h and Z_h( U_h(q_h)) defined as above, there holdsγ Q_h- q_h_τ ^2 ≤[ q_h- Q_h , Z_h( U_h(q_h))-z_h]_τ.It follows from(<ref>) and (<ref>), similarly from (<ref>) and (<ref>), that( -Δ_h) ( U_h(q_h)- U_h) = q_h-Q_h(∂̅_τ^α -Δ_h) ( Z_h( v_h)- Z_h) =v_h -U_h.Together with (<ref>), these identities imply[ q_h- Q_h ,Z_h- Z_h( U_h(q_h))]_τ =[( - Δ_h)( U_h (q_h)- U_h),Z_h- Z_h( U_h(q_h))]_τ= [ U_h (q_h)- U_h, (∂̅_τ^α - Δ_h)( Z_h- Z_h( U_h(q_h)))]_τ= -U_h (q_h)- U_h _τ^2≤0.Next, since (<ref>) holds pointwise in time, i.e., q_h(t_n-1) = P_U_ad(-γ^-1z_h(t_n-1)), we have(q_h(t_n-1)+γ^-1z_h(t_n-1),χ-q_h(t_n-1))≥ 0,∀ χ∈ L^2(Ω)a≤χ≤ b.Upon setting v=q_h(t_n-1) in (<ref>) and χ=Q_h^n-1 in (<ref>), we deduceγ Q_h- q_h_τ ^2 =γ[ Q_h - q_h,Q_h]_τ- γ[ Q_h - q_h,q_h]_τ≤ [ q_h- Q_h ,Z_h]_τ - [ q_h- Q_h ,z_h]_τ= [ q_h- Q_h ,Z_h- Z_h( U_h(q_h))]_τ + [ q_h- Q_h , Z_h( U_h(q_h))-z_h]_τ.Now invoking (<ref>) completes the proof of the lemma. The next result gives an error estimate for the approximate state 𝐔_h(q_h).Let f,u_d∈ H^1(0,T;L^2(Ω)). For any ϵ∈(0,min(1/2,α)), there holdsU_h(q_h) -u_h_τ≤ cτ^1/2+min(1/2,α-ϵ) . By the triangle inequality, we haveU_h(q_h) -u_h_τ≤ U_h(q_h)- U_h(q_h)_τ +U_h(q_h)- u_h_τ,where 𝐔_h(q_h)=(U_h^n(q_h))_n=1^N is the solution toU_h^n(q_h)-Δ_h^nU_h^n(q_h)= f_h^n + q̃_h^n,n =1,2,…,NU_h^0(q_h)=0 ,with q̃_h^n=P_hq_h(t_n) (and q̃_h=(q̃_h^n)_n=1^N). That is, U_h^n(q_h) is the fully discrete solution of problem (<ref>) with g=f+q_h. By Lemmas <ref> and <ref>, we haveU_h(q_h)- U_h(q_h)_τ≤ c q_h-q̃_h_τ≤ cτ^min(1/2+α-ϵ,1)q_h_H^1/2+α-ϵ(0,T;L^2(Ω)).Further, Theorem <ref> (with s=min(1,1/2+α-ϵ)∈(1/2,1)) impliesU_h(q_h) -u_h_τ ≤ c P_hf+q_h _ H^s(0,T;L^2(Ω))τ^s≤ c(P_hf_H^s(0,T;L^2(Ω))+q_h _H^s(0,T;L^2(Ω))) τ^s .The last two estimates and Lemma <ref> (with s=1/2-ϵ) yield the desired assertion. Now we can give an ℓ^2(L^2(Ω)) error estimate for the approximation (U_h^n,Z_h^n,Q_h^n).For f∈H^1 (0,T;L^2) and u_d∈H^1(0,T;L^2(Ω)), (u_h,z_h,q_h) and (U_h^n,Z_h^n,Q_h^n) be the solutions of problems (<ref>)-(<ref>) and (<ref>)-(<ref>), respectively. Then there holds for any small ϵ>0u_h-U_h_τ +z_h-Z_h_τ +q_h-Q_h_τ≤ c τ ^1/2+min(1/2,α-ϵ) ,where the constant c depends on f_H^1(0,T;L^2(Ω)) and u_d_H^1(0,T;L^2(Ω)). By Lemma <ref> and the triangle inequality, we deduceQ_h- q_h_τ ≤ cZ_h( U_h(q_h))-Z_h( 𝐮_h)_τ+ cZ_h(𝐮_h)-z_h_τ.It suffices to bound the two terms on the right hand side. Lemmas <ref> and <ref> implyZ_h( U_h(q_h))-Z_h( 𝐮_h)_τ≤ c U_h(q_h) -u_h _τ≤ cτ^r.with r=1/2+min(1/2,α-ϵ). Further, since 𝐙_h(𝐮_h) is a fully discrete approximation to z_h(u_h), by Theorem <ref> (with s=r) and Lemma <ref>, we haveZ_h( u_h)-z_h_τ≤ c u_h-P_hu_d _H^r (0,T;L^2(Ω))τ^r ≤ cτ ^r.Thus, we obtain the estimate Q_h- q_h_τ≤ c τ^r. Next, by Lemmas <ref> and <ref>, we deduceU_h -u_h _τ ≤ U_h - U_h ( q_h)_τ+ U_h ( q_h) -u_h _τ≤ c Q_h -q_h _τ+ U_h ( q_h) -u_h _τ≤ c τ^r.Similarly, Z_h -z_h _τ can be bounded byZ_h -z_h _τ ≤ Z_h - Z_h ( u_h)_τ+ Z_h ( u_h) -z_h _τ≤ c U_h -u_h _τ+ Z_h ( u_h) -z_h _τ≤ cτ^r.This completes the proof of Theorem <ref>. Last, we give a pointwise-in-time error estimate for the approximation (U_h^n,Q_h^n,Z_h^n).For f,u_d∈ W^1,p (0,T;L^2)∩ H^1(0,T;L^2(Ω)), p>1/α with α∈(0,1/p'), let (u_h,z_h,q_h) and (U_h^n,Z_h^n,Q_h^n) be the solutions of problems (<ref>)-(<ref>) and (<ref>)-(<ref>), respectively. Then there holds for any small ϵ>0max_1≤ n≤ N(u^n_h- U^n_h _L^2(Ω) + z^n-1_h- Z^n-1_h _L^2(Ω) + q^n-1_h- Q^n-1_h _L^2(Ω))≤ c τ^α-ϵ ,where the constant c depends on f_W^1,min(p,2)(0,T;L^2(Ω)) and u_d_W^1,min(p,2)(0,T;L^2(Ω)). It follows from (<ref>) and (<ref>) that U_h^0-U_h^0(q_h) =0 and(U_h^n-U_h^n(q_h)) -Δ_h(U_h^n-U_h^n(q_h)) =Q_h^n-1-q_h(t_n-1), n=1,…,N .By Lemma <ref> and the inverse inequality (in time), we obtain for any 1/α< p_1<∞(U_h^n-U_h^n(q_h))_n=1^N_ℓ^p_1(L^2(Ω)) ≤ c (Q_h^n-1-q_h(t_n-1))_n=1^N_ℓ^p_1(L^2(Ω))≤ cτ^min(0,1/p_1-1/2)(Q_h^n-1-q_h(t_n-1))_n=1^N_ℓ^2(L^2(Ω)).This and Theorem <ref> imply(U_h^n-U_h^n(q_h))_n=1^N_ℓ^p_1(L^2(Ω))≤cτ^min(1/p_1,1/2) +min(1/2,α-ϵ).By choosing p_1>1/α sufficiently close to 1/α and discrete embedding <cit.>,(U_h^n-U_h^n(q_h))_n=1^N_ℓ^∞(L^2(Ω)) ≤ c (U_h^n-U_h^n(q_h))_n=1^N_ℓ^p_1(L^2(Ω))≤ cτ^min(1/p_1,1/2)+min(1/2,α-ϵ)≤ cτ^α-ϵ,where the last inequality follows from the inequality min(1/p_1,1/2)+min(1/2,α-ϵ)≥α-ϵ, due to the choice of p_1. Further, by the definition of U_h^n(q_h) in(<ref>), choosing p_2>1/α sufficiently large so that α∈(0,1/p_2') and applying (<ref>) and Lemma <ref>, we getu_h(t_n)-U_h^n(q_h)_L^2(Ω)≤cτ^α-ϵf+q_h_W^1/p_2+α-ϵ,p_2(0,T;L^2(Ω)) ≤ cτ^α-ϵ(q_h_W^1/p_2+α-ϵ,p_2(0,T;L^2(Ω)) +c) ≤ cτ^α-ϵ.Last, by choosing p_3>1/α sufficiently close to 1/α, Lemmas <ref>, <ref>, and <ref>, and discrete embedding <cit.>, we obtain(U_h^n(q_h)-U_h^n(q_h))_n=1^N_ℓ^∞(L^2(Ω)) ≤ c (U_h^n(q_h)-U_h^n(q_h))_n=1^N_ℓ^p_3(L^2(Ω))≤ c(q_h(t_n-1)-q_h(t_n))_n=1^N_ℓ^p_3(L^2(Ω))≤ cτ^α-ϵ.The last three estimates yield the desired bound on u_h(t_n)-U_h^n_L^2(Ω). The bound on z_h(t_n-1)-Z_h^n-1_L^2(Ω) follows similarly, and that on q_h(t_n-1)-Q_h^n-1_L^2(Ω) by the contraction property of P_. § NUMERICAL RESULTS AND DISCUSSIONSNow we present numerical experiments to illustrate the theoretical findings. We perform experiments on the unit interval Ω=(0,1). The domain Ω is divided into M equally spaced subintervals with a mesh size h = 1/M. To discretize the fractional derivatives _0∂_t^α u and _t∂_T^α z, we fix the time stepsize τ = T/N. We present numerical results only for the fully discrete scheme by the Galerkin FEM in space and the L1 scheme in time, sinceBE-CQ gives nearly identical results.We consider the following two examples to illustrate the analysis. (a) f≡0 and u_d(x,t)=e^t x(1-x).(b) f=(1+cos(t))χ_(1/2,1)(x) and u_d(x,t)=5e^tx(1-x).Throughout, unless otherwise specified, the penalty parameter γ is set to γ=1, and the lower and upper bounds a and b in the admissible setto a=0 and b=0.05. The final time T is fixed at T=0.1. The conditions from Theorems <ref>, <ref> and <ref> are satisfied for both examples, and thus the error estimates therein hold.In Tables <ref> and <ref>, we present the spatial error e_h(u) in the L^∞ (0,T;L^2(Ω))-norm for the semidiscrete solution u_h, defined bye_h(u) = max_1≤ n≤ Nu_h(t_n)-u(t_n)_L^2(Ω),and similarly for the approximations z_h and q_h. The numbers in the bracket denote the theoretical rates. Since the exact solution to problem (<ref>) is unavailable, we compute reference solutions on a finer mesh, i.e., the continuous solution u(t_n) with a fixed time step τ=T/1000 and mesh size h=1/1280. The empirical rate for the spatial error e_h is of order O(h^2), which is consistent with the theoretical result in Theorem <ref>. For case (a), the box constraint is inactive, and thus the errors for the control q and adjoint z are identical (since γ=1).Next, to examine the convergence in time, we compute the ℓ^2(L^2(Ω)) and ℓ^∞ (L^2(Ω)) temporal errors e_τ,2(u) and e_τ,∞(u) for the fully discrete solutions U_h^n, respectively, defined bye_τ,2(u) = (U_h^n-u_h(t_n))_n=1^N_ℓ^2(L^2(Ω)) e_τ,∞(u) = max_1≤ n≤ NU_h^n-u_h(t_n)_L^2(Ω),and similarly for the approximations Z_h^n and Q_h^n. The reference semidiscrete solution u_h is computed with h=1/50 and τ=1/(64×10^4). Numerical experiments show that the empirical rate for the temporal discretization error is of order O(τ^min(1/2+α,1)) and O(τ^α) in the ℓ^2(L^2(Ω)) and ℓ^∞(L^2(Ω))-norms, respectively, cf. Tables <ref>–<ref> and <ref>–<ref>, for cases (a) and (b). These results agree well with the theoretical predictions from Theorems<ref> and <ref>, and thus fully support the error analysis in Section <ref>. In Fig. <ref>, we plot the optimal control q, the state u and the adjoint z. One clearly observes the weak solution singularity at t=0 for the state u and at t=T for the adjoint z. The latter is especially pronounced for case (b). The weak solution singularity is due to the incompatibility of the source term with the zero initial/terminal data. § CONCLUSIONS In this work, we have developed a complete numerical analysis of a fully discrete scheme for a distributed optimal control problem governed by a subdiffusion equation, with box constraint on the control variable, and derived nearly sharp pointwise-in-time error estimates for both space and time discretizations. These estimates agree well with the empirical rates observed in the numerical experiments. The theoretical and numerical results showthe adverse influence of the fractional derivatives on the convergence rate when the fractional order α is small. § PROOF OF LEMMA <REF> By Sobolev embedding, W^s,p(0,1;L^2(Ω))↪ C([0,1];L^2(Ω)) for s∈ (1/p,1], and thus we can define an interpolation operator Π by Π v(t̂) = v(1), for t̂∈(0,1), for any v∈ W^s,p(0,1;L^2(Ω)). The operator E =I-Π is bounded from W^s,p(0,1;L^2(Ω)) to L^2(0,1;L^2(Ω)):Ev_L^2(0,1;L^2(Ω)) =(I-Π)v_L^2(0,1;L^2(Ω))≤ cv _W^s,p(0,1;L^2(Ω)).By the fractional Poincaré inequality (cf. <cit.>), we haveE v_L^p(0,1;L^2(Ω)) = inf_p∈ℝ E(v - p) _L^p(0,1;L^2(Ω))≤ c inf_p∈ℝ v- p_W^s,p(0,1;L^2(Ω))≤c | v|_W^s,p(0,1;L^2(Ω)) ,where the seminorm |·|_W^s,p(0,T;L^2(Ω)) is defined in (<ref>). By Hölder's inequality, we obtain(v(t_n) - v̅^n)_n=1^N_ℓ^p(L^2)^p= τ∑_n=1^N v(t_n)-τ^-1∫_t_n-1^t_nv(t)ṭ_L^2(Ω)^p = τ^1-p∑_n=1^N ∫_t_n-1^t_n(v(t_n)-v(t))ṭ_L^2(Ω)^p ≤∑_n=1^N ∫_t_n-1^t_nv(t_n) - v(t)_L^2(Ω)^p ṭ.Let v_n(t̂)=v(t_n-1+τt̂), for t̂∈[0,1], n=1,…,N. Then v_n(t̂) ∈ W^s,p(0,1;L^2(Ω)) and by (<ref>), we have(v(t_n) - v̅^n)_n=1^N_ℓ^p(L^2)^p≤τ∑_n=1^N ∫_0^1Πv_n-v_n_L^2(Ω)^p( t̂) ̣̂t ≤ c τ∑_n=1^N| v_n|_W^s,p(0,1;L^2(Ω))^p ≤ c τ∑_n=1^N ∫_0^1 ∫_0^1 v_n(t̂)-v_n(ξ̂)_L^2(Ω)^p/|t̂-ξ̂|^1+ps ̣̂t ̣̂ξ= c τ^ps∑_n=1^N ∫_t_n-1^t_n∫_t_n-1^t_nv(t)-v(ξ)_L^2(Ω)^p/|t-ξ|^1+ps ṭξ̣ ≤ c τ^ps∫_0^T∫_0^T v(t)-v(ξ)_L^2(Ω)^p/|t-ξ|^1+ps ṭξ̣=c τ^ps|v|_W^s,p(0,T;L^2(Ω))^p ,which implies the desired assertion. abbrv | http://arxiv.org/abs/1707.08808v2 | {
"authors": [
"Bangti Jin",
"Buyang Li",
"Zhi Zhou"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170727100554",
"title": "Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint"
} |
2.0mm | http://arxiv.org/abs/1707.08523v1 | {
"authors": [
"Yizhuang Liu",
"Ismail Zahed"
],
"categories": [
"hep-ph",
"hep-th",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170726161743",
"title": "Hydrodynamical corrections to electromagnetic emissivities in QCD"
} |
Exploiting Web Images for Weakly Supervised Object Detection Qingyi Tao Hao Yang Jianfei CaiNanyang Technological University, Singapore ============================================================================================= 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.§ INTRODUCTIONWith 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 <cit.>. However, state-of-the-art object detection methods still require bounding box annotations which cost extensive human labour. To alleviate this problem, weakly supervised object detection approaches <cit.> 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 <ref>, 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 multi-object 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 <cit.> 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 <cit.>, 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 <cit.>. 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 <cit.>, 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 <cit.> address this problem with multiple instance learning (MIL) <cit.>, 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. <cit.> 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, <cit.> also uses the two stream structure and involves the context feature in the localization stream.In this research, we use WSDDN <cit.> 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 <cit.> 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. <cit.> used this easy-to-hard 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 <cit.>. In <cit.>, they proposed aclassifier-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 <cit.>. In <cit.>, 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, <cit.> 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 <cit.> 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. §.§ WSDDNWe first introduce weakly supervised deep detection network, or WSDDN <cit.>, 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 <cit.> to obtain SPP feature of each region of interest (RoI). As shown in Figure <ref>, 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 fc8 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 (<ref>). L(y_ci, x_i | w) = -log(y_ci(Φ_c(x_i|w)-12)+12) 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_ci={-1, 1} for class c in image i. Output from the last sum pooling layer is denoted as Φ_c^y(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 DatasetIn 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 DiversifyFree 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 <ref>. 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. <ref> 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 <ref>, 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 fc7 feature from a pretrained vgg-f model with PCA dimension reduction to represent each image. As shown in Figure <ref>, 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 RegularizerIncorporating 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 <cit.>, 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) = ∑_i=1^n∑_c=1^CL(y_i , x_i | w) · f(u_i,v_i), f(u_i,v_i) = σ (u_i) ·ψ (v_i),where u_i is the relevance variable indicating whether the training sample is relevant as discussed in <ref>. 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 <cit.> 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 LearningWe 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. <ref> 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 of five overlapped regions with gradually increased image complexity. Table <ref> shows the detection result (`CurrWSDDN') 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[https://github.com/hbilen/WSDDN], which is slightly different from the result of 34.5 reported in <cit.>. §.§ Results Regarding Constructed Web Dataset We now evaluate the usefulness of our constructed web image dataset for WSD. As mentioned in Section <ref>, 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 <ref> 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 <ref> and <ref>, 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 <cit.>, which containsmore 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 <ref>) 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. <ref> gives some examples of the two relevance metrics. For the semantic relevance, we use the classification scores by the outlier detector described in Section <ref>, 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. <ref>. Table <ref> 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, the training of WSD tends to drift to optimize small clusters of training samples. Although additional training instances 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.§.§ More Comparison Results Table <ref> lists 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 Table <ref> 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 (CurrWSDDN), simply combining our web images with VOC images with the semantic relevance for training (WebRel), combining our web images with VOC images for easy-to-hard 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-to-curriculum training (WebRelETC), where we train easy web images first and then train VOC images in a more detailed curriculum.The results of CurrWSDDN, WebETH and WebRelETH have been discussed previously w.r.t. Tables <ref>, <ref> and <ref>, 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 <ref> 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. <ref> 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. <ref>), and missing objects in WSDDN can also be detected by our model in some test images (see the bottom rows of Fig. <ref>).§ 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. ieee | http://arxiv.org/abs/1707.08721v2 | {
"authors": [
"Qingyi Tao",
"Hao Yang",
"Jianfei Cai"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727065839",
"title": "Exploiting Web Images for Weakly Supervised Object Detection"
} |
firstpage–lastpage 2017 The [Y/Mg] clock works for evolved solar metallicity stars Based on spectroscopic observations made with two telescopes: the Nordic Optical Telescope operated by NOTSA at the Observatorio del Roque de los Muchachos (La Palma, Spain) of the Instituto de Astrofísica de Canarias and the Keck I Telescope at the W.M. Keck Observatory (Mauna Kea, Hawaii, USA) operated by the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.D. Slumstrup 1 F. Grundahl 1 K. Brogaard 1,2 A. O. Thygesen 3 P. E. Nissen 1 J. Jessen-Hansen 1 V. Van Eylen 4 M. G. Pedersen 5 December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We study the photoevaporation of molecular clumps exposed to a UV radiation field including hydrogen-ionizing photons (hν > 13.6 eV) produced by massive stars or quasars. We follow the propagation and collision of shock waves inside clumps and take into account self-shielding effects, determining the evolution of clump size and density with time. The structure of the ionization-photodissociation region (iPDR) is obtained for different initial clump masses (M=0.01 - 10^4) and impinging fluxes (G_0=10^2 - 10^5 in units of the Habing flux). The cases of molecular clumps engulfed in the HII region of an OB star and clumps carried within quasar outflows are treated separately. We find that the clump undergoes in both cases an initial shock-contraction phase and a following expansion phase, which lets the radiation penetrate in until the clump is completely evaporated. Typical evaporation time-scales are ≃ 0.01 Myr in the stellar case and 0.1 Myr in the quasar case, where the clump mass is 0.1and 10^3 respectively. We find that clump lifetimes in quasar outflows are compatible with their observed extension, suggesting that photoevaporation is the main mechanism regulating the size of molecular outflows.ISM: clouds, evolution, photodissociation region - quasars: general § INTRODUCTIONThe diffuse interstellar medium (ISM) is characterized by a turbulent multi-phase structure, showing a broad range of densities, temperatures and chemical compositions. In some regions, gravitational forces and pressure compress the gas to sufficiently high densities, so that the formation of molecules such as H_2 and CO is allowed. CO maps have revealed that Giant Molecular Clouds (GMCs) contain a very rich internal structure featuring filaments and clumps <cit.>. The typical sizes of the detected clumps range from 1 to 10 pc. Temperature and density of the gas can be estimated by combining line intensities with radiative transfer calculations. Such studies yield kinetic temperatures in the range T=15-200 K, associated with H_2 densities of n=10^3-4 cm^-3 <cit.>. A correlation between clump temperature and Hα flux suggests that denser clumps are warmer because of a larger UV radiation intensity, likely provided by external sources or internal star-formation activity.Dense molecular clumps have also been detected within the Photo Dissociation Regions (PDRs) of OB stars, through observations in the infrared and millimiter bands <cit.>. Detections of fine-structure lines of [CI] and [CII], high-J CO rotational lines, and J=3-2 lines of HCN and HCO, show that PDRs are made of a low-density, more diffuse component (n≃ 10^2-4 cm^-3), and high-density structures (n≃ 10^6-7 cm^-3), such as in M17SW <cit.>, and in the Orion bar <cit.>. These clumps must have sizes as small as one tenth or a hundredth of pc, often showing elongated shapes. The presence of such clumps affects significantly the emission spectrum of stellar PDRs.According to recent observations <cit.>, molecular clumps are also detected in outflowing gas around quasars. The radiation pressure drives a powerful wind (v∼ 0.1c with c speed of light) which collides with the ISM, so that a shock propagates forward into the ISM and a reverse shock propagates back into the wind <cit.>. The outflowing gas is heated by the shock to very high temperatures (T∼ 10^7 K), so that it is expected to be completely ionized. Nevertheless, detections of the CO, OH and H_2O lines <cit.> show that the outflow is in molecular form up to a radius of 1-10 kpc. To reach such distances, the molecular gas has to be structured in clumps, able to provide sufficient self-shielding against the strong quasar radiation field. The structure of a molecular clump is significantly determined by the presence of an ionizing/photo-dissociating radiation field, since incident photons with different wavelengths alter the chemical composition of the gas and its physical properties. Far ultraviolet (FUV) radiation (6 eV< h ν< 13.6 eV) is responsible for the dissociation of molecules, determining the formation of a PDR <cit.> at the surface of the clump itself. Furthermore, radiation above the Lyman limit (hν > 13.6eV) ionizes neutral atoms, and it is completely absorbed within a shallow layer. The goal of this paper is to understand the evolution of radius and the density profile of molecular clumps exposed to a UV radiation field including hydrogen-ionizing photons produced by massive stars or quasars. The key point is that an ionized shell and an atomic shell form at the edge of the clump. The dynamics of this layered structure is determined by the fact that each layer is at a different temperature and pressure. For a clump with initial density n_0≃10^5 cm^-3, typical temperatures deep into the clump are T≃ 10-100 K, while an atomic (ionized) region can be heated up to around T≃ 10^3 K (T≃ 10^4 K). We denote this type of ionization/photodissociation regions as iPDR.In particular, we apply our model to two scenarios. * Stellar case: a molecular clump is in pressure equilibrium within a GMC in the proximity of an OB star, and it is suddenly engulfed by the expanding HII region. * Quasar case: a clump forms as a result of thermal instabilities within the outflow, finds itself embedded in the ionized wind and exposed to the quasar radiation.Previous works in the literature concentrated mostly on the effects of non-ionizing photons on photoevaporation of clouds <cit.>. The evolution in their case is simplified by the fact that the clump is exposed only to radiation below the Lyman limit. As a result, the clump develops a single shell structure.The paper is organized as follows. In Sec. <ref> we describe the model adopted for the structure of molecular clumps, together with the physics of the dynamical and thermal processes involved. In Sec. <ref> we present the results for the evolution of the clump radius, and compute the evaporation time of a clump for different parameters of the system. The results are presented separately for clumps located near stellar or quasar sources. In Sec. <ref> we summarize our results.§ MODEL §.§ Gaseous environmentThe gaseous environment where clumps are located plays a crucial role in determining properties such as temperature, density and confining pressure. We now describe the interclump medium (ICM) which surrounds clumps in the stellar and quasar case. The ICM properties are summarized in Tab. <ref>.We assume that clumps in a stellar surrounding are in pressure equilibrium with atomic gas, whose temperature depends on the distance and luminosity from the stellar source. We take n_at≃ 10^3 as a typical density for this surrounding gas, and we compute the corresponding temperature according to the FUV flux, using the same method used to compute the temperature of the atomic phase in the clump (see Sec. <ref>). Such temperature ranges between 10^2 K and 10^3 K. The clump is exposed to the radiation of the massive star when it is engulfed in the growing HII region, for which the density is taken to be n_≃ 10.In the case of quasars, molecular clumps likely form from thermal instabilities within the outflow. Clumps detach from the hot phase at the discontinuity between the fast wind and the ISM <cit.>, starting from the distance at which the outflow has become energy-driven. This critical radius has been estimated by <cit.>:R_c = 520σ_200 M_8^1/2 v_0.1 pc,where σ_200 is the velocity dispersion in the host galaxy in units of 200 km s^-1, M_8 is the mass of the SMBH in units of10^8 and v_0.1 is the wind velocity in units of 0.1 times the speed of light. In the outflow, the gas is heated to T_≃ 2.2× 10^7 K and has a typical density n_≃ 60 cm^-3 <cit.>. The outflow fragments <cit.> because of thermal instabilties, so that one component cools to a low temperature. The existence of an equilibrium between a 10^4 K and a 10^7 K phase has been studied by <cit.>, while <cit.> show that an atomic clump requires a very short time to cool and turn to molecular form. Molecule formation can occur in the overdensities generated via thermal instabilities, since radiation is efficiently self-shielded and the gas deep into the clump is allowed to cool[This conclusion has been re-examined by <cit.>, who pointed out that molecule formation is problematic due to the efficient dust destruction by the outflow shock.]. When a clump starts to cool, we assume that it maintains pressure balance with the ICM until its temperature is T 10^4 K. Below such temperature the cooling time-scale is very short, and the evaporation process detailed in the next Sections happens before the clump can readjust to the external pressure. The final temperature of the molecular gas is about 100 K, in agreement with detections with CO and water vapour line emission <cit.>.§.§ Radiation fieldRadiation affects the structure of a clump according to the shape of the emitted spectrum. In particular we are interested in ionizing (energy hν≥ 13.6 eV) and FUV photons (6 eV < h ν< 13.6 eV), whose flux G_0 is measured in units of the Habing flux[The Habing flux (1.6 × 10^-3ergs^-1 cm^-2) is the average interstellar radiation field of our Galaxy in the range [6 eV, 13.6 eV] <cit.>.].For stellar sources, we use black body spectra with different effective temperatures T_eff. In terms of solar luminosity (), OB stars have typical luminosities ranging between 10^3 and 10^5. Then, the effective temperature is given byT_eff = ( L4π R_⋆^2 σ_sb)^1/4,where σ_sb is the Stefan-Boltzmann constant, R_⋆ is the star radius and L is the bolometric luminosity. We compute R_⋆ through the mass-luminosity and radius-luminosity relations by <cit.>, which for an OB star giveR_⋆ = 1.33( L1.02)^0.142,whereis the solar radius. Integrating the black body spectrum in the FUV band, typical values of the FUV flux are G_0=10^2 - 10^4 for gas at 0.3 pc from sources with luminosities in the range L=10^3 - 10^5. In the same way, we integrate the spectrum for hν≥ 13.6 eV to obtain the ionizing flux. In the case of quasars, the fundamental difference is the wide extension of the spectrum to the X-rays, so that ionizing radiation is much more intense in this case. An analytical expression for the ionizing flux can be found with the same approach as in <cit.>, thus obtaining that the specific (ionizing) luminosity for ν > ν_L≃ 3.3× 10^15 Hz is:L_ν = 6.2× 10^-17(νν_L)^α-2 (Lergs^-1) ergs^-1Hz^-1,where α= 0.5 for a radio-quiet quasar <cit.>. We assume eq. <ref> to be valid for energies below the cut-off value E_c=300 keV <cit.>. Furthermore, we can easily infer a relation between the bolometric and FUV luminosity, setting ν = ν_L in eq. <ref>. The spectrum is almost flat in the FUV band, with typical values G_0≃ 10^3 - 5 at 1 kpc, for L=10^45-47ergs^-1.We investigate the evolution of clumps irradiated by stars or quasars, with L as the only free parameter determining the flux in the bands we are interested in.§.§ Clump structure In this paper a clump is modeled as a Bonnor-Ebert (BE) sphere <cit.>, which is isotropically affected by an external impinging radiation field. Given the clump mass, the clump temperature and the confining pressure, the BE sphere model allows to compute the radial density profile inside the clump and its radius. In Fig. <ref> we show the density profile for clumps of different mass, with the same temperature T=10 K and a confining pressure P=10^-11 erg cm^-3 (corresponding to a confining medium with T_ = 100 K and n_ = 10^3). The clumps have different radii and same outer density, set by the pressure equilibrium between the clump and the surrounding gas. The density increases towards the centre, with a steeper profile for larger values of the mass. A clump undergoes a collapse if its mass is larger than the BE mass:M_be≃ 1.18 c_s^4√(P_0 G^3),where P_0 is the confining pressure, c_s is the isothermal sound speed and G is the gravitational constant. We are considering only the thermal contribution to pressure, not accounting for turbulent and magnetic pressure. We underline that, apart from the use of a BE density profile, gravity is not included in the hydrodynamical equations for the clump evolution presented in Sec. <ref>.Given the clump density profile, we assume that the impinging radiation induces a shell-like structure, before any dynamical response of the gas to the photo-heating occurs (sudden heating approximation, see Fig. <ref>). The FUV radiation is responsible for the formation of an atomic layer (HI shell). The more energetic part of the spectrum partially or totally ionizes an outer shell (HII shell), depending on the intensity of the source. This sets up the initial condition for the subsequent hydrodynamical evolution of the clump.The sudden heating approximation means that radiation dissociates and ionizes particles and heats the gas to its final temperature instantaneously, while the clump shape is unaltered. This situation is often referred to as a R-type ionization front <cit.>. To justify this assumption, we compare the sound-crossing time-scale t_cross with the ionization timescale t_i and the heating time-scale t_h. The Strömgren theory adapted for plane geometry (which can be assumed when the radius of the clump is much smaller than the distance from the source) allows to compute the HII shell thickness, δ_ (t)=δ_ (1-e^-nα_bt),where n is the gas number density, and α_b is the case B recombination coefficient<cit.>. Thus, we have t_i = 1/nα_b. On the other hand, the sound-crossing time-scale is t_cross = r_c/c_s where r_c is the clump radius and c_s∼√(k_bT/m_p) is the sound speed. Plugging in typical values, it is easily seen that the condition t_i≪ t_cross is always satisfied for physically reasonable values of r_c (0.01- 1 pc) and c_s (0.1 1 km s^-1 in the cold phase). Regarding the gas heating time-scale (t_h), a simple estimate gives us:t_h = k_B T_f Γ(T_f),where k_B is the Boltzmann constant, T_f is the final gas temperature and Γ(T_f) is the heating rate (in erg/s), mainly due to photoionization. To give some examples, Γ / n varies from ∼ 10^-23 erg cm^3 s^-1 at a distance of 1 pc from a star with L=10^3, to ∼ 10^-16 erg cm^3 s^-1 at 0.5 kpc from a quasar with L=10^47 erg s^-1 <cit.>. The result is that for any value of r_c and c_s of interest, the condition t_h≪ t_cross holds. In what follows, we discuss how we compute the thickness and the temperature of each shell in the clump. The thickness of the HI shell (δ_) is defined as the depth at which hydrogen is found in molecular form. <cit.> find how such depth (expressed as hydrogen column density) scales with the gas density and the FUV flux:N_H∝ n^-4/3G_0^4/3,and that the thickness for n=2.3 × 10^5 cm^-3 and G_0=10^5 is δ_ =1.5× 10^16 cm. Then from eq. <ref> we find for the shell thicknessδ_= 0.034(n10^5cm^-3)^-7/3 ( G_010^5)^4/3pc.<cit.> outline that the PDR temperature is rather constant before it drops to the low values of the molecular core. They plot the temperature for different values of the density and the FUV flux, and we use a fit of their model to estimate the temperature of the HI shell. The outer shell presents a partial or total ionization, depending on its density and the intensity of the impinging radiation field. We compute the equilibrium temperature as a function of depth into the shell by balancing photoionization heating andrecombination cooling, line cooling and bremsstrahlung. The presence of a radiation field alters the heating and cooling rate: 1) the ionized fraction of each species is modified and thus the cooling rate by line emission is changed accordingly; 2) injection of photoionized electrons in the gas provides an extra heating term. Approximate heating and cooling functions, assuming collisional equilibrium but non-zero radiation field, are provided by <cit.>. We assume a fiducial value for metallicity, i.e. the mass fraction of elements heavier than helium, of Z=0.02 <cit.>, noticing that metals are important for the energetics of the gas, but their contribution to its dynamics (determined by gas pressure) is negligible. Moreover, we account for Compton heating, which we expect to be important for hard radiation fields:H_C = σ_T Fm_e c^2 (⟨ hν⟩ - 4k_bT),where σ_T is the Thomson cross section, m_e is the electron mass, F is the total flux and ⟨ hν⟩ is the average photon energy beyond the Lyman limit.Once the temperature profile is computed, we obtain the ionization profile by balancing photoionization, collisional ionization and recombination:γ(T) n_e n_p + n_h∫_ν_L^∞F_νhνe^-τ_νa_ν(T) dν = α_b(T) n_e n_p ,where n_e, n_p and n_h are the electron, proton and neutral hydrogen density, respectively; F_ν is the specific flux from the source, τ_ν is the optical depth, a_ν and γ are the photoionization cross section and the collision ionization coefficient, respectively <cit.>. In Fig. <ref> we plot the ionization fraction x=n_e/n=n_p/n, as a function of the depth into the clump, for different source luminosities. The ionization profile varies smoothly throughout the HII shell, between the edge of the clump and the PDR, both in the stellar and quasar case. Nevertheless, the region where x is varying is of the order of 10^-5 pc, which is negligible with respect to typical clump radii (0.01 - 1 pc). Then we adopt a reference value for the HII shell thickness δ_, computed with the approximation of a sharp boundary between ionized and phase, in the same way as done for the Strömgren radius for a stellar HII region. Further assuming that the clump radius is much smaller than the distance from the source and that it is illuminated isotropically, the HII shell depth isδ_ = 1x^2_maxn^2 α_b(T)∫_ν_L^∞F_νh ν dν,where x_max is the maximum ionization fraction, at the edge of the clump. The thickness δ_ is shown with dashed lines in Fig. <ref>.We restrict our analysis to clumps where the shell thickness is much smaller than the molecular core radius. This allows us to determine the densities of the HI and the HII shells by using the outer density of the BE sphere. Furthermore, it simplifies our calculations, because we can consider separately the evolution of the shells and the core, since the dynamical time-scale of the formeris much shorter than the core one. In Fig. <ref> we plot the ratio of the HI shell thickness (left panels) and the HII shell thickness (right panels) to the total clump radius r_c as a function of the clump mass, for stars and quasars with different luminosities. The atomic shell is always thicker than the ionized shell, showing a large self-shielding effect of the ionized gas. The typical masses of clumps are different for stars and quasars, and we consider only masses smaller than the BE mass for collapse. The distances of clumps from the source are fixed in the two scenarios, and are reasonable for molecular gas engulfed by an expanding stellar HII regions (the distance scale is given by the Strömgren radius) and clumps forming in quasar outflows (critical radius given in eq. <ref>).Clumps presenting a ratio δ_ /r_c=1 are completely dissociated on a time-scale t_i, and the analysis of this paper restricts to clumps where δ_≪ r_c. From the plot, we see that such condition is usually verified and breaks only in the stellar case for small clumps (M <0.01) and very intense sources (L≃ 10^5 ).§.§ Shock dynamics inside the clump Having set the initial conditions on a clump, i.e. a core-double shell structure, now we can study its dynamical evolution for t>0. The different layers in the clumps have different pressures, so that a shock or rarefaction waves originate, enforcing a continuous value of pressure and velocity across the contact discontinuity between two layers. The cooling time-scale of a gas at temperature T ist_cool = k_B TΛ(T),with Λ being the cooling function given by <cit.> for molecular gasand by <cit.> for PDRs. For the range of temperatures and n, r_c and c_s values of interest here, t_cool≪ t_cross. Thus the fluid motion and the propagation of any disturbance in the gas (as shock and rarefaction waves) can be safely considered as isothermal processes. A qualitative diagram of the possible outcomes at an arbitrary discontinuity is shown in Fig. <ref>. In the situation considered in the upper inset, a gas has a pressure P_r to the right of an interface, and a pressure P_l to the left (with P_l>P_r). The velocity to the right is v_r=0, while we consider different values v_l,1, v_l,2 and v_l,3 for the velocity to the left. The solid lines connect to the initial state all the possible final states of the gas, when it is crossed by a rarefaction wave (RW) or a shock wave (SW). For example, the points on the blue line represent the possible final states of the gas to the right when it is crossed by a rarefaction wave.The solution of the discontinuity problem is obtained when the lines departing from the two initial states of the gas to the left and to the right intersect, since the final values of P and v must be the same across the discontinuity. This also determines which kind of wave is required, i.e. a SW or a RW. The solution of the problem for given values of the initial pressure, density and velocity across the discontinuity is obtained numerically, imposing the final pressure and velocity to be continuous. The post-shock values are obtained solving the isothermal Rankine-Hugoniot conditions <cit.> ρ_0v_0 =ρ_1v_1 ρ_0v_0^2+P_0 =ρ_1v_1^2+P_1T_0= T_1 where the subscript 0 is used for pre-shock values and the subscript 1 for post-shock values, with v velocity in the shock front frame. Rearranging the equations <ref>, it is possible to write the following relations ρ_1 =ρ_0ℳ^2 P_1 =P_0ℳ^2 v_1 =v_0/ℳ^2 with ℳ = v_0/c_s being the shock Mach number. On the other hand, rarefaction waves are not discontinuities and values of flow variables across such waves are obtained following <cit.> and adapting the calculations to the isothermal case. Consider a wave originating at x=0 and propagating toward x>0, such that the final velocity after the wave has completely passed is v_f=-U. The profile between the wave head, moving at the initial sound speed c_s,0 in the gas, and the wave tail, moving at speed v_tail=c_s,0 - (γ+1)U/2, is v(x) =-(c_s,0-x/t ) ρ(x) =ρ_0exp(x/c_s,0t-1 )P(x) =P_0exp(x/c_s,0t-1 ) where ρ_0 and P_0 are the values of density and pressure before the rarefaction has passed.Since shock waves are discontinuities, an interaction between two shocks can be treated as an arbitrary discontinuity between post-shock values of flow variables. To simplify our analysis, we also consider interactions involving rarefactions as discontinuities, by accounting only for the post-rarefaction values of flow variables. In App. <ref> we compare this approach with a numerical solution of the fluid dynamics equations, showing that the two results differ negligibly.To compute the shock speed inside the clump, we have to account for the spherical geometry and for the density gradient given by the BE profile. Following <cit.>, the flow equations can be written as∂_t ρ + ∂_r(ρ v) + ρ v ∂_r A(r)A(r) = 0∂_t v + v ∂_r v + 1ρ∂_r P - 1ρ_0(r)∂_r P_0(r)=0 where r is the radial coordinate, A(r)=4π r^2 in the spherical case, ρ_0(r) and P_0(r) are the initial density and pressure profiles for a BE sphere. Eq. <ref> and eq. <ref> can be combined to give the equivalent equation valid along the curves dr/dt = r+c_s in the (r,t) plane (called the C_+ characteristics):dP + ρ c_s d v + ρ c_s^2 vv+cA'(r)A(r) - ρ c_sv + c_s1ρ_0(r) P'_0(r)=0,where the prime denotes the derivative with respect to r.According to <cit.> the shock trajectory in the (r,t) plane is approximately a C_+ characteristic, so that eq. <ref> can be applied along the shock. Then we can write eq. <ref> as a function of the Mach number ℳ, substituting the post-shock values P, ρ, v from eq. <ref>:dℳdr = - 12ℳ^2ℳ^2 - 1A'(r)/A(r) + 12ℳ^3ℳ+1P_0'(r)P_0(r),which is a differential equation for ℳ as a function of r. In Fig. <ref>, the solid lines show the the numerical solution of eq. <ref> for a molecular clump with mass M=0.1 and radius r_c=0.025pc, for different values of the initial Mach numbers ℳ_0 of the shock at the edge of the clump, assuming an isothermal shock. For comparison, the dashed line is the classical analytical solution obtained by <cit.> for a homogeneous density distribution, and in the limit of a strong adiabatic shock:ℳ(r) = ℳ_0 (r_0r)^n(γ),where r_0 is the radius of the bubble, and n(γ) is an exponent depending on the adiabatic coefficient γ (e.g. n(5/3)≃ 0.543 for monoatomic gas and n(7/5)≃ 0.394 for diatomic gas). As opposed to Guderley solution, the isothermal shock speeds up considerably only at a smaller radius. After the shock wave has reached the centre, a reflected shock will travel outwards. The velocity as a function of radius has the same profile of the focusing shock.§ RESULTSIn our model, a clump exposed to UV radiation develops an ionized PDR (iPDR) at its surface. First, we inspect the qualitative behaviour of the structure simply using the arbitrary discontinuity criterion. Fig. <ref> shows a diagram of shock and rarefaction waves propagating inside the clump because of the pressure difference between adjacent layers.The HII shell pressure (P_) is higher both than the pressure of the HI shell (P_) and the pressure of the confining ICM (P_), hence two rarefaction waves cross the HII shell, originating from its edges. Since δ_≪ r_c, the evolution of the HII shell has a much shorter time-scale than the clump evaporation time. The rarefaction waves which propagate into it interact and reflect at its edges, determining a complex density profile. Nevertheless, the global effect is that the HII shell expands decreasing its density, eventually becoming completely transparent to the ionizing radiation (i.e. the mean free path of photons is much larger than the shell thickness) . On the other hand, P_ is lower than P_, but higher than P_h_2. Thus, a shock is driven from the HII shell into the HI shell, and a rarefaction wave propagates from the discontinuity with the core (see Fig. <ref>). Once the shock has crossed the HI shell, it reaches the core surface and speeds up its contraction. As a result, the inner boundary of the HI shell moves faster than the outer boundary, and the HI shell is also expanding and becoming transparent on a time-scale shorter than the core evolution time-scale. The cold (T_2 ≃ 10- 100 K) molecular core is compressed because of the shock wave originating at the discontinuity with the atomic shell and propagating towards the centre. In addition, the shock wave originating at the HII/HI boundary reaches the core surface and catches up with the shock already propagating in the core, resulting in a single stronger converging shock wave. The shock wave is reflected at the centre of the clump, and eventually gets back to the core edge. The contraction is almost halted, and since the core has a much higher density than the surrounding medium, it starts to expand. The expansion velocity v_exp is computed considering the discontinuity between the core compressed by the reflected shock wave and the ICM at rest, using the arbitrary discontinuity algorithm.We have explicitly verified that the core is so dense (n≃ 10^5-6 cm^-3) during the contraction phase that the FUV radiation penetrates to a negligible depth with respect to its radius. Thus we can ignore photoevaporation during the contraction phase. When the clump starts expanding, we have computed for each time t the thickness δ_ (t) of an HI shell (see eq. <ref>) for the corresponding core gas density. We get the core radius at t by subtracting δ_ (t) to the radius R(t) = R_0 + v_expt (R_0 is the core radius at the end of the shock-contraction phase).§.§ Stellar caseAs a first application of our analysis, we consider molecular clumps photoevaporating because of stellar radiation. We consider a cold clump (T_2 = 10 K) located in the surrounding of a star, embedded in an atomic region with density n_at = 10^3 cm^-3. Then we assume that the expanding HII region of the massive star engulfs the clump (the density of the HII region gas is n_ = 10, see Tab. <ref>), and we apply the machinery we developed in Sec. <ref>. We consider stars with bolometric luminosities L=1× 10^3, L=1× 10^4, L=5× 10^4 and L=1× 10^5. We assume the clump is located 0.3 pc from the source, since this distance is smaller than the Strömgren radius for every star in our set (for the fainter star the Strömgren radius is R_Str≃ 0.75 pc for a gas density n=10). The BE masses for the collapse of molecular clumps at 10 K are around few tenths of solar masses, and for each luminosity we consider only clumps with mass below that limit.The time evolution of the molecular core radius is shown in Fig. <ref> (left panel), where a clump of initial mass M=0.1 is exposed to the stellar radiation field for the different luminosities considered. The radius has a similar evolution for the different luminosities, with a shorter time-scales for larger luminosities. Consider for example the (brown) curve for L=10^4. A clump with mass 0.1 at the distance of 0.3 pc has an initial radius of 0.023 pc when it is in pressure equilibrium with the ICM. In the shock contraction phase, the radius reduces to 6× 10^-4 pc in about 6000 yr because of the shock waves driven by the heated HI and HII shells. Then the expansion phase follows, and the core expands allowing the impinging radiation to penetrate and dissociate the molecules. This occurs significantly after the radius reaches its maximum value r≃ 0.025 pc. While the contraction phase has almost the same duration for the tracks of the three more intense sources, we see that it takes more time for the 10^3 star. In fact for this source the ionization fraction is low (see Fig. <ref>), since the temperature of the HII shell is only about 900 K. The shock driven by the HII shell is weak for this star, and needs more time to reach the centre of the core.The lifetime of a clump (t_c) is defined as the time when the core radius goes to zero. In Fig. <ref> (right panel) we show t_c as a function of the clump mass and the source luminosity, at the same distance to the source (0.3 pc). <cit.> compute the lifetime of clumps located in a stellar PDR, in the absence of ionizing radiation. They account for photoevaporation by assuming that the clump continuously loses mass at a rated M d t = - 4πρ_c r^2_c(t)c_,where ρ_c is the mean mass density of the clump, and c_ the sound speed in the PDR of the clump. This implies that the clump loses mass also in the shock-compression phase. 02cit do not account for the shock reflection at the centre of the clump, so that the core does not expand after the compression phase. On the other hand, in our treatment, photoevaporation is negligible while the clump is being compressed, and the shock reflection allows for the following expansion of the core. As a result, radiation is allowed to penetrate and dissociate the molecules only when the gas is sufficiently expanded and diluted. Furthermore, 02cit findthat under certain initial conditions[According to <cit.>, a clump undergoes a shock-compression only if its mean column density isnr_c < 2.7×10^21cm^-2 (c_/c_2)^3. ] there is no shock-compression, since the shock suddenly stalls just after its formation, and the clump directly expands and photoevaporates. In our treatment, we do not recover this scenario, since we always allow the shock to reach the centre of the core. Magnetic and turbulent contribution to pressure are included by 02cit, but not in this work. In Fig. <ref> we compare our predictions for the photoevaporation time (without ionizing radiation) with those from the 02cit model. A range of clump masses between 0.01 and the BE mass is considered, at a distance of 0.3 pc, for three different source luminosities, A modification in our code for iPDR is required, since the HII shell is not present when ionizing radiation is absent. In the 02cit model, the 10^4 and the 10^5 sources induce a shock-compression in the clump, while instead they predict an initial expansion for the 10^3 case. The evaporation time-scale for the low-L case differs by an order of magnitude with respect to the other two. In our model we do not find such dichotomy, and the evaporation time smoothly increases with L. However, the lifetimes are in agreement within a factor of 2 with those found in 02cit.Finally, we make a comparison between the evaporation times obtained with our full iPDR model (see Fig. <ref>) and our model without ionizing radiation, i.e. with no HII shell (see Fig. <ref>). Clump lifetimes are always shorter when we consider the ionizing part of the spectrum, generally by a factor between 2 and 4 depending on clump mass and luminosity. This behaviour is expected, since the outer shell of the clump is heated to an high temperature and a stronger and faster shock propagates into the clump, decreasing its evolution time-scale.§.§ Quasar caseWe now describe the evolution of a clumps forming in the ionized outflows of quasars. We choose 1 kpc as a typical distance of a molecular clump from the source, that is of the order of the critical radius (eq. <ref>). The mass of clumps has been estimated by <cit.> to be around 8600, thus we consider masses up to 10^4. The ICM is the hot ionized medium of the wind, with temperature T_ ¸≃ 2.2× 10^7 K, and density n_≃ 60 cm^-3 at 1 kpc from the source. As explained in Sec. <ref>, we assume that the clump is in pressure equilibrium with the ICM until its temperature reaches 10^4 K (when the clump gas is still in atomic form). Afterwards, the gas turns into molecular form (T_2=100 K) in a very short time-scale, so that the density profile remains unchanged with respect to the 10^4 K BE sphere. We apply our model to quasars with bolometric luminosities L=1× 10^45ergs^-1, L=1× 10^46ergs^-1, L=5× 10^46ergs^-1 and L=1× 10^47ergs^-1, with a spectral energy distribution given by eq. <ref>.The evolution of the molecular core radius of the clump is shown in the left panel of Fig. <ref>, while the right panel shows the lifetime of a clump at a distance of 1 kpc from a quasar, as a function of clump mass and source luminosity. Similarly to clumps around stars, the core radius presents a contraction phase followed by an expansion phase, where the core is dissociated and then ionized. Notice that the evaporation time is about ten times longer than in the stellar case, although the clump mass considered is about 10^4 times larger. This is because both the FUV and ionizing radiation field are much more intense since the quasar spectrumextends to very high energies (see Table <ref> for reference values of the fluxes). This implies a higher temperature of the HII and HI shells, and a stronger compression ratio of the shock waves originating at the discontinuities.Observations have detected molecular gas only up to a maximum distance of few kpc from quasars <cit.>. Photoevaporation has sometimes been invoked as an explanation for such limited extension. We have slightly modified our code toaccount for the fact that radiation intensity decreases as the clump moves away from the source, being carried by the outflow.have shown that molecular clumps forming at the base of the adiabatic outflow are ablated in a short time because of the friction by hot flowing gas. Therefore, we analyse the alternative scenario in which clumps form within the outflow, so that they are at rest with the outflow and they are not subject to a strong acceleration. In this way our model is able to predict the distance travelled by a clump during its lifetime, and we can compare this length with observations of molecular outflow extensions. We consider a subset of active galactic nuclei (AGNs) listed in 14cit (Table <ref>). In Fig. <ref> we plot the outflow extension obtained with our photoevaporation model as a function of source luminosity. We study clumps with 90% of their BE mass for collapse (i.e. the most massive clumps that do not collapse), forming at the contact discontinuity (CD) between the quasar wind and the surrounding ISM. According to <cit.> model, the initial position of the CD coincides with the critical radius R_c given in eq. <ref>, and it moves at a speedv_cd = 875σ_200^2/3km s^-1in the energy-driven phase. In Fig. <ref> the dotted lines correspond to the position of the CD at different times, while the shaded regions represent the maximum distance that clumps with a range of masses (0.1-0.9 times their Bonnor-Ebert mass) can travel before photoevaporating. The observed extension of the outflows in the considered quasar sample exceed the maximum distance travelled by clumps before they are photoevaporated, if they form at R_c. This implies that there is no mechanism more efficient than photoevaporation in destroying molecular clumps. On the other hand, the existence of outflows with large extensions (up to 1 kpc) suggests that clumps continue to form within the outflow, when the CD moves outwards from its initial position. This can bee see from the other two cases shown in Fig. <ref> where we consider alsoclumps formed at a later times when the CD has moved to a radius R_c +v_cdΔ t, with Δ t=0.3 ,0.6 Myr. It appears that such delayed formation via thermal instabilities in the outflow can match the observed extensions.§ CONCLUSIONSWe have studied the evolution of molecular clumps exposed to radiation having both a far ultraviolet (FUV) and an ionizing component, determining the formation of an ionization/photodissociation region (iPDR) at the surface of clumps. The cases of a clump forming in the surroundings of an OB stars and a clump forming in the fast outflow of a quasar are studied separately. The clump is assumed to be an isothermal Bonnor-Ebert sphere with a mass lower than the critical mass for collapse. We assume a sudden heating scenario, inducing a shell structure in the clump, and then we analyse the evolution of its radius and density profile as a function of time, finally computing the clump lifetime (i.e. the time at which the molecular gas in the clump is completely dissociated). The clump evolution is solely determined by two parameters: its mass, M, and the bolometric luminosity L of the source. We show that the pressure difference between adjacent layers causes the propagation of shock and rarefaction waves into the clump. The core shrinks until the shock wave hits the centre and reflects back, while the external layers expand and become eventually transparent to radiation. The dense core is thus surrounded by a diluted medium and it starts an expansion phase. As a result, the core density decreases and the radiation propagates in the interior, progressively evaporating the whole core. In this analysis we have not included gravity effects which could limit the expansion following the shock-contraction phase of clumps. Gravity may also play a role for the clumps that become gravitationally unstable during the contraction phase, possibly triggering star formation <cit.>.In the stellar case, we find that a higher luminosity speeds up considerably the shock-contraction phase: clumps of 0.1at 0.3 pc from the source evaporate in 0.01 Myr for the brightest star considered (10^5), while it takes 0.06 Myr in the case of the 10^3 star. Indeed, the radiation from the fainter star is not able to completely ionize the surface layer of the clump, resulting in a lower pressure of the HII shell and a weaker shock-induce contraction phase. Our model agrees within a factor of 2 with the <cit.> model, in the case of clumps embedded in the PDR of a massive star and in absence of ionizing radiation. The main difference between the two models is the evaporation channel. In 02cit evaporation is due to a constant mass flow from the clump surface; in our model the clump evaporates as a consequence of the expansion and dilution driven by the reflected shock wave. We also notice that, in the absence of ionizing radiation,evaporation times are always longer by a factor 2-4 with respect to the full iPDR model including both FUV and ionizing radiation. Therefore, considering ionizing radiation is important, since the evolution history of clumps is significantly modified.In the context of high-redshift galaxies, this is significant for far infrared (FIR) emission, as [CII]. Indeed, most of the [CII] emission from high-redshift galaxy seems to be due to molecular clumps <cit.> and because of the high radiation field observed in such galaxies <cit.>, photoevaporation can play an important role. While <cit.> analyses the effect of clump photoevaporation using a time evolution based on 02cit, we argue that shorter photoevaporation time-scales obtained with our iPDR model could further affect the detectability ofhigh-redshift galaxies. However, we underline that other effects are also important: the contrast with cosmic microwave background (CMB) attenuates the observed FIR emission for redshift z≳ 5 <cit.>, which is relevant for low density gas <cit.>, while CO destruction by cosmic rays may enhance [CI] and [CII] emission <cit.>.The evolution in the quasar context is characterized by a similar behaviour. The duration of the contraction phase is roughly constant for different L, since all the quasars in the set are able to completely ionize and heat to about 10^4 K the outer shell of the clump. We obtain evaporation times of 0.21 Myr for the 10^47ergs^-1 quasar and 0.51 Myr for the 10^45ergs^-1 quasar. With comparison to the stellar case, the evaporation times are longer only by a factor ∼ 10, even though the clumps in the quasar case are ≃ 10^4 times more massive. This is consistent with the higher UV fluxes produced by quasars in spite of the larger spatial scales of the problem (see Tab. <ref>). Applying our algorithm to clumps embedded in quasar outflows, we have been able to predict the outflow extension. This is set by the maximum distance travelled by clumps before photoevaporating, assuming that they form at the contact discontinuity (CD) between the quasar wind and the ISM. We find that the observed molecular outflow extensions are always larger than the distance travelled by clumps forming at the initial position of CD, but they are compatible with clumps forming at the CD with a time delay Δ t≃ 0 - 0.6 Myr after the outflow has entered the energy-driven phase. Therefore, we argue that: * photoevaporation must be a crucial mechanism involved in the evolution of molecular gas structures in quasars, since none of the observed outflows has a smaller extension than what predicted with our photoevaporation model;* clumps need to form continuously within outflows, when the CD has moved farther from the quasar, in order to explain the most extended outflows.A more comprehensive analysis of quasar outflows should consider a distribution of clump masses, the contribution of scattered light in a clumpy medium and the possible occurrence of star formation within the outflow <cit.>. style/mnras § COLLISION OF RAREFACTION WAVESThe pressure and density profile of a fluid crossed by a rarefaction wave is given by eq. <ref> (assuming an isothermalprocess). Since the profile is not a discontinuity, as for shock waves, collisions between rarefaction waves cannot be studied analytically. The MacCormack method <cit.> is a discretization algorithm for solving numerically hyperbolic differential. We apply this method to find a solution of fluid dynamics equation, where the initial condition is set by two approaching rarefaction wave, travelling in the opposite direction. In Fig. <ref> we have simulated the collision of the two rarefaction waves generating at the edges of the HII shell of a clump exposed to the UV radiation of a quasar (see Fig. <ref> and description in Sec. <ref>). The two rarefaction waves are set so that they travel 3× 10^-4 pc before colliding, a distance of the order of the HII shell thickness (see Fig. <ref>). The lines in figure represent the pressure and density profile at different intervals of time. The total length considered (3× 10^-2 pc) has been divided in a 10^5-cell grid, while the timesteps are chosen according to the Courant condition <cit.>. The result of the collision is two rarefaction waves travelling in the opposite direction, with an equilibrium pressure in the central region P_mc≃ 1.0× 10^-7ergcm^-3. The same prediction for the outcome can be obtained with the arbitrary discontinuity algorithm discussed in Sec. <ref>. We approximate the rarefaction waves to jumps in the flow variables (P, ρ, v), i.e. we consider a contact discontinuity between the post-rarefaction states of the fluid. We obtain for the equilibrium pressure a value P_ad≃ 9.8× 10^-8ergcm^-3,in agreement with the value found previously within a few percent. Therefore, the discontinuity approximation introduces a small error, but it is more convenient from a computational point of view. | http://arxiv.org/abs/1707.08574v1 | {
"authors": [
"Davide Decataldo",
"Andrea Ferrara",
"Andrea Pallottini",
"Simona Gallerani",
"Livia Vallini"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170726180001",
"title": "Molecular clumps photoevaporation in ionized regions"
} |
Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks Srdjan Ostojic December 30, 2023 ==========================================================================================================It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems.Data assimilation techniques have been designed to exploit this phenomena, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors.Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the Assimilation in the Unstable Subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace.Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with non-negative exponents.Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations and the evolution of forecast error in linear models with additive model error.We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision and observational accuracy.Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors.Furthermore, we numerically explore the relationship between observational design, dynamical instability and filter boundedness.Additionally, we include a detailed introduction to the Multiplicative Ergodic Theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous re-injection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large.In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors. Kalman filter, data assimilation, model error, Lyapunov vectors, control theory 93E11, 93C05, 93B07, 60G35, 15A03 § INTRODUCTION The seminal work of Lorenz <cit.> demonstrated that, even in deterministic systems, infinitesimal perturbations in initial conditions can rapidly lead to a long-term loss of predictability in chaotic, physical models.In weather prediction, this understanding led to the transition from single-trajectory forecasts to ensemble-based, probabilistic forecasting <cit.>.Historically, ensembles have been initialized in order to capture the spread of rapidly growing perturbations <cit.>.Data assimilation methods have likewise been designed to capture this variability in the context of Bayesian and variational data assimilation schemes; see e.g., Carrassi et al. for a recent survey of data assimilation techniques in geosciences <cit.>.The ensemble Kalman filter, particularly, has been shown to strongly reflect these dynamical instabilities <cit.>, and its performance depends significantly upon whether these rapidly growing errors are sufficiently observed and corrected. The Assimilation in the Unstable Subspace (AUS) methodology of Trevisan et. al. <cit.> has provided a robust, dynamical interpretation of these observed properties of the ensemble Kalman filter.For deterministic, linear, Gaussian models, Trevisan et al. hypothesized that the asymptotic filter error concentrates in the span of the unstable-neutral backward Lyapunov vectors (BLVs), and this has recently been mathematically proven.Gurumoorthy et. al. <cit.> demonstrated that the null space of the forecast error covariance matrices asymptotically contain the time varying subspace spanned by the stable BLVs.This result was generalized by Bocquet et. al. <cit.>, proving the asymptotic equivalence of reduced rank initializations of the Kalman filter with the full rank Kalman filter: as the number of assimilations increases towards infinity, the covariance of the full rank Kalman filter converges to a sequence of low rank covariance matrices initialized only in the unstable-neutral BLVs.The convergence of the Kalman smoother error covariances onto the span of the unstable-neutral BLVs, and stability of low rank initializations, was established by Bocquet & Carrassi <cit.>; this latter work also numerically extended this relationship to weakly nonlinear dynamics and ensemble-variational methods.The works of Bocquet et al. <cit.> and Bocquet & Carrassi <cit.> relied upon the sufficient hypothesis that the span of the unstable and neutral BLVs remained uniformly-completely observed.This hypothesis has recently been refined to a necessary and sufficient criterion for the exponential stability of continuous time filters, in perfect models, in terms of the detectability of the unstable-neutral subspace <cit.>.The present study is concerned with extending the limits of the results developed in deterministic dynamics (perfect models), now to the presence of stochastic model errors.This manuscript and its sequel <cit.> seek to: (i) determine the extent to which stable dynamics confine the uncertainty in the sequential state estimation problem in models with additive noise, and (ii) to use these results to interpret the properties, and suggest design, of ensemble-based Kalman filters with model error.This manuscript studies the asymptotic properties of the full rank, theoretical Kalman filter <cit.>, and the unfiltered errors in the stable BLVs.The sequel <cit.> utilizes these results to interpret filter divergence for reduced rank, ensemble-based Kalman filters.In <ref> we present a detailed introduction to the BLVs.In section <ref> develop novel bounds on the forecast error covariance, describing the evolution of uncertainty as the growth of error, due to dynamic instability and model imprecision, with respect to the constraint of observations.Together, the rate of dynamic instability and the observational precision form an inverse relationship which we use to characterize the boundedness of forecast errors.In <ref> and <ref>, we prove a necessary criterion for filter boundedness in autonomous and time varying systems: the observational precision, relative to the background uncertainty, must be greater than the leading instability which forces the model error.Our results derive from the bounds provided in <ref> for autonomous dynamics and <ref> for time varying systems.An important consequence is that under generic assumptions, forecast errors in the span of the stable BLVs remain uniformly bounded independently of filtering.Described in <ref>, this extends the intuition of AUS now to the presence of model errors: filters need only target corrections to the span of the unstable and neutral BLVs to maintain bounded errors.However, the intuition of AUS needs additional qualifications when interpreting the role of model errors in reduced rank filters. Unlike perfect models, uncertainty in the stableBLVs does not generically converge to zero as a consequence of re-introducing model errors.Moreover, while stability guarantees that unfiltered errors remain uniformly bounded in the stable BLVs, the uncertainty may still be impractically large due: even when a Lyapunov exponent is strictly negative, positive realizations of the local Lyapunov exponents can force transient instabilities which strongly amplify the forecast uncertainty.The impact of stable modes on forecast uncertainty differs from similar results for nonlinear, perfect models by Ng. et. al. <cit.>, and Bocquet et. al. <cit.>, where the authors demonstrate the need to correct stable modes in the ensemble Kalman filter due to sampling errors induced by nonlinearity.Likewise, this differs from the EKF-AUS-NL of Palatella & Trevisan <cit.>, that accounts for truncation errors in the estimate of the forecast uncertainty in nonlinear models.In <ref>, we derive the mechanism for the transient instabilities amplifying perturbations as a linear effect in the presence of model errors.We furthermore provide a computational framework to study the variance of these perturbations. In <ref>, we study the filter boundedness and stability criteria of Bocquet et al. <cit.> and Frank & Zhuk <cit.> in their relation to bounding forecast errors in imperfect models.Likewise, we explore their differences in the context of dynamically selecting observations, similar to the work of Law et al. <cit.>.With respect to several observational designs as benchmarks, we numerically demonstrate that the unconstrained growth of errors in the stable BLVs of high variance can be impractically large compared to the uncertainty of the full rank Kalman filter. These results have strong implications for ensemble-based filtering in geosciences and weather prediction, where ensemble sizes are typically extremely small relative to the model dimension.In perfect models, an ensemble size large to correct the small number unstable and neutral modes might suffice.However, our results suggest the need to further increase the rank of ensemble-based gains.The significance of this result for ensemble-based Kalman filters and their divergence is further elaborated on in the sequel <cit.>. § LINEAR STATE ESTIMATION The purpose of recursive data assimilation is estimating an unknown state with a sequential flow of partial and noisy observations; we make the simplifying assumption that the dynamical and observational models are both linear and the error distributions are Gaussian.In this setting, given a Gaussian distribution for the initial state, the distribution of the estimated state is Gaussian at all times.Formulated as a Bayesian inference problem, we seek to estimate the distribution of the random vector _k ∈ℝ^n evolved via a linear Markov model,_k = _k_k-1 + _k , with observations _k ∈ℝ^d given as_k = _k _k + _k.The model variables and observation vectors are related via the linear observation operator _k: ℝ^n ↦ℝ^d.Let _n denote the n × n identity matrix.We denote the model propagator from time t_l-1 to time t_k as _k:l≜_k ⋯_l, where _k:k≜_n.For all k, l ∈ℕ, the random vectors of model and observation noise, _k, _l∈ℝ^n and _k,_l∈ℝ^d, are assumed mutually independent, unbiased, Gaussian white sequences.Particularly, we define𝔼[_k_l^] = δ_k,l_kand𝔼[_k_l^] = δ_k,l_k,where 𝔼 is the expectation, _k∈ℝ^d× d is the observation error covariance matrix at time t_k, and _k ∈ℝ^n× n stands for the model error covariance matrix. The error covariance matrix _k can be assumed invertible without losing generality.For simplicity we assume the dimension of the observations d≤ n will be fixed.For two positive semi-definite matrices,and , the partial ordering is defined ≤ if and only if - is positive semi-definite. To avoid pathologies, we assume that the model error and the observational error covariance matrices are uniformly bounded, i.e., there are constants q_inf, q_sup, r_inf, r_sup∈ℝ such that for all k,≤ q_inf_n ≤_k ≤ q_sup_n,< r_inf_d ≤_k ≤ r_sup_d.Rather than explicitly computing the evolution of the distribution for _k, the Kalman filter computes the forecast and posterior distributions parametrically via recursive equations for the mean and covariance of each distribution. The forecast error covariance matrix _k of the Kalman filter satisfies the discrete-time dynamic Riccati equation <cit.>_k+1 = _k+1_n+_k _k^-1_k_k+1^ +_k+1 ,where _k ≜_k^_k^-1_k is the precision matrix of the observations. Equation <ref> expresses the error covariance matrix, _k+1, as the result of a two-step process: (i) the assimilation at time t_k yielding the analysis error covariance,^a_k = _n+_k _k^-1_k;and (ii) the forecast, where the analysis error covariance is forward propagated by_k+1 = _k+1^ a_k_k+1^ + _k+1 .Assuming that the filter is unbiased, such that the initial error is mean zero, it is easy to demonstrate that the forecast and analysis error distributions are mean zero at all times.In this context, the covariances _k,^a_k represent the uncertainty of the state estimate defined by the filter mean.As we will focus on the evolution of the covariances, we neglect the update equations for the mean state and refer the reader to Jazwinski <cit.> for a more complete discussion. The classical conditions for theboundedness of filter errors, and the independence of the asymptotic filter behavior from its initialization, are given in terms of observability and controllability.Observability is the condition that given finitely many observations, the initial state of the system can be reconstructed. Controllability describes the ability to move the system from any initial state to a desired state given a finite sequence of control actions — in our case the moves are the realizations of model error.These conditions are described in the following definitions, beginning with the information and controllability matrices.We define _k:j to be the time varying information matrix and _k:j to be the time varying controllability matrix, where_k:j ≜∑_l=j^k_k:l^-_l _k:l^-1, _k:j≜∑_l=j^k_k:l_l _k:l^.For γ≥ 0 let us define the weighted controllability matrix as^γ_k:j≜∑_l=j^k 1/1+γ^k-l_k:l_l _k:l^. Note that, ^0_k:j≡_k:j.We recall from section 7.5 of Jazwinski <cit.> the definitions of uniform complete observability (respectively controllability).Suppose there exists N_Φ, a,b > 0 independent of k such that k> N_Φ implies0< a _n ≤_k:k - N_Φ≤ b _n,then the system is uniformly completely observable.Likewise suppose there exists N_Υ, a,b >0 independent of k for which k> N_Υ implies0< a _n ≤_k:k- N_Υ≤ b _n,then the system is uniformly completely controllable.Assume that the system of equations <ref> and <ref> is* uniformly completely observable; * uniformly completely controllable.We will explicitly refer to <ref> whenever it is used.When we refer <ref> alone, we refer to both parts (a) and (b).At times, we will explicitly only use either part (a) or (b) of <ref>. Suppose the system of equations <ref> and <ref> satisfies <ref> and _0 >0.Then there exists constants p^a_inf and p^a_sup independent of k such that the analysis error covariance is uniformly bounded above and below,0 < p^a_inf_n ≤^a_k ≤ p^a_sup_n < ∞. Given any two initializations of the prior error covariance _0,_0>0, with associated sequences of analysis error covariances ^a_k,^a_k, the covariance sequences converge, lim_k→∞ł|^a_k - ^a_k |̊ = 0,exponentially in k.These are classical results of filter stability, see for example Theorem 7.4 of Jazwinski <cit.>, or Bougerol's work with random matrices <cit.> for a generalization.The square root Kalman filter is a reformulation of the recurrence in equation <ref> which is used to reduce computational cost and obtain superior numerical precision and stability over the standard implementations see, e.g., <cit.>.The advantage of this formulation to be used in our analysis is to explicitly represent the recurrence in equation <ref> in terms of positive semi-definite, symmetric matrices. Let _k be a solution to the time varying Riccati equation <ref> and define _k ∈ℝ^n× n to be a Cholesky factor of _k, such that_k = _k ^_k.The root _k in equation <ref> can be interpreted as an ensemble of anomalies about the mean as in the ensemble Kalman filter <cit.>.In operational conditions, it is standard that the forecast error distribution is approximated with a sub-optimal, reduced rank surrogate <cit.>.Using a reduced rank approximation, the estimated covariance and exact error covariance are not equal, and this can lead to the systematic underestimation of the uncertainty <cit.>.However, in the following we will assume that _k is computed as an exact root.The sequel to this work explicitly treats the case of reduced rank, sub-optimal filters <cit.>. We order singular values σ_1 > ⋯ > σ_n such that,0 ≤σ_n_k^_k _k_n ≤_k^_k _k ≤σ_1_k^_k _k_n <∞.We defineα≜inf_k {σ_n_k^_k _k}≥ 0, β≜sup_k {σ_1_k^_k _k}≤∞,and we write 0≤α_n ≤_k^_k _k ≤β_n ≤∞ for all k. Equation <ref> is closely related to the singular value analysis of the precision matrix by Johnson et al. <cit.> and the analysis of the conditioning number for the Hessian of the variational cost function by Haben et al. <cit.> and Tabeart et al. <cit.>.These works study the information gain from observations, relative to the background uncertainty, due to the assimilation step.The primary difference between these earlier works and our study here is that the background error covariance is static in these variational formulations, while in the present study the root _k is flow dependent.In this flow dependent context, the constant α (respectively β) is interpreted as the minimal (respectively maximal) observational precision relative to the maximal (respectively minimal) background forecast uncertainty.The constant α is nonzero if and only if the principal angles between the column span of _k and the kernel of _k are bounded uniformly below.Generally, we thus take α=0 unless observations are full dimensional.A nonzero value for α can be understood as an ideal scenario.Using <ref> and the matrix shift lemma <cit.> we re-write the forecast Riccati equation <ref> as _k= _k(_n+ _k-1_k-1)^-1_k-1_k^ + _k = _k _k-1(_n + ^_k-1_k-1_k-1)^-1^_k-1_k^ + _kfrom which we infer1/1+ β_k _k-1_k^ + _k ≤_k ≤1/1+ α_k _k-1_k^ + _k.Iterating on the above inequality, we obtain the recursive bound1/1+ β^k _k:0_0_k:0^ + ^β_k:1≤_k ≤1/1+ α^k _k:0_0_k:0^ + ^α_k:1. Equation <ref> holds if there is no filtering step, setting β = α =0.The bounds in equation <ref> explicitly describe the previously introduced uncertainty as dynamically evolved to time k, relative to the constraint of the observations.We will utilize the BLVs vectors to extract the dynamic information from the sequences of matrices _k:l, _k:l^.§ LYAPUNOV VECTORSThis section contains a short introduction to Lyapunov vectors and the Multiplicative Ergodic Theorem (MET).For a more comprehensive introduction, there are many excellent resources at different levels of complexity, see for example <cit.>.There is inconsistent use of the terminology for Lyapunov vectors in the literature, so we choose to use the nomenclature of Kuptsov & Parlitz <cit.> for its accessibility and self-consistency.Consider the growth or decay of an arbitrary, norm one vector v_0 ∈ℝ^n to its state at time t_k via the propagator _k:0.This is written as ł| v_k |̊ =ł| _k:0 v_0 |̊ =√( v_0^_k:0^_k:0 v_0),so that the eigenvectors of the matrix _k:0^_k:0 describe the principal axes of the ellipsoid defined by the unit disk evolved to time t_k.Using the above relationship for the reverse time model, we see the growth or decay of the unit disk in reverse time asł| u_-k|̊ =ł| ^-1_0:-k u_0 |̊ = √( u_0^_0:-k^-^-1_0:-k u_0).The principal axes of the past ellipsoid that evolves to the unit disk at the present time are thus precisely the eigenvectors of the matrix ^-_0:-k^-1_0:-k. There is no guarantee in general that there is consistency between the asymptotic forward and reverse time growth and decay rates, i.e., in equations<ref> and <ref> as k →∞.Generally, models may have Lyapunov spectrum defined as intervals of lower and upper growth rates, see e.g., Dieci & Van Vleck <cit.>.However, as we are motivated by the tangent-linear model for a nonlinear system, we may assume some “regularity” in the dynamics.The anti-symmetry of the forward/reverse time, regular and adjoint models' growth and decay is known as Lyapunov-Perron regularity (LP-regularity) <cit.>, and is equivalent to the classical Oseledec decomposition <cit.>[see Theorem 2.1.1].LP-regularity guarantees that: (i) the Lyapunov exponents are well defined for the linear model as point-spectrum; (ii) the linear space is decomposable into subspaces that evolve covariantly with the linear propagator; and (iii) each such subspace asymptotically grows or decays according to one of the point-spectrum rates.We summarize the essential results of Oseledec's theorem for use in our work in the following, see Theorem 2.1.1 of Barreira & Pesin <cit.> for a complete statement.[Oseledec's Theorem] The model _k = _k _k-1 is LP-regular if and only if there exists real numbers λ_1 > ⋯> λ_p, for 1 ≤ p ≤ n, and subspaces _k^i ⊂ℝ^n, ^i_k = κ_i, such that for every k,l ∈ℤ⊕_i=1^p ^i_k = ℝ^n _k ± l:k ^i_k = ^i_k ± l,and ∈^i_k implieslim_l→∞1/ l log(ł| _k ± l:k|̊) = ±λ_i .For p≤ n, the Lyapunov spectrum of the system <ref> is defined as the set{λ_i : κ_i }_i=1^p where λ_1 > ⋯ > λ_p and κ_i corresponds to the multiplicity (degeneracy) of the exponent λ_i. We separate non-negative and negative exponents, λ_n_0≥ 0 > λ_n_0 +1, such that each index i> n_0 corresponds to a stable exponent.The subspaces ^i_k are denoted Oseledec spaces, and the decomposition of the model space into the direct sum is denoted Oseledec splitting. For arbitrary linear systems LP-regularity is not a generic property — it is the MET that shows that this is a typical scenario for a wide class of nonlinear systems.A point will be defined to be LP-regular if the tangent-linear model along its evolution is LP-regular.We state a classical version of the MET <cit.> but note that there are more general formulations of this result and more general forms of the associated covariant-subspace decompositions.These results go beyond the current work, see e.g., Froyland et al. <cit.> and Dieci et al. <cit.> for a stronger version of the MET and related topics.[Multiplicative Ergodic Theorem] If f is a C^1 diffeomorphism of a compact, smooth, Riemannian manifold M, the set of points in M which are LP-regular has measure 1 with respect to any f-invariant Borel probability measure ν on M.If ν is ergodic, then the Lyapunov spectrum is constant with ν-probability 1.Loosely, the MET states that, with respect to an ergodic probability measure (that is compatible with the map f and the usual topology), there is probability one of choosing initial conditions for which the Lyapunov exponents are well defined and independent of initial condition.This form of the MET has a wide range of applications in differentiable dynamical systems, but the MET is not limited to this setting.The strong version of the MET has been applied in, e.g., hard disk systems, the truncated Fourier expansions of PDEs, and with non-autonomous ODEs and their transfer operators <cit.>.For the rest of this work, we will take the hypothesis that our model satisfies LP-regularity.The model defined by the deterministic equation _k = _k _k-1,is assumed to be LP-regular.The deterministic evolution in equation <ref> comes naturally in the formulation of the Kalman filter, where the mean state is evolved via the deterministic component in the forecast step.For Gaussian error distributions, the evolution of the forecast error distribution is interpreted in terms of Oseledec's theorem as the evolution of deviations from the mean, propagated via the equations for perturbations.While Oseledec's theorem guarantees that a decomposition of the model space exists, constructing such a decomposition is non-trivial.Motivated by equations <ref> and <ref>, we define the following operators as in equations (13) and (14) of Kuptsov & Parlitz <cit.>.We define the far-future operator as^+(k) ≜ lim_l→∞ ^_k+l: k_ k+l: k ^1/2l,and the far-past operator as^-(k) ≜ lim_l→∞ ^-_k: k-l^-1_ k: k-l ^1/2l.In the classical proof of the MET, the far-future/past operators are shown to be well defined positive definite, symmetric operators <cit.>.As they are diagonalizable over ℝ, we order the eigenvalues of ^+(k) as μ^+_1(k)>⋯ >μ^+_p(k) and the eigenvalues of ^-(k) as μ^-_p(k) > ⋯ > μ^-_1(k). By the MET, the eigenvalues μ^±_i(k) are independent of k and satisfy the relationshiplog(μ^±_i) = ±λ_i.Let the columns of the matrix _k, respectively _k, be any orthonormal eigenbasis for the far-future operator ^+(k), respectively far past operator ^-(k).Order the columns block-wise, such that for each i =1,⋯, p and each j =1,⋯, κ_i, ^i_j_k is an eigenvector for μ^+_i and ^i_j_k is an eigenvector for μ^-_i.We define ^i_j_k to be the i_j-th forward Lyapunov vector (FLV) at time k and _k^i_j to be the i_j-th backward Lyapunov vector (BLV). Let the columns of _k form any basis such that for each i=1,⋯, p and each j=1,⋯, κ_i, ^i_j_k ∈^i_k.Then we define ^i_j_k to be the i-th covariant Lyapunov vector (CLV) at time k. The CLVs are defined only by the Oseledec spaces, and therefore, are independent of the choice of a norm — any choice of basis subordinate to the Oseledec splitting is valid.On the other hand, the FLVs and the BLVs are determined specifically with respect to a choice of a norm and the induced metric.The choice of basis in each case can be made uniquely (up to a scalar and the choice of a norm) only when p=n.For the remaining work we will focus on the BLVs; for a general survey on constructing FLVs, BLVs and CLVs, see e.g., Kuptsov & Parlitz <cit.> and Froyland et. al. <cit.>.The Oseledec spaces and Lyapunov vectors can also be defined in terms of filtrations, i.e., chains of ascending or descending subspaces of ℝ^n.This forms an axiomatic approach to constructing abstract Lyapunov exponents used by, e.g., Barreira & Pesin <cit.>.The BLVs describe an orthonormal basis for the ascending chain of Oseledec subspaces, the backward filtration <cit.>.For all 1 ≤ m≤ p we obtain the equality⊕_i=1^m ^i_k = ⊕_i=1^m {_k^i_j}_j=1^κ_i,by equation (17) by Kuptsov & Parlitz <cit.>, and the decomposition of the backward filtration in equations (1.5.1) and (1.5.2) of Barreira & Pesin <cit.>.Note that equation <ref> does not imply ^i_j_k ∈^i_k for i> 1, as the BLVs are not themselves covariant with the model dynamics.However, the BLVs are covariant with the QR algorithm. Outside of a set of Lebesgue measure zero, a choice of i_j linearly independent initial conditions for the recursive QR algorithm converges to some choice for the leading i_j BLVs.For any k, the BLVs satisfy the relationship _k _k-1 =_k _k, ⇔_k = _k _k _k-1^where _k is an upper triangular matrix. Moreover, for any i_j and any k,lim_l → -∞1/k-llogł|^_k:l^ i_j_k|̊ = λ_i.The covariance of the BLVs with respect to the QR algorithm in equation <ref> can be derived from equations <ref> and <ref>.For all 1≤ m ≤ p,_k ⊕_i=1^m {_k-1^i_j}_j=1^κ_i = ⊕_i=1^m {_k^i_j}_j=1^κ_i,due to the covariance of the Oseledec spaces.Therefore the transformation _k represented in a moving frame of BLVs is upper triangular.When the spectrum is degenerate, p<n, there is non-uniqueness in the choice of the BLVs.However, given an initial choice of the BLVs at some time k-1, the choice of BLVs at time k can be defined directly via the relationship in <ref>.This is the relationship derived in equation (31) by Kuptsov & Parlitz <cit.>, and is the basis of the recursive QR algorithms of Shimada & Nagashima <cit.> and Benettin et. al. <cit.>.A choice of BLVs gives a special choice of the classical Perron transformation <cit.>[see Theorems 3.3.1 & 3.3.2], and in particular, it is proven by Ershov & Potapov <cit.> that outside of a set of Lebesgue measure zero, the recursive QR algorithm converges to some choice of BLVs.Note that the far-future/past operators arealso well defined for the propagator of the adjoint model _k = ^-_k _k-1. Equation <ref> thus follows from the far-past operator for the adjoint model, defined^∗ -(k) ≜lim_l →∞ _k:k-l_k:k-l^ ^1/2l.It is easy to verify that the BLVs defined by the adjoint model agree with those defined via the regular model — in each case, the left singular vectors of _k:k-l converge to a choice of the BLVs as l→∞.Notice that the eigenvalues of ^∗ -(k) are reciprocal to those of ^-(k), i.e., μ^∗ -_i = 1/μ^-_i.Thus by equation <ref>, logμ^∗ -_i =λ_i. Equation <ref> describes the dynamics in the moving frame of BLVs, where the transition map from the frame at time t_k-1 to time t_k is given by _k.Applying the change of basis sequentially for the matrix _k:l, we recover_k =_k _k _k-1^⇒_k:l =_k _k:l_l^,where we define _k:l≜_k ⋯_l.We note that _k:k=_n implies _k:k≡_n.Let _i_j denote the i_j-th standard basis vector, such thatł|^_k:l^i_j_k |̊^2 = _i_j^_k:l_k:l^_i_j=ł|^_k:l^i_j|̊^2where ^_k:l^i_j denotes the i_j-th column of ^_k:k-l, i.e., the i_j-th row of _k:k-l.For any k and any ϵ>0, there exists some N_ϵ, k such that if k-l is taken sufficiently large, equation <ref> guaranteese^2(λ_i - ϵ)l≤ł|^_k:k-l^i_j|̊^2 ≤ e^2(λ_i + ϵ)l.For each k>l, i=1,⋯, p and j=1, ⋯, κ_i we define the i_j-th local Lyapunov exponent (LLE) from k to l as 1/k-llog| T^i_j_k:l| where T^i_j_k:l is defined to be the i_j-th diagonal entry of _k:l. For any fixed l,lim_k→∞1/k-llog| T^i_j_k:l| = λ_iThis is also discussed by Ershov & Potapov <cit.>, in demonstrating the convergence of the recursive QR algorithm.For a discussion on the numerical stability and convergence see, e.g., Dieci & Van Vleck <cit.>. Perturbations of model error to the mean equation for the Kalman filter are not governed by the asymptotic rates of growth or decay, but rather, the LLEs. While the LLE 1/k-llog| T^i_j_k:l| approaches the value λ_i as k-l approaches infinity,its behavior on short time scales can be highly variable.Particularly, for an arbitrary LP-regular system, the rate of convergence in equation <ref> may depend on k.An important class of such systems is, e.g., non-uniformly hyperbolic systems <cit.>[see chapter 2].To make the LLEs tractable, we make an additional assumption, compatible with the typical assumptions for partial hyperbolicity <cit.>.We adapt the definition of partial hyperbolicity from Hasselblatt & Pesin <cit.> to our setting. Let λ_n_0= 0.For every k we define the splitting into unstable, neutral and stable subspaces: E^u_k ≜⊕_i= 1^n_0 - 1^i_k,E^c_k ≜^n_0_k andE^s_k ≜⊕_i= n_0 + 1^p^i_k.Suppose there exists constants C>0 and 0 < η_s ≤ν_s < η_c ≤ν_c <η_u ≤ν_uindependent of k such that ν_s <1 < η_u and for any l>0, ∈ E^m_k, ł| |̊=1, and m∈{s,c,u}(η_m)^l/C≤ł| _k+l:k|̊≤ (ν_m)^l C .Then the model <ref> is (uniformly) partially hyperbolic (in the narrow sense). Partially hyperbolic systems, as in <ref>, have LLEs which are bounded uniformly with respect to rates defined on the subspaces in equation <ref>.When C is taken large the definition permits transient growth of stable modes and transient decay of unstable modes.The neutral subspace encapsulates diverse behaviors which always fall below prescribed rates of exponential growth or decay.We will make a slightly stronger assumption on these uniform growth and decay rates that is equivalent to fixing a uniform window of transient variability on each Lyapunov exponent.Let ϵ>0 be given.We assume that for each i there exists some N_i,ϵ, independent of k and j, such that for any ^i_j_k whenever k-l> N_i,ϵ-ϵ < 1/k-llogł| ^_k : l^i_j_k |̊- λ_i < ϵ ,i.e., the growth and decay is uniform (translation invariant) in k.Unless specifically stated otherwise, we assume <ref> for the remaining of this paper.However, our results may be generalized to all systems satisfying <ref> by using only the uniform rates of growth or decay on the entire unstable, neutral and stable subspaces in equation <ref>.Our results also apply to systems without neutral exponents, i.e. λ_n_0>0, as a trivial extension.§ DYNAMICALLY INDUCED BOUNDS FOR THE RICCATI EQUATION§.§ Autonomous systemsConsider the classical theorem regarding the existence and uniqueness of solutions to the stable Riccati equation for autonomous dynamics.This is paraphrased from Theorem 2.38, Chapter 7, of Kumar & Varaiya <cit.> in terms of the forecast error covariance recurrence in equation <ref>. The autonomous system is defined such that for every k_k ≡ , _k ≡ , _k ≡ , _k ≡and_k ≡.Let = ^ for some ∈ℝ^n× n, the stable Riccati equation is defined as= (_n + ^ )^-1^^ +Let the autonomous system defined by equations <ref>, <ref> and <ref> satisfy <ref>.There is a positive semi-definite matrix, ≡^, which is the unique solution to the stable Riccati equation <ref>.For any initial choice of _0, if _k satisfies the recursion in equation <ref>, then lim_k→∞_k =.Slightly abusing notation, take α and β to be defined by the solution to the stable Riccati equation <ref>,α≜σ_n^≥ 0 β≜σ_1^ < ∞.Then for any k we recover the invariant recursion for the stable limit(1+ β)^-k^k^^k + ^β_k:1≤≤ (1+ α)^-k^k^^k + ^α_k:1.Assume equations <ref> and <ref> satisfy <ref> and define α,β for the stable Riccati equation as in equation <ref>.For any 1≤ i≤ p, if there exists ϵ > 0 such that e^2(λ_i + ϵ)/1+ α < 1,choose N_i,ϵ as in <ref>. For the eigenvalue μ_i of ^, where |μ_i | = e^λ_i, choose any eigenvector _i_j.Then_i_j^_i_j ≤_i_j^_i_j/1-e^2λ_i/1+α.Moreover, if ^i_j is the i_j-th BLV, then^i_j^^ i_j ≤^ i_j^^α_N_i,ϵ:0^ i_j +e^2(λ_i +ϵ)/1+ α^N_i,ϵ +1q_sup/1-e^2(λ_i+ϵ)/1+α.For every 1≤ i≤ p, any ϵ>0, and associated N_i,ϵ as in <ref>,_i_j^_i_j/1-e^2λ_i/1+β≤_i_j^_i_jand ^ i_j^^β_N_i,ϵ:0^ i_j +e^2(λ_i - ϵ)/1+ β^N_i,ϵ +1q_inf/1-e^2(λ_i-ϵ)/1+β≤^ i_j^^ i_j . Note that time invariant propagators trivially satisfy <ref> and it is easy to verify the relationship |μ_i | = e^λ_i directly from the definition of the Lyapunov exponents.We begin by proving equations <ref> and <ref> for eigenvectors of ^.If _i_j is an eigenvector of ^ associated to μ_i, equation <ref> implies_i_j^_i_j≤|μ_i|^2 /1+ α^k+1_i_j^_i_j + ∑_l=0^k|μ_i |^2 /1+α^l _i_j^_i_jfor every k.For λ_i < 0 generally, or for any λ_i such that α > e^2λ_i - 1,lim_k→∞ |μ_i|^2 /1+ α^k+1_i_j^_i_j +∑_l=0^k|μ_i |^2 /1+α^l _i_j^_i_j = _i_j^_i_j/1-|μ_i |^2 /1+αand_i_j^_i_j≤_i_j^_i_j/1-e^2λ_i/1+α . The stable Riccati equation <ref> implies ≤.Therefore, using the left side of <ref> demonstrates that for any eigenvector _i_j∑^k_l=0|μ_i |^2/1+ β^l _i_j^_i_j≤_i_j^_i_jfor all k.In particular, for every eigenvector _i_j we obtain_i_j^_i_j/1 - |μ_i|^2/1 + β ≤_i_j^_i_j . The above argument does not have a straightforward extension to the generalized eigenspaces so we coarsen the bound to obtain a closed limiting form in terms of the BLVs which retain the important growth characteristics under ^.For i> n_0, or for any λ_i such that α > e^2λ_i - 1, there is a choice of ϵ as in equation <ref> and N_i,ϵ as in <ref>.Let ≤p_sup_n, then from the right side of equation <ref> we derive≤p_sup^k+1^^k+1/(1+ α)^k+1+ ∑_l=0^k^l^^l/(1+α)^l≤p_sup^k+1^^k+1/(1+ α)^k+1+ ^α_N_ϵ,i:1 + q_sup∑_l=N_ϵ,i+1^k^l^^l/(1+α)^l,which implies ^ i_j^^ i_j can be bounded above byp_supł| ^^k+1^ i_j|̊^2/(1+ α)^k+1 + ^ i_j^^α_N_i,ϵ:1^ i_j + q_sup∑_l=N_i,ϵ + 1^kł| ^^l ^ i_j|̊^2/(1+α)^l.Utilizing equation <ref> we bound ^ i_j^^ i_j byp_supe^2(λ_i +ϵ)/1+ α^k+1+ ^ i_j^^α_N_i,ϵ:1^ i_j + q_sup∑_l=N_i,ϵ +1^ke^2(λ_i+ϵ)/1+α^lfor every k > N_i,ϵ.Taking the limit of equation <ref> as k→∞ yields^ i_j^^ i_j ≤^i_j^^α_N_i,ϵ:1^ i_j + e^2(λ_i +ϵ)/1+ α^N_i,ϵ +1q_sup/1-e^2(λ_i+ϵ)/1+α,The lower bound is demonstrated by similar arguments with the lower bound in equation <ref>, utilizing the property < ∞. <ref> is similar results in perfect models <cit.>, but with some key differences.Once again that the estimation errors are dissipated by the dynamics in the span of the stable BLVs, but the recurrent injection of model error prevents the total collapse of the covariance to the unstable-neutral subspace.In equation <ref>, we see that for very strong decay, when e^2λ_i≈ 0, or high precision observations, i.e., when the system is fully observed and as α→∞, the stable limit of the forecast uncertainty reduces to what is introduced by the recurrent injection of model error.The SEEK filter of Pham et. al. <cit.> has exploited these properties by neglecting corrections in the stable eigenspaces and only making corrections in the unstable directions.This is likewise the motivation for AUS of Trevisan et. al. <cit.>, though the work of AUS was concerned with nonlinear, perfect models.The upper bounds in equations <ref> and <ref> generally hold for i≤ n_0 only when the system is fully observed.Therefore, these bounds can be considered an ideal bound for the unstable-neutral modes.However, the lower bound in equation <ref> hold generally for i<n_0.By assuming the existence of an invariant solution to the stable Riccati equation <ref>, we will recover a necessary condition for its existence. Assume there exists a solutionto the stable Riccati equation <ref>.Choose the smallest index i such that 1≤ i ≤ n_0 and there exists some generalized eigenvector _i_j of ^ for which _i_j∉null. Then it is necessary thate^2λ_i/1+σ_1^2^-1/2 <1.Let _i_1 be an eigenvector for ^ and _i_1≠ 0.Then by the definition of β in equation <ref>, the equation <ref> holds for all k if and only if equation <ref> holds.More generally, suppose {_i_j}_j=1^κ_i are (possibly complex) generalized eigenvectors forming a Jordan block for ^.Let j be the smallest index for which _i_j≠ 0.Recall that for each j∈{1,⋯,κ_i} the Jordan basis satisfies^ -μ_i _n_i_j = _i_j-1where _i_0≡ 0.Therefore, for any m≥ 1, the vector ^ -μ_i _n^m _i_j is in the span of {_i_1, ⋯, _i_j-1}. Let us define ≜^ -μ_i _n so that∑_l=0^k+1^^l _i_j = ∑_l=0^k+1+ μ_i _n^l_i_j = ∑_l=0^k+1∑_m=0^l μ_i^l-mlm^m_i_j = ∑_l=0^k+1μ_i^l _i_j.Multiply equation <ref> on the left with _i_j^$̋ (the conjugate transpose) and the right with_i_j.Combining this with the equality in equation <ref>, proves the result.<ref> shows that it is necessary for the existence of the stable Riccati equation that observations are precise enough, relative to the background uncertainty, to counteract the strongest dynamic instability forcing the model error.The quantity in <ref> thus represents the stabilizing effect of the observations, similar to the bounds on the conditioning number provided by Haben et al. <cit.> and Tabeart et al. <cit.>, but in <ref> expressly in response to the system's dynamic instabilities.§.§ Time varying systemsIn the following, we will extend the results of <ref> and <ref> to time-varying systems, and derive a uniform bound on the unfiltered errors in the stable BLVs in <ref>. Assume equations <ref> and <ref> satisfy <ref> (b).For any 1 ≤ i ≤ p, if there exists ϵ > 0 such that e^2(λ_i + ϵ)/1+ α < 1,choose N_i,ϵ as in <ref>.Then there exists a constant 0≤ C_α,N_i,ϵ such thatlim sup_k→∞^ i_j_k^_k^ i_j_k≤C_α, N_iϵ +e^2(λ_i +ϵ)/1+ α^N_i,ϵ +1q_sup/1-e^2(λ_i+ϵ)/1+α.If <ref> (a) is also satisfied, then for every 1 ≤ i ≤ p, any ϵ>0 and associated N_i,ϵ, there exists 0≤ C_β,N_i,ϵ such thatC_β, N_iϵ +e^2(λ_i - ϵ)/1+ β^N_i,ϵ +1q_inf/1-e^2(λ_i-ϵ)/1+β≤lim inf_k→∞^ i_j_k^_k^ i_j_k .If the system satisfies <ref> (b) then^α_k:k-N_i,ϵ ≤^0_k:k-N_i,ϵ≡_k:k-N_i,ϵ≤ b_N_i,ϵ_n,where b_N_i,ϵ is independent of k.Therefore, there exists a constant depending on α and N_i,ϵ, but independent of k, such that^α_k:k-N_i,ϵ≤ C_α,N_i,ϵ_n. Let _0 ≤ p_0 _n bound the prior covariance.Equation <ref> implies_k ≤ p_0 _k:0^_k:0/(1+α)^k +q_sup∑_l=1^k_k:l^_k:l/(1+α)^k-l.From the above, we bound ^ i_j_k^_k ^i_j_k with p_0 ł| ^_k:0^ i_j_k|̊^2/(1+α)^k +^ i_j_k^^α_k:k-N_i,ϵ^ i_j_k + q_sup∑_l=0^k-N_i,ϵ-1ł| ^_k:l^i_j_k|̊^2/(1+α)^k-l, thus ^ i_j_k^_k ^i_j_k ≤ p_0 e^2(λ_i + ϵ)/1+α^k +C_α,N_i,ϵ + q_sup∑_l=N_i,ϵ+1^ke^2(λ_i + ϵ)/1+α^l .Taking the lim sup in equation <ref> as k →∞ yields equation <ref>.Suppose that <ref> (a) and (b) are both satisfied, then by <ref> there exists a uniform bound on _k such that _k must also be uniformly bounded; together with uniform boundedness of _k and _k, this implies β < ∞.Note that^β_k:k-N_i,ϵ ≥1/1 + β^N_i,ϵ^0_k:k-N_i,ϵ≥1/1 + β^N_i,ϵ a_N_i,ϵ_nfor some constant a_N_i,ϵ independent of k.This implies^β_k:k-N_i,ϵ≥ C_β,N_i,ϵ_nfor a constant C_β,N_i,ϵ depending on β and N_i,ϵ but independent of k.Utilizing the recursion in equation <ref>, choosing ϵ and an appropriate N_i,ϵ, and finally bounding the weighted controllability matrix with equation <ref> allows one to recover the lower bound in equation <ref> in a similar manner to the upper bound.The above proposition shows that there is a uniform upper and lower bound on the forecast error for the Kalman filter, in the presence of model error, which can be described in terms of inverse, competing factors: the constantα(respectivelyβ) is interpreted as the minimal (respectively maximal) observational precision relative to the maximal (respectively minimal) background forecast uncertainty, represented in the observation variables. AdditionallyC_β, N_iϵ, C_α, N_iϵrepresent the lower and upper bounds on local variability of the evolution of model errors, before perturbations adhere within anϵthreshold to their asymptotic behavior.Assume equations <ref> and <ref> satisfy <ref> (b), and there exists uniform bound to the forecast error Riccati equation <ref> for all k.Then it is necessary that e^2λ_1/1+sup_kσ_1^2_k^-1/2_k _k <1.If the forecast error Riccati equation <ref> is uniformly bounded, there is a 0<p_sup<∞ such that we have the inequality, _k≤ p_sup_n for all k, and β< ∞.Using the lower bound in equation <ref>, for all k we have1/1 + β^k _k:0_0 ^_k:0 + ^β_k:1 ≤ p_sup_n.The summands in equation <ref> are positive semi-definite such that for any k> N_Υ+1, truncating _k:1^β verifies∑_l=1^N_Υ + 11/1 + β^k-l_k:l_l _k:l^≤_k:1^β≤ p_sup_n.Note that by <ref>, if k>N_Υ+2 _k:N_Υ + 1_N_Υ +1 :1_k:N_Υ + 1^= ∑_l=1^N_Υ +1_k:l_l _k:l^,and therefore, for every k>N_Υ+2 1/1+β^k -1 _k:N_Υ + 1_N_Υ +1 :1_k:N_Υ + 1^ ≤ p_sup_n.Using <ref> (b), for every k>N_Υ+2 we derive1/1+β^k-N_Υ - 1_k:N_Υ +1_k:N_Υ +1^ ≤p_sup( 1+β)^N_Υ/b_n,using the inequality in <ref>.For any j, multiplying equation <ref> on the left by ^1_j_k^ and on the right by ^1_j_k and taking the limit as k→∞ shows that it is necessary for equation <ref> to hold for the left side to be bounded away from ∞. In contrast to <ref> for autonomous systems, <ref> uses the <ref> (b) to simplify the arguments — this moreover guarantees the necessary criterion is with respect toλ_1, as the controllability matrix is guaranteed to be positive definite and thus nonvanishing on every Oseledec space.There is, however, a more direct analogue to the statement of <ref> where the adjoint-covariant Lyapunov vectors will play the role of the eigenvectors of^.It is easy to demonstrate that the adjoint-covariant Lyapunov vectors have the desired covariance and growth/decay with respect to the reverse time adjoint model,^_k.There exist, under the condition of integrally separated Oseledec spaces, classical constructions for covariant and adjoint-covariant bases that decompose the model propagator into a block-upper-triangular form <cit.>[see Theorem 5.4.9]. This decomposition makes the derivation of a precise statement like <ref> analogous in time varying models, with respect to the adjoint-covariant Laypunov vectors and adjoint-covariant Oseledec spaces.However, the above arguments require significant additional exposition which we feel unnecessary, as <ref> is sufficiently general. Assume equations <ref> and <ref> satisfy <ref> (b) and suppose _k _k ≡ for every k such that α = β = 0.Let k≥ 1 and choose ∈{^i_j_k : n_0 < i ≤ p,1≤ j ≤κ_i} such that ł| |̊ =1.There is a C>0 independent of k such that^_k≤ C <∞.The inequality in equation <ref> is an equality for the unfiltered forecast where β = α=0.Thus the corollary is clear for any stable BLV directly from <ref> and the conclusion extends to norm one linear combinations.<ref> extends the intuition of AUS to the presence of model error: corrections may be targeted along the expanding modes while the uncertainty in the stable modes remains bounded by the system's dynamic stability alone.Particularly, without filtering uncertainty remains uniformly bounded in the span of the stable BLVs. This is analogous to the results of Bocquet et. al. <cit.>, where in perfect models, the stable dynamics alone are sufficient to dissipate forecast error in the span of the stable BLVs.Withα=0, the uniform bound in <ref> may be understood by the two components which equation <ref> is composed of, the bound on_k:k-N_i,ϵand q_sup e^2(λ_i +ϵ)N_i,ϵ +1/1-e^2(λ_i+ϵ).The controllability matrix_k:k-N_i,ϵrepresents the newly introduced uncertainty from model error that is yet to be dominated by the dynamics.Equation <ref> represents an upper bound on the past model errors that have already been dissipated by the stable dynamics.Nevertheless, this uniform bound is uninformative about the local variability.In the following sections, we study the variance of the unfiltered uncertainty in the stable BLVs compared to the uncertainty of the Kalman filter.§ NUMERICAL EXPERIMENTS§.§ Experimental setup To satisfy <ref>, we construct a discrete, linear model from the nonlinear Lorenz-96 (L96) equations <cit.>, commonly used in data assimilation literature see, e.g., <cit.>[and references therein]. For eachm∈{1,⋯,n}, the L96 equations readd/d t ≜(),L^m()=-x^m-2x^m-1 + x^m-1x^m+1 - x^m + Fsuch that the components of the vectorare given by the variablesx^mwith periodic boundary conditions,x^0=x^n,x^-1=x^n-1andx^n+1=x^1.The termFin L96 is the forcing parameter.The tangent-linear model <cit.>is governed by the equations of the Jacobian matrix,∇(),∇ L^m() = 0, ⋯, -x^m-1, x^m+1 - x^m-2, -1, x^m-1,0, ⋯, 0.We fix the model dimensionn≜10and the forcing parameter as the standardF=8, as the model exhibits chaotic behavior, while the small model dimension makes the robust computation of Lyapunov vectors numerically efficient.The linear propagator_kis generated by computing the discrete, tangent-linear model <cit.> from the resolvent of the Jacobian equation <ref> along a trajectory of the L96, with an interval of discretization atδ≜0.1.We numerically integrate the Jacobian equation with a fourth order Runge-Kutta scheme with a fixed time step ofh ≜0.01.ForF=8, the 10 dimensional nonlinear L96 model has a non-degenerate Lyapunov spectrum and we replace the superscripti_jwithifor the BLVs.The model has three positive, one neutral and six negative Lyapunov exponents, such thatn_0 = 4.The Lyapunov spectrum for the discrete, linear model is computed directly via the relationship in <ref>, where the average is taken over10^5iterations of the recursive QR algorithm, after pre-computing the BLVs to convergence.In our simulations, before our analysis, we pre-compute the BLVs and the FLVs over10^5iterations of the recursive QR algorithm for the forward model, or respectively, for the reverse time adjoint model <cit.>[see section 3].We note that the computed Lyapunov spectrum for the discrete, linear model as in simulations is related to the spectrum of the nonlinear L96 model by rescaling the linear model's exponents by1/δ. §.§ Variability of recurrent perturbations While <ref> guarantees that the uncertainty in the stable BLVs is uniformly bounded, this bound strongly reflects the scale of the model error and the local variance of the Lyapunov exponents.If model errors are large, or the stable Lyapunov exponents have high variance, this indicates that the uniform bound can be impractically large for forecasting. Assume no observational or filtering constraint, i.e.,_k_k≡.Suppose that the model error statistics are uniform in time and spatially uncorrelated with respect to a basis of BLVs:_k ≜_k _k^, where∈ℝ^n×nis a fixed diagonal matrix with thei_j-th diagonal entry given byD_i_j.Denote_0 ≡_0, then equation <ref> becomes^ i_j_k^_k ^i_j_k = ∑_l=0^k ^ i_j_k^_k:l_l ^_k:l^ i_j_k=∑_l=0^k _i_j^_k:l^_k:l_i_j = D_i_j∑_l=0^k ł| ^_k:l^i_j|̊^2 ,whereł| ^_k:l^i_j |̊is the norm of thei_j-th row of_k:l.In equation <ref>,_krepresents the freely evolved uncertainty at timek, and thus∑_l=0^k ł| ^_k:l^i_j |̊^2describes the variance of the free evolution of perturbations in the direction of^i_j_k.For each 1 ≤ i ≤ p, each 1 ≤ j ≤κ_i and any k, we defineΨ_k^i_j≜∑_l=0^k ł| ^_k:l^i_j|̊^2to be the free evolution of perturbations in the direction of ^i_j_k. Assuming the errors are uncorrelated in the basis of BLVs is a strict assumption, but studying the free evolution of perturbations has general applicability:_k^_k _k ≤q_sup_n, and therefore, equation <ref> may be interpreted in terms of an upper bound on the variance of the freely evolved forecast uncertainty in thei_j-th mode.<ref> describes our recursive approximation of the free evolution, given by equation <ref>, fork∈{1,⋯,m}via the QR algorithm.We assume that the QR algorithm has been run to numerical convergence for the BLVs at time0. Equation <ref> implies ł| ^^i_j|̊^2 decays exponentially in k-l and the inner loop of <ref> needs only be computed to the first l such that ł| ^^i_j|̊^2 is numerically zero. The approximation of <ref> with <ref> is numerically stable for allkand anyi > n_0, precisely due to the upper triangular dynamics in the BLVs.The upper triangularity of all_kmeans the lower right block of_k:lis given as the product of the lower right blocks of the sequence of matrices{_j}_j= l+1^k.Therefore, computing the stable block of_k:lis independent of the unstable exponents, and the row norms of_k:lconverge uniformly and exponentially to zero by <ref>.In <ref> we plotΨ_k^5andΨ_k^6as in <ref> and the LLEs for^5_kand^6_kfork∈{1, ⋯, 10^4}.Assuming that_k ≤q_sup_n,Ψ^i_kbounds the variance in thei-th stable mode at timek, up to the scaling factor ofq_sup.Asn_0=4, the exponentλ_5is the stable exponent closest to zero.The left side of <ref> corresponds to the exponentλ_5 ≈-0.0433while the right side corresponds to the exponentλ_6 ≈-0.0878.The upper row in <ref> plots the evolution ofΨ^i_kfor^5_kand^6_k, whilethe bottom row shows the corresponding time series of LLEs.The mean of the LLEs are approximately equal to their corresponding Lyapunov exponent, while the standard deviation is given by0.142forλ_5and0.133forλ_6respectively.WhileΨ_k^5is uniformly bounded, <ref> illustrates that it can be on the order of𝒪10^3, with a mean value of approximately 808 over the10^4iterations.This is in contrast to perfect models where the projection of the unfiltered forecast error into any stable mode converges to zero at an exponential rate <cit.>.Moreover, the frequency and scale of positive realizations of LLEs of^5_khas a strong impact the variance of the unfiltered error.The fewer, and weaker, positive realizations of the LLEs of^6_kcorrespond to the lower overall uncertainty represented byΨ^6_k.The maximum ofΨ_k^6is on the order ofØ(10^2), with a mean value of approximately 28. §.§ Unfiltered versus filtered uncertainty In the following, we compare the variance of the unfiltered error in the stable BLVs, represented byΨ_k^i, fori∈{5,⋯,10}, with the uncertainty in the Kalman filter.Assuming that_k ≜_n, in this caseΨ_k^iis equal to the variance of the unfiltered error along^i_k.While the error in the Kalman filter depends on the observational configuration, the value ofΨ_k^idepends only on the underlying dynamics.Therefore, we benchmark the variance of the unfiltered error over a range of observational designs to determine under what conditions the unfiltered error in the stable BLVs will exceed the uncertainty of the full rank Kalman filter.This analysis allows us to evaluate how many of the stable BLVs can remain unfiltered while achieving an acceptable forecast performance.This comparison has a special significance when considering reduced rank, sub-optimal filters, which is the subject of the sequel <cit.>.The recent works of Bocquet et al. <cit.> and Frank & Zhuk <cit.>, weaken <ref> to criteria on the observability, or detectability, of the unstable-neutral subspace to obtain filter stability and boundedness in perfect models.The results in <ref>, <ref> and <ref> similarly suggest that the sufficient condition for filter boundedness, <ref>, may be weakened in the presence of model errors. For this reason, we will study the variance of the filtered error with respect to observations satisfying the criteria discussed by Bocquet et al. <cit.> and Frank & Zhuk <cit.>.Given a fixed dimension of the observational spaced < n, consider finding an observational operator,_k, which minimizes the forecast uncertainty.Suppose the singular value decomposition of an arbitrary choice of_kis given as_k = _k _k _k^.For a given observation error covariance matrix, the size of the singular values of_kcorrespond to the precision of observations relative to the uncertainty in the precision matrix,_k ≜_k^^-1_k_k. Imposing that all singular values of_kmust be equal to one, then up to an orthogonal transformation of^-1_k, we equate the choice of an observational operator_kwith the selection of an orthogonal matrix_k ∈ℝ^n ×d.For perfect models,_k ≡0, we write the forecast error Riccati equation in terms of a choice of_k ≜_k^as_k+1 = _k+1_k _n + _k^-1/2^_k _k^_k^-1/2_k^_k ^-1^_k_k+1^.The Frobenius norm,ł| _k+1 |̊_= √(_k+1^2),is bounded by{_k+1_k _n + _k^-1/2^_k _k^_k^-1/2_k^_k ^-1^_k_k+1^} ≤ { _n + _k^-1/2^_k _k^_k^-1/2_k^_k ^-1}^_k_k+1^_k+1_k.Equation <ref> attains its smallest values when the eigenvalues of_n + _k^-1/2^_k _k^_k^-1/2_k^_kare as large as possible, similar to maximizing the denominator of equation <ref>.For a fixed sequence of observation error covariances, finding the largest eigenvalues of equation <ref> can be studied by finding the subspace for which the matrix of orthogonal projection coefficients^_k _khas the largest singular values.In perfect models, the forecast error covariance for the Kalman filter asymptotically has support confined to the span of the unstable and neutral BLVs <cit.>.This is likewise, evidenced for the ensemble Kalman filter in weakly-nonlinear models <cit.>, suggesting that the columns of_kshould be taken as the leadingdBLVs. Given d ≥ 1, let ^1:d_k ∈ℝ^n× d denote the matrix comprised of the first d columns of _k.We define the observation operator ^ bd_k ≜_k^1:d^. <ref> is a formalization of the AUS observational paradigms <cit.> utilizing “bred vectors” as proxies for the BLVs.The breeding method of Toth & Kalnay <cit.> simulates how the modes of fast growing error are maintained and propagated through the successive use of short range forecasts in weather prediction.The bred vectors are formed by initializing small perturbations of a control trajectory and forecasting these in parallel along the control. Upon iteration, the span of these perturbations generically converge to the leading BLVs.For a discussion of variants of this algorithm, and the convergence to the BLVs, see e.g., Balci et al.<cit.>.The choice of observation operator in <ref> is also related to the numerical study of targeted observations for the L96 model of Law et al. <cit.>.Law et al. target observations with the eigenvectors of the operator^_k+1 _k+1, but note that for a small intervalδ≜t_k+1 - t_k, the difference between the linearized equations defining^_k+1 _k+1and_k+1^_k+1is negligible <cit.>[see Remark 5.1].Law et al. suggest that the eigenvectors of either^_k+1 _k+1and_k+1^_k+1may be sensible depending on whether the filter should take into account the principle axes of growth from the past to the current time or from the present to future time.It is clear from equations <ref> and <ref> that asδbecomes large, the eigenvectors of_k+1^_k+1approach the BLVs, whereas^_k+1 _k+1approach the FLVs. Given d ≥ 1, let ^1:d_k ∈ℝ^n× d denote the matrix comprised of the first d columns of _k.We define the observation operator ^ fd_k ≜_k^1:d^. Note that the observation operator_k^b4uniformly-completely observes the span of the unstable and neutral BLVs, and thus ford≥4,_k^bdsatisfies the sufficient criterion for filter stability in perfect dynamics discussed by Bocquet et al. <cit.>.The operator_k^b4likewise satisfies the necessary and sufficient detectability criterion for filter stability perfect dynamics of Frank & Zhuk <cit.>.On the other hand, the operator^fd_kobserves the span of the leadingdFLVs.Unlike the BLVs, the FLVs define a QL decomposition of the span of the covariant Lyapunov vectors <cit.>[see equation (53)].This implies that the columns of the operator_k^f4actually spans the orthogonal complement to the stable Oseledec spaces.Therefore,_k^f4satisfies the criterion of Frank & Zhuk <cit.>, but will not generally satisfy the condition of Bocquet et al. <cit.>.We perform parallel experiments, fixing the sequence of linear propagators_k, and the initial prior error covariance_0 ≜_n, while varying the choice of the observation operator and the observational dimensiond.In each parallel experiment, we study the average forecast uncertainty for the full rank Kalman filter as described by Frobenius norm of the forecast error covariance_k, averaged over10^5assimilations, neglecting a separate filter stabilization period of10^4assimilations.For eachd∈{ 4, ⋯, 9}, we compare the following choices of observation operators: (i)_k^bd; (ii)_k^fd; (iii)_k ≜_k^for randomly drawn orthogonal matrices,_k ∈ℝ^n ×d; and (iv) a fixed network of observations, given by the leadingdrows of the identity matrix, i.e.,_k ≜_d×n.We also compute the average Frobenius norm of the forecast error covariance for full dimensional observations, with_k ≜_n.In each experiment, the observational and model error covariances are fixed as_k ≜_dand_k ≜_n.For eachi, the value ofΨ^i_kis averaged over the10^5assimilations.In <ref>, we plot the average Frobenius norm of the Kalman filter forecast error covariance matrix as a function of the number of observations,d.We consider the observation configurations^bd_k,^_kand_n(plotted horizontally).The average values ofΨ_k^ifori=7, ⋯,10are also plotted horizontally.While the observational dimensiond < 7, the average uncertainty for the Kalman filter with random observations, or observations in the BLVs, is greater than the average variance of the unfiltered error along^7_k.Similarly, in <ref> we consider the configurations with observations defined by_k^bd,^fd_kand_d ×n.The average values ofΨ_k^ifori=5, ⋯,8are plotted horizontally.The variance of the unfiltered error in^5_kexceeds the uncertainty of the Kalman filter in every configuration.The Kalman filter with observations fixed, or in the FLVs, do not obtain comparable performance with the unfiltered error in^6_kuntild≥6.Only the LLEs of^i_kfori=8,9,10are sufficiently stable to bound the unfiltered errors below the Kalman filter with a fully observed system.Our results have strong implications for the necessary rank of the gain in ensemble-based Kalman filters.In perfect, weakly nonlinear models, the ensemble span typically aligns with the leading BLVs <cit.>.From the above results, we conclude that the effective rank of the ensemble-based gain must be increased to account for weakly stable BLVs of high variance in the presence of model errors.The perturbations of model errors excited by transient instabilities in these modes can lead to the unfiltered errors becoming unacceptably large compared to the filtered errors.In <ref> and <ref>, the choice of observations in the span of the leading BLVs dramatically outperforms the observations in the span of the leading FLVs, or fixed observations.Likewise^bd_kmakes a slight reduction to the overall forecast error over a choice ofdrandom observations.As the span of the leadingn_0FLVs is orthogonal to the trailing, stable Oseledec spaces, this choice is can be considered closer to the minimum necessary observational constraint on the forecast errors.Particularly, the kernel of^f n_0_kis identically equal to the sum of the stable Oseledec spaces.This suggests that a necessary and sufficient condition for filtered boundedness can be described in terms of the observability of then_0leading FLVs, similar to the criterion of Frank & Zhuk <cit.>.While it is not necessary, the sufficient condition of Bocquet et al. <cit.> leads to a lower filter uncertainty as the span of the leadingn_0BLVs generally contains the largest projection of the forecast error.This suggest that observing the leading eigenvectors of_k+1^_k+1may generally outperform observing the leading eigenvectors of^_k+1 _k+1when the time between observationsδ= t_k+1 -t_kleads to significant differences in the linear expansions, as was noted as an alternative design by Law et al. <cit.>.For operational forecasting, this supports the use of the breeding technique <cit.> to target observations, over using the axes of forward growth.§ CONCLUSION This work formalizes the relationship between the Kalman filter uncertainty and the underlying model dynamics, so far understood in perfect models, now in the presence of model error.Generically, model error prevents the collapse of the covariance to the unstable-neutral subspace and our <ref> and <ref> characterize the asymptotic window of uncertainty.We provide a necessary condition for the boundedness of the Kalman filter forecast errors for autonomous and time varying dynamics in <ref> and <ref>: the observational precision, relative to the background uncertainty, must be greater than the leading instability which forces the model error.Particularly, <ref> proves that forecast errors in the span of the stable BLVs remain uniformly bounded, in the absence of filtering, by the effect of dynamic dissipation alone.The uniform bound on the errors in the span of the stable BLVs extends the intuition of AUS to the presence of model error, but with qualifications.Studying this uniform bound with <ref>, we identify an important mechanism for the growth of forecast uncertainty in sub-optimal filters: variability in the LLEs for asymptotically stable modes can produce transient instabilities, amplifying perturbations of model error.The impact of stable modes close to zero differs from similar results for nonlinear, perfect models by Ng. et. al. <cit.>, and Bocquet et. al. <cit.>, where the authors demonstrate the need to correct weakly stable modes in the ensemble Kalman filter due to the sampling error induced by nonlinearity.Likewise, this differs from the EKF-AUS-NL of Palatella & Trevisan <cit.>, that accounts for the truncation errors in the estimate of the forecast uncertainty in perfect, nonlinear models.Our work instead establishes the fundamental impact of these transient instabilities as a linear effect in the presence of model errors.In addition to our necessary criterion for filter boundedness, in <ref> we discuss the criteria of Bocquet et al. <cit.> and Frank & Zhuk <cit.> in relation to dynamically targeted observations.Our numerical results suggest how these sufficient, and respectively necessary and sufficient, criteria can be extended to the presence of model errors. Moreover, we distinguish between the minimal necessary observational constraints for filter boundedness and more operationally effective, sufficient designs.Particularly, our results suggest that while it may be necessary that the observations uniformly completely observe the span of the unstable-neutral FLVs, it is sufficient and improves performance to uniformly completely observe the span of unstable-neutral BLVs.In terms of operational forecasting, this strongly supports the use of bred vectors to target observations to constrain the forecast errors.<ref>, <ref>, <ref> and the results of <ref> suggest that as a theoretical framework for the ensemble Kalman filter, AUS may be extended to the presence of model errors.By uniformly completely observing and correcting for the growth of uncertainty in the span of the unstable, neutral and some number of stable BLVs, reduced rank filters in the presence of model errors may obtain satisfactory performance. In practice, one may compute off-line the typical uncertainty in the stable BLVs via <ref> and determine the necessary observational and ensemble dimension at which unfiltered forecast error has negligible impact on predictions.However, computational limits on ensemble sizes may make this strategy unattainable in practice — the impact of these unfiltered errors on the performance of a reduced rank, sub-optimal filter is the subject of the direct sequel to this work <cit.>.§ ACKNOWLEDGMENTSThe authors thank two anonymous referees, and their colleagues Karthik Gurumoorthy, Amit Apte, Erik Van Vleck, Sergiy Zhuk and Nancy Nichols, for their valuable feedback and discussions on this work.CEREA is a member of the Institut Pierre-Simon Laplace (IPSL). plain | http://arxiv.org/abs/1707.08334v4 | {
"authors": [
"Colin Grudzien",
"Alberto Carrassi",
"Marc Bocquet"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170726092811",
"title": "Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error"
} |
[email protected]^1Department of Physics, Columbia University, New York, NY, USA ^2Department of Physics, University of Seoul, Seoul 02504, Korea ^3Centre for Advanced 2D Materials, National University of Singapore, Singapore 117546, Singapore ^4Deparment of Physics, Faculty of Science, National University of Singapore, Singapore 117542, Singapore ^5National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan ^6Yale-NUS College, Singapore 138527, Singapore ^7National High Magnetic Field Laboratory, Tallahassee, FL, USA Heterostructures of atomically-thin materials have attracted significant interest owing to their ability to host novel electronic properties fundamentally distinct from their constituent layers <cit.>. In the case of graphene on boron nitride, the closely-matched lattices yield a moiré superlattice that modifies the graphene electron dispersion <cit.> and opens gaps both at the primary Dirac point (DP) and the moiré-induced secondary Dirac point (SDP) in the valence band <cit.>. While significant effort has focused on controlling the superlattice period via the rotational stacking order <cit.>, the role played by the magnitude of the interlayer coupling has received comparatively little attention. Here, we modify the interaction between graphene and boron nitride by tuning their separation with hydrostatic pressure.We observe a dramatic enhancement of the DP gap with increasing pressure, but little change in the SDP gap. Our surprising results identify the critical role played by atomic-scale structural deformations of the graphene lattice and reveal new opportunities for band structure engineering in van der Waals heterostructures.Dynamic band structure tuning of graphene moiré superlattices with pressure Cory R. Dean^1 December 30, 2023 ===========================================================================Heterostructures fabricated from the mechanical assembly of atomically-thin van der Waals (vdW) crystals represent an exciting new paradigm in materials design. Owing to weak interlayer bonding, 2D crystals with wide ranging characteristics and composition – such as graphene, boron nitride (BN), and the transition metal dichalcogenides – can be readily mixed and matched, without the usual interfacial constraints of conventional crystal growth <cit.>. Moreover, atomic-scale crystalline alignment between the layers often plays a critical role in the resulting device characteristics leading to additional and controllable degrees of freedom. For example, electronic coupling processes that are sensitive to momentum mismatch, such as interlayer tunneling <cit.> or exciton binding <cit.>, can be sensitively tuned by varying the rotational order.For crystals with closely matched lattice constants, moiré interference at zero-angle alignment can additionally result in long-range superlattice potentials, which in turn can lead to entirely new electronic device characteristics <cit.>. BN-encapsulated graphene provides a model example of the variety of electronic properties that can be realized on-demand in a single type of vdW heterostructure. At large relative twist angles, BN acts as a featureless dielectric for graphene, minimizing coupling to extrinsic disorder but otherwise remaining effectively inert <cit.>. However, at small twist angle, coupling to the resulting moiré superlattice (MSL) profoundly alters the graphene bandstructure, giving rise to secondary Dirac cones at finite energy <cit.> while also modifying the Fermi velocity near the Dirac point <cit.>. As a consequence several unusual electronic properties have been observed, such as density-dependent topological valley currents at zero magnetic field <cit.> and the fractal Hofstadter butterfly spectrum at high field <cit.> – recently identified to host integer and fractional Chern insulating states <cit.> and charge density waves <cit.>. Additionally, the MSL opens band gaps at both the primary and secondary graphene Dirac points, which are of particular interest for graphene-based digital logic applications. Numerous techniques have been developed to control the rotational alignment in graphene/BN and related vdW heterostructures <cit.>. Equally important is the spacing between the layers, which dictates the magnitude of interlayer interactions. However, littleexperimental work has been done to characterize or control this parameter, which is often not well-known and considered invariable. Here we demonstrate that by applying hydrostatic pressure to BN-encapsulated graphene, we are able to decrease the interlayer spacing by more than 5%. At small rotation angles the resulting increase in the effective MSL potential substantially modifies the electronic device characteristics. Most dramatically, we observe a divergence in the moiré-induced band gap at the DP with increasing pressure and near doubling over the pressure ranges studied, yielding the largest gap so far demonstrated in pristine monolayer graphene. By contrast, the SDP gap shows little change with pressure. This unexpected result provides new insight into the precise influence of the MSL on the graphene layer and suggests that in addition to electrostatic coupling, lattice-scale deformations play an important role. Our findings reveal that interlayer spacing in vdW heterostructures is both an important and tunable degree of freedom that provides a new route to bandstructure engineering.Fig. 1a shows a cartoon schematic of our experimental setup. We fabricate BN-encapsulated graphene devices using the vdW assembly technique <cit.> and mount them into a piston-cylinder pressure cell with electrical feedthroughs, capable of reaching temperatures below 1 K and magnetic fields above 18 T (see Methods and Supplementary Information 1 for details). The pressure cell sample space is filled with an oil solution which results in a uniform transfer of pressure to the sample. However, since the Young's modulus in the the out-of-plane stacking direction (c-axis) of the vdW crystal is typically a few orders of magnitude smaller than in-plane <cit.>, the pressure primarily results in a c-axis compression. We first examine the effect of applying pressure on a misaligned heterostructure where no MSL effects are present, and find that the oil environment and application of pressure have virtually no effect on the electronic properties of the graphene. Fig. 1b shows low temperature transport acquired from a misaligned device (> 2^∘ relative alignment) at zero magnetic field in both vacuum (with no oil) and under high pressure (1 GPa). The high-density resistance grows by small amount (less than 25 Ω) with pressure in this device. Other similar devices showed no measurable change or even slightly decreasing resistance under pressure, indicating the field effect mobility of the encapsulated device is largely insensitive to the application of pressure (see Supplementary Information 2). The DP resistance is also not strongly modified, and there is no significant pressure dependence on the contact resistance (Fig. 1b inset). Fig. 1c shows similar transport measurements but acquired from an aligned MSL device. Under ambient pressure the device shows excellent transport characteristics with moiré coupling evident by the appearance of two resistance peaks symmetrically located about the DP at roughly ±3.3× 10^12 cm^-2, corresponding to a relative rotation angle of ∼ 0.6^∘. Moreover, the large resistivities exceeding 25 kΩ at the DP and SDP suggest sizable band gaps at those points in the band structure. Notably, as we apply high pressure up to 2.3 GPa, we find that the positions of the secondary peaks move symmetrically towards the DP. To understand this effect, we track the back gate capacitance per unit area C_g = en/V_g = ϵϵ_0/t, where e is the charge of the electron, ϵ is dielectric constant of BN, ϵ_0 is the vacuum permittivity, t is the thickness of the BN, V_g is the applied gate bias, and n is the carrier density. We measure the density in two ways; both from the Hall effect and magnetoresistance oscillations (Figs. 2a and b, see Methods). Notably, all devices exhibit a universal increase of C_g with pressure of roughly 9% per GPa independent of their relative alignment (Fig. 2c), which must arise due to a decrease in t and/or an increase in ϵ. To deconvolve the two, we have performed LDA ab initio simulations of bulk BN multilayers under pressure and find approximately 2.5% compression per GPa (green curve in the top inset of Fig. 2c), in remarkable quantitative agreement with previous X-ray diffraction measurements from Ref. <cit.> (reproduced in the red curve). The remaining increase in gate capacitance is therefore attributed to an increase in the dielectric constant of the BN of roughly 6% per GPa (blue curve in the bottom inset of Fig. 2c). Our ab initio simulations also confirm that the BN dielectric constant should grow with pressure, though the effect is predicted to be a few times smaller (green curve). Taken together, this suggests that pressure is able to sensitively tune both the interlayer spacing between layered 2D materials and their dielectric properties <cit.>. Returning to the transport measurements of the MSL device, we find that the three resistance peaks align exactly at all pressures when replotted against charge carrier density n (Fig. 1c inset), suggesting that while the graphene and BN layers move closer together, the relative rotation angle and moiré period remains fixed under pressure.A second notable feature of the MSL transport is that the resistance at the DP grows strongly with pressure, increasing by roughly 100 kΩ between 0 GPa and 2.3 GPa. We investigate the DP response in more detail by measuring its temperature dependence. Fig. 3a shows an Arrhenius plot of the DP conductivity, σ_DP, versus inverse temperature for various pressures, where for each pressure a linear fit to the simply activated regime (red dashed lines) gives a measure of the activation gap (see Methods). The resulting DP gap, Δ_p, is shown versus pressure in Fig. 3b (square markers). The gap is found to diverge with increasing pressure, and is enhanced by nearly a factor of 2 at the highest pressure studied for the ∼ 0.6^∘ device presented here. Similar gap enhancement was observed in other nearly-aligned samples as well, and we find that the order and direction of pressure cycling do not influence the measurement (see Supplementary Figure 3a). We similarly measure the pressure dependence of the valence band SDP gap, Δ_s, plotted with circle markers in Fig. 3b. In significant contrast to the DP gap, the SDP gap is nearly unresponsive to pressure. The insets of Fig. 3a schematically illustrate the inferred band structure modifications with interlayer spacing, showing a growing Δ_p but a fixed Δ_s. As a simple approximation, we may consider the increasing DP gap with pressure to result from increasing MSL coupling owing to the decreasing interlayer spacing. In this case, we expect that the SDP behavior might respond in a similar way, and from this perspective its insensitivity to pressure is surprising. We note, however, that despite considerable effort <cit.> consensus is still lacking as to the exact origin of these band gaps. Lattice scale deformations (in-plane strains and out-of-plane corrugations) of the graphene layer are expected to play an important role <cit.>, but the exact equilibrium structure of graphene in contact with BN remains poorly understood, including whether these deformations even exist in fully BN-encapsulated devices <cit.>. In an effort to understand our observed behavior, we therefore first consider a rigid graphene lattice and examine the effects of pressure on the heterostructure using a combination of ab initio and analytical models (see Supplementary Information 4 - 6). The interlayer electronic coupling between the graphene and BN, V_0 = Ve^-β (z_0-z_r), is highly sensitive to pressure, with ab initio predictions indicating that the average interlayer spacing z_0 should decrease by ∼0.07 Å/GPa. Here V≈ 18 meV is the interlayer electronic coupling, z_r ≈ 3.35 Å is the equilibrium average interlayer spacing between the graphene and BN, and β≈ 3.2 Å^-1 quantifies the rate of increase of interlayer coupling when the spacing is reduced. Generically, both Δ_p and Δ_s should scale proportionally to V_0 under applied pressure, however this is in stark contrast to our experimental observations where Δ_s in particular exhibits little pressure dependence. Even at 0 GPa, a rigid model does not properly capture the experimentally observed hierarchy of the gaps, predicting a large Δ_s and a Δ_p ≈ 0 (Fig. 4a).Next we consider the potential role of atomic-scale graphene deformations. By considering realistic values of the graphene elastic deformation potential, our atomic structure models suggest that it is favorable for the graphene lattice to both corrugate out-of-plane and strain in-plane on the moiré scale to minimize the overall stacking potential with the BN substrate (Fig. 4b). These deformations break the sublattice symmetry of the graphene, resulting in a finite average mass term in the Hamiltonian which opens a sizable Δ_p. There are also moiré-induced electrostatic potentials and pseudomagnetic fields which contribute to Δ_s in addition to the mass term. Including corrugations alone opens a small Δ_p at 0 GPa, but still does not recover the observed gap hierarchy of Δ_p > Δ_s (Fig. 4a). By additionally including strains, we are able to recover the observed 0 GPa gap hierarchy (Fig. 4a) as well as the diverging Δ_p and flat Δ_s response with pressure. One such example, whose corrugations and strains are obtained for BN-encapsulated graphene by a model partly informed by ab initio calculation inputs (see Supplementary Information 4 - 6), is shown in Fig. 4c, and predicts gaps in remarkably good quantitative agreement with our experimental data (dashed curves in Fig. 3b).While more experimental and theoretical work is necessary to understand the exact equilibrium structure of aligned graphene on BN, the varying evolution of the gaps with pressure directly rule out the possibility of a rigid graphene lattice in these structures, and moreover demonstrates that these gaps are of fundamentally different origin as the SDP cone is not a true replica of the DP cone. This suggests the possibility to independently control the magnitude of the two gaps, as well as other features of the moiré band structure and magnetoresponse, by selectively engineering specific lattice deformations. Additionally, sufficiently strong enhancement of the interlayer coupling could drive a phase-transition to the fully commensurate (lattice-matched) stacking configuration <cit.>, marked by the absence of a MSL but the emergence of strong sublattice-symmetry breaking in the graphene and a gap many times room temperature at the Dirac point <cit.>. More generally, our results indicate that a wide variety of vdW heterostructure properties may be tunable by controlling the interlayer coupling strength with pressure. For example, the band structures of Bernal-stacked and twisted bilayer graphene depend critically on interlayer hopping terms <cit.>, as do the strength of proximity-induced spin-orbit interactions in graphene resting on heavy transition metal dichalcogenides <cit.>.§ METHODS§.§ Application of pressure In order to control the interlayer spacing in vdW heterostructures, we fabricate graphene devices encapsulated in BN with a graphite bottom gate on Si/SiO_2 wafers (Fig. 1a) and make one-dimensional electrical contact using standard reactive ion etching and electron beam patterning and deposition techniques <cit.>. Alignment to one of the encapsulating BN layers is achieved through either optical matching of crystalline edges <cit.> or through thermal self-rotation <cit.>. We then mount the device on a stage with a pre-wired feed-through and affix wires onto the electrodes by hand using silver paint. The stage is then enclosed by a Teflon cup filled with a hydrostatic pressure medium (Daphne 7373 or 7474 oil) <cit.>, and the cup is fit into the inner bore of a piston-cylinder pressure cell and loaded to the desired pressure using a hydraulic press (Fig. 1a). Finally, the pressure cell is affixed to a probe for electrical characterization at low temperature and high magnetic field (see Supplementary Information 1 for further details of the experimental setup). The pressure was determined by measuring the photoluminescence of a ruby crystal at both room- and low-temperature.§.§ Extraction of capacitance and band gaps We extract n(V_g) for each pressure through either the dispersion of the quantum Hall states in high magnetic field as n = ν e B/h, where ν is the filling factor (Fig. 2a), or from the low field Hall resistance R_xy before the onset of strong Shubnikov-de Haas oscillations as n = 1/eR_xy (Fig. 2b). Quantum Hall states move symmetrically closer to the DP with increasing pressure, and the slope of the Hall resistance decreases with pressure, both implying a growing n(V_g) with increasing pressure. In Fig. 2c, C_g is normalized to its measured value at 0 GPa to account for the different thicknesses of the bottom BNs across the different devices. Additionally, the ab initio value of ϵ at 0 GPa is normalized to match the average experimental value of ≈ 3. We extract the band gaps Δ of the DP and SDP at each pressure according to the thermally activated response where σ_DP(T) ∝ e^-Δ/2 k T, where k is the Boltzmann constant.§ ACKNOWLEDGEMENTS We thank Pablo San-Jose, Justin Song, Andrey Shytov, Leonid Levitov, John Wallbank, and Pilkyung Moon for valuable theoretical discussions. This work was supported by the National Science Foundation (DMR-1462383). CRD acknowledges partial support from the David and Lucille Packard foundation. We acknowledge Stan Tozer for use of his 16 T PPMS which is partially supported as part of the Center for Actinide Science and Technology (CAST), an Energy Frontier Research Center (EFRC) funded by the Department of Energy, Office of Science, Basic Energy Sciences under Award Number DE-SC0016568. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-0654118, the State of Florida and the U.S. Department of Energy and additionally provided support for pressure cell development through User Collaboration Grant Program (UCGP) funding. JJ and NL have been supported by the Korean NRF through the grant NRF-2016R1A2B4010105 and the Korean Research Fellowship grant NRF-2016H1D3A1023826, and BLC has been supported by the grant NRF-2017R1D1A1B03035932. EL and SA are supported by the National Research Foundation of Singapore under its Fellowship program (NRF-NRFF2012-01). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and JSPS KAKENHI Grant Numbers JP15K21722.§ AUTHOR CONTRIBUTIONS M.Y. and C.R.D. conceived the experiment. M.Y. fabricated the samples, analyzed the data and wrote the paper. M.Y. and D.G. performed the experiments. J.J., E.L. and S.A. developed the theory. N.L. and B.L.C. calculated the ab initio potentials. K.W. and T.T. grew the hBN crystals. C.R.D. advised on the experiments. § SUPPLEMENTARY INFORMATION§.§ Details of pressure experiments All devices in this study consist of monolayer graphene encapsulated between two layers of boron nitride. The BNs were generally between 20 - 60 nm thick, though our results did not depend on this in any noticeable way. The encapsulated stack sits on a flake of graphite which acts as a local back gate. Fig. <ref>a shows an image of a completed stack on a transfer slide. For aligned samples, the graphene (outlined with a dotted white line) was either intentionally aligned to one of the BNs using straight edges, or rotated to an aligned position with thermal heating during the transfer process. The final stack was partially etched into a Hall bar geometry, leaving some of the bottom BN unetched to prevent the metal contacts from shorting to the graphite gate (Fig. <ref>b). The Hall bar was intentionally kept small (≈ 6 μm by 2 μm) to keep the pressure as uniform as possible across the entire device. The entire device sits on a Si/SiO_2 wafer, which must be diced to approximately 2 mm by 2 mm to fit into the inner bore of the pressure cell (Fig. <ref>c).To prepare the pressure cell, a clean metal stage (Fig. <ref>d) is threaded with insulated copper 75 μm wires with tinned ends, which are epoxied in place using Stycast 2850 FT using 24LV catalyst (Fig. <ref>e). A ruby crystal is glued to the tip of a thin optical fiber that is fixed in place by the wires for in situ pressure calibration. The sample is then glued above the fiber (Fig. <ref>f). Flexible 15 μm Pt wires are soldered to the copper wires and affixed by hand to the gold sample contacts using Dupong 4929N silver paste (Fig. <ref>g). Next, a Teflon cup is filled with the pressure medium (Daphne 7373 or 7474 oil) and carefully fitted over the sample and onto the stage (Fig. <ref>h). The sample is now completely encapsulated in oil. For the special case of vacuum measurements (0 GPa), no oil is loaded into the Teflon cup. The stage/teflon cup is then fitted into the inner bore of a piston cylinder cell (Fig. <ref>i), and a hydraulic press is used to compress the top of the Teflon cup (Fig. <ref>j). The oil medium hydrostatically increases in pressure as the cup is compressed. A top locking nut is tightened as more force is applied on the hydraulic press until the desired load is reached, at which point the hydraulic press is backed off, and the threads of the locking nut hold the pressure in the cell. The cell is then affixed to the end of a probe (Fig. <ref>k) for low-temperature, high-field measurements. Changing the pressure requires warming the cell to room temperature, adding or removing load with the hydraulic press, and re-cooling the cell.Although the oil does not influence the electronic properties of the device, special care must be taken to account for its presence during measurements. The oil freezes at around 200 K as it is cooled at ambient pressure, and the freezing point moves to higher temperatures at higher starting loads. The pressure in the cell also drops as the oil cools inside the cell, with a larger relative drop in pressure at smaller initial loads. For example, a starting load below ∼ 0.3 GPa at room temperature will result in nearly ambient conditions at low temperature, while at pressures above 2 GPa there is virtually no change between the room temperature and low temperature pressures. We have found that the primary consequence of this effect is that starting at room temperature loads of roughly 0.5 GPa or less is dangerous for the device, as the stack is typically torn when the oil freezes below these pressures. However, above these pressures the devices always survive the cooling, and we have not noticed any effect of the oil freezing in transport measurements. Finally, special care is taken to account for the large thermal load of the pressure cell when performing temperature sweeps to measure band gaps. The temperature is swept slowly to keep the sample as close to equilibrium as possible, ideally at 0.5 K/min, and no faster than 1.5 K/min. §.§ Pressure dependent transport in other devices The behavior of high density resistance of devices under pressure varies slightly across devices. For Device P2 shown in Fig. 1b of the main text, an increase of 15-25 Ohms is observed at high density under pressure. However, this is not always the case, as is shown for Device P3 in Fig. <ref>. This device exhibits virtually no pressure dependent hole-side resistance, and shows a decrease in the electron-side resistance at high pressure. We have measured a total of four gapped devices as a function of pressure. Fig. <ref>a plots the gap of the primary DP, where the data from Device P1 is copied from the main text. Error bars are left off for clarity, but are similar in magnitude to those in Fig. 3b of the main text. Devices P3 and P4 are slightly misaligned, such that the SDPs are outside the accessible density range in the device - therefore the rotation angle is unknown, but must be larger than ∼ 2^∘. Previous work has demonstrated that devices with misalignment angles as large as ∼ 5^∘ can exhibit band gaps in transport <cit.>. In that study, the misalignment angle was determined by scanning tunneling topography measurements, however this is not possible in our devices due to the encapsulating top BN layer. Nevertheless, these devices show clear activation gaps, with insulating behavior at low temperatures and a decrease of the device conductivity by nearly an order of magnitude in the thermally activated regime at high temperatures (Fig. <ref>a - c). The magnitude of the gap grows with pressure in all three samples.The fourth gapped device (P5) was very well aligned, with an estimated misalignment angle of ∼0.1^∘. However, the device was significantly more disordered than all the other devices in this study, and as a result the onset of the variable range hopping regime occurred at significantly higher temperature. Consequentially, the simply-activated regime spanned only a small range of temperature, and therefore the gap extraction is significantly more uncertain than the other gapped devices examined. Nevertheless, the primary DP showed an unambiguous enhancement with pressure, while the SDP appeared relatively insensitive to pressure (Fig. <ref>b). This behavior is completely consistent with the clean aligned device (P1) presented in the main text, providing further evidence of the fundamental difference between the primary and secondary DPs. Fig. <ref> additionally shows the gap extraction for the SDP of the aligned device P1 in the main text Fig. 3b, demonstrating that this gap does not change with pressure.The enhancement of the band gap does not depend on the history of the pressure the device has been exposed to, rather it responds directly to the pressure the device is under at the time of measurement. To illustrate this, Fig. <ref>a annotates the order in which the gaps were acquired. In no case does the measured gap fall out of the anticipated sequence. This is an especially robust effect, as it does not depend on whether the device becomes more or less insulating with pressure. Fig. <ref> tracks the CNP conductivity across a number of devices down to the lowest temperatures measured (∼ 2 K). Typically the devices become more insulating at low temperatures for higher pressures, however this is not always the case. For Devices P1 and P3, the low temperature conductivity becomes out of sequence upon unloading pressure, or in reloading pressure after previously pressure cycling and returning to 0 GPa (see the annotations in Fig. <ref>a for the order of curve acquisition). However, even in Device P4, where the pressure was loaded up uniformly, the low temperature conductivity still depends non-monotonically on pressure. This suggests that small changes in the amount of disorder in the device and potentially even the details of its microscopic organization can ultimately influence how insulating the device becomes. However, no matter how the DP conductivity behaves at low temperature, the band gap is always enhanced by pressure, suggesting that the gap enhancement is a robust property of the heterostructure and depends most critically on the interlayer interaction strength between the graphene and the BN rather than small changes in device disorder. §.§ Effects of pressure in the quantum Hall regime The disorder in these devices can also be characterized by considering the amount of density n necessary to switch the Hall resistance R_xy from positive to negative (Fig. <ref>a), as this gives a measure of the effective magnitude of the electron-hole puddles in the bulk. Fig. <ref>b shows that the device disorder is not strongly dependent on pressure, growing by less than the extraction uncertainty over the applied pressure range. Curiously, pressure serves to qualitatively improve the quantum Hall response at high magnetic field. Fig. <ref>c shows one such example, where both integer and fractional quantum Hall states develop more clearly under pressure. This effect is observed in every device examined. It is even more surprising that this improvement persists even after the pressure is released (Fig. <ref>d), as this suggests the need for an explanation describing an irreversible effect. At present, we may only speculate as to the nature of this effect. First, it is important to note that despite the high electronic quality of the devices, only the main sequence of IQH gaps at ν = -2, -6, -10, etc., are typically clearly developed in initial measurements in vacuum, with the symmetry broken gaps still developing (Fig. <ref>d). This is not an intrinsic property of the graphene – rather it is symptomatic of the metal contacts sitting above the graphite gate in the device geometry employed here (Fig. <ref>b). While the reason for this is currently not well understood, the situation may be analogous to previous observations in GaAs, where a partial reduction of the carrier concentration in the graphene just in front of the contacts can lead to a depletion region along the boundary, which may impede the ability to observe well-developed quantum Hall plateaus <cit.>. This problem has been addressed previously by leaving Si-gated regions of the graphene Hall bar leads to act as contacts, where the density at the boundary can vary more smoothly <cit.>. However, the devices in this study were intentionally kept small to ensure the pressure is as uniform as possible across the device, and so this geometry was not utilized.Applying pressure may, for instance, provide a way to reduce this depletion region at the contacts, permitting better coupling to the QH edge modes. This could arise due to self-cleaning of contaminants along the contact boundaries, or if the metal making edge contact to the graphene forms a better bond to the graphene edge and lowers the work function mismatch at the boundary. Both of these effects could in principle persist even after the pressure is released. We find that the contact resistance is not significantly modified by pressure at B = 0 T (Fig. 1b of the main text), pointing to an improvement arising from the detailed electrostatics at the contact barrier in high magnetic field, or to a reduction of non-local contamination which may be most relevant in the QH regime. Further study is necessary to understand the exact nature of this effect.Finally, we examine the effects of pressure on the Hofstadter butterfly spectrum of the aligned Device P1. Fig. <ref> shows the high field response at ambient conditions (panel a) and at 2.3 GPa (panel b). As pressure enhances the effective strength of the moiré potential, it should also influence the relative size of the LL gaps originating from the secondary DPs. The qualitative behavior of the two butterfly maps seems to be in qualitative agreement with this expectation, with stronger features originating from the secondary DPs under pressure. However, this must be deconvolved with the overall change in the QHE behavior with pressure, and a full investigation of this effect is outside the scope of this work.§.§ First principle calculations and parametrizations The atomic and electronic structures of graphene and hexagonal boron nitride (G/BN) interfaces are captured using input from ab-initio density functional theory (DFT) calculations. We rely on exact exchange and random phase approximation (EXX+RPA) input for atomic structure, while our treatment of the electronic structure is based on local-density approximation (LDA). §.§.§ Calculation of the dielectric constant from DFTOne of the striking features of the experiment is an increase of the dielectric constant as a function of pressure.Our DFT calculations for bulk confirm this behavior showing increases on the order of 3% for the applied pressures up to 2.5 GPa as shown in Fig. <ref>.When the pressure is modified from P=0 to P=2.5 GPa, the EXX+RPA equilibrium interlayer distance is squeezed by ∼ 0.2 as we will discuss in the following subsection.First principles calculations based on density functional theory (DFT) were performed using Quantum Espresso <cit.> under LDA within the plane wave basis set. Norm-conserving Vanderbilt pseudopotentials were used.The structures are fully optimized without anysymmetry constraint by using conjugate gradient method. The convergence criterion for the force on each ion is taken to be less than 0.005 eV/ while the energy is converged with a tolerance of 10^-5 eV. In an insulating BN, the optical properties are dominated with the direct inter band contributions to the absorptive or imaginary part of dielectric function <cit.>. The matrix elements for a given interband transition β→α for a set of plane wave Bloch function: | ψ_k,n⟩= e^iG · r u_k,n= 1/√(V)∑_G a_n,k,G e^i(k + G) · r is given by: M̂_α ,β= ( ∑_G a_n,k,G^* a_n',k,G G_α)( ∑_G a_n,k,G^* a_n',k,G G_β)These matrix elements account only for the interband transitions, i.e the electric-dipole approximation where the momentum transfer is zero. The imaginary part of the dielectric tensor ε_2α ,β is a response function that comes from a perturbation theory within adiabatic approximation. All of the possible transitions from the occupied to the unoccupied states without local field effects are given by the imaginary dielectric function: ε _2α ,β= 4π e^2 /Ω N_k m^2 ∑_n,n'∑_k M̂_α ,β/(E_k,n'- E_k,n )^2 {f(E_k,n )/E_k,n'- E_k,n+ ħω+ iħΓ + f(E_k,n )/E_k,n'- E_k,n- ħω- iħΓ},where Γ is the adiabatic parameter which must be zero due to conservation of energy. So, the equation can be rewritten in terms of Dirac delta functions: ε_2α ,β= 4π e^2 /Ω N_k m^2 ∑_n,n'∑_k M̂_α ,β f(E_k,n )/(E_k,n'- E_k,n )^2 [ δ (E_k,n'- E_k,n+ ħω ) + δ (E_k,n'- E_k,n- ħω )],However, interaction with the electromagnetic field which also take place even in the absence of photons, i.e spontaneous emission, brings intrinsic broadening to all excited states that leads to a finite life time (Γ > 0). In the limit of non-vanishing Γ, the dielectric tensor takes the Drude-Lorentz form <cit.>: ε _2α ,β= 4π e^2 /Ω N_k m^2 ∑_n,kdf(E_k,n )/dE_k,nηωM̂_α ,β/ω^4- η ^2 ω ^2+ 8π e^2 /Ω N_k m^2 ∑_nn'∑_k M̂_α ,β/E_k,n'- E_k,nΓω f(E_k,n )/[(ω _k,n'- ω_k,n )^2- ω^2 ]^2+ Γ^2 ω ^2 ,and the real part of the dielectric function is derived from the Kramers-Kronig transformation:ε_1α ,β= 1 + 2/π∫_0^∞ω 'ε _2α ,β (ω ')/ω'^2- ω ^2 dω'.Calculations of the dielectric function were carried out using the EPSILON packagedistributed with Quantum Espresso <cit.>. §.§.§ DFT deformation with pressure The mechanical behavior of nearly aligned graphene and boron nitride heterostructure (G/BN) is influenced by the encapsulating top BN layer. The interlayer interaction of nearly aligned G/BN is described using the assumption that they can be viewed as a collection of different stacking registries to be treated independently, from which the atomic structure of the whole sample can be obtained.We obtain the equilibrium spacing for each stacking point z_𝐬 (𝐬∈{AA, AB, BA}, see Fig. <ref>) by minimizing the total potential energy as a function of the separation distance between the top and bottom BN layer d. We also assume that the elastic resistance of the graphene sheet is negligible in the out-of-plane direction.The interlayer potential energy of the graphene sheet considering the effects of bottom and top BN layers is given by:V(z_AA,z_AB,z_BA,d) = 1/3∑_𝐬[V_𝐬(z_𝐬) + V̅(d - z_𝐬)]where V̅(d-z) = 1/3∑_𝐬 V_𝐬(d-z) accounts for the averaging of the interlayer potential between the top BN and the graphene sheet due to their misalignment.We use for the interlayer interaction potentials the exact exchange and random phase approximation (EXX+RPA) for G/BN and BN/BN interactions as implemented in VASP. <cit.> In equilibrium, the local pressure exerted by the encapsulating layer is equally balanced by that coming from the underlying substrate, i.e. P_𝐬(z_𝐬) = P̅(d - z_𝐬). Defining the interlayer distance between graphene and the top layer z'_𝐬 = d - z_𝐬, we can write the equilibrium condition as 1/3∑_𝐬 P_𝐬(z_𝐬) = 1/3∑_𝐬P̅(z'_𝐬) = P, where P is the external pressure applied on the system. Fig. <ref> illustrates how the corrugation profiles of a graphene sheet change with pressure under the influence of two BN layers.§.§.§ Electronic coupling parameters and elastic properties of G/BN In our treatment of the moiré bands in G/BN, we consider modifications in the low-energy regime of graphene electronic structure which are introduced by the interlayer tunneling. The modification is captured by two interlayer coupling terms V and Ṽ whose values can be obtained from first-principle calculations. V and Ṽ can be calculated from the following expressions <cit.>:V = 1/2[t_NC^2/|ϵ_N|-t_BC^2/|ϵ_B|]≈ 0.04meV,Ṽ = √(3)/2[t_NC^2/|ϵ_N|+t_BC^2/|ϵ_B|]≈ 10meV, where t_NC (t_BC) stands for the interlayer hopping term between C–N (C–B) atoms which highly depends on the interlayer separation, and ε_N (ε_B) are the on-site energies of N (B) atoms. Both parameters are calculated from the parametrizations of the interlayer terms in Ref. <cit.>, at a separation of z_r = 3.35.To fully account for the modification in the electronic couplings, ab-initio method in Ref. <cit.> introduces three exponential factors {β_AA,β_AB,β_BB}, each quantifies the rate of increase of each sublattice term (β_i > 0). The values of β_i's are determined by the amount overlap between carbon orbitals with the orbitals of the underlying BN atoms, for which it ranges between 3.0–3.3 ^-1 <cit.>.As sample deformation is expected to play an important role in determining the band features, we model the structural relaxation within Born-von Karman plate theory in which the elastic properties of graphene are fully characterized by two Lamé constants: λ_g = 3.25eV ^-2 and μ_g = 9.57eV ^-2 <cit.>. One mechanism through which deformation leaves an imprint on the bands is the intralayer effects which are induced by the change of the carbon on-site energy and nearest-neighbor hoppings as each carbon atom is displaced with respect to its neighbors. They give rise to additional terms in the moiré couplings which are equivalent to the electrostatic potentials and the pseudomagnetic fields, quantified by γ_V≈ 4.0 eV and γ_B≈ 4.5 eV respectively <cit.>. §.§ Theoretical Model Graphene and BN are two-dimensional materials with hexagonal crystal structures, but with different lattice constants. When graphene is deposited on a BN substrate with near perfect alignment (small twist angles), the slight lattice mismatch(ε = a_G-a_BN/a_BN = -1.7%), leads to the formation of a moiré structure which is characterized by a wavelength that is one or two orders of magnitude larger than graphene's unit cell. As a consequence, the dynamics of the low energy Dirac electrons of graphene is governed by the corresponding long-wavelength component of the electronic couplings. Similarly, the interlayer potentials and deformations in G/BN are also largely characterized by this long-wavelength theory.It is by now established that the dominant contributions are captured by the first harmonics of the moiré structure characterized by a set of vectors G⃗_m, that is related with the original graphene's lattice vectors g⃗_m:g⃗_m =R̂_2π (m-1)/6 (0,g) ,G⃗_m = [(1+ε)-R̂_θ]g⃗_m≈εg⃗_m-θẑ×g⃗_m,where m ∈{1,2,...,6}, R̂_θ denotes a rotation by θ, g = 4π/3a is the length of graphene lattice vectors with a≈ 1.42 stands for the carbon-carbon distance, and the approximate sign indicates the approximation within small twist angle limit (θ≪ 1).We also show in Fig. <ref>, that G⃗_m defines the moiré Brillouin Zone (mBZ), whose lateral dimension is scaled by ε̃=√(ε^2+θ^2)with respect to graphene BZ that amounts to values ≲ 5% for twist angles θ≲ 2^∘. In G/BN system which possesses triangular symmetry, we can define two periodic functions within the first harmonics: (1) f_1(r⃗) = ∑_mexp(iG⃗_m·r⃗) which satisfies inversion and hexagonal symmetries, and (2) f_2(r⃗) = -i∑_m(-1)^mexp(iG⃗_m·r⃗) which is asymmetric under inversion.§.§.§ Definition of symmetric and antisymmetric deformations Deformation of the sample results from the minimization of the energy functional, leading to the following equation of motion for the in-plane components:ε+θ(ẑ×1)/ε̃^2A_g ∇⃗[U_1f_1(r⃗) + U_2 f_2(r⃗)] ≈ 2μ_g[∇⃗^2u⃗-1/2(ẑ×∇⃗) (∂_xu_y-∂_yu_x)] +λ_g∇⃗(∇⃗·u⃗),Similar to the electronic couplings, the interlayer potential consists of symmmetric and asymmetric components which are quantified by U_1 and U_2. We treat the deformation within the first harmonics, which dominates the relaxation profile for small atomic displacements (|u⃗|≪ a) which accounts for the approximate sign in <ref>. Since the interlayer potential is completely defined by the first harmonics (Eq. <ref>, LHS), the solution for the displacement vector u⃗(r⃗) = u_x x̂ + u_y ŷ + h(r⃗) ẑ is also constrained by moiré periodicity, containing only the first harmonics components. The displacement and the corresponding in-plane expansion of carbon site at r⃗ is thus given by:u⃗(r⃗) = k_s|ε^3|/G^2ε̃^2 [∇⃗+χ_Rθ/ε(ẑ×∇⃗)][U_1f_1(r⃗)+U_2f_2(r⃗)] + [z_1f_1(r⃗) + z_2f_2(r⃗)] ẑ, a/a_0≈ 1+1/2Tr[u_ij(r⃗)] = 1 + 1/2[∂ u_x/∂ x+∂ u_y/∂ y] =1 -k_s|ε^3|/2ε̃^2 [U_1f_1(r⃗)+U_2f_2(r⃗)],where U_1 (U_2) parametrize the inversion-symmetric (asymmetric) components of in-plane deformation, and z_1 (z_2) quantifies similar components for the deformation in the out-of-plane direction, i.e. corrugation. Meanwhile, the coefficient k_s which relates u⃗(r⃗) with the potentials, and the factor χ_R which sets the magnitude of the curl-like terms are as follows k_s = 2/3√(3)ε^2a^2(λ_g+2μ_g) = 0.029meV^-1, χ_R = [2+λ_g/μ_g] ≈ 2.34,It can be seen in the the subsequent analysis that k_s serves as the coefficient which encodes the contribution of in-plane deformation which is parametrized by U_1 and U_2, on the resulting gaps. We also assume the following definition for the symmetric and asymmetric components of the deformation:[ U_0; U_1; U_2 ] = M[ U_AA; U_AB; U_BA ] = 1/18[ 6(U_AA+U_AB+U_BA); 2U_AA-U_AB-U_BA; √(3)(U_AB-U_BA) ],in which we take AA stacking point as the origin of our coordinate system, and the matrix elements of M are obtained in a way consistent with our definitions of f_1(r⃗) and f_2(r⃗), and in accordance with the convention in the literature <cit.>. The symmetric and antisymmetric profiles are plotted in Fig. <ref>. We also use similar definition of symmetric and antisymmetric components for the corrugation and moiré electronic couplings. Within this choice of coordinates, we can expect from symmetry consideration alone that a sizable primary gap only opens up when there is a relatively large symmetric component in the deformation (the asymmetric terms vanish after doing the mBZ average).In general, since in our choice of coordinate system the inversion-asymmetric electronic couplings dominate (Ṽ≫ V), this implies that the dominant contribution to theglobal mass term results from symmetric deformations.Since the secondary Dirac cone gap is not a simple average over the mBZ, we expect both the symmetric and asymmetric contributions to be equally important.§.§.§ Moiré couplings and gaps in G/BN The dynamics of low-energy electrons in G/BN can be understood as a relativistic Dirac particle subject to three types of perturbations: (1) H_0(r⃗), describing the periodic electrostatic potential, (2) H_z(r⃗), acting as a local mass term which arises from the sublattice asymmetry, and (3) H_xy(r⃗), quantifying the gauge field due to asymmetry in hoppings induced by the interaction with the BN substrate. For graphene electrons on valley K, the equation of motion can be expressed as followsH_K(r⃗) =υp⃗·σ⃗ +H_0(r⃗)1 +H_z(r⃗)σ_3+ [ε(ẑ×1)-θ/ε̃G]∇⃗H_xy(r⃗)·σ⃗,It is important to note that only the H_xy contribution is scaled by factors which depend on the twist angle. The term in Eq. <ref> which is proportional to θ can be gauged away so that the influence of H_xy on the electronic structures is scaled by χ_θ = ε/ε̃. The three pseudospin components can be written in a compact manner using the following matrix notation:[H_0(r⃗) H_xy(r⃗)H_z(r⃗) ] = [ 1 f_1(r⃗) f_2(r⃗) ]W,where W stands for the electronic coupling matrix. Typically each pseudospin term for zero pressure has oscillating amplitudes on the order of ∼50 meV. Around AA, DFT parametrizations of the interlayer couplings lead to a dominating inversion-asymmetric electronic coupling (Ṽ≫ V). In addition, the exponential coefficients for all psuedospin terms also agree within 10%. Therefore, in this work we develop an analytical model containing only the inversion-asymmetric term Ṽ, which is scaled according to a single exponential coefficient β when pressure is applied on the system. Treating the problem at θ = 0^∘, W is given by the following expression:W =[ 1+ k_s[0 6U_1 6U_2;0U_1 -U_2;0 -U_2 -U_1 ]] CW_0 + [000;-Γ_VU_1 -Γ_B U_20;-Γ_VU_2Γ_B U_10 ],which is related to the corrugation matrix C and the coupling matrix in a rigid sample W_0:C =M^-1[ e^-3β z_1 0 0; 0 e^-2√(3)β z_2 0; 0 0e^2√(3)β z_2; ]M, W_0 = e^-β (z_0-z_r)[ 0 0 0; 0 2 -√(3); 1 0 0 ]Ṽ/2,The coupling terms appearing in W_0 are based on the special ratio between psuedospin components <cit.>, while the prefactors for z_1 and z_2 are estimated from the full numerical treatment of the Fourier components within the first harmonics. In Eq. <ref>, the first change in the electronic couplings as captured by the first term, is induced by shifts in the positions of the carbon atoms with respect to the underlying BN substrates. In-plane deformation of graphene brings additional contributions to the electrostatic potentials (Γ_V) and pseudomagnetic fields (Γ_B), both independent on the interlayer spacings.The primary gap arises primarily from the mass term Δ_p∼ 2|ω_z|, while the the secondary gap that appears on the valence band on valley K' at the edges of the mBZ (see Fig. <ref>): K⃗' = 1/3(G⃗_2 + G⃗_3) = 1/3(G⃗_4 + G⃗_5) = 1/3(G⃗_1 + G⃗_6) results from an interplay between the pseudospin terms H_0, H_z and H_xy <cit.>.These gaps have the following analytical form in a fully relaxed sample:Δ_p= 2√(3)Ṽe^-β (z_0-z_r)[cosh(2√(3)β z_2)-e^-3β z_1 + k_s(2U_1e^-3β z_1+U_1cosh(2√(3)β z_2) + √(3)U_2sinh(2√(3)β z_2))], Δ_s= √(3)Ṽ/6e^-β (z_0-z_r)[7e^-3β z_1 + 2cosh(2√(3)β z_2) + 12sinh(2√(3)β z_2) ] - Γ_G(√(3)U_1-U_2) +δ_s,whereδ_s is a small additional contribution to the secondary gap which results from the interplay between corrugation and in-plane deformation (assumed here to be zero).We define Γ_G = (√(3)/2)(Γ_V + 2Γ_B), for Γ_V,B= |ε^3|k_sγ_V,B/ε̃^2.The first principle calculations at θ = 0^∘ discussed above, gives a sizable Γ_G≈ 5.6meV^-1.This is important, because for generic band parameters, in the case of Γ_G→ 0 the bands that form the sDC are possibly mixed with the surrounding moiré minibands such that no gap could be observed experimentally.Since graphene sites have an affinity towards the BA configuration, where one carbon atom sits on top of the boron atom, both U_1 and U_2 are expected to acquire positive values ∼3 meV, such that the BA domain is enlarged compared to the rigid case. The exponential prefactor e^-β (z_0-z_r) in the primary gap gives the increase in gap arising from the increased coupling between the graphene and BN when the two layers are closer together under pressure.This multiplies two terms, the first depends only on the corrugations and to leading order is given by the symmetric component of the out-of-plane deformations ∼3 β z_1.The second term proportional to k_s depends on both in-plane and out-of-plane deformations, but for realistic parameters depends only on the symmetric deformation ∼ 3 k_s U_1.This shows that to leading order, the primary gap is determined mostly by the symmetric in-plane and out-of-plane deformations and vanishes in the case of rigid graphene.The secondary gap is finite even for the rigid case without any deformations Δ_s = (√(27)/2) Ṽ e^-β (z_0-z_r) that grows with increasing pressure.The experimental observation that the primary gap is larger than the secondary gap even at zero pressure implies that the second (negative) term in Eq. <ref> proportional to Γ_G must be of comparable magnitude.In contrast to the primary gap, the secondary gap is controlled by both the symmetric and antisymmetric components of the deformation i.e. ∼ (U_2 - √(3)U_1) for in-plane deformations, and ∼ ((8 √(3)/7) z_2 - z_1) for corrugations.In both cases, for the secondary gap to remain constant under pressure, the difference between the asymmetric deformations and symmetric deformations must increase with pressure. We note that these experimental observations consistent with DFT expectations.The EXX-RPA calculations discussed above show that with increasing pressure, (i) The difference between the asymmetric and symmetric components of in-plane deformations increases, while (ii) The asymmetric part of out-of-plane corrugation decreases.This suggests that the dominant contribution to the flatness of the secondary gap comes from corrugations.Physically, the AA points (carbon-carbon on BN) start further away form the underlying BN substrate. With increased pressure, they compress more than either the AB (carbon on nitrogen) or BA (carbon on boron).While all potentials increase with pressure, the BA increases the least making this region expand.This naturally explains the growing primary gap.Similarly, under pressure the G/BN spacing decrease at all stacking points (so that the graphene becomes flatter). But the AA points which were further away to begin with decrease faster than either the BA or AB points, which is consistent with the Δ_s being constant.§.§ Reverse Engineering §.§.§ Fixed corrugation, free in-plane deformation Since the observed secondary gaps are smaller than the primary gaps at all pressures, this implies that Γ_G term originating from the in-plane strain-induced pseudomagnetic fields and potentials is large. We rely on DFT (EXX+RPA) calculations to determine the corrugation, assuming that the forces exerted by a top BN layer completely fixes the shapes of graphene corrugations under pressure. Since the corrugation is known at all pressures, the in-plane deformation parameters U_1 and U_2 are then uniquely determined by the observed primary and secondary gaps.Here, the graphene sample is relatively flat δ z ≲ 0.1 and the insensitivity of the secondary gap is accounted by an increase in U_1 with respect to U_2 as shown in the main text and in the top row of Fig. <ref>.§.§.§ Fixed in-plane deformation, free corrugationTo explore the role of corrugation in the experimental observation, we consider the opposite scenario where the in-plane deformation is assumed to be constant under pressure.We fix U_1 = 3 meV and U_2 = 2 meV, comparable to ab-initio estimates <cit.>. The insensitivity of the secondary gap under pressure constrains both the symmetric (z_1) and asymmetric components (z_2) of the corrugation, resulting in a significant transformation of the corrugation profile as shown in the middle row Fig. <ref>.One striking feature is that the experimental observations can be explained by a decrease in the asymmetric corrugations i.e. the separation between AB and BA becomes smaller under pressure.Since this contradicts DFT expectations we think this scenario is unlikely.§.§.§ Self-consistent corrugation and in-plane deformation Another possible approach to address the gap behaviors is to construct interlayer potentials on G/BN special stacking points, which simultaneously define in-plane and out-of-plane deformation in G/BN and capture the increasing and constant trend in the primary and secondary gaps respectively. We assume a uniform pressure across the sample, and it is expected that there exists a certain potential profile which can account for the observed gaps. Inspired by the DFT results which suggest Lennard-Jones (LJ)-like potential profiles around the minima, we limit our search for the self-consistent potentials U_𝐬 which assume the LJ form at each stacking point (𝐬∈{AA,AB,BA}), which also leads to an interlayer distance of z_𝐬(P) at pressure P:U_𝐬(z) = σ_𝐬[ 1/z^2n_𝐬-1- (2n_𝐬-1/n_𝐬-1) 1/z_0,𝐬^n_𝐬 z^n_𝐬-1] + U_0,𝐬, z_𝐬(P) = [1/2z_0,𝐬^n_𝐬 + √(1/2z_0,𝐬^n_𝐬+4PA_g/(2n_𝐬-1)σ_𝐬) ]^1/n_𝐬The coefficient σ_𝐬 accounts for the strength of the potentials, and z_0,𝐬 denotes the equilibrium distance at zero pressure. The power coefficient n_𝐬 determines the behaviors of the profile on the small and large z tails, while A_g denotes the area of graphene's unit cell. We therefore try to obtain the best estimates for the potentials which account for the observed experimental trend. The LJ-like potentials which lead to remarkable agreement with the experiment can be found, i.e. <10% error for each data point, with the fitting parameters are shown in Table <ref>. As expected, this solution suggests the need for asymmetric components of the corrugation to decrease rapidly under pressure.However, this leads to a striking conclusion: Although BA is initially at a minimum spacing at zero pressure, AB overtakes BA as the closest point from the underlying BN substrate at P ≳ 1.5 GPa. This is accompnied by a slight increase in the asymmetric component of the in-plane deformation. Again, in disagreement with the DFT, the self-consistent solution is dominated by the rapid change in the asymmetric component of the corrugations (z_AB-z_BA).The EXX-RPA ab initio potentials (which can also be fit with Lennard-Jones-like functions) are shown in the inset for comparison with the potentials determined self-consistently.§.§.§ Generic features of the deformation While the analysis was done in the simplified semi-analytical model, we have checked that the conclusions remain robust in a full numerical simulation of the first-Harmonic approximation.The relatively large primary gap suggests there is a strong presence of either in-plane relaxation or corrugation. The fact that secondary gap is always smaller than the primary gap requires large in-plane deformation at all pressures. Since the increase in the moiré couplings as the layers get closer would give larger secondary gaps, the experimental observation of a flat secondary gap behavior implies that one or both of the following must be happening: (1) There is a significant decrease in the corrugation asymmetry z_2, which places AB closer to BA in the out-of-plane deformation when pressure is applied on G/BN, (2) There is a rapid transformation of the in-plane deformation profile under pressure such that the difference between U_1 and U_2 balances the effects of stronger electronic couplings. | http://arxiv.org/abs/1707.09054v2 | {
"authors": [
"Matthew Yankowitz",
"Jeil Jung",
"Evan Laksono",
"Nicolas Leconte",
"Bheema L. Chittari",
"K. Watanabe",
"T. Taniguchi",
"Shaffique Adam",
"David Graf",
"Cory R. Dean"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727214434",
"title": "Dynamic band structure tuning of graphene moiré superlattices with pressure"
} |
Active and Passive Transport of Cargo in a Corrugated Channel: A Lattice Model Study Moumita Das December 30, 2023 ====================================================================================In this work, we study the pointwise and ergodic iteration-complexity of a family of projective splitting methods proposed by Eckstein and Svaiter, for finding a zero of the sum of two maximal monotone operators. As a consequence of the complexity analysis of the projective splitting methods, we obtain complexity bounds for the two-operator case of Spingarn's partial inverse method. We also present inexact variants of two specific instances of this family of algorithms and derive corresponding convergence rate results. Keywords. splitting algorithms; maximal monotone operators; complexity; Spingarn method. AMS Classification: 47H05, 49M27, 90C60, 65K05.§ INTRODUCTION A wide variety of problems, such as optimization and min-max problems,complementarity problems and variational inequalities, can be posed as the monotone inclusion problem (MIP) associated to a maximal monotone point-to-set operator. An important tool for the design and analysis of several implementable methods for solving MIPs is the proximal point algorithm (PPA), proposed by Martinet <cit.> and generalized by Rockafellar<cit.>.Even though the PPA has good global and local convergence properties <cit.>, its major drawback is that it requires the evaluation of the resolvent mappings (or proximal mappings) associated with the operator. The difficulty lies in the fact that evaluating a resolvent mapping, which is equivalent to solving a proximal subproblem, can be as complicated as finding a root of the operator. One alternative to surmount this difficulty is to decompose the operator as the sum of two maximal monotone operators such that their resolvents are considerably easier to evaluate. Then, one can devise methods that use independently these proximal mappings.In this work, we are concerned with MIPs defined by the sum of two maximal monotone operators.We are also interested in the casewhere the problems of finding zeros of these operators separately are easier than solving the MIP for the sum.A typical instance of this situation is the variational inequality problem associated with a maximal monotone operator A and a closed convex subset C, whose solutions are precisely the zeros of the sum of A and the normal operator associated with C, known to be maximal monotone.Splitting methods (or decomposition methods) for problems of the above-mentioned type attempt to converge to a solution of the MIP by solving, in each iteration, subproblems involving one of the operators, but not both. Peaceman-Rachford and Douglas-Rachford methods are examples of this type of algorithms. Thesewere first introduced in <cit.> and <cit.> for the particular case oflinear mappings, and then generalized in <cit.> by Lions and Mercier to address MIPs. Forward-backward methods <cit.>,which generalize standard gradient projection methods for variational inequalities and optimizationproblems, are also examples of splitting algorithms.Recently, a new family of splitting methods for solving MIPs given by the sum of two maximal monotone operators was introduced in <cit.> by Eckstein and Svaiter. Through a generalized solution set in a product space, whose projection onto the first space is indeed the solution set of the problem, the authors constructed a class of decomposition methods with quite solid convergence properties. These algorithms are essentially projection methods, in the sense that in each iteration a hyperplane is constructed separating the current iterate from the generalized solution set, and then the next iterate is taken as a relaxed projection of the current one onto this separating hyperplane. In order to construct such hyperplanes, two proximal subproblems are solved, each of which involves only one of the two maximal monotone operators, which ensures the splitting nature of the methods.In this work we study the iteration-complexity of the family of methods proposed in <cit.>, to be referred as projective splitting methods (PSM) in the sequel.We start our analysis by introducing a projective algorithm that generalizes the PSM. We then consider a termination criterion for this general algorithm in terms of theϵ-enlargements of the operators, which allows us to obtain convergence rates for thePSM measured by the pointwise and ergodic iteration-complexities.Using the complexity analysis developed for the PSM, we also study the complexityof Spingarn's splitting methodfor solving inclusion problems given by the sum of two maximal monotone operators.In <cit.>, Spingarn introduced a splitting method for finding a zero of the sum of m maximal monotoneoperators using the concept of partial inverses. For the two-operator case, Eckstein and Svaiter provedin <cit.> that Spingarn's method is a special case of a scaled variant of the PSM. This will allow us to establish iteration-complexity results for Spingarn's method for the case of the sum of two maximal monotone operators.The general projective method that we introduce in this work is also used to construct inexact variants of two special cases of the family of PSM. For two specific instances of the PSM, we consider a relative error condition forapproximately evaluating the resolvents. The error criterion considered in this work is different from the oneused in <cit.>, where was generalized the projective splitting framework for MIPs given by the sum of m maximalmonotone operators. Indeed, we will use the notion of approximate solutions of a proximal subproblem presented in<cit.>, which yields a more flexible error tolerance criterion and allows evaluation of the ϵ-enlargement.We also derive convergence rate results for these two novel algorithms. The remainder of this paper is organized as follows. Section <ref> reviews the definitions and some basic properties of a point-to-set maximal monotone operator and its ϵ-enlargements. Section <ref> presents a relaxed projection method that extends the framework introduced in <cit.>. It also proves some properties regarding this general scheme and establishes the stopping criterion that will be considered for such method and its instances. Section <ref> presents the PSM introduced in <cit.> and derives global convergence rate results for these methods. Subsection <ref> specializes these general complexity bounds for the case where global convergence for the family of PSM was obtained in <cit.>. Section <ref> studies theiteration complexity of the two-operator case of Spingarn’s method of partial inverses <cit.>. Finally, sections <ref> and <ref> propose inexact versions of two special cases of the PSM and establish iteration-complexity results for them. § PRELIMINARIESThroughout this paper, we let ^n denote an n-dimensional space with inner product and induced norm denoted by ·· and ·, respectively. We also define the spaces _+ and 𝔼 as _+:=x∈ : x≥0 and 𝔼:=^n×^n×_+.In what follows in this section, we will review some material related to a point-to-set maximal monotone operator and its ϵ-enlargements that will be needed along this work. A point-to-set operator T:ℝ^n⇉ℝ^n is a relation T ⊆^n×^n and T(z) := {v∈ℝ^n:(z,v)∈ T} z∈ℝ^n.Given T:ℝ^n⇉ℝ^n its graph is the setT:={(z,v)∈^n×^n:v∈ T(z)}.An operator T:ℝ^n⇉ℝ^n is monotone, if z-z'v-v'≥0∀ (z,v),(z',v')∈T,and it is maximal monotone if it is monotone and maximal in the family of monotone operators of ℝ^n into ℝ^n, with respect to the partial order of inclusion. This is, if S:ℝ^n⇉ℝ^n is a monotone operator such that T⊆S, then S=T.The resolvent mapping of a maximal monotone operator T with parameter λ>0 is (I+λ T)^-1, where I is the identity mapping. It follows directly from the definition that z'=(I+λ T)^-1(z), if and only if z' is the solution of the proximal subproblem 0∈λ T(z') + (z'-z). The ϵ-enlargement of a maximal monotone operator was introduced in <cit.> by Burachik, Iusem and Svaiter. In <cit.>, Monteiro and Svaiter extended this notion to a generic point-to-set operator as follows. Given T:ℝ^n⇉ℝ^n and ϵ∈ℝ, define the operator ϵ-enlargement of T, T^ϵ:ℝ^n⇉ℝ^n, byT^ϵ(z) := {v∈ℝ^n:z'-zv'-v≥-ϵ,∀ (z',v') ∈ T},∀ z∈ℝ^n.The following proposition presents some important properties of T^ϵ. Its proof can be found in <cit.>. Let T:ℝ^n⇉ℝ^n. Then, (a) if ϵ'≤ϵ, we have T^ϵ'(z)⊆ T^ϵ(z) for all z∈ℝ^n;(b) T is monotone if and only if T⊆ T^0;(c) T is maximal monotone if and only if T=T^0.Observe that items (a) and (c) above imply that, if T:^n⇉^n is maximal monotone, then T(z)⊆ T^ϵ(z) for all z∈^n and ϵ≥0. Hence, T^ϵ(z) is indeed an enlargement of T(z).We now state the weak transportation formula <cit.> for computing points in the graph of T^ϵ. This formula will be used in the complexity analysis of some ergodic iterates generated by the algorithms studied in this work (see subsection <ref>). Assume that T:ℝ^n⇉ℝ^n is a maximal monotone operator. Let z_i,v_i∈ℝ^n and ϵ_i,α_i∈ℝ_+, for i=1,…,k, be such thatv_i∈ T^ϵ_i(z_i), i=1,…,k, ∑_i=1^kα_i=1,and define z: = ∑_i=1^kα_iz_i, v:=∑_i=1^kα_iv_i, ϵ:=∑_i=1^kα_i(ϵ_i+z_i-zv_i).Then, ϵ≥0 and v∈ T^ϵ(z). § THE GENERAL PROJECTIVE SPLITTING FRAMEWORK The monotone inclusion problem (MIP) of interest in this work consists of finding z∈ℝ^n such that0 ∈ A(z) + B(z),where A,B:ℝ^n⇉ℝ^n are maximal monotone operators. The framework presented in <cit.> reformulates problem (<ref>) intermsofaconvex feasibility problem, whichisdefinedbya certain closed convex extended solution set. To solve the feasibility problem, the authors introduced successive projectionalgorithms that use, at each iteration, independent calculations involving each operator. Our goals in this section are to present a scheme that generalizes the methods in <cit.>, and to study its properties. This general framework will allow us to derive convergence rates for the family of PSM and Spingarn's method. In addition, using this general method, we construct inexact versions of two specific instances of the PSM and study their complexities.Consider S_e(A,B)⊂^n×^n the extended solution set of (<ref>) defined in <cit.> asS_e(A,B):=(z,w)∈^n×^n : w∈ B(z),-w∈ A(z).The following result establishes two important properties of AB. Its proof can be found in <cit.>.If A,B:ℝ^n⇉ℝ^n are maximal monotone operators, then the following statements hold. (a) A point z∈ℝ^n is a solution of (<ref>), if and only if there is w∈ℝ^n such that (z,w)∈AB.(b) S_e(A,B) is a closed and convex subset of ^n×^n.According to the above lemma, problem (<ref>) is equivalent to theconvex feasibility problem of finding a point in S_e(A,B). In order to solve this feasibility problem by successive orthogonal projection methods, we need to construct hyperplanes separating points (z,w)∉ S_e(A,B) from S_e(A,B). For this purpose, in <cit.> it was used points in the graph of A and B to define affine functions, which were called decomposable separators, such that AB was contained in the non-positive half-spaces determined by them. Here, we generalize this concept using points in the ϵ-enlargements of A and B.Given two triplets (x,b,ϵ^x), (y,a,ϵ^y)∈𝔼 such that b∈ B^ϵ^x(x) and a∈ A^ϵ^y(y), the decomposable separator associated with (x,b,ϵ^x) and (y,a,ϵ^y) is the affine function ϕ:^n×^n→ℝϕ(z,w) := z-xb-w + z-ya+w -ϵ^x - ϵ^y.The non-positive level set of ϕ isH_ϕ:=(z,w)∈^n×^n : ϕ(z,w) ≤ 0.If ϕ is the decomposable separator associated with (x,b,ϵ^x) and (y,a,ϵ^y)∈𝔼, where b∈ B^ϵ^x(x) and a∈ A^ϵ^y(y), and H_ϕ is its non-positive level set, then (a) S_e(A,B)⊆ H_ϕ;(b) either ∇ϕ≠ 0 or ϕ≤ 0 in ℝ^n×ℝ^n;(c) either H_ϕ is a closed half-space or H_ϕ=^n×^n. Item (a) is a direct consequence of the definitions of the ϵ-enlargement of a point-to-set operator and the set S_e(A,B). Rewriting ϕ(z,w)asϕ(z,w) = z-ya+b + w-bx-y -ϵ^x -ϵ^y∀(z,w)∈^n×^n,and noting that ∇ϕ=(a+b,x-y) and ϵ^x, ϵ^y≥0, then (b) and (c) follow immediately. We now present the general projection scheme for finding a point in S_e(A,B) that will be studied in this work.Algorithm <ref> below generalizes the framework introduced in <cit.>, since we use the notion of decomposable separator introduced in Definition <ref>. Note that the general form of Algorithm <ref> is not sufficient to guarantee convergence of the sequence {(z_k,w_k)} to a point in AB.For example, if the separation between the point (z_k-1,w_k-1)∉ S_e(A,B) and S_e(A,B) by ϕ_k is not strict, then the next iterate is in fact (z_k-1,w_k-1) itself, which might lead to a constant sequence.Hence, to ensure convergence it is necessary to impose additional conditions on the decomposable separators, see <cit.> and sections <ref>, <ref> and <ref> below. However, since Algorithm <ref> is a relaxed projection type method, it is possible to establish Fejérmonotone convergence to S_e(A,B) and boundedness of its generated sequence, as well as other classical properties of this kind of algorithms (see for example <cit.>, <cit.>). §.§ The Generated SequencesWe will now analyze some properties of the sequences {(z_k,w_k)}, {ϕ_k}, {γ_k} and {ρ_k} generated by Algorithm <ref>, which will be needed in our complexity study. To this end, let us first prove the following technical result.For any (z,w)∈^n×^n and k≥1 we have12(z,w)-(z_k,w_k)^2 + 12∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 = 12(z,w)-(z_0,w_0)^2 + ∑_j=1^kρ_jγ_jϕ_j(z,w). First we observe that for j=1,2,…, and any (z,w)∈^n×^n it holds that12(z,w)-(z_j,w_j)^2= 12(z,w)-(z_j-1,w_j-1)+ρ_jγ_j∇ϕ_j^2 = 12(z,w)-(z_j-1,w_j-1)^2 + (z,w)-(z_j-1,w_j-1)ρ_jγ_j∇ϕ_j + 12ρ_j^2γ_j^2∇ϕ_j^2= 12(z,w)-(z_j-1,w_j-1)^2 + ρ_jγ_j(z,w)-(y_j,b_j)∇ϕ_j + ρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)∇ϕ_j + 12ρ_j^2γ_j^2∇ϕ_j^2,where the first equality above follows from the update rule in step 3 of Algorithm <ref>.Equation (<ref>) with ϕ=ϕ_j implies thatϕ_j(z,w) = (z,w)-(y_j,b_j)∇ϕ_j - ϵ^x_j - ϵ^y_j∀(z,w) ∈^n×^n.Therefore, adding and subtracting ρ_jγ_j(ϵ^x_j + ϵ^y_j) on the right-hand side of the last equality in (<ref>) and combining with the identity above, we obtain 12(z,w)-(z_j,w_j)^2 = 12(z,w)-(z_j-1,w_j-1)^2 + ρ_jγ_jϕ_j(z,w) - ρ_jγ_jϕ_j(z_j-1,w_j-1) + 12ρ_j^2γ_j^2∇ϕ_j^2. If we assume that γ_j>0, then the definition of γ_j in step 2 of Algorithm <ref> yields that ϕ_j(z_j-1,w_j-1)=γ_j∇ϕ_j^2. Hence, substituting this expression into the equality above and rearranging, we have12(z,w)-(z_j,w_j)^2 + 12ρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 = 12(z,w)-(z_j-1,w_j-1)^2 + ρ_jγ_jϕ_j(z,w).It is clear that this latter equality also holds if γ_j=0. Thus, adding equation above from j=1 to k we obtain (<ref>). In what follows we assume that problem (<ref>) has at least one solution, which implies that S_e(A,B) is a non-empty set in view of Lemma <ref>. Next theorem, which follows directly from Lemma <ref>, establishes boundedness of the sequence {(z_k,w_k)} calculated by Algorithm <ref>, and it also shows that the sum appearing on the left-handside of (<ref>) is bounded by the distance of the initial point to the set S_e(A,B).Take(z_0,w_0)∈^n×^n and let{(z_k,w_k)}, {ϕ_k}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>. Then, for every integer k≥1, we have∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2≤ d_0^2and(z_k,w_k)-(z_0,w_0)≤ 2d_0,where d_0 is the distance of (z_0,w_0) to AB.Take (z^∗,w^∗) the orthogonal projection of (z_0,w_0) onto S_e(A,B). From Lemma <ref>(a) it follows that ϕ_j(z^∗,w^∗)≤0 for all integer j≥1.Hence, specializing equality (<ref>) with (z^∗,w^∗) we obtain the first bound in (<ref>) and the following inequality(z_k,w_k)-(z^∗,w^∗)≤ d_0.Since (z_0,w_0)-(z^∗,w^∗)=d_0, the second estimate in (<ref>) follows from the latter two relations and the triangle inequality for norms. It is important to say that Theorem <ref> can be proven using standard arguments of relaxed projection algorithms, see for instance <cit.>.We have chosen the above approach since it will be more convenient for our subsequent analysis. §.§ The Ergodic SequencesBesides the pointwise complexity of specific instances of Algorithm <ref>, we are also interested in deriving their convergence rates measured by the iterationcomplexity in an ergodic sense. To do this, we consider sequences obtained by weightedaverages of the sequences {x_k} and {y_k}, generated by Algorithm <ref>, and study their properties. Let {x_k}, {y_k}, {γ_k} and {ρ_k} be the sequences computed with Algorithm <ref>, for every integer k≥1 assume that γ_k>0 anddefine x_k and y_k asx_k := 1/Γ_k∑_j=1^kρ_jγ_jx_j,y_k := 1/Γ_k∑_j=1^kρ_jγ_jy_j,whereΓ_k := ∑_j=1^kρ_jγ_j.The following lemma is a direct consequence of the weak transportation formula, Theorem <ref>.Let {(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>.For every integer k≥1, suppose that γ_k>0 and consider x_k, y_k and Γ_k given as in (<ref>). Define alsob_k := 1/Γ_k∑_j=1^kρ_jγ_jb_j, ϵ^x_k := 1/Γ_k∑_j=1^kρ_jγ_j(ϵ^x_j+x_j-x_kb_j), a_k := 1/Γ_k∑_j=1^kρ_jγ_ja_j, ϵ^y_k := 1/Γ_k∑_j=1^kρ_jγ_j(ϵ^y_j+y_j-y_ka_j).Then, we haveϵ^x_k≥0, b_k∈ B^ϵ^x_k(x_k),ϵ^y_k≥0, a_k∈ A^ϵ^y_k(y_k). The lemma follows from Theorem <ref> and the inclusions b_j∈ B^ϵ^x_j(x_j) and a_j∈ A^ϵ^y_j(y_j). We will refer to the sequences {(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)}, defined in (<ref>)-(<ref>), as the ergodic sequences associated with Algorithm <ref>.Next lemma presents alternative expressions for a_k+b_k, x_k-y_k and ϵ^x_k+ϵ^y_k, which will be used for obtaining bounds on their sizes. Let {(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>. Assume that γ_k>0 for all k≥1, and define the sequences {x_k}, {y_k}, {Γ_k}, {b_k}, {a_k}, {ϵ^x_k} and {ϵ^y_k} as in (<ref>), (<ref>) and (<ref>). Then, for every integer k≥1, we havea_k+b_k = 1/Γ_k(z_0-z_k), x_k-y_k = 1/Γ_k(w_0-w_k), ϵ^x_k + ϵ^y_k = -1/Γ_k∑_j=1^kρ_jγ_jϕ_j(y_k,b_k). Direct use of the definitions ofx_k, y_k, b_k and a_k yields(a_k+b_k,x_k-y_k) = 1/Γ_k∑_j=1^kρ_jγ_j(a_j+b_j,x_j-y_j).Since ∇ϕ_j=(a_j+b_j,x_j-y_j) for all integer j≥1, in view of the update rule in step 3 of Algorithm <ref>, the definition of Γ_k and the equation above, we have(z_k,w_k) = (z_0,w_0) - ∑_j=1^kρ_jγ_j(a_j+b_j,x_j-y_j) =(z_0,w_0) - Γ_k(a_k+b_k,x_k-y_k).The relation above clearly implies the first two identities in (<ref>).To prove the last equality in (<ref>) we first note that∑_j=1^kρ_jγ_jϕ_j(y_k,b_k)= ∑_j=1^k ρ_jγ_j(y_k-x_jb_j-b_k+y_k-y_ja_j+b_k -ϵ^x_j -ϵ^y_j)= ∑_j=1^k ρ_jγ_j(y_kb_j + x_jb_k-b_j +y_k-y_ja_j -y_jb_k -ϵ^x_j -ϵ^y_j).Next, we multiply the equation above by 1/Γ_k and use the definitions of y_k and b_k to obtain1/Γ_k∑_j=1^kρ_jγ_jϕ_j(y_k,b_k)= 1/Γ_k∑_j=1^k ρ_jγ_j(x_jb_k-b_j+y_k-y_ja_j -ϵ^x_j -ϵ^y_j).Now, we observe that ϵ^x_k can be rewritten asϵ^x_k = 1/Γ_k∑_j=1^kρ_jγ_j(ϵ^x_j + x_jb_j-b_k).Thus, adding ϵ^x_k and ϵ^y_k and combining with (<ref>) and the equation above, we deduce the third equality in (<ref>). The following result establishes bounds for the quantities a_k+b_k, x_k-y_k and ϵ^x_k+ϵ_k^y.Assume the hypotheses of Lemma <ref> and let d_0 be the distance of (z_0,w_0) to AB. Then, for all integer k≥1, we havea_k+b_k≤2d_0/Γ_k,x_k-y_k≤2d_0/Γ_k,ϵ^x_k + ϵ^y_k≤1/Γ_k[1/Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2+4d_0^2]. We combine the first two identities in (<ref>) with the second inequality in (<ref>) to obtain(a_k+b_k,x_k-y_k) = 1/Γ_k(z_0,w_0)-(z_k,w_k)≤2d_0/Γ_k.Thus, the bounds in (<ref>) follow immediately from the equation above.Now, taking (z,w)=(y_k,b_k) in equation (<ref>) and rearranging the terms we have-∑_j=1^kρ_jγ_jϕ_j(y_k,b_k) =12(y_k,b_k)-(z_0,w_0)^2 - 12(y_k,b_k)-(z_k,w_k)^2 - 1/2∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2.Since ρ_j∈]0,2[ for all integer j≥1, the equation above implies-∑_j=1^kρ_jγ_jϕ_j(y_k,b_k) ≤12(y_k,b_k)-(z_0,w_0)^2.Next, we define (z_k,w_k):=1Γ_k∑_j=1^kρ_jγ_j(z_j-1,w_j-1) and use the triangle inequality for norms to obtain12(y_k,b_k)-(z_0,w_0)^2 ≤(y_k,b_k)-(z_k,w_k)^2 + (z_k,w_k)-(z_0,w_0)^2 ≤1Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2+ 1Γ_k∑_j=1^kρ_jγ_j(z_j-1,w_j-1)-(z_0,w_0)^2 ≤1Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2 + 4d_0^2,where the second and the third inequalities above are due to the convexity of ·^2 and the second bound in (<ref>), respectively.Combining (<ref>) with (<ref>) we deduce that -∑_j=1^kρ_jγ_jϕ_j(y_k,b_k) ≤1Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2 + 4d_0^2.Relation above, together with the last equality in (<ref>), now yields (<ref>). §.§ Stopping Criterion In order to analyze the complexity properties of instances of Algorithm <ref>, we define a termination condition for this general method in terms of the ϵ-enlargements of operators A and B. This criterion will enable the obtention of complexity bounds, proportional to the distanceof the initial iterate to the extended solution set S_e(A,B), for all the schemes presented in this work.We consider the following stopping criterion for Algorithm <ref>. Given an arbitrary pair of scalars δ, ϵ>0, Algorithm <ref> will stop when it finds a pair of points (x,b,ϵ^x), (y,a,ϵ^y) ∈𝔼 such that b ∈ B^ϵ^x(x),a ∈ A^ϵ^y(y), max{a+b,x-y}≤δ, max{ϵ^x,ϵ^y}≤ϵ.We observe that if δ=ϵ=0, in view of Proposition <ref>, the above termination criterion is reduced to b∈ B(x), a∈ A(y), x=y and b=-a, in which case (x,b)∈ S_e(A,B).Based on the termination condition (<ref>) we can define the following notion of approximate solutions of problem (<ref>). For a given pair of positive scalars (δ,ϵ), a pair (x,y)∈^n×^n is called a (δ,ϵ)-approximate solution (or (δ,ϵ)-solution) of problem (<ref>), if there exist b,a∈ℝ^n and ϵ^x,ϵ^y∈ℝ_+ such that the relations in (<ref>) hold.§ THE FAMILY OF PROJECTIVE SPLITTING METHODS Our goal in this section is to establish the complexity analysis of the family of PSM developed in<cit.> for solving (<ref>). First, we will observe that the PSM is an instance of the generalAlgorithm <ref> with the feature of solving two proximal subproblems exactly, one involving only A and the other one only B, for obtaining the decomposable separator in step 2 of Algorithm <ref>. This will allow us to use the results of section <ref> to derive general iteration-complexity bounds for the PSM. Such bounds will be expressed in terms of the parameter sequences {λ_k},{μ_k}, {ρ_k} and {α_k}, calculated at each iteration of the method (see the PSM below). Insubsection <ref>, we will specialize these results for the case where global convergence was obtainedin <cit.>. We start by stating the family of projective splitting methods (PSM). [PSM]Choose (z_0,w_0)∈^n×^n. For k=1,2,… 1. Choose λ_k, μ_k>0 and α_k∈ℝ such that μ_k/λ_k-(α_k/2)^2>0,and find (x_k,b_k)∈B and (y_k,a_k)∈A such thatλ_kb_k + x_k = z_k-1 + λ_kw_k-1,μ_ka_k + y_k = (1-α_k)z_k-1 + α_kx_k - μ_kw_k-1. 2. If a_k+b_k + x_k-y_k=0 stop. Otherwise, setγ_k = z_k-1-x_kb_k-w_k-1 + z_k-1-y_ka_k+w_k-1/a_k+b_k^2 + x_k-y_k^2. 3. Choose a parameter ρ_k∈]0,2[ and setz_k =z_k-1 - ρ_kγ_k(a_k+b_k),w_k =w_k-1 - ρ_kγ_k(x_k-y_k). Several remarks are in order. The PSM is the same as <cit.>, except for the stopping criterion in step 2 above and boundednessconditions imposed on the parameters ρ_k, λ_k and μ_k in <cit.>.Note that if a_k+b_k+x_k-y_k=0 for some k, then x_k=y_k, b_k=-a_k and, since the points (x_k,b_k) and (y_k,a_k) are chosen in the graph of B and A, respectively, we have (x_k,b_k)∈AB. Therefore, when the PSM stops in step 2, it has found a point in the extended solution set.Observe also that, since B is maximal monotone, Minty's theorem <cit.> implies that the resolvent mappings (I+λ_kB)^-1 are everywhere defined and single valued for all integer k≥1. Hence, by (<ref>) we have that the points x_k=(I+λ_k B)^-1(z_k-1+λ_kw_k-1) and b_k=(1/λ_k)(z_k-1-x_k)+w_k-1 exist and are unique. Similarly, the maximal monotonicity of A, together with (<ref>), guarantees theexistenceanduniquenessofy_k=(I+μ_kA)^-1((1-α_k)z_k-1+α_kx_k-μ_kw_k-1)and a_k=(1/μ_k)((1-α_k)z_k-1+α_kx_k)-w_k-1.Moreover, if for k=1,2,…, we denote by ϕ_k the decomposable separator (see Definition <ref>) associated with the triplets (x_k,b_k,0) and (y_k,a_k,0), calculated in step 1 of the PSM,then the update rule in step 3 of the PSM can be restated as(z_k,w_k) = (z_k-1,w_k-1) - ρ_kγ_k∇ϕ_k.Consequently, the family of PSM falls within the general framework of Algorithm <ref>, and the results of section <ref> apply.Finally, note that the PSM generates, on each iteration, a pair (x_k,y_k) and vectorsb_k, a_k∈^n such that the inclusions in (<ref>) hold with (x,b,ϵ^x)=(x_k,b_k,0) and (y,a,ϵ^y)=(y_k,a_k,0). Hence, we can try to develop bounds for the quantities a_k+b_k and x_k-y_k to estimate when an iterate (x_k,y_k) is bound to satisfy the termination criterion (<ref>).Before establishing the iteration-complexity results for the PSM, we need the following technical result.It presents two lower bounds for ϕ_k(z_k-1,w_k-1).Let {(x_k,b_k)}, {(y_k,a_k)}, {(z_k,w_k)}, {λ_k}, {μ_k}, {α_k} and {ρ_k} be the sequences generated by the PSM, and {ϕ_k} be thesequence of decomposable separators associated with the PSM. Then, for all integer k≥1, the following inequalities holdϕ_k(z_k-1,w_k-1) ≥θ_k/δ_k(a_k+b_k^2 + x_k-y_k^2),ϕ_k(z_k-1,w_k-1) ≥θ_k/μ_k(z_k-1-y_k^2 + w_k-1-b_k^2),where δ_k:=μ_k+(1-α_k)λ_k>0 and θ_k>0 is the smallest eigenvalue of the matrix ([ 1 -λ_k|α_k|/2; -λ_k|α_k|/2λ_kμ_k ]). Inequality (<ref>) was obtained in <cit.> as part of the convergence proof of Algorithm 2 in <cit.>, as were the assertions that θ_k, δ_k>0under assumption (<ref>). Therefore, we only need to prove here relation (<ref>).If we subtract y_k from both sides of (<ref>) and rearrange the terms we obtainx_k - y_k = z_k-1-y_k + λ_k(w_k-1-b_k).Now, adding μ_kb_k to both sides of (<ref>) and rearranging we haveμ_k(a_k+b_k)= (1-α_k)z_k-1 + α_kx_k - y_k - μ_k(w_k-1-b_k)= α_k(x_k-y_k) + (1-α_k)(z_k-1-y_k) - μ_k(w_k-1-b_k).Next, we substitute (<ref>) into (<ref>) and divide by μ_k to obtaina_k+b_k = α_k/μ_k(z_k-1-y_k + λ_k(w_k-1-b_k)) + (1-α_k)/μ_k(z_k-1-y_k) - (w_k-1-b_k) = 1/μ_k(z_k-1-y_k) + (α_kλ_k/μ_k-1)(w_k-1-b_k).Sinceϕ_k(z_k-1,w_k-1) =a_k+b_kz_k-1-y_k + x_k-y_kw_k-1-b_k,equation above, together with (<ref>) and (<ref>), yieldsϕ_k(z_k-1,w_k-1) = 1/μ_kz_k-1-y_k^2 + α_kλ_k/μ_kz_k-1-y_kw_k-1-b_k + λ_kw_k-1-b_k^2 ≥1/μ_kz_k-1-y_k^2 - |α_k|λ_k/μ_kz_k-1-y_kw_k-1-b_k + λ_kw_k-1-b_k^2=1/μ_k([ z_k-1-y_k; w_k-1-b_k ])^T([ 1 -λ_k|α_k|/2; -λ_k|α_k|/2λ_kμ_k ]) ([ z_k-1-y_k; w_k-1-b_k ]),where the inequality in the above relation follows from the Cauchy-Schwartz inequality. Finally, (<ref>) follows from the expression above and the definition of θ_k. For simplicity, the convergence rate results presented below suppose that the PSM never stops in step 2, i.e. they are assuming that ∇ϕ_k>0 for all integer k≥1. However, they can easily be restated without assuming such condition by saying that either the conclusion stated below holds or(x_k,b_k) is a point in AB.Next result estimates the quality of the best iterate among (x_1,y_1),…,(x_k,y_k) in terms of the stopping criterion (<ref>).We refer to these estimates as pointwise complexity bounds for the PSM. Let {(x_k,b_k)}, {(y_k,a_k)}, {λ_k}, {μ_k}, {α_k}, {γ_k} and {ρ_k} be the sequences generated by the PSM.Then, for every integer k≥1, we haveb_k∈ B(x_k), a_k∈ A(y_k),and there exists an index 1≤ i ≤ k such thata_i+b_i^2+x_i-y_i^2≤d_0^2/∑_j=1^kρ_j(2-ρ_j)(θ_j/δ_j)^2,where d_0 is the distance of the first iterate (z_0,w_0) to AB, and θ_k and δ_k are defined in Lemma <ref>.The assertions that b_k∈ B(x_k) and a_k∈ A(y_k) are direct consequences of step 1 in the PSM. The definition of γ_j in step 2 of the method, together with the definition of ϕ_jand inequality (<ref>), yieldsγ_j=ϕ_j(z_j-1,w_j-1)/∇ϕ_j^2≥θ_j/δ_jfor j=1,2,….Therefore,γ_j^2≥(θ_jδ_j)^2for j=1,2,….Multiplying both sides of the inequality above by ρ_j(2-ρ_j)∇ϕ_j^2, adding from j=1 to k and using the first bound in (<ref>), we haved_0^2≥∑_j=1^k∇ϕ_j^2ρ_j(2-ρ_j)(θ_j/δ_j)^2.Taking i such thati∈min_j=1,…,k(∇ϕ_j^2),and using the previous inequality we obtaind_0^2≥∑_j=1^kρ_j(2-ρ_j)(θ_j/δ_j)^2∇ϕ_i^2.Bound (<ref>) now follows from the above relation and noting that ∇ϕ_i=(a_i+b_i,x_i-y_i). We will now derive alternative complexity bounds for the PSM.Using the sequences of ergodic iterates associated with the PSM, defined as in subsection <ref>, we will obtain convergence rates for the methods in the ergodic sense.We refer to these kind of estimates as ergodic complexity bounds.Define the sequences of ergodic means {(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)}, associated with the sequences {(x_k,b_k,0)}, {(y_k,a_k,0)}, {γ_k} and {ρ_k} generated by the PSM, as in (<ref>), (<ref>) and (<ref>). According to Lemma <ref>, we can attempt to bound the size of a_k+b_k, x_k-y_k, ϵ^x_k andϵ^y_k, in order to know when the ergodic iterates {x_k} and {y_k} will meet the stopping condition (<ref>).Assume the hypotheses of Theorem <ref>. In addition, consider the sequences of ergodic iterates {(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)} associated with the sequences generated by the PSM, defined as in (<ref>), (<ref>) and (<ref>). Then, for every integer k≥1, we haveb_k∈ B^ϵ^x_k(x_k),a_k∈ A^ϵ^y_k(y_k),anda_k+b_k≤2d_0/Γ_k,x_k-y_k≤2d_0/Γ_k,ϵ^x_k + ϵ^y_k≤d_0^2(ς_k+4)/Γ_k,where ς_k:=max_j=1,…,k{μ_j/θ_j(2-ρ_j)Γ_k}. Inclusions in (<ref>) are a consequence of Lemma <ref>. The first two inequalities in (<ref>) are obtained by applying Theorem <ref>.Now, we observe that relation (<ref>), together with the equality in (<ref>), impliesμ_j/θ_j∇ϕ_j^2γ_j≥(y_j,b_j)-(z_j-1,w_j-1)^2forj=1,2,….Multiplying the inequality above by 1Γ_kρ_jγ_j and adding from j=1 to k, we obtain1/Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2≤1/Γ_k∑_j=1^kμ_j/θ_jρ_jγ_j^2∇ϕ_j^2= 1/Γ_k∑_j=1^kμ_j/θ_j(2-ρ_j)ρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 ≤(max_j=1,…,k{μ_j/θ_j(2-ρ_j)Γ_k})∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 ≤(max_j=1,…,k{μ_j/θ_j(2-ρ_j)Γ_k})d_0^2,where the last inequality above follows from the first estimate in (<ref>). We combine the relation above with (<ref>) and the definitionof ς_k to deduce the last bound in (<ref>).§.§ Specialized Complexity Results In this subsection, we will specialize the general pointwise and ergodic complexity bounds derived for the PSM in Theorems <ref> and <ref>, respectively, for the case where global convergence was obtained in <cit.>.In <cit.>, it was proven convergence of the sequences {(x_k,b_k)}, {(y_k,-a_k)} and {(z_k,w_k)}, calculated by the PSM, to a point of S_e(A,B) under the following assumptions:(A.1) there exist λ and λ such that, λ≥λ>0 and λ_k,μ_k∈[λ,λ] for all integer k≥1;(A.2) there exists ρ∈[0,1[ such that ρ_k∈[1-ρ,1+ρ] for all integer k≥1;(A.3) ν:=inf_k{μ_kλ_k-(α_k2)^2}>0. Under hypotheses (A.1)-(A.3) we will show that the PSM has 𝒪(1/√(k)) pointwise convergence rate, while the rate in the ergodic sense is 𝒪(1/k).Let {(x_k,b_k)}, {(y_k,a_k)}, {λ_k}, {μ_k}, {α_k}, {γ_k} and {ρ_k} be the sequences generated by the PSM under assumptions (A.1)-(A.3). If d_0 denote the distance of (z_0,w_0) to the extended solution set AB. Then, for all integer k≥1, we haveb_k∈ B(x_k), a_k∈ A(y_k),and there exists an index 1≤ i≤ k such thata_i+b_i≤d_0υ/√(k)(1-ρ)andx_i-y_i≤d_0υ/√(k)(1-ρ),where υ:=2λ(1+λ^2)(1+√(λ/λ))/λ^2ν. The inclusions in the statement of the theorem follow from (<ref>). Now, we note that condition (A.2) impliesρ_j(2-ρ_j)≥(1-ρ)^2for j=1,2,….Next, we observe that relation (<ref>) in step 1 of the PSM yields |α_j|≤2√(μ_j/λ_j). Hence, assumption (A.1) implies |α_j|≤2√(λ/λ)for j=1,2,….The inequality above, together with the definition of δ_j in Lemma <ref> and assumption (A.1), yieldsδ_j≤ 2λ(1+√(λ/λ)).Moreover, in <cit.> it was proven thatθ_j: = 1/2(1+λ_jμ_j-√((1+λ_jμ_j)^2-4(λ_jμ_j-(λ_jα_j/2)^2))) ≥λ_j^2(μ_j/λ_j-(α_j/2)^2)/1+λ_jμ_j.Thus, under hypotheses (A.1)-(A.3) we have θ_j ≥λ^2ν/1+λ^2,and combining (<ref>) with (<ref>) we obtainθ_j/δ_j≥λ^2ν/(1+λ^2)2λ(1+√(λ/λ))=1/υfor j=1,2,….Now, from inequalities (<ref>) and (<ref>) we deduce thatρ_j(2-ρ_j)(θ_j/δ_j)^2≥(1-ρ)^2/υ^2for j=1,…,k. Hence, adding equation above from j=1to k we have∑_j=1^kρ_j(2-ρ_j)(θ_j/δ_j)^2≥ k(1-ρ)^2/υ^2.The theorem follows combining the above expression with inequality (<ref>). Assume the hypotheses of Theorem <ref>. Consider the sequences of ergodic iterates {(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)} associated with the sequences generated by the PSM, defined as in (<ref>), (<ref>) and (<ref>). Then, for every integer k≥1, we have b_k∈ B^ϵ^x_k(x_k),a_k∈ A^ϵ^y_k(y_k),anda_k+b_k≤2d_0υ/k(1-ρ),x_k-y_k≤2d_0υ/k(1-ρ),ϵ^x_k + ϵ^y_k≤d_0^2υ(υ'_k+4)/k(1-ρ),whereυ'_k:=λ(1+λ^2)υ/λ^2ν(1-ρ)^2k. The inclusions in (<ref>) follow from Lemma <ref>. The definition of Γ_k, together with assumption (A.2) and equation (<ref>), yieldsΓ_k≥ (1-ρ)∑_j=1^kθ_j/δ_j.Therefore, by (<ref>) and the inequality above we haveΓ_k≥(1-ρ)k/υ.The first two bounds in (<ref>) now follow from (<ref>) and the first two inequalities in (<ref>).To conclude the proof we observe that the definition of ς_k, hypothesis (A.1), (<ref>) and (<ref>) imply ς_k ≤λ(1+λ^2)υ/λ^2ν(1-ρ)^2k.Thus, combining the above relation with the last inequality in (<ref>), the definition of υ'_k and (<ref>), we obtain the last bound in (<ref>).We emphasize here that the derived bounds obtained in Theorem <ref> are asymptotically better than the ones obtained in Theorem <ref>. Indeed, the bounds for x_k, y_k, a_k and b_k are 𝒪(1/√(k)), whereas for x_k, y_k, b_k, a_k, ϵ_k^x and ϵ_k^y the bounds are 𝒪(1/k). However, the iterates calculated by the PSM are points in the graph of A and B, while the ergodic iterates are in some outer approximation of the operators, namely they are points in an ϵ-enlargement of A and B. The following result, which is an immediate consequence of Theorems <ref> and <ref>, presents complexity bounds for the PSM to obtain (δ,ϵ)-approximate solutions of problem (<ref>). Assume the hypotheses of Theorem <ref>. Then, the following statements hold.(a) For every δ>0 there exists an indexi = 𝒪(d_0^2υ^2/δ^2)such that the iterate (x_i,y_i) is a (δ,0)-solution of problem (<ref>).(b) For every δ, ϵ>0 there exists an index k_0 = 𝒪(max{d_0υ/δ,d_0^2υ/ϵ}) such that, for any k ≥ k_0, the ergodic iterate (x_k,y_k) is a (δ,ϵ)-solution of problem (<ref>).§ SPINGARN'S SPLITTING METHOD Inthis section, we study the iteration-complexityof the two-operator case ofSpingarn's splitting algorithm.In <cit.>, Spingarn describes a partial inverse method for solving MIPs given by the sum of m maximal monotone operators. Spingarn's method computes, at each iteration,independent proximal subproblems on each of the m operators involved in the problem and then finds the next iterate by essentially averaging the results.This algorithm is actually a special case of the Douglas-Rachford splitting method <cit.>,and it is also a particular instance of the general projective splitting methods for sums of m maximal monotone operators, which were introduced in <cit.>. Eckstein and Svaiter proved in <cit.> that the m=2 case of the Spingarn splitting method is a special case of a scaled variant of the PSM.Interpreting Spingarn's algorithm as an instance of the PSM allows us to use the analysis developed in the previous section for obtaining its complexity bounds. For this purpose, let us begin with a brief discussion of the reformulation of problem (<ref>) studied in <cit.>, obtained via including a scale factor. If η>0 is a fixed scalar, multiplying both sides of (<ref>) by η gives the problem 0∈η A(z) + η B(z).This simple reformulation leaves the solution set unchanged, but it transforms the set S_e(A,B). Indeed, the extended solution set associated with operators η A and η B has the formη Aη B = {(z,η w):(z,w)∈AB}. If we apply the PSM to η A and η B, and consider η a_k, η b_k and η w_k, respectively, in place of a_k, b_k and w_k, after some algebraic manipulations we obtain a scheme identical to the PSM, except that (<ref>)-(<ref>) are modified toλ_kη b_k + x_k = z_k-1 + λ_kη w_k-1,b_k∈ B(x_k),μ_kη a_k + y_k = (1-α_k)z_k-1 + α_kx_k - μ_kη w_k-1,a_k∈ A(y_k),γ_k=z_k-1-x_kb_k-w_k-1 + z_k-1-y_ka_k+w_k-1/ηa_k+b_k^2 + 1/ηx_k-y_k^2, z_k =z_k-1 - ρ_kγ_kη(a_k+b_k), w_k =w_k-1 - ρ_kγ_k/η(x_k-y_k).The general pointwise and ergodic complexity bounds for the method above are obtained as a direct consequence of Theorems <ref> and <ref>, replacing a_i, b_i, a_k, b_k, ϵ^x_k and ϵ^y_k by η a_i, η b_i, ηa_k, ηb_k, ηϵ^x_k and ηϵ^y_k, respectively.If η>0, in our notation, the Spingarn splitting method is reduced to the following set of recursions:η b_k + x_k = z_k-1 + η w_k-1,b_k∈ B(x_k),η a_k + y_k = z_k-1 - η w_k-1,a_k∈ A(y_k), z_k =(1-ρ_k)z_k-1 + ρ_k/2(x_k+y_k), w_k = (1-ρ_k)w_k-1 + ρ_k/2(b_k-a_k). Note that if we take λ_k=μ_k=1 and α_k=0 in (<ref>)-(<ref>) for all integer k≥1, then (<ref>)-(<ref>) and (<ref>)-(<ref>) are identical. Moreover, the remaining calculations, (<ref>), (<ref>) and (<ref>), can be rewritten into the form (<ref>)-(<ref>), as it is established in the next result.Assume that the following condition is satisfied: (B)λ_k=μ_k=1 and α_k=0 in (<ref>)-(<ref>) for every integer k≥1.Then, the recursions (<ref>)-(<ref>) and (<ref>)-(<ref>) are identical. Hence, Spingarn's method is a special case of the PSM. This result was proven in <cit.>. The following theorem derives the global convergence rate of Spingarn's splitting method in terms of the termination criterion (<ref>). Let η>0 and let {(x_k,b_k)}, {(y_k,a_k)} and {ρ_k} be the sequences generated by Spingarn's splitting method (<ref>)-(<ref>).For every k≥1, defineP_k := ∑_j=1^kρ_j,andx_k := 1/P_k∑_j=1^kρ_jx_j,b_k := 1/P_k∑_j=1^kρ_jb_j,ϵ^x_k:=1/P_k∑_j=1^kρ_jx_j-x_kb_j,y_k := 1/P_k∑_j=1^kρ_jy_j,a_k := 1/P_k∑_j=1^kρ_ja_j,ϵ^y_k:=1/P_k∑_j=1^kρ_jy_j-y_ka_j.Assume hypothesis (A.2) and set d_0:=(z_0,η w_0)η Aη B. Then, the following statements hold. (a) For every integer k≥1 we haveb_k∈ B(x_k), a_k∈ A(y_k),and there exists an index 1≤ i≤ k such that a_i+b_i≤2d_0/η√(k)(1-ρ),x_i-y_i≤2d_0/√(k)(1-ρ). (b) For every integer k≥1 we have b_k∈ B^ϵ^x_k(x_k),a_k∈ A^ϵ^y_k(y_k), anda_k+b_k≤4d_0/η k(1-ρ),x_k-y_k≤4d_0/k(1-ρ), ϵ^x_k + ϵ^y_k≤2d_0^2/η k(1-ρ)(2/(1-ρ)^2+4).(a) By the definitions of δ_k and θ_k in the statement of Lemma <ref> and hypothesis (B), we have that δ_k=2 and θ_k=1 for every integer k≥1. Therefore, since Theorem <ref> gives that Spingarn's algorithm is a special case of (<ref>)-(<ref>) under assumption (B), the claims in (a) follow from assumption (A.2) and Theorem <ref> applied to (<ref>)-(<ref>) with α_k=0 and λ_k=μ_k=1. (b) The first assertions in (b) follow from the definitions of P_k, x_k, b_k, ϵ^x_k, y_k, a_k, ϵ^y_k in (<ref>), (<ref>) and (<ref>), the inclusions in (<ref>), (<ref>) and Theorem <ref>.Now, we observe that Theorem <ref> implies that the sequences {(x_k,η b_k)} and {(y_k, η a_k)} can be viewed as generated by the PSM applied to the operatorsη A and η B, with λ_k=μ_k=1 and α_k=0 for k=1,2,…. Moreover, in <cit.> it was proven under assumption (B) that γ_k, given in (<ref>), is equal to 1/2 for every integer k≥1. Therefore,the sequences of ergodic iterates associated with {(x_k,η b_k)}, {(y_k, η a_k)}, {ρ_k} and {γ_k}, which are obtained by equations (<ref>), (<ref>) and (<ref>)with Γ_k=(1/2)P_k, are exactly as defined in (<ref>) and (<ref>), but with ηb_k, ηϵ^x_k, ηa_k and ηϵ^y_k instead of b_k, ϵ^x_k, a_k and ϵ^y_k, respectively.Hence, applying Theorem <ref> we have η(a_k+b_k)≤2d_0/(1/2)P_k,x_k-y_k≤2d_0/(1/2)P_k,η(ϵ^x_k + ϵ^y_k) ≤d_0^2(ς_k+4)/(1/2)P_k,where d_0 is the distance of (z_0,η w_0) to S_e(η A,η B) and ς_k=max_j=1,…,k{μ_jθ_j(2-ρ_j)(1/2)P_k}.Next, we note that condition (A.2) yields ρ_j ≥ 1-ρfor every j, therefore by the definition of P_k we have P_k≥ k(1-ρ). Furthermore, since in this case μ_j=1, θ_j=1and2-ρ_j≥ 1-ρfor all integerj≥1, the definition of ς_k and (<ref>) implyς_k≤2/(1-ρ)^2k≤2/(1-ρ)^2,fork=1,2,….Hence, the remaining claims in (b) follow combining the bounds in (<ref>) with the inequality above and (<ref>).Consider the sequences {x_k} and {y_k} generated by Spingarn's method and the sequences {x_k} and {y_k} defined in (<ref>) and (<ref>), respectively. Then, the following statements hold.(a) For every δ>0 there exists an indexi = 𝒪(max{d_0^2/η^2δ^2,d_0^2/δ^2})such that the pair (x_i,y_i) is a (δ,0)-solution of problem (<ref>).(b) For every δ, ϵ>0 there exists an index k_0 = 𝒪(max{d_0/ηδ, d_0/δ,d_0^2/ηϵ}) such that, for any k ≥ k_0, the pair (x_k,y_k) is a (δ,ϵ)-solution of problem (<ref>).§ A PARALLEL INEXACT CASEThe PSM has to solve two proximal subproblems on each iteration, in order to construct decomposable separators. Since finding the exact solution of subproblems (<ref>) and (<ref>) could be a challenging task, one might wish to allow approximate evaluations of the resolvent mappings, while maintaining convergence of the method.Our main goal in this section is to propose an inexact version of the PSM in the special case of taking α_k=0 for all iteration k, which possibly allows the subproblems to be performed in parallel. It is customary to appeal to the theory of approximation criteria for the PPA and related methods, when attempting to approximate solutions of proximal subproblems. The first inexact versions of the PPA were introduced in <cit.> by Rockafellar and are based on absolute summable error criteria. For instance, one of the approximation conditions proposed in <cit.> isz_k+1-(I+λ_kT)^-1(z_k)≤ s_k, ∑_k=1^∞s_k<∞.This kind of approximation criteria, which involves a theoretical sequence {s_k}⊂[0,∞[such that ∑_k=1^∞s_k<∞, has as a practical disadvantage that there is no direct guidance as to how to select it when solving a specific problem. Therefore, it is useful to construct error conditions for approximating proximal subproblems that could be computable during the progress of the algorithm.Inexact versions of the PPA, which use relative error tolerance criteria of this kind,were developed in <cit.>. To solve subproblems (<ref>) and (<ref>) inexactly, we will use the notion of approximate solutions of a proximal subproblem proposed in <cit.> by Solodov and Svaiter.The general projective splitting framework for the sum of m≥2 maximal monotone operators <cit.> admits a relative error condition for approximately evaluating resolvents.The criterion used in <cit.> is a generalization for the case of m maximal monotone operators of the relative error tolerance of the hybrid proximal extragradient method <cit.>.We have preferred the framework developed in <cit.> since it yields a more flexible error tolerance criterion and evaluation of the ϵ-enlargements of the operators.We now present the notion of inexact solutions of a proximal subproblem introduced in <cit.>. Let T:ℝ^n⇉ℝ^n be a maximal monotone operator,λ>0 and z∈ℝ^n. Consider the proximal system{[ w∈ T(z'),; λ w + z' - z=0, ].which is clearly equivalent to the proximal subproblem (<ref>).Given σ∈[0,1[, a triplet (z',w,ϵ)∈𝔼 is called a σ-approximate solution of (<ref>) at (λ,z), if w∈ T^ϵ(z'),λ w+z'-z^2 +2λϵ≤σ(λ w^2+z'-z^2).We observe that if (z',w) is the exact solution of (<ref>) then, taking ϵ=0, the triplet (z',w,ϵ) satisfies the approximation criterion(<ref>) for all σ∈[0,1[. Conversely, if σ=0, only the exact solution of (<ref>), with ϵ=0, will satisfy (<ref>).The method that will be studied in this section is as follows.Note that for all iteration k, the triplet (x_k,b_k,ϵ^x_k) calculated in step 1 of Algorithm <ref> is a σ-approximate solution of (<ref>) at (λ_k,z_k-1), where T=B-w_k-1. Similarly, (y_k,a_k,ϵ^y_k) is a σ-approximate solution of (<ref>) (with T=A+w_k-1) at point (μ_k,z_k-1). Observe also that taking σ=0 in Algorithm <ref> yields exactly the PSM with α_k=0 for all integer k≥1, since condition (<ref>) is satisfied.Let us denote by ϕ_k the decomposable separator associated with the pair (x_k,b_k,ϵ^x_k) and (y_k,a_k,ϵ^y_k), for every integer k≥1 (see Definition <ref>). It will be shown in the following lemma that if Algorithm <ref> stops in step 2 at iteration k, then it has found a point in S_e(A,B). Otherwise we will have ∇ϕ_k>0, which gives ϕ_k(z_k-1,w_k-1)>0. This clearly implies that Algorithm <ref> falls within the general framework of Algorithm <ref>. Let {(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {(z_k,w_k)}, {λ_k}, {μ_k} and {ρ_k} be the sequences generated by Algorithm <ref>, and{ϕ_k} be the sequence of decomposable separators associated with Algorithm <ref>. Then, for every integer k≥1, we haveϕ_k(z_k-1,w_k-1)≥1-σ/4ξ_k(a_k+b_k^2+x_k-y_k^2)≥0,where ξ_k:=min{λ_k,1/λ_k,μ_k,1/μ_k}.If ∇ϕ_k>0, then it follows that ϕ_k(z_k-1,w_k-1)>0. Furthermore, ∇ϕ_k=0 if and only if (x_k,b_k)=(y_k,-a_k)∈AB.From the definition of ϕ_k and direct calculations it follows thatϕ_k(z_k-1,w_k-1) =z_k-1-x_kb_k-w_k-1 + z_k-1-y_ka_k+w_k-1 - ϵ^x_k - ϵ^y_k=1/2λ_k(z_k-1-x_k^2 + λ_k(b_k-w_k-1)^2 - r^x_k^2 - 2λ_kϵ^x_k) + 1/2μ_k(z_k-1-y_k^2 + μ_k(a_k+w_k-1)^2 - r^y_k^2 - 2μ_kϵ^y_k).The identity above, together with the error criteria (<ref>) and (<ref>), implies ϕ_k(z_k-1,w_k-1)≥1-σ/2λ_k(z_k-1-x_k^2 + λ_k(b_k-w_k-1)^2)+ 1-σ/2μ_k(z_k-1-y_k^2 + μ_k(a_k+w_k-1)^2).If we interpret this last expression as a quadratic form applied to the ℝ^4 vector(z_k-1-x_k,b_k-w_k-1,z_k-1-y_k,a_k+w_k-1)^T, we obtain ϕ_k(z_k-1,w_k-1) ≥1-σ/2([ z_k-1-x_k; w_k-1-b_k; z_k-1-y_k; w_k-1+a_k ])^T([ 1/λ_k 0 0 0; 0 λ_k 0 0; 0 0 1/μ_k 0; 0 0 0 μ_k ]) ([ z_k-1-x_k; w_k-1-b_k; z_k-1-y_k; w_k-1+a_k ]) ≥1-σ/2ξ_k(z_k-1-x_k^2+b_k-w_k-1^2+z_k-1-y_k^2+a_k+w_k-1^2),where ξ_k, defined in (<ref>), is the smallest eigenvalue of the matrix in (<ref>).Now, we combine the second inequality in (<ref>) with relations z_k-1-x_k^2+z_k-1-y_k^2≥1/2x_k-y_k^2, b_k-w_k-1^2+a_k+w_k-1^2≥1/2a_k+b_k^2;to obtain the first inequality in (<ref>). Since ξ_k>0 and σ∈[0,1[, the second inequality in (<ref>) follows directly. Furthermore, these last relations, together with equation (<ref>), clearly imply that ϕ_k(z_k-1,w_k-1)>0 whenever ∇ϕ_k>0.To prove the last assertion of the lemma we rewrite ϕ_k(z_k-1,w_k-1) asϕ_k(z_k-1,w_k-1) = a_k+b_kz_k-1-y_k + x_k-y_kw_k-1-b_k - ϵ^x_k - ϵ^y_k.Then, if ∇ϕ_k=0, it follows that x_k=y_k, b_k=-a_k and ϕ_k(z_k-1,w_k-1)= - ϵ^x_k - ϵ^y_k. From equation (<ref>), the equality above and the fact that ϵ^x_k, ϵ^y_k≥0, we obtain ϵ^x_k=ϵ^y_k=0. Hence, b_k∈ B(x_k), a_k∈ A(y_k) and we conclude that (x_k,b_k)=(y_k,-a_k)∈AB. For deriving complexity bounds for Algorithm <ref> we will assume, as was done in the preceding section, that the method does not stop in a finite number of iterations. Thus, from now on we suppose that ∇ϕ_k>0 for all integer k≥1.Next result establishes pointwise complexity bounds for Algorithm <ref>.Take (z_0,w_0)∈^n×^n and let{(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {λ_k}, {μ_k}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>.Let d_0 be the distance of (z_0,w_0) to the set AB and, for all integer k≥1, define ξ_k by (<ref>). Then, for every integer k≥1, we have b_k∈ B^ϵ^x_k(x_k), a_k∈ A^ϵ^y_k(y_k),and there exists an index 1≤ i≤ k such that a_i+b_i^2 + x_i-y_i^2≤16d_0^2/(1-σ)^2(1-ρ)^2ξ_i∑_j=1^kξ_j,ϵ^x_i + ϵ^y_i≤4σ d_0^2/(1-σ)^2(1-ρ)^2∑_j=1^kξ_j. The inclusions in (<ref>) are due to step 1 of Algorithm <ref>. Since γ_k=ϕ_k(z_k-1,w_k-1)∇ϕ_k^2, using (<ref>) we have γ_k≥1-σ/4ξ_kfork=1,2,….Thus, squaring both sides of (<ref>) and multiplying by ∇ϕ_k^2 we obtainγ_k^2∇ϕ_k^2≥(1-σ/4)^2ξ_k^2∇ϕ_k^2.Now, we observe that the error criteria (<ref>) and (<ref>) implyϵ^x_k≤σ/2λ_k(z_k-1-x_k^2 + λ_k(b_k-w_k-1)^2)andϵ^y_k≤σ/2μ_k(z_k-1-y_k^2 + μ_k(a_k+w_k-1)^2),respectively.Adding these two inequalities and combining with relation (<ref>) we obtain ϵ^x_k + ϵ^y_k≤σ/1-σϕ_k(z_k-1,w_k-1)= σ/1-σγ_k∇ϕ_k^2.Multiplying the latter inequality by γ_k, using (<ref>) and multiplying both sides of the resulting expression by 1-σσ, we have(1-σ)^2/4σξ_k(ϵ^x_k + ϵ^y_k) ≤γ_k^2∇ϕ_k^2for k=1,2,…. Now, we defineψ_k := max{(1-σ/4)^2ξ_k∇ϕ_k^2,(1-σ)^2/4σ(ϵ^x_k+ϵ^y_k)},and combine (<ref>) with (<ref>) to obtainξ_kψ_k≤γ_k^2∇ϕ_k^2for k=1,2,….Next, adding the inequality above from j=1 to k, using the assumption that ρ_k∈[1-ρ,1+ρ] for all integer k≥1 and the first inequality in (<ref>), we have∑_j=1^kξ_jψ_j≤d_0^2/(1-ρ)^2, and consequently(min_j=1,…,k{ψ_j})∑_j=1^kξ_j≤d_0^2/(1-ρ)^2.The theorem now follows from this last inequality and the definition of ψ_k. If{(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {γ_k} and {ρ_k} are the sequences generated by Algorithm <ref>, we consider their associated sequences of ergodic iterates {(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)}, defined as in (<ref>), (<ref>) and (<ref>). Since Algorithm <ref> is a special instance of Algorithm <ref>, the results of subsection <ref> hold for its ergodic sequences.Thus, combining Theorem <ref> and Lemma <ref> we can derive ergodic complexity estimates for the method.Let {(x_k,b_k,ϵ^x_k)}, {(y_k,a_k,ϵ^y_k)}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>. Let{(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)} be the sequences of ergodic iterates associated withAlgorithm <ref>, defined as in (<ref>)-(<ref>), and consider ξ_k given by (<ref>). Then, for all integer k≥1, we haveb_k∈ B^ϵ^x_k(x_k),a_k∈ A^ϵ^y_k(y_k)anda_k+b_k≤2d_0/Γ_k,x_k-y_k≤2d_0/Γ_k,ϵ^x_k + ϵ^y_k≤d_0^2(φ_k+4)/Γ_k,where d_0 is the distance of (z_0,w_0) to AB and φ_k:=(2/1-σ)max_j=1,…,k{1/ξ_j(2-ρ_j)Γ_k}. The inclusions in (<ref>) follow from Lemma <ref>. The first two bounds in (<ref>) are due to (<ref>) in Theorem <ref>. Now, we note that the second inequality in (<ref>) implies(z_j-1-y_j^2+b_j-w_j-1^2)1-σ/2ξ_j≤ϕ_j(z_j-1,w_j-1)forj=1,2,….The relation above, together with the definition of γ_j, yieldsz_j-1-y_j^2+b_j-w_j-1^2≤2/(1-σ)ξ_jγ_j∇ϕ_j^2forj=1,2,….Multiplying the above inequality by 1Γ_kρ_jγ_j and adding from j=1 to k, we obtain1/Γ_k∑_j=1^kρ_jγ_j(y_j,b_j)-(z_j-1,w_j-1)^2≤1/Γ_k∑_j=1^k2/(1-σ)ξ_jρ_jγ_j^2∇ϕ_j^2= 1/Γ_k∑_j=1^k2/(1-σ)ξ_j(2-ρ_j)ρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 ≤φ_k∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2 ≤φ_kd_0^2,where the second and the third inequalities are due to the definition of φ_k and the first bound in (<ref>), respectively.Substituting equation above into (<ref>) we obtain the last bound in (<ref>). Theorems <ref> and <ref> provide general complexity results for Algorithm <ref>. Observe that the derived bounds are expressed in terms of ξ_k and Γ_k. Next result, which is a direct consequence of these theorems, presents iteration-complexity bounds for Algorithm <ref> to obtain (δ,ϵ)-approximate solutions of problem (<ref>). Assume the hypotheses of Theorem <ref>. Assume also condition (A.1) anddefine ξ:=min{λ,1λ}. Then, for all δ, ϵ>0, the following statements hold. (a) There exists an indexi = 𝒪(max{d_0^2/ξ^2δ^2,d_0^2/ξϵ})such that the iterate (x_i,y_i) is a (δ,ϵ)-solution of problem (<ref>).(b) There exists an index k_0 = 𝒪(max{d_0/ξδ,d_0^2/ξϵ}) such that, for any k ≥ k_0, the ergodic iterate (x_k,y_k) is a (δ,ϵ)-solution of problem (<ref>).We first note that assumption (A.1) impliesξ_j≥ξfor j=1,2,….Now, we combine the definition of Γ_k in (<ref>) with (<ref>) and (<ref>) to obtain Γ_k≥ k(1-ρ)ξ(1-σ)/4.Furthermore, the inequality above, together with (<ref>) and the definition of φ_k, yieldsφ_k≤8/(1-ρ)^2(1-σ)^2ξ^2.We conclude the proof combining Theorems <ref> and <ref> with these three relations above.§ A SEQUENTIAL INEXACT CASE In this section, we propose an inexact variant of a sequential case of the PSM and study its iteration-complexity. We observe that, unless α_k=0, subproblems (<ref>) and (<ref>) cannot be solved in parallel. For example, if we specialize the PSM by setting α_k=1 for all k, we have to perform on each iteration the following stepsλ_kb_k + x_k = z_k-1 + λ_kw_k-1, b_k∈ B(x_k),μ_ka_k + y_k = x_k - μ_kw_k-1, a_k∈ A(y_k).Therefore, the first problem above must be solved after the second one; these steps cannot be performed simultaneously like the proximal subproblems of Algorithm <ref>. However, this choice of α_k could be an advantage since the second subproblem uses more recent information, that is x_k instead of z_k-1. In this section, we are assuming that the resolvent mappings of operator B are easy to evaluate, but this is not the case for the proximal mappings associated with operator A. Such situations are typical in practice even in the case of convex optimization. Indeed, if A=∂ f and B=∂ g are the subdifferential operators of functions f and g, where f and g are proper, convex and lower semicontinuous, then the solutions of the MIP (<ref>) are minimizers of the sum f+g. In this case,in order to evaluatethe resolvent mapping (I+λ A)^-1=(I+λ∂ f)^-1, it is necessary to solve a strongly convex minimization problem and, if f has a complicated algebraic expression, such problem could be hard to solve exactly. Therefore, it is desirable to admit inexact solutions of the proximal subproblems associated with this operator.With these assumptions, we propose the following modification of the specific case of the PSM where α_k=1 for all iteration k. Specifically, in Algorithm <ref> below we allow the solution of the second proximal subproblem to be approximated, provided that the approximate solution satisfies the relative error condition of Definition <ref>.We note that the maximum tolerance for the relative error in the resolution of (<ref>)-(<ref>) is 1/2, instead of 1 as in Algorithm <ref>. We also note that the proximal parameter in step 1 of Algorithm <ref> is not allowed to change from one subproblem to another within an iteration.For every integer k≥1 denote by ϕ_k the decomposable separator associated with the triplets (x_k,b_k,0) and (y_k,a_k,ϵ^y_k), calculated in step 1 of Algorithm <ref> (see Definition <ref>). It is thus clear that if ϕ_k(z_k-1,w_k-1)>0 for all integer k≥1, then Algorithm <ref> is an instance of the general scheme presented in section <ref>.The following lemma implies that Algorithm <ref> stops in step 2 when it has found a point in the extended solution set AB.Let {(x_k,b_k)}, {(y_k,a_k,ϵ^y_k)}, {(z_k,w_k)}, {λ_k} and {ρ_k} be the sequences generated by Algorithm <ref>,and {ϕ_k} be the sequence of decomposable separators associated with Algorithm <ref>. Then, for all integer k≥1, we haveϕ_k(z_k-1,w_k-1)≥1-2σ/2τ_k(a_k+b_k^2+x_k-y_k^2)≥0,where τ_k:=min{λ_k,1/λ_k}.If ∇ϕ_k>0, then it follows that ϕ_k(z_k-1,w_k-1)>0. Furthermore, ∇ϕ_k=0 if and only if (x_k,b_k)=(y_k,-a_k)∈AB. Since ϕ_k(z_k-1,w_k-1)=z_k-1-x_kb_k-w_k-1+ z_k-1-y_ka_k+w_k-1 - ϵ^y_k,adding and subtracting x_ka_k+w_k-1 on the right-hand side of this equation and regrouping the terms, we obtainϕ_k(z_k-1,w_k-1)=z_k-1-x_kb_k+a_k+ x_k-y_ka_k+w_k-1 - ϵ^y_k= λ_kb_k-w_k-1b_k+a_k + 1/2λ_k[x_k-y_k^2+λ_k(a_k+w_k-1)^2] - 1/2λ_k[r_k^2 + 2λ_kϵ^y_k],where we have used in the last equality the identity in (<ref>) and r_k is given in (<ref>). We observe that λ_kb_k-w_k-1b_k+a_k = λ_k/2[b_k-w_k-1^2+b_k+a_k^2-a_k+w_k-1^2].Hence, combining equality above with (<ref>) and the error criterion (<ref>) we haveϕ_k(z_k-1,w_k-1)≥λ_k/2b_k-w_k-1^2 + λ_k/2a_k+b_k^2 + 1-σ/2λ_kx_k-y_k^2 - σλ_k/2a_k+w_k-1^2.Since a_k+w_k-1^2≤2a_k+b_k^2+2b_k-w_k-1^2, we deduce thatϕ_k(z_k-1,w_k-1)≥λ_k(1-2σ)/2b_k-w_k-1^2 + λ_k(1-2σ)/2a_k+b_k^2 + 1-σ/2λ_kx_k-y_k^2.The inequalities in (<ref>) now follow from the relation above, the definition of τ_k and noting that 1-σ≥1-2σ>0.The claim that ∇ϕ_k>0 implies ϕ_k(z_k-1,w_k-1)>0 is obtained as a direct consequence of (<ref>). To prove the remaining assertion of the lemma we observe that if ∇ϕ_k=0, then x_k=y_k, b_k=-a_k and it follows from (<ref>), the first equality in (<ref>) and the fact that ϵ^y_k∈_+, that ϵ^y_k=0. Thus, we have (x_k,b_k)∈AB. From now on we assume that Algorithm <ref> generates infinite sequences {x_k} and {y_k}, which is equivalent to ∇ϕ_k>0 for every integer k≥1. We are now ready to establish pointwise iteration-complexity bounds for Algorithm <ref>. The theorem below will be proven in much the same way as Theorem <ref>, using Lemma <ref> instead of Lemma <ref>. Take (z_0,w_0)∈^n×^n and let{(x_k,b_k)}, {(y_k,a_k,ϵ^y_k)}, {λ_k}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>.Let d_0 be the distance of (z_0,w_0) to AB and, for all integer k≥1, let τ_k be given by (<ref>). Then, for every integer k≥1, we haveb_k∈ B(x_k), a_k∈ A^ϵ^y_k(y_k),and there exists an index 1≤ i≤ k such that a_i+b_i^2 + x_i-y_i^2≤4d_0^2/(1-2σ)^2(1-ρ)^2τ_i∑_j=1^kτ_j,ϵ^y_i≤4σ d_0^2/(1-2σ)^2(1-ρ)^2∑_j=1^kτ_j. The inclusions in (<ref>) are due to step 1 of Algorithm <ref>. It follows from the definition of γ_k and inequality (<ref>) thatγ_k≥(1-2σ/2)τ_kfork=1,2,….Squaring both sides of the above inequality and multiplying by ∇ϕ_k^2 we obtainγ_k^2∇ϕ_k^2≥(1-2σ/2)^2τ_k^2∇ϕ_k^2,fork=1,2,….Now, we note that the error criterion (<ref>) impliesϵ^y_k≤σ/2λ_k[x_k-y_k^2+λ_k(a_k+b_k)^2].Consequently, we haveϵ^y_k≤σ/2λ_kx_k-y_k^2+σλ_ka_k+w_k-1^2 +σλ_kb_k-w_k-1^2.The above inequality, together with (<ref>), yieldsϵ^y_k≤2σ/1-2σϕ_k(z_k-1,w_k-1).Next, multiplying the above relation by γ_k and combining with (<ref>), after some manipulations, we obtain(1-2σ)^2/4στ_kϵ^y_k≤γ_k^2∇ϕ_k^2.Finally, definingψ_k:=max{(1-2σ)^2/4τ_k∇ϕ_k^2,(1-2σ)^2/4σϵ^y_k}and using (<ref>) and (<ref>), we can conclude the proof proceeding analogously to the proof of Theorem <ref>. The following theorem presents complexity estimates in the ergodic sense for Algorithm <ref>.Let {(x_k,b_k)}, {(y_k,a_k,ϵ^y_k)}, {γ_k} and {ρ_k} be the sequences generated by Algorithm <ref>. Let{(x_k,b_k,ϵ^x_k)} and {(y_k,a_k,ϵ^y_k)} be the associated sequences of ergodic iterates, defined as in (<ref>)-(<ref>), and consider τ_k given by (<ref>). Then, for all integer k≥1, we haveb_k∈ B^ϵ^x_k(x_k),a_k∈ A^ϵ^y_k(y_k),anda_k+b_k≤2d_0/Γ_k,x_k-y_k≤2d_0/Γ_k,ϵ^x_k + ϵ^y_k≤d_0^2(ϑ_k+4)/Γ_k,where d_0 is the distance of (z_0,w_0) to AB and ϑ_k:=max_j=1,…,k{8/τ_j(1-2σ)(2-ρ_j)Γ_k}. Since Algorithm <ref> is an instance of Algorithm <ref>, Lemma <ref> and Theorem <ref> apply, therefore the inclusions in (<ref>) and the first two inequalities in (<ref>) follow.We derive now an estimate for the sum on the right-hand side of (<ref>). We note that (<ref>) impliesϕ_j(z_j-1,w_j-1)≥λ_j(1-2σ/2)b_j-w_j-1^2for all integer j≥1. We also note that z_j-1-y_j=z_j-1-x_j+x_j-y_j=λ_j(b_j-w_j-1)+x_j-y_j,where the last identity is due to the equality in (<ref>). This last expression and the triangle inequality for norms yieldz_j-1-y_j≤λ_jb_j-w_j-1+x_j-y_j.Moreover, squaring both sides of the inequality above and making some manipulations, we obtain1/2λ_jz_j-1-y_j^2 ≤λ_jb_j-w_j-1^2+1/λ_jx_j-y_j^2≤2/1-2σϕ_j(z_j-1,w_j-1),where the last inequality follows from (<ref>). Now, adding (<ref>) and (<ref>) we have1/2λ_jz_j-1-y_j^2+λ_jb_j-w_j-1^2≤4/1-2σϕ_j(z_j-1,w_j-1).The above relation, together with the definitions of γ_j and τ_j, impliesb_j-w_j-1^2 + z_j-1-y_j^2≤8/(1-2σ)τ_jγ_j∇ϕ_j^2.Multiplying both sides of the above inequality by 1Γ_kρ_jγ_j and adding from j=1 to k, we obtain the desired estimate, i.e.1Γ_k∑_j=1^kρ_jγ_j[b_j-w_j-1^2 + z_j-1-y_j^2]≤1Γ_k∑_j=1^k8/(1-2σ)τ_jρ_jγ_j^2∇ϕ_j^2= 1Γ_k∑_j=1^k8/(1-2σ)τ_j(2-ρ_j)ρ_j(2-ρ_j)γ_j^2∇ϕ_j^2≤ϑ_k∑_j=1^kρ_j(2-ρ_j)γ_j^2∇ϕ_j^2≤ϑ_kd_0^2,where the second and the third inequalities above follow from the definition of ϑ_k and (<ref>), respectively. The proof of the last bound in (<ref>) now follows combining the above relation with (<ref>). Next result provides complexity bounds for Algorithm <ref> to find a (δ,ϵ)-approximate solution of problem(<ref>).It may be proven in much the same way as Theorem <ref> and for the sake of brevity we omit the proof here.Assume the hypotheses of Theorem <ref>. Suppose also that there exist λ and λ such that λ≥λ>0 and λ_k∈[λ,λ], for all integer k≥1, and define τ:=min{λ,1λ}. Then, for every δ, ϵ>0, the following claims hold. (a) There exists an index i = 𝒪(max{d_0^2/τ^2δ^2,d_0^2/τϵ})such that the point (x_i,y_i) calculated by Algorithm <ref> is a (δ,ϵ)-approximate solution of problem (<ref>).(b) There exists an indexk_0 = 𝒪(max{d_0/τδ,d_0^2/τϵ})such that, for any k ≥ k_0, the ergodic iterate (x_k,y_k) is a (δ,ϵ)-approximate solution of problem (<ref>).§ CONCLUSIONSWe introduced a general projective splitting scheme for solving monotone inclusion problems given by the sum of two maximal monotone operators, which generalizes the family of projective splitting methods (PSM) proposed by Eckstein and Svaiter. Using this general framework we analyzed the iteration-complexity of the family of PSM and, as a consequence, we obtained the iteration-complexity of the two-operator case of the Spingarn partial inverse method. We introduced two inexact variants of two special cases of the family of PSM, which allow the resolvent mappings to be solved inexactly. We also proved the iteration-complexity for the above-mentioned methods.AcknowledgmentsThis work is part of the author's Ph.D. thesis, written under the supervision of Benar Fux Svaiter at IMPA, and supported by CAPES and FAPERJ.spmpsci_unsrt | http://arxiv.org/abs/1707.08655v2 | {
"authors": [
"Majela Pentón Machado"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170726221857",
"title": "On the complexity of the projective splitting and Spingarn's methods for the sum of two maximal monotone operators"
} |
We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality λand a superstable-like forking notion for models of cardinality λ^+, then orbital types over models of cardinality λ^+ are determined by their restrictions to submodels of cardinality λ. By a superstable-like forking notion, we mean here a good frame, a central concept of Shelah's book on AECs. It is known that locality of orbital types together with the existence of a superstable-like notion for models of cardinality λ implies the existence of a superstable-like notion for models of cardinality λ^+, and here we prove the converse. An immediate consequence is that forking in λ^+ can be described in terms of forking in λ. [ L. N. Pfeiffer December 30, 2023 =====================§ INTRODUCTION Good frames are the central notion of Shelah's two volume book <cit.> on classification theory for abstract elementary classes (AECs). Roughly speaking, an AEC is a concrete category (whose objects are structures) satisfying several axioms (for example, morphisms must be monomorphisms and the class must be closed under directed colimits). It generalizes the notion of an elementary class (i.e. a class axiomatized by an _ω, ω-theory, where the morphisms are elementary embeddings) and also encompasses many infinitary logics such as _∞, ω (i.e. disjunctions and conjunctions of arbitrary, possibly infinite, length are allowed). An AEChas a good λ-frame if its restriction to models of cardinality λ is reasonably well-behaved (e.g. has amalgamation, no maximal models, and is stable) and it admits an abstract notion of independence (for orbital types of elements over models of cardinality λ) that satisfies some of the basic properties of forking in a superstable elementary class: monotonicity, existence, uniqueness, symmetry, and local character. Here, local character is described not as “every type does not fork over a finite set” but as “every type over the union of an increasing continuous chain of models of cardinality λ does not fork over some member of the chain”. The theory of good frames is used heavily in several recent results of the classification theory of AECs, including the author's proof of Shelah's eventual categoricity conjecture in universal classes <cit.>, see also <cit.> for a survey.The reason for restricting oneself to models of cardinality λ is that the compactness theorem fails in general AECs, and so it is much easier in practice to exhibit a local notion of forking than it is to define forking globally for models of all sizes. In fact, Shelah's program is to start with a good λ-frame and only then try to extend it to models of bigger sizes. For this purpose, he describes a dividing line[The nonstructure side is described by <cit.>, showing that failure of successfulness implies the AEC has many non-isomorphic models.], being successful, and shows that if a good λ-frame is successful, then there is a good λ^+-frame on an appropriate subclass of _λ^+.A related approach is to outright assume some weak amount of compactness. Tameness <cit.> was proposed by Grossberg and VanDieren to that end: λ-tameness says that orbital types are determined by their restrictions of cardinality λ. This is a nontrivial assumption, since in AECs syntactic types are not as well-behaved as one might wish, so one defines types purely semantically (roughly, as the finest notion of type preserving isomorphisms and the -substructure relation). It is known that tameness follows from a large cardinal axiom (see Fact <ref>) and some amount of it can be derived from categoricity (see Fact <ref>). The present paper gives another way to derive some tameness.Grossberg conjectured in 2006 that[See the introduction of <cit.> for a detailed history.], assuming amalgamation in λ^+, a good λ-frame extends to a good λ^+-frame if the class is λ-tame. He told his conjecture to Jarden who could prove all the axioms of a good λ^+-frame, except symmetry. Boney <cit.> then proved symmetry assuming a slightly stronger version of tameness, Jarden <cit.> proved symmetry from a certain continuity assumption, and Boney and the author finally settled the full conjecture <cit.>, see Fact <ref>. In this context, forking in the good λ^+-frame can be described in terms of forking in the good λ-frame. Let us call this result the upward frame transfer theorem.This paper discusses the converse of the upward frame transfer theorem. Consider the following question: if there is a good λ-frame and a good λ^+-frame, can we say anything on how the two frames are related (i.e. can forking in λ^+ be described in terms of forking in λ?) and can we conclude some amount of tameness? We answer positively by proving the following converse to the upward frame transfer theorem:Corollary <ref>. Letbe an AEC and let λ≥ (). Assume 2^λ < 2^λ^+. If there is a categorical good λ-frameon _λ and a good λ^+-frameon _λ^+, thenis (λ, λ^+)-tame. Moreover, _λ^+λ^+_λ^+ = λ^+_λ^+ (see Definition <ref>).In the author's opinion, Corollary <ref> is quite a surprising result since it shows that we cannot really study superstability in λ and λ^+ “independently”: the two levels must in some sense be connected (this follows from a canonicity result for good frames, see the paragraph after next). Put another way, two successive local instances of superstability already give a nontrivial amount of compactness. In fact after the initial submission of this article, Corollary <ref> was used by the author <cit.> as a key tool to prove a class has some amount of tameness and deduce strong versions of Shelah's eventual categoricity conjecture in several types of AECs.In Corollary <ref>, “categorical” simply means thatis assumed to be categorical in λ. We see it as a very mild assumption, since we can usually restrict to a subclass of saturated models if this is not the case, see the discussion after Definition <ref>. As for (λ, λ^+)-tameness, it means that types over models of cardinality λ^+ are determined by their restrictions of cardinality λ. In fact, it is possible to obtain a related conclusion by starting with a good λ^+-frame on the class of saturated models inof cardinality λ^+. In this case, we deduce thatis (λ, λ^+)-weakly tame, i.e. only types over saturated models of cardinality λ^+ are determined by their restrictions to submodels of cardinality λ. We deduce that weak tameness is equivalent to the existence of a good λ^+-frame on the saturated models of cardinality λ^+, see Corollary <ref>.An immediate consequence of Corollary <ref> is that forking in λ^+ (at least over saturated models) can be described in terms of forking in λ. Indeed, the upward frame extension theorem gives a good λ^+-frame with such a property, and good frames on subclass of saturated models are canonical: there can be at most one, see Fact <ref>. In fact, assuming that forking in λ^+ is determined by forking in λ is equivalent to tameness (see <cit.>) because of the uniqueness and local character properties of forking. In Corollary <ref> we of course do not start with such an assumption: forking in λ^+ is any abstract notion satisfying some superstable-like properties for models of cardinality λ^+.The proof of Corollary <ref> goes as follows: we use 2^λ < 2^λ^+ to derive that the good λ-frame is weakly successful (a dividing line introduced by Shelah in Chapter II of <cit.>). This is the only place where 2^λ < 2^λ^+ is used. Being weakly successful imply that we can extend the good λ-frame from types of singletons to types of models of cardinality λ. We then have to show that the good λ-frame is also successful. This is equivalent to a certain reflecting down property of nonforking of models. Jarden <cit.> has shown that successfulness follows from (λ, λ^+)-weak tameness and amalgamation in λ^+, and here we push Jarden's argument further by showing that having a good λ^+-frame suffices, see Theorem <ref>. A key issue that we constantly deal with is the question of whether a union of saturated models of cardinality λ^+ is saturated. In Section <ref>, we introduce a new property of forking, being decent, which characterizes a positive answer to this question and sheds further light on recent work of VanDieren <cit.>. The author believes it has independent interest.Tameness has been used by Grossberg and VanDieren to prove an upward categoricity transfer from categoricity in two successive cardinals <cit.>. In Section <ref>, we revisit this result and show that tameness is in some sense needed to prove it. Although this could have been derived from the results of Shelah's books, this seems not to have been noticed before. Nevertheless, the results of this paper show that if an AEC is categorical λ and λ^+ and has a good frame in both λ and λ^+, then it is categorical in λ^++, see Corollary <ref>.To read this paper, the reader should preferably have a solid knowledge of good frames, including knowing Chapter II of <cit.>, <cit.>, as well as <cit.>. Still, we have tried to give all the definitions and relevant background facts in Section <ref>.The author thanks John T. Baldwin, Will Boney, Rami Grossberg, Adi Jarden, Marcos Mazari-Armida, and the referee for comments that helped improve the quality of this paper.§ PRELIMINARIES§.§ Notational conventions Given a structure M, write write |M| for its universe and M for the cardinality of its universe. We often do not distinguish between M and |M|, writing e.g. a ∈ M or ∈<αM instead of a ∈ |M| and ∈<α|M|. We write M ⊆ N to mean that M is a substructure of N. §.§ Abstract elementary classesAn abstract class is a pair = (K, ), where K is a class of structures in a fixed vocabulary τ = τ () andis a partial order, MN implies M ⊆ N, and both K andrespect isomorphisms (the definition is due to Grossberg). Any abstract class admits a notion of -embedding: these are functions f: M → N such that f: M ≅ f[M] and f[M]N.We often do not distinguish between K and . For λ a cardinal, we will write _λ for the restriction ofto models of cardinality λ. Similarly define _≥λ or more generally _S, where S is a class of cardinals. We will also use the following notation: Foran abstract class and N ∈, write (N) for the set of M ∈ with MN. Similarly define λ(N), <λ(N), etc. For an abstract class , we denote by () the number of models inup to isomorphism (i.e. the cardinality of /_≅). We write (, λ) instead of (_λ). When () = 1, we say thatis categorical. We say thatis categorical in λ if _λ is categorical, i.e. (, λ) = 1.We say thathas amalgamation if for any M_0M_ℓ, ℓ = 1,2 there is M_3 ∈ and -embeddings f_ℓ : M_ℓ→ M_3, ℓ = 1,2.has joint embedding if any two models can be -embedded in a common model.has no maximal models if for any M ∈ there exists N ∈ with MN and M ≠ N (we write MN). Localized concepts such as amalgamation in λ mean that _λ has amalgamation.The definition of an abstract elementary class is due to Shelah <cit.>: An abstract elementary class (AEC) is an abstract classin a finitary vocabulary satisfying:* Coherence: if M_0, M_1, M_2 ∈, M_0 ⊆ M_1M_2 and M_0M_2, then M_0M_1.* Tarski-Vaught chain axioms: if M_i : i ∈ I is a -directed system and M := ⋃_i ∈ I M_i, then: * M ∈.* M_iM for all i ∈ I.* Smoothness: if N ∈ is such that M_iN for all i ∈ I, then MN. * Löwenheim-Skolem-Tarski (LST) axiom: there exists a cardinal λ≥ |τ ()| + ℵ_0 such that for any N ∈ and any A ⊆ |N|, there exists M ∈≤ (|A| + λ) (N) with A ⊆ |M|. We write () for the least such λ. We say that an abstract classis an AEC in λ ifsatisfies coherence, = _λ, λ≥ |τ ()|, andsatisfies the Tarski-Vaught chain axioms whenever |I| ≤λ. Ifis an AEC in λ, then by <cit.> there exists a unique AEC ^∗ such that _λ = _λ^∗, _<λ^∗ = ∅, and (^∗) = λ. We call ^∗ the AEC generated by .§.§ Types In any abstract class , we can define a semantic notion of type, called Galois or orbital types in the literature (such types were introduced by Shelah in <cit.>). For M ∈, A ⊆ |M|, and ∈<∞M, we write _ ( / A; M) for the orbital type ofover A as computed in M (usuallywill be clear from context and we will omit it from the notation). It is the finest notion of type respecting -embeddings, see <cit.> for a formal definition. Whenis an elementary class, ( / A; M) contains the same information as the usual notion of _ω, ω-syntactic type, but in general the two notions need not coincide <cit.>.The length of ( / A; M) is the length of . For M ∈ and α a cardinal, we write _^α (M) = ^α (M) for the set of types over M of length α. Similarly define ^<α (M). When α = 1, we just write (M). We define naturally what it means for a type to be realized inside a model, to extend another type, and to take the image of a type by a -embedding. §.§ Stability and saturation We say that an abstract class , is stable in λ (for λ an infinite cardinal) if | (M)| ≤λ for any M ∈_λ. Ifis an AEC, λ≥ (),is stable in λ andhas amalgamation in λ, then we will often use without comments the existence of universal extension <cit.>: for any M ∈_λ, there exists N ∈_λ universal over M. This means that MN and any extension of M of cardinality λ -embeds into N over M.Foran AEC and λ >(), a model N ∈ is called λ-saturated if for any M ∈_<λ with MN, any p ∈ (M) is realized in N. N is called saturated if it is N-saturated.We will often use without mention the model-homogeneous = saturated lemma <cit.>: it says that when _<λ has amalgamation, a model N ∈ is λ-saturated if and only if it is λ-model-homogeneous. The latter means that for any M ∈<λ (N), any M' ∈_<λ -extending M can be -embedded into N over M. In particular, assuming amalgamation and joint embedding, there is at most one saturated model of a given cardinality. We write λ for the abstract class of λ-saturated models in(ordered by the appropriate restriction of ). §.§ Frames Roughly, a frame consists of a class of models of the same cardinality together with an abstract notion of nonforking. The idea is that if the frame is “sufficiently nice”, then it is possible to extend it to cover bigger models as well. This is the approach of Shelah's book <cit.>, where good frames were introduced. There the abstract notion of nonforking is required to satisfy some of the basic properties of forking in a superstable elementary class. We redefine here the definition of a frame (called pre-frame in <cit.>, <cit.>). We give a slightly different definition, as we do not include certain monotonicity axioms as part of the definition. Shelah assumes that nonforking is only defined for a certain class of types he calls the basic types. For generality, we will work in the same setup (except that we define the basic types from the nonforking relation), but the reader will not lose much by pretending that all types are basic (In Shelah's terminology, that the frame is type-full). Note that any weakly successful good frame can be extended to a type-full one, see <cit.>. Let λ be an infinite cardinal, and α≤λ^+ be a non-zero cardinal. A (<α, λ)-frame consists of a pair = (, ), where:*is an abstract class with = _λ.*is a 4-ary relation on pairs (, M_0, M, N), where M_0MN and ∈<αN. We write M_0MN instead of (, M_0, M, N).*respects -embeddings: if f: N → N' is a -embedding and M_0MN, ∈<αN, then M_0MN if and only if f[M_0]f ()f[M]N'. We may write _ = (K_, ) forand _ for . We say thatis on ^∗ if _ = ^∗. A (≤β, λ)-frame (for β≤λ) is a (<β^+, λ)-frame. A λ-frame is a (≤ 1, λ)-frame. We say a (<α, λ)-frameextends a (<β, λ)-frameif β≤α, _ = _, and for M_0MN and ∈<βN, _ (M_0, , M, N) if and only if _ (M_0, , M, N). Sincerespects -embeddings, it respects types. Therefore we can define: Letbe a (<α,λ)-frame. For M_0M and p ∈^<α (M), we say that p does not -fork over M_0 if M_0MN whenever p =( / M; N). Whenis clear from context, we omit it and just say that p does not fork over M_0. We will call a type basic if it interacts with the nonforking relation, i.e. if it does not fork over a -substructure of its base. Note that basic types are respected by -embeddings (by the definition of a frame). Letbe a (<α, λ)-frame. For M ∈_, a type p ∈^<α (M) will be called -basic (or just basic whenis clear from context) if there exists M_0M such that p does not -fork over M_0. We say thatis type-full if for any M ∈_, every type in ^<α (M) is basic.We will often consider frames whose underlying class is categorical: We say that a (<α, λ)-frameis categorical if _ is categorical (i.e. it contains a single model up to isomorphism). We will consider the following properties that forking may have in a frame: Letbe a (<α, λ)-frame.*has non-order if whenever M_0MN, , ∈<αM and A := () =(), then M_0MN if and only if M_0MN. In this case we will write M_0AMN.*has monotonicity if whenever M_0M_0'MNN', ∈<αN', I ⊆ (), and M_0NN', we have that M_0' IMN'.*has density of basic types if whenever MN and β < α, there exists ∈βN \ M such that ( / M; N) is basic.*has disjointness if M_0MN and ∈<αM imply ∈<αM_0.*has existence if whenever MN, any basic p ∈^<α (M) has a nonforking extension to ^<α (N).*has uniqueness if whenever MN, if p, q ∈^<α (N) both do not fork over M and pM = qM, then p = q.*has local character if for any β≤min (α, λ), any limit ordinal δ < λ^+ with δ≥β, any -increasing continuous chain M_i : i ≤δ, and any basic p ∈^<β (M_δ), there exists i < δ such that p does not fork over M_i.*has symmetry if the following are equivalent for any MN, , ∈<αN such that ( / M; N) and ( / M; N) are both basic:* There exists M_, N_ such that NN_, MM_ N_, ∈<αM_, and ( / M_; N_) does not fork over M.* There exists M_, N_ such that NN_, MM_ N_, ∈<αM_, and ( / M_; N_) does not fork over M. * When α = λ^+,has long transitivity if it has non-order and whenever γ < λ^+ is an ordinal (not necessarily limit), M_i : i ≤γ, N_i : i ≤γ are -increasing continuous, and M_iN_iM_i + 1N_i + 1 for all i < γ, we have that M_0N_0M_γN_γ.Shelah's definition of a good frame (for types of length one) says that a frame must have all the properties above and its underlying class must be “reasonable” <cit.>. The prototypical example is the class of models of cardinality λ of a superstable elementary class which is stable in λ. Taking this class with nonforking gives a good λ-frame (even a good (≤λ, λ)-frame). We use the definition from <cit.> (we omit the continuity property since it follows, see <cit.>). We add the long transitivity property from <cit.> when the types have length λ. We say a (<α, λ)-frameis good if:* _ is an AEC in λ (see the end of Definition <ref>), and _≠∅. Moreover _ is stable in λ, has amalgamation, joint embedding, and no maximal models.*has non-order, monotonicity, density of basic types, disjointness, existence, uniqueness, local character, symmetry, and (when α = λ^+) long transitivity.We also call the AEC generated by _ (see Remark <ref>) the AEC generated by . We will use the following conjugation property of good frames at a crucial point in the proof of Theorem <ref>:[III.1.21 in <cit.>] Letbe a categorical good λ-frame (see Definition <ref>). Let MN and let p ∈ (N). If p does not fork over M, then p and pM are conjugate. That is, there exists an isomorphism f: N ≅ M such that f (p) = pM.The results of this paper also carry over (with essentially the same proofs) in the slightly weaker framework of semi-good λ-frames introduced in <cit.>, where only “almost stability” (i.e. | (M)| ≤λ^+ for all M ∈_λ) and the conjugation property are assumed. For example, in Corollary <ref> we can assume only that there is a semi-good λ-frame on _λ with conjugation (but we still should assume there is a good λ^+-frame on λ^+_λ^+). Several sufficient conditions for the existence of a good frame are known. Assuming GCH, Shelah showed that the existence of a good λ-frame follows from categoricity in λ, λ^+, and a medium number of models in λ^++ (see Fact <ref>). Good frames can also be built using a small amount of tamenessthat follows from amalgamation, no maximal models, and categoricity in a sufficiently high cardinal (see Section <ref> and Fact <ref>).Note that on a categorical good λ-frame, there is only one possible notion of nonforking with the required properties. In fact, nonforking can be given an explicit description, see <cit.>. We will use this without comments:[Canonicity of categorical good frames] Ifandare categorical good λ-frames with the same basic types and _ = _, then =.§.§ Superlimits As has been done in several recent papers (e.g. <cit.>), we have dropped the requirement thathas a superlimit in λ from Shelah's definition of a good frame:[I.3.3 in <cit.>] Letbe an AEC and let λ≥ (). A model N is superlimit in λ if:* N ∈_λ and N has a proper extension.* N is universal: any M ∈_λ -embeds into N.* For any limit ordinal δ < λ^+ and any increasing chain N_i : i < δ, if N ≅ N_i for all i < δ, then N ≅⋃_i < δ N_i. We say that M is superlimit if it is superlimit in M. Again, for a prototypical example consider a superstable elementary class which is stable in a cardinal λ and let M be a saturated model of cardinality λ. Then because unions of chains of λ-saturated are λ-saturated, M is superlimit in λ. In fact, an elementary class has a superlimit in some high-enough cardinal if and only if it is superstable <cit.>.There are no known examples of good λ-frames that do not have a superlimit in λ. In fact most constructions of good λ-frames give one, see for example <cit.>. When a good λ-framehas a superlimit, we can restrictto the AEC generated by this superlimit (i.e the unique AEC ^∗ such that _λ consists of isomorphic copies of the superlimit and is ordered by the appropriate restriction of ) and obtain a new good frame that will be categorical in the sense of Definition <ref>. Thus in this paper we will often assume that the good frame is categorical to start with.When one has a good λ-frame, it is natural to ask whether the frame can be extended to a good λ^+-frame. It turns out that the behavior of the saturated model in λ^+ can be crucial for this purpose. Note that amalgamation in λ and stability in λ indeed imply that there is a unique, universal, saturated model in λ^+ (amalgamation in λ^+ is not needed for this purpose). It is also known that there are no maximal models in λ^+ (see <cit.> or <cit.>). A key property is whether union of chains of saturated models of cardinality λ^+ are saturated. In fact, it is easy to see that this is equivalent to the existence of a superlimit in λ^+. We will use it without comments and leave the proof to the reader: Letbe a good λ-frame generating an AEC . The following are equivalent:*has a superlimit in λ^+.* The saturated model inof cardinality λ^+ is superlimit.* For any limit δ < λ^++ and any increasing chain M_i : i < δ of saturated model in _λ^+, ⋃_i < δ M_i is saturated. §.§ Tameness In an elementary class, types coincides with sets of formulas so are in particular determined by their restrictions to small subsets of their domain. One may be interested in studying AECs where types have a similar behavior. Such a property is called tameness. Tameness was extracted from an argument of Shelah <cit.> and made into a definition by Grossberg and VanDieren <cit.> who used it to prove an upward categoricity transfer theorem <cit.>. Here we present the definitions we will use, and a few sufficient conditions for tameness (given as motivation but not used in this paper). We refer the reader to the survey of Boney and the author <cit.> for more on tame AECs. For an abstract class , a class of types Γ, and an infinite cardinal χ, we say thatis (<χ)-tame for Γ if for any p, q ∈Γ over the same set B, pA = qA for all A ⊆ B with |A| < χ implies that p = q. We say thatis χ-tame for Γ if it is (<χ^+)-tame for Γ. We will use the following variation from <cit.>: An AECis (<χ, λ)-weakly tame ifis (<χ)-tame for the class of types of length one over saturated models of cardinality λ. When we omit the weakly, we mean thatis (<χ)-tame for the class of types of length one over any model of cardinality λ. Define similarly variations such as (χ, <λ)-weakly tame, or χ-tame (which means (χ, λ)-tame for all λ≥χ). A consequence of the compactness theorem is that any elementary class is (<ℵ_0)-tame (for any class of types). Boney <cit.>, building on work of Makkai and Shelah <cit.> showed that tameness follows from a large cardinal axiom: Ifis an AEC and χ >() is strongly compact, thenis (<χ)-tame for any class of types. Recent work of Boney and Unger <cit.> show that this can, in a sense, be reversed: the statement “For every AECthere is χ such thatis χ-tame” is equivalent to a large cardinal axiom. It is however known that when the AEC is stability-theoretically well-behaved, some amount of tameness automatically holds. This was observed for categoricity in a cardinal of high-enough cofinality by Shelah <cit.> and later improved to categoricity in any big-enough cardinal by the author <cit.>: Letbe an AEC with arbitrarily large models and let λ≥ (). If _<λ has amalgamation and no maximal models andis categorical in λ, then there exists χ <() such thatis (χ, <λ)-weakly tame. Very relevant to this paper is the fact that tameness allows one to transfer good frames up <cit.>: Letbe a good λ-frame generating an AEC . Ifis (λ, λ^+)-tame and has amalgamation in λ^+, then there is a good λ^+-frameon _λ^+. Moreover -nonforking can be described in terms of -nonforking as follows: for MN, p ∈ (N) does not -fork over M if and only if there exists M_0 ∈λM so that for all N_0 ∈λN with M_0N_0, pN_0 does not -fork over M_0.[<cit.>] Letbe a good λ-frame generating an AEC . We write _λ^+ for the frame on _λ^+ with nonforking relation described by Fact <ref>. We write λ^+_λ^+ for its restriction to λ^+_λ^+. § WHEN IS THERE A SUPERLIMIT IN Λ^+? Starting with a good λ-frame generating an AEC , it is natural to ask whenhas a superlimit in λ^+, i.e. when the union of any increasing chains of λ^+-saturated models is λ^+-saturated. We should say that there are no known examples when this fails, but we are unable to prove it unconditionally. We give here the following condition on forking characterizing the existence of a superlimit in λ^+. The condition is extracted from the property (∗∗)_M_1^∗, M_2^∗ in <cit.>. After the initial submission of this article, John T. Baldwin pointed out that the condition is similar, in the first-order case, to the fact (valid in any stable theory) that λ^+-saturated models are strongly λ^+-saturated (in Shelah's terminology, F_λ^+^a-saturated), see <cit.>.Recall that wheneveris a good λ-frame, there is always a unique saturated model in _λ^+ (amalgamation in λ^+ is not needed to build it), so the definition below always makes sense. Letbe a good λ-frame generating an AEC . We say thatis decent if for any two saturated model MN both in _λ^+, any two λ-sized models M_0N_0N with M_0M, and any basic p ∈ (M_0), the nonforking extension of p to N_0 is already realized inside M. In other words, there exists a ∈ M such that (a / N_0; N) is the nonforking extension of p to N_0. Note that the type p in the above definition will of course be realized inside N (since N is saturated and contains N_0). The point is that it will also be realized inside the smaller model M, even though M itself will not necessarily contain N_0. Letbe a good λ-frame generating an AEC . The following are equivalent:* has a superlimit model in λ^+.*There exists a partial order ≤^∗ on Kλ^+_λ^+ such that: * Whenever M_0M_1 are both in λ^+_λ^+, there exists M_2 ∈λ^+_λ^+ such that M_1M_2 and M_0 ≤^∗ M_2* For any increasing chain M_i : i < ω in λ^+_λ^+ such that M_i ≤^∗ M_i + 1 for all i < ω, ⋃_i < ω M_i is saturated. * is decent.* (<ref>) implies (<ref>): Trivial (take ≤^∗ to be ).* (<ref>) implies (<ref>): Assume (<ref>). Let δ < λ^++ be a limit ordinal and let M_i : i < δ be an increasing chain of saturated models in _λ^+. We want to show that M_δ := ⋃_i < δ M_i is saturated. Without loss of generality, δ = δ < λ^+. Let M^0 ∈λ(M_δ). Build M_i^0 : i < δ increasing chain in _λ such that for all i < δ, M_i^0M_i and |M^0| ∩ |M_i| ⊆ |M_i + 1^0|. This is possible using that each M_i is saturated. Let M_δ^0 := ⋃_i < δ M_i^0. Then M^0M_δ^0. Let p ∈ (M_δ^0). By local character, there exists i < δ such that p does not fork over M_i^0. Sinceis decent, p is realized in M_i, hence in M_δ. This shows that M_δ realizes all basic types over its -substructures of cardinality λ, which in turns implies that M_δ is saturated <cit.>. * (<ref>) implies (<ref>): Assume (<ref>) and suppose for a contradiction thatis not decent. We build M_i : i ≤ω, M_i^0 : i ≤ω increasing continuous and a type p ∈ (M_0^0) such that for all i < ω: (writing p_M for the nonforking extension of p to (M)):* M_i ∈λ^+_λ^+.* M_i≤^∗ M_i + 1.* M_i^0 ∈λ(M_i).* p_M_i + 1^0 is not realized in M_i. This is enough: By assumption, M_ω is saturated. Therefore p_M_ω^0 is realized inside M_ω, and therefore inside M_i for some i < ω. This means in particular that p_M_i + 1 is realized in M_i, a contradiction.This is possible: For i = 0, pick M_0, M_1', M_0^0, M_1^0, p ∈ (M_0^0) witnessing the failure of being decent (i.e. M_0M_1' are both saturated in _λ^+, M_0^0M_0, M_0^0M_1^0M_1', M_0^0, M_1^0 ∈_λ, and p_M_1^0 is omitted in M_)), and then use the properties of ≤^∗ to obtain M_1 ∈λ^+_λ^+ such that M_0 ≤^∗ M_1 and M_1'M_1. Now given M_j^0 : j ≤ i + 1, M_j : j ≤ i + 1, M_i and M_i + 1 are both saturated and so must be isomorphic over any common submodel of cardinality λ. Let f: M_i ≅_M_i^0 M_i + 1 and let g : M_i + 1≅ M_i + 2' be an extension of f (in particular M_i + 1 M_i + 2'). Since g is an isomorphism, g (p_M_i + 1^0) is not realized in g[M_i] = M_i + 1. Pick M_i + 2^0 in λ (M_i + 2') such that M_i + 1^0M_i + 2^0 and g[M_i + 1^0]M_i + 2^0. Finally, pick some M_i + 2∈λ^+_λ^+ such that M_i + 1≤^∗ M_i + 2 and M_i + 2'M_i + 2. This is possible by the assumed properties of ≤^∗.Theorem <ref> can be generalized to the weaker framework of a λ-superstable AEC (implicit for example in <cit.>; see <cit.> for what forking is there and what properties it has). For this we ask in the definition of decent that M be limit, and we ask for example that M_i + 1 is limit over M_i in the proof of (<ref>) implies (<ref>) of Theorem <ref>. VanDieren <cit.> has shown that λ-symmetry (a property akin to the symmetry property of good λ-frame, see <cit.>) is equivalent to the property that reduced towers are continuous, and if there is a superlimit in λ^+, then reduced towers are continuous. Thus decent is to “superlimit in λ^+” what λ-symmetry is to “reduced towers are continuous”. In particular, being decent is a strengthening of the symmetry property. Note also that taking ≤^∗ in (<ref>) of Theorem <ref> as “being universal over”, we obtain an alternate proof of the main theorem of <cit.> which showed that λ and λ^+-superstability together with the uniqueness of limit models in λ^+ imply that the union of a chain of λ^+-saturated models is λ^+-saturated (see also the proof of Lemma <ref> here). A property related to being decent is what Shelah calls . We show thatimplies decent. We do not know whether the converse holds but it seems (see Fact <ref>) that wherever Shelah uses , he only needs decent.[III.1.3 in <cit.>] Letbe a good λ-frame generating an AEC . We say thatisif the following is impossible:There exists increasing continuous chains in _λ M_i : i < λ^+, N_i : i < λ^+, a basic type p ∈ (M_0), and a_i : i < λ^+ such that for any i < λ^+:* M_iN_i.* a_i + 1∈ |M_i + 2| and (a_i + 1, M_i + 1, M_i + 2) is a nonforking extension of p, but (a_i + 1, N_0, N_i + 2) forks over M_0.* ⋃_j < λ^+ M_j is saturated. Letbe a good λ-frame. Ifis , thenis decent. Letbe the AEC generated by . Suppose thatis not decent. Fix witnesses M, N, M^0, N^0, p. We build increasing continuous chains in _λ, M_i : i < λ^+, N_i : i < λ^+ and a_i :i < λ^+ such that for all i < λ^+:* M_0 = M^0, N_0 = N^0.* M_iM, N_iN.* M_iN_i.* M_i + 1 is universal over M_i.* a_i ∈ M_i + 1.* (a_i / M_i; M_i + 1) is a nonforking extension of p. This is possible since both M and N are saturated. This is enough: M_λ^+ := ⋃_i < λ^+ M_i is saturated (we could also require that M_λ^+ = M but this is not needed) and for any i < λ^+, (a_i + 1 / N_0; N_i + 2) forks over M_0. If not, then by the uniqueness property of nonforking, we would have that a_i + 1 realizes the nonforking extension of p to N_0 = N^0, which we assumed was impossible. § FROM WEAK TAMENESS TO GOOD FRAME In this section, we briefly investigate how to generalize Fact <ref> to AECs that are only (λ, λ^+)-weakly tame. The main problem is that the class λ^+_λ^+ may not be closed under unions of chains (i.e. it may not be an AEC in λ^+). Indeed, this is the only difficulty.Letbe a good λ-frame generating an AEC . Ifis (λ, λ^+)-weakly tame, the following are equivalent:* is decent and λ^+_λ^+ has amalgamation.*_λ^+λ^+_λ^+ (see Definition <ref>) is a good λ^+-frame on λ^+_λ^+.*There is a good λ^+-frame on λ^+_λ^+. Moreover in (<ref>), ifis type-full, then we can conclude that the good λ^+-frame described in (<ref>) is also type-full. (<ref>) implies (<ref>) is exactly as in the proof of Fact <ref>. (<ref>) implies (<ref>) is trivial. (<ref>) implies (<ref>) is by Fact <ref> and Theorem <ref>, since by definition the existence of a good λ^+-frame on λ^+_λ^+ implies that λ^+_λ^+ has amalgamation and λ^+_λ^+ is an AEC in λ^+, hence that the union of an increasing chain of λ^+-saturated models is λ^+-saturated, i.e. thathas a superlimit in λ^+. A note on the definitions: statements of the form “there is a good λ^+-frame on λ^+_λ^+” imply in particular (from the definition of a good frame) that λ^+_λ^+ is an AEC in λ^+, i.e. that the union of an increasing chain of λ^+-saturated models is λ^+-saturated. We will often use this without comments. It is natural to ask whether the good λ^+-frame in Theorem <ref> is itself decent. We do not know the answer, but can answer positively assuming 2^λ < 2^λ^+ (see Corollary <ref>) and in fact in this caseis also . We can show in ZFC that beingtransfers up. The proof adapts an argument of Shelah <cit.>. Letbe aλ-frame generating an AEC . Ifis (λ, λ^+)-weakly tame and λ^+_λ^+ has amalgamation, then _λ^+λ^+_λ^+ is aλ^+-frame on λ^+_λ^+. By Lemma <ref>,is decent, so by Theorem <ref> := _λ^+λ^+_λ^+ is a good λ^+-frame on λ^+_λ^+. Moreover by definition -nonforking is described in terms of -nonforking as in the statement of Fact <ref>.Suppose thatis not . Let M_i : i < λ^++, N_i : i < λ^++, p, a_i : i < λ^++ witness thatis not . Let M_j' : j < λ^+ be increasing continuous in _λ such that M_0 = ⋃_j < λ^+ M_j' and let N_j' : j < λ^+ be increasing continuous in _λ such that N_0 = ⋃_j < λ^+ N_j' and M_j'N_j' for all j < λ^+.By a standard pruning argument, there is j^∗ < λ^+ and an unbounded S ⊆λ^++ of successor ordinals such that for all i ∈ S and all M' ∈λ (M_i) with M_j^∗'M', (a_i / M'; M_i + 1) does not -fork over M_j^∗'. Now by assumption for all i ∈ S, (a_i / N_0; N_i + 1) -forks over M_0, so by a pruning argument again, there is an unbounded S' ⊆ S and j^∗∗∈ [j^∗, λ^+) such that for all i ∈ S', (a_i / N_j^∗∗'; N_i + 1) -forks over M_j^∗'. We build i_j : j < λ^+ and M_j^∗ : j < λ^+, N_j^∗ : j < λ^+ increasing continuous in _λ such that for all j < λ^+:* i_j ∈ S'.* M_j^∗ N_j^∗.* M_j + 1^∗ is universal over M_j^∗.* M_0^∗ = M_j^∗', N_0^∗ = N_j^∗∗'.* M_j^∗ M_i_j, N_j^∗ N_i_j.* a_i_j∈ M_j + 1^∗. This is enough: Then by construction of S', j^∗, and j^∗∗, M_j^∗ : j < λ^+, N_j^∗ : j < λ^+, (a_i_0 / M_0^∗; M_1^∗) and a_i_j : j < λ^+ witness thatis not .This is possible: Let M_λ^++ := ⋃_i < λ^++ M_i, N_λ^++ := ⋃_i < λ^++ N_i. We are already given M_0^∗ and N_0^∗ and for j limit we take unions. Now assume inductively that M_k^∗ : k ≤ j, N_k^∗ : k ≤ j and i_k : k < j are already given, with M_j^∗ M_λ^++ and N_j^∗ N_λ^++. Let i_j ∈ S' be big-enough such that N_i_j contains N_j^∗, M_i_j contains M_j^∗, and i_k < i_j for all k < j. Such an i_j exists since S' is unbounded. Now let M^∗∈λ^+ M_λ^++ contain M_j^∗ and a_i_j and be saturated. Such an M^∗ exists since M_λ^++ is saturated by assumption. Now pick M_j + 1^∗∈λ (M^∗) so that a_i_j∈ M_j + 1^∗ and M_j + 1^∗ is universal over M_j^∗. This is possible since M^∗ is saturated. Finally, pick any N_j + 1^∗∈λ (N_λ^++) containing M_j + 1^∗ and N_j^∗. We end this section by noting that in the context of Fact <ref>,is decent and hence by Theorem <ref> there is a good λ^+-frame on λ^+_λ^+. In fact: Letbe a good λ-frame generating an AEC . If there is a good λ^+-frame on _λ^+, thenis decent. The proof uses the uniqueness of limit models in good frames, due to Shelah <cit.> (see <cit.> for a proof):Letbe a good λ-frame, δ_1, δ_2 < λ^+ be limit ordinals. Let M_i^ℓ : i ≤δ_ℓ, ℓ = 1,2, be increasing continuous. If for all ℓ = 1,2, i < δ_ℓ, M_i + 1^ℓ is universal over M_i, then M_δ_1^1 ≅ M_δ_2^2.By Theorem <ref>, it suffices to show thathas a superlimit in λ^+. We could apply two results of VanDieren <cit.> but we prefer to give a more explicit proof here.Let N_i : i ≤λ^+ be increasing continuous in _λ^+ such that N_i + 1 is universal over N_i for all i < λ^+. This is possible since by definition of a good λ^+-frame,is stable in λ^+ and has amalgamation in λ^+. Clearly, N_λ^+ is saturated. Moreover by Fact <ref>, N_λ^+≅ N_ω. Thus N_ω is also saturated. We chose N_i : i ≤ω arbitrarily, therefore (<ref>) of Theorem <ref> holds with ≤^∗ being “universal over or equal to”. Thus (<ref>) there holds:is decent, as desired. There are other variations on Lemma <ref>. For example, ifis a good λ-frame generating an AEC ,has amalgamation in λ^+, andis (λ, λ^+)-weakly tame, thenis decent (to prove this, we transfer enough of the good λ-frame up, then apply results of VanDieren <cit.>). § FROM GOOD FRAME TO WEAK TAMENESS In this section, we look at a sufficient condition (due to Shelah) implying that a good λ-frame can be extended to a good λ^+-frame and prove its necessity.It turns out it is convenient to first extend the good λ-frame to a good (≤λ, λ)-frame. For this, the next technical property is of great importance, and it is key in Chapter II and III of <cit.>. The definition below follows <cit.> (but as usual, we work only with type-full frames). Note that we will not use the exact content of the definition, only its consequence. We give the definition only for the benefit of the curious reader.Letbe an abstract class and λ be a cardinal.* amalgam For M_0M_ℓ all in _λ, ℓ = 1,2, an amalgam of M_1 and M_2 over M_0 is a triple (f_1, f_2, N) such that N ∈_λ and f_ℓ : M_ℓ N.* equivalence of amalgam Let (f_1^x, f_2^x, N^x), x = a,b be amalgams of M_1 and M_2 over M_0. We say (f_1^a, f_2^a, N^a) and (f_1^b, f_2^b, N^b) are equivalent over M_0 if there exists N_∗∈_λ and f^x : N^x → N_∗ such that f^b ∘ f_1^b = f^a ∘ f_1^a and f^b ∘ f_2^b = f^a ∘ f_2^a, namely, the following commutes: N^a @.>[r]^f^aN_∗M_1 [ru]^f_1^a[rr]|>>>>>f_1^b N^b @.>[u]_f^bM_0 [u] [r] M_2 [uu]|>>>>>f_2^a[ur]_f_2^b Note that being “equivalent over M_0” is an equivalence relation (<cit.>).* Letbe a good (<α, λ)-frame. * A uniqueness triple inis a triple (, M, N) such that MN, ∈<αN and for any M_1M, there exists a unique (up to equivalence over M) amalgam (f_1, f_2, N_1) of N and M_1 over M such that (f_1 (a) / f_2[M_1] ; N_1) does not fork over M.*has the existence property for uniqueness triples if for any M ∈_ and any basic p ∈^<α (M), one can write p =( / M; N) with (, M, N) a uniqueness triple.* We say thatis weakly successful if its restriction to types of length one has the existence property for uniqueness triples. As an additional motivation, we mention the closely related notion of a domination triple: Letbe a good (≤λ, λ)-frame. A domination triple inis a triple (, M, N) such that MN, ∈≤λN, and whenever M'N' are such MM', NN', then MM'N' implies MNM'N'. The following fact shows that domination triples are the same as uniqueness triples once we have managed to get a type-full good (≤λ, λ)-frame. The advantage of uniqueness triples is that they can be defined already inside a good λ-frame.[11.7, 11.8 in <cit.>] Letbe a type-full good (≤λ, λ)-frame. Then inuniqueness triples and domination triples coincide. The importance of weakly successful good frames is that they extend to longer types. This is due to Shelah: Letbe a categorical good λ-frame. Ifis weakly successful, then there is a unique type-full good (≤λ, λ)-frame extending . The uniqueness is <cit.>. Existence is the main result of <cit.>, although there Shelah only builds a nonforking relation for models satisfying the axioms of a good (≤λ, λ)-frame. How to extend this to all types of length at most λ is done in <cit.>. An outline of the full argument is in the proof of <cit.>. It is not clear whether the converse of Fact <ref> holds, but see Fact <ref>.Shelah proved <cit.> that a categorical good λ-frame (generating an AEC ) is weakly successful whenever 2^λ < 2^λ^+ < 2^λ^++ andhas a “medium” number of models in λ^++. Shelah has also shown that being weakly successful follows from some stability in λ^+ and 2^λ < 2^λ^+, see <cit.> for a proof: Letbe a categorical good λ-frame generating an AEC . Assume 2^λ < 2^λ^+. If for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M, thenis weakly successful. Once we have a good (≤λ, λ)-frame, we can define a candidate for a good λ^+-frame on the saturated models in _λ^+. Nonforking in λ^+ is defined in terms of nonforking in λ (in fact one can make sense of this definition even if we only start with a good λ-frame), but the problem is that we do not know that the class of saturated models in _λ^+ has amalgamation. To achieve this, the orderingis changed to a new ordering ^+ so that nonforking “reflects down”. The definition is due to Shelah <cit.> but we follow <cit.>: Let = (_, ) be a type-full good (≤λ, λ)-frame generating an AEC . We define a pair ^+ = (_^+, _^+) as follows:* _^+ = (K_^+, ^+), where: * K_^+ is the class of saturated models in _λ^+.* For M, N ∈ K_^+, M ^+ N holds if and only if there exists increasing continuous chains M_i : i < λ^+, N_i : i < λ^+ in _λ such that:* M = ⋃_i < λ^+ M_i.* N = ⋃_i < λ^+ N_i.* For all i < j < λ^+, M_iN_iM_jN_j.* For M_0 ^+ M ^+ N and a ∈ N, _^+ (M_0, a, M, N) holds if and only if there exists M_0' ∈λ (M_0) such that for all M' ∈λ (M) and all N' ∈λ (N), if M_0'M'N' and a ∈ N', then _ (M_0', a, M', N'). For a weakly successful categorical good λ-frame , we define ^+ := ^+, whereis the unique extension ofto a type-full good (≤λ, λ)-frame (Fact <ref>). As a motivation for the definition of ^+, observe that if M_i : i < λ^+, N_i : i < λ^+ are increasing continuous in _λ such that M_iN_i for all i < λ^+, then it is known that there is a club C ⊆λ^+ such that M_j ∩ N_i = M_i for all i ∈ C and all j > i. We would like to conclude the stronger property that M_iN_iM_jN_j for i ∈ C and j > i. If we are working in a superstable elementary class this is true as nonforking has a strong finite character property, but in the more general setup of good frames this is not clear. Thus the property is built into the definition by changing the ordering. This creates a new problem: we do not know whether _^+ is an AEC (smoothness is the problematic axiom). Note that there are no known examples of weakly successful good λ-framewhere ^+ is not .The following general properties of ^+ are known. They are all proven in <cit.> but we cite from <cit.> since the proofs there are more detailed: Letbe a weakly successful categorical good λ-frame generating an AEC .* <cit.> If M, N ∈_^+ and M ^+ N, then MN.*<cit.>, _^+ is an abstract class which is closed under unions of chains of length strictly less than λ^++.*<cit.> _^+ satisfies the following strengthening of the coherence axiom: if M_0, M_1, M_2 ∈_^+ are such that M_0 ^+ M_2 and M_0M_1M_2, then M_0 ^+ M_1.* <cit.> _^+ has no maximal models and amalgamation. *<cit.> Let M ^+ N_ℓ and let a_ℓ∈ N_ℓ, ℓ = 1,2. Then (a_1 / M; N_1) =(a_2 / M; N_2) if and only if _^+ (a_1 / M; N_1) = _^+ (a_2 / M; N_2). In particular, ^+ is a λ^+-frame.We will use the following terminology, taken from <cit.>: Letbe a type-full good (≤λ, λ)-frame generating an AEC . We say thatreflects down if MN implies M ^+ N for all M, N ∈_^+. We say thatalmost reflects down if _^+ is an AEC in λ^+ (see the bottom of Definition <ref>). We say that a weakly successful good λ-frame [almost] reflects down if its extension to a type-full good (≤λ, λ)-frame [almost] reflects down, see Fact <ref>. Shelah uses the less descriptive “successful”:[III.1.1 in <cit.>] We say that a good λ-frameis successful if it is weakly successful and almost reflects down. The point of this definition is that if it holds, thencan be extended to a good λ^+-frame. Moreoveris successful when there are few models in λ^++:[III.1.9 in <cit.>] Ifis a successful categorical good λ-frame, then ^+ is aλ^+-frame. [II.8.4, II.8.5 in <cit.>, or see 7.1.3 in <cit.>] Letbe a weakly successful categorical good λ-frame generating the AEC . If (, λ^++) < 2^λ^++, thenis successful. The downside of working only with a successful good λ-frame is that the ordering of _^+ may not beanymore. Thus the AEC generated by _^+ may be different from the one generated by _. For example it may fail to have arbitrarily large models even if the original one does. This is why we will focus on good frames that reflect down, i.e. ^+ is just . Several characterizations of this situation are known. (<ref>) implies (<ref>) below is due to Jarden and all the other implications are due to Shelah (but as usual we mostly quote from <cit.>): Letbe a categorical good λ-frame. The following are equivalent:* extends to a type-full good (≤λ, λ)-frame which reflects down.* is successful and(see Definition <ref>).* is successful and decent (see Definition <ref>).* is weakly successful and generates an AECwhich is (λ, λ^+)-weakly tame and so that λ^+_λ^+ has amalgamation.* (<ref>) implies (<ref>): By <cit.>,is weakly successful. Since it reflects down, it is successful and ^+ is the restriction ofto λ^+_λ^+. By adapting the proof of <cit.> (see <cit.>), we get thatis .* (<ref>) implies (<ref>): By Lemma <ref>.* (<ref>) implies (<ref>): By definition of successful,is weakly successful, so by Fact <ref> extends to a unique type-full good (≤λ, λ)-frame. Sinceis decent, we have that the ordering ≼_λ^+^⊗ from <cit.> is the same as . By assumptionis successful, so all of the equivalent conditions of <cit.> are false, so in particular for M, N ∈_^+, MN implies M ^+ N. Thereforereflects down.* (<ref>) implies (<ref>): we have already argued thatis successful, and by definition ^+ is just the restriction of , i.e. _^+ = λ^+_λ^+. Now weak tameness follows from <cit.> (a similar argument already appears in <cit.>) and amalgamation is because by Fact <ref> ^+ is a good λ^+-frame.* (<ref>) implies (<ref>): By <cit.>.The aim of this section is to add another condition to Fact <ref>: the existence of a good λ^+-frame on λ^+_λ^+. Toward this, we will use the following ordering on pairs of saturated models, introduced by Jarden <cit.>. The idea is as follows: call a pair (M, N) of saturated models in _λ^+ bad if MN but M ^+ N. Jarden's ordering of these pairs works in such a way that higher pairs interact non-trivially with lower pairs. Jarden showed that there are maximal such pairs, and we will show that if there is a good λ^+-frame then on the other hand it is always possible to extend these pairs. This will be a contradiction showing that there are no bad pairs after all. Letbe a weakly successful categorical good λ-frame generating an AEC . For pairs (M, N), (M', N') in _^+ with MN, M'N', we write (M, N)(M', N') if:* M ^+ M'.* NN'.* M' ∩ N ≠ M. We write (M, N)(M', N') if (M, N)(M', N') or (M, N) = (M', N').Letbe a decent weakly successful categorical good λ-frame generating an AEC .*For any MN both in _^+, there exists a pair (M', N') such that (M, N)(M', N') and (M', N') is -maximal.*<cit.> MN are both in _^+, and M ^+ N, then there exists M_i : i ≤λ^+ increasing continuous in _^+ such that M_λ^+ = M and M_i ^+ N for all i < λ^+.* By the proof of <cit.>. We use decency here to make sure (via Theorem <ref>) that λ^+_λ^+ is an AEC in λ^+, hence closed under unions of short chains (Jarden works with the whole of _λ^+).* The definition of decent says that the ordering ≼_λ^+^⊗ from <cit.> is the same as the usual orderingon λ^+_λ^+. Thus if MN are both in _^+ but M ^+ N, then condition (2) in the equivalence proven in <cit.> holds, hence condition (3) there also holds, which is what we wanted to prove.We have arrived to the main theorem of this section: Letbe a weakly successful categorical good λ-frame generating an AEC . If there is a good λ^+-frame on λ^+_λ^+, thenreflects down. Letbe the good λ^+-frame on λ^+_λ^+. Note that sinceis good, λ^+_λ^+ must be an AEC in λ^+, i.e. unions of chains of λ^+-saturated models are λ^+-saturated, so by Fact <ref>has a superlimit in λ^+ which by Theorem <ref> implies thatis decent. Therefore we will later be able to apply Fact <ref>. We first show:Claim: If MN are both in _^+, a ∈ |N| \ |M| and (a / M ; N) is realized in a ^+-extension of M, then (M, N) is not -maximal.Proof of Claim: Let p :=(a / M; N). Let N' be such that M ^+ N' and p is realized by b ∈ N' (i.e. p =(b / M; N')). By amalgamation in λ^+_λ^+ (which holds since we are assuming there is a good frame on this class) and the fact that (a / M; N) =(b / M; N'), there exists N”∈λ^+_λ^+ and f: N'N” with NN” such that f (b) = a. Consider the pair (f[N'], N”). Since ^+ is invariant under isomorphisms and M ^+ N', M ^+ f[N']. Moreover, a ∈ |N| \ |M|, so b ∈ |N'| \ |M|, and so a = f (b) ∈ |f[N']| \ |M|. Since we also have that a ∈ |N|, this implies that f[N'] ∩ N ≠M, so (M, N)(f[N'], N”), hence (M, N) is not -maximal. †_ClaimLet M, N ∈_^+ be such that MN. We have to show that M ^+ N. Suppose not. By Fact <ref>(<ref>), there exists (M', N') such that (M, N)(M', N') and (M', N') is -maximal. Observe that M' ^+ N': if M' ^+ N', then since also M ^+ M' and _^+ is an abstract class (Fact <ref>(<ref>)), we would have that M ^+ N' so by Fact <ref>(<ref>), M ^+ N, a contradiction. By Fact <ref>(<ref>), there exists M_i : i ≤λ^+ increasing continuous in _^+ such that M_λ^+ = M' and M_i ^+ N' for all i < λ^+. Since M' ^+ N', we have in particular that M' ≠ N'. By density of basic types, let a ∈ N' \ M' and let p :=(a / M'; N') be basic. By local character in(the good λ^+-frame on λ^+_λ^+), there is i < λ^+ such that p does not -fork over M_i. By conjugation in(Fact <ref>), this means that pM_i and p are isomorphic. However since M_i ^+ N', pM_i is realized inside some ^+-extension of M_i, hence p is also realized inside some ^+-extension of M'. Together with the claim, this contradicts the -maximality of (M', N'). We are not using all the properties of the good λ^+-frame. In particular, it suffices that local character holds for chains of length λ^+. We obtain a new characterization of when a weakly successful frame reflects down. We emphasize that only (<ref>) implies (<ref>) below is new. (<ref>) implies (<ref>) and (<ref>) implies (<ref>) are due to Shelah while (<ref>) implies (<ref>) is due to Jarden. Letbe a weakly successful categorical good λ-frame generating the AEC . The following are equivalent:* is successful and decent.* is (λ, λ^+)-weakly tame and λ^+_λ^+ has amalgamation.*There is agood λ^+-frame on λ^+_λ^+. Moreover, if it exists then the good λ^+-frameon λ^+_λ^+ is unique, can be enlarged to be type-full, and coincides (on the appropriate class of basic types) with ^+ and _λ^+λ^+_λ^+ (see Definition <ref>). * (<ref>) is equivalent to (<ref>): By Fact <ref>.* (<ref>) implies (<ref>): By Facts <ref> and <ref>.* (<ref>) implies (<ref>): By Theorem <ref> and Fact <ref>. For the moreover part, assume thatis a good λ^+-frame on λ^+_λ^+. By the proof of (<ref>) implies (<ref>), ^+ is in fact a good λ^+-frame on λ^+_λ^+. By weak tameness and Theorem <ref>, _λ^+λ^+_λ^+ is also a good λ^+-frame on λ^+_λ^+. Without loss of generality (Fact <ref>(<ref>)),is type-full so by the moreover part of Theorem <ref>, _λ^+λ^+_λ^+ is also type-full, and by canonicity (Fact <ref>) is the desired extension of . We can combine our result with the weak GCH to obtain weak tameness from two successive good frames. This gives a converse to Theorem <ref>. Letbe an AEC, let λ≥ () and assume that 2^λ < 2^λ^+. Letbe a categorical good λ-frame on _λ. The following are equivalent:*There is aλ^+-frame on λ^+_λ^+.*There is a good λ^+-frame on λ^+_λ^+.* is (λ, λ^+)-weakly tame,is decent, and λ^+_λ^+ has amalgamation.* is (λ, λ^+)-weakly tame and for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M.* is successful and .* (<ref>) implies (<ref>): Trivial.* (<ref>) implies (<ref>): By Fact <ref>,is weakly successful. Now apply Corollary <ref>.* (<ref>) implies (<ref>): By Theorem <ref>.* (<ref>) implies (<ref>): Assume (<ref>). Then by definition of a good λ^+-frame, for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M. Further we proved already that (<ref>) holds. Thereforeis (λ, λ^+)-weakly tame.* (<ref>) implies (<ref>): By Fact <ref>,is weakly successful. Now apply (<ref>) implies (<ref>) in Fact <ref>.* (<ref>) implies (<ref>): By Fact <ref> and sincereflects down (Fact <ref>).§ FROM WEAK TO STRONG TAMENESS Assuming amalgamation in λ^+, we are able to conclude (λ, λ^+)-tameness in (<ref>) of Corollary <ref>. To prove this, we recall that the definition of ^+ (Definition <ref>) can be extended to all of _λ in the following way: Let = (_, ) be a good (≤λ, λ)-frame generating an AEC . We define a pair ^∗ = (_^∗, _^∗) as follows:* _^∗ = (K_^∗, ^∗), where: * K_^∗ = K_λ^+.* For M, N ∈ K_^∗, M ^∗ N holds if and only if there exists increasing continuous chains M_i : i < λ^+, N_i : i < λ^+ in _λ such that:* M = ⋃_i < λ^+ M_i.* N = ⋃_i < λ^+ N_i.* For all i < j < λ^+, M_iN_iM_jN_j.* For M_0 ^∗ M ^∗ N and a ∈ N, _^∗ (M_0, a, M, N) holds if and only if there exists M_0' ∈λ (M_0) such that for all M' ∈λ (M) and all N' ∈λ (N), if M_0'M'N' and a ∈ N', then _ (M_0', a, M', N'). For a weakly successful categorical good λ-frame , we define ^∗ := ^∗, whereis the unique extension ofto a good (≤λ, λ)-frame (Fact <ref>). The parts of <cit.> referenced by Fact <ref> apply to the whole of _λ^+ (i.e. there is no need to restrict to saturated models). Thus we have: Letbe a weakly successful categorical good λ-frame.*<cit.> ^∗ is a λ^+-frame and _^∗ has amalgamation.*<cit.> if M_0, M_1, M_2 ∈_^∗ are such that M_0 ^∗ M_2 and M_0M_1M_2, then M_0 ^∗ M_1.We will also use that every model of _λ^+ has a saturated ^∗-extension.[7.1.10, 7.1.12(a) in <cit.>] Letbe a weakly successful categorical good λ-frame. For any M ∈_^∗, there exists N ∈_^+ such that M ^∗ N.Similarly to Definition <ref>, we name what it means for ^∗ to be trivial: Letbe a good (≤λ, λ)-frame generating an AEC . We say thatstrongly reflects down if MN implies M ^∗ N for all M, N ∈_^∗. We say that a weakly successful good λ-frame strongly reflects down if its extension to a good (≤λ, λ)-frame strongly reflects down, see Fact <ref>. We establish the following criteria for strongly reflecting down: Letbe a weakly successful categorical good λ-frame generating an AEC . The following are equivalent: * reflects down andhas amalgamation in λ^+.* strongly reflects down.(<ref>) implies (<ref>) is Fact <ref>(<ref>). Assume now that (<ref>) holds. Let M, N ∈_^∗ be such that MN. By Fact <ref>, there exists M' ∈_^+ such that M ^∗ M'. By Facts <ref> and <ref>, ^+ is a good λ^+-frame. In particular, there exists N' ∈_^+ such that M' s^+ N' and N' is universal over M' in _^+. Sincehas amalgamation in λ^+ and ^+ is the same asby definition of reflecting down, N' is universal over M. Moreover, M ^∗ N' by transitivity of ^∗. Now let f: NN' by a -embedding. By Fact <ref>(<ref>), M ^∗ f[N], so by invariance M ^∗ N, as desired. It is known that strongly reflecting down gives tameness. This is <cit.>. There is a mistake in the statement given by Jarden and Shelah (≼ there should be ≼_NF) but the proof is still correct. Letbe a weakly successful categorical good λ-frame generating an AEC . Ifstrongly reflects down, thenis (λ, λ^+)-tame. We obtain that tameness is equivalent to being able to extend the frame. Note that only (<ref>) implies (<ref>) and (<ref>) implies (<ref>) are new. Letbe a weakly successful categorical good λ-frame generating an AEC . The following are equivalent:* strongly reflects down.* has amalgamation in λ^+ and is (λ, λ^+)-weakly tame.* has amalgamation in λ^+ and is (λ, λ^+)-tame.*There is a good λ^+-frame on _λ^+.* (<ref>) implies (<ref>): By Facts <ref>(<ref>) and <ref>.* (<ref>) implies (<ref>): Trivial.* (<ref>) implies (<ref>): By Corollary <ref>,is successful and decent. By Fact <ref>,reflects down. By Theorem <ref>,strongly reflects down.* (<ref>) implies (<ref>): By Fact <ref>.* (<ref>) implies (<ref>): Assume that (<ref>) holds. By Lemma <ref>,is decent so by Theorem <ref>,has a superlimit in λ^+. Thus by Fact <ref> we can restrict the good λ^+-frame on _λ^+ to the class λ^+_λ^+ and still obtain a good λ^+-frame. By Theorem <ref>,reflects down. Since we are assuming there is a good λ^+-frame on _λ^+,has amalgamation in λ^+, so by Theorem <ref>,strongly reflects down.Assuming 2^λ < 2^λ^+, we do not need to assume thatis weakly successful. We do not repeat all the equivalences of Corollary <ref> and only state our main result: Letbe an AEC, let λ≥ () and assume that 2^λ < 2^λ^+. If there is a categorical good λ-frameon _λ and a good λ^+-frameon _λ^+, thenis (λ, λ^+)-tame. Moreover, λ^+_λ^+ = _λ^+λ^+_λ^+ (see Definition <ref>). Letbe a categorical good λ-frame on _λ. By Fact <ref>,is weakly successful. Now apply Corollary <ref>. The moreover part follows from the moreover part of Corollary <ref>. § ON CATEGORICITY IN TWO SUCCESSIVE CARDINALS Grossberg and VanDieren have shown <cit.> that in a λ-tame AEC with amalgamation and no maximal models, categoricity in λ and λ^+ imply categoricity in all μ≥λ. In <cit.>, the author gave a more local conclusion as well as a more abstract proof using good frames. Here, we give a converse: assuming the weak GCH, if we can prove categoricity in λ^++ from categoricity in λ and λ^+, then we must have some tameness. This follows from combining results of Shelah[Let us sketch how (of course Corollary <ref> gives another proof but it uses the results of the present paper). By Fact <ref>(<ref>), there is a successful good λ-frameon _λ. In fact, the basic types of this good frames are the minimal types. Thusis(as nonforking corresponds to disjoint amalgamation). It then follows from <cit.> and categoricity thatis (λ, λ^+)-tame.] but seems not to have been noticed before. Only (<ref>) implies (<ref>) below really uses the results of this paper. Letbe an AEC and let λ≥ (). Assume 2^λ < 2^λ^+ < 2^λ^++ and assume thatis categorical in λ and λ^+. The following are equivalent: *There is a successful good λ-frame on _λ.*There is a good λ-frame on _λ and a good λ^+-frame on _λ^+.* is stable in λ,is (λ, λ^+)-tame,has amalgamation in λ^+, and _λ^++≠∅.* is categorical in λ^++.*1 ≤ (, λ^++) < μ_unif (λ^++, 2^λ^+).* is stable in λ,is stable in λ^+, and 1 ≤ (, λ^++) < 2^λ^++.On μ_unif, see <cit.> for a definition and <cit.> for what is known. It seems that for all practical purposes the reader can take μ_unif (λ^++, 2^λ^+) to mean 2^λ^++.Note that if the AECof Corollary <ref> has arbitrarily large models, then stability in λ would follow from categoricity in λ^+ <cit.>. Thus we obtain thatis categorical in λ^++ if and only ifis (λ, λ^+)-tame and has amalgamation in λ^+.To prove Corollary <ref>, we will use several facts:[I.3.8 in <cit.>] Letbe an AEC and let λ≥ (). Assume 2^λ < 2^λ^+. Ifis categorical in λ and (, λ^+) < 2^λ^+, thenhas amalgamation in λ. Letbe an AEC and let λ≥ (). Assume 2^λ < 2^λ^+ and assume thatis categorical in both λ and λ^+.*If _λ^++≠∅ andis stable in λ, then there is an almost good λ-frame (see <cit.>) on _λ.*If _λ^++≠∅,has amalgamation in λ^+,is stable in λ, andis stable in λ^+, then there is a weakly successful good λ-frame on _λ.*If 2^λ^+ < 2^λ^++, 1 ≤ (, λ^++) < 2^λ^++,is stable in λ, andis stable in λ^+, then there is a successful good λ-frame on _λ.*If 2^λ < 2^λ^+ < 2^λ^++ and 1 ≤ (, λ^++) < μ_unif (λ^++, 2^λ^+), then there is a successful good λ-frame on _λ.* By Fact <ref>,has amalgamation in λ. We check that the hypotheses of <cit.> are satisfied. The only ones that we are not explicitly assuming are: * The extension property in _λ, i.e. for every MN both in _λ and every p ∈ (M), if p is not algebraic (i.e. not realized inside M), then p has a nonalgebraic extension to (N): holds by <cit.>.* The existence of an inevitable type in _λ: holds by <cit.> and the density of minimal types (see the proof of (∗)_5 in <cit.>). * By (<ref>), there is an almost good λ-frameon _λ. The proof of <cit.> goes through even for almost good λ-frames and gives thatis weakly successful (note that Shelah's proof of Fact <ref> still goes through in almost good λ-frames). By <cit.>,is in fact a good λ-frame.* By Fact <ref>,has amalgamation in λ and λ^+. By (<ref>), there is a weakly successful good λ-frameon _λ. By Fact <ref>,is successful.* By <cit.>, there is an almost good λ-frameon _λ. By <cit.>,has existence for a certain relative of uniqueness triples. Thus by <cit.>,is actually a good λ-frame. By <cit.>,is weakly successful. By Fact <ref>,is successful. Letbe an AEC and let λ≥ (). Ifis categorical in both λ and λ^+ and there is a successful good λ-frame on _λ, thenis categorical in λ^++. Letbe a successful good λ-frame on _λ. Sinceis categorical in λ^+, there is a superlimit in λ^+, hence by Theorem <ref>is decent, hence by Fact <ref> is . Now combine <cit.> with <cit.>.By categoricity in λ and λ^+ all good λ-frames on _λ are categorical, λ^+_λ^+ = _λ^+, and (λ, λ^+)-weak tameness is the same as (λ, λ^+)-tameness. By Theorem <ref>, any good λ-frame on _λ is decent. We will use this without comments.* (<ref>) implies (<ref>):By Fact <ref>, the definition of reflecting down (Definition <ref>) and the definition of ^+ (Definition <ref>), _^+ = λ^+_λ^+ = _λ^+, so the result follows from Fact <ref>.* (<ref>) implies (<ref>): By definition of a good λ-frame,is stable in λ and by definition of a good λ^+-framehas amalgamation in λ^+. Since a good λ^+-frame has no maximal models in λ^+, _λ^++≠∅. By Corollary <ref>,is (λ, λ^+)-tame.* (<ref>) implies (<ref>): By Fact <ref>,has amalgamation in λ. By Fact <ref>, there is an almost good λ-frame on _λ. By the proof of Fact <ref>,is stable in λ^+. By Fact <ref>, there is a weakly successful good λ-frameon _λ. By Fact <ref>,is successful.* (<ref>) implies (<ref>): By Fact <ref>.* (<ref>) implies (<ref>): Trivial.* (<ref>) implies (<ref>): By Fact <ref>.* (<ref>) implies (<ref>): (<ref>) trivially implies that 1 ≤ (, λ^++) < 2^λ^++. We have also seen that (<ref>) implies (<ref>) implies (<ref>) which by definition implies stability in λ and λ^+.* (<ref>) implies (<ref>): By Fact <ref>. amsalpha | http://arxiv.org/abs/1707.09008v5 | {
"authors": [
"Sebastien Vasey"
],
"categories": [
"math.LO",
"03C48 (Primary), 03C45, 03C52, 03C55, 03C75 (Secondary)"
],
"primary_category": "math.LO",
"published": "20170727192304",
"title": "Tameness from two successive good frames"
} |
SUPA, Physics Department and Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, AB24 3UE (UK)Faculty of Information Studies in Novo Mesto, 8000 Novo Mesto (Slovenia) SUPA, Physics Department and Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, AB24 3UE (UK)Consiglio Nazionale delle Ricerche, Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze (Italy)Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019 Sesto Fiorentino (Italy)Experimental evidence of an absorbing phase transition, so far associated with spatio-temporal dynamics is provided in a purely temporal optical system. A bistable semiconductor laser, with long-delayed opto-electronic feedback and multiplicative noise shows the peculiar features of a critical phenomenon belonging to the directed percolation universality class. The numerical study of a simple, effective model provides accurate estimates of the transition critical exponents, in agreement with both theory and our experiment. This result pushes forward an hard equivalence of non-trivial stochastic, long-delayed systems with spatio-temporal ones and opens a new avenue for studying out-of-equilibrium universality classes in purely temporal dynamics.Evidence of a critical phase transition in a purely temporal dynamics with long-delayed feedback Giovanni Giacomelli December 30, 2023 ================================================================================================The concepts of scaling and universality play a prominent role in statistical physics <cit.>.Starting with early – and pioneering – scaling ideas <cit.>up to renormalization group theory <cit.>, they have been successfully applied to develop a comprehensive theory of critical phenomena in equilibrium <cit.> and, to a large extent in non-equilibrium systems <cit.>. Diverging correlation lengths and a scaling of relevant quantities ruled by universal exponents are the signature of such phenomena. In the framework of spatially extended media, universality is uncovered as the system is inspected at increasing length scales, and often characterized via spatially-resolved measurements of the significant quantities (e.g., correlation functions).An important question is, to what extent the universality classes predicted and observed in spatio-temporal systems can also hold in a purely temporal dynamics, without explicit spatial degrees of freedom. In this Letter, we bring the first answer to the above issue by investigating the behavior of a stochastic, long-delayed bistable system. A delayed feedback sets an infinite-dimensional phase space for the dynamics <cit.>. A special case is the so-called long-delay limit, i.e. when the delay time τ is much longer than any other, internal timescale. Here, a suitable representation <cit.> permits to unveil the role of the involved multiple time-scales acting as effective spatial variables (for a recent review see <cit.>) and a thermodynamic limit is defined as τ→∞. This correspondence has been shown to hold in deterministic systems <cit.> and even put on rigorous grounds close to a Hopf bifurcation <cit.>. However, in the presence of non-trivial stochastic processes only few numerical studies in simple models have been reported <cit.>. In particular, whether this equivalence is preserved as a critical transition point is approached and correlation lengths diverge has never been tested experimentally.Here, we address this fundamental question showing that a stochastic, long-delayed bistable laser – where effective spatial degrees of freedom emerge from different, well separated timescales – undergoes a genuine out-of-equilibrium active-to-absorbing critical phase transition, belonging to the Directed Percolation (DP) universality class in one-spatial dimension <cit.>. Non-equilibrium models related to epidemic spreading <cit.>, the gravity-driven percolation of fluids through a porous medium <cit.> intermittent, interface-depinning and synchronization phenomena <cit.> and to the transition from laminar to turbulent flows <cit.> are other well-known examples of this universality class, so far mainly associated to genuine spatiotemporal dynamics. Recently, DP and non-equilibrium phase transitions into absorbing states have been investigated in open many-body quantum systems <cit.>. Due to its prominence, DP is commonly regarded as the Ising model of non-equilibrium critical phenomena, butexperimental evidence has long been elusive. Only recently, measurements in systems displaying spatio-temporal turbulence <cit.> have provided the first clear evidence of DP critical behavior in one and two spatial dimensions, and sparkled renewed attention for this ubiquitous non-equilibrium phenomenon.Experiment. The experimental setup (Fig.<ref>) is based on a bistable Vertical Cavity Surface Emitting Laser (VCSEL) which, for a particular choice of the pump current, displays the coexistence of two linear polarization states of the emitted optical field <cit.>. The two polarizations are separated by means of an half-wave plate and a polarizing beamsplitter and then their intensities are monitored by photodetectors. The signal corresponding to the upper state is then acquired, delayed by a time τ=190 ms and subsequently fed back into the VCSEL through the pump current using a summer-circuit. A Gaussian noise, software-generated as a sequence of zero-mean, delta-correlated numbers is then multiplied to the (non-delayed) main polarization signal and re-injected as well. Due to its multiplicative nature, this noise vanishes on the lower state, while it affects the other polarization inducing spontaneous fluctuations towards the lower state. The delayed feedback is realized sampling the electric signal from the detector with a A/D-D/A board hosted by a PC driven by a Real-Time Linux OS. The data are treated with a custom software which allow to choose the initial condition, the gain and the amount of multiplicative noise.We begin our experimental test by a visual investigation of evolution of the polarized intensity signal I(t) (in the following, denoted as intensity). The transformation t=n τ+σ with integer n and real σ∈[0, τ) is used to obtain the corresponding Pseudo Spatio-Temporal (P-ST) representation in the plane (σ, n).The original timeseries is cut in consecutive segments of length τ, each labeled by the pseudo-time (PT) n and, inside each slide σ marks a pseudo-space (PS) position in a 1D (one-dimensional) space.Due to causality, the information transfer processes are strongly asymmetric in this representation <cit.>, yielding a non-zero drift term with velocity (in P-ST units), α≪τ. For visualization purposes, we adopt the comoving transformation t=n' (τ+α) +σ' with σ' ∈[0, τ+α) <cit.>. In this representation, the VCSEL intensity is characterized by nucleation, propagation and annihilation of fronts that, in the absence of multiplicative noise leads to coarsening <cit.>. Notably, the P-ST description allows to unfold and display the features of the dynamics over a range of peculiar and independent timescales: the width of the fronts separating the upper and lower polarization states (bandwidth-limited at few μ s), the PS correlation length, the delay τ, and the PT correlation length. All of them play a role in our setup. The ratio between τ and the fronts width corresponds to the aspect-ratio, as defined in spatially extended systems <cit.>. The PS and PT correlation lengths determine the features of the patterns of the active state and are known to scale with the distance from the critical point.Typical P-ST patterns of the intensity as the pump current is increased and for a fixed multiplicative noise amplitude are shown in Fig.<ref>a-c. The system is initialized with a sequence of length τ close to the upper state (with a random Gaussian statistics). We observe a complicated P-ST dynamics with a relaxation towards either the lower (Fig.<ref>a) or upper (Fig.<ref>c) state. For intermediate values of the pump, the system apparently evolves slowly on longer PT-scales. Such behavior is immediately reminiscent of the active-to-absorbed phase transition, as observed in a large class of reaction diffusion systemswith an absorbing state (i.e. a state able to trap the dynamics indefinitely).One can readily identify the lower state, preserved by our choice of the multiplicative noise, with such an absorbing state: with other experimental noises carefully minimized, the upper states cannot nucleate spontaneously inside PS homogeneous patches of lower states, which can only be displaced by the invasion of moving fronts. On the contrary, fluctuations can easily nucleate lower state patches inside the upper ones. Moreover, once the laser sets down in the lower state for at least one full delay, its emission has no more chances to jump back to the upper polarized state, being effectively absorbed.According to the Janssen-Grassberger conjecture <cit.>, in the absence of additional symmetries and/or quenched randomness, any spatio-temporal dynamics displaying such a transition from a fluctuating active (i.e. not dynamically frozen) phase into a unique absorbed state is expected to belong to the DP universality class. One should thus expect a critical, power-law behavior described by three independent critical exponents, numerically known with high accuracy in 1D <cit.>. We use as a control parameter the coarsening velocity v in the absence of the multiplicative noise estimated at the beginning and the end of each measurement. While this velocity is strictly related to the pump current, this procedure allows for a more precise determination of the actual working point of the laser (see Supplemental Material). In particular, we retain only those measurements whose final-initial relative difference in the speed is smaller than 3%. As an order parameter, we introduce the delay-averaged intensity ρ(n) = ⟨ I(σ,n) ⟩_σ, normalized between 0 and 1 and its PT-asymptotic value ρ_∞. In Fig.<ref>d the PT evolution of ρ(n) is reported for different values of the corresponding coarsening velocity. A clear transition is present from asymptotically non-zero values (the active phase) to an exponential decrease towards zero (absorbing phase). The bold green and blue curves correspond to the patterns shown in Fig.<ref>a and (c) respectively.A near-critical curve, displaying a power law decay over more than one decade, is also plotted alongside the known DP asymptotic scaling ρ(n) ∼ n^-δ (with δ= 0.159464(6) <cit.>), showing a satisfactory agreement between our experiment and DP scaling theory.We further define the normalized control parameter Δ= (v -v_c)/v_c where v_c is the empirical critical coarsening velocity, and plot the PT-asymptotic value of the order parameter ρ_∞ as a function of Δ. Fig.<ref>e clearly shows the signature of a transition between the absorbed (Δ<0) and active phases (Δ>0). One would now ideally proceed to measure the scaling of ρ_∞ as the critical point is approached from the active phase, ρ_∞∼Δ^β with β= 0.276486(6) <cit.>. Unfortunately, when initialized in the upper state the system is prone to non-negligible fluctuations in the working point, which prevented us from reaching the large PTs needed for a clear testing of this latter scaling law.However, we report in Fig.<ref> the results of another set of measurements: the so called single seed behavior. For every value of the control parameter, the system is prepared in the initial delay cell close to the lower state, except for a set of small intervals – equally spaced along the delay length – which are set in the higher state. This procedure creates an ensemble of 10^2 active seeds states which evolve independently as long as their ensuing activity remains separated in PS. The relevant quantities are evaluated as averages over such ensembles. Working point fluctuations are milder for (mainly) lower-state initial conditions and we achieve a better control over the critical dynamics.In DP scaling theory, single seeds initial conditions may decay into the absorbing state or survive and spread with a time-dependent probabilityP(n). In the absorbing phase one expects P(n) ∼ n^-δexp (-n/ξ_∥), where the temporal correlation length divergesas the critical point is approached: ξ_∥∼Δ^-ν_∥ with ν_∥=β/δ=1.733847(6) being a second independent exponent<cit.>. Thus at the critical point the survival probability decays as a power law P(n) ∼ n^-δ.A third independent exponent, the so-called initial slip exponent θ, can be finally deduced from the initial growth of activityat the critical point when starting from a single seed (or sparsely active) initial condition, ρ(n) ∼ n^θ for ρ(n) ≪ 1, with θ= 0.313686(8) <cit.>.In Fig.<ref>a-c, we present three single-seed sample patterns showing the onset of a near-critical behavior in Fig.<ref>b. We further report in Fig.<ref>d-e the growth of the delay-averaged signal (d) and the survival probability (SP) of a seed (e) as a function of PT for different values of Δ. The bold green and blue again denote the sub- and super-critical behaviors corresponding to the patterns in Fig.<ref>a-c;the bold black curve corresponds to the critical case of Fig.<ref>b. The dashed lines superimposed in the density and SP plots are the power lawsexpected for DP at criticality, showing an excellent agreement with the spatio-temporal theory. In the inset of Fig.<ref>e, we plot the PT correlation length ξ_∥ estimated in the sub-critical case. In spite of the large errors due to residual working point fluctuations, the results are compatible with the DP power-law scaling as depicted by the dashed line.Model.In order to corroborate our experimental findings, we introduce a stochastic effective model, derived from the deterministic description of Ref. <cit.>,dx_t = [ g x_t-τ +F_a(x_t) ] dt + bx_tdW_twhere the real variable x_t represents the intensity, dW_t is the increment of a Wiener process, F_a(x) =-d/dx U_a(x)≡ - x(x-1)(x- a) a force term derived from a bi-stable quartic potential and τ the delay time. The dynamics (<ref>) is controlled by three real parameters, the delayed feedback coupling g, the multiplicative noise amplitude b and the potential asymmetry awith a>g>0.The deterministic dynamics (b=0) has two stable fixed points, x_t=y_0≡0 and x_t = y_1≡ [(1+a) +Γ]/2, with Γ≡√((1-a)^2 + 4 g), separated by the unstable fixed point x_t = y_u≡ [(1+a) -Γ]/2. In the absence of delay (τ→ 0), the deterministic dynamics only consists in a relaxation to equilibria on intrinsic timescales of order t_0 = (a-g)^-1 and t_1 =Γ^-1 (1+a+Γ)^-1. In the case of a long delay τ≫ t_0,t_1, quasi-heteroclinic fronts joining the two stable fixed points can be observed as a transient phenomena. Indeed, the relative stability of the fixed points controlled by system parameters, in particular by the potential asymmetry a, determines the coarsening dynamics <cit.>. In the following, we investigate numerically the full stochastic dynamics for long delays, interpreting equation (<ref>) in the Itô sense and adopting a simple Euler-Maruyama integration method with a timestepping Δ t=0.01 <cit.>. We fix the noise amplitude b=1/√(7) and the delayed feedback coupling g=0.22, using the potential asymmetry a as our main control parameter. We have however verified that analogous results hold using, for instance, g as control parameter. We focus on the range a ∈ [0.5,1], preparing our system with the initial conditions x_t = y_1 for t ∈ [0,τ ). Our numerical results for the delay-averaged intensity are reported in the P-ST plots of Fig. <ref>a-c. For values of the asymmetric parameter a close to 1, the P-ST dynamics quickly drops from the active state to the absorbing one (we consider a state as absorbed when x_t<y_u for one full delay). As a is lowered, the system goes through a phase transition located at a_c ≈ 0.8946(1) to reach an active phase. The critical exponentsδ and β can be estimated within a 6% accuracy as shown in Fig. <ref>d. A third independent exponent can be evaluated by measuring the finite-size scaling of the typical time T_ abs needed for a finite size system to be absorbed at the critical point. Scaling theory predicts T_ abs∼τ^z with the dynamical exponent z=1.580745(1), in very good agreement with our numerical simulations (seeFig. <ref>e).Single seed simulations, reported in Fig. <ref>, further confirm the identification of our stochastic dynamics (<ref>) with the DP universality class. The slight deviation between the size asymptotic critical point a_c^ss≈ 0.89447(2) and the finite size one reported above for fully active initial conditions is indeed compatible with the expected PS finite size scaling (see Supplemental Material) <cit.>.Discussion. To summarize, we have experimentally shown and numerically confirmed the existence of a DP critical phase transition in the long-delayed dynamics of a bistable system with multiplicative noise. Our system is purely temporal, and the effective spatial variables involved emerge from the multiple time-scales of the dynamics. While the onset of non-equilibrium critical phenomena in systems with long-delay has been previously put forward in simple model systems <cit.>, this work represents the first experimental evidence of such behavior. We show that the mapping between long-delayed dynamics and spatio-temporal systems is preserved for stochastic dynamics even as a critical point is approached, so that the former and latter systems may share the same universality class.Our result opens a new avenue for studying experimentally a number of out-of-equilibrium universality classes– for instance the Kardar-Parisi-Zhang class <cit.> – in purely temporal, long-delayed setups. Moreover, the occurrence of critical phenomena and their scalings in higher effective spatial dimensions could be investigated by means of different types of delayed feedbacks with multiple, hierarchically long delays <cit.>. Furthermore, absorbing phase transitions such as DP may also take place in non-equilibrium many-body quantum systems <cit.>, for instance, in interacting gases of Rydberg atoms <cit.>. Moreover, an interesting connection between superradiance in cold atom systems and standard lasing has been recently argued in <cit.>.In this context, we expect our system – where in appropriate conditions quantum fluctuations due to spontaneous emission could have an impact on the transition– to attract the interest of a larger community interested in phase transition in quantum simulators. We wish to thankS. Lepri and A. Politi for useful discussions. MF and FG acknowledge support from EU Marie Curie ITN grant n. 64256 (COSMOS). 24natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURLKadanoff90 L.P. Kadanoff, Physica A 163, 1 (1990).Widom65 B. Widom, J. Chem. Phys. 43, 3892 (1965); 43, 3898 (1965).Kadanoff66 L.P. Kadanoff, Physics 2, 263 (1966).HH69 B.L. Halperin, P.C. 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Yanchuk, G. Giacomelli, Phys. Rev. Lett. 112 174103 (2014).Marcuzzi2015 M. Marcuzzi et al., New J. Phys. 17, 072003 (2015).Kirton P. Kirton and J. Keeling,New J. Phys. 20, 015009 (2018). | http://arxiv.org/abs/1707.08327v2 | {
"authors": [
"Marco Faggian",
"Francesco Ginelli",
"Francesco Marino",
"Giovanni Giacomelli"
],
"categories": [
"cond-mat.stat-mech",
"physics.optics"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170726090925",
"title": "Evidence of a critical phase transition in a purely temporal dynamics with long-delayed feedback"
} |
firstpage–lastpage Sequential Inverse Approximation of a Regularized Sample Covariance Matrix Tomer LancewickiEBay Inc.625 6th AveNew York, NYEmail: [email protected] December 30, 2023 ======================================================================================= Energy-conserving, angular momentum-changing collisions between protons and highly excited Rydberghydrogen atoms are important for precise understanding of atomic recombination at the photon decoupling era, and the elemental abundanceafter primordial nucleosynthesis. Early approaches to ℓ-changing collisions used perturbation theory for only dipole-allowed (Δℓ=± 1) transitions. An exact non-perturbative quantum mechanical treatment is possible, but it comes at computational cost for highly excited Rydberg states. In this note we show how to obtaina semi-classical limit that is accurate and simple, and develop further physical insights afforded by the non-perturbative quantum mechanical treatment.cosmology: observations–primordial nucleosynthesis – ISM: abundances– atomic data § INTRODUCTIONThe dynamics of atomic recombination and its impact on the cosmic background radiation are crucial to constrain variants of Big Bang models <cit.>. The recombination cascade of highly excited Rydberg H atoms is influencedby energy-changing <cit.> and angular momentum-changing collisional processes (<cit.> - PS64 thereafter; <cit.> - VOS12 thereafter), and is a major source of systematic error for an accurate determination of the recombination history. Moreover, primordial nucleosynthesis is studied by determining the He/H abundance ratio. This is obtained by determining the ratio ofemission lines of He I and H I, and using the most accurate models for the recombination rate coefficients <cit.>.Besides cosmology, recombination rate coefficients for hydrogen and helium are alsoimportant in studying radio emission from nebulae <cit.>,and in the study of cold and ultracold laboratory plasmas <cit.>.In particular, there is a pending puzzle in the determination of elemental abundance andelectron temperature in planetary nebulae, as optical recombination lines and collisionally inducedlines provides significantly different values <cit.>.Dipole ℓ-changing collisions n ℓ→ n ℓ± 1 betweenenergy-degenerate states within an n-shell are dominant in the dynamics of proton-Rydberg hydrogenatom collisions, and have been addressed long ago by Pengelly and Seaton in the framework of the Betheapproximation in a perturbative framework (PS64). More recently, we examined (VOS12) the problemobtaining non-perturbative results for arbitrary n ℓ→ n ℓ' energy-conserving transitions, including the dipole allowed transitions, which produce rate coefficients smaller compared with PS64.This results in the estimation of higher densities for available spectroscopic data, which is of relevance at least in cosmology as different H I emissivities are derived using the two models, withdifferences of up to 10% <cit.>. This in turn impacts the precision required on the primordial He/H abundanceratio to constrain cosmological models. The exact quantum expression obtained in VOS12 was complemented by a simplified classical limit transition rate that was in good quantitative agreement with the quantum rate and also with the results of Monte Carlo classical trajectory simulations for arbitrary Δℓ. For dipole allowed transitions, Δℓ = ±1,Monte Carlo computations in VOS12 predicted a finite cross section instead of a logarithmically divergent one, due to a discontinuity in the classical transition probability at large impact parameters.In a recent publication <cit.> recommended that the PS64 rates should be preferred over the classical results in VOS12 due to how PS64 employed an ad hoc density-dependent cutoff procedure to treat the dipole-allowed angular momentum changing collisions. In a series of papers <cit.> investigated the influence of differently calculated ℓ-changing rate coefficients in CLOUDY simulations of emissivity ratios, concluding that the quantum VOS12 treatment is more appropriate when modeling recombination through Rydberg cascades.In this note, we provide further validations and insights on our model and show how a slightly different classical limit is constructed to provide non-perturbative expressions that are uniformly consistent with the quantum behaviorfor all impact parameters. In this way, the deficiency of the classical transition rates discussed by <cit.> is effectively eliminated.§ PROTON-HYDROGEN ATOM COLLISIONS AT LARGE IMPACT PARAMETERConsider an ion projectile with electric charge, in elementary units, of Z moving at speed v smaller or comparable with that ofthe target Rydberg electron v_n = e^2/nħ in a state with principal quantum numbern and angular quantum number ℓ. Results for collisions with proton are obtained by setting Z=1. Even when the impact parameter b is larger than the size of the Rydberg atom, a_n = n^2 a_0, with n theprincipal quantum number and a_0 = 0.53 × 10^-10m the Bohr radius, the weak electric field created by theprojectile lifts the degeneracy of the Rydberg energy shell and mixes angular momentum states within the shell.At the end of the slow and distant collision with the ion, the Rydberg atom is in a different angularmomentum state with the same initial energy. Therefore collisions that change angular momentum, without anyenergy transfer, have extremely large cross sections and rate coefficients.The rate coefficient q of this process scales as q_nℓ→ℓ'∼ n^4/√(T)Δℓ^3 (VOS12) with temperature T, and change in angular momentum Δℓ = ℓ' - ℓ.Since the angular momentum changing collisions are most probable at large impact parameters it is safe toassume that the dipole term in the interaction energy dominates over the other multipolar contributions, which can be therefore neglected. Moreover, as the projectile has a much greater angular momentum than that of the target atom, it can be assumed that the projectile's angular momentum is conserved and theprojectile moves along a straight line trajectory defined by the projectile position vector R(t).According to these assumptions, the Hamiltonian of the Rydberg electron contains a time-dependent interaction potential term given byV(t) ≈ - Z e^2r· R(t)/| R(t)|^3,where r is the electron position.At extremely large impact parameter b>> n^2 a_0 the interaction potential (<ref>) may be treatedas a perturbation and the collision can be treated in the first Born approximation for the transition probabilityP^(B)_nℓ→ nℓ±1= 1/ħ^21/2ℓ+1∑_mm'| ∫_-∞^∞⟨ nℓ'm'|V(t)|nℓ m ⟩ dt|^2= 3 (Ze^2a_0/ħ bv)^2 ℓ_>/2ℓ + 1 n^2(n^2 - ℓ_>^2)where ℓ_>=max(ℓ,ℓ±1). This result has been obtained in the pioneering work PS64 and forms the basis for PS64 rate coefficientfor angular momentum changing collisions. Although simple and easy to use, the expression (<ref>)leads to a number of severe difficulties at smaller b. Various proposals were published attempting to improve the theory beyondthe perturbation theory: close-coupling channel approximation <cit.>, infinite level <cit.>, and rotatingframe approximations <cit.>. This also stimulated experimental investigations, in which theredistribution of Na(28) Rydberg atom ℓ states in collisions with slow Na^+ ions was measured <cit.>.Specifically, the difficulties that stem from using perturbative solutions for the potential (<ref>) are: * The perturbative solution is derived from the matrix elements of (<ref>) with respect to unperturbed states, and therefore only results for ℓ→ℓ± 1 transitions can be obtained with this theory, as prescribed by the dipole selection rule. * The transition probability (<ref>) diverges as b → 0, violating P_nℓ→ nℓ±1 < 1,reflecting unitarity. This difficulty is handled in the PS64 formulation by introducing a cutoff impactparameter R_1 such that the probability for transitions at b ≤ R_1 is exactly 1/2:P^(PS)_ℓ→ℓ± 1(v, b ≤ R_1) = 1/2. The justification for this adjustment was that for b < R_1, P(b) is an oscillatory functionwith a mean value close to 1/2. This assumption is quite reasonable for collisions involvingenergy transfer, when the cutoff R_1 is about the size of the atom.However, the probability for angular momentum changing collisions are dominated by very large impact parameters (b >>n^2 a_0) and probabilities for collision at small impact parameters are much smaller than 1/2. In order to address this difficulty an extension to PS64 method was recently proposed <cit.> in which the constant 1/2 is replaced with 1/4 (model PS-M in that paper). The overall trend of P(b), as explained in the next sections, is to grow linearly with b. This is the reason why the PS64 rates are overestimated.* As b →∞, P^(B)_nℓ→ nℓ±1∼ 1/b^2, leading to a cross sectionσ_nℓ→ nℓ' = 2π∫_0^R_c P_nℓ→ nℓ'bdbwhich diverges logarithmically as log(R_c) when the cut-off parameter R_c→∞. The divergence of the cross section can be understood in the context of the dynamics of degenerate quantum systems, such as the ℓ-levels shell in a Rydberg atom. The transition between degenerate states under the influence of a perturbation that have non-zero couplingmatrix elements is possible no matter how weak this perturbation is. The time scale governing transition probabilities isdefined by the Rabi frequency, which for a degenerate system is given simply by |V_ab|/ħ, where V_abare the transition matrix elements of the perturbation V between degenerate states a and b.Therefore, for weak electric fields, either produced during a very distant collision with an ion, or microfields generated by thesurrounding plasma, the ℓ→ℓ± 1 dipole transitions between Rydberg levels have rates proportional to theintensity of the perturbation.§ EXACT NON-PERTURBATIVE TRANSITION PROBABILITYBy taking advantage of the symmetries in the problem, an exact non-perturbative solution for the Rydberg atom dynamics under the interaction potential (<ref>) can be obtained <cit.> and expressed as successive physical rotations, withdirect interpretations both in quantum <cit.> and classical <cit.> contexts. Like in other physical situations, for example the precession of a magnetic moment in magnetic field, the sourceof similarities between quantum and classical motions is the group of symmetry operations for the given system,which for the hydrogen atom is SO(4).The exact result for the non-perturbative transition probability isP_nℓ→ nℓ'= 2 ℓ' + 1/2 j + 1∑_L = |ℓ' - ℓ|^2j (2 L + 1){=1pt [ ℓ'ℓL;jjj ]}^2H_jL^2(χ)Here {⋯} is Wigner's six-j symbol, and H_jL is the generalized character function for irreducible representationsof rotations defined byH_jL(χ) = ∑_m C^jm_jmL0 e^-2imχ = L! √((2j+1)(2j-L)!/(2j+L+1)!) (2sinχ)^L C^(L+1)_2j-L(cosχ)where C^jm_j_1m_1j_2m_2 are the Clebsch-Gordan coefficients and C^(a)_n(x) are ultraspherical (Gegenbauer) polynomials. The effective rotation angle χ issinχ = 2α/1+α^2[ 1 + α^2 cosπ/2√(1+α^2)]^1/2sin( π/2√(1+α^2))with α a parameter that characterizes the dynamics of the ion projectile incoming at speed vα = 3 Z nħ/2 m_e v bThis parameter can be expressed as the product of the Stark precession frequency and the collision time. Here m_e is electron mass.The probability (<ref>) eliminates all the difficulties associated with the perturbative expression (<ref>) as it is not restricted to dipole transitions, it is well behaved in the b→ 0 limit, has simplerclassical and semi-classical limits, as explained in the next sections, beyond the perturbative approximation.The large b→∞ (or small α→ 0) limit for the ℓ→ℓ± 1 transition probability(<ref>) can be obtained from the first L=1 term in the summation and by observing thatlim_α→ 0 H_j1(χ) = 2j+1/3√(j(j+1)) 4αand that the six-j symbol has a particularly simple form in this case{=3pt [ ℓ± 1ℓ1;jjj ]}^2 = l_>(n^2-ℓ_>^2)/n (n^2-1) (4ℓ_>^2 - 1)The result for the limitlim_α→ 0 P_nℓ→ nℓ± 1 = 4/3ℓ_>/2ℓ+1(n^2-ℓ_>^2)α^2is identical with the perturbative result (<ref>).Equation (<ref>) can be efficiently implemented for the computation of approximation-freetransition rates for angular momentum changing collisions for use in astrophysical models, beyond the PS64 result. However, for n ≳ 100, the direct summation becomes inefficient and it might lead to accumulation of truncation errors due to summation of large alternatingsign numbers. For these cases, and also with the goal of obtaining more physics insight into this process, it is useful to investigatethe limit n→∞ of (<ref>). This can be done in two different ways, as explained in the next sections: onewhich applies for general transitions and impact parameters up to a critical value, and another one that applies only to dipole allowed transitions and very large b.§ CLASSICAL LIMITThe Bohr's correspondence principle asserts that quantum calculations tend to reproduce results obtained by using classical mechanics in the limit of large quantum numbers. In the case of the probability (<ref>) this limit is obtained by transforming the summation into an integral and allowing quantum numbers to have continuous values, lim_n→∞ P_nℓ→ nℓ' = 2ℓ'n ∫_0^1{=1pt [ ℓ'ℓL;jjj ]}^2H_jL^2(χ) d(L/n)^2 The classical limit of Wigner's six-j symbol <cit.> is given by 1/24π√(V_T) in terms of the volume V_T ofa tetrahedron made by the six angular momentum quantum numbers. By using the Cayley-Menger determinant to calculate this volume,one gets for arbitrary transitions thatlim_n→∞ L/n < ∞π n^3 {=1pt [ ℓ'ℓL;jjj ]}^2= lim_n→∞ L/n < ∞ (2/n^6 | =1pt [01111;10j^2j^2j^2;1j^20ℓ^2 ℓ'^2;1j^2ℓ^20L^2;1j^2 ℓ'^2L^20;]|)^-1/2= 1/√(sin(η_1 + η_2)^2 - (L/n)^2)1/√((L/n)^2 - sin(η_1 - η_2)^2)Here the limit is taken such that the ratio L/n remains finite, as well as the ratios for the initial and finalangular momenta defined through cosη_1 = ℓ/n and cosη_2 = ℓ'/n. This classical limit is valid only for values that make the arguments of the square root positive, which limits the integration range in L/n. For example, L/n > sin(η_1 - η_2), which depends on the change Δℓ of angular momentum in transition.The generalized character function H_jL is the solution of a differential equation that can be interpreted as Schrödinger's equation for a particle confined by a 1/sin^2 χ potential well, that has infinite barriers at χ=0 and χ=π and a minimum at χ=π/2. A WKB approximation for this problem is obtained aslim_n→∞ L/n < ∞H_jL(χ) = 1/√(2sinχ) (sin^2χ - (L/n)^2)^-1/4and is in excellent agreement with the exact solution at any χ, except at the classical turning points (|sinχ| = L/n) where the WKB approximation diverges, showing that classically the particle tends to be found with infinite probability at the turningpoints. Beyond the turning points, the classical probability is zero while the exact solution decreases to zero gradually. This contradictory behavior is characteristic to the WKB approximation, and leads in the present case to a discontinuity in the transition probability as a function of b, as shown in Figures 1 and 2. The nature of this discontinuity is discussed below.Figure 1 demonstrates graphically that probability (<ref>) converges to (<ref>) in the n→∞ limit, showing a linear increase up to a maximum impact parameter, followed by a sharp drop. By combining equations (<ref>) and (<ref>), we see that classical probability is nonzero only whensinχ < L/n < |sin(η_1 -η_2|. Otherwise, integration (<ref>) has analytic results in termsof elliptical integrals (see VOS12 for details). It is interesting to note that the same resultwas obtained directly from the classical solution of the motion under potential (<ref>) and by defining the transition probabilities as ratios of phase space volumes <cit.>.The resulting classical limit agrees very well with the non-perturbative result (<ref>) as seenin the inset in Fig 2, forall b, except at very large b, where the probability drops to zero abruptly, instead of showing the 1/b^2 decay of (<ref>).For 1<<b< b_max, which means small α and χ, only small angular momentum changes are possible andone can approximate sinχ≈ 2α, sin(η_1 - η_2) ≈Δℓ/√(n^2-ℓ^2) and sin(η_1+η_2) ≈ 2ℓ√(n^2-ℓ^2)/n^2, to provide a much simplified transition probabilityP^(C)_nℓ→ nℓ' = {[ b/2b_max b ≤ b_max/Δℓ;0 b > b_max/Δℓ ].where the classical cutoff radius b_max = 3n a_0 √(n^2 - ℓ^2) Ze^2/ħ v is obtained from the cusp relation sinχ = |sin(η_1 - η_2)|. This linear increase for b<b_max is in contrast with the ad-hoc PS64 assumption that the probability is 1/2 for b<R_1, and it explains why the PS64rate coefficient is larger than the quantum VOS12 rate coefficient.The abrupt discontinuity in b at b_max displayed by equation (<ref>) is problematic, reflecting the deficiencyof the WKB approximation to describe quantum tunneling. The most significant difficulty for (<ref>) is for dipoleallowed |Δℓ|=1 transitions that have logarithmically divergent cross sections. Instead, by using probability (<ref>) in integrating (<ref>) the result is a finite cross section, denoted as σ_C for future reference. For all other |Δℓ| > 1 transitions, the sharp discontinuity has a minor effect since both the classical andquantum transitions have finite cross sections and rate coefficients, and the approximation (<ref>) works surprisingly well. The next section shows how to address the deficiency of classical probability (<ref>) for Δℓ=1 atb = b_max by taking the classical limit differently. This procedure is akin to the textbook prescription oftreating the WKB singularity at the turning points, by developing a local approximation around those points andthen "stitching" together approximations over various intervals.§ SEMICLASSICAL LIMITInstead of the classical approximation (<ref>) valid over a wide range of χ values, we use a local approximation <cit.>lim_n→∞α→ 0, α n < ∞1/n H_jL(χ) = j_L(2 α n)valid only for small α, as long as the product α n is finite. Here j_L(x) is the spherical Bessel function. By using this approximation in the integration (<ref>), and working only for dipole transitions ℓ'=ℓ±1, we obtain a semiclassical transition probability as the integralP^(SC) = 2ℓ/π∫_1^n j_L^2(nα)dL/√(4ℓ^2[1 - (ℓ/n)^2] - L^2)which is dominated by values around the L=1 end of the integration range. Since j_1(x) ≈ x/3 +O(x^3), this semiclassical transition probability has the correct asymptotic ∼ 1/b^2 at b→∞ limit. The integral can be approximated to getP^(SC)≈3/2 j_1^2( 2 α√(n^2-ℓ^2))Figure 2 shows the PS64 perturbation theory (<ref>), classical approximation (<ref>) and semiclassical approximation (<ref>) for a dipole allowed transition as compared with the quantum probability (<ref>). The classical limit agrees well with the exact result for low and moderate impact parameters (as shown in inset), displaying the abrupt classical discontinuity at b_max. On the other hand, the semiclassical approximation does well at very large b,but fails at small b < b_S, as shown in the figure by a dashed line.In order to take advantage of the good agreement of the classical and semiclassical transition probabilities in their respectiveranges and obtain an accurate approximation for the cross section, we combine them in an effective transition probability defined as:P^(E)_nℓ→ nℓ' = {[ b/2b_maxb ≤ b_S; 3/2 j_1^2( b_max/b )b > b_S ].with the matching b_S = γ b_max defined as the smallest impact parameter for which the classical and semiclassicalapproximations are equal, ensuring the continuity of the probability, and γ = 0.3235133 is the solution to thetranscendental equation j_1^2(1/x) = x/3.The cross section is calculated by using Eq. (<ref>) to get the semiclassical cross sectionσ^(SC)_nℓ→ nℓ' = π b_max^2/3{[(R_c/b_max^2)^3, R_c ≤ b_S; γ^3 + [ T(R_c/b_max) - T(γ)], R_c > b_S ].where the function T is T(x) = -Ci(2/x) + 3*x^4(3+2x^2)/8 -x^2(2-3x^2 + 6x^4)cos(2/x)/8 +x(2-x^2-6x^4) sin(2/x)/4and Ci(z) = - ∫_z^∞cos(t)/tdt is the cosine integral function.Figure 3 shows calculations of the cumulativetransition cross section as a function of the cutoff parameter R_c used to regularize the logarithmic singularity. The PS64 result overestimates the non-perturbative quantum cross section derived from Eq. (<ref>) by amounts that depend on the cutoff parameter R_c. As explained in section 2, the PS64 rates are overestimated because the probability of transition is assumed to be 1/2 for 0 < b < R_1, while the non-perturbative calculation demonstrates that the probability increases linearly with b. Asymptotically, both PS64 and the semiclassical cross sections (<ref>) diverge logarithmically as∼ + π b_max^2 ln(R_c)/3 with R_c →∞, but with the PS64 constant approximately twice as large as the semiclassical one. Therefore, even for high temperature and density considered by <cit.> the PS64 rate overestimates the ℓ-changing rate by a constant amount. This difference is independent of R_c, and therefore the ratio of the two rates approaches unity in the R_c→∞. The PS-M model also has the linear increase with b and the same asymptotic behavior, but as noted in their paper, the agreement with the quantum VOS12 model is reasonable good in general, similar with the results derived from Eq. (<ref>), but deficient in some extreme cases, such as low ℓ values.Recent papers <cit.> argued that quantum formula (<ref>) is computationally expensive, while the classical limit (<ref>) has an abrupt drop, instead of the 1/b^2 decay as b→∞, and therefore the PS64 perturbative rates should be still preferable. Figure 3 addresses this concern by showing that semiclassical cross sections, and by extension the transition rate coefficients, are consistent with quantum non-perturbative results, but easier to use in practical calculations due to the simplicity of the effective probability (<ref>). § CONCLUSIONSWe have contrasted two different models for the evaluation of proton-Rydberg atom angular changingcollision, with particular emphasis on the anatomy of their assumptions and approximations, and thecomparison to the full quantum-mechanical setting at small principal quantum numbers. We argue that parameters of astrophysical interest derived from diverging cross-sectionscontain a degree of arbitrariness in principle reflected in large and unknown systematic errors.In the absence of full quantum calculations or of precision laboratory measurements, it is more meaningful to use models with clearer physical interpretation, less assumptions, and controllable approximations.We believe that this pluralistic approach is even more imperative in astrophysics, since the modelsinvolved in the extraction of astrophysical parameters from observations are typically the major sourceof systematic error, as already extensively advocated in <cit.>.It was advocated in <cit.> that VOS12 quantum rates to be used when high accuracies are required and faster PS64 when that accuracy is not needed to speed the calculations. The results introduced here, derived from improved semiclassical limit (<ref>), are accurate over the whole range of impact parameters and computationally inexpensive, eliminating the dilemma of having to choose speed over accuracy.§ ACKNOWLEDGMENTSThis work was supported by the National Science Foundation through a grant to ITAMP at the Harvard-Smithsonian Center for Astrophysics. One of the authors (DV) is also grateful for the support received from the National Science Foundation through grants for the Center for Research on Complex Networks (HRD- 1137732), and Research Infrastructure for Science and Engineering (RISE) (HRD-1345173). We thank G. Ferland, and his collaborators, for fruitful and stimulating dialog on this topic. 99 [Bellomo1998]Bellomo1998 Bellomo P., Stroud C. R., Farrelly D., T. Uzer, 1998, Phys. Rev. A, 58, 3896 [Benjamin, Skillman & Smits1999]Benjamin1999 Benjamin R. A., Skillman E. D., Smits D. 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"authors": [
"D. Vrinceanu",
"R. Onofrio",
"H. R. Sadeghpour"
],
"categories": [
"physics.atom-ph",
"astro-ph.CO"
],
"primary_category": "physics.atom-ph",
"published": "20170727143324",
"title": "On the treatment of $\\ell$-changing proton-hydrogen Rydberg atom collisions"
} |
Replica resummation of the Baker-Campbell-Hausdorff series Anatoli Polkovnikov December 30, 2023 ==========================================================empty empty Evaluating aesthetic value of digital photographs is a challenging task, mainly due to numerous factors that need to be taken into account and subjective manner of this process. In this paper, we propose to approach this problem using deep convolutional neural networks. Using a dataset of over 1.7 million photos collected from Flickr, we train and evaluate a deep learning model whose goal is to classify input images by analysing their aesthetic value. The result of this work is a publicly available Web-based application that can be used in several real-life applications, e.g. to improve the workflow of professional photographers by pre-selecting the best photos. § INTRODUCTION Predicting aesthetic value of photographs is a challenging task for a computer-based system. Humans experience a lot of difficulties when explaining why a given picture is perceived as aesthetically pleasing. This is why it does not seem to be possible to solve this challenge by defining a set of rules (such as if a photo has a lot of blue color it is considered beautiful). Instead, we believe that this problem can be addressed by referring to a crowd-sourced dataset of photographs with corresponding popularity score which we treat as aesthetic metric proxy. Training a machine learning algorithm using this dataset appears to be a more practical approach to estimating the aesthetic value of a photograph and we follow this methodology here.More precisely, in this paper, we assess the aesthetic value of a photograph using only the values of its pixels. Building up on the successful applications of deep convolutional neural networks in other related domains, such as image recognition <cit.>, we propose to use this approach to address this problem. Our method is strongly inspired by previous research on similar problems, which concluded that deep convolutional neural networks perform well in such cases <cit.>.One could imagine a wide variety of applications for a system solving the problem stated in the paper. First of all, such a system can significantly improve the workflow of every photographer. By preselecting or suggesting the best photos from a defined set we save a lot of time, storage space and network traffic. Usefulness of the system could be furthermore improved by combining it with some means of detecting similar photos. That results in easy removal of duplicates, which saves photographer the time that he would spend on selecting the best frame out of a set of similar ones. To provide another example, one could even imagine camera or post processing software using such system to suggest and perform automatic image enhancements, like exposure compensation or even cropping and framing a photograph.Our paper provides the following contributions: * a machine learning system capable of automatic assessment of aesthetic value of digital photographs basing only on the image content,* publicly available web interface that allows anyone to test the system on their own photos,* dataset, published online, consisting of images and labels that we use to train and evaluate our system. The remainder of this paper is organized in the following way: firstly, we show a brief overview of the related work. Then, after more formal problem definition, we describe our method. The next part contains thorough description of experiments conducted in order to evaluate and improve the performance of the system. Later, we briefly describe the implementation of our web application. Lastly, we conclude the paper, mentioning also future research possibilities.§ RELATED WORK Training machine learning systems on photos seems to be sensible in the wake of ever-growing popularity of online photo sharing websites, like Flickr or Instagram. The ubiquity of image data available nowadays is not only making machine learning systems easier to train, but also increasing their significance. The growing volume of image data is constantly making it harder to maintain. Machine learning systems doing for example image classification or face recognition are making this data more useful to us.While so far there is little research on the topic of this paper, numerous papers tackling similar problems are published. One example is provided by an attempt to address the problem of popularity prediction for online images available on Flickr published in 2014 by Khosla et al. <cit.>. The authors use a dataset of over 2 milion images from the mentioned service to extract various visual features from them and train a set of Support Vector Machines on the resulting data. In their research, they prove that pre-trained deep learning convolutional neural networks are the best extractors of data for SVMs from the images. This fact encourages us to evaluate convolutional neural networks in our paper. In the discussed article, the authors use the 'ImageNet network' <cit.> trained on images from the ImageNet <cit.> challenge. Concretely, they extract features from the fully connected layer before the classification layer, which outputs a vector of size 4096.Another paper tackling the problem of popularity prediction for online images was published in 2015 by Gelli et al. <cit.>. In their approach, they explore additional cues that could help to predict the popularity of a photo. Similarly to Khosla et al., the proposed model is trained on object, context and user features. Object features are extracted by passing every image through a deep convolutional neural network (with 16 layers), resulting in 4,096-dimensional representation from the 7th fully connected layer. Examples of context features in this case are tags, descriptions and location data of images. User features contain data like the mean views of the images of the photo author. However, compared to Khosla et al., the authors propose to use three new context features and – most importantly – visual sentiment features. To discover which visual emotions are associated with a particular image, a visual sentiment concept classification is performed based on the Visual Sentiment Ontology. This additional input allows their model to perform better than that proposed by Khosla et al., especially when classifying images coming from the same user. A paper published by Karayev et al. in 2013 provides another interesting and related approach <cit.>. The authors try to train a classifier which could recognize a visual style that characterizes a given painting or photograph. They use different datasets, including photos from Flickr, like in the previous example. This problem seems quite related to the task that we tackle in this particular paper. Again, deep learning convolutional neural network used in a similar way like in the paragraph above, proved to be the best method of all of the tested ones. This tells us that deep convolutional architectures are the current state of the art in image-related machine learning problems.Research which is the greatly related to our problem was published by Deng et al. in 2016 <cit.>. In the paper, the authors summarize different state-of-the-art techniques used today in the assessment of image aesthetic quality. In addition to that, they also publish results of the evaluation of those methods on various datasets. The best results are obtained with deep learning models, which is consistent with research discussed above.. That further justifies the usage of such models in our research.While the next example is not a published research, it is worth mentioning because of high correlation to the topic of this paper. An article <cit.>, published on NVIDIA Developer Blog in 2016, is discussing exactly the same topic as in our paper. The article describes that the solution is also based on a convolutional neural network. However, the research is not reproducible because authors do not publish the dataset nor the implementation details. Moreover, they use a private dataset of photographs manually curated by award-winning photographers, which is expected to provide training data of very high quality, hence greatly improving the performance of the trained model.§ METHOD In the following sections, we describe our proposal in detail. First section contains formal problem definition, whereas in the next one we describe the dataset. §.§ Problem definitionIn order to simplify the problem, we define it as a binary classification task. That means that our dataset contains photos classified either as aesthetically pleasing or not. We define our objective as a task of photo classification as aesthetically pleasing or not, using only visual information.Formally, we define a set of N samples, {x_i, l_i} where x_i is a photo and l_i ∈{0,1} is a corresponding aesthetic label. The label is equal to 1, if the system perceives the photo as aesthetically pleasing, or equal to 0 in the opposite case. Therefore, we aim to train to classifier C that given a sample (photograph) x_i ∈ X predicts a label l̂_̂î = f_C(x_i, θ), where θ is a set of parameters of a classifier C. Going further, our concrete goal is to minimize the loss function defined below:min_θ ∑_i=1^N[ l_i log f_C(x_i, θ) + (1-l_i)log (1-f_C(x_i, θ)) ]The function defined above is widely known as a multinominal logistic loss function, which is an equivalent to the cross entropy loss function. §.§ Dataset Our dataset consists of 1.7 milion photos downloaded from Flickr – a popular image and video hosting service. In order to classify the photos, we use various metadata associated with them. Basing on work of Khosla et al., we use number of views for a given photo as a main measure of image quality. Furthermore, as was shown <cit.> in 2010, we know that visual media on the Internet tend to receive more views over time. To suppress this effect, we normalize each metric using the upload date, i.e. each value is divided by number of days since the photo was uploaded.Concretely, our aesthetics score is defined by the following equation:score = log_2(n_views + 1 / n_days + 1)where: score is photo aesthetics score,n_views is number of views for a given photo,n_days is number of days since the given photo was uploaded. This results in distribution plotted on Figure <ref>. The score defined above is computer for all of the photos in the dataset – the computed value is used to sort the examples from the dataset. Photos with the biggest value of the score are classified as aesthetically pleasing, whereas photos with the lowest value are classified as not pleasing. Moreover, we focus on top and bottom 20% of the dataset – our goal is to distinguish very good photos from those of low quality. That amounts to exactly 513 382 photos in the training dataset and 85 562 photos in the test dataset. In order to determine if our score is in fact correlated to the quality of the photo, we perform a sanity check. To do so, we sort the photos by their aesthetics score calculated using the formula defined above. Next, we manually assess the aesthetic value of the 100 best and 100 worst photos, i.e. with the highest and lowest score. We do so in order to ensure that our dataset contains valuable examples. Manual examination of the photos on Figure <ref> proves that the photos are classified properly, i.e. the positive dataset is containing mostly aesthetically pleasing photos, whereas the negative contains mostly not aesthetically pleasing photos . §.§ ModelThe model we propose is based on deep convolutional neural network called AlexNet <cit.>. The network is made up of 5 convolutional layers, max-pooling layers, dropout layers, and 3 fully connected layers. We use ReLU activations, like in the original AlexNet. The main modification is the size of the last fully connected layer – as we use our network to perform binary classification, we decrease the size of the last layer from 1000 to 2 neurons. We use cross entropy loss function as our optimization objective.In our experiments, we use modified AlexNet in two scenarios. In the first one, we use this neural network as a feature extractor and combine it with a Support Vector Machine (SVM) and a random forests classifier. We treat this approach as a baseline for the second approach, where we fine-tune a pre-trained AlexNet for the purposes of aesthetic value assessment.§ EXPERIMENTS Dataset described in Section <ref> allows us to perform various experiments related to the researched topic. Based upon previous work <cit.>, we focus on convolutional neural networks as they are proven in tackling computer vision problems, like image classification. In all of our experiments we use the Caffe framework in order to perform feature extraction or fine tuning.In this section, we describe the experiments conducted. First part presents the baseline methods that use a convolutional neural network as a feature extractor for various classifiers. The next part describes the results obtained after fine-tuning of our convolutional neural network for the purpose of our problem. The last part contains a brief analysis of the model created to solve the problem stated in the paper. §.§ Baseline As a baseline of our experiments, we use an AlexNet trained on ImageNet classification problem as a feature extractor. We perform forward propagation using images from our dataset and extract activations from the sixth layer. This is similar to the approach used in the previous work <cit.>. We determine the optimal hyperparameters for the SVM classifier by conducting experiments on a smaller set of data. That results in selecting the RBF kernel with C parameter equal to 10 and γ parameter equal to 10^-6. The number of decision trees is set to 10 considering the RF classifier. We then train SVM and RF classifiers on the full dataset. Both of the models are implemented using the scikit-learn library available for Python. The results are presented in Table <ref>. §.§ Fine-tuning AlexNet for aesthetic value assessmentAfter defining the baseline we conduct multiple experiments using fine-tuning approach. Like in the previous experiments, we use the AlexNet convolutional neural network. We define the last layer as containing two neurons. During training, we use stochastic gradient descent solver. As we fine-tune the network, starting from the weights learned on ImageNet dataset, we use smaller learning rate on the fine-tuned weights and bigger learning rate on weights which are randomly initialized (the last layer of the network). After trying various values for the hyperparameters, we are able to pick optimal ones, which results in the training illustrated on Figure <ref>. We train the model for 450 000 iterations, which equals to over 43 epochs with batch size of 50 examples. The trained model achieves accuracy of 70.9%. The comparison to the baseline is presented in Table <ref>.Comparing our method to the baselines, we notice an improvement of 3.4 percentage points over SVM classifier and 10.1 percentage points over RF classifier. §.§ AnalysisIn order to fully understand at which features of the photos the model is looking while performing the inference, we sort the test set by the output values of the second neuron in the last layer. This value equals to our photo aesthetics score, i.e. probability that the input photo belongs to aesthetically pleasing class, according to our model. After that, we can select the best 100 (with the highest aesthetics score) and worst 100 photos (with the lowest aesthetics score), according to the model. That allows us to analyze which properties seem important to the model. Looking at the sorted photos, we can easily distinguish different properties of the photos that result in classifying the photo as aesthetically pleasing or not. The model is classifying the photo as a good one, when it has the following properties:* saturated colors,* high sharpness – at least in some part of the photo,* main subject standing out from the background,* high contrast. On the other hand, these properties of the photo are likely resulting in classifying it as not aesthetically pleasing: * flat backgrounds,* small main object,* low contrast,* wrong white balance,* poor lightning or exposure. § IMPLEMENTATION This chapter describes the implementation of the application that was the main aim of this paper. It starts with the schema of the architecture, followed by a brief description of the building blocks of the application.The system is deployed as an web application, in order to ensure that the application is easily available to any user without problematic installation.§.§ Architecture §.§ Frontend and web serverBoth of these layers are hosted on Heroku platform. Heroku is a cloud Platform-as-a-Service (PaaS) supporting several programming languages. Concretely, we use Python in our case to create the web server. It is implemented using Flask microframework. The web server serves a minimalistic webpage to the user, along with the frontend JavaScript code that runs in the browser.The interface allows the user to upload any photo for analysis using our system. The image is uploaded using HTML5 File API. After the upload, the photo is displayed on the webpage and resized to smaller size if necessary. The resize process is done in JavaScript, fully in the browser. That saves some bandwith and speeds up the process, as the photo needs to be sent to the machine learning backend, which is remote. The photo is sent with AJAX call to the Flask web server, which in turn uses ZeroMQ library – exactly its socket implementation – to send the photo to our machine learning backend for processing. The backend responds with a aesthetics score for the photo that was sent. §.§ Backend and neural network modelThe backend is also implemented using Python and deployed on a machine located at the Faculty of Electronics and Information Technology at Warsaw University of Technology. The machine is equipped with a NVIDIA GeForce 780 Ti GPU which is used during the inference process. The same GPU was used to train the neural network that we use. For communication with the Flask web server we use ZeroMQ library, which is sending data through an SSH tunnel.The backend application is using Python bindings for the Caffe library to create AlexNet neural network in GPU memory. The weights for the deployed model are loaded during startup from a file. After receiving an image, the backend performs inference on a GPU using our model and returns a photo aesthetics score via ZeroMQ. § CONCLUSIONSIn this paper, we analyzed what makes photos aesthetically pleasing to the viewers. Specifically, we proposed a method to create a classifier that scores a photograph basing on its aesthetic value using only visual cues. The classifier itself could have numerous applications, which are listed in the introduction of this paper. However, there is also another benefit that comes from the analysis of the results. Namely, the convolutional neural network trained on our dataset provided us with some interesting remarks, which helped us to understand what makes a photograph aesthetically pleasing to the viewer.However, our paper provides also future research opportunities. One could for example try to investigate if the neural network will perform better if we provide higher resolution photographs to its inputs. Other than that, future researchers may try to use our dataset to train more sophisticated deep convolutional neural networks. Recent architectures, like for example a Residual Network (ResNet) developed by Microsoft Research <cit.> or GoogleNet created by Google <cit.> could achieve better performance. Moreover, it is easy to spot that different photography genres have different rules, e.g. landscape photography often benefits from large depth of field, whereas portrait photography is often associated with shallow depth of field. One could use this fact to train different models for different photography genres, which could bring interesting results. § ACKNOWLEDGMENTS The authors would like to thank the Faculty of Electronics and Information Technology at Warsaw University of Technology for providing the necessary hardware. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research. § APPENDIX The web interface of the application is accessible here: <http://photo-critic.herokuapp.com>, whereas the dataset is available under the following links: * training set: <https://goo.gl/yYw18a>,* test set: <https://goo.gl/Q7AVLZ>.IEEEtran | http://arxiv.org/abs/1707.08985v2 | {
"authors": [
"Maciej Suchecki",
"Tomasz Trzcinski"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727181510",
"title": "Understanding Aesthetics in Photography using Deep Convolutional Neural Networks"
} |
Electronic mail: [email protected] ^1School of Basic Sciences, Indian Institute of Technology Mandi, Kamand 175005, Himachal Pradesh, India ^2School of Engineering, Indian Institute of Technology Mandi, Kamand 175005, Himachal Pradesh, India Theoretically, NaAuS is predicted as topological insulator, while no detail electronic structure study has been done for this compound. Here, we report the structural and electronic properties of NaAuS by using LDA, PBEsol, PBE and revPBE exchange correlation functionals. The calculated values of equilibrium lattice constant for LDA, PBEsol, PBE and revPBE exchange correlation functionals are found to be ∼6.128 Å, ∼6.219 Å, ∼6.353 Å and ∼6.442 Å, respectively. The bulk modulus predicted by LDA, PBEsol, PBE and revPBE exchange correlation functionals is ∼66.6, ∼56.4, ∼46.5 and ∼39.3 GPa, respectively. Hence, the order of calculated values of bulk modulus is consistent with the order of calculated values of equilibrium lattice parameters for these exchange correlation functionals. The spread of total density of states below the Fermi level decreases as the exchange correlation functional changes from LDA to PBEsol to PBE to revPBE, which is also found to be consistent with the order of bulk modulus for these exchange correlation functionals. In presence of spin-orbit coupling, a direct band gap is observed in NaAuS compound, which is found to be ∼0.26, ∼0.25, ∼0.24 and ∼0.23 eV for LDA, PBEsol, PBE and revPBE exchange correlation functionals, respectively. Here, NaAuS is found to be topological insulator as it shows band inversion at Γ point. The calculated values of band inversion strength for LDA (PBEsol) and PBE (revPBE) exchange correlation functionals are ∼1.58 eV (∼1.57 eV) and ∼1.50 eV (∼1.47 eV), respectively.71.70.Gm, 71.20.-b, 71.15.MbStudying the effect of different exchange correlation functionals on the structural and electronic properties of a half-Heusler NaAuS compound Sudhir K. Pandey^2 December 30, 2023 ==============================================================================================================================================§ INTRODUCTIONHeusler alloys are taken great importance to study their various physical properties after discovered by Fredrick Heusler in 1903 <cit.>. Generally, the Heusler alloys are divided into two categories, one is called half-Heusler alloy with chemical formula MM'X and another is called full-Heusler alloy with chemical formula M_2M'X. These alloys have special feature that their properties differ completely from those of the contained elements. For example, Cu_2MnAl (full-Heusler alloy) is ferromagnetic, where Cu and Al atoms are non-magnetic and Mn atom is anti-ferromagnetic by themselves <cit.>. Other example is TiNiSn (half-Heusler alloy), which issemiconducting, even though it is made of three components that are metals <cit.>. New trend comes to find the topological insulation property from ternary half-Heusler materials in condensed matter physics <cit.>. Half-Heusler compounds are usually nonmagnetic and semiconducting when the number of total valence electrons is 18,V(M)+V(M')+V(X)=18 which is known as 18-electron rule <cit.>. This rule in general is used for searching the topological insulators (TIs)<cit.>. Hence, half-Heusler material takes great attention to predict TIs. First-principles calculations have been widely used to predict topological insulators with great success <cit.>.The materials which are insulating in their interior but can support the flow of electrons on their surface, are called TIs. TIs are currently creating a new division of research activities in condensed matter physics <cit.>. First TI was theoretically predicted by Bernevig et al. and experimentally observed by Konig et al. in the HgTe quantum well <cit.>. Bernevig et al. have shown that the band inversion property driven by spin-orbit coupling (SOC) is used to discover the TIs. Sawai et al. have observed that the topology of the electronic band structures can be described by band inversion between Γ_6 and Γ_8 energy levels at the Γ symmetry point in the Brillouin zone and they define the band inversion strength Δ as the energy difference between these two states, i.e., Δ=[E_Γ_8-E_Γ_6], where the E_Γ_6 and E_Γ_8 are the energy levels for Γ_6 and Γ_8 at the Γ point <cit.>. As compared with an ordinary superconductor, TIs have an important role for creating topological quantum computation <cit.>. In order to find the new topological insulators, Lin et al. have studied theoretically more than 2000 half-Heusler compounds <cit.>. Out of these, only LiAuS and NaAuS compounds have found to be TIs with band gap ∼0.20 and ∼0.19 eV, respectively. They have studied both compounds using only PBE and hybrid density functionals in face-centered-cubic (FCC) phase. As per our knowledge, the synthesis of NaAuS compound (which we have chosen to study in the present manuscript) in the FCC phase is not available in the literature. However, this compound is synthesized experimentally in orthorhombic phase <cit.>. In general, half-Heusler compounds are described by space group F-43m. Hence, in NaAuS half-Heusler with space group F-43m, Na, Au and S atoms are located at Wyckoff positions (1/2, 1/2, 1/2), (1/4, 1/4, 1/4) and (0,0,0), respectively. The detailed structural and electronic study of NaAuS compound is not found in the literature <cit.>. Hence, it will be interesting to see the effect of different exchange correlation functionals on the structural and electronic behaviour of this compound in more detail.Here in the present work, we study the structural and electronic behaviour of NaAuS compound by using density functional theory. The order of calculated values of equilibrium lattice parameter for LDA>PBEsol>PBE>revPBE exchange correlation functionals, which is consistent with the order of bulk modulus predicted by these exchange correlation functionals. Among these exchange correlation functionals, the total density of states below the Fermi level are spread in more region for LDA and less region for revPBE, which indicates that the bulk modulus predicted by LDA is largest and revPBE is smallest. The order of calculated values of direct band gap is found to be LDA>PBEsol>PBE>revPBE exchange correlation functionals. The band inversion is found at Γ point for NaAuS indicates that this compound is a topological insulator. The order of magnitude of calculated values of band inversion strength is found to LDA>PBEsol>PBE>revPBE exchange correlation functionals.§ COMPUTATIONAL DETAILSThe electronic structure calculation of NaAuS compound is performed by using the full-potential linearized-augmented plane-wave (FP-LAPW) method as implemented in Elk code <cit.>. Here we have employed LDA, PBEsol, PBE and revPBE exchange correlation functionals <cit.>. SOC is also considered to see its effect on the electronic properties of this compound. The muffin-tin sphere radii used for Na, Au and S atoms are 2.0, 2.5 and 2.0 bohr, respectively. The values of rgkmax and gmaxvr are set to be 8 and 14, respectively. These values are sufficient to get nice parabolic energy versus volume curves. 10× 10× 10 k-point mesh size has been used in the present calculations. Total energy to reach convergence has been set below 10^-4 Hartree/cell.The equilibrium lattice parameters are computed by fitting the total energy versus unit cell volume data to the universal equation of state <cit.>. The universal equation of state is defined as,P = [3B_0(1 - χ)/χ^2]e^3/2(B'_0-1)(1-χ),P = -(∂E/∂V) where P, E, V, B_0 and B_0^' are the pressure, energy, volume, bulk modulus and pressure derivative of bulk modulus, respectively and χ = (V/V_0)^1/3. § RESULTS AND DISCUSSION The total energy difference between the energies (function of volume) and energy corresponding to the equilibrium volume [ΔE=E(V)-E(V_ eq)] per formula unit versus primitive unit cell volume of NaAuS for LDA, PBEsol, PBE and revPBE exchange correlation functionals have been plotted in Fig. 1. A parabolic behavior is observed in each curves for every exchange correlation functional. It is also clear from the figure that the equilibrium volume is shifted towards the higher value as the exchange correlation functional changes from LDA →PBEsol →PBE → revPBE. In order to determine the equilibrium volume and bulk modulus for the different exchange correlation functional, we have fitted the data of the total energy-volume by using the universal equation of state. The equilibrium volume corresponding to minimum energy predicted by LDA, PBEsol, PBE and revPBE exchange correlation functionals are ∼388.2, ∼405.7, ∼432.7 and ∼450.9 bohr^3, respectively. The equilibrium lattice constant for different exchange correlation functionals is obtained from the equilibrium volumes of respective exchange correlation functional. The calculated values of equilibrium lattice constant (bulk modulus) of NaAuS for LDA, PBEsol, PBE and revPBE are ∼6.128 Å (∼66.6 GPa), ∼6.219 Å (∼56.4 GPa), ∼6.353 Å (∼46.5 GPa) and ∼6.442 Å (∼39.3 GPa), respectively. These values of equilibrium lattice constant and bulk modulus are shown in Table 1. It is clear from the table that bulk modulus (equilibrium lattice constant) decreases (increases) as exchange correlation functional changes from LDA to PBEsol to PBE to revPBE. The reason of opposite behaviour of bulk modulus as compared to equilibrium lattice constant for these exchange correlation functionals will be discussed after the SOC discussion. The SOC is expected to play an important role in NaAuS compound because of the heavier Au atom. Hence, we have included the SOC in our calculations. The plot of ΔE per formula unit versus primitive unit cell volume is shown in Fig. 2. It is clear from the figure that a parabolic behavior is also seen here for these exchange correlation functionals, which is similar to that observed without SOC. The equilibrium volume and bulk modulus of NaAuS compound for every exchange correlation functional are also computed by fitting total energy-volume data using the universal equation of state. The equilibrium volume corresponding to minimum energy predicted by LDA, PBEsol, PBE and revPBE exchange correlation functionals are ∼384.3, ∼401.2, ∼427.5 and ∼445.6 bohr^3, respectively. Hence, by including SOC, a small decrease in equilibrium volumes is observed for every exchange correlation functional. The calculated values of equilibrium lattice constant (bulk modulus) of NaAuS for LDA, PBEsol, PBE and revPBE are ∼6.107 Å (∼69.0 GPa), ∼6.196 Å (∼58.7 GPa), ∼6.328 Å (∼48.4 GPa) and ∼6.416 Å (∼41.1 GPa), respectively. These values of equilibrium lattice constant and bulk modulus are also shown in Table 1. It is clear from the table that the values of equilibrium lattice constant and bulk modulus by including SOC does not differ too much from without SOC values for every exchange correlation functional. The values of equilibrium lattice constant differ only about 0.3∼0.4% for both calculation for every exchange correlation functional, which indicates that SOC does not show any important role to calculate equilibrium lattice constant for NaAuS. However, by including SOC, the values of bulk modulus is about 3.5∼4.5% greater than without SOC values for every exchange correlation functional. It is also clear from the table that among these exchange correlation functionals, LDA (revPBE) gives the lowest (highest) value of equilibrium lattice constant and highest (lowest) value of the bulk modulus. Generally, LDA overestimates, PBE underestimates and PBEsol slightly overestimates the bulk modulus <cit.>. Now, we discuss the cause of opposite behavior of bulk modulus as compared to equilibrium lattice constant, when exchange correlation functional changes from LDA to PBEsol to PBE to revPBE. As we know that the pressure is defined as the rate of change of energy with respect to volume and the bulk modulus is directly proportional to the rate of change of pressure with respect to volume. We have plotted ΔE per formula unit versus total primitive unit cell volume difference between the primitive unit cell volume and the corresponding equilibrium volume [ΔV=V-V_ eq] for NaAuS using every exchange correlation functional in Fig. 3. It is evident from the figure that LDA (revPBE) gives the most (least) steeper slope, which indicates that LDA (revPBE) gives higher (lower) rate of change of energy with respect to volume. Hence, LDA (revPBE) gives higher (lower) bulk modulus as shown in Table 1. This is due to the direct relationship between energy and bulk modulus.The plot of the total density of states (TDOS) of NaAuS for every exchange correlation functional is shown in the Fig. 4. It is clear from the figure that the every exchange correlation functional gives almost similar behaviour of the TDOS, when SOC is excluded in the calculations. However, a very small TDOS at the Fermi level is obtained for every exchange correlation functional, which indicates a soft band gap. Here, it is important to note that in the valance band (VB), the peaks of TDOS shifted towards the Fermi energy as the exchange correlation functional changes from LDA to PBEsol to PBE to revPBE. Below Fermi level, the TDOS is spread upto -5.9 eV (-5.6 eV) and -5.1 eV (-4.8 eV) for LDA (PBEsol) and PBE (revPBE), respectively. It is interesting to note that the order of bulk modulus and the spread of TDOS below Fermi level is similar for these exchange correlation functional. Hence, in general one can predict the order of bulk modulus by looking the TDOS of these exchange correlation functionals.The plot of partial density of states (PDOS) of Na, Au and S atoms of NaAuS with and without including SOC in the calculations only for PBEsol exchange correlation functional is shown in the Fig. 5(a-f). This is due to the fact that among LDA, PBEsol, PBE and revPBE exchange correlation functionals, PBEsol is newest one and show almost similar PDOS as compared to other exchange correlation functionals. Here, we discuss the PDOS of Na, Au and S atoms of NaAuS without including SOC in the calculations. Below Fermi level, the dominant electronic contribution comes from 3sand 3p states of Na atom, which are ∼45% and ∼55%, respectively. For Au atom, the most dominant electronic contribution comes from 5d state (∼93%), while the small contribution comes from other states. For S atom, the dominant electronic character comes from 3p state (∼95%) as compared to 3s state. Now, we discuss the PDOS of Na, Au and S atoms above the Fermi level. The electronic contribution of 3s and 3p states to PDOS of Na atom are ∼66% and ∼34%, respectively. For Au atom, the contribution of 6s and 5d states are negligible to PDOS. The dominant electronic character for S atom comes from 3p state (∼75%) as compared to 3s state. Almost a similar electronic contribution from various states to PDOS of Na, Au and S atoms of NaAuS is observed by including the SOC in the calculations and is shown in Fig. 5(d-f).The spin unpolarised dispersion curves along the high symmetry direction of first Brillouin zone obtained in calculations for NaAuS compound for above mentioned exchange correlation functionals is shown in Fig. 6. The high symmetric k-points for FCC structure are W, L, Γ and X, respectively. Firstly, we discuss the band structure of NaAuS for LDA exchange correlation functional. Top of the energy band of VB and bottom of the energy band of conduction band (CB) touches at Γ point. The first two energy bands of VB from Fermi level are doubly degenerate from Γ to L and Γ to X directions. Along L to W and X to W directions, the degeneracy of energy bands is completely lifted. Γ to L direction, 4 and 5 energy bands of VB from Fermi level are doubly degenerate, while Γ to X and L to W directions both energy bands are non degenerate. However, both energy bands are crossing to each other with 3 energy band almost middle of X and W points. 6, 7 and 8 energy bands are triply degenerate at Γ point. Along Γ to L and Γ to X directions, only 6 and 7 energy bands remains degenerate. While, from L to W and X to W, the degeneracy of 6 and 7 energy bands is lifted. Now, we discuss the various energy bands from the Fermi level of CB. 2' and 3' energy bands of CB are degenerate at Γ point. However, the degeneracy of these bands is lifted from Γ to L and Γ to X direction. Also, it is clear from the figure that almost similar behaviour of energy bands of both VB and CB is observed for PBEsol, PBE and revPBE exchange correlation functionals. However, energy bands of VB are shifted towards the Fermi level, when the exchange correlation functional changes from LDA→PBEsol→PBE→revPBE. Near the Fermi level, all energy bands of VB of every exchange correlation functional are overlapped with each other. Almost similar behaviour of energy bands of VB and CB is observed for every exchange correlation functional with including SOC in calculations. Here, we discuss the band properties of NaAuS compound for PBEsol exchange correlation functional as it is the newest among all these exchange correlation functionals. It is clear from Fig. 7 that the every energy band of VB and CB are splitted into two states by including SOC. The calculated values of direct band gap at Γ point is predicted by LDA (PBEsol) and PBE (revPBE) exchange correlation functionals are ∼0.26 eV (∼0.25 eV) and ∼0.24 eV (∼0.23 eV), respectively as shown in Table 1.At last, we discuss the band inversion property of NaAuS at Γ point as it is the most common tool to identify the topological insulating behavior of a compound. Here, we focus on Γ_6 and Γ_8 point when SOC is included in calculations, which is shown in Fig. 7. After the detailed analysis of the bands indicate that Γ_6 point is rich with Au 6s state and the contribution of S 3p state is greater than Au 5d state at the Γ_8 point when SOC is not included in calculations. However, when the effect of SOC is included, Γ_6 point does not change its own contributor, where the contribution of Au 5d state is increased at Γ_8 point as compared to S 3p state. From our basic knowledge, we know that outer electronic state will stay at higher energy with respect to inner state. Au 5d is a inner state than Au 6s state. But, after including SOC Au 6s state goes to lower energy level than the Au 5d state. Hence, Au s-d band inversion between Γ_6 and Γ_8 point is observed here. The band inversion strength for LDA, PBEsol, PBE and revPBE exchange correlation functionals are calculated by using relation, Δ=[E_Γ_8-E_Γ_6] <cit.>. The calculated values of band inversion strength (shown in Table 1) for LDA, PBEsol, PBE and revPBE exchange correlation functionals are ∼1.58, ∼1.57, ∼1.50 and ∼1.47 eV, respectively. § CONCLUSIONSThe detail electronic structure study of half-Heusler NaAuS compound has not been done theoretically. Here, we have studied the various comparative physical properties such as, structural and electronic for NaAuS by using LDA, PBEsol, PBE and revPBE exchange correlation functionals. The calculated values of equilibrium lattice constant (bulk modulus) for LDA, PBEsol, PBE and revPBE exchange correlation functionals are found to be ∼6.128 Å (∼66.6 GPa), ∼6.219 Å (∼56.4 GPa), ∼6.353 Å (∼46.5 GPa) and ∼6.442 Å (∼39.3 GPa), respectively. Hence, the order of calculated values of bulk modulus for these exchange correlation functionals are consistent with the order of calculated values of equilibrium lattice parameters because of the inverse relationship between bulk modulus and lattice parameter. Among these functionals, the total density of states below the Fermi level were found to be spread in more region for LDA and less region for revPBE, which was also found to be consistent with the order of bulk modulus for these exchange correlation functionals. In presence of spin-orbit coupling, a direct band gap was observed for NaAuS compound, which was found to be ∼0.26, ∼0.25, ∼0.24 and ∼0.23 eV for LDA, PBEsol, PBE and revPBE exchange correlation functionals, respectively. The band inversion was observed at Γ point, which indicates that NaAuS compound shows topological insulating behaviour. The calculated values of band inversion strength for LDA (PBEsol) and PBE (revPBE) exchange correlation functionals were found to be ∼1.58 eV (∼1.57 eV) and ∼1.50 eV (∼1.47 eV), respectively. § ACKNOWLEDGEMENTS S.L. is thankful to UGC, India, for financial support. 99Starck Heusler F, Starck W and Haupt E 1903 Verh. d. DPG vol 5 220Heusler Heusler F 1903 Verh. d. DPG vol 5 219Geiersbach Geiersbach U, Bergmann A and Westerholt K 2002 J. Magn. Magn. Mater. 240 546 Aliev Aliev F G, Brandt N B, Moshchalkov V V, Kozyrkov V V, Skolozdra R V and Belogorokhov A I 1989 Z. Phys. 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Phys. 82 3045Moore Moore J E 2010 Nature 464 194 Qi Qi X-L and Zhang S-C 2011 Rev. Mod. Phys. 83 1057BernevigBernevig B A, Hughes T L and Zhang S C 2006 Science 314 1757Konig Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L, Qi X-L and Zhang S-C 2007 Science 318 766Al Sawai W A, Lin H, Markiewicz R S, Wray L A, Xia Y, Xu S -Y, Hasan M Z and Bansil A 2010 Phys. Rev. B 82 125208Fu Fu L and Kane C L 2008 Phys. Rev. Lett. 100 096407Axtell Axtell E A, Liao J-H and Kanatzidis M G 1998 Inorg. Chem. 37 5583elk http://elk.sourceforge.netPerdew Perdew J P and Zunger A 1981 Phys. Rev. B 23 5048 Pbesol Perdew J P, Ruzsinszky A, Csonka G I, Vydrov O A, Scuseria G E, Constantin L A, Zhou X and Burke K 2008 Phys. Rev. Lett. 100 136406GGA Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865Yang Zhang Y and Yang W 1998 Phys. Rev. Lett. 80 890Vinet Vinet P, Rose J H, Ferrante J and Smith J R 1989 J. Phys.: Condens. Matter 1 1941Csonka Csonka G I, Perdew J P, Ruzsinszky A, Philipsen P H T, Lebegue S, Paier J, Vydrovan O A and Angyan J G 2009 Phys. Rev. B 79 155107Sharma Sharma S and Pandey S K 2014 Comput. Mater. Sci. 85 340 | http://arxiv.org/abs/1707.08457v1 | {
"authors": [
"Antik Sihi",
"Sohan Lal",
"Sudhir K. Pandey"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726141715",
"title": "Studying the effect of different exchange correlation functionals on the structural and electronic properties of a half-Heusler NaAuS compound"
} |
[ [ December 30, 2023 ===================== We propose a general and model-free approach for Reinforcement Learning (RL) on real robotics with sparse rewards. We build upon the Deep Deterministic Policy Gradient (DDPG) algorithm to use demonstrations. Both demonstrations and actual interactions are used to fill a replay buffer and the sampling ratio between demonstrations and transitions is automatically tuned via a prioritized replay mechanism. Typically, carefully engineered shaping rewards are required to enable the agents to efficiently explore on high dimensional control problems such as robotics. They are also required for model-based acceleration methods relying on local solvers such as iLQG (e.g. Guided Policy Search and Normalized Advantage Function). The demonstrations replace the need for carefully engineered rewards, and reduce the exploration problem encountered by classical RL approaches in these domains. Demonstrations are collected by a robot kinesthetically force-controlled by a human demonstrator. Results on four simulated insertion tasks show that DDPG from demonstrations out-performs DDPG, and does not require engineered rewards. Finally, we demonstrate the method on a real robotics task consisting of inserting a clip (flexible object) into a rigid object. § INTRODUCTION The latest generation of collaborative robots are designed to eliminate cumbersome path programming by allowing humans to kinesthetically guide a robot through a desired motion. This approach dramatically reduces the time and expertise required to get a robot to solve a novel task, but there is still a fundamental dependence on scripted trajectories. Consider the task of inserting a wire into a connector: it is difficult to imagine any predefined motion which can handle variability in wire shape and stiffness. To solve these sorts of tasks, it is desirable to have a richer control policy which considers a large amount of feedback including states, forces, and even raw images. Reinforcement Learning (RL) offers, in principle, a method to learn such policies from exploration, but the amount of actual exploration required has prohibited its use in real applications. In this paper we address this challenge by combining the demonstration and RL paradigms into a single framework which uses kinesthetic demonstrations to guide a deep-RL algorithm. Our long-term vision is for it to be possible to provide a few minutes of demonstrations, and have the robot rapidly and safely learn a policy to solve arbitrary manipulation tasks.The primary alternative to demonstrations for guiding RL agents in continuous control tasks is reward shaping. Shaping is typically achieved using a hand-coded function, such as Cartesian distance to a goal site, which provides a smoothly varying reward signal for every state the agent visits. While attractive in theory, reward shaping can lead to bizarre behavior or premature convergence to local minima, and in practice requires considerable engineering and experimentation to get right <cit.>.By contrast, it is often quite natural to express a task goal as a sparse reward function, e.g. +1 if the wire is inserted, and 0 otherwise. Our central contribution is to show that off-policy replay-memory-based RL (e.g. DDPG) is a natural vehicle for injecting demonstration data into sparse-reward tasks, and that it obviates the need for reward-shaping. In contrast to on-policy RL algorithms, such as classical policy gradient, DDPG can accept and learn from arbitrary transition data. Furthermore, the replay memory allows the agent to maintain these transitions for long enough to propagate the sparse rewards throughout the value function.We present results of simulation experiments on a set of robot insertion problems involving rigid and flexible objects. We then demonstrate the viability of our approach on a real robot task consisting of inserting a clip (flexible object) into a rigid object. This task is realized by a Sawyer robotic arm, using demonstrations collected by kinesthetically controlling an arm by the wrist. Our results suggest that sparse rewards and a few human demonstrations are a practical alternative to shaping for teaching robots to solve challenging continuous control tasks.§ BACKGROUND This section provides mathematical background for Markov Decision Processes (MDPs), DDPG, and deep RL techniques such as prioritized replay and n-step return. We adopt the standard Markov Decision Process (MDP) formalism for this work <cit.>. An MDP is defined by a tuple <S, A, R, P, γ>, which consists of a set of states S, a set of actions A, a reward function R(s,a), a transition function P(s'|s,a), and a discount factor γ. In each state s ∈ S, the agent takes an action a ∈ A. Upon taking this action, the agent receives a reward R(s,a) and reaches a new state s', determined from the probability distribution P(s'|s,a). A deterministic and stationary policy π specifies for each state which action the agent will take. The goal of the agent is to find the policy π mapping states to actions that maximizes the expected discounted total reward over the agent's lifetime. This concept is formalized by the action value function: Q^π(s,a)=𝔼^π[∑_t=0^+∞γ^tR(s_t,a_t)], where 𝔼^π is the expectation over the distribution of the admissible trajectories (s_0,a_0,s_1, a_1,…) obtained by executing the policy π starting from s_0=s and a_0=a. Here, we are interested in continuous control problems, and take an actor-critic approach in which both components are represented using neural networks.These methods consist in maximizing a mean value J(θ)=𝔼_s∼μ[Q^π(.|θ)(s,π(s|θ))] with respect to parameters θ that parameterise the policy and where μ is an initial state distribution. To do so, a gradient approach is considered and the parameters θ are updated as follows: θ←θ+α∇_θ J(θ). Deep Deterministic Policy Gradient (DDPG) <cit.> is an actor-critic algorithm which directly uses the gradient of the Q-function w.r.t. the action to train the policy. DDPG maintains a parameterized policy network π(.|θ^π) (actor function) and a parameterized action-value function network (critic function) Q(.|θ^Q). It produces new transitions e=(s,a,r=R(s,a),s'∼ P(.|s,a)) by acting according to a=π(s|θ^π)+𝒩 where 𝒩 is a random process allowing action exploration. Those transitions are added to a replay buffer B. To update the action-value network, a one-step off-policy evaluation is used and consists of minimizing the following loss:L_1(θ^Q)=𝔼_(s,a,r,s')∼ D[R_1-Q(s,a|θ^Q)]^2,where D is a distribution over transitions e=(s,a,r=R(s,a),s'∼ P(.|s,a)) contained in a replay buffer and the one-step return R_1 is defined as: R_1=r+γ Q'(s',π'(s')|θ^π')|θ^Q').Here Q'(.|θ^Q') and π'(.|θ^π') are the associated target networks of Q(.|θ^Q) and π(.|θ^π) which stabilizes the learning (updated every N' steps to the values of their associated networks). To update the policy network a gradient step is taken with respect to:∇_θ^πJ(θ^π)≈𝔼_(s,a)∼ D[∇_aQ(s,a|θ^Q)_|a=π(s|θ^Q)∇_θ^ππ(s|θ^π)].The off-policy nature of the algorithm allows the use of arbitrary data such as human demonstrations.Our experiments made use of several general techniques from the deep RL literature which significantly improved the overall performance of DDPG on our test domains.As we discuss in Sec. <ref>, these improvements had a particularly large impact when combined with demonstration data.§ DDPG FROM DEMONSTRATIONS Our algorithm modifies DDPG to take advantage of demonstrations. The demonstrations are of the form of RL transitions: (s,a,s',r). DDPGfD loads the demonstration transitions into the replay buffer before the training begins and keeps all transitions forever.DDPGfD uses prioritized replay to enable efficient propagation of the reward information, which is essential in problems with sparse rewards. Prioritized experience replay <cit.> modifies the agent to sample more important transitions from its replay buffer more frequently. The probability of sampling a particular transition i is proportional to its priority, P(i) = p_i^α/∑_k p_k^α, where p_i is the priority of the transition. DDPGfD uses p_i = δ_i^2 + λ_3 |∇_aQ(s_i,a_i|θ^Q) |^2 + ϵ + ϵ_D, where δ_i is the last TD error calculated for this transition, the second term represents the loss applied to the actor, ϵ is a small positive constant to ensure all transitions are sampled with some probability, ϵ_D is a positive constant for demonstration transitions to increase their probability of getting sampled, and λ_3 is used to weight the contributions. To account for the change in the distribution, updates to the network are weighted with importance sampling weights, w_i = (1/N·1/P(i))^β. DDPGfD uses α = 0.3 and β = 1 as we want to learn about the correct distribution from the very beginning. In addition, the prioritized replay is used to prioritize samples between the demonstration and agent data, controlling the ratio of data between the two in a natural way.A second modification for the sparse reward case is to use a mix of 1-step and n-step returns when updating the critic function. Incorporating n-step returns helps propagate the Q-values along the trajectories. The n-step return loss consists of using rollouts (forward view) of size n of a policy π close to the current policy π(.|θ^π) in order to evaluate the action-value function Q(.|θ^Q). The idea is to minimize the difference between the action-value at state (s=s_0,π(s)=a_0) and the return of a rollout (s_i,a_i=π(s_i), s_i'∼ P(.|s_i,a_i),r_i)_i=0^n-1 of size n starting from (s,π(s)) and following π. The n-step return has the following form: R_n = ∑_i=0^n-1γ^ir_i + γ^n Q(s'_n-1,π(s'_n-1);θ^Q'). The loss corresponding to this particular rollout is then: L_n(θ^Q)= 1/2(R_n-Q(s,π(s)|θ^Q))^2.A third modification is to do multiple learning updates per environment step. If a single learning update per environment step is used, each transition will only be sampled as many times as the size of the minibatch. Choosing a balance between gathering fresher data and doing more learning is in general a complicated trade-off. If our data is stale, the samples from the replay buffer no longer represent the distribution of states our current policy would experience. This can lead to wrong Q values in states which were not previously visited and potentially cause our policy and values to diverge. However in our case we require data efficiency and therefore we need to use each transition several times. In our experiments, we could increase the number of learning updates to 20 without affecting the per-update learning efficiency. In practice, we used the value of 40 which provided a good balance between learning from previous interaction (data efficiency) and stability.Finally, L2 regularization on the parameters of the actor and the critic networks are added to stabilize the final learning performance.The final loss can be written as:L_Critic(θ^Q)= L_1(θ^Q) + λ_1 L_n(θ^Q) +λ_2 L^C_reg(θ^Q) ∇_θ^πL_Actor(θ^π) = - ∇_θ^πJ(θ^π) + λ_2 ∇_θ^πL^A_reg(θ^π) To summarize, we modified the original DDPG algorithm in the following ways: * Transitions from a human demonstrator are added to the replay buffer.* Prioritized replay is used for sampling transitions across both the demonstration and agent data.* A mix of 1-step L_1(θ^Q) and n-step return L_n(θ^Q) losses are used.* Learning multiple times per environment step.* L2 regularization losses on the weights of the critic L_reg^C(θ^Q) and the actor L_reg^A(θ^π) are used. § EXPERIMENTAL SETUPOur approach is designed for problems in which it is easy to specify a goal state, but difficult to specify a smooth distance function for reward shaping that does not lead to sub-optimal behavior. One example of this is insertion tasks in which the goal state for the plug is at the bottom of a socket, but the only path to reach it, and therefore the focus of exploration, is at the socket opening. While this may sound like a minor distinction, we found in our initial experiments that DDPG with a simple goal-distance reward would quickly find a path to a local minimum on the outside of the socket, and fail to ever explore around the opening.We therefore sought to design a set of insertion tasks that presented a range of exploration difficulties.Our tasks are illustrated in Fig. <ref>.The first (Fig. <ref>) is a classic peg-in-hole task, in which both bodies are rigid, and the plug is free to rotate along the insertion axis.The second (Fig. <ref>) models a drive-insertion problem into an ATX-style computer chassis.Both bodies are again rigid, but in this case the drive orientation is relevant.The third task (Fig. <ref>) models the problem of inserting a two-pronged deformable plastic clip into a housing.The clip is modeled as three separate bodies with hinge joints at the base of each prong.These joints are spring-loaded, and the resting state pinches inwards as is common with physical connectors to maintain pressure on the housing.The final task (Fig. <ref>) is a simplified cable insertion task in which the plug is modeled as a 20-link chain of capsules coupled by ball-joints.This cable is highly under-actuated, but otherwise shares the same task specification as the peg-in-hole task.We created two reward functions for our experiments. The first is a sparse reward function which returned +10 if the plug was within a small tolerance of the goal site(s) on the socket:r = 0 , ∑_i ∈ sites W_g ||g_i - x_i||_2 > ϵ 10 , ∑_i ∈ sites W_g ||g_i - x_i||_2 < ϵwhere x_i is the position of the i^th tip site on the plug, g_i is the i^th goal site on the socket, W_g contains weighting coefficients for the goal site error vector, and ϵ is a proximity threshold.If this tolerance was reached, the robot received the reward signal and the episode was immediately terminated.The second reward function is a shaped reward which composes terms for two movement phases: a reaching phase c_o to align the plug to the socket opening, and an inserting phase c_g to reach the socket goal.Both terms compute a weighted ℓ_2-distance between the plug tip(s) and their respective goal site(s). The distance from the goal to the opening site (i.e. the maximum value of c_g) is added to c_o during the reaching phase, such that the reward monotonically increases throughout an insertion:c_g= min(∑_i ∈ sites W_g^T ||g_i-x_i||_2, ∑_i ∈ sites W_o^T ||g_i-o_i||_2) c_o= I(c_g > ∑_i ∈ sites W_o^T ||g_i-o_i||_2) ∑_i ∈ sites W_o^T ||o_i-x_i||_2r= min(1, max(0, -αlog(β (c_o + c_g))) - 1where g_i is the i^th goal site, o_i is the i^th opening site, W_g and W_o are weighting coefficients for the goal and opening site errors, respectively, I is the indicator function, and α and β are scaling parameters for log-transforming these distances into rewards ranging from 0 to 1. Note that tuning the weighting of each dimension in W_g and W_o must be done very carefully for the agent to learn the real desired task. In addition, the shaping of both stages must be balanced out in a delicate manner.r0.45 0.45< g r a p h i c s >Real-robot experiment setup for deformable-clip insertion task.The clip is made of deformable nylon, and is rigidly attached to the robot gripper.All tasks utilized a single vertically mounted robot arm. The robot was a Sawyer 7-DOF torque-controlled arm from Rethink Robotics, instrumented with a cuff for kinesthetic teaching. We utilized the Mujoco simulator <cit.> to simulate the Sawyer using publicly available kinematics and mesh files. In the simulation experiments the actions were joint velocities, the rewards were sparse or shaped as described above, and the observations included joint position and velocity, joint-torque feedback, and the global pose of the socket and plug. In both the simulation and real world experiments the object being inserted was rigidly attached to the gripper, and the socket was fixed to a table top.In addition to the four simulation tasks, we also constructed a real world clip insertion problem using a physical Sawyer robot.In the real robot experiment the clip was rigidly mounted to the robot gripper using a 3D printed attachment.The socket position was provided to the robot, and rewards were computed by evaluating the distance from the clip prongs (available via the robot's kinematics) to the goal sites in the socket as described above. In real robot experiments the observations included the robot joint position and velocity, gravity-compensated torque feedback from the joints, and the relative pose of the plug tip sites in the socket opening site frames. §.§ Demonstration data collectionTo collect the demonstration data in simulated tasks, we used a Sawyer robotic arm. The arm was kinesthetically force controlled by a human demonstrator. In simulation an agent was running a hard-coded joint space P-controller to match the joint positions of the simulated Sawyer robot to the joint positions of the real one. This agent was using the same action space as the DDPGfD agent which allowed the demonstration transitions to be added directly to the agent's replay buffer.For providing demonstration for the real world tasks we used the same setup, this time controlling a second robotic arm. Separating the arm we were controlling and the arm which solved the task ensured that the demonstrator did not affect the dynamics of the environment from the agent's perspective. For each experiment, we collected 100 episodes of human demonstrations which were on average about 25 steps (≈5s) long. This involved a total of 10-15 minutes of robot interaction time per task.§ RESULTSIn our first experiment we compared our approach to DDPG on sparse and shaped variants of the four simulated robotic tasks presented in Sec. <ref>. In addition, we show rewards for the demonstrations themselves as well as supervised imitation of the demonstrations. The DDPG implementation utilized all of the optimizations we incorporated into DDPGfD, including prioritized replay, n-step returns, and ℓ-2 regularization. For each task we evaluated the agent with both the shaped and sparse versions of the reward, with results shown in Figure <ref>. All traces plot the shaped-reward value achieved, regardless of which reward was given to the agent.All of these experiments were performed with fixed hyper-parameters, tuned in advance.We can see that in the case where we have hand-tuned shaping rewards all algorithms can solve the task. The results show that DDPGfD always out-performs DDPG, even when DDPG is given a well-tuned shaping reward. In contrast, DDPGfD learns nearly as well with sparse rewards as with shaping rewards. DDPGfD even out-performs DDPG on the hard drive insertion task, where the demonstrations are relatively poor.In general, DDPGfD not only learns to solve the task, but learns to solve it more efficiently than the demonstrations, usually learning to insert the object in 2-4x fewer steps than the demonstrations. DDPGfD also learns more reliably, as the percentile plots are much wider for DDPG. Doing purely supervised learning of the demonstration policy performs poorly in every task.In our second experiment we examined the effect of varying the quantity of demonstration data on agent performance. Fig. <ref> compares learning curves for DDPGfD agents initialized with 1, 2, 3, 5, 10, and 100 expert trajectories on the sparse-reward clip-insertion task. DDPGfD is capable of solving this task with only a single demonstration, and we see diminishing returns with 50-100 demonstrations. This was surprising, since each demonstration contains only one state transition with non-zero reward.Finally, we show results of DDPGfD learning the clip insertion task on physical Sawyer robot in Figure <ref>. DDPGfD was able to learn a robust insertion policy on the real robot. DDPGfD with sparse rewards outperforms shaped DDPG, showing that DDPGfD achieves faster learning without the extra engineering.A video demonstrating the performance can be viewed here: <https://www.youtube.com/watch?v=WGJwLfeVN9w>§ RELATED WORK Imitation learning is primarily concerned with matching expert demonstrations. Our work combines imitation learning with learning from task rewards, so that the agent is able to improve upon the demonstrations it has seen. Imitation learning can be cast into a supervised learning problem (like classification) <cit.>. One popular imitation learning algorithm is DAGGER <cit.> which iteratively produces new policies based on polling the expert policy outside its original state space. This leads to no-regret over validation data in the online learning sense. DAGGER requires the expert to be available during training to provide additional feedback to the agent.Imitation can also been achieved through inverse optimal control or inverse RL. The main principle is to learn a cost or a reward function under which the demonstration data is optimal. For instance, in <cit.> the inverse RL problem is cast into a two-player zero-sum game where one player chooses policies and the other chooses reward functions. However, it doesn't scale to continuous state-action spaces and requires knowledge of the dynamics. To address continuous state spaces and unknown dynamics, <cit.> solve inverse RL by combining classification and regression. Yet it is restricted to discrete action spaces. Demonstrations have also been used for inverse optimal control in high-dimensional, continuous robotic control problems <cit.>. However, these approaches only do imitation learning and do not allow for learning from task rewards.Guided Cost Learning (GCL) <cit.> and Generative Adversarial Imitation Learning (GAIL) <cit.> are the first efficient imitation learning algorithms to learn from high-dimensional inputs without knowledge of the dynamics and hand-crafted features. They have a very similar algorithmic structure which consists of matching the distribution of the expert trajectories. To do so, they simultaneously learn the reward and the policy that imitates the expert demonstrations. At each step, sampled trajectories of the current policy and the expert policy are used to produce a reward function. Then, this reward is (partially) optimized to produce an updated policy and so on. In GAIL, the reward is obtained from a network trained to discriminate between expert trajectories and (partial) trajectories sampled from a generator (the policy), which is itself trained by TRPO<cit.>. In GCL, the reward is obtained by minimization of the Maximum Entropy IRL cost<cit.> and one could use any RL algorithm procedure (DDPG, TRPO etc.) to optimize this reward.Control in continuous state-action domains typically uses smooth shaped rewards that are designed to be amenable to classical analysis yielding closed-form solutions. Such requirements might be difficult to meet in real world applications. For instance, iterative Linear Quadratic Gaussian (iLQG) <cit.> is a method for nonlinear stochastic systems where the dynamics is known and the reward has to be quadratic (and thus entails hand-crafted task designs). It uses iterative linearization of the dynamics around the current trajectory in order to obtain a noisy linear system (where the noise is a centered Gaussian) and where the reward constraints are quadratic. Then the algorithm uses the Ricatti family of equations to obtain locally linear optimal trajectories that improve on the current trajectory.Guided Policy Search <cit.> aims at finding an optimal policy by decomposing the problem into three steps. First, it uses nominal or expert trajectories, obtained by previous interactions with the environment to learn locally linear approximations of its dynamics. Then, it uses optimal control algorithms such as iLQG or DDP to find the locally linear optimal policies corresponding to these dynamics. Finally, via supervised learning, a neural network is trained to fit the trajectories generated by these policies. Here again, there is a quadratic constraint on the reward that must be purposely shaped.Normalized Advantage Functions (NAF) <cit.> with model-based acceleration is a model-free RL algorithm using imagination rollouts coming from a model learned with the previous interactions with the environment or via expert demonstrations. NAF is the natural extension of Q-Learning in the continuous case where the advantage function is parameterized as a quadratic function of non-linear state features. The uni-modal nature of this function allows the maximizing action for the Q-function to be obtained directly as the mean policy.This formulation makes the greedy step of Q-Learning tractable for continuous action domains. Then, similarly as GPS, locally linear approximations of the dynamics of the environment are learned and iLQG is used to produce model-guided rollouts to accelerate learning. The most similar work to ours is DQfD <cit.>, which combines Deep Q Networks (DQN) <cit.> with learning from demonstrations in a similar way to DDPGfD. It additionally adds a supervised loss to keep the agent close to the policy from the demonstrations. However DQfD is restricted to domains with discrete action spaces and is not applicable to robotics.§ CONCLUSIONIn this paper we presented DDPGfD, an off-policy RL algorithm which uses demonstration trajectories to quickly bootstrap performance on challenging motor tasks specified by sparse rewards. DDPGfD utilizes a prioritized replay mechanism to prioritize samples across both demonstration and self-generated agent data. In addition, it incorporates n-step returns to better propagate the sparse rewards across the entire trajectory.Most work on RL in high-dimensional continuous control problems relies on well-tuned shaping rewards both for communicating the goal to the agent as well as easing the exploration problem. While many of these tasks can be defined by a terminal goal state fairly easily, tuning a proper shaping reward that does not lead to degenerate solutions is very difficult. This task only becomes more difficult when you move to multi-stage tasks such as insertion. In this work, we replaced these difficult to tune shaping reward functions with demonstrations of the task from a human demonstrator. This eases the exploration problem without requiring careful tuning of shaping rewards.In our experiments we sought to determine whether demonstrations were a viable alternative to shaping rewards for training object insertion tasks. Insertion is an important subclass of object manipulation, with extensive applications in manufacturing. In addition, it is a challenging set of domains for shaping rewards, as it requires two stages: one for reaching the insertion point, and one for inserting the object. Our results suggest that Deep-RL is poised to have a large impact on real robot applications by extending the learning-from-demonstration paradigm to include richer, force-sensitive policies.§ REAL ROBOT SAFETYTo be able to run DDPG on the real robot we needed to ensure that the agent will not apply excessive force. To do this we created an intermediate impedance controller which subjects the agent's commands to safety constraints before relaying them to the robot. It modifies the target velocity set by the agent according to the externally applied forces.u_control = u_agent k_a + f_applied k_fWhere u_agent is agent's control signal, f_applied are externally applied forces such as the clip pushing against the housing, and k_a and k_f are constants to choose the correct sensitivity. We further limit the velocity control signal u_control to limit the maximal speed increase while still allowing the agent to stop quickly. This increases the control stability of the system.This allowed us to keep the agent's control frequency, u_agent, at 5Hz while still having a physically safe system as f_applied and u_control were updated at 1kHz. | http://arxiv.org/abs/1707.08817v2 | {
"authors": [
"Mel Vecerik",
"Todd Hester",
"Jonathan Scholz",
"Fumin Wang",
"Olivier Pietquin",
"Bilal Piot",
"Nicolas Heess",
"Thomas Rothörl",
"Thomas Lampe",
"Martin Riedmiller"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170727111653",
"title": "Leveraging Demonstrations for Deep Reinforcement Learning on Robotics Problems with Sparse Rewards"
} |
Department of Physics, Southeast University, Nanjing 211189, China Beijing Computational Science Research Center, Beijing 100084, ChinaDepartment of Physics, Southeast University, Nanjing 211189, ChinaSynergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, CAS, Hefei 230026, China Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, CAS, Hefei 230026, China Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1, [email protected] Department of Physics, Southeast University, Nanjing 211189, China Beijing Computational Science Research Center, Beijing 100084, China State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically it is vulnerable to imperfections such as decoherence. Here, we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We have simulated noisy channels by implementing space-multiplexed dual interferometers with quantum photonic inputs. We have demonstrated the advantage of entanglement-assisted protocols in phase estimation experiment run with either single-probe or multi-probe approach. These results establish that entanglement with ancillæ is a valuable approach for delivering quantum-enhanced metrology. Our new approach to entanglement-assisted quantum metrology via a simple linear-optical interferometric network with easy-to-prepare photonic inputs provides a path towards practical quantum metrology. Entanglement-enhanced quantum metrology in a noisy environment Peng Xue December 30, 2023 ==============================================================Introduction:-Quantum metrology <cit.> exploits nonclassicality to surpass classical limits to interferometric parameter estimation <cit.>. Quantum metrological enhancement is achieved by employing quantum probes for detecting physical properties with resolution beyond the reach of classical approaches <cit.>. Without noise, entangling the measurement system with ancillary quantum degrees of freedom provides no advantage to scaling of measurement precision with number of particles <cit.>. Contrariwise, in the presence of noise, which deleteriously affects measurement precision, entangling with ancillæ is suggested to deliver higher precision than not using entanglement with ancillæ <cit.>.We demonstrate experimentally that entangling probes with ancillæ significantly enhances the performance of noisy quantum metrology as quantified by the quantum Fisher information (QFI) for parameter estimation (Fig. <ref>). Through entanglement with ancillæthe probe state is less sensitive to noise. Information from probes is limited by the Holevo bound <cit.> whereas enlarging the Hilbert space by entangling with ancillæ allows more information to be accessed by measurements that exploit the larger dimension of Hilbert space. The QFI is obtained by tracing over the auxiliary space, which maximizes over all mixed states. That might make the QFI larger than that without ancillæ <cit.>. The enlargement enhances the precision only for certain noisy channels, for which the input states entangled between the space of probes and ancillæ are optimal <cit.>.Based on these theoretical proposals, we experimentally investigate whether entangled ancillæ can deliver enhanced metrological precision in the presence of noise <cit.> realized as simulated decohering quantum channels <cit.>, and herein establish that indeed entangling with ancillæis advantageous for efficiently inferring the unknown parameter measuring for a wide range of noise values. We develop space-multiplexed noisy channels via a dual interferometric network <cit.> and inject hyperentangled photonic states entangled in their polarizations and spatial modes <cit.>. Theory:-First, we use a single-probe scheme as an example. Entanglement-assisted parameter estimation comprises three stages: preparation in which a probe (a photonic qubit in our case) shares entanglement with an ancilla; parametrization where the probe evolves in a channel and the parameter to be estimated is encoded in the probe whereas the ancilla does not participate; and measurement in which a joint measurement is performed on both the probe and ancilla to yield a precise estimate of the parameter. We focus on a two-level probe detecting a phase shift modelled by the unitary map𝒰_ϕ(ρ) =U_ϕρ U_ϕ^†, U_ϕ=|0⟩⟨0|+e^iϕ|1⟩⟨1|for ρ the initial state. The noise map ℰ acts after 𝒰_ϕ: ϕ is encoded into the probe state ρ_ϕ=Λ_ϕρ for Λ(ϕ)=ℰ∘𝒰_ϕ.We use QFI <cit.>J(ρ(ϕ))=Tr(ρ(ϕ) A^2), ∂ρ(ϕ)/∂ϕ=Aρ(ϕ)+ρ(ϕ) A/2,to quantify the metrological precision, with A the symmetric logarithmic-derivative operator. QFI is an appropriate measure as it serves as an asymptotic measure of the amount of information inherent in how much the system parameters can be acquired by measurement. The quantum Cramér-Rao bound <cit.> is a lower bound for the precision Δϕ of the estimate of ϕ: Δϕ≥1/√(ν J(ρ(ϕ))) for ν the number of repetitions of the phase-estimate procedure. The best bound is found by maximizing the QFI, which depends on both ρ and ϕ.For a single-probe instance, noise diminishes the measurement precision evident through reducing the output-state QFI after passing through ℰ. Entangling with an ancilla enhances precision for noisy channels and the state transformation is (Λ_ϕ⊗) ρ with the ancilla unchanged. Here, ρ denotes the probe+ancilla state whereas ρ denotes the single-probe state. We consider three decoherence processes encountered in quantum-enhanced metrology: amplitude-damping (spontaneous emission and photon scattering inside the interferometer), general-Pauli (most general lossless channel) and depolarizing (most symmetric Pauli channel assuming uncorrelated noise) channels <cit.>, which are typically utilized when accounting for decoherence in optical interferometry <cit.>.We start with the amplitude-damping channel <cit.>∑_=0^1 A_ρ A^†_, A_0=[10;0 √(1-η) ], A_1=[0 √(η);00 ]for η the probability of decay |1⟩↦|0⟩. For a single-probe input state, the optimized QFI is 1-η and the optimal state is |+⟩:=(|0⟩+|1⟩)/√(2). For the entanglement-assisted approach, the QFI is 2(1-η)/(2-η) for an entangled state of the probe and ancilla |Φ⟩:=(|00⟩+|11⟩)/√(2) and is always greater than that of the case without assisted entanglement for arbitrary η∈(0,1) <cit.>.For Ξ=(,X,Y,Z) the Pauli matrices, the general-Pauli channel is the mapℰ_GPC(ρ)=∑_i=0^3p_iΞ_iρΞ_i,∑_i p_i=1, 0≤ p_i≤1,and the depolarizing channel p_1=p_2=p_3=p/4 is a special case. For a single-qubit probe, |+⟩ is the optimal state, and the optimal QFI is (1-p)^2 <cit.>. If the joint-probe ancilla state is |Φ⟩, the QFI is 2(1-p)^2/(2-p). For arbitrary p∈(0,1), the QFI is always greater than that of the case without assisted entanglement <cit.>.The depolarizing channel can be regarded as a time-sharing combination of a noiseless channel and a noisy channel in which the state will evolve to a maximally mixed state <cit.>. For either of the two channels, the entanglement-assisted approach does not provide any advantage. However, somewhat surprisingly, assisted entanglement improves QFI for the depolarizing channel. We can test for the general-Pauli channel (the depolarizing channel is a special case) which can be implemented in a time-sharing way <cit.>. Each Pauli operator is applied over a specific activation time, respectively, and the total decoherence process lasted over an activation cycle, achieving a time-sharing general-Pauli channel. To explain the advantages of entanglement-assisted quantum metrology, we rather implement a new type of general-Pauli channel, namely a space-multiplexed Pauli channel.Our method can be extended to a more complicated case — an N-probe approach. In the absence of noise, an N-probe approach with an optimal N-qubit input state (e.g., a N00N state) achieves the Heisenberg limit scaling, which provides improvement over classical limits. However, the advantages are destroyed by noise. Our entanglement-assisted approach in which N probes are entangled with noiseless ancillæ protects against noise and the effect caused by noise can be eliminated by assisted entanglement. Even in the presence of noise, the entanglement-assisted approach beats the shot-noise limit and even maintains the Heisenberg limit scaling for some special noisy channel.We use a two-probe approach as an example. A two-qubit N00N state (one of the Bell states) |Φ^+⟩=(|00⟩+|11⟩)/√(2) with both qubits being probes is optimal only in the noiseless case. The phase ϕ to be estimated is obtained via the unitary map applied in parallel𝒰^2_ϕ(ϱ)=U_ϕ⊗ U_ϕϱ U_ϕ^†⊗ U_ϕ^†with ϱ=|Φ^+⟩⟨Φ^+|. Through a collective noisy channel in parallel, the probe state becomes ϱ_ϕ=Λ_ϕ^⊗2ϱ.A four-qubit entangled state ϱ̃=(|0000⟩+|1111⟩)(⟨0000|+⟨1111|)/2 of two probes and two ancillæ beats the optimal state of two probes ϱ in the presence of noise. Taking the collective damping channel as an example, its QFI is8(η-1)^2{2(η-1)^2cos8ϕ+(η-2)η[(η-2)η+2]+2}/[(η-2)η+2]^3and is larger than that of ϱ, even though this particular four-qubit entangled state is not necessarily optimal. Realization of noisy channels:-The experimental setup in Fig. <ref> involves the three stages of state preparation, parametrization and measurement. In the preparation stage, we prepare single photons in polarization-spatial hyperentangled states for entanglement-assisted single-probe approach <cit.>. Whereas, for entanglement-assisted two-probe approach, polarization-entangled photon pairs are used to prepare the four-qubit hyperentangled state <cit.>.The probe state is transformed according to the noisy channel, whereas the ancilla qubit is not evolving. The noise is introduced in a controlled way only on the probe. The efficiency of the optimal estimation is shown to outperform quantum process tomography (QPT).We now present the experimental implementation of a single-qubit amplitude-damping channel. As the noisy channel is only applied to the probe state, i.e., the polarization degree of freedom of the photons, the longitudinal spatial modes of the photons (|U⟩ and |D⟩) are not affected. The photons on either of the modes encounter the same noisy channel. In the polarization basis, the amplitude-damping map is realized by the dual interferometer setup implemented by splitting the two polarization components and putting independent polarization controls inside a beam displacer (BD) interferometer <cit.>.First a BD whose optical axis is perpendicular to that of the one which is used for preparing hyperentangled states in the state preparation stage splits the two polarization components by directly transmitting the vertically polarized photons and shifting the horizontally polarized photons by a lateral displacement. A half-wave plate (HWP) at 45^∘ rotates |H⟩ to |V⟩ and another HWP (H_A) at θ_A with cos2θ_A=-√(1-η) applies a rotation [ -√(1-η)√(η);√(η)√(1-η) ] on the polarization of photons. The following BD splits and combines the photons due to their polarizations, and the HWPs with certain setting angles are used to rotation the polarization of the photons.A quartz crystal (QC) with thickness of 28.77mm <cit.> is inserted to reduce the spatial coherence of the photons with different polarizations.The sandwich-type HWP-BD-HWP setup works as a 50:50 beamsplitter recombining the photons. Accordingly, with probability 1/2, the state emerging from the output port is the desired output state. Furthermore, we can also create a single-qubit space-multiplexed general-Pauli channel (<ref>) with five BDs and twelve HWPs. Six HWPs (H_l at θ_l, l=1,…,6) control the ratio of photons in different lateral spatial modes, and three of them at 45^∘ (in front of the fifth BD) flip the polarizations and then change the spatial modes of the corresponding photons. Therefore, after the fifth BD, the photons are distributed into four lateral spatial modes according to the parameters p_i. For a given desired channel the setting angles θ_l of the HWPs (H_l) are chosen to satisfy the relations√(p_0) =cos2θ_1sin2θ_3=cos2θ_2cos2θ_4sin2θ_6, √(p_1) =sin2θ_1=-cos2θ_2cos2θ_4cos2θ_6, √(p_2) =cos2θ_1cos2θ_3cos2θ_5=sin2θ_2, √(p_3) =cos2θ_1cos2θ_3sin2θ_5=-cos2θ_2sin2θ_4.Then the last three HWPs at 0^∘ and 45^∘, respectively, are inserted into different spatial modes and act as Pauli operators Ξ on the probe qubit.Two nonpolarizing beamsplitters (NBSs) recombine the photons in the four lateral spatial modes. To reduce the spatial coherence of the photons, the optical distance ς between the photons in the different lateral spatial modes should satisfy L_coh<ς<cΔ t=0.9m. In our experiment, maxς≈0.6m. Hence, we realize the space-multiplexed general-Pauli channel.To compare the approaches with and without assisted entanglement, we realize noisy channels on the probe qubit, which does not share entanglement with an ancilla. In our experiment, in both the state preparation and process tomography stages, the BDs and some WPs are removed from the setup in Fig. <ref> as no ancillary spatial mode is needed. In the parametrization stage, the photons are not distributed into different longitudinal spatial modes.Experimental results of QFI:-We present our experimental results for noisy channels and compared the QFI for the single-probe approach with and without assisted entanglement. Our experimental process matrices χ_exp are reconstructed using process fidelity <cit.>F=Tr(χ^†_thχ_exp)/√(Tr(χ^†_expχ_exp)Tr(χ^†_thχ_th))to characterize the experimental realization of the noisy channels <cit.>. Figure <ref> shows the experimentally reconstructed χ_exp for the amplitude-damping channel with η=0.5 and the depolarizing channel with p=0.4. Our results exhibit F≈1. Without assisted entanglement, all the fidelities of the amplitude-damping channel with various parameters are great than 0.9949± 0.0007 and those of the depolarizing channel are greater than 0.9700±0.0041. Whereas with entanglement sharing between the probe and ancilla, all the fidelities of the amplitude-damping channel are greater than 0.9647± 0.0003 and those of the depolarizing channel are greater than 0.9593± 0.0016.To calculate the QFI, we use the diagonal form of the output state ρ_exp(ϕ)=∑_iλ_i|ψ_i⟩⟨ψ_i|+ρ_noise, where λ_i and |ψ_i⟩ are the eigenvalues and eigenstates, ρ_noise is the irrelevant part of the density matrix and is independent of ϕ <cit.>. With this formula, we calculate the matrix elements of A in the basis {|ψ_i⟩}.We use the amplitude-damping and depolarizing channels as examples as usual for decoherence in optical interferometry. For the amplitude-damping channel, the optimized QFI of the output state is [2ρ_exp^12(ϕ)]^2/ρ_exp^11(ϕ)+ρ_exp^22(ϕ), and [2ρ_exp^14(ϕ)]^2/ρ_exp^11(ϕ)+ρ_exp^44(ϕ) for a single-probe input state and for the entanglement-assisted approach, respectively, with ρ_exp^ij a matrix element of ρ_exp. For the depolarizing channel, without assisted entanglement, the optimized QFI for a single probe is [2ρ_exp^12(ϕ)]^2/[ρ_exp^11(ϕ)+ρ_exp^22(ϕ)]. With assisted entanglement, the QFI of the output state of the probe+ancilla system is then [2ρ_exp^14(ϕ)]^2/[ρ_exp^11(ϕ)+ρ_exp^44(ϕ)]+[2ρ_exp^23(ϕ)]^2/[ρ_exp^22(ϕ)+ρ_exp^33(ϕ)].As we reconstruct all noisy-channel information via QPT <cit.>, the output state for each case is reconstructed. By setting ϕ=0, we calculate experimental QFI values of the output states. In Fig. <ref>, experimental values of the QFI for the amplitude-damping and depolarizing channels either with or without the assisted entanglement are shown. Our experimental results agree well with theoretical calculations.Evidently, for a single probe, in the presence of amplitude-damping noise and depolarizing noise, an entanglement-assisted scheme improves the QFI compared to the unentangled case for all ranges of noise regimes. To illustrate this, we also realize the general-Pauli channel with p_0=p_2=0.5 and p_1=p_3=0. The experimental value for QFI for the entanglement-assisted approach is 0.984±0.045, which agrees with the theoretical prediction 1, whereas the optimized QFI for a single probe is 0. This represents the case of orthogonal noise when the ancilla approach recovers almost the full information on the phase even in the presence of noise.Phase estimation:-For the single-probe approach, the phase ϕ to be estimated has been obtained via a unitary map via an additional HWP inserting in the interferometer which causes the optical path difference between photons with different polarizations. The optimal measurement strategy around ϕ∼ 0 consists in projecting in the polarization-spatial hyperentangled states (|HU⟩±i|VD⟩)/√(2). Since no information on ϕ is carried on the other bases, for convenience, we choose |HD⟩ and |VU⟩. The projective measurements are realized via a BD, a quarter-wave plate (QWP) at 0, HWPs at 45^∘ and 22.5^∘ respectively, and a polarizing beamsplitter (PBS). Coincidences between the outputs and the trigger are detected by single photon avalanche photodiodes (APDs) <cit.>.For the amplitude-damping channel, the outcome probabilities of the projective measurements are P[(|HU⟩±i|VD⟩)/√(2)]=[2-η±2v√(1-η)sinϕ]/4, P(|HD⟩)=η/2 and P(|VU⟩)=0, where v is the visibility of the interferometer. The optimal measurement is identified by optimising the highest QFI 2v^2(1-η)/(2-η), which proves that the measurement achieves the quantum Cramér-Rao bound for the input state. Whereas, for depolarizing channel, the outcome probabilities are P[(|HU⟩±i|VD⟩)/√(2)]=[2-p±2v(1-p)sinϕ]/4, P(|HD⟩)=p/4 and P(|VU⟩)=p/4 and the corresponding QFI is 2v^2(1-p)^2/(2-p), which is always above the single-probe QFI.For the two-probe approach, we use the amplitude damping channel as an example. The input state is prepared in a two-photon N00N state (|HH⟩+|VV⟩)/√(2). Each probe is affected by an individual amplitude damping channel with the noise parameter η. With ancillary degree of freedom—spatial modes of two photons, the entanglement-assisted state becomes (|HUHU⟩+|VDVD⟩)/√(2). The optimal measurement strategy around ϕ∼ 0 consists in projecting in the polarization-spatial hyperentangled states (|HUHU⟩±i|VDVD⟩)/√(2). No information on ϕ is carried on the other 14 bases. The outcome probabilities of the projective measurements are P[(|HUHU⟩±i|VDVD⟩)/√(2)]=[2-2η+η^2∓2v(1-η)sin2ϕ]/4, P(|HDHD⟩=η^2/2), P(|HDVD⟩)=η(1-η)/2, P(|VDHD⟩)=η(1-η)/2, and zero for the other projective measurements. The optimal measurement is identified by optimising the highest QFI 8v^2(1-η)^2/[1+(1-η)^2], which is always above the two-probe approach without assisted entanglement 4v^2(1-η)^2/[1-η+η^2].To realize the entanglement-assisted single-probe approach, for each of the various noise parameters, data is accumulated for collecting time of 10s, corresponding to a coincidence count rate of about 20,000 events per acquisition. Whereas for the entanglement-assisted two-probe approach, the coincidence count rate is about 2,000 events per acquisition. Totally 100 values of the phase ϕ are collected. The standard deviation of the sample δϕ is expected to converge to the ultimate limit established by the quantum Cramér-Rao bound in the limit of a large number of repetitions. We use the standard deviation of the sample multiplied by √(ν) (here, ν is the average number of the events) to indicate the error √(ν)δϕ.Figure <ref> shows the experimental results of the error √(ν)δϕ as a function of the noise parameters for different approaches in different noisy channels. For the single-probe approach, due to experimental imperfections such as imperfect interferometric visibility of the setup, it is difficult to observe the advantages of the entanglement-assisted approach at low noise. With the noise parameter increasing, the advantages are more obvious. For the two-probe case, the approach of a two-qubit N00N state beats the shot-noise limit both in the noiseless case and at low noise level. The advantage over the classical metrology is affected by noise. Assisted entanglement protects against the noise, especially at high noise level.Discussion:-We experimentally realized entangled-assisted quantum metrology and demonstrated its efficacy through the QFI for single-qubit amplitude-damping, depolarizing and general-Pauli noisy channels. Compared to the approach without assisted entanglement, we observe an enhancement over the noisy cases. Our achievement relies on replacing time-sharing noisy channels by space-multiplexed noisy channels using a practical, linear-optical interferometric network. Our demonstration serves as a foundation for future experimental simulations employing networks of multi-qubit channel simulations.We use polarization-spatial hyperentangled states encoded in photons, which are easier to create and control. Our new approach to entanglement-assisted quantum metrology via a simple linear-optical interferometric network with easy-to-prepare photonic inputs provides a path towards practical quantum metrology.Note:-After completing this work, we learned of related work by the group of Marco Barbieri <cit.>.Acknowledgments We thank Lorenzo Maccone for helpful discussions and appreciate elucidating correspondence with Carlton M. Caves regarding how and why assisted entanglement is an advantage. We acknowledge support by NSFC (Nos. 11474049, 11674056 and GG2340000241), NSFJS (No. BK20160024), the Scientific Research Foundation of the Graduate School of Southeast University and the Open Fund from State Key Laboratory of Precision Spectroscopy of East China Normal University. 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Preprint at https://arxiv.org/abs/1707.08792 (2017).§ SUPPLEMENTAL MATERIAL FOR “ENTANGLEMENT-ENHANCED QUANTUM METROLOGY IN A NOISY ENVIRONMENT” In this Supplemental Material, we discuss extended-channel quantum Fisher information, optimal probe states under the dynamics with depolarization, intuitive understanding why assisted entanglement helps against noise, as well as some experimental details. § EXTENDED-CHANNEL QUANTUM FISHER INFORMATIONThe action of a quantum channel Λ_ϕ=ℰ∘𝒰_ϕ can always be expressed as its operator-sum representation, Λ_ϕρ =∑_iK_i(ϕ)ρ K_i^†(ϕ) with Kraus operator K_i(ϕ) satisfying ∑_iK_i(ϕ)K_i^†(ϕ)=. Evidently, this representation is not unique; different sets of linearly independent Kraus operators can be related by unitary transformations <cit.>K_i(ϕ)=∑_j u_ij(ϕ)K_i(ϕ),where u_ij is the element of a unitary matrix u(ϕ) possibly depending on ϕ.The single-channel quantum Fisher information is equal to the smallest quantum Fisher information of its purifications Λ_ϕρ = Tr_E(Ψ_ϕ) with |Ψ_ϕ⟩ the state of input+environment and the subscript E for tracing out environment <cit.>J (Λ_ϕρ)=min_|Ψ_ϕ⟩ J(|Ψ_ϕ⟩)by minimizing over the state of input+environment |Ψ_ϕ⟩.For a pure input state (not an unreasonable constraint as the optimal input state is always pure <cit.>), different purifications correspond to different Kraus representations of the channel. Moreover, it is enough to parameterize equivalent Kraus representations in Eq. (S1) with a Hermitian matrix h, which is the generator of infinitesimal rotations; i.e., u(ϕ)=e^-ih(ϕ-ϕ_0), in the vicinity of the real value ϕ_0. This formulation simplifies the optimization problem Eq. (S2) by revising it as a minimization problem over h. Therefore, we obtain the maximal quantum Fisher information after performing the input optimization as <cit.>max_ρJ (Λ_ϕρ)= 4max_ρmin_hTr(ρ∑_iK̇_i^†(ϕ)K̇_i(ϕ))with K̇_i(ϕ)=∂_ϕK_i(ϕ).By considering an ancillary system with extended input states involving probe and ancilla, we acquire full information available about ϕ imprinted by the map Λ_ϕ on the extended output state. Then quantum Fisher information of the extended-channel is calculated in a similar way. The map becomes ρ(ϕ)=Λ_ϕ⊗ρ, where ρ denotes the initial pure state of the probe+ancilla system. The quantum Fisher information ismax_ρJ (Λ_ϕ⊗ρ)= 4max_ρ_Amin_hTr(ρ_A∑_iK̇_i^†(ϕ)K̇_i(ϕ)),where ρ_A=Tr_A(ρ) is obtained by tracing over the auxiliary space, which leads to the maximization over all mixed states ρ_A. Equation (S4) is exactly Eq. (S3) with the pure input state replaced by a general mixed one. By maximizing over all mixed states, the extended channel quantum Fisher information can be larger than the unextened one.If and only if the optimal ρ_A is a pure state, assisted entanglement does not help.§ OPTIMAL PROBE STATES UNDER THE DYNAMICS WITH DEPOLARIZATION The depolarizing channel is described by Kraus operatorsK_0=√(1-3p/4)Ξ_0,K_1,2,3=√(p/4)Ξ_1,2,3,where Ξ=(,X,Y,Z) are the Pauli matrices. Using the method of semi-definite programming <cit.>, we find the optimal generatorh=1/2([000ξ;00 -i0;0i00;ξ000 ]), ξ=√((4-3p))/2-p. For the single-probe approach, the optimal input state is ρ=|+⟩⟨+|, where |±⟩=(|0⟩±|1⟩)/√(2). Substituting the optimal state and generator into Eq. (S3), we obtain the maximal quantum Fisher information of the single probemax_ρJ (Λ_ϕρ)=(1-p)^2. For the entanglement-assisted approach, the optimal reduced state is the maximally mixed state ρ_A=(|0⟩⟨0|+|1⟩⟨1|)/2. The optimal entangled input state in this case is any pure state ρ with the reduced state equal to ρ_A. The simplest choice of the optimal input state is the maximally entangled state ρ=(|00⟩+|11⟩)(⟨00|+⟨11|)/2 <cit.>, and the corresponding maximal quantum Fisher information ismax_ρJ (Λ_ϕ⊗ρ)=2(1-p)^2/(2-p),which is always greater than that of the single-probe approach for arbitrary p∈ (0,1).§ INTUITIVE UNDERSTANDING WHY ASSISTED ENTANGLEMENT HELPS AGAINST NOISE The intuitive understanding of how and why the ancilla qubit helps is crucial to making progress on entanglement-assisted metrology. Here, we provide it for the case of a depolarizing channel.Figure S1(a) shows the single-probe approach. A Hadamard operator creates the state of the probe qubit |+⟩. With U_ϕ=e^-iZϕ/2, the depolarizing channel isℰ⊙ = (1-3p/4)⊙ + p/4(Z⊙ Z+X⊙ X+Y⊙ Y),where ⊙ is a placeholder for the operator which the quantum operation acts on, and the measurement is in the Y basis. Figure S1(b) shows the entanglement-assisted protocol. The Hadamard and controlled-NOT operators together create the entangled state |Φ^+⟩=(|00⟩+|11⟩)/√(2), and the final measurement is a controlled-NOT followed by Y⊗ Z, i.e., Y on the probe qubit and Z on the ancilla qubit.Then we use the convention that tensor products are written in the order lower-upper. Figure S1(c) shows the second form of the circuit in Fig. S1(b), in which the first controlled-NOT is moved through the rotation U_ϕ, then moved through the depolarizing channel, combining the second controlled-NOT and then converting the channel to a two-qubit quantum operationℱ⊙= (1-3p/4)⊗⊙⊗ + p/4(Z⊗⊙ Z⊗+X⊗ X⊙ X⊗ X+Y⊗ X⊙ Y⊗ X).The finial measurement is then of Y⊗ Z.The effect of the single-qubit circuit on the state |+⟩ isℰ∘𝒰_ϕ(|+⟩⟨+|) = (1-p)U_ϕ|+⟩⟨+|U_ϕ^†+p/2;i.e., the rotation is applied with probability 1-p, and the qubit is mapped to the maximally mixed state with probability p.The effect of the ancilla-assisted circuit on the state |+⟩|0⟩ isℱ∘𝒰_ϕ⊗(|+⟩|0⟩⟨0|⟨+|)=[(1-p)U_ϕ|+⟩⟨+|U_ϕ^†+p/4]⊗|0⟩⟨0|+p/4⊗|1⟩⟨1|=(1-p/2)[(1-q)U_ϕ|+⟩⟨+|U_ϕ^†+q/2]⊗|0⟩⟨0|+p/4⊗|1⟩⟨1|,whereq=p/2/1-p/2⟺ 1-q=1-p/1-p/2. Evidently, the form of ℱ shows that X and Y errors map the main qubit to the maximally mixed state, wiping out the information about ϕ. This happens just as for the single-qubit circuit, except that a record of when an X or Y error occurs is stored in the ancilla qubit. By monitoring the ancilla qubit, one can discard the random data that results from X or Y errors.The upshot is that, with probability 1-p/2, the entanglement-assisted quantum circuit works just like the single-qubit circuit. Compared to the single-qubit circuit, the entanglement-assisted quantum circuit achieves a successful rotation with probability (1-p/2)(1-q), and with probability (1-p/2)q/2, maps to the maximally mixed state and with a record stored in the outcome 0 of the ancilla qubit. As the single-qubit circuit achieves an estimator variance 1/(1-p)^2, the entanglement-assisted circuit achieves an estimator variance1/1-p/21/(1-q)^2 = 1-p/2/(1-p)^2,which is smaller than 1/(1-p)^2. That means assisted entanglement helps to achieve an smaller estimator variance compared to the single-probe approach. The term 1-p/2 in the denominator of the first expression comes from the reduction in the number of trials when one discards the trials that give outcome 1 on the ancilla qubit.§ STATE PREPARATION We prepare single photons in polarization-spatial hyperentangled states for entanglement-assisted single-probe approach <cit.>. The source consists of a β-barium-borate (BBO) nonlinear crystal pumped by a CW diode laser, and polarization-degenerate photon pairs at 801.6nm are generated by a type-I spontaneous parametric down-conversion (SPDC) process. The photon pairs have a coherence length of L_coh=214.2μm and spectral bandwidth Δλ=3nm.Upon detection of a trigger photon, the signal photon is heralded in the measurement setup. This trigger-signal photon pair is registered by a coincidence count at two single-photon APDs with a Δ t=3ns time window. Total coincidence counts are about 20,000 over a collection time of 10s. The probe is encoded in the horizontal |H⟩ and vertical |V⟩ polarizations of the heralded single photons.After passing through a PBS followed by a HWP and a QWP, the single photons are prepared in an arbitrary single-qubit state. The longitudinal spatial modes |U⟩ and |D⟩ represent the basis states of the ancilla. A birefringent calcite BD whose optical axis is cut so that horizontally polarized light is directly transmitted and vertical light undergoes a longitudinal displacement into a neighboring mode, acts as an effective controlled-NOT gate on the polarizations and the spatial modes and prepare the initial state into a polarization-spatial hyperentangled state α|HU⟩+β|VD⟩ (|α|^2+|β|^2=1 and α,β≠ 0).Whereas, for entanglement-assisted two-probe approach, polarization-entangled photon pairs are used to prepare the four-qubit hyperentangled state. Similarly, entangled photons in (|HH⟩+|VV⟩)/√(2) are also generated via type-I SPDC. Two β-BBO crystals and a following titled HWP (H_C) placed right after two joint α-BBO crystals are used to compensate the walk-off between photons with horizontal and vertical polarizations. Each photon passes through a BD and then a four-qubit polarization-spatial hyperentangled state (|HUHU⟩+|VDVD⟩)/√(2) is generated. Total coincidence counts are about 2,000 over a collection time of 10s.§ ACCURACY OF THE NOISY CHANNEL SIMULATION To verify accuracy of the noisy channel simulation, we reconstruct the process matrices of the channels via two-qubit QPT <cit.>. The action of a generic channel operating on a probe qubit isℰ(ρ)=∑_n,m,n',m'=0^3 χ_nmn'm'(Ξ_n⊗Ξ_m)ρ(Ξ_n'⊗Ξ_m'),where χ_nmn'm' completely characterizes the process. To determine ℰ we first choose some fixed states {ρ}, which form a basis for the set of operators acting on the state space of the probe+ancilla system. Each state is then subject to the process ℰ⊗, and quantum state tomography is used to determine the output state (ℰ⊗) ρ.A total of sixteen initial states ρ_l, l=1,…,16, and sixteen measurements on a two-qubit state of the probe+ancilla system are needed. These states are generated by PBS, BD and WPs. The HWP (H_S1),and QWP (Q_S1) are used to control the ratio and relative phase between the photons in the upper and lower modes, respectively, whereas H_S2 is used to control the ratio between the photons with different polarizations and Q_S2 is for the relative phase. Measurements are performed in the bases{|H⟩,|V⟩,|H⟩-i|V⟩/√(2), |H⟩+|V⟩/√(2)} ⊗{|U⟩,|D⟩,|U⟩-i|D⟩/√(2), |U⟩+|D⟩/√(2)}. After reconstructing the process matrices, we use process fidelity in Eq. (<ref>) to characterize the experimental realization of the noisy channels.§ PROJECTIVE MEASUREMENTS FOR REALIZING PHASE ESTIMATION For entanglement-assisted single-probe approach, the optimal measurement strategy around ϕ∼0 is projecting the output state into four basis states:{1/√(2)(|HU⟩±i|VD⟩),|HD⟩,|VU⟩},respectively. The projective measurements can be realized via a BD, a QWP, several HWPs and a PBS. A sandwich-type setup, i.e., HWP(at 45^∘)-BD-HWP(at 45^∘) separate the photons in the states |VU⟩ and |HD⟩ into the uppermost and lowest modes, and combine the photons in the states |HU⟩ and |VD⟩ into the middle mode. In the middle mode, a QWH at 0^∘ following by a HWP at 22.5^∘ applies a rotation on the polarization states, i.e.,1/√(2)(|H⟩-i|V⟩)⟶|H⟩, 1/√(2)(|H⟩+i|V⟩)⟶|V⟩.Finally the PBS projects the photons in the middle mode into two basis states (|HU⟩±i|VD⟩)/√(2). Coincidences between the outputs and the trigger are detected by APDs. The outcome probabilities of projecting the state in the basis {(|HU⟩±i|VD⟩)/√(2),|HD⟩,|VU⟩} depend on the coincidences between two of APDs (D_0, D_R), (D_0, D_L), (D_0, D_H) and (D_0, D_V), respectively.For entanglement-assisted two-probe approach, the optimal measurement strategy around ϕ∼0 is projecting the output state into sixteen basis states:{ 1/√(2)(|HUHU⟩±i|VDVD⟩),|HUHD⟩,|HUVU⟩,|HUVD⟩,|HDHU⟩,|HDHD⟩,|HDVU⟩,|HDVD⟩,|VUHU⟩,|VUHD⟩,|VUVU⟩,|VUVD⟩,|VDHU⟩,|VDHD⟩,|VDVU⟩},respectively. Similar to the entanglement-assisted single-probe approach, the projective measurements here are realized via BDs, WPs, NBSs, and a PBS. We use a multi-channel coincidence counting system that records all possible combinations of two-photon detection events occurring coincidentally across 12 APDs (D_1,…,D_12). The outcome probabilities of projecting the state in the bases depends on the combinations of coincidences between pair of APDs (D_1,…,D_12). The corresponding relation is shown in Table S1. | http://arxiv.org/abs/1707.08790v2 | {
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Gaussian Processes for Individualized Continuous Treatment Rule Estimation Pavel Shvechikovand Evgeniy Riabenko The reported study was funded by RFBR, according to the research project No. 16-07-01192 А. Department of Computer Science, Higher School of Economics December 30, 2023 =====================================================================================================================================================================================================empty§ INTRODUCTION§.§ Screening mammographyBreast cancer is the most common cancer in women and it is the main cause of death from cancer among women in the world <cit.>. Screening mammography was shown to reduce breast cancer mortality by 38–48% among participants <cit.>. In the EU 25 of the 28 member states are planning, piloting or implementing screening programs to diagnose and treat breast cancer in an early stage <cit.>. During a standard mammographic screening examination, X-ray images are captured from 2 angles of each breast. These images are inspected for malignant lesions by one or two experienced radiologists. Suspicious cases are called back for further diagnostic evaluation. Screening mammograms are evaluated by human readers. The reading process is monotonous, tiring, lengthy, costly and most importantly prone to errors.Multiple studies have shown that 20-30% of the diagnosed cancers could be found retrospectively on the previous negative screening exam by blinded reviewers <cit.>. The problem of missed cancers still persists despite modern full field digital mammography (FFDM) <cit.>.The sensitivityand specificity of screeningmammography is reported to be between 77-87% and 89-97% respectively. These metrics describe the average performance of readers, and there is substantial variance in the performance of individual physicians, with reported false positive rates between 1-29%, and sensitivities between 29-97% <cit.>.Double reading was found to improve the performance of mammographic evaluation and it had been implemented in many countries <cit.>. Multiple reading can further improve diagnostic performance up to more than 10 readers, proving that there is room for improvement in mammogram evaluation beyond double reading <cit.>. §.§ Computer-aided detection in mammographic screeningComputer-aided detection (CAD) solutions were developed to help radiologists in reading mammograms. These programs usually analyze a mammogram and mark the suspicious regions, which should be reviewed by the radiologist <cit.>. The technology was approved by FDA and had spread quickly.By 2008, in the US, 74% of all screening mammograms in the Medicare population were interpreted with CAD, the cost of CAD usage is over $400 million a year. <cit.>. The benefits of using CAD are controversial. Initially several studies have shown promising results with CAD <cit.>. A large clinical trial in the United Kingdom has shown that single reading with CAD assistance has similar performance to double reading. <cit.> However, in the last decade multiple studies concluded that currently used CAD technologies do not improve the performance of radiologists in everyday practice in the United States. <cit.>. These controversial results indicate that CAD systems need to be improved before radiologists can ultimately benefit from using the technology in everyday practice. Currently used CAD approaches are based on describing the X-ray image with meticulously designed hand crafted features, and machine learning for classification on top of these features <cit.> . In the field of computer vision, since 2012, deep convolutional neural networks (CNN) have significantly outperformed these traditional methods<cit.>. Deep CNN-s have reached or even surpassed human performance in image classification and object detection. <cit.>. These models have tremendous potential in medical image analysis.Several studies have attempted to apply Deep Learning to analyze mammograms <cit.>, but the problem is still far from being solved. §.§ The Digital Mammography DREAM Challenge The Digital Mammography DREAM Challenge (DM challenge) <cit.> asked participants to write algorithms which can predict whether a breast in a screening mammography exam will be diagnosed with cancer. The dataset consisted of 86000 exams, with no pixel level annotation, only a binary label indicating if breast cancer was diagnosed in the next 12 months after the exam. Each side of the breasts were treated as a separate case that we will call breast-level prediction in this paper. The participants had to upload their programs to a secure cloud platform, and they were not able to download or view the images, neither interact with their program during training or testing. The DM challenge provided an excellent opportunity to compare the performance of competing methods in a controlled and fair way instead of self-reported evaluations on different or proprietary datasets. § MATERIAL AND METHODS§.§ DataMammograms with pixel level annotations were needed to train a lesion detector and test the classification and localization performance. We have trained the model on the public Digital Database for Screening Mammography(DDSM) <cit.> and a dataset from the Semmelweis University in Budapest, and tested it on the public INbreast <cit.> dataset.The images used for training contain either histologically proven cancers or benign lesions which were recalled for further examinations, but later turned out to be nonmalignant. We expect that training with both kinds of lesions helps our model to find more lesions of interest, and differentiate between malignant and benign examples. The DDSMdataset contains 2620 digitized film-screen screening mammography exams, with pixel-level ground truth annotation of lesions.Cancerous lesions have histological proof. We have only used the DDSM database for training our model and not evaluating it. The quality of digitized film-screen mammograms is not as good as full field digital mammograms therefore, evaluation on these cases is not relevant. We have converted the lossless jpeg images to png format, mapped the pixel values to optical density using the calibration functions from the DDSM website, and rescaled the pixel values to the 0-255 range. The dataset from the Department of Radiology at the Semmelweis University in Budapest, Hungary contains 847 FFDM images of 214 exams from 174 patients, recorded with a Hologic LORAD Selenia device. Institutional board approval was obtained for the dataset. This dataset was not available for the full period of the DM challenge, it is used only for improvement in the second stage of the DM challenge, after the pixel level annotation by the authors. The INbreast dataset contains 115 FFDM cases with pixel-level ground truth annotations, and histological proof for cancers <cit.>. We have adapted the INbreast pixel level annotations to suit our testing scenario.We have ignored all benign annotations, and converted the malignant lesion annotations to bounding boxes. We have excluded 8 exams which had other findings, artifacts, previous surgeries or ambiguous pathological outcome. The images have low contrast therefore, we have adjusted the window of the pixel levels. The pixel values were clipped to be minimum 500 pixel lower and maximum 800 pixels higher than the mode of the pixel value distribution (excluding the background) and were rescaled to the 0-255 range. §.§ Data Availability The DDSM dataset is available online at <http://marathon.csee.usf.edu/Mammography/Database.html>.The INBreast dataset can be requested online at <http://medicalresearch.inescporto.pt/breastresearch/index.php/Get_INbreast_Database>.For the dataset from the Semmelweis university (http://semmelweis.hu/radiologia/) restrictions apply to the availability of these data, which were used under special licence from Semmelweis University, and so are not publicly available. Data are however available from the authors upon reasonable request and permission of the Semmelweis University. §.§ MethodsThe heart of our model is a state of the art object detection framework, Faster R-CNN<cit.>. Faster R-CNN is based on a convolutional neural network with additional components for detecting, localizing and classifying objects in an image.Faster R-CNN has a branch of convolutional layers, called Region Proposal Network (RPN), after the last convolutional layer of the original network, which is trained to detect and localize objects on the image, regardless of the class of the object.There are default detection boxes with different sizes and aspect ratios in order to find objects with varying sizes and shapes. The highest scoring default boxes are called the region proposals for the other branch of the network.The other branch of the neural network evaluates the signal coming from each proposed region of the last convolutional layer, resampled to a fix size.Both branches try to solve a classification task to detect the presence of objects and a bounding-box regression task in order to refine the boundaries of the object present in the region.From the detected overlapping objects, the best predictions are selected using non-maximum suppression. Further details about Faster R-CNN can be found in the original article <cit.>. An outline of the model can be seen in Fig. <ref>.The base CNN used in our model was a VGG16 network, which is 16 layer deep CNN <cit.>.The final layer can detect 2 kinds of objects in the images, benign lesions or malignant lesions. The model outputs a bounding box for each detected lesion, and a score, which reflects the confidence in the class of the lesion.To describe an image with one score, we take the maximum of the scores of all malignant lesions detected in the image. For multiple images of the same breast, we take the average of the scores of individual images.For the DM challenge we have trained 2 models using shuffled training datasets. When ensembling these models, the score of an image was the average score of the individual models. This approach has been motivated by a previous study on independent human readers, and it has proven reasonably effective, while simple and flexible <cit.>.We have used the framework developed by the authors of Faster R-CNN <cit.>, which was built in the Caffe framework for deep learning <cit.>. During training we optimized the object detection part and the classifier part of the model in the same time, this is called the joint optimization <cit.>. We used backpropagation and stochastic gradient descent with weight decay. The initial model used for training has been pretrained on 1.2 million images from the ImageNet dataset <cit.>. We have found that higher resolution yields better results, therefore the mammograms were rescaled isotropically to longer side smaller than 2100 pixels or shorter side smaller than 1700 pixels. This resolution is close to the maximum size which fits in the memory of the graphics card used. The aspect ratio was selected to fit the regular aspect ratio of Hologic images.We applied vertical and horizontal flipping to augment the training dataset.Mammograms contain fewer object than ordinary images, therefore negative proposals dominate minibatches. The Intersection over Union (IoU) threshold for foreground objects in the region proposal network was relaxed from 0.7 to 0.5 to allow more positive samples in each minibatch. Relaxation of positive examples is also supported by the fact that lesions on a mammogram have much less well defined boundaries than a car or a dog on a tradiotional image.The IoU threshold of the final non maximum suppression (nms) was set to 0.1, because mammograms represent a smaller and compressed 3D space compared to ordinary images, therefore overlapping detections are expected to happen less often than in usual object detection.The model was trained for 40k iterations, this number was previously found to be close to optimal by testing multiple models on the DM challenge training data. The model was trained and evaluated on an Nvidia GTX 1080Ti graphics card.Our final entry in the DM challenge was an ensemble of 2 models.§ RESULTS §.§ Cancer classificationWe also evaluated the model's performance on the public INbreast dataset with the receiver operating characteristics (ROC) metric, Fig. <ref>. The INbreast dataset has many exams with only one laterality, therefore we have evaluated predictions for each breast. The system achieved AUC =0.95, (95 percentile interval: 0.91 to 0.98, estimated from 10000 bootstrap samples).To our best knowledge this is the highest AUC score reported on the INbreast dataset with a fully automated system based on a single model. §.§ FROC analysisIn order to test the model's ability to detect and accurately localize malignant lesions, we evaluated the predictions on the INbreast dataset using the Free-response ROC (FROC) curve <cit.>. The FROC curve shows the sensitivity (fraction of correctly localized lesions) as a function of the number of false positive marks put on an image Fig. <ref>. A detection was considered correct if the center of the proposed lesion fell inside a ground truth box. The same criteria is generally used when measuring the performance of currently used CAD products <cit.>. The DM challenge dataset has no lesion annotation, therefore we can not use it for an FROC analysis. §.§ ExamplesTo demonstrate the characteristics and errors of the detector, we have created a collection of correctly classified, false positive and missed malignant lesions of the INbreast dataset, see in Fig. <ref>. The score threshold for the examples was defined at sensitivity =0.9 and 0.3 false positive marks per image.After inspecting the false positive detections, we have found that most were benign masses or calcifications. Some of these benign lesions were biopsy tested according to the case descriptions of the INbreast dataset.While 10% of the ground truth malignant lesions were missed at this detection threshold, these were not completely overlooked by the model. With a score threshold which corresponds to 3 false positive marks per image, all the lesions were correctly detected (see Fig. <ref>). Note that the exact number of false positive and true positive detections slightly varies with with different samples of images, indicated by the area in Fig. <ref>.§ DISCUSSIONWe have proposed a Faster R-CNN based CAD approach, which achieved the 2nd position in the Digital Mammography DREAM Challenge with an AUC = 0.85 score on the final validation dataset. The competition results have proved that the method described in this article is one of the best approach for cancer classification in mammograms.Our method was the only one of the top contenders in the DM challenge which is based on the detection of malignant lesions, and whole image classification is just a trivial step from the detection task. We think that a lesion detector is clinically much more useful than a simple classifier. A classifier only gives a single score for a case or breast, but it is not able to locate the cancer which is essential for further diagnostic tests or treatment. We have evaluated the model on the publicly available INbreast dataset. The system is able to detect 90% of the malignant lesions in the INbreast dataset with only 0.3 false positive marks per image. It also sets the state of the art performance in cancer classification on the publicly available INbreast dataset. The system uses the mammograms as the only input without any annotation or user interaction. An object detection framework developed to detect objects in ordinary images shows excellent performance. This result indicates that lesion detection on mammograms is not very different from the regular object detection task.Therefore the expensive, traditional CAD solutions, which have controversial efficiency, could be replaced with the recently developed, potentially more accurate, deep learning based, open source object detection methods in the near future. The FROC analysis results suggest that the proposed model could be applied as a perception enhancer tool, which could help radiologists to detect more cancers. A limitation of our study comes from the small size of the publicly available pixel-level annotated dataset. While the classification performance of the model has been evaluated on a large screening dataset, the detectionperformance could only be evaluated on the small INbreast dataset. § AUTHOR CONTRIBUTIONS STATEMENT D.R., I.C. and P.P. contributed to the conception and design of the study. A.H. and Z.U. contributed to the acquisition, analysis and interpretation of data. All authors reviewed the manuscript.§ ADDITIONAL INFORMATION §.§ Competing financial interestsThe authors declare no competing financial interests.§ ACKNOWLEDGEMENTS This work was supported by the Novo Nordisk Foundation Interdisciplinary Synergy Programme [Grant NNF15OC0016584]; andNational Research, Development and Innovation Fund of Hungary, [Project no. FIEK_16-1-2016-0005]. The funding sources cover computational and publishing costs. | http://arxiv.org/abs/1707.08401v3 | {
"authors": [
"Dezső Ribli",
"Anna Horváth",
"Zsuzsa Unger",
"Péter Pollner",
"István Csabai"
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"title": "Detecting and classifying lesions in mammograms with Deep Learning"
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xx xx 2017 Simitev & BusseBaroclinically-driven flows and dynamosBaroclinically-driven flows and dynamo actionin rotating spherical fluid shells RADOSTIN D. SIMITEV†^∗^∗Corresponding author. Email: [email protected] and FRIEDRICH H. BUSSE †School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8SQ, UKInstitute of Physics, University of Bayreuth, 95440 Bayreuth,GermanyDecember 30, 2023December 30, 2023 ======================================================================================================================================================================================================================================================================================================The dynamics of stably stratified stellar radiative zones is of considerable interest due to the availability of increasingly detailed observations of Solar and stellar interiors. This article reports the first non-axisymmetric and time-dependent simulations of flows of anelastic fluids driven by baroclinic torques in stably stratified rotating spherical shells – a system serving as an elemental model of a stellar radiative zone. With increasing baroclinicity a sequence of bifurcations from simpler to more complex flows is found in which some of the available symmetries of the problem are broken subsequently. The poloidal component of the flow grows relative to the dominanttoroidalcomponent with increasing baroclinicity. The possibility of magnetic field generation thus arises and this paper proceeds to provide some indications for self-sustained dynamo action in baroclinically-driven flows. We speculate that magnetic fields in stably stratified stellar interiors are thus not necessarily of fossil origin as it is often assumed.Stably-stratified stellar interiors, baroclinic flows, dynamo action. § INTRODUCTION Stellar radiation zones are typically supposed to be motionless in standard models of stellar structure and evolution <cit.>, but this assumption is poorly justified <cit.>. Indeed, increasingly detailed evidence is emerging, e.g. from helio- and asteroseismology, of dynamical processes such as differential rotation, meridional circulation, turbulence, and internal waves in radiative zones<cit.>. Thesetransport processes are important in mixing of angular momentum, evolution of chemical abundances, and magnetic field sustenance <cit.>. The problem of fluid motions in radiative zones is thus of wide significance.Historically, this problem hasbeen of much interest ever since it became apparent that rotating stellar interiors cannot be in a static equilibrium <cit.>, a statement known as von Zeipel's paradox.Two different forms of flow have been hypothesized, namely (a) steady low-amplitude meridional circulations<cit.>, and (b) particular forms of strong differential rotation with minimal meridional circulations <cit.>. It was demonstrated that hypothesis (a) is not an acceptable solution to von Zeipel's paradox <cit.>.Hypothesis (b) has since gained support with self-consistent quasi-one-dimensional solutions first derived by <cit.> and their baroclinic instabilities studied <cit.>.In recent years a number of numerical studies of axisymmetric and steady baroclinically-driven flows of finite amplitudes in rotating, stably stratified spherical shells have been published<cit.>. In <cit.> the dynamics of the radiative zone of the Sundriven by the differential rotation of theconvective zone is investigated, but the flow is essentially driven by the boundary conditions and baroclinicity is not the main driving force.The cited papers by Rieutord and coworkersoffer perhaps the most detailed studies of the problem to date. However, the analysis is strongly restricted in two respects: first, two-dimensional and steady axisymmetric flows are studied, and second, the Boussinesq approximation is used which does not account for the strong density variations typical for stably stratified regions of stars.In this paper we present a model of baroclinic flows in rotating spherical fluid shells. The model is based on the anelastic approximation <cit.> of the fully-compressible fluid equations which is widely adopted for numerical simulation of convection in Solar and stellar interiors <cit.>. While this approximation is strictly only valid for a system close to an adiabatic state it, nevertheless, relaxes the assumptions of the Boussinesq approximation that has been used previously.In the basic state the density is stably stratified in the radial direction and axisymmetric shear driven motions are realized. Starting with this basic axisymmetric baroclinic state we explore the onset of non-axisymmetric and time dependent states and investigate their nonlinear properties. Further, we consider the possibility of magnetic field generation by these flows. Although a prime motivation for our study is to understand the motions in stellar radiative zones, we have not strived to adjust the values of the dimensionless parameters in our model to physically realistic ones. This is not possible in any case since the Reynolds number of flows in stars is huge and turbulent mixing occurs. Only the large scale flows can be simulated numerically while the influence of turbulence is taken into account through the use of eddy diffusivities in the equations of motion. For details of this approach we refer to the work of <cit.> and references therein. Using this approach we shall overcome the restrictions of axisymmetric steady flows which are likely to be unstable.As we shall show non-axisymmetric and time dependent flows must be expected instead with properties that could give rise to dynamo action. § MATHEMATICAL MODELWe consider a perfect gas confined to a spherical shell rotating with a fixed angular velocity $̨ and with a positive entropy contrastΔSimposed between its outer and inner surfaces at radiir_oandr_i, respectively. We assume a gravity field proportional to g/r^2. To justify this choice consider the Sun, the star with the best-known physical properties. The Solar density drops from 150 g/cm^3 at the centre to 20 g/cm^3 at the core-radiative zone boundary (at 0.25 R_⊙) to only 0.2 at the tachocline (at 0.7 R_⊙). A crude piecewise linear interpolation shows that most of the mass is concentrated within the core. In this setting a hydrostatic polytropic reference stateexists with profiles of density= ρ_cζ^n, temperature=T_cζand pressure= P_c ζ^n+1, whereζ= c_0+c_1/randc_0=(2ζ_o-η-1)/(1-η),c_1=(1+η)(1-ζ_o)/(1-η)^2,ζ_o=(η+1)/(ηexp(/n)+1), see <cit.>. The parametersρ_c,P_candT_care reference values of density, pressure and temperature at mid-shell. The gas polytropic indexn, the density scale heightN_ρand the shell radius ratioηare defined further below.Following <cit.> we neglect the distortion of the isopycnals caused by the centrifugal force. The governing anelastic equations of continuity, momentum and energy (entropy) take the form ∇·=̆0, _t +̆ (∇×)̆×=̆ -∇ -τ(×̨)̆+/S/r^2 + F̌_ν = -(S+S) ×̨(×̊)̨, _t S + ·̆∇(S+S)= 1/∇·∇ S + c_1 /Q_ν,whereSis the deviation from the background entropy profileS=(ζ(r)^-n-ζ_o^-n)/(ζ_o^-n-ζ_i^-n)and$̆ is the velocity vector, ∇ includes all terms that can be written as gradients,and =̊r is the position vector with respect to the center of the sphere <cit.>.The viscous force F̌_ν = (ρ_c/)∇·Ŝ̌ and the viscous heating Q_ν=Ŝ̌ :ě are defined in terms of the deviatoric stress tensor Ŝ_ij=2(e_ij-e_kkδ_ij/3) with e_ij=(_i u_j +_j u_i)/2, where double-dots (:) denote a Frobenius inner product, and ν is a constant viscosity. The governing equations (<ref>) have been non-dimensionalized with the shell thickness d=r_o-r_i as unit of length, d^2/ν as unit of time, and Δ S, ρ_c and T_c as units of entropy, density and temperature, respectively.The system is then characterized by seven dimensionless parameters: the radius ratio η=r_i/r_o,the polytropic index of the gas n, the density scale number N_ρ=ln((r_i)/(r_o)), the Prandtl number =ν/κ, the Rayleigh number =-gd^3Δ S/(νκ c_p),the baroclinicity parameter=^2 d^4 Δ S/(ν^2 c_p), and the Coriolis number τ = 2 d^2/ν, where κ is a constant entropy diffusivity and c_p is the specific heat at constant pressure. Note, that the Rayleigh number assumes negative values in the present problem since the basic entropy gradient is reversed with respect to the case of buoyancy driven convection. The poloidal-toroidal decompositionu⃗ = ∇× ( ∇× r) + ∇× r^2is used to enforce the solenoidality of the mass flux $̆. This has the further advantages that the pressure gradient iseliminated and scalar equations for the poloidal and the toroidal scalar fields,vandw, are obtained by taking·∇×∇×and·∇×of equation (<ref>). Except for the term with the baroclinicity parameterthe resulting equations are identical to those described by <cit.>. Assuming that entropy fluctuations are damped by convection in the region above r=r_o we choose the boundary conditionS =0atr ={[ r_i≡η/(1-η) ,; r_o≡1/(1-η) , ].while the inner and the outer boundaries of the shell are assumed stress-free and impenetrable for the flow = 0, _r^2= '/ r_r (r), _r (r )= '/atr ={[ r_i ,; r_o . ].§ NUMERICAL SOLUTIONFor the numerical solution of problem (<ref>–<ref>)a pseudo-spectral method described by <cit.> was employed. A code developed and used by us for a number of years <cit.> and extensively benchmarked for accuracy <cit.>was adapted.Adequate numerical resolution for the runs has been chosen as described in <cit.>. To analyse the properties of the solutions we decompose the kinetic energy density into poloidal and toroidal components and further into mean (axisymmetric) and fluctuating (nonaxisymmetric) components and into equatorially-symmetric and equatorially-antisymmetric components, E_p = E_p^s +E_p^a=⟨(∇× ( ∇×r⃗) )^2/(2)⟩,E_t = E_t^s +E_t^a= ⟨(∇ r ×r⃗)^2 /(2)⟩,E_p = E_p^s +E_p^a = ⟨(∇× ( ∇×r⃗))^2/(2) ⟩,E_t = E_t^s +E_t^a = ⟨( ∇ r ×r⃗)^2 /(2) ⟩,whereangular brackets⟨ ⟩denote averagesover the volume of the spherical shell. § PARAMETER VALUES AND INITIAL CONDITIONSIn the simulations presented here fixed values are used for all governing parameters except for the baroclinicitynamelyη=0.3,n=2,N_ρ=2,=0.1,τ=300and=-5 ×10^4. The value for the shell thickness represents the presence of a stellar core geometrically similar to that of the Sun.The values fornandN_ρare not far removed from estimates n=1.5 and = 4.6052 for the solar radiative zone.The strength of the baroclinic forcing, measured by , is limited from above so that< (1-η)^3||/.This restriction guarantees that the apparent gravity does not point outward such that the model excludes standard thermal convection instabilities. Unresolved subgrid-scales in convective envelopes are typically modelled by the assumption of approximately equal turbulent eddy diffusivities and the choice of=1is often made in the literature <cit.>. However, in the presence of the minute Prandtl number values based on molecular diffusivities the eddy diffusivities are not likely to yield an effective Prandtl number of the order unity.Furthermore, in a stably stratified system turbulence is expected to be anisotropic <cit.>. With this in mind we have chosen=0.1in our study.The valueτ= 300offers a good compromise in which the effect of rotation is strong enough to govern the dynamics of the system, but not too strong to cause a significant increase of the computational expenses; similar values are used e.g. by <cit.> and in cases F1–4 of <cit.>. A negative=-5 ×10^4is assumed to model a convectivelystable situation. Initial conditions of no fluid motion are used at vanishingly small values of, while at finite values ofthe closest equilibrated neighbouring case is used as initial condition to help convergence and reduce transients. To ensure that transient effects are eliminated from the sequence presented below, all solutions have been continued for at least 15 time units. § BAROCLINIC FLOW INSTABILITIES The baroclinically-driven problem is invariant under a group of symmetry operations including rotations about the polar axis i.e. invariance with respect to the coordinate transformationφ→φ+α, reflections in the equatorial planeθ→π-θ, and translations in timet →t+a, see <cit.>. As baroclinicityis increased the available symmetries of the solution are broken resulting in a sequence of states ranging from simpler and more symmetric flow patterns to more complex flow patterns of lesser symmetry as illustrated in figure<ref>.In this respect the system resembles Rayleigh-Benard convection <cit.>. The sequence starts with the basic axisymmetric, equatorially symmetric and time-independent state with a dominant wave number k=1 in the radial direction labelled A in figures <ref> and<ref>. An instability occurs at about =8.1 × 10^4 leading to a stateB characterized by a dominant azimuthal wavenumberm=2in the expansion of the solution in spherical harmonicsY_l^m, i.e. while the full axisymmetry is broken, a symmetry holds with respect to the transformationφ→φ+π. Simultaneously, the symmetry about the equatorial plane is also broken in state B.At =13.25 × 10^4 a further transition to a pattern labelled D occurs. While state D continues to have a m=2 azimuthal symmetry, the symmetry about the equatorial plane is now restored. In addition,a dominantradial wave number k=2 develops as is evident, for instance, from the concentric two-roll meridional circulation in state D. Remarkably, both states B and D coexist with a steady pattern C that can be found for the range of values > 12 × 10^4. State C is axisymmetric, equatorially symmetric and time-independent but differs from state A in that it keeps a dominant radial wave number k=2. Which one of the coexisting branches will be found in a given numerical simulation is determined by the specific initial conditions used.While all patterns in the sequence presented here remain time-independent in their respective drifting frames of reference, we expect that time-dependent solutions will befound for lower values of the Prandtl number since the Reynolds number is increasing at a fixed baroclinicityand as a result breaking of additional symmetries is likely to occur. The bifurcations and their abrupt nature are also evident in figure <ref> where the time-averaged kinetic energy densities are plotted with increasing baroclinicity.At=0all energies vanish corresponding to the state of rigid body rotationwith vanishing velocity field. In the range <6 × 10^4 velocities grow linearly withand the corresponding growth of kinetic energies is well described by the empirical relations= 5.647×10^-8 ^2,= 2.377× 10^-10 ^2,indicating the dominance of toroidal motions as argued in earlier theoretical analyses <cit.>. While the particular numerical factors in these expressions depend on the parameter values used, the quadratic form of this dependence is expected to be generic. The abrupt jumps in energies at intermediate values ofcorrespond to the breaking of spatial symmetries. In state A, for instance, only the axially-symmetric poloidal (i.e. meridional circulation) and toroidal (i.e. differential rotation) kinetic energies have non negligible values. In state B the non-axisymmetric components emerge with the equatorially asymmetric ones being larger than the corresponding equatorially symmetric ones. In state D equatorially asymmetric components become negligible againand in state C all non-axisymmetric energy components decay.In the regions of hysteretic bistability12×10^4 < < 13.25×10^4and> 13.25×10^4either of states Bor C orstates D or C may be realized, respectively,depending on initial conditions.§ BAROCLINIC DYNAMOSApproximately 10% of intermediate-mass and massive stars which are mainly radiative are known to be magnetic <cit.>. The most popular hypothesis is that these are fossil fields remnants of an early phase of the stellar evolution <cit.>. The possibility of dynamos generated in stably stratified stellar radiation regions has received only limited support in the literature <cit.>, because it is well known that dynamo action does not exist in a spherical system in the absence of a radial component of motion <cit.>. The latter is, indeed, rather weak in the basic state A of low poloidal kinetic energy as discussed above. However, with increasing baroclinicity, thegrowing radial component of the velocity field strengthens the probability of dynamo action.To investigate this possibility the Lorentz force(1/) (∇×)×has been added to the right-hand side of equation (<ref>) and the equation of magnetic induction_t= ∇×(×̆)+^-1∇^2 ,has been added to equations (<ref>) to attain the full magnetohydrodynamic form as used in the works of <cit.> and <cit.>. Here,is the magnetic flux density and =ν/λ is the magnetic Prandtl number with λ being the magnetic diffusivity.After adding vanishingly small random magnetic fields as initial conditions to equilibrated purely hydrodynamic solutions, a number of solutions with growingmagnetic fields of dipolar character have been obtained. As an example, figures <ref> and <ref> demonstrate the magnetic field generated at =17 × 10^4 and for the magnetic Prandtl number =16 which has been started from an equilibrated neighbouring case to help convergence and reduce transients. The dynamo solution quickly attains a stationary state that we have followed for nearly 2 magnetic diffusion time units, t/, as shown in figure <ref>. The ratio of poloidal to toroidal magnetic energy is E_p^magn/E_t^magn=0.059 and the ratio of magnetic to kinetic total energy is E^magn/E^kin=0.127.Furthermore, the energy density of the magnetic field E^magn is comparable to the kinetic energy density of the poloidal component of the velocity field, E_p^kin. The magnetic field has a negligible quadrupolar component and a large-scale dipolar topology shown in figure <ref> that does not change in time, resembling in this respect the surface fields of Ap-Bp stars <cit.>. The surface structure of the magnetic field is characterised by two prominent patches of opposite polarity situated at approximately 45^∘ in latitude. The azimuthally-averaged toroidal field shows a pair of hook-shaped toroidal flux tubes largely filling the outer layer of the shell at lower latitudes as well as a pair of toroidal flux tubes in the polar regions and parallel to the rotation axis. While a large scale dipolar component emerges outside of the spherical shell, the azimuthally-averaged poloidal field also shows apair of octupole poloidal flux tubes confined to the interior of the shell. Azimuthally-averaged kinetic and magnetic helicities are plotted in the last column of figure <ref>. These latter quantities are important in modelling the electromotive force in mean-field dynamo theories, and in estimating the topological linkage of magnetic field lines, respectively.§ CONCLUSIONIn summary, we have described the bifurcations leading to non-axisymmetric and non-equatorially symmetric flow states with increasing baroclinicity. The observed sequence of baroclinic instabilities is rather different from the typical sequence of convective instabilities familiar from the literature <cit.>. Furthermore, we have demonstrated the possibility of dynamo action in baroclinically-driven flows.We remark that the details of the bifurcation sequence depend on the choice of parameter values. But preliminary additional computations indicate that the general picture described here persists when parameter values are changed by up to a factor of 10.While the dynamo solution discussed above is obtained for an unphysically large value of the magnetic Prandtl number, we argue that magnetic fields in stably stratified stellar interiors may not necessarily be of fossil origin as often assumed and that dynamo action may possibly occur in the radiative zones of rotating stars. Even in the solar interior below the convection zone magnetic fields may be generated. 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"authors": [
"Radostin D. Simitev",
"Friedrich H. Busse"
],
"categories": [
"physics.flu-dyn",
"astro-ph.SR"
],
"primary_category": "physics.flu-dyn",
"published": "20170727125510",
"title": "Baroclinically-driven flows and dynamo action in rotating spherical fluid shells"
} |
Polarization Transfer Observables in Elastic Electron-Proton Scattering at 𝐐^2= 2.5, 5.2, 6.8 and 8.5 GeV^2 L. Zhu December 30, 2023 ===========================================================================================================§ INTRODUCTION Dark matter (DM) direct detection experiments are conventionally interpreted in terms of effective operators, which parametrize the details of the underlying interactions of the DM particle in terms of an effective suppression scale Λ <cit.>. This is a good approximation as long as the particle mediating the interaction is heavy compared to the typical momentum transfer in direct detection experiments, which is of order 1–100 MeV. Nevertheless, it is perfectly conceivable that the mediator mass is close to or below this scale, so that the effective operator description is no longer valid <cit.>. This scenario has for example been considered in the context of self-interacting DM <cit.>. In these models the exchange of light mediators induces large DM self-scattering rates <cit.>, which can potentially resolve the tension between the predictions of collisionless cold DM and observations on small astrophysical scales <cit.>.Current experimental limits e.g. from searches for rare decays <cit.> put strong constraints on the mediator coupling to Standard Model (SM) particles. Nevertheless, the smallness of the mediator mass leads to a huge enhancement of direct detection cross sections, so that an observation of DM scattering may be possible in spite of the small couplings. In fact, if the mediator has sizeable couplings to DM, direct detection experiments can probe regions of parameter space inaccessible to other low-energy searches. As the recoil spectrum depends sensitively on both the DM and mediator mass, given an observation of DM scattering via a light mediator with sufficient statistics at a direct detection experiment, it may be possible to reconstruct both masses.Cryogenic direct detection experiments, such as SuperCDMS <cit.>, CRESST <cit.> and EDELWEISS <cit.>, are particularly well-suited for this task due to their excellent energy resolution and low threshold <cit.>. In fact, a low energy threshold is even more important in the case of a light mediator than for a heavy one, because the recoil spectrum falls even more steeply and therefore the sensitivity can be considerably improved by lowering the threshold <cit.>. The excellent energy resolution, on the other hand, makes it possible to extract the maximum amount of information on the particle physics properties of DM from a successful discovery. In other words, cryogenic detectors are not only well-suited to explore models with light DM particles (see e.g. <cit.>), but also to probe light mediators.The projected progress for the low-threshold technology implies that parameter points that are currently consistent with all experimental constraints may predict up to thousands of events in near-future detectors. In this paper we study the amount of information that can be extracted from such a signal, taking into account background uncertainties, astrophysical uncertainties and degeneracies with other particle physics parameters. We demonstrate that cryogenic experiments can probe the mediator mass precisely in the regions of parameter space relevant for DM self-interactions, potentially enabling us to infer the behaviour of DM on astrophysical scales with laboratory experiments.Direct detection experiments in the context of self-interacting DM have been studied previously <cit.>, most notably in ref. <cit.>. Our work differs from these earlier studies in that we do not attempt to derive existing constraints but rather to explore the potential of future low-threshold detectors to infer the properties of self-interacting DM. For this purpose, we implement several present and future direct detection experiments in a realistic and efficient manner, in order to perform parameter reconstruction with a number of nuisance parameters. For similar studies in the context of effective operators see refs. <cit.>.This paper is structured as follows. In section <ref> we discuss the phenomenology of direct detection experiments in the presence of light mediators. We review current and proposed low-threshold experiments and calculate their sensitivity to long-range interactions in comparison to conventional direct detection experiments. Section <ref> focusses on the potential of low-threshold experiments to determine the particle physics parameters of the DM particle and its interactions. We discuss the impact of experimental, theoretical and astrophysical uncertainties, introduce suitable nuisance parameters to represent them and assess their impact on our results. Finally, in section <ref> we connect our results to the idea of self-interacting DM. Additional details are provided in appendices <ref> and <ref>.§ DIRECT DETECTION WITH LIGHT MEDIATORSWe consider a DM particle of mass m_DM scattering off nuclei via the exchange of a mediator with mass m_med. Throughout this paper we will focus on the case that the mediator has spin-independent couplings to both nucleons and DM. The differential event rate with respect to recoil energy E_R for DM scattering off a given target isotope T with mass m_T and mass fraction ξ_T is then given bydR_T/dE_R = ρ_0ξ_T/2 πm_DMg^2 F_T^2(E_R)/( 2 m_T E_R + m_med^2 )^2 η ( v_min (E_R))with ρ_0 = 0.3 GeV/cm^3 being the local DM density.As long as the assumption of spin-independent interactions holds, the functional form of the differential event rate does not depend on the spin of the DM particle or the mediator nor on whether or not the DM particle is its own anti-particle. The numerical pre-factors, however, may differ for these different scenarios. We assume that these pre-factors have been absorbed into the definition of the effective low-energy coupling g, i.e. we take g to be defined via eq. (<ref>). The precise definition of g in terms of the fundamental parameters of the specific models that we discuss will be provided below.In eq. (<ref>) the factor F_T^2(E_R) denotes the nuclear response function, which depends on the ratio of the mediator couplings to neutrons and protons, f_n / f_p. We parametrise this ratio via θ≡arctan f_n / f_p, such that models with f_n = 0 (such as a dark photon with kinetic mixing <cit.>) have θ = 0, whereas models with f_n = f_p (such as a light scalar mixing with the Higgs boson <cit.>) have θ = π / 4. In the limit of zero momentum transfer the nuclear response function is then given by F_T^2(0) = (Z_T cosθ + (A_T - Z_T) sinθ)^2, where Z_T and A_T denote the charge and mass number of the target isotope T. For non-zero momentum transfer, F_T^2(E_R) decreases due to a loss of coherence, which we parametrise by the standard Helm form factors <cit.>.The final factor in eq. (<ref>) denotes the velocity integral, which is given byη(v_min) = ∫_v_min^∞d^3v f(𝐯)/v ,where v_min = √(m_T E_R/2μ_T^2). Unless stated otherwise, we assume the velocity distribution f(𝐯) to be given by an isotropic Maxwell-Boltzmann distribution in the Galactic rest frame with v_0 = 220km/s, cut off at the Galactic escape velocity v_esc = 544km/s and transformed into the solar rest frame with v_obs =232km/s.For the purpose of this work, we will be most interested in the impact of the term ( 2 m_T E_R + m_med^2 )^2 = (q^2 + m_med^2)^2 in the denominator, where q denotes the momentum transfer in the scattering process. This momentum transfer is bounded by , wherein terms of the low-energy threshold E_th andwith . If q^max is small compared to m_med, we recover the limit of contact interactions conventionally considered in the effective operator approach. Conversely, if m_med is negligible compared to q^min, scattering with large momentum transfer is suppressed by an additional factor of 1/q^4, leading to very steeply falling recoil spectra. In the intermediate regime we may hope to infer the properties of the mediator from the detailed shape of the recoil spectra. These interesting mediator masses are typically in the MeV range. For example, for m_DM = 4 GeV a germanium experiment (m_T ≈ 68 GeV) with E_th = 100 eV has q^min≈ 3.7 MeV and q^max≈ 20.7 MeV.In real direct detection experiments, the detected recoil energy E_D may differ from the true physical recoil energy E_R due to the finite energy resolution of the detector. Assuming this resolution to be described by a Gaussian distribution with energy-dependent standard deviation σ(E_R), we can calculate the probability for a scattering event with energy E_R to be observed in the interval E_1 ≤ E_D≤ E_2:p(E_R, E_1, E_2) = 1/2[(E_2 - E_R/√(2)σ(E_R)) - (E_1 - E_R/√(2)σ(E_R))] ,wheredenotes the error function. Given the total exposure of an experiment κ(E_R), which may again depend on the recoil energy, we can then calculate the total number of events expected in the interval [E_1, E_2]:N(E_1, E_2) = ∑_T ∫ p_T(E_R, E_1, E_2)κ_T(E_R)dR_T/dE_RdE_R,where the sum is over all isotopes T in the target, weighted with appropriate factors ξ_T, and we allow both p and κ to depend on the isotope.The two most sensitive cryogenic direct detection experiments are CRESST-II <cit.> and CDMSlite <cit.>. In the near future CRESST-III <cit.> and SuperCDMS SNOLAB <cit.> plan to significantly improve sensitivity.[A proposal for a similar effort with optimized EDELWEISS-III detectors was very recently put forward in ref. <cit.>. The suggested approach is very similar to the one employed by the SuperCDMS collaboration and we expect qualitatively similar results.] We describe our implementation of these experiments in appendix <ref>. For comparison, we also consider bounds from Xenon1T <cit.>, as well as projections for the future sensitivity of LZ <cit.>. We note that all projections have similar time scales and correspond to the sensitivity that may be achievable within the next five to ten years.As a validation of our implementation, we show in figure <ref> the (projected) constraints on standard spin-independent interactions that we obtain from present and future direct detection experiments. For this purpose, we take m_med≫ q^max, set θ = π / 4 and g^2 = 2 π σ_p m_med^4 / μ_p^2, where μ_p ≡ m_DMm_p / (m_DM + m_p) and m_p denotes the proton mass. Our recalculation of existing constraints is in good agreement with the respective published exclusion limits. For SuperCDMS and CRESST-III we have chosen a rather conservative low-energy threshold of 100 eV in both cases, so that our projected exclusion limits are somewhat weaker than the official projections for small DM masses. Nevertheless, we find good agreement at larger DM masses, where the precise value of the threshold does not matter.Having validated our implementation, we now show the corresponding constraints on the effective coupling g for smaller mediator masses (figure <ref>). As expected, we observe that with decreasing mediator mass direct detection experiments become sensitive to smaller values of the effective coupling g, with particularly large enhancement factors found for low-threshold experiments. Another important observation is that for m_med≪ q^min the shape of the recoil spectra (and hence the exclusion limit) becomes independent of m_med. We will show in more detail in section <ref> that the ability of direct detection experiments to reconstruct the mediator mass is limited to the range q^min≲ m_med≲ q^max.We note that the constraints shown in figure <ref> depend on the assumed value of the coupling ratio θ. Here we have chosen θ = 0 corresponding for example to a vector mediator with kinetic mixing. In terms of the kinetic mixing parameter ϵ and the mediator-DM coupling g_DM the effective coupling g is then simply given by g = eϵg_DM, where e = √(4 πα) is the electromagnetic coupling. Different values of θ would typically enhance the sensitivity of heavy targets like tungsten relative to light targets like oxygen, except for specific values of θ that lead to destructive interference between proton and neutron contributions (see section <ref>).§ RECONSTRUCTING PARTICLE PHYSICS PARAMETERSFrom figure <ref> we make two central observations: first, if DM-nucleon scattering is due to the exchange of light mediators, cryogenic experiments will have the best sensitivity to such interactions up to DM masses of around 10 GeV. And second, compared to current bounds this sensitivity will improve by up to two orders of magnitude in terms of the effective coupling g, corresponding to up to four orders of magnitude in terms of the scattering rate. These observations immediately raise the question what we can hope to learn from a DM signal in low-threshold direct detection experiments. In this section we will answer this question by generating mock data and using this data to perform a parameter reconstruction.To determine the compatibility of a specific particle physics hypothesis (characterized by a set of parameters 𝐱) with a given set of data, we construct a likelihood function ℒ(𝐱) as follows. For each individual experiment α, we calculate the Poisson likelihood-2 logℒ^α(𝐱, 𝐲) = 2 ∑_i [ R^α_i(𝐱, 𝐲) + B^α_i(𝐲) - N^α_i + N^α_i logN^α_i/R^α_i(𝐱, 𝐲) + B^α_i(𝐲)] ,where the sum is over all bins, and R^α_i, B^α_i, and N^α_i denote the number of predicted signal events, predicted background events and observed events, respectively. In addition to the particle physics parameters 𝐱 we have introduced a number of nuisance parameters 𝐲, which represent for example astrophysical or experimental uncertainties and may affect both signal and background predictions. These nuisance parameters may be constrained by an additional likelihood function ℒ^n. The total profile likelihood is then given by the product of the individual likelihoods maximised with respect to the nuisance parameters:ℒ(𝐱) = max_𝐲ℒ^n(𝐲) ∏_αℒ^α(𝐱, 𝐲) . For the purpose of parameter estimation, the next step is to determine the value of the particle physics parameters 𝐱 that maximise the profile likelihood, called 𝐱_0. We can then construct the likelihood ratio ℛ(𝐱) = ℒ(𝐱) / ℒ(𝐱_0), which by definition is smaller than unity. Under random fluctuations in the data, the quantity -2 logℛ(𝐱) is expected to follow a χ^2 distribution with number of degrees of freedom n equal to the number of parameters 𝐱. We can therefore exclude a hypothetical set of parameters 𝐱 at confidence level 1-p if1 - CDF_χ^2(n, -2 logℛ(𝐱)) < p ,where CDF_χ^2(n, x) denotes the cumulative distribution function for the χ^2 distribution with n degrees of freedom. For the case of two parameters, the 95% confidence level (CL) bound is therefore given by -2 logℛ < 5.99.In the following we will focus on m_DM≲ 5 GeV, where cryogenic detectors have better sensitivity than liquid xenon experiments (see figure <ref>). We will first focus on one specific benchmark case, namely m_DM = 2 GeV, m_med = 3 MeV and θ = 0, and then discuss alternative benchmarks in section <ref>. The assumed value of g is chosen to be compatible with existing direct detection constraints. Choosing g close to current exclusion limits will lead to an optimistic scenario, in which thousands of events can be observed in future experiments, whereas smaller values of g imply smaller statistics and less precise parameter reconstruction. In the following, we will consider two alternative choices, namely g = 2 · 10^-11 (referred to as the low-statistics case) and g = 6 · 10^-11 (the high-statistics case). For our benchmark scenario, these choices correspond to around 900 and 8000 events across the set of future experiments that we consider (with SuperCDMS SNOLAB predicted to observe about four times as many events as CRESST-III).We can now generate mock data for our benchmark scenario and the two possible coupling choices and then determine which alternative choices of m_DM and m_med are compatible with this data. For the purpose of parameter reconstruction it is sufficient to consider mock data sets without Poisson fluctuations. Although in this case the best-fit point will have a very high likelihood, ℒ≈ 1, we nevertheless obtain reasonable estimates of the likelihood ratio ℛ(𝐱) expected in a typical realization of the experiments. We will return to the issue of Poisson fluctuations in the context of goodness-of-fit estimates in section <ref>.Figure <ref> shows the regions of parameter space compatible with the mock data generated for our benchmark scenario. For the purpose of these plots we are not interested in reconstructing the effective coupling g, i.e. we will simply treat it as a nuisance parameter and fix it to the value that maximises the likelihood. Nevertheless, the assumed value of g does play an important role as it determines the number of events that we expect to observe. The left (right) panel corresponds to the low-statistics (high-statistics) case. Red and blue contours correspond to the parameter reconstruction based only on data from SuperCDMS SNOLAB and CRESST-III respectively, while the grey region indicates the combined constraints. Note that in these plots we do not yet take into account nuisance parameters related to background or astrophysical uncertainties; these will be discussed later in this section.A striking feature in the left panel of figure <ref> is the accuracy of the parameter reconstruction from SuperCDMS SNOLAB as compared to CRESST-III. This happens because of two reasons: first, SuperCDMS SNOLAB is predicted to observe about four times more events than CRESST-III and hence has much better statistics. Second, several target elements contribute to the observed event rates in CRESST-III, leading to different ways in which a good fit to the observed data can be obtained. While for the benchmark case that we consider the event rate is dominated by scattering off oxygen (because tungsten recoils are below threshold), very similar recoil spectra are obtained for heavier masses and scattering off tungsten. This observation also explains the two different `branches' found for heavy mediator masses. With sufficient statistics it becomes possible to distinguish between the two possible scenarios and reject the solution with dominant scattering off tungsten (see right panel on figure <ref>).Another interesting feature is that all reconstructed parameter regions exhibit a characteristic `tilt' in the sense that lighter mediators are necessary for fitting heavier DM masses and vice versa. The origin of this shape is that heavier DM masses predict flatter recoil spectra, while lighter mediators predict steeper recoil spectra. Increasing the DM mass while decreasing the mediator mass and the effective coupling g may therefore leave the recoil spectra approximately unchanged. This degeneracy disappears once the mediator becomes so light (or so heavy) that direct detection experiments are effectively in the massless mediator limit (or the contact interaction limit). The recoil spectra then no longer depend on the precise value of the mediator mass.Finally we make the crucial observation that combining data from SuperCDMS SNOLAB and CRESST-III allows for a much more precise reconstruction of the mediator mass than considering the individual experiments. The primary reason behind this is that the degeneracy between DM mass, mediator mass and effective coupling strength g discussed above depends on the target element and therefore on the experiments (see eq. (<ref>)). This degeneracy is therefore effectively removed when combining data from several experiments. Nevertheless, it is of course conceivable that the degeneracy will reappear (or new degeneracies will arise) once we include various sources of uncertainties. We will therefore now discuss such uncertainties in detail and assess their impact on our results. §.§ Background uncertaintiesIn the parameter reconstruction performed above we have assumed exact knowledge of the shape and normalization of the experimental background(s). This is an overly optimistic assumption and we will now discuss a more conservative approach. Of course, if we were to allow arbitrary background shapes and normalizations, any kind of DM signal could be absorbed into the background, making it impossible to claim anything but exclusion limits. Any parameter reconstruction therefore necessarily requires some knowledge on the distribution of backgrounds. Here we assume that the shape of the background in each experiment is known, but we keep the normalization completely free. In other words, we introduce a nuisance parameter y^α for each experiment α such that the background predictions in eq. (<ref>) are given by B^α_i(𝐲) = y^α B^α_i. We do not impose any constraints on the parameters y^α apart from the trivial requirement that they must be strictly positive. As explained in more detail in appendix <ref>, we assume the backgrounds to be flat in energy both for CRESST-III and SuperCDMS SNOLAB. However, this assumption can easily be modified within our framework once more detailed informations about the future experiments are available.We show the impact of including background uncertainties in figure <ref>. As expected, these additional nuisance parameters visibly increase the size of the allowed parameter regions, in particular for SuperCDMS SNOLAB, where only a relatively small range of recoil energies is used to constrain the DM properties. For CRESST-III we observe that the second branch with scattering dominantly on tungsten now appears also in the high-statistics case. In principle it may be possible to distinguish these two branches, because the light yield of tungsten recoils in CRESST-III differs from the one for recoils on oxygen and calcium. Including this information (for example by constructing a two-dimensional likelihood in recoil energy and light yield) may hence make it possible to remove the second branch and obtain only one best-fit region. We leave this interesting possibility for future work.In principle, given additional information on the different background contributions, it would be straight-forward to extend our approach and introduce individual nuisance parameters for each background source, such that not only the normalization of the background but also its shape can be varied. However, such detailed information is not presently available for the future projections that we consider. Nevertheless, we note in passing that the likelihood function defined in eq. (<ref>) has an interesting property: as long as we only introduce nuisance parameters that rescale the signal prediction or (parts of) the background prediction, -2 logℒ is a convex function of these parameters, so that any local minimum is necessarily a global one <cit.>. It is therefore numerically trivial to maximize the likelihood with respect to these nuisance parameters. §.§ Degeneracies with coupling ratiosSo far we have made the assumption that only the mass of the mediator is unknown but its coupling structure is fixed. However, even in the simplest models of light mediators there are a number of different possible coupling structures <cit.>. In fact, it is well-motivated to assume that the ratio of couplings to neutrons and protons is essentially a free parameter <cit.>. If the effect of varying the mediator mass can be compensated by changing the coupling ratio, our lack of knowledge concerning the coupling structure of the mediator may affect our ability to determine its mass. In this section we discuss how the parameter reconstruction is affected if we do not make any assumptions on θ.For experiments consisting only of a single target element with charge number Z_T and mass number A_T, the differential event rate is approximately proportional to[Strictly speaking, this relation only holds exactly for zero momentum transfer, because at finite momentum transfer there may be differences in the form factors for protons and neutrons <cit.>, but taking these effects into account is beyond the scope of this work.] (Z_T cosθ + (A_T - Z_T) sinθ)^2. This rescaling factor takes different values for different elements, so that varying θ will affect the combination of information from different experiments. In fact, varying θ can even change the shape of the recoil spectrum in a single experiment that consists of several different elements.In contrast to the nuisance parameters introduced to parametrize the background uncertainties, the parameter θ enters into the likelihood in a more complicated way. The reason is that for θ < 0 there will be destructive interference between proton and neutron contributions, which for a given experiment will be maximal if θ = arctan(- Z_T / (A_T - Z_T)). Consequently, there will generally be a number of different values of θ that maximize ℒ(θ) locally. We therefore perform an explicit scan over the full range -π/2 < θ < π / 2 to determine the global maximum of ℒ(θ).If we consider SuperCDMS SNOLAB alone, the likelihood is essentially independent of θ, because the recoil spectra for the different isotopes look so similar that a change in θ can almost entirely be compensated by a change in the effective coupling g. Indeed, we will see below that the allowed parameter regions for SuperCDMS SNOLAB remain essentially unchanged when including θ as a nuisance parameter. For CRESST-III, on the other hand, the situation is very different. Since the recoil spectrum for scattering of tungsten is typically much steeper than the recoil spectrum for scattering of oxygen and calcium, the shape of the recoil spectrum depends sensitively on θ. Figure <ref> shows the best-fit value of θ depending on the assumed values of m_DM and m_med. We can make a number of interesting observations:* The best-fit value for θ typically differs significantly from the value assumed to generate the mock data, covering almost the entire range - π/2 ≤θ≤π/2. * If the assumed DM mass is large compared to the DM mass used to generate the mock data, the predicted recoil spectrum will be too flat, so the best fit can be obtained if scattering occurs exclusively on tungsten, i.e. for θ = -π/4. * If the assumed mediator mass is small compared to the mediator mass used to generate the mock data, the predicted recoil spectrum will be too steep, so the best fit can be obtained if scattering occurs exclusively on oxygen and calcium, i.e. for θ∼ -0.59. It should be clear from these observations that the ability of CRESST-III to reconstruct the mediator mass and the DM mass is significantly reduced when allowing for arbitrary values of θ. This is confirmed explicitly in figure <ref>. Clearly, in this case it is essential to combine the information from CRESST-III with data from SuperCDMS SNOLAB to perform a precise parameter reconstruction. In the absence of data from SuperCDMS SNOLAB it could also be interesting to attempt a discrimination between the different target elements in CRESST-III or to perform a combination of CRESST-III with the existing exclusion limits from Xenon1T, which would disfavour solutions with heavy DM scattering on tungsten. §.§ Astrophysical uncertaintiesThere are two kinds of astrophysical uncertainties that may affect the interpretation of direct detection experiments: uncertainties in the local DM density ρ_0 and uncertainties in the DM velocity distribution f(𝐯). The local DM density ρ_0 enters linearly into the differential event rate, so varying this quantity is equivalent to varying the effective coupling g. Since we are not interested in determining g and simply treat it as a nuisance parameter, our approach therefore already accounts also for uncertainties in the local DM density. The velocity distribution, on the other hand, enters in a more complicated way, giving rise to additional uncertainties that we will now discuss.The differential event rate depends on the velocity distribution via the velocity integral η(v_min), which in turn depends on the DM mass and the recoil energy via v_min = √(m_T E_R/2 μ_T^2). Changes in the velocity distribution may therefore change the shape of the recoil spectrum and thereby limit our ability to extract information on the particle physics parameters. One possible way to deal with astrophysical uncertainties are so-called halo-independent methods <cit.>, which aim to combine information from different experiments in such a way that the dependence on η(v_min) drops out. This approach has been successfully applied to many different models and in particular to models with light mediators <cit.>.However, as pointed out in ref. <cit.>, there is a fundamental limitation of this approach in the case of low-mass DM. If m_DM≪ m_T for all target elements under consideration, we find that v_min≃√(m_T E_R/2 m_DM^2) = q/2 m_DM. The velocity integral hence depends on the same combination of m_T and E_R that also enters in the factor for light mediator exchange, eq. (<ref>). In other words, for low-mass DM, any change in the mediator mass can be compensated for by an appropriate change in the velocity integral for all target elements simultaneously. As a result, it will not be possible to infer any information on the mediator mass if we allow for completely arbitrary velocity distributions.We will therefore take a different approach and consider only velocity distributions of the Maxwell-Boltzmann form. This assumption is supported by recent studies involving numerical simulations of Milky Way like galaxies <cit.>.We can then study the impact of astrophysical parameters by varying the underlying parameters v_0, v_esc and v_obs. In this case we can actually use the fact that m_DM≪ m_T to our advantage. As shown in appendix <ref>, if we simultaneously rescale all three velocities by a common factor z, this change is fully equivalent to rescaling the DM mass by a factor z. We therefore introduce a new nuisance parameter z and, rather than calculating the differential event rate as a function of m_DM, we calculate the differential event rate as a function of z m_DM. We restrict z to lie in the range consistent with observations. At 95% CL v_0 is constrained to lie in the range [180 km s^-1, 280 km s^-1] while the range for v_esc is approximately [450 km s^-1, 650 km s^-1], see <cit.> and references therein. These ranges can be reproduced if we require 0.8 ≤ z ≤ 1.2 at 95% CL. We implement this by means of a likelihood function for z given byℒ^z = 1/√(2π)σ_zexp(- (z - 1)^2/2σ_z^2)with σ_z = 0.1 and include this extra factor in the total likelihood.As in the case of θ it is possible that the likelihood has several local maxima for different values of z, making it necessary to explicitly scan over all values of z within the relevant range. Nevertheless, the simple way in which ℒ depends on z means that it is not in fact necessary to perform a two-dimensional scan over both θ and z, but rather that two separate one-dimensional scans are sufficient. Figure <ref> shows the impact of including astrophysical uncertainties in addition to the uncertainties discussed above. As expected, the effect of varying z is essentially to reduce our ability to reconstruct the DM mass, while not affecting our ability to reconstruct the mediator mass. Figure <ref> constitutes our central result for the benchmark scenario: even when including a number of different nuisance parameters, an accurate reconstruction of the DM and mediator masses is possible given sufficient statistics. §.§ Alternative benchmark scenariosIn the discussion above we have introduced a number of nuisance parameters that should be taken into account for a realistic assessment of the power of future low-threshold direct detection experiments. In addition to the two parameters that we are interested in reconstructing (the DM mass and the mediator mass), we have introduced two particle physics nuisance parameters (the coupling strength g and the coupling ratio parameter θ), one astrophysics nuisance parameter (the rescaling factor z) and one experimental nuisance parameter for each experiment (the background normalizations y^α).[Even in the presence of these nuisance parameters it only takes a few seconds on a single CPU to calculate the profile likelihood for a single grid point and around an hour to perform the parameter reconstruction on a grid with 10^4 points. A significant amount of computing time can be saved by reusing the information from neighbouring grid points to profile out the nuisance parameter related to astrophysical uncertainties.]In this section we present our results for a number of different hypotheses on the particle physics properties of DM and discuss the physics interpretation.In figure <ref> we perform a parameter reconstruction of the DM and mediator masses for two different assumptions on the true DM mass, namely 1 GeV (left panel) and 3GeV (right panel). In both cases we fix the mediator mass to m_med = 3MeV and the effective coupling to g = 6 · 10^-11.[It should be noted that this procedure leads to somewhat different numbers of events in the left and right panel. In particular the event rate in SuperCDMS is significantly suppressed for m_DM = 1 GeV.] For m_DM = 1 GeV all observed events are very close to the low-energy threshold (i.e. within the first two or three bins). As a result the parameter reconstruction becomes more difficult and neither of the two experiments can individually constrain the mass of the mediator. For CRESST-III one furthermore finds a second branch of solutions corresponding to scattering off tungsten. Combining the information from both experiments leads to a somewhat better reconstruction, but the allowed parameter region still extends to arbitrarily heavy mediators. For heavier DM masses, on the other hand, an accurate parameter reconstruction is possible (see the right panel of figure <ref>).In figure <ref> we investigate the effect of varying the assumed mediator mass while keeping the DM mass fixed to 2 GeV. For each mediator mass we choose the value of the coupling g such that the predicted number of events is comparable to the high-statistics case discussed previously. In the top-left panel, the mediator mass is set to 0.1 MeV, i.e. effectively massless for the experiments under consideration. The combined fit to both experiments then places an upper bound on the mediator mass of about m_med < 0.8MeV. As expected, the mock data is compatible with arbitrarily light mediators so that no lower bound can be placed. Conversely, if the assumed mediator mass is larger than about 10 MeV, it is no longer possible to distinguish our scenario from the case of contact interactions and the allowed parameter region extends up to arbitrarily high mediator masses (see bottom row of figure <ref>). An accurate reconstruction of the mediator mass is possible only if the mass falls between these two extremes, as illustrated in the top-right panel for m_med = 1 MeV.To conclude this section we note that we have also studied the effect of making different assumptions on the value of θ used to generate mock data. If the effective coupling g is adjusted in such a way that the event numbers are comparable to the ones discussed above, we find very similar results for different choices of θ. An example with θ≠ 0 will be discussed in section <ref>. §.§ Goodness-of-fit estimatesSo far we have focussed on the issue of parameter estimation, i.e. we have constructed likelihood ratios to determine the parameter regions compatible with a given set of mock data. Another interesting topic that we can study in our framework are goodness-of-fit estimates, i.e. the question whether a specific choice of parameters yields a good description of the data. To answer this question we need to consider the absolute value of the likelihood rather than a likelihood ratio. Clearly, it is then no longer possible to neglect Poisson fluctuations in the data, because doing so would exaggerate the likelihood, i.e. would suggest unrealistically good agreement between data and model. In this section we therefore briefly discuss the effect of Poisson fluctuations and give a few examples for questions that can be answered with goodness-of-fit estimates.In the limit of large bin counts the likelihood function defined in eq. (<ref>) approaches a χ^2 test statistic. We therefore expect that for the true parameters of nature 𝐱_0 the likelihood ℒ(𝐱_0) follows a χ^2-distribution with n = n_b - n_y degrees of freedom, where n_b denotes the total number of bins across all experiments and n_y denotes the total number of nuisance parameters that have been profiled out. For the two experiments that we consider n_b = 29, so if we profile out the effective coupling strength g and the normalization of the background in both experiments, we expect to find n = 26. We confirm that this is indeed the case by performing a Monte Carlo (MC) simulation, i.e. by considering a large ensemble of mock data sets with Poisson fluctuations.We can make use of this observation to study whether a given data set may enable us to exclude specific hypotheses about the particle physics nature of DM. For example, any DM signal observed in future direct detection experiments will first be interpreted under the assumption of a heavy mediator, i.e. contact interactions between DM and nuclei. A question of great interest would therefore be whether this hypothesis can be confidently excluded if the mediator is in fact light. To answer this question, we can use an ensemble of mock data sets generated under the assumption of a light mediator and calculate the likelihood under the incorrect assumption of a heavy mediator. For this purpose, we treat both m_DM and θ as nuisance parameters, i.e. we fix them to the value that maximizes the likelihood, such that n = 24.[Note that due to the degeneracy between the DM mass and the nuisance parameter z parametrizing astrophysical uncertainties, including z as an additional nuisance parameter would not increase the likelihood further and therefore does not reduce the relevant number of degrees of freedom.] For any specific mock data set, the heavy-mediator assumption can then be excluded at 95% CL if -2 logℒ > 36.4.Considering the same benchmark scenario as before (m_DM = 2 GeV, m_med = 3 MeV and θ = 0) we find that in the low-statistics case it is typically not possible to exclude the heavy-mediator hypothesis at 95% CL, whereas in the high-statistics case a 95%-CL exclusion is possible for about 98% of the mock data sets. In the latter case it is also possible to exclude the hypothesis of a very light mediator (with m_med≪ q^min) at 95% CL for more than 99.9% of the samples.§ CONNECTION TO SELF-INTERACTING DARK MATTERThe analysis performed in section <ref> applies to any model of DM interacting with nucleons via a light mediator, provided that the couplings of the mediator are spin-independent. In this section we apply these results to a particle physics model of particular interest, namely the case of self-interacting DM. It is well known that the presence of a light mediator can significantly enhance the rate of DM self-scattering in astrophysical systems, not only because of the long-range nature of the interactions but also because of additional non-perturbative effects due to multiple mediator exchange <cit.>. For a mediator with spin-independent interactions, these effects can be calculated by solving the non-relativistic Schroedinger equation for a Yukawa potential. The resulting scattering rate exhibits a characteristic velocity dependence such that the largest effects are expected on small scales, just as required in order to resolve the potential small-scale problems of collisionless cold DM <cit.>. The quantity of interest for astrophysical observables is the momentum transfer cross section σ_T, which should lie in the range 0.1cm^2/g≲σ_T / m_DM≲ 10cm^2/g in order to induce sizeable effects consistent with observations.Out of the various possible particle physics realizations of this general scenario, the most attractive model consists of a fermionic DM particle ψ (here assumed to be a Dirac fermion) and a scalar mediator ϕ <cit.>. For this model, DM self-annihilation is suppressed at small velocities due to CP conservation, evading the strong constraints from the Cosmic Microwave Background on many DM models with light mediators <cit.>. The effective coupling g is then given by g ≈ 1.6 × 10^-3y_ψy_SM, where y_ψ denotes the DM-mediator coupling and y_SM denotes the rescaled Yukawa coupling of the mediator to Standard Model (SM) fermions (i.e. the mixing of the scalar mediator with the SM Higgs boson).A recent analysis of this model <cit.> concluded that for DM masses in the GeV region and mediator masses in the MeV region it is possible to reproduce the observed DM relic abundance and at the same time obtain large DM self-interactions if y_ψ∼ 0.1. Bounds from direct detection experiments then require y_SM≲ 10^-6 so that constraints from Higgs measurements and flavour physics are easily satisfied.[We note however that for such small values of y_SM the lifetime of the mediator will be so large that it only decays after the onset of primordial nucleosynthesis. Nevertheless, the mass of the mediator is so small that it can only decay electromagnetically and with an energy small compared to typical nuclear binding energies. The dominant constraint is therefore expected to come from the entropy injected by the mediator decays, which modify the effective relativistic degrees of freedom N_eff. The magnitude of this effect depends however on additional aspects of the model, which are not relevant for the direct detection phenomenology. A detailed analysis of constraints from primordial nucleosynthesis is challenging and will be discussed elsewhere <cit.>.] This mass range is precisely the parameter region that we have identified above to be the most interesting for cryogenic direct detection experiments. In fact, these experiments already place the strongest bounds on the SM-mediator coupling in this scenario. For the purpose of this section, we will therefore focus on scalar mediators and set θ = π/4 as expected from Higgs mixing <cit.>, both for the generation of mock data and for the subsequent parameter reconstruction.While the DM momentum transfer cross section depends only on the DM-mediator coupling y_ψ, the DM-nucleon scattering cross section depends also on the mediator-SM coupling y_SM, which arises for example from mixing of ϕ with the SM Higgs boson. The effective coupling g relevant for direct detection experiments can therefore be treated as an independent parameter in the same way as we have done above. Thus, if we can use direct detection experiments to infer the mediator mass and the DM mass, we can effectively determine the DM momentum transfer cross section (under the assumption that DM is a thermal relic <cit.>). This is illustrated in figure <ref>. In the left panel we show a plot very similar to the ones shown in section <ref>, except that we also indicate the combinations of DM mass and mediator mass that lead to a momentum transfer cross section in the desired range <cit.>. In the right panel, we show the range of momentum transfer cross sections σ_T compatible with the inferred range of masses as a function of the DM relative velocity v. Since the DM self-interaction cross section depends very sensitively on the mediator mass, a precise determination of σ_T is clearly impossible. Nevertheless, figure <ref> shows that it may be possible to demonstrate that both the magnitude and the velocity dependence of the DM self-scattering rates are broadly in agreement with astrophysical requirements, at least for the specific assumptions that we have made. Direct detection experiments may hence help us to determine not only the properties of the DM particle itself, but also to learn about its role in the formation of small-scale structures.§ CONCLUSIONSLight mediators communicating the interactions between DM and nuclei offer an interesting alternative to the effective operator approach conventionally adopted for the analysis of direct detection experiments. In this work we have demonstrated that cryogenic direct detection experiments are particularly well-suited for exploring this scenario. Their low threshold enhances the sensitivity to steeply falling recoil spectra and their excellent energy resolution allows for a precise reconstruction of the underlying particle physics from a potential signal. Present searches based on this technology already place stringent constraints on models with light mediators and significant improvements are expected for the next few years.To illustrate the potential of future low-threshold detectors, we have performed parameter reconstructions from mock data sets in two planned experiments: SuperCDMS SNOLAB and CRESST-III. A strong emphasis has been placed on a realistic implementation of the experimental details. We include detector effects such as energy resolution, nuisance parameters for the background contributions as well as additional uncertainties reflecting both the unknown particle physics properties of DM and its astrophysical distribution. Even when including all of these uncertainties, low-threshold direct detection experiments maintain a remarkable potential to reconstruct the properties of the mediator.For mediator masses in the range 1–10 MeV, the possibility to reconstruct mediator masses is mostly limited by statistics. Given sufficient exposure, it should be possible to reconstruct the mediator mass to better than a factor of 2. If the mediator is either lighter than about 1 MeV (and hence indistinguishable from truly long-range interactions) or heavier than about 10 MeV (and hence indistinguishable from contact interactions), a precise reconstruction of its mass is typically not possible. Nevertheless, with enough statistics one can clearly rule out the hypothesis of contact interactions for a very light mediator (and vice versa) using goodness-of-fit estimates.The ability to reconstruct mediator masses relies strongly on the combination of different target materials. We have focussed on the combination of experiments based on CaWO_4 and germanium targets, but we have checked that a similarly good reconstruction can be obtained by combining data from a germanium experiment with results based on silicon detectors, which are being developed by the SuperCDMS collaboration. Another interesting, albeit challenging, possibility would be to exploit the differences in light yield for the different target elements in CRESST-III to statistically distinguish between models with dominant scattering on tungsten and those with dominant scattering on oxygen and calcium.Throughout this work we have focussed on mediators with spin-independent interactions. Spin-dependent interactions are more challenging, since neither oxygen nor calcium nuclei carry spin and therefore the sensitivity of CRESST-III is significantly suppressed. It may be possible to gain complementary information with bubble chamber experiments like PICO-500 <cit.>, but these lack the excellent energy resolution of cryogenic detectors. The most promising avenue appears to be the combination of germanium and silicon detectors, both of which possess at least some sensitivity to spin-dependent interactions. A detailed investigation of a wider range of interactions and a broader set of experimental technologies is left for future work.Finally, we have pointed out that the range of mediator and DM masses that can be studied with cryogenic experiments coincides with parameter regions that have been considered in the context of self-interacting DM. This observation implies that, within specific model assumptions, one can translate between signals in direct detection experiments and astrophysical observables, such as core sizes in dwarf galaxies. By correlating these very different signatures, we can therefore hope to ultimately obtain a coherent picture of both the microscopic and macroscopic properties of DM.We thank Achim Gütlein for early collaboration and David Cerdeño, Josef Jochum and Kai Schmidt-Hoberg for useful discussions. FK is supported by the DFG Emmy Noether Grant No. KA 4662/1-1, SK is supported by the `New Frontiers' program of the Austrian Academy of Sciences and by FWF project number V592-N27, and SW is supported by the ERC Starting Grant `NewAve' (638528). SK thanks DESY for hospitality during the completion of this work.§ IMPLEMENTATION OF PRESENT AND FUTURE EXPERIMENTSIn this appendix we describe the experimental information and the assumptions that we make to construct the likelihood functions for the various experiments.§.§ CRESST-II The latest CRESST results <cit.> are based on 52.2 kg days using the Lise detector module. We smear the physical recoil spectrum assuming a Gaussian energy resolution function with σ = 62eV <cit.>, considering only fluctuations up to 3σ in order to avoid unphysical results far below threshold. The cut survival probabilities as a function of the detected recoil energy are taken from the recent data release <cit.>. We then group the events observed in the acceptance region into 47 equidistant bins between 0.3 and 5.0 keV. For the calculation of the upper limits we conservatively set the number of expected background events to zero, which gives results similar to the optimum interval method <cit.> employed in ref. <cit.>. §.§ CRESST-III Our implementation of the future CRESST detector is based on the projections for the final state of CRESST-III presented in ref. <cit.>. The expected exposure is 1000 kg days with an energy threshold of 100eV. We group the events in the energy range between 0.1 and 2keV into 19 bins of width 0.1 keV. Furthermore, we assume a background level of , corresponding to 3.5 events in each of the bins. The expected number of signal events is calculated assuming an efficiency of unity down to threshold, and using a Gaussian energy resolution of 20eV. We take into account only recoils with a true energy above 60eV in order to avoid unphysical upward fluctuations from very small recoil energies. §.§ CDMSLite CDMSLite has analysed data from an exposure of 70.14 kg days <cit.>. We take the energy-dependent signal efficiency from <cit.> and follow the procedure outlined there to convert nuclear recoil energies (eVnr) into electron equivalent energies (eVee), using k = 0.157. We perform a fit to the width of various electron-capture peaks to determine the energy-dependent detector resolution. We then consider the energy range [60 eVee, 500 eVee], which we divide into 10 bins of increasing size, such that the number of observed events in each bin is greater than one. Following the analysis in ref. <cit.>, we do not assume a background model, such that all observed events can potentially be DM signals. This procedure allows us to calculate a likelihood function that yields a bound very similar to the one obtained from the optimum interval method.§.§ SuperCDMS SNOLAB We estimate the sensitivity of SuperCDMS SNOLAB following ref. <cit.>, focussing on the high-voltage germanium detectors. Although the SuperCDMS collaboration expects that a threshold as low as 40eV (nuclear recoil energy) can be achieved, we consider a low-energy threshold of 100 eV, so that it is a good approximation to assume the signal efficiency to be constant (at 85%) and the background level to be flat (at ) <cit.>. We also limit ourselves to recoil energies below 300 eV in order to avoid backgrounds from electron capture lines. This restriction will reduce the sensitivity of SuperCDMS SNOLAB to heavy DM but has no effect in the low-mass region that we are interested in. We divide the search window into 10 equally-spaced bins and include an energy resolution of 10 keV. SuperCDMS expects to achieve a total exposure of 1.6 · 10^4kg days over the course of five years of data taking. To compare CRESST-III and SuperCDMS on similar time-scales, however, we consider the sensitivity that can be achieved with a single year of data taking, corresponding to an exposure of approximately 3200kg days.[Incidentally, this is also the exposure for which backgrounds become non-negligible (we expect about 2 events per bin), so increasing the exposure by a factor of 5 will improve the sensitivity by significantly less than a factor of 5.] §.§ Xenon1T The first results of Xenon1T are based upon an exposure of 35636 kg days <cit.>. To calculate the acceptance function, we simulate fluctuations in the S1 and S2 signal making use of the scintillation and ionization yields from ref. <cit.> and taking into account the anti-correlation between the two signals. Rather than attempting to model the detector response, we determine the light collection efficiency and the electron extraction probability by fitting to the nuclear recoil band shown in ref. <cit.>. We then consider events with S1 ≥ 3 phe and require the S2 signal to lie below the mean of the nuclear recoil band. For low recoil energies, this procedure essentially reproduces the acceptance function shown in ref. <cit.>, where no cut on the S2 signal is applied, whereas for large recoil energies we obtain an acceptance that is approximately a factor of 2 smaller. This result is in agreement with the expectation that the nuclear recoil cut has almost 100% signal acceptance close to the threshold, where the signal results exclusively from upward fluctuations in the S1 signal, and around 50% signal acceptance away from the threshold. The total number of expected background events below the mean of the nuclear recoil band is 0.36, and no events have been observed. We therefore set a Poisson upper bound on the signal contribution of 1.94 events at 90% CL, noting that this procedure leads to a slightly weaker bound than the asymptotic limit assumed in ref. <cit.>. §.§ LZ Our implementation of LZ is based upon the LZ conceptual design report <cit.>. Specifically, we assume a total exposure of 5.6 · 10^6kg days and consider the search region 3 phe≤ S1 ≤ 30 phe. We calculate the energy-dependent acceptance by considering Poisson fluctuations in the S1 signal based on the effective scintillation yield for liquid xenon taken from the LUX analysis <cit.> and assuming a light collection efficiency of 7.5%. We multiply this acceptance function by an overall factor of 1/2 to account for the requirement that the S2 signal must lie below the mean of the nuclear recoil band. Finally, we calculate the expected sensitivity by assuming that the number of observed events is equal to the expected background (2.37 events).§ ASTROPHYSICAL UNCERTAINTIES FOR LOW-MASS DARK MATTERIn the Galactic rest frame the standard Maxwell-Boltzmann velocity distribution is given byf(𝐯, v_0, v_esc) = 1/N[ exp(-v^2/v_0^2) - exp(-v^2/v_esc^2) ] ,where N is an appropriate normalization constant. In the laboratory frame we need to account for the velocity of the Earth 𝐯_E relative to the Galactic rest frame. Neglecting the motion of the Earth relative to the Sun, we take this velocity to be time-independent, 𝐯_E = 𝐯_obs. We then obtainf(v, θ_v, v_0, v_esc, v_obs) = 1/N[exp(-v^2 + 2 v v_obscosθ_v + v_obs^2/v_0^2)- exp(-v^2 + 2 v v_obscosθ_v + v_obs^2/v_esc^2) ] ,where θ_v denotes the angle between 𝐯_obs and the velocity vector 𝐯.Let us now consider two different velocity distributions of the Maxwell-Boltzmann form, f(v) and f̃(v), which are related byṽ_0 = v_0/z , ṽ_esc = v_esc/z , ṽ_obs = v_obs/z ,where z is an arbitrary rescaling factor.[We note that for an isothermal halo the velocity dispersion and the circular velocity are directly proportional to each other and it is therefore well-motivated to rescale v_0 and v_obs simultaneously. The escape velocity is introduced by hand and is therefore in principle an independent parameter. Since our results do not depend sensitively on v_esc we use the same rescaling factor for simplicity.] It then follows immediately from eq. (<ref>) that the two velocity distributions satisfy the relationf̃(v) = z^3 f(z v) ,where the pre-factor ensures that f̃(v) is normalized, ∫ f(v)d^3 v= ∫f̃(v)d^3 v = 1. The velocity integral η̃(v_min) corresponding to f̃(v) is then given byη̃(v_min) = ∫_v_min^∞f̃(v)/vd^3 v = ∫_v_min^∞ z^3 f(z v)/vd^3 v = ∫_ṽ_min^∞αf(ṽ)/ṽd^3 ṽ = z η(ṽ_min),where ṽ_min = z v_min.The differential event rate resulting from the rescaled velocity distribution therefore becomesdR̃_T/dE_R = ρ_0ξ_T/2 πm_DMg^2 F_T^2(E_R)/( 2 m_T E_R + m_med^2 )^2z η (ṽ_min (E_R, m_DM))The crucial observation is now that for m_DM≪ m_T the minimum velocity is given by v_min(E_R, m_DM) = 1/m_DM√(m_T E_R/2) and hence ṽ_min(E_R, m_DM) = v_min(E_R, m̃_DM) with m̃_DM = m_DM / z. In terms of this rescaled mass, the differential event rate becomesdR̃_T/dE_R = ρ_0ξ_T/2 π m̃_DMg^2 F_T^2(E_R)/( 2 m_T E_R + m_med^2 )^2 η (v_min(E_R, m̃_DM)) ,which is identical to eq. (<ref>) with m_DM replaced by m̃_DM. 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Yellin, http://dx.doi.org/10.1103/PhysRevD.66.032005Finding an upper limit in the presence of unknown background,Phys. Rev. D66 (2002) 032005, [http://arxiv.org/abs/physics/0203002physics/0203002].Akerib:2015rjg LUX, D. S. Akerib et al., http://dx.doi.org/10.1103/PhysRevLett.116.161301Improved Limits on Scattering of Weakly Interacting Massive Particles from Reanalysis of 2013 LUX Data,Phys. Rev. Lett. 116 (2016), no. 16 161301, [http://arxiv.org/abs/1512.035061512.03506]. | http://arxiv.org/abs/1707.08571v2 | {
"authors": [
"Felix Kahlhoefer",
"Suchita Kulkarni",
"Sebastian Wild"
],
"categories": [
"hep-ph",
"astro-ph.CO",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170726180000",
"title": "Exploring light mediators with low-threshold direct detection experiments"
} |
STRUCTURAL INSTABILITY OF A BIOCHEMICAL PROCESS] STRUCTURAL INSTABILITY OF A BIOCHEMICAL PROCESS577.3 05.45.-a, 05.45.Pq,05.65.+b V.GrytsayBy the example of a mathematical model of a biochemical process, the structural instability of dynamical systems is studied by calculating the full spectrum of Lyapunov indices with the use of the generalized Benettin algorithm. For the reliability of the results obtained, the higher Lyapunov index determined with the orthogonalization of perturbation vectors by the Gram–Schmidt method is compared with that determined with the overdetermination of only the norm of a perturbation vector. Specific features of these methods and the comparison of their efficiencies for a multidimensional phase space are presented. A scenario of the formation of strange attractors at a change of the dissipation parameter is studied. The main regularities and the mechanism of formation of a deterministic chaos due to the appearance of a fold or a funnel, which leads to the uncertainty of the evolution of a biosystem, are determined. [ V. Grytsay December 30, 2023 ===================== § INTRODUCTION One of the main physical problems is the appearance of ordered structures from the chaos in systems different by their nature due to the self-organization. The synergy was follow from theoretical physics [1]. The first model of synergy was the Turing model (“Morphogenesis model”) [2]. The next was the Prigogine's “Brusselator”, where self-organization regimes were considered in an abstract chemicothermal system [3]. Synergy allows one to find common physical rules for the self-organization of opened nonlinear systems [4–7]. To a significant extent, the problem concerns the question of the self-origination of life, the evolutionary development of the alive, and the basic mechanisms of structural-functional regularities of transformations in various biochemical systems. In the general case, biochemical systems are described by ordinary nonlinear differential equations of the form d X dt = f (X ,a ), where X = (X_1,...X_n)∈R^nis a vector of variables of states (phase variables), and a = (a_1,...,a_k) ∈R^k) is the vector of parameters of the system. The results of numerical solutions of the equations can be compared with experiments and would clarify self-organization laws.Works [8–18] considered the mathematical model of a bioreactor transforming steroids [19] under flow conditions depending on a change of the dissipation, the kinetic membrane potential of cells, and the input flows of a substrate and oxygen. Various scenarios of the transition from stationary modes to self-oscillatory modes with different multiplicities were presented, and the regions of the formation of strange attractors were determined. It is worth noting an experiment that proved the existence of self-oscillations in a population of Arthrobacter globiformis cells [20].The studies were performed with the use of the method of phase portraits. The determined regions with qualitatively identical phase portraits and the points of bifurcation do not characterize the dynamics of a biosystem sufficiently completely. The most complete information about the stability of various modes is contained in the full spectrum of Lyapunov indices. But since a mathematical model of the given biochemical system contains a lot of variables and parameters, the limitations on the solution of such problems on a computer arise due to a small volume of the work memory for the processing of a matrix of small perturbations. In addition, any error made in the programming will essentially influence the overdetermination of perturbation vectors and their orthogonalization.To attain the reliable results, we carried out the independent calculations of both the higher Lyapunov index with the same parameters, by using the Benettin algorithm with the overdetermination of only the norm of perturbation vectors, and the full spectrum of Lyapunov indices with the orthogonalization of these vectors by the Gram–Schmidt method [21–23]. The higher Lyapunov indices obtained were practically identical, which confirms the correctness of the developed computer program.The essence of the calculation of a higher Lyapunov index with the overdetermination of only the norm of perturbation vectors consists in the determination of the evolution of an arbitrarily small deviation from a studied trajectory of the system λ = 1 nτ∑_k=1^nln∥u_k∥ε. After each step of calculations, it is necessary to overdetermine a deviation so that its direction will remain the same, and the norm will be equal to the input value ε, namely: u_0k = εu_k∥u_k∥.The algorithm of calculations of the full spectrum of Lyapunov indices consisted in the following. Taking some point on the attractor X_0 as the initial one, we traced the trajectory outgoing from it and the evolution of N perturbation vectors. In our case, N = 10 (the number of variables of the system [18]). The initial equations of the system supplemented by 10 complexes of equations in variations were solved numerically. As the initial perturbation vectors, we set the collection of vectors b^0_1, b^0_2,... b^0_10 which are mutually orthogonal and normed by one. In some time T, the trajectory arrives at a point X_1, and the perturbation vectors become b^1_1, b^1_2,... b^1_10, Their renormalization and orthogonalization by the Gram–Schmidt method are performed by the following scheme:b^1_1 = b_1/∥b_1∥, b^'_2 = b^0_2 - (b^0_2,b^1_1) b^1_1,b^1_2 = b^'_2/∥b^'_2∥, b^'_3 = b^0_3 - (b^0_3,b^1_1) b^1_1 - (b^0_3,b^1_2) b^1_2,b^1_3 = b^'_3/∥b^'_3∥, b^'_4 = b^0_4 - (b^0_4,b^1_1) b^1_1 - (b^0_4,b^1_2) b^1_2 - (b^0_4,b^1_3) b^1_3,b^1_4 = b^'_4/∥b^'_4∥, ............................................................................... .......Then the calculations are continued, by starting from the point X_1 and perturbation vectors b^1_1, b^1_2,... b^1_10. After the next time interval T, a new collection of perturbation vectors b^2_1, b^2_2,... b^2_10 is formed and undergoes again the orthogonalization and renormalization by the above-indicated scheme. The described sequence of manipulations is repeated a sufficiently large number of times, M. In this case in the course of calculations, we evaluated the sumsS_1 =∑_i=1^M ln∥ b^' i_1 ∥,S_2 =∑_i=1^M ln∥ b^' i_2 ∥,..., S_10 =∑_i=1^M ln∥ b^' i_10∥,which involve the perturbation vectors prior to the renormalization, but after the normalization.The estimation of 10 Lyapunov indices was carried out in the following way:λ_j = S_j MT, i=1,2,...10. As the test calculations for the verification of a program, we reproduced the well-known results for the finite-dimensional Lorentz system. § MATHEMATICAL MODELA mathematical model of the process under flow conditions in a bioreactor was developed by the general scheme of metabolic processes in Arthrobacter globiformis cells at a transformation of steroids [8-18]. dG/dt =G_0/N_3 + G + γ_2Ψ - l_1 V(E_1)V(G) - α_3G ,dP/dt =l_1 V(E_1)V(G) - l_2V(E_2)V(N)V(P) - α_4P,dB/dt = l_2 V(E_2)V(N) V(P) - k_1V(Ψ)V(B) - α_5B,dN/dt = - l_2V(E_2)V(P)V(N) - l_7V(Q)V(N) + + k_16V(B) Ψ/K_10 + Ψ + N_0/N_4 + N - α_6N,dE_1/dt = E_10G^2/β_1 + G^2(1 - P + mN/N_1 + P+mN) - - l_1V(E_1)V(G) + l_4V(e_1)V(Q) - a_1E_1,de_1/dt = - l_4V(e_1)V(Q) + l_1V(E_1)V(G) - α_1e_1,dQ/dt = 6lV(Q^0 + q^0 - Q)V(O_2)V^(1)(Ψ) - - l_6V(e_1)V(Q) - l_7V(Q)V(N),dO_2/dt = O_20/N_5 + O_2 - lV(O_2)V(Q^0 + q^0 - Q)V^(1)(Ψ) - - α_7O_2,dE_2/dt = E_20P^2/β_2 + P^2N/β + N(1- B/N_2 + B - - l_10V(E_2)V(N)V(P) - α_2E_2,dΨ/dt = l_5V(E_1)V(G) + l_9V(N)V(Q) - αΨ. where: V(X) =X/(1 + X);V^(1)(Ψ) =1/( 1 + Ψ^2); V(X) is a function involving the adsorption of an enzyme in the region of a local bond; V^(1)(Ψ) is a function characterizing the influence of the kinetic membrane potential on the respiratory chain.In the modeling, it is convenient to use the following dimensionless quantities [1–11] which are set as follows: l = l_1 = k_1 = 0.2; l_2 = l_10 = 0.27; l_5 = 0.6; l_4 = l_6 = 0.5; l_7 = 1.2; l_9 = 2.4; k_2 = 1.5; E_10 = 3; β_1 = 2; N_1 = 0.03; m = 2.5; α = 0.0033; a_1 = 0.007; α_1= 0.0068; E_20 = 1.2; β = 0.01; β_2 = 1; N_2 = 0.03; α_2 = 0.02; G_0 = 0.019; N_3 = 2; γ_2 = 0.2; α_5 = 0.014; α_3 = α_4 = α_6 = α_7 = 0.001; O_20 = 0.015; N_5 = 0.1; N_0 = 0.003; N_4 = 1; K_10 = 0.7.Equations (1)–(9) describe a change in the concentrations of (1) – hydrocortisone (G); (2) – prednisolone (P); (3) – 20β-oxyderivative of prednisolone (B); (4) – reduced form of nicotinamideadeninedinucleotide (N); (5) – oxidized form of 3-ketosteroid-△-dehydrogenase (E_1); (6) – reduced form of 3-ketosteroid-△-dehydrogenase (e_1); (7) – oxidized form of the respiratory chain (Q); (8) – oxygen (O_2); (9) – 20β-oxysteroid-dehydrogenase (E_2). Equation (10) describes a change in the kinetic membrane potential (Ψ).The initial parameters of the system are as follows: G^0 = 0.17; P^0 = 0.844; B^0 = 0.439; N^0 = 1.789; E^0_1 = 0.216; e^0_1 = 1.835; Q^0 = 2.219; O^0_2 = 0.309; E^0_2 = 1.645; Ψ^0 = 0.300.The reduction of parameters of the system to dimensionless quantities is given in works [8,9].To solve this autonomous system of nonlinear differential equations, we applied the Runge–Kutta–Merson method. The accuracy of solutions was set to be 10^-12. To get the reliable results, namely in order that the system, being in the initial transient state, approach the asymptotic attractor mode, we took the duration of calculations to be 100000. For this time interval, the trajectory “sticks” the corresponding attractor. § RESULTS OF STUDIESIn work [18], the diagram of states of the system in the parametric space of input flows of the substrate and oxygen was constructed. By varying the input flow, it was established that the scenario of the formation of autoperiodic and chaotic modes is regularly repeated. Numerical calculations showed that the same scenario is preserved for fixed flows, but under a change of the dissipation of a kinetic membrane potential. Such a scenario is presented in Table 1 together with the spectra of Lyapunov indices for the given modes. As the dissipation coefficient decreases from 0.84 down to 0.04131, the stationary state is destroyed, the attractor of a single-valued autoperiodic mode is formed as a result of the Andronov–Hopf bifurcation, and then the bifurcations with the doubling of the period from a single cycle to an 8-fold cycle appear. A subsequent decrease in the dissipation coefficient leads to the formation of strange attractors with the corresponding multiplicity between regular attractors. After the attainment of the 14-fold period, a single cycle is formed again, and then it passes to the stationary state. The examples of the corresponding attractors and the kinetic curves are given in Fig. 1,a–d. The shape of all regular attractors n*2^0is analogous to those in Fig. 1,b, where n = 1, 2,...,14. Strange attractors n*2^∞, where n = 8, 9,...,13, are analogous to those in Fig. 1,c. It is of interest that, after the appearance of a strange attractor 13*2^∞ at α = 0.032160, a regular attractor 22*2^∞ is formed (see Fig. 1,d), after which the attractor 14*2^0 appears further. One more specific feature of the dynamics of biosystems is shown in Fig. 1,a. It is revealed at small input flows of the substrate and oxygen. We observe the appearance of a strange attractor which differs from the previous ones and possesses a complicated structure. It arises regularly at various input flows on the boundary of the transition from a stationary state to 1*2^0. At small input flows, in addition to oscillations due to the desynchronization of the processes of transformation and accumulation of substrates in the biosystem, there appears the desynchronization between the processes of respiration and transformation of the substrate. Two unstable points appear. The trajectories rotate chaotically around them, by passing from one center of rotation to another one. By comparing Fig. 1,a and Fig. 1,c, it is worth noting that the chaotic mode in this biosystem is formed by two means: in the first case, the attractor creates folds inside itself, whereas a funnel is formed in the second case. Due to this circumstance, the chaotic motion mixes the trajectories in the phase space.=1In addition to the phase portraits, the figures show the kinetics of one of the variables of the system. It is seen that the curves differ from one another in different modes. For strange attractors, the plots represent irregular oscillations. We indicate a combination of oscillations and jumps. The figures demonstrate also the dependence of the chaotic kinetics on the initial conditions.An important role in the analysis of the scenario of the formation of various modes is played by Lyapunov indices. For characteristic modes, Table 1 presents their full spectra λ_1, λ_2,...,λ_10, and the value of their sum Λ = ∑_j=1^10λ_j. Figure 2,a-d gives the plots of the dependence of λ_1, λ_2, λ_3, and Λon the dissipation coefficient α in the interval from 0.0321 to 0.033.By analyzing the results obtained, we note that all autoperiodic modes corresponding to regular attractors n*2^∞have higher Lyapunov indices practically equal to zero. But the chaotic modes corresponding to strange attractors n*2^∞ have higher Lyapunov indices which are positive and greater by one order. It is seen in Fig. 2,a how the “windows of periodicity” are formed at λ_1<0. At the given α, the regular attractors appear, whereas the strange attractors arise outside them. The most pronounced chaotic modes correspond to maximal peaks of λ_1. By the given plot, it is possible to choose beforehand the corresponding mode of functioning for a bioreactor making no calculations again.In Fig. 2,b, we show the plot for the second Lyapunov index. There, λ_2 changes accordingly to λ_1. For strange attractors, n*2^∞, λ_1>0, whereas λ_2≈ 0. That is, the diverging trajectories hold themselves closely to the given limiting cycle, which preserves the multiplicity of a strange attractor. The transition from the limiting cycle to the chaotic mode occurs by means of the intermittence. The kinetics of the variable in Fig. 2,b, shows how the periodic limiting cycle is suddenly broken by the chaotic motion; but then the periodicity is restored again. On the phase portrait, we can separate a clearly pronounced region, whose shape is close to that of the disappeared limiting cycle, relative to which a chaotic trajectory was formed.The variation of the third Lyapunov index is shown in Fig. 2,. The behavior of λ_3is characteristic by that this index is changed oppositely to λ_1 and λ_2. When these two indices grow, λ_3decreases, and vice versa. For all α, the index λ_3is negative.We may imagine that a 10-dimensional parallelepiped P^10 is constructed in the given phase space R^10 at the beginning of a trajectory on the base of the perturbation vectors b^0_1, b^0_2,... b^0_10, which are orthogonal to one another and are normed by one. The value of each index λ_jcharacterizes a deformation of this parallelepiped along the corresponding perturbation vector b^i_1, b^i_2,... b^i_10, after i steps of the motion along the trajectory. The parallelepiped spreads along the given vector for a positive Lyapunov index and shrinks for a negative Lyapunov index.The sum of all indices Λ as a function of α is given in Fig. 2,d and in Table 1. This quantity determines the flow divergence and, hence, the evolution of a phase volume along a trajectory. For the given dissipative system, the divergence is negative and increases with the growth of the dissipation α. This means that the phase volume element shrinks, on the whole, along a trajectory for all values of α. The greater the dissipation, the greater the contraction. For the regular attractors n*2^0, the contraction is greater than that for the corresponding strange attractor n*2^∞.Since we deal with a dissipative system, whose divergence must be negative in any modes, the phase volume must always shrink. But b^i_1 > 0 in the modes of a strange attractor, and there occurs the exponential spreading in this direction. Because two adjacent orbits cannot permanently exponentially diverge, the strange attractor organizes itself so that it creates a fold (Fig. 1,a) or a funnel (Fig. 1,c) in itself, where the mixing of trajectories is realized. Even a slight deviation of the initial data influences essentially the evolution of the trajectory, namely the deterministic chaos is created. Such a chaos characterizes the appearance of a random nonpredictable behavior of a system controlled by deterministic laws. We note that, in real biosystems, the fluctuations are permanently present and, in unstable modes, create chaotic states. Thus, the given mathematical model adequately describes stable autoperiodic modes, as well as unstable chaotic ones.=1§ CONCLUSIONS Due to the successful development of an algorithm of calculations of the full spectrum of Lyapunov indices on an ordinary personal computer for a multidimensional phase space not bounded by the number of variables, we manage to reliably calculate these indices. This allows one to extend the possibilities to forecast the dynamics of complicated systems. By the example of a mathematical model of biosystems, we have found two different scenarios of the formation of the modes of a strange attractor: the creation of a fold or a funnel, where the formation of a deterministic chaos is realized. The self-organization of the phase flow of a strange attractor occurs under the action of two mutually competitive processes: the exponential extension (of one of the components, in the given case) and the dissipative contraction of the whole phase space. Any fluctuation which has appeared there causes the nonpredictability of the evolution of the system on the whole. 99 1 L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1975).2 A.M. Turing, Phil. Trans. Roy. Soc. Lond. B 237, 37 (1952).3 G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations (Wiley, New York, 1977).4 Yu.M. Romanovskii, N.V. Stepanova, and D.S. Chernavskii, Mathematical Biophysics (Nauka, Moscow, 1984) (in Rissian).5 T.S. Akhromeyeva, S.P. Kurdyumov, G.G. Malinetskii, and A.A. Samarskii, Physics Reports 176, 189 (1989).6 T. Reichenbach, M. Mobilia, and E. Frey, Physical Reviev E 74, 051907 (2006).7 J. W.Pröß, R. Schnaubelt, and R. Zacher, Mathematische Modelle in der Biologie (Basel, Birkhäuser, 2008).8 V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechnology and Bioengineering 33, 661 (1989).9 V.P. Gachok, V.I. Grytsay, A.Yu. Arinbasarova, A.G. Medentsev, K.A. Koshcheyenko, and V.K. Akimenko, Biotechnology and Bioengineering 33, 668 (1989).10 V.I. Grytsay, Dopov.NAN Ukr. No 2, 175 (2000).11 V.I. Grytsay, Dopov.NAN Ukr. No 3, 201 (2000).12 V.I. Grytsay, Dopov.NAN Ukr. No 11, 112 (2000).1.5mm13 V.I. Grytsay, Ukr. Fiz. Zh. 46, 124 (2001).1.5mm14 V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17,57 (2005).1.5mm15 V.V. Andreev and V.I. Grytsay, Matem. Modelir. 17, 3 (2005).1.5mm16 V.I. Grytsay and V.V. Andreev, Matem. Modelir. 18, 88 (2006).1.5mm17 V.I. Grytsay, Biofiz. Visn. No 2, 25 (2008).1.5mm18 V.I. Grytsay, Biofiz. Visn. No 2, 77 (2009).1.5mm19 A.A. Akhrem and Yu.A. Titov, Steroids and Microorganisms (Nauka, Moscow, 1970) (in Russian).1.5mm20 A.G. Dorofeev, M.V. Glagolev, T.F. Bondarenko, and N.S. Panikov, Mikrobiol. 61, 33 (1992).1.5mm21 I. Shimada and T. Nagashima, Progr. Theor. Phys. 61, 1605 (1979).1.5mm22 M. Holodniok, A. Klic, M. Kubicek, M. Marek, Metody Analýzy Nelinearnich Dynamickych Modelu (Academia, Praha, 1986).1.5mm23 S.P. Kuznetsov, Dynamical Chaos (Nauka, Moscow, 2001) (in Russian).1.5mmReceived 30.09.09 | http://arxiv.org/abs/1707.08724v1 | {
"authors": [
"V. I. Grytsay"
],
"categories": [
"nlin.CD",
"q-bio.CB"
],
"primary_category": "nlin.CD",
"published": "20170727070200",
"title": "Structural Instability of a Biochemical Process"
} |
The SAMI Galaxy Survey: Data Release One]The SAMI Galaxy Survey: Data Release One with Emission-line Physics Value-Added Products IFSintegral-field spectroscopy IFUintegral-field unitAAO [email protected] ^†How each author contributed to the paper is listed at the end. SIfA CAASTRO SIfA ICRAR RSAA Cahill Center for Astronomy and Astrophysics, California Institute of Technology, MS 249-17, Pasadena, CA 91125, USA Hubble Fellow CAASTRO RSAA AAO SIfA CAASTRO SIfA SIfA CAASTRO RSAAMax Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, GermanyRSAA UQ CAASTRO AAO AAO SIfA RSAA AAO AAO MQ AAO AAO AAO UNSW ICRAR Dept. Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA RSAA AAO ICRAR RSAA AAO AAO RSAA AAO English Language and Foundation Studies Centre, University of Newcastle, Callaghan NSW 2308, Australia AAO Atlassian 341 George St Sydney, NSW 2000, Australia AAO SIfA Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany AAO MQ AAO Swin ICRAR AAO MQ AAO SIfA CAASTRO SUPA School of Physics & Astronomy, University of St Andrews, KY16 9SS, Scotland SIfA AAO CAASTRO Swin CAASTRO ICRAR CAASTRO UMelb UMelb AAO We present the first major release of data from the SAMI Galaxy Survey. This data release focuses on the emission-line physics of galaxies. Data Release One includes data for 772 galaxies, about 20% of the full survey.Galaxies included have the redshift range 0.004 < z < 0.092, a large mass range (7.6 < log/ < 11.6), and star-formation rates of ∼10^-4 to ∼10^1. For each galaxy, we include two spectral cubes and a set of spatially resolved 2D maps: single- and multi-component emission-line fits (with dust extinction corrections for strong lines), local dust extinction and star-formation rate. Calibration of the fibre throughputs, fluxes and differential-atmospheric-refraction has been improved over the Early Data Release. The data have average spatial resolution of meanSeeing arcsec (FWHM) over the 15 arcsec diameter field of view and spectral (kinematic) resolution R=specR-red (σ=30) around . The relative flux calibration is better than 5% and absolute flux calibration better than ±0.22 mag, with the latter estimate limited by galaxy photometry. The data are presented online through the Australian Astronomical Observatory's Data Central. [ T. ZafarAAO January 15, 2018 ====================§ INTRODUCTION Our textbooks provide a reasonable picture of how the first dark matter structures assembled out of the primordial matter perturbations <cit.>. But just how gas settled into these structures to form the first stars and galaxies, and how these evolved to provide the rich diversity of galaxies we see around us today, remains an extremely difficult problem to unravel.Over the past twenty years, imaging surveys from the Hubble Space Telescope (far field) and the Sloan Digital Sky Survey (near field) have been particularly effective in identifying evolution of galaxy parameters with cosmic time and with environment across large-scale structure. This has been matched by extensive surveys using multi-object spectroscopy <cit.> that have usually provided a single spectrum within a fixed fibre aperture at the centre of each galaxy; spatial information must be drawn from multi-wavelength broadband images.It has long been recognized that large-scale multi-object spectroscopic surveys do not provide a complete picture of galaxies. The complexity of galaxies cannot be captured with a single average or central spectrum. Three-dimensional imaging spectroscopy, or IFS is needed to quantify each galaxy.Driven by pioneering work using Fabry-Perot interferometry <cit.> and lenslet arrays <cit.>, IFS has exploited the plunging costs of large-area detectors to dominate extra-galactic studies today <cit.>.The first generation of IFS surveys, sampling 10s to 100s of galaxies, have only recently completed.Examples include ATLAS^3D <cit.>, CALIFA <cit.> and SINS <cit.>.These surveys demonstrated that there is much to learn from both the stellar and gaseous components in data of this kind.However, these surveys all used instruments that target individual galaxies one at a time and are, therefore, not optimal for surveying thousands of galaxies.To move beyond catalogues of a few hundred requires effective multiplexing. Multiplexing IFS has only recently become possible.The FLAMES instrument on the VLT <cit.> was the first, with 15 integral-field units (IFUs) each having 20 spatial resolution elements in a 2×3 arcsec field of view.The two main approaches to IFS are fibre-based and slicer-based systems.Slicers have higher sensitivity below 400 nm and in the infrared as shown by the KMOS instrument on the VLT <cit.>; they also have excellent performance over narrow fields of view, particularly when assisted by adaptive optics <cit.>.However, fibres ease deployment of IFUs over wide fields of view and allow the spectrograph to be mounted on the floor rather than on the telescope, simplifying design and improving stability.Fibre based systems are therefore preferred for wide-field, multi-object IFS in the optical bands.With the aim of carrying out IFS surveys targeting thousands of galaxies, we developed the Sydney/AAO Multi-object Integral-field spectrograph <cit.> on the 3.9m Anglo-Australian Telescope.SAMI provides a multiplex of × 13 with each IFU having a diameter of 15 arcsec and uses compact fused fibre bundles with minimised cladding between the fibre cores <cit.>.The MaNGA Survey <cit.> operating on the Apache Point 2.3m Telescope, has also begun a similar project, with an IFU multiplex of × 16.Meanwhile, the high-redshift KMOS-3D and KROSS Surveys <cit.> are making spatially resolved observations of high redshift galaxies.Large-scale IFS surveys are uniquely positioned to address a number of the outstanding questions regarding galaxy formation and evolution <cit.>, including: * What is the physical role of environment in galaxy evolution?* What is the interplay between gas flows and galaxy evolution?* How are mass and angular momentum built up in galaxies? Mass is thought to be the primary discriminant driving the huge variety of galaxies observed, setting their star formation rate <cit.>, metallicity <cit.>, and morphology.However, in addition to mass, the environment of a galaxy also plays a central role in controlling such properties (e.g., and , respectively). Despite the wealth of data at hand, the physical processes that drive environmental differences are still uncertain. The processes are likely to depend on whether a galaxy is the central galaxy or a satellite in its parent halo, the mass of the parent halo, and local galaxy–galaxy interactions <cit.>. With the broad range of observables available to SAMI, we can directly test which physical processes are at play in environmental transformations.Gas flow (or lack thereof) in and out of a galaxy controls its evolution with time. Inflows have formed disks, fuelled generation upon generation of new stars, and fed supermassive black holes. In current galaxy-formation theory, galactic-scale outflows explain the difference between the theoretical cold-dark-matter mass function and the observed stellar-mass function <cit.>. A feedback process with strong mass dependence is needed to resolve this problem.Outflows offer the most promising solution <cit.>, and are clearly detected by combining gaseous emission-line ionisation diagnostics with kinematics <cit.>.Gas inflows can be traced using the measurement of misalignment between gas and stellar kinematics <cit.> and by searching for flattened metallicity gradients <cit.>.The mass and angular momentum of a galaxy are most directly probed by its kinematic state. A galaxy's accretion and merger history is central to defining its character, and aspects of this history are encoded in the line-of-sight velocity distributions. By studying the detailed kinematics of galaxies across the mass and environment plane, we unlock a new view of galaxy evolution <cit.>.IFS has defined a new set of morphological classifications in terms of dynamical properties <cit.>, such as the separation into fast rotators (rotation dominated) and slow rotators (dispersion dominated).We aim to understand how these kinematic properties are distributed across the mass–environment plane, and to make direct comparison to simulations that are now becoming available to measure more complex dynamical signatures <cit.>.IFS surveys have arrived at an auspicious time.Cosmological-scale hydrodynamic simulations can now form thousands of galaxies with realistic properties in ∼100^3 volumes <cit.>.These simulations allow study of how gas enters galaxies <cit.> and the impact of feedback <cit.>.Those at higher resolution <cit.> are probing details of disk formation, gas flows and feedback, though not yet within a full cosmological context.Direct, detailed comparison of spatially resolved data to these simulations is required to advance our understanding.In this paper, we present Data Release One (DR1) of the SAMI Galaxy Survey, building on our Early Data Release (EDR) in 2014 <cit.>.We provide data cubes for nsample galaxies and value-added products based on detailed emission-line fitting.Future releases will provide more galaxies and products.In Section <ref> we review the SAMI Galaxy Survey itself, including the selection, observations, data reduction and analysis.In Section <ref> we describe the Core data being released, with discussion of data quality in subsection <ref>. The emission-line-physics value-added products are described in Section <ref>. The online database is introduced in Section <ref>. We summarise in Section <ref>.Where required, we assume a cosmology with Ω_m=0.3, Ω_Λ=0.7 and H_0=70^-1.§ BRIEF REVIEW OF THE SAMI GALAXY SURVEYThe SAMI Galaxy Survey is the first integral-field spectroscopic survey of enough galaxies to characterise the spatially-resolved variation in galaxy properties as a function of both mass and environment.Specific details concerning the survey can be found in papers describing the SAMI instrument <cit.>, the SAMI-GAMA Sample Target Selection <cit.>, the SAMI Cluster Sample Target Selection <cit.>, data reduction <cit.> and the Early Data Release <cit.>.Below we review key aspects of the survey.§.§ The SAMI instrumentSAMI is mounted at the prime focus of the Anglo-Australian Telescope and has 1-degree-diameter field of view. SAMI uses 13 fused optical fibre bundles <cit.> with a high (75 percent) fill factor. Each bundle combines 61 optical fibres of 1.6 arcsec diameter to form an IFU of 15-arcsec diameter. The 13 IFU and 26 sky fibres are inserted into pre-drilled plates using magnetic connectors. Optical fibres from SAMI feed into AAOmega, a bench-mounted double-beam optical spectrograph <cit.>. AAOmega provides a selection of different spectral resolutions and wavelength ranges. For the SAMI Galaxy Survey, we use the 580V grating at 3700-5700Å and the 1000R grating at 6250-7350Å. With this setup, SAMI delivers a spectral resolution of R=specR-blue (σ=specDeltaSigma-blue) for the blue arm, and R=specR-red (σ=specDeltaSigma-red) for the red arm at their respective central wavelengths <cit.>. A dichroic splits the light between the two arms of the spectrograph at 5700Å. §.§ Target SelectionIn order to cover a large dynamic range in galaxy environment, the SAMI Galaxy Survey is drawn from two regions with carefully matched selection criteria.The majority of targets are from the Galaxy And Mass Assembly (GAMA) Survey <cit.>, and we denote this as the SAMI-GAMA Sample.However, the volume of the SAMI-GAMA region does not contain any massive galaxy clusters, so a second set of targets are drawn from specific cluster fields.This we denote as the SAMI Cluster Sample <cit.>.DR1 includes galaxies only from the SAMI-GAMA Sample and the selection for these targets is described by <cit.>. Briefly, the sample is drawn from the 4×12-degree fields of the initial GAMA-I survey <cit.>, but uses the deeper spectroscopy to r<19.8 of the GAMA-II sample <cit.>.The high completeness of the GAMA sample (98.5 per cent) leads to high-reliability group catalogues <cit.> and environmental metrics <cit.>.The GAMA regions also provide broad-band imaging from the ultraviolet to far-infrared <cit.>.The selection limits for the SAMI-GAMA Sample, shown in Figure <ref>, consist of a set of volume-limited samples with stellar-mass limits stepped with redshift. We select using stellar masses determined from only g- and i-band photometry and redshift, using the relationship given in Eq. 3 of <cit.>.This determination is based on the relationship between mass-to-light ratio and colour derived by <cit.>, and assumes a <cit.> initial-mass function. §.§ Observing strategy<cit.> describe the process of allocating target galaxies to fields for observation.Our standard observing sequence consists of a flat-field frame (from the illuminated AAT dome) and arc frame, followed by seven object frames each of 1800 s exposure.A flat field and arc are taken to end the sequence.The seven object exposures are offset from one another in a hexagonal dither pattern <cit.>, with the subsequent frames radially offset from the first exposure by 0.7" in each of six directions 60 degrees apart.This offset is applied based on the most central guide star in the field, using an offset in pixels on the guide camera.Variations in atmospheric refraction and dispersion between different exposures causes the effective offsets to differ for different galaxies on the same field plate.However, the high fill factor of SAMI hexabundles minimises the effect on data quality (see especially Section <ref>).The change in offset across the field is measured as part of the alignment process during data reduction as described in <cit.>.Where possible, twilight-sky frames are taken for each field to calibrate fibre-throughput. Primary spectrophotometric standards are observed each night that had photometric conditions to provide relative flux calibration (i.e. the relative colour response of the system).§.§ Data reductionRaw telescope data is reduced to construct spectral cubes and other core data products in two stages that are automated for batch processing using the “SAMI Manager”, part of thepython package <cit.>.The specifics of both stages are detailed in <cit.>.Subsequent changes and improvements to the process are described in section 3 of <cit.> and in Section <ref> below.The first stage of data reduction takes raw 2D detector images to partially calibrated spectra from each fibre of the instrument, including spectral extraction, flat-fielding, wavelength calibration and sky subtraction.Processing for this stage uses the 2dfdr fibre data reduction package <cit.> provided by the Australian Astronomical Observatory[Different versions of 2dfdr are available, along with the source code for more recent versions at <http://wwww.aao.gov.au/science/software/2dfdr>]. This stage outputs the individual fibre spectra as an array indexed by fibre number and wavelength, and referred to as “row-stacked spectra” (RSS). In the second stage, the row-stacked spectra are sampled on a regular spatial grid to construct a 3-dimensional (2 spatial and 1 spectral) cube. Processing for the second stage is done within thepython package <cit.>. This stage includes telluric correction, flux calibration, dither registration, differential atmospheric refraction correction and mapping input spectra onto the output spectral cube.The last of these stages uses a drizzle-like algorithm <cit.>.The spectral cubes simplify most subsequent analysis because the cube can be read easily into various packages and programming languages, and spatial mapping of the data is straightforward. However, in creating the spectral cube, additional covariance between spatial pixels is introduced that must be correctly considered when fitting models and calculating errors <cit.>.§.§ Comparing SAMI with other large IFS Surveys Spatial resolutionThe SAMI Galaxy Survey has less spatial resolution elements per galaxy than most first generation IFS surveys. First generation surveys were based on instruments with a single IFU with a large field of view on the sky and many spatial samples.For example, CALIFA uses the PPAK fibre bundle <cit.> that contains 331 science fibres and uses this bundle to target a single galaxy at a time. In contrast, SAMI has 793 target fibres, a factor of ×2.4 more, but distributes them over 13 targets, with a much smaller field of view per IFU.The ATLAS^3D and CALIFA Surveys target lower redshift galaxies better matched in size to their larger IFUs, leading to higher spatial resolution. Therefore, these first generation surveys continue to serve as a benchmark for local (<100) galaxies, while second generation surveys will provide much larger samples of slightly more distant galaxies (typically >100).Spectral resolutionIn the neighbourhood of theemission line, the SAMI Galaxy Survey has higher spectral resolution than most other first- and second-generation surveys.In the blue arm the large number of spectral features visible drives the survey design to broad wavelength coverage (3700–5700Å), leading to a resolution of R≃specR-blue.However, in the red arm, by limiting spectral coverage to a ∼1100Å region around theemission line we can select a higher spectral resolution, R≃specR-red.This selection is distinct from most other surveys, such as CALIFA and MaNGA, with R≃850 and R≃2000 respectively around theline. Therefore, analyses based on SAMI data can better separate distinct kinematic components <cit.>, can more accurately measure the gas velocity dispersion in galaxy disks <cit.>, and can investigate the kinematics of dwarf galaxies.The trade-off for the higher spectral resolution in the red arm is more limited spectral coverage, that only extends to ∼7400Å, whereas MaNGA reaches to ∼1 μm.Environment measuresThe SAMI Galaxy Survey also benefits from more complete and accurate environmental density metrics than other IFS surveys.The GAMA Survey has much greater depth (r<19.8 vs r<17.8) and spectroscopic completeness (>98 per cent vs ≃ 94) than the SDSS on which the MaNGA Survey is based ( and , respectively). Therefore, GAMA provides several improved environmental metrics over SDSS, including group catalogues and local-density estimates ( and , respectively).For example, 58 per cent of primary Survey targets are members of a group identified from GAMA <cit.>, but only 15 per cent are members of a group identified from SDSS <cit.>.Range in massThe SAMI Survey provides a broader range in mass of galaxies than MaNGA at the expense of more variability in the radial coverage of galaxies. Our target selection aims to be 90 percent complete above the stellar-mass limit for each redshift interval targeted while covering a large range in stellar mass (8 ≲log(M_* / ) ≲ 11.5).This selection results in a more extensive sampling of low-mass galaxies than previous surveys. It also differs from the MaNGA selection, which targets galaxies in a relatively narrow luminosity range at each redshift. The MaNGA selection leads to less variability in the radial extent of the data relative to galaxy size. Sampling of galaxy clustersThe Survey's cluster sample is also unique among IFS surveys.Massive clusters are rare, so volume-limited samples typically include few galaxies belonging to these extreme environments. However, only in clusters are the extremes of environmental effects demonstrated on galaxy evolution. With the Survey's cluster sample, one can trace in detail the evolution of galaxies in the densest environments.Other programs have targeted individual clusters for IFS observations <cit.>, but the SAMI cluster sample is the most comprehensive IFS study of clusters yet attempted.The SAMI Galaxy Survey sample includes eight different clusters (APMCC0917, A168, A4038, EDCC442, A3880, A2399, A119 and A85), allowing investigation of variability between clusters. Part of the Survey includes new (single-fibre) multi-object spectroscopy of these clusters to ascertain cluster membership, mass, and dynamical properties <cit.>.§ CORE DATA RELEASEThe galaxies included in DR1 are drawn exclusively from the SAMI-GAMA Sample. The included core data products are the regularly gridded flux cubes (spectral cubes). All of the core data included have met minimum quality standards, and the quality of the final data has been measured with care.§.§ Galaxies included in DR1Galaxies in DR1 are drawn from all nsample0 galaxies observed in the SAMI-GAMA sample through June, 2015 (AAT semesters 2013A to 2015A). This includes all galaxies in the Survey's EDR (but the data for those galaxies have been reprocessed for this Release). Table <ref> shows how the DR1 galaxy numbers compare to the current progress of the SAMI Galaxy Survey in the GAMA regions. The distribution of these targets in the stellar mass–redshift plane, on the sky and in the star formation rate–stellar mass plane can be seen in Figures <ref>, <ref> and <ref>, respectively. We have not included some observed galaxies in DR1 for quality control reasons.From the nsample0 galaxies, we removed those with: * fewer than 6 individual exposures meeting the minimum standard of transmission greater than 0.65 and seeing less than 3 arcsec FWHM (48 galaxies removed); and* individual observations that span more than one month for a single field and have differences in their heliocentric velocity frames of greater than 10 km/s (12 galaxies removed).After removing observations that did not meet these data quality requirements, nsample galaxies remain. Galaxies included in DR1 may have a small bias towards denser regions over the full field sample. The order in which galaxies are observed over the course of the Survey is set by the tiling process, which allocates galaxies to individual observing fields. Tiling is based only on the sky distribution of galaxies—not their individual properties.Initial tiles are allocated preferentially to regions with higher sky density to maximize the efficiency of the Survey over all. Figure <ref> shows the three GAMA-I fields (G09, G12 and G15) and the sky distribution of galaxies in this data release compared with the overall SAMI field sample. DR1 galaxies are distributed across the full range of the primary sample in redshift, stellar mass and effective radius as illustrated in Figure <ref>. A Kolmogorov–Smirnov test indicates that the DR1 sample has the same effective radius distribution as the SAMI field sample (D-statistic=0.025, p-value=0.85). However there is a difference in the distribution of stellar mass (D-statistic=0.08, p-value=0.001), such that lower mass galaxies are slightly over represented in the DR1 sample.§.§ Changes in data reduction methods since the Early Data ReleaseFor DR1 we use thepython package snapshot identified as Mercurial changeset , and 2dfdr version 5.62 with custom modifications. The version of 2dfdr is the same as for our Early Data Release <cit.>, and all of the modifications are described by <cit.>. These changes have been integrated into subsequent public release versions of 2dfdr. Changes in thepackage are described in the rest of this section. §.§.§ Fibre throughput calibrationTo achieve good flux calibration and uniform image quality, the relative throughput of each of the fibres fibres (including 26 sky fibres) must be normalised to a common value. We have improved the approach for normalising the fibre throughputs over that used in our EDR.The fibre-throughput calibration used in our EDR had two shortcomings that limited data quality, particularly from the blue arm of the spectrograph.In our EDR, the relative throughput of individual fibres was primarily determined from the integrated flux in the night-sky lines for long exposures, and from the twilight flat-fields for short exposures.However, the blue data (3700–5700Å) include only one strong night sky line, 5577Å, so are particularly susceptible to two problems.First, sky lines are occasionally impacted by cosmic rays, leading to poor throughput estimates for individual fibres.Second, the limited photon counts in the sky line limits the estimates of the relative throughput to ≃ 1-2 per cent. For DR1 the relative fibre throughputs were calibrated from either twilight flat-field frames, or from dome flat-field frames for fields where no twilight flat was available. The night sky spectrum was then subtracted using this calibration. If the residual flux in sky spectra was excessive (mean fractional residuals exceeded 0.025), then the fibre throughputs were remeasured using the integrated flux in the night-sky lines (as in the EDR). If all sky lines in a fibre were affected by bad pixels (typically only an issue for the blue wavelength range, which covers only a single sky line), then the mean fibre-throughput calibration derived from all other frames of the same field was adopted. The sky subtraction was then repeated with the revised throughput values. The method that provided the final throughput calibration is listed with the cubes in the online database. This approach ensures that, for the calibration options available, the best option is used to calibrate the fibre throughputs. §.§.§ Flux calibrationThe flux calibration process has been improved over our EDR to better account for transparency changes between individual observations of a field and improve overall flux calibration accuracy.In our EDR, the absolute flux calibration was applied after forming all cubes for a field of 12 galaxies and 1 secondary standard star. All objects in the field were scaled by the ratio of the field's secondary standard star observed g-band flux to the SDSS photometry after combining individual observations into cubes <cit.>.For DR1 this scaling has also been applied to each individual RSS frame for a given field before forming cubes, i.e., the scaling is now applied twice. This additional scaling ensures that differences in transparency between individual observations are removed before the cube is formed, which improves the local flux calibration accuracy and removes spatial `patchiness' in the data. The accuracy of the overall flux calibration is discussed in Section <ref>. §.§.§ Differential atmospheric refraction correctionFor DR1 we have improved the correction for differential atmospheric refraction over that in our EDR. The atmospheric dispersion is corrected by recomputing the drizzle locations of the cube at regular wavelength intervals <cit.>. In our EDR the drizzle locations were recomputed when the accumulated dispersion misalignment reached 1/10th of a spaxel (0.05 arcsec). We found that this frequency caused unphysical `steps' in the spectra within a spaxel. In DR1 we recalculated the drizzle locations when the accumulated dispersion misalignment reached 1/50th of a spaxel, i.e., five times more often than in the Early Data Release. This significantly reduced the impact of atmospheric dispersion on the local flux calibration within individual spaxels. Section <ref> elaborates on how atmospheric dispersion affects the quality of the data. §.§ Core Data Products includedSeveral Core data products are included in DR1: flux spectral cubes with supporting information, GAMA catalogue data used for the target selection, and Milky Way extinction spectra. §.§.§ Spectral CubesThe position–velocity spectral flux cubes are the products most users will value. These cubes are presented with the following supporting data, all sampled on the same regular grid: variance The uncertainty of the intensities as a variance, including detector-readout noise and Poisson-sampling noise propagated from the raw data frames.spatial covariance co-variance between adjacent spatial pixels introduced by drizzle mapping onto the regular grid. The co-variance and the format of this five-dimensional array are described in section 5.7 of <cit.>.weights The effective fractional exposure time of each pixel, accounting for gaps between individual fibres, dithering, etc. These are described in section 5.3 of <cit.>. A world-coordinate system (WCS) for each cube is included. This WCS maps the regular grid onto sky- (right ascension and declination) and wavelength-coordinates. The origin of the spatial coordinates in the WCS is defined using a 2D Gaussian fit to the emission in the first frame of the observed dither sequence. The wavelength coordinates are defined in the data-reduction process from arc-lamp frames. The accuracy of the spatial coordinates is discussed in Section <ref> and that of the wavelength coordinate in section 5.1.3 of <cit.>.Also provided for each spectral cube are estimates of the point-spread function (PSF) of the data in the spatial directions. The PSF is measured simultaneously with data collection using the secondary standard star included in each SAMI field. We provide the parameters of a circular-Moffat-profile fit to that star image (i.e. the flux calibrated red and blue star cubes summed over the wavelength axis). The Moffat profile has formf = β - 1/πα^2(1 + (r/α)^2)^-β.where α and β parameterize the fit and r^2 = x^2 + y^2 is the free variable denoting spatial position <cit.>. The reported PSF is the luminosity weighted average over the full (i.e. red + blue) SAMI wavelength range. With the parameters of the Moffat-profile fit, we also provide the corresponding FWHM, W, as given byW = 2 α√(2^(1/β) - 1),measured in arcseconds. The distribution of measured PSF is discussed in section 5.3.2 of <cit.>, and is unchanged in DR1. Finally, for convenience, we include the exact versions of the GAMA data used in the sample selection of the SAMI field sample. Note that in some cases, newer versions of these data are available from the GAMA Survey and should be used for scientific analysis. §.§.§ Milky Way dust-extinction correctionSAMI spectral cubes are not corrected for dust extinction, either internal to the observed galaxy or externally from Milky Way dust. However, we do provide a dust-extinction-correction curve for each galaxy to correct for the latter.Using the right ascension and declination of a galaxy, we determined the interstellar reddening, E(B-V), from the Planck v1.2 reddening maps <cit.> and the <cit.> extinction law to provide a single dust-correction curve for each spectral cube. Note that this curve has not been applied to the spectral cubes. To correct a SAMI cube for the effects of Milky Way dust, the spectrum of each spaxel must be multiplied by the dust-correction curve.§.§ Data QualityWe now discuss data quality measurements for the Core data released. <cit.> discusses the quality of the data in our EDR, including fibre cross-talk, wavelength calibration, flat-fielding accuracy, and other metrics. Where data quality does not differ between our EDR and DR1, we have not repeated the discussion of <cit.>.Instead, we discuss the data-quality metrics potentially affected by changes in the data reduction. §.§.§ Sky Subtraction AccuracyThe changes to fibre throughput calibration (see Section <ref>) removes occasional (less than one fibre per frame) catastrophically bad throughputs.It does not change the overall average sky subtraction accuracy, as presented by <cit.>. The lack of change in sky subtraction precision suggests that fibre throughput and photon counting noise in the blue 5577Å line is not currently a limiting factor in the precision of sky subtraction.Residuals after subtracting sky-continuum may instead arise from scattered light in the spectrograph. The residuals are shown as a function of wavelength and sky fibre number in Figure <ref>. To clarify the impact of sky subtraction errors, we sum the residual flux in wavelength bins (20 uniform bins per spectrograph arm).The sum reveals sky residuals that would otherwise be dominated by CCD read noise and photon counting errors in a single 0.5 to 1Å-wide wavelength channel.Figure <ref> shows that across most of both the blue and red arm CCDs, residuals of the sky-continuum subtraction are ∼1 percent.However, a strong residual appears at the short wavelength corners of the blue CCD.This is due to a ghost in the spectrograph caused by a double bounce between the CCD and air-glass surfaces of the AAOmega camera corrector lens (Ross Zhelem, private communication).The ghost results in poor fitting of the fibre profiles, which in turn results in poor extraction and then sky subtraction.A solution to this using twilight sky flats to generate fibre profiles has now been developed, but has not been applied to the data in DR1.§.§.§ Point spread functionThe spatial PSF is measured by fitting a Moffat function to the reconstructed image of the secondary-standard star in each SAMI field. SAMI fibres have diameter 1.6arcsec, therefore in seeing ≲ 3arcsec, the PSF in the individual dithered exposures is under-sampled. Stacking images introduces additional uncertainty from mis-alignment of the seven frames <cit.>, and from combining exposures with slightly different seeing.Therefore, the PSF of the final spectral cube is degraded from the PSF of the individual frames. In Figure <ref> we compare the FWHM of the reconstructed stellar image (output FWHM) to the mean FWHM of the individual exposures (input FWHM). For small input FWHM (≈ 1arcsec), output FWHM increases by 50%.This regime is likely dominated by PSF under-sampling.When input FWHM exceeds ≈ 1.5arcsec, output FWHM is typically 10% larger.No stars have FWHM > 3.0arcsec as such data is excluded by a quality control limit. In summary, DR1 spectral cubes have a mean PSF of meanSeeing arcsec (FWHM). §.§.§ Flux CalibrationThe relative flux calibration as a function of wavelength in DR1 is consistent with that in the EDR.By comparing SAMI data with SDSS g- and r-band images, <cit.> showed that SAMI derived g-r colours have 4.3 percent scatter, with a systematic offset of 4.1 percent, relative to established photometry.To test the absolute flux calibration, Figure <ref> shows the distribution of g-band magnitude differences between the SAMI galaxies and the corresponding Petrosian magnitudes from SDSS. To avoid aperture losses and extrapolations, the distribution is only shown for the 127 SAMI galaxies having Petrosian half-light radius () < 2arcsec from SDSS. The median offset is -0.07mag, and the standard deviation is 0.22mag.This is an improvement over the standard deviation of 0.27mag in the EDR.As pointed out by <cit.>, there is a 0.14mag scatter between SDSS Petrosian and model magnitudes for our sample, so a considerable fraction of the 0.22mag scatter is likely due to the inherent limitations in galaxy photometry. §.§.§ WCS and Centring of Fibre Bundles in CubesThe accuracy of the WCS is limited by the stability and accuracy of the single Gaussian fit on the observation chosen as the reference (typically the first frame, see Section <ref> and section 5.2 of ).By fitting to the individual observed galaxies we lose some robustness. However, we minimize the impact of mechanical errors (plate manufacturing, movement of the connectors within the drilled holes, and uncertainty of the bundle positions) on the WCS accuracy.Examining the data, we have identified three possible failure modes of our approach: * The fit may identify a bright star within the field of view of the hexabundle instead of the galaxy of interest. Examples include galaxies 8570 and 91961. * The catalogue coordinate may not correspond to a peak in the surface brightness of the object, such as one with a very disturbed morphology, or for objects where the catalogue coordinate has been intentionally set to be between two galaxies (galaxies with BAD_CLASS=5 in the target catalogue), see <cit.> for details. Examples include galaxy 91999.* Finally, the circular Gaussian distribution may not represent the true flux distribution well, leading to some instability or bias in the fit result. Examples include large, extended galaxies such as 514260.In these cases the WCS origin may not be very accurate, and the hexabundle field of view may not be well centred in the output spectral cube. We carry out two tests to characterise uncertainties in the WCS.The first is an internal check that considers offsets at different stages of the alignment process to constrain the expected WCS uncertainties. The second cross-correlates the reconstructed SAMI images with SDSS broad-band images to measure the offset between SAMI and SDSS coordinates. These two tests, which we detail in the following paragraphs, suggest that the WCS accuracy is ≲ 0.3 arcsec for most galaxies, except for the failures noted above.The internal tests to examine WCS uncertainties use alignment offsets to infer bounds on the typical size of the WCS uncertainties.The first dither pointing of an observation aims to centre each galaxy in its bundle.The dither-alignment transformation aligns the galaxy centroid positions in a dither with the galaxy centroid positions in the first (`reference') frame of an observation. Figure <ref> shows the RMS of the residuals for all bundles in a dither after the dither was aligned with the reference frame.The residuals are shown for transformations that are translation-only, translation and rotation, and using the full transformation of a translation, rotation and scaling.At least translation is necessary because the dithers are deliberately spatially offset.However rotation is also important in aligning the dither frames to the centre of the cubes as the SAMI instrument plate holder has a small (∼0.01 degrees) bulk rotation away from its nominal orientation.This rotation suffices to generate offsets from the nominal bundle centres of up to ∼1 arcsec at the edge of the field of view.A further improvement is gained using the modification of the plate scale, due to differential atmospheric refraction causing small positional shifts over the course of an observation.The mean RMS of ∼11 (0.16 arcsec) for the full transformation reflects how accurately the data are spatially combined for a typical galaxy and hence provides a lower limit to the WCS uncertainty.The cross-correlation test of the WCS accuracy compares the spatial flux distribution of the final, reconstructed SAMI cubes to SDSS g-band images.Each cube is multiplied by the SDSS-g-band-filter response and then summed spectrally.The resulting image is then cross-correlated with an SDSS g-band image. These SDSS images are centred on the expected coordinates of the galaxy (based on the GAMA input catalogue), are 36×36 arcsec in size, and have been re-sampled to the same 0.5 arcsec pixel scale as the SAMI cubes. The cross-correlation offset (measured using a fit to the peak in the cross-correlation image) is then the difference between the SAMI WCS and the SDSS WCS. These differences are shown in Figure <ref>.Outliers in most cases are caused by the cross-correlation centring on bright stars that are present in the SDSS image, but not in the SAMI field of view. Visual checks of outliers also identified five galaxies with gross errors in their SAMI cube WCS, caused by the data reduction centroiding on a bright star in the SAMI field of view rather than the target galaxy (catalogue IDs 8570, 91961, 218717, 228104 and 609396). When outliers are removed using an iterative 5σ clipping (that removes 7.7 per cent of coordinates), the mean of the remaining differences is -0.074±0.020 arcsec in right ascension and -0.048±0.037 arcsec in declination.The root-mean-square scatter is 0.18 arcsec in right ascension and 0.27 arcsec in declination. This test suggests a typical radial WCS error of 0.32 arcsec. Given that the result of the measurement of the WCS uncertainty in the cross-correlation test is consistent with the bounds suggested by the internal tests, we expect that it is representative of the actual uncertainty in our WCS for most targets. The targets subject to one of the failures mentioned above will have a much larger error in their WCS (no attempt has been made to correct these failures).§.§ Impact of aliasing from sampling and DAR on SAMI data The combined effects of DAR and limited, incomplete spatial sampling can cause the PSF of IFS data to vary both spatially and spectrally within a spectral cube, an effect we call “aliasing”.We describe this in the Appendix, but <cit.> also provide an excellent discussion. Aliasing can cause issues in comparing widely separated parts of the spectrum on spatial scales comparable to, or smaller than, the size of the PSF. Examples are spectral colour and ratios of widely spaced emission lines. We therefore check the impact of aliasing on our data and discuss options for reducing this impact.To test the impact of aliasing in SAMI data, we check the variation in colour within galaxies expected to have uniform colour across their extent. Uniform colour galaxies are chosen to be passive (no significant emission lines) and to have weak (or flat) stellar population gradients.Using only spaxels in the blue SAMI cubes that have a median S/N >15, we smooth them with a Gaussian kernel in the spectral direction (σ=15Å) to reduce noise, and then sum the flux in two bands at wavelengths 3800–4000Å and 5400–5600Å. These bands are chosen to be narrower than typical broad-band filters, but be more sensitive to the size of the aliasing effects (see Appendix).For each galaxy we then estimate the RMS scatter in the colour formed by the ratio of the flux in these two bands.Figure <ref> shows the distribution of RMS scatter measurements in the spaxel-to-spaxel spectral colour for 29 galaxies.For the default 0.5×0.5-arcsec spaxels (solid line in Figure <ref>) the median scatter is 0.052 and the 5th–95th percentile range is 0.033-0.093.Summing spaxels 2×2 within the cubes so that we have 1.0×1.0-arcsec spaxels (dotted line in Figure <ref>) leads to a reduced RMS with median value of 0.035 and the 5th–95th percentile range is 0.012-0.061.The reduction in scatter when the data are binned to larger spaxels is consistent with the scatter being caused by aliasing in DAR re-sampling.Aliasing from DAR re-sampling can also affect line-ratios. The ratio of theandemission lines is typically used to estimate dust attenuation. Variations in the PSF at these two wavelengths causes the ratio to reflect not only the true ratio of the two lines, but also the difference in the PSF between the two wavelengths. The later effect will be most pronounced where there is a sharp change in flux with spatial position in either of the two lines (such as near an unresolved H2 region). In such a region, there will be variations pixel-to-pixel (smaller than the PSF) that are larger than would be indicated by the variance information of the data alone.One possible method for reducing the impact of aliasing on SAMI data is to smooth it.For example, smoothing the - line ratio map by a 2D Gaussian kernel of Gaussian-σ of 0.5 arcsec (one spatial pixel) and truncated to 5×5 pixels removes most of the variation caused by aliasing without greatly affecting the output spatial resolution.This smoothing brings the noise properties of the - line ratio into agreement with Gaussian statistics and significantly reduces variation in the normalised spectra for (point-source) stars.The best choice for the smoothing kernel σ probably ranges between 0.2 and 1 arcsec, depending on the science goal and the level of DAR aliasing associated with the galaxy properties and observational conditions.Smoothing should only be necessary when no other averaging is implicit in the analysis (e.g. smoothing is not necessary for measuring radial gradients).Alternative data reconstruction schemes may reduce the effects of aliasing from the DAR re-sampling.Smoothing options are discussed further in A. Medling et al. (submitted) as they pertain to the emission-line Value Added Products (described briefly in Section <ref>). In general only results that depend on the highest possible spatial resolution are likely to be sensitive to aliasing. § EMISSION-LINE PHYSICS VALUE-ADDED DATA PRODUCTSWith the Core Data Products described above, our DR1 also includes Value-Added Products based on the ionized-gas emission lines in our galaxies.We provide fits for eight emission lines from five ionisation species, maps of Balmer extinction, star-formation masks, and maps of star-formation rate for each galaxy. Examples of these products are shown in Figure <ref> for a selection of galaxies spanning the range of stellar masses in DR1. §.§ Single- and multi-component emission-line fitsWe have fit the strong emission lines ( 3726,3729, , 4959,5007,6300,6548,6583, , and6716,6731) in the spectral cubes with between one and three Gaussian profiles. We fit with the LZIFU software package detailed in <cit.>.These fits include corrections for underlying stellar-continuum absorption. produces both a single component fit and a multi-component fit for each spatial pixel of the spectral cube. The latter fits select the optimum number of kinematic components in each spatial pixel.All lines are fit simultaneously across both arms of the spectrograph. The blue and red spectral cubes have FWHM spectral resolutions of specfwhmblue and specfwhmred, respectively. Assuming that the kinematic profiles are consistent for all lines, the higher resolution in the red helps to constrain the fits in the blue, where individual kinematic components may not be resolved.LZIFU first fits underlying stellar continuum absorption using the penalized pixel-fitting routine <cit.>, then uses MPFIT <cit.> to find the best-fit Gaussian model solution.Our continuum fits combine template spectra of simple stellar populations from the Medium resolution INT Library of Empirical Spectra <cit.>. These spectra are based on the Padova isochrones <cit.>. The selected templates have four metallicities ([M/H] = -0.71, -0.40,0.0, +0.22) and 13 ages (logarithmically spaced between 63.1 Myr and 15.8 Gyr). In fitting the template spectra to our observed data, Legendre polynomials (orders 2-10) are added (not multiplied) to account for scattered light and other possible non-stellar emission within the observed spectral cubes, and a reddening curve parametrised by <cit.> is applied. Note that the MILES templates have slightly lower spectral resolution than the red arm of our spectra; therefore, in low-stellar-velocity-dispersion galaxies (σ<30), the template may under-estimate the absorption.To account for this and other systematic errors from mis-matched templates, we calculate the expected uncertainty in the Balmer absorption from the uncertainty in stellar-population age as measured from the size of the D_ n4000 break.This uncertainty is added into the Balmer-emission-flux uncertainty in quadrature. Each emission line in each spaxel is fit separately with one, two, and three Gaussian components. In each case, a consistent velocity and velocity dispersion are required for a given component across all lines.For each galaxy, DR1 includes two sets of fits: one that uses a single Gaussian for each line in each spatial pixel (“single component”), another that includes one to three components for each spatial pixel (“recommended components”). Examples of these two fits are shown in Figure <ref>. For the fits with recommended components, the number of fits included for each spatial pixel is chosen by an artificial neural network trained by SAMI Team members <cit.>. For the recommended components, we also require that each component has S/N ≥5 in ; if this condition is not met, we reduce the number of components until it does. The single-component fits include eight maps of line fluxes, and a map each of ionized gas velocity and velocity dispersion. The 3726,3729 doublet is summed because the blue spectral resolution prevents robust independent measurements of it's components. Flux maps of4959 and6548 are omitted because they are constrained to be exactly one-third of5007 and6583, respectively. The recommended-component fits include maps of the total line fluxes (i.e. the sum of individual components) for each emission line. Additionally, for theline, three maps show fluxes of the individual fit components, and there are three maps each of the velocity and velocity dispersions, which correspond to the individual components of the Hα emission line.The maps showing individual components offlux, velocity, and velocity dispersion are ordered by component width, i.e.first corresponds to the narrowest line and third to the widest. Where there are fewer than three components, higher numbered components are set to the floating point flag , as are all maps without a valid fit.Figure <ref> illustrates the value of the emission line fits and the richness of our DR1. It shows how the nature of gas emission changes within galaxies as a function of their stellar mass and star-formation rate. At lower stellar masses, emission is driven by star formation, and the gas typically has lower metallicity, which is represented by lower / ratios (blue). At higher stellar masses, low-star-formation-rate galaxies often host AGN, often resulting in the prominent peak in / ratio at the centre of the galaxy (red).Our DR1 includes total-flux model spectral cubes (continuum model plus all fitted emission lines) for direct comparison with the spectral cubes, and maps of quality flags to highlight issues such as bad continuum fits or poor sky subtraction.§.§.§ Accuracy of GAMA redshifts and systemic velocities from emission line fits LZIFU derived velocities are with reference to the catalogued GAMA redshifts that are listed in the SAMI input catalogue <cit.>.The GAMA redshifts are on a heliocentric frame and sourced from various surveys such as the main GAMA spectroscopic program <cit.>, SDSS <cit.>, and 2dFGRS <cit.>. To check the velocity scale of the SAMI cubes, we construct aperture spectra by summing across an 1R_ e ellipse.For SAMI cubes that do not extend to 1R_ e, we sum over the whole SAMI cube. Each aperture spectrum is then fit with LZIFU using exactly the same process as the individual cube spaxels.Figure <ref> shows the velocity difference between the assumed GAMA redshifts and that measured in the aperture spectra.The median difference is -1.6 and a robust 1σ range based on the 68–percentile range is 43.9.The GAMA redshifts used in the SAMI input catalogue were measured using the runz code, and GAMA reports an error on individual runz-derived emission-line redshifts of 33 from repeat observations <cit.>. By subtracting the two in quadrature, we estimate an intrinsic scatter of 35 for our DR1.This number is an upper limit to the true scatter in SAMI velocity measurements, because it also accounts for differences due to the spatial distribution of .For example, with single-fibre observations targeting a location that is not the dynamical centre of a galaxy, or the SAMI aperture spectrum being dominated by strongflux in the outer parts of galaxies in some cases.The distribution of velocity differences is well described by a Lorentzian distribution, as found by <cit.> for the GAMA velocity uncertainties.The best fit Lorentzian is shown by the red dashed line in Figure <ref>.The galaxies in the wings of the distribution of velocity differences tend to be those that have lower S/N ratio in the emission line flux.§.§ Star Formation Value-Added ProductsIncluded with DR1 are value added products necessary for understanding the spatially-resolved star formation. These are: * Maps ofextinction: these are derived by assuming a Balmer decrement (/ ratio), unphysical ratios have extinction corrections set to 1 (no correction). Uncertainties in the extinction correction are also provided.* Masks classifing each spaxel's total emission-line flux as `star-forming' or `other': these are derived using the line-ratio classification scheme of <cit.>.* Maps of star-formation rate: these are derived from luminosities and include the extinction and masking above. The conversion factor used is 7.9× 10^-42 ()^-1 from <cit.>, which assumes a Salpeter initial mass function <cit.>.These data products will be described in detail in a companion paper by Anne Medling, et al.§ ONLINE DATABASEThe data of this Release are presented via an online database interface available from the Australian Astronomical Observatory's Data Central[Data Central's URL is http://datacentral.aao.gov.auhttp://datacentral.aao.gov.au]. Data Central is a new service of the Observatory that will ultimately deliver various astronomical datasets of significance to Australian research. Users of the service can find summary tables of the galaxies included in our DR1, browse the data available for individual galaxies, and visualize data interactively online. The service provides for downloading individual and bulk data sets, and a programmatic interface allowing direct access to the data through the HTTP protocol. Also provided are extensive documentation of DR1, the individual datasets within it, and the formatting and structure of the returned data.Data Central presents data in an object-oriented, hierarchical structure. The primary entities of the database are astronomical objects, such as stars or galaxies. These entities have various measurements and analysis products associated with them as properties. For example, each galaxy in our DR1 is an entity in the database, with properties such as red and blue spectral cubes, LZIFU data products, and star-formation maps. In future, these galaxies may also have data from other surveys associated as properties. This structure is designed to provide an intuitive data model readily discoverable by a general astronomer.Before deciding to use Data Central to host the Survey's data, the SAMI Team worked on developing our own solution, samiDB <cit.>. We developed this solution because, at the time, there were no compelling options available to us for organising and making public a data set such as ours. samiDB is designed to require minimum setup and maintenance overhead while providing a long-term stable format.The solution also provides a hierarchical organisation of the data, which has proved valuable as an organisational model. The Team ultimately decided not to use samiDB to present the data because Data Central offers ongoing support for the data archive from the Australian Astronomical Observatory, and hence a better chance that the Survey's data will remain generally and easily available even after the Team has dissolved. However, the hierarchical data model of samiDB has become a central part of the Data Central design.Further development of Data Central is planned. Most relevant to the SAMI Galaxy Survey will be addition of all data products of the GAMA Survey, enabling seamless querying of SAMI and GAMA as a single data set. Also planned are more tools for interacting with the data online. As this development progresses, the online user interface is expected to continue to evolve, but the data of DR1 (and their provenance), are stable and in their final form on the Data Central service.§ SUMMARY AND FUTUREThe SAMI Galaxy Survey is collecting optical integral-field spectroscopy for ∼3,600 nearby galaxies to characterise the spatially-resolved variation in galaxy properties as a function of mass and environment. The Survey data are collected with the Sydney/AAO Multi-object Integral-field Spectrograph (SAMI) instrument on the Anglo-Australian Telescope. Survey targets are selected in two distinct samples: a field sample drawn from the GAMA Survey fields, and a cluster sample drawn from eight massive clusters.With this paper, we release spectral cubes for 772 galaxies from the GAMA sample of the Survey, one-fifth of the ultimate product. We also release Value-Added products for the same galaxies, including maps of emission-line fits, star-formation rate, and dust extinction. These data are well suited to studies of the emission-line physics of galaxies over a range of masses and rates of star formation. The spectral cubes enable a multitude of science in other areas.The next public data release of the SAMI Galaxy Survey is planned for mid 2018, and will include further data and value-added products.§ ACKNOWLEDGEMENTSThe SAMI Galaxy Survey is based on observations made at the Anglo-Australian Telescope. The Sydney/AAO Multi-object Integral-field spectrograph (SAMI) was developed jointly by the University of Sydney and the Australian Astronomical Observatory. The SAMI input catalogue is based on data taken from the Sloan Digital Sky Survey, the GAMA Survey and the VST ATLAS Survey. The SAMI Galaxy Survey is funded by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020, and other participating institutions. The SAMI Galaxy Survey website is <http://sami-survey.org/>. JTA acknowledges the award of a SIEF John Stocker Fellowship.MSO acknowledges the funding support from the Australian Research Council through a Future Fellowship (FT140100255).BG is the recipient of an Australian Research Council Future Fellowship (FT140101202).NS acknowledges support of a University of Sydney Postdoctoral Research Fellowship.SB acknowledges the funding support from the Australian Research Council through a Future Fellowship (FT140101166).JvdS is funded under Bland-Hawthorn's ARC Laureate Fellowship (FL140100278).SMC acknowledges the support of an Australian Research Council Future Fellowship (FT100100457).Support for AMM is provided by NASA through Hubble Fellowship grant #HST-HF2-51377 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.CF gratefully acknowledges funding provided by the Australian Research Council's Discovery Projects (grants DP150104329 and DP170100603).BC is the recipient of an Australian Research Council Future Fellowship (FT120100660).§ CONTRIBUTIONSAWG and SMC oversaw DR1 and edited the paper.SMC is the Survey's Principal Investigator.JBH and SMC wrote the introduction. JB oversaw the target selection, and wrote those parts of the paper. NS wrote sections on the changes to the data reduction, and oversaw the data reduction with JTA and RS. ITH oversaw the emission line fits and produced Figures <ref> and <ref>. AMM ran quality control on the emission line fits, produced the higher-order value-added data products, and coordinated ingestion of these to the database. BG helped coordinate preparation of value-added products for release.MJD and LC oversaw the formatting and preparation of all data for inclusion in the online database. JvdS prepared the survey overview diagram, Figure <ref>. ADT and SMC measured the accuracy of the WCS information and wrote the corresponding Section <ref>. RMM provided heliocentric velocity corrections. FDE and JTA created Figures <ref> and <ref> and contributed to the data reduction software and to the assessment of the data quality, Section <ref>. AWG, EM, LH, SO, MV, KS, and AMH built the online database serving the data.Remaining authors contributed to overall Team operations including target catalogue and observing preparation, instrument maintenance, observing at the telescope, writing data reduction and analysis software, managing various pieces of team infrastructure such as the website and data storage systems, and innumerable other tasks critical to the preparation and presentation of a large data set such as this DR1.andyapj§ ALIASINGCAUSED BY DIFFERENTIAL ATMOSPHERIC REFRACTION CORRECTION AND LIMITED RESOLUTION AND SAMPLING The effects of differential atmospheric refraction can combine with limited spatial resolution and incomplete sampling to introduce aliasing into the spectra on scales comparable to the PSF. This aliasing is not unique to IFS, though the generally poorer sampling in both resolution and completeness tend to exacerbate the effect. We will use the much simpler case of a long-slit spectrograph to explain the effect.To understand the impact of aliasing on spectral data in the presence of differential atmospheric refraction, we consider a simple long-slit image[For our purposes, a long-slit image is an image of a set of simultaneously observed spectra with spatial coordinate along the slit (chosen to be oriented along the paralactic angle for our examples) on the vertical axis and wavelength coordinates along the horizontal axis.] of a white continuum source (i.e. one with a flat spectral-energy distribution in wavelength space). The slit has been aligned with the parallactic angle so that atmospheric refraction acts along the length of the slit. For illustrative purposes, we'll consider the fairly extreme example of an object observed at a zenith distance of 60 degrees. Throughout this section, we assume the seeing is Gaussian, with one arcsecond FWHM.Consider a long-slit image of this object with a spatial scale of one arcsecond per pixel. This image is shown (before correction for DAR) on the left of Figure <ref>a. Note that the PSF, even before correction, varies considerably along the wavelength axis due to the poor spatial sampling of the data. A correction for DAR is applied by shifting the pixels by the amount of the refraction along the spatial direction and rebinning to the original regular grid. After correction, the image of the object no longer shows a position shift with wavelength (shown on the right in Figure <ref>a). However, aliasing of the rebinning and sampling are readily visible, causing the individual spectra at each spatial location (shown below the image) to vary within the PSF, and the PSF (shown above the image) to vary with wavelength. Now, let us extend our example to be a close, 2D analogy to our own 3D spectral cubes. This extended example is shown graphically in Figure <ref>b. First, we observe the source at several dither positions and air-masses. Second, we introduced gaps in the spatial coverage that are smaller than and within the 1-arcsec pixels (and therefore not readily apparent in the individual frames on the left). The dithering ensures information falling in the gaps in one frame will be picked up in another frame. It also tends to smooth out the aliasing because individual dithers will each have a slightly different aliasing PSF, which will be averaged out in the combination. Finally, to bring our long-slit example closer to the actual process used in SAMI, we add another complication: up-sampling. SAMI fibres are 1.6 arcsec, but we sample the multiple observations onto a 0.5-arcsecond output grid.Note that, in combining these six individual frames, it is also necessary to track the weights of the individual output pixels, which account for the gaps in the input data. This extended example has all the same characteristics and similar sampling dimensions of our actual SAMI data, except that we are working with only one spatial dimension instead of two.Reviewing the resulting combined, DAR-corrected long-slit image shows that, despite its seeming smoothness, the PSF exhibits subtle but important variations with wavelength and spatial position.This long-slit image is shown on the right of Figure <ref>b. The image is fairly smooth because up-sampling and several dither positions and airmasses have averaged out some of the aliasing. Yet the subtle differences in the PSF at different wavelengths are still present. These differences are much more apparent in the plot of individual spectra, where the spectrum at each spatial position has been normalised to highlight the relative differences. The spatial location of each of these spectra is shown by the corresponding coloured tick on the right edge of the image. Spectra further from the centre of the PSF (and with lower total flux) tend to have larger relative deviations from the actual spectral shape (this trend matches our analysis of observations of individual stars with SAMI).Pixelated (discretely sampled) data observed with DAR present show effects of aliasing. These effects are exacerbated by poor spatial resolution and incomplete sampling. Combining observations with many dithers and different airmasses helps to average the aliasing out. Up-sampling combined with sub-pixel dithering of the observations can also reduce the severity of the aliasing. Aliasing is not typically seen in long-slit data because the PSF is typically well sampled. However, the tension in IFS between spatial sampling and sensitivity, and the incomplete sampling present in many designs has led to noticeable aliasing in IFS data. Although we have only demonstrated the effect in 2D, long-slit data, DAR is only a 2D effect, so our treatment of aliasing readily extends to 3D IFS data.The general impact of aliasing is that the PSF varies both with spatial and spectral position within either (2D) long-slit images or (3D) spectral cubes. This effect is subtle, and in many cases can be safely ignored without affecting results. There are, however, two important exceptions. The first exception is cases where the PSF must be known to very high accuracy. The second is when comparing data that are widely separated in wavelength, for example emission-line ratios or spatially resolved colours. Any analysis that averages over scales larger than the PSF will not be affected by aliasing, such as measures of radial gradients in galaxies and analysis that requires spatial binning to bring out faint signals. | http://arxiv.org/abs/1707.08402v1 | {
"authors": [
"Andrew W. Green",
"Scott M. Croom",
"Nicholas Scott",
"Luca Cortese",
"Anne M. Medling",
"Francesco D'Eugenio",
"Julia J. Bryant",
"Joss Bland-Hawthorn",
"J. T. Allen",
"Rob Sharp",
"I-Ting Ho",
"Brent Groves",
"Michael J. Drinkwater",
"Elizabeth Mannering",
"Lloyd Harischandra",
"Jesse van de Sande",
"Adam D. Thomas",
"Simon O'Toole",
"Richard M. McDermid",
"Minh Vuong",
"Katrina Sealey",
"Amanda E. Bauer",
"S. Brough",
"Barbara Catinella",
"Gerald Cecil",
"Matthew Colless",
"Warrick J. Couch",
"Simon P. Driver",
"Christoph Federrath",
"Caroline Foster",
"Michael Goodwin",
"Elise J. Hampton",
"A. M. Hopkins",
"D. Heath Jones",
"Iraklis S. Konstantopoulos",
"J. S. Lawrence",
"Sergio G. Leon-Saval",
"Jochen Liske",
"Angel R. Lopez-Sanchez",
"Nuria P. F. Lorente",
"Jeremy Mould",
"Danail Obreschkow",
"Matt S. Owers",
"Samuel N. Richards",
"Aaron S. G. Robotham",
"Adam L. Schaefer",
"Sarah M. Sweet",
"Dan S. Taranu",
"Edoardo Tescari",
"Chiara Tonini",
"T. Zafar"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170726121110",
"title": "The SAMI Galaxy Survey: Data Release One with Emission-line Physics Value-Added Products"
} |
Anisotropic EM Segmentation by 3D Affinity Learning and Agglomeration Toufiq Parag^1, Fabian Tschopp^4, William Grisaitis^2, Srinivas C Turaga^2, Xuewen Zhang^5Brian Matejek^1,Lee Kamentsky^1, Jeff W. Lichtman^3, Hanspeter Pfister^1^1School of Engg and Applied Sciences, Harvard University, Cambridge, MA ^2Janelia Research Campus, Ashburn, VA ^3Dept of Molecular and Cellular Biology, Harvard University, Cambridge, MA^4 Institute of Neuroinformatics, University of Zurich and ETH Zurich, Switzerland. ^5 Chester F. Carlson Center for Imaging Science, RIT, Rochester, NY. email: [email protected] 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty The field of connectomics has recently produced neuron wiring diagrams from relatively large brain regions from multiple animals. Most of these neural reconstructions were computed from isotropic (e.g., FIBSEM) or near isotropic (e.g., SBEM) data. In spite of the remarkable progress on algorithms in recent years, automatic dense reconstruction from anisotropic data remains a challenge for the connectomics community. One significant hurdle in the segmentation of anisotropic data is the difficulty in generating a suitable initial over-segmentation. In this study, we present a segmentation method for anisotropic EM data that agglomerates a 3D over-segmentation computed from the 3D affinity prediction. A 3D U-net is trained to predict 3D affinities by the MALIS approach. Experiments on multiple datasets demonstrates the strength and robustness of the proposed method for anisotropic EM segmentation.§ INTRODUCTION In past few years, connectomics has grown to become a mature field of study in neuroscience. Reconstruction of neural circuits from Electron Microscopic (EM) images of animal brain is not a hypothetical concept anymore – multiple attempts in this field have already furnished the neuroscience community with wiring diagrams from different animals <cit.>. These studies, and others e.g., <cit.>, report crucial biological discoveries stemming from the computed wiring diagrams. While electron microscopy is capable of providing the most exhaustive knowledge about the cellular anatomy and connectivity among all other imaging techniques, it also produces an enormous amount of data that is too large to process manually. All the aforementioned works adopt a semi-automated strategy where the results of automated algorithms are manually corrected afterwards.Extraction of neural shapes entails a 3D segmentation of EM data volume, i.e., tracing of cellular processes within and across different sections/planes of the EM volume. With the same resolution in all x, y, z dimensions, isotropic images can capture the continuity in z dimension (or in depth) more than other imaging approaches. In an isotropic EM volume, a cellular process almost never overlaps (to a significant extent) with that from another neuron across different sections. This characteristic of isotropic recording offers a fundamental advantage for the automatic 3D segmentation methods. Another benefit of the isotropic, or near isotropic, imaging techniques such as FIBSEM <cit.> and SBEM <cit.> is that they typically give rise to little or no staining and imaging artifacts. Not surprisingly, most of the successful efforts for neural reconstruction were performed on isotropic or near isotropic data. SBEM has a voxel resolution of 16 × 16 × 25nm in x, y, z respectively which we consider to be very close to being isotropic for practical reconstruction purposes. The success of these efforts can be largely contributed to the progress in the 3D segmentation algorithms that have been developed recently <cit.>. In addition to the improved methods,the profound improvement in automatic processing accuracy in the studies of <cit.>, that led to more than 5 times speed up in overall reconstruction time compared to <cit.>, can also be partially attributed to the transition to isotropic FIBSEM images. However, isotropic imaging has its limitations. FIBSEM, for example, is not ideal for large scale imaging in the range of several hundreds cubic microns <cit.> . There have been experiments <cit.> for scaling up the volume that FIBSEM can capture successfully, but have not yet delivered a large scale connectome. On the other hand, anisotropic approaches such as MSEM <cit.> are capable of imaging volume in cubic millimeter scale and therefore is suitable for large scale connectomics. Anisotropic imaging records images from tissue sections thicker than those isotropic methods (e.g., FIBSEM) can capture. While this strategy enlarges the brain region that can be captured by the same volume of data, it decreases the continuity in z dimension significantly. As a result, the area that pertains to one cellular process on any particular plane can overlap with multiple processes across different planes. This gives an additional challenge to the segmentation process – without an additional mechanism specifically designed to handle such situation, applying 3D segmentation in a straightforward fashion will inaccurately merge many neurons into one large body. Perhaps a natural idea to operate on this data is to apply 2D segmentation on each section <cit.> and then link the 2D segments on different sections with a linkage or cosegmentation algorithm <cit.>. However, as the results from the EM segmentation challenges SNEMI (<brainiac2.mit.edu/SNEMI3D>), CREMI (<cremi.org>) and many experiments across different research groups suggest, such an approach typically lead to a level of over-segmentation that is not very favorable for efficient reconstruction. Our observation indicates that if the input over-segmentation is largely fragmented or under-segmented, one cannot expect a high quality solution from these algorithms despite their solid conceptual and theoretical foundation. It appears to us that generating the initial over-segmentation remains a challenge for the anisotropic EM volumes.In this paper, we present a method to compute a 3D over-segmentation of anisotropic data bylearning 3D affinities directly using a deep neural network. Given such an input over-segmentation we apply the agglomeration method of <cit.> to generate the final segmentation results. We train a 3D U-net <cit.> using MALIS <cit.> loss to predict the 3D affinities in x, y, z dimensions. This particular loss function emphasizes on learning the sparse locations that are more important than other to preserve neuron topology rather than imposing equal weight on all pixels. In addition to <cit.>, multiple other studies <cit.> demonstrated the importance on training at sparse topologically important locations for EM connectomics. In addition, the experiments in <cit.> and <cit.>suggests that agglomerative clustering can achieve equivalent or better segmentation accuracy than linkage algorithms <cit.> given a 3D over-segmentation with minimal or no false merges. Extensive experiments on multiple anisotropic dataset exhibit superior performance of the 3D affinity learning and agglomeration compared to standard methods. The proposed method produces impressive improvement over the existing approaches both qualitatively and quantitatively as we report them in the result section. Another study by Funke et.al. also reports similar findings independently on different datasets. This approach uses a different agglomeration technique as well, however the quality of their segmentation is also very impressive. § 3D AFFINITY LEARNING AND OVER-SEGMENTATION Given an anisotropic volume, we learn and predict the 3D affinities among the voxels within the volume. In particular, instead of identifying whether or not one particular voxel belong to cell boundary (or membrane),we decide whether or not any pair of voxels { {x, y, z}, {x+1, y, z} } – or{ {x, y, z}, {x, y+1, z} } and { {x, y, z}, {x, y, z+1} } – reside within a cell and therefore needs to be connected together to form the segments. Learning affinities has been popularized in connectomics by the past works of <cit.>. We apply the MALIS training method proposed in <cit.> in this work.The MALIS learning algorithm emphasizes on learning the affinities that are critical to preserve the neural anatomy. Instead of training on affinities between voxels from all possible pairs in a volume, it locates the the edge that either incorrectly splits the path between two voxels from one cell or incorrectly merges voxels from different cells. These edges are often called the maximin edges (conversely, minimax orbottleneck edges of they are defined on costs) and can be efficiently computed for all possible paths between two voxels within a volume from a minimal spanning tree (MST) <cit.>. Topologically, these edges are more important than others for correct segmentation of neural shapes and therefore must be emphasized in the learning algorithm. In connectomics, multiple other studies have also discovered this phenomenon and trained a classifier with more (sometimes exclusive) attention to sparse but topologically important samples <cit.>. Although the mechanism by which these samples are selected and learned is different in each of these works.We train a 3D U-net <cit.> for learning the affinities with MALIS loss. The particular architecture we utilize extends the Caffe model for 3D convolutions and bypass connections in U-nete and is publicly available at <https://github.com/naibaf7/PyGreentea>. One notable aspect of the output ( and target label) of the U-net is we only learn the affinities of 3 consecutive sections. That is, the z-affinities we compute only connects one voxel to the adjacent voxels in the preceding and following sections only. In our experiments, training z-affinities only for the two neighboring sections resulted in the similar segmentation results as those produced by long range (>3 sections) z-affinities.Given the x, y, z affinities produced by the U-net, we apply the Z-watershed algorithm <cit.> with suitable size parameters for different datasets. § 3D AGGLOMERATION The 3D over-segmentation generated by Z-watershed is then refined by an agglomeration algorithm presented in <cit.>. Given an over-segmented volume with minimal or no false merges in it, an agglomeration algorithm repeatedly merges two neighboring supervoxels or fragmented regions into one.A supervoxel boundary classifier decides which supervoxels should be merged to reduce the fragmentation as well as which ones should be left separated for accurate identification of neuron structure. The classifier is trained to make a decision based on features computed on each of the supervoxels and the boundary separating them. The process is hierarchical, that is, after each merge operation, the boundaries and the featuresare recomputed for the newly created supervoxel. The study of <cit.> reported experimental evidence that the hierarchical agglomeration leads to better segmentation accuracy that the optimization based non hierarchical method of <cit.>. Another paper <cit.> also suggested using agglomerative clustering of <cit.> produced improved segmentation results compared to fusion method <cit.>. § EXPERIMENTS AND RESULTSWe have applied our method multiple datasets. Our findings from these experiments are summarized in the following sections. In all of our experiments, we used a 3D U-net with depth 3, very similar to the author's architecture <cit.>. The most important difference between our network and the author's version is we only pool in x and y dimensions only, and not in z, in each downsampling layer. The input and output to the network are volumes of grayscale EM stacks of sizes [204, 204, 33] and[116, 116, 3] respectively. Network trained with more z sections (input 44 and output 16) did not increase the accuracy. From different iterations of the MALIS training, we pick the iteration at which the network produces the least under-segmented output. For supervoxel training, the features were computed on all 3 affinities predicted in the x, y, z dimensions and we used the features, training strategy of <cit.> (without mitochondria information). §.§ Mouse Neocortex data <cit.>We have tested this method on the MSEM (single beam microscope) images collected from mouse cerebral cortex as discussed in <cit.>. The pixel resolution for these dataset was 3 × 3 × 30nm in x, y, z respectively. However, for our experiments, the images was downsampled by a factor of 2 in x, y dimensions. For training, a volume of 1024 × 1024 × 100 pixels (approx 6 × 6 × 3 μ^3 ) was cropped from the manually annotated parts of <cit.> data. Both the affinity predictor 3D U-net and the supervoxel boundary classifier were trained on this volume. In the first experiment, we examine the quality of the segmentation yielded by only the 3D affinity prediction and the Z-watershed <cit.> method to understand whether or not a subsequent agglomeration is necessary. A volume of 1024 × 1024 × 150 pixels (approx 6 × 6 × 4 μ^3 ) was utilized for this test. The segmentation performance is evaluated using the split-VI mesaures, i.e., the two quantities of the Variation of Information (VI) metric that correspond to uner- and over-segmentation errors. Following <cit.>, the VI quantities pertaining to under and over-segmentation are plotted on x and y axes respectively. An ideal result should have a (0, 0) error and should be placed at the origin of this plot. Please note that the under-segmentation error is more pronounced in the spli-VI plots since false merges are far more detrimental to the overall neural reconstruction accuracy than over-segmentation. In Figure <ref>, the segmentation errors from only Z-watershed outputand that of Z-watershed and agglomeration at different thresholds are plotted with red and blue respectively. The split-VI values suggests that one could achieve significantly better segmentation by applying 3D agglomeration on the result of Z-watershed.The study of <cit.> annotateda relatively large volume for their biological analyses. For a more comprehensive test, we collected 4 volumes of size 6 × 12 × 3 μ^3 and 2 volumes of size 6 × 12 × 2.4 μ^3 manually labeled volumes and compared our method with a baseline algorithm similar to that presented in the Rhoana pipeline paper <cit.>. For the baseline method, a 2D U-net was trained to predict the pixel membrane probabilities. These 2D probabilities were then used to generate a 3D over-segmentation by waterhsed on dilated membrane predictions. The parameters for the dilation and watershed were tuned to minimize the false merges. This over-segmentation is thenrefined by 3D agglomeration of <cit.>. The paper <cit.> reported improved result with agglomeration over fusion based method <cit.> on anisotropic data. For comparison, we plot the differences in split-VI <cit.> between the baseline and the the proposed method, VI_baseline - VI_proposed in Figure <ref>. Because it will be difficult to compare the curves at different agglomeration thresholds, we selected the parameters with the lowest error for both methods. Also, the plot is stretched more in the x-axis than y to emphasize the under-segmentation error more than the false splits.If the difference lies in the first quadrant (clockwise), the result from proposed method is more accurate than those of baseline method in terms of both over- and under-segmentation.In addition, we also compared our results on these six volumes with those produced by the VD2D3D affinity prediction <cit.> followed by Z-watershed and agglomeration. The VD2D3D network and the supervoxel classifier were trained on the same 6 × 6 × 3 μ^3 volume that the proposed method uses. We plot the split-VI difference with VD2D3D prediction and agglomeration in Figure <ref>. As the plot suggests, the proposed method achieves superior accuracy to both baseline and VD2D3D in both under- and over-segmentation in almost all the volumes. Figure <ref> compares 3D views of neural reconstructions from the proposed and the baseline algorithms (based on Rhoana) on one of the 6 × 12 × 3 μ^3. On each row, the left column shows the segmented volume of the proposed method. These reconstructions exhibit how the proposed method was able to correctly trace thinner processes and therefore can capture the topology more effectively than the baseline algorithm.§.§ Rat Visual Cortex (V1) dataWe have recently collected a relatively larger block of data of size 100 × 100 × 100 μ^3 from rat visual cortex. The images was collected at a resolution of 4 × 4 × 30nm. Twp volumes of size 6 × 6 × 3 and 6 × 6 × 4 μ^3 were used as training and test sets for this experiment respectively. We compared the quality of the output of the proposed method with a segmentation algorithm based on membrane prediction. Given a membrane prediction from a 3D U-net, the over-segmentation is generated in 3D by dilating the membrane probabilities to avoid potential false merges due to anisotropy. The 3D over-segmentation is then agglomerated by <cit.>. This particular membrane predictor based segmentation method is of interest due to the the efficiency it offers us – we have experimented rigorously to improve its accuracy.Our experiments on the membrane probability based technique included manipulating the input size (working on downsampled version), modifying the network architecture (alternating between 2D-3D convolutions), and testing different types of filters (with or without zero-padding) to attain the desired speed and accuracy. We generated the final segmentation by applying watershed and agglomeration on the output each of these membrane predictors.In Figure <ref>, we show the split-VI (Variation of Information) measure for each of the different predictors we tested. The x and y axes correspond to the under and over-segmentation error, as before. Each point on the curve corresponds to a threshold for the agglomeration process.Our tests suggest that the proposed 3D affinity learning with agglomeration (blue curve) achieves the best result on this dataset. The closest technique in terms of accuracy seems to be aUnet 2/3D with downsampled images. By 2/3D, we imply that different layers of this deep network applied the convolution with different dimensionality, i.e., some layers applies a 2D filter and some other layers the convolution is 3D. The real valued output on the downsampled images are scaled up before the subsequent operations. In order to test the performance of the proposed technique on longer processes, we applied the proposed segmentation algorithm on a 4 × 4 × 100 μ^3 volume of this data. We divided this tall volume into smaller (overlapping) blocks and then stitched the segmentation across these blocks using the segmentation overlap. The 3D representation of a few largest reconstructions are demonstrated in Figure <ref>. Although the results show some false splits of small processes on these results,we observed negligible false merges between two cellular processes. The membrane based method under-segmented more significantly compared to the proposed approach and very few of the largest reconstructions resembled a neuron part.On this dataset, we have also investigated whether or not standard affinity learning can achieve the same level of accuracy. Recall that, MALIS learning strategy identifies a sparse set of locations that are more important to maintain the neuron morphology and emphasized learning affinities at these locations. On the other hand, standard affinity learning has no such bias and tries to learn all affinities accurately. We used the same training and test volumes for this experiment. The segmentation produced by the proposed method with MALIS affinity resulted in a higher accuracy than that produced by the affinities learned in a standard fashion. § CONCLUSION This paper presents an algorithm for anisotropic EM volume segmentation. Through rigorous experimentation, we have demonstrated that 3D affinities learned directly from the anisotropic images by U-net using MALIS leads to very accurate over-segmentation. This over-segmentation can then be agglomerated to produce a segmentation result that is significantly superior in quality than those generated by several existing techniques. We believe the EM connectomics community will benefit profoundly be using the proposed segmentation approach on anisotropic datasets. ieee | http://arxiv.org/abs/1707.08935v2 | {
"authors": [
"Toufiq Parag",
"Fabian Tschopp",
"William Grisaitis",
"Srinivas C Turaga",
"Xuewen Zhang",
"Brian Matejek",
"Lee Kamentsky",
"Jeff W. Lichtman",
"Hanspeter Pfister"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727170828",
"title": "Anisotropic EM Segmentation by 3D Affinity Learning and Agglomeration"
} |
The Advantage of Evidential Attributes in Social Networks Salma Ben Dhaou^a, Kuang Zhou^c,Mouloud Kharoune^b,Arnaud Martin^b, and Boutheina Ben Yaghlane^aa. LARODEC, Higher Institute of Management, 41 Rue de la Liberte Cite Bouchoucha, 2000 Tunis, Tunisia b. DRUID, IRISA, University of Rennes 1, Rue E. Branly, 22300 Lannion, France c. Northwestern Polytechnical University, Xi'an, Shaanxi 710072, PR China. December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================== Nowadays, there are many approaches designed for the task of detecting communities in social networks. Among them, some methods only considerthe topological graph structure, while others take use of both the graph structure and the node attributes. In real-world networks, there are many uncertain and noisy attributes in the graph. In this paper, we will present how we detect communities in graphs with uncertain attributes in the first step. The numerical, probabilistic as well as evidential attributes are generated according to the graph structure. In the second step,some noise will beadded to the attributes. We perform experiments on graphs with different types of attributes and compare the detection results in terms ofthe Normalized Mutual Information (NMI) values. The experimental results show that the clustering with evidential attributes gives better results comparing to those with probabilistic and numerical attributes. This illustrates the advantages of evidential attributes. Belief function theory, uncertain attributes, community detection.§ INTRODUCTIONRecently, social network analysis has become an important research topic. In fact, social network analysis can be defined as a distinct research perspective within the social and behavioral sciences. It is also based on the assumption of the importance of relationships among interacting units.In social network analysis <cit.>, the observed attributes of social actors are understood in terms of patterns or structures of ties among the units. These ties may be any existing relationship between units; for example friendship, material transactions, etc.In this context, many studies use the concept of social network analysis to classify, for example the users's opinions <cit.> or to determine the true nature of a received message <cit.>.However, using the techniques of the social network analysis in a real application can be a difficult task. Indeed, we will find ourselves in front of imprecise and uncertain information. This is the case of data collected through automated sensors for example <cit.>. For these reasons, it will be more interesting to use an attributed graph which is composed of weighted vertices and edges. Although the majority of existing works in the literature focused on the study of weighted networks where the weights take integer values, recently, there has been studies of capturing nodes attributes via evidential models as presented in <cit.>.However, as we are manipulating social data, there is always a probability to get errors in the observations or missing data. In fact, the attributes can be constructed using statistical methods or maybe we are not totally sure about the type of the attribute. Indeed, for any node, we can have a vector of values composing its attribute. Hence, it will be interesting to use uncertain attributes in social networks.In the same context, many studies focus on modeling the uncertain social network. In fact, they represent an uncertain network by weighting the nodes or links with values in [0,1] to model uncertainties. Then, it will be more easier to monitor the behavior of the social network <cit.>. Nowadays, we can no longer talk about social networks without stating the concept of community detection. Indeed, in all social networks, there is a group of individuals who are closely related to each other more than to others. This may be due to shared interests, practice, apprenticeship or preferences regarding particular topics.According to Santo Fortunato <cit.>, communities, also called clusters or modules, represent groups of vertices which probably share common properties and/or play similar roles within the graph. He argues also that the word community itself refers to a social context. In fact, people naturally tend to form groups, within their work environment, family or friends.The community detection task becomes important since it allows us to classify the nodes according to their structural position and/or their attributes. Indeed, the clusters obtained by the community detection algorithm contain similar objects.The aim of this paper is to show how, from clustering uncertain attributes of the nodes, we can detect the existing communities in the graph. We will also show that, after adding some noisy uncertain attributes, the evidential generation of the attributes gives the best NMI values comparing to the numerical and the probabilistic versions.This paper is structured as follows. In section <ref>, we remind some basic concepts of the theory of belief functions and some community detection methods. Section <ref> will be dedicated to our contribution. Finally, section <ref> will be devoted to the experimentations and section <ref> will conclude the paper. § BACKGROUND In this section, some basis of the belief functions theory will be recalled first. Then, we will present a definition of an attributed graph. Finally, we will compare some community detection methods. §.§ Belief Functions Theory The belief functions theory allows explicitly to consider the uncertainty of knowledge using mathematical tools <cit.>. It is a useful and effective way in many fields of applications such as classification, decision making, representation of uncertainand not accurate information, etc.In fact, it is a suitable theory for the representation and management of imperfect knowledge. It allows to handle the uncertainty and imprecision of the data sets, to fuse evidence and make decisions.The principle of the theory of belief functions consists on the manipulation of functions defined on subsets rather than singletons as in probability theory. These functions are called mass functions and range from 0 to 1.Let Ω be a finite and exhaustive set whose elements are mutually exclusive, Ω is called a frame of discernment. A mass function is a mappingm:2^Ω→ [0,1]such that∑_X ∈ 2^Ω m(X)=1m(∅)=0 The mass m(X) expresses the amount of belief that is allocated to the subset X.We call X a focal element if m(X) > 0.A consonant mass function is a mass function which focal elements are nested A_1 ⊂ A_2 ⊂…⊂Ω. §.§ Attributed GraphsAccording to <cit.>, an attributed graph G_a=(V_a,E_a) can be defined as a set of attributed vertices V_a={v_1,…,v_p,…,v_q,…,v_n} and a set of attributed edges E_a={…,e_pq,…}. The edge e_pq connects vertices v_p and v_q with an attributed relation.§.§ Some Community Detection Methods with only Graphs StructuresIn this section, we recall some methods which aim to find communities based on the network structure. In the literature, there are several studies such as the hierarchical clustering <cit.> which is a method based on the development of a measure of similarity between pairs of vertices using the network structure. The disadvantage of this technique consists on ignoring the number of communities that should be used to get the best division of the network.The second type of methods is the algorithms based on edge removal. We present here two techniques: The algorithm of Girvan and Newman <cit.> which is a divisive method, in which edges are progressively removed from a network. In addition, the edges to be removed are chosen by computing the betweenness scores. The final step consists on recomputing the betweenness scores following the removal of each edge. This algorithm does not provide any guide to how many communities a network should split into. it is also slow. The algorithm of Radicchi et al. <cit.> is also based on iterative removal of edges but uses a different measure. It is based on counting short loops of edges in the network. This method has a principle disadvantage which consists on failing to find communities if the network containing few triangles in the first place. The last method is an approach which aim is to discover community structure based on the modularity Q <cit.>. The quality is high for good community divisions and low for poor ones. §.§ Some Community Detection Methods with Graphs Structures and AttributesIn this section, we introduce some community detection methods based on graph structure and attributes. The presented model in <cit.> uses both informations. In fact, a unified neighborhood random walk distance measure allows to measure the closeness of vertex on an attribute augmented graph. Then, the authors uses a K-Medoids clustering method to partition the network into k clusters. A second method presented in <cit.> consists on a model dedicated to detect circles that combines network structure and user profile. The authors learns for each circle, its members and the circle-specific user profile similarity metric. A third methodpresented in <cit.> consists on dealing with the uncertainty that occurs in the attribute values within the belief function framework in the case of clustering. It is important to consider both structure information and attributes in order to detect the network communities. In fact, if one source of information is missing or noisy, the other can solve the problem.§ PROPOSED PROCESS§.§ Graphs with Uncertain AttributesGenerally, a social network is modeled by a graph G=(V,E) where V is a set of vertices and E a set of edges. However, such a representation does not take into account imperfections resulting from inaccurate and uncertain data.Therefore, it will be interesting to combine the theory of graphs with the theories dealing with uncertainty like probability <cit.>, possibility or theory of belief functions <cit.> in order to provide a general framework for an intuitive and clear graphical representation of real-world problems <cit.>.Therefore, an uncertain social network will be represented by the classic notation of a simple graph in addition of attributes defined in [0,1] on the nodes and links.In this paper, we want to show how we can detect communities for graphs with uncertain attributes. We precise that this is not a new method of community detection, but a way to consider these kind of data.§.§ Algorithm In the algorithm <ref>, we propose a method of generating numerical, probabilistic and evidential attributes in order to find communities and show how different attributes make it possible to place each node in its true community.In the first step, we give a numerical attribute to each node (a single value x ∈[0,1]) which indicates the membership of that node to the community according to the number of communities. We consider the node's class C_i among the set of n possible classes according to the value of x:x∈[ i-1/n,i/n].First scenario: We randomly generate the values of the attributes for each node v ∈ C_i of the graph. We consider three kind of attributes: numerical, probabilistic and evidential.* Numerical attribute: We generate a value x in [ i-1/n,i/n] for v. * Probabilistic attribute: We generate a value x in [ i-1/n,i/n] corresponding to the probability p(v ∈ C_i). For the n-1 other probabilities, we generate n-1 values in [0, 1-x] that we associate randomly to the other classes. In order to normalize the probability we divide by the sum of the generated values. This process generates n values x_i. * Evidential attribute: We generate consonant mass function. First, we generate a value x in [ i-1/n,i/n] corresponding to the probability m(C_i). Then the mass of the 2^n-1 other focal elements containing C_i are generated in [0, 1-x] and randomly associated to the focal elements. At last, we normalize the mass function as in the probabilistic case. This process generates 1+2^n-1 values x_i. Second scenario: In order to avoid the arbitrary level of value on the real class, we affect the highest value to the real class.* Numerical attribute: In that case, we have only one value, so this second scenario cannot concern the numerical attributes. * Probabilistic attribute: We search the maximum of the n values x_i, and we swap the values. * Evidential attribute: We search the maximum of the 1+2^n-1 values x_i, and we swap the values.After the generation of the attributes of each node, the community detection is made by the K-Medoids algorithm which is robust in the presence of noise. Moreover, this algorithm is interesting and effective in the case of small data. In the case of evidential attributes, we use the distance of Jousselme <cit.> between the attributes.After that, we compare the obtained clusters with the real clusters. In order to measure the clustering quality in each cluster, we use the Normalized Mutual Information (NMI), a measure that allows a compromise between the number of clusters and their quality <cit.>. The NMI is given by:NMI(A,B) = H(A)+H(B)/H(A,B)withH(A) = - ∑_a P_A(a) log P_A(a)H(A,B) = - ∑_a,b P_A,B(a,b) log P_A,B (a,b)In a second step, in order to evaluate the robustness of the proposed approach, we select randomly few nodes of the graph and modify their class. Then, we compute again the NMI and compute the Interval of Confidence.Algorithm <ref> shows the outline of the process followed for evidential attributes in the second scenario. §.§ General ExampleLet G be a network with 2 communities: * C_1={1,2,3}* C_2={4,5,6,7}§.§.§ Generation First Scenario: We start first by generating 3 types of attributes: * Numerical Attributes: for the nodes of C_1, we set the value of x be a real number in [0,0.5] and for the nodes of C_2, the attributes will be set in [0.5,1]. For example, a node v in C_1 can have an attribute value equal to 0.2.* Probabilistic Attributes: for the nodes of C_1, we consider two values, x which is in [0,0.5] and the second one is 1-x. For the elements of C_2 we do the same except that we choose the first value x in [0.5,1]. For example, a node v in C_2 can have an attribute values equal to (0.6,0.4). * Evidential Attributes: for the nodes of C_1, we consider a mass function with only two focal elements: x on C_1, picked in [0,0.5] and one on Ω. The second value will be equal to 1-x. For example, a node v in C_1 can have as attributes (0,0.4,0,0.6).For the elements of C_2, we follow the same process, except that we generate the first value in [0.5,1]. After that we use the K-medoids algorithm to cluster the nodes according to the three types of attributes. Then, we compare the obtained clusters with the real ones according to the NMI values. And finally, we compute the intervals of confidence.Second Scenario: In this scenario, we only consider probabilistic and evidential attributes. From the previous generation, we affect the highest generated value to the real class. Let's consider a node v in C_1 in the case of the probabilistic attributes. Let's assume that it has initially a couple of values (0.2,0.8). Hence, in this scenario, the node v ∈ C_1 will have a new values (0.8,0.2). In the case of evidential attributes, the node vwill have new sorted values (0,0.6,0,0.4). §.§.§ Noisy AttributesOnce the generation is done, we will select randomly few nodes of the network and we will modify their attributes in both of scenarios.First Scenario: We consider the random generation and choose for example to modify the attributes of one node v ∈ C_1. Initially this node has (0.2,0.8) as probabilistic attributes. Now, we modify that by selecting randomly a first value on C_1 in the interval [0.5,1] instead of [0,0.5]. Hence, the node v will have a new attributes values, for example (0.7,0.3). We do the same thing for the evidential attributes. After that, we use the K-medoids to cluster the nodes and compute the NMI value.Second Scenario: We consider the sorted attributes. The same process as for the random generation will be followed for both probabilistic and evidential attributes. § EXPERIMENTATIONS In this section we will perform someexperiments on real networks from the UCI data sets, such asthe Karate Club network, the Dolphins network and the Books about US Politics network.The Zachary Karate Club is a well-known social networkstudied by Zachary <cit.>. The study was carried out over a period of three years from 1970 to 1972. In this network, we find: * 34 nodes that represent the members ofKarate Club.* 78 pairwise links between members who are interacted outside the club. During the study a conflict arose between the administrator “John A" and instructor “Mr. Hi", which led to the split of the club into two. Half of the members formed a new club around Mr. Hi, members from the other part found a new instructor or gave up karate.The Dolphins, animals social network introduced by Lusseau et al. <cit.> is composed of 62 bottle-nose dolphins living in Doubtful Sound, New Zealand and social ties established by direct observations over a period of several years. During the course of the study, the dolphins group split into two smaller subgroups following the departure of a key member of the population.The network of books <cit.> is composed of 105 nodes that represent books dealing with US politics sold by the on-line bookseller Amazon.com. The edges represent frequent co-purchasing of books by the same buyers.§.§.§ Process of ExperimentationsThis experimentations allow us to show: * How we can detect communities for graphs with uncertain attributes.* To what extent the uncertain attributes make it possible to find the communities after adding noisy data.In this experimentation, we start first by generating attributes based on the structure of each network: * Numerical Attributes: We remind that for this type of attribute, we generate a single value. * Karate Club: This network has 2 communities, so we give a single value of attribute to each node belonging to C_1 in the interval [0,0.5] and a value in [0.5,1] if the node belongs to C_2.* Dolphins Network: This network has also 2 communities. We choose the same intervals as for the Karate Club: if the node belongs to C_1, we generate an attribute in [0,0.5] in [0.5,1] if the node belongs to C_2.* Books about US Politics Network: This network has 3 communities: For the node belonging to C_1, we give an attribute in [0,0.33]. Each node belonging to C_2 has an attribute in [0.33,0.66]. Finally, for the nodes of C_3, they have an attribute in [0.33,1]. * Probabilistic Attributes: For this type of attributes, we generate 2 or 3 values depending on the type of network. * Karate Club: For the nodes belonging to C_1, they have a first value picked randomly in the interval [0,0.5] and the second value is deduced from that (1-x). For the elements of C_2, the first values of attributes was picked randomly from the interval [0.5,1] and the second one is deduced from that (1-x).* Dolphins Network: Same things as for the karate club, the nodes of C_1 have a first value of attribute in [0,0.5] and the second value is deduced from that. The nodes of C_2 have a first value in [0.5,1] and the second one is deduced from the first one.* Books about US Politics Network: For the nodes of C_1, their first value of attributes will be picked in the interval [0,0.33], the second and third values will be generated randomly from [0,(1-x)]. After that, we normalize by dividing the second and the third prob by the sum of the first, second and third probabilities.For the nodes of C_2, we followed the same process and for the elements of C_1, except that we picked the first values of the attributes in the interval [0.33,0.66]. We note the same thing for the elements of C_3 except that we picked the first value in [0.66,1]. * Evidential Attributes: For this type of attributes we generate 2 and 4 values, depending on the type of network. * Karate Club: This network has 2 communities so, Ω={C_1,C_2} and 2^Ω={∅, C_1, C_2, C_1 ∪ C_2}. We choose to put 2 values on C_1 and Ω for the nodes belonging to C_1. For the rest of hypothesis, we put 0. For the value of C_1, itwas picked in the interval [0,0.5] and the second value on Ωwas deduced from the first value. We remind that the sum should be equal to 1. For the nodes of C_2, we put 2 values on C_2 and Ω. The first value of C_2 is picked in [0.5,1] and the second one is deduced of the first value.* Dolphins Network: We did the same thing with the nodes of this network as the Karate Club.* Books about US Politics: this network has 3 communities so,Ω={C_1,C_2, C_3} and2^Ω={∅, C_1, C_2, C_1 ∪ C_2, C_3, C_1 ∪ C_3, C_2 ∪ C_3, C_1 ∪ C_2 ∪ C_3}. We choose to put 4 values on C_1, C_1 ∪ C_2, C_1 ∪ C_3 and Ω when the nodes belongs to C_1. For the rest of the hypothesis, we put the value 0. The value of C_1 is picked from [0,0.33] and the rest of values is deduced from the first one. We used the same principle as deducing the rest of probabilities presented previously, except that we generated 3 other probabilities instead of 2.For the second community, we did the same thing, except that we put values on C_2 and the subsets containing C_2. The value of C_2 is picked in [0.33,0.66] and the rest of the values was deduced as explained before. For the third community, the values are generated on C_3, and each subset contain C_3. The value of C_3 is picked in [0.66,1] and the rest of values is deduced as explained before. Once the attributes generated, we use the K-medoids algorithm to cluster the nodes according to their attributes. After that, we use the NMI method to compare the detected clusters with the real clusters of each network. Then, we compute the confidence interval. These experimentations are repeated 100 times.In a second time, we sort the generated matrices by putting the highest values on C_1 and C_2 in the case of the Karate Club and Dolphins network and on C_1, C_2 and C_3 in the case of the Books about US Politics network. After that, we cluster again the nodes according to their new attributes and compute the NMI average.The second part of the experimentation consists on adding some noisy attributes by modifying the attributes of some nodes of C_1, C_2 and C_3. For each noisy attribute, we choose its value outside the interval set for its class. Then we cluster the nodes according to their attributes and compute the NMI and the interval of confidence. We perform this experimentation for the random and the sorted matrix of attributes.We precise that we use the sorted attributes matrices in the case of the probabilistic and the evidential generation.In the results below, we present the average of NMI computed for 100 executions of the experimentation and the interval of confidence for the numerical, probabilistic and evidential attributes. §.§ Comparison between the different versions of the labeled networks§.§.§ Karate Club (First Scenario)In this section, we show the results of the NMI computation of the random generated attributes. We present below the results of the average values of NMI for 100 runs of random attributes generation.NMI-AverageInterval of Confidence Numerical 0.776 [0.596,0.955] Probabilistic 0.778 [0.59,0.967] Evidential 1 [1,1] The results show that the evidential generated attributes give better results than the probabilistic and the numerical ones. In fact, we obtained a value of the NMI average equal to 1 which means that the K-medoids is able to classify the nodes according to their evidential attributes in the right cluster even when the generation is random.§.§.§ Dolphins (First Scenario) We present below the average values of NMI for 100 runs of random generated attributes in the Dolphins network.NMI-Average Interval of ConfidenceNumerical 0.782 [0.587,0.976] Probabilistic 0.765 [0.554,0.976] Evidential 1 [1,1] We notice that the average evidential NMI is the highest value comparing to the probabilistic and the numerical ones. Same thing, the K-medoids is able to classify the nodes in their right cluster based on their evidential attributes.§.§.§ Books about US Politics: First Scenario In this section, we show the obtained results of the NMI average values in the case of 100 runs of random generated attributes.NMI-Average Interval of ConfidenceNumerical 0.699 [0.551,0.848] Probabilistic 0.758 [0.668,0.848] Evidential 1 [1,1] The results show that the clustering based on the generated evidential attributes gives better results than the probabilistic and the numerical ones. In fact, the evidential NMI average is equal to one which means that all the nodes were classified in their right cluster.§.§.§ Karate Club (Second Scenario)We executed the generation of the attributes several time and we sorted the matrix of attributes (We put the highest value on the attribute C_1 or C_2 depending on the belonging of the node to C_1 or C_2). We obtain the results of the average values of NMI for 100 executions below: NMI-AverageInterval of ConfidenceProbabilistic 0.7843 [0.602,0.966] Evidential 1 [1,1] The results show that the evidential version gives an average NMI value equal to 1, which means that each node was detected in the right cluster. We notice that after sorting the probabilistic attributes, the K-medoids was not able to affect all the nodes in their right cluster.§.§.§ Dolphins (Second Scenario) We proceed to sort the matrix of generated attributes and we compute the average values of NMI for 100 executions.NMI-AverageInterval of ConfidenceProbabilistic 0.79 [0.597,0.983]Evidential 1 [1,1] We notice that the evidential version gives an average NMI value equal to 1 comparing to the probabilistic and numerical versions. We also notice that the K-medoids was not able to classify the nodes in their right clusters based on their probabilistic attributes.§.§.§ Books about US Politics (Second Scenario) We executed the generation of the attributes several times and we sorted the matrix of attributes (We put the highest value on the attribute C_1, C_2 or C_3 depending of the belonging of the node to C_1, C_2 or C_3). We obtain the results of the average values of NMI for 100 below: NMI-Average Interval of Confidence Probabilistic 0.895 [0.828,0.962] Evidential 1 [1,1] The results show that the evidential version gives an average NMI value equal to 1 comparing to the probabilistic one which means that all the nodes were classified in their right clusters.§.§ Comparison between the different versions of the labeled networks after adding the noisy attributes In this section, we present the obtained results after adding some noisy attributes. To do so, we choose randomly 1 to 9 nodes of the networks on which we add some noise. Hence, we modify their attributes values and we compute each time the NMI average values. This experimentation is repeated 100 times for each number of modified nodes, for cross-validation.At first, we consider the first scenario and we present the results obtained on Karate Club dataset in figure <ref>, on Dolphins dataset in figure <ref> and on Books about US Politics dataset in figure <ref>. From the different curves, we deduce that the evidential attributes allow the K-medoids to cluster the nodes in their right clusters better than the numerical and the probabilistic attributes. In fact, we can notice that with the evidential attributes, almost all the nodes are classified in their right clusters even when the number of the noisy nodes is equal to 9. In addition, the intervals of confidence show that the evidential attributes are better than the probabilistic and numerical ones. For example, for 3 noisy nodes, the interval of confidence in the case of the karate club network is equal to: [0.408,0.809] for the numerical version, [0.356,0.618] for the probabilistic version and [0.711,0.967] for the evidential version.In the case of the Dolphins network, the interval of confidence is equal to: [0.425,0.784] for the numerical version, [0.467,0.908] for the probabilistic version and [0.862,1] for the evidential version.Moreover, in the case of the Books about US Politics network, the interval of confidence is equal to: [0.411,0.685] for the numerical version, [0.533,0.646] for the probabilistic version and [0.965,1] for the evidential version. Now, we consider the second scenario and we present the results obtained on Karate Club dataset in figure <ref>, on Dolphins dataset in figure <ref> and on Books about US Politics dataset in figure <ref>. The results show that the clustering based on the evidential attributes gives better results than the probabilistic attributes. Indeed, the nodes with the evidential attributes are almost all classified in their right clusters. In addition, the intervals of confidence show that the evidential attributes are better than the probabilistic ones. For example, for 3 noisy nodes, the interval of confidence in the case of the karate club network is equal to: [0.467,0.793] for the probabilistic version and [0.946,1] for the evidential version. In the case of the Dolphins network, the interval of confidence is equal to: [0.515,0.791] for the probabilistic version and [1,1] for the evidential version. And in the case of the Books about US Politics network, the interval of confidence is equal to: [0.629,0.75] for the probabilistic version and [0.963,1] for the evidential version. § CONCLUSION Throughout this paper, we reminded in the second section some basic concepts of the theory of belief functions and presented some communities detection methods. In the third section, we introduced our contribution consisting on communities detection based on the uncertain attributes of the vertices. In addition, we introduced some noisy attributes and compute the NMI values and the interval of confidence in order to show how uncertain attributes on nodes can be useful to find communities and show which type of attributes gives the best results.We applied our algorithm on three real social networks: the Karate Club, the Dolphins network and the Books about US Politics.Finally, in the experimentations section, we showed the results of each step used in our work. In both scenarios, the evidential attributes generation gives a good results comparing to the probabilistic and the numerical attributes. In fact, in the random and sorted generation of attributes, when we cluster the nodes with evidential attributes, we obtained an average of NMI for 100 runs equal to one which means thatthe nodes are affected to their real cluster. In addition, when we introduced the noisy data, in the case of the evidential attributes, we obtained an average of NMI almost equal to 1. We can conclude that the theory of belief functions is a strong tool to model imprecise and uncertain attributes in the social networks.1IEEEhowto:karate “The UCI network data repository,” http://networkdata.ics.uci.edu/index.php.book Shafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton university press.book4 Scott, J. (2012). Social network analysis. Sage.jour5 Shafer, G. (1990). Perspectives on the theory and practice of belief functions. 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F., Maintz, J. A., Viergever, M. A.,Pluim, J. P. (2006). Normalized mutual information based registration using k-means clustering and shading correction. Medical image analysis, 10(3), 432-439.jour20 Jousselme, A. L., Grenier, D.,Bossé, É. (2001). A new distance between two bodies of evidence. Information fusion, 2(2), 91-101.Jour21 Dabarera, R., Premaratne, K., Murthi, M. N.,Sarkar, D. (2016). Consensus in the Presence of Multiple Opinion Leaders: Effect of Bounded Confidence. IEEE Transactions on Signal and Information Processing over Networks, 2(3), 336-349. | http://arxiv.org/abs/1707.08418v2 | {
"authors": [
"Salma Ben Dhaou",
"Kuang Zhou",
"Mouloud Kharoune",
"Arnaud Martin",
"Boutheina Ben Yaghlane"
],
"categories": [
"cs.AI",
"cs.SI"
],
"primary_category": "cs.AI",
"published": "20170726130442",
"title": "The Advantage of Evidential Attributes in Social Networks"
} |
Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Kraków, Poland[][email protected] of TheoreticalPhysics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Kraków, Poland Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Kraków, Poland We propose a nanodevice for single-electron spin initialization. It is based on a gated planar semiconductor heterostructure with a quantum well and with potentials generated by voltages applied to local gates. Initially we insert an electron with arbitrary spin into the nanodevice. Next we perform a sequence of spin manipulations, after which the spin is set in a desired direction (e.g., the growth direction). The operations are done all-electrically, do not require any external fields and do not depend on the initial spin direction. All-electric single electron spin initialization G. Skowron December 30, 2023 ================================================ Since the emergence of first quantum algorithms, which have shown that there exist problems, that can be solved much faster by a quantum computer <cit.>, an intense research on quantum computing and its possible physical implementations have begun. Qubits can be defined by nuclear spins of molecules, operated on using NMR <cit.>, internal states of isolated atomic ions <cit.> or spins of electrons (holes) trapped in quantum dots (QD), or more generally, quantum semiconductor nanostructures <cit.>.Regardless of the choice of a qubit carrier, one has to be able to perform the most fundamental operations <cit.>: initialization, manipulation and readout. For each qubit implementation, these operations require appropriate methods, which affect the fidelities and duration of operations. Fidelity near 100% is being achieved for qubits realised with ion traps <cit.>, for which the initialization and manipulation are performed using pulses of light. In QDs, in which confinement is created by heterojunctions of semiconductors, spin can also be controlled with photons. In self-assembled QDs, it is possible to initialize spin of an electron <cit.> or a hole <cit.> along the direction of magnetic field using optical transitions through trionic states. Similarly spin can be initialized in QDs formed by heterojunctions in catalytically grown nanowires <cit.>. The achieved fidelities exceed 99% <cit.>.In electrostatic QDs the lateral confinement is generated in a quantum well (QW) <cit.> or a quantum wire <cit.> using voltages applied to gates. In such systems it is impossible to initialize spin optically through trionic states, since the electrostatic potential, which is attractive for electrons, is repulsive for holes and cannot create a stable excitonic state. Instead, spin is initialized using Pauli blockades <cit.>. This method allows to set spin of the second electron in a double QD in the same direction as spin of the previously trapped electron. To deliberately set spin of the first electron, one has to apply a strong magnetic field and wait until the electron relaxes to the ground state <cit.>. Unfortunately, this operation is neither exact nor fast. In the literature we can find theoretical proposals of fast spin initialization <cit.> but none has been verified experimentally. In our article, we present a nanodevice, which allows to initialize the electron spin in a few hundred picoseconds with fidelity exceeding 99%. In this work we propose a nanodevice capable of initializing spin of a single electron using exclusively the electric field. As a quantum bit carrier we assume spin of a single electron confined inside a QW. The qubit basis states correspond to spin states with orientation parallel and antiparallel to the z-axis. During operation of the proposed nanodevice, an arbitrary spin state of the electron is turned into a state with spin parallel to the z-axis. The entire process is divided into two stages. In the first stage we separate the electron wavepacket into two parts of opposite spins (parallel to the y-axis in the left half of the nanodevice and antiparallel in the right). In the second stage, due to spatial separation of both parts, we rotate their spins independently in opposite directions by 90^∘. As a result, spins of both parts become parallel to the z-axis, which is the goal of the spin initialization procedure.The proposed device is based on a planar semiconductor heterostructure with a QW parallel to its surface. We assume InSb for the QW material due to its strong spin-orbit coupling. The potential barriers for the QW are created by presence of two adjacent layers of Al_xIn_1-xSb on both sides of the QW. It is so, because for the assumed x=25 % the bottom of the conduction bands in Al_xIn_1-xSb and InSb are shifted by about 300 meV <cit.> with respect to one another. The growth direction of the heterostructure must be chosen as the [111] crystallographic direction. In such a case the Rashba and Dresselhaus spin-orbit interactions (SOI) are described by operators of the same form and can be merged <cit.>. On the nanostructure substrate, which consists of highly donor-doped Al_xIn_1-xSb (n^++), we first deposit a 230 nm wide barrier layer of Al_xIn_1-xSb, next a 20 nm wide InSb layer constituting the QW and a second 50 nm wide barrier layer of Al_xIn_1-xSb (see Fig. <ref>).On this layer we place metallic gates (see Fig. <ref>). The entire nanodevice is covered with an additional 100 nm wide dielectric layer of AlN. Finally we cover the top of the dielectric with another metallic gate, subsequently referred to as top gate U^top (not shown in the figures). If no external electric fields are present, electrons from the substrate fill the QW forming a two-dimensional electron gas (2DEG). Electrons trapped in the QW have two motional degrees of freedom (x,y). If we now apply appropriate negative voltages to the gates, the gas can be depleted until only one electron remains. It is confined between two inner gates U^rail subsequently referred to as rails. The rails can block electron movement along the y direction and define a path along which the electron can move. Note that all voltages are applied with respect to the substrate.Initially, we apply a voltage U_bias=-400 mV to the top gate U^top and the rails U^rail. The same voltage is applied to two lateral gates denoted by U_0, hence we have U_0(t=0)=U_bias. Voltages applied to remaining gates grow proportionally to the square of the distance from the center of the nanodevice calculated along the x-axis. Since the gates are of equal widths, this translates directly into proportionality to the square of the gate index. Thus we have U_i^a,b,c,d(t=0)=U_bias+i^2U_par with U_par = -5 mV. The potential inside the nanodevice is calculated using the generalized Poisson's equation <cit.> at every time step <cit.>. The obtained potential takes into account applied variable voltages, the device geometry and time-dependent charge distribution inside the QW <cit.>. It also includes the image charge induced on the gates, self-focusing the electron wavefunction, as well as dielectric polarization counteracting this effect <cit.>. In this way we can generate a nearly perfect parabolical confinement potential along the x-axis, which is necessary to generate coherent states of the harmonic potential <cit.>. The electron confined in the QW has two motional degrees of freedom (x,y) and its Hamiltonian takes the formĤ=[-ħ^2/2m∇^2-|e|φ_el(x,y,z_0,t)]𝐈_2+Ĥ_SO,where ∇^2=∂_x^2+∂_y^2 and m=0.014m_e is the electron effective mass in InSb, -|e|φ_el is the confinement potential felt by the electron and z_0 is the z-coordinate of the QW plane. The potential is calculated at every time step using the Poisson's equation with boundary conditions taking into account voltages applied to the gates <cit.>. Note that 𝐈_2 in Eq. (<ref>) is a 2× 2 identity matrix. The wavefunction takes the form of a spinor Ψ= (Ψ_↑(x,y,t),Ψ_↓(x,y,t))^T.The last term in (<ref>) is the sum of the Rashba and Dresselhaus SOI of the following formĤ_SO = (α(x,y,t)+β)(σ_xp̂_y-σ_yp̂_x),with Pauli matrices σ_x, σ_y. The Dresselhaus coupling β=γ(π/d)^2/ħ is calculated for the QW width d=20 nm and Dresselhaus coefficient for InSb γ=228.3 eV Å^3 <cit.>. The Rashba coupling is calculated for the z-component of the electric field E_z within the QW as α(x,y,t)=α_SO|e|E_z(x,y,z_0,t)/ħ, with Rashba coefficient α_SO=523 Å^2 adequate for InSb <cit.>.For the initial state of the electron trapped in our nanodevice we assume its ground state in the confinement potential. We examine the time evolution of the electron wavefunction solving the time-dependent Schrödinger equation iteratively. To include time-dependency of voltages and the charge distribution in the QW we solve the Poisson's equation at every time step <cit.>. During the first phase of time evolution we change all the gate voltages sinusoidally according to the formula U_i^a,b,c,d,rail,top(t)=U_i^a,b,c,d,rail,top(0)+Δ Usin(Ω t), with the amplitude Δ U=350 mV and frequency of oscillations Ω tuned to the natural frequency of the harmonic confinement potential along the x-axis. In our simulations the optimal value is 70 GHz (ħΩ=0.2895 meV). At this stage of simulation we shift all the voltages by the same value, thus the shape of the potential along the x-axis does not change over time but the level of its bottom does. Oscillating voltages applied to the gates result in oscillations of E_z in the QW, which in turn introduces oscillations to the Rashba coupling<cit.>. According to the SOI Hamiltonian (<ref>), such oscillations can induce electron motion in the x and y directions. However, negative voltages applied to the rails ensure strong confinement in the y-direction, hence the amplitude of oscillations in this direction is very small. Motion along the x-axis is less restrained. If the electron spin is parallel or antiparallel to the y-axis, the electron starts to oscillate in the coherent state of the harmonic oscillator <cit.> with increasing amplitude <cit.>. If spin is parallel (antiparallel) to the y-axis, the change of the SOI coupling initially push the electron to the right (left). As a result electrons with ⟨σ_y⟩=1 and ⟨σ_y⟩=-1 oscillate with opposite phases.The red and blue curves (Fig. <ref>a) show the x-component of the expectation value of position of the electron ⟨ x⟩ for spins parallel and antiparallel to the y-axis, respectively.Now, let us assume that the spin is set in parallel to the z-axis. The spin wavefunction χ_z can be expressed as a linear combination of wavefunctions corresponding to spin parallel and antiparallel to the y-axis denoted as χ_y and χ_-y respectively χ_z=[ 1; 0 ]=1/2[ 1; i ]+1/2[1; -i ]=1/√(2)(χ_y+χ_-y).For this state ⟨σ_y⟩=0. Generally we can express the full wavefunction asΨ(x,y,t)=Ψ_y(x,y,t)χ_y+Ψ_-y(x,y,t)χ_-y.Let us now define the electron density ρ=ρ(x,t) and spin density ρ_σ=ρ_σ(x,t) (along the x-axis) asρ=∫_0^L_ydyΨ^†Ψ=∫_0^L_ydy(|Ψ_y|^2+|Ψ_-y|^2), ρ_σ=∫_0^L_ydyΨ^†σ_yΨ=∫_0^L_ydy(|Ψ_y|^2-|Ψ_-y|^2).Given these definitions we can describe the electron behavior.Initially both spatial components Ψ_y and Ψ_-y are identical and equal to the spatial wavefunction of the initial state of the electron—being the ground state in the confining potential. During time evolution they behave accordingly to their spin and their ⟨ x⟩ oscillate as shown in Fig. <ref>a. The black curve in Fig. <ref>b shows ⟨σ_y⟩ calculated only for the right half of the nanodevice according to the formula⟨σ_y⟩_right=∫_L_x/2^L_xdxρ_σ(x,t). Initially ⟨σ_y⟩_right = 0, because both spatial components of the wavefunction are identical. After a while they no longer overlap and if Ψ_y is shifted to the right (Ψ_-y to the left), the value of ⟨σ_y⟩_right is positive. For an opposite shift, ⟨σ_y⟩_right is negative. Note that the amplitude of oscillations of ⟨σ_y⟩_right grows until it reaches 0.5. The value of 0.5 indicates full spatial separation of both spin components of the wavefunction. Fig. <ref>c shows the wavepacket at this very moment. The dotted orange and solid green curves denotes the electron and spin densities along the x-axis, ρ and ρ_σ respectively. According to (<ref>,<ref>), in the area where the spin is parallel to the y-axis both densities overlap, while in the area with antiparallel spin the densities have opposite signs. Fig. <ref>c shows a situation with spin parallel to the y-axis in the right half of the nanodevice and antiparallel in the left. This is a Schrödinger's cat-like state. A similar state has been obtained experimentally in ion traps <cit.>. Now if we cease the oscillations of voltages and rise a potential barrier between two parts of the wavefunction of opposite spins, we can separate them permanently. We achieve this by lowering voltages applied to U_0 by 875 mV. The obtained potential and the electron and spin densities are visible in Fig. <ref>d. The potential energy has two minima separated by a barrier. It has to be high enough and the minima sufficiently deep to allow for operations on both spin parts independently. The voltages are switched fast but the switching moment should be chosen, so that the positions of the potential minima coincide with centers of both wavepacket parts. This way we avoid an unnecessary rise of the electron energy. We thus obtained a state with spatially separated spin parts. If we start the separation from a different superposition of states parallel and antiparallel to the y-axis, full separation will take the same amount of time but the final value of ⟨σ_y⟩_right will be different.We can now proceed to the next operation, namely, spin rotation about the x-axis. If we rotate the spin in the left half of the device clockwise about the x-axis, and in the right half counterclockwise, by 90^∘, both wavepacket parts will gain spins directed along the z-axis. The Hamiltonian (<ref>) implies that motion in the x-direction results in spin rotation about the y-axis while motion in the y-direction rotates spin about the x-axis. Therefore we need to put the electron into motion along the y-axis. The voltages applied to the rails U^rail stabilizes the wavepacket in the middle between them. Fig. <ref> shows the charge density at the very beginning of the simulation. We can induce small oscillations of the electron in the y-direction by applying sinusoidal voltages U_as(t)=U_as^0sin(ω t) between upper U^a,c and lower U^b,d gates, introducing a potential asymmetry in this direction. The black curve in Fig. <ref> shows the expectation value of the electron position along the y-axis, calculated in the right half of the nanodevice as⟨y⟩_right=∫_L_x/2^L_xdx∫_0^L_ydyΨ^†yΨ/∫_L_x/2^L_xdx∫_0^L_ydyΨ^†ΨThis is not a resonant process and the frequency ω can be arbitrary, yet not too high as the oscillations cease to be adiabatic which significantly reduces the fidelity of spin initialization. In our simulations we assumed ω=60 GHz (ħω=0.24 meV). This is the highest value for which the achieved spin initialization accuracy exceeded 99%. Electron oscillations along a straight line are not enough to rotate the spin, since after every period the spin reverts to the initial value. This situation changes when the SOI coupling depends on time <cit.>. We can accomplish this by applying an additional oscillatory voltage, with a phase shift of π/2 with respect to U_as, namely U_off(t)=U_off^0cos(ω t) , to all the gates including U^top. This way the electron moves forwards with a different coupling than when it moves backwards and the spin rotations gradually accumulate. The voltage U_off is identical for all the gates. On the other hand U_as must be applied in a way, so that spin in the left and right halves of the nanodevice rotate in opposite directions. The values of ⟨y⟩_right and analogous ⟨y⟩_left oscillate with opposite phases as shown in the diagram above Fig. <ref>. We thus apply the following voltages:U^top,rail(t) =U_bias+U_off(t),U_0(t) =U_bias+U_off(t)+U_barr,U_i^a,d(t) =U_bias+U_off(t)+U_as(t)+i^2U_par,U_i^b,c(t) =U_bias+U_off(t)-U_as(t)+i^2U_parwith U_bias=-500 mV (new bias voltage), U_as^0=250 mV, U_off^0=250 mV, U_par=-5 mV, U_bar=-875 mV. Please note the signs in front of U_as.Finally, Fig. <ref> shows the spin z-component ⟨σ_z⟩ evolutions for several different initial spin orientations. The blue curves correspond to spins lying in the yz plane, while the red ones in the xy plane. The evolutions differ only in the first phase of the device operation, lasting about 125 ps. After this time the wavepacket is separated into two parts of definite spins. One is parallel to the y-axis and the other antiparallel. At this moment the spin z-component equals zero. In the latter phase of operation, in which the spins are being rotated, the courses of ⟨σ_z⟩ overlap regardless of the initial spin orientation. At t=403 ps they all reach values close to 1. At this moment we merge both wavepacket parts removing the potential barrier that separates them spatially. To do this we again put U_barr = 0. The barrier should not be removed instantly and it is worth to spend a dozen of picoseconds for this part to avoid unnecessary energy rise of the final electron state, resulting in wavepacket oscillations. The final value of spin slightly depends on the initial spin orientation but for all the simulated cases the fidelity falls between 99.3% and 99.7%. According to our calculations the highest fidelity is obtained for initial states being equally weighted linear combinations of spins parallel and antiparallel to the y-axis, hence for initial spin parallel to the z or x axis. The lowest fidelity has been obtained for highly non-equal linear combination weights. It is possible to decrease the frequency of oscillations below 50 GHz for the price of a proportional increase of the initialization time. This way, however, the fidelities may rise.Thus far, we managed to perform a simulation of spin initialization in time sufficiently lower than the spin coherence time only for InSb material parameters, for which the SO coupling is very strong: α_SO = 523 Å^2. Our simulations have been performed for an ideal QW without any doping or structural imperfections. Currently fabricated InSb QWs are not yet ideal and their imperfection can result in randomly fluctuating fields giving contributions to the Rashba SO interaction <cit.>. The electron wavefunction in our nanodevice is spatially stretched over a distance of about d=600 nm along the x-axis, which is much greater than the fluctuation correlation length λ=30 nm. Therefore, their influence is effectively averaged (proportionally to λ/d) and significantly reduced. Because these fluctuations are time independent (do not depend on gate voltages), their influence on the second stage of the simulation, namely on spin rotations, is negligible. However the first stage, in which two wavepacket parts change their positions, might be disturbed and the fidelity of spin initialization reduced. It should be possible to avoid such fluctuations in nanostructures made of Si/SiGe. In Si the Rashba coupling α_SO is about 5000 times lower than in InSb <cit.> but the coherence time is several orders longer <cit.> and the electron effective mass is many times higher, which gives hope that the initialization time will be sufficiently lower than the coherence time.The nanodevice presented in Figs. <ref> and <ref> might be multiplied by putting copies of the device to the left and to the right. This way we obtain a multiple QD. However, if we only duplicate the nanodevice we can confine two electrons inside, then set the spins of each one of them separately and merge. This way also singlet-triplet qubits can be initialized. We have designed a nanodevice capable of initializing spin of a single electron regardless of its initial orientation. By applying a sequence of voltages to the local gates, we can set the spin parallel to the z-axis. The process does not depend on the initial spin and the outcomes are virtually identical. The operation of the device does not require any external fields, microwaves or photons. The goal is achieved all-electrically through application of voltages to the gates. The calculations are done for realistic material constants and the assumed geometry details. Our approach includes the fundamental electrostatic effects important for operation of the device. The performed simulations take into account subtle effects not accounted for using perturbation methods and model potentials. Interface imperfections have not been taken into account. Short range interactions are averaged due to spatial stretching of the wavefunction along the x-axis over a distance of about 600 nm. However, charges trapped at the interfaces can be a source of potential nonparabolicity, which can affect the operation of the nanodevice. This work has been supported by National Science Centre, Poland, under UMO-2014/13/B/ST3/04526. | http://arxiv.org/abs/1707.09056v2 | {
"authors": [
"S. Bednarek",
"J. Pawłowski",
"M. Górski",
"G. Skowron"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727214955",
"title": "All-electric single electron spin initialization"
} |
[][email protected] QUIT group, Dipartimento di Fisica, Università degli Studi di Pavia, via Bassi 6, 27100 Pavia, Italy Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia, Italy [][email protected] QUIT group, Dipartimento di Fisica, Università degliStudi di Pavia, via Bassi 6, 27100 Pavia, Italy Istituto Nazionale di FisicaNucleare, Gruppo IV, via Bassi 6, 27100 Pavia, Italy [][email protected] group, Dipartimento di Fisica, Università degliStudi di Pavia, via Bassi 6, 27100 Pavia, Italy Istituto Nazionale di FisicaNucleare, Gruppo IV, via Bassi 6, 27100 Pavia, ItalyQuantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation—the so called Weyl walk—one finds a non linear realisation of the Poincaré group, which recovers the usual linear representation in the small wave-vector limit.In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincaré group and the group of dilations. Quantum Walks, Weyl equation and the Lorentz group Paolo Perinotti December 30, 2023 ================================================== § INTRODUCTIONThe conjecture, originally advanced by Feynman <cit.>, that the laws of physics can be ultimately modelled by finite algorithms is a very inspirational proposal <cit.>.There are many reasons why this might prove to be the case and, thus, for adopting this conjecture as a standpoint for a research program.The primary reason is stated by Feynman himself: “It always bothers me that according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space and no matter how tiny a region of time". A similar concern is that in an arbitrarily small region of a continuous space-time it is in principle possible to store an infinite amount of bits of information. The only alternative to this situation is that the dynamics of systems in a finite region of space-time is perfectly computed by a finite algorithm running on a finite memory. Furthermore, the idea that the dynamical laws could be reconstructed within a (quantum) computational framework appears as a natural continuation of the research on quantum foundations from the information perspective (see e.g. Refs. <cit.> and for a comprehensive historical overview see Refs.<cit.>).As long as we accept that the best microscopic theory at our disposal is quantum theory, the most natural computational model for the description of physical laws is a quantum cellular automaton <cit.>. The approach to the foundations of quantum field theory based on quantum cellular automata was explored for various decades <cit.> and it is gathering increasing interest<cit.>. Nevertheless, the idea that a discrete quantum computer can exactly compute the evolution of elementary physical systems is seemingly at clash with continuous symmetries <cit.>. In recent years, free relativistic field equations were derived starting from the requirements of homogeneity, locality, linearity and isotropy <cit.>. The free quantum field theory (Weyl, Dirac, and Maxwell) is achieved by restricting to evolutions that are linear in the field–i.e. a quantum walk–in the limit of small wave-vectors, namely for states so delocalised that the discrete underlying structure cannot be resolved. It is remarkable that Lorentz-invariant equations can be derived without imposing the relativity principle, and not even mechanical notions. However, the Lorentz symmetry has no direct interpretation in the above framework, where the geometry of space-time is not assumed a priori. The achievement of Weyl, Dirac and Maxwell's equations is a clear indication that an alleged conflict between discrete dynamics and continuous symmetries was drawn based only on naive intuition.In Ref. <cit.> the notion of inertial reference frame has been formulated in terms of representation of the dynamics parameterised by the values of the constants of motion. Such notion is suitable to the study of dynamical symmetries, without the need of resorting to a space-time background. In this way the Galileo principle of relativity is formulated by identifying the notion of change of inertial frame with the change of representation that leaves the eigenvalue equation of the quantum walk invariant. In the same Ref. <cit.> it has been shown that such changes of representations for the Weyl quantum walk encompass a non-linear realization of the Poincaré group. This result, besides embodying a microscopic model of Doubly Special Relativity (DSR) <cit.>, represents a proof of principle of the coexistence of a discrete quantum dynamics with the symmetries of classical space-time.In this paper we review and extend the results of Ref. <cit.> classifying the full symmetry group of the Weyl quantum walk, which is a semidirect product of the group of diffeomorphic dilations of the null mass shell by the Poincaré group. § WEYL QUANTUM WALKA quantum cellular automaton gives the evolution of a denumerable set of cells, each one corresponding to a quantum system. We consider the case in which each quantum system is described by the algebra generated by a set of field operators. Following the definition of Ref.<cit.>, a quantum cellular automaton is an automorphism of the quasi-local algebra. The restriction to non interacting dynamics corresponds to consider algebra automorphism that are linear in the field operators (i.e. each field operator is mapped to a linear combination of field operators). In the same way the dynamics of a free field is specified by its single particle sector, a linear quantum cellular automaton is specified by a quantum walk describing the evolution of a single particle. A quantum walk <cit.> on a discrete lattice Γ of sites ∈Γ is given by a unitary operator A ∈ℒ(ℋ) where ℋ := ℓ^2(Γ) ⊗ℂ^𝗌 where ℓ^2(Γ) is the space of square summable functions on Γ and ℂ^𝗌 corresponds to some internal degree of freedom.If |⟩, |i⟩ are orthonormal basis for ℓ^2(Γ) and ℂ^𝗌 respectively, a (pure)state in ℋ is a vector |ψ⟩ = ∑_∈Γ , i ∈𝗌ψ(,i) |⟩|i⟩ where ∑_∈Γ ,i ∈𝗌|ψ(,i)|^2 =1.The quantum walk A is usually assumed to be local, i.e., for any , we have that ⟨|⟨i|A |'⟩|i'⟩≠ 0 only if ' belongs to a finite neighboring set[ For example, if Γ is the one dimensional lattice which we identify with the set of integers ℤ, we may require |x-x'| ≥ n ⇒⟨x|⟨i|A |x'⟩|i'⟩ = 0 for some n≥ 1. More synthetically we can say that the unitary matrix A is block-sparse.]. As it shown in Ref. <cit.> (which we refer to for a complete discussion), in the three-dimensional case with minimal dimension 𝗌=2 the assumptions of locality, homegeneity, and isotropy single out only one lattice, the body centered cubic one, and four admissible quantum walks (modulo a local change of basis) A^(±), B^(±).These quantum walks are given by the following unitaryoperatorsA^(±) = ∑_h∈ S T_h⊗ A^(±)_h B^(±) = ∑_h∈ S T_h⊗ B^(±)_hB^(±)_h= (A^(±)_h )^TwhereS is a set of generators of the BCC latticeS:= {±h_1, ±h_2, ±h_3, ±h_3 } with_1=1/√(3)[ 1; 1; 1 ], _2=1/√(3)[1; -1; -1 ],_3=1/√(3)[ -1;1; -1 ], _4=1/√(3)[ -1; -1;1 ],T_h are the translation operators T_h|⟩ = |-h⟩,and the matrices A^(±)_h are defined as follows:A^(±)__1 = [ ζ^* 0; ζ^* 0 ],A^(±)__2 = [ 0 ζ^*; 0 ζ^* ],A^(±)__3 = [0 -ζ^*;0ζ^* ],A^(±)__4 = [ζ^*0; -ζ^*0 ],A^(±)_-_1 = [0 -ζ;0ζ ],A^(±)_-_2 = [ζ0; -ζ0 ], A^(±)_-_3 = [ ζ 0; ζ 0 ],A^(±)_-_4 = [ 0 ζ; 0 ζ ]ζ=1± i/4.From Eq. (<ref>) one immediately sees that the quantum walk commutes with the lattice translations generated by the vectors h_i, i.e. [A^±,T_h_i⊗ I]=[B^±,T_h_i⊗ I]=0. It is therefore convenient to consider the Fourier transform basis|=1/√(|B|)∑_∈Γe^-i·|, |=1/√(|B|)∫_Bd e^i·|,=∑_j=1^3k_j_j, _j·_l=δ_jl .where B is the first Brillouin zone of the BCC lattice (see Fig. <ref>). In the Fourier basis the quantum walks of Eq. (<ref>) becomesA^(±)= ∫_B d ||⊗ A^(±)_,A_ǩ = I λ^(±)(ǩ) -i ň^(±)(ǩ) ·σ̌^(±)λ^(±) (ǩ) :=c_x c_y c_z ∓ s_x s_y s_z ň^(±) (ǩ)=[ n_x^(±); n_y^(±); n_z^(±) ]:= [ s_x c_y c_z ±c_x s_y s_z; c_x s_y c_z ∓s_x c_y s_z; c_x c_y s_z± s_x s_y c_z ]c_i = cos( k_i/√(3))s_i = sin( k_i/√(3)) σ̌^(±) :=( σ_x , ∓σ_y , σ_z)^T.It is possible to show that the matrices A^(±) can be written as A^±_=e^-i k_x/√(3)σ_xe^∓ i k_y/√(3)σ_ye^-i k_z/√(3)σ_z.from which one can immediately see that, in the limit of small wave-vector → 0,the quantum walk A^(+) recovers (up to a rescaling /√(3)→) the Weyl equation for right-handed spinors, i.e. (i ∂_t - k·σ) ψ=0. Therefore, in order to lighten the notation, it is useful to make the rescaling/√(3)→.We can also verify that, in the limit → 0, the quantum walk A^(-) recovers, up to the change of basis induced by the conjugation with the σ_y matrix, the Weyl equation for left-handed spinors i.e. (i∂_t + k·σ) ψ=0. For this reason, the quantum walks A^(±), B^(±) are called Weyl quantum walks. The Weyl equation is also recovered when | k - k_i | → 0where k_1 := π2(1,1,1), k_2 := -π2(1,1,1), k_3 := π(1,0,0). For →k_2 we have the same chirality as for→k_0 :=0 while for →k_1, k_3 the chirality changes. We have then that a single quantum walk describes four different kind of massless particles, two left-handed and two right-handed. This fact can be interpreted as an instance of the known phenomenon of fermion doubling<cit.> but with a different discrete framework. In the following we will use the expression “small wave-vector” to denote the neighborhoods of the vectors k_i, i=0,…3. §.§ The map n() Before discussing the symmetries and the change of inertial frame for the Weyl Quantum Walks, we are going to describe some features of the mapsn^(±)() defined in Eq. (<ref>).The results we are going to show, will be used for the characterization of the symmetry transformations of the Weyl Quantum Walks. For sake of simplicity, we focus on the map n^(+)()=: n() but the same analysis can be carried out for the map n^(-). Moreoverthe map n() is a smooth analytic map from the Brillouin zone B to ℝ^3. Its Jacobian J_ň(ǩ) is given byJ_ň(ǩ):=[∂_in_j(ǩ)]=cos(2k_y)λ(ǩ),and it vanishes on the set :=∪, where:= {ǩ∈ B | cos(2 k_y)=0},:={ǩ∈ B | λ(ǩ) = 0}.Let us then define the open setsB'_0:={ǩ∈ B | λ(ǩ) > 0, cos(2 k_y) > 0}, B'_1:={ǩ∈ B | λ(ǩ) < 0, cos(2 k_y) > 0}, B'_2:={ǩ∈ B | λ(ǩ) > 0, cos(2 k_y) < 0}, B'_3:={ǩ∈ B | λ(ǩ) < 0, cos(2 k_y) < 0 }.and let us denote with n_i() the restriction of n() to the set B'_i.Since J_ň(ǩ) ≠ 0 for ∈ B'_i the map n_i() defines an analytic diffeomorphism between B'_i and its image n_i(B'_i ). An expression for the inverse map n^-1_i : ℝ^3 →B'_i can be obtained exploting the following identities:2(λ n_x - n_y n_z ) = sin 2 k_x cos 2 k_y, 2(λ n_z - n_y n_x ) = sin 2 k_z cos 2 k_y 1-2(n^2_x+n^2_y) = cos 2 k_y cos 2 k_x, 1-2(n^2_z+n^2_y) = cos 2 k_y cos 2 k_z2( λ n_y + n_x n_z) =sin 2 k_y, λ^2 = 1 - n_x^2 -n_y^2 - n_z^2.The ambiguities emerging from the inverse trigonometric functions are solved by the requirement thatn^-1_i(n) ∈ B'_i. One can see that the domain of the inverse function coincides with the unit ball in ℝ^3 except for theimage n() of the critical pointsof n. This set is easily characterized as follows:H':= \ n(), n()= {m̌∈𝖴 | m_x = ± m_z , 2m_x^2 + 2m_y^2 = 1 }, 𝖴 := {m̌∈ℝ^3| m^2<1},namely the unit ball minus two ellipses (see Fig. <ref>). The map n_i then defines an analytic diffeomorphism between B'_i and 𝖧'. We can easily see that 𝖧' is connected but not simply connected. For our purposes we will need to restrict the range of the function n to a star-shaped (and then simply connected) region. The largest star-shaped region including H' is := \ ',':={m̌∈𝖴 | m_x = ± m_z , 2m_x^2 + 2m_y^2 ≥ 1 },and we also restrict the domain of n_i (see Fig. <ref>) to the counter imageB_i :=n^-1_i(𝖧). Let us summarize what we have shown so far.We have defined four different sets B_i such that their union is the whole Brillouin zone B except a null-measure set. We introduced the set 𝖧 which is star shaped and differs from the unit ball in ℝ^3 by a null measure set.For each i=0,… , 3, the map n_i() defines an analytic diffeomorphism between B_i and 𝖧.We can verify that each of the vectors _i, which were defined at the end of the previous section, belongs to a different set B_i, namely _i ∈ B_i. In the following we will see that wecan interpret the four regions B_i as the momentum space of four different massless fermionic particles. § CHANGE OF INERTIAL FRAMEIt is now convenient to express the dynamics of the Weyl quantum walk through its eigenvalue equationA_ǩψ(ω,ǩ)=e^iωψ(ω,ǩ),whose solution set provides an equivalent way to present the walk operator A. In order to lighten the notation we will focus only on the walk A_ǩ:= A^(+)_ǩ.However, the following derivation holds for any of the admissible Weyl quantum walks. If we consider the real and imaginary part of A_ǩ separately,Eq. (<ref>) splits into two equations as follows:{ [cosω-λ(ǩ)]ψ(ω,ǩ)=0,[sinω I-()·σ̌]ψ(ω,ǩ)=0, .where λ (ǩ) and () were defined in Eq. (<ref>). Notice that the two equations are not independent, as one can easily verify by applying sinω I+()·σ̌ to the left of the second equation, and then reminding that by unitarity λ()=1-()^2. The second equation can be easily rewritten in relativistic notation as followsn_μ(k)σ^μψ(k) = 0,where we introduced the four-vectors k:=(ω,), n(k):=(sinω,ň()), and we defined σ:=(I,σ̌). The eigenvalues ω of Eq. (<ref>) then necessarily obey the dispersion relationcosω= λ (ǩ),with two branches of eigenvalues, namely ω = ±arccosλ (ǩ).In the small wave-vector limit, Eq. (<ref>) is approximated by the usual relativistic dispersion relation ω^2 =^2.Following the analogy with quantum field theory, we can interpret and the two solutions of Eq. (<ref>) as particles for ω >0 and anti-particles for ω<0. Let us now restrict the domain of the function () to one of the four region B_i defined in Eq. (<ref>). Since the following considerations won't be affected by the choice ofB_i we will omit the subscript i. The solutions of equation (<ref>) are preserved if we multiply the left hand side by an arbitrary function f(k) such that f(k)() can be inverted as a function on B_i. In particular, we choose an arbitrary rescaling function f(k) such that f(k)() maps B_i to the full ℝ^3. This is achieved by any rescaling function f that, besides preserving invertibility of f(k)() on the regions B_i, is singular at the border of the region . In particular, we consider C^∞ functions f. The eigenvalue equation thus becomes p^(f)_μ(k) σ^μψ(k) = 0, p^(f)=𝒟^(f)(k):=f(k)n(k) . The valuesand ω provide a representation of the state space in terms of constants of motion of the quantum walk dynamics. We now define a change of inertial frame as a change of representation that preserve the set of solutions of the eigenvalue equation. We conveniently use the expression of the eigenvalue equation in Eq. (<ref>).A change of representation of the dynamics in terms of the constants of motion is given by a functionk': k =[ ω; ]↦ k'(k) := [ ω';' ].We remark that by definition, since p^(f)_μ(k)=f(k)n_μ(k) and n_μ(k)n^μ(k)=(n_μ(k)σ^μ)=sin^2ω-()^2, for ω= ±arccosλ (ǩ) one has p^(f)_μ(k)p^(f)μ(k)=0. On the other hand, for ω≠±arccosλ (ǩ) the eigenvalue equation must have trivial solution ψ(k)=0, and then one has p^(f)_μ(k)p^(f)μ(k)≠0. Thus, for every invertible map k' one can define M(k)∈(2,ℂ) such that M(k)ψ(k)=α(k')ψ(k'), with α(k)∈ℂ. For values of k on the mass shell k=(ω(),)^T, this linear transformation can be expressed in the space ℓ^2(Γ) ⊗ℂ^2 asT:=∫_B|'()|⊗ M().Let usrestrict ourselves to those transformations k'(k) for which there exists an M∈(2,ℂ) independent of k and a rescaling α(k) such that Mψ(k)=α(k')ψ(k'). The above arguments motivate the following definition: A change of inertial reference frame for the Weyl walk is a quadruple (k', a, M, M̃) wherek' : k = [ ω; ]↦ k'(k) := [ ω';' ]a: B×[-π,π]→[-π,π]M , M̃∈(2,ℂ)such that the eigenvalue equation (<ref>) is preserved, i.e.p^(f')_μ[k'(k)]σ^μ= M̃ p^(f)_μ(k)σ^μM^-1,and the eigenvectors are transformed asψ'(k')=e^ia(k)Mψ(k). Notice that the change of f to f' in Eq. (<ref>) allows to take α(k') as a phase e^i a(k).A special case of change of inertial frame is given by the trivial map k'=k along with the matrices M=M̃=I. As we will discuss in the next section, the above subgroup of changes of inertial frame, that only involves the phases e^i a(k), recovers the group of translations in the relativistic limit. The set of all the admissible changes of inertial frame forms a group, which is the largest group of symmetriesof the Weyl walk. In order to classify this group, we now observe that a map acting as in Eq. (<ref>) transforms the four Pauli matrices linearly σ^μ↦ L^μ_νσ^ν, and in turn this implies that p^(f')_μ(k')=L_μ^ν p^(f)_ν(k). Moreover, the set of invertible linear transformations represented by L_μ^ν must preserve the mass-shell p^(f)_ν p^(f)ν=0. By the Alexandrov-Zeeman theorem<cit.> this implies that the transformations L_μ^ν must be a representation of the Lorentz group. Thus, a general change of inertial frame (k',a,M,M̃) for the right-handed Weyl walks must be of the formk'(k)= g^-1∘ L_β∘ f,M=Λ_β,M̃=Λ̃_β,where L_β, Λ_β and Λ̃_β are the (12,12), (0,12) and (12,0) representations of the Lorentz group, respectively. The only difference in the case of left-handed Weyl walks is that the representations Λ_β and Λ̃_β are exchanged. Notice thatf ∘ g^-1=M_f∘ n∘ n^-1∘ M_g^-1= M_f∘ M_g^-1,whereM_f(m)=f(n^-1(m))mone has f ∘ g^-1(m)=h(m)m,and thus(g'^-1∘ L_β'∘f')∘( g^-1∘ L_β∘ f)=(g'^-1∘ L_β'∘f'∘ g^-1∘ L_β'^-1) ∘ L_β'∘L_β∘ f=g”∘ L_β'∘β∘ fg” := g'^-1∘ L_β'∘f'∘ g^-1∘ L_β'^-1It is then sufficient to prove that a function f with the desired properties exists, otherwise the group of symmetries of the walk would be trivial.We have already shown in Section <ref> that the restriction n_i() of n() to B_i define an analytic diffeomorphism between B_iand the manifold 𝖧⊂𝖴. Let us consider the solutions of Eq. (<ref>), and define the function g(ω, r m ) := f(ω, n^-1(r m) ), where g is monotonic versus r ≥ 0 for every m∈ H. We notice that the function g(ω, r m ) is well defined since 𝖧 is star-shaped. Furthermore, if g(ω, r m ) diverges on the boundary of 𝖧, we have that the map 𝒟^(f)(k) defines a diffeomorphism between the set C_i:={ k = (ω, ) |∈B_i , cosω = λ} and the null mass shell 𝖪 := { p∈ℝ^4, s.t. p^μ p_μ = 0 }. A possible choice of f(k) which satisfies all the previous requirements is given byf(ω, ) :=f'(ň()) ,f̃'̃(r,θ,ϕ) := 1+ r ∫_0 ^rds ( 1/a(s) + 1/b(s, θ, ϕ)),a(r) :=1- r^2 , b(r,θ, ϕ) := cos^22ϕ + (12 - r^2 (1- cos^2θsin^2ϕ))^2where we used spherical coordinates n_x = r cosθcosϕ, n_y = r sinθ, n_z = r cosθsinϕ for the argument in the definition of the function f': H→ℝ, with the convention that for =0 one has ϕ=0. In order to classify the most general transformation leaving the walk invariant, it is still possible to allow for transformations of the kindk'=∑_i n_j(i)^-1n_i,a=0, M=M̃=I,where the region B_i is mapped to the region B_j(i). Notice that this corresponds to a permutation of the four regions B_i, which however must fulfil the constraint that i and j(i) must labe lregions corresponding to walks with the same chirality ({B_0,B_2} and {B_3,B_4}). This part of the group thus corresponds to ℤ_2×ℤ_2.By considering the case f=g in Eq. (<ref>), we haveℒ_β:= 𝒟^(f)^-1∘ L_β∘𝒟^(f)which is a non linear representation of the Lorentz group as the ones considered within the context of doubly special relativity <cit.>. It is easy to observe that, if f'(0)=1 and ∂_μ f' = 0 where f(ω,)=f'(sinω, n()) as in Eq. (<ref>), the Jacobian J_ℒ_β of ℒ_β coincides with L_β. In the limit of small wave-vector we have that ℒ_β = L_β + O(||^2) that is the non linear Lorentz transformations recover the usual linear one.In Fig. <ref> we show the numerical evaluation of some wave-vector orbits under the subgroup of rotationsof the nonlinear representation of the Lorentz group. We see how the distortion effects, which are negligible for small wave-vector, become evident at larger wave-vectors.§ CONCLUSION The analysis of the previous section can be in priniciple applied to any quantum walk dynamics for which we know a complete set of constant of motion.In particular we could consider the Dirac quantum walk of Ref. <cit.>, whose eigenvalue equation is (p_μ(ω, k, m) γ^μ - m I)ψ(ω, k, m) = 0 where γ_μ are the Dirac γ matrices in the chiral representation, m is the particle mass and p(ω, k, m):=(sinω, √(1-m^2)n() ). In this case we may generalize Definition <ref> and allow for maps that change the value of m. We can then consider the invariance of the whole family of Dirac quantum walks parametrized by m. One could prove that the symmetry group of the Dirac walks include a non-linear representation of the De Sitter group 𝕊𝕆(1,4). Since the frequency (or energy) ω and the wave-vector (or momentum)are the constant of motion of the quantum walk dynamics,the scenario we discussed so far deals with the changes of reference frame in the energy-momentum (ω , ) space. In particular we saw that the Lorentz group is recovered and one could wonder how to give a time-position description of the deformed relativity framework that we obtained in energy-momentum space. It is believed that the nonlinear deformations of the Lorentz group in momentum space have profound consequences on our notion of space-time. In particular we may have the emergence of relative locality<cit.>, i.e. the coincidence of events in space-time becomes observer dependent. This would imply that not only the coordinates on space-time are observer dependent, as in ordinary special relativity, but also that different observer may infer different space-time manifolds for the same dynamics. Non-commutative space-time and Hopf algebra symmetries<cit.> have been also considered for a time-position space formulation of deformed relativity.This publication was made possible through the support of a grant from the John Templeton Foundation under the project ID# 60609 Quantum Causal Structures. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.apsrev4-1 | http://arxiv.org/abs/1707.08455v1 | {
"authors": [
"Alessandro Bisio",
"Giacomo Mauro D'Ariano",
"Paolo Perinotti"
],
"categories": [
"quant-ph",
"hep-th"
],
"primary_category": "quant-ph",
"published": "20170726141159",
"title": "Quantum Walks, Weyl equation and the Lorentz group"
} |
Department of Computer Engineering, Istanbul Medeniyet University, Kadikoy, Istanbul, Turkey [email protected] In this paper, we study the subset-sum problemby usinga quantum heuristic approach similar to the verification circuit of quantum Arthur-Merlin games <cit.>. Under described certain assumptions, we show that the exact solution of the subset sum problem my be obtained in polynomial time andthe exponential speed-up over the classical algorithms may be possible. We give a numerical example and discuss the complexity of the approach and its further application to theknapsack problem. A Quantum Approach to Subset-Sum and Similar Problems Ammar Daskin Received: date / Accepted: date =====================================================§ INTRODUCTION Subset-sum <cit.> is a widely-studied NP-complete problem formally expressed as follows: Given a set of integer elements V = {v_1, …, v_n} and a target value W, determine if there is a subset, S, of V whose sum is equal to W. In the associated optimization problem, the subset S with the maximum sum less thanW is searched. The exact solution for this problem can be found by first computing the sum of elements for each possible S and then selecting the maximum among those whose sum is less than W. Clearly, this algorithm would take exponential-time in the number of elements. Another means to solve this problem is through dynamic programming which requires O(nW) time. This is also exponential in the required number of bits to represent W: If W=2^m and m ≈ n, then the running time is O(n2^m) =O(n2^n). There are also many forms of polynomial time approximation algorithms applied to the subset problem. For an overall review of subset-sum problems and the different algorithms, we recommend the book by Keller et al. <cit.>. Quantum algorithms in general provides computational speed-up over the classical counter parts. Quantum walk algorithm presented for element distinctness <cit.> is applied to the subset problems <cit.>. The computational complexity of this algorithm is shown to be bounded by O(n^|L|), where n is the number of items and |L| is the subset size. In quantum computing, the cases where exponential speedups are possible are generally related to hidden subgroup problems: a few examples of these cases are the factoring <cit.>, the dihedral hidden subgroup problem <cit.> and some lattice problems <cit.>. A review of the algorithms giving the exponential speedups in the solutions of algebraic problems are given in Ref.<cit.>. There are also quantum optimization algorithms such as the ones in the adiabatic quantum computation <cit.> applied to different NP problems <cit.> and the quantum approximate optimization algorithm <cit.>applied to the NP-hard problems. For a further review on general quantum algorithms, please refer to Ref. <cit.>, or to the introductory books <cit.>. It is known that having the ability of a post-selected quantum computing (an imaginary computing model), one can obtain the result of a Grover search problem in O(1)<cit.>. This ability would also lead a quantum computer to solve NP-complete problems in polynomial time. Although this model is imaginary, it still provides an insight to see one of the differences between quantum and classical computers on the solution of NP-complete problems: i.e., mainly a quantum computer can generate all the solution space and mark the correct answer in polynomial time, which is not possible on classical computers. However, this information, the marked item, can only be obtained by an observer withan exponential overhead (which makes the computational complexity exponential in the number of qubits.). This motivates to do research on the applications of the algorithmssuch as Grover's search algorithm <cit.> to the special cases of NP-complete problems so as togain at least some speed-up over the classical algorithms. The Grover search algorithmin quantum computing provides quadratic computational-speedup over the classical brute force search algorithm. It is well-known that the employment of this algorithm in general yields a quadratic speed-up also in theexact solutions of NP-complete problems (please see the explanation for Hamiltonian cycle given in page 263 of Ref.<cit.>).The algorithm also plays important role in quantum Arthur-Merlin games <cit.> and applied along with the phase estimation algorithm to NP-complete problems such as 3-SAT and k-local Hamiltonian problems<cit.>. The similar idea is also used to prepare the ground state of the many-body quantum systems <cit.>.Here, we study the subset-sum problems by using a quantum heuristic approach similar to the verification circuits of quantum Arthur-Merlin games<cit.>.The approach may provide exponential speed-up for the solution of the subset problems over the classical algorithms under the following assumptions: Let |L| be the number of possible subset-sums less than W and|L'| be the number of possible subset-sums greater than or equal to W. In this paper, we will assume that |L'|/|L| = poly(n). Under this assumption, the probability of having a subset-sum less than W is 1/poly(n). However, we may still have O(2^n/poly(n)) number of possible subsets which gives a sum less than W. Whence we can easily make the following remark: The subset-sum problem under this assumption is as difficult as without this assumption. Therefore, the maximization version of the problem is still NP-hard when W = O(2^n). In addition, the correctness of the solution produced by our heuristic is determined from the distribution (which can be guessed from the distribution of the input elements) of the feasible subset-sums. It yields the exact answer with a high-probability if the following condition is satisfied. Let ϕ_max be the maximum subset-sum less than W. Let m-qubits in the output register of our algorithm represents the binary value of ϕ_max. If the bit value of any tth qubit is 1 in the binary value of ϕ_max = (b_0… b_m-1)_2; then after measuring the first (t-1) number of qubits with the correct values (b_0b_1… b_t-2)_2, the probability of seeing |1⟩ on the tth qubit is not exponentially small (i.e. the probability is 1/poly(n).) in the normalized-collapsed state. This assumption (condition)only affects the accuracy of the output. As we shall show in the following sections, it does not change the polynomial running time.In addition, to the best of our knowledge, this assumptiondoes not simplify the original problem for the current classical algorithms:For instance, this condition is likely to hold when we have a random uniformly distributed set of input elements.As mentioned, generating possible subset-sums alone takes exponential time for any classical algorithm and the computational complexity of finding the solution is still bounded by O(2^n) for any classical algorithm when W=O(2^n). In the following sections, after preliminaries, we list the algorithmic steps and explain each step in the subsections. Then, we discuss the complexity analysis and show how the approach takes O(poly(n)) time under the above assumptions. We also discuss the application to the knapsack problem. Finally, we present a numerical example and conclude the paper. § PRELIMINARIES In this section, brief descriptions of the quantum algorithms used in this paper are given. For a broader understanding of these algorithms, the reader should refer to the introductory book by Chuang and Nielsen <cit.>. §.§ Notes on Notations Throughout the paper, we will use |...⟩ to represent a quantum state (a vector) and ⟨...| to conjugate transpose of a vector. Bold faces such as |0⟩ indicates the vector is at least a two-dimensional vector. Using a value inside the ket-notation such as |ϕ⟩ indicates the basis vector associated with the binary representation of ϕ. Other than W which is a given value, the capital letters generally represent matrices (operators). The quantum state |1⟩ on a qubit represents 1 as a bit value and |0⟩ is 0. The indices start from 0. §.§ Quantum Phase Estimation Algorithm Quantum phase estimation algorithm (PEA)<cit.> is a well-known eigenvalue solver which estimates the phases of the eigenvalues of a unitary matrix, U∈ C^⊗ n: i.e. the eigenvalues of U comes in the form of e^iϕ 2π with an associated eigenvector |ψ⟩. Thealgorithm estimates the value of ϕ for a given approximate eigenvector |ψ⟩. The accuracy of the estimation is determined by the overlap of the approximate and actual eigenvectors and the number of qubits used to represent the phase value. PEA in general requires two registers to hold the value of the phase and the eigenvector. The algorithm starts with an initial approximation of the eigenvector on the second register and |0⟩ state on the first register: |0⟩|ψ⟩. Then, the quantum Fourier transform is applied to the first register. It then the controlled-operators U^2^js are applied to the second register in consecutive order: here, 0≤ j< m, each U^2^j is controlled by the jth qubit of the first register andm is the size of this register which determines the decimal precision of the estimation. At this point in the first register the Fourier transform of the phase is obtained. Therefore,applying the inverse Fourier transform and measurement on the first register yields the estimation of ϕ. In general, the computational complexity of PEA is governed by the number of gates used to implement each U^2^j. When they can be implemented in polynomial time, then the complexity of the algorithm can be bounded by some polynomial time, O(poly(n)). §.§ Amplitude Amplification Algorithm The amplitude amplification (AA) is based on the Grover search algorithm <cit.> and used to amplify the part of a quantum statewhich is considered as “good". The algorithm is mainly composed of two operators. The first operator, F, marks (negates the signs of) the “good" states and the second operator, S, amplifies the amplitudes of the marked states. For |ψ⟩ = α|ψ_good⟩ + β|ψ_bad⟩; F|ψ⟩ = α|ψ_good⟩ - β|ψ_bad⟩. The implementation of F depends on the function that describes the good part of the states. Through this paper, F is simply some combination of controlled Z and X gates (X and Z are Pauli spin matrices.). If |ψ⟩ = A |0⟩ for someunitary matrix A ∈ C^N, then S = 2|ψ⟩⟨ψ| - I = A U_0^⊥A^*, where “*" represents the conjugate transpose. The amplification is done by applying the iterator G = SF consecutively to |ψ⟩. The number of iteration depends on β and is bounded by O(1/β). For further details andvariants of AA, please refer to Chapter 8 of Ref.<cit.>.§ ALGORITHM The approach uses a qubit and a rotation phase gate for each element of V, to encode the possible subset sums as the eigen-phases of a diagonal unitary matrix U. Then, it applies the phase estimation algorithmto obtain the possible sums and associate eigenvectors on two quantum registers. Marking the states with phases less than W, it eliminates the states with phases greater W through the amplitude amplification. Finally, it again employs the amplitude amplification in measurement processes to obtain the maximum phase and its associated eigenvector which indicates the solution of the problem. Here, first the algorithmic steps in general are listed, and then the explanations and more details for each steps are given in the following subsections. The steps are generalized as follows: * Encode the integer values as the phases of the rotation gates aligned on different qubits. * Apply the phase estimation algorithm to the equal superposition state so as to produce the phases and the associated eigenvectors on quantum registers. * Apply the amplitude amplification to eliminate the states where ϕ_j ≥ W. Now, the superposition of the sums and the eigenvectors are obtained with equal probabilities. * Find the maximum ϕ_j in the first register. Then, measure the second register to attain the solution. The quantum circuit representing the above steps is drawn in Fig.<ref>. §.§ Encoding the Values into the Phases First, the values are scaled so that ∑_j =0^n-1v_j ≤ 0.5. For each value v_j with 0≤ j<n, a rotation gate in the following form is put on the (j+1)st qubit: R_j = ( 1 0 0 e^iv_j2π).The n-qubit circuit formed with these rotation gates can be then represented by the following unitary matrix: U= R_n-1⊗…⊗ R_0. Here, U is a diagonal matrix with the diagonal elements (eigenvalues): [ 1, e^iv_0, e^iv_1, e^iv_0+v_1, …, e^i(v_0+ … + v_n-1)] = [ e^iϕ_0, e^iϕ_1,…, e^i ϕ_n-1]. As seen in the above, the phases of the eigenvalues associated with the eigenvectors forming the standard-basis-set encode all the possible sum and the subset information: i.e. the j vector in the standard basis indicates the phase ϕ_j and the elements of the jth subset. §.§ Generating All Possible Sums and Subsets on RegistersConsider the phase estimation algorithm applied to U with the following initial state: |ψ_0⟩ = |0⟩1/√(2^n)∑_j=0^2^n-1|j⟩.Here, |ψ_0⟩ can be simply generated by( I^⊗ m⊗ H^⊗ n) |0⟩|0⟩, where I represents an identity matrix and H is the Hadamard matrix.Since the eigenvectors of U are of the standard basis, the final output of the phase estimation holds the equal superposition of the eigenvector and phases: |ψ_1⟩ = U_pea|ψ_0⟩ = 1/√(2^n)∑_j=0^2^n-1|ϕ_j⟩|j⟩, where U_pea represents the phase estimation algorithm applied to U andforms the first part of the circuit in Fig.<ref>. Obviously if we are able to efficiently find the index of the maximum ϕ_j less thanW in the above, then we solve the maximum subset-sum problem efficiently. §.§ Eliminating the Subsets with ϕ_j≥ WBefore searching for the solution, we divide the quantum state in Eq.(<ref>) into two parts: |ψ_1⟩ = 1/√(2^n)∑_j∈ L|ϕ_j⟩|j⟩ + 1/√(2^n)∑_j∈ L'|ϕ_j⟩|j⟩ . = √(|L|/2^n)|ψ_good⟩ + √(|L'|/2^n)|ψ_bad⟩ where L ={j:ϕ_j<W} andL' = {j:ϕ_j≥ W} with 0≤ j<2^n. This equation includes all the possible eigenpairs. To eliminate the ones included in L', we apply the amplitude amplification algorithm defined by the iterator G =S(F_ϕ⊗ I^⊗ n) as shown in Fig.<ref>. Here, F_ϕ operates on the first register and flips the sign of the states with ϕ_j< W: (F_ϕ⊗ I^⊗ n)|ψ_1⟩ = -√(|L|/2^n)|ψ_good⟩ +√(|L'|/2^n)|ψ_bad⟩.S = 2|ψ_1⟩⟨ψ_1|-I and can be implemented as follows: S = (I^⊗ m⊗ H ^⊗ n)U_pea U_0^⊥(I^⊗ m⊗ H ^⊗ n)U_pea^*,whereU_0^⊥ = I - 2|0⟩⟨0|.After each iteration, the amplitudes of the “good" states are amplified. The number of iterations in the algorithm (the number of applications of G) is determined by the initial probability andis bounded by O( √(2^n/|L|)). In the worst case where |L|<<|L'|, the complexity becomes O(√(2^n)).However, in the other cases, the number of iterations is bounded by O(poly(n)). In addition, using the quantum counting one can estimate the value of √(|L|/2^n) and |L| in polynomial time (see quantum counting in Chapter 8 of Ref.<cit.>). As explained in the Introduction, in this paper we make the assumption given in Assumption <ref>: i.e., mainly |L'|/|L| = O(poly(n)). As a result of this assumption, this part of the algorithm takes O(poly(n)) time. And at the end of the amplitude amplification, the final quantum statebecomes: |ψ_2⟩≈1/√(|L|)∑_j ∈ L|ϕ_j⟩|j⟩.§.§ Finding the Maximum Sum with Its Subset Grover search algorithm <cit.> is able to find a maximum or minimum element of a list of |L| items in O(√(|L|)) times <cit.>. It can be directly applied to Eq.(<ref>) to find the maximum of ϕ_js and the value of j. However, this makes the running time of the whole algorithm exponential because of Assumption <ref>. The elements of the set {ϕ_j:0 ≤ j<2^n} is partially sorted and mostly ϕ_j ≤ϕ_j+x for a considerably large x. Therefore, in some cases, quantum binary search algorithm (e.g. <cit.>) can be used to produce an approximate solution. This will require O(lg|L|) = O(poly(n)) time complexity. Below, similarly to the binary search algorithm and the verification circuit given <cit.>, a polynomial time method for finding the maximum is presented by applying a sequence of conditional amplitude amplifications: Let us assume the maximum ϕ_j in Eq.(<ref>) is ϕ_max = (b_0… b_m-1)_2. If we try to maximize the measurement outcome of the first register, then we attain a value close to ϕ_max. This maximization can be done by starting the measurement from the most significant qubits while trying to measureas many qubits in |1⟩ state as possible. Therefore, we will measure a qubit: if the outcome is not |1⟩, then we apply the amplitude amplification to amplify the states where this qubit is in |0⟩ state and then do the measurement again. If the qubit does not yield |1⟩ after a few iterations, then we will assume |0⟩ as the qubit value and move on to the next qubit. And we repeat this process for the all qubits in the first register. This is explained in more details below and indicated in Fig.<ref> (Note that the measurements after each amplitude amplification is omitted in the figure.): * We measure the first qubit (representing the most significant bit, b_0): * If it is |1⟩, then we set |b_0⟩ = |1⟩ and move on to the next qubit in the collapsed state. * Otherwise, we apply the amplitude amplification by flipping the signs of the states in which the first qubit is in |1⟩ state. While the flipping can be done by simply using the Pauli Z-gate, the amplification operator,S_1, can be implemented in a way similar to Eq.(<ref>): S1 = U_1 U_0^⊥U_1^*, where U_1 represents all the quantum operations up to this point. Here, we only apply theiterator G_1 = S_1 (Z⊗ I^⊗ mn-1)a few times until the measurement yields |1⟩. If it does not, then |b_0⟩ is set to |0⟩. * In the second qubit, we repeat the same process. However, this time G_2 = S_2 (Ẑ⊗ I^⊗ mn-2); where, S_2 involves all the operations done up to this point, and the gate Ẑ is the controlled-Z gate acting on the second qubit and controlled by the first qubit: if the first qubit was |0⟩, then the control bit is |0⟩ (i.e. the gate acts when the first qubit is |0⟩.). Otherwise, it is set to |1⟩. * Similarly, using Z gates controlled by the previous qubits, the measurements along with the amplitude amplifications are repeated for the remaining qubits. Here, the control-bits are either 0 or 1 determined from the measurement results of the previous qubits. The above maximization method is able to amplify the amplitude of ϕ_max if at any point the probability to measure |1⟩ on the qubit is not exponentially small. Otherwise, the number of the amplitude amplification required to see |1⟩ on that qubit becomes exponentially large. Since the amplitude amplificationis only applied a few times (when |1⟩ is not encountered, the qubit is assumed to be |0⟩), this will cause an error in the result. Let us assume that the condition given in Assumption <ref> holds: i.e., if the bit value of any tth qubit is 1 in the binary value of ϕ_max = (b_0… b_m-1)_2; then after the individual measurements of the first (t-1) number of qubits with the values (b_0b_1… b_t-2)_2, the probability of seeing |1⟩ on the tth qubit is not exponentially small in the normalized-collapsed state. This assumptionaffects the accuracy of the result rather than the running time since the amplitude amplification on a qubit is applied only a few times and when |1⟩ is not encountered, the related bit value of ϕ_max is assumed 0. To further simplify the circuit in Fig.<ref> and the numerical simulations, we will also make following remark: S_1 can be used in places of S_2… S_m to simplify the implementation of the amplitude amplifications. The circuit in accordance with the above remark is presented in Fig.<ref> (The measurements on the qubits are also explicitly indicated in this figure however not in Fig.<ref>).Sect. VI gives a numerical example based on this circuit. Now, we will explainhow this circuit may yield the solution by going through the measurements of the first two qubits on the circuit: Let us first divide the state given in Eq.(<ref>) (the state after U_1 in Fig.<ref>) into four parts with the same length: |ψ_2⟩ = ( x_0 x_1 x_2 x_3). The probabilities of measuring |0⟩ and |1⟩ on the first qubit are P_0 = (||x_0||^2 + ||x_1||^2 ) and P_1 = (||x_2||^2 + ||x_3||^2 ), respectively. If P_1 is not exponentially less than P_0, then with the help of the amplitude amplification (G_1 = S_1Ẑ), the first qubit, can be measured in |1⟩. Therefore, b_0 becomes 1. Let us assume we obtain b_0=1 after the measurement. If we use a qubitin place of the first qubit and initialize it in |1⟩ state, we obtain the following normalized-sate: |ψ_3⟩ = |1⟩⊗( ζx_2 ζx_3), with ζ = 1/√(||x_2||^2+||x_3||^2). For the second qubit,the part represented by x_3 is marked by the controlled Z gate, and then S_1 is applied: S_1(Ẑ⊗ I)|ψ_3⟩ =(2|ψ_2⟩⟨ψ_2|-I) ( 0 0 ζx_2-ζx_3) =ζ( 2d_x x_0 2d_xx_1 (2d_x-1)x_2 (2d_x+1) x_3), where d_x = ||x_2||^2- ||x_3||^2. Due to ||x_2|| ≥ ||x_3||, d_x≥ 0: * If dx ≥ 0.5, then all of the amplitudes in the above quantum states are unmarked. Therefore, in the subsequent iteration of AA only x_3 willbe marked, and the amplitudes corresponding to x_3 will be amplified. * If dx = (||x_2||- ||x_3|| )< 0.5, that means the probability difference between |0⟩ and |1⟩ is small; thus, we are very likely to measure |1⟩after the first amplitude amplification. After the measurement, the state collapses to ζ(0 2d_xx_1 0 (2d_x+1) x_3). § COMPLEXITY ANALYSIS We will follow the circuit in Fig.<ref> to analyze the complexity of the whole approach under Assumption 1 and 2. The algorithm starts with two quantum registers of respectively m and n qubits, and then later in the maximum finding part of the algorithm it uses another register with m qubits which is implicitly indicated in Fig.<ref> but notin Fig.<ref>. Therefore, the total number of qubits employed in the whole running of the algorithm is (2m+n).Since U involves only phase gates described in Eq.(<ref>), using U^2^j, 0 ≤ j<m in the phase estimation requires only n number of controlled phase gates (Note that the power of the unitary can be taken by simply changing the angles of the rotation gates.). Therefore, including the complexity of the quantum Fourier transform <cit.>, U_pea, the phase estimation part, requires O(n+mlgm) =O(poly(n)) number of quantum gates. In the amplitude amplification part, the operator F_ϕ can bedesigned in O(poly(n)) time by using a logical circuit: i.e., the circuit is composed of X and Z gates and marks all of the states less than W.Moreover, the implementation of S defined in Eq.(<ref>) involves the Hadamard gates, U_pea and U_0^⊥ all of whichcan be implemented in polynomial time. The remaining part of the circuit is for finding maximum and involvesS_js and controlled-Z gates. The implementation of any S_j is similar to the operator S: they involve the repetitions of the circuit up to their location, and hence requires more computations. However, because of Assumption 2, this part of the circuit and the whole processes are still bounded by some polynomial time, O(poly(n)).§ APPLICATION TO 1/0-KNAPSACK PROBLEMThe maximum subset-sum problem is related to many other problems. One of theseis the 1/0-knapsack problem <cit.> described as: For a given items with weights {w_0 … w_n-1} and values {v_0, v_n-1}, determine which items should be included in a subset to maximize the total subset-value while keeping the total-weight less than W. This problem can be solved in a similar fashion to the subset-problem by adding one additional register to the algorithm: * |sw⟩ – The first register holds the sum of the weights. * |sv⟩ – The second register holds the values. * |j⟩ – The third register indicates the items included in the subset: |j⟩ describes the jth vector in the standard basis. The algorithm starts with constructing the superposition of the possible sums of weights and values as a quantum state by using the phase gates where the least-significant-bits encode the item-values while the most-significant bits are used for the weights. After the phase estimation, the following quantum state is generated: |ψ_1⟩ = 1/√(N)∑_j=1^N|sw_j⟩|sv_j⟩|j⟩ Then, applying the amplitude amplification, the states where sw_j ≥ W are eliminated andthe probability of the states where|sw_j⟩ < W are put into the equal superposition: |ψ_2⟩ = 1/√(|L|)∑_j∈ L|sw_j⟩|sv_j⟩|j⟩ Now, the maximum finding is done on the second register:After finding |sv_j⟩ with the maximum decimal value, the solution to the knapsack problem is obtained from the corresponding |j⟩ which indicates the involved items. § NUMERICAL EXAMPLE In this section, we present a random numerical example based on the circuit in Fig.<ref>. Let us assume that given the set of values V = { 0.10937500, 0.10546875, 0.10156250, 0.09375000, 0.05468750, 0.02343750, 0.00390625 }, which are normalized so that the maximum possible sum is at most 0.5, we are asked to find the subset which gives the maximum possible sum less than W = 0.19921875. If we use 9 bits of precision: i.e. m=9 (the size of the first register in PEA), then W=(001100101)_2). We start with the construction of U which requires a qubit and a single rotation gate for each element of V: that means n=7 (the size of the second-register in PEA) and U∈ C^⊗ n. The eigen-phases of U shown in Fig.<ref> represents the solution space, the possible subset-sums of V.Fig.<ref> depicts the distribution of these phases. After PEA is applied with an initial superposition state on the second register; using AA, the amplitudes of the states in which the phase value on the first register is less than W=(001100101)_2 are marked and amplified. After one iteration of AA, the probability of the eigenvector-phase pairs are presented in Fig<ref>. At this point, we have ∑_j, ϕ_j<W|ϕ_j⟩|j⟩. In the maximum-finding part, we start doing the measurement from the most significant qubits: * Qubit-1 and qubit-2 yields |0⟩s with probability ≈1s. Therefore, |b_0b_1⟩ = |00⟩. After the normalization, we have the solution space where all ϕ_js are less than W. This is shown in Fig.<ref>. * The measurement on qubit-3 yields either |0⟩ or |1⟩ with probabilities, respectively, 0.4390 and 0.5610. Therefore,it is very likely |1⟩ is measured after a few attempts. In that case, we have |b_0⟩|b_1⟩|b_2⟩=|0⟩|0⟩|1⟩ on the first three qubits and the collapsed-state on the remaining qubits. The normalized probabilities at this point are drawn on Fig.<ref>. * The probabilities of |0⟩ and |1⟩ for qubit-4 is 0.8261 and 0.1739 in the normalized state. After a few measurements, if we see |1⟩, we set |b_4⟩ = |1⟩ and continue on the fifth qubit. Otherwise, we apply the amplitude amplification. This changes the probabilities for qubit-4 to0.1991 and 0.8009, respectively. However, it also brings back some of the eliminated states with some small probabilities. This is shown in Fig.<ref>. After the measurement of qubit-4 in |1⟩, the state collapses into four remaining eigenpairs with equal probabilities shown in Fig.<ref>. * In all of these four states, qubit-5 and qubit-6 are in |0⟩ state. Therefore, the measurements on these qubits yield |0⟩ states with probability 1. * The probability of seeing qubit-7 in |1⟩ state is 0.25. An iteration of AA amplifies this to 0.3488 as shown in Fig.<ref>. At this point, after a few measurements, we are likely to encounter |1⟩ on qubit-7. This collapses the state into the solution. * And finally, qubit-8 and qubit-9 are measured in |0⟩ with probability 1. The above maximization yields 0.1953125 as the maximum phase with |0011000⟩ as the corresponding eigenvector, which is the exact solution. § CONCLUSIONIn this paper we have studied the subset-sum and similar problems: e.g. the knapsack problem. In particular, we have generated the possible sums by using the phase estimation and the amplitude amplification algorithms. Then, we have used a maximum-finding procedure to obtain the solution. The approach requires polynomial time if the number of possible sums less than or equal to the given value are not exponentially smaller than the number of possible sums greater than the value. In addition, it yields the exact answerif the probability of seeing the correct bit valueon the tth qubit is not exponentially small in the normalized-collapsed state after the first (t-1) number of most significant bit values are correctly measured. The approach is general and can be further improved for the similar NP-complete problems. unsrtnat | http://arxiv.org/abs/1707.08730v4 | {
"authors": [
"Ammar Daskin"
],
"categories": [
"quant-ph",
"cs.CC",
"cs.DS"
],
"primary_category": "quant-ph",
"published": "20170727073305",
"title": "A Quantum Approach to Subset-Sum and Similar Problems"
} |
1 Center for Computational Astrophysics, Flatiron Institute, New York, 10010, NY,USA 2 Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel E-mail:[email protected], [email protected] One of the puzzles in the recent observations of gravitational waves from binary black hole mergers is the observed low (projected) spins of the progenitor black holes.In two of the four events, GW150914,and the recent GW170104,the observed spinsare most likely negative (but consistent with zero). In the third case LVT151012 it is practically zero and only in theforth case, GW151226,the spin is positive but low. These observations are puzzling within the field binary scenario in which positivehigher spins are expected. Considering the most favorable Wolfe Rayet(WR) progenitors we estimate theexpected spin distribution for differentevolution scenarios and compare it to the observations.With typical parameters one expects a significant fraction (≥ 25%) of the mergersto have high effective spin values.However due touncertainties in theoutcome of the common envelope phase (typical separation and whether the stars are rotating or not) and in the late stages of massive star evolution(the strength of the winds) one cannot rule our scenarios in which the expected spins are low. While observations of high effective spin events will support this scenario, further observations of negative spin events would rule it out.§ INTRODUCTIONGravitational-wave astronomy begunin Sept 14th 2015 with LIGO's discovery <cit.> ofGW150914, a binary black hole (BBH) merger. An additional BBH merger, GW151226, as well as a merger candidate, LVT151012 were discovered in LIGO's O1 run. A forth event, GW170104 <cit.>, was discovered in the O2 run that begun in the late fall of 2016 and still continues now. All BBH mergers discovered so far involved rather massive BHs with the lightest one (observed in GW151226) is of7.5m_⊙.Among the remarkable features of all four events are the relatively low values of χ_ eff, the mass-wighted average of the dimensionless spin components χ≡ a/m, projected along the orbital angular momentum, of theindividual BHs before they merged.While in none of the cases χ_ eff is large, in two cases the best fit values are negative (but the error bars don't exclude zero), in one case it is practically zero and only in the forth case this value is positive butsmall. These values are best fitted bya low-isotropic spin distribution ( and see also ) and are at some “tension" with the expectationsfromfield binary evolution scenarios that suggest that the individual spins should be aligned with the orbital angular momentum axis and at least in a significant fraction of the events the spins should be large <cit.>.We compare here the observed distribution with the one expected in the most favorable evolutionary scenarios involving WR stars and discuss whether the observations disfavorfield binary evolution models <cit.>andsupportcapture models<cit.> in which the spins are expected to be randomly oriented. The essence of the argument concerning that suggests alarge χ_ eff in field binary BH is the following: (i) To merge within a Hubble time, t_ H, the initial semi-major axis of the BBH at the moment of the formation of the second BH, a,should not be too large.(ii)With a relatively small a the stars feel a significant tidal force andtheir spin tends to be synchronized with the orbital motion. (iii) If synchronized, the shortorbital period implies that the progenitor's spin χ_*≡ S c/Gm^2 (where m is the BH's mass, S its spin angular momentum, G is the gravitational constant, and c is the speed of light) is large:χ≥χ_*(t_H), where χ_*(t_H) is thespin parameter of astar in abinary that will merge in a Hubble time. Therefore, χ_*(t_ H)/2≲χ_ eff≲ 1/2 if only the secondaryhas been synchronized. χ_*(t_ H)≲χ_ eff≲ 1 if both progenitors have been synchronized. We discuss first, in <ref>, the gravitational wave observations as well as observations ofgalactic X-ray binaries containing BHs. In <ref>, following this chain of arguments, weexpress the initial semi-major axis, a, in terms of the the merger time, t_c, and we estimateχ_*(t_c).Weexpress the dimensionless BH spin χ in terms of the progenitor's parameters.In <ref>, using these estimates we calculate the expected spin distribution in different scenarios and compare it to the gravitational-wave observations.§ OBSERVATIONS Binary BH Mergers: Some basic observed properties of the BBH merger events are summarized in Table I. The most interesting ones for our purpose are the BHs' masses, theirχ_ eff values and the final BH spin. This latter quantity isdefined as:χ_ eff≡ m_1 χ_1 + m_2 χ_2 /m_ tot where χ_1,2≡cS⃗_1,2·L̂/ G m_1,2^2,and m_tot=m_1+m_2 and L̂ is a unit vector in the direction of the system's orbital angular momentum L⃗. The limits on χ_ eff are obtained from the observations of the gravitational wave signals before the merger. The lack of extended ringdown phases also puts limits on the spins of the final BHs, a_f. The fact that those areof order 0.6-0.7 and not close to unity is an independent evidence that the initial aligned spins of the BHs were not close to unity. Had the initial aligned spins been large, the final spin of the merged BHs would have been very close to unity and would have had a long ringdown phase. Thus the final spin and the initial spins estimates are consistent.Indeed, the final spin is slightly larger (0.74^+0.06_-0.06) for GW151226, the only case for which the nominal value of χ_ eff =0.21^+0.20_-0.10 is positive. Fig. <ref>describes the observed χ_ eff distribution in terms ofthe corresponding fourGaussians describing approximately the χ_ eff posterior distributions of the observed events and the resulting combined spin distribution for the whole sample. GalacticBHs in X-ray binaries:Observations of X-ray binaries involving BHs, albeit smaller mass ones, can also shed some light on the problem at hand.In particular observations of two such systems that include massive (>10 m_⊙) BHs, Cyg X-1 and GRS 1915+105, provide a good evidence that these massive BHs formed in situ,in a direct implosion and without a kick <cit.>.For example, Cyg X-1 moves at 9 ± 2 km/s relative to the stellar association Cygnus OB3, indicating that it could have lost at most 1 ± 0.3 m_⊙ at formation.Furthermore, the minuscule eccentricity of Cyg X-1, 0.018 ± 0.0003, <cit.>suggests that the orbit has been circularized during the binary evolution and the collapse didn't give the system a significant kick that disturbed the circular orbit.In addition, <cit.> shows that large natal kicks, >80 km/s, are not required toexplain the observed positions of low-mass X-ray binaries.Estimates of the spins of the relevant BHs<cit.> suggest that in these two systemsa/m>0.95. Three other BHs, LMC X-1, M33 X-7, and 4U 1543-47, whose masses are larger than 9 m_⊙, have χ>0.8.Only one BH with a mass > 9 m_⊙, XTE J1550-564 has a significantly lower values (χ = 0.34^+.20_-.28).It is important to note that these large spins must be obtained at birth as accretion cannot spin up a massive BH to such a high spin value. § MERGER TIME, ORBITAL SEPARATION AND SYNCHRONIZATION. Assuming circular orbits the merger time, due to gravitational radiation driven orbital decay,is: t_c ≈ 10 Gyr (2 q^2/1+q)(a/44R_⊙)^4 (m_2/30M_⊙)^-3 ,where q≡ m_2/m_1. Note thatwe assume circular orbits here and elsewhere. This simplifying assumption is based on the expectation that the orbit will be circularized during the binary evolution and that it won't be affected by the collapse on the secondary. It is supported by the observations of binaries containing massive BHs, reported earlier (see <ref>). Tidal forces exerted by the primary will tend to synchronize the secondary. If fully synchronized the final stellar spin would equal: χ_2≈0.5 q^1/4(1+q/2)^1/8(ϵ/0.075) (R_2/2R_⊙)^2 (m_2/30M_⊙)^-13/8(t_c/1Gyr)^-3/8 ,where ϵ characterizes the star's moment of inertiaI_2 ≡ϵ m_2 R_2^2.The progenitor's spin,χ, increases with the progenitors size and decreases when t_c increases.Thus, a compact progenitor star that formed at a high redshift producesa low spin BH while a large progenitorformed recently collapses to a large spin BH (see ). The synchronization process takes place over t_ syn:t_ syn ≈ 20 Myr (ϵ/0.075) (E_2/10^-5)^-1(R/2 R_⊙)^-7 ((1+q)^31/24/q^33/8) (m_2/30M_⊙)^47/8(t_c/1Gyr)^17/8 ,where E_2, is a dimensionless quantityintroduced by <cit.> characterizing the inner structure of the star. E_2 is∼ 10^-7–10^-4 formassive main sequence stars and Wolf-Rayet (WR) stars <cit.>.The characteristic values used in Eq. (<ref>) correspond to a WR star.For WR stars, t_ synWR can be expressed, following <cit.>:t_ synWR≈ 10 Myr q^-1/8(1+q/2q)^31/24(t_c/1 Gyr)^17/8. Because of their short stellar lifetime, WR stars are not necessarily synchronized in binary systems even with t_c of a few hundreds Myr. Therefore the final stellar spin depends on (i) χ_i, the spins of the stars at the beginning of the WR phase (ii)on the ratio of t_ syn and the lifetime of the WR star, t_ WR and (iii) on theangular momentumloss timescale during the WR phase, t_ wind <cit.>. With these parameters, we solve the following equation toobtain the stellar spin parameter at the end of the WR phase <cit.>:χ̇_* = χ_ syn/t_ syn(1-χ_*/χ_ syn)^8/3 - χ_*/t_ wind,where χ_ syn is the stellar spin parameter in the synchronized state. § COLLAPSEANDTHE BH SPIN One can expect that, unless there is too much angular momentum (that is for χ_* ≤ 1),the collapsing star implodes and the BH that formsswallows all the collapsing stellar mass[The original stellar mass could be larger but this lost in an earlier phase due to winds <cit.>.]. If χ _*> 1a fraction of the matter will be ejected carryingthe excess angular momentum andleading to a BH with χ≤ 1<cit.>. Thus we expect that χ_ BH≈1 χ_* ≥ 1 , χ_* χ_* < 1 . One may wonder if there are caveats to this conclusion. First, is it possible that matter is ejected during the collapse to a BH even if χ_* < 1? This will, of course, change the relation between the progenitor's spin and the BH's spin. Second is mass ejected isotropically?If not the BH will receive a kick and the BBH will be put into an elliptical orbit (that will merge faster). The kick may also change the resulting BH spin. Since the initial spin is in the direction of the orbital angular momentum the kick may reduce the spin component along this direction. Clearly these issues can be addressed by a detailed numerical study of collapse to a BH. However, as discussed in <ref> observations of binaries containing massive(>10 m_⊙) BHs, Cyg X-1 and GRS 1915+105 provide a good evidence that massive BHs form in situin a direct implosion and without a kick<cit.>. Estimates of spins of accreting massive BHs give an independent support to this conclusion.§ A COMPARISON WITH OBSERVATIONS AND IMPLICATIONS<cit.> examinedthe expected χ_ eff for BBH binaries of different types (see their Fig. 2). Either red or blue giantprogenitors are easily ruled out. Even regular main sequence stars would lead to progenitors' spin values that are much larger than unity. The only possible candidates are Pop III stars, that have in fact been predicted to produce massive BBHs<cit.> and WR stars, stripped massive stars that have lost their H and He envelopes.As mentioned earlier, the observed spin values are low even for those. The “tension" appeared already in the first detection of GW150914 and it was intensified with the additional observations and in particular with the observation of GW170104.To clarify this issue we turn now tocompare these results with predictions from a χ_ eff distribution of BBHs (see ). Here we focus onWR stars.To form massive BHsevolutionary scenarios requirelow metallicityprogenitors(otherwise the mass loss would be very significant). Long GRBs (LGRBs) are known to arise preferably in low metallicity hosts[From a theoretical point of view it has been suggested thatstrong winds that arise in higher metallicity progenitors will prevent fast rotation which is probably needed to produce a LGRB.].Therefore we use the redshift distribution of long GRB rate <cit.> to estimate the rate of formation of BBH progenitors.We also consider BBH formation rates that follow the cosmic star formation rate (SFR, ) and a constant BBH formation rate. The resulting distribution of the SFR model is not very different from the one that follows LGRBs. On the other handthe results for a constant BBH formation rate are quite different as this produces a significant fraction ofbinaries formed at low redshifts that have to merge rapidly and hence have small initial separations.We expect that the rate ofmergers follows the BBH formationrate with a time delay t_c whose probability is distributedas ∝ t_c^-1.We consider a minimal time delay of 1 Myr(corresponding to an initial separation of 3 · 10^11cm), 10 Myr (5.4 · 10^11cm), or 100 Myr (10^12cm) between the formation of the BBH and its merger. These differences are important as the synchronization time depends strongly on the separation and hence on t_c (see Eqs. <ref> and <ref>).We also consider different timescales of t_ wind≡ J_s/J̇_s, where J_s is the spin angular momentum of the star, with stronger winds corresponding to shorter t_ wind values. With these assumptions we obtain several probability distributions for the observed χ_ eff values. In general, the field binary scenario predicts a bimodal χ_ eff distribution with low and high spin peaks (seefor simple models andfor population synthesis studies).Here the high spin peak corresponds to tidally synchronized binaries.Fig. <ref>depictsthe integrated observed distribution of χ_ eff compared with several WR models.One can see the large variety of the resulting χ_ eff distribution: some modelsgive a very large fraction of high χ_ eff mergers, while for others χ_ eff is concentrated around zero.The models with the lowestχ_ eff distributions are those in which the progenitors (i)are not synchronized at the beginningof the WR phase (χ_i=0); (ii)have a strongwind[Note however that such winds might not be consistent with very massive remnants.] (t_ wind=0.1 Myr) and (iii) have a long minimal time delay (t_c, min = 100 Myr) - corresponding to a large initial separation.The question whether one or two of theprogenitors is influenced by the tidal interaction is secondary as it determines the largest χ_ effvalues (> 0.4 or >0.8) that havenot been observed so far.Models with χ_i=1, amoderate wind (t_ wind = 0.3Myr) and a longdelay (100 Myr) in which the BBH formation rate follows theSFR or LGRBs rates are consistent with the data (apart from the nominal negative values, of course, but those could be due to the large measurement errors). The top four panels compare different models to a fiducial model in which both progenitors are spinning rapidly at the end of the common envelope phase χ_i=1,t_ wind=0.3 Myr, t_c, min=10Myr, and the BBH formation rate following the cosmic SFR.The lower two panels depictmore extreme models. Here we find that if all the above conditions are satisfied then the resulting χ_ eff distribution (for SFR or LGRB rate) is too narrowly centered around zero. A better fit to the data is obtained under these conditions if the BBH formation rate is a constant (see bottom left panel of Fig. <ref>). Even if one of the progenitor stars is synchronized at the end of the common envelope phase (χ_i=1) astrong enough wind (t_ wind=0.1 Myr) can lead to sufficient loss of angular momentum so that the final χ_ eff distributions would be very low (see bottom right panel of Fig. <ref>).With just four observations it is difficult to obtain a quantitative estimate for the quality of the fit.Evenwith the two negative nominal χ_ eff values the models that look qualitatively fine areconsistent with the data.The KS measure, D_KS=0.5, of these modelsyields chance probabilityof∼ 20%, which is roughlyconsistent with the ∼15% probability that χ_ eff of GW 170104 is positive <cit.>.Other models that assume that even one of the stars is synchronized early on, or that t_ wind is large(0.3 or 1 Myr) or that the minimal merger time is short are qualitatively inconsistent with the observed distributionand their quantitative chance probability < 0.5% even with this small number of events. The error bars of the χ_ eff estimates are not taken into account in this KS analysis. In order to take the relatively large measurement errors of χ_ eff into account when comparingdifferent models with the data, we evaluate the odds ratios between the marginal likelihoods of different field evolutionmodelsfollowing<cit.>.Here we calculate the marginal likelihood of each model for the four events, p_i(d|M) and then combine them as p(d|M)=∏_ip_i(d|M). Tables 2 and 3 list the odds ratios of these different models to the low-isotropic spin model of<cit.>, p(d|M)/p(d| Low Iso). Note that this low-isotropic spin model is the most favorable one among the simple modelsused in <cit.>. None of the field binary evolution models has an odds ratio larger than unity so that the low-isotropic spin model is more consistent with the observed χ_ eff distribution than our aligned WR binary models. However, many of the models satisfying the conditions mentioned above have p(d|M)/p(d| Low Iso)≳ 0.1. These cannot be ruled out with the current χ_ eff distribution of the four observed events. § CONCLUSIONS Before discussing the implications of these findings we turn, once more, to possible caveats. We have already argued that observations of Galactic binaries including massive BHs provide a good evidence for our model for the formation of massive BHs (no kick and no mass loss).It seems that the main open issues are (i) the question of the spins of the BHs at the end of the common envelope phase,(ii)the separation at the end of the common envelope phase, that determines thetidal locking process and (iii)the effect ofwinds on the final spin. Turning now to the results,Clearly the negative observed values are inconsistent with themodel (unless there are significant kicks at the formation of the BHs). However,the large error bars of these measurements don't allow us to rule out any scenario.The observed low aligned spin values are at some “tension" with the expectations of the standard evolutionary scenario if one takes the fiducial values we considered here<cit.>. However, even for these parameters the small number statistics is insufficientto make any clear conclusions. More importantare the uncertainties in the outcome of the common envelope phase (whether the stars are synchronized or not at the end of this phase), in the strength of the wind at the late stages of the evolution of these massive stars, in the minimal time delay for mergers (corresponding to the minimal separation and hence to the importance of the tidal locking process) and finally in the tidal synchronization process itself.For example, it is clear that strong enough winds would reduce the final spin of the stars. However, one may wonder if such strong winds are consistent with the very massive BHs observed.Note that t_ wind=0.1 Myr roughly corresponds to a mass loss of10^-4.5 m_⊙/yrwhich is at the level of the strong winds of the observed WR stars <cit.>. Both the SFR andLGRB rates are favorable as proxies for the BBH formation rate. In both cases most of the formation takes place at early times, allowing for large initial separations. The resulting χ_ eff distributions arising from these two scenarios are practically indistinguishable. A constant BBH formation rate implies more recent formation events and hence shorter merger times leading tolarger χ_ eff values. Still withextreme parameters even this distribution can be made consistent with the current data. High redshift WR stars are the best candidates for being progenitors with low χ_ eff. They gain from having a long merger times, that allows them to begin with a relatively large separation that implies much weaker synchronization. However, this is not enough and strong or moderate winds (forprogenitors that are non-rotating at the end of the common envelope phase)are essentialfor consistency with the current distribution. A longerminimal time delay (corresponding to larger separations at the birth ofBBHs) helps, but is insufficient to lead to consistency.To concludewe note that a comparison of the currently observed O1 and O2 χ_ eff values with the models show some tension, however it does not rule out evolutionary models based on WR stars.In fact somemodels are almost as consistent as the best fitted low-isotropic spin model of <cit.>. While many models predict a significant fraction (>25%) of large (>0.4 for singly synchronized and>0.8 for double synchronization) χ_ eff events, someproduce distributions that are concentrated around positive very low χ_ eff values.The question which models are consistent depends on largely unexplored late stage evolution of very massive stars.Given these results it seems that while a significant fraction of high χ_ eff mergers will strongly support the field evolutionary scenario, lack of those will be hard to interpret. It may indicate another scenario or, for example, strong winds that remove the spin angular momentum. On the other hand a significant fraction of negative χ_ eff merger will be difficult to reconcile with this scenario, unless the BHs' angular momentum is dominated by very strong natal kicks. § ACKNOWLEDGMENTS KH is supported by the Flatiron Fellowship at the Simons Foundation. 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"authors": [
"Kenta Hotokezaka",
"Tsvi Piran"
],
"categories": [
"astro-ph.HE",
"gr-qc"
],
"primary_category": "astro-ph.HE",
"published": "20170727180200",
"title": "Are the observed black hole mergers spins consistent with field binary progenitors?"
} |
Chiral symmetry constraints on resonant amplitudes [ December 30, 2023 ==================================================emptyWereview the categorical approach to the BPS sector of a 4d 𝒩=2 QFT, clarifying many tricky issues and presenting a fewnovel results. To a given 𝒩=2 QFT one associatesseveral triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than onwhat we can learn from a single description. A second general theme is viewing the cluster category as a sort of `categorification' of 't Hooft's theory of quantum phases for a 4d non-Abelian gauge theory.The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of thegiven 𝒩=2 theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry.For class 𝒮 theories, all the relevant triangle categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangle categories andthe WKB approach of GMN. We also review how the VEV's of UV line operators are related to cluster characters. July, 2017§ INTRODUCTION AND OVERVIEW The BPS objects of a supersymmetric theory are naturally described in terms of (-linear) triangle categories <cit.> and their stability conditions <cit.>. The BPS sector of a given physical theory 𝒯 is described by a plurality of different triangle categories 𝔗_(a) depending on:[ The index a take values in some index set I.] a) the class of BPS objects (particles, branes, local or non-local operators,...) we are interested in;b) the physical picture (fundamental UV theory, IR effective theory,...);c) the particular engineering of 𝒯 in QFT/string/M-/F-theory.The diverse BPS categories 𝔗_(a) are related by a web of exact functors, 𝔗_(a)𝔗_(b), which express physical consistency conditions between the different physical pictures and BPS objects. The simplest instance is given by two different engineerings of the same theory: the duality 𝒯↔𝒯^' induces equivalences of triangle categories 𝔗_(a)𝔗^'_(a) for all objects and all physical descriptions a∈ I. An exampleis mirror symmetry betweenIIA and IIB string theories compactified on a pair of mirror Calabi-Yau 3-folds, ℳ, ℳ^∨ which induces on the BPS branes homological mirror symmetry, that is, the equivalences of triangle categories <cit.>D^b(𝖢𝗈𝗁 ℳ)≅ D^b(𝖥𝗎𝗄 ℳ^∨), D^b(𝖢𝗈𝗁 ℳ^∨)≅ D^b(𝖥𝗎𝗄 ℳ).In the same way, the functor relating the IR and UV descriptions of the BPS sector may be seen as homological Renormalization Group, while the functor relating particles and branes may be seen as describing properties of the combined system. A duality induces a family of equivalences 𝖽_(a), one for each category 𝔗_(a), and these equivalences should be compatible with the functors 𝖼_(a,b), that is, they should give an equivalence of the full web of categories and functors. Our philosophy is that the study of equivalences of the full functorial web is a very efficient toolto detectdualities. We shall focus on the case of 4d 𝒩=2 QFTs, but the strategy has general validity. We are particularly concerned with S-dualities, i.e. auto-dualities of the theory 𝒯 which act non trivially on the UV degrees of freedom.Building on previous work by several people[ References to previous work are provided in the appropriate sections of the paper.], we present our proposal for the triangle categories describing different BPS objects,both from the UV and IR points of view, and study the functors relating them. This leads, in particular, to a categorical understanding of the S-dualitygroups and of the vev of UV line operators. The categorical language unifies in a systematic wayall aspects of the BPS physics, and leads to new powerful techniques to compute susy protected quantities in 𝒩=2 4d theories. We check in many explicit examples that the results obtained from this more abstract viewpoint reproduce the ones obtained by more traditional techniques. However the categorical approach may also be used to tackle problems which look too hard for other techniques.Main triangle categories and functors. The basic example of a web of functors relating distinct BPS categories for 4d 𝒩=4 QFT is the following exact sequence of triangle categories (Theorem 5.6 of <cit.>):0 → D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ)→ 0,where (see . <ref> for precise definitions and details): * Γ is the Ginzburg algebra <cit.> of aquiver with superpotential <cit.>associated to the 𝒩=2 theory at hand;* D^bΓ is the bounded derived category of Γ. D^bΓ may be seen as a “universal envelope” ofthe categories describing, in the deep IR, the BPS particle spectrum in the several BPS chambers. To discuss states in the IRwe need to fix a Coulomb vacuum u; this datum defines a stability condition Z_u on D^bΓ. The category which describes the BPS particles in the u vacuum isthe subcategory of D^bΓ consisting of objects which are semi-stable for Z_u. The BPS particlesarise from (the quantization of the moduliof) the simple objects in this subcategory. Its Grothendieck group K_0(D^bΓ) is identified with the Abelian group of the IR additiveconserved quantum numbers (electric, magnetic, and flavor charges) which take value in the lattice Λ≅ K_0(D^bΓ). D^bΓis a 3-Calabi-Yau (3-CY)[ See . <ref> for precise definitions. Informally, a triangle category is k-CY iff it behaves as the derived category of coherent sheaves, D^b 𝖼𝗈𝗁 ℳ_k, on a Calabi-Yau k-fold ℳ_k. ] triangle category, which implies thatits Euler form χ(X,Y)≡∑_k∈ (-1)^k Hom_D^bΓ(X,Y[k]), X, Y∈ D^bΓis a skew-symmetric form Λ×Λ→ whose physical meaning is the Dirac electro-magnetic pairing between thecharges [X], [Y]∈Λ carried by the states associated to the stable objects X, Y∈ D^bΓ;* 𝒞(Γ) is the cluster category of Γwhich describes[ This is slightly imprecise. Properlyspeaking, the line operators correspond to the generic objects on theirreducible components of the moduli spaces of isoclasses of objects of 𝒞(Γ).] the BPS UV line operators. This identificationis deeply related to the Kontsevitch-Soibelmann wall-crossing formula <cit.>, see <cit.>. The Grothendieck group K_0(𝒞(Γ))then corresponds to the, additive as well as multiplicative,UV quantum numbers of the line operators. These quantum numbers, in particular the multiplicative ones follow from the analysis by 't Hooft of the quantum phases of a 4d non-Abelian gauge theory being determined by the topology of the gauge group <cit.>. 't Hooft arguments are briefly reviewed in . <ref>: the UV line quantum numbers take value in a finitely generated Abelian group whose torsion part consists of two copies of the fundamental group of the gauge group while its free part describes flavor. The fact that K_0(𝒞(Γ)) is automatically equal to the correct UV group, as predictedby 't Hooft (detecting the precise topology of the gauge group!), yields convincing evidence for the proposed identification, see . <ref>.𝒞(Γ) is a 2-CY category, and hence its Euler form induces a symmetric form on the additive UV charges, which roughly speaking has the formK_0(𝒞(Γ))/K_0(𝒞(Γ))_torsion ⊗K_0(𝒞(Γ))/K_0(𝒞(Γ))_torsion→,but whose precise definition is slightly more involved[ The subtleties in the definition are immaterial when the QFT is UV superconformal (as contrasted toasymptotically-free) and all chiral operators have integral dimensions.] (see . <ref>). We call this pairing the Tits form of 𝒞(Γ). Its physical meaning is simple: while in the IR the masses break (generically) the flavor group to its maximal torus U(1)^f, in the deep UV the masses become irrelevant and the flavor group getsenhanced to its maximal non-Abelian form F. Then the UV category should see the full F and not just its Cartan torus. The datum of the group F may be given as its weight lattice together with its Tits form; the cluster Tits form is equal to the Tits form of the non-Abelian flavor group F, and we may read F directly from the cluster category. In fact the cluster category also detects the global topology of the flavor group, distinguishing (say) SO(N) and Spin(N) flavor groups.[ In facts, the cluster Grothendieck group K_0(𝒞(Γ)) should contain even more detailed informations on the flavor. For instance, in SU(2) gauge theory with N_f flavors the states of even magnetic charge are in tensor representations of the flavor SO(2N_f) while states of odd magnetic charge are in spinorial representation of Spin(8); K_0(𝒞(Γ)) should know the correlation between the parity of the magnetic charge and SO(2N_f) vs. Spin(2N_f) flavor symmetries (and it does). ]For objects of 𝒞(Γ) there is also a weaker notion of `charge', taking value in the lattice Λ of electric/magnetic/flavor charges, namely the index, which is the quantity referred to as `charge' in many treatments. Since 𝒞(Γ) yields an UV description of the theory, there must exist relations between its mathematical properties and the physical conditions assuring UV completeness of the associated QFT. We shall point of some of them in . <ref>;* 𝔓𝔢𝔯 Γ is the perfect derived category of Γ. From eqn.(<ref>) we see that, morally speaking, the triangle category 𝔓𝔢𝔯 Γ describes allpossible BPS IR object generated by the insertion of UV line operators, dressed (screened) by particles, in all possible vacua. This rough idea is basically correct. Perhaps the most convincing argument comes from consideration ofclass 𝒮 theories, where we have a geometric construction of the perfect category 𝔓𝔢𝔯 Γ <cit.> as well as a detailedunderstanding of the BPS physics <cit.>. In agreement with this identification, the Grothendieck group K_0(𝔓𝔢𝔯 Γ) is isomorphic to the IR group Λ. 𝔓𝔢𝔯 Γ is not CY, instead the Euler form defines a perfect pairingK_0(D^bΓ)⊗ K_0(𝔓𝔢𝔯 Γ)→; * the exact functor 𝗋 in eqn.(<ref>) may be seen as the homological (inverse) RG flow.Dualities. The (self)-dualities of an 𝒩=2 theory should relate BPS objects to BPS objects of the same kind, and hence should be (triangle) auto-equivalences of the above categories which are consistent with the functors relating them (e.g. 𝗌, 𝗋 in eqn.(<ref>)). We may describe the physical situation from different viewpoints. In the IR picture one would have the putative `duality' group Aut D^bΓ; however a subgroup acts trivially on all observables <cit.>, and the physical IR `duality' group is[ For the precise definition of Auteq D^bΓ, see . 5. Aut(Q) is the group of automorphisms of the quiver Q modulo the subgroup which fixes all nodes.]𝒮_IR≡Aut D^bΓ/{physically trivial autoequivalences} = Auteq D^bΓ⋊Aut(Q).In the UV (that is, at the operator level) the natural candidate `duality' group is𝒮_UV≡Aut 𝒞(Γ)/{physically trivial} From the explicit description of AutD^bΓ (see .5) we learn that 𝒮_IR extends to a group of autoequivalences of 𝔓𝔢𝔯 Γ which preserve D^bΓ (by definition). Hence the exact functor 𝗋𝔓𝔢𝔯 Γ→𝒞(Γ)in eqn.(<ref>) induces a group homomorphism𝒮_IR𝒮_UV,whose image is𝕊= r(Auteq D^bΓ)⋊Aut Q.𝕊 is a group of auto-equivalences whose action is defined at the operator level, that is, independently of a choice of vacuum. They are equivalences of the full web of BPS categories in eqn.(<ref>). Thus 𝕊 is the natural candidate for the role of the (extended) S-duality group of our 𝒩=2 model. Indeed, in the examples where we know theS-duality group from moreconventional considerations, it coincides with our categorical group 𝕊. In this survey we take equation (<ref>) as the definition of theS-duality group.Clearly, the essential part of 𝕊 is the group r(Auteq D^bΓ). It turns out that precisely this group is an object of central interest in the mathematical literature which provides anexplicit combinatorial description of it <cit.>. This combinatorial description is the basis of an algorithm for computer search of S-dualities, see . <ref>. If our 𝒩=2 theory is not too complicated (that is, the ranks of the gauge and flavor group are not too big) the algorithm may be effectively implemented on a laptop, see . <ref> for explicit examples. For class 𝒮 theories, the above combinatorial description of S-duality has a nice geometric intepretation as the (tagged) mapping class group of the Gaiotto surface, in agreement with the predictions of <cit.>(see also <cit.>), see . 5.2. More generally, for class 𝒮 theories all categorical constructions have a simple geometric realization which makes manifest their physical meaning. The IR group 𝒮_IR may be understood in terms of duality walls, see .<ref>. Cluster characters and vev of line operators. The datum of a Coulomb vacuum u defines a map⟨- ⟩_u GenOb(𝒞(Γ)) →,given by taking the vev in the vacuum u of the UV line operator associated to a given generic object of the cluster category 𝒞(Γ). Physically, the renormalization group implies that themap ⟨ - ⟩_u factors through the (Laurent) ring [ℒ] of line operatorsin the effective (Abelian) IR theory. The associated mapGenOb(𝒞(Γ)) →[ℒ]is called a cluster character and is well understood in the mathematical literature. Thus the theory of cluster character solves(in principle) the problem of computing the vev of arbitrary BPS line operators (see .<ref>). Organization of the paper. The rest of this paper is organized as follows. In section <ref> we review the mathematics of derived DG categories, cluster categories, and related topics in order to provide the reader with a language which allows to unify and generalize several previous analysis of the BPS sector of a 𝒩=2 4d theory. In section <ref> we recall some general physical properties that our categories should enjoy in order to be valid descriptions of the various BPS objects. Here we stress the various notions of charge, with particular reference to the 't Hooft charges in the UV description. Section <ref> is the core of the paper, where we present our physical interpretation of the categories described mathematically in section <ref> and show that they satisfy all physical requirements listed in section <ref>. In section <ref> we discuss S-duality from the point of view of triangle categories, present first examples and describe the relation to 3d mirror symmetry. In section <ref> we introduce our combinatorial algorithm to find S-dualities and give a number of examples. In section <ref> we consider class 𝒮[A_1] theories and describe all the relevant triangle categories in geometric terms à la Gaiotto. In particular, this shows that our categorical definition of S-duality is indeed equivalent to more conventional physical definitions. In section <ref> we discuss vev's of line operators from the point of view of cluster characters. Explicit computer codes are presented in the appendices.§ MATHEMATICAL BACKGROUND In this section we recall the basic definitions of DG categories <cit.>,cluster categories <cit.>, stability conditions for Abelianand triangulated categories <cit.> and show some concrete examples.We then specialize these definitions to the Ginzburg algebra <cit.> Γ associatedto a BPS quiver with (super)potential (Q,W) <cit.>. Some readers may prefer to skip this sectionand refer back to it when looking for definitions and/or details on some mathematical tool used in the main body of this survey.§.§ Differential graded categories The main reference for this section is <cit.>. Let k be a commutative ring,[ In all our physical applications k will be the (algebraically closed) field of complex numbers ℂ.] for example a field or the ring of integers . We will write ⊗ for the tensor product over k.A k-algebra is a k-module A endowed with a k-linear associative multiplication A ⊗_k A → A admitting a two-sided unit 1 ∈ A. For example, a -algebra is just a (possibly non-commutative) ring. A k-categoryis a “k-algebra with several objects”. Thus, it is the datum of a class of objects obj(), of a k-module (X, Y ) for all objects X, Y of , and of k-linear associative composition maps(Y, Z) ⊗(X, Y ) →(X, Z), (f, g) ↦ fgadmitting units 1_X ∈(X, X). For example, we can interpret k-algebras as k-categories with only one object. The category 𝗆𝗈𝖽 A of finitely generated right A-modules over a k-algebra A is an example of a k-category with many objects. It is also an example of a k-linear category (i.e. a k-category which admits all finite direct sums). A graded k-module is a k-module V together with a decomposition indexed by the positive and the negative integers:V =⊕_p∈V^p.The shifted module V [1] is defined by V [1]^p = V^ p+1, p ∈. A morphism f : V → V of graded k-modules of degree n is a k-linear morphism such that f (V^p) ⊂ V^p+n for all p ∈.The tensor product V ⊗ W of two graded k-modules V and W is the graded k-module with components(V ⊗ W )^n =⊕_p+q=nV^p ⊗ W^q ,n ∈.The tensor product f ⊗ g of two maps f : V → V and g : W → W of graded k-modules is defined using the Koszul sign rule: we have(f ⊗ g)(v ⊗ w) = (-1)^pq f (v) ⊗ g(w)if g is of degree p and v belongs to V^q. A graded k-algebra is a graded k-module A endowed with a multiplication morphism A ⊗ A → A which is graded of degree 0, associative and admits a unit 1 ∈ A^0.An “ordinary” k-algebra may be identified with a graded k-algebra concentrated in degree 0. A differential graded (=DG) k-module is a -graded k-module V endowed with a differential d_V, i.e. a map d_V V → V of degree 1 such that d^2_V = 0. Equivalently, V is a complex of k-modules. The shifted DG module V [1] is the shifted graded module endowed with the differential -d_V . The tensor product of two DG k-modules is the graded module V ⊗ W endowed with the differential d_V ⊗ 1_W + 1_V ⊗ d_W.A differential graded k-algebra A is a DG k-module endowed with a multiplication morphism A ⊗ A → A gradedof degree 0 and associative. Moreover, the differential satisfies the graded Leibnitz rule:d(ab)=(da) b+(-1)^(a) a(db), ∀a,b∈ Aandahomogeneous. The cohomology of a DG algebra is defined as H^*(A):=kerd/im d. Let 𝗆𝗈𝖽 A denote the category of finitely generatedDG modules over the DG algebra A.The derived category D(A):=D(𝗆𝗈𝖽 A) is the localization of the category 𝗆𝗈𝖽 A with respect to the class of quasi-isomorphisms. Thus, the objects of D(A) are the DG modules and its morphisms are obtained from morphisms of DG modules by formally inverting all quasi-isomorphisms. The bounded derived category of 𝗆𝗈𝖽 A, denoted D^bA,is the triangulated subcategory of D(A) whose objects are quasi-isomorphic to objects with bounded cohomology.The perfect derived category of a DG algebra A, 𝔓𝔢𝔯 A, is the smallest full triangulated subcategory of D(A) containing A which is stable under taking shifts, extensions and direct summands. §.§ Quivers and mutations In this section we follow <cit.>. Let k be an algebraically closed field. A (finite) quiver Q is a (finite) oriented graph (possibly with loops and 2-cycles). We denote its set of vertices by Q_0 and its set of arrows by Q_1. For an arrow a of Q, let s(a) denote its source node and t(a) denote its target node. The lazy path corresponding to a vertex i will be denoted by e_i. The path algebra kQ̂ is the associative unital algebra whoseelements are finite compositions of arrows of Q, where the composition of a,b ∈ Q_1 is denoted ab andit is nonzero iff s(b)=t(a). The complete path algebra kQ is the completion of the path algebra withrespect to the ideal I generated by the arrows of Q. Let I be the ideal of kQ generated by the arrows of Q. A potential W on Q is an element of the closure of the space generated by all non trivial cyclic paths of Q. We say two potentials are cyclically equivalent if their difference is in the closure of the space generated by all differences a_1 ... a_s - a_2 ... a_sa_1, where a_1 ... a_s is a cycle.Let u,p and v be nontrivial paths of Q such that c=upv is a nontrivial cycle. For the path p of Q, we define∂_p : kQ → kQas the unique continuous linear map which takes a cycle c to the sum ∑_ c=upv vu taken over all decompositions of the cycle c (where u and v are possibly lazy paths).Obviously two cyclically equivalent potentials have the same image under ∂_p. If p = a is an arrow of Q, we call ∂_a the cyclic derivative with respect to a. Let W be a potential on Q such that W is in I^2 and no two cyclically equivalent cyclic paths appear in the decomposition of W. Then the pair (Q, W) is called a quiver with potential. Two quivers with potential (Q, W) and (Q', W') are right-equivalent if Q and Q' have the same set of vertices and there exists an algebra isomorphism ϕ : kQ → kQ' whose restriction on vertices is the identity map and ϕ(W) and W' are cyclically equivalent. Such an isomorphism ϕ is called a right-equivalence. The Jacobian algebra of a quiver with potential (Q, W), denoted by J(Q, W), is the quotient of the complete path algebra kQ by the closure of the ideal generated by ∂_a W, where a runs over all arrows of Q:J(Q,W):=kQ/∂_aW.We say that the quiver with potential (Q,W) is Jacobi-finite if the Jacobian algebraJ(Q,W) is finite-dimensional over k.It is clear that two right-equivalent quivers with potential have isomorphic Jacobian algebras. A quiver with potential is called trivial if its potential is a linear combination of cycles of length 2 and its Jacobian algebra is the product of copies of the base field k.§.§.§ Quiver mutationsLet (Q, W) be a quiver with potential. Let i∈ Q_0 a vertex. Assume the following conditions: * the quiver Q has no loops;* the quiver Q does not have 2-cycles at i;We define a new quiver with potential μ̃_i(Q, W) = (Q', W') as follows. The new quiver Q' is obtained from Q by * For each arrow β with target i and each arrow α with source i, add a new arrow [αβ] from the source of β to the target of α.* Replace each arrow α with source or target i with an arrow α^* in the opposite direction.If we represent the quiver with its exchange matrix B_ij, i.e.the matrix such thatB_ij=#{ arrows fromitoj}-#{ arrows fromjtoi} then the transformation that B_ij undergoes isB'_ij= - B_ij, i=kor j=kB_ij+max[-B_ik,0] B_kj+B_ikmax[B_kj,0]otherwise.The new potential W' is the sum of two potentials W'_1 and W'_2. The potential W'_1 is obtained from W by replacing each composition αβ by [αβ], where β is an arrow with target i. The potential W'_2 is given by <cit.>W'_2 =∑_α,β[αβ]β^*α^*,the sum ranging over all pairs of arrows α and β such that β ends at i and α starts at i.Let I be the ideal in kQ generated by all arrows. Then, a quiver with potential is called reduced if ∂_aW is contained in I^2 for all arrows a of Q.One shows that all quivers with potential (Q,W) are right-equivalent to the direct sum of areduced quiver with potential and a trivial one.[ In terms of the corresponding SQM system, the process of replacing the pair (Q,W) by its reduced part (Q_red.,W_red.) corresponds to integrate away the massive Higgs bifundamentals.]We can now give the definition of the mutated quiver: we define μ_i(Q, W) as the reduced partof μ̃_i(Q, W), and call μ_i the mutation at the vertex i. An example will clarify all these concepts. [A_3 quiver] Consider the quiver A_3 given by Q: ∙_1 α←∙_2 β→∙_3 with W=0.Let us consider the quiver μ_1(Q):∙_1 α^*→∙_2 β→∙_3 with W=0.Now apply the mutation at vertex 2: we get ∙_2[dl]_α∙_1[rr]^[αβ]∙_3 [ul]_β^* μ_2(μ_1(Q))with potential W=αβ^*[αβ].We conclude this subsection with the following-12pt * The right-equivalence class of μ̃_i(Q, W) is determined by the right-equivalence class of (Q, W).* The quiver with potential μ̃^2_i(Q, W) is right-equivalent to the direct sum of (Q, W) with a trivial quiver with potential.* The correspondence μ_i acts as an involution on the right equivalence classes of reduced quivers with potential. §.§ Cluster algebrasWe follow <cit.>. Let Q be a 2-acyclic quiver with vertices 1, 2, ..., n, and let F = (x_1, ... , x_n) be the function field in n indeterminates over . Consider the pair (x⃗, Q), where x⃗ = {x_1,... , x_n}. The cluster algebra C(x⃗, Q) will be defined to be a subring of F. The pair (x⃗, Q) consisting of a transcendence basis x⃗ for F over the rational numbers , together with a quiver with n vertices, is called a seed. For i = 1, ... , n we define a mutation μ_i taking the seed (x⃗, Q) to a new seed (x⃗', Q'), where Q' = μ_i(Q) as discussed in <ref>, and x⃗' is obtained from x⃗ by replacing x_i by a new element x'_i in F. Here x'_i is defined byx_ix'_i = m_1 + m_2,where m_1 is a monomial in the variables x_1, ... , x_n, where the power of x_j is the number of arrows from j to i in Q, and m_2 is the monomial where the power of x_j is the number of arrows from i to j. (If there is no arrow from j to i, then m_1 = 1, and if there is no arrow from i to j, then m_2 = 1.) Note that while in the new seed the quiver Q' only depends on the quiver Q, then x' depends on both x and Q. We haveμ^2_i(x⃗, Q) = (x⃗, Q).The procedure to get the full cluster algebra is iterative. We perform this mutation operation for all i = 1, ... , n, then we perform it on the new seeds and so on. Either we get new seeds or we get back one of the seeds already computed. The n-element subsets x⃗, x⃗', x⃗”, ... occurring are by definition the clusters, the elements in the clusters are the cluster variables, and the seeds are all pairs (x⃗', Q') occurring in the iterative procedure. The corresponding cluster algebra C(x⃗, Q), which as an algebra only depends on Q, is the subring of F generated by the cluster variables.Let Q be the quiver 1 → 2 → 3 and x⃗ = {x_1, x_2, x_3}, where x_1, x_2, x_3 are indeterminates, and F = (x_1, x_2, x_3). We have μ_1(x⃗, Q) = (x', Q'), where Q' = μ_1(Q) is the quiver 1 ← 2 → 3 and x⃗' = {x'_1, x_2, x_3}, where x_1x'_1 = 1+x_2, so that x'_1 =1+x_2/x_1. And so on. The clusters are:{x_1, x_2, x_3},{1+x_2/x_1, x_2, x_3},{ x_1, x_1+x_3/x_2, x_3},{x_1, x_2,1+x_2/x_3},{1+x_2/x_1,x_1+(1+x_2)x_3/x_1x_2, x_3},{1+x_2/x_1, x_2,1+x_2/x_3},{x_1+(1+x_2)x_3/x_1x_2,x_1+x_3/x_2, x_3}, {x_1,x_1+x_3/x_2, (1+x_2)x_1+x_3/x_2x_3}, {x_1,(1+x_2)x_1+x_3/x_2x_3, 1+x_2/x_3}, {1+x_2/x_1, x_1+(1+x_2)x_3/x_1x_2, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3},{1+x_2/x_1, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,1+x_2/x_3}, {x_1+(1+x_2)x_3/x_1x_2, x_1+x_3/x_2, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3}, {(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,x_1+x_3/x_2,(1+x_2)x_1+x_3/x_2x_3}, {(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,(1+x_2)x_1+x_3/x_2x_3, 1+x_2/x_3}, and the cluster variables are:x_1,x_2, x_3,1+x_2/x_1,x_1+x_3/x_2,1+x_2/x_3,x_1+(1+x_2)x_3/x_1x_2,(1+x_2)x_1+x_3/x_2x_3,(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3. §.§.§ The cluster exchange graph (CEG)If Q' is a quiver mutation equivalent to Q, then the cluster algebras C(Q') and C(Q) are isomorphic. The n-regular connected graph whose vertices are the seeds of C(x⃗,Q) (up to simultaneous renumbering of rows, columns and variables) and whose edges connect the seeds related by a single mutation is called cluster exchange graph (=CEG). The CEG for Example <ref> is represented in figure <ref>. §.§ Ginzburg DG algebrasGiven a quiver Q with potential W, we can associate to it the Jacobian algebra J(Q,W):=kQ/∂ W, where kQis the quiver path algebra (see section <ref>). It is also possible to extend the path algebra kQ to a DG algebra: the Ginzburg algebra.(Ginzburg <cit.>). Let (Q, W) be a quiver with potential. Let Q̂ be the graded quiver with the same set of vertices as Q and whose arrows are: * the arrows of Q (of degree 0);* an arrow a^*: j → i of degree -1 for each arrow a : i → j of Q;* a loop t_i: i → i of degree -2 for each vertex i ∈ Q_0.The completed Ginzburg DG algebra Γ(Q, W) is the DG algebra whose underlying graded algebra is the completion[ The completion is taken with respect to the I-adic topology,where I is the ideal of the path algebra generated by all arrows of the quiver.] of the graded path algebra kQ̂. The differential of Γ(Q, W) is the unique continuous linear endomorphism homogeneous of degree 1 which satisfies the Leibniz rule (i.e. d(uv) = (du)v + (-1)^p udv for all homogeneous u of degree p and all v), and takes the following values on the arrows of Q̂:d(a)= 0d(a^*)= ∂_aW, ∀ a ∈ Q_1; d(t_i)= e_i(∑_a∈ Q_1[a, a^*])e_i, ∀ i ∈ Q_0.We shall write Γ(Q,W) simply as Γ, unless we wish to stress its dependence on (Q,W).From the definition of Γ and d, one sees that H^0Γ≅ J(Q,W). To the DG algebra Γ weassociate three important triangle categories which we are now going to define and analyze in detail. §.§ The bounded and perfect derived categories The DG-category 𝗆𝗈𝖽 Γ is the category whose objects are finitelygenerated graded Γ-modules and the morphisms spaces have the structure of DG modules(cfr. section <ref>).The derived category DΓ:=D(𝗆𝗈𝖽 Γ)<cit.> is the localization of 𝗆𝗈𝖽 Γat quasi-isomorphisms (the cohomology structure is given by the differential d of the Ginzburg algebra).Thus, the objects of DΓ are DG modules. There are two fundamental subcategories associated to DΓ: * The bounded derived category D^b Γ: it is the full subcategory of DΓ such that its objectsare graded modules M for which, given a certain N>0, H^n (M)=0 for all |n| > N.This category is 3-CY (see below).* The perfect derived category 𝔓𝔢𝔯 Γ, i.e. the smallest full triangulated subcategoryof DΓ which contains Γ and is closed under extensions, shifts in degree and taking direct summands.Both 𝔓𝔢𝔯 Γ and D^b Γ are triangulated subcategories of DΓ and in particular,𝔓𝔢𝔯 Γ⊃ D^b Γ as a full subcategory (as explained in <cit.>).Furthermore,[ See <cit.> for more details.] the category D^bΓhas finite-dimensional morphism spaces (even its graded morphism spaces are of finite total dimension) and is 3-Calabi-Yau (3-CY), by which we mean that we have bifunctorial isomorphisms[ More generally, we say that a triangle category is ℓ-CY (for ℓ∈ℕ) iff we have the bifunctorial isomorphism DHom(X,Y)≅Hom(Y,X[ℓ]).]D Hom(X, Y ) ≅Hom(Y, X[3]) ,where D is the duality functor Hom_k(-, k) and [1] the shift functor. The simple J(Q, W)-modules S_i can be viewed as Γ-modules via the canonical morphismΓ→ H^0(Γ). [A_2 quiver] Consider the A_2 quiver 1→ 2. The following is a graded indecomposable Γ-module:t_1^*(k[-1]⊕ k[-3]) a^*a⇌ kt_2^*,where a=0, a^*:k1↦k[-1], t_2^*=0, and t_1^*:k[-1]1↦ k[-3].This object can be generated from S_1[-1], S_1[-3] and S_2 by successive extensions. Moreover, the modules S_i, i =1,2and their shifts are enough to generate[ In the triangulated category 𝒯, a set of objects S_i ∈𝒯is a generating set if all objects of 𝒯 can be obtained from the generating set via an iterated cone construction.] all(homologically finite) graded modules. §.§.§ Seidel-Thomas twists and braid group actionsSimple Γ-modules S_i become 3-spherical objects in D^bΓ (hence also in DΓ), that is, Hom(S,S[j])≅ k (δ_j,0+δ_j,3). They yield the Seidel-Thomas <cit.> twist functors T_S_i. These are autoequivalences of DΓ such thateach object X fits into a triangleHom^∙_D(S_i, X) ⊗_k S_i → X → T_S_i(X) → .By construction, T_S_i restricts to an autoequivalence of the subcategory D^bΓ⊂ DΓ. From the explicit realization of T_S_i as a cone in DΓ, eqn.(<ref>), it is also clear that it restricts to an auto-equivalence of 𝔓𝔢𝔯 Γ.As shown in <cit.>, the twist functors give rise to a (weak) action on DΓ of the braid group associated with Q, i.e. the group with generators σ_i, i ∈ Q_0, and relationsσ_iσ_j = σ_jσ_iif i and j are not linked by an arrow in Q andσ_iσ_jσ_i = σ_jσ_iσ_jif there is exactly one arrow between i and j (no relation if there are two or more arrows).We write Sph(D^bΓ)⊂Aut(D^bΓ) for the subgroup of autoequivalencesgenerated by the Seidel-Thomas twists associated to all simple objects S_i∈ D^bΓ.§.§.§ The natural t-structure and the canonical heart The category DΓ admits a natural t-structure whose truncation functors are those of the natural t-structure on the category of complexes of vector spaces (because Γ is concentrated in degrees ≤ 0). Thus, we have an induced natural t-structure on D^bΓ. Its heartis canonically equivalent to the category 𝗇𝗂𝗅 J(Q, W) of nilpotent modules[ If (Q,W) is Jacobi-finite (as in our applications), 𝗇𝗂𝗅 J(Q, W)≡𝗆𝗈𝖽J(Q, W).] <cit.>. In particular, the inclusion ofinto D^bΓ induces an isomorphism of Grothendieck groupsK_0() ≅ K_0(D^bΓ)≅⊕_i[S_i]. The skew-symmetric form. Notice that the lattice K_0(D^bΓ) carries the canonical Euler form defined byX,Y=∑_i=0^3(-1)^iHom_D(Γ)(X,Y[i]).It is skew-symmetric thanks to the 3-Calabi-Yau property (<ref>). Indeed it follows from the Calabi-Yau property and from the fact that Ext^i_(L, M) = Hom_D^bΓ(L, M[i]) for i = 0 and i = 1 (but not i > 1 in general) that for two objects L and M of ⊂ D^bΓ, we haveL, M = Hom(L, M) - Ext^1(L, M) + Ext^1(M, L) - Hom(M, L).Since the dimension of Ext^1(S_i, S_j ) equals the number of arrows in Q from j to i (Gabriel theorem <cit.>), we obtain that the matrix of -,- in the basis of the simples ofhas its(i, j)-coefficient equal to the number of arrows from i to j minus the number of arrows from j to i in Q, that is,(cfr. eqn.(<ref>)) ⟨ S_i,S_j⟩= B_ij. §.§.§ Mutations at category levelThe main reference for this subsection is <cit.>. Let k be a vertex of the quiver Q notlying on a 2-cycle and let (Q', W') be the mutation of (Q, W) at k. Let Γ' be the Ginzburg algebra associated with (Q', W'). Let ' be the canonical heart in D^bΓ'. There are two canonical equivalences DΓ' → DΓgiven by functors Φ^± related byT_S_k∘Φ^- →Φ^+.where, again, T_S_k is the Seidel-Thomas twist generated by the spherical object S_k. If we put P_i = Γ e_i, i ∈ Q_0, and similarly for Γ', then both Φ^+ and Φ^- send P'_i to P_i for i ≠ k; the images of P'_k under the two functors fit into trianglesP_k →⊕_k → i P_i →Φ^-(P'_k) →andΦ^+(P'_k) →⊕_j→ k P_j → P_k .The functors Φ^± send ' onto the hearts μ_k^±() of two new t-structures. These can be described in terms ofand the subcategory[ Here and below, given a (collection of) object(s) 𝒪 of a linear category 𝔏, by 𝖺𝖽𝖽 𝒪 we mean the additive closure of 𝒪 in 𝔏, that is, the full subcategory over the direct summands of finite direct sums of copies of 𝒪.]𝖺𝖽𝖽 S_k as follows: Let S^⊥_k be the right orthogonal subcategory of S_k in [ Its objects are those M's with Hom(S_k, M) = 0. It is a full subcategory of .]. Then μ^+_k() is formed by the objects X of D^bΓ such that the object H^0(X) belongs to S^⊥_k, the object H^1(X) belongs to 𝖺𝖽𝖽 S_k and H^p(X) vanishes for all p ≠0, 1. Similarly, the subcategory μ^-_k() is formed by the objects X such that the object H^0(X) belongs to the left orthogonal subcategory ^⊥ S_k, the object H^-1(X) belongs to 𝖺𝖽𝖽 S_k and H^p(X) vanishes for all p ≠ -1, 0. The subcategory μ^+_k() is the right mutation ofand μ^-_k() is its left mutation. By construction, we haveT_S_k(μ^-_k()) = μ^+_k().Since the categoriesand μ^±() are hearts of bounded, non degenerate t-structures on D^bΓ, their Grothendieck groups identify canonically with that of D^bΓ. They are endowed with canonical basis given by the simples. Those ofidentify with the simples S_i, i ∈ Q_0, of 𝗇𝗂𝗅 J(Q, W). The simples of μ^+_k() are S_k[-1], the simples S_i ofsuch that Ext^1(S_k, S_i) vanishes and the objects T_S_k(S_i) where Ext^1(S_k, S_i) is of dimension ≥ 1. By applying T^-1_S_k to these objects we obtain the simples of μ^-_k(). We saw that D^bΓ⊂𝔓𝔢𝔯 Γ as a full subcategory <cit.>:what is then the meaning of the Verdier quotient <cit.> of these two triangulated categories? §.§ The cluster category The next result is the main step in the construction of new 2-CY categories with cluster-tilting object which generalize the acyclic cluster categories introduced by Buan-Marsh-Reineke-Todorov to categorify the cluster agebras of Fomin and Zelevinski.Let A be a DG-algebra with the following properties: * A is homologically smooth (i.e. A∈𝔓𝔢𝔯(A⊗ A^op)),* H^p(A) = 0 for all p ≥ 1,* H^0(A) is finite dimensional as a k-vector space,* A is bimodule 3-CY, i.e. Hom_D(A)(X, Y ) ≅ DHom_D(A)(Y, X[3]),for any X ∈ D(A) and Y ∈ D^b(A).Then the triangulated category 𝒞(A) = 𝔓𝔢𝔯 A/D^b A is Hom-finite, 2-CY, i.e. Hom_𝒞(A)(X, Y ) ≅ DHom_𝒞(A)(Y, X[2]), X,Y∈𝒞(A). and the object A is a cluster-tilting object[ See Definition <ref>.] with End_𝒞(A)(A) ≅ H^0(A). The category 𝒞(A) is called the generalized cluster category andit reduces to the standard cluster category <cit.> in the acyclic case.It is triangulated since it is the Verdier quotient of triangulatedcategories.[ The main references for these categorical facts are <cit.>. We recall the definition of Verdier quotient of triangle categories:Lemma. Let D be a triangulated category. Let D' ⊂ D be a full triangulated subcategory. Let S⊂𝖬𝗈𝗋(D) be the subset of morphisms such that there exists a distinguished triangle (X,Y,Z,f,g,h)∈ D with Z isomorphic to an object of D^'.Then S is a multiplicative system compatible with the triangulated structure on D.Definition.Let D be a triangulated category. Let B be a full triangulated subcategory.We define the (Verdier) quotient category D/B by the formula D/B=S^-1D, where S is the multiplicativesystem of D associated to B via the previous lemma.] §.§.§ The case of the Ginzburg algebra of (Q,W)In particular, we may specialize to the case where A=Γ, i.e. the Ginzburg algebra of a quiver withpotential (Q,W), and write the following sequence:0 → D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ) → 0the above theorem states that this sequence is exact and 𝗋(Γ)=T, where T is the canonical cluster-tiltingobject[ See Definition <ref>.] of 𝒞(Γ). The first map in eqn.(<ref>) is theinclusion map: see <cit.> for details. Moreover, an object M∈𝔓𝔢𝔯 Γ belongs to the subcategory D^bΓ if and only if the space Hom_𝔓𝔢𝔯 Γ(P, M) is finite-dimensional for each P∈𝔓𝔢𝔯 Γ.In particular, this implies that there is a duality between the simple objects S_i ∈ D^bΓand the projective objects Γ e_i ∈𝔓𝔢𝔯 Γ⟨Γ e_i, S_j⟩=δ_ij. The completed Ginzburg DG algebra Γ(Q, W) is homologically smooth and bimodule 3-Calabi-Yau.We have already shown that Γ(Q, W) is non zero only in negative degrees, and that H^0(Γ(Q, W)) ≅J(Q, W). Therefore by the theorem above we get the followingLet (Q, W) be a Jacobi-finite quiver with potential. Then the category𝒞(Γ(Q,W)):= 𝔓𝔢𝔯 Γ(Q,W)/D^b(Γ(Q, W))is Hom-finite, 2-Calabi-Yau, and has a canonical cluster-tilting object[ See Definition <ref>.] whose endomorphism algebra is isomorphic to J(Q, W). We shall write 𝒞(Γ(Q,W)) simply as 𝒞(Γ) leaving (Q,W) implicit.§.§.§ The cluster category of a hereditary category The above structure simplifies in the case of a cluster category arising from a hereditary (Abelian) category ℋ (with a Serre functor and a tilting object) <cit.>. Physically this happens for the following list of complete 𝒩=2 QFTs <cit.>: i) Argyres-Douglas of type ADE, ii) asymptotically-free SU(2) gauge theories coupled to fundamental quarks and/or Argyes-Douglas models of type D, and iii) SCFT SU(2) theories with the same kind of matter. In terms of quiver mutations classes, they correspond (respectively) to ADE Dynkin quivers of the finite, affine, and elliptic type[ In the elliptic type we are restricted to the four types D_4, E_6, E_7 and E_8, corresponding to the four tubular weighted projective lines <cit.>. Elliptic D_4 is SU(2) with N_f=4 <cit.>.] <cit.>. In all these case we have an hereditary (Abelian) category ℋ, with the Serre functor S=τ [1] where τ is the Auslander-Reiten translation. That is, in their derived category we haveHom_D^b(ℋ)(X,Y)≅ DHom_D^b(ℋ)(Y,τ X[1])τ is an auto-equivalence of D^b(ℋ). The cluster category can be shown to be equivalent to the orbit category <cit.>𝒞(ℋ)≅ D^b(ℋ)/⟨τ^-1[1]⟩^. For future reference, we list the relevant categories ℋ (further details may be found in <cit.>): * For Argyres-Douglas of type ADE, we have ℋ≅𝗆𝗈𝖽 k 𝔤⃗, where 𝔤⃗ is a quiver obtained by choosing an orientation to the Dynkin graph of type 𝔤∈ ADE (all orientations being derived-equivalent). τ satisfies the equation (for more refined results see <cit.>)τ^h=[-2],where h is the Coxeter number of the associated Lie algebra 𝔤;* for SU(2) gauge theories coupled to Argyres-Douglas systems of types[ In our conventions, p_i=1 means the empty matter system, while p_i=2 is a free quark doublet.] D_p_1,⋯,D_p_s, we have ℋ=𝖼𝗈𝗁 𝕏(p_1,…,p_s), the coherent sheaves over a weighted projective line of weights (p_1,…, p_s) <cit.>[ For a review of the category of coherent sheaves on weighted projective lines and corresponding cluster categoriesfrom a physicist prospective, see <cit.>.]. τ acts by multiplication by the canonical sheaf ω, and hence is periodic iff ω=0; in general, ω is minus the Euler characteristic of 𝕏(p_1,…,p_s), χ=2-∑_i(1-1/p_i). However, τ is always periodic of period lcm(p_i) when restricted to the zero rank sheaves (`skyskrapers' sheaves).§.§ Mutation invarianceWe have already stated that mutations correspond to Seiberg-like dualities. Therefore, our categorical construction makes sense only if it is invariant by mutations: indeed, we do not want the categories representing the physics to change when we change the mathematical description of the same dynamics. The following two results give a connection between the DG categories we just analyzed and quivers with potentials linked by mutations.Let (Q, W) be a quiver with potential without loops and i ∈ Q_0 not on a 2-cycle in Q. Denote by Γ := Γ(Q, W) and Γ':= Γ(μ_i(Q, W)) the completed Ginzburg DG algebras. * <cit.> There are triangle equivalences𝔓𝔢𝔯Γ[r]^∼𝔓𝔢𝔯 Γ^' D^bΓ[r]^∼@^(->[u]D^bΓ^'@^(->[u] Hence we have a triangle equivalence 𝒞(Γ) ≅𝒞(Γ').* <cit.>We have a diagram𝔓𝔢𝔯Γ[rrr]^∼[d]^H^0𝔓𝔢𝔯Γ^'[d]^H^0𝗆𝗈𝖽J(Q,W) @<.>[rrr]^mutation 𝗆𝗈𝖽J(μ_i(Q,W)) Let 𝒞 be a Hom-finite triangulated category. An object T ∈𝒞 is called cluster-tilting (or 2-cluster-tilting) if T is basic (i.e. with pairwise non-isomorphic direct summands) and if we have𝖺𝖽𝖽 T = {X ∈𝒞 | Hom_𝒞(X, T [1]) = 0} = {X ∈𝒞 | Hom_𝒞(T, X[1]) = 0}.Note that a cluster-tilting object is maximal rigid (the converse is not always true, see <cit.>), and that the second equality in the definition always holds when 𝒞 is 2-Calabi-Yau.If there exists a cluster-tilting object in a 2-CY category 𝒞, then it is possible to construct others by a recursive process resumed in the following:Let 𝒞 be a Hom-finite 2-CY triangulated category with a cluster-tilting object T. Let T_i be an indecomposable direct summand of T ≅ T_i ⊕ T_0. Then there exists a unique indecomposable T^*_i non isomorphic to T_i such that T_0 ⊕ T^*_i is cluster-tilting. Moreover T_i and T^*_i are linked by the existence of trianglesT_i u→Bv→T^*_i w→ T_i[1] and T^*_iu'→B'v'→T_iw'→T^*_i[1]where u and u' are minimal left 𝖺𝖽𝖽 T_0-approximations and v and v' are minimal right 𝖺𝖽𝖽 T_0-approximations.These triangles allow to make a mutation of the cluster-tilting object: they are called IY-mutations.Let 𝒞 be a 2-CY triangulated category with a cluster-tilting object T. Then the functorF_T = Hom_𝒞(T, -)𝒞→𝗆𝗈𝖽 End_𝒞(T)induces an equivalence 𝒞/𝖺𝖽𝖽 T [1] ≅𝗆𝗈𝖽 End_𝒞(T ).If the objects T and T^' are linked by an IY-mutation, then the categories 𝗆𝗈𝖽 End_𝒞(T ) and 𝗆𝗈𝖽 End_𝒞(T^') are nearly Morita equivalent, that is, there exists a simple End_𝒞(T)-module S, and a simple End_𝒞(T^')-module S^', and an equivalence of categories 𝗆𝗈𝖽 End_𝒞(T)/𝖺𝖽𝖽 S ≅𝗆𝗈𝖽 End_𝒞(T^')/𝖺𝖽𝖽S^'.Moreover, if X has no direct summands in 𝖺𝖽𝖽 T[1], then F_TX is projective (resp. injective) if and only if X lies in 𝖺𝖽𝖽 T (resp. in 𝖺𝖽𝖽 T[2]).Thus, from Theorem <ref> and the above Proposition, we get that in the Jacobi-finite case, for any cluster-tilting object T ∈𝒞(Γ) which is IY-mutation equivalent to the canonical one, we have: 𝒞(Γ) [dl]_F_T^'[dr]^F_T𝗆𝗈𝖽End_𝒞(Γ)(T) @<.>[rr]^mutation𝗆𝗈𝖽 End_𝒞(Γ)(T^')§.§ Grothendieck groups, skew-symmetric pairing, and the index §.§.§ Motivations from physicsIn a quantum theory there are two distinct notions of `quantum numbers': thequantities which are conserved in all physical processes and, on the other hand, the numbers which are used to label (i.e. to distinguish) states and operators. If a class of BPS objects is described (in a certain physical set-up) by the triangle category 𝔗,these two notions of `quantum numbers' get identified as follows: * conserved quantities:numerical invariants of objects X∈𝔗 which only depend on their Grothendieck class [X]∈ K_0(𝔗).[ In general, the conserved quantum numbers take value in the numeric Grothendieck group K_0(𝔗)_num. For the categories we consider in this paper, the Grothendieck group is a finitely generated Abelian group and the two groups coincide.] This is the free Abelian group over the isoclasses of objects of 𝔗 modulo the relations given by distinguished triangles of 𝔗;* labelingnumbers: correspond tonumerical invariants of the objects X∈𝔗 which are well-defined, that is, depend only on its isoclass (technically, on their class in the split-Grothendieck group). Of course, conserved quantities are in particular labelingnumbers. Depending on the category 𝔗 there may be or not be enough conserved quantities K_0(𝔗) to label all the relevant BPS objects. In the categorical approach to the BPS sector of a supersymmetrictheory, the basic problem takes the form:Given a class of BPS objects A in a specified physical set-up, determine the corresponding triangulated category 𝔗_A.The Grothendieck group is a very handy tool to solve this Problem. Indeed, the BPS objects of A carry certain conserved quantum numbers which satisfy a number of physical consistency requirements. The allowed quantum numbers take value in an Abelian group 𝖠𝖻_A, and the consistency requirements endow the group with some extra mathematical structures. Both the group 𝖠𝖻_A and the extra structures on it are known from physical considerations (we shall review the ones of interest in . <ref>). Then suppose we have a putative solution 𝔗_A of the above problem. We compute its Grothendieck group; if K_0(𝔗_A)≇𝖠𝖻_A, we can rule out 𝔗_A as a solution of the above Problem. Even ifK_0(𝔗_A)≅𝖠𝖻_A, but K_0(𝔗_A) is not naturally endowed with the required extra structures, we may rule out 𝔗_A. On the other hand, if we find that K_0(𝔗_A)≅𝖠𝖻_A and the Grothendieck group is canonically equipped with the physically expected structures, we gain confidence on the proposed solution, especially if the requirements on K_0(𝔗_A) are quite restrictive.Therefore, as a preparation for the discussion of their physical interpretation in section 4, we need to analyze in detail the Grothendieck groups of the three triangle categories D^bΓ, 𝔓𝔢𝔯 Γ, or 𝒞(Γ). Thesecategories are related by the functors 𝗌, 𝗋 which, being exact, induce group homomorphisms between the corresponding Grothendieck groups.In all three cases K_0(𝔗) is a finitely generated Abelian group carrying additional structures; later in the paper we shall compare this structures with the one required by quantum physics. §.§.§ The lattice K_0(D^bΓ) and the skew-symmetric formThe group K_0(D^bΓ) is easy to compute using the followingThe Abelian category 𝗇𝗂𝗅 J(Q,W) is the heart of a bounded t-structure in D^bΓ.Hence, since we assume (Q,W) to be Jacobi-finite, 𝗇𝗂𝗅 J(Q,W)≅𝗆𝗈𝖽 J(Q,W) andK_0(D^bΓ)≃ K_0( J(Q,W)) is isomorphic to the free Abelian group over the isoclasses [S_i] of the simple Jacobian modules S_i, that is,K_0(D^bΓ)≅^n (n being the number of nodes of Q). D^bΓ is 3-CY, and then the lattice K_0(D^bΓ) is equipped with an intrinsic skew-symmetric pairing given by the Euler characteristics, see discussion around eqn.(<ref>). This pairing has an intepretation in terms of modules of the Jacobian algebra B≡ J(Q,W)≅End_𝒞(Γ)(Γ).LetX,Y∈𝗆𝗈𝖽 B. Then theform⟨ X, Y⟩_a=Hom(X,Y)-Ext^1(X,Y)-Hom(Y,X)+Ext^1(Y,X)descends to an antisymmetric form on K_0(𝗆𝗈𝖽 B). Its matrix in the basis of simples {S_i} is the exchange matrix B of the quiver Q (cfr. eqn.(<ref>)).In conclusion, for the 3-CY category D^bΓ, the Grothendieck group is a rank n lattice equipped with a skew-symmetric bilinear form ⟨-,-⟩. We shall refer to the radical of this form as the flavor lattice Λ_flav=rad ⟨-,-⟩. §.§.§ K_0(𝔓𝔢𝔯 Γ)≅ K_0(D^bΓ)^∨ More or less by definition, K_0(𝔓𝔢𝔯 Γ) is the free Abelian group over the classes [Γ_i] of indecomposable summands of Γ. Since the general perfect object has infinite homology, there is no well-defined Euler bilinear form. However, eqn.(<ref>) implies that for X∈𝔓𝔢𝔯 Γ, Y∈ D^bΓ,Hom_𝔓𝔢𝔯(X,Y[k])=Hom_𝔓𝔢𝔯(Y,X[k])=0for k<0 or k>3and hence we have a Euler pairingK_0(𝔓𝔢𝔯 Γ)× K_0(D^bΓ)→,under which⟨Γ_i, S_j⟩=-⟨ S_j, Γ_i⟩=δ_ij.Thus [S_i] and [Γ_i] are dual basis and both Grothendieck groups are free (i.e. lattices) of rank n. Then we have two group isomorphisms^n→ K_0(D^bΓ) (m_1,m_2,…,m_n) ⟼⊕_i=1^n m_i [S_i]K_0(𝔓𝔢𝔯 Γ)→^n[X]⟼(⟨ X, S_1⟩,⟨ X, S_2⟩,…, ⟨ X, S_n⟩). The image of K_0(D^bΓ) inside K_0(𝔓𝔢𝔯 Γ)≅ Z^n is isomorphic to the image of B^n→^n where B is the exchange matrix of the quiver Q.[ Note that this image is invariant under quiver mutation.] We have the obvious isomorphismK_0(𝔓𝔢𝔯 Γ)≅ K_0(𝖺𝖽𝖽 Γ). §.§.§ The structure of K_0(𝒞(Γ)) From the basic exact sequence of categories (<ref>) we get0 [r] K_0(D^b Γ) [r]^s K_0(𝔓𝔢𝔯 Γ) [r]^r K_0(𝒞(Γ))[r] 0hence K_0(𝒞(Γ))≅^n/B·^n.K_0(𝒞(Γ)) is not a free Abelian group (in general) but has a torsion part which we denote as 𝗍𝖧 (and call the 't Hooft group)K_0(𝒞(Γ))= K_0(𝒞(Γ))_free⊕𝗍𝖧≅^f⊕𝖠⊕𝖠where f=corank B and 𝖠 is the torsion group𝖠= ⊕_s /d_s , d_s| d_s+1where the d_s are the positive integers in the normal form of B <cit.>B0⊕ 0⊕⋯⊕ 0 ^f summands ⊕[0d_1; -d_10 ]⊕[0d_2; -d_20 ]⊕⋯⊕[0d_ℓ; -d_ℓ0 ] §.§.§ The index of a cluster object Since the rank of the Abelian group K_0(𝒞(Γ)) is (in general) smaller than n, the Grothendieck class is not sufficient to label different objects (modulo deformation). We need to introduce other `labeling quantum numbers' which do the job. This corresponds to the math concept of index (or, dually, coindex).For each object L∈𝒞(Γ) there is a triangleT_1→ T_0→ L→withT_1,T_0∈𝖺𝖽𝖽 Γ.The difference[T_0]-[T_1]∈ K_0(𝖺𝖽𝖽 Γ)does not depend on the choice of this triangle. The quantity𝗂𝗇𝖽(L)≡ [T_0]-[T_1]∈ K_0(𝖺𝖽𝖽 Γ)≡ K_0(𝗉𝗋𝗈𝗃 J(Q,W))≅ K_0(𝔓𝔢𝔯 Γ)≅Λ^∨is called the index of the object L∈𝒞(Γ). It is clear from the Lemma that the class [L]∈ K_0(𝒞(Γ)) is the image of 𝗂𝗇𝖽(L) under the projectionΛ^∨→Λ^∨/B·Λ. As always, we use the canonical cluster-tilting object Γ; the modules F_ΓΓ_i∈𝗆𝗈𝖽 J(Q,W);are the indecomposable projective modules (cfr. Proposition <ref>). We write S_i≡𝖳𝗈𝗉 F_ΓΓ_i∈𝗆𝗈𝖽 J(Q,W)for the simple with support at the i-th node.Let X∈𝒞(Γ) be indecomposable. Then𝗂𝗇𝖽X=-[Γ_i]X≅Γ_i[1] ∑_i=1^n ⟨ F_Γ X,S_i⟩ [Γ_i]otherwise,where ⟨ -,-⟩ is the Euler form in 𝗆𝗈𝖽 J(Q,W).The dual notion to the index is the coindex <cit.>. For X∈𝒞(Γ) one has𝗂𝗇𝖽 X=-𝖼𝗈𝗂𝗇𝖽 X[1] 𝖼𝗈𝗂𝗇𝖽 X-𝗂𝗇𝖽 X=∑_i=1^n ⟨ S_i, F_Γ X⟩_a [Γ_i] 𝖼𝗈𝗂𝗇𝖽 X-𝗂𝗇𝖽 Xdepends only on F_Γ X∈𝗆𝗈𝖽 J(Q,W).From (<ref>) it is clear that the projections in K_0(𝒞(Γ)) of the index and coindex agree. The precise mathematical statement corresponding to the rough idea that the `index yields enough quantum numbers to distinguish operator' is the followingTwo rigid objects of 𝒞(Γ) are isomorphic if and only if their indices are equal.We shall show in .<ref> how this is related to UV completeness of the corresponding QFT.§.§ Periodic subcategories,the normalized Euler and Titsforms We have seen that the group K_0(D^bΓ) has an extra structure namely a skew-symmetric pairing. It is natural to look for additional structures on the group K_0(𝒞(Γ)). The argument around (<ref>) implies that the Euler form of the 2-CY category 𝒞(Γ) if defined is symmetric:⟨ X,Y⟩_𝒞(Γ) ≡∑_k∈ (-1)^k Hom_𝒞(Γ)(X,Y[k])==∑_k∈ (-1)^2-k Hom_𝒞(Γ)(Y, X[2-k])=⟨ Y, X⟩_𝒞(Γ).However the sum in the rhsis typically not defined, since it is not true (in general) that Hom_𝒞(Γ)(X,Y[k])=0 fork≪ 0. In order to remediate this, weintroduce an alternative concept. We say that a full subcategory ℱ(p)⊂𝒞(Γ), closed under shifts, direct sums and summands, is p-periodic (p∈ℕ) iff the functor [p] restricts to an equivalence in ℱ(p), and ℱ(p) is maximal with respect to these properties. Note that we do not require p to be the minimal period.A p-periodic sub-category, ℱ(p)⊂𝒞(Γ), is triangulated and 2-CY[ ℱ(p) is linear, Hom-finite, and 2-CY. However, it is not necessarily a generalized cluster category since it may or may not have a tilting object.The prime examples of such a category without a tilting object are the cluster tubes, see <cit.>. Sometimes the term `cluster categories' is extended also to such categories.] and the inclusion functor ℱ(p)𝒞(Γ) is exact.Since ℱ(p) is closed under shifts, direct sums, and summands in 𝒞(Γ), it suffices to verify that X,Y∈ℱ(p) implies Z∈ℱ(p) for all triangles X→ Y→ Z→ in 𝒞(Γ). Applying [p] to the triangle, one gets Z[p]≃ Z.Let ℱ(p)⊂𝒞(Γ) be p-periodic. We define the normalized Euler form as⟨⟨ X,Y⟩⟩=⟨⟨ Y,X⟩⟩ =1/p∑_k=0^p-1(-1)^kHom_𝒞(Γ)(X,Y[k]),X,Y∈ℱ(p).Note that it is independent of the chosen p as long as Y[p]≅ Y.If p is odd, ⟨⟨-,-⟩⟩≡0.The normalized Euler form ⟨⟨ -,-⟩⟩ induces a symmetric form on the groupK_0(ℱ(p))/K_0(ℱ(p))_torsion,which we call the Tits form of ℱ(p).We shall see in . <ref> the physical meaning of the periodic sub-categories and their Tits form. §.§.§ Example: cluster category of the projective line of weights (2,2,2,2) As an example of Tits form in the sense of the above Proposition, we consider the cluster category (see .<ref>)𝒞= D^b(ℋ)/⟨τ^-1[1]⟩^,where ℋ=𝖼𝗈𝗁 𝕏(2,2,2,2)which corresponds to SU(2) SQCD with N_f=4 <cit.>. We may think of this cluster category as having the same objects as 𝖼𝗈𝗁 𝕏(2,2,2,2) and extra arrows <cit.>. In this case ω=0, and hence the category 𝒞 is triangulated and periodic of period p=2 in the sense of Definition <ref>, so ℱ(2) is the full cluster category 𝒞. We write 𝒪 for the structure sheaf and 𝒮_i,0 for the unique simple sheaf with support at the i-th special point such that Hom_𝖼𝗈𝗁 𝕏(𝒪,𝒮_i,0)≅ k. The cluster Grothedieck group K_0(𝒞) is generated by [𝒪] and [𝒮_i,0] (i=1,2,3,4) subjected to the relation <cit.>2[𝒪]=∑_i=1^4[𝒮_i,0].Thus we may identifyK_0(𝒞)≅{ (w_1,w_2,w_3,w_4)∈(12)^2 |w_i=w_j 1}≡Γ_weight, 𝔰𝔭𝔦𝔫(8).by writing a class as ∑_iw_i[𝒮_i,0]. The Tits pairing is⟨⟨ [𝒮_i,0], [𝒮_j,0]⟩⟩=δ_i,j,Then K_0(𝒞) equipped with this pairing is isomorphic to the 𝔰𝔭𝔦𝔫(8) weight lattice equipped with its standard inner product(valued in 12) dual to the even one given on the root lattice by the Cartan matrix. We remark that a class in K_0(𝒞) is a spinorial 𝔰𝔭𝔦𝔫(8) weight iff it is of the form k[𝒪]+∑_i m_i[𝒮_i,0] (m_i∈) with k odd. The physical meaning of this statement and eqn.(<ref>) will be clear in . <ref>. §.§ Stability conditions for Abelian and triangulated categories We start with the Abelian category case, since it all boils down to it. The main reference for this part is <cit.>. Letbe an Abelian category and K_0() its Grothendieck group.A Bridgeland stability condition on an Abelian category 𝒜 is a group homomorphismZ : K_0( ) →,satisfying certain properties:[ If [X]∈ K_0() is the class of X∈, we write simply Z(X) for Z([X]). ] * Z() ⊂ℍ∖ℝ_>0, the closed upper half plane minus the positive reals;* If Z(E) = 0, then E = 0. This allows to define the mapZ(-) K_0()∖{0}→ (0,π]; * The Harder-Narasimhan (HN) property. Every object E ∈ admits a filtration0 = E_0 ⊂ E_1 ⊂ E_2 ⊂···⊂ E_n = E,such that, for each i: * E_i+1/E_i is Z-semistable;[ See below Definition <ref> ofsemistability of objects in an abelian category.]* Z(E_i+1/E_i) >Z(E_i+2/E_i+1).We also have the followingAn object E∈ is called Z-stable if for all nonzero proper subobjects E_0 ⊂ E,Z(E_0) <Z(E).If ≤ replaces <, then we get the definition of Z-semistable.We are now going to give the corresponding definitions for the triangulated categories. The definition is more involved since there is no concept of subobject.A slicingof a triangulated categoryis a collection of full additive subcategories (ϕ) for each ϕ∈ satisfying * (ϕ + 1) =(ϕ)[1];* For all ϕ_1 > ϕ_2 we have Hom((ϕ_1), (ϕ_2)) = 0;* For each 0 ≠ E ∈ there is a sequence ϕ_1 > ϕ_2 > ··· > ϕ_n of real numbers and a sequence of exact triangles0=E_0 [rr]E_1 [r][dl] ⋯⋯[r]E_n-1 [rr]E_n=E [dl]A_1 @.>[ul] ⋯ A_n @.>[ul]with A_i∈(ϕ_i) (which we call the Harder-Narasimhan filtration of E). We call the objects in (ϕ) semistable of phase ϕ. And finally, the definition of stability conditions in a triangulated category.A stability condition on a triangulated categoryis a pair (Z, ) where Z : K_0() → is a group homomorphism (called central charge) andis a slicing, so that for every 0 ≠ E ∈(ϕ) we haveZ(E) = m(E) e^iπϕfor some m(E) ∈>0.Indeed, the following proposition shows that to some extent (once we identify a t-structure), stability is intrinsically defined. It also describes how stability conditions are actually constructed:Giving a stability condition (Z, ) on a triangulated categoryis equivalent to giving a heartof a bounded t-structure with a stability function Z_: K_0( ) → such that ( Z_ , ) have the Harder-Narasimhan property, i.e. any object inhas a HN-filtration by Z_-stable objects.We will focus on how to obtain a stability condition from the datum ( Z_ , ), as this is how stability conditions are actually constructed:Ifis the heart of a bounded t-structure on , then we have K_0() = K_0( ), so clearly Z and Z_ determine each other. Given ( Z_ , ), we define (ϕ) for ϕ∈ (0, 1] to be the Z_-semistable objects inof phase ϕ(E) = ϕ. This is extended to all real numbers by (ϕ + n) =(ϕ)[n] ⊂[n] for ϕ∈ (0, 1] and 0 ≠ n ∈ℤ. The compatibility condition 1/πZ(E)=ϕis satisfied by construction, so we just need show thatsatisfies the remaining properties in our definition of slicing. The Hom-vanishing condition in definition <ref> follows from the definition of heart of a bounded t-structure. Finally, given E ∈, its filtration by cohomology objects A_i ∈[k_i], and the HN-filtrations 0 → A_i1→ A_i2→…→ A_im_i = A_i given by the HN-property insidecan be combined into a HN-filtration of E: it begins with0 → F_1 = A_11[k_1]→ F_2 = A_12[k_1]→⋯→ F_m_1 = A_1[k_1] = E_1,i.e. with the HN-filtration of A_1. Then the following filtration steps F_m_1+i are an extensions of A_2i[k_2] by E_1 that can be constructed as the cone of the composition A_2i[k_2]→ A_2[k_2] → ^[1] E_1 (the octahedral axiom shows that these have the same filtration quotients as 0 → A_21[k_2] → A_22[k_2]⋯ ); continuing this way we obtain a filtration of E as desired. Conversely, given the stability condition, we set =((0, 1]) as before; by the compatibility condition, the central charge Z(E) of any -semistable object E lies in ℍ∖_>0; since any object inis an extension of semistable ones, this follows for all objects inby the additivity. Finally, it is fairly straightforward to show that Z-semistable objects inare exactly the semistable objects with respect to . § SOME PHYSICAL PRELIMINARIES In the next section we shall relate the various triangle categories introduced in the previous section to the BPS objects of a 4d 𝒩=2 QFT as describedfrom two different points of view: i) the microscopic UV description (i.e. in terms of a UV complete Lagrangian description or a UVfixed-point SCFT), and ii) the effective Seiberg-Witten IR description. Before doing that, we discuss some general properties of these physical systems. As discussed in . <ref>, the categories 𝔗_A which describe the BPS objectsshould enjoy the categorical versions of these physical properties in order to be valid solutions to the Problem in . <ref>. §.§ IR viewpoint §.§.§ IR conserved charges The Seiberg-Witten theory <cit.> describes, in a quantum exact way, the low-energy physics of our 4d 𝒩=2 model in a given vacuum u along its Coulomb branch. Assuming u and the mass deformations to be generic, the effective theory is an Abelian gauge theory U(1)^r coupled to states carrying both electric and magnetic charges. The flavor group is also Abelian U(1)^f, so that the IR conserved charges consist of r electric, r magnetic, and f flavor charges. In a non-trivial theory the gauge group is compact, and the flavor group is always compact, so these charges are quantized. Then the conserved charges take value in a lattice Λ (a free Abelian group) of rankn=2r+f.The lattice Λ is equipped with an extra structure, namely a skew-symmetric quadratic form⟨-,-⟩Λ×Λ→,given by the Dirac electro-magnetic pairing. The radical of this form,Λ_flav=rad ⟨-,-⟩≡{λ∈Λ | ⟨μ,λ⟩=0∀ μ∈Λ}⊂Λ, is the lattice of flavor charges and has rank f. The effective theory has another bosonic complex-valued conserved charge, namely the central charge of the 4d 𝒩= 2 superalgebra Z := ϵ^αβϵ_AB{Q^A_α, Q^B_β}. Z is not an independent charge but a linear combination of the charges in Λ with complex coefficients which depend on all IR data, and in particular on the vacuum u. Hence, for a given u, the susy central charge is a linear map (group homomorphism)Z_uΛ→.Any given state of charge λ∈Λ has mass greater than or equal to |Z_u(λ)|. BPS states are the ones which saturate this bound. In the case of a 4d 𝒩=2 with a UV Lagrangian formulation, r and f are the ranks of the (non-Abelian) gauge G and flavor F groups, respectively. At extreme weak coupling, the IR electric and flavor charges are the weights under the respective maximal tori.§.§.§ The IR landscape vs. the swampland The IR 𝒩=2 theories we consider are not generic Abelian gauge theories with electric and magnetic charged matter. They belong to the landscape (as opposed to the swampland), that is, they have a well defined UV completion. Such theories have special properties.One property which seems to be true in the landscape, is that there are “enough” conserved IR charges to label all BPS states, so we don't need extra quantum numbers to distinguish the BPS objects in the IR description. This condition is certainly not sufficient to distinguish the landscape from the swampland, but it plays a special role in our discussion.To support the suggestion that being UV complete is related to Λ being large enough to label IR objects, we mention a simple fact. Let the UV theory consists of a 𝒩=2 gauge theory with semi-simple gauge group G and quark half-hypermultiplets in the (reducible) quaternionic representation H. Assume that the beta-functions of all simple factor of G are non-positive. In the IR theory along the Coulomb branch, consider the BPS hypermultiplets h_i with zero magnetic charge and write [h_i] for their IR charges in Λ. Then [h_i]=[h_j] and h_i≠ h_j⇒ [h_i]∈Λ_flav.That is, the charges in Λ are enough to distinguish (zero magnetic charge) hypermultiplets unless they carry only flavor charge (i.e. are electrically neutral).The hypermultiplets with purely flavor charge (called “everywhere light” since their mass is independent of the Coulomb branch parameters) just decouple in the IR, so in a sense they are no part of the IR picture. To show the above fact, just list for all possible gauge group all representations compatible with non-positivity of the beta-function. Check, using Weyl formula, that the multiplicities of all weights for these representations is 1 except for the zero weight.§.§ UV line operators and the 't Hooft group§.§.§ 't Hooft theory of quantum phases of gauge theoriesWe start by recalling the classical arguments by 't Hooft on the quantum phases of a 4d gauge theory <cit.>.The basic order operator in a gauge theory is the Wilson line associated to a(real) curve C in space time and a representation R of the gauge group G,W_R(C)= tr_R e^-∫_C A.Here C is either a closed loop or is stretched out to infinity. In the second case we don't take the trace and hence the operator depends on a choice of a weight w of the representation R modulo che action of Weyl group. In the 𝒩=2 case, the Wilson line (<ref>) is replaced by its half-BPS counterpart <cit.> which, to preserve half supersymmetries should be stretched along a straight line L; we still denote this operator as W_w(L).[ The half-BPS lines are also parametrized by an angle ϑ which specifies which susy subalgebra leaves them invariant. We suppress ϑ from the notation.]What are the quantum numbers carried by W_w(L)? This class of UV line operators is labelled by (the Weyl orbit of) the weight w, so gauge weightsare useful quantum numbers. However, these numbers do not correspond to conserved quantities in a general gauge theory. For instance, consider pure (super-)Yang-Mills theory and let R be the adjoint representation. Since an adjoint Wilson line may terminate at the location of a colored particle transforming in the adjoint representation, a gluon (gluino) particle-antiparticle pair may be dynamically created out of the vacuum, breaking the line, see figure <ref>. If our gauge theory is in the confined phase,breaking the line L is energetically favorable, so the line label w does not correspond to a conserved quantity.On the contrary, a Wilson line in the fundamental representation cannot break in pure SU(N) (S)YM, since there is no dynamical particle which can be created out of the vacuum where it can terminate. The obstruction to breaking the line is the center Z(SU(N))≅_N of the gauge group under which all local degrees of freedom are inert while the fundamental Wilson line is charged. Stated differently, the gluons may screen all color degrees of freedom of a physical statebut the center of the gauge group. The conclusion is that the conserved quantum numbers of the line operators W_R(L) consist ofthe representation R seen as a representation of the center of the gauge group, Z(G), which take value in the dual group Z(G)^∨≅Z(G). On the other hand, in SU(N) (S)QCD we have quarks transforming in the fundamental representation; hence a quark-antiquark pair may be created to break a fundamental Wilson line. Then, in presence of fundamental matter, Wilson lines do not carry any conserved quantum number. In general, the conserved quantum numbers of the Wilson lines of a gauge theory with gauge group G take value in the finite Abelian group π_1(G_eff)^∨≅π_1(G_eff), whereG_eff isthe quotient group of G which acts effectively on the microscopic UV degrees of freedom.For clarity of presentation, the above discussion was in the confined phase. This is not the case of the 𝒩=2 theory which we assume to be realized in its Coulomb phase. In the physically realized phase the line W_w(L) may be stable; thenits labeling quantum number w becomes an emergent conserved quantity of the IR description (see .<ref>). However, from the UV perspective, the only strictly conserved quantum numbers are still the (multiplicative) characters of π_1(G_eff) which take value in the groupπ_1(G_eff)^∨≡Hom(π_1(G), U(1))≅π_1(G_eff). More generally, we may have Wilson-'t Hooft lines <cit.> which carry both electric and magnetic weights. Their multiplicative conserved quantum numbers take value in the (Abelian) 't Hooft group𝗍𝖧= π_1(G_eff)^∨⊕π_1(G_eff),equipped with the canonical skew-symmetric bilinear pairing (the Weil pairing)[ As always, μ denotes the group of roots of unity. The name `Weil pairing' is due to its analogy with the Weil pairing in the torsion group of a polarized Abelian variety which arises in exactly the same way.]𝗍𝖧×𝗍𝖧→μ, (x,y)× (x^',y^')↦ x(y^') x^'(y)^-1. The 't Hooft multiplicative quantum numbers of a line operator, written additively, are just its electric/magnetic weights (w_e,w_m) modulo the weight lattice of G_eff. The best way to understand the proper UV conserved quantum numbers of line operators is to consider the different sectors in which we may decompose the microscopic path integral of the theory which preserve the symmetries of a line operator stretched in the 3-direction in space (that is, rotations in the orthogonal plane and translations). In a 4d gauge theory quantized on a periodic 3-box of size L we may defined the 't Hooft twisted path integral <cit.> (see <cit.> for nice reviews)e^-β F(e⃗, m⃗, θ, μ_s,β)≡Tr_e⃗, m⃗[e^-β H+iθν+μ_s F_s],e⃗∈(π_1(G_eff)^∨)^3, m⃗∈π_1(G_eff)^3,where e⃗, m⃗ are 't Hooft (multiplicative) electric and magnetic fluxes, θ is the instanton angle, and μ_s are chemical potentials in the Cartan algebra of the flavor group F. Imposing rotational invariance in the 1-2 plane and taking the Fourier transform with respect to the μ_s we remain (at fixed θ) with the quantum numbers(e_3, m_3, w)∈π_1(G_eff)^∨⊕π_1(G_eff)⊕(weight lattice of F).We shall call the vector (e_3, m_3, w) the 't Hooft charge and the group in the rhs the extended 't Hooft group.We stress that the structure of the Weil pairing is required in order to relate the Euclidean path integral in given topological sectorsto the free energy F(e⃗, m⃗, θ, μ_s,β) with fixed non-abelian fluxes <cit.>. The boundary condition on the Euclidean box which corresponds to a given 't Hooft charge does no break any supercharges, that is, we do not need to specify a BPS angle ϑ to define it.§.§.§ Non-Abelian enhancement of the flavor group in the UV Consider a UV complete 𝒩=2 gauge theory. In the IR theory the flavor group is (generically) Abelian of rank f. In the UV the masses are irrelevant and the flavor group enhances from the Abelian group U(1)^f to some possibly non-Abelian rank f Lie group F. The free part of the 't Hooft group (<ref>) is then the weight lattice of F. This weight lattice is equipped with a quadratic form dual to the Cartan form on the root lattice. From the quadratic form we recover the non-Abelian Lie group F.In conclusion:The UV conserved quantities are encoded in the extended 't Hooft group, a finitely generated Abelian groupof the formπ_1(G_eff)^∨⊕π_1(G_eff) ⊕Γ_flav,weight, whose free part has rank f. The extended 't Hooft group (<ref>) is equipped with two additional structures: i) the Weil pairing on the torsion part, ii) the dual Cartan symmetric form on the free part. Moreover, iii) the UV lines carry an adjoint action of the half quantum monodromy 𝕂 (see . <ref>) which acts on the 't Hooft group as -1. Finer structures on the 't Hooft group. The 't Hooft group(<ref>) detects the global topology of the gauge group G_eff; it also detects the topology of the flavor group F, e.g. it distinguishes between the flavor groups SO(N) and Spin(N), since they have different weight lattices[ Γ_𝔰𝔭𝔦𝔫(N): Γ_𝔰𝔬(N) ]=2.But there even finer informations on the flavor symmetry which we should be able to recover from the relevant categories. To illustrate the issue, consider SU(2) SQCD with N_f fundamental hypers. In the perturbative sector (states of zero magnetic charge) the flavor group is SO(2N_f), but non-perturbatively it gets enhanced to Spin(2N_f). More precisely, states of odd (resp. even) magnetic charge are in spinorial (resp. tensorial) representations of the flavor group Spin(2N_f). This is due to the zero modes of the Fermi fields in the magnetic monopole background <cit.>, which is turn are predicted by the Atiyah-Singer index theorem. The index theorem is an integrated version of the axial anomaly, so the correlation between magnetic charge and flavor representations should emerge from the same aspect of the category which expresses the U(1)_R anomaly (and the β-function).§.§.§ The effective `charge' of a UV line operator We have two kinds of quantum numbers: conserved quantities and labeling numbers. In the IR we expect (see .<ref>) that conserved quantities are (typically) sufficient to label BPS objects. However, the UV group of eqn.(<ref>) is too small to distinguish inequivalent BPS line operators.We may introduce a different notion of `charge' for UV operators which takes value in a rank n=2r+f lattice. This notion, albeit referred to UV objects, depends on a IR choice, e.g. the choice of avacuum u. Suppose that in this vacuum we have n species of stable lines L_i (i=1,…, n) which are preserved by the the same susy sub-algebra preserving L andcarry emergent IR quantum numbers [L_i] which are ℚ-linearly independent. We may consider the BPS state |{n_i}⟩ in which we have a configuration of parallel stable lines with n_1 of type L_1, n_2 of type L_2, and so on. Suppose that for our BPS line operator L⟨{n_i^'}| L |{n_i}⟩≠ 0It would be tempting to assign to the operator L the `charge'∑_i(n_i^'-n_i)[L_i]∈⊕_i [L_i].Such a charge would be well-defined on UV operators provided two conditions are satisfied: i) for all L we can find a pair of states |{n_i}⟩, |{n_i^'}⟩ such that eqn.(<ref>) holds, and moreover ii) we can show that n^'_i-n_i does not depend on the chosen |{n_i}⟩, |{n_i^'}⟩. The attentive reader may notice that this procedure is an exact parallel to the definition of the index of a cluster object (Definition <ref>). However the i-th `charge' n^'_i-n_i is PCT-odd only if the lines L, L_i carry `mutually local charge', that is, have trivial braiding; the projection of the `charge' so defined in the 't Hooft group (<ref>) is, of course, independent of all choices. This follows from the fact that the action of PCT onthe UV lines is given by the half quantum monodromy (see . <ref>) which does not act as -1 on the present `effective' charges; of course, it acts as -1 on the 't Hooft charges as it should. §.§ The quantum monodromy There is one more crucial structure on the UV BPS operators, namely the quantum monodromy <cit.>. Let us consider first the case in which the UV fixed point is a good regular SCFT. At the UV fixed point the U(1)_r R-symmetry is restored. Let e^2π i r be the operator implementing a U(1)_r rotation by 2π (it acts on the supercharges as -1). e^2π i r acts on a chiral primary operator of the UV SCFT as multiplication by e^2π i Δ, where Δ is the scaling dimension of the chiral operator. Suppose that for all chiral operators Δ∈ℕ, then e^2π i r=(-1)^F acts as 1 on all UV observables. More generally, if all Δ∈ m ℕ for some integer m, the operator (e^2π i r)^m acts as 1 on observables <cit.>.If the theory is asymptotically-free, meaning that the UV fixed point is approached with logarithmic deviations from scaling, the above relations get also corrected, in a way that may be described rather explicitly, see<cit.>. Now suppose we deform the SCFT by relevant operators to flow to the original 𝒩=2 theory. We claim that, although the Abelian R-charge r is no longer conserved,e^2π ir remain a symmetry in this set up[ For the corresponding discussion in 2d, see <cit.>.] <cit.>. This is obvious when the dimensions Δ are integral, since e^2π ir commutes with the deforming operator. The quantum monodromy 𝕄 is theoperator induced in the massive theory from e^2π i r in this way <cit.>. It is well defined only up to conjugacy,[ When the UV fixed point SCFT is non degenerated, the operator 𝕄 is semisimple, and its conjugacy class is encoded in its spectrum, that is, the spectrum of dimensions of chiral operators Δ1.] and may be written as a Kontsevitch-Soibelmann (KS) product of BPS factors ordered according to their phase[ In eqn.(<ref>) we use the notations of <cit.>: the product is over the BPS stable states of charge λ∈Λ and spin s_λ taken in the clockwise order in their phase Z_u(λ); ψ(z;q)=∏_n≥0(1-q^n+1/2z)^-1 is the quantum dilogarithm, and the X_λ are quantum torus operators, i.e. they satisfy the algebra X_λ X_λ^'=q^⟨λ,λ^'⟩/2 X_λ+λ^' with ⟨-,-⟩ the Dirac pairing. ] <cit.>𝕄=∏^↺_λ∈BPSΨ(q^s_λX_λ;q)^(-1)^2s_λ.The KS wall-crossing formula <cit.> is simply the statement that the conjugacy class of 𝕄, being an UV datum, is independent of the particular massive deformation as well as of the particular BPS chamber we use to compute it (see <cit.>).We may also define the half-monodromy 𝕂, such that 𝕂^2=𝕄 <cit.>. The effect of the adjoint action of 𝕂 on a line operator L is to produce its PCT-conjugate. Then 𝕂 inverts the 't Hooft charges.We summarize this subsection in the followingIf our 𝒩=2 has a regular UV fixed-point SCFT and the dimension of all chiral operators satisfy Δ∈ mℕ for a certain integer m, then 𝕂^2m acts as the identity on the line operators. 𝕂 acts as -1 on the 't Hooft charges. § PHYSICAL MEANING OF THE CATEGORIES D^BΓ, 𝔓𝔢𝔯 Γ, 𝒞(Γ) We start this section by reviewing as quivers with (super)potentials arise in the description of the BPS sector of a (large class of) 4d 𝒩=2 theories, see <cit.>. §.§ 𝒩=2 BPS spectra and quivers We consider the IR physics of a 4d 𝒩 = 2 model at a generic vacuum u along its Coulomb branch.We have the IR structures described in. <ref>: a charge lattice Λ of rank n=2r+f,equipped with an integral skew-symmetric form given by the Dirac electro-magnetic pairing, and a complex linear form given by the 𝒩=2 central charge:-,-Λ×Λ→, Z_uΛ→.A 4d 𝒩 = 2 model has a BPS quiver at u iff there exists a set of n hypermultiplets, stable in the vacuum u, such that <cit.>: i) their charges e_i∈Λ generate Λ, i.e. Λ≅⊕_i e_i, and ii) the charge of each BPS states (stable in u), λ∈Λ, satisfiesλ∈Λ_+ or-λ∈Λ_+,where Λ_+ = ⊕_i _+ e_i is the convex cone of `particles' [ As contrasted with `antiparticles' whose charges belong to -Λ_+.].The BPS quiver Q is encoded in the skew-symmetric n× n exchange matrixB_ij :=e_i, e_j, i,j=1,⋯,n.The nodes of Q are in one-to-one correspondence with the generators {e_i} of Λ. If B_ij≥ 0 then there are |B_ij| arrows from node i to node j; viceversa for B_ij<0. To find the spectrum of particles with given charge λ = ∑_i m_i e_i ∈Λ_+ we may study the effective theory on their world-line. This is a SQM model with four supercharges <cit.>, corresponding to the subalgebra of 4d susy which preserves the world-line. A particle is BPS in the 4d sense iff it is invariant under 4 supersymmetries, that is, if it is a susy vacuum state of the world-line SQM. The 4-supercharge SQM is based on the quiver Q defined in eqn.(<ref>) <cit.>: to the i–th node there correspond a 1d U(m_i) gauge multiplet, while to an arrow i→ j a 1d chiral multipletin the (m_i,m_j) bifundamental representation of the groups at its two ends. To each oriented cycle in Q there is associated a single-trace gauge invariant chiral operator, namely the trace of the product of the Higgs fields along the cycle. The (gauge invariant) superpotential of the SQM is a complex linear combination of such operators associated to cycles of Q <cit.>. Since we are interested only in the susy vacua, we are free to integrate out all fields entering quadratically in the superpotential. We remain with a SQM system described by a reduced quiver with (super)potential (Q,W) in the sense of section 2.Then the solutions of the SQM F-term equations are exactly the modules X of the Jacobian algebra[ From now on the ground field k is taken to be .]J(Q, W) with dimension vector X = λ∈Λ.The D-term equation is traded for the stability condition <cit.>. Given the central charge Z_u(-), we canchoose a phase θ∈ [0,2π) such that Z_u(Λ_+) lies inside[ ℍ denotes the upper half-plane ℍ:={z∈ | Im z>0}.] ℍ_θ:=e^iθℍ.Given a module X∈𝗆𝗈𝖽 J(Q, W), we define its stability function as ζ(X) := e^-iθZ_u(X) ∈ℍ. The module X is stable iffζ(Y)<ζ(X), ∀Y⊂ Xproper submodule.A stable module X is always a brick, i.e. End_𝗆𝗈𝖽 J(Q,W)X≅ <cit.>.Keeping into account gauge equivalence, the SQM classical vacuum space is the compact Kähler variety <cit.>M_λ:={ X ∈ J(Q,W)|X stable,X = λ}/∏_iGL(m_i , ),that is, the space of isoclasses of stable Jacobian modules of the given dimension λ. The space of SQM quantum vacua is then H^∗(M_λ,) which carries a representation R of SU(2) by hard Lefschetz <cit.>, whose maximal spin is M_λ/2; the space-time spin content of the charge λ BPS particle is[ The Cartan generator of SU(2)_R acting on a BPS particle described by a (p,q)-harmonic form on M_λ is (p-q); however, it is conjectured that only trivial representations of SU(2)_R appear <cit.>.] (0⊕2)⊗R. For example, the charge λ BPS states consist of a (half) hypermultiplet iff the corresponding moduli space is a point, i.e. if the module X is rigid.The splitting between particles and antiparticles is conventional: different choices lead to different pairs (Q,W). However all these (Q,W) should lead to equivalent SQM quiver models. Indeed, distinct pairs are related by a chain of 1d Seiberg dualities <cit.>. The Seiberg dualities act on (Q,W) as the quiver mutations described in section 2. Indeed, the authors of <cit.> modeled their construction onSeiberg's original work <cit.>. The conclusion of this subsection is that to a (continuous family of) 4d 𝒩=2 QFT (with the quiver property) there is associated a full mutation-class of quivers with potentials (Q,W). All (Q,W) known to arise from consistent QFTs are Jacobi-finite, and we assume this condition throughout. Using the mathematical constructions reviewed in . 2, to such an 𝒩=2 theory we naturally associate the three triangle categories D^bΓ, 𝔓𝔢𝔯 Γ, and 𝒞(Γ), together with the functors 𝗌, 𝗋 relating them. We stress that the association is intrinsic, in the sense that thecategories are independent of the choice of (Q,W) in themutation-classmodulo triangle equivalence (cfr. Theorem <ref>).Our next task is to give a physical interpretation to these three naturally defined categories. We start from the simpler one, D^bΓ.§.§ Stable objects of D^bΓ and BPS statesLet Γ be the Ginzburg algebra associated to the pair (Q,W). Keller proved <cit.> that the Abelian category 𝗆𝗈𝖽 J(Q,W) is the heart of a bounded t-structure in D^bΓ. In particular, its Grothendieck group isK_0(D^bΓ)≅ K_0(𝗆𝗈𝖽 J(Q,W))≡Λ,that is the lattice of the IR conserved charges (. <ref>). Thus, given a stability condition on the Abelian category 𝗆𝗈𝖽 J(Q,W), we canextend it to the entire triangular category D^bΓ. In particular, since the semi-stable objects of D^bΓ arethe elements of (ϕ) (cfr. the proof of Proposition <ref>), we have two possibilities: * ϕ∈ (0,1], then the only semistable objects are the semistable objects of 𝗆𝗈𝖽 J(Q,W) in the sense of “Abelian category stability” plus the zero object of D^bΓ;* ϕ∉(0,1], then the only semistable objects are the shifts of the semistable objects of 𝗆𝗈𝖽 J(Q,W)in the sense of “Abelian category stability”.In other words, a generic object E ∈ D^bΓ is unstable if it has a nontrivial HN filtration. Thus, up to shift [n], the only possible semistable objects in D^bΓ are those objects belonging to the heart 𝗆𝗈𝖽 J(Q,W) that are “Abelian”-stable in it.We have already seen that the category 𝗆𝗈𝖽 J(Q,W) describes the BPS spectrum of our 4d 𝒩=2 QFT: by what we just concluded, the isoclasses of stable objects X of D^bΓ with Grothendieck class [X]=λ∈Λ are parametrized, up to even shifts[ Since the shift by [1] acts on the BPS states as PCT, it is quite natural to identify the BPS states associated to stable objects differing by even shifts.], by the Kähler manifolds M_λ≅ M_-λ in eqn.(<ref>) whose cohomology yields the BPS states.The category 𝒫(ϕ) is an Abelian category in its own right. The stable objects with BPS phase e^iπϕ are the simple objects in this category; in particular they are bricks in 𝒫(ϕ) hence bricks in 𝗆𝗈𝖽 J(Q,W), that is,X stable⇒End_𝗆𝗈𝖽 J(Q,W)(X)≅. §.§ Grothendieck groups vs. physical charges When the triangle category 𝒯 describes a class of BPS objects, the Abelian group K_0(𝒯) is identified with the conserved quantum numbers carried by those objects. In particular, the group K_0(𝒯) should carry allthe additional structures required by the physics of the corresponding BPS objects, as described in . <ref>.Let us pause a while to discuss the Grothendieck groups of the three triangle categories K_0(𝒯), where 𝒯=D^bΓ, 𝔓𝔢𝔯 Γ, or 𝒞(Γ), and check that they indeed possess all properties and additional structures as required by their proposed physical interpretation. §.§.§ K_0(D^bΓ)Since D^bΓ describes BPS particles, K_0(D^bΓ) is just the IR charge lattice Λ, see eqn.(<ref>). Physically, the charge lattice carries the structure of a skew-symmetric integral bilinear form, namely the Dirac electromagnetic pairing. This matches with the fact that,since D^bΓ is 3-CY,its Euler form (<ref>) is skew-symmetric and is identified with the Dirac pairing (compare eqn.(<ref>) and the last part of Proposition <ref>). We stress that the pairing is intrinsic (independent of all choices) as it should be on physical grounds.§.§.§ K_0(𝒞(Γ)): structureThe structure of the group K_0(𝒞(Γ)) was described in . <ref>. We haveK_0(𝒞(Γ))= ^f⊕𝖠^∨⊕𝖠where 𝖠 is the torsion group[ Of course, 𝖠^∨≅𝖠; however it is natural to distinguish the group and its dual.]𝖠= ⊕_s /d_s , d_s| d_s+1where the d_s are the positive integers appearing in the normal form of B, see eqn.(<ref>).The physical meaning of the Grothendieck group (<ref>) is easily understood by considering the case of pure 𝒩=2 super-Yang-Mills with gauge group G.Then one shows <cit.>𝖠=Z(G)≡the center of the (simply-connected) gauge group Gthat isK_0(𝒞(Γ_SYM,G))≅Z(G)^∨⊕Z(G).This is exactly the group of multiplicative quantum numbers labeling the UV Wilson-'t Hooft line operators in the pure SYM case <cit.>, as reviewed in . <ref>. This strongly suggests the identification of the cluster Grothendieck group K_0(𝒞(Γ)) with the group of additive and multiplicative quantum numbers carried by the UV line operators. This is confirmed by of the example of 𝒩=2 SQCD with (semi-simple) gauge group G and quark hypermultiplets in a (generally reducible) representation R. One finds <cit.>K_0(𝒞(Γ_SQCD))≅^rankF⊕π_1(G_eff)^∨⊕π_1(G_eff),where F is the flavor group and G_eff is the quotient of G acting effectively on the UV degrees of freedom. Again, this corresponds to the UV extended 't Hooft group as defined in . <ref>. More generally one has: In all 𝒩=2 theories with a Lagrangian formulation (and a BPS quiver) we haveK_0(𝒞(Γ))≅(the extended 't Hooft group of . <ref>). This is already strong evidence that the cluster category 𝒞(Γ) describes UV line operators. For 𝒩=2 theories without a Lagrangian, we adopt the above Fact as the definition of the extended 't Hooft group.From Fact <ref> we know that the physical 't Hooft group has three additional structures. Let us show that all three structures are naturally present in K_0(𝒞(Γ)).§.§.§ K_0(𝒞(Γ)): action of half-monodromy and periodic subcategories There is a natural candidate for the half-monodromy: on X∈𝒞(Γ), 𝕂 acts as X↦ X[1] and hence the full monodromy as X↦ X[2].Then 𝕂 acts on K_0(𝒞(Γ)) as -1, as required. Let us check that this action has the correct physical properties e.g. the right periodicity as described in Fact <ref>. [Periodicity for Argyres-Douglas models] We use the notations of . <ref>. We know that the quantum monodromy 𝕄 has a periodicity[ For the relation of this fact with the Y-systems, see <cit.>.] equal to (a divisor of) h+2 <cit.>, corresponding to the fact that the dimension of the chiral operators Δ∈1h+2ℕ. Indeed, from the explicit description of the cluster category, eqn.(<ref>), we have 𝒞(𝔤)= D^b(𝗆𝗈𝖽 𝔤)/⟨τ^-1[1]⟩^, so that τ≅ [1] in 𝒞(𝔤). Hence,𝕄^h+2≡ [h+2] ≅τ^h[2]=Id,where we used eqn.(<ref>).Under the identification 𝕂↔ [1], we may rephrase Fact <ref> in the form: Let 𝒞(Γ) be the cluster category associated to a 𝒩=2 theory with a regular UV fixed-point SCFT such that all chiral operators have dimensions Δ∈ mℕ. Then 𝒞(Γ) is periodic with minimal period p| 2m.If the theory has flavor charges, p is even (more in general: p is even unless the 't Hooft group is a vector space over 𝔽_2). In particular, for 𝒩=2 theories with a regular UV fixed-point the cluster Tits form ⟨⟨ [X],[Y]⟩⟩ is well-defined.Asymptotically-free theories. It remain to discuss theasymptotically-free theories. The associate cluster categories 𝒞(Γ) are not periodic. However, from the properties of the 't Hooft group, we expect that, whenever our theory has a non-trivial flavor symmetry, 𝒞(Γ) still contains a periodic sub-cluster category of even period. We give an informal argument corroborating this idea which may be checked in several explicit examples.Sending all non-exactly marginal couplings to zero, our asymptotically-free theory reduces to a decoupled system of free glue and UV regular matter SCFTs. Categorically, this means that cluster category of each matter SCFT, 𝒞_mat embeds as an additive sub-category in 𝒞(Γ) closed under shifts (by PCT). The embedding functor ι is not exact (in general), so we take the triangular hull of the full subcategory over the objects in its image 𝖧𝗎_((ι 𝒞_mat)_full)⊂𝒞(Γ). If the model has non-trivial flavor, at least one matter subsector has non trivial flavor, and the corresponding category 𝒞_matter is periodic of even period p. Its objects satisfy X[p]≅ X and this property is preserved by ι. The triangle category 𝖧𝗎_((ι 𝒞_mat)_full) is generated by these periodic objects and hence is again periodic of period p. Then setℱ(p)=𝖧𝗎_((ι 𝒞_mat)_full).Again, the flavor Tits form is well defined. The above discussion shows that the presence of a p-periodic subcategory ℱ(p)⊂𝒞 is related to the presence of a sector in the 𝒩=2 theory describedby susy protected operators of dimensionΔ= 2pℕ.Let us present some simple examples.[Pure SU(2) SYM] The cluster category 𝒞_SU(2) is not periodic; this is a manifestation of the fact that the β-function of the theory is non zero <cit.>. However, let us focus on the perturbative (≡ zero magnetic charge) sector in the g_YM→0 limit. The chiral algebra is generated by a single operator of dimension Δ=2, namely tr(ϕ^2). Hence we expect that the zero-magnetic charge sector is described by a subcategory of𝒞_SU(2) which is 1-periodic. Indeed, this is correct, ℱ(1) being a ℙ^1 family of homogenous cluster tubes. [SU(2) SYM coupled to D_p Argyres-Douglas] In this case the matter is an Argyres-Douglas theory of type D_p; the matter half quantum monodromy 𝕂_matter has order (h(D_p)+2)/(2,h(D_p))=p as we may read from the spectrum of chiral ring dimensions of the Argyres-Douglas model <cit.>. Thus the matter corresponds to a periodic subcategory ℱ(p)⊂𝒞. This category is a cluster tube of period p <cit.>. See also <cit.>.Equivalently, we may understand that the presence of a non-trivial flavor group implies the existence of a 2-periodic subcategory ℱ(p)⊂𝒞 by the fact that the corresponding conserved super-currents have canonical dimension 1 which cannot be corrected by RG.§.§.§ K_0(𝒞(Γ)): non-Abelian enhancement of flavorAs discussed in . <ref>, the IR flavor symmetry U(1)^f gets enhanced in the UV to a non-Abelian group F. The identification of K_0(𝒞(Γ)) with the extended 't Hooft group requires, in particular, that its free part is equipped with the correct dual Cartan form for the flavor group F.In . <ref> we defined a Tits form associated to (a periodic subcategory of) 𝒞(Γ). This is a symmetric form on the free part of the Grothendieck group, and is the natural candidate for the dual Cartan form of the physical flavor group F.Let us check in a couple of examples that this identification yields the correct flavor group: the cluster category knows the actual non-Abelian group. [SU(2) with N_f≥1 fundamentals] We use the same notations[ However we often write simply 𝕏 instead of 𝕏(p_1,…,p_s) leaving the weights implicit.] as in . <ref>. The cluster category is𝒞_N_f=D^b(𝖼𝗈𝗁 𝕏( 2,⋯,2^N_f 2's ))/⟨τ^-1[1]⟩^.For N_f≠4 this category is not periodic since the canonical sheaf has non-zero degree (in the physical language: the β-function is non-zero). We are in the situation discussed at the end of . <ref>, and the present example is also an illustration of that issue.The 2-periodic triangle 2-CY subcategory ℱ(2)𝒞(N_f) is given by the orbit category of the derived category of finite-length sheaves.It consists of a ℙ^1 family of cluster tubes; in ℙ^1 there are N_f special points whose cluster tubes have period 2. Let 𝒮_i,k, k∈/2, be the simples in the i-th special cluster tube, satisfying𝒮_i,k[1]≅τ𝒮_i,k≅𝒮_i,k+1and let 𝒮_z be the simple over the regular pointz∈ℙ^1, τ𝒮_z≅𝒮_z. Thus [𝒮_z]=0and K_0(ℱ(2))is generated by the [𝒮_i,0] (i=1,…, N_f).The image of K_0(ℱ(2))in K_0(𝒞(N_f)) has index 2; indeed in K_0(𝒞(N_f)) we have an extra generator [𝒪] and a relation <cit.>2[𝒪]=∑_i=1^N_f[𝒮_i,0] Then as in . <ref> (for the special case N_f=4) we have K_0(𝒞(N_f))≅{(w_1,⋯,w_N_f)∈(12)^N_f |w_i=w_j1}≡Γ_weight, 𝔰𝔭𝔦𝔫(2N_f) with ⟨⟨ [𝒮_i,0],[𝒮_j,0]⟩⟩=δ_i,j, that is, K_0(𝒞(N_f)) is the 𝔰𝔭𝔦𝔫(2N_f) weight lattice equipped with the dual Cartan pairing which is the correct physical extended ' t Hooft group for this modelwhich has π_1(G_eff)=1 and F=Spin(2N_f), as expected. The case of N_f=4 was already presented in . <ref>. In that case 𝒦=0 (i.e. β=0), the theory is UV superconformal, and the cluster category is periodic. The correlation between magnetic charge and Spin(8) representation becomes the fact that the modular group PSL(2,) acts on the flavor by triality <cit.>, see <cit.> for details from the cluster category viewpoint. §.§.§ Example <ref>: Finer flavor structures,U(1)_r anomaly, Witten effectThe cluster category contains even more detailed information on the UV flavor physics of the corresponding 𝒩=2 QFT. Let us illustrate the finer flavor structures in the caseof SU(2) SYM coupled to N_f flavors[ Or, more generally, to several Argyres-Douglas systems of type D.] (Example <ref>).Note that the sublatticeK_0(ℱ(2))⊂ K_0(𝒞(N_f)) is the weight lattice of SO(2N_f); since ℱ(2) is the cluster sub-category of the `perturbative' (zero magnetic charge) sector, we recover the finer flavor structures mentioned at the end of . <ref>. In facts, eqn.(<ref>) is the image in the Grothendieck group of the equation which is the categorical expression of the U(1)_r anomaly <cit.>. Indeed, in the language of coherent sheaves, the U(1)_r anomaly is measured by the non-triviality of the canonical sheaf 𝒦(think of a (1,1) σ-model: 𝒦 trivial means the target space is Calabi-Yau, which is the condition of no anomaly). The coefficient of the β-function, b, is (twice) its degree,[ Notice that 𝒦=0 does not mean that 𝒦 is trivial but only that it is a torsion sheaf in the sense that 𝒦^m≅𝒪 for some integer m.] 𝒦=-χ(𝕏) <cit.>. As a preparation to the examples of . 6, we briefly digress to recall how this comes about. β-function and Witten effect. The AR translation τ acts on 𝖼𝗈𝗁 𝕏 as multiplication by the canonical sheaf <cit.> τ𝒜↦𝒜⊗𝒦≡𝒜⊗𝒪(ω⃗).Hence the U(1)_R anomaly and β-function may be read from the action of τ on the derived category D^b𝖼𝗈𝗁 𝕏 which we may identify as the IR category of BPS particles.[ Indeed, for N_f≤ 3, the triangle category D^b𝖼𝗈𝗁 𝕏 admits 𝗆𝗈𝖽 𝔤̂ as the core of a t-structure (here 𝔤̂ is an acyclic affine quiver in the mutation class of the model <cit.>; see also Example <ref>.] Now, in the cluster category of a weighted projective line, 𝒞(𝖼𝗈𝗁 𝕏)≡ D^b(𝖼𝗈𝗁 𝕏)/⟨τ^-1[1]⟩, one has τ≅[1], while [1] acts in the UV as the half-monodromy, that is, as a UV U(1)_r rotation by π. In the normalization of ref.<cit.> (see their eqn.(4.3)), the complexified SU(2) Yang-Mills coupling at weak coupling, a→∞, isθ/π+8π i/g^2= -b/π ilog a+⋯,Under a U(1)_r rotation by π, a→ e^π ia, the vacuum angle shifts as θ→θ-bπ. Since a dyon of magnetic charge m carries an electric charge m θ/2π mod 1 (the Witten effect <cit.>), under the action of τ the IR electric/magnetic charges (e,m) should undergo the flow τ (e,m)→ (e-m b/2,m).For an object of D^b(𝖼𝗈𝗁 𝕏) the magnetic (electric) charge correspond to its rank (degree); then comparing eqns.(<ref>),(<ref>) we get b=-2𝒦=2 χ(𝕏). Finer flavor structures (. <ref>). The Grothendieck group of𝖼𝗈𝗁 𝕏(2,…,2) is generated by[𝒪], [𝒮_0], [𝒮_i,j] (i=1,…,N_f, j∈/2) subjected to the relations[𝒮_0]=[𝒮_i,0]+[𝒮_i,1] ∀ i, see Proposition 2.1 of<cit.>. The action of τ in K_0(𝖼𝗈𝗁 𝕏) is[τ𝒮_i,j]=[𝒮_i,j+1], [τ𝒪]-[𝒪]=(N_f-2)[𝒮_0]-∑_i=1^N_f[𝒮_i,0].The difference [τ𝒪]-[𝒪] measures the non-triviality of the canonical sheaf, that is, the β-function/U(1)_r anomaly. In the cluster category, for all sheaf [τ𝒜]=-[𝒜], so that [𝒮_i,0]=0 and the second eqn.(<ref>) reduces to (<ref>). Hence, as suggested by the physical arguments at the end of . <ref>, the non-perturbative flavor enhancementSO(2N_f)→Spin(2N_f) follows from the counting of the Fermi zero-modes implied by the axial anomaly.§.§.§ K_0(𝒞(Γ))_torsion: the Weil pairing Let X∈𝒞(Γ) The projection⟨ S_i, F_Γ X⟩∈^n/B^n,depends only on [X]. Rewrite the integral vector ⟨ S_i, F_Γ X⟩ in the -basis where B takes the normal form (<ref>)(⟨ S_1, F_Γ X⟩,⋯, ⟨ S_n, F_Γ X⟩) (w_1, w_2,⋯, w_f, u_1,1,u_2,1,⋯, u_1,s, u_2,s,⋯)and see its classas an element of (ℚ^2/^2)^r(w_1, w_2,⋯, w_f, u_1,1,u_2,1,⋯, u_1,s, u_2,s,⋯)↦(u_1,1/d_1,u_2,1/d_1,⋯, u_1,s/d_s,u_2,s/d_s⋯) ∈ (ℚ^2/ℤ^2)^r.The skew-symmetric matrix B then defines a skew-symmetric pairing2π i∑_s=1^r ϵ^ab u_a,s u^'_b,s/d_s∈ 2π i ℚ/ℤ.The exponential of this expression is the canonical Weil pairing. Let us check one example.[Pure SU(2)] The basis [P_1], [P_2] is canonical. Then the Weil pairing is(Z/2)^2× (Z/2)^2∋ (e,m) × (e^',m^')↦ (-1)^em^' -me^'.§.§ The cluster category as the UV line operators We have seen that for a 𝒩=2 theory (with quiver property) the Grothendieck group K_0(𝒞(Γ)) is the extended 't Hooft group of additive and multiplicative conserved quantum numbers of the UV line operators and that this group is naturally endowed with all the structures required by physics, including the finer ones.This amazing correspondence makes almost inevitable the identification of the the cluster category 𝒞(Γ) of the mutation-class of quivers with (super)potentials associated to a 4d 𝒩=2 model with the triangle category describing its UV BPS line operators. This identification has been pointed out by several authors working from different points of view <cit.>. In particular, the structure of the mutations of the Y–seeds in the cluster algebras lead to the Kontsevich-Soibelman wall crossing formula <cit.> (see <cit.> fordetails).This is just the action of the shift [1] on the cluster category which implements the quantum half monodromy 𝕂 (cfr. . <ref>).In section <ref> below wecheck explicitly this identification by relating the geometrical description of the cluster category of a surface as given in the mathematical literature with the WKB analysis of line operators by GMN <cit.>. For BPS line operators we also had a notion of `charge' which is useful to distinguish them, see . <ref>. We already mentioned there that both the definition and the properties of this `charge' have a precise correspondent in the mathematical notion of the index of a cluster object. Now we may identify these two quantities. Note that, while the 't Hooft charge is invariant under quantum monodromy (i.e. under the shift [2]), the index is not. This is the effect of non-trivial wall-crossing and, essentially, measures it <cit.>.In . <ref> we saw that the index is fine enough to distinguish rigid objects of the cluster category. This is reminiscentof our discussion in .<ref> about a (necessary) condition for UV completeness.In section . <ref>, building over refs.<cit.>, we discuss how the interpretation of the cluster category 𝒞(Γ) as describing UV BPS line operators L_ind X(ζ) (labeled by the index of the corresponding cluster object X and the phase ζ of the preserved supersymmetry) leads to concrete expressions for their vacuum expectation values in the vacuum u⟨ L_ind X(ζ)⟩_u.§.§ The perfect derived category 𝔓𝔢𝔯 Γ To complete the understanding of the web of categories and functorsdescribing the BPS physics ofa 4d 𝒩=2 theory, it remains to discuss the physical meaning of the perfect category𝔓𝔢𝔯 Γ. To the best of our knowledge, an interpretation of the perfect category of a Ginzburg DG algebra has not appeared before in the physics literature.We may extract some properties of the BPS objected described by the perfect category already from its Grothendieck group K_0(𝔓𝔢𝔯 Γ) and the basic sequence of functors0→ D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ)→ 0.The Grothendieck group K_0(𝔓𝔢𝔯 Γ) is isomorphic to the IR charge lattice Λ, so 𝔓𝔢𝔯 Γ is a category of IR BPS objects whose existence (i.e. “stability”) depends on the particular vacuum u. 𝔓𝔢𝔯 Γ yields the description of these physical objects from the viewpoint of the Seiberg-Witten low-energy effective Abelian theory. This is already clear from the fact that 𝔓𝔢𝔯 Γ contains the category describing the IR BPS particles i.e. D^bΓ; BPS particles then form part of the physics described by 𝔓𝔢𝔯 Γ. A general object in 𝔓𝔢𝔯 Γ∖ D^bΓ differs from an object in the category D^bΓ in one crucial aspect: its total homology has infinite dimension, so (typically) infinite susy central charge and hence infinite energy. Then 𝔓𝔢𝔯 Γ is naturally interpreted as the category yielding the IR description ofhalf-BPS branes of some kind. They may have infinite energy just because their volume may beinfinite.Although their central charge is not well defined, its phase is: it is just the angle θ corresponding to the subalgebra of supersymmetries under which the brane is invariant.On the other hand, the RG functor 𝗋 in (<ref>) associates to each IR object in 𝒪∈𝔓𝔢𝔯 Γ∖ D^bΓ a non-trivial UV line operator 𝗋(𝒪).This suggests a heuristic physical picture: let𝒪∈𝔓𝔢𝔯 Γ∖ D^bΓ describe a BPS brane which is stable in the Coulomb vacuum u; this brane should be identified with the “state” obtained by acting with the UV line operator 𝗋(𝒪) on the vacuum u as seen in the low-energy Seiberg-Witten effective Abelian theory.In order to make this proposal explicit, in the next section we shall consider a particular class of examples, namely the class 𝒮[A_1] theories <cit.>. In this case all three categories D^bΓ, 𝔓𝔢𝔯 Γ and 𝒞(Γ) are explicitly understood both from the Representation-Theoretical side (in terms of string/band modules <cit.>) as well as in terms of the geometry of curves on the Gaiotto surface C. In this setting BPS objects are also well understood from the physical side since WKB is exact in the BPS sector.Comparing the mathematical definition of the various triangle categories associated to a class 𝒮[A_1] model, and the physical description of the BPS objects, we shall check that the above interpretation of 𝔓𝔢𝔯 Γ is correct.§.§.§ “Calibrations” of perfect categoriesTo complete the story we need to introduce a notion of “calibration” on the objects of 𝔓𝔢𝔯 Γ which restricts in the full subcategory D^bΓ to the Bridgeland notion of stability. The specification of a “calibration” requires the datum of the Coulomb vacuum u and a phase θ=πϕ∈. Given an u (corresponding to specifying a central charge Z), the ϕ-calibrated objects form a full additive subcategory of 𝔓𝔢𝔯 Γ, 𝒦(ϕ), such that𝒫(ϕ)⊂𝒦(ϕ)⊂𝔓𝔢𝔯 Γ,∀ ϕ∈. We use the term “calibration” instead of “stability” since it is quite a different notion with respect to Bridgeland stability (in a sense, it has “opposite” properties), and it does not correspond to the physical idea of stability. These aspects are already clear from the fact that the central charge Z is not defined for general objects in 𝔓𝔢𝔯 Γ. In the special case of the perfect categories arising from class 𝒮[A_1] QFTs, where everything is explicit and geometric, the calibration condition may be expressed in terms of flows of quadratic differentials, see . <ref>. We leave a more precise discussion of calibrations for perfect categories to future work. Here we limit ourselves to make some observations we learn from the class 𝒮[A_1] example.A phase πϕ∈ is called a BPS phase if the slice 𝒫(ϕ)⊂ D^bΓ contains non-zero objects. A phase πϕ is generic if it is not a BPS phase nor an accumulation point of BPS phases.In a class 𝒮[A_1] theory, assume there is no BPS phase in the range [πϕ,πϕ^']. Then𝒦(ϕ)≅𝒦(ϕ^').Moreover, let πϕ be a generic phase. Then the ϕ-calibrated category𝒦(ϕ)⊂𝔓𝔢𝔯 Γ has the form 𝒦(ϕ)≅𝖺𝖽𝖽 𝒯_ϕfor an object 𝒯_ϕ∈𝔓𝔢𝔯 Γ such that𝗋(𝒯_ϕ)∈𝒞(Γ) is cluster-tilting.In other words, the generic 𝒯_ϕ is a silting object of 𝔓𝔢𝔯 Γ. We conjecture that something like the above Fact holds for general 4d 𝒩=2 theories. § CLUSTER AUTOMORPHISMS AND S-DUALITY §.§ Generalities A duality between two supersymmetric theoriesinduces a (triangle) equivalence between the triangle categories describing itsBPS objects. The celebrate example is mirror symmetry betweenIIA and IIB string theories compactified on a pair of mirror Calabi-Yau 3-folds, ℳ, ℳ^∨. At the level of the corresponding categories of BPS branes, mirror symmetry duality induces homological mirror symmetry, that is the equivalences of triangle categories <cit.>D^b(𝖢𝗈𝗁 ℳ)≅ D^b(𝖥𝗎𝗄 ℳ^∨), D^b(𝖢𝗈𝗁 ℳ^∨)≅ D^b(𝖥𝗎𝗄 ℳ). In fact, since to a supersymmetric theory 𝒯 we associate a family of triangle categories, {𝔗_(a)}_a∈ I, depending on the class of BPS objects and the physical picture (e.g. IR versus UV), a dual pair of theories 𝒯, 𝒯^∨, yields a family of equivalences of categories labeled by the index set I 𝔗_(a)𝔗_(a)^∨ a∈ I.The several categories associated to the theory, {𝔗_(a)}_a∈ I, are related by physical compatibility functors having the schematic form𝔗_(a)𝔗_(b) a,b∈ I(e.g. the `inverse RG flow' functor 𝗋 in eqn.(<ref>)). Physical consistency of the duality then require that we have commutative diagrams of functors of the form𝔗_(a)[r]^𝖽_(a)[d]_𝖼_(a,b) 𝔗_(a)^∨[d]^𝖼_(a,b)^∨ 𝔗_(b)[r]^𝖽_(b) 𝔗_(b)^∨The philosophy of the present review is that the dualities are better understood in terms of such diagrams of exact functorsbetween the relevant triangle categories. This idea may be applied to all kinds of dualities; here we are particularly interested in auto-dualities, that is, dualities of the theory with itself. The prime examples of auto-dualities is S-duality in 𝒩=2^* SYM and Gaiotto's 𝒩=2 generalized S-dualities <cit.>.One of the motivation of this paper is to use categorical methods to compute the group𝕊 of S-dualities which generalize the PSL(2,) group for𝒩=2^* as well as the results by Gaiotto. An auto-duality induces a family of exact functors 𝖽_(a)𝔗_(a)→𝔗_(a), one for each BPS category 𝔗_(a), such that: a) for all a∈ I, 𝖽_(a) is anautoequivalence of the triangle category 𝔗_(a);b) the {𝖽_(a)} satisfy physical consistency conditions in the form of commutative diagrams 𝔗_(a)[r]^𝖽_(a)[d]_𝖼_(a,b) 𝔗_(a)[d]^𝖼_(a,b) 𝔗_(b)[r]^𝖽_(b) 𝔗_(b) 1) The group 𝔖 of generalized auto-dualities is the group of families 𝖽_(a) of autoequivalences satisfying eqn.(<ref>) modulo itssubgroup acting trivially on the physical observables.2) The group 𝕊 of (generalized) S-dualities is the quotient group of 𝔖 which acts effectively on the (UV) microscopic local degrees of freedom of the theory.With our definition of the S-duality group, the Weyl group of the flavor group is always part of the duality group 𝕊. It action on the free part of the cluster Grothendieck group is the natural one on the weight lattice.With this definition, the group 𝕊 for SU(2) SQCD with N_f=4 is <cit.> 𝕊_SU(2), N_f=4=SL(2,)⋊Weyl(SO(8)). We shall see in Example <ref> that with this definition the S-duality group of a class 𝒮[A_1] theory is the tagged mapping class group of its Gaiotto surface, in agreement with the geometric picture in <cit.>, see also<cit.>. §.§ Specializing to 𝒩=2 in 4d We specialize the discussion to the case of a 4d 𝒩=2 theory having the BPS quiver property. Such a theory is associated to a mutation-class of quivers with potential, hence to the three categories D^bΓ, 𝔓𝔢𝔯 Γ, 𝒞(Γ), discussed in the previous sections. They are related by the compatibility functors 𝗌, 𝗋 as in the exact sequence(<ref>).Applying Definition <ref> to the present set-up, we are lead to consider thediagram of triangle functors 0 [r] D^bΓ[d]_d_D[r]^𝗌 𝔓𝔢𝔯 Γ[d]_d_𝔓[r]^𝗋 𝒞(Γ)[d]_d_𝒞[r]00 [r] D^bΓ[r]^𝗌 𝔓𝔢𝔯 Γ[r]^𝗋 𝒞(Γ)[r]0 having exact rows and commuting squares, whered_D∈Aut D^bΓ, d_𝔓∈Aut 𝔓𝔢𝔯 Γ, d_𝒞∈Aut 𝒞(Γ).The group 𝔖 is the group of such triples (d_D,d_𝔓,d_𝒞) modulo the subgroup which acts trivially on the observables. The S-duality group 𝕊 is the image of 𝔖 under the homomorphismr𝔖→Aut 𝒞(Γ)/Aut 𝒞(Γ)_trivial, (d_D,d_𝔓,d_𝒞)↦ d_𝒞. §.§.§ The trivial subgroup (Aut D^bΓ)_0 We start by characterizing the subgroup (Aut D^bΓ)_0⊂Aut D^bΓ of `trivial' auto-equivalences, i.e. the ones which leave the physical observables invariant. Since the Grothendieck group is identified with the IR charge lattice Λ, and charge is an observable, (Aut D^bΓ)_0 is a subgroup of the kernel Aut D^bΓ→Aut K_0(D^bΓ). Next all ϱ∈(Aut D^bΓ)_0 should leave invariant the stability condition, that is the slicing 𝒫(ϕ), and hence the canonical heart 𝗆𝗈𝖽 J(Q,W) of D^bΓ. Since ϱ acts trivially on the Grothendieck group, it should fix all simples S_i. Hence the projection Aut D^bΓ→Aut D^bΓ/(Aut D^bΓ)_0 factors through the quotient groupAutphD^bΓ :=AutD^bΓ/ {autoequivalences preserving the simples S_i (element-wise)}.An equivalence in the kernel of the projection Aut D^bΓ→Autph D^bΓ preserves (Q,W), the central charge Z, and the Grothendieck class λ. Hence it maps stable objects of charge λ into stable objects of charge λ. Comparing with eqn.(<ref>), we see that the net effect of an autoequivalence in the kernel is to produce an automorphism of projective varieties M_λ→ M_λ for each λ. Since the BPS states are the susy vacua of the 1d sigma-model with target space M_λ, this is just a change of variables in the SQM path integral, which leave invariant all physical observables[The simplest example of such a negligible equivalence is the case of pure SU(2) whose quiver is the Kronecker quiver, 𝖪𝗋= ∙⇉∙. The stable representations associated to the W boson are the simples in the homogeneous tube which form a ℙ^1 family (i.e. M_W boson≡ℙ^1) since the W boson belongs to a vector superfield. Then a negligible auto-equivalence is just a projective automorphism of ℙ^1.]. Since the auto-duality groups are defined modulo transformations acting trivially on the observables, AuteqD^b is the proper auto-duality group 𝒮_IR at the level of the BPS category D^bΓ. The automorphisms of the quiver extend to automorphisms of D^bΓ; let Aut(Q) be the group of quiver automorphisms modulo the ones which fix the nodes. Clearly,Autph D^bΓ=Auteq D^bΓ⋊Aut(Q),whereAuteqD^bΓ :=AutD^bΓ/ {autoequivalences preserving the simples S_i (as a set)}. §.§.§ The duality groups 𝔖 and 𝕊With the notation of section <ref>, Bridgeland in <cit.> and Goncharov in <cit.> showed that the following sequence0 →Sph D^bΓ→AuteqD^bΓ→Aut_Q(CEG)→ 0is exact.Here CEG stands for the cluster exchange graph (cfr. . <ref>): the clusters of the cluster algebra C_Γ are the vertices of the CEG and the edges are single mutations connecting two seeds; Aut_Q(CEG) is the graph automorphism group that sends the quiver to itself up to relabeling of the vertices. By construction this graph is connected. One has Aut_Q(CEG)⊂Aut 𝒞(Γ),i.e. the graph automorphisms (see <cit.>) are a subgroup of the autoequivalences of the cluster category. Note that Auteq D^bΓ≡Auteq 𝔓𝔢𝔯 Γ, the quotient group of Aut 𝔓𝔢𝔯 Γ by the subgroup fixing the Γ_i (as a set). Indeed, all autoequivalences of 𝔓𝔢𝔯 Γ preservethe subcategory D^bΓ and hence restrict to autoequivalences of the bounded category; an autoequivalence ϱ∈Aut 𝔓𝔢𝔯 Γ which does not preserve the Γ_i's restricts to an element ϱ̅∈Aut D^bΓ which does not preserve the S_i's. Hence the restriction homomorphism Auteq 𝔓𝔢𝔯 Γ→Auteq D^bΓ, is injective. On the other hand, from eqn.(<ref>) we see that all autoequivalences in Auteq D^bΓ extend to autoequivalences in Auteq 𝔓𝔢𝔯 Γ:indeed, the objects which are spherical in the subcategory D^bΓ remain spherical and 3-CY in the larger category 𝔓𝔢𝔯 Γ (cfr. eqn.(<ref>)), so the auto-equivalences is Sph D^bΓ extend to 𝔓𝔢𝔯 Γ; the autoequivalences in Aut_Q(CEG) are induced by quiver mutations, and hence induce auto-equivalences of 𝔓𝔢𝔯 Γ.Comparing with our discussion around eqn.(<ref>) we conclude: For a 4d 𝒩=2 theory with the BPS quiver property𝔖 ≅Auteq D^bΓ⋊Aut(Q), 𝕊 ≅Aut_Q(CEG)⋊Aut(Q).§.§.§ Example: the group 𝕊 for SU(2) 𝒩=2^*The mutation class of SU(2) 𝒩=2^* contains a single quiver, the Markoff one Q_Mar≡ ∙_1 @2->[r] ∙_2 @2->[ld] ∙_3 @2->[u]which is the quiver associated to the once punctured torus <cit.>. Clearly Aut(Q_Mar)≅_3, while all mutations leave Q_Mar invariant up to a permutation of the nodes. Consider the covering graph CEG of CEG where we do not mod out the permutations of the nodes. Then CEG is the trivalent tree whose edges are decorated by {1,2,3}, the number attached to an edge corresponding to the nodes which gets mutated along that edge. One can check that there are no identifications between the nodes of this tree.One may compare this ({1,2,3}-decorated) trivalent treewith the ({1,2,3}-decorated) standard triangulation of the upper half-plane ℍ given by the reflections of thegeodesic triangle of vertices 0,1,∞ (see ref.<cit.>). One labels the nodes of a triangle of the standard triangulation by elements of {1,2,3}, and then extends (uniquely) the numeration to all other vertices so that the vertices of each triangle get different labels. The sides of a triangle are numbered as their opposite vertex. The dual of this decorated triangulation is our decorated trivalent graph CEG, see figure <ref>. The arithmetic subgroup of the hyperbolic isometry group, PGL(2,)⊂ PGL(2,) preserves the standard triangulation of ℍ while permuting the decorations {1,2,3}. Since permutations are valid S-dualities, we get𝕊≅ PGL(2,)≅ PSL(2,)⋊_2where the extra _2 may be identified with the Weyl group of the flavor SU(2). Thus we recover as S-duality group in the usual sense (≡ the kernel of𝕊→Weyl(F)) the modular group PSL(2,) <cit.>.In the case of SU(2) 𝒩=2^* we haveK_0(𝒞_Mar)≡coker B≅_2⊕_2⊕,as expected for a quark in the adjoint representation (since π_1(G_eff)=_2), with the free part the weight lattice of SU(2)_flav. Hence the flavor Weyl group acts on K_0(𝒞_Mar) as -1, that is, as the cluster auto-equivalence [1]. Notice that the cluster category is 2-periodic, as expected for a UV SCFT with integral dimensions Δ.§.§ Relation to duality walls and 3d mirrors The UV S-duality group 𝕊 has a clear interpretation: it is the usual S-duality group of the 𝒩=2 theory (twisted by the flavor Weyl group). What about its IR counterpart 𝔖?For Argyres-Douglas models we can put forward a precise physical interpretation based on the findings of <cit.>. Similar statements should hold in general.Given an element of the S-duality group, σ∈𝕊 we may construct a half-BPS duality wall in the 4d theory <cit.>: just take the theory for x_3<0 to be the image through σ of the theory for x_3>0 and adjust the field profiles along the hyperplane x_3=0 in such a way that the resulting Janus configuration is 12-BPS.It is a domain wall interpolating between two dual 𝒩=2 theories in complementary half-spaces. On the wall live suitable 3d degrees of freedom interacting with the bulk 4d fields on both sides <cit.>.In this construction we may use a UV duality as well as an IR one <cit.>. Hence we expect to get duality walls for all elements of 𝔖. An element 𝔰∈𝔖 acts non-trivially on the central charge Z so, in general, as we go from x_3=-∞ to x_3=+∞ we induce a non-trivial flow of the central charge Z in the space of stability functions. If lim_x_3→±∞Z is such that all the bulk degrees of freedom get an infinite mass and decouple, we remain with a pure 3d 𝒩=2 theory on the wall. Of course this may happen only forspecial choices of 𝔰. Thus we may use (suitable) 4d dualities to engineer 3d 𝒩=2 QFTs.The engineering of 3d 𝒩=2 theories as a domain wall in a 4d 𝒩=2 QFT, by central-charge flow in the normal direction, is precisely the set-up of ref.<cit.>.In that paper one started from a 4d Argyres-Douglas of type 𝔤∈ ADE. The Z-flow along the x_3-axis was such thatasymptotic behaviors as x_3→-∞ and x_3→+∞ were related in the UV by the action of the quantum half-monodromy 𝕂, that is, in the categorical language by the shift [1]∈𝕊. Two choices of IR duality elements, 𝔰, 𝔰^'∈𝔖, which produce the half-monodromy in the UV,differ by an element of the spherical twist group (cfr. eqn.(<ref>))𝔰^'𝔰^-1∈Sph D^b.The arguments at the end of . <ref> imply that for Argyres-Douglas of type 𝔤 the group Sph D^b is isomorphic to the Artin braid group of type 𝔤, ℬ_𝔤.As the title of ref.<cit.> implies, the explicit engineering of a 3d 𝒩=2 theory along those lines requires a specification of a braid, i.e. of an element of ℬ_𝔤. More precisely, in .5.3.2 of ref.<cit.> is given an explicit map (for 𝔤=A_r)(a braid in ℬ_𝔤)⟷(a 3d𝒩=2 Lagrangian).So the Lagrangian description/Z-flow engineering of the 3d theories are in one-to-one correspondence with the 𝔰∈𝔖 such that r(𝔰)=[1]. It is natural to think of the 3d Lagrangian theory associated to 𝔰∈𝔖 as the duality wall associated to the IR duality 𝔰.Distinct 𝔰 lead to 3d theories whichsuperficially look quite different. However,in this context, 3d mirror symmetry is precisely the statement that two theories defined by different IR dualities 𝔰,𝔰^'∈𝔖 which induce the same UV duality, r(𝔰^')=r(𝔰) produce equivalent 3d QFTs. From this viewpoint 3d mirror symmetry is a bit tautological, since the condition r(𝔰^')=r(𝔰) just says that the two 3d theories have the same description in terms of 4dmicroscopic degrees of freedom. §.§ S-duality for Argyres-Douglas and SU(2) gauge theories When (Q,W) is in the mutation-class of an ADE Dynkin graph (corresponding to an Argyres-Douglas model <cit.>) or of an ADE acyclic affine quiver (corresponding to SU(2) SYM coupled to matter such that the YM coupling is asymptotically-free <cit.>) to get 𝕊 we can equivalently study the automorphism of the transjective component of the AR quiver associated to the cluster category 𝒞(Γ): the inclusion above is due to the fact that we only consider the transjective component:Let C be an acyclic cluster algebra and Γ_tr the transjective component of the Auslander-Reiten quiver of the associated cluster category 𝒞(Γ). Then Aut^+C is the quotient of the group Aut Γ_tr of the quiver automorphisms of Γ_tr, modulo the stabilizer Stab(Γ_tr)_0 of the points of this component. Moreover, if Γ_tr≅Δ, where Δ is a tree or of type  thenAutC = Aut^+C ⋊_2 and this semidirect product is not direct.In order to understand why this is the relevant component, we first recall that the Auslander-Reiten quiver of a cluster-tilted algebra always has a unique component containing local slices, which coincides with the whole Auslander-Reiten quiver whenever the cluster-tilted algebra is representation-finite. This component is called the transjective component and an indecomposable module lying in it is called a transjective module. With this terminology, the main result is:Let C be a cluster-tilted algebra and M, N be indecomposable transjective C-modules. Then M is isomorphic to N if and only if M and N have the same dimension vector.Therefore, since the dimension vector is the physical charge, we focus our attention to this class of autoequivalences. The classification results are summarized in table <ref> where H_p,q:=r,s|r^p=s^q, sr=rsG=τ,σ,ρ_1,ρ_n|ρ_1^2=ρ_n^2=1, τρ_1=ρ_1τ,τρ_n=ρ_nτ, τσ=στ, σ^2=τ^n-3, ρ_1σ=σρ_n, σρ_1 = ρ_nσ [SU(2) with N_f≤3] SU(2) SQCD with N_f=0,1,2,3 correspond, respectively, to the following four affine 𝒩=2 theories <cit.>Â_1,1,Â_2,1,Â_2,2,D̂_4.A part for the flavor Weyl group Weyl(𝔰𝔭𝔦𝔫(2N_f)) (cfr. Example <ref>) we get a duality groupgenerated by the shift [1]. As discussed around eqn.(<ref>), this is equivalent to the shift of the Yang-Mills angle θθ→θ-4π +N_fπ.The case N_f=0 is special; physically one expects that the shift of θ by -2π should also be a valid S-duality. This shift should correspond to an auto-equivalence ξ of the N_f=0 cluster category with ξ^2=τ. Indeed, this is what one obtains from the automorphism of the transjective component see figure <ref>. Alternatively, we may see the cluster category of pure SU(2) as the category of coherent sheaves on ℙ^1 endowed with extra odd morphisms <cit.>. In this language τ acts as the tensor product with the canonical bundle τ↦⊗𝒦 (cfr. eqn.(<ref>)). Let ℒ be the unique spin structure on ℙ^1; we have the obvious auto-equivalence ξ↦⊗ℒ. From ℒ^2=𝒦 we see that ξ^2=τ. § COMPUTER ALGORITHM TO DETERMINE THE S-DUALITY GROUP The identification of the S-duality group with Aut_Q(CEG) yield a combinatoric characterization of S-dualities which leads to an algorithm to search S-dualities for an arbitrary 𝒩=2 model having a BPS quiver. This algorithm is similar in spirit to the mutation algorithm to find the BPS spectrum <cit.> but in a sense more efficient. The algorithm may be easily implemented on a computer; if the ranks of the gauge and flavor groups are not too big (say < 10), running the procedure on a laptop typically produces the generators of the duality group in a matter of minutes. §.§ The algorithm The group Aut_Q(CEG)may be defined in terms of the transformations under quiver mutations of the d-vectors which specify the denominators of the generic cluster variables <cit.>. The actions of the elementary quiver mutation at the k–th node, μ_k, on the exchange matrix B and the d-vector d_i areμ_k(B)_ij = - B_ij, i=kor j=kB_ij+max[-B_ik,0] B_kj+B_ik max[B_kj,0]—– otherwise. μ_k(d)_l ={[ d_l, l≠ k; -d_k+max[∑_imax[B_ik,0]d_i,∑_imax[-B_ik,0]d_i]l=k ].A quiver mutation μ=μ_k_sμ_k_s-1⋯μ_k_1 is the compositionof a finite sequence of elementary quiver mutations μ_k_1, μ_k_2,⋯,μ_k_s. We write 𝖬𝗎𝗍 for the set of all quiver mutations. Aut_Q(CEG) is the group of quiver mutations whichleave invariant the quiver Q up to a permutation π of its nodes,modulo the ones which leave the d-vector invariant up to π: Aut_Q(CEG)={μ∈𝖬𝗎𝗍 | ∃ π∈S_nμ(B)_i,j=B_π(i),π(j)}/{μ∈𝖬𝗎𝗍 | ∃ π∈S_nμ(B)_i,j=B_π(i),π(j) and μ(d)_i=d_π(i)},while 𝕊=Aut_Q(CEG)⋊Aut(Q). [A_2 cluster automorphisms]Consider the quiver ∙_1→∙_2. The CEG is the pentagon in figure <ref>: every vertex is associated to a quiver of the form ∙_1→∙_2 or ∙_2→∙_1. Thus, in this case every sequence of mutations gives rise to a cluster automorphism. For example, consider μ_1: the quiver nodes get permuted under π=(12). We explicitly check – for example using Keller applet[See <https://webusers.imj-prg.fr/ bernhard.keller/quivermutation/>.] – that(μ_source)^5=μ_1μ_2μ_1μ_2μ_1=1since (μ_source)^5 leaves the d-vectors invariant. From figure <ref> one sees that _5 is indeed the full automorphism group of the CEG of A_2. This result is coherent with the analysis leading to table <ref>, as well as with the tagged mapping class group of the associated Gaiotto surface, see Example <ref>. The explicit expression (<ref>) of the S-duality group is the basis of a computer search for S-dualities. Schematically: let the computer generate a finite sequence of nodes of Q, k_1,⋯, k_s, then construct the corresponding mutation μ_k_sμ_k_s-1⋯μ_k_1=μ, and check whether it leaves the exchange matrix B invariant up to a permutation π. If the answer is yes, let the machine check whetherμ(d)_i≠ d_π(i). If the answer is again yes the computer has discovered a non-trivial S-duality and prints it. Then the computer generates anothersequence and go cyclically throughthe same steps again and again. After running the procedure for some time t, we get a print-out with a list ℒ_t of non-trivial S-dualities of our 𝒩=2 theory. A Mathematica Code performing this routine is presented in Appendix <ref>. If the S-duality group is finite (and not too huge) ℒ_t will contain the full list of S-dualities. However, the most interesting S-duality groups are infinite, and the computer cannot find all its elements in finite time. This is not a fundamental problem for the automatic computation of the S-duality group. Indeed, the S-duality groups, while often infinite, are expected to be finitely generated, and in fact finitely presented. If this is the case, we need only that the finite list ℒ_t produced by the computer contains a complete set of generators of 𝕊. Taking various products of these generators, and checking which products act trivially on the d-vectors, we may find the finitely many relations. The method works better if we have some physical hint on what the generators and relations may be.Of course, the duality group obtained from the computer search is a priori only a subgroup of the actual 𝕊 because there is always the possibility of further generators of the group which are outside our range of search. However, pragmatically, running the procedure for enough time, the group one gets is the full one at a high confidence level.§.§ Sample determinations of S-duality groups We present a sample of the results obtainedby running our Mathematica Code.[SU(2) 𝒩=2^* again] The CEG automorphism group for this model was already described in . <ref>. Recall that PSL(2,ℤ) is the quotient of the braid group over three strands, ℬ_3 by its center Z(ℬ_3)PSL(2,ℤ)≅ℬ_3/Z(ℬ_3). Running our algorithm for a short time returns a list of dualities which in particular contains the two standard generators of the braid group σ_1,σ_2∈ℬ_3, which correspond tothe following sequences of elementary quiver mutations:σ_1:=μ_1μ_2,andσ_2:=μ_1μ_3,with permutation π=(1 3 2).One easily checks the braid relationσ_1σ_2σ_1=σ_2σ_1σ_2up to permutation,as well as that the generator of the center Z(ℬ_3), (σ_2σ_1)^3, acts trivially on the cluster category: indeed, it sends the initial dimension vector d⃗=-Id_3 × 3 to itself. From eqn.(<ref>) we conclude that the two S-dualities σ_1, σ_2 generate a PSL(2,) duality (sub)group.In facts, 𝕊/PSL(2,)≅_2 where the class of the non-trivial _2 element may be represented (say) by μ_1. Indeed the map𝕊→_2≡Weyl(F_flav) send the mutation μ to (-1)^ℓ(μ), where the length ℓ(μ) of μ≡μ_k_sμ_k_s-1⋯μ_k_1 is s (length is well defined mod 2). [SU(2) with N_f=4] We use the quiver in figure <ref> where for future reference we also draw the corresponding ideal triangulation of the sphere with 4 punctures <cit.>.The following two even-length sequences of mutations leave the quiver invariant:S=μ_2μ_3μ_2μ_0μ_2μ_5μ_3μ_0,T= μ_5μ_2μ_0μ_3μ_5μ_3μ_4μ_2μ_4μ_1μ_4μ_2μ_4μ_5μ_1μ_2.These sequences of mutations satisfy the following relations:S^4=1, (ST)^6=1, Thas infinite order.Moreover, T and S commute with S^2 and (ST)^3. Write _2×_2 for the subgroup generated by S^2 and (ST)^3. Then we have1 →_2×_2→⟨S, T ⟩→ PSL(2,)→ 1.Againthis shows that the duality sub-group ⟨ S,T ⟩ generated by S and T is equal to thethe mapping class group of the sphere with four punctures (cfr. Proposition 2.7 of <cit.>). In fact one has Aut_Q(CEG)/⟨ S,T ⟩≅_2; geometrically (see next section) the extra _2 arises because for class 𝒮[A_1] theories Aut_Q(CEG)is the tagged mapping class group of the corresponding Gaiotto surface (Bridgeland theorem<cit.>);the extra _2 is just the changein tagging. This extra _2 is also detected by the computer program which turns out dualities of order 12 and 8 which are not contained in ⟨ S,T ⟩ but in its _2 extension.Taking into account the S_4 automorphism of the quiver, we recover PSL(2,)⋉Weyl(𝔰𝔭𝔦𝔫(8)) with the proper triality action of the modular group on the flavor weights <cit.>. For an alternative discussion of the S-duality group of this model as the automorphism group of the corresponding cluster category, see ref.<cit.>. [E_6 Minahan-Nemeschanski] This SCFT is the T_3 theory, that is, the Gaiotto theory obtained by compactifying the 6d (2,0) SCFT of type A_2 on a sphere with three maximal punctures <cit.>. Since the three-punctured sphere is rigid, geometrically we expect a finite S-duality group. The homological methods of <cit.> confirm this expectation. The computer search produced a list of group elements of order 2,3,4,5,6,8,9,10,12 and 18. Since, with our definition, the S-duality group should contain theWeyl group of E_6, we may compare this list with the list of orders of elements of Weyl(E_6),{2,3,4,5,6,8,9,10,12}.We see that the two lists coincide, except for 18. Thus the S-duality group is slightly larger than the Weyl group, possibly just Weyl(E_6)⋊_2, where _2 is the automorphism of the Dynkin diagram. Notice that this is the largest group which may act on the free part of the cluster Grothendieck group (since it should act by isometries of the Tits form).[Generic T_𝔤 theories] By the same argument as in theprevious Example, we expect the S-duality group to be finite for all T_𝔤 (𝔤∈ ADE) theories.We performed a few sample computer searches getting agreement with the expectation. [E_7 Minahan-Nemeschanski] The computer search for this example produced a list of group elements of order 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18 and 30. Since our S-duality group contains theWeyl group of the flavor E_7, wecompare this list with the list of orders of elements of Weyl(E_7), {2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 30}. We see that the two lists coincide. It is reasonable to believe that the full S-duality group coincides with Weyl(E_7). This is also the largest group preserving the flavor Tits form. §.§ Asymptotic-free examples As an appetizer, let us consider a𝒩=2 gauge theorywith a gauge group of the form SU(2)^k coupled to (half-)hypermultiplets in some representation of the gauge group so that all Yang-Mills couplings g_i (i=1,…,k) have strictly negative β-functions. As discussed in . <ref>, the fact that the theory is asymptotically-free means that its cluster category 𝒞 is not periodic. However, its Coulomb branch is parametrized by k operators whose dimension in the UV limit g_i→0 becomes Δ=2. As in Example <ref>, this implies the existence of a 1-periodic sub-category ℱ(1)⊂𝒞. Iff all YM couplings g_i are strictly asymptotically-free, the category ℱ(1) consists of k copies of the 1-periodic sub-category of pure SU(2), Example <ref>. In such an asymptotic-free theory the S-duality group is bound to be `small' since all auto-equivalence σ of the cluster category should preserve the 1-periodic sub-category ℱ(1); therefore, up to (possibly)permutations of the various SU(2) gauge factors, σ should restrict to a subgroup of autoequivalences of the periodic category ℱ(1)_pure of pure SU(2) SYM. As we saw in Example <ref>, the S-dualities corresponding to shifts of the Yang-Mills angle θ preserve[ Physically this is obvious. Mathematically, consider e.g. the shift shift θ→θ-4π+N_fπ in SU(2) with N_f flavors. It corresponds to the auto-equivalence ↦[1], which acts trivially on the 1-periodic subcategory.] the subcategory ℱ(1)_pure. Thus besides shifts of the various theta angles, permutations of identical subsectors, and flavor Weyl groups/Dynkin graph automorphism, we do not expect additional S-dualities in these models. Let us check this expectaction against the computer search for dualities in a tricky example.[SU(2)^3 with 12(2,2,2)]A quiver for this model is given in figure <ref>. In this case the cluster Grothendieck group K_0(𝒞_pris)is pure torsion, since a single half-hyper carries no flavor charge. The three SU(2) gauge couplings g_i are asymptotically-free and the cluster category 𝒞_pris is not periodic but it contains the 1-periodic subcategory ℱ(1)⊂𝒞_pris described above[ Notice that there is no periodic sub-category associated to the quark sector; this is related to the absence of conserved flavor currents in this model.]. The S-duality group is then expected to consists of permutations of the three SU(2)'s and the three independent shifts of the Yang-Mills angles θ_i→θ_i-2π, that is, 𝕊=S_3⋉^3.The computer algorithm produced the following three commuting generators of the cluster automorphism group of infiniteθ_1=μ_23μ_22μ_11,θ_2=μ_21μ_23μ_12,θ_3=μ_22μ_21μ_13.These three generators are identified with the three θ-shifts.Since the model is of class 𝒮[A_1] (with irregular poles), the S-duality group may also be computed geometrically (see section 7). The computer result is of course consistent with geometry: each θ_i translation correspond to a twist around one of the three holes on the sphere: their order is clearly infinite and the three twists commute with one another.§.§ Q-systems as groups of S-duality The above discussion may be generalized to all 𝒩=2 QFTs having a weakly coupled Lagrangian formulation. If the gauge group G is a product of k simple factors G_i, we expect the S-duality group to contain a universal subgroup ^k consisting of shifts θ_i→θ_i-b_iπ, with b_i the β-function coefficient of the i-th YM coupling. One may run the algorithm and find the universal subgroup; however, just because it is universal, its description in terms of quiver mutations also has a universal form which is easy to describe.We begin with an example. [Pure SYM: simply-laced gauge group] If the gauge group G is simply-laced, the exchange matrix of its quiver may be put in the form <cit.>[ In particular, coker B=Z(G)^∨⊕Z(G) is the correct 't Hooft group for pure SYM.]B=([0 -C; -C0 ])≡ C⊗ iσ_2, C≡the Cartan matrix of G.For instance, the quiver for SU(N) SYM is represented in figure <ref>.These quivers are bipartite: we may color the nodes black and white so that a node is linked only to nodes of the opposite color. Quiver mutations at nodes of the same color commute, so the productν=∏_i whiteμ_iis well-defined. Moreover interchanging (black) ↔ (white) yields the opposite quiver Q^opp which is isomorphic to Q via the node permutation π=1⊗σ_1. The effect of the canonical mutation ν on the quiver is to invert all arrows i.e. it gives back the same quiver up to the involution π. Thus ν corresponds to an universal duality of pure SYM. One checks that it has infinite order, i.e. generates a subgroup of S-dualities isomorphic to .This sub-groupof S-dualities has different physical interpretations/applications in statistical physics <cit.> as well as in the context of the Thermodynamical Bethe Ansatz <cit.>. Indeed, consider its index 2 subgroup generated by the square of νν^2=∏_jblackμ_j ∏_i whiteμ_i.The repeated application of the S-duality ν^2 generates a recursion relation for the cluster variables which is known as the Q-system of type G. It has deep relation with the theory of quantum groups; moreover it generates a linear recursion relation of finite length and has many other “magical” properties <cit.>.We claim that the duality ν^2 corresponds to a shift of θ. Indeed, the cluster category in this case is the triangular hull of the orbit category of D^b(𝗆𝗈𝖽 Â_1,1⊗ G) and ν^2 corresponds to the auto-equivalence τ⊗Id <cit.>. Comparing the action of τ⊗Id in the covering category with the Witten effect (along the lines of .<ref>) one gets the claim. [SYM with non-simply laced gauge group] The authors of ref.<cit.> defined Q-systems also for non-simply laced Lie groups. To a simple Lie group G one associates a quiver and a mutation ν^2 which generates a groupwhich has all the required “magic” properties. In ref.<cit.>it was shown that the non-simply-laced Q-system does give the quiver description of the BPS sectors of the corresponding SYM theories. The Q-system group is again the group of S-dualities corresponding to θ-shifts. [General 𝒩=2 SQCD models] We may consider the general Lagrangian case in which the gauge group is a product of simple Lie groups, ∏_j G_j and we have hypermultiplets in some representation of the gauge group. The quivers for such a theory may be found in refs.<cit.>. For instance figure <ref> shows the quiver for SU(M)× SU(N) gauge theory coupled to 2 flavors of quark in the (N,1) and a quark bifundamental in the (N̅,M). It is easy to check that the two canonical mutations of the subquivers associated to the two simple factors of the gauge group ν_∘=∏_i=∘μ_i,ν_□=∏_i=□μ_i,leave the quiver invariant up to the permutation∘↔∙ and, respectively, □↔▪. The construction extends straightforwardly to any number of gauge factors G_j and all matter representations. The conclusion is thatwe have a canonicalsubgroup of the S-duality group per simple factor of the gauge group. It corresponds toshifts of the corresponding θ-angle. If the matter is such that the β-function vanishes, the full cluster category becomes periodic, and we typically get a larger S-duality group.One can convince himself that the sequence of mutations μ does not change the quiver and that its order, in all the above cases, is infinite, as it is for the shift is the θ_j's.§ CLASS 𝒮 QFTS: SURFACES, TRIANGULATIONS, AND CATEGORIESIn this section we focus on aspecial class of 𝒩=2 theories: the Gaiotto 𝒮[A_1] models <cit.>. We study them for two reasons: first of all they are interesting for their own sake, and second for these theories the three categories D^bΓ, 𝔓𝔢𝔯 Γ,and 𝒞(Γ) have geometric constructions, directly related to the WKB analysis of <cit.>. Comparing the categorical description with the results of refs.<cit.> we check the correctness of our physical interpretation of the various categories and functors.Class 𝒮[A_1] theories are obtained by the compactification of the 6d (2,0) SCFT of type A_1 over a complex curve C having regular and irregular punctures <cit.>.If there is at least one puncture, these theory have the quiver property <cit.>, and their quivers with superpotentials are constructed in terms of an ideal triangulation of C <cit.>. In the geometrical setting of Gaiotto curves, we can interpret the categories defined in section <ref> as categories of (real) curves on the spectral cover of the Gaiotto curve C. When only irregular punctures are present, the quiver with potential arising from these theories <cit.> has a Jacobian algebra which is gentle <cit.>, and hence all triangle categories associated to its BPS sector, eqn.(<ref>), have a simple explicit description.[ In facts, there is a systematic procedure, called gentling in ref.<cit.> which allow to reduce the general class 𝒮[A_1] model to one having a gente Jacobian algebra.]When only regular punctures are present, the 𝒩=2 theory has a Lagrangian formulation (which is weakly coupled in some corner of its moduli space) and is UV superconformal. In particular the corresponding cluster category is 2-periodic, as the arguments of . <ref> imply.§.§ UV and IR descriptionsThe main reference for this part is <cit.>. In the deep UV a class 𝒮[A_1] 𝒩=2 theory is described by the Gaiotto curve C, namely a complex curve of genus g with a number of punctures x_i∈ C. Punctures are of two kinds: regular punctures (called simply punctures) and irregular ones (called boundaries). The i–th boundary carries a positive integer k_i≥1 (the number of its cilia); sometimes it is convenient to regard regular punctures asboundaries with k_i=0. Iff k_i≤ 2 for all i, the 𝒩=2 theory is a Lagrangian model with gauge group[ When g=0, the theory is defined only if b≥ 1 or b=0 and p≥ 3; in case p=0, b=1 we require k≥ 4; when g=1 we need p+b≥ 1. Except for the case p=0, b=1, corresponding to Argyres-Douglas of type A, m in eqn.(<ref>) is ≥0. m=0 only for Argyres-Douglas of type D <cit.>.]𝒢=SU(2)^m, m=3g-3+p+2bwherep=#{regular punctures}b=#{boundaries}. If b=0 the theory is superconformal in the UV, and the space of exactly marginal coupling coincides with the moduli space of genus g curves with p punctures, ℳ_g,p, whose complex dimension is m ≡ the rank of the gauge group 𝒢. Instead, if b≥ 1 (and m≥ 2), b out of the m SU(2) factors in the Yang-Mills group 𝒢 have asymptotically free couplings; these b YM couplings go to zero in the extreme UV, so that the UV marginal couplings are again equal in number to the complex deformations ℳ_g,p+b of C.If some of the boundaries have k_i≥ 3, we have a gauge theory with the same gauge group SU(2)^m coupled to “matter” consisting, besides free quarks (in the fundamental, bi-fundamental, and three-fundamental of 𝒢), in an Argyres-Douglas SCFT of type D_k_i for each boundary[ Argyres-Douglas of type D_1 is the empty theory and the one of type D_2 a fundamental quark doublet.] <cit.>. The space of exactly marginal deformations is as before.The IR description of the model is given by the Seiberg-Witten curve Σ which, for class 𝒮[A_1], is a double cover of C. More precisely, one considers in the total space of the ℙ^1-bundleℙ(K_C⊕𝒪_C)→ Cwhere K_C canonical line bundle on C 𝒪_C trivial line bundle on Cthe curve Σ≡{ y^2=ϕ_2(x) z^2 |(y,z) homogeneous coordinates in the fiber}→ C,where ϕ_2(x) is a quadratic differential on C with poles of degree at most k_i+2 at x_i. The Seiberg-Witten differential is the tautological oneλ= y/z dxwhose periods in Σ yield the 𝒩=2 central charges of the BPS states.The dimension of the space of IR deformations is then[ This formula holds under the condition ℳ_g,p+b=3g-3+p+b≥0.] [ Here the asymptotically-free gauge couplings are counted as IR deformations.]s = H^0(C, P K_C^2)≡ 3g-3+2p+2b+∑_i k_iwhere P=∑_i(k_i+2)[x_i],so that the total space of parameters, UV+IR, has dimensionn=m+s= 6g-6+3p+3b+∑_ik_i.There are two kinds of IR deformations, normalizable and unnormalizable ones. The unnormalizable ones correspond to deformations of the Lagrangian, while the normalizable onesto moduli space of vacua (that is, Coulomb branch parameters); their dimensions are[ As written, these equations hold even if the condition in footnote <ref> is not satisfied. Notice that we count also the dimensions of the internal Coulomb branches of the matter Argyres-Douglas systems.]s_nor ≡(Coulomb branch)=3g-3+p+b+∑_i (k_i-[k_i/2]),s_un-nor =s-s_nor. The double cover Σ→ C is branched over the zerosw_a∈ C of the quadratic differential ϕ_2(x). Their number ist= 4g-4+2p+2b+∑_i k_i,but their positions are constrained by the condition that the divisor ∑_a[w_a] is linear equivalent to PK_C^2, so that their positionsdepend on only t-g parameters; ϕ_2(x) depends on one more parameters=t-g+1since the positions of its zeros fix ϕ_2(x) only up to an overall scale (that is, up to the overall normalization of the Seiberg-Witten differential, which is the overall mass scale).Therefore, up to the overall mass scale, giving the cover Σ→ C is equivalent to specifying the zeros w_a∈ C of the quadratic differential. Indeed, double covers are fixed, up to isomorphism, by their branching points.We shall refer to the points w_a∈ C as decorated points on the Gaiotto curve C. In summary: the UV description of a class 𝒮[A_1] amounts to giving the datum of the Gaiotto curve C, that is, a complex structure of a genus g Riemann surface and a number of marked points x_i∈ C together with a non-negative integer k_i at each marked point. To get the IR description we have to specify, in addition, the decorated points w_a∈ C (whose divisor is constrained to be linear equivalent to PK_C^2). We may equivalently state this result in the form: In theories of class 𝒮[A_1], to go from the IR to the UV description we simply delete (i.e. forget) the decorated points of C. We shall see in .<ref> below that this `forget the decoration' prescription is precisely the map denoted 𝗋 in the exact sequence of triangle categories of eqn.(<ref>).§.§ BPS states In class 𝒮[A_1] 𝒩=2 theories, the natural BPS objects are described by (real) curves η on the Seiberg-Witten curve Σ which are calibrated by the Seiberg-Witten differential <cit.>λ= y/z dx≡√(ϕ_2(x)) dx,that is, they are required to satisfy the condition (we set ϕ≡ϕ_2 (dx)^2), √(ϕ)|_η=e^iθ dt,here t ∈,for some real constant θ, and are maximal with respect to this condition. Being maximal, η may terminate only at marked or decorated points.[ We call marked/decorated points in Σ the pre-images of marked/decorated points on C.]BPS particles have finite mass, i.e. they correspond to calibrated arcs η with |Z(η)|<∞ wherethe central charge of the would be BPS state η,is Z(η)=∫_ηλ.In this case, the parameter θ in eqn.(<ref>) is given by θ= Z(η). Arcs η associated to BPS particles may end only at decorations. All other maximal calibrated arcs have infinite mass and are interpreted as BPS branes <cit.>.There are two possibilities for BPS particles: * they are closed arcsconnecting zeros of ϕ. These calibrated arcs are rigid and hence correspond to BPS hypermultiplets;* they are loops. Such calibrated arcs form ℙ^1-families and give rise to BPS vector multiplets.Conserved charges. From eqn.(<ref>) we see that the central charge of an arc η factors through its homology class η∈ H_1(Σ,). More precisely, since the Seiberg-Witten differential λ is odd under the covering group _2≅𝖦𝖺𝗅(Σ/C), Z factorsthrough the free Abelian groupΛ≡ H_1(Σ,)_odd,rank Λ=2(g(Σ)-g(C))+#{k_i even}.Applying the Riemann-Hurwitz formula[ Compare eqns.(6.26)-(6.28) in ref.<cit.>.] to the cover Σ→ C, we see that the rank of Λ is equal to the number n of UV+IR deformations, see eqn.(<ref>). In turn n is the number of conserved charges (electric, magnetic, flavor) of the IR theory. Hence the group homomorphism ZΛ→, [η]↦∫_ηλ,is the map which associates to the IR charge γ∈Λ of a state of the 𝒩=2 theory the corresponding central charge Z(γ). An arc with homology [η]∈Λ then has `mass'∫_η |λ|≥ |Z(η)|with equality iff and only if it is calibrated, that is, BPS. To get the correspondingUV statements, we apply to these results our RG principle, that is, we forget the decorations. BPS particles then disappear (as they should form the UV perspective), while BPS branes project to arcs on the Gaiotto curve C. Several IR branes project to the same arc on C. The arcs on C have the interpretation of UV BPS line operators, and the branes which project to it are the objects they create in the given vacuum (specified by the cover Σ→ C) which may be dressed (screened) in various ways by BPS states, so that the IR-to-UV correspondence is many-to-one in the line sector. The UV conserved charges is the projection of Λ; over ℚ all electric/magnetic charges are projected out by the oddness condition, and we remain with just the flavor lattice. However overthe story is more interesting and we get <cit.>Λ_UV≅^#{k_i even}⊕2-torsion. Comparing with our discussion in the Introduction, we see that Λ and Λ_UV should be identified with the Grothedieck group of the triangle categories D^bΓ and 𝒞(Γ), respectively. We shall check these identifications below. §.§ Quadratic differentials We want to study the BPS equations to find the BPS spectrum of the theory. We start by analyzing the local behavior of the flow of the quadratic differential. The quadratic differential near a zero can be locally analyzed in a coordinate patch where ϕ∼ w; thus we have to solve the following equation:√(w) dw/dt=e^i θ,which gives w(t)=(3/2te^i θ+w_0^3/2)^2/3. We plot here the solution:The three straight trajectories, which all start at the zero of ϕ, end at infinity in the poles of ϕ, i.e. the marked points on the boundaries of C. Since all zeros of ϕ are simple by hypothesis, it is possible to associate a triangulation to a quadratic differential by selecting the flow lines connecting the marked points as shown in the following figure: Moreover, by construction, it is clear that to each triangle we associate a zero of ϕ. These are the decorating points Δ of section <ref>. If we make θ vary, we deform the triangulation up to a point in which the triangulation jumps: at that value of θ=θ_c, two zeros of ϕ are connected by a curve η: this curve is the stable BPS state. From what we have just stated, it will be clear that the closed arcs of section <ref> will correspond to BPS states. Before and after the critical value of θ=θ_c, the triangulation undergoes a flip. Flips of the triangulation correspond to mutations at the level of BPS quiver (see section <ref> on how to associate a quiver to a triangulation). This topic is develop in full details in <cit.>.Moreover, the second class of BPS objects, i.e. loops representing vector-multiplets BPS states, appears in one-parameter families and behave as in example <ref>; the map X of section <ref> will allow us to write the corresponding graded module X(a) ∈ D^b(Γ). What about the curves connecting punctures or marked points (but not zeros of the quadratic differential)?[ These are the objects in the perfect derived category 𝔓𝔢𝔯 Γ] To answer this question, we have to take a detour into line operators.We point out that the space of quadratic differential is isomorphic to the Coulomb branch of the theory. In a recent paper of Bridgeland <cit.> it is stated that the space of stability conditions of D^b Γ satisfy the following equation:Stab^0(D^b Γ)/Sph(D^bΓ)≅Quad(S,M).Thus, the Coulomb branch is isomorphic to the space of stability conditions, up to a spherical twist. Indeed, a stability condition is a pair (Z,𝒫), where Z is the stability function (i.e. the central charge) and the category 𝒫 is the category of stable objects, i.e. stable BPS states (see section <ref> for more details). §.§ Geometric interpretation of defect operatorsThe main reference for this section is <cit.> and also <cit.>. There it is explained how to use M-theory to construct vertex operators, line operators and surface operators by intersecting an M5 brane with an M2 one. In particular,* a vertex operator corresponds to a point in the physical space: it means that the remaining two dimensions of the M2 brane are wrapped on the Gaiotto surface S, forming a co-dimension 0 object.* a line operator corresponds to a one-dimensional object in the physical space: the remaining one dimension of the M2 brane is wrapped around the surface S as a 1-cycle (i.e. non self-intersecting closed loop) or as an arc connecting two punctures or marked points.* a surface operator has two spacial dimensions and thus it is represented by a puncture over the surface C.These operators are physically and geometrically related. A vertex operator, since it is a sub-variety of codimension 0 over the complex curve S, can be interpreted as connecting line operators (i.e the loops at the boundary of the sub-variety representing the vertex operator). Moreover, a line operator γ can be interpreted as acting on a surface operator x and transporting the point corresponding to the surface operator around the curve S; the action on its dual Liouville field (<cit.>) is the monodromy action along the line associated to the line operator γ. Finally, as explained in <cit.>, when we describe the curve S via a quadratic differential ϕ, we can interpret the poles of ϕ as surface operators and the arcs connecting marked points and the closed loops correspond to Verlinde line operators[ Traditionally, they are defined by composing a sequence of elementary operations on conformal blocks, each corresponding to a map between spaces of conformal blocks which may differ in the number or type of insertions. Roughly speaking, one inserts an identity operator into the original conformal block, splits into two conjugate chiral operators ϕ_a and ϕ̅_a, transports ϕ_a along γ and then fuses the operators ϕ_a and ϕ̅_a back to the identity channel.]. If we consider two curves γ_1, γ_2 that intersect at some point, the line operators L(γ_1,ζ),L(γ_2,ζ) corresponding to these curves do not commute (see section <ref>). In the case of self-intersecting arcs, we get more general Verlinde operators: as line operators, they can be decomposed into a linear combination of non self-intersecting line operators by splitting the intersection in all possible pairs.§.§ Punctures and tagged arcsThe main reference, for this short section, is <cit.>. In there it is pointed out how the arcs over the Gaiotto curve C are tagged arcs: at each singularity (extrema of the arc) we have to choose the eigenvalue of the monodromy operator around that singular point. In particular, in section 8 of <cit.>, we discover that for irregular singularity, the tagging is not necessary, since it is equal to an overall rotation of the marked points around that boundary component. For punctures, on the other hand, it is not the case: we have to specify a _2-tagging. This boils down to a tagged triangulation, as explained in <cit.>. Therefore, when considering surfaces with punctures, we have to consider tagged arcs and not simple arcs. This observation will be important when we discover that the S-duality group is the tagged mapping class group of the Gaiotto surface C (see section <ref>). §.§ Ideal triangulations surfaces and quivers with potential Let C be the Gaiotto curve of a class 𝒮[A_1] model. Theinvariant of the family of QFTs obtained by continuous deformations of it is the topological type of C. More precisely, we define the underlying topological surface S of C by the following steps: i) forget the complex structure, and ii) replace each irregular punture with a boundary component ∂ S_i with k_i≥ 1 marked points (ordinary punctures on C remain punctures on S). S is then the invariant datum which describes the continuous family of 𝒮[A_1] theories. An ideal triangulation of S is a maximal set of pairwise non-isotopic arcs ending in punctures and marked points which do not intersect (except at the end points) and are not homotopic to a boundary arc.[ A boundary arc is the part of a boundary component between two adjacent marked points.] All ideal triangulations have the same number of arcs <cit.>n=6g-6+3p+3b+∑_ik_i.Note that is the same number as the number of UV+IR deformations, eqn.(<ref>), as well as the number of IR conserved charges rank Λ, eqn.(<ref>).To an ideal triangulation T of the surface of S we associate a quiver with superpotential (Q,W).The association S↔ (Q,W) is intrinsic in the following sense: Let (Q,W) be the quiver with potential associated to an ideal triangulation T of the surface S. A quiver with potential (Q^',W^') is mutation equivalent to (Q,W) if and only if it[ This is slightly imprecise since, in presence of regular punctures W contains free parameters <cit.>. The statement in the text refers to the full family of allowed W's.] arises from an ideal triangulation T^' of the same surface S.In view of Corollary <ref>, an important result is: The quiver with superpotential of an ideal triangulation is always Jacobi-finite. Thus an ideal triangulation T defines a Jacobi-finite Ginzburg DG algebra Γ≡Γ(Q,W) and therefore also the three triangle categories D^bΓ, 𝔓𝔢𝔯 Γ and 𝒞(Γ) described in . 2.5, 2.6. Then Up to isomorphism, the three triangle categories D^bΓ, 𝔓𝔢𝔯 Γ and 𝒞(Γ) are independent of the chosen triangulation T, and hence are intrinsic properties of the topological surface S. It remains to describe the quiver with potential (Q,W) associated to the ideal triangulation T. The nodes of Q are in one-to-one correspondence with the arcs γ_i of T (their number being equal to the number of IR charges, as required for the BPS quiver of any 𝒩=2 theory). Giving the quiver, is equivalent to specifying its exchange matrix:For any triangle D in T={γ_i}_i=1^n which is not self-folded, we define a matrix B^D = (b^D)_ij ,1≤ i ≤ n,1≤ j≤ n as follows. * b^D_ij = 1 and b^D_ji = -1 in each of the following cases: * γ_i and γ_j are sides of D with γ_j following γ_i in the clockwise order;* γ_j is a radius in a self-folded triangle enclosed by a loop γ_l, and γ_i and γ_l are sides of D with γ_l following γ_i in the clockwise order;* γ_i is a radius in a self-folded triangle enclosed by a loop γ_i, and γ_l and γ_j are sides of D with γ_j following γ_l in the clockwise order; * b^D_ij = 0 otherwise.Then define the matrix B^T := (b_ij ),1≤ i ≤ n,1≤ j≤ n by b_ij =∑_D b^D_ij , where the sum is taken over all triangles in T that are not self-folded. The matrix B^T is a skew-symmetric matrix whose incidence graph is the quiver Q associated to the triangulation.The superpotential. The superpotential W is the sum of two parts. The first one is a sum over all internal triangles of T (that is, triangles having no side on a boundary component). The full subquiver over the three nodes of Q associated with an internal triangle of T has the form∙[dl]_α ∙[rr]_β∙[ul]_γSuch a triangle contributes a term γβα to W. The second part of W is a sum over the regular punctures. Let γ_1,γ_2⋯, γ_nbe the set of arcs ending at the puncture p taken in the clockwise order. The full subquiver of Q over the nodes corresponding to this set of arcs: it is an oriented n-cycle. The contribution to W from the puncture p is λ_p times the associated n-cycle, where λ_p≠0 is a complex coefficient. No regular puncture: gentle algebras. Suppose S has no regular puncture. Since an arc of T belongs to two triangles (which may be internal or not), at a node of Q end (start) at most 2 arrows.The superpotential is a sum over the internal triangles ∑_iγ_iβ_iα_i and the Jacobi relations are of the formthe arrows α, β arise from the same internal triangle ⟹ αβ=0.A finite-dimensional algebra whose quiver satisfies (<ref>) and whose relations have the form (<ref>) is called a gentle algebra <cit.>, a special case of a string algebra. Thus, in absence of regular punctures, the Jacobian algebra J(Q,W) is gentle. Indecomposable modules of a gentle algebra may be explicitly constructed in terms of string and band modules <cit.> (for a review in the physics literature see <cit.>). A gentle algebra is then automatically tame; in particular, the BPS particles are either hypermultiplets or vector multiplets, higher spin BPS particles being excluded. How to reduce the general case of a class 𝒮[A_1] theory to this gentle situation is explained in ref. <cit.>.We give here an example of the quiver associated to an ideal triangulation T, whose incidence matrix is B^T: §.§ Geometric representation of categories The main reference is <cit.>. There is a precise dictionary between curves over a decorated marked surface and the objects in the category DΓ. Then, let Δ be the set of decorated points: to each triangle of T, choose a point in the interior of that triangle. With respect to a quadratic differential ϕ of section <ref>, the decorated points correspond to the simple zeros of ϕ. The marked points, on the other hand, correspond to poles of ϕ of order m_i+2. Let S_Δ be the surface S with the decorated points. The basic correspondence between geometry and categories is, on the one hand, between objects in D(Γ) and curves over S_Δ and on the other hand between morphisms in D(Γ) and intersections between curves. Here follow the complete dictionary: * Recall that an object S∈ D^b (Γ) is spherical iffHom_D^bΓ(S,S[j])≅ k (δ_j,0+δ_j,3).A spherical object in the category D^b (Γ) corresponds to a simple[ A simple arc does not have self intersections.] closed[ A closed arc starts and ends in Δ.] arc (CA) between 2 points in Δ. In particular, simple objects are elements of the dual triangulation and are all spherical. These are some of the BPS hypermultiplets. We can describe these curves as elements of the relative homology with -coefficients H_1(S,Δ,ℤ) over the curve S and the set of points Δ: indeed, the operation of “sum” is well defined and it corresponds to the relation that defines the Grothendieck group K_0(D^bΓ). We now describe the map that associates a graded module (or equivalently a complex) to a closed arc. Consider a closed arc γ that is in minimal position with respect to the triangulation: every time γ intersects the triangulation, we add to the complex a simple shifted module S_i[j] and we connect it to the complex with a graded arrow: the grading of the map depends on how the curve and the decorated points are related. In particular, the grading – corresponding to the Ginzburg algebra grading – is defined in the following figure[ The figure is taken from <cit.>.]:We call X both the map X: CA(S_Δ) → D^b (Γ) and X:OA(S_Δ) →𝔓𝔢𝔯 Γ, where CA(S_Δ) are the arcs of the decorated surface S_Δ connecting at most two points in Δ and OA(S_Δ) are the arcs connecting marked points. Notice that for the open arc (OA) case, we also have to take into account the tagging at the punctures. In particular, the situation is the following: * For an open curve ending on a puncture inside a monogon: if the curve is not tagged, then it intersects only the monogon boundary; if the curve is tagged, then the curve intersect the ray inside the monogon.* For an open curve ending on a puncture that is not inside a monogon, then the untagged curve does intersect the curve it would intersect as if it were untagged; if the curve is tagged, then it is as if the curve made a little loop around the puncture and so changes the intersection.Let us consider the quiver A_2 again. Then, consider the curves γ_1, γ_2 and γ_3 as in the picture. By applying the rules above we get:X(γ_1):S_1 a=1→S_2≅ k1→k X(γ_2):S_1[-1] a^*=1←S_2≅ k[-1]1←k X(γ_3):S_2 t_1=1→S_2[-2]≅ 0 0→k⊕ k[-2] 1This map also works for the graded modules corresponding to closed loops: both those starting at the zeros of the quadratic differential and those not. As we will see in the example <ref> of the Kronecker quiver, only certain loops are in the category D^b Γ. All other possible loops belong to 𝔓𝔢𝔯 Γ. Moreover, even in the case with punctures, the algorithm to get (graded) modules form the curves is the same (thanks to proposition 4.4 of <cit.>).* Since every simple object is a spherical objects in D^b Γ – in particular, since D^b Γ is 3-CY, the simple objects are 3-spherical – we can define the Thomas-Seidel twist T_S associated to such a spherical object S by the following triangleHom_D^b Γ^∙(X,S)⊗ S → X → T_S(X) →.These twists are autoequivalences of DΓ. Geometrically, these spherical twists correspond to braid twists associated to the simple closed arc γ_S: let BT(S_Δ) denote the full group generated by all the braid twists. The action of the braid twist is like in figureMoreover, being T^* the dual triangulation of the triangulation T,CA(S_Δ)=BT(S_Δ)· T^*as shown in <cit.>.* In general, braid twists of S_Δ correspond to spherical twists of D^b Γ: BT(T_S_Δ)=Sph(D^b (Γ)). Moreover, the quiver representing the braid relations is exactly the quiver Q: to each vertex we associate a twist T_S_i and to each single arrow i → j a braid relation T_S_iT_S_jT_S_i=T_S_jT_S_iT_S_j. If there is no arrow between i and j, then [T_S_i,T_S_j]=0. * Rigid and reachable objects in 𝔓𝔢𝔯 Γ correspond to the simple open arcs, i.e. simple curves connecting marked points. The other objects in 𝔓𝔢𝔯 Γ correspond to generic arcs: both those arcs connecting two different punctures or marked points and closed loops encircling decorations or boundaries or punctures. We can describe these curves as elements of the relative homology H_1(S_Δ,M,ℤ) over the curve S_Δ (where the points in Δ are topological points in S_Δ) and the set of marked points M: indeed, the operation of “sum” is well defined and it corresponds to the relation that defines the Grothendieck group K_0(𝔓𝔢𝔯 Γ). * Let T be the triangulation of the surface. The arcs of the triangulations are associated to the Γ e_i objects in 𝔓𝔢𝔯 Γ and the elements of the dual triangulations are the simple objects in D^bΓ. This is the geometrical version of the simple-projective duality:Hom^j(Γ e_i,S_k[l])=δ_lj δ_ki.The choice of a heart in D^bΓ corresponds to the choice of the simple objects; thus, via the simple-projective duality it also corresponds to the choice of a triangulation T. The relation between the Grothendieck groups of D^bΓ and 𝔓𝔢𝔯 Γ is via the Euler form (which corresponds to the intersection form, as pointed out in item <ref>) and it corresponds to Poincaré duality of the relative homology groups. * Flips of the triangulation (forward and backward) correspond to right and left mutations μ_i^±. Two different flips of the same arc are connected by a braid twist assocaited to that simple closed arc: T_S_i=μ^+_i(μ_i^-)^-1.* Hom spaces correspond to intersection numbers:[ CA=closed arc, OA=open arc.] the full proof of the following facts can be found in <cit.>:Hom(X(CA),X(CA))=2 Int(CA,CA) (X(OA),X(CA))= Int(OA,CA)The intersection numbers between arcs in S_Δ are defined as follows:* For an open arc γ and any arc η, their intersection number is the geometric intersection number in S_Δ -M:Int(γ, η) = min{|γ^'∩η^'∩(S_Δ - M)||γ^'≅γ, η^'≅η}. * For two closed arcs α, β in CA(S_Δ), their intersection number is an half integer in 1/2ℤ and defined as follows:Int(α,β) = 1/2 Int_Δ (α,β) + Int_S-Δ(α,β),whereInt_S_Δ-Δ(α,β) = min{|α^'∩β^'∩S_Δ -Δ| | α≅α^', β≅β^'}andInt_Δ(α,β) = ∑_Z∈Δ|{t | α(t) = Z}| ·|{r| β(r) = Z}|.Let T_0 be a triangulation and η any arc; it is straightforward to see Int(T_0,η) ≥ 2 for a loop, and the equality holds if and only if η is contained within two triangles of T_0 (in this case, η encircles exactly one decorating point). [Sphere with three holes and one marked point per each boundary component.] In this particular case (the following results do not hold in general), the matrix corresponding to the bilinear form Int(-,-) can be obtained by taking the Cartan matrix of the quiver with potential (Q,W), i.e. the matrix whose columns are the dimensions of the projective modules, inverting and transposing it.In particular, the incidence matrix for a sphere with three boundary components with m_i=1, ∀ i ∈{1,2,3} – over the basis of simples (corresponding to the edges of the dual triangulation) – is 0.35! !<0cm,0cm>;<0.5cm,0cm>:<0cm,0.5cm>:: !(0,0) *+∙="b1" !(5,0) *+∙="b3" !(2.5,-2.5) *+∙="b2" !(-1,0) *+*<15pt>="b4" !(5.8,0.7) *+*<15pt>="b5" !(2.5,-3.6) *+*<15pt>="b6" !(2.5,-1.5) *+*<105pt>="center" "b1" -@[blue] "b2"^3 "b2" -@[blue] "b3"^2 "b3" -@[blue] "b1"_1 "b1" -@`(10,7.5),(9,-6)@[blue] "b3"^4 "b3" -@`(2.5,-12),(-2,-3)@[blue] "b2"^5 "b2"-@`(-11,0),(2,4)@[blue]"b1" ^6 Int=( [1/2 -1/21/2000;1/21/2 -1/2000; -1/21/21/2000; -1001/21/2 -1/2;0 -10 -1/21/21/2;00 -11/2 -1/21/2;])The Euler characteristic of D^b (Γ), as a bilinear form defined in <ref>, on the other hand, is an antisymmetric integral matrix that is the antisymmetric part of Int(-,-):χ(-,-)=( [01 -1 -100; -1010 -10;1 -1000 -1;1000 -11;01010 -1;001 -110;]). Notice that the skew-symmetric matrix we have just found corresponds to the matrix B^T associated to the ideal triangulation of figure <ref>. * Relations amongst the exchange graphs (EG) of the surface S (to each vertex of this graph we associate a triangulation and to each edge of the graph a flip of the triangulation) and the cluster exchange graph (CEG) of the cluster algebra (to each vertex of the graph we associate a cluster and to each edge a mutation):EG(S) = CEG(Γ) EG(D^b (Γ))/Sph(D^b (Γ))=CEG(Γ) * A distinguished triangle in D^b (Γ) corresponds to a contractible triangle in S_Δ whose edges are 3 closed arcs (α,β,η) as in the figure, such that the categorical triangle is X(α) → X(η) → X(β) →.Exploiting the group structure of the homology group H_1(S,Δ,ℤ), we see that we have the following relation:[α]-[η]+[β]=0;this is the defining relation of the Grothendieck group K_0(D^bΓ).* The triangulated structure of 𝔓𝔢𝔯 Γ is less easy to represent geometrically. Before proceeding with an example, we define the left and right mutations in 𝔓𝔢𝔯 Γ starting from the silting set. A silting set ℙ in a category D is an Ext^>0-configuration, i.e. a maximal collection of non-isomorphic indecomposables such that Ext^i(P, T) = 0 for any P, T ∈ℙ and integer i > 0. The forward mutation μ^-_P at an element P ∈ℙ is another silting set ℙ^-_P, obtained from ℙ by replacing P withP^- = Cone(P →⊕_T ∈ℙ {P}DHom_irr(P, T)⊗ T ),where Hom_irr(X, Y ) is the space of irreducible maps X → Y , in the additive subcategory 𝖺𝖽𝖽 ⊕_T ∈ℙ T of D. The backward mutation μ^+_P at an element P ∈ℙ is another silting set ℙ^+_P, obtained from ℙ by replacing P withP^+ = Cone( ⊕_T ∈ℙ {P}Hom_irr(T, P) ⊗ T → P)[-1]Notice that equations (<ref>) and (<ref>) are exactly the same as (<ref>) and (<ref>): the notation of the latter is more straightforward for the next computations. We give now an example to show how one can get the triangulated structure of per Γ and we relate it to the group structure of H_1(S_Δ,M,). [A_2 example] The silting set for the quiver A_2: ∙_1 →∙_2 is ℙ={Γ e_1,Γ e_2} which we denote as {Γ_1,Γ_2}. We apply the left mutation corresponding to a forward flip: it gives the following triangleΓ_2 →Γ_1 →Γ_2^-→ .The element Γ_2^- is an infinite complex of the form S_1 t_1→Γ_1[-2]. At a geometrical level it corresponds to the red curve in this figure:As one can see, these three open arcs do not seem to be geometrically easily related. The next step, which is fundamental for consistency of the geometric representation, is to consider the following triangle:Γ_1[-2]→Γ_2^- → S_1 →This triangle is exact and moreover S_1 ∈ D^bΓ. This implies that in the cluster category, both Γ_2^- and Γ_1[-2] map to the same curve. We can exploit the geometric effect of the shift (see item <ref>) to compute Γ_1[-2]: it is the green curve in the next figureWhen we map to the cluster category via the forgetful functor (see item <ref>), we have that both Γ_1[-2] and Γ_2^- are sent to the same object: S_2. Indeed, the corresponding cluster category is made of 5 indecomposables which form the following periodic AR diagram Γ_1 [dr]P_1[dr]Γ_2[ur]S_2 [ur] S_1 where we can read the corresponding triangles in the cluster category. The geometric picture is We can now generalize the above results by stating that the triangulated structure is consistent with the group structure of the relative homology group H_1(S_Δ,M,ℤ) paired with the relative homology of H_1(S,Δ,ℤ): indeed, we see that in H_1(S_Δ,M,ℤ) – after choosing opposite direction for the red and green path of figure <ref>– we have[Γ_1[-2]]-[Γ_2^-]+[S_1]=0.The relations defining the Grothendieck K_0(𝔓𝔢𝔯 Γ), such as[Γ_2]-[Γ_1]+[Γ_2^-]=0,are less obvious from a homological viewpoint: there is no other way but compute them explicitly when needed.*The Amiot quotient 𝔓𝔢𝔯 Γ/ D^b (Γ) <cit.> – through which the cluster category is defined – corresponds to the forgetful map F:S_Δ→ S. For a short reminder of triangulated quotients, see section <ref>. In particular, we recover easily the results of <cit.>: the indecomposable objects of the cluster categories are string modules or band modules. Geometrically a string is an open arc and the procedure to associated a module to it is the same as the one described in item <ref> for closed arcs. So indeed, since open curves are indecomposable objects of 𝔓𝔢𝔯 Γ, as pointed out in item <ref>, the following diagram – at least for string modules – commutes:𝒪A(S_Δ) [r]^F [d]^X OA(S) [d]^X 𝔓𝔢𝔯 Γ[r]^π𝒞(Γ) We expect no difference in the case of band modules (which correspond to families of loops). *In the cluster category 𝒞(Γ) and in the perfect derived category 𝔓𝔢𝔯 Γ, the shift [1] corresponds to a global anticlockwise rotation of all the marked points on each boundary component. For punctures, the action of [1] corresponds to a change in the tagging. In 𝒞(Γ), it is equivalent to the operation τ (the AR translation), as defined in <cit.>. In particular we see that in presence of only regular punctures, [2] flips twice the tagging getting back to the original situation, that is, in this case the cluster category is 2-periodic, as expected on physical grounds. Let us consider here a simple example that will allow us to clarify some aspects. [A_2 again]Let us consider the following curves over a disk with 5 marked points on the boundary and no punctures. The triangulation is made of the black lines 1 and 2 and the boundary arcs:The curves a and d correspond to graded modules in D^b Γ as shown in the example <ref>; indeed, the curve d corresponds to the graded module S_1 ← S_2[-1] in D^b Γ. The red curves b and c', on the other hand, cannot be associated to any closed curve, since they cannot intersect any closed curve in minimal position: they are elements only of 𝔓𝔢𝔯 Γ and not of D^b Γ. When we take the Amiot quotient, the green dots disappear and so do the curves a and d. Moreover, the curve c is homotopic to a boundary arc and b is homotopic to 2. Thus the quotient does what we expect: only the curves in 𝔓𝔢𝔯 Γ that are not in D^b Γ do not vanish.The intersection form in this example is Int= ( [1 -1;01;])and thus the Euler characteristic is χ= ( [01; -10;])We can thus construct the Thomas-Seidel twists associated to the simple modules:T_S_1=Id-|S_1⟩⟨S_1|·χ=( [1 -1;01;]) T_S_2=Id-|S_2⟩⟨S_2|·χ=( [ 1 0; 1 1; ])And we can explicitly check the braid relation T_S_1· T_S_2· T_S_1=T_S_2·T_S_1· T_S_2 both on K_0(D^b Γ)and geometrically. Moreover it is clear from the matrix representation that T_S_1 and T_S_2 generate SL_2(ℤ). We can also act with these twists on the graded modules:T_S_1· X(S_2)=X((-1,1))The graded module whose dimension vector is (-1,1) is k[-1] a^*=1←k. This is consistent with the geometric picture, as one can verify:[Kronecker quiver]The surface corresponding to the Kronecker quiver is an annulus with one marked point on each boundary component. This theory corresponds to a pure SU(2) SYM. Thus, we shall find closed curves corresponding to BPS vector bosons. They must also appear a one-parameter family. In the following figure, the green loops around the two points represent a band module, the black dots the marked points; the black lines are the flow lines associated to the quadratic differential ϕ described in section <ref>.The module corresponding to (one of) the green curve can be computed via the map X: a ↦ X(a) ∈ D^b Γ, starting at a generic point along the curve. We getX(a): S_1 a=1,b=λ⟹ S_2 ≅ k a=1,b=λ⟹ k .This module is stable (see section <ref>) since it is equivalent to the regular module in the homogeneous tube of -k 𝖪𝗋.§.§ SummaryWe give here a sketchy summary of what we have written so far; recall that, given a quadratic differential ϕ(z) dz⊗ dz there are three kind of markings: the zeros of ϕ (decorations), the simple poles of √(ϕ) (punctures) and the irregular singularities of ϕ, which generate the marked points on the boundary segments. * BPS vector multiplets correspond to loops, not crossing the separating arcs of the flow of ϕ, for θ= Z(BPS)=θ_c. They belong to the category D^b Γ via the maps X.* BPS hypermultiplets correspond to arcs connecting two zeros of ϕ. They belong to the category D^b Γ via the map X and can be identified with elements in the relative homology H_1(S,Δ,), where Δ are the zeros of ϕ.* Surface operators corresponds to punctures and marked points.* The objects in the perfect derived category 𝔓𝔢𝔯 Γ are the screening states created by line operators acting on the vacuum. They correspond to arcs connecting marked points and punctures and thus belong to H_1(S_Δ,M,).* UV line operators correspond to arcs over S connecting marked points and punctures (but not zeros of ϕ).They belong to the cluster category 𝒞(Γ) and can also be identified with elements of the relative homology H_1(S,M,ℤ), where M is the set of markings of the surface S. With this dictionary in mind, we can now exploit the categorical language to compute physical quantities (such as vacuum expectation values of UV line operators). A generalization of these concepts, in particular towards ideal webs and dimer models can be found in <cit.>. There it is argued that the 3-CY category D^bΓ is the physical BPS states category, in accordance with our analysis. Moreover, in the case in which we have a geometrical interpretation via bipartite graphs, the mapping class group of the punctured surface, is a subgroup of the full S-duality group. Here we give a simple example. [Pure SU(3)] The pure SU(3) theory can be described as a S[A_2] Gaiotto theory over a cylinder with one full punctures per each boundary <cit.>. The mapping class group is generated by a single Dehn twist around the cylinder. It acts on the bipartite graph associated to the SU(3) theory and it must be a subgroup of the full S-duality group. It is isomorphic to ℤ. Since we also have that pure SU(3) theory can be described by the quiver𝖪𝗋⊠ A_2,We explicitly find – using he techniques of <cit.> – that a cluster automorphism is given by τ_𝖪𝗋⊗τ_A_2, which generates a free group, thus isomorphic to ℤ.§ CLUSTER CHARACTERS AND LINE OPERATORS§.§ Quick review of line operators The main reference for this part is <cit.>. We are going to study, in the following section, the algebra of line operators: we shall discover that this algebra is closely related to the cluster algebra of Fomin and Zelevinski <cit.>. Recall that an IR line operator (also called framed BPS state <cit.>) is characterize by a central charge ζ and a charge γ. Similarly for a UV line operator. The starting point is to consider the RG flow:RG(·,α,ζ):{UVlineoperators}→{IRlineoperators}L(α,ζ) ↦∑_γ∈ΓΩ̅(α,ζ,γ,u,y) L(γ,ζ),where L(α,ζ) is a supersymmetric UV line operator of UV charge α (see <ref>). We can think of it as a supersymmetric Wilson line operator:L(α,ζ):=exp(i α∫_time A+1/2(ζ^-1ϕ +ζϕ̅)).where A is the gauge connection and ϕ and ϕ̅ are the supersymmetric partners. The idea is that cluster characters provide the coefficients Ω̅(α,ζ,γ,u,y); moreover, the OPE's of line operators can be identified with the cluster exchange relations. The physical definition of Ω̅(α,ζ,γ,u,y) as supersymmetric index is the following. Define the Hilbert space of our system with a line operator in it polarizing the vacuum: H_L,ζ,u=⊕_γ∈Γ_uH_L,ζ,u,γ, where Γ_u is the charge lattice and u a point in the Coulomb branch. When we restrict only to BPS states we have H^BPS_L,ζ,u. We now define the following index (i.e. a number that counts the line operators):Ω̅(α,ζ,γ,u,y)=Tr_H_L,ζ,u,γ(y^2J_3(-y)^2I_3),where the I and J operators are the Cartan generators associated to the unbroken Lorentz symmetry SO(3) and unbroken R-symmetry SU(2)_R by the presence of the line operator which moves along a straight line in the time direction. In particular, if y=1:Ω̅(α,ζ,γ,u,1)=Tr_H_L,ζ,u,γ(1^2J_3(-1)^2I_3)=∑_m1^2m(-1)^0= H_L,ζ,u,γ^BPS.This index corresponds to the Poincaré polynomial of the quiver Grassmannian Gr_γ(α), where we interpret the UV line operator L of charge α as the quiver representation of which we compute the cluster character (see <ref> for more details). §.§.§ Algebra of UV line operatorsLet the OPE's of UV line operators be defined as followsL(α, ζ)L(α',ζ)=∑_β c(α,α',β)L(β,ζ).From now on, let us fix the generic point of the Coulomb branch u. We also define the generating functions for the indexes Ω̅(L, γ)= Ω̅(L,γ,u=fixed,y=1):F(L)=∑_γΩ̅(L,γ)X_γ,where the formal variable X_γ is such that X_γ X_γ'=X_γ+γ'. One can check that F(LL')=F(L)F(L'). This equality gives a recursive formula to compute F(LL'). Furthermore, we can study the wall-crossing of UV line operators via the formula of KS <cit.>:F^+(L_γ_c)=F^-(L_γ_c)∏_γ∏_m=-M_γ^M_γ(1+(-1)^mX_γ)^|γ,γ_c|a_m,γwhere γ_c is the charge of the wall we are crossing, and M_γ is the maximal value of the operator 𝒥_3=J_2+I_3 and the a_m,γ are the coefficients of the index:Ω̅(α, ζ,γ,u,y)=∑_m=-M_γ^M_γ a_m,γy^m.We can transfer this formula on the newly defined variables to implement the wall-crossing more efficiently: X_γ'=X_γ∏_γ∏_m=-M_γ^M_γ(1+(-1)^mX_γ)^γ,γ_ca_m,γ.Moreover, the transformation of the indices Ω̅(L, γ) is an automorphism of the OPE algebra. Thus, the algebra obeyed by the generating functionals is in fact an invariant of the UV sector theory. The properties of the KS wall-crossing formula of <ref> and the fact that these generating functionals are invariants of the UV theory (i.e. of the cluster category C(Γ)), tell us that F(L) are exactly the cluster character. The non-commutative generalization is done via the star product <cit.>:L(α, ζ)*_yL(α',ζ)=∑_β c(α,α',β,y)L(β,ζ)and X_γ*_yX_γ'=y^γ,γ'_DX_γ+γ'. We define again the generating functions F(L)=∑_γΩ̅(L,γ,y)X_γ, and then verify that F(L*_yL')=F(L)*_yF(L').Indeed we discover that the non-commutative version of F behaves exactly like a quantum cluster character (usual cluster characters are obtained by setting y=1). We can thus analyze these objects from a purely algebraic point of view and study cluster algebras and cluster characters: we shall do this is the next section.§.§ Cluster charactersThe main references for this section are <cit.>. We begin by recalling some basic definitions and properties of cluster characters. Let 𝒞:=C(Γ) be a cluster category.A cluster character on 𝒞 with values in a commutative ring A is a mapX : obj(𝒞) → Asuch that * for all isomorphic objects L and M, we have X(L) = X(M),* for all objects L and M of 𝒞, we have X(L ⊕ M) = X(L)X(M),* for all objects L and M of 𝒞 such that Ext^1_𝒞(L, M) = 1, we haveX(L)X(M) = X(B) + X(B'),where B and B' are the middle terms of the non-split trianglesL →B → M → andM → B^'→L →with end terms L and M.[ If B^' does not exist, then X(B^')=1.] Let T=⊕_i T_i be a cluster tilting object, and let B=Ent_𝒞 T. The functorF_T:𝒞→-B, X ↦Hom(T,X) is the projection functor that induces an equivalence between 𝒞/𝖺𝖽𝖽 T[1] →𝗆𝗈𝖽-B. Then the Caldero-Chapoton map<cit.>, X^T_?: ind 𝒞→ℚ(x_1, . . . , x_n)is given byX^T_M ={[x_i ifM ≅Σ T_i; ∑_e χ(Gr_e F_TM)∏^n_i=1 x_i^S_i,e_D-S_i,FMelse, ].where the summation is over the isoclasses of submodules[ Recall that a module N is a submodule of M iff there exists an injective map N → M.] of M and S_i are the simple B-modules. Moreover, the Euler form and Dirac form in the formula above, are those of 𝗆𝗈𝖽-B. We now recall some properties of quiver Grassmannians[ Gr_e(FM):={N ⊂ FM|N =e}, i.e. it is the space of subrepresentations of M with fixed dimension e.] and in particular their Euler Poincaré characteristic χ (with respect to the étale cohomology).Let Λ be a finite dimensional basic ℂ-algebra. For a Λ-module M we define the F-polynomial to be the generating function for the Euler characteristic of all possible quiver grassmanians, i.e.F_M := ∑_eχ(Gr_e(M))y^e ∈ℤ[y_1, . . . , y_n]where the sum runs over all possible dimension vectors of submodules of M.Moreover, we assume that S_1, . . . , S_n is a complete system of representatives of the simple Λ-modules, and we identify the classes [S_i] ∈ K_0(Λ) with the natural basis of ℤ^n. Let Λ be a finite dimensional basic ℂ-algebra. Then the following holds: * If0 → L i→ M π→ N → 0is an Auslander-Reiten sequence in Λ-mod, thenF_LF_N = F_M + y^ N . * For the indecomposable projective Λ-module P_i with top S_i we haveF_P_i = F_radP_i + y^ P_ifor i = 1, . . . , n.* For the indecomposable injective Λ-module I_j module with socle S_j we haveF_I_j = y_jF_I_j/S_j + 1for j = 1, 2, . . . , n. The recursive relations of cluster characters and F-polynomials are the key tools to find a computational recipe: the next section is devoted to pointing out this algorithm. All aspects will be clarified in Example <ref>. §.§.§ Computing cluster characters The best way to compute cluster characters, is to exploit the results in <cit.>. The idea is to associate a Laurent polynomial to a path in the quiver. If the algebra is gentle, to a path we can associate a string module: computing the cluster character associated to string modules (up to an overall monomial factor) becomes a simple combinatorics problem. For any locally finite quiver Q, we define a family of matrices with coefficients in ℤ[x_Q] = ℤ[x_i|i ∈ Q_0] as follows. For any arrow β∈ Q_1, we setA(β) :=[[ x_t(β)0;1 x_s(β) ]]and A(β^-1) :=[[ x_t(β)1;0 x_s(β) ]].Let c = c_1 ... c_n be a walk of length n ≥ 1 in Q. For any i ∈{0, . . . , n} we setv_i+1 = t(c_i)(still with the notation c_0 = e_s(c)) andV_c(i): = [[ ∏_α∈ Q_1(v_i,-), α≠ c_i^± 1,c_i-1^± 1x_t(α) 0; 0 ∏_α∈ Q_1(-,v_i), α≠ c_i^± 1,c_i-1^± 1x_s(α) ]].We then setL_c =1/x_v_1... x_v_n+1[1, 1]V_c(1) ∏_i=1^nA(c_i)V_c(i + 1) [[ 1; 1 ]] ∈ℒ (x_Q).If c = e_i is a walk of length 0 at a point i, we similarly setV_e_i(1): = [[ ∏_α∈ Q_1(v_i,-)x_t(α) 0; 0 ∏_α∈ Q_1(-,v_i)x_s(α) ]].andL_e_i =1/x_i[1, 1]V_e_i[[ 1; 1 ]] ∈ℒ (x_Q).In other words, if c is any walk, either of length zero, or of the form c = c_1 ... c_n, we haveL_c =1/∏^n_i=0 x_t(c_i)[1, 1]∏^n_i=0A(c_i)V_c(i + 1)[[ 1; 1 ]] ∈ℒ (x_Q).with the convention that A(c_0) is the identity matrix. In general, we have the following result:X_M=1/x^n_ML_c,where M is the string module associated to the path c and the monomial x^n_M is the normalization coefficient. Let us consider the cluster category of A_4: its AR quiver is the followingWe have also made the choice of tilting objects. The algebra EndT is given by the following quiver:[ The vertices are the T_i and the arrows j → i correspond to Hom_𝒞(T_i,T_j).] with relations βα = γβ = αγ = 0. The Dirac form is the following:-,-_D=( [0100; -10 -11;010 -1;0 -110;]),whereas the Euler form is a,b:=Hom_𝒞(a,b)-Hom_𝒞(a,b[1]). Consider the B-module F_TM=(1,1,0,0). Its submodules are 0, S_1 and F_TM itself. The corresponding path is just the arrow c:2 → 1. By applying the formulas above we get:L_c = 1/x_1x_2[1,1] A(c_0)· V_c(1)· A(c)· V_c(2)[[ 1; 1 ]]= 1/x_1x_2[1,1] Id[[ x_3 0; 0 x_4 ]] [[ x_1 1; 0 x_2 ]] [[ 1 0; 0 1 ]][[ 1; 1 ]]= x_1x_3+x_4+x_2x_4/x_1x_2.Notice that the denominator is exactly x^ FM: this is a general feature for the decategorification process <cit.>. Moreover, we know that the Euler characteristic of a point is 1 and thus χ(Gr_0F_TM)=1=χ(Gr_F_TMF_TM). We then exploit the AR sequence0 → S_1 → F_TM → S_2 → 0and get the recursive relationF_S_1F_S_2=F_F_TM+y_2,which is equivalent to the following polynomial equation:1+y_1 χ_S_1+y_2 χ_S_2+y_1y_2 χ_S_1χ_S_2=1+χ_S_1y_1+χ_FMy_1y_2+y_2,which implies that χ_S_1=χ_S_2=1 and this is consistent with the previous result. One can check this and many other computations using appendix <ref>. One final remark is needed: we could have computed the cluster characters by a sequence of mutations of the standard seed of the cluster algebra associated to the quiver of B=End T. For the non-commutative case, i.e. whenx^α x^β=q^1/2α,β_D x^α+β,this procedure is the only one we know to compute quantum cluster characters. From the physics point of view, this is the important quantity: since cluster variables behave like UV line operators, they must satisfy the same non-commutative algebra. §.§ Cluster characters and vev's of UV line operatorsLet us start by recalling how the vacuum expectation values of line operators are computed in <cit.>. The idea is associate to a loop over a punctured Gaiotto surface a product of matrices. In the case of irregular singularities, since these singularities can be understood as coming from a collision of punctures, loops can get pinched and become laminations. Thus to both string modules (associated to laminations) and band modules (associated to loops), we can associate a rational function in some shear variables Y_i. Their expression turns out to be equal to cluster characters: for string modules we can use section <ref>, whereas for band modules we can use the bangle basis of <cit.> and the multiplication formula or the Galois covering technique of <cit.>. We shall now give some detailed examples in which we apply what we just described. [A_2 quiver] The computations of <cit.> of the vev's of the UV line operators can be found in section 10.1. They have been made using the “traffic rule”. The idea is to follow the lamination path and create a sequence of matrices according to the traffic rule. In the end, one takes the trace of the product of matrices (loop case) or contract the product of matrices with special vectors (open arcs case). For example,The matrix product is the following:L_1=(B_R· R· M_Y_2· L· E_R) (B_R· R· M_Y_2· L· E_R),where the matrices areL=([ 1 1; 0 1 ]) , R=([ 1 0; 1 1 ]) , M_X=([ √(X)0;0 1/√(X) ]),and the vectors areB_R=(10),B_L=(0 1),E_L=(10)^t, E_R=(01).Then we get:R.M_Y_2.L= ( [√(Y_2)√(Y_2);√(Y_2) √(Y_2)+1/√(Y_2); ]) ,and finallyL_1=√(Y_2)√(Y_2)=Y_2.The other four line operators, corresponding to the four remaining indecomposable objects of the cluster category of 𝒞(Γ_A_2) (or equivalently the remaining four cluster variables) are:L_2= Y_1 + Y_2Y_1,L_3=1/Y_2+Y_1/Y_2+ Y_1, L_4 = 1/Y_2+1/Y_2Y_1,L_5 =1/Y_1.On the other hand, the cluster characters of A_2 are:x_1, x_2, 1/x_2+x_1/x_2, 1/x_1+x_2/x_1,1/x_1x_2+1/x_1+1/x_2.The following map (Y_1,Y_2) ↦ (1/x_2,x_1) transforms one set to the other. This map is the tropicalization map of Fock and Goncharov <cit.>:Y_i=∏_jx_j^B_ij.Notice that this result was expected from the general algebraic properties of the line operators algebra and the cluster algebras.We now proceed to a more interesting example: the pure SU(2) theory. The computations of section <ref> has to be modified a bit: as we will see, it is convenient to exploit the Galois covering techniques of <cit.>. [Kronecker quiver] Let us focus our attention to the non rigid modules, i.e. those belonging to the homogeneous tubes of the AR quiver of the cluster category C(Γ_Kr). There is a ℙ^1 family of these modules and, amongst them, two of them are string modules (those of the form 11λ=0⇒ 2). By the theorems of <cit.>, the cluster characters do not depend of the value of the parameter λ and we are thus free to choose the simplest one to compute the character. Geometrically, this family of modules corresponds to loops around the cylinder (see figure <ref>). With the traffic rule techniques – with the slight modification of taking the trace instead of using the B and E vectors – we compute the vev of the line operator whose e.m. charge is (1,1):L_(1,1)=√(Y_1Y_2)+√(Y_1/Y_2)+1/√(Y_1Y_2).We can reproduce this result using cluster characters. The only observation is that we cannot simply use section <ref> to compute the character associated to the module dim M=(1,1): we have a path ambiguity. We thus have to construct a _2 Galois cover <cit.>, compute the character on the cover, and then project it down to the Kr quiver. The reason is that the Kronecker quiver has a double arrow and we have to be able to specify the path we follow uniquely. On the _2 cover the ambiguity is lifted and the character can be computed. The _2 cover is: ∙_3∙_2'∙_1[ur]_c [dr]∙_2 [ul] [dl] π→∙_4∙_1' @=>[uu]where the covering map π sends 1,2 ↦ 1^' and 3,4 ↦ 2'. The character corresponding to the string c with respect to the covering quiver is1+x_1x_2+x_3x_4/x_1x_3Therefore, if we identify the cluster variables following the covering map π, we get1+x_1'^2+x_2'^2/x_1'x_2'.This result is consistent with what we find in literature (e.g. <cit.>). Also in this case, we find that the tropicalization map [ Id est Y_i=∏_jx_j^B_ij.] sends the rational function <ref> to <ref>:(Y_1,Y_2) ↦ (x_2^-2,x_1^2). In this final example we show how to compute the cluster character associated to a band module in a more complicated quiver. [SU(2) with N_f=4] We are interested in the module M corresponding to the purple loop in the following figureUsing the traffic rule, it is rather straightforward to compute the VEV of the line operator associated to the module M. The result – which is similar to the ones computed in <cit.> – is:L_M=Tr(L· M_Y_1· R· M_Y_2· R.M_Y_4· L· M_Y_5)=1 + Y_4 + Y_2 Y_4 + Y_4 Y_5 + Y_2 Y_4 Y_5 + Y_1 Y_2 Y_4 Y_5/√(Y_1Y_2Y_4Y_5)The cluster character computation is more involved than the simple application of section <ref>: we exploit the techniques of <cit.>. The idea is similar to the traffic rule: we find a path over the hexagonal graph of <cit.> that is homotopic to the path considered. Then, to each edge of the hexagonal graph we associate a matrix with the following rule:The matrices D and F are:D(x)=( [0x; -1/x0;]), F(x_α/x_β x_γ)=( [ 1 0; x_α/x_β x_γ 1; ]).In our example we find:Tr(D(x_1)F(x_3/x_1x_2)F(x_3/x_4x_2)D(x_4)F^-1(x_6/x_4x_5)F^-1(x_6/x_1x_5)) =x_2x_5x_1^2+x_2x_5x_4^2+x_3x_6x_1^2+x_3x_6x_4^2+2 x_1x_4x_3x_6/x_1x_2x_4x_5.Again we can check that the result (<ref>) is the tropicalization of (<ref>). § ACKNOWLEGMENTSWe have benefit from discussions with Michele Del Zotto, Dirk Kussin and Pierre-Guy Plamondon. SC thanks the Simons Center for Geometry and Physics, where this work was completed, for hospitality. § CODE FOR CLUSTER CHARACTERS This is a short Mathematica code that computes the L_c polynomials of section <ref>. Up to an overall normalization factor, the L_c polynomials are the cluster characters. The algorithm follows precisely the procedure described in section <ref>. § CODE FOR CLUSTER AUTOMORPHISMS This short Mathematica script is useful to find generators and relations for the automorphisms of the cluster exchange graph. The formulas used to implement the mutations for the exchange matrix B_ij and the dimension vectors d_l (where l is an index that runs over the nodes) are the following:μ_k(B)_ij = - B_ij, i=kor j=kB_ij+max[-B_ik,0] B_kj+B_ik max[B_kj,0]—– otherwise. μ_k(d)_l ={[ d_l, l≠ k; -d_k+max[∑_imax[B_ik,0]d_i,∑_imax[-B_ik,0]d_i]l=k ]. The procedure of this script is explained in section <ref>. § WEYL GROUP OF E_6 With this short Mathematica script, we explicitly construct the Weyl group of E_6 over the basis of simple roots.We directly checked that the longest elements has length 36, that the order of the Weyl group is51840=2^7 3^45and the order of each element belongs to this set:{1,3,2,5,4,6,12,8,10,9}. 10alday2012wilson L. F. Alday. Wilson loops in supersymmetric gauge theories. Lecture Notes, CERN Winter School on Supergravity, Strings, and Gauge Theory, 2012.alday2010loop Luis F. Alday, Davide Gaiotto, Sergei Gukov, Yuji Tachikawa, and Herman Verlinde. Loop and surface operators in gauge theory and Liouville modular geometry. Journal of High Energy Physics, 2010(1):1–50, 2010.alim2013bps Murad Alim, Sergio Cecotti, Clay Córdova, Sam Espahbodi, Ashwin Rastogi, and Cumrun Vafa. BPS Quivers and Spectra of Complete N=2 Quantum Field Theories. 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"authors": [
"Matteo Caorsi",
"Sergio Cecotti"
],
"categories": [
"hep-th",
"math-ph",
"math.MP",
"math.RT"
],
"primary_category": "hep-th",
"published": "20170727180321",
"title": "Categorical Webs and $S$-duality in 4d $\\mathcal{N}=2$ QFT"
} |
^1Department of Physics, University of Konstanz, Germany^2Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8., H-111 Budapest, Hungary^3 Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary ^4Center for Computational Materials Science, Institute for Applied Physics, Vienna University of Technology, Wiedner Hauptstrasse 8, A-1060, Vienna, Austria^5MTA-BME Condensed Matter Research Group, Budafoki út 8., H-111 Budapest, Hungary Based on a multi-scale calculations, combining ab-initio methods with spin dynamics simulations, we perform a detailed study of the magnetic behavior ofNi_2MnAl/Febilayers. Our simulations show that such a bilayerexhibits a small exchange bias effect when the Ni_2MnAl Heusler alloy is in a disordered B2 phase. Additionally, we present an effective way to control the magnetic structure of theNi_2MnAl antiferromagnet, in the pseudo-ordered B2-I as well as the disordered B2 phases, via a spin-flop coupling to the Fe layer.75.50.Ss, 75.60.Jk, 75.70.Cn, 75.30.GwInterfacial exchange interactions and magnetism of Ni_2MnAl/Fe bilayers R. Yanes^1, E. Simon^2, S. Keller^1, B. Nagyfalusi ^3, S. Khmelevsky^4, L. Szunyogh^2,5and U. Nowak^1 December 30, 2023 =========================================================================================================== § INTRODUCTION Antiferromagnets build a class of materials which is used in magnetic multilayer devices, such as GMR sensors or magnetic tunnel junctions, to stabilize and control the magnetization of a ferromagnetic compound. This fact has increased the demand of antiferromagnets and it has led to an increasing interest in novel antiferromagnetic materials, with Heusler alloys as promising candidates for that <cit.> . Heusler alloys are ternary inter-metallic compounds with the general formula X_2YZ, in which X and Y are typically transition metals and Z is an main group element. This kind of alloys has been in the center of intensive studies in the last decades, mainly due to the wide range of their multifunctional properties. These include magnetic shape memory effects, magneto-caloric and spintronic effects, as well as thermoelectric properties amongst others. <cit.>Heusler alloys can be categorized in two distinct groups by their crystalline structures: half Heusler alloys with the form of XYZ in the C_1b structure and full Heusler alloys with the form of X_2YZ in the L2_1 structure <cit.>. The unit cell of the L2_1 structure consists of four interpenetrating face-centered cubic (fcc) lattices, while that of the C_1b structure is formed by removing one of the X sites. The L2_1 structuretransforms into the so-called disordered B2 phase when the Y and Z atoms are mixed, replacing each other at random.The majority of magnetic Heusler alloys are ferromagnetic though it has been reported that some of them are ferrimagnets or even antiferromagnets. In particular, those compounds with 3d elements where onlythe Mn atoms carry magnetic moments at Y site are antiferromagnets in the disordered B2 phase. In this context Ni_2MnAl is especially interesting since it has been reported to exhibit an antiferromagneticbehavior in the disordered B2 phase <cit.> as well as in the pseudo-order phase B2-I <cit.>. The latter is certain limit of the disordered B2 phase, where all the Mn atoms are located in the same (001) plane. Ni_2MnAl in the disordered B2 phase has a perfectly compensated antiferromagnetic ground state, where the Al and Ni atoms posses no net magnetic moment and the site and anti-site Mn atoms are equivalent <cit.>. Ni_2MnAl has been in the center of former studies regardingshape memory applications <cit.> because of its capability to change its magnetic order along with its chemical order. The existence of exchange bias (EB) was reported for Heusler alloys which undergomartensitic phase transitions<cit.>, for Ru_2MnGe/Fe bilayers <cit.> and, recently, Tsuchiya et al. published an experimental study of EB in Ni_2MnAl/Fe bilayers <cit.>. In general, EB is related to the coupling between a ferromagnet (FM) and an antiferromagnet (AF) and its strength depends on the exchange interaction across the interface and the stability provided by the AF. This calls for a detailed study of the interfacial exchange interactions in Ni_2MnAl/Fe bilayers.This paper is organized as follows: first we introduce a spin model which is based on first-principles calculations. Then, we analyze the exchange interactionsin the bulk and across the interface for the Ni_2MnAl(B2-I; B2)/Fe interfaces. In the next section, we present spin-dynamics simulations and analyze the possibility to control the magnetic state of the Ni_2MnAl layer via the Fe layer. We finish with a discussion of the origin of a small in-plane EB found in the Ni_2MnAl(B2)/Fe system.§ MODEL AND NUMERICAL APPROACH In the following we study the magnetic properties of Ni_2MnAl(B2-I)/Fe andNi_2MnAl(B2)/Fe interfaces in the spirit of a multi-scale model, linking ab initio calculations with dynamical spin model simulations. In terms of the fully relativistic Screened Korringa-Kohn-Rostoker (SKKR) Green's function method <cit.> we perform self-consistent calculations of the Ni_2MnAl/Fe bilayers in the disordered local magnetic moment approach <cit.>. We used the general gradient approximation (GGA) <cit.> in connection with the atomic sphere approximation and an angular momentum cut-off of l_max=3. We derive the exchange interactions between the magnetic moments by using the spin-cluster expansion (SCE) technique <cit.> that provides a systematic parametrization of the adiabatic energy of an itinerant magnetic system. Combining this method with the relativistic disordered local moment (RDLM) scheme <cit.>, the parameters of the spin-Hamiltonian below can be determined on a quite general level <cit.>. It is important to note that, due to the relativistic spin-orbit coupling, the exchange interactions between two spins form a 3 × 3 matrix. Furthermore, since the RDLM-SCE scheme relies on the paramagnetic state as reference, a priori knowledge of the magnetic ground state is not required, which makes it suitable for interface calculations. The magnetic properties of our system are well described by the following generalized spin model,ℋ=-1/2∑_i,js⃗_iJ_ijs⃗_j -∑_is⃗_iK_is⃗_i-∑_iμ_iH⃗_As⃗_i,where the s⃗_i represent classical spins, i. e. unit vectors along the direction of each magnetic moment at sites i.The first term stands for the exchange contribution to the energy, with J_ij denoting the tensorial exchange interaction between moment i and j. The second term comprises the on-site anisotropy as well as the magneto-static energy, where K_i is called the anisotropy matrix.In the presence of an external magnetic field, H⃗_A, the last term adds a Zeeman contribution to the Hamiltonian, where μ_i is the magnetic moment of the atom i.The exchange tenors J_ij can be further decomposed into three parts,J_ij=J_ij^isoI+J_ij^S+J_ij^A <cit.>, with the isotropic exchange interaction J_ij^iso=1/3Tr[J_ij],the traceless symmetric (anisotropic) part J_ij^S=1/2(J_ij+J_ij^T)-J_ij^isoI, and the antisymmetric part J_ij^A=1/2(J_ij-J_ij^T). The latter one is clearly related to the Dzyaloshinskii-Moriya (DM) interaction, s⃗_iJ_ij^As⃗_j=D⃗_ij·(s⃗_i×s⃗_j),with the DM vector D⃗_ij.The DM interaction arises due to spin-orbit coupling and favors a perpendicular alignment of the spins s⃗_i and s⃗_j <cit.>.Our first principle calculations show that the Nickel as well as the Alumina atoms have negligible magnetic moments in both phases of the Ni_2MnAl compound, the pseudo-ordered B2-I phase as well as the disordered B2 phase. Therefore we restrict our spin dynamics analysis to the evolution of Fe and Mn moments only.To study ground state properties along with spin dynamics at zero and finite temperatures we solve the stochastic Landau-Lifshitz-Gilbert (SLLG) equation,∂s⃗_i/∂ t= -γ/(1+α^2)μ_ s s⃗_i ×H⃗_i -γα/(1+α^2)μ_ ss⃗_i ×(s⃗_i ×H⃗_i),by means of Langevin dynamics, using a Heun algorithm<cit.>. The SLLG equation includes the gyromagnetic ratio γ, a phenomenological damping parameter, α, and the effective fieldH⃗_i = ζ⃗_i(t) - ∂ℋ/∂s⃗_i,which considers also the influence of a temperature T by adding a stochastic noise term ζ⃗_i(t), obeying the properties of white noise<cit.> with⟨ζ⃗_i(t)⟩=0, ⟨ζ_i^η(t)ζ_j^θ(t')⟩=2k_BTαμ_s/γδ_ijδ_ηθδ(t-t').Here i, j denote lattice sites and η and θ Cartesian components of the stochastic noise. § RESULTS AND DISCUSSIONS §.§ Ab initio resultsFor the two cases investigated in this work, the Ni_2MnAl(B2-I)/FeandNi_2MnAl(B2)/Fe bilayer, we first calculated the exchange interactions, the magnetic moment and the on-site anisotropy layered resolvedwith the methods described above.In Fig.<ref> the isotropic contribution of the exchange interaction between Mn-Mn and Mn-Fe neighbors are presented as a function of the distance between spin pairs. For the isotropic Mn-Mn exchange interactions our results indicate a similar behavior for the pseudo-orderedB2-I and the disordered B2 phase. The dominant nearest neighbor Mn-Mn exchange interaction, J_1, Mn-Mn≈-15meV, supports antiferromagnetic order while the magnitude of the exchange interactions between Mn atoms in successive shells decay rapidly (Fig. <ref>(a)). The exchange interactions between Mn and Fe atoms across the interface are plotted in Fig. <ref>(b).The dominant Mn-Fe exchange interaction is again the nearest neighbor one, favoring antiferromagnetic alignment. Remarkably, this interaction is even larger in magnitude than the nearest neighbor Mn-Mn interaction in the bulk. It should also be mentioned that the magnitude of the nearest neighbor exchange interaction in bulk Fe,J_1, Fe-Fe≈ 50meV, is again much larger in magnitude than the above interactions. A summary of the most relevant isotropic exchange parameters is given in Table <ref>.Another important parameter, which influences the magnetic behavior of a magnetic bilayer and which can lead to the existence of exchange bias is the magnetic anisotropy energy (MAE). It has been reported <cit.> that bulk Ni_2MnAl(B2-I) has a small in-plane anisotropy with a magnitude of0.19meV per spin, while in the case of the perfectly disordered B2 phase the MAE is negligible. Close to the AF/FMinterface, however, the magnetic anisotropy is modified. In case of the Ni_2MnAl(B2-I)/Fe interface the preferred magnetic orientation is in-plane with an energy of 0.03 meV in the interface Mn layer and 0.06 meV in the Fe layer. Similarly, an easy plane anisotropy was determined for the Ni_2MnAl(B2)/Fe interface, with a MAE of0.05 meV and 0.10 meV in the interface Mn and Fe layers, respectively.§.§ Spin dynamics simulations For our spin-dynamics simulations we use the model parameters as determined above from first principles. We suppose the Ni and Al atoms to be nonmagnetic and only consider the dynamics of the Mn and Fe moments.The antiferromagnetis hence modeled by the Mn sub-lattice, forming in total 30×30× t_AF unit cells and the ferromagnet by 30×30× 3 unit cells, t_AF denoting the number of Ni_2MnAl atomic monolayers perpendicular to the interface (in the following labeled [Ni_2MnAl(B2I;B2)]_t_AF/[Fe]_3). We consider open boundary conditions. For the case of the disordered B2 phase, the Mn atoms are statistically distributed. The magnitudes of the magnetic moments of Mn and Fe atoms were taken uniformly in the sample using the values given in Table <ref>. Additionally we approximate the effects of the magneto-static interaction in the FM layer as an uniaxial shape anisotropy with K_Fe=-0.134 meV and the magnetic hard axis perpendicularto the FM/AFM interface. In the following sections we will analyze the magnetic properties of the two types of bilayers described above. We evaluate the in-plane hysteresis loops and explore the existence of EB and the switching of the magnetic structure of the Ni_2MnAl layer. §.§.§ Hysteresis in the pseudo-ordered Ni_2MnAl(B2-I)/Fe bilayer To study the magnetic behavior of this bilayer we calculate hysteresis loops as a succession of quasi-equilibrium states determined by the numerical integration of the SLLG equation applied to the spin model described above. The tensorial exchange interactions are considered up to 11th neighbor. Initially we prepare the system similar to experiments by simulating a field-cooling process. This process starts from a random spin configuration in the AFM at an initial temperature T above the Néel temperature of the AFM but below the Curie temperature of the FM, and proceeds to a final temperatureunder the influenceofan in-plane magnetic H_FC.After the field cooling process Mn as well as Fe magnetic moments are oriented in-plane which is in correspondence with the calculated in-plane magnetic anisotropy. Importantly, the direction of the Mn moments is nearly perpendicular to that of the Fe moments, a consequence of the so-called spin-flop coupling <cit.>.Near the interface, the Mn moments are slightly tilted from this perpendicular (x-) direction, leading to a very small net magnetic moment, anti-parallel to the Fe moments. This configuration follows from the strong antiferromagneticexchange interaction between Mn-Fe moments and the fact that theinterface between Ni_2MnAl(B2-I)/Fe is compensated (equal number of Mn moments in both magnetic sub-lattice). This spin configuration is shown in Fig. <ref>.We investigate theswitching mechanismand the possible existence of EB for different values of the thickness t_AF of the Ni_2MnAl(B2-I) layer. Our findings indicate that for perfect bilayers there is no EB within our numerical error of ± 75Oe. However, the behavior of the AF changes drastically as t_AF is increased. As an example in Fig. <ref> we show hysteresis loops for two different thicknesses of the AF, focusing on the evolution of the magnetization of the FM along the direction of the applied field, M_x(FM),the total magnetization of the system along the direction of the applied field, m_h(Tot), as well as the in-plane antiferromagnetic order parameter, M^st_y, perpendicular to the applied field. We observe that during hysteresis the Fe moments rotate coherently, staying mostly in-plane.For the smaller thickness, due to the strong exchange interactions between Mn-Fe moments, the small net magnetic moment of the AF close to the interface also rotates,maintaining the antiferromagnetic order. Finally the AF switches following the FM (see Fig. <ref> (a)). When the thickness of the AF increases, and concomitantly the relevance of the on-site MAE of the AF, the AF cannot switch anymore and the antiferromagnetic order parameter, M^st_y, remains close to unity (see Fig. <ref> (b)). Nevertheless, the small canting of the Mn moments at the interface switches with the FM so that the magnetic moment of the AF maintains its direction antiparallel to the Fe moments. These results indicate that for sufficiently thin layers it is possible to manipulate the magnetic order of the antiferromagnetic Ni_2MnAl layer through the magnetization of the Fe layer. A similar control of the AF magnetization by the FM layer has been reported for NiFe/IrMn/MgO/Pt heterostructure<cit.> as a key point to the use of that system in an AF-based tunnel junction. Our finding opens hence the door for new Heusler-alloy-based antiferromagnetic spintronic devices.§.§.§ Hysteresis in the disordered Ni_2MnAl(B2)/Fe bilayer Calculations similar to the ones described above were performed for the disordered Ni_2MnAl(B2)/Fe system. First of all, it is important to note that, as a result of the chemical disorder in the B2 phase and its low effective anisotropy, much more complex spin structures appear in the AF after the field cooling process (see Fig. <ref>). As before, the Fe moments are aligned along the x-direction, the direction of the field during cooling. Again, we observe a kind of spin-flop coupling with the AF ordered mostly perpendicular to the FM and in-plane. However, the canting of the Mn moments at the interface is much more pronounced as compared to theNi_2MnAl(B2-I)/Fe system (see the red and yellow Mn moments in the first layer of Fig. <ref> b)). The reason for this much stronger canting is the structural disorder in Mn moment positions. Due to the statistical distribution of the Mn moments with some probability clusters of moments within the same sub-lattice appear. In these clusters the moments have a smaller connectivity to Mn moments of the other Mn sub-lattice where the antiferromagnetic exchange would counteract the canting. As a consequence, larger tilting angles and with that a larger net magnetization antiparallel to the Fe magnetization appears. However, the effective coupling between Mn and Fe layers is still smaller since only 50 % of the sites of the layer which is closest to the Fe are occupied. For comparison, in the pseudo-orderedNi_2MnAl(B2-I)/Fe interface 100% of these sites are occupied by Mn atoms.In Fig. <ref> (a) hysteresis curves are presented. These hysteresis loops are shifted horizontally, corresponding to an exchange bias field of H_EB=200 ± 50Oe. From the difference between the magnetization of the FM and the total magnetization one can see that not only the Fe moments contribute to the hysteresis loops but also the Mn moments. Furthermore, the sub-lattice magnetization of the AF switches as well indicating that the AF follows the FM as in the case of the pseudo-ordered bilayer for the thin AF layer (see Fig.<ref> (c)). During the switching process the Fe moments rotate again mainly in plane, as we can see in Fig. <ref>(b) where the x,y and z components of the Fe magnetization are plotted. However a small out-of-plane component of the magnetization appears as well during the switching in both branches of the hysteresis loops.The coercive field is much smaller than in the previous case of the pseudo-ordered bilayer. This smaller coercive field is due to the fact that the effective interface coupling is smaller because of the smaller occupancy with magnetic Mn atoms at the interface. Furthermore, the anisotropy of the AF is smaller which leads to a smaller stability of the AF against switching.For an investigation of the thermal stability of the EB effect mean hysteresis loops where calculated as anaverage over 5 hysteresis loops performed using the same spacial distribution of Mn-Al atoms. The EB we find is not only rather small and but also unstable against thermal fluctuations (see Fig. <ref>). Our results suggest a blocking temperature below 100K. Over all our simulations indicate that the EB is related to the disorder — the lack of perfect compensation due to the random distributions of the Mn and Al atoms into theY-Z positions in the Heusler alloy — in combination with the anisotropy in the AF. As a consequence a small part of the interface magnetization of the AF becomes frozen and does not switch with the FM which leads to the EB. This conclusion is supported by the fact that the EB vanishes for increasing lateral size. § CONCLUSION In summary, by means of a multi-scale modeling we investigate the interfacial magnetic interactions, the magnetic state andthe hysteresis loops of Ni_2MnAl(B2-I;B2)/Fe bilayers. Based on first principles calculations we find a strong negative Mn-Fe interface interaction, exceeding the antiferromagnetic interactions within theNi_2MnAl. For the disordered Ni_2MnAl(B2)/Fe bilayer we find a small EB at low temperatures in agreement with recent measurements <cit.>. The existence of such an exchange bias is related to the disorder in the AF and with that to a lack of perfect compensation at the interface. More importantly, we have shown that it is possible to switch the magnetic structure of the antiferromagnetic Ni_2MnAl layer in both, the pseudo-ordered B2-I and disordered B2 phase, via a spin-flop coupling to the ferromagnetic Fe capping layer. This open the doors for the control of antiferromagnetic Heusler alloys in spintronic devices with antiferromagnetic components.This work was supported by the European Commission via the Collaborative Project HARFIR (Project No. 604398) and the National Research, Development and Innovation Office of Hungary under Project No. K115575.widest-label hirohataIEEE15 A. Hirohata, H. Sukegawa, H. Yanagihara, I. Zutic, T. Seki, S. Mizukami, and R. Swaminathan, IEEE Trans. Magn. 51, 1 (2015) tanjaPSSC11 T. Graf, C. Felser and S.S.P. Parkin, Prog. Solid Stat. Chem.39, 50 (2011). websterBook88P. J. Webster and K. 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Materials 10, 347 (2011). | http://arxiv.org/abs/1707.08651v1 | {
"authors": [
"Rocio Yanes",
"Eszter Simon",
"Sebastian Keller",
"Balazs Nagyfalusi",
"Sergii Khmelevsky",
"Laszlo Szunyogh",
"Ulrich Nowak"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726213042",
"title": "Interfacial exchange interactions and magnetism of Ni2MnAl/Fe bilayers"
} |
1]Ajay [email protected] 2]Kengo [email protected] 3]Hiroki [email protected] [1]Department of StatisticsApplied Probability, National University of Singapore, SG [2]Department of Engineering Science, Osaka University, JP [3]Faculty of Mathematics, Kyushu University, JP Bayesian inference for Stable Lévy driven Stochastic Differential Equations with high-frequency data [ This version: December 30, 2023 ====================================================================================================In this article we consider parametric Bayesian inference for stochastic differential equations (SDE) driven by a pure-jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available, so we use a quasi-likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions thatthere is a Bernstein-von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference and assisted by our theoretical results, we show how to scale Metropolis-Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio going to zero in the large data limit. Our algorithm is presented on numerical examples that help to verify our theoretical findings.Keywords: Markov Chain; Monte Carlo; Lévy process; Bayesian inference; high-frequency data§ INTRODUCTION Stochastic differential equations (SDE) are found in a wide variety of real applications, such as finance and econometrics (see for instance <cit.> and the references therein), mathematical biology, movement ecology, turbulence, signal processing, to mention just a few. In this article we are concerned with Bayesian parameter estimation of discretely observed SDE driven by a pure-jump stable Lévy process with high-frequency data, which is often found in the afore-mentioned applications. Among others, we refer to <cit.>, <cit.>, and <cit.> for a comprehensive account of stable distribution and stochastic processes driven by a stable .Often, the main challenge with Bayesian or classical inference with Lévy driven SDE is the lack of tractability of the transition density of the process, hence one cannot evaluate a non-negative and unbiased estimate of the transition density. The latter is often required for inference methods. In such scenarios, one often has to resort to time-discretization. We focus on the time discretization induced by the quasi-likelihood approach, which, under suitable regularity conditions, enables us to deduce several desirable properties, such as consistency and asymptotic (mixed) normality; see <cit.> and the references therein. As a result, one expects rather favorable properties of the associated posterior distribution in similar settings. In addition, one must construct inference techniques, in this article MCMC, which will turn out to be robust in the high-frequency limit. That is, algorithms that will not collapse in some sense.Bayesian inference for diffusions and jump diffusions have been considered in many articles. This includes the fully observed (jump) diffusion case <cit.> and the partially observed jump diffusioncase <cit.>. In particular the work <cit.> considered some related, but different classes of models to this article, except using approximate Bayesian computation <cit.> methods with MCMC.We note that most of the previous works do not consider quasi-likelihood and/or the large data limit of the performance of MCMC. Our contributions of this article are roughly summarized as follows: * To construct an approximate posterior distribution with favorable theoretical properties. That is, under assumptions, that there is a Bernstein-von Mises theorem associated to the posterior.* To develop an MCMC algorithm and assisted by our theoretical results, show how to scale Metropolis-Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio going to zero in the large data limit. This article is structured as follows. In Section <ref>, the model setup and an associated Bernstein-von Mises theorem are given. Our MCMC algorithm is also described. In Section <ref>, our theoretical results for MCMC are stated. Section <ref> provides some numerical simulations which confirm our theoretical findings. We also analyze a real data set from the NYSEwhich features properties often captured well by the processes we study in this article.The appendix features a variety of proofs for our technical results. § STABLE-LÉVY SDES, MODEL AND ALGORITHM§.§ Setup Let (Ω,ℱ,(_t)_t∈[0,T],ℙ) be a complete stochastic basis.Let {X_t;t∈ [0,T]} be a solution to the one-dimensional stochastic differential equationX_t=a(X_t,α) t+c(X_t-,γ) J_t,where {J_t}_t is the stable Lévy-process independent of the initial variable X_0 and such that𝔼(e^iuJ_1)=e^-|u|^β,where it is assumed throughout that _t=(J_s: s≤ t)∨(X_0) and that 1≤β <2. Denote by ϕ_β the β-stable density of (J_1). We will write ℙ_θ for the image measure of (X_t) associated with the parameter valueθ:=(,)∈Θ_×Θ_=Θ⊂^p,with Θ_⊂^p_ and Θ_⊂^p_ being bounded convex domains. The true value is denoted by =(_0,_0)∈Θ. For brevity, we will also writefor _θ_0 as well. Let Θ denote the closure of Θ, and write a≲ b if a≤ Cb for some universal constant C>0. [Regularity of the coefficients] * The functions a(·,_0) and c(·,γ_0) are globally Lipschitz and of class ^2(), and c(x,γ) is positive for every (x,γ);* a(x,·)∈^3(Θ_) and c(x,·)∈^3(Θ_) for each x∈;* sup_θ∈Θ{max_0≤ k≤ 3max_0≤ l≤ 2( |_^k_x^la(x,)|+ |_^k_x^lc(x,)|) + c^-1(x,)}≲ 1+ |x|^C.[Identifiability conditions] The random functions t↦(a(X_t,), c(X_t,)) and t↦(a(X_t,_0), c(X_t,_0))on [0,T] a.s. coincide if and only if θ=.The process X is observed only at discrete time t=nh (j=0,…, N) where h=T/N with the terminal sampling time T>0 being fixed. Let X^N:={X_nh;n=0,…, N}, the available data set. The Euler-Maruyama discretization under ℙ_θ is then given byΔ^N_nX ≈ a(X_(n-1)h,α)h+c(X_(n-1)h,γ)Δ^N_n J,where Δ^N_n X:=X_nh-X_(n-1)h and Δ^N_nJ:=J_nh-J_(n-1)h. This suggests to consider the following stable quasi-likelihood <cit.> (the multiplicative constant “h^-N/β” omitted):𝕃_N(θ|X^N)=∏_n=1^N1/c(X_(n-1)h,γ)ϕ_β(Δ^N_nX-a(X_(n-1)h,α)h/c(X_(n-1)h,γ)h^1/β).We define the stable quasi-maximum likelihood estimator by any element=(,)∈ argmax_θ∈Θlog_N(θ|X^N).Note that 𝕃_N(θ|X^N) is the true likelihood only when both a(x,) and c(x,) are constants.It follows from <cit.> that under Assumptions <ref> and <ref> the estimatoris asymptotically mixed-normally distributed: on a suitably extended probability space (Ω̃,ℱ̃,ℙ̃) carrying a p-dimensional standard Gaussian random variableη independent of , we have(√(n)h^1-1/β(-_0), √(n)(-_0)) _st→ I()^-1/2η,where _st→ stands for the -stable convergence in law (see <cit.> for details) and where I()=( I_(),I_(_0)) denotes the a.s. positive definite quasi Fisher-information matrix specified byI_()=C_(β)Σ_T,(θ_0),I_(_0)= C_(β)Σ_T,(_0)withC_(β) := ∫(∂ϕ_β/ϕ_β(y))^2ϕ_β(y) y, C_(β):=∫(1+ y∂ϕ_β/ϕ_β(y))^2ϕ_β(y) y,Σ_T,(θ_0) :=1/T∫_0^T{∂_a(X_t,_0)}^⊗ 2/c^2(X_t,_0) t, Σ_T,(_0) :=1/T∫_0^T{∂_c(X_t,_0)}^⊗ 2/c^2(X_t,_0) t.It is known that the stable quasi-maximum likelihood estimator θ̂_N attains the Hajék-Le Cam minimax lower bound in some special cases, see <cit.> and <cit.>. The convergence (<ref>) shows that we do have the conventional Studentization (asymptotic standard normality) result without a finite-variance property as well as any stability condition such as ergodicity. §.§ Posterior and Markov chain Monte Carlo methods §.§.§ Quasi-posterior distributionThe quasi-posterior distribution is given byΠ_N(θ|X^N)∝𝕃_N(θ|X^N)Π(θ),where Π(θ)=π(θ)θ denotes the prior distribution of θ. The β-stable density ϕ_β does not have a closed form for β∈(1,2), hence cannot be computed pointwise and neither then can the posterior up-to a constant; this is required for stochastic simulation algorithms such as MCMC. In general, ϕ_β can be only computed via certain numerical integration, the iteration of which may be rather time-consuming <cit.>. However, we have access to a non-negative unbiased estimate of the un-normalized posterior, which does suffice to apply (e.g.) MCMC and it may be constructed as follows. Let F_β( v)=f_β(v) v denote the positive β/2-stable distribution with Laplace transform ∫_0^∞ e^-vtF_β( v)=e^-|2t|^β/2 (t≥ 0).Then, the stable distribution (J_1) is the law of √(V)W where W∼ N(0,1) and V∼ F_β,and we have the normal variance-mixture representation:ϕ_β(x)=∫_0^∞1/√(2π v)exp(-x^2/2v)F_β( v).If V_1,…, V_N ∼ F_β,and ξ_1,…, ξ_N∼ N(0,1), then ℒ(h^1/β√(V_n)ξ_n)=ℒ(Δ^N_nJ). Invoking (<ref>) we may expect that the formal distributional approximationΔ^N_nX≈ a(X_(n-1)h,α)h+c(X_(n-1)h,γ)h^1/β√(V_n)ξ_nunder ℙ_θ would be meaningful.Building on the above observation, we will make use of the “complete” quasi-likelihood 𝕃_N(θ|X^N,V^N)=∏_n=1^N1/c(X_(n-1)h,γ)√(V_n)ϕ(Δ^N_nX-a(X_(n-1)h,α)h/c(X_(n-1)h,γ)√(V_n)h^1/β)f_β(V_n),where ϕ denotes the standard normal density. The corresponding posterior distribution isΠ_N(θ,V^N|X^N)∝𝕃_N(θ|X^N,V^N)Π(θ).Note that we can still not compute Π_N(θ,V^N|X^N) up to a constant. Nevertheless, we will still be able to devise a Metropolis-within-Gibbs MCMC algorithm which mitigates this issue.§.§.§ MCMC Algorithm Let θ∈Θ. For n=1,…, N,generate V_n from F_β( v|ϵ_n(θ)) whereF_β( v|x)= .1/√(2π v)exp(-x^2/2v)F_β( v)/ϕ_β(x)and ϵ_n(θ)= Δ^N_nX-a(X_(n-1)h,α)h/c(X_(n-1)h,γ)h^1/β.Then update θ^*←θ+(noise) and accept θ^* with probability α(θ,θ^*)=min{1,𝕃_N(θ^*|X^N,V^N)π(θ^*)/𝕃_N(θ|X^N,V^N)π(θ)}.This noise should be properly scaled by the rate matrixD_N := diag( √(N)h^1-1/βI_p_, √(N)I_p_), as specified in (<ref>). The random variable V_n∼ F_β( v|ϵ_n(θ)) can be sampled via rejection sampling for each n∈{1,…,N}: Generate v∼ F_β( v) and accept it with probability.{v^-1/2exp(-x^2/2v)}/ {|x|^-1/2exp(-|x|/2)}.The algorithm is presented in Algorithm <ref> (We write N_p(μ,Σ) for the p-dimensional Gaussian distribution with mean μ and covariance matrix Σ).A minor modification is to consider a (correlated) pseudo-marginal algorithm <cit.> to avoid rejection sampling.In this case, update θ as in Algorithm in <ref> together withV_n^*←ρ^2/β V_n + (1-ρ)^2/βξ_n,ξ_n∼ F_β,as in <cit.> and <cit.>.Then accept (θ^*, V^N*)where V^N*={V_n^*}_n=1,… N with probability α((θ,V^N),(θ^*,V^N*))=min{1,𝕃_N(θ^*|X^N,V^N*)π(θ^*)/𝕃_N(θ|X^N,V^N)π(θ)}.Here, ρ is a tuning parameter and we recommend to take ρ close to 1. §.§ Bernstein-von Mises theoremIn this section we give a Bernstein-von Mises theorem associated with the stable quasi-likelihood (<ref>). Different from the classical version <cit.>, we will look at the convergence of the posterior distribution of the rescaled parameter centered not at , but at . The prior distribution Π admits a bounded Lebesgue density π which is continuous and positive at .Let _N(θ):=log_N(θ|X^N)be the logarithmic quasi-likelihood function. Following <cit.>, we introduce the quasi-likelihood ratio random field_N(u):=exp{_N(θ_0+D_N^-1u) - _N(θ_0) }defined on the set _N=_N():=D_N(Θ-)⊂^p. For convenience, we extend the domain of _N into the whole ^p with _N≡ 0 on ^p∖_N.Then, the quasi-posterior distribution Π̅_N(du|X^N) of the rescaled parameter u=D_N(θ-) admits the densityu ↦_N(u)π(+D_N^-1u)/∫_N(v)π(+D_N^-1v) v.Let Δ_N():=D_N^-1_θ_N() and denote bythe convergence in probability. For an -measurable random variable (μ,Σ)∈^p×^p× p, we keep denoting by N_p(μ,Σ) the -conditional Gaussian distribution associated with the characteristic function ξ↦[exp{iμ[ξ]-(1/2)Σ [ξ,ξ] }]. Let ν_TV=sup_|f|≤ 1|∫ f(x)ν( x)| denote the total variation norm of a signed measure ν.Under Assumptions <ref> to <ref>, we haveΠ̅_N(·|X^N)-N_p(I()^-1Δ_N(),I()^-1)_TV 0.The proof of Theorem <ref> is given in Appendix <ref>. § LARGE SAMPLE PROPERTIES OF THE MARKOV CHAIN MONTE CARLO METHOD In this section, we will first introduce a general framework for developing some stability properties of MCMC algorithms, and then apply it to theMCMC algorithms proposed in Section <ref>. In particular, we show how to scale the proposals when the frequency of the data grows. §.§ Local consistency property For a moment we step away from the main context. Notations and terminologies in this section generally follow those of <cit.>.§.§.§ Some general notions Let E be a normed space equipped with the Borel σ-algebra ℰ.For a probability measures μ and ν on (E,ℰ), let μ-ν_BL=sup_f∈BL_1|∫ f(x)μ( x)-∫ f(x)ν( x)| be the bounded Lipschitz distance between μ and ν whereBL_1 denotes the set of any ℰ-measurable real-valued function f such thatsup_x∈ E|f(x)|≤ 1 and |f(x)-f(y)|≤x-y (x, y∈ E). Letℰ_N={𝒳^(N),𝔄^(N),𝐏_θ^(N),θ∈Θ}be a family of statistical experiments and x^N be the corresponding observation. Here, the parameter spaceΘ⊂ℝ^p be an open set equipped with the Borel σ-algebra Ξ.Let 𝕃_N(θ|x^N) be the 𝒳^(N)⊗Ξ-measurable quasi-likelihood function which approximates the true likelihood 𝐏_θ^(N)(x^N).Let Π(θ) be the prior distribution for the experiments, and Π_N(θ|x^N) be the quasi-posterior distribution defined by Π_N(θ|x^N)=𝕃_N(θ|x^N)Π(θ)/∫_Θ𝕃_N(θ|x^N)Π(θ).We shall refer to ℳ_N the Markov chain Monte Carlo method for the experiment ℰ_N if the algorithm generates a Markov chain {θ^N_m;m=0,1,…} for each x^N. We shall refer to F^N_M the empirical distribution of{θ^N_m;0≤ m≤ M-1}, that is, F^N_M(A)=1/M∑_m=0^M-11_A(θ_m^N)for each Borel set A of ℝ^p. Let D_N be anon-degenerate p× p-matrix for each N∈ℕ.In practice, we expect that Markov chain Monte Carlo methods are robust and does not collapse as N→∞.We formalize this favourable property as follows. A family ℳ_N is called locally consistent ifas N, M→∞ Π_N(θ_0 + D_N^-1 u|x^N)-F^N_M(θ_0 + D_N^-1 u)_BL→ 0 in 𝐏_θ_0^(N)-probability. In the next subsection, we introduce a sufficient condition for this property under the general framework.§.§.§ Sufficient conditions Let Q_N(θ,ϑ|x^N) (n∈ℕ) be a probability transition kernelfrom 𝒳^(N)×Θ to Θ,and let Π_N(θ|x^N) be a probability transition kernel from 𝒳^(N) to Θ.Let A_N:𝒳^(N)×Θ^2→[0,1] be a measurable map. Let P_N(θ,ϑ|x^N) =Q_N(θ,ϑ|x^N)A_N(θ,ϑ|x^N)+R_N(θ|x^N)δ_θ(ϑ),where R_N(θ|x^N)=1-∫_Θ Q_N(θ,ϑ|x^N)A_N(θ,ϑ|x^N).As probability measures on Θ^2, we assume that for each x^N∈𝒳^(N), Π_N(θ|x^N)Q_N(θ,ϑ|x^N)A_N(θ,ϑ|x^N) = Π_N(ϑ|x^N)Q_N(ϑ,θ|x^N)A_N(ϑ,θ|x^N).Then for each x^N, the probability transition kernel P_N(θ,ϑ|x^N) is Π_N(θ|x^N)-reversible, that is, Π_N(θ|x^N)P_N(θ,ϑ|x^N)=Π_N(ϑ|x^N)P_N(ϑ,θ|x^N)as probability measures on Θ^2.Let ℳ_N be the Markov chain Monte Carlo method associated with the transition kernel P_N,that is, for each x^N, the algorithm generates a Markov chain{θ_m^N}_mwith respect to the probability transition kernel P_N(θ,ϑ|x^N).For simplicity, the initial point is generated from the quasi-posterior distribution, that is,θ_0^N∼Π_N(θ|x^N). Let q∈ℕ. Let Π( u|s)=π(u|s) u be a probability transition kernel from an open set 𝒮⊂ℝ^q to ℝ^p and let Q(u,v|s)=q(u,v|s) v be a probability transition kernel from 𝒮×ℝ^p to ℝ^p.Let A(u,v|s) be a [0,1]-valued function. For each s∈𝒮 we define a probability transition kernelP(u, v|s) =Q(u, v|s)A(u,v|s)+R(u|s)δ_u( v), R(u|s) =1-∫ Q(u, v|s)A(u,v|s). [Regularity of the limit kernel] For each u,v∈ℝ^p, π(u|s), q(u,v|s) and A(u,v|s) are continuous in s, and sup_s∈ Ksup_u,v∈ℝ^pπ(u|s)+q(u,v|s)<∞for any compact set K of 𝒮.Lets_N:𝒳^(N)→𝒮 be a sequence of random variables. The following proposition illustrates a sufficient condition forlocal consistency. See Appendix <ref> for the proof.In the following, we call that a transition probability kernel P is ergodic if it is irreducible and positive recurrent.Suppose that P(u, v|s) is ergodic with invariant probability measure Π( u|s) for each s∈𝒮.Suppose that s_N is 𝐏_θ_0^(N)-tight for each θ_0∈Θ. Then, under Assumption <ref>, the family ℳ_N is locally consistent if A_N(θ_0 + D_N^-1 u,θ_0 + D_N^-1 v|x^N)- A(u,v|s_N)→ 0 in probability for each u, v∈ℝ^p and Π_N(θ_0 + D_N^-1 u|x^N)Q_N(θ_0 + D_N^-1 u,θ_0 + D_N^-1 v|x^N)-Π( u|s_N)Q(u, v|s_N)_TVconverges in 𝐏_θ_0^(N)-probability to 0. §.§ Main result We now go back to the main context given in Section <ref>. In Algorithm <ref>, the proposal distribution for observation X^N is Q_N(θ_0 + D_N^-1 u,θ_0 + D_N^-1 v|X^N)=Q(u, v|s_N)=N_p(0,Σ)for some non-degenerate matrix Σ∈ℝ^p× p.We will set s_N=(Δ_N(θ_0), I(θ_0), I^*(θ_0)) where I^*() is defined in Section <ref>. By this setting of the proposal distribution, the following convergence is equivalent tothe condition (<ref>):Π_N(θ_0 + D_N^-1 u|X^N)-Π( u|s_N)_TV=o_p(1).The acceptance probability ofAlgorithm <ref> is A_N(θ,ϑ|X^N)=𝔼[.min{1,𝕃(ϑ|X^N,V^N)π(ϑ)/𝕃_N(θ|X^N,V^N)π(θ)}|X^N,θ].Let s=(Δ, I, I^*)∈ℝ^p×ℝ^p× p×ℝ^p× p.We assume that I and I^* are p× p-symmetric matrices.In the following theorem, we will show that the scaled version of the acceptance ratio converges toA(u,u^*|Δ,I,I^*)=𝔼[min{1,η}]where η=Δ[u^*-u]+W[u^*-u]-I[(u^*)^⊗ 2-u^⊗ 2]-1/2I^*[(u^*-u)^⊗ 2]with W∼ N(0, I^*).By showing this, we prove that Algorithm <ref> generates a locally consistent family of Markov chain Monte Carlo methods.In particular, the algorithm will not collapse in the large data limit. Under Assumptions <ref> to <ref>, A_N(θ_0+D_N^-1u,θ_0+D_N^-1 v|X^N)-A(u,v|Δ_N(θ_0), I(θ_0), I^*(θ_0))→ 0 in probability for each u, v∈ℝ^p. In particular, the family of Markov chain Monte Carlo methods ℳ_N defined by Algorithm <ref> is locally consistent.The proof of the theorem is given in Section <ref>.§ SIMULATIONS§.§ Simulated ExampleWe generated seven data sets from the time discretized models with a(x,α)=α_1(x-α_2), c(x,γ)=exp(γcos(x)), β=1.5, h_N=1/N and N∈{10,50,100,250,500,1000,2000}. α_1, α_2, γ are independently standard normal variables in the prior. Algorithm <ref> was run for 10,000 iterations. All simulations are repeated 100 times. The results are given in Figure <ref>.Figure <ref> illustrates the average acceptance rate against the number of data. As can be seen the average acceptance rate is stable from small data to large data. In particular, the data increases the algorithm does not collapse and the acceptance rate is very reasonable. The run time of the algorithm for N=2000 is only about five minutes and was coded in R (code is available upon request); the code could be substantially improved to further reduce the computation time. Note that the update on V^N|⋯ can be parallelized to improve the running speed. §.§ Application to IBM Stock Data We return to the data of NYSE.In the model, we set the same model with above, andβ=1.411, h_N=T/N, where T is measured in minutes (T=390× 3) and N=1156. The stable index β was estimated by applying `stableFit' function in R-package `fBasics' <cit.> to {Δ_n^N X}_n=1,…, N. The data contains some missing values which are ignored for the purposes of the analysis. α_1, α_2,γ are independently normal variables in the prior with mean 0 and standard deviation 1. Algorithm <ref> was run for 100,000 iterations.The algorithm ran in approximately 1 hour and the acceptance rate for the move on the parameters was 0.34.In Figures <ref> and <ref> we can observe our results.Figure <ref> is a log-likelihood andtime average drift andjump coefficients: 1/N∑_n=1^Na(X_(n-1),h,α), 1/N∑_n=1^Nc(X_(n-1),h,γ). For this reasonable size data set, the algorithm performs well, with good mixing over the parameters. The acceptance rate is very reasonable as is the run-time - recall one can improve the code or coding language.Figure <ref> is a p-p plot of the posterior expected value of the standardized residual:Δ_n^N X-ha(X_(n-1)h,α)h/c(X_(n-1)h,γ)h^1/βagainst β-stable distribution. This provides an idea of the ability of this model to fit the real NYSE data. We can see that the model, to some extent, can exhibit the behaviour in the real data.§.§.§ AcknowledgementsWe acknowledge JST CREST Grant Number JPMJCR14D7, Japan, for supporting the research. AJ was additionally supported by Ministry of Education AcRF tier 2 grant, R-155-000-161-112, and KK was additionally supported by JSPS KAKENHI Grant Number JP16K00046.§ PROOFS FOR THE BERNSTEIN-VON MISES THEOREMThe purpose of this section is to prove Theorem <ref>.First, we introduce the quadratic random field^0_N(u) =exp( Δ_N()[u] - 1/2I()[u,u] ).By integrating the Gaussian density,( ∫_N^0(u)π() u )^-1 = 1/π()exp( 1/2I()^-1[Δ_N()^⊗ 2] ) = O_p(1).To complete the proof it suffices to show_N := ∫|_N(u)π(+D_N^-1u) - _N^0(u)π() | u0,since the left-hand side of (<ref>) can be bounded from above by the quantity∫|_N(u)π(+D_N^-1u)/∫_N(u)π(+D_N^-1u) u - _N^0(u)π()/∫^0_N(u)π() u|u ≤ 2 ( ∫_N^0(u)π()u )^-1_N.We only prove (<ref>) for β∈(1,2), since the remaining case of β=1, where we have the single localization rate √(N)(=√(N)h^1-1/β), is completely analogous and simpler. First we will introduce a good event G_N∈ whose probability can get arbitrarily close to 1 for N→∞ (see (<ref>) below). Let_1,N(u_1;):= exp{_N(, _0+u_1/√(N)) - _N(, _0) },_2,N(u_2):= exp{_N(_0 + u_2/√(N)h^1-1/β, _0) - _N(_0, _0) },both being defined to be zero for u=(u_1,u_2)∈^p∖_N. Also define_1,N(θ):=1/N{_N(,) - _N(,_0)},_1():= 1/T∫_0^T∫[log{c(X_t,_0)/c(X_t,)ϕ_β(c(X_t,_0)/c(X_t,)z)} -logϕ_β(z)]ϕ_β(z) z t,_2,N():= 1/Nh^2(1-1/β){_N(,_0) - _N(_0,_0)},_2():= -1/2T∫_0^T( a(X_t,_0)-a(X_t,)/c(X_t,_0))^2 t ·∫(ϕ_β/ϕ_β(z))^2ϕ_β(z) z.The random functions _1 and _2 represent quasi-Kullback-Leibler divergences for estimatingand , respectively; we can estimate _0 more quickly than _0 in case of β∈(1,2). For any matrix A we will write |A| for the Frobenius norm of A. For later use we mention the following statements, which are given in <cit.> or can be directly deduced from the arguments therein. * We have (recall (<ref>)) under (Δ_N(),-D_N^-1_θ^2_N()D_N^-1) →( I()^1/2η,I()),r̅_N:=sup_θ| D_N^-1_θ^3_N(θ)D_N^-1| = O_p(1).* There exists an a.s. positive random variable χ_0 such that for each κ>0,sup_;|-_0|≥κ_1() ∨sup_;|-_0|≥κ_2() ≤ -χ_0κ^2a.s.* In case of β>1, we havesup_| 1/√(N)__N(,_0) | = O_p(1), sup_| -1/N_^2_N(,_0) - C_(β)Σ_T,(_0) |=o_p(1).In addition, we will also need the following uniform laws of large numbers with convergence rates: * There exists a constant q∈(0,1) for which(√(N))^qsup_θ|_1,N(θ)-_1()| ∨ (√(N)h^1-1/β)^qsup_|_2,N()-_2()|0. A sketch of derivation of (<ref>) will be given at the end of this section.Set_N=N^-cfor 0<c<q/2(1/β-1/2)with q∈(0,1) given in (<ref>). Then, since√(N)h^1-1/β∼ N^1/β-1/2up to a multiplicative constant, we see that _N√(N)h^1-1/β↑∞ and _N^-2(√(N)h^1-1/β)^-q=O(1); since β≥ 1, the latter implies that _N^-2N^-q/2=O(1). It is straightforward to deduce from (<ref>) that_N^-2sup_θ|_1,N(θ)-_1()| ∨_N^-2sup_|_2,N()-_2()|0.Let _min(A) denote the minimum eigenvalue of a square matrix A. We now introduce the event G_N=G_N(M,) for positive constants M and :G_N := { |Δ_N()| ∨sup_| 1/√(N)__N(,_0) | ≤ M, | -D_N^-1_θ^2_N()D_N^-1 - I() | ∨sup_| -1/N_^2_N(,_0) - C_(β)Σ_T,(_0) | <,_N^-2sup_θ|_1,N(θ)-_1()| ∨_N^-2sup_|_2,N()-_2()| < /2, _min(I_()) ∧_min(I_()) ≥ 4, χ_0≥ 4, r_N<3/_N}.Given any '>0, we can find a triple (M_1,_1,_1) and an N_1∈ such thatsup_N≥ N_1{G_N(M,)^c}<'holds for every M≥ M_1 and ∈(0,_1]. Since the objective here is the convergence in probability, we may and do focus on the event G_N with M andbeing sufficiently large and small, respectively.We divide the domain of the integration in the definition of _N: ^p=A_N⊔ A_N^c whereA_N:={u;|D_N^-1u|≤_N},and then denote the associated integrals by '_N and ”_N, respectively: _N = '_N + ”_N with'_N := ∫_A_N|_N(u)π(+D_N^-1u) - _N^0(u)π() | u,”_N := ∫_A_N^c|_N(u)π(+D_N^-1u) - _N^0(u)π() | u.We will deal with these terms separately.First we show that '_N 0. We have '_N≤'_1,N+'_2,N, where'_1,N := ∫_A_N_N(u)| π(+D_N^-1u)-π()| u,'_2,N := π()∫_A_N| _N(u)-_N^0(u) | u.By the third order Taylor expansion_N(u)= exp{Δ_N()[u] - 1/2( I() + ( -D_N^-1_θ^2_N()D_N^-1 - I() )-1/3(D_N^-1_θ^3_N(θ̌_n(u))D_N^-1)[D_N^-1u] )[u,u]}for some random point θ̌_n(u) on the segment joiningand , we see that _N(u) ≤exp(M|u| - |u|^2) for u∈ A_N on G_N. Hence, under Assumption <ref> we obtain'_1,N≤sup_u ∈ A_N| π(+D_N^-1u)-π()| ∫ e^M|u| - |u|^2 u0.To handle '_2,N, we introduce the random functionM_N(u) := I_G_N(ω) I_A_N(u) | _N(u)-_N^0(u) |.To deduce the required convergenceM_N:=∫ M_N(u) u0,we make use of the subsequence argument: fix any infinite sequence '⊂. In view of the estimateM_N(u)≤ I_G_N(ω) I_A_N(u)( _N(u) + _N^0(u)) ≤ e^M|u| - |u|^2 + e^M|u| - 2|u|^2≲ e^M|u| - |u|^2a.s.and the dominated convergence theorem, it suffices to show that there exists a further subset ”={N”}⊂' along which M_N”(u) → 0 a.s. for each u. We have | log_N(u) - log^0_N(u) | ≲ |u|^2( | -D_N^-1_θ^2_N()D_N^-1 - I() | + r̅_N_N) =: |u|^2R_N.Since | -D_N^-1_θ^2_N()D_N^-1 - I() | ∨r̅_N_N→ 0,it is possible to pick a further subset ”={N”}⊂' along which R_N”→ 0 a.s.; note that the random sequence (R_N) is free from the variable u. With this ”, for each u we haveM_N”(u)≤_N”^0(u)| exp{log_N(u) - log^0_N(u)}-1 | ≤_N”^0(u) |u|^2R_N” e^|u|^2R_N”→ 0a.s.,followed by M_N”→ 0 a.s., hence by (<ref>). We conclude that '_N 0.Now we turn to the proof of ”_N 0, still focusing on the event G_N of (<ref>). Note that A_N^c↓∅ under (<ref>) since |D_N^-1u|>_N implies that |u| ≳_N√(N)h^1-1/β↑∞. Under the condition (<ref>) we have _N/|D_N^-1| ∼ C_N√(N)h^1-1/β→∞, so that∫_A_N^c_N^0(u) u ≤∫_|u| ≥_N/|D_N^-1|exp(M|u| - 2|u|^2)u → 0on G_N. Since π is bounded, we are left to show that ∫_A_N^c_N(u) u0.Write u=(u_2,u_1)∈^p_×^p_ and observe that (recall (<ref>) and (<ref>))_N(u) ≤(sup__1,N(u_1;)) _2,N(u_2) =: _1,N(u_1)_2,N(u_2).Since A_N^c⊂{u; |u_2|≥ (_N/2)√(N)h^1-1/β}∪{u; |u_1|≥ (_N/2)√(N)}, we obtain∫_A_N^c_N(u) u≤∫_|u_1|≥(_N/2)√(N)_1,N(u_1) u_1·∫_2,N(u_2) u_2 + ∫_1,N(u_1) u_1·∫_|u_2|≥ (_N/2)√(N)h^1-1/β_2,N(u_2) u_2.We will only show that∫_|u_1|≥(_N/2)√(N)_1,N(u_1) u_10,∫_1,N(u_1) u_1 = O_p(1).Indeed, in view of the definition (<ref>) of the good event G_N, in order to deduce∫_|u_2|≥(_N/2)√(N)h^1-1/β_2,N(u_2) u_2 0 and∫_2,N(u_2) u_2 = O_p(1),we can follow exactly the same route as in the proofs of (<ref>) and (<ref>) below, with _1,N(θ) and _1() replaced by _2,N() and _2(), respectively.On G_N we have _N^-2sup_;|-_0|≥_N/2_1()≤ -, so thatsup_|u_1|≥ (_N/2)√(N)log_1,N(u_1)≲ N ( sup_θ|_1,N(θ)-_1()| +sup_;|-_0|≥_N/2_1()) ≤ -/2(√(N)_N)^2≲ -N^1-2c.Therefore, recalling that _1,N(u_1)=0 outside the set _1,N:=√(N)(Θ_-_0)⊂^p_ with Θ_ being bounded, we obtain the following estimate for some positive constants C_0 and C_1: on G_N,∫_|u_1|≥(_N/2)√(N)_1,N(u_1) u_1 ≤ e^-C_0N^1-2c∫_{|u_1|≥ (_N/2)√(N)}∩_1,N u_1≲ e^-C_0N^1-2c∫_|u_1|≤ C_1√(N) u_1≲ N^p_/2e^-C_0N^1-2c→ 0.Hence (<ref>) is obtained.It follows from (<ref>) that there exist N_1∈ and K>0 such that for every M_1>1,sup_N≥ N_1( G_N∩{∫_1,N(u_1) u_1 > M_1}) ≤sup_N≥ N_1( G_N∩{∫_|u_1|≥ K_1,N(u_1) u_1 > M_1/2})+ sup_N≥ N_1( G_N∩{sup_|u_1|≤ K_1,N(u_1)≳M_1/2K^p_}) ≤sup_N≥ N_1( sup_|u_1|≤ K_1,N(u_1)≳M_1/2K^p_).For every K>0 the sequence {sup_|u_1|≤ K_1,N(u_1)}_N is tight in , hence we can make the last upper bound arbitrarily small by taking a sufficiently large M_1. This verifies (<ref>), and we are left to deduce (<ref>).Proof of (<ref>). Leta_n-1(α) = a(X _ (n - 1) h, α),c_n-1(γ) = c(X _ (n - 1) h, γ),Δ^N_n X = X _ nh - X _ (n-1)h,ϵ_n(θ) = ϵ_N,n(θ) := Δ^N_nX - h a_n-1(α)/h ^ 1 / βc_n-1(γ).Let z_n:=h^-1/β(J_nh-J_(n-1)h)∼ J_1 and b_n-1(θ):=c^-1_n-1(){a_n-1(_0)-a_n-1()}. Below we will repeatedly make use of several statements in <cit.>, hence at this stage it should be noted that, by the localization procedure <cit.>, without loss of generality we may and do suppose that𝔼(sup_t≤ T|X_t|^q)≲ 1,sup_t∈ [s,s+h]∩ [0,T]𝔼(|X_t-X_s|^q|ℱ_s)≲ h(1+|X_s|^C)for any q≥ 2 and s∈ [0,T].This fact in particular implies that sup_t∈ [s,s+h]∩ [0,T]𝔼(|X_t-X_s|)≲√(h).For convenience, for a sequence of random functions {f_N(·)} on Θ and a positive sequence (b_N) we will write f_N(θ)=O^∗_p(b_N) if sup_θ|f_N(θ)|=O_p(b_N).We can write _1,N(θ) as_1,N(θ)= 1/N∑_n=1^Nlogc_n-1(_0)/c_n-1() +1/N∑_n=1^Nlogϕ_β(_n(θ)) -1/N∑_n=1^Nlogϕ_β(_n(,_0)).The function y↦logϕ_β(y) fulfills the conditions on η in Lemmas 6.2 and 6.3 of <cit.>. Applying the two lemmas to the second and third terms in the right-hand side of (<ref>), we can deduce that_1,N(θ)= 1/N∑_n=1^Nlogc_n-1(_0)/c_n-1() +1/N∑_n=1^N{logϕ_β( c_n-1(_0)/c_n-1()z_n+h^1-1/βb_n-1(θ))|_(n-1)h} -1/N∑_n=1^N{logϕ_β( z_n+h^1-1/βb_n-1(,_0))|_(n-1)h}+O^∗_p(1/√(N)∨ h^2-1/β).Proceeding as in the second equality in Eq.(6.19) of <cit.>, we obtain_1,N(θ)= 1/N∑_n=1^N[ logc_n-1(_0)/c_n-1() +∫{logϕ_β(c_n-1(_0)/c_n-1()z) - logϕ_β(z)}ϕ_β(z)dz]+O^∗_p(h^1-1/β∨1/√(N)∨ h^2-1/β).Using the estimate (<ref>) combined with the inequality given in the proof of <cit.>, we can deduce from (<ref>) thatsup_θ|_1,N(θ)-_1()|≤ O_p(√(h)∨ h^1-1/β∨1/√(N)∨ h^2-1/β) = O_p(h^1-1/β)=O_p(N^-(1-1/β)).Hence (√(N))^qsup_θ|_1,N(θ)-_1()|0 holds for q∈(0, 2(1-1/β)).As for _2,N, following a similar line to the case of _1,N we can derive_2,N()= 1/2N∑_n=1^Nb^2_n-1(,_0) (_y^2logϕ_β(y))|_y=_n() + O^∗_p(1/√(N)h^1-1/β∨ h^1-1/β).Here, we also made use of <cit.> (the function y↦_ylogϕ_β(y) is odd) and the arguments in <cit.>. Then it is not difficult to arrive at sup_|_2,N()-_2()|=O_p(N^-s) for any sufficiently small s>0. In view of (<ref>) we conclude that (√(N)h^1-1/β)^qsup_|_2,N()-_2()|0 for q>0 small enough. The proof of (<ref>) is thus complete. § PROOF FOR THE LOCAL CONSISTENCY OF THE METROPOLIS-HASTINGS ALGORITHMWe prove Proposition <ref> in this section.Without loss of generality, we can assume s_N→ s in probability for some random variable s.For notational simplicity, we write F̅^N for the rescaled version of the function or measure F^N by u↦θ_0+D_N^-1u, and write ν̅_N( u, v|x^N)=Π̅( u|x^N)Q̅_N(u, v|x^N),μ̅_N( u, v|x^N)=Π̅( u|x^N)P̅_N(u, v|x^N)and ν( u, v|s)=Π( u|s)Q(u, v|s),μ( u, v|s)=Π( u|s)P(u, v|s).The equation (<ref>) becomes ν̅_N(·|x^N)-ν(·|s_N)_TV→ 0.By Lemmas 2 and 3 of <cit.> the following convergence is sufficient for local consistency:δ_N:=μ̅_N(·|x^N)-μ(·|s)_TV=o_p(1).First, we prove the convergence of δ_N':=μ(·|s_N)-μ(·|s)_TV. By triangular inequality, δ_N' is dominated above by the sum of δ_N,1':= ν(·|s)A(·|s)-ν(·|s_N)A(·|s_N)_TV, δ_N,2':= Π( u|s)R(u|s)δ_u( v)-Π( u|s_N)R(u|s_N)δ_u( v)_TV.For the former, by Assumption <ref>, δ_N,1'=2∫(π(u|s)q(u,v|s)A(u,v|s)-π(u|s_N)q(u,v|s_N)A(u,v|s_N))^+ u v=o_p(1)by the dominated convergence theorem, where x^+=max{0,x}. We can prove δ_N,3':=ν(·|s)-ν(·|s_N)_TV=o_p(1) in the same way.On the other hand, by triangular inequality, δ_N,2' =Π( u|s)R(u|s)-Π( u|s_N)R(u|s_N)_TV={ν(·×ℝ^p|s)-∫ν(·× v|s)A(·,v|s)}-{ν(·×ℝ^p|s_N)-∫ν(·× v|s_N)A(·,v|s_N)}_TV≤δ_N,3'+δ_N,1'=o_p(1).Hence, δ_N'≤δ_N,1'+δ_N,2'→ 0 in probability. Next, we prove the convergence of δ_N”:=μ̅_N(·|x^N)-μ(·|s_N)_TV. By the same argument as above, it is sufficient to show the convergence ofδ_N,1”:=ν̅_N(·|x^N)A̅_N(·|x^N)-ν(·|s_N)A(·|s_N)_TV and δ_N,3”:=ν̅_N(·|x^N)-ν(·|s_N)_TV.We only proves the former.By triangular inequality, δ_N,1” ≤{ν̅_N(·|x^N)-ν(·|s_N)}A̅_N(·|x^N)_TV+ν(·|s_N)(A̅_N(·|x^N)-A(·|s_N))_TV≤ν̅_N(·|x^N)-ν(·|s_N)_TV+∫ |A̅_N(u,v|x^N)-A(u,v|s_N)|ν( u, v|s_N).The first term converges to 0 by assumption.Since ν(·|s_N) converges to ν(·|s),and s_N is tight,for any ϵ>0, we can choose a compact set K_p⊂ℝ^p×ℝ^p and K_q⊂𝒮 so that lim sup_N→∞𝐄_θ_0^(N)[ν(K_p^c|s_N)]<ϵ/2,lim sup_N→∞𝐏_θ_0^(N)(s_N∈ K_q^c)<ϵ/2.Thus, 𝐄_θ_0^(N)[∫ |A̅_N(u,v|x^N)-A(u,v|s_N)|ν( u, v|s_N)]≤ϵ + 𝐄_θ_0^(N)[∫_K_p |A̅_N(u,v|x^N)-A(u,v|s_N)|ν( u, v|s_N), s_N∈ K_q] + o(1)≤ϵ + c∫_K_p𝐄_θ_0^(N)[|A̅_N(u,v|x^N)-A(u,v|s_N)|] u v+ o(1)=ϵ +o(1),where c=sup_s∈ K_qsup_u,vπ(u|s)q(u,v|s) is the upper bound of the probability density function of ν( u, v|s_N) when s_N∈ K_q.This completes the proof of δ_N”→ 0 in probability, and henceδ_N≤δ_N'+δ_N”=o_p(1). § CONVERGENCE OF THE ACCEPTANCE RATIO §.§ Setting and notation We keep using some notation introduced in Section <ref>.We consideran extended probability spaceΩ̃ = Ω× A, ℱ̃ = ℱ⊗𝒜, ℙ̃ = ℙ(ω)ℚ_ω( a)where (A,𝒜) is a measurable space, and ℚ_ω( a) is a probability transition kernel from Ω to A. We consider a stochastic processX̃_t(ω, a) = X _ t(ω)=ω_t.Fix u ∈ℝ ^ d. Setθ_N=(α_N,γ_N)=θ_0+D_N^-1u.We now consider random variables V_n, n=1,…, N, which correspond to the pseudo-data generated in Algorithm <ref> from parameterθ=θ_N.Thus, for each ω∈Ω, the random variables V_n(ω,·):A→ℝ_+ (n = 1, …, N) are independent, and ℙ̃( V _ n ∈ v |ω) = ℚ_ω(V_n∈ v)=F_β( v|ϵ_n(θ_N)). The log likelihood and the augmented-data (pseudo + observed data) log likelihood areℍ_N(θ)=-∑_n=1^Nlog c_n-1(γ)+∑_n=1^Nlogϕ_β(ϵ_n(θ)), ℍ^†_N(θ)= -∑_n=1^Nlog c_n-1(γ)-1/2∑_n=1^Nϵ_n(θ)^2/V_n.Here we omit terms that do not include θ.We define the Fisher information matrix for the augmented-data model by I ^ †(θ _ 0) := diag( C _ α ^ † (β) Σ _ T, α(θ _ 0),C _ γ ^ † (β) Σ _ T, γ(γ _ 0) ),withC_α^†(β):=∫_ℝ_+1/vF_β( v),C^†_γ(β):=2and that for the pseudo-data model by I ^ *(θ _ 0) :=I ^ †(θ _ 0)-I(θ _ 0)= diag( C _ α ^ * (β) Σ _ T, α(θ _ 0),C _ γ ^ * (β) Σ _ T, γ(γ _ 0) ), withC_α^*(β):=C_α^†(β)-C_α(β),C^*_γ(β):=C^†_γ(β)-C_γ(β).In the next section, we will use the following law of large numbers.[Proposition 6.5 of <cit.>] If η(x)∈𝒞^1(ℝ) and π(x,θ)∈𝒞^1(ℝ×Θ) satisfy sup_x∈ℝ|η(x)|+|η'(x)|≲ 1 and sup_θ|π(x,θ)|+|∂_θπ(x,θ)|≲ 1+|x|^C for some C>0, thenN^-1∑_n=1^Nη(ϵ_n(θ_N))π(X_(n-1)h,θ_N)=O_p(1),N^-1∑_n=1^Nη(ϵ_n(θ_N))π(X_(n-1)h,θ_N)= 1/T∫η(x)ϕ_β(x) x∫_0^Tπ(X_t,θ_0) t+o_p(1). We also use the following fact: Let β∈ (1,2). If π(x,θ) satisfies the condition in Proposition <ref>, then1/Nh^1-1/β∑_n=1^N g_β(ϵ_n(θ_N))π(X_(n-1)h,θ_N)=O_p((√(N)h^1-1/β)^-1).This convergence comes from Corollary 6.6. of <cit.> together with the estimate|g_β(ϵ_n(θ_N))-g_β(ϵ_n(α_0,γ_N))|≲ |ϵ_n(θ_N)-ϵ_n(α_0,γ_N))|≲1/√(N)h^1-1/β(1+|X_(n-1)h|^C). §.§ Some properties of F_β Let g_β(x):=∂/∂ xlogϕ _ β(x), h_β(x):=∂^2/∂ x^2logϕ_β(x)-1/x∂/∂ xlogϕ_β(x).Recall that by the property of stable law (see pp.88–89 of <cit.>), we have ϕ_β(x)∼ c|x|^-(β +1) as|x|→∞ for some c>0. Moreover, the probability density functionf_β(v) of F_β is bounded above, and as v→+∞, we havef_β(v)∼ c|v|^-β/2-1 for some c>0,and as v→ 0 the density of the positive stable distribution f_β(v) converges to 0 exponentially fast.Thus ∫ v^-kF_β( v|x) is continuous at x=0 for k≥ 0.Moreover, we have the following identities and an estimate.Fix β∈ [1,2).*g_β(x)=-∫x/vF_β( v|x),h_β(x)=∫(x/v+g_β(x))^2F_β( v|x). *∫ x g_β(x)ϕ_β(x) x=-1, ∫ h_β(x)ϕ_β(x) x=C_α^*(β), ∫ x^2h_β(x)ϕ_β(x) x=C_γ^*(β).* For any k≥ 0, ∫_0^∞ v^-kF_β( v|x)≲ |x|^-2k.The expression of g_β(x) can be obtained via simple interchange of the derivative and the integral in the equation (<ref>).For the expression of h_β(x), we haveh_β(x) =ϕ_β”(x)/ϕ_β(x)-g_β(x)^2-1/xg_β(x).By (<ref>), the second derivative of ϕ_β(x) is∂^2/∂ x^2∫_ℝ_+1/√(2π v)exp(-x^2/2v)F_β( v)= ∫_ℝ_+(x^2/v^2-1/v)1/√(2π v)exp(-x^2/2v)F_β( v)=ϕ_β(x)∫_ℝ_+(x^2/v^2-1/v)F_β( v|x).This equation yields the expression in (<ref>) by using the identity ∫ v^-1F_β( v|x)=-g_β(x)/x. Next, we prove identities (<ref>). Applying the change of variable u=x/√(v), we have∫ x g_β(x)ϕ_β(x) x =-∫ x{x/vF_β( v|x)}ϕ_β(x) x=-∫x^2/v1/√(2π v)exp(-x^2/2v) xF_β( v)= -∫ u^2ϕ(u) u=-1where ϕ(u) is the probability density function of the standard normal distribution.In the same way, by the change of variable, we obtain∫(x/v)^2F_β( v|x)ϕ_β(x) x=C_α^†(β), ∫ x^2(x/v)^2F_β( v|x)ϕ_β(x) x=3.Then we have∫ h_β(x)ϕ_β(x) x=∫(x/v)^2F_β( v|x)ϕ_β(x) x-∫ g_β(x)^2ϕ_β(x) x= C_α^*(β)and ∫ x^2h_β(x)ϕ_β(x) x =∫ x^2(x/v)^2F_β( v|x)ϕ_β(x) x-∫ x^2g_β(x)^2ϕ_β(x) x=3-∫ x^2g_β(x)^2ϕ_β(x) x=2-∫ (1+xg_β(x))^2ϕ_β(x) x=C_γ^*(β).Finally we check (<ref>). By the property of stable law, ∫_0^∞x^2k/v^kF_β( v|x) =ϕ_β(x)^-1∫_0^∞x^2k/v^k1/√(2π v)exp(-x^2/2v)F_β( v)≲∫_0^∞(x^2/v)^(2k+β+1)/2exp(-x^2/2v) v/v= ∫_0^∞(1/u)^(2k+β+1)/2exp(-1/2u) u/u<∞by the change of variable u =v/x^2.Thus, the claim follows. §.§ Estimates for the likelihood functions Let Δ_N^†(θ_N)=D_N^-1∂_θℍ^†_N(θ_N) and set Δ_N^*(θ_N)=Δ_N^†(θ_N)-Δ_N(θ_N). ℒ(Δ_N^*(θ_N)|ω)- N_k(0, I^*(θ_0))_BL→ 0 in ℙ-probability. Observe that ∂_αϵ_n(θ)=-h^1-1/β∂_α a_n-1(α)/c_n-1(γ), ∂_γϵ_n(θ)=-ϵ_n(θ)∂_γ c_n-1(γ)/c_n-1(γ). By this fact, we have Δ_N^†(θ_N)= N^-1/2∑_n=1^N [ ϵ_n(θ_N)/V_n∂_α a_n-1(α_N)/c_n-1(γ_N); {ϵ_n(θ_N)^2/V_n - 1 }∂_γ c_n-1(γ_N)/c_n-1(γ_N) ], and hence Δ_N^*(θ_N)= N^-1/2∑_n=1^N [{ϵ_n(θ_N)/V_n +g_β(ϵ_n(θ_N))}∂_α a_n-1(α_N)/c_n-1(γ_N); {ϵ_n(θ_N)/V_n+g_β(ϵ_n(θ_N)) }ϵ_n(θ_N) ∂_γ c_n-1(γ_N)/c_n-1(γ_N) ] =:N^-1/2∑_n=1^Nξ_n. Recall the definition of the extended probability space (<ref>). In this setting, the pseudo-data variables V_1,…, V_N are independent conditioned on ω∈Ω and generated from ℚ_ω. Fix ω∈Ω. Since ∫ (x/v+g_β(x))F_β( v|x)=0,the random variable Δ_N^*(θ_N)(ω,·):A→ℝ^p is a sum of independent variablesξ_n(ω,·) in ℚ_ω. By the expression of h_β, the covariance matrix of ξ_n conditioned on ω becomes 𝔼̃[ξ_n^⊗ 2|ω]= [ h_β(ϵ_n(θ_N)) (∂_α a_n-1(α_N)/c_n-1(γ_N))^⊗ 2 h_β(ϵ_n(θ_N))ϵ_n(θ_N) (∂_α a_n-1(α_N)/c_n-1(γ_N))⊗(∂_γ c_n-1(γ_N)/c_n-1(γ_N)); Sym.h_β(ϵ_n(θ_N))ϵ_n(θ_N)^2 (∂_γ c_n-1(γ_N)/c_n-1(γ_N))^⊗ 2 ]. This conditional covariance can be written as 𝔼̃[ξ_n^⊗ 2|ω]=η(ϵ_n(θ_N))[π(X_(n-1)h,θ_N)] where η(x)=h_β(x)[ 1 x; x x^2 ], π(x,θ)=[∂_α a_n-1(α_N)/c_n-1(γ_N),∂_γ c_n-1(γ_N)/c_n-1(γ_N)]^⊗ 2. By (<ref>) and Assumption <ref>, η(x) and π(x,θ) satisfy the condition of Proposition <ref>. On the other hand,for the forth moment on ξ_n, we have 𝔼̃[|ξ_n|^4|ω]^1/4≲ {∫|ϵ_n(θ_N)/v|^4F_β( v|ϵ_n(θ_N))}^1/4|∂_α a_n-1(α_N)/c_n-1(γ_N)|+ {∫|ϵ_n(θ_N)^2/v|^4F_β( v|ϵ_n(θ_N))}^1/4|∂_γ c_n-1(α_N)/c_n-1(γ_N)|≲1+|X_(n-1)h|^C for some C>0 by (<ref>) and Assumption <ref> together with the Minkowski and Jensen inequalities. Therefore by Proposition <ref>, 𝔼̃[(Δ_N^*(θ_N))^⊗ 2|ω]=N^-1∑_n=1^N𝔼̃[ξ_n^⊗ 2|ω]→ I^*(θ_0), N^-2∑_n=1^N𝔼̃[|ξ_n|^4|ω]→ 0 in ℙ̃-probability. For any infinite elements ℕ'⊂ℕ there exists a further subsequence ℕ”={N”}⊂ℕ' such that the above convergence satisfies in almost surely in ω∈Ω. Therefore, by the central limit theorem, ℒ(Δ_N”^*(θ_N)|ω)→ N_k(0, I^*(θ_0)) almost surely. Thus, the claim follows. -D_N^-1∂^2_θℍ^†_N(θ_N)D_N^-1→ I^†(θ_0) in ℙ̃-probability, and sup_Θmax_k=1,…, p|∂_θ_kD_N^-1∂^2_θℍ^†_N(θ)D_N^-1|=O_p(1). By calculation, (Nh^2(1-β))^-1∂_α^2 ℍ_N^† (θ_N) =1/Nh^1-1/β∑_n=1^Nϵ_n(θ_N)/V_n(∂_α^2 a_n-1(α_N)/c_n-1(γ_N))-1/N∑_n=1^N1/V_n(∂_α a_n-1(α_N)/c_n-1(γ_N))^⊗ 2.The first term of the right-hand side of (<ref>) is -1/Nh^1-1/β∑_n=1^Ng_β(ϵ_n(θ_N))(∂_α^2 a_n-1(α_N)/c_n-1(γ_N)) +1/Nh^1-1/β∑_n=1^N(ϵ_n(θ_N)/V_n + g_β(ϵ_n(θ_N)))(∂_α^2 a_n-1(α_N)/c_n-1(γ_N)).Then, the first term is negligible by Proposition <ref> for β=1 and by (<ref>) for β∈ (1,2), and the second term is also negligible since it is a sum of independent variables conditioned on ω, and its variance is o_p(1).Also, the second term of (<ref>) is 1/N∑_n=1^N(g_β(ϵ_n(θ_N))/ϵ_n(θ_N))(∂_α a_n-1(α_N)/c_n-1(γ_N))^⊗ 2 - 1/N∑_n=1^N(1/V_n+g_β(ϵ_n(θ_N))/ϵ_n(θ_N))(∂_α a_n-1(α_N)/c_n-1(γ_N))^⊗ 2,and again, the second term is negligible since it is a sum of conditionally independent random variables. Thus, we obtain-(Nh^2(1-β))^-1∂_α^2 ℍ_N^† (θ_N) = -1/N∑_n=1^N(g_β(ϵ_n(θ_N))/ϵ_n(θ_N))(∂_α a_n-1(α_N)/c_n-1(γ_N))^⊗ 2+o_p(1) = C_α^†(β)Σ_T,α(θ_0)+o_p(1)by Proposition <ref>. Similarly, we have (Nh^(1-β))^-1∂_α∂_γℍ_N^† (θ_N) = -2/N∑_n=1^Nϵ_n(θ_N)/V_n(∂_α a_n-1(α_N)/c_n-1(γ_N))⊗(∂_γ c_n-1(γ_N)/c_n-1(γ_N))= 2/N∑_n=1^Ng_β(ϵ_N(θ_N))(∂_α a_n-1(α_N)/c_n-1(γ_N))⊗(∂_γ c_n-1(γ_N)/c_n-1(γ_N))-2/N∑_n=1^N(g_β(ϵ_N(θ_N))+ϵ_n(θ_N)/V_n)(∂_α a_n-1(α_N)/c_n-1(γ_N))⊗(∂_γ c_n-1(γ_N)/c_n-1(γ_N)).The first term is o_p(1) by Proposition <ref> together with the fact ∫ g_β(x)ϕ_β(x) x=0,and the second term converges to 0 in L^2 since it is a sum of conditionally independent random variables. Then(Nh^(1-β))^-1∂_α∂_γℍ_N^† (θ_N) → 0 in probability. In the same way, decomposing the sum into the main term and the sum of independent variables, we haveN^-1∂_γ^2 ℍ_N^† (θ_N) = -2/N∑_n=1^Nϵ_n(θ_N)^2/V_n(∂_γ c_n-1(γ_N)/c_n-1(γ_N))^⊗ 2+ 1/N∑_n=1^N(ϵ_n(θ_N)^2/V_n-1) {(∂_γ^2 c_n-1(γ_N)/c_n-1(γ_N)) -(∂_γ c_n-1(γ_N)/c_n-1(γ_N))^⊗ 2}= 2/N∑_n=1^Nϵ_n(θ_N)g_β(ϵ_n(θ_N)) (∂_γ c_n-1(γ_N)/c_n-1(γ_N))^⊗ 2+ 1/N∑_n=1^N(-ϵ_n(θ_N)g_β(ϵ_n(θ_N))-1) {(∂_γ^2 c_n-1(γ_N)/c_n-1(γ_N)) -(∂_γ c_n-1(γ_N)/c_n-1(γ_N))^⊗ 2}+o_p(1).The first term in the right-hand side converges to -C_γ^†(β)Σ_T,γ(θ_0) and the second term is o_p(1) by Lemma <ref> together with the fact that ∫ xg(x)ϕ_β(x) x=-1.Then, we obtain-N^-1∂_γ^2 ℍ_N^† (θ_N) → C_γ^†(β)Σ_T,γ(θ_0)in probability. Almost the same arguments give convergence of the thrice derivatives. We omit the detail. §.§ Proof of Theorem <ref> The proof of Theorem <ref> is an application of Proposition <ref>. Since Π(u|s)=N_p(I(θ_0)^-1s, I(θ_0)^-1) and Q(u, v|s)=N_p(u,Σ),Assumption <ref> is easy to check. Ergodicity is also obvious by irreducibility.The Bernstein-von Mises theorem is already proved in Theorem <ref> which is a sufficient condition for (<ref>) in this case.We will complete the proofby showing the convergence of the acceptance ratio A_N.Let θ_N=θ_0+D_N^-1u and θ_N^*=θ_0+D_N^-1v.By (<ref>) and (<ref>) together with Taylor's expansion, we have the difference of the log quasi-posterior density satisfies ξ_N:={ℍ_N^†(θ^*_N)+logπ(θ_N^*)}-{ℍ_N^†(θ_N)+logπ(θ_N)}=∂_θℍ^†_N(θ_N^*-θ_N) +1/2∂_θ^2ℍ_N^†[(θ_N^*-θ)^⊗ 2]+o_p(1)=Δ^†_N(θ_N)[v-u]-1/2I^†(θ_0)[(v-u)^⊗ 2]+o_p(1).Now we rewrite the two terms in the right-hand side of the above equation.First, observe that difference of the score function at θ_N and θ_0 is Δ_N(θ_N)-Δ_N(θ_0)=D_N^-1∂_θ^2ℍ_N(θ_0)D_n^-1[u]+o_p(1)=-I(θ_0)[u]+o_p(1)by Taylor's expansion. Also, by I^†(θ_0)=I^*(θ_0)+I(θ_0), we have the following identity among Fisher information matrices: I(θ_0)[u, v-u] +1/2I^†(θ_0)[(v-u)^⊗ 2]=I(θ_0)[v^⊗ 2-u^⊗ 2]+1/2I^*(θ_0)[(v-u)^⊗ 2].By these facts, we can rewrite ξ_N byξ_N= Δ_N(θ_N)[v-u]+Δ(θ_N)[v-u]-1/2I^†(θ_0)[(v-u)^⊗ 2]+o_p(1)= Δ_N(θ_N)[v-u]+Δ(θ_0)[v-u]-I(θ_0)[u,v-u]-1/2I^†(θ_0)[(v-u)^⊗ 2]+o_p(1)= η_N(Δ_N^*(θ_N))+o_p(1)where η_N(w)=Δ_N(θ_0)[v-u]+w[v-u]-I(θ_0)[v^⊗ 2-u^⊗ 2]-1/2I^*(θ_0)[(v-u)^⊗ 2]for w∈ℝ^p. By the expression of the difference of the log quasi-posterior density together with the function ψ(x)=min{1,exp(x)}, we have the expression of the acceptance probabilities: A_N(θ_N,θ^*_N|X^N)=𝔼̃[ψ(ξ_N)|ω], A(u, v|Δ_N(θ_0), I(θ_0), I^*(θ_0))=𝔼[ψ(η_N(W))|w]where W∼ N(0, I^*(θ_0)).On the other hand,we have|𝔼̃[ψ(ξ_N)|ω]-𝔼[ψ(η_N(W))|w]| ≤ |𝔼̃[ψ(ξ_N)|ω]-𝔼[ψ(η_N(Δ_N^*(θ_N))|w]|+ |𝔼̃[ψ(η_N(Δ_N^*(θ_N))|ω]-𝔼[ψ(η_N(W))|w]| ≤o_p(1)+ℒ(Δ^*_N,T(θ_N)|w)-N(0, I^*(θ_0))_BL=o_p(1).Hence the claim follows. 99bns Barndorff-Nielsen, O. E. & Shephard, N. (2001). Non-Gaussian OU-based models and some of their uses in financial economics (with discussion). J. Roy. Statist. Soc. B, 63, 167–241.BKP71 Borwanker, J., Kallianpur, G., & Rao, B. P. (1971). The Bernstein-von Mises theorem for Markov processes. The Annals of Mathematical Statistics, 43, 1241–1253.CleGlo15 Clément, E. and Gloter, A. (2015). Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes. Stochastic Process. Appl. 125, no. 6, 2316–2352.cont Cont, R. & Tankov, P. (2004). Financial modelling with Jump Processes. Chapman & Hall: London.2015arXiv151104992D Deligiannidis, G., Doucet, A., and Pitt, M. K. (2015).The Correlated Pseudo-Marginal Method. arXiv:1511.04992v2 (v2). gander Gander, M. P. S. & Stephens, D. A. (2007). Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika,94, 627–646.golight Golightly, A., & Wilkinson, D. J. (2008). 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Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Ann. Inst. Statist. Math., 63, 431–479. | http://arxiv.org/abs/1707.08788v1 | {
"authors": [
"Ajay Jasra",
"Kengo Kamatani",
"Hiroki Masuda"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170727092213",
"title": "Bayesian inference for Stable Levy driven Stochastic Differential Equations with high-frequency data"
} |
]Richard S. Laugesen and Shiya Liu Department of Mathematics, University of Illinois, Urbana, IL 61801, [email protected] [email protected][2010]Primary 35P15. Secondary 11P21, 52C05 Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. Take a decreasing concave (or convex) curve in the first quadrant and construct a family of curves by rescaling in the coordinate directions while preserving area. Consider the curve in the family that encloses the greatest number of the shifted lattice points: we seek to identify the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead we show it degenerates.Our results handle the p-circle x^p+y^p=1 when p>1 (concave) and also when 0<p<1 (convex). Rescaling the p-circle generates the family of p-ellipses, and so in particular we identify the asymptotically optimal p-ellipses associated with shifted integer lattices. The circular case p=2 with shift -1/2 corresponds to minimizing high eigenvalues in a symmetry class for the Laplacian on rectangles, while the straight line case (p=1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.Shifted lattices and asymptotically optimal ellipses [ December 30, 2023 ====================================================§ INTRODUCTION Among all ellipses centered at the origin with given area, consider the one enclosing the maximum number of positive integer lattice points. Does it approach a circular shape as the area tends to infinity? Antunes and Freitas <cit.>showed the answer is yes. We tackle a variant of the problem in which the lattice is translated by some increments in the x- and y-directions, and show the asymptotically optimal ellipse is no longer a circle but an ellipse whose semi-axis ratio depends explicitly on the translation increments. This optimal ratio succeeds in “balancing” the horizontal and vertical empty strip areas created by the translation of the lattice; see <ref>. The precise statement is given in <ref>. Generalized ellipses obtained by stretching a (concave or convex) smooth curve can be handled by our methods too, in <ref>. The results hold for all positive translations, and for small negative translations that satisfy a computable, curve-dependent criterion. When the curve is a straight line, one arrives at an open problem for right triangles that contain the most lattice points. The shape of these triangles exhibits a surprising clustering behavior as the area tends to infinity, asrevealed by our numerical investigations in<ref>. This clustering conjecture has been investigated recently in the unshifted case by Marshall and Steinerberger <cit.>.<ref> motivates this paper by connecting to spectral minimization problems for the Dirichlet Laplacian, and raises conjectures for the quantum harmonic oscillator and for a whole family of such Schrödinger eigenvalue problems. The recent advances on high eigenvalue minimizationbegan with work of Antunes and Freitas <cit.>, and continued with contributions from van den Berg, Bucur and Gittins <cit.>, van den Berg and Gittins <cit.>, Bucur and Freitas <cit.>, Gittins and Larson <cit.>, Larson <cit.>, and Marshall <cit.>. We show in <ref> that the original result of Antunes and Freitas does not extend to the subclass of symmetric eigenvalues. Instead, the optimal rectangle degenerates in the limit.The lattice point counting estimates in this paper are similar to those used for the Gauss circle problem, which aims for accurate asymptotics on the counting function inside the circle (and other closed curves) as the area grows to infinity. The best known error estimate on the Circle Problem is due to Huxley <cit.>. The lattice counting formulas in our paper differ somewhat from that work, because we consider only one quadrant of lattice points and our regions contain empty strips due to the translation of the lattice. Further, we focus on proving suitable inequalities (rather than asymptotics) for the counting function, in order to prevent the maximizing shape from degenerating. In essence, we develop inequalities that trade off the empty regions in the vertical and horizontal directions. After degeneration has been ruled out, we can invoke asymptotic formulas with error terms that need not be as good as Huxley's in order to prove convergence to a limiting shape.§ RESULTS Consider a strictly decreasing curve Γ lying in the first quadrant with x- and y-intercepts at L. Represent the curve as the graph of y= f(x) where f is strictly decreasing for x∈[0,L]. Denote the inverse function of f as g(y) for y ∈ [0,L]. Now compress the curve by a factor s>0 in the x-direction and stretch it by the same factor in the y-direction:Γ(s) = graph ofsf(sx). Next scale the curve Γ(s) by a factor r>0:rΓ(s)= graph ofrsf(sx/r) . Given numbers σ, τ>-1, consider the translated or shifted positive-integer lattice(+σ) × (+τ) ,which lies in the open first quadrant. Define the shifted-lattice counting function under the curve sΓ(s) to be N(r,s)=number of shifted positive-integer lattice points lying inside or on rΓ(s)=#{ (j,k)∈ℕ×ℕ:k+τ≤ rsf ( (j+σ)s/r ) }. The set S(r) consists of s-values that maximize N(r,s), that is, S(r)= _s>0 N(r,s),r>0 .Writex^- = 0 , x ≥ 0, |x| , x < 0. Our first theorem will say that the maximizing set S(r) is bounded, under either of the following conditions on the shift parameters σ,τ>-1.Γ is concave and strictly decreasing, withmax{f(1-σ^-/2-σ^-L),g(1-τ^-/2-τ^-L)} <2(1/2-σ^–τ^-)L.Γ is convex and strictly decreasing, with min{(1-σ^-)f(1-σ^-/2-σ^-L), (1-τ^-)g(1-τ^-/2-τ^-L)} > 2(σ^- + τ^-) L and μ_f(σ)def=min{ (1+σ)f(1+σ/2+σx)-f(x) : 1+σ/2+σL ≤ x ≤ L} > 0,μ_g(τ)def=min{ (1+τ)g(1+τ/2+τy)-g(y) : 1+τ/2+τL ≤ y ≤ L} > 0.When σ,τ≥ 0, conditions <ref> and <ref> hold automatically (using that 0<f(x)<L and 0<g(y)<L when x,y ∈ (0,L)) and conditions <ref> and <ref> also hold (using that f and g are strictly decreasing and positive). Thus the Parameter Assumptions are significant only when σ<0 or τ<0. They are used to obtain an upper bound on the counting function: see the comments after <ref> and <ref>. If the curve Γ and the shift parameters σ,τ> -1/2 satisfy Parameter Assumption <ref> or <ref>, then for each >0 one has S(r) ⊂[B(τ,σ)^-1-,B(σ,τ)+] for all large r,where B(σ,τ)= 2+σ+τ + √( (2+σ + τ)^2 - 4(σ+1/2)τ ))/2(σ+1/2).The bounding constant B(σ,τ) depends only on the shift parameters, not on the curve Γ. The bounding constant B(0,0)=4 in the unshifted case is consistent with our earlier work <cit.>. <ref> is proved in <ref>. Note it does not assume the curve is smooth. If the curve is smooth, then the optimal stretch set S(r) for maximizing the lattice count is not only bounded but converges asymptotically to a computable value, as stated in the next theorem. First we state the smoothness conditions to be used. Γ is concave, and for some (α ,β) ∈Γ with α,β>0 one has f∈ C^2[0,α], g ∈ C^2[0,β], with f'<0 on (0,α], f”<0 on [0,α], f” monotonic on [0, α],g'<0 on (0,β], g”<0 on [0,β], g” monotonic on [0, β]. Γ is convex, and for some (α ,β) ∈Γ with α,β>0 one has f∈ C^2[α,L], g ∈ C^2[β,L], with f'<0 on [α,L), f”>0 on [α,L], f” monotonic on [α,L],g'<0 on [β,L), g”>0 on [β,L], g” monotonic on [β,L].If the curve Γ and shift parameters σ, τ>-1/2 satisfy either Parameter Assumption <ref> and Concave Condition <ref>, or Parameter Assumption <ref> and Convex Condition <ref>, then the stretch factors maximizing N(r,s) approachs^*=√(τ + 1/2/σ+1/2)as r →∞, with S(r) ⊂ [s^* -O(r^-1/6), s^* + O(r^-1/6)],and the maximal lattice count has asymptotic formulamax_s>0 N(r, s) = r^2 (Γ)-2rL √((σ+1/2)(τ+1/2)) + O(r^2/3).In particular, when the shift parameters σ and τ are equal, the optimal stretch factors for maximizing N(r,s) approach s^*=1 as r →∞. The theorem follows from <ref> below, which has weaker hypotheses. We call the optimally stretched curve (s=s^*) “asymptotically balanced” in terms of the shift parameters, because the optimal shape balances the areas of the empty strips that are created by translation of the lattice: a horizontal rectangle of width rL/s^* and height τ+1/2 has the same area as a vertical rectangle of height rs^*L and width σ+1/2. (The “+1/2” arises from thinking of each lattice point as the center of a unit square.) Further, subtracting these two areas, each of which equals rL √((σ+1/2)(τ+1/2)), gives a heuristic derivation of the order-r correction term in the theorem. [Sufficient condition on shift parameters for the circle] When the curve Γ is the portion of the unit circle in the first quadrant, one takes L=1, f(x) = √(1-x^2), and α=β=1/√(2). Notice f is smooth and concave, with monotonic second derivative. By symmetry it suffices to consider σ≤τ. When σ≤τ≤ 0, Parameter Assumption <ref> says √(1-(1+σ/2+σ)^2) < 2σ+2τ + 1.When σ≤ 0≤τ, equality in Parameter Assumption <ref> would give a straight line. The resulting allowable region of (σ,τ)-shift parameters for <ref> is plotted on the left side of <ref>.[Sufficient condition on shift parameters for p-circle with p=1/2] Suppose Γ is the part of the 1/2-circle lying in the first quadrant, so that L=1, f(x) = (1-x^1/2)^2, and take α=β=1/4. Notice f is smooth and convex, with monotonic second derivative f”(x)=1/2x^-3/2. The region of allowable shift parameters for <ref> can be found numerically from Parameter Assumption <ref>, as shown on the right side of <ref>.Next we define weaker smoothness conditions. Let (α ,β) be a point on the curve Γ with α,β>0. Suppose Γ is concave, and:* f ∈ C^2(0,α], f'<0, f”<0, and a partition 0=α_0<α_1 < … < α_l = α exists such that f” is monotonic on (α_i-1,α_i) for each i=1,2,…,l;* g ∈ C^2(0,β], g'<0, g”<0, and a partition 0=β_0<β_1 < … < β_m = β exists such that g” is monotonic on (β_i-1,β_i) for each i=1,2,…,m; * positive functions δ(r) and ϵ(r) exist such that δ(r) = O(r^-2a_1) , f”(δ(r))^-1 = O(r^1-4a_2) ,ϵ(r) = O(r^-2b_1) , g”(ϵ(r))^-1 = O(r^1-4b_2) , as r →∞, for some numbers a_1,a_2,b_1,b_2>0;* and define a_3=1/2, b_3=1/2.(The second condition in <ref> says that f”(x) cannot be too small as x → 0.)Suppose Γ is convex, and:* f ∈ C^2[α,L), f'<0, f”>0, and a partition α=α_0<α_1 < … < α_l = L exists such that f” is monotonic on (α_i-1,α_i) for each i=1,2,…,l; * g ∈ C^2[β,L), g'<0, g”>0, and a partition β=β_0<β_1 < … < β_m = L exists such that g” is monotonic on (β_i-1,β_i) for each i=1,2,…,m; * positive functions δ(r) and ϵ(r) exist such that δ(r) = O(r^-2a_1) , f”(L-δ(r))^-1 = O(r^1-4a_2) ,ϵ(r) = O(r^-2b_1) , g”(L-ϵ(r))^-1 = O(r^1-4b_2) , as r →∞, for some numbers a_1,a_2,b_1,b_2>0;* and suppose f(x) = L+O(x^2a_3) as x → 0, and g(y) = L+O(y^2b_3) as y → 0, for some numbers a_3,b_3>0.(The last condition says Γ cannot approach the axes too rapidly near the intercept points.) Concave Condition <ref> implies Weaker Concave Condition <ref>, by choosing δ(r)=ϵ(r)=r^-1, a_1 = b_1 = 1/2 and a_2 = b_2 = 1/4, and noting that f”(0) ≠ 0, g”(0) ≠ 0. The same reasoning shows Convex Condition <ref> implies Weaker Convex Condition <ref> with a_3 = b_3 = 1/4, since g(L)=0,g'(L) ≤ 0,g”(L)>0 ⟹ g(L-y) ≥ cy^2for small y>0where c>0, and substituting y=√(x/c) gives L-f(x) ≤√(x/c) for small x>0, and similarly for g. Thus <ref> follows immediately from the next result. If the curve Γ and shift parameters σ, τ>-1/2 satisfy either Parameter Assumption <ref> and Weaker Concave Condition <ref>, or Parameter Assumption <ref> and Weaker Convex Condition <ref>, then the stretch factors maximizing N(r,s) approachs^*=√(τ + 1/2/σ+1/2)as r →∞, withS(r) ⊂[ s^*-O(r^-), s^*+O(r^-) ] where =min{16,a_1,a_2,a_3,b_1,b_2,b_3 } .Further, the maximal lattice count has asymptotic formula max_s>0 N(r, s) = r^2 (Γ)-2rL √((σ+1/2)(τ+1/2)) + O(r^1-2). The proof in <ref> relies on lattice point counting propositions developed in <ref>, <ref> and <ref>.[p-circles] Suppose Γ is the part of the p-circle |x|^p+|y|^p=1 lying in the first quadrant. When p>1 the curve is concave, and satisfies Weaker Concave Condition <ref> by <cit.>. When 0<p<1 it is convex and satisfies Weaker Convex Condition <ref> by <cit.>, noting that a_3=b_3=p/2 since f(x)=1+O(x^p) as x→ 0 and g(y)=1+O(y^p) as y → 0. Thus <ref> applies to each p-circle, p ≠ 1. The allowable shift parameters can be determined numerically from Parameter Assumption <ref> or <ref>, as in <ref> and <ref>. Next we show there can be no “universal” allowable region of negative shifts for <ref>. Specifically, for each choice of negative shifts σ, τ < 0, no matter how close to zero, a curve exists whose optimal stretch parameters grow to infinity or shrink to 0 as r →∞. That is, the optimal curve degenerates in the limit.If -1<σ<0, τ > -1, then a concave C^2-smooth curve Γ exists, with intercepts at L=1, such that for each ϵ∈ (0,1) one has S(r) ⊂ (0,r^ϵ-1) ∪ (r^1-ϵ,∞)for all large r. The construction is given in <ref>. The point of the theorem is that as soon as one of the shift parameters is negative, a concave curve exists for which the maximizing stretch parameters approach either 0 or ∞ as r →∞. For convex curves, we do not know an analogue of <ref>: does a universal allowable region of (σ,τ) parameters exist in which <ref> holds for all C^2-smooth convex decreasing curves? The “bad” curve in <ref> can even be a quarter circle:If the curve Γ is the quarter unit circle, and σ , τ>-1 with either σ≤ -2/5 or τ≤ -2/5, then for each ϵ∈ (0,1) one hasS(r) ⊂ (0,r^ϵ-1) ∪ (r^1-ϵ,∞) for all large r.The proof is in <ref>. And in <ref> we apply this result to Laplacian eigenvalue minimization on rectangles. § CONCAVE CURVES — COUNTING FUNCTION ESTIMATES In order to prove <ref> we need to estimate the counting function. The curve Γ is takento be concave decreasing in the first quadrant, throughout this section. Denote the horizontal and vertical intercepts by x=L and y=M respectively, where L and M are positive but not necessarily equal. Allowing unequal intercepts is helpful for some of the results below. We start with a preliminary r-dependent bound on the maximizing set S(r). The proof of this bound also makes clear why N(r,s) attains its maximum as a function of s, for each fixed r, so that the set S(r) is well defined. If σ, τ > -1 thenS(r)⊂[(1+τ)/rM, rL/(1+σ)] whenever r ≥ (2+σ+τ)/√(LM). The curve rΓ(s) with the particular choice s=√(L/M) has horizontal and vertical intercepts equal to r√(LM). That intercept value is ≥ (2+σ+τ), by assumption on r in this lemma. Hence by concavity, rΓ(s) encloses thepoint (1+σ,1+τ) and so N(r,s)>0 for this particular value of s, which means the maximum of s ↦ N(r,s) is greater than 0. When s> rL/(1+σ), the x-intercept of rΓ(s) is less than 1+σ and so no shifted lattice points are enclosed, meaning N(r,s)=0. Thus the maximum is not attained for such s-values. Arguing similarly with the y-intercept shows the maximum is also not attained when s<(1+τ)/rM. The lemma follows. The last lemma required only that Γ be concave decreasing. Smoothness was not needed. Smoothness is not used in the next proposition either, which gives an upper bound on the counting function and so extends a result from the unshifted case <cit.>.Let σ, τ > -1. The number N(r, s) of shifted lattice points lying inside rΓ(s) satisfies N(r,s) ≤ r^2 (Γ)-C_1rs+σ^- τ^- for all r ≥ (1-σ^-)s/L and s ≥ 1, whereC_1 = C_1(Γ, σ, τ) = 1/2(M-f(1-σ^-/2-σ^-L))-σ^-M-τ^-L.The constant C_1 might or might not be positive. Parameter Assumption <ref> consists of the assumption C_1>0 along with the corresponding inequality for g, in the situation where L=M.First suppose σ≤ 0, τ≤ 0. Write N for the number of shifted lattice points under Γ, and suppose L≥ 1+σ so that ⌊ L-σ⌋≥ 1. Extend the curve Γ horizontally from (0, M) to (σ, M), so that f(σ)=M. Construct triangles with vertices at (i-1+ σ, f(i-1+σ)), (i+σ, f(i+σ)), (i-1+σ, f(i+σ)) for i = 1,… ,⌊ L -σ⌋, as illustrated in <ref>. The rightmost vertex of the final triangle has horizontal coordinate ⌊ L-σ⌋ + σ, which is less than or equal to L. These triangles lie above the unit squares with upper right vertices at shifted lattice points, and lie below the curve Γ due to concavity. HenceN + (triangles) ≤(Γ)-σ(M-τ)-τ(L-σ) -στ, where the correction terms on the right side of the inequality represent the areas of the rectangular regions outside the first quadrant. Letting k = ⌊ L-σ⌋≥ 1, we compute(triangles) =∑_i=1^k1/2(f(i-1+σ) - f(i+σ)) =1/2(M-f(k+σ)) ≥1/2(M- f(1+σ/2+σL))because f is decreasing and k+σ≤ L< k+1+σ implies k+σ > k+σ/k+1+σ L ≥1+σ/2+σ L .Combining <ref> and <ref> provesN ≤(Γ) - σ M -τ L -1/2(M- f(1+σ/2+σL)) +στ . Now we replace Γ with the curve rΓ(s), meaning we replace N, L, M, f(x) with N(r,s), rs^-1L, rsM, rsf(sx/r) respectively, thereby obtaining the desired estimate <ref> (noting that L/s ≤ Ls since s ≥ 1). The restriction L≥ 1+σ becomes r ≥ (1+σ)s/L under the rescaling, and so we have proved the proposition in the case σ≤ 0, τ≤ 0. When σ> 0, τ> 0, the number of shifted lattice points inside rΓ(s) is less than or equal to the number when there is no shift (σ=τ=0), simply because the curve is decreasing. Thus this case of the proposition follows from the “σ,τ≤ 0” case above. When σ > 0, τ≤ 0, the number of shifted lattice points inside rΓ(s) is less than or equal to the number for σ = 0 with the same τ value, and so this case of the proposition follows also from the “σ,τ≤ 0” case above. A similar argument holds when σ≤ 0, τ > 0.Let σ, τ > -1. If s is bounded above and bounded below away from 0, as r →∞, then the number N(r, s) of shifted lattice points lying inside rΓ(s) satisfies N(r,s) ≤ r^2(Γ)-r(s^-1τ L + s(σ+1/2)M )+ o(r) .Take c>1 and suppose c^-1<s<c throughout the rest of the proof. Suppose σ,τ≤ 0. Let K ≥ 1. Repeat the proof of <ref> except with the initial supposition L ≥ 1+σ replaced by L ≥ K + σ, and do not assume s ≥ 1. One finds N(r,s) ≤ r^2 (Γ)-D_K rs-τ L r s^-1 + στfor all r ≥ (K+σ)s/L, where D_K = D_K(Γ, σ) = 1/2(M-f(K+σ/K+1+σL))+σ M.We deduce lim sup_r →∞sup_s<c1/r( N(r,s) - r^2(Γ) + r(s(σ+1/2)M +s^-1τ L ) ) ≤c/2 f(K+σ/K+1+σL) . The last expression can be made arbitrarily small by choosing K sufficiently large (recall f(L)=0), and so the left side is ≤ 0. That proves the corollary when σ,τ≤ 0. Suppose σ >0, τ≤ 0. We will relate this case to the previous one. To emphasize the dependence of the counting function on the shift parameters, write N_σ,τ(r,s) for the counting function that was previously written N(r,s). Addingcolumns of shifted lattice points at x=σ-⌈σ⌉ + 1, …, σ-1, σ gives the counting function N_σ,τ(r,s) where σ=σ-⌈σ⌉∈ (-1,0]. This counting function is related to the original one by N_σ,τ(r,s) =N_σ,τ(r,s) + ∑_i=0^⌈σ⌉ - 1⌊ rsf(s(σ-i)/r)-τ⌋,=N_σ,τ(r,s) + ⌈σ⌉ rsM+o(r), as r→∞, since s is bounded above and f is continuous with f(0)=M. Since σ,τ≤ 0, we may apply <ref> with σ replaced by σ to obtainN_σ,τ(r,s) ≤ r^2(Γ)-r(s^-1τ L + s(σ-⌈σ⌉ +1/2)M )+ o(r) as r→∞.Combining the above two formulas, we prove the corollary for σ>0, τ≤ 0. When σ≤ 0, τ > 0, simply add the appropriate rows instead of columns and argue like above using ⌈τ⌉ instead of ⌈σ⌉, and using the boundedness of s^-1. Similarly, one can treat the case σ>0, τ > 0.The next proposition gives an asymptotic approximation to N(r,s), assuming the curve is concave decreasing and has suitably monotonic second derivative.Let σ, τ> -1 and 0 ≤ q < 1. If Weaker Concave Condition <ref> holds and s+s^-1 = O(r^q) thenN(r,s)=r^2 (Γ)-r( s^-1(τ + 1/2)L+s(σ+1/2)M)+ O(r^Q) as r →∞, whereQ = max{23, 12+32q , 1-2a_1+q,1-2a_2+32q, 1-2b_1+q, 1-2b_2+32q }.Special cases: (i) If q=0 then Q = 1-2e where e = min{16, a_1, a_2,b_1, b_2 }. (ii) If Concave Condition <ref> holds then Q=max{23, 12+32q }.The numbers a_1,a_2,b_1,b_2 come from Weaker Concave Condition <ref>. That Condition also involves a point (α,β) ∈Γ with α,β>0, which we use in the following proof.The idea is to translate and truncate the curve rΓ(s) as in <ref>, in order to reduce to an unshifted lattice problem.Then we invoke known results from our earlier paper <cit.> (which builds on work of Krätzel <cit.> and a theorem of van der Corput). Step 1 — Translating and truncating. Notice rs →∞ and rs^-1→∞ as r →∞, since s=O(r^q) and s^-1=O(r^q) with q<1. Thus by taking r large enough, we insurers^-1 g ( s^-11+τ/r) > rs^-1α > 1+σ ,rs f ( s 1+σ/r) > rs β > 1+τ .For all large r one also has δ(r)<α and ϵ(r)<β, by Weaker Concave Condition <ref>. Given a large r satisfying the above conditions, and a corresponding s>0, we letα = rs^-1α - (1+σ) , β = rs β - (1+τ) ,andL = rs^-1 g ( s^-11+τ/r) - (1+σ) , M = rs f ( s 1+σ/r) - (1+τ) ,so that 0 < α < L,0 < β < M .Consider the point O=(1+σ,1+τ) in the first quadrant. Regard this point as the new origin, and let Γ be the portion of rΓ(s) lying in the new first quadrant — see <ref>. That is, Γ is the graph of f(x) = rs f ( s x+1+σ/r) - (1+τ) ,0 ≤ x ≤L ,and also of its inverse function g(y) = rs^-1 g ( s^-1y+1+τ/r) - (1+σ) ,0 ≤ y ≤M .Notice (α,β) ∈Γ, since f(α)=β. Write N for the number of positive-integer lattice points under the curve Γ. That is, N = #{ (j,k) ∈× : k ≤f(j) } .This N does not count the lattice points in the first column or row, which arise from j=0 or k=0.Weaker Concave Condition <ref> guarantees that f is C^2-smooth on the interval [0,α], with f^' < 0 and f^''<0 there, and similarly g is C^2-smooth on [0,β] with g^' < 0 and g^''<0 there. Next, we partition the interval [0,α] as 0=α_0 < α_1 < … < α_l = α where the interior partition points are chosen to be the elements of{ rs^-1α_i - (1+σ) : i=1,…,l-1 }that happen to lie between 0 and α. Observe f^'' is monotonic on each subinterval of the partition, by Weaker Concave Condition <ref>. Similarly, g^'' is monotonic on each subinterval of the corresponding partition 0=β_0 < β_1 < … < β_m = β of the interval [0,β].Let δ = [ rs^-1δ(r) - (1+σ) ]^+ , ϵ = [ rs ϵ(r) - (1+τ) ]^+ ,so that 0 ≤δ<α and 0 ≤ϵ<β. To relate some of these old and new quantities, we denote antiderivatives of f, g byF(x) = ∫_0^x f(t)t, G(y) = ∫_0^y g(t)t ,and observe that ()=r^2(Γ) - r^2 (F((1+σ) s/r)+ G((1+τ) s^-1/r))+ (1+σ)(1+ τ),'(x) = s^2f'(sx+1+σ/r),”(x)= s^3/rf”(sx+1+σ/r), ∫_0^ | ”(x)|^1/3 x= r^2/3∫_(1+σ)s/r^α | f ”(x)|^1/3 x ≤ r^2/3∫_0^α | f ”(x)|^1/3 x, ∑_i=1^l1/|”(_i)|^1/2 ≤∑_i=1^lr^1/2s^-3/2/|f”(α_i)|^1/2, and similarly forexcept with s replaced by s^-1.Step 2 — Estimating the counting function. Applying part (a) of <cit.> to the curveand using the preceding relationships, we get |-r^2 (Γ)+ r^2 (F((1+σ) s/r)+ G((1+τ) s^-1/r)) + r/2(s f((1+σ)s/r) + s^-1g((1+τ)s^-1/r) ) |≤ 6r^2/3(∫_0^α |f”(x)|^1/3 x+∫_0^β |g”(y)|^1/3 y)+175r^1/2(s^-3/2/|f”(δ(r))|^1/2+s^3/2/|g”(ϵ(r))|^1/2)+525r^1/2(∑_i=1^ls^-3/2/|f”(α_i)|^1/2+∑_j=1^ms^3/2/|g”(β_j)|^1/2) +1/4(∑_i=1^l s^2|f'(α_i)|+∑_j=1^m s^-2|g'(β_j)|)+r/2(s^-1δ(r)+sϵ(r))+l+m + 1/2(1+σ)+1/2(1+τ)+(1+σ)(1+τ)+1 ,where we dealt with the term involving |”(δ)|^-1/2 in <cit.> as follows. One has ”(δ)=r^-1s^3 f”(z) where z = r^-1s(δ+1+σ) ≥δ(r), and so by monotonicity of f” on each subinterval of the partition (as assumed in Weaker Concave Condition <ref>) one concludes|”(δ)|≥ r^-1s^3 min{ |f”(δ(r))|,|f”(α_1)| , …,|f”(α_l )| }.Thus the term involving |”(δ)|^-1/2 can be estimated by the sum of terms involving |f”(δ(r))|^-1/2 and |f”(α_i)|^-1/2. The right side of <ref> already has the desired order O(r^Q), by direct estimation and using that s+s^-1=O(r^q) and 2q < 12+32q since q<1. Step 3 — Understanding the left side of inequality <ref>.It remains to deal with the terms on the left of <ref>. Clearly N(r,s) andcount the same lattice points, except that N(r,s) also counts the points in the first row and column. That is,= N(r,s) - ⌊ rsf((1+σ)s/r)-τ⌋ - ⌊ rs^-1 g((1+τ)s^-1/r)-σ⌋ +1= N(r,s) -rsf((1+σ)s/r) -τ - rs^-1 g((1+τ)s^-1/r)-σ + ρ(r,s) for some number ρ(r,s) ∈ [1,3]. Substitute this formula into the left side of <ref>. Substitute also the following expressions, which are obtained from <ref>:rsf((1+σ)s /r)=rsM +O(s^2),r^2F((1+σ) s/r)=rs (1+σ) M +O(s^2), and similarly for g and G.The proposition now follows straightforwardly, since O(s^2)=O(r^2q). If f is decreasing and concave on [0,L] then f(x)=f(0) +O(x),F(x)=f(0)x + O(x^2), as x → 0,where F(x) = ∫_0^x f(t)t is the antiderivative of f(x). The difference quotient (f(x)-f(0))/x is a decreasing function of x since f is concave, and it is less than or equal to 0 since f is decreasing. Hence the difference quotient is bounded, and so f(x)=f(0) +O(x). Integrating completes the proof.§ CONVEX CURVES — COUNTING FUNCTION ESTIMATES Assume the curve Γ is convex decreasing, throughout this section. We will prove estimates for convex curves analogous to the work in <ref> for concave curves. <ref> below is an improvedr-dependent bound on the optimal stretch factors, generalizing Ariturk and Laugesen's lemma from the unshifted situation <cit.>. By “improved” we refer to the upper and lower bounds: for instance, when σ =0 the upper bound in <ref> improves on the bound in <ref> by a factor of 2. This tighter bound on the optimal stretch factor gives us more flexibility when deriving the two-term counting estimate in <ref>. In the next lemma we assume for simplicity that the x- and y-intercepts are both L, so that we need not change the definitions of μ_f(σ) and μ_g(τ) in <ref>. If σ, τ > -1 with μ_f(σ)> 0 and μ_g(τ) > 0, thenS(r)⊂[ 2+τ/rL , rL/2+σ]wheneverr ≥max((2+σ)√(2(1+τ)/Lμ_f(σ)), (2+τ)√(2(1+σ)/Lμ_g(τ))) .Claim 1: N(r,s)=0 if s ∈( 0,(1+τ)/rL ] or s ∈[ rL/(1+σ),∞). Indeed, the curve rΓ(s) has x- and y-intercepts at rL/s and rsL, respectively, and so if rL/s ≤ 1+σ or rsL ≤ 1+τ then the point (1+σ,1+τ) is not enclosed by the curve and so the lattice count N(r,s) is zero. Claim 2: if <ref> holds and s∈( rL/(2+σ),rL/(1+σ) ) then N(r,s) < N(r,1+σ/2+σ s) .To prove this claim, notice the x-intercept satisfies 1+σ < rL/s < 2+σ ,and so only the first column of shifted lattice points (the points with x-coordinate at 1+σ) can contribute to the count inside rΓ(s). Hence N(r,s)= ⌊ rsf((1+σ)s/r) - τ⌋. Meanwhile, if we count shifted lattice points in the first two columns (where x=1+σ and x=2+σ) we findN(r,1+σ/2+σ s) ≥⌊ rs1+σ/2+σf( (1+σ)^2s/(2+σ )r)- τ⌋ + ⌊ rs1+σ/2+σf( (1+σ)s/r) - τ⌋> rs1+σ/2+σf( (1+σ)^2s/(2+σ )r) + rs1+σ/2+σf( (1+σ)s/r) -2τ -2 = rsf((1+σ)s/r) + rs/2+σ( (1+σ)f( (1+σ)^2s/(2+σ )r) -f( (1+σ)s/r) ) -2(1+τ)≥ rsf((1+σ)s/r) + rs/2+σμ_f(σ) -2(1+τ) >rsf((1+σ)s/r) ≥ N(r,s) , where to get the final line we use that rs/2+σμ_f(σ)>2(1+τ), which follows from s>rL/(2+σ) and the lower bound on r in <ref>. The proof of Claim 2 is complete. Claim 3: if <ref> holds and s ∈( (1+τ)/rL , (2+τ)/rL ) then N(r,s) < N(r,2+τ/1+τ s ) .The proof is analogous to Claim 2, except counting in rows instead of columns.Claim 4: if <ref> holds then the maximizing s-values for N(r,s) lie in the interval [ (2+τ)/rL, rL/(2+σ) ]. To see this, note that N(r,s^')>0 for some s^'>0, by the strict inequality in Claim 2, and so the maximum does not occur in the intervals considered in Claim 1. The maximum does not occur in the interval considered in Claim 2, as that claim itself shows, and similarly for Claim 3. Thus the maximum must occur in the remaining interval, which proves Claim 4 and thus finishes the proof of the lemma. The next bound generalizes work of Ariturk and Laugesen <cit.> from the unshifted situation (σ=τ=0) to the shifted case. Let σ,τ> -1. The number N(r, s) of shifted lattice points lying inside rΓ(s) satisfies N(r,s) ≤ r^2 (Γ)-C_2rs + σ^- τ^- for all r ≥ (2-σ^-)s/L and s≥ 1, where C_2 = C_2(Γ, σ, τ) = 1/2(1-σ^-)f(1-σ^-/2-σ^-L)-σ^-M-τ^-L .The constant C_2 need not be positive. That is why hypothesis <ref> in Parameter Assumption <ref> includes (for L=M) the assertion that C_2>0.First consider σ≤0, τ≤0. Write N for the number of shifted lattice points under Γ. Suppose L≥ 2+σ. Extend the curve horizontally from (0, M) to (σ, M), so that f(σ)=M. Construct a trapezoid (see <ref>) with vertices at ( σ,f(1+σ) ), ( 1+σ, f(1+σ) ), (0,h), (σ,h) where h=f(1+σ)-(1+σ) f'(1+σ). Also construct triangles with vertices (i-1+σ, f(i+σ)), (i+σ, f(i+σ)), (i-1+ σ, f(i+σ)-f'(i+σ)), where i = 2,… ,⌊ L -σ⌋. These triangles lie above the squares with upper right vertices at the shifted lattice points, and likebelow the curve by convexity, as <ref> illustrates. Hence N + (trapezoid and triangles) ≤(Γ)-σ(M-τ)-τ(L-σ) -στLet k = ⌊ L-σ⌋≥ 2, so that k+σ≤ L < k+σ+1. Then (trapezoid)= 1/2(base+top) · (height)= - 1/2 (1-σ) · (1+σ) f'(1+σ) ≥1/2 (1+σ) ( f(1+σ) - f(2+σ) ) by convexity, and using that 1-σ≥ 1. Further, convexity implies (triangles) = - 1/2∑_i=2^kf'(i+σ) ≥1/2∑_i=2^k-1(f(i+σ)-f(i+1+σ) ) + 1/2(f(k+σ)-f(L) )=1/2f(2+σ) . Hence (trapezoid) + (triangles) ≥1/2(1+σ)f(1+σ) - 1/2σ f(2+σ) ≥1/2(1+σ)f(1+σ/2+σL) - 1/2σ f(2+σ/2+σL) since f is decreasing and L/(2+σ) ≥ 1. Combining <ref> and <ref> and using f(L)=0 provesN ≤(Γ) - σ M -τ L -1/2(1+σ)f(1+σ/2+σL) +στ .Now replace Γ with the curve rΓ(s), meaning replace N, L, M, f(x) with N(r,s), rs^-1L, rsM, rsf(sx/r) respectively. Using s ≥ 1, we know L/s ≤ Ls; the assumption L ≥ 2+σ becomes r ≥ (2+σ)s/L. Thus we obtain <ref> in the case σ≤ 0, τ≤ 0. One may now deduce the remaining cases as was done in the proof of <ref>. Let σ, τ > -1. If s is bounded above and bounded below away from 0, as r →∞, then the number N(r, s) of shifted lattice points lying inside rΓ(s) satisfies N(r,s) ≤ r^2(Γ)-r(s^-1τ L + s(σ+1/2)M )+ o(r) .Fix c>1 and assume c^-1<s<c in the rest of the proof. Suppose σ,τ≤ 0, and let K ≥ 2. Repeat the proof of <ref> except with the initial requirement L ≥ 2+σ replaced by L ≥ K + σ, and do not assume s ≥ 1. The argument gives N(r,s) ≤ r^2 (Γ) -E_K rs - τ L rs^-1 +στ .for all r ≥ (K+σ)s/L, where E_K = E_K(Γ, σ) = 1/2(1+σ)f(1+σ/K+σL) - 1/2σ f(2+σ/K+σL) + σ M .Hence lim sup_r →∞sup_s<c1/r( N(r,s) - r^2(Γ) + r(s(σ+1/2)M +s^-1τ L ) ) ≤c/2| M - (1+σ)f(1+σ/K+σL) + σ f(2+σ/K+σL) | . The last expression can be made arbitrarily small by choosing K sufficiently large (recall f(0)=M), and so the left side is ≤ 0, which proves the corollary when σ,τ≤ 0. By arguing as in the proof of <ref>, one handles the other three cases for σ and τ. In the next proposition we state a two-term asymptotic for lattice point counting under convex curves. Let σ, τ> -1. If Weaker Convex Condition <ref> holds and s+s^-1 = O(1) thenN(r,s)=r^2 (Γ)-r( s^-1(τ + 1/2)L+s(σ+1/2)M)+ O(r^1-2) as r →∞, where = min{16, a_1, a_2, a_3,b_1, b_2, b_3 }. In particular, if Convex Condition <ref> holdsthen <ref> holds with =16.<ref> does not assume the intercepts L and M are equal, and so we modify Weaker Convex Condition <ref> by taking each occurrence of “L” that relates to the function g and changing it to “M”, and changing the a_3-condition to f(x) = M+O(x^2a_3). We use the idea from <ref>: translate and truncate the curve rΓ(s) to reduce to an unshifted lattice problem, and then use results from Ariturk and Laugesen's paper <cit.>. Assume rΓ(s) does not pass through any point in the shifted lattice. This assumption will be removed in the final step of the proof. Step 1 — Translating and truncating. Keep the notation from the proof of <ref>, except redefine the quantities δ and ϵ to beδ = [ L + 1+σ - rs^-1 (L - δ(r))]^+ , ϵ = [ M + 1+τ - rs (M - ϵ(r))]^+ .Arguing as in Step 1 of that proof, we have 0 < α < ⌊L⌋,0 < β < ⌊M⌋ , by taking r large enough, and also0 ≤δ< ⌊L⌋-α, 0 ≤ϵ< ⌊M⌋-β . Step 2 — Estimating the counting function. Recall F represents the antiderivative of f, defined in <ref>. Applying part (a) of <cit.> to the curveand using the relationships between the unshifted and shifted quantities as in the proof of <ref>, we get |-r^2 (Γ) + r^2(F((1+σ)s/r) + G((1+τ)s^-1/r))+ r/2(sf((1+σ)s/r) +s^-1g((1+τ)s^-1/r)))|≤ 6r^2/3(∫_α^L f”(x)^1/3 x+∫_β^M g”(y)^1/3 y) +175r^1/2(s^-3/2/f”(L- δ(r))^1/2+ s^3/2/g”(M-ϵ(r))^1/2)+700r^1/2(∑_i=0^l-1s^-3/2/f”(α_i)^1/2+∑_j=0^m-1 s^3/2/g”(β_j)^1/2)+1/4( ∑_i=0^l-1s^2|f'(α_i)|+∑_j=0^m-1s^-2|g'(β_j)| ) +1/2r(s^-1δ(r)+sϵ(r))+ l+m +1/2(1+σ)+1/2(1+τ)+(1+σ)(1+τ)+5+rs^-1g((1+τ)/rs)-(1+σ)/rsf((1+σ)s/r)-(1+τ)+rsf((1+σ)s/r)-(1+τ)/rs^-1g((1+τ)/rs)-(1+σ),where we estimated the term involving ”(-δ)^-1/2 as follows. One has ”(-δ) =r^-1s^3 f”(z) where z=r^-1s (L-δ+1+σ) ≤ L-δ(r) ,and so by monotonicity of f” on each subinterval of the partition (as assumed in Weaker Convex Condition <ref>) one concludes”(-δ)≥ r^-1s^3 min{ f”(L-δ(r)),f”(α_0),…,f”(α_l-1)}.Thus the term involving ”(-δ)^-1/2 can be estimated by the sum of terms involving f”(L-δ(r))^-1/2 and f”(α_i)^-1/2.The right side of <ref> has the form O(r^1-2e), by arguing directly with s+s^-1=O(1) and the assumptions in Weaker Convex Condition <ref>, and estimating the last two terms in <ref> byrs^-1g((1+τ)/rs)-(1+σ)/rsf((1+σ)s/r)-(1+τ)=s^-1L-o(1)/sM-o(1) = O(1)and similarly with f and g interchanged. Step 3 — Understanding the left side of inequality <ref>. The terms on the left of <ref> are dealt with in the same manner as in Step 3 of <ref>, except replacing <ref> with the last assumption in Weaker Convex Condition <ref>, as follows. Substituting x=(1+σ)s/r into f(x)=M+O(x^2a_3) and into F(x)=Mx+O(x^1+2a_3) givesrsf((1+σ)s /r)=rsM +O(r^1-2a_3),r^2F((1+σ) s/r)=rs (1+σ) M +O(r^1-2a_3),since s+s^-1=O(1). One argues similarly for g and G. Thus we have finished the proof under the assumption that rΓ(s) passes through no lattice points. Step 4 — Finishing the proof. Now drop the assumption that rΓ(s) passes through no lattice points. Notice the counting function N(r,s) is increasing in the r-variable. Fix the r and s values, and modify the functions δ(·) and ϵ(·) to be continuous at r. For sufficiently small η>0 we have N(r+η,s)=N(r,s), because the r-variable would have to increase by some positive amount for the curve rΓ(s) to reach any new lattice points. Since no lattice points lie on the curve (r + η)Γ(s), Steps 1–3 above apply to that curve. Hence by continuity as η→ 0, the conclusion of the proposition holds also for rΓ(s).§ LOWER BOUND ON THE COUNTING FUNCTION FOR DECREASING Γ We need a rough lower bound on the counting function, in order to prove boundedness of the maximizing set in <ref>. Assume the curve Γ is strictly decreasing in the first quadrant, and has x- and y-intercepts at L and M. The intercepts need not be equal, in the next lemma.The number N(r, s) of shifted lattice points lying inside rΓ(s) satisfiesN(r,s)≥ r^2(Γ)-r(s^-1(1+τ)L+s(1+σ)M),r,s>0. We split the proof into two cases, and later rescale to handle the general curve. Write N for the number of shifted lattice points under Γ. Case I: The point (1+σ,1+τ) lies outside the curve Γ, and so N=0. Then the rectangles with vertices (0,0),(L,0),(L,1+τ),(0,1+τ) and (0,0),(1+σ,0),(1+σ,M),(0,M) cover Γ since the curve is decreasing, and so by comparing areas one hasN +(1+τ)L + (1+σ)M≥(Γ).Case II: The point (1+σ,1+τ) lies inside the curve. We shift the origin to =(1+σ, 1+τ) and draw new axes, denoting the x- and y-intercepts on the new axes byand ; see <ref>. The part of Γ lying in the new first quadrant is .Each lattice point corresponds to a square whose lower left vertex sits at that point. These squares coversince the curve is strictly decreasing. The remaining area under Γ is covered by the two rectangles described in Case I. The sum of the areas of the squares and rectangles must exceed the area under Γ, and so <ref> holds once again. To complete the proof, simply replace the curve Γ with rΓ(s), meaning that in <ref> we replace N, L, M with N(r,s), rs^-1L, rsM respectively. The lemma follows.§ PROOF OF <REF> We prove the theorem in two parts: first for concave curves, and then for convex curves. When Γ is concave, we will utilize the bound on S(r) in <ref> and the two-term upper bound on the counting function in <ref>, along with the improved upper bound in <ref> and the rough lower bound on the counting function in <ref>. Recall the intercepts are assumed equal (L=M) in this theorem.§.§ Part 1: Γ is concave and Parameter Assumption <ref> holds The proof has two steps. Step 1 shows S(r) is bounded above and below away from 0, for large r. Step 2 uses this boundedness to improve the asymptotic bound on S(r), revealing that it depends only on σ and τ and not the curve Γ. Step 1. Take s∈ S(r) and suppose r ≥ (2+σ+τ)/L. Then <ref> says s≤ rL/(1+σ), so that r ≥(1+σ)s/L≥(1-σ^-)s/L.If s≥ 1 then <ref> implies N(r,s)≤ r^2 (Γ)-C_1rs + σ^- τ^- . Parameter Assumption <ref> guarantees here that C_1>0.The lower bound in <ref> with “s = 1” saysN(r,1) ≥ r^2 (Γ)-(2+σ+τ)Lr. Since s∈ S(r) is a maximizing value, one has N(r,s) ≥ N(r,1), and so the preceding two inequalities give s ≤(2+σ+τ)L/C_1 + σ^- τ^- L/(2+σ+τ)C_1when r ≥ (2+σ+τ)/L and s ≥ 1. Thus S(r) is bounded above for all large r. Similarly if s∈ S(r) then s^-1 is bounded above, by interchanging the roles of the horizontal and vertical axes in the argument above. Thus the set S(r) is bounded below away from 0, for large r.Step 2. The number s = lim sup_s ∈ S(r), r →∞ sis finite and positive by Step 1. Combining the inequality N(r,s) ≥ N(r,1) with estimate <ref> and <ref> (which relies on the boundedness of S(r)) we obtain(σ+1/2) s^2 - (2+σ+τ) s + τ≤ 0after letting r →∞. Notice σ+1/2>0 by hypothesis in <ref>. Hence s is bounded above by the larger root of the quadratic; that is,s≤ B(σ,τ)= 2+σ+τ + √( (2+σ + τ)^2 - 4(σ+1/2)τ)/2(σ+1/2) .Similarly lim sup_r→∞ s^-1≤ B(τ,σ), by interchanging the roles of the axes. The proof of <ref> is complete, in the concave case. §.§ Part 2: Γ is convex and Parameter Assumption <ref> holdsTake s ∈ S(r) and suppose r satisfies (<ref>), recalling there that μ_f(σ) and μ_g(τ) are positive by Parameter Assumption <ref>. Now proceed as in Part 1 of the proof, simply replacing <ref>, <ref> and <ref> with <ref>, <ref> and <ref>, respectively. § PROOF OF <REF>Recall the intercepts are equal, L=M, in this theorem. The optimal stretch parameters are bounded above and bounded below away from 0 as r →∞, by <ref>. (It suffices to use the curve-dependent bound from Step 1 of that proof; we do not need the curve-independent bound B(σ,τ) from Step 2.) Hence by <ref> (if Γ is concave) or <ref> (if Γ is convex),N(r,s) =r^2 (Γ)-rL(s^-1(τ + 1/2)+s(σ+1/2))+ O(r^1-2) when s ∈ S(r); this estimate holds also when s>0 is any fixed value. Thus for s ∈ S(r) and s^*=√((τ + 1/2)/(σ+1/2)) we have N(r,s) ≤ r^2 (Γ)-rL(s^-1(τ+ 1/2)+s(σ+1/2))+ O(r^1-2), N(r,s^*)≥ r^2 (Γ)-2rL√((τ+1/2)(σ + 1/2)) + O(r^1-2), as r →∞. Notice N(r, s^*)≤ N(r,s) because s ∈ S(r) is a maximizing value, and so s^-1(τ + 1/2)+s(σ+1/2) ≤ 2√((τ + 1/2)(σ+1/2))+O(r^-2) . Therefore s =s^*+ O(r^-), by <ref> below with a = τ + 1/2, b = σ+1/2. For the final statement of the theorem, when s∈ S(r) one has 2√((τ + 1/2)(σ+1/2))≤ s^-1(τ + 1/2)+s(σ+1/2)≤ 2√((τ + 1/2)(σ+1/2))+O(r^-2)by the arithmetic–geometric mean inequality and <ref>. Multiplying by rL and substituting into <ref> gives the asymptotic formula <ref>.When a, b, s >0 and 0 ≤ t ≤√(ab), s^-1a + sb ≤ 2√(ab) +t ⟹| s - √(a/b)| ≤ 3(ab)^1/4/ b√(t) . By taking the square root on both sides of the inequality ( (s^-1a)^1/2-(sb)^1/2)^2 = s^-1a + sb - 2√(ab)≤ tand then using that the number (ab)^1/4 lies between (s^-1a)^1/2 and (sb)^1/2 (because it is their geometric mean), we find| (ab)^1/4 - (sb)^1/2 | ≤ t^1/2 .Hence (ab)^1/4 - t^1/2≤ (sb)^1/2≤ (ab)^1/4 + t^1/2. Squaring and using that t ≤ (ab)^1/4t^1/2 (when t ≤√(ab)) proves the lemma.§ PROOF OF <REF> AND <REF>§.§ Proof of <ref>Fix σ∈ (-1,0) and τ>-1. Since 0<1+σ<1, we may choose m ∈ large enough that (1+σ)^2m < 1/2m+1 . Defining ϕ(x) = 1-x^2m for 0 ≤ x ≤ 1, one checks ϕ(1+σ) > area under graph of ϕ. Thus one may choose 0<δ<1 small enough that the functionf(x) = 1-δ x^2 - (1-δ) x^2m ,0 ≤ x ≤ 1 ,satisfiesf(1+σ) > area under graph of f .Observe f is smooth and strictly decreasing, with f” < 0 on [0,1], so that its graph Γ is concave. The inverse function g satisfies the same conditions. The curve rΓ(r) is the graph of r^2f(x) for 0 ≤ x ≤ 1. This curve contains only the first column of shifted lattice points (the points with x-coordinate 1+σ), and soN(r,r)= ⌊ r^2 f(1+σ) - τ⌋≥ r^2 f(1+σ) - τ - 1.Now fix 0<ϵ<1. If s ∈ [r^ϵ-1,r^1-ϵ] then s + s^-1 = O(r^1-ϵ), and so <ref> with q=1-ϵ and L=M=1 gives that N(r,s)= r^2 (Γ) - r(s(σ+1/2) + s^-1(τ + 1/2)) + O(r^2-3ϵ/2)= r^2 (Γ) + o(r^2) . Since (Γ) < f(1+σ), we conclude that for all large r,N(r,s) < N(r,r)and so s ∉ S(r), which proves the theorem.§.§ Proof of <ref>By symmetry, we may suppose σ≤ -2/5.The argument is the same as for <ref>, except now the curve is a quarter circle, described by f(x) = √(1-x^2). The only point to check in the proof is that f(1+σ) > (Γ)when -1<σ≤-2/5, which reduces to the fact that 4/5 > π/4.§ NUMERICAL EXAMPLES, AND CONJECTURES FOR TRIANGLES (P=1) <ref>(a) illustrates the convergence of s ∈ S(r) to s^*, when Γ is a quarter circle and the shifts are positive. The convergence is erratic, while still obeying the decay rate O(r^-1/6) as promised by <ref>. <ref>(b) shows the degeneration that can occur when the shifts are negative, as explained in <ref>. Quite different behavior occurs when Γ is a straight line with slope -1, in other words, when the curve is the 1-ellipse described by f(x)=1-x, which is not covered by our results in <ref>. Here N(r,s) counts the shifted lattice points inside the right triangle with vertices at (r/s,0), (0,rs) and the origin. <ref> insures the maximizing set S(r) is bounded above and below, being contained in [B(τ,σ)^-1-,B(σ,τ)+] for all large r. This boundedness depends on Parameter Assumption <ref> holding, which in this case says(2-max(σ^-,τ^-))(1-2σ^- - 2τ^-) > 1 .In particular, S(r) is bounded for the 1-ellipse if σ=τ>-0.117. <ref> for convergence of S(r) does not apply, though, to the 1-ellipse. The numerical plots in <ref> suggest S(r) might not converge, and might instead cluster at many different heights. Are those heights determined by a number theoretic property of some kind? (Such behavior would be particularly interesting when the shifts are σ=τ=-1/2, since those shifted lattice points correspond to energy levels of harmonic oscillators in 2-dimensions, as explained in the next section.) For a more detailed discussion and precise conjecture on this open problem for p=1 in the unshifted case, see our work in <cit.> and the partial results of Marshall and Steinerberger <cit.>.The numerical method that generated the figures is described in <cit.> for p=1. It adapts easily to handle other values of p, in particular p=2 (the circle), and the code is available in <cit.>.§ FUTURE DIRECTIONS — OPTIMAL QUANTUM OSCILLATORS§.§ Literature on spectral minimizationAntunes and Freitas <cit.> investigated the problem of maximizing the number of first-quadrant lattice points in ellipses with fixed area, and showed that optimal ellipses must approach a circle as the radius approaches infinity. In terms of eigenvalues of the Dirichlet Laplacian on rectangles having fixed area, their result says that the rectangle minimizing the n-th eigenvalue must approach a square as n →∞. Their intuition is that high eigenvalues should be asymptotically minimal for the “most symmetrical” domain. Besides Antunes and Freitas's work on eigenvalue minimization <cit.>, we mention that van den Berg and Gittins <cit.> showed the cube is asymptotically minimal in 3-dimensions as n →∞, while Gittins and Larson <cit.> handle all dimensions ≥ 2. Van den Berg, Bucur and Gittins <cit.> proved an analogous asymptotic maximization result for eigenvalues of the Neumann Laplacian on rectangles. Bucur and Freitas <cit.> showed for general domains in 2 dimensions that eigenvalue minimizing regions become circular in the limit, under a perimeter normalization. Larson <cit.> shows among convex domains that the disk asymptotically minimizes the Riesz means of theeigenvalues, for Riesz exponents ≥ 3/2. Eigenvalue minimizing domains have been studied numerically by Oudet <cit.>, Antunes and Freitas <cit.>, and Antunes and Oudet <cit.>, <cit.>. Incidentally, Colbois and El Soufi <cit.> proved subadditivity of n ↦λ_n^* (the minimal value of the n-eigenvalue), from which it follows that the the famous Pólya conjecture λ_n ≥ 4π n / in 2-dimensions would be a corollary of the conjecture that the eigenvalue minimizing domain approaches a disk as n →∞. A different way of extending the work of Antunes and Freitas is to investigate lattice point counting inside more general curves, not just ellipses. Laugesen and Liu <cit.> and Laugesen and Ariturk <cit.> showed this can be done for p-ellipses with p ≠ 1, and for more general concave and convex curves in the first quadrant too. Marshall <cit.> has extended the results to strongly convex domains in all dimensions, using somewhat different methods. For more on the literature see <cit.>.§.§ Spectral application of <ref>This paper sheds new light on the rectangular result of Antunes and Freitas. Consider the family of rectangles defined by[0,2π s^-1] × [0,2π s]for s>0. The “even–even” eigenfunctions (that are symmetrical with respect to the two axes through the center) have the formu = sin( s(j-1/2)x ) sin( s^-1(k-1/2)y )with corresponding eigenvaluesλ = ( s(j-1/2) )^ 2 + ( s^-1(k-1/2) )^ 2 ,for j,k ≥ 1. These even–even eigenvalues have counting function#{λ≤ r^2 } = number of points in the shifted lattice ( - 1/2) × (-1/2) lyinginside or on the ellipse (sx)^2+(s^-1y)^2 ≤ r^2 = N(r,s) where the shift parameters are σ=τ=-1/2 and the curve Γ is the quarter-circle.<ref> says the set S(r) of s-values that maximize the counting function does not approach 1 as r →∞. Instead, the maximizing s-values approach 0 or ∞. Thus the even–even symmetry class of eigenvalues on the rectangle behaves quite differently from the full collection of eigenvalues studied by Antunes and Freitas. The asymptotically optimal rectangle for maximizing the counting function as r →∞ (or equivalently, minimizing the n-th eigenvalue as n →∞) is not the square but rather the degenerate rectangle.§.§ Open problem for harmonic oscillatorsA quantum analogue of the Antunes–Freitas theorem for rectangles would be to minimize the n-th energy level among the following family of harmonic oscillators. For each s>0, consider-Δ u + 1/4( (sx)^2 + (s^-1y)^2 ) u = E u ,x,y ∈ , with boundary condition u → 0 as |(x,y)| →∞. Write s_n for an s-value that minimizes the n-th eigenvalue E_n. By analogy with Antunes and Freitas's theorem for Dirichlet rectangles, one might conjecture that s_n → 1 as n →∞. In fact, the behavior is quite different, as we now explain. Let us translate the harmonic oscillator problem into a shifted lattice point counting problem. The 1-dimensional oscillator equation -u^'' + 1/4 x^2 u = E u has eigenvalues E=j-1/2 for j=1,2,3,.…. By separating variables and rescaling, one finds that equation <ref> has spectrum { E_n } = { s(j-1/2) + s^-1(k-1/2) : j,k = 1,2,3,…} . Hence the number of harmonic oscillator eigenvalues less than or equal to r equals the number of points in the shifted lattice ( - 1/2) × ( - 1/2) lying below the straight line sx+s^-1y=r, which is given by our counting function N(r,s) where Γ is the straight line y=1-x (the 1-ellipse) and the shift parameters are σ=τ=-1/2. To minimize the eigenvalues we should maximize the counting function. The numerical evidence in the left part of <ref> suggests that the s-values maximizing the counting function N(r,s) do not converge to 1 as r →∞. Rather, the optimal s-values seem to cluster at various heights. (For a precise such clustering conjecture in the unshifted case, see <cit.>.) Thus the family of harmonic oscillators exhibits strikingly different spectral behavior from the family of Dirichlet rectangles.§.§ Interpolating family of Schrödinger potentialsThe family of Schrödinger potentials |sx|^q + |s^-1y|^q, where 2<q<∞ and s>0, interpolates between the harmonic oscillator (q=2) and the infinite potential well (q=∞) that corresponds to the Dirichlet Laplacian on a rectangular domain. We conjecture that when 2<q<∞, the set S(r) of values maximizing the eigenvalue counting function will converge to 1 as r →∞. This conjecture would provide a 1-parameter family of quantum oscillators for which the analogue of the Antunes–Freitas theorem holds true, with the family terminating in an exceptional endpoint case: the harmonic oscillator. The difficulty is that the eigenvalues of the 1-dimensional oscillator with potential |x|^q do not grow at a precisely regular rate. Hence to tackle the conjecture, one will need to extend the current paper from shifted lattices, where each row and column of the lattice is translated by the same amount, and find a way to handle deformed lattices, where the amount of translation varies with the rows and columns. This challenge remains for the future.§ ACKNOWLEDGMENTSThis research was supported by grants from the SimonsFoundation (#429422 to Richard Laugesen) and the Universityof Illinois Research Board (award RB17002).The material in this paper forms part of Shiya Liu's Ph.D. dissertation at the University of Illinois, Urbana–Champaign <cit.>. 99AF12 P. R. S. Antunes and P. Freitas. Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154 (2012), 235–257.AF13P. R. S. Antunes and P. Freitas. Optimal spectral rectangles and lattice ellipses. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2150, 20120492, 15 pp.AF16 P. R. S. Antunes and P. Freitas. Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction. Appl. Math. Optim. 73 (2016), no. 2, 313–328.Freitas_oudet P. R. S. Antunes and É. Oudet. Numerical minimization of Dirichlet–Laplacian eigenvalues of four-dimensional geometries. SIAM J. Sci. Comput., to appear.AL17 S. Ariturk and R. S. Laugesen. Optimal stretching for lattice points under convex curves. Port. Math., to appear. 1701.03217BBG16a M. van den Berg, D. Bucur and K. Gittins. Maximizing Neumann eigenvalues on rectangles. Bull. Lond. Math. Soc. 48 (2016), no. 5, 877–894.BBG16b M. van den Berg and K. Gittins. Minimising Dirichlet eigenvalues on cuboids of unit measure. Mathematika 63 (2017), no. 2, 469–482.BF13 D. Bucur and P. Freitas. Asymptotic behaviour of optimal spectral planar domains with fixed perimeter. J. Math. Phys. 54 (2013), no. 5, 053504.colbois_soufi B. Colbois and A. El Soufi. Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces. Math. Z. 278 (2014), 529–549. f16 P. Freitas. Asymptotic behaviour of extremal averages of Laplacian eigenvalues. J. Stat. Phys. 167 (2017), no. 6,1511–1518.GL17 K. Gittins and S. Larson Asymptotic behaviour of cuboids optimising Laplacian eigenvalues1703.10249 H17 A. Henrot, ed. Shape Optimization and Spectral Theory. De Gruyter Open, to appear, 2017.Hux96 M. N. Huxley. Area, Lattice Points, and Exponential Sums. London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications. Hux03 M. N. Huxley. Exponential sums and lattice points. III.Proc. London Math. Soc. (3) 87 (2003), 591–609.kratzel00 E. Krätzel. Analytische Funktionen in der Zahlentheorie. Teubner–Texte zur Mathematik, 139. B. G. Teubner, Stuttgart, 2000. 288 pp.kratzel04 E. Krätzel. Lattice points in planar convex domains. Monatsh. Math. 143 (2004), 145–162. Larson S. Larson. Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. 1611.05680Lau_Liu17 R. S. Laugesen and S. Liu. Optimal stretching for lattice points and eigenvalues. Ark. Mat., submitted, .thesis S. Liu. Asymptotically optimal shapes for counting lattice points and eigenvalues. Ph.D. dissertation, University of Illinois, Urbana–Champaign, 2017.marshall N. F. Marshall. Stretching convex domains to capture many lattice points. 1707.00682.marshall_steinerberger N. F. Marshall and S. Steinerberger. Triangles capturing many lattice points. 1706.04170.Oud04 É. Oudet. Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10 (2004), 315–330. | http://arxiv.org/abs/1707.08590v1 | {
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Department of Physics and Astronomy, University of Exeter, Stocker Road, EX4 4QL Exeter, UK; [email protected] Convection in astrophysical systems must be maintained against dissipation. Although the effects of dissipation are often assumed to be negligible, theory suggests that in strongly stratified convecting fluids, the dissipative heating rate can exceed the luminosity carried by convection. Here we explore this possibility using a series of numerical simulations. We consider two-dimensional numerical models of hydrodynamic convection in a Cartesian layer under the anelastic approximation and demonstrate that the dissipative heating rate can indeed exceed the imposed luminosity. We establish a theoretical expression for the ratio of the dissipative heating rate to the luminosity emerging at the upper boundary, in terms only of the depth of the layer and the thermal scale height. In particular, we show that this ratio is independent of the diffusivities and confirm this with a series of numerical simulations. Our results suggest that dissipative heating may significantly alter the internal dynamics of stars and planets. § INTRODUCTIONConvection occurs in the interiors of many astrophysical bodies and must be sustained against viscous and ohmic dissipation. This dissipation is often neglected in astrophysical models, e.g., in standard stellar 1D evolution codes <cit.> though its effects have lately been considered in a few specific contexts <cit.>.Astrophysical convection often occurs over many scale heights. While for incompressible fluids the contribution of dissipative heating to the internal energy budget is negligible <cit.>, <cit.> (hereafter HMW) showed that in strongly stratified systems, it is theoretically possible for the rate of dissipative heating to exceed the luminosity. This was supported numerically by <cit.> for the case of a compressible liquid with infinite Prandtl number, Pr, (the ratio of viscous and thermal diffusivities), appropriate for models of the Earth's interior.In this study we aim to establish the magnitude of dissipation for conditions more akin to those encountered in stellar interiors. Specifically, we consider dissipation in a stratified gas at finite Pr, and examine how the total heating changes as system parameters are varied. To begin, we briefly review some relevant thermodynamic considerations that underpin our work. §.§ Thermodynamic constraints on dissipative heatingFor a volume V of convecting fluid enclosed by a surface S with associated magnetic field 𝐁, in which the normal component of the fluid velocity 𝐮 vanishes on the surface, and either all components of 𝐮, or the tangential stress, also vanish on the surface, local conservation of energy gives that the rate of change of total energy is equal to the sum of the net inward flux of energy and the rate of internal heat generation (e.g., by radioactivity or nuclear reactions). This implies ∂/∂t(ρe+1/2ρu^2. .+B^2/2μ_0-ρΨ)=-∇·(ρ(e+1/2u^2-Ψ)𝐮..+(𝐄×𝐁)/μ_0+P𝐮-τ·𝐮-k∇T)+Hwhere ρ is the fluid density, e is the internal energy of the fluid, Ψ is the gravitational potential that satisfies 𝐠=∇Ψ, P is the pressure, τ_ij is the contribution to the total stress tensor from irreversible processes, k is the thermal conductivity, T is the temperature, H is the rate of internal heat generation, and 𝐄×𝐁/μ_0 is the Poynting flux (𝐄 is the electric field and μ_0 is the permeability of free space).Integrating (<ref>) over V gives the global relation ∫_Sk∂T/∂x_i dS_i+∫_VH dV=0,assuming both a steady state and that the electric current, 𝐣, vanishes everywhere outside V. Equation (<ref>) implies that the net flux out of V is equal to the total rate of internal heating. Viscous and ohmic heating do not contribute to the overall heat flux: dissipative heating terms do not appear in equation (<ref>).To examine dissipative heating, we consider the internal energy equation:ρ(∂e/∂t+(𝐮·∇)e)=∇(k∇T)-P(∇·𝐮)+τ_ij∂u_i/∂x_j+j^2/σ+Hwhere σ is the conductivity of the fluid. Integrating over V, and assuming a steady state, (<ref>) becomes∫_V(𝐮·∇)P dV+Φ =0.Here Φ=∫_Vτ_ij∂u_i/∂x_j+j^2/σ dVis the total dissipative heating rate including viscous and ohmic heating terms. Equation (<ref>) implies that the global rate of dissipative heating is cancelled by the work done against the pressure gradient. Equation (<ref>) is only equivalent to HMW's equation (22) when considering an ideal gas (so that αT=1, where α is the coefficient of thermal expansion); however, in arriving at (<ref>), we made no assumption about the fluid being a gas. <cit.> note that this inconsistency arises because HMW assume c_p to be constant in their derivation, which is not valid when α T≠1.Alternatively, from the first law of thermodynamics, we haveTds=de-P/ρ^2dρwhere s is the specific entropy, so (<ref>) can also be written as Φ=∫_VρT(𝐮·∇)s dV=-∫_Vρs(𝐮·∇)T dVwhere we have invoked mass continuity in a steady state (∇·(ρ𝐮)=0). Hence the global dissipation rate can also be thought of as being balanced by the work done against buoyancy <cit.>.HMW used the entropy equation to derive an upper bound for the dissipative heating rate in a steadily convecting fluid that is valid for any equation of state or stress-strain relationship.For the case of convection in a plane layer, that upper bound is Φ/L_u<T_max-T_u/T_uwhere L_u is the luminosity at the upper boundary, T_max is the maximum temperature and T_u is the temperature on the upper boundary. One consequence of this bound is that, for large enough thermal gradients, the dissipative heating rate may exceed the heat flux through the layer; this is perhaps counter-intuitive, but is thermodynamically permitted, essentially because the dissipative heating remains in the system's internal energy <cit.>. The above considerations should hold for both ohmic and viscous dissipation. However, HMW further considered the simple case of viscous heating in a liquid (neglecting magnetism) and showed that the viscous dissipation rate is not only bounded by (<ref>) but thatE≡Φ/L_u=d/H_T(1-μ/2)where d is the height of the convective layer, H_T is the (constant) thermal scale height and 0≤μ≤1 is the fraction of internal heat generation. Interestingly, the theoretical expression (<ref>) is dependent only on the ratio of the layer depth to the thermal scale height and the fraction of internal heat generation.As expected, (<ref>) implies that the dissipative heating rate is negligible when compared with the heat flux in cases where the Boussinesq approximation is valid (i.e., when the scale heights of the system are large compared to the depth of the motion). But it follows from (<ref>) that Φ is significant compared to L_u if d is comparable to H_T, i.e., if the system has significant thermal stratification. Stellar convection often lies in this regime, so it is not clear that dissipative heating can be ignored.This paper explores these theoretical predictions using simulations of stratified convection under conditions akin to those encountered in stellar interiors. Previous numerical simulations conducted by HMW considered only 2D Boussinesq convection and neglected inertial forces (infinite Pr approximation); later work by <cit.> within the so-called anelastic liquid approximation considered stronger stratifications but likewise assumed a liquid at infinite Pr.We extend these by considering an ideal gas (so that αT=1) at finite Pr, so inertial effects are important and compressibility is not negligible. In section <ref>, we describe the model setup before presenting results from numerical simulations. In section <ref> we offer a discussion of the most significant results that emerge before providing conclusions. § SIMULATIONS OF DISSIPATIVE CONVECTION §.§ Model setupWe consider a layer of convecting fluid lying between impermeable boundaries at z=0 and z=d. We assume thermodynamic quantities to be comprised of a background, time-independent, reference state and perturbations to this reference state. The reference state is taken to be a polytropic, ideal gas with polytropic index m given by T̅=T_0(1-β z), ρ̅=ρ_0(1-β z)^m, p̅=ℛρ_0T_0(1-β z)^m+1,where β=g/c_p,0T_0. Here, g is the acceleration due to gravity, c_p is the specific heat capacity at constant pressure, ℛ is the ideal gas constant and a subscript 0 represents the value of that quantity on the bottom boundary. β is equivalent to the inverse temperature scale height and so is a measure of the stratification of the layer, although we shall use the more conventional N_ρ=-mln(1-β d)to quantify the stratification, with N_ρ the number of density scale heights across the layer. We assume a polytropic, monatomic, adiabatic, ideal gas, therefore m=1.5. Here we consider only the hydrodynamic problem; i.e., all dissipation is viscous.We use anelastic equations under the Lantz-Braginsky-Roberts (LBR) approximation <cit.>; these are valid when the reference state is nearly adiabatic and when the flows are subsonic <cit.>, as they are here.The governing equations are then∂𝐮/∂t +(𝐮·∇)𝐮=-∇p̃+gs/c_p𝐞̂_̂𝐳̂+ν[1/ρ̅∂/∂x_j(ρ̅(∂u_i/∂x_j+∂u_j/∂x_i))-2/3ρ̅∂/∂x_i(ρ̅∂u_j/∂x_j)]∇·(ρ̅𝐮)=0 ρ̅T̅(∂s/∂t+(𝐮·∇)s)=∇·(κρ̅T̅∇s)+τ_ij∂u_i/∂x_j+H,where 𝐮 is the fluid velocity, p̃=p/ρ̅ is a modified pressure and ν is the kinematic viscosity. The specific entropy, s, is related to pressure and density bys=c_vlnp-c_plnρ.We assume the perturbation of the thermodynamic quantities to be small compared with their reference state value. Therefore the entropy is obtained froms=c_vp/p̅-c_pρ/ρ̅and the linearised equation of state isp/p̅=T/T̅+ρ/ρ̅.In (<ref>) κ is the thermal diffusivity andτ_ij=νρ̅(∂u_i/∂x_j+∂u_j/∂x_i-2/3δ_ij∇·𝐮)is the viscous stress tensor (δ_ij is the Kronecker delta). Here, we only consider cases with H=0 (i.e., no internal heat generation), and instead impose a flux (F) at the bottom boundary. Note the LBR approximation diffuses entropy (not temperature); see <cit.> for a discussion of the differences. We assume a constant ν and κ. We solve these equations using the Dedalus pseudo-spectral code <cit.> with fixed flux on the lower boundary and fixed entropy on the upper boundary.We assume these boundaries to be impermeable and stress-free. We employ a sin/cosine decomposition in the horizontal, ensuring there is no lateral heat flux. We employ the semi-implicit Crank-Nicolson Adams-Bashforth numerical scheme and typically use 192 grid points in each direction with dealiasing (so that 128 modes are used). In some cases, 384 (256) grid points (modes) were used to ensure adequate resolution of the solutions. For simplicity, and to compare our results with those of HMW, we consider 2D solutions so that 𝐮=(u,0,w) and ∂/∂y≡0. This also allows us to reach higher supercriticalities and N_ρ with relative ease.As we neglect magnetism, the total dissipation rate, Φ, is given by (<ref>) with 𝐣=0 and τ_ij as given by (<ref>).An appropriate non-dimensionalisation of the system allows the parameter space to be collapsed such that the dimensionless solutions (in particular E) are fully specified by m, N_ρ, Pr, together with F̂_̂0̂= Fd/κc_p,0ρ_0T_0 (a dimensionless measure of the flux applied at the lower boundary) and a flux-based Rayleigh number <cit.>Ra=gd^4F_u/νκ^2ρ_0c_p,0T_0.The parameters used in our simulations are given in Table <ref>.In a steady state, an expression for the luminosity L at each depth z=z' can be obtained by integrating the internal energy equation (<ref>) over the volume contained between the bottom of the layer and the depth z=z':L= FA=∫_V_z'∇·(ρ̅T̅s𝐮) dV+∫_V_z'-∇·(κρ̅T̅∇s) dV+∫_V_z'-sρ̅(𝐮·∇)T̅ dV+∫_V_z'-τ_ij∂u_i/∂x_j dV,where A is the surface area. The divergence theorem allows the first two integrals to be transformed into surface integrals givingL= FA=∫_S_z'ρ̅T̅sw dS_L_conv=AF_conv+∫_S_z'-κρ̅T̅∂s/∂z dS_L_cond=AF_cond+∫_V_z'-sρ̅(𝐮·∇)T̅ dV_L_buoy=A∫_0^z'Q_buoy dz+∫_V_z'-τ_ij∂u_i/∂x_j dV_L_diss=A∫_0^z'Q_diss dz,where the surface integrals are over the surface at height z=z'. The first and second terms define the horizontally-averaged heat fluxes associated with convection (F_conv) and conduction (F_cond) respectively, along with associated luminosities. The third and fourth terms define additional sources of heating and cooling (Q_diss and Q_buoy) associated with viscous dissipation and with work done against the background stratification, respectively. These two terms must cancel in a global sense i.e., when integrating from z=0 to z=d, but they do not necessarily cancel at each layer depth. An alternative view of the heat transport may be derived by considering the total energy equation (<ref>), which includes both internal and mechanical energy.In a steady state (with entropy diffusion), the local balance gives∇·(ρ̅(e+1/2u^2-Ψ)𝐮+p𝐮-τ·𝐮-κρ̅T∇s)=Hwhich when integrated over the volume for an ideal gas gives <cit.>L= FA=∫_S_z'ρ̅c_pwT' dS_L_e=AF_e+∫_S_z'-κρ̅T̅∂s/∂z dS_L_cond=AF_cond+∫_S_z'1/2ρ̅|u^2|w dS_L_KE=AF_KE+∫_S_z'-(τ_iju_i)·ê_𝐳 dS_L_visc=AF_visc,defining the horizontally-averaged enthalpy flux (F_e), kinetic energy flux (F_KE) and viscous flux (F_visc). Note that (<ref>) and (<ref>) are equivalent; whether decomposed in the manner of (<ref>) or the complementary fashion of (<ref>), the transport terms must sum to the total luminosity L. L_visc represents the total work done by surface forces, whereas L_diss represents only the (negative-definite) portion of this that goes into deforming a fluid parcel and hence into heating. §.§ Relations between global dissipation rate and convective fluxFor the model described in section <ref>, equation (<ref>) becomesΦ= -∫_Vρ̅s(𝐮·∇)T̅ dV= g/c_p,0∫_Vsρ̅w dV=gA/c_p,0∫_0^dF_conv/T̅ dz,Often it is assumed that in the bulk of the convection zone, the total heat flux is just equal to the convective flux as defined above (i.e., F_conv≈F). We show later that this a poor assumption in strongly stratified cases, but it is reasonable for approximately Boussinesq systems. In the case F_conv≈F, (<ref>) becomesΦ=gAF/c_p,0T_0∫_0^d1/1-βz dz=-L_uln(1-βd)and E=-ln(1-βd)=βd+…≈d/H_T,0.However, in strongly stratified cases F≈F_conv+F_other where F_other=∫_0^z'(Q_buoy+Q_diss) dz from (<ref>), or alternatively, F_other=F_p+F_KE+F_visc from (<ref>) (the conductive flux is small in the bulk convection zone). Here F_p=1/A∫_S_z'wp dS is the difference between the enthalpy flux F_e and the convective flux F_conv.Physically, F_other is equivalent to the steady-state transport associated with processes other than the convective flux as defined above. In this case, (<ref>) becomesΦ=gAF/c_p,0∫_0^d(1-F_other/F)1/T̅ dz,where we note that in general F_other is a function of depth and (1-F_other/F)≥1. A complete theory of convection would specify F_other a priori, and thereby constrain the dissipative heating everywhere. In the absence of such a theory,we turn to numerical simulations to determine the magnitude of Φ for strong stratifications. §.§ Dissipation in simulations: determined by stratificationWe examine the steady-state magnitude of Φ for different values of N_ρ and Ra.Figure <ref> shows the ratio of the global dissipation rate to the luminosity through the layer, E=Φ/L_u, for varying stratifications. First, we highlight the difference between simulations in which the dissipative heating terms were included (red squares) and those where they were not (black circles). At weak stratification,there is not much difference in the dissipative heating rate between these cases, but differences become apparent as N_ρ is increased. Including the heating terms in a self-consistent calculation leads to a much larger value of E than if Φ is only calculated after the simulation has run (i.e., if heating is not allowed to feedback on the system). When heating terms are included, the global dissipative heating rate exceeds the flux passing through the system (i.e., E>1) when N_ρ>1.22. As expected, the expression for E, in the Boussinesq limit, given by (<ref>), is a good approximation to E for small N_ρ, but vastly underestimates E at large N_ρ (see Figure <ref>, dash-dot line). In the cases where the heating terms are not included, E cannot exceed unity for all N_ρ. This might have been expected, since in this case none of the dissipated heat is returned to the internal energy of the system; instead, the dissipated energy is simply lost (i.e., energy is not conserved). This has the practical consequence that the flux emerging from the top of the layer is less than that input at the bottom. In these cases E is very well described by the dashed line which is given by d/H_T,0, the leading order term from the expression for E in (<ref>).The theoretical upper bound derived by HMW is shown on Figure <ref> by the solid black line. It is clear that all of our cases fit well within this upper bound, even at strong stratifications. This upper bound is equivalent to d/H_T,u in this system, where H_T,u is the value of H_T on the upper boundary.Cases in which the heating terms were included are well described by E=d/H̃_̃T̃,whereH̃_̃T̃ = H_T,0H_T,u/H_T,z^*is a modified thermal scale height involving H_T at the top, bottom and at a height z^*, defined such that half the fluid (by mass) lies below z^* and half sits above; for a uniform density fluid, z^*=d/2. This expression resembles that originally proposed by HMW, on heuristic grounds, for a gas (E≈d/H_T); in our case H_T is not constant across the layer and we find that the combination H̃_̃T̃ is the appropriate “scale height" instead. Like HMW's suggestion, it depends only on the layer depth and temperature scale heights of the system.For 2D convection, at Pr=1 and the Ra considered here, the solutions are steady (time-independent) <cit.>; the convection takes the form of a single stationary cell occupying the layer. To assess if the same behaviour occurs for chaotic (time-dependent) solutions, we have included some cases at Pr=10 (orange triangles), since then the flow is unsteady. In the cases included here, this unsteady flow is characterised by the breakup of the single coherent convection cell (seen at Pr=1); these time-dependent solutions seem also to be well described by the line given by (<ref>). This behaviour is sampled in Figure <ref>, Supplementary Material, which shows the velocity and entropy fields in a simulation with Pr=10, N_ρ=1.31, Ra=4.13×10^8 and F̂_0=0.14. At higher Ra, the solutions transition to turbulence <cit.>. §.§ Dissipation in simulations: independent of diffusivitiesThe results of section <ref>, specifically equation (<ref>), suggest that the amount of dissipative heating is determined by the stratification, not by other parameters such as Ra. To probe this further, we consider how/if E changes as Ra is varied. Figure <ref> shows the results for three different stratifications. For N_ρ≈0.1, the fluid is close to being Boussinesq and it is clear that E remains constant (and equal to the value given by (<ref>)) for many decades increase in Ra. This result complements that of HMW obtained from Boussinesq simulations at infinite Pr. For increasing N_ρ, we find that for large enough Ra, E approaches the constant given by (<ref>). That E becomes independent of Ra at large enough Ra for all N_ρ was also found by <cit.>, albeit for liquids at infinite Pr. Figure <ref> indicates that the solutions have to be sufficiently supercritical in order for the theory to be valid. It also suggests that stronger stratifications require simulations to be more supercritical in order to reach the asymptotic regime. (All the simulations displayed in Figure <ref> approach this asymptotic regime, except possibly the uppermost point at N_ρ=2.8. That simulation has Ra/Ra_c ≈ 9 ×10^5, but it is likely that still higher Ra would yield somewhat greater values of E at this stratification.)§ DISCUSSION AND CONCLUSIONWe have demonstrated explicitly that the amount of dissipative heating in a convective gaseous layer can, for strong stratifications, equal or exceed the luminosity through the layer.A principal conclusion is that the ratio of the global viscous heating rate to the emergent luminosity is approximated by a theoretical expression dependent only on the depth of the layer and its thermal scale heights.This ratio, akin to one originally derived for a simpler system by HMW, is given (for the cases studied here) by (<ref>). Interestingly, this relation does not depend on other parameters such as the Rayleigh number. Our simulations confirm that this expression holds for 2D convection in an anelastic gas, provided the convection is sufficiently supercritical.This regime is attainable in our 2D simulations, and is surely reached in real astrophysical objects, but may be more challenging to obtain in (for example) 3D global calculations <cit.>.The dissipative heating appears in the local internal energy (or entropy) equation, in the same way as heating by fusion or radioactive decay.Where it is large, we therefore expect it will modify the thermal structure, just as including a new source of heating or cooling would have done.It must be reiterated, though, that in a global sense this heating is balanced by equivalent cooling terms; i.e., L_diss and L_buoy in equation (<ref>) cancel in a global sense; no additional flux emerges from the upper boundary. Stars are not brighter because of viscous dissipation. Locally, however, these terms do not necessarily cancel, as explored in Figure <ref>.There we show the net heating and cooling at each depth in two simulations; in Figure <ref>a, the fluid is weakly stratified, and in (b) is has a stratification given by N_ρ=2.08. In both cases the sum of the terms must be zero at the top and bottom of the layer, but not in between. Furthermore, in (a) the terms are small compared to the flux through the layer (typically a few %) but in the strongly stratified case, the local heating and cooling become comparable to the overall luminosity.In general, stronger stratifications lead to stronger local heating and cooling in the fluid. In a steady state the imbalance between this local heating and cooling is equivalent to certain transport terms as discussed in section <ref>; these are assessed for our simulations in figure <ref> where the terms are plotted as luminosities and labelled correspondingly. Turning first to Figure <ref>a, we show the components of the total flux of thermal energy (as described by (<ref>)), namely L_conv, L_cond, L_buoy and L_diss. The conductive flux is small throughout the domain except in thin boundary layers and the dissipative heating (L_diss) is comparable to the convective flux (L_conv) throughout the domain. The sum of the four transport terms is shown as the black line (L) and is constant across the layer depth, indicating thermal balance. Figure <ref>b assesses the total energy transport using the complementary analysis of (<ref>), using L_KE, L_cond, L_e and L_visc. The primary balance is between the positive L_e and the negative L_KE. Viewed in this way, the viscous flux (L_visc) is small except near the lower boundary, but (as discussed in section <ref>) this does not necessarily mean the effect of viscous dissipation is also small. In figure <ref>c we highlight the equivalence of some transport terms, by showing the term AF_other together with its different constituent terms from either the total or thermal energy equations. As expected, AF_other is the same in both cases; it is the sum of L_diss and L_buoy, or equivalently, it is the sum of L_p, L_KE and L_visc. That is, changes in the dissipative heating are reflected not just in Q_diss (if analysing internal energy) or F_visc (if analysing total energy); the other transport terms (F_KE, F_p, F_e, F_conv, Q_buoy) also change in response. To emphasise the importance of dissipative heating in modifying the transport terms, we include in Figure <ref>d, L_KE^nh , L_e^nh , L_cond^nhand L_visc^nh i.e., the kinetic energy, enthalpy, conductive and viscous fluxes (expressed as luminosities) respectively, in the case where heating terms were not included. It is clear that these are much smaller than in the equivalent simulation with heating (Figure <ref>b), demonstrating explicitly that the inclusion of dissipative heating influences the other transport terms. In particular, the maximum value of the kinetic energy flux is 3.2 times larger when the heating terms are included. The black line in Figure <ref>d shows that when heating is not included the flux emerging at the upper boundary is smaller than the flux imposed at the lower boundary; in this case it is approximately 27% of L.The local heating and cooling (or, equivalently, the transport term F_other that must arise from this in a steady state) described above is not included in standard 1D stellar evolution models, and we do not yet know what effects (if any) would arise from its inclusion.In some contexts they may be negligible; the total internal energy of a star is enormously greater than its luminosity L⋆, so even internal heating that exceeds L⋆ may not have a noticeable effect on the gross structure. If, however, this heating is concentrated in certain regions (e.g., because of spatially varying conductivity) or occurs in places with lower heat capacity, its impact may be more significant.If the results explored here also apply to the full 3D problem with rotation and magnetism – which clearly must be checked by future calculation – then the total dissipative heating is determined non-locally, dependent as it is on the total layer depth.Simple modifications to the mixing-length theory (which is determined locally) may not then suffice to capture it. We have begun to explore these issues by modification of a suitable 1D stellar evolution code, and will report on this in future work.We acknowledge support from the European Research Council under ERC grant agreements No. 337705 (CHASM). The simulations here were carried out on the University of Exeter supercomputer, a DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS and the University of Exeter. We also acknowledge PRACE for awarding us access to computational resources Mare Nostrum based in Spain at the Barcelona Supercomputing Center, and Fermi and Marconi based in Italy at Cineca. We thank the referee for a thoughtful review that helped to improve the manuscript.§ SIMULATION PARAMETERSCCCCCC Simulation parameters used in figures <ref>-<ref>6 1 0ptPr N_ρ Ra F̂_̂0̂ E Figure1 0.1050 3.83 ×10^5 3.26 ×10^-4 0.0630 21 0.1050 3.83 ×10^6 3.26 ×10^-3 0.0662 21 0.1050 2.63 ×10^7 2.24 ×10^-2 0.0678 21 0.1050 6.13 ×10^7 5.22 ×10^-2 0.0682 21 0.1050 3.83 ×10^8 3.26 ×10^-1 0.06891-31 0.2776 6.58 ×10^7 5.60 ×10^-2 0.182811 0.3828 8.77 ×10^7 7.47 ×10^-2 0.255711 0.5819 8.01 ×10^7 1.07 ×10^-3 0.401411 0.7060 6.65 ×10^4 3.62 ×10^-3 0.415921 0.7060 6.65 ×10^5 3.62 ×10^-2 0.459421 0.7060 9.36 ×10^6 1.24 ×10^-4 0.487521 0.7060 1.05 ×10^8 2.64 ×10^-3 0.500821 0.7060 2.72 ×10^8 3.62 ×10^-3 0.503821 0.7060 4.88 ×10^8 4.40 ×10^-3 0.50571-21 0.7967 1.03 ×10^8 1.37 ×10^-3 0.577011 0.9887 1.20 ×10^8 1.60 ×10^-3 0.753311 1.3104 8.45 ×10^7 1.12 ×10^-3 1.090811 2.0846 1.33 ×10^5 7.24 ×10^-3 1.583021 2.0846 1.33 ×10^6 2.68 ×10^-3 1.872621 2.0846 1.63 ×10^7 2.17 ×10^-4 2.088221 2.0846 5.44 ×10^7 7.24 ×10^-4 2.16561-41 2.7938 1.23 ×10^8 1.63 ×10^-3 3.59511 10 0.1050 2.63 ×10^7 1.43 ×10^-1 0.0668 110 0.2776 1.15 ×10^8 9.78 ×10^-3 0.1822 110 0.3828 3.25 ×10^6 1.59 ×10^-1 0.2454 110 0.9887 1.37 ×10^9 4.66 ×10^-1 0.7413 110 1.3104 4.13 ×10^8 1.40 ×10^-1 1.0594 1 [Alboussiere & Ricard(2013)]AlboussiereRicard2013 Alboussiere, T., & Ricard, Y. 2013, JFM, 725, R1 [Alboussiere & Ricard(2014)]AlboussiereRicard2014 Alboussiere, T., & Ricard, Y. 2014, JFM, 751, 749[Aubert et al.(2017)]Aubertetal2017 Aubert, J., Gastine, T., & Fournier, A. 2017, JFM, 813, 558 [Backus(1975)]Backus1975 Backus, G. 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"authors": [
"Laura K. Currie",
"Matthew K. Browning"
],
"categories": [
"astro-ph.SR",
"physics.flu-dyn"
],
"primary_category": "astro-ph.SR",
"published": "20170727132803",
"title": "The magnitude of viscous dissipation in strongly stratified two-dimensional convection"
} |
Ignas Snellen [email protected] Leiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The NetherlandsAnton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, the NetherlandsSRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands Anton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, the NetherlandsDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA NASA Sagan FellowAstronomy Department, University of Washington, USALeiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The NetherlandsLeiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Delft, The NetherlandsMax-Planck-Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, GermanyMax-Planck-Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, GermanySRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the NetherlandsSRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the NetherlandsObservatoire de Genève, Université de Genève, 51 chemin des Maillettes, 1290 Versoix, SwitzerlandAnton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, the NetherlandsLeiden Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The NetherlandsSchool of Physics, University of Exeter, Exeter, UK School of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge MA 02138, USA Anton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, the Netherlands Center for Astrophysics and Space Astronomy, University of Colorado at Boulder, Boulder, CO 80309, USA NASA Hubble Fellow Exoplanet Proxima b will be an important laboratory for the search for extraterrestrial life for the decades ahead. Here we discuss the prospects of detecting carbon dioxide at 15 μm using a spectral filtering technique with the Medium Resolution Spectrograph (MRS) mode of the Mid-Infrared Instrument (MIRI) on the James Webb Space Telescope (JWST). At superior conjunction, the planet is expected to show a contrast of up to 100 ppm with respect to the star. At a spectral resolving power of R=1790–2640, about 100 spectral CO_2 features are visible within the 13.2-15.8 μm (3B) band, which can be combined to boost the planet atmospheric signal by a factor 3–4, depending on the atmospheric temperature structure and CO_2 abundance. If atmospheric conditions are favorable (assuming an Earth-like atmosphere),with this new application to the cross-correlation technique carbon dioxide can be detected within a few days of JWST observations. However, this can only be achieved if both the instrumental spectral response and the stellar spectrum can be determined to a relative precision of ≤1× 10^-4 between adjacent spectral channels. Absolute flux calibration is not required, and the method is insensitive to the strong broadband variability of the host star. Precise calibration of the spectral features of the host star may only be attainable by obtaining deep observations of the system during inferior conjunction that serve as a reference. The high-pass filter spectroscopic technique with the MIRI MRS can be tested on warm Jupiters, Neptunes, and super-Earths with significantly higher planet/star contrast ratios than the Proxima system.§ INTRODUCTIONThe discovery of the exoplanet Proxima b through long-term radial velocity monitoring <cit.> is exciting for two reasons. First, it confirms that low-mass planets are very common around red dwarf stars, a picture that was already emerging from both transit and radial velocity surveys <cit.>. Second, the proximity of this likely temperate rocky planet at a mere 1.4 parsec from Earth makes it most favourable for atmospheric characterisation, making Proxima b an important laboratory for the search for extraterrestrial life for the decades ahead.Proxima b is found to orbit its host star in 11.2 days, placing it at an orbital distance of 0.0485 au.Since the luminosity of the host star is only 0.17% of that of our Sun, the level of stellar energy the planet receives is 30% less than the Earth, but nearly 70% more than Mars. This means that in principle it could have surface conditions that sustain liquid water - generally thought as a prerequisite for the emergence and evolution of biological activity <cit.>. Although other habitats can be envisaged outside the so called 'habitable zone' , such as under the icy surface of Jupiter's moon Europa <cit.>, it is rather unlikely that signs of biological activity under such conditions could be detected in extrasolar planet systems <cit.>.It is highly debatable whether Earth-mass planets in the habitable zones of red dwarf stars, such as Proxima b, could sustain or have ever sustained life. First, it is expected that the pre-main sequence of red dwarf stars lasts up to a billion years during which the stellar luminosity is significantly higher than during the main-sequence lifetime of the star. This means that the planet will have had a significantly hotter climate early on, during which it may have lost most or maybe all of its potential water content <cit.>. Second, Proxima, as a large fraction of red dwarf stars, is a flare star that actively bombards the planet atmosphere with highly energetic photons and particles <cit.>, possibly causing a large fraction of the planet atmosphere to be lost. Thirdly, planets in the habitable zone of red dwarf stars are expected to be tidally locked, and may be synchronously rotating – always faced with the same side to the host star <cit.>. It is not clear whether a habitable climate can be sustained with such a configuration <cit.>. However, several theoretical endeavours, also in the wake of the Proxima b discovery, show that despite these possible drawbacks the planet may well host an atmosphere with liquid water on its surface <cit.>. Several studies have investigated the potential detectability of Earth-like atmospheres of planets orbiting late M-dwarfs using high-dispersion spectroscopy. <cit.> calculated whether the transmission signature of molecular oxygen of a twin earth-planet in front of a mid-M dwarf could be observed using high-dispersion spectroscopy <cit.>and showed that it would require a few dozen transits with the European Extremely Large Telescope (E-ELT) to reach a detection. However, it is very unlikely that Proxima b is transiting <cit.>. A more promising avenue is to combine high-dispersion spectroscopy with high-contrast imaging <cit.>. <cit.> simulated observations with the E-ELT using optical HDS+HCI of a then still hypothetical Earth-like planet around Proxima, showing that detection of such planet would be possible within one night. Recently, <cit.> argued that if the new ESPRESSO high-dispersion spectrograph at the ESO Very Large Telescope (VLT) can be coupled with the high-contrast imager SPHERE, and the latter has a major upgrade in adaptive optics and coronagraphic capabilities, a detection of Proxima b is within reach. Since the next-generation of extremely large ground-based telescopes is at least 5 to 10 years away, the James Webb Space Telescope (JWST) could be a more immediate option to detect an atmospheric signature of Proxima b. Unfortunately, simple diffraction arguments tell us that the JWST is not large enough to spatially separate the planet from its host star - with a maximum elongation of37 mas (∼1λ/D at 1 μm). Several studies <cit.> show that atmospheric characterisation of super-Earths transiting small M-dwarf stars is within range of the JWST. However, the probability that Proxima b transits its host star is only 1.3%.Instead, <cit.> discuss the possibility of detecting the thermal phase curve with the JWST Mid-Infrared Instrument MIRI, using its slitless 5 - 12 μm Low Resolution (LRS) mode . Since the planet is expected to be tidally locked,the night-to-dayside temperature gradient will result in a variation in apparent thermal flux as function of orbital phase. Depending on the orbital inclination, and on whether the planet has an atmosphere (which affects the redistribution of absorbed stellar energy around the planet), variations of up to ∼35 ppm are expected in the LRS wavelength regime. They show that in the ideal case of photon-limited precision, one can indeed detect the phase variation over a planet orbit (11.2 days - 268 hrs). However, a particular concern is the intrinsic variability of Proxima Cen - which is known to be a flare star. A month-long observation program of the MOST satellite <cit.> detected on average two strong optical flares a day.Extrapolating their result to lower energies and mid-infrared wavelength implies that the star exhibits ∼50 flares a day at levels >500 ppm - an order of magnitude stronger than the expected phase variation amplitude of the planet.Hence it will be challenging to discern the planet signal with such observations.In this paper we discuss a new application to the cross-correlation technique used to probe exoplanet atmospheres <cit.>, targeting the 15 μmCO_2 planet signal with the Medium Resolution Spectrograph (MRS) mode of MIRI. Importantly, it is unaffected by broadband flux variations of the host star. Furthermore, a detection of CO_2 would constitute conclusive evidence that Proxima b contains an atmosphere, and provide constraints to its temperature structure.If Proxima b is indeed a terrestrial planet that formed at or near its current semi-major axis, it would have been subjected to the super-luminous phase of the star for up to 160 My <cit.>. In this time it may have undergone ocean loss and a runaway greenhouse, or had all but the heaviest molecules in its atmosphere stripped early on, with the possibility of longer-term replenishment by volcanic outgassing over its 5 Gyr history <cit.>. All of these evolutionary processes would have increased the likelihood that the planetary atmosphere currently contains CO_2.In Section 2 we describe the details of the method, of the MRS mode of MIRI, and present simulations, including atmospheric modelling. The results are presented and discussed in Section 3.§ A HIGH-PASS SPECTRAL FILTERING TECHNIQUE The key to any exoplanet atmospheric observation is the ability to separate planet signatures from the overwhelming flux of the host star.The cross-correlation technique used to probe exoplanet atmospheres does not work in the case of Proxima b with the JWST, since the spectral resolution is not sufficient to use the change in the radial component of the planet orbital velocity to filter out the planet light <cit.>, neither is the angular resolution sufficient to separate the planet from the star (in which case the HDS+HCI technique could be used). In this case we propose to use a new version of these techniques in which a spectral feature is targeted in the integrated planet+star spectrum that can be differentiated from the stellar spectrum and attributed to the planet.Generally, this would not work because it constitutes an absolute spectrophotometric measurement requiring instrumental calibration and knowledge of the stellar spectrum to a level significantly better than the planet signal. However, if the instrument spectral resolution is sufficiently high, the requirements on calibration and stellar spectrum knowledge can be significantly relaxed by probing the high-pass spectrally filtered signature instead of the absolute spectrophotometric signal. In this case the planet signal is the difference in flux density relative to a low-resolution mean. E.g. in the case of a molecular band composed of a series of distinct absorption lines, the high-pass filtered spectral signature consists of series of peaks and valleys in the spectrum (Fig. 1). This has the advantage that the planet signal is spread out over many pixels and consists of positive and negative high-frequency components. In particular low-frequency components in the spectrum are notoriously difficult to calibrate due to required accuracies in the spectrum of the calibration source, and in stray-light and background corrections. However, these are not important in this case. We arguethat the spectral filtering technique will work well for the MRS mode of MIRI targeting CO_2 at 15 μm. In the case of Proxima b, its variations in the radial component of its orbital velocity of up to 50 km sec^-1 corresponds to a shift of ±1 wavelength step, which may also be detected and subsequently constrain its orbital inclination.§.§ MRS mode of MIRI The Medium Resolution Spectrograph (MRS) mode of the Mid-Infrared Instrument <cit.> on JWST utilises an integral field spectrograph (IFS) which has four image slices producing dispersed images of the sky on two 1024x1024 infrared detector arrays, which provide R = 1300-3600 integral field spectroscopy over a λ = 5 - 28.3 μm wavelength range <cit.>. The spectral window is divided in four channels covered by four integral field units: (1) 4.96-7.77 μm, (2) 7.71-11.90 μm, (3) 11.90-18.35 μm, and (4) 18.35-28.30 μm. Two grating and dichroic wheels select the wavelength coverage within these four channels simultaneously, dividing each channel into three spectral sub-bands indicated by A, B, and C respectively. To obtain a complete spectrum over the whole MIRI band one has to combine exposures in the three spectral settings, A, B, and C. Since we are primarily interested in the 15 μm CO_2 feature, only one setting will be sufficient: 3B covering the 13.2 - 15.8 μm range. Note that the same setting will deliverthe 1B (5.6 – 6.7 μm), 2B (8.6 – 10.2 μm) & 4B (20.4 – 24.7 μm) wavelength ranges for free. Of these, 2B is particularly interesting since it contains the ozone absorption feature. This is briefly discussed in Section 3.5. The IFS of Channel 3 consists of 16 slices (width = 0.39”), each containing 26 pixels (0.24”) providing a field of view of ∼ 6”× 6”. The spectrum is dispersed over 1024 pixels (1 pix = 2.53 nm = 52 km sec^-1) at a spectral resolving power of R∼1790–2640 (168 - 113 km sec^-1).§.§ Modelling Proxima b and its atmosphere Exoplanet Proxima b is found to orbit its host star in 11.186^+0.001_-0.002 days <cit.>. The amplitude of its radial velocity variations corresponds to a minimum mass of 1.27^+0.19_-0.17 M_Earth. If the mean density of the planet is the same as that of the Earth, and its orbit is nearly edge-on, it will have a radius of ∼1.1 R_Earth. Proxima Cen has an estimated mass of 0.123±0.006 M_Sun, implying an orbital semi-major axis of 0.0485^+0.0041_-0.0051 AU, corresponding to a maximum angular separation of 37.5 mas. Proxima Cen has an effective temperature of T_Eff= 3042±117 K, radius of 0.141±0.007 R_Sun,and bolometric luminosity of L=0.0017 L_Sun <cit.>. Due to the close vicinity of the planet to its host star, it is generally assumed that Proxima b is tidally locked, meaning that the same dayside hemisphere is eternally facing the star. The planet effective dayside temperature will strongly depend on its Bond albedo and global circulation patterns. If due to atmospheric circulation the absorbed stellar energy is homogeneously distributed over the planet, and the Bond albedo is similar to that of Earth (A_B=0.306), the dayside equilibrium temperature of Proxima b is 235 K. If there is effectively nocirculation and the absorbed stellar energy is instantaneously reradiated, its observed dayside temperature could be as high as 300 K (and even up to 320 K for a moon-like albedo) . In the other extreme case in which the planet has an albedo such as Venus (A_B=0.9) with a very effective atmospheric circulation, its dayside effective temperature could be as low as 145 K. For the calculations below we assume a continuum brightness temperature of 280 K at 15 μm and a near-transiting orbital inclination, corresponding to a planet/star contrast ratio of 6× 10^-5 at superior conjunction.Simulated high-resolution emission spectra of Proxima b were generated by the Spectral Mapping Atmospheric Radiative Transfer (SMART) model assuming it to have an atmosphere such as Earth, using opacities from the Line-By-Line ABsorption Coefficient (LBLABC) tool <cit.>.The HITRAN 2012 line database <cit.> was used as input to LBLABC, which generates opacities at ultra-fine resolution (resolving each line with >10 resolution elements within the half-width) on a grid of pressures and temperatures that spans a range relevant to Earth's atmosphere.Following this, SMART – which has been extensively validated against moderate- to high-resolution observations of Earth <cit.> – was used to simulate spectra at 5×10^-3 cm^-1 resolution (corresponding to R > 10^5 at the simulated wavelengths).Our spectral simulations used a standard Earth atmospheric model for temperatures and gas mixing ratios <cit.>, the spectra of which are shown in Figure 2.To bound certain extremes in thermal emission, model runs were performed for both clear sky conditions (the `clear atmosphere model') and for an opaque high-altitude cirrus cloud <cit.>, called the `optically-thick cirrus model'.Also, to explore a situation with large thermal contrast between the surface and stratosphere, a case where Earth's stratospheric temperatures were artificially made isothermal and equal to the tropopause temperature (210 K) was simulated (`isothermal stratosphere model').§.§ Simulated observations First, an estimate of the expected signal-to-noise (S/N) for Proxima Cen with MIRI is obtained from the beta version of the JWST exposure time calculator[http://jwst.etc.stsci.edu]. The 12 μmand 22 μm flux densities of Proxima have been determined by the NASA Wide-field Infrared Survey Explorer (WISE) to be 924 mJy (m_W3 = 3.838±0.015) and 278 mJy (m_W4=3.688±0.025) respectively, which are fitted to a 3000 K Planck spectrum and subsequently interpolated to 13, 14, and 15 μmflux densities of 816, 713, and 630 mJy respectively. These fluxes are fed to the exposure time calculator for channel 3B of the MIRI MRS mode. A detector setup of 5 groups and fast readout gives an integration time of 16.6 seconds delivering an SNR of 200 for a total exposure time of 38.85 seconds.This extrapolates to a S/N of2000 per hour assuming that calibration uncertainties do not contribute to the noise.The model planet spectra are smoothed to the spectral resolving power of R=2200(the mean of the MRS 3B band) and subsequently binned to the wavelength steps of the 3B MRS channel. The resulting clear-atmosphere model spectrumnormalised by the stellar spectrum (which for clarity is assumed to be featureless) is shown in the top panel of Fig. <ref>. Since the spectral filtering technique is only sensitive to the high-frequency signals, the low-frequency components are removed by subtracting a 25 wavelength-step sliding average (a rather arbitrary width) from the spectrum, resulting in the spectral differential spectrum shown in the middle panel. Subsequently, random noise is added to this differential spectrum at a level expected for the total simulated integration time, as shown in the bottom panel of Fig. <ref>.At this stage it is determined at what statistical significance level the differential model spectrum is preferred to be present in the datawith respect to pure noise. This is done by calculating the chi-squared of the observed spectrum, with its sliding average and differential model spectrum removed - for a range of planet/star contrasts and radial velocities.The minimum chi-squared is assigned as the best fit and the Δχ^2 interval is used to determine the statistical uncertainties of a possible CO_2 detection. These simulations were repeated for the three different models, and for a range star/planet contrasts corresponding to different orbital phases or different effective dayside temperatures.§ RESULTS AND DISCUSSION§.§ Detectability Our simulations show that the 15 μm CO_2 high-pass filtered signal of the Earth-mass planet can be detected within a limited amount of observing time. The MRS mode of MIRI at the JWST will in 24 hours integration time (excluding overheads) deliver a R=1790–2640 spectrum of Proxima Cen between 13.2 and 15.8 μm at a S/N of ∼10,000 per wavelength step (assuming photon noise). This corresponds to a 1σ contrast limit of ∼ 1 × 10^-4. While the high-frequency features in the filtered planet spectrum are typically at a 1–3× 10^-5, there are about 100 within the targeted wavelength range - combining to a detection at a ∼2σ level. It means that while the continuum planet/star contrast is at a level of 6× 10^-5 (∼0.6σ per wavelength step),the combined spectrally filtered signal over the 3B band is about a factor 3–4 higher.The top-right panel of Fig. <ref> shows the statistical confidence intervals for 5×24 hrs of observations if no CO_2 signal is present, while the bottom-right panel shows the same, but then with the clear-atmosphere CO_2 model spectrum for a face-on planet injected, indicating it can be detected at nearly 4σ within this exposure time. Hence, while the individual CO_2 features are not visible in the simulated spectrum, their combined signal can be clearly detected. Results are very similar for the Isothermal Stratosphere model and the Optically-Thick Cirrus model.§.§ Important prerequisites §.§.§ The stellar spectrum and its variability We have made several assumptions that are vital for the high-pass spectral filtering technique to succeed in detecting CO_2 in Proxima b.First, it is assumed that the high-frequency components of the spectrum of the host star itself are perfectly known. Although low-resolution (R=600) mid-infrared spectra of M-dwarfs taken with the Spitzer Space Telescope <cit.> seem featureless, Phoenix model spectra[https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011_2015/SPECTRA/] <cit.> show that the 13.2-15.8 μm wavelength region of an M5V dwarf star harbours thousands of H_2O lines, collectively resulting in wavelength-to-wavelength variations of ∼ 2% in the MIRI MRS spectrum.It means that these features need to be calibrated to better than a relative precision of 1% for them not to interfere with the planet CO_2 signal and not to act as an extra noise source. The star is also expected to have numerous but much weaker lines from hot CO_2in this MRS band, resulting in wavelength-to-wavelength fluctuations of a few times 10^-4 - hence which need to be calibrated to a relative precision of ∼10%.It is rather unlikely that spectral modelling could be sufficient to calibrate the high-frequency components of the mid-infrared spectrum to ≤ 10^-4. This means thata deep stellar spectrummay need to be obtained near or at inferior conjunction when the contribution from planet emission from the system is the smallest, which would limit the sensitivity of this method for low orbital inclinations and would take as much time as the observations at superior conjunction. Since the data at superior and inferior conjunction must be taken at least ∼5 days apart, variability of spectral features should also be below ≤ 10^-4 on this time scale. We used archival data of Proxima from the UVES spectrograph (R=100,000) at the Very Large Telescope separated by 4 days (October 10 & 14, 2009) to assess the optical variability of the star. For each of the two nights, a few dozen spectra were combined and the 868 – 878 nm wavelength range extracted which is dominated by hundreds of TiO lines but is free of telluric lines. The averaged spectrum was subsequently convolved with a Gaussian to mimic the resolution of MIRI and subsequently binned to match its wavelength steps (in Δλ / λ). After dividing out a linear trend with wavelength, the standard deviation of the ratio of the resulting spectra of the two nights is 4× 10^-4. Since these data are possibly limited by flat fielding uncertainties, and variability in the mid-infrared is expected to be lower, this result is encouraging.§.§.§ Instrument calibration Another prerequisite is that the spectral responses of the MRS pixels of MIRI can be adequately calibrated. Neither absolute flux calibration nor the low-frequency spectral response are important, but the sensitivity of one wavelength relative to the next is crucial - e.g. thespectral pixel-to-pixel calibration of the flat field. Potentially challenging is fringing, a common characteristic of infrared spectrometers. It is caused by interference at plane-parallel surfaces in the light-path of the instrument.Experiences with data from ISO and Spitzer show that it can be removed down to the noise level <cit.>. <cit.> have characterised the fringing of the MIRI detectors in the laboratory and identify three fringe components with scale lengths (in wave number) of 2.8, 0.37 and 10-100 cm^-1, originating from the detector substrate, dichroic, and fringe beating respectively. The planet CO_2 features also show a regular pattern, but with a characteristic scale length of ∼1.6 cm^-1, which fortunately is significantly different from these fringe components. A potentially unwelcome source of error may be fringing in combination with dithering. Small residuals left over after defringing, combined from different dither positions, may be challenging to calibrate. In the signal-to-noise calculations presented above we assumed that the instrument calibration is perfect. For it not to add an extra source of noise, the wavelength-to-wavelength precision of the flat fielding and fringe removal must be ≤ 10^-4. If this level can be reached for individual IFS pixel, a tailored dithering strategy will subsequently push the calibration noise to below 10% of the noise budget for a 24h observation. A single observation will have the star light mostly distributed over 2 slices × 2 pixels in the IFS, and by moving the star stepwise over the field of view of the IFS about 80 independent positions can be obtained. This will reduce the calibration noise by a factor √(2×2×80)≈ 15. §.§.§ Planet spectrumIn addition, temporal stellar atmospheric disturbances can modify the chemical composition of a planet atmosphere, meaning that flares could temporarily change the CO_2 abundance of Proxima b. Such effects has been investigated by <cit.>, who find that although the abundances of some chemical species can be significantly altered deep in to the atmosphere (∼1 bar), CO_2 is expected to only be affected very high in the atmosphere at < 6 × 10^-4 mbar - suggesting that the 15 μm CO_2 will hardly be affected. In principle, the observations could also be sensitive to other planets in the system. Since such planet would likely be in a significantly wider orbit and be colder, the expected signal would be smaller. The CO_2 signal of such hypothetical planet could be distinguished from that of Proxima b since its superior conjunctions would occur on different epochs. §.§ Phase variations Adetection of CO_2 will provide us with both the strength of the planet signal and its radial velocity. These can be used to constrain the orbital inclination of Proxima b. An example of such observation is shown in Fig. <ref>, showing in the left panel the expected variation in contrast and radial velocity and their uncertainties for 9×24 hrs exposures for each three measurements at orbital phase ϕ=0.25, 0.5, and 0.75 - assuming an orbital inclination of near 90^o. These represent significantly longer exposures than that presented in Fig. 3.The right panel shows the same but for an inclination of i = 30^o, resulting in a smaller variation in contrast and radial velocity as function of phase. The observations at inferior conjunction may need to serve as a reference for the stellar spectrum (see Sect. 3.2.1). In both cases it is assumed that all thermal flux originated from the dayside hemisphere of the planet with an effective temperature of 280 K. Since the strength of the signal at ϕ=0.25 and 0.75 can be up to a factor two lower than that at ϕ=0.5, calibration of the instrumental response and the stellar spectrum willbe even more important. If the orbital inclination is low, the planet will never be seen entirely face-on, reducing the maximum signal at superior conjunction. However, it would also mean that the mass of the planet is higher, meaning that the planet radius could be larger than assumed above (in particular if such more massive Proxima b is volatile rich), possibly counteracting the reduction in expected planet surface brightness. §.§ Atmospheric characterisation We performed our MRS MIRI simulations for three different atmospheric model spectra, 1) for a planet with an isothermal stratosphere, 2) for a planet with an inversion layer and clear atmosphere, and 3) with inversion layer combined with optically thick cirrus. Independent of which model we use, the increase in S/N over that expected for one wavelength step are very similar for all models at a factor 3–4. We also experimented by using the Earth transmission spectrum instead, which is similar to the isothermal stratosphere spectrum, but with significant differential signal at the heart of the CO_2 band. This provides a S/N increase of a factor of ∼5 over the S/N at at single wavelength step, which can be treated as an upper limit to the differential gain. Retrieving an injected signal from one model using one of the other spectra results in a significant decrease in signal to noise - not surprising since a large number of the spectral features appear as either emission or absorption in the models. The brightness temperature of the planet atmosphere at a certain wavelength roughly corresponds to the atmospheric temperature at the τ = 1 surface. In the center of the strongest CO_2 lines, where the opacity is greatest, we probe the atmosphere at the highest altitudes. Therefore, in the case of a strong thermal inversion (the clear atmosphere and the optically-thick cirrus models) the atmosphere will be warmer at such low pressure - resulting in emission lines in stead of absorption lines when the atmosphere is cooler at higher altitudes. This implies that a detection will also constrain the temperature structure of the upper atmosphere, giving additional insights in high-altitude atmospheric processes. It will not just merely be a detection of the planet atmosphere, which can be compared with theoretical models. E.g. <cit.> argue that Earth-like planets orbiting M-dwarfs are likely to have relatively cool, bordering on isothermal stratospheres – even with O_3 present.Several features from other molecules are present in the 13.2 - 15.8 μm wavelength range, such as C_2H_2 at 13.7 μm and HCN at 14 μm, which may be included in the atmospheric spectral template if needed (and if expected to be present in the planet atmosphere).§.§ Prospects of detecting ozone WhenCO_2 is targeted in the MRS 3B band,the 1B, 2B, and 4B bands are observed simultaneously. Interestingly, the 2B band, ranging from 8.6 to 10.3 μm covers the 9.6μm ozone band. While CO_2 in the atmosphere of Proxima b may be likely, if the planet did undergo ocean loss early it its history to generate a massive O_2 atmosphere <cit.>, it is also possible that O_3, photochemically-produced from the O_2, is also present in higher abundances than seen on Earth <cit.>. O_3 is of course also of high interest as it can be used as a proxy for the O_2 biosignature from a photosynthetic biosphere. Detection of O_3 with JWST would therefore provide an intriguing first hint that life might be present on an extrasolar planet, although O_3 production via abiotic O_2 from ocean loss would first have to be ruled out.The expected S/N per wavelength step in a 24 hour observation is 2× 10^4, about a factor of two higher than in the 3B band because of the high stellar flux. On the other hand, the expected continuum planet/star contrast is a factor ∼2 lower compared to that expected at 15 μm. Unfortunately, the individual lines within the 9.6 μm band are more tightly packed than the lines in the 15 μm CO_2 band, i.e. the ozone band is not fully resolved at the MRS resolution of R=2800 in this wavelength range. This means that if ozone is present in the atmosphere of Proxima b, it's spectral differential signal will be about a factor 3–4 smaller than that of CO_2.We estimate that in the best case one could expect a 2σ result in 20 days of JWST observing.We also note that the prerequisite for spectral calibration is also more stringent by a factor 2 compared to the CO_2 case. Hence, although observations of ozone come for free when the CO_2 band is targeted, it is unlikely this could result in a firm detection and is probably beyond the limit of what the JWST can achieve.§.§ Strategy for proof of concept and other prospects We envisage two ways to show proof of concept for the high-pass spectral filtering technique with the MRS mode of MIRI. First, the method can be used on exoplanet targets with significantly higher planet/star contrasts. For example, a T=1000 K hot Jupiter orbiting a solar type star will have a 15 μm contrast of 10^-3, a factor 20 higher than Proxima b. It means that for a host star whose 15 μm flux is 40 times (4 magnitudes) fainter than Proxima b, CO_2 will still be detected 10x faster. Also, one could aim for cool Neptunes or super-Earths orbiting nearby M-dwarfs, such as Gliese 687b. If the spectrum of this particular planet exhibits a CO_2 absorption feature it can be detected a factor 4 faster than in the case of Proxima b.From a theoretical point of view it will be important to identify those planets that are expected to have CO_2 in their atmospheres and select those with the most favourable stellar magnitudes and planet/star contrasts. Ultimately, one should target Proxima itself, gradually increasing the integration time and validating at each step that the expected S/N limits are being reached. Detecting CO_2 in the atmosphere of Proxima b will be a major step forward in our quest for potential habitats and signs of extraterrestrial life. The detection of specific spectral features expected in a planet atmosphere becomes orders of magnitude more powerful if the planet can also be angularly separated from the planet <cit.>. In such case the stellar spectrum can be effectively removed from all pixels in the IFU (since it is identical everywhere), after which the residual spectra can be searched for the planet features. Hoeijmakers et al. (2017; in prep.) show that this technique is very effective for the SINFONI and OSIRIS IFU spectrographs, located atthe VLT and Keck Telescopes respectively, which have spectral resolving powers similar to that of MIRI (and the NIRSPEC IFU). If we could point the JWST directly at α Centauri A using the MIRI MRS, only 1–1.5” away the starlight is at the level of that of Proxima implying that an Earth-size planet in the habitable zone of α Cen A could possibly be detected in 24 hrs. Unfortunately, α Centauri A will saturate the MIRI detectors within a small fraction of a second, irreversibly damaging the instrument.Possible ways to mitigate this issue need to be investigated. § ACKNOWLEDGMENTSSnellen acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 694513, and from research program VICI 639.043.107, which is financed by The Netherlands Organisation for Scientific Research (NWO).Désert acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement nr 679633; Exo-Atmos). Robinson and Birkby gratefully acknowledge support from the National Aeronautics and Space Administration (NASA) through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. Support for this work was provided in part by NASA through Hubble Fellowship grant HST-HF2-51336 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. Meadows and Robinson are members of the NASA Astrobiology Institute's Virtual Planetary Laboratory Lead Team, supported by NASA under Cooperative Agreement No. NNA13AA93A. | http://arxiv.org/abs/1707.08596v2 | {
"authors": [
"I. Snellen",
"J. -M. Desert",
"L. Waters",
"T. Robinson",
"V. Meadows",
"E. van Dishoeck",
"B. Brandl",
"T. Henning",
"J. Bouwman",
"F. Lahuis",
"M. Min",
"C. Lovis",
"C. Dominik",
"V. Van Eylen",
"D. Sing",
"G. Anglada-Escude",
"J. Birkby",
"M. Brogi"
],
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"astro-ph.EP"
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"published": "20170726181602",
"title": "Detecting Proxima b's atmosphere with JWST targeting CO2 at 15 micron using a high-pass spectral filtering technique"
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"authors": [
"Miguel Aparicio Resco",
"Antonio L. Maroto"
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"published": "20170727165338",
"title": "Parametrizing growth in dark energy and modified gravity models"
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Mining Device-Specific Apps Usage Patterns from Large-Scale Android UsersHuoran Li, Xuan LuPeking University{lihr, luxuan}@pku.edu.cnReceived: December 30, 2023/ Accepted: date ========================================================================= When smartphones, applications (a.k.a, apps), and app stores have been widely adopted by the billions, an interesting debate emerges: whether and to what extent do device models influence the behaviors of their users? The answer to this question is critical to almost every stakeholder in the smartphone app ecosystem, including app store operators, developers, end-users, and network providers. To approach this question, we collect a longitudinal data set of app usage through a leading Android app store in China, called Wandoujia. The data set covers the detailed behavioral profiles of 0.7 million (761,262) unique users who use 500 popular types of Android devices and about 0.2 million (228,144) apps, including their app management activities, daily network access time, and network traffic of apps. We present a comprehensive study on investigating how the choices of device models affect user behaviors such as the adoption of app stores, app selection and abandonment, data plan usage, online time length, the tendency to use paid/free apps, and the preferences to choosing competing apps. Some significant correlations between device models and app usage are derived, leading to important findings on the various user behaviors. For example, users owning different device models have a substantial diversity of selecting competing apps, and users owning lower-end devices spend more money to purchase apps and spend more time under cellular network.H.1.2Information System ApplicationsUser/Machine Systems § INTRODUCTION Since Apple announced the iPhones in 2007, smartphones have been playing an indispensable role in people's daily lives. A great variety of applications (a.k.a, apps) such as Web browsers, social network apps, media players, and games make smartphones become the main access channels to Internet-based services rather than communication tools. With the ever-increasing amount of smartphone users and apps, comprehensive and insightful knowledge on what, when, where, and how the apps are used by the users is extremely important <cit.>. Many significant research efforts have been made in the past few years on portraying the users and understand their behaviors in term of apps <cit.>.Like all Internet users, smartphone users can be classified based on various facts, including demographics such as location, gender and age, and behaviors such as preferences to apps <cit.>, content consumed within apps <cit.>, and so on. Actually, in the current app ecosystem, a user is naturally identified by his/her device <cit.>. Many such classifications of smartphone users boil down to classifications of devices. In other words, much variance of user behaviors may be explained by the devices they use. Indeed, when users interact with their smartphones, download apps from online app stores, and use the apps for different purposes, their experiences are usually affected by various parameters of the device models they use, such as brands, hardware specifications, etc. Understanding how device models affect user behaviors can help app store operators know their users better and improve their recommender systems by considering device models. For instance, one may recommend apps with fancy graphical effects to devices that are equipped with a powerful GPU and high-resolution screen <cit.>. Device-specific ads is another big opportunity. For example, Facebook customizes mobile ads according to device model types since 2014 <cit.>. App developers are thrilled to know through which kind of device models they can gain more users and more clicks, so that they can invest their effort in customizing the ads for those models, e.g., by designing banners of proper sizes or placing videos at proper positions on the screen. Furthermore, device models may be more informative in the behavioral analyses of Android users, due to the large diversity and heavy fragmentation of Android devices <cit.>. Some existing efforts have been made to investigate how apps usage is affected by device models[In this paper, the term “device model" refers to the device with specific product type with hardware specifications, e.g., , , and so on] <cit.>. Unfortunately, due to the lack of sufficient user behavioral data at scale, most existing studies suffer from serious selection bias, including specific user groups (e.g., in-school students) <cit.>, fixed device models or apps <cit.>, and limited metrics (e.g., screen size) <cit.>. In this paper, we present a comprehensive user study exploring whether, how, and how much the device models can really influence the user behaviors on using smartphones and apps. We collect the behavioral profiles of about 0.7 million anonymized Android users[Our study has been approved by the research ethnics board of the Institute of Software, Peking University. The data is legally used without leaking any sensitive information. The details of user privacy protection are presented later in the data set description. We plan to open the data set when the manuscript is published.] by a leading app store operator in China, called Wandoujia[http://www.wandoujia.com]. Besides the largest data set to date, our study differs from existing efforts further in two aspects. ∙We conduct the study from a new perspective, i.e., the sensitivity of device's price against the app usage. In our opinion, the price of a device model can generally reflect the level of hardware specifications when the device is released. Additionally, such a metric can imply the users' economic background, which influences user behavior at demographics level <cit.>. In this way, we try to categorize the users into different economic groups and explore the sensitivity of the device against the user behavior. ∙ Second, we explore comprehensive behavioral profiles that contain various useful information, including the apps selection, apps management activities (e.g., download, update, and uninstallation), data plan usage per app, and online time length per app, etc. In addition, we focus on only the behavioral profiles from long-term users who steadily contribute to our study. This provides solid ground for the findings from our study. The major contributions made by this paper can be summarized as follows.* We collect app usage from over 0.7 million users in a period of five months (May 1, 2014 to September 30, 2014). Our data set covers 500 popular Android devices and over 0.2 million Android apps. The detailed user behavioral profiles include the activity log of downloading, updating, and uninstalling apps, the daily traffic and access time of every app through both Wi-Fi and cellular. Based on such a large-scale data set, we explore a comprehensive study on how the device models can impact the app usage.* We find significant correlations between the choice of device models and app usage spanning the adoption of app stores, the selection and abandonment of apps, the online access time and data traffic of apps, the revenue of apps, and the preferences against competing apps. Some findings can be quite interesting, e.g., the users holding lower-end devices are likely to spend more money on purchasing apps and spend more time under cellular network, the selection of the apps with similar functionalities presents a substantial diversity among users, etc. The findings can be leveraged to understand the user requirements better, preferences, interests, or even the possible background such as economic or profession.* We derive some implications that are directly helpful to several stakeholders in the app-centric ecosystem, e.g., how app store operators can improve their recommendation systems, how the app developers can identify problematic issues and gain more revenues, and how the network service providers can explore more personalized services <cit.>.The remainder of this paper is organized as follows. We first present the related work in the area of user behavior analysis of smartphone users. Next we describe the Wandoujia and the five-month data set, and present our measurement approach alongside the research questions and hypotheses. Then we conduct the correlation analysis on how the choice of device models affects the user behaviors on app usage. In addition, we propose the underlying reasons leading to such significant correlations. We also discuss about our implications for relevant stakeholders, and describe the limitation of our study and threats to the generalization of our results. Finally, we conclude the study and some outlooks to future work. § RELATED WORKPrecise classification of users and understanding their behaviors of using apps are significant to every stakeholder in the app ecosystem, including app store operators, content providers, developers, advertisers, network providers, etc. Several efforts have been made in the fields.One straightforward way to understand the user behavior is conducting field study. Usually, the field studies are conducted over some specific user groups. Rahmati et al. <cit.> made a four-month field study of the usage of smartphone apps of 14 users, and summarized the influences of long-term study and short-term study. Lim et al. <cit.> made a questionnaire-based study to discover the diverse usages from about 4,800 users across 15 top GDP countries. The results show that the country differences can make significant impacts on the app store adoption, app selection and abandonment, app review, and so on. Falaki et al. <cit.> performed a study of smartphone usage based on detailed traces from 255 volunteers, and found the diversity of users by characterizing user activities. A number of other studies have been made in similar ways <cit.>. To have more comprehensive behavioral data, the monitoring tools/apps are more appreciated other than questionaire. Yan et al. <cit.> developed an app, called AppJoy, and deployed such an app to collect the usage logs from over 4,000 users and find the possible patterns in selecting and using apps.Besides the general analysis, some studies aim to investigate the diversity of user behaviors from specific perspectives. Raptis et al. <cit.> performed a study of how the screen size of smartphones can affect the users' perceived usability, effectiveness, and efficiency. Rahmati et al. <cit.> explored how users in different socio-economic status groups adopted new smartphone technologies along with how apps are installed and used. They found that users with relatively low socio-economic background are more likely to buy paid apps. Ma et al. <cit.> proposed an approach for conquering the sparseness of behavior pattern space and thus made it possible to discover similar mobile users with respect to their habits by leveraging behavior pattern mining. Song et al. <cit.> presented a log-based study on about 1 million users' search behavior from three different platforms: desktop, mobile, and tablet, and attempted to understand how and to what extent mobile and tablet searchers behave differently compared with desktop users. Some field studies were made towards specific apps. Böhmer et al. <cit.> made a field study over three popular apps such as Angry Bird, Facebook, and Kindle. Patro et al. <cit.> deployed a multiplayer RPG app game and an education app, respectively, and collected diverse information to understand various factors affecting the app revenues. Due to the difficulty of involving a large volume of users, the preceding field studies may suffer from relatively limited users and apps. At large-scale, Xu et al. <cit.> presented usage patterns by analyzing IP-level traces of thousands of users from a tier-1 cellular carrier in U.S. They identified traffic from apps based on HTTP signatures and present aggregate results on their spatial and temporal prevalence, locality, and correlation. Our previous work <cit.> was conducted over a one-month data collected by Wandoujia, and evidenced that some app usage patterns in terms of app selection, management, network activity, and so on. To understand the diversity of user behaviors better, the study made in this paper is based on a more comprehensive data set that consists of the 5-month behavioral profiles from various users, and focuses on the impact made by the choice of device models. § THE DATA SETIn this section, we present the data set collected from Wandoujia, by describing the detailed information that shall be used to conduct our empirical study.§.§ About WandoujiaOur data is from Wandoujia[Visit its official site via <http://www.wandoujia.com>.], a free Android app marketplace in China. Wandoujia was founded in 2009 and has grown to be a leading Android app marketplace. Like other marketplaces, third-party app developers can upload their apps to Wandoujia and get them published after authenticated. Compared to other marketplaces such as Google Play, apps on Wandoujia are all free, although some apps can still support “in-app purchase". Users have two channels to access the Wandoujia marketplace, either from the Web portal, or from the Wandoujia management app. The Wandoujia management app is a native Android app, by which people can manage their apps, e.g., downloading, searching, updating, and uninstalling apps. The logs of these management activities are all automatically recorded. Beyond these basic features, the Wandoujia management app is developed with some advanced but optional features that can monitor and optimize a device. These features include network activity statistics, permission monitoring, content recommendation, etc. All features are developed upon Android system APIs and do not require “root" privilege. Users can decide whether to enable these features. However, these advanced features are supported only in the Chinese version of Wandoujia management app. §.§ Data Collection Each user of Wandoujia is actually associated with a unique Android device, and each device is allocated a unique anonymous ID. In this study, we collected usage data ranging from May 1, 2014 to September 30, 2014. The data set consists three categories of information:∙ Device Model and Price. The Wandoujia management app records the type of each user's device model, e.g., , etc. In our raw data set, there are more than 10,000 different Android device models. Such a fact implies the severe fragmentation of Android devices. However, from our previous work <cit.>, the distribution of users against device models typically complies with the Pareto Principle, i.e., quite small set of device models accounts for substantial percentage of users. Hence, we choose the top 500 device models according to their number of users. We then look up the first-release price of these device models from [<http://shouji.jd.com>, Jd is the largest e-commerce for electronic devices in China]. Indeed, the price of a device model always decline over time due to the Moore's Law.However, such a price can somewhat reflect the hardware specifications and the target users at the time when the device model is released on market. Considering the large volume of users, we choose to rely on this type of price as our classification criteria. ∙App Management Activities. The Wandoujia management app can monitor the users' management activities such as searching, downloading, updating, and uninstalling their apps. Each management action is recorded as an entry in the log file that can be uploaded to the Wandoujia servers. With the management activities, we can know which apps ever appeared on the user's device, and when these apps are downloaded, updated, and installed. ∙App Network Activities. Besides the basic management functionalities, the Wandoujia management app provides advanced features to record the daily network activities of an app. The network statistic features are optional, and are enabled when and only when the users explicitly grant the permissions to the Wandoujia management app[Due to space limit, the details of how Wandoujia management app works can be referred to our previous work <cit.>].If the app generates network connections either from Wi-Fi or cellular (2G/3G/LTE), the network usage can be monitored and recorded at the TCP-level, including the data traffic and network connection time. Note that the network statistic feature works as a system-wide service, thereby network usage from all apps can be captured, even if the app is not installed via the Wandoujia management app. The Wandoujia management app does not record the details of every interaction session, due to the concerns of system overhead. Instead, the Wandoujia management app aggregates the total network usage of an app. In particular, the data traffic and access time generated from foreground and background are distinguished, so that we can have more fine-grained knowledge of network usage. In other words, we can use eight dimensions of the daily network usage per app, i.e., 2 metrics (access time and traffic) * 2 modes (Wi-Fi and Cellular) * 2 states (foreground and background).To conduct a comprehensive and longitudinal measurement, we should process and filter the collected data guided by the following principles. We choose only the users who explicitly granted Wandoujia to collect their usage data from the preceding top 500 device models (denoted as the user set 𝕌). For these users, we further take into account those who continuously contribute their daily usage data for five months. Such a step assures that we can have rather complete behavioral data of these users. In this way, we obtain 761,262 users (denoted as the subset 𝕌') and their usage data of 228,144 apps. §.§ User PrivacyCertainly, the user privacy is a key issue to conduct such a measurement study based on large-scale user behavior data. Besides collecting the network activity data from only the users who explicitly have granted the permission, we take a series of steps to preserve the privacy of involved users in our data set. First, all raw data collected for this study was kept within the Wandoujia data warehouse servers (which live behind a company firewall). Second, our data collection logic and analysis pipelines were completely governed by three Wandoujia employees[One co-author is the head of Wandoujia product. He supervised the process of data collection and de-identification.] to ensure compliance with the commitments of Wandoujia privacy stated in the Term-of-Use statements. Finally, the Wandoujia employees anonymized the user identifiers. The data set includes only the aggregated statistics for the users covered by our study period. §.§ Limitation of the Data SetThe preceding data set may have some limitations. First, the management activities come from only the apps that are operated by the Wandoujia management app. The management activities of apps that are downloaded and updated in other channels such as directly from the app's websites or uninstalled via the default uninstaller of Android system, cannot be captured by our data set. Second, we can take into account only the network usage as an indicator that an app is exactly used. Some apps such as calculator and book readers are often used offline, so we cannot know whether these apps have been launched and how they are used. However, our data set is the largest to date and comprehensive enough to conduct a longitudinal user study.§ MEASUREMENT APPROACHIn practice, it is a common fact that device models with similar price usually have similar hardware specifications and target user groups. Our measurement study then aims to evaluate whether and to what extent the choice of device models, or more specifically, the price of smartphones, can affect app usage in various metrics. According to the preceding data set of user behavioral profiles, we propose some research questions that are significant to stakeholders.* RQ1: Does the choice of device models impact the usage of app stores? If such an impact exists, which users are more likely to adopt the app stores (e.g., downloading new apps and updating existing apps), and what about the various requirements of different users when they use app stores?* RQ2: Does the choice of device models impact the online time spent by users? If such an impact exists, which apps do different users tend to spend their time on?* RQ3: Does the choice of device models impact the traffic used by users, especially the data plan of cellular? If such an impact exists, which apps are more likely to consume the cellular data plan of different users?* RQ4: Does the choice of device models impact the purchase behavior of users and thus the app revenue? If such an impact exists, which group of users are more likely to pay for apps, and which apps are more likely to be paid for by different users?* RQ5: Does the choice of device models impact the choice of apps with similar functionalities or purposes? If such an impact exists, which apps are more likely to be adopted by different users? To answer the above RQs, our measurement study is conducted from two aspects.§.§.§ User Group AnalysisAs we assume that the price of device models can possibly reflect the economic background of the users, we first study the overall user behaviors by categorizing the users into groups according to the first-release price. In China, the price systems of popular e-commerce web sites such as , , and , the price of device models is usually segmented at every 1,000 RMB level, i.e.,⩽ 1,000 RMB, 1,000 RMB-2,000 RMB, 2,000 RMB-3,000 RMB, 3,000 RMB-4,000 RMB, and ⩾4,000 RMB. Hence, we roughly categorize the device models into three groups according to their on-sale price information that is published on the , i.e., the High-End (⩾4,000 RMB, about 600 USD), the Middle-End (1,000 RMB-4,000 RMB, about 150-600 USD), and the Low-End (⩽ 1,000 RMB, about 150 USD). We list the categorization results in Table <ref>. We manually check the price history evolution of the top 500 device models onas well as look up some third-party data sources such as [<https://itunes.apple.com/us/app/dong-dong-gou-wu-zhu-shou/id868597002?mt=8>, is an app for inquiring history price of products on .] and [<http://www.xitie.com>, is a website for inquiring price history of products on popular e-commerce sites.]. Most of the device models were first released to market after 2012, and can still fall into the above coarse-grained groups as of May 1, 2014 (the start time of our data set). Very few device models cannot meet such criteria, e.g., the first-release price ofwas 4,399 RMB, but the price fell down to about 3,200 RMB as of May 1, 2014. For this case, we still categorize the device models by the first-release price. Luckily, suchis rare in our data set. In this way, the dynamics of prices can have the side effects as little as possible. For each RQ, we use the box-plot to report the distribution of user behaviors over the corresponding metrics in each group. To further evaluate the statistical significance among the groups, we employ the Mann-Whitney U test (U-Test for short) <cit.> among the three groups. The U-Test is widely adopted in large data set to test whether the two groups have significant difference. In particular, the U-Test can be applied onto unknown distributions, and fit our data set very well. §.§.§ App Category AnalysisWe then study whether the choice of device models can impact the user behavior on specific apps. To this end, we run the regression analysis between the price of device model and the corresponding metrics, by organizing the 𝕌' according to the first-release price of each device model. Investing each RQ essentially relates to the user's preferences and requirements of specific apps. Hence, we categorize the apps according to the classification system of Wandoujia, e.g., NEWS_AND_READER, GAME, VIDEO, and so on. For each RQ, we summarize the related metrics over each app category for every single user, and thus compute the Pearson correlation coefficient between the metrics and the price of the device model. In this way, we can explore whether the choice of device models can be statistically significant to the app usage.In the next section, we conduct our analysis in the following workflow. We first present the motivations of each research question, respectively. Then we try to examine the impact caused by the choice of device models,by synthesizing the correlation analysis results at the granularity of each device model and the groups. Finally, we summarize the findings and try to explore the underlying reasons leading to the diverse user behaviors on app usage.§ ANALYSIS AND RESULTSIn this section, we explore the research questions and validate all the hypotheses, respectively. We show the results of user group U-Test in Table <ref> and the results of app-specific regression analysis in Table <ref>. Generally, most of the pairwise U-Test results in Table <ref> at p < 0.0001 level. The only exception is for RQ4, indicating that the difference on expenditure of paid apps is not quite significant between middle-end group and low-end group. Such an observation indicates that the price of device model has a statistically significant correlation of the behaviors of users from different groups. Hence, we can focus more on the distribution variance among groups (in the box-plots) and the results reported in Table <ref>. §.§ Effect on App Management ActivitiesFirst, we focus on RQ1, i.e., whether users holding different device models perform differently in terms of app downloads, updates, and uninstallations. This research question is motivated by three folds. First, the behaviors of app downloads can indicate the adoption of app stores by users, i.e., whether the users prefer seeking apps from app stores and which users are more active. Second, we can identify the most popular/unpopular apps for a given device model, the app stores operators can accurately recommend the apps. Third, we can explore which apps are more likely to be abandoned by the users holding a specific device model, and such knowledge can help app developers identify the possible problems such as device-specific bugs.§.§.§ Adoption of App StoresIn Figure <ref>, we first report the number of 5-month management activities in terms of download and update. On average, we can see that most users do not frequently access the app stores. However, in each group, the standard deviation of management activities is quite significant. Such an observation indicates that users can perform quite variously in terms of management. We can observe that the users holding higher-level devices are more likely to access the app stores. §.§.§ App SelectionWe next investigate whether the choice of device models can impact the app selection. From our previous study of the global distribution of apps <cit.>, we find that users can have quite high overlap in selecting the popular apps, such as , , . Hence, we first explore the similarity of apps selection. We aggregate the apps that are downloaded and updated by at least 1,00 unique users of each device model, and compute the pairwise cosine similarity between the three groups. The cosine similarity values are 0.81 (high-end and middle-end), 0.86 (high-end and low-end), and 0.81 (middle-end and low-end), respectively. Such an observation evidences our preliminary findings. Then we explore the diversity of app selection. For simplicity, we cluster each app according to its category information provided by Wandoujia, e.g., Game, NEWS_AND_READING, etc. We compute the contributions of downloads and updates from every single device model against a specific app category. For example, if there are 1,000,000 downloads of GAME apps and 50,000 of these downloads and updates come from the device model , we assign the contributions made by this device model is 5%. Then we make the correlation analysis of app selection and the price of device, by means of Pearson correlation co-efficient. We find that as the price of device models increases, the users are more likely to choose apps from the categories of TRAFFIC (r=0.376, p = 0.000) , LIFESTYLE(r = 0.469, p = 0.000), NEWS_AND_READING(r = 0.471, p = 0.000), SHOPPING(r = 0.488, p = 0.000), FINANCE(r = 0.513, p = 0.000), and TRAVEL(r = 0.560, p = 0.000). In contrast, the correlation analysis show that as the prices of device models increases, the users are less likely to choose the apps from GAME (r = -0.567, p = 0.000) and MUSIC (r = -0.407, p = 0.000).Such observations imply that the choice of device models can significantly impact the app selections, and infer the characteristics and requirements of the users. For example, users with high-end smartphones are more likely to have higher positions and better economic background, so they care about the apps from NEWS AND READING, FINANCE, TRAVEL, and SHOPPING. Users holding low-end device models care more about the entertainment such as Game and Music. §.§.§ App AbandonmentThe uninstallation can indicate the users' negative attitudes towards an app, i.e., the user does not like or require the app any longer. From Figure <ref>, it is observed that the median of uninstallation activities differ quite marginally among the three groups. We then perform the correlation analysis in a similar way of downloads and updates. In most categories, the Pearson co-efficient is not quite significant. However, we find that the correlation in TRAVEL (r = 0.406, p = 0.000) is significant. Although the travel apps are most likely to be downloaded by higher-end users, these apps are also very possible to be uninstalled by these users. Although the uninstallation does not take significant correlations to device models at the level of app category, investigating the individual apps that are possibly abandoned by a specific device model is still meaningful. To this end, we explore the apps which have been uninstalled for more than 100 times in our data set, and get 3,123 apps. We then draw the distribution of uninstallations according to the device model. An interesting finding is that, the manufacturer-customized or preloaded apps are more possibly uninstalled on their own lower-end device models. For example, the app(the package ) is a preloaded app on almost all device models produced by . This app has 20,985 uninstallations, while 17,641 uninstallations come from the lower-end devices. The similar findings can be found in other device models produced by , and . Such an observation implies that the lower-end users are less likely to adopt these customized or preloaded apps. Besides the preloaded apps, some apps are also more likely to be uninstalled by a specific device model. For example, theaccounts for more than 80% uninstallations of two camera apps. Such a finding implies that these apps can suffer from device-specific incompatibility or bugs.§.§ Effect on Online TimeWe next intend to validate RQ2, i.e., the choice of device models impacts on how long the users spend their time on the Internet. Such a research question is motivated by understanding whether the choice of device models can lead to various preferences toward cellular and Wi-Fi usage. If the app developers can know that some users from a specific device model spend more cellular time rather than Wi-Fi, they can provide customized features to these models. For example, developers can optimize the data plan usage by providing pre-downloading contents when these users are in Wi-Fi network. In addition, if users holding specific device models spend much time on a few categories of apps, the developers, web content providers, and advertisers can leverage such knowledge to customize more accurate advertisements to audience. The Wandoujia management app can record the daily foreground/background connection time under both cellular and Wi-Fi. Since the foreground time is computed only when the users interact with the app (by checking the stack of active apps in Android system), we exclude the background time in the online time analysis. Figure <ref> describes the distribution of online time. For online time, we are surprised to find that users rely less on the cellular network as the price of device model increases.In other words, the higher-end users typically spend less time under cellular network. For the average daily online time, the low-end users (⩽ 1,000-RMB device models) spend about 10 minutes more than the high-end users (⩾ 4,000-RMB device models) under cellular, while the high-end users spend 1 hour more than the low-end users under Wi-Fi. Immediately, we can infer that the network conditions vary a lot among different users, i.e., the lower-end users are less likely to stay in the places where Wi-Fi are covered. In contrast, the higher-end users tend to have better Wi-Fi connections. Such a difference can further imply the possible locations where different users may stay, e.g., the high-end users are more likely to stay in offices. We then investigate whether the choice of device models affect the usage of “network-intensive" apps. Similar to the preceding analysis in the management activities, we compare the online time distribution of device models over each app category, under cellular and Wi-Fi, respectively. For cellular, the online time of apps have no significant correlation with the price of device models, except the categories of TRAVEL (r = 0.3427, p = 0.000) SHOPPING (r = 0.452, p = 0.000), and EDUCATION (r = -0.305, p =0.000). This result can support the findings of RQ1. On the other hand, it is interesting to see that users holding lower-end smartphones are more likely to use EDUCATION (r = -0.305, p = 0.000) apps under the cellular. Such a finding suggests that a considerable proportion of lower-end users may be students. The correlation between the choice of device model and online time under Wi-Fi seems not to be quite significant, either. Only in the category of SHOPPING (r = -0.304, p = 0.000), the choice of device models seems to take significant correlation with the price of device. Such an observation is not surprising, as higher-end users are supposed to have better economic background and more likely to spend more on shopping. We can infer that users share quite similar preferences in app usage under Wi-Fi. §.§ Effect on Traffic Consumption Another important network-related issue for smartphone users is the traffic. We next focus on RQ3, i.e., whether users holding different device models differ in cellular and Wi-Fi traffic usage. Generally, users do not care about the traffic from Wi-Fi, but do concern the traffic from cellular that they need to pay for, especially for low-end users who usually have relatively limited budgets. However, exploring the traffic generated from Wi-Fi can be meaningful. Intuitively, we can explore which apps are more “bandwidth-sensitive" on specific device models, so the app developers and network service provider can consider techniques to compensate for bandwidth variability.The distribution of traffic consumption among device models is shown in Figure <ref>. Interestingly, although the higher-end users are observed to spend the least time under the cellular network in RQ2, they spend the most traffic. In other words, we can infer that the higher-end users are more likely to use those “traffic-heavy" apps. On average, a high-end user can spend more 100 MB cellular data plan than a low-end user. In China, such a difference of data plan does matter very much.Since the gaps between different device models are substantial, identifying which apps consume most traffic on specific device models can be quite meaningful. Similar to preceding analysis, we compute the Pearson correlation coefficients between the choice of device model from every single user and the apps on which the traffic is consumed. The cellular data plan consumed over the apps from SHOPPING (r = 0.423, p = 0.000) and TRAVEL (r = 0.445, p = 0.000) presents a quite significant positive correlation to the price of device models. In contrast, the correlations seem to be significantly negative in GAME (r = -0.371, p = 0.000) and MUSIC (r = -0.333, p = 0.000) apps. In these app categories, users with lower-end smartphones tend to spend more cellular traffic. These findings are consistent with the app download and update preferences in RQ1. The traffic generated under Wi-Fi presents significant correlations with the device models in some categories. The lower-end users tend to spend a large number of Wi-Fi traffic on the VIDEO apps, In contrast, the higher-end users are more likely to rely on the apps of COMMUNICATION, PRODUCTIVITY, SYSTEM_TOOL, TRAVEL, BEAUTIFY, NEWS_AND_READING, LIFESTYLE, and SHOPPING under Wi-Fi. Such a difference in the Wi-Fi traffic usage can indicate the requirements and preferences of users holding different device models, and thus implies the possibly different background of the users. §.§ Effect on App Revenue The above three research questions have revealed the general correlations between the choice of device models and app usage. To further explore the underlying reasons that can be more relevant to other stakeholders such as app developers, we try to focus on the RQ4, i.e., whether the choice of device models can affect the app revenue. It should be mentioned that all apps directly downloaded from Wandoujia are free, despite the in-app purchase of some apps. However, since the Wandoujia management app act as a system-wide service to monitor all apps that are installed on a device, we can still identify which apps should be paid ones that are downloaded from other channels. To this end, we write a crawler program to retrieve the package names of paid Android apps from other app stores including Google Play, 360safe App Store, and Baidu App Store. Then we extract the logs that are related to these paid apps from our data set. In addition, we record the fee of each paid app. By such a step, we finally obtained 27,375 users that have used paid apps.As the price of a device model can serve as an indicator of its owner's economic background, it is not uncommon to suppose users with expensive devices to pay more money in purchasing paid apps. We are interested in three facets, (1) the possibility to buy paid apps, i.e., are users with high-end devices more likely to use paid apps? (2) the expenditure of apps, i.e., do the high-end users pay more money on buying apps? (3) the diversity of paid apps, i.e., do users with different models pay for different purposes? First, we investigate the possibility of buying apps. From Table <ref>, we can observe that the higher-end users account for about 47% of all users who have ever purchased paid apps, and 45% of purchased records. We further perform a Chi-square test to confirm the significant correlations between the proportion of buyers and group (χ^2=2875.05, p = 0.000).When moving to the expense of apps, we perform the pairwise T-test of average expenditure of users. The T-Test can identify the positive or negative correlation of the tendency of buying apps against the groups. The T-test values are -4.281 (high-end v.s. low-end) and -5.740 (middle-end v.s. low-end), respectively, given the p = 0.000. To our surprise, buyers with higher-end device models are likely to spend even less money than lower-end users. Interestingly, although the lower-end users account for quite small proportion of all buyers and the purchase orders, they are likely to spend more money on average. Such a finding indicates that the choice of device models impact the wish to buy apps, which affects the revenue of apps accordingly.Finally, the choice of device models can impact the preferences of paid apps. The lower-end users are more likely to pay for GAME (r = -0.221, p = 0.000), BEAUTIFY (r = -0.086, p = 0.000 such as themes), and SPORTS(r = -0.131, p = 0.000). In contrast, the higher-end users tend to buy PRODUCTIVITY (r = 0.306, p = 0.000, such as office suite). Such a difference can reflect the different requirements of apps which the users would like to pay for.§.§ Effect on Choices of Competing AppsWe finally move to the RQ5, i.e., whether the choice of device models can affect selecting the apps of the same or similar functionalities (we name such apps as “competing apps" in the follows) Such a research question is motivated by the existence of a number of “competing apps" on the app stores. For example, there are a number of competing browsers such as , maps such as , and so on. Although these apps can provide very similar or even the same functionalities, they can perform quite variously such as the data traffic and energy drain, given the same user requests <cit.>. In addition, besides the common functionalities, the competing apps are likely to provide some differentiated features such as adjustable color and light for display. End-users often feel confusing to select the apps that are more adequate to their own preferences and requirements.Unlike the correlation analysis of RQ1-RQ4, we do not directly conduct correlation analysis to all app categories. Instead, we choose three typical apps: News reader, Video player, and Browser, as they are observed to be commonly used in daily life. For each app, we select the top-5 apps according to the online time that the users spend on them. The reason why we employ the online time instead of the number of downloads is that the online time can be computed only when the users interact with the app. The selected competing apps are as follows. The News contains , and ; the Video Player contains , and ; the Browser contains , and .First, we want to figure out the distribution of the user preferences against the app according to the device model. We employ the cumulative distribution function (CDF) to demonstrate such distributions, as shown in Figure <ref>. For each app, the X-Axis represents the price of device models that are sorted in ascending order, and the Y-Axis refers to the percentage of the app's users holding such a device model. An app tends to have be used by more higher-end users if the curve draws near the bottom. Obviously, we can observe that the choice of device models significantly impact the selection of competing apps. For the news readers, we can see that theandare more likely to be adopted by higher-end users. In contrast, thetend to be more preferred by the lower-end users. The difference among device models is even more significant for the Video players. Thetakes a very significant difference compared to other 4 apps, indicating most of its users are lower-end. One possible reason is theis a preloaded app that is used mainly on smartphones manufactured by , and most of these smartphones are categorized into middle-end and low-end groups. Finally, in the browser group, the similar findings can be observed. The most preferred browser of higher-end users is , followed by thebrowser,browser, and thebrowser. Thebrowser are the most likely to be adopted by the low-end users. We can confirm that the device models can affect the choice of competing apps. In addition, we are interested in the possible underlying reasons. Referencing the app similarity model proposed by Chen et al. <cit.>, we explore the profile of these competing apps, including the textual descriptions, vendors, and features. Some immediate findings are derived. First, we can infer that the lower-end users prefer the “local" apps more than the “international apps". For example, thebrowser and thebrowser are both developed by local app vendors in China. Second, some special features can be more attractive to different users. For example, from the profile theand UC Web claim that they are more “traffic-friendly" by compressing the Web page on a front-end proxy before the Web page is delivered to the users. Hence, it is not surprising that these two browsers are more appreciated by the lower-end users, given that these users have limited budgets. Third, the content providers can be an possible impact factor. For example,is famous for the fast channel of entertainment news in China, and thus the lower-end users, who are possibly in-school students and low-position employees, are more likely to take theas the favored app. In contrast, theis provided by the [http://ir.ifeng.com/], which is famous for the in-depth, objective reviews of economic, finance, and politics. As a result, the users holding higher-end device models theas a preference, since they may care more about the related topics. §.§ Summary of FindingsSo far we have answered the 5 research questions proposed previously. Although users holding different device models share similarities in some categories of apps such as Communication, significant diversities are observed. The correlation analysis and hypotheses testing confirm that the choice of device models exactly impacts the user behaviors in terms of adoption of app stores, app selection, app abandonment, online time, data plan, app revenue, and selection of competing apps. In summary, we can conclude that the choice of device models can significantly impact the app usage, which in turn reflects the classification of the user requirements, preferences, and possible background. Besides the correlation analysis, we also derive some possible reasons why such diversity exists. We will further explore how our derived knowledge can benefit the research community of mobile computing.§ IMPLICATIONS In this section, we present some implications and suggestions that might be helpful to relevant stakeholders in the app-centric ecosystem.§.§ App Store Operators App stores play the central role in the ecosystem. Intuitively, the recommender systems deployed on the app stores should accurately suggest the proper apps of users. From our findings in RQ1-RQ5, the choice of device models can significantly impact the user preferences of app selection, especially in some competing apps. It is reported that most app stores mainly rely on the similarity-based recommendations such as the apps frequently downloaded by users, the apps developed by the vendors, or apps with similar purposes <cit.>. Some advanced recommendation techniques can further improve the recommendation quality, such as those based on the similarity between apps from various aspects such as app profile, category, permission, images, and updates <cit.>. However, synthesizing the user requirements, preferences, and even economic background inferred from our study can be further helpful. For example, the app stores can recommend a lower-end user with the browsers such asandinstead of , if the user cares about the data plan. To the best of our knowledge, very few app stores take into account the impacts of device model as an influential factor, including Wandoujia. In practice, we plan to integrate the device model as a dimension to improve the recommendation quality.From RQ1, we can find that a large number of users (at least in China) do not frequently download and update their apps from app stores. Although the Android users can use other third-party app stores (e.g., those provided by the device manufacturers) other than Wandoujia, it is still worth reporting that the lower-end users are less likely to access the app stores. In this way, the app stores have to carefully consider how to expand the desires from these users. §.§ App DevelopersDevelopers can also learn some lessons from our study when designing and publishing their apps. From RQ1, we can find that some apps are more frequently uninstalled on some device models. It implies that there can be some problems such as compliance with hardware or the device-specific APIs. As reported on the StackOverFlow <cit.>, some camera related bugs have been found on . Our finding can validate such problematic issues. When having the distribution of uninstallations according device models, the developers, OS-vendors, and device manufacturers candraw attentions and prevent problematic issues.As we presented in RQ2, the choice of device models makes quite significant impacts on theonline time of apps. For example, the higher-end user can spend more time on NEWS_AND_READING and TRAVEL. Due to the fragmentation of Android devices <cit.>, currently more and more in-app ads networks allow the developers to customize the title, banner, and content of ads according to the device model <cit.>. Hence, the developers of these apps should consider customizing some device-specific ads networks to these “heavy" users and gain the potential revenue of ads clicking. From RQ4, we observe that lower-end users are likely to pay more for apps than higher-end users. Such a finding can assist developers to locate target audiences from whom the revenue can be gained. To attract the lower-end users who are less likely to buy their apps but probably with less budgets, the developers can consider the “try-out-and-buy" model to increase the user interests. In addition, they can further explore some new features or in-app ads to increase more revenue. From RQ5, we can find that the selection of competing apps can vary a lot among users holding different levels of device models. Developers can further explore why their apps are less adopted by users from some specific device models, and fix possible bugs, optimize the design, or provide advanced features. For example, it is said that thecan save traffics for users by compressing images and refactoring the Web page layout on the sever before the page is downloaded by the device. Such optimized features can be leveraged to improve the user experiences of apps. §.§ Network Service ProvidersThe network service providers or carriers such as T-Mobile, China Mobile, and China UniCom, play an important role in providing the communication infrastructure and service delivery. From RQ2 and RQ3, we can derive the network usage patterns of users choosing different device models. From RQ2, the lower-end are more likely to connect to cellular network than the higher-end users, especially in using some specific apps. However, the findings of RQ3 suggest that lower-end users typically spend less data plan in cellular network since they may have relatively limited budgets. Such an observation indicates that the lower-end users are less possible to use the “traffic-heavy" apps such as online music and video players under cellular network. To increase the data plan usage of these lower-end users, the network service providers do need to concern some special business models by case. For example, the network service providers can negotiate with the vendors and provide special packages of data plan, such as the “buy-out" ones that are customized to specific apps, e.g., music or video players. Such packages provide the users the unlimited cellular traffic that can be used but only to access the specific apps. As the customized packages are independent from the regular data plan, they are possibly appreciated by users for specific purposes. The network service providers can exploit such a business model to the device manufacturers and app store operators. For example, the device manufacturers commonly preload some apps in their devices. The network service providers can bind these apps with the customized data package, and share the commissions from the revenue of device manufacturers. In addition, from RQ2, we can also infer that the lower-end users are less likely to be covered by Wi-Fi, and tend to spend more network access time under cellular. By synthesizing another findings of RQ2, i.e., the lower-end users spend more time on EDUCATION apps under cellular, we can infer that these users are possibly in-school students. As the network service providers can easily obtain the device model information and location distribution of the connected devices, e.g., from the tier-1 cellular carriers, they can estimate and allocate the radio resources around the places where lower-end users are more likely to stay.§ THREATS TO VALIDITYConsiderable care and attention should be addressed to ensure the rigor of our study. As with any chosen research methodology, it is hardly without limitations. In this section, we will present the potential threats for validating our results.One potential limitation of our work is that the data set is collected from a single app marketplace in China. The users under study are mainly Chinese, and the region differences should be considered. In addition, the investigated apps are only Android apps. Hence, some results may not be fully generalized to other app stores, platforms, or countries. Such an limitation can hardly be addressed due to the difficulty of acquiring the similar behavioral profile from large-scale users. However, the measurement approach itself can be generalized to other similar data sets from other app stores. Additionally, China has become the biggest market of mobile devices and apps all over the world, so our findings derived from over 0.7 million Android users can provide some comprehensive and representative knowledge to the research communities of modern Internet. Choosing the price as indicator can introduce threat since the price of devices can change quite frequently. To address such a threat, we make a coarse-grained categorization of device models. We manually checked the first-to-market price and the latest price as of September 2014 (the end point of our data set) of each model. The price of most device models can still fall into our category. In addition, it would be interesting that if we categorize the device models based on other levels of price. Although there can be some bias caused by the varying price of device models, we believe that our measurement approach and findings can be generalized to any other data sets like ours. § CONCLUSION AND FUTURE WORK In this paper, we have presented the correlation analysis of choice of device models against the user behaviors of using Android apps. Our study was conducted over the largest to date set collected from over 0.7 million users of Wandoujia. We reported how the choice of device models can impact the adoption of app stores, app selection and abandonment, online time, data plan usage, revenues gained from apps, and comparisons of competing apps. The results revealed the significance of device models against app usage. We summarized our findings and presented implications for relevant stakeholders in the app-centric ecosystem.Currently, we plan to take into account the device models as an important impact factor in the recommendation systems of Wandoujia <cit.>. The analytical techniques shall be developed as an offline learning kernel and improve more personalized recommendation of apps.We are now developing features to collect much finer-grained information in the Wandoujia management apps, such as the traffic/access time per session in apps and the click-through logs. We believe that such detailed information can explore more diversity among users and thus improve our study.SIGCHI-Reference-Format2 | http://arxiv.org/abs/1707.09252v1 | {
"authors": [
"Huoran Li",
"Xuan Lu"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170727074755",
"title": "Mining Device-Specific Apps Usage Patterns from Large-Scale Android Users"
} |
Prior specification for binaryMarkov mesh modelsXin Luo and Håkon Tjelmeland Department of Mathematical Sciences, Norwegian University of Science and TechnologyWe propose prior distributions for all parts of the specification of a Markov mesh model. In the formulation we define priors for the sequential neighborhood, for the parametric form of the conditional distributions and forthe parameter values. By simulating from the resulting posterior distribution when conditioning on an observedscene, we thereby obtain an automatic model selection procedure for Markov mesh models. To sample from such a posterior distribution, we construct a reversible jump Markov chain Monte Carlo algorithm (RJMCMC). We demonstrate theusefulness of our prior formulation and the limitations of our RJMCMC algorithm in two examples.Key words: Markov mesh model, prior construction, pseudo-Boolean functions, reversible jump MCMC,sequential neighborhood.§ INTRODUCTIONDiscrete Markov random fields (MRFs) and Markov mesh models (MMMs) defined on rectangular lattices are popular model classes in spatial statistics, see for example <cit.> and <cit.> for MRFs, and <cit.> and <cit.> for MMMs.Discrete MRFs are frequently used tomodel available prior information about an unobserved scene x of a discrete variable. Thisprior is then combined with a likelihood function describing the relation between x and some observed data y into a posterior distribution, and this posterior is the basis for making inference about x. When specifying the MRF prior, the most frequent approach is tofix the neighborhood and parametric model structures and also to specify the values of the model parameters a priori. However, some authors have also explored a more fully Bayesian approach<cit.>. In particular, <cit.> formulate a prior for all parts of the MRF prior and demonstrate how MCMC sampling from the corresponding posterior distribution when conditioning on an observedscene produces MRFs that give realizations with a similar spatial structure as present in the scene used to define the posterior. The class of Markov mesh models, and the partially ordered Markov model (POMM) generalization<cit.> of this class, is much less used in the literature. We think the main reason for this is that it is much harder to manually choose an MMM than an MRF that reflects given prior information. It is neither an easy task to specify an MRF that isconsistent with given prior information, but except for boundary effects it is for an MRF easy to ensure that the field is stationary. This is an important practicalargument when completely specifying the prior a priori,but it is not so important when a fully Bayesian model is adopted as in <cit.>.It should also be noted that MRFs contain a computationally intractable normalizing constant which severely limits the practicability of MRFs in a fully Bayesian context, see for example the discussion in <cit.>. In contrast, the normalizing constant for an MMM is explicitly given in an easy to compute form. Also for this reason an MMM is much better suited as a prior than an MRF when adopting the fully Bayesian approach.Our goal in the present article is to formulate a fully Bayesian MMM. In particular, we would like the hyper-prior to include distributions for the neighborhood structure, for the interaction structure of the conditional distributions defining the MMM, and for the parameter values. We should thereby obtain a flexible prior that is able to adapt to a wide varietyof scenes. To specify the MMM hyper-prior, we adapt the general strategy used in<cit.> for the MRF to our MMM situation.Given such a Bayesian model, we also want to formulate an MCMC algorithmto simulate from the resulting posterior distribution conditioned on an observedscene. It should thereby be possible to learn both the form of the parametric model and the values of the model parameters from an observed scene. For simplicity we here limit our attentionto binary MMMs, but our approach can quite easily be generalized to asituation where each node has more than two possible values. The remainder of this article is organized as follows. In Section <ref> we introduce most of the notations we use for defining our Bayesian Markov mesh model, and in particular discuss pseudo-Boolean functions. In Section <ref> we use this to formulatethe Markov mesh model class. In Section <ref> we construct our prior distribution, and in Section <ref> we formulate proposaldistributions that we use in a reversible jump Markov chain Monte Carlo algorithm tosimulate from the corresponding posterior when conditioning on an observed scene. In Section <ref> we present two simulation examples and lastly we givesome closing remarks in Section <ref>. § PRELIMINARIESIn this section we first introduce the notation we use to represent a rectangular lattice, thevariables associated to this lattice and some quantities we use to formulate ourMarkov mesh model defined on this lattice. Thereafter, we define the class of pseudo-Booleanfunctions and explain how a pseudo-Boolean function can be used to represent a conditionaldistribution for binary variables.§.§ NotationConsider a rectangular m× n lattice. Let v=(i,j) denote a node in this lattice, wherei and j specify the vertical and horizontal positions of the node in the lattice, respectively. We let i=1 be at the top of the lattice and i=m at the bottom, and j=1and j=n are at the left and right ends of the lattice, respectively. We use lowercase Greekletters to denote sets of nodes, and in particular we let χ={(i,j):i=1,…,m,j=1,…,n} be the set of all nodes in the lattice. Occasionally we also consideran infinite lattice ℤ^2, where ℤ={ 0,± 1,± 2,…} is the set ofall integers, and we use v=(i,j)∈ℤ^2 also to denote a node in such an infinite lattice. We use λ,λ^⋆⊆ℤ^2 to denote arbitrary sets of nodes. To translate a node (i,j)∈ℤ^2 by an amount (k,l)∈χ, we adopt the notation(i,j) ⊕ (k,l) = (i+k,j+l).One should note that even if (i,j)∈χ, (i,j)⊕ (k,l) may fall outside the finitem× n lattice.To translate all nodes in a set λ⊆ℤ^2 by the same amount (k,l)∈χ, we writeλ⊕ (k,l) = { (i,j) ⊕ (k,l): (i,j) ∈λ}.To denote sets of subsets of nodes, we use uppercase Greek letters, and in particular, we letΩ(χ)={λ:λ⊆χ} denote the set of all subsets of χ, oftencalled the power set of χ. One should note that Ω(χ) in particular includes the empty set and χ itself. We use Λ,Λ^⋆⊆Ω(χ) to denotearbitrary sets of subsets of nodes.To define a Markov mesh model one must, for each node v=(i,j), define a so-called predecessor setand a sequential neighborhood. After numbering the nodes in the lattice from one to mn in the lexicographicalorder, we let the predecessor set of a node (i,j) consist of all nodes with a lower number thanthe number of (i,j). We let ρ_v = ρ_(i,j)⊂χ denote the predecessor set of a node v=(i,j)∈χ, i.e.ρ_(i,j) = {(k,l)∈χ: nk+l < ni+j},see the illustration in Figure <ref>(a).We let ν_v=ν_(i,j)⊆ρ_(i,j) denote the sequential neighborhood for node v=(i,j)∈χ as illustrated in Figure <ref>(b). In Section <ref> we consider a Markov mesh model where all the sequential neighborhoods are defined by a translation of a template sequential neighborhood τ. The τ can be thought of as thesequential neighborhood of node (0,0) in the infinite lattice. More precisely, τ is required to include a finite number of elements and τ⊂ψ = { (i,j): i∈ℤ,j∈ℤ^-}∪{ (0,j): j∈ℤ^-},where ℤ^- = { -1,-2,…} is the set of all negative integers.The sequential neighborhood of a node v∈χ is thendefined as ν_v = ( τ⊕ v) ∩χ.As illustrated in Figure <ref>, sequential neighborhoods for all nodes sufficientlyfar away from the lattice borders then have the same form, whereas nodes close to the borders have fewer sequential neighbors. One can note that with this construction one always has ν_(1,1)=∅.To each node v=(i,j)∈χ, we also associate a corresponding binary variable which we denote byx_v=x_(i,j)∈{0,1}. The collection of all these binary variables we denote by x=(x_v; x∈χ)and we let x_λ=(x_v;v∈λ) represent the collection of variables associated to the nodes in a set λ⊆χ. In particular, x_ν_v is the collection of variables associated tothe sequential neighborhood of node v. If x_v=1 we say node v is on, and if x_v=0 we say thenode is off. We let ξ(x)⊆χ denote the set of all nodes that are on, i.e. ξ(x) = { v∈χ: x_v=1}.In particular, the set of nodes in the sequential neighborhood of node v that is on is then, using (<ref>) and that ξ(x)⊆χ,ξ(x)∩ν_v = ξ(x)∩ (τ⊕ v).In the next section, we define the class of pseudo-Boolean functions which we in Section <ref> use to define the class of Markov mesh models.§.§ Pseudo-Boolean functionsWhen defining pseudo-Boolean functions, we reuse some of the symbols introduced when discussing concepts related to the rectangular m× n lattice above. In particular, we define a pseudo-Boolean functionwith respect to some finite set, denoted by τ. In the definition, this τ has no relation to thetemplate sequential neighborhood τ introduced above. However, when applying a pseudo-Boolean function to represent the conditional distribution of x_v for a node v∈χ given the values of the nodes in the sequential neighborhood ν_v, the set τ used to define a pseudo-Boolean function is equal to the template sequential neighborhood τ. In particular, the elements of τ is then the nodes in the lattice χ, and therefore we use λ and λ^⋆ to represent subsets of τ alsowhen discussing pseudo-Boolean functions in general. A pseudo-Boolean function θ(· ) defined on a finite set τ is a function that associates a real value to each subset of τ, i.e. θ: Ω(τ) →ℝ,where Ω(τ) is the power set of τ. Thereby, for any λ⊆τ the value ofthe pseudo-Boolean function is θ(λ). Equivalently, one may think of a pseudo-Booleanfunction as a function that associates a real value to each vector z∈{0,1}^|τ|, where |τ| is the number of elements in the set τ. To see the correspondence, one should set anelement in z equal to one if and only if the corresponding element in τ is in the set λ. This last formulation of pseudo-Boolean functions is the more popular one, see for example <cit.> and<cit.>, but in the present article we adopt the formulation in (<ref>) as this givessimpler expressions when formulating our Markov mesh model in Section <ref> and thecorresponding prior distribution in Section <ref>.<cit.> show that any pseudo-Boolean functioncan be uniquely represented by a collection ofinteraction parameters (β(λ),λ∈Ω(τ)) by the relationθ(λ) = β(λ) + ∑_λ^⋆⊂λβ(λ^⋆)The corresponding inverse relation is given byβ(λ) = θ(λ) + ∑_λ^⋆⊂λ (-1)^|λ∖λ^⋆|θ(λ^⋆)The one-to-one relation in (<ref>) and (<ref>) is known as Moebious inversion, see for example <cit.>.If one or more of the interaction parameters β(λ) are restricted to be zero, a reduced representationof the pseudo-Boolean function can be defined. For some Λ⊆Ω(τ) assume now that one restricts β(λ)=0 for all λ∉Λ. One can then represent the pseudo-Boolean function θ(· ) by the interaction parameters {β(λ),λ∈Λ}, and the relation in (<ref>) becomesθ(λ) = ∑_λ^⋆∈Λ∩Ω(λ)β(λ^⋆)where Ω(λ) is the power set of λ. We then say that θ(·) is represented onΛ. Moreover, we say that the set Λ is dense if for all λ∈Λ, all subsets ofλ is also included in Λ, and that the template sequential neighborhood τ is minimal for Λ if all nodes v∈τ are included in at least one of the elements of Λ. One should note that if Λ is dense and τ is minimal for Λ then there is a one-to-onerelation between the elements in τ and the sets λ∈Λ which contains only one node, {{v}: v∈τ} = {λ∈Λ: |λ| = 1}.Throughout this paper, we restrict attention to pseudo-Boolean functions that are represented on a Λ that is dense and the template sequential neighborhood τ that is minimal for this Λ. A λ∈Ω(τ) we term an interaction, we say the interaction is active if λ∈Λ and otherwise we say it is inactive. The Λ is thereby the set of active interactions. As also discussed in <cit.>, the set of active interactions Λ can be visualized by adirected acyclic graph (DAG), where we have one vertex for each active interaction λ∈Λ and avertex λ∈Λ is achild of another vertex λ^⋆∈Λ if and only if λ=λ^⋆∪{ v} for somev∈τ∖λ^⋆. Figure <ref>shows such a DAG for Λ={∅, {(0,-1)}, {(-1,0)}, {(-1,-1)}, {(-1,1)}, {(0,-1),(-1,0)}, {(0,-1),(-1,1)}}, which is based on τ={(0,-1),(-1,-1),(-1,0),(-1,1)}. This τ can be used to define the sequential neighborhoods for nodes in a rectangular lattice asdiscussed in Section <ref>. In the vertices of the DAG shown in the figure, node (0,0) is represented by the symbol⊠, whereas each of the nodes in λ∈Λ is represented by the symbol □. Thinking of τ as a finite set of nodes in a lattice, the position of the □ representing node (i,j)∈λ is placedat position (i,j) relative to ⊠.As also discussed in <cit.>, one should note that a pseudo-Boolean function θ(·) thatis represented on a dense set Λ⊆Ω(τ) can be uniquely specified bythe values of {θ(λ):λ∈Λ}. The remaining values of the pseudo-Boolean function, θ(λ),λ∈Ω(τ)∖Λ, are then given by (<ref>) and (<ref>) and the restriction β(λ)=0 for λ∉Λ. Moreover, as the relations in(<ref>) and (<ref>) are linear, each θ(λ), λ∈Ω(τ)∖Λ is a linear function of{θ(λ):λ∈Λ}.§ MARKOV MESH MODELIn this section we formulate a homogeneous binary Markov mesh model <cit.> for a rectangular m× n lattice. We adopt the notation introduced in Section <ref>,so in particular χ denotes the set ofall nodes in the m× n lattice and x = (x_v,v∈χ) is the collection ofthe binary variables associated to χ. In a Markov mesh model thedistribution of x is expressed asf(x) = ∏_v∈χ f(x_v|x_ρ_v),where f(x_v|x_ρ_v) is the conditional distribution for x_v given the values of thevariables in the predecessor nodes. Moreover, one assumes the Markov property f(x_v|x_ρ_v) = f(x_v|x_ν_v),i.e. the conditional distribution of x_v given the values in all predecessors of vonly depends on the values in the nodes in the sequential neighborhood of v. As discussed inSection <ref>, we assume the sequential neighborhoods ν_v,v∈χ to be definedas translations of a template sequential neighborhood τas specified in (<ref>) and (<ref>). Using the result in (<ref>), the conditional distributionf(x_v|x_ν_v) can then be uniquely represented by a pseudo-Boolean function θ_v(λ),λ⊆τ by the relation f(x_v|x_ρ_v)=exp{ x_v·θ_v( ξ(x) ∩ (τ⊕ v))}/1 + exp{θ_v( ξ(x) ∩ (τ⊕ v))}.In general, one may have one pseudo-Boolean function θ_v(λ) foreach v∈χ, but in the following we limit the attention to homogeneous models, sowe require all θ_v(·),v∈χ to be equal. We let θ(·) denote this common pseudo-Boolean function, i.e. θ_v(λ)=θ(λ) for allλ⊆τ and v∈χ and, without loss of generality, we assumeθ(· ) to have a dense representation on a set Λ⊆Ω(τ) and τ to be minimal for Λ. Thus, the distribution of our homogeneous binary Markov mesh model isf(x) = ∏_v∈χexp{ x_v·θ( ξ(x) ∩ (τ⊕ v))}/1 + exp{θ( ξ(x) ∩ (τ⊕ v))}.Assuming, as we do, the Markov mesh model to be homogeneous is convenient in that we do not need to specify a separate pseudo-Boolean function for each node v∈χ, and it is also statistically favorable as it limits the number of parameters in the model. However, one should note that this choice implies that for a node v∈χ close to theboundary of the lattice so that the set (τ⊕ v)∖χ is non-empty, theconditional distribution f(x_v|x_ν_v) is as if the nodes (for the infinite lattice) inthe translation of τ that fall outside the lattice χ are all zero. Thus,even if the model is homogeneous it is not stationary, and in particular one shouldexpect strong edge effects since we in some sense are conditioning on everything outsidethe lattice χ to be zero. When estimating or fitting the model to an observed scene, it is crucial to take this edge effect into account. Having defined our class of homogeneous Markov mesh models as above, a model is specified by the template sequential neighborhood τ, the set of active interactions Λ⊆Ω(τ) on whichthe pseudo-Boolean function θ(·) is represented, and the parameter values {θ(λ):λ∈Λ}. Thus, toadopt a fully Bayesian approach, we need to formulate prior distributions forτ, Λ and {θ(λ):λ∈Λ}, and this isthe focus of the next section.§ PRIOR DISTRIBUTIONWhen constructing our prior distribution for the template sequential neighborhood τ, the set of activeinteractions Λand the parameter values {θ(λ):λ∈Λ}, we have two properties in mind. Firstly, the prior should be vague so that the Markov mesh model manages to adapt to a large variety of scenes. To obtain this, the number of elements in τ should be allowed to be reasonably large and higher-order interactions should be allowed inthe model. Secondly, to avoid overfitting, the prior should favor parsimonious Markov mesh models, and in particular this implies that the highest prior probabilities should be assigned to models with just a few higher-order interactions. We define the prior as a product of three factorsf(τ,Λ,{θ(λ): λ∈Λ}) = f(τ) f(Λ|τ) f({θ(λ):λ∈Λ}|τ,Λ),where f(τ) is a prior for the template sequential neighborhood τ, f(Λ|τ) is a prior for the set of activeinteractionsΛ when τ is given, and f({θ(λ):λ∈Λ}|Λ) is a prior for the parameter values givenτ and Λ. In the following we discuss each of these factors in turn. §.§ Prior for the template sequential neighborhood τWe restrict the template sequential neighborhood to be a subset of a given finite set τ_0⊂ψ, where ψ is defined in (<ref>). The τ_0 can be thoughof as a set of possible sequential neighbors for node (0,0). To get a flexible prior it is important that the number ofelements in τ_0 is not too small, and it is natural to let τ_0 include nodes close to (0,0). For example, one may letψ include all nodes that are inside the circle centered at (0,0) with some specified radius r. In the examples discussed in Section <ref> we use this with r=5, see the illustration in Figure <ref>. Given the set τ_0 we specify the prior for τ⊆τ_0 by first choosing a prior distribution for the number of elements in τ, andthereafter a prior for τ given the number of elements in τ. Letting n_τ=|τ| denote the number of elements in τ we thereby havef(τ) = f(n_τ) f(τ|n_τ).For simplicity we choose both f(n_τ) and f(τ|n_τ) to be uniform distributions. The possible values for n_τ are all integers from0 to |τ_0|, so we getf(n_τ) = 1/n_τ + 1.Moreover, having chosen τ to be uniform given n_τ=|τ|, we getf(τ|n_τ) = 1|τ_0|n_τ,where the binomial coefficient in the numerator is the number of possible sets τ's with n_τ elements.One should note that our choice of the two uniforms above is very different from adopting a uniform prior for τ directly.A uniform prior on τ would have resulted in very high a priori probabilities for n_τ being close to |τ_0|/2 and verysmall a priori probabilities for values of n_τ close to zero, which is clearly not desirable.One can easily construct other reasonable priors for τ than the one defined above. For example, one could want to build into theprior f(τ|n_τ) that nodes close to (0,0) are more likely to be in τ than nodes further away. Recalling that we want to simulate from a corresponding posterior distribution by a reversible jump Markov chain Monte Carlo algorithm (RJMCMC) <cit.>, thechallenge is to formulate a prior with this property so that we are able to compute the (normalized) probability f(τ|n_τ), as this is needed to evaluate the Metropolis–Hastings acceptance probability. For thedata sets discussed in Section <ref>, we have also tried a prior f(τ|n_τ) in which we split the nodes inτ_0 into two or three zones dependent on their distances from (0,0) and have a different prior probability fora node to be in τ dependent on which zone it is in. As long as the number of zones is reasonably small, it is thenpossible to compute the normalizing constant of f(τ|n_τ) efficiently. However, in our simulation examples this gave essentially the same posterior results as the very simple double uniform prior specified above.§.§ Prior for the set of active interactions ΛTo specify a prior for the set of active interactions Λ, we first split Λ into several subsets dependent on how many nodes an element λ∈Λ contains. More precisely, for k=0,1,…,|τ| we define Ω_k(τ) = {λ∈Ω(τ): |λ|=k}Λ_k = {λ∈Λ: |λ|=k}.Thus, Ω_k(τ) contains all k'th order interactions, and Λ_k⊆Ω_k(τ) is the set of all k'th order active interactions.As we have assumed τ to be minimal for Λ, τ is uniquelyspecifying Λ_1={λ∈Λ: |λ|=1}, see the discussion inSection <ref> and in particular (<ref>).Moreover, we restrict∅ always to be active, i.e. ∅∈Λ with probability one, which implies that we force the pseudo-Booleanfunction θ(·) always to include a constant term. As we have already assumed Λ to bedense and τ to be minimal for Λ this is only an extra restriction when τ=∅. Thus, for given τ the sets Λ_0 and Λ_1 areknown, so to formulate a prior for Λ we only need to define a prior for Λ_k,k=2,…,|τ|. We assume a Markov property for these sets in that f(Λ|τ) = ∏_k=2^|τ| f(Λ_k |Λ_k-1).Thus, to choose a prior f(Λ|τ) we only need to formulate f(Λ_k|Λ_k-1), and to do so we adopt thesame strategy for all values of k. In the specification process of f(Λ_k|λ_k-1) we should remember that we have alreadyrestricted Λ to be dense, so the chosen prior needs to be consistent with this. For a given Λ_k-1, an interaction λ∈Ω_k(τ) can then be active only if allk-1'th order interactions λ^⋆∈Ω_k-1(λ) are active. We let Π_k denote this set ofpossible active k'th order interactions, i.e. we must haveΛ_k ⊆Π_k = {λ∈Ω_k(τ): λ^⋆∈Λ_k-1λ^⋆⊂λ}.We assume each interaction λ∈Π_k to be active with some probability p_k, independently of each other, and getf(Λ_k|Λ_k-1)=p_k^|Λ_k|(1-p_k)^|Π_k|-|Λ_k|Λ_k⊆Π_k.One should note that if Λ_k-1=∅ one gets Π_k=∅ and thereby also f(Λ_k=∅|Λ_k-1)=1.The probabilities p_k,k=2,…,|τ| should be chosen to get a reasonable number of higher-order active interactions. To obtain a parsimonious model, one need to adopt a small value for p_k if the number of elements in Π_k is large, but to favor a model to include some higher-order interactions, the value of p_k can be large when the number of elements in Π_k is small. We choosep_k =p^⋆|Π_k| ≤ |Λ_k-1|,p^⋆·|Λ_k-1||Π_k|where p^⋆∈ (0,1) is a hyper-parameter to be specified. One should note that this choice in particular ensures theexpected number of active k'th order interactions to be smaller than |Λ_k-1|.§.§ Prior for the parameter values {θ(λ):λ∈Λ}Given τ and the set of active interactions Λ, the set of model parameters for which we need to formulate a prior is {θ(λ):λ∈Λ}. From the model assumptions in (<ref>) and (<ref>), we have that eachθ(λ),λ∈Λ have a one-to-one correspondence with the conditional probabilityp(λ) = f(x_v=1|x_ρ_v) = exp{θ(λ)}/1+exp{θ(λ)}.Since the θ(λ)'s define probabilities conditioning on different values for x_ρ_v, we findit reasonable, unless particular prior information is available and suggests otherwise, to assume theθ(λ),λ∈Λ to be independent. In the following we adopt this independence assumption. Moreover, as we do not have a particular class of scenes in mind but want the prior to be reasonable for a widevariety of scenes, we adopt the same prior density for all parameters θ(λ),λ∈Λ.To formulate a reasonable and vague prior for θ(λ), we use the one-to-one correspondence between θ(λ) andthe probability p(λ). The interpretation for p(λ) is much simpler than that of θ(λ), so our strategy is first to choose a prior for p(λ) and from this derive the corresponding prior for θ(λ). As we do not have a particular class of scenes in mind but want our prior to be reasonable for a wide variety of scenes, we find it most natural to adopt a uniform prior on [0,1] for p(λ). However, as previously mentioned we want to explore acorresponding posterior distribution by running a reversible jump Metropolis–Hastings algorithm, and in particular wewant to use adaptive rejection sampling <cit.> to update θ(λ). For this to work, the full conditional for θ(λ) needs to be log-concave. Adopting the uniform on [0,1] prior for p(λ) the resulting posterior full conditional becomes log-concave, but the second derivative of the log full conditional converges to zero when θ(λ) goes toplus or minus infinity. As this may generate numerical problems when running the adaptive rejection sampling algorithm, we adopt a prior for p(λ) slightly modified relative to the uniform and obtaina posterior distribution where the second derivative of the log full conditional for θ(λ) converges toa value strictly less than zero. More precisely, we adopt the following prior for θ(λ),f(θ(λ)|τ,Λ) ∝e^θ(λ)(1+e^θ(λ))^2· e^-θ(λ)^2/2σ^2,where the first factor is the prior resulting from assuming p(λ) to be uniform, the second factor is the modification we adopt to avoid numerical problems when running the adaptive rejection sampling algorithm, andσ > 0 is a hyper-parameter to be specified. The resulting priors for p(λ) and θ(λ)when σ = 10 are shown in Figure <ref>.We see that the prior for p(λ) is close to the uniform. One can also note that f(θ(λ)|Λ) have heavier tailsthan a normal distribution with the same variance. One should note that the normalizing constant in (<ref>) isrequired when updating Λ in a reversible jump Metropolis–Hastings algorithm targeting a corresponding posterior distribution, but since(<ref>) is a univariate distribution this normalizing constant can easily be found by numerical integration. Letting c(σ) denote the normalizing constant of f(θ(λ)|τ,Λ) the complete expression for the prior for{θ(λ):λ∈Λ} isf({θ(λ):λ∈Λ}|τ,Λ)= ∏_λ∈Λ[ c(σ) ·e^θ(λ)(1+e^θ(λ))^2· e^-θ(λ)^2/2σ^2]. Having specified priors for τ, Λ and {θ(λ):λ∈Λ} we formulate in the next section a reversible jump Metropolis–Hastings algorithm for simulating from the corresponding posterior when a scene x is observed. § SIMULATION ALGORITHMIn the section we assume we have observed a complete scene x=(x_v;v∈χ) and assume this to be a realization fromthe Markov mesh model defined in Section <ref>. We adopt the prior defined in Section <ref> andwant to explore the resulting posterior distribution f(τ,Λ,{θ(λ):λ∈Λ}|x) ∝f(τ,Λ,{θ(λ):λ∈Λ}) f(x|τ,Λ,{θ(λ):λ∈Λ}),by a reversible jump Markov chain Monte Carlo algorithm (RJMCMC), see <cit.>. We combine two types of updates.In the first update class, we keep τ and Λ unchanged and update the parametervector {θ(λ):λ∈Λ} by a Gibbs step along a direction sampled uniformly at random.In the second update class,we propose a trans-dimensional move by adding an inactive interaction to Λ or removing an active interaction fromΛ, and proposing corresponding changes for the parameter vector {θ(λ):λ∈Λ}.It is clearly of interest to consider also the resulting posterior distribution when parts of the scene x isunobserved or when x is an unobserved latent field. The former is of interest if one wants to reduce theboundary effects of the Markov mesh model by letting x include an unobserved boundaryaround the observed area, and the latter is a common situation in image analysis applications.However, to simplify the discussion of the simulation algorithm in this section,we assume the complete scene x to be observed. In Section <ref>, where we present a number ofsimulation examples, we describe how to adapt the simulation algorithm to situation in which a part of x is unobserved. In the following we describe each of the two update types in turn, starting with the Gibbs update for the parameter values. We only discuss the proposal distribution, as the acceptance probabilities is then given by standard formulas.§.§ Gibbs update for the parameter values{θ(λ):λ∈Λ} Let τ, Λ and {θ(λ):λ∈Λ} be the current state. In this update, we keep τ andΛ unchanged and generate new parameter values {θ^⋆(λ):λ∈Λ}.To generate the new parameter values we first draw a random direction {Δ(λ):λ∈Λ} by sampling Δ(λ) from a standard normal distribution, independently for each λ∈Λ. We then setθ^∗(λ) = θ(λ) + αΔ(λ),where α∈ℝ is sampled from the full conditional f(α|τ,Λ,{θ(λ) + αΔ(λ):λ∈Λ},x)∝ f({θ(λ) + αΔ(λ):λ∈Λ}|τ,Λ) · f(x|τ,Λ,{θ(λ)+αΔ(λ):λ∈Λ}).As α is sampled from its full conditional, this is a Gibbs update and the Metropolis–Hastings acceptanceprobability is one. The full conditional (<ref>) for α is not of a standard form, butin Appendix <ref> we show that it is log-concave, so to generate samples from it we adopt the adaptive rejection sampling algorithm of <cit.>. §.§ Updating the set of active interactionsAgain let again τ, Λ and {θ(λ):λ∈Λ} be the current state. In this update we modify Λ, and possibly also τ, by adding an inactive interaction to Λ or by removingan active interaction from Λ. We let τ^⋆ and Λ^⋆ denote the potential new values forτ and Λ, respectively. With a change in Λ, the number of parameter values{θ(λ):λ∈Λ} is also changed, and to try to obtain a high acceptance rate, we in factpropose a change also in some of the parameter values that are in both the current and potential new states.We let {θ^⋆(λ):λ∈Λ^⋆} denote the set of potential parameter values.To generate τ^⋆, Λ^⋆ and {θ^⋆(λ):λ∈Λ^⋆}, we first draw at randomwhether to add an inactive interaction to Λ or to remove an active interaction from Λ.In the following we specify in turn our procedures for proposing to remove and add an interaction. §.§.§ Proposing to remove an active interaction from ΛHaving decided that an interaction should be removed, the next step is to decide whatinteraction λ^⋆∈Λ to remove. As the potential new Λ^⋆=Λ∖{λ^⋆} should be dense, we first find the set of active interactions λ^⋆ that fulfill this requirement, Λ_r={λ∈Λ∖{∅} : Λ∖{λ}}.Thereafter we draw what interaction λ^⋆∈Λ_r to be removed, with probabilitiesq(λ^⋆) = exp{-ν d(λ^⋆,τ,Λ,{θ(λ):λ∈Λ})}/∑_λ∈Λ_rexp{ -ν d(λ,τ,Λ,{θ(λ):λ∈Λ})}λ^⋆∈Λ_r,where ν≥ 0 is an algorithmic tuning parameter to be specified, andd(λ^⋆,τ,Λ,{θ(λ):λ∈Λ}) is a function that should measure the difference between the current pseudo-Boolean function defined byτ, Λ and {θ(λ):λ∈Λ} and the potential new pseudo-Boolean functiondefined by τ^⋆, Λ^⋆ and {θ^⋆(λ):λ∈Λ^⋆}. The precise formula we use for d(λ^⋆,τ,Λ,{θ(λ):λ∈Λ}) we specify below, after having specified how to set the potential new parameter values {θ^⋆(λ):λ∈Λ^⋆}. By setting the algorithmic tuning parameter ν=0, we draw Λ^⋆ uniformly at random from the elements inΛ_r. With a larger value for ν, we get higher probability for proposing to remove aninteraction λ^⋆ that gives a small change in the pseudo-Boolean function. If it should happen that Λ_r=∅,we simply propose an unchanged state. Assuming we have sampled a λ^⋆ to remove, we have two possibilities.If λ^⋆ is a higher-order interaction the sequential neighborhood is unchanged, i.e. τ^⋆=τ, whereas if λ^⋆ isa first-order interaction the sequential neighborhood is reduced to τ^⋆ = τ∖λ^⋆. Having decided τ^⋆ and Λ^⋆, the next step is to specify the potential new parameter values{θ^⋆(λ):λ∈Λ^⋆}. To understand our procedure for doing this, one should remember that there is a one-to-onerelation between the current parameter values {θ(λ):λ∈Λ} and a set of current interaction parameters {β(λ):λ∈Λ}, where the relation is given by (<ref>) and (<ref>). Moreover, together with the restriction β(λ)=0 for λ∉Λ, this defines a pseudo-Boolean function {θ(λ):λ∈Ω(τ_0)}. Correspondingly, there is a one-to-one relationbetween the potential new parameter values {θ^⋆(λ):λ∈Λ} and a set of potentialnew interaction parameters {β^⋆(λ):λ∈Λ^⋆}, and together with the restrictions β^⋆(λ)=0 for λ∉Λ^⋆ this defines a potential new pseudo-Boolean function{θ^⋆(λ):λ∈Ω(τ_0)}. To get a high acceptance probability for the proposed change, it is reasonable to choose the potential new parameter values {θ^⋆(λ):λ∈Λ^⋆} so thatthe difference between the two pseudo-Boolean functions {θ(λ):λ∈Ω(τ_0)} and{θ^⋆(λ):λ∈Ω(τ_0)} is small. One may consider the potential new pseudo-Boolean function{θ^⋆(λ):λ∈Ω(τ_0)} as an approximation to the current{θ(λ):λ∈Ω(τ_0)} and, adopting a minimum sum of squares criterion, minimize({θ^⋆(λ):λ∈Λ^⋆}) =∑_λ∈Ω(τ_0)( θ^⋆(λ) - θ(λ))^2with respect to {θ^⋆(λ):λ∈Ω(τ_0)}. <cit.> solved this minimization problem. Expressed in terms of the corresponding interaction parameters{β(λ): λ∈Λ}, the optimal potential new parameter values are β^⋆(λ) = {[ β(λ) -(-1/2)^|λ^⋆|-|λ|β(λ^⋆) ; β(λ)].and the obtained minimum sum of squares ismin{({θ^⋆(λ):λ∈Λ})} = β(λ^⋆)/2^|λ^⋆|.We use the latter to define the function d(λ^⋆,τ,Λ,{θ(λ):λ∈Λ}), used in (<ref>) to define the distribution for what interaction λ^⋆ to remove. We simply setd(λ^⋆,τ,Λ,{θ(λ):λ∈Λ}) = β(λ^⋆)/2^|λ^⋆|.Combining theexpression in (<ref>) with the one-to-one relations in (<ref>) and (<ref>),one can find the potential new parameters {θ^⋆(λ):λ∈Λ^⋆} in terms of the current parameters {θ(λ):λ∈Λ}. In particular, we see that this relation islinear and we have a |Λ|× |Λ| matrix A so that [[θ^⋆; β(λ^⋆) ]] = A θ⇔θ = A^-1[ [θ^⋆; β(λ^⋆) ]],where θ^⋆=(θ^⋆(λ):λ∈Λ)^T andθ = (θ(λ):λ∈Λ)^T are column vectors of the potential new and current parameter values, respectively. As the number of elements in θ^⋆ is one less than the number of elements in θ, weuse β(λ^⋆) to obtain the one-to-one relation we need for a reversiblejump proposal. The Jacobian determinant in the expression forthe corresponding acceptance probability is clearly (A), and in Appendix <ref> we show that the absolutevalue of this determinant is always equal to one, i.e. |(A)| = 1.§.§.§ Proposing to add an inactive interaction to Λ If it is decided that an inactive interaction should be added to Λ, the next step is to decide what interactionλ^⋆∈Ω(τ_0)∖Λ to add. We do this in two steps, first we draw at random whether afirst-order or a higher-order interaction should be added to Λ. If a first-order interaction should be added, we draw uniformly at random a node v^⋆ from τ_0∖τ and set λ^⋆={v^⋆}. Thenτ^⋆ = τ∪λ^⋆ and Λ^⋆ = Λ∪{λ^⋆}. If τ=τ_0, so that no suchv^⋆ exist, we simply propose an unchanged state. If a higher-order interaction should be added we need to ensure thatΛ∪{λ^⋆} is dense. We therefore first findΛ_a = {λ∈Ω(τ_0)∖Λ: |λ| > 1 Λ∪{λ}}and thereafter draw λ^⋆ uniformly at random from Λ_a. Then τ^⋆=τ andΛ^⋆=Λ∪{λ^⋆}. If it should happen that Λ_a=∅, we again simply propose an unchanged state. Having decided τ^⋆ and Λ^⋆, the next step is to generate the potential new parameter values {θ^⋆(λ): λ∈Λ^⋆}. When doing this, one should remember that this addinga potential new interaction proposal must be one-to-one with the reverse removing an interaction proposaldiscussed in Section <ref>. Therefore, the proposal distribution for the potential new parameter values {θ^⋆(λ): λ∈Λ^⋆} must conform with (<ref>), and thereby also with (<ref>). A natural way to achieve this is to draw a value β^⋆(λ^⋆) from some distribution and define the potential new interaction parameters by the inverse transformation of(<ref>), i.e. β^⋆(λ) = {[ β(λ) + (-1/2)^|λ^⋆|-|λ|β^⋆(λ^⋆) ,;β(λ) ].It now just remains to specify from what distribution to sample β^⋆(λ^⋆). The potential new parameter values {θ^⋆(λ):λ∈Λ^⋆} are linear functions of β^⋆(λ^⋆), and by setting β^⋆(λ^⋆)=α it can be expressed as in(<ref>) for the Gibbs update. The difference between what we now have to do and what is done in the Gibbs update is that in the Gibbs update the values Δ(λ) are sampled independentlyfrom a Gaussian distribution, whereas here these are implicitly defined by (<ref>) together with the one-to-one relations (<ref>) and (<ref>). It is tempting to sample α=β^⋆(λ^⋆) from the resulting full conditional, as this would give a high density for values of β^⋆(λ^⋆) that corresponds to models witha high posterior probability. As discussed in Section <ref> for the Gibbs update, it iscomputationally feasible to sample from this full conditional by adaptive rejection sampling. However, thenormalizing constant of this full conditional is not computationally available, and for computing the associatedacceptance probability the normalizing constant of the distribution of β^⋆(λ^⋆) must be available.To construct a proposal distribution for β^⋆(λ^⋆)=α, we thereforeinstead first generate r (say) independent samples α_1,…,α_r from the full conditional for α, by adaptive rejection sampling, and thereafter drawα=β^⋆(λ^⋆) from a Gaussian distribution with mean value α̅=1/n∑_i=1^rα_i andvariance s_α^2 = 1/r-1∑_i=1^r (α_i-α̅)^2. Our proposal distributionfor β^⋆(λ^⋆) is thereby an approximation to its full conditional.As this is a reversible jump proposal, the associated acceptance probability includes a Jacobian determinant.By construction the Jacobian determinant for this proposal is the inverse of the Jacobian determinant for the removing an interaction proposal discussed in Section <ref>. As we have|(A)|=1, we also get |(A^-1)|=1. § SIMULATION EXAMPLESIn this section we investigate our prior and proposal distributions on two binary example scenes.Firstly, we consider a mortality map forliver and gallbladder cancers for white males from 1950 to 1959 in the eastern United States, compiled by <cit.>.Using Markov random field models, this data set has previously been analyzed by <cit.>, <cit.> and <cit.>, seealso <cit.>. Secondly, we consider a data setpreviously considered by <cit.>. They also fitted a Markov mesh model to this data set, but with manually chosen neighborhood and interaction structures. In the following we first discuss some general aspects relevant for both the two examples and thereafter present details of each of the two examples in turn.As also briefly discussed in Section <ref>, we reduce the boundary effects of the Markov mesh model by letting x include an unobserved boundary around the observed area. We choose the unobserved boundary large enough sothat each of the observed nodes are at least 20 nodes away from the extended lattice boundary. We let χ denote the set of nodes in the extended lattice and let x = (x_v,v∈χ) be the corresponding collection of binary variables. We assume x to be distributedaccording to the Markov mesh model defined in Section <ref>, and for τ, Λ and{θ(λ) : λ∈Λ} we adopt the prior specified in Section <ref>. We letχ_o⊂χ denote the set of nodes for which we have observed values. Thereby χ_u=χ∖χ_o is the set of unobserved nodes. Correspondingly, we let x_o=(x_v, v∈χ_o) be the observed values andx_u=(x_v,v∈χ_u) the unobserved values. The posterior distribution of interest is therebyf(τ,Λ,{θ(λ),λ∈Λ}|x_o). To simplify the posterior simulation, we include x_u asauxiliary variables and adopt the reversible jump Metropolis–Hastings algorithm to simulate from f(τ,Λ,{θ(λ),λ∈Λ},x_u|x_o) ∝ f(τ,Λ,{θ(λ),λ∈Λ}) f(x_o,x_u|τ,Λ,{θ(λ),λ∈Λ}).To simulate from this distribution, we adopt the updates discussed in Section <ref> to update τ, Λ and {θ(λ),λ∈Λ} conditioned on x=(x_o,x_u), and we use single-siteGibbs updates for each unobserved node v∈χ_u given τ, Λ, {θ(λ),λ∈Λ}and x_χ∖{ v}. We define one iteration of the algorithm to include |χ_u| single-site Gibbs updates for randomly chosen nodes in χ_u followed by either one Gibbs update of the parameter values{θ(λ),λ∈Λ} as discussed in Section <ref> or one update of the activeinteractions as discussed in Section <ref>. In each iteration we independently updatethe parameter values or the active interactions with probabilities 0.55 and 0.45 respectively.The prior defined in Section <ref> contains three hyper-parameters, the radius r which defines the set ofpossible neighbors, the probability p^⋆ in (<ref>), and the parameter σ in (<ref>). In both examples, we use r=5 which gives the 34 possible neighbors shown in Figure <ref>. To get a prior where the probability for a Markov mesh model with higher-order interactions is reasonably high, we set the value of p^⋆ as high as 0.9, and to get an essentially uniform prior distribution for p(λ), we set σ=100. The proposal distribution discussed in Section <ref> has onealgorithmic tuning parameter, ν, and based on simulation results in preliminary runs we set ν=0.5.In the following we present the example scene and discuss corresponding simulation results for each of our twoexamples. We start with the cancer mortality map compiled by <cit.>.§.§ Cancer mortality mapThe cancer mortality map data are shown in Figure <ref>, where black (x_v=1) and white (x_v=0) pixels representcounties with high and low cancer mortality rates, respectively. The gray area aroundthe observed map represents unobserved nodes which we included in the model to reduce the boundary effects of the Markov mesh model.Adopting the Markov mesh and prior models discussed in Sections <ref> and <ref>, respectively, with the hyper-parameters defined above, we use the RJMCMC setup discussed above to explore the resultingposterior distribution. We run the Markov chain for 2 500 000 iterations, and study trace plots of differentscalar quantities to evaluate the convergence and mixing properties of the simulated Markov chain. Figure <ref> shows trace plots of the first 25 000 iterations for the number of interactions and for the logarithm of the posterior density. From these two and the other trace plots we have studied, we conclude that the simulated chain has converged at least within the first 10 000-15 000 iterations. As an extra precaution we discard the first25 000 iterations when estimating posterior properties.To study the posterior distribution we first estimate, for each of the 34 apriori potential neighbors in τ_0, the posterior probability for v∈τ_0 to be a neighbor. To estimate this we simply use the fraction of simulated models where v is in the template sequential neighborhood τ. The result is shown in Figure <ref>, where we use a gray scale to visualize the probabilities. Nodes (0,-1) and (-1,0) have high estimated posterior probabilities, equal to 0.999819 and 0.990577, respectively.The third and fourth most probable neighbor nodes are (-1,-1) and (-1,2), where the estimatedprobabilities are 0.049388 and 0.030353, respectively. From the data set shown in Figure <ref>, we see that the dependencebetween neighbor nodes seems to be quite weak, so the low number of simulated neighbors should come as nosurprise.Next we correspondingly estimate the posterior probabilities for each possible interaction to be included in the model.Table <ref> shows the top 10 a posteriori most likely interactions and the corresponding estimated probabilities. We see that the first four interactions have high posterior probabilities while the others have low probabilities.In addition, the four most likely interactions only include the high probability neighbor nodes (0,-1) and (-1,0).We also estimate the a posteriori marginal distributions for the parameter values θ(·) corresponding tothe four high probable interactions. Note that some of the interactions do not exist in some of the simulated models,but the θ(·) value is still well defined and can be computed as discussed in Section <ref>.Figure <ref> depicts the histograms of the simulated parameter values θ(·).From the simulation we also estimate the posterior probability for each of the possible models. The two most probable models are shown in Figure <ref>. These two models have posterior probabilities as high as 0.475 and0.381 while the remaining probability mass is spread out on a very large number of models.Finally, we generate realizations from simulated Markov mesh models. Figure <ref> contains realizationssimulated from four randomly chosen models simulated in the Markov chain (after the specified burn-in).As in Figure <ref>, showing the observed data set, black and white nodes v represent x_v=1 and 0, respectively. Comparing the realizations with the data set in Figure <ref>, we can get a visual impression of to what degree the simulated models have captured the dependence structure in the data set. To study this also more quantitatively, we consider the 16 possible configurations in a 2×2 block of nodes. For each of theseconfigurations, we find in a realization the fraction of such blocks that has the specified configuration. By repeating this for a large number of realizations we estimate the posterior distribution for thefraction of 2× 2 blocks with a specified configuration in a realization. This distribution should be compared with the corresponding fraction in the observed data set.Figure <ref> shows the estimated density for each of the 16 configurations. The corresponding fractionsfor the observed data set are marked by vertical dotted lines. Note that for most of these distributions the correspondingfractions for the observed data set are centrally located in the distribution. The exceptions are (g) and partly (i) and (j),where the observed quantity is more in the tail of the distribution.§.§ Sisim data setIn this example we reconsider a data set previously studied in<cit.>. The scene, shown in Figure <ref>, is simulated by the sequential indicator simulation procedure <cit.> and it is a much used example scene in the geostatistical community. We name the data set "sisim". The sisim scene is represented on a 121× 121 lattice. To reduce the boundary effects of the Markov mesh model we again include unobserved nodes around the observed area, shown as gray in Figure<ref>.Again adopting the Markov mesh and prior models defined in Sections <ref> and <ref> andthe hyper-parameters defined above, we use the RJMCMC setup discussed above to explore the resultingposterior distribution. For this data set each iteration of the algorithm requires more computation time thanin the cancer mortality map data, so we run the Markov chain for only 1 250 000 iterations.To evaluate the convergence properties of the simulated Markov chain, we study trace plots of different scalar quantitiesin the same way as in Section <ref>. Figure <ref> shows trace plots of the first 50 000 iterations for the number of interactions and for the logarithm of the posterior density. At first glance at these two and the other trace plots we have studied, we assertedthat the simulated chain had converged at least within the first 30 000-40 000 iterations. As an extra precaution we discardedthe first 250 000 iterations when estimating posterior properties.As in Section <ref>, we estimate the posterior probability for v∈τ_0 to be in the template sequential neighborhood τ. The results are shown in Figure <ref>,where we use a gray scale to visualize the probabilities. There are five nodes whose estimated posteriorprobabilities are essentially equal to 1, and these are (0,-1), (-1,0), (-1,2), (0,-3) and (-1,4). Four more nodes haveestimated posterior probabilities higher than 0.1. These are (-2,3), (-3,-1), (-2,-3) and (-1,3) with estimated probabilities 0.444608, 0.425779, 0.323181 and 0.182879, respectively. It is interesting to note the spatial locations of the high probability nodes. At least for a part of the area every second node is chosen as a neighbor with high probability. To understand this effect, we must remember that the values of two nodes that are lying next to each other are highly correlated, so one would not gain much extra information by includingboth of them in the template sequential neighborhood. Moreover, the prior prefers parsimonious models, which we obtain by not including too many nodes in the template sequential neighborhood. Next, as for the cancer mortality map data set, we correspondingly estimate the posterior probabilities for each possibleinteraction to be included in the model. Table <ref> shows the top 20 a posteriori most likely interactions and corresponding estimated probabilities. We see that many interactions have high posterior probabilities. We also estimate the a posteriori marginal distributions for the parameter values θ(·) corresponding to the topeight most likely interactions. Figure <ref> depicts the histograms of thesimulated parameter values θ(·).From the simulation we also estimate the posterior probability for each of the possible models. The most probable model isshown in Figure <ref>. This model has posterior probability equal to 0.13802.The remaining probability mass is spread out on a very large number of models.As in the cancer mortality data set example, we also now generate realizations fromthe simulated Markov mesh models. Figure <ref> containsrealizations simulated from four randomly chosen models simulated in the Markov chain (after the specified burn-in).As in Figure <ref>, showing the observed data set, black and white nodes v represent x_v=1 and 0, respectively.Also now we estimate the distribution of values in a 2×2 block of nodes. Figure <ref> shows the estimated density for each of the 16 configurations. The corresponding fractionsfor the observed data set are marked by vertical dotted lines. Note that for most of these distributions, thecorresponding fractions for the observed data set are centrally located in the distribution.The exceptions are (c), and partly (e), (f), (i) and (j), where the observed quantities are more in the tail of the distribution. In the cancer mortality data set example, essentially all of the posterior probability mass was concentrated in a few models. In the sisim data set example, the probability mass is spread out on a very large number of models. Inparticular, as also discussed above, the most probable model has a posterior probability estimated to be aslow as 0.13802. Using thesimulated models to understand the posterior model distribution is then more difficult. As a first step in describing the posterior model distribution, our focus here is on whether it has one or several modes. To do this wefirst need to define what we should mean by a mode in this complicated model space. We start by defining two models to beneighbors if one of them can be obtained from the other by including one extra interaction. Thus, our proposal distribution in Section <ref>, proposing to change the set of active interactions, isalways generating a potential new model that is a neighbor of the current model. To explore whether we have several modes in our posterior distribution, we first subsample the simulated Markov chain, keeping a realization every 50 iterations after the burn-in period. This leave us with20 000 realizations.From these we first find the most frequent model,visualized in Figure <ref>, and then all neighbor models to this most probable model, allneighbor models to the neighbors, and so on until the process stops. The sum of the estimatedposterior probabilities of the models in the resulting cluster of models is 0.80755, giving a clear indication that the posterior model distribution have more than one mode. To find a secondmode we limit the attention to the simulated models that was not included in the first cluster of models and repeat the process. Thus, we first find the a posteriori most probable model not included in the first model cluster. This model is shown in Figure <ref>. Then we find all neighbors of this model, all neighbors of the neighbors and so on. The estimated posterior probability in this second cluster of models is 0.146563. Thus, these two first clusters contain more than 95% of the simulated models, and we therefore choose not to search for a third cluster. Knowing that we have two important clusters ormodes it is natural to reconsiderthe convergence and mixing properties of our Markov chain. Figure <ref>shows a trace plot of the visited clusters for the subsampled models, where 1and 2 on the y-axis represent thefirst and second clusters found, respectively, and 3 represent all remaining models. We then see that the second cluster is in factvisited only once, giving a clear indication of poor mixing. We should thereby not trust the estimated probabilities for the two clusters given above, but that the chain is first moving from the first cluster to the second and thereafter back again clearly shows that both of them have a significant posterior probability mass. § CLOSING REMARKSIn this article we propose a prior distribution for a binary Markov mesh model. The specification of a Markov mesh model has three parts. First a sequential neighborhood is specified, next the parametric form of the conditional distributions is defined, and finally we assign values to the parameters. We formulate prior distributions for all these threeparts. To favor parsimonious models, our prior in particular assigns positive prior probabilities for some interaction parameters to be exactly zero. A corresponding prior formulation has previously been proposed for Markov random fields <cit.>. The advantage of using it for a Markov mesh model is that an explicit and easy to compute expression is available for the resulting posterior distribution, whereas the posterior based on a Markov random field will include the computationally intractable normalizing constant of the Markov random field.To sample from the resulting posterior distribution when conditioning on an observed scene,we adopt the RJMCMC setup. We propose an algorithm based on thecombination of two proposal distributions, a Gibbs proposal for the parameter values and a reversible jump proposalchanging the sequential neighborhood and parametric form of the conditional distributions.To explore the performance of the specified prior distribution and the corresponding RJMCMC posterior simulationalgorithm, we consider two scenes. The first is an observed cancer mortality map data with small spatial coupling between neighboring nodes. For this scene the RJMCMC algorithm converges quickly and has good mixing properties.Most of the posterior mass ends up in models with only two nodes in the sequential neighborhood.The second scene we tried is a frequently used scene in the geostatistical community. It has more spatial continuity than the first scene. Theconvergence of the RJMCMC algorithm becomes much slower when conditioning on this scene. In particular the posterior seems to have at least two modes and the mixing between the modes is slow. Our simulation results indicate that the a posteriori most likely model has six nodes in the sequential neighborhood and the conditional distributions has a parametric form with as much as twelve parameters. This shows that the specified prior isflexible in that the model complexity favored by the corresponding posterior adapts to the the complexity ofthe scene conditioned on.In this article we have focused on binary Markov mesh models and thereby binary scenes. Our strategy for prior specification and posterior simulation, however, can easily be extended to a situation with more than two colors.The main challenge in this generalization does not lie in the specification of the prior, but is computational in that one should expect the convergence and mixing of a corresponding RJMCMC algorithm to be slower for amulti-color model. A direction for future research is therefore to improve the proposaldistributions to obtain better convergence and mixing for the RJMCMC algorithm, both in the binary andmulti-color cases. In particular we think a promising direction here is to define an MCMC algorithm whereseveral Metropolis–Hastings proposals can be generated in parallel and where the proposals may haveadded and removed more than just one interaction relative to the current model.jasa§ LOG-CONCAVITY OF THE FULL CONDITIONAL FOR ΑIn this appendix we prove that the full conditional f(α|τ,Λ,{θ(λ)+αΔ(λ):λ∈Λ},x) defined in (<ref>) is log-concave, so that we can use adaptive rejection sampling to generate samples from it. Defining g(α) = ln[ f(α|τ,Λ,{θ(λ)+αΔ(λ):λ∈Λ},x)] and using (<ref>) we haveg(α)= ln[f({θ(λ) + αΔ(λ):λ∈Λ}|τ,Λ)]+ ln[ f(x|τ,Λ,{θ(λ)+αΔ(λ):λ∈Λ})]+ C,where C is the logarithm of the normalizing constant in (<ref>). Inserting expressions forthe prior and likelihood in (<ref>) and (<ref>), respectively, we getg(α)=∑_λ∈Λ[ c(λ) +θ(λ)+αΔ(λ)-2ln(1+e^θ(λ)+αΔ(λ)) -(θ(λ)+αΔ(λ))^2/2σ^2] + ∑_v∈χ[ x_v (θ(ξ(x)∩(τ⊕ v)) + αΔ(ξ(x)∩(τ⊕ v))) . - .ln(1+e^θ(ξ(x)∩(τ⊕ v)) + αΔ(ξ(x)∩(τ⊕ v)))] + C.Grouping terms of the same functional form, we getg(α)= C_0 + C_1 α - 1/2σ^2∑_λ∈Λ (θ(λ)+αΔ(λ))^2- 2∑_λ∈Λln(1+e^θ(λ)+αΔ(λ))- ∑_v∈χln(1+e^θ(ξ(x)∩(τ⊕ v)) + αΔ(ξ(x)∩(τ⊕ v))),where C_0 = C + ∑_λ∈Λθ(λ) +∑_v∈χx_vθ(ξ(x)∩ (τ⊕ v))C_1 = ∑_λ∈ΛΔ(λ) + ∑_v∈χΔ(ξ(x)∩ (τ⊕ v))are constants as a function of α. The second derivative of the constant and linear terms in(<ref>) are of course zero. Since the coefficients of the quadratic terms are all negative, the secondderivative of all of these are less or equal to zero, and unless Δ(λ) equals zero forall λ∈Λ the second derivative of the sum of these terms is even strictly less than zero.The remaining terms in (<ref>) all have the same functional formas a function of α, namelyh(α) = - a ln(1+e^b+cα),a term in the sum over λ∈Λ has a=2, b=θ(λ) and c=Δ(λ), whereas a term in thesum over v∈χ has a=1, b=θ(ξ(x)∩ (τ⊕ v)) and c=Δ(ξ(x)∩ (τ⊕ v)). To provethat the second derivative of all of these terms are negative, it is thereby sufficient to show that h^''(α) < 0 for all a>0 and α,b,c∈ℝ. Simple differentiation givesh^''(α) = - ac^2 e^b+cα/(1+e^b+cα)^2.Thus, h^''(α)<0 for all a>0 and α,b,c∈ℝ, and thereby g(α) is concave and the full conditionalf(α|τ,Λ,{θ(λ)+αΔ(λ):λ∈Λ},x) is log-concave. § JACOBIAN DETERMINANT FOR THE PROPOSAL INSECTION <REF> The Jacobi determinant for our removing an active interaction from Λ proposal is (A), where A is defined by (<ref>). The exact form of the matrix A depends on how we define the vectors θ and θ^⋆ used in (<ref>). The vector θ should contain the set of current parameters {θ(λ):λ∈Λ}, but so far wehave not specified what order to use when arranging this set of parameters into the vector θ. Correspondingly, we have not specified what order to use when arranging the set of potential new parameters{θ^⋆(λ):λ∈Λ^⋆} into the vector θ^⋆. However, even if the elements ofA depends on how we construct θ and θ^⋆, the absolute value of the determinant of A is the samefor all arrangements of θ and θ^⋆. To find (A) we arrange the vector θ so that parameters corresponding to lower order interactions comes first.The first element of the vector θ is thereby θ(∅), thereafter follows parameters corresponding tothe first order interactions {θ(λ):λ∈Λ, |λ|=1} (in an arbitrary order), then all parameters corresponding to second order interactions {θ(λ):λ∈Λ, |λ|=2} (again in an arbitrary order),and so on. We arrange θ^⋆ correspondingly, parameters corresponding to lower order interactions comes first.As also touched on in Section <ref>, the transformation in (<ref>) can be done in three steps. First θ is transformed into a vector β of the corresponding current interaction parameters{β(λ):λ∈Λ}. This relation is given in (<ref>) and is in particular linear so we canwriteβ = A_1θ.Arranging also the vector β so that lower order interactions comes first, it is easy to see from(<ref>) that A_1 is a lower triangular matrix with all diagonal elements equal to one. Thus (A_1)=1. The second step in the transformation is to use (<ref>) to define a vector β^⋆ containing the set of potential new interaction parameters {β^⋆(λ):λ∈Λ^⋆}. As the proposalis to remove an interaction, the number of elements in β^⋆ is one less than the number of elements inβ. To obtain a one-to-one relation as required in the reversible jump setup, we include the current value β(λ^⋆) in a vector together with β^⋆. We let β(λ^⋆) be the last element in the vector and we arrange also the vector β^⋆ so that lower order interactionparameters come first. As the relation in (<ref>) is linear we can then write [[β^⋆; β(λ^⋆) ]] = A_2β,where the elements of the square matrix A_2 is defined by (<ref>). To find the determinant of A_2, letr denote the number of elements in β before β(λ^⋆), so that element number r+1 in βis β(λ^⋆). From (<ref>) it then follows that A_2 has the block structure,A_2 = [ [ I_r× r A_2^12; 0_(|Λ|-r)× r A_2^22 ]],where I_r× r is the r× r identity matrix, A_2^12 is an r× (|Λ|-r) matrix, 0_(|Λ|-r)× r is a (|Λ|-r)× r matrix of only zeros, and A_2^22 is the (|Λ|-r)× (|Λ|-r) permutation matrix where the elements (i,i+1) for i=1,…,|Λ|-r-1 and (|Λ|-r,1) equals one and all other elements arezero. Thereby we have (A_2) = (I_r× r) ·(A_2^22) = (A_2^22), and as A_2^22 is a permutation matrix its determinant is plus or minus one. Thus, |(A_2)|=1. The third step inthe transformation from θ to θ^⋆ is to use (<ref>) to transform the vector of potential newinteraction parameters, β^⋆, to a corresponding vector θ^⋆ of potential new parameter values. As the relation in(<ref>) is also linear, we can write[[θ^⋆; β(λ^⋆) ]] = A_3 [[β^⋆; β(λ^⋆) ]],where the elements of the matrix A_3 is defined by (<ref>). Recalling that we have arranged the elements in both θ^⋆ and β^⋆ sothat parameters corresponding to lower order interactions come first, itis easy to see from (<ref>) that A_3 is an upper triangular matrix with all diagonal elements equal to one. Thus (A_3)=1. Setting thethree steps in the transformation together we have A=A_1 A_2 A_3 and thereby |(A)| = |(A_1)|· |(A_2)|· |(A_3)| = 1. | http://arxiv.org/abs/1707.08339v1 | {
"authors": [
"Håkon Tjelmeland",
"Xin Luo"
],
"categories": [
"stat.ME",
"stat.CO"
],
"primary_category": "stat.ME",
"published": "20170726094518",
"title": "Prior specification for binary Markov mesh models"
} |
empty FERMILAB-PUB-17-285-T FTUAM-17-14IFT-UAM/CSIC-17-070 a]Valentina De Romeri, b ,c]Enrique Fernandez-Martinez, b, c]Julia Gehrlein, d]Pedro A. N. Machado e]and Viviana Niro[a] AHEP Group, Instituto de Física Corpuscular, C.S.I.C./Universitat de València,Calle Catedrático José Beltrán, 2 E-46980 Paterna, Spain[b] DepartamentodeFísica Teórica,UniversidadAutónomadeMadrid,CantoblancoE-28049Madrid,Spain [c] InstitutodeFísicaTeóricaUAM/CSIC, Calle Nicolás Cabrera13-15,CantoblancoE-28049Madrid,Spain [d] Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL, 60510, USA[e]Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, [email protected], [email protected], [email protected], [email protected],[email protected] Inverse Seesaw naturally explains the smallness of neutrino masses via an approximate B-L symmetry broken only by a correspondingly small parameter. In this work the possible dynamical generation of the Inverse Seesaw neutrino mass mechanism from the spontaneous breaking of a gauged U(1) B-L symmetry is investigated. Interestingly, the Inverse Seesaw pattern requires a chiral content such that anomaly cancellation predicts the existence of extra fermions belonging to a dark sector with large, non-trivial, charges under the U(1) B-L. We investigate the phenomenology associated to these new states and find that one of them is aviable dark matter candidate with mass around the TeV scale, whose interaction with the Standard Model is mediated by the Z' boson associated to the gauged U(1) B-L symmetry.Given the large charges required for anomaly cancellation in the dark sector, the B-L Z' interacts preferentially with this dark sector rather than with the Standard Model.This suppresses the rate at direct detection searches and thus alleviates the constraints on Z'-mediated dark matter relic abundance. The collider phenomenology of this elusive Z' is also discussed.Dark Matter and the elusive 𝐙'in a dynamical Inverse Seesaw scenario [=======================================================================§ INTRODUCTION The simplest and most popular mechanism to accommodate the evidence for neutrino masses andmixings <cit.>and to naturally explain their extreme smallness, calls upon the introduction of right-handed neutrinos through the celebratedSeesaw mechanism <cit.>. Its appeal stems from the simplicity of its particle content, consisting only of the right-handed neutrinos otherwise conspicuously missing from the Standard Model (SM) ingredients.In the Seesaw mechanism, the smallness of neutrino masses is explained through the ratio of their Dirac masses and the Majorana mass term of the extra fermion singlets. Unfortunately, this very same ratio suppresses any phenomenological probe of the existence of this mechanism. Indeed, either the right-handed neutrino masses would be too large to be reached by our highest energy colliders, or the Dirac masses, and hence the Yukawa interactions that mediate the right-handed neutrino phenomenology, would be too small for even our more accurate precision probes through flavour and precision electroweak observables. However, a large hierarchy of scales is not the only possibility to naturally explain the smallness of neutrino masses. Indeed, neutrino masses are protected by theB-L (Baryon minus Lepton number) global symmetry, otherwise exact in the SM. Thus, if this symmetry is only mildly broken, neutrino masses will be necessarily suppressed bythe small B-L-breaking parameters. Conversely, the production and detection of the extra right-handed neutrinos at colliders as well as their indirect effects in flavour and precision electroweak observables are not protected by the B-L symmetry and therefore not necessarily suppressed, leading to a much richer and interesting phenomenology. This is the rationale behind the popular Inverse Seesaw Mechanism <cit.> (ISS) as well as the Linear <cit.> and Double Seesaw <cit.> variants.In the presence of right-handed neutrinos, B-L is the only flavour-universal SM quantum number that is not anomalous, besides hypercharge. Therefore, just like the addition of right-handed neutrinos, a very natural plausible SM extension is the gauging of this symmetry. In this work these two elements are combined to explore a possible dynamical origin of the ISS pattern from the spontaneous breaking of the gauged B-L symmetry. Previous models in the literature have been constructed using the ISS idea or gauging B-L to explain the smallness of the neutrino masses,see e.g. <cit.>.A minimal model in which the ISS is realised dynamically and where the smallness of the Lepton Number Violating (LNV) term is generated at the two-loop level was studied in <cit.>. Concerning U(1)_B-L extensions of the SM with an ISSgeneration of neutrino masses, several models have been investigated <cit.>. A common origin of both sterile neutrinos andDark Matter (DM) has been proposed in <cit.>.An ISS model which incorporates a keV sterile neutrino as a DM candidate was constructed in e.g. <cit.>.Neutrino masses break B-L, if this symmetry is not gauged and dynamically broken, a massless Goldstone boson, the Majoron, appears in the spectrum. Such models have been investigatedfor example in <cit.>.Interestingly, since the ISS mechanism requires a chiral pattern in the neutrino sector, the gauging of B-Lpredicts the existence of extra fermion singlets with non-trivial charges so as to cancel the anomalies. We find that these extra states may play the role of DM candidates as thermally producedWeakly Interacting Massive Particles (WIMPs) (see for instance <cit.> for a review). Indeed, the extra states would form a dark sector, only connected to the SM via the Z' gauge boson associated to the B-L symmetry and, more indirectly, through the mixing of the scalar responsible for the spontaneous symmetry breaking of B-L with the Higgs boson. For the simplest charge assignment, this dark sector would be constituted by one heavy Dirac and one massless Weyl fermion with large B-L charges. These large charges make the Z' couple preferentially to the dark sector rather than to the SM, making it particularly elusive.In this work the phenomenology associated with this dark sector and the elusive Z' is investigated. We find that the heavy Dirac fermion of the dark sector can be a viable DM candidate with its relic abundance mediated by the elusive Z'. Conversely, the massless Weyl fermion can be probed through measurements of the relativistic degrees of freedom in the early Universe. The collider phenomenology of the elusive Z' is also investigated andthe LHC bounds are derived. The paper is structured as follows. In Sec. <ref> we describe the features of the model, namely its Lagrangian and particle content. In Sec. <ref> we analyse the phenomenology of the DM candidate and its viability. The collider phenomenology of the Z' boson is discussed in Sec. <ref>. Finally, in Secs. <ref> and <ref> we summarise our results and conclude. § THE MODEL The usual ISS model consists of the addition of a pair of right-handed SM singlet fermions (right-handed neutrinos) for each massiveactive neutrino <cit.>. These extra fermion copies, say N_R and N_R', carry a globalLepton Number (LN) of +1 and -1, respectively, and this leads to the following mass Lagrangian- ℒ_ ISS =L̅ Y_νH N_R + N_R^c M_N N_R' + N_R'^cμN_R' +h.c., where Y_ν is the neutrino Yukawa coupling matrix, H=iσ_2 H^* (H being the SM Higgs doublet) and L is the SM lepton doublet.Moreover, M_Nis a LN conserving matrix, while the mass matrix μ breaks LN explicitly by 2 units.The right-handed neutrinos can be integrated out,leading to the Weinberg operator <cit.> which generates masses for the light, active neutrinos of the form:m_ν∼ v^2 Y_ν M_N^-1μ (M_N^T)^-1 Y^T_ν. Having TeV-scale right-handed neutrinos (e.g. motivated by naturalness <cit.>) and 𝒪(1) Yukawa couplings would require μ∼𝒪(). In the original ISS formulation <cit.>, the smallness of this LNV parameter arises from a superstring inspired E6 scenario. Alternative explanations call upon other extensions of the SM such as Supersymmetry and Grand Unified Theories (see for instance <cit.>).Here a dynamical origin for μ will be instead explored. The μ parameter is technically natural: since it is the only parameter that breaks LN, its running is multiplicative and thusonce chosen to be small, it will remain small at all energy scales.To promotethe LN breaking parameter μ in the ISS scenario to a dynamical quantity, we choose to gauge the B-L number <cit.>.The spontaneous breaking of this symmetry will convey LN breaking, generate neutrino masses via a scalar vev, and give rise to a massive vector boson, dubbed here Z'. B-L is an accidental symmetry of the SM, and it is well motivated in theories in which quarks and leptons are unified <cit.>. In unified theories, the chiral anomalies cancel within each family, provided that SM fermion singlets with charge +1 are included. In the usual ISS framework, this is not the case due to the presence of right-handed neutrinos with charges +1 and -1. The triangle anomalies that do not cancel are those involving three U(1)_B-L vertices, as well as one U(1)_B-L vertex and gravity. Therefore, to achieve anomaly cancellation for gauged B-L we have to include additional chiral content to the model with charges that satisfy∑ Q_i=0⇒∑ Q_iL-∑ Q_iR=0,∑ Q_i^3=0⇒∑ Q_iL^3-∑ Q_iR^3=0,where the first and second equation refer to the mixed gravity-U(1)_B-L and U(1)_B-L^3anomalies, respectively. The index i runs through all fermions of the model. In the following subsections we will discuss the fermion and the scalar sectors of the model in more detail.§.§ The fermion sectorBesides the anomaly constraint, the ISS mechanism can only work with a certain number of N_R and N_R' fields (see, e.g., Ref. <cit.>).We find a phenomenologically interesting and viable scenario which consists of the following copies of SM fermion singlets and their respective B-L charges:3 N_R with charge -1; 3 N_R' with charge +1; 1 χ_R with charge +5; 1 χ_L with charge +4and 1 ω with charge +4[Introducing 2 N_R and 3 N_R' as for example in <cit.> leads to a keV sterile neutrino as a potentially interestingwarm DM candidate <cit.> in the spectrum due to the mismatch between the number of N_R and N_R'. However, therelic abundance of this sterile neutrino, if thermally produced via freeze out, is an order of magnitude too large. Thus, in order to avoidits thermalisation, very small Yukawa couplings and mixings must be adopted instead.] Some of these right-handed neutrinos allow for a mass term,namely, M_N N_R^c N_R', but to lift the mass of the other sterile fermions and to generate SM neutrino masses, two extra scalars are introduced.Thus, besides the Higgs doublet H, the scalar fields ϕ_1 with B-L charge +1 and ϕ_2with charge +2 are considered. The SM leptons have B-L charge -1, while the quarks have charge 1/3. Thescalar and fermion content of the model, related to neutrino mass generation, is summarised inTable <ref>. The most general Lagrangian in the neutrino sector is then given by[Notice that acoupling ϕ_1^* ω Y_ωχ_R, while allowed, can always be reabsorbedinto ϕ_1^*χ_L Y_χχ_R through a rotation between ω and χ_L.]- ℒ_ν =L̅ Y_νHN_R +N_R^c M_N N_R' +ϕ_2N_R^c Y_N N_R + ϕ_2^* (N_R')^cY'_N N_R'+ϕ_1^*χ_LY_χχ_R + h.c., where the capitalised variables are to be understood as matrices (the indices were omitted). The singlet fermion spectrum splits into two parts, an ISS sector composed by ν_L, N_R, and N'_R, and a dark sectorwith χ_L and χ_R, as can be seenin the following mass matrix written in the basis (ν_L^c, N_R,N_R',χ^c_L,χ_R): M=( [ 0Y_νH 0 0 0;Y_ν^TH^†Y_Nϕ_2 M_N 0 0; 0 M_N^T Y_N'ϕ_2^* 0 0; 0 0 0 0Y_χϕ_1^*; 0 0 0Y_χ^Tϕ_1 0 ]). The dynamical equivalent of the μ parameter can be identified with Y_N' ϕ_2^*[The analogous term Y_Nϕ_2 - also dynamically generated -contributes to neutrino masses only at the one-loop level and is therefore typically sub-leading.].After ϕ_1 develops a vacuum expectation value (vev) a Dirac fermion χ=(χ_L,χ_R) and a massless fermion ω are formed in the dark sector. Although the cosmological impact of this extra relativistic degree of freedom may seem worrisome at first, we will show later that the contribution toN_ eff is suppressed as this sector is well secluded from the SM.To recover a TeV-scale ISS scenario with the correct neutrino masses and 𝒪(1) Yukawa couplings,v_2≡ϕ_2∼≪ v (where v=H=246 is the electroweak vev) and M_R∼ are needed. Moreover, the mass of theB-L gauge boson will be linked to the vevs of ϕ_1 and ϕ_2, and hence to lift its mass above the electroweak scale willrequire v_1≡ϕ_1≳. In particular, we will show that a triple scalar coupling ηϕ_1^2ϕ_2^* can induce a smallv_2 even when v_1 is large, similar to what occurs in the type-II seesaw <cit.>.After the spontaneous symmetry breaking, the particle spectrum would then consist of a B-L gauge boson, 3 pseudo-Dirac neutrino pairs and a Dirac dark fermion at the TeV scale, as well as a massless dark fermion. The SM neutrinos would in turn develop small masses via the ISS in the usual way. Interestingly, both dark fermions only interact with the SM via the new gauge boson Z' and via the suppressed mixing of ϕ_1 with the Higgs.They are also stable and thus the heavy dark fermion is a natural WIMP DM candidate. Since all new fermions carry B-L charge, they all couple to the Z', but specially the ones in the dark sector which have larger B-L charge. §.§ The scalar sectorThe scalar potential of the model can be written asV =m_H^2/2 H^† H+λ_H/2 (H^† H)^2 + m_1^2/2ϕ_1^*ϕ_1 + m_2^2/2ϕ_2^*ϕ_2 + λ_1/2(ϕ_1^*ϕ_1)^2 + λ_2/2(ϕ_2^*ϕ_2)^2 + λ_12/2(ϕ_1^*ϕ_1)(ϕ_2^*ϕ_2)+λ_1H/2(ϕ_1^*ϕ_1)(H^† H)+λ_2H/2(ϕ_2^*ϕ_2)(H^† H)-η(ϕ_1^2ϕ_2^*+ϕ_1^*2ϕ_2).Both m_H^2 and m_1^2 are negative, but m_2^2 is positive and large. Then, for suitable values of the quartic couplings, the vev of ϕ_2, v_2, is only induced by the vev of ϕ_1, v_1, through η and thus it can be made small.With the convention ϕ_j=(v_j+φ_j+ia_j)/√(2) and the neutral component of the complex Higgs field given by H^0=(v+h+i G_Z)/√(2) (where G_Z is the Goldstone associated with the Z boson mass), the minimisation of the potential yieldsm_H^2= -1/2(λ_1Hv_1^2+λ_2Hv_2^2+2λ_H v^2)≃-1/2(λ_1Hv_1^2+2λ_H v^2),m_1^2= -1/2(2λ_1 v_1^2 + λ_1Hv^2-4√(2)η v_2+λ_12v_2^2)≃-1/2(2λ_1 v_1^2 + λ_1Hv^2),m_2^2= (√(2)η/v_2-λ_12/2)v_1^2 -λ_2 v_2^2 - λ_2H/2v^2≃√(2)η v_1^2/v_2,or, equivalently,v_2≃√(2)η v_1^2/m_2^2 .Clearly, when η→0 or m_2^2→∞, the vev of ϕ_2 goes to zero. For example, to obtain v_2∼𝒪(), one could have m_2∼ 10 TeV, v_1∼ 10,and η∼ 10^-5. The neutral scalar mass matrix is then given byM_0^2≃( [λ_H v^2λ_1Hv_1 v/20;λ_1Hv_1 v/2λ_1 v_1^2 -√(2)η v_1;0 -√(2)η v_1 η v_1^2/√(2) v_2 ]). Higgs data constrain the mixing angle between Re(H^0) and Re(ϕ_1^0) to be below ∼30% <cit.>. Moreover, since η≪ m_2,v_1, the mixing between the new scalars is also small. Thus, the masses of the physical scalars h, φ_1 and φ_2 are approximately m_h^2=λ_H v^2, m_φ_1^2=λ_1 v_1^2, and m_φ_2^2=m_2^2/2,while the mixing angles α_1 and α_2 between h-φ_1 and φ_1-φ_2, respectively, aretanα_1≃λ_1H/λ_1v/2v_1, andtanα_2≃2v_2/v_1. If v_1∼ and the quartics λ_1 and λ_1H are 𝒪(1), the mixing α_1 is expected to be small but non-negligible. A mixing between the Higgs doublet and a scalar singlet can only diminish the Higgs couplings to SM particles. Concretely, the couplings of the Higgs to gauge bosons and fermions, relative to the SM couplings, areκ_F=κ_V=cosα_1,which is constrained to becosα_1>0.92 (or equivalently sinα_1<0.39) <cit.>. Since the massless fermion does not couple to any scalar, and all other extra particles in the model are heavy, the modifications to the SM Higgs couplings are the only phenomenological impact of the model on Higgs physics. The other mixing angle, α_2, is very small since it is proportional to the LN breaking vev and thus is related to neutrino masses. Its presence will induce a mixing between the Higgs and φ_2, but for the parameters of interest here it is unobservable. Besides Higgs physics, the direct production of φ_1 at LHC via its mixing with the Higgs would be possible if it is light enough. Otherwise, loop effects that would change the W mass bound can also test this scenario imposing sinα_1≲ 0.2 for m_φ_1=800GeV <cit.>. Apart from that, the only physical pseudoscalar degree of freedom is A=1/√(v_1^2 + 4 v_2^2)[2 v_2 a_1-v_1 a_2]and its mass is degenerate with the heavy scalar mass, m_A≃ m_φ_2.We have built this model in SARAH 4.9 <cit.>.This Mathematica package produces the model files for SPheno 3.3.8 <cit.> and CalcHep <cit.>which are then used to study the DM phenomenology with Micromegas 4.3 <cit.>.We have used these packages to compute the results presented in the following sections. Moreover, we will present analytical estimations to further interpret the numerical results.§ DARK MATTER PHENOMENOLOGY As discussed in the previous section, in this dynamical realisation of the ISS mechanism we have two stable fermions. One of them is a Dirac fermion, χ=(χ_L,χ_R), whichacquires a mass from ϕ_1, and therefore is manifest at the TeV scale. The other, ω, is massless and will contribute to the number of relativistic species in the early Universe. First we analyse if χ can yieldthe observed DM abundance of the Universe.§.§ Relic densityIn the early Universe, χ isin thermal equilibrium with the plasma due to its gauge interaction with Z'. The relevant part of the Lagrangian is ℒ_DM = - g_ BLχ̅γ^μ ( 5P_R+4 P_L) χ Z'_μ + 1/2M_Z'^2Z'_μ Z^'μ - m_χχ̅χ,whereM_Z'= g_ BL√(v_1^2+4v_2^2)≃ g_ BL v_1, and m_χ = Y_χ v_1/√(2),and P_R,L are the chirality projectors. The main annihilation channels of χ are χχ̅→ f f̅ via the Z' boson exchangeand χχ̅→ Z' Z' - if kinematically allowed (see fig. <ref>). The annihilation cross section to a fermion species f, at leading order in v, reads:σv_ff≃ n_c (q_χ_L+q_χ_R)^2 q^2_f_L+q^2_f_R/8πg_ BL^4 m_χ^2/(4m_χ^2-M_Z'^2)^2+Γ^2_Z'M_Z'^2 +𝒪(v^2 ),see e.g. <cit.>,where n_c is the color factor of the final state fermion (=1 for leptons), q_χ_L=4 and q_χ_R=5 and q_f_L,R are the B-L charges of the left- andright-handed components of the DM candidate χ and of the fermion f, respectively. Moreover, the partial decaywidth of the Z' into a pair of fermions (including the DM, for which f=χ) is given byΓ_Z'^ff =n_c g_ BL^2 (6 q_f_L q_f_R m^2_f + ( q^2_f_L+q^2_f_R) (M_Z'^2 - m_f^2 ) ) √(M^2_Z' - 4 m_f^2)/24 π M^2_Z' .When M_Z'^2 < m_χ^2, the annihilation channel χχ̅→ Z'Z' is also available. The cross section for this process (lower diagrams in fig. <ref>)is given by(to leading order in the relative velocity) <cit.>σv_Z'Z' ≃1/256π m_χ^2 M_Z'^2(1-M_Z'^2/m_χ^2)^3/2(1-M_Z'^2/2 m_χ^2)^-2(8 g_BL^4 (q_χ_R+q_χ_L)^2 (q_χ_R-q_χ_L)^2 m_χ^2+( (q_χ_R-q_χ_L)^4+(q_χ_R+q_χ_L)^4 . . ..-6(q_χ_R-q_χ_L)^2(q_χ_R+q_χ_L)^2) g_BL^4 M_Z'^2 ) , The χχ̅→φ_1→ Z' Z' (upper right diagram in fig. <ref>) channel is velocity suppressed and hence typically subleading.Further decay channels likeχχ̅→φ_1 φ_1 and χχ̅→ Z' φ_1 open when 2 m_χ>m_φ_1+m_φ_1 (m_φ_1+m_Z^').With m_χ= Y_χ/√(2)v_1, m_φ_1=√(λ_1)v_1, m_Z^'=g_ BLv_1 and the additional constraint from perturbativityY_χ≤ 1 we get onlysmall kinematically allowed regions which play a subleading role for the relic abundance.The cross section for the annihilation channel χχ̅→ Z' h^0 is also subleading due to the mixing angle α_1 between φ_1 - h^0which is small althoughnon-negligible (cf. Eq. (<ref>)).The relic density of χ has been computed numerically with Micromegas obtaining also, for several points of the parameter space, the DM freeze-out temperature at which the annihilation rate becomes smaller than the Hubble rate σv n_χ≲ H. Given the freeze-out temperature and the annihilation cross sections of Eqs. (<ref>) and (<ref>), the DM relic density can thus be estimated by <cit.>: Ω_χ h^2 = 2.5· 10^28 m_χ/T^ f.o._χ M^2_Pl√(g_⋆)σv,where g_⋆ is the number of degrees of freedom in radiation at the temperature of freeze-out of the DM(T^ f.o._χ), σv is its thermally averaged annihilation cross section and M_Pl = 1.2 · 10^19 GeV is the Planck mass. In Sec. <ref> we will use this estimation of Ω_χ h^2 together with its constraint Ω_χ h^2 ≃ 0.1186 ± 0.0020 <cit.> to explore the regions of the parameter space for which the correct DM relic abundance is obtained.§.§ Direct DetectionThe same Z' couplings that contribute to the relic abundance can give rise to signals in DM direct detection experiments. The DM-SM interactions in the model via the Z' are either vector-vector or axial-vector interactions.Indeed, the Z'- SM interactions are vectorial (with the exception of the couplings to neutrinos) while χ has different left- and right-handed charges.The axial-vector interaction does not lead to a signal in direct detection and the vector-vectorinteraction leads to a spin-independent cross section <cit.>. Thecross section for coherent elastic scattering on a nucleon isσ^ DD_χ=μ_χ N^2/π(9/2g_ BL^2/M_Z'^2)^2 where μ_χ N is the reduced mass of the DM-nucleon system. The strongest bounds on the spin-independent scattering cross section come from LUX <cit.> and XENON1T <cit.>. The constraint on the DM-nucleon scattering cross section isσ^ DD_χ<10^-9 pb for m_χ=1 TeVand σ^ DD_χ<10^-8 pbfor m_χ=10 TeV.The experimental bound on the spin-independent cross section (Eq. (<ref>)) allows to derive a lower bound on the vev of ϕ_1:v_1 [GeV]> (2.2· 10^9/σ^ DD_χ [pb])^1/4 . This bound pushes the DM mass to be m_χ≳ TeV.For instance, for g_ BL = 0.25 and m_Z' = 10 TeV, a DM mass m_χ = 3.8 TeV is required to haveσ^ DD_χ ∼ 9 × 10^-10 pb. In turn, this bound translates into a lower limit on the vev of ϕ_1: v_1 ≳ 40 TeV (with Y_χ≳ 0.1).Next generation experiments such as XENON1T <cit.> and LZ <cit.> are expected to improve the current bounds by an order of magnitude and could test the parameter space of this model, as it will be discussed in Sec. <ref>.§.§ Indirect DetectionIn full generality, the annihilation ofχ today could lead also to indirect detection signatures, in the form of charged cosmic rays, neutrinos andgamma rays.However, since the main annihilation channel of χ is via the Z' which couples dominantly to the dark sector, the bounds from indirect detection searches turn out to be subdominant.The strongest experimental bounds come from gamma rays produced through direct emission from the annihilation of χ into τ^+ τ^-. Both the constraints from the Fermi-LAT Space Telescope (6-year observation of gamma rays from dwarf spheroidal galaxies) <cit.> and H.E.S.S. (10-year observation of gamma rays from the Galactic Center) <cit.> are not very stringent for the range of DM masses considered here.Indeed, the current experimental bounds on the velocity-weighted annihilation cross section <σ v> (χχ̅→τ^+τ^-) range from 10^-25 cm^3 s^-1 to 10^-22 cm^3 s^-1 for DM masses between 1 and 10 TeV. These values are more than two orders of magnitude above the values obtained for the regions of the parameter space in which we obtain the correct relic abundance (notice that the branching ratio of the DM annihilation to χ into τ^+ τ^- is only about 5%). Future experiments like CTA <cit.> could be suited to sensitively address DM masses in the range of interest of this model (m_χ≳ 1 TeV).§.§ Effective number of neutrino species, N_ effThe presence of the massless fermion ω implies a contribution to the number of relativistic degrees of freedom in the early Universe. In the following, we discuss its contribution to the effective number of neutrino species, N_ eff, which has been measured to be N_ eff^exp=3.04± 0.33 <cit.>.Since the massless ω only interacts with the SM via the Z', its contribution to N_ eff will be washed out through entropy injection to the thermal bath by the number of relativistic degrees of freedom g_⋆(T) at the time of its decoupling:Δ N_ eff=(T^ f.o._ω/T_ν)^4 = (11/2 g_⋆(T^ f.o._ω))^4/3 ,where T^ f.o._ω is the freeze-out temperature of ω and T_ν is the temperature of the neutrino background.The freeze-out temperature can be estimated when the Hubble expansion rate of the Universe H = 1.66 √(g_⋆) T^2/M_Pl overcomes the ω interaction rate Γ = <σ v> n_ω leading to:(T^ f.o._ω)^3 ∼2.16 √(g_⋆)M^4_Z'/M_Pl g_ BL^4 ∑_f (q^2_f_L + q^2_f_R) . With the typical values that satisfy the correct DM relic abundance: m_Z'∼𝒪(10 TeV) and g_ BL∼𝒪(0.1) ω would therefore freeze out at T^ f.o._ω∼ 4 GeV, before the QCD phase transition. Thus, the SM bath will heat significantly after ω decouples and the contribution of the latter to the number of degrees of freedom in radiation will be suppressed:Δ N_ eff≈ 0.026which is one order of magnitude smaller than the current uncertainty on N_ eff.For gauge boson masses between 1-50 TeV and gauge couplings between 0.01 and 0.5, Δ N_ eff∈[0.02,0.04].Nevertheless, this deviation from N_ eff matches the sensitivity expected from a EUCLID-like survey <cit.> and would bean interesting probe of the model in the future. § COLLIDER PHENOMENOLOGYThe new gauge boson can lead to resonant signals at the LHC. Dissimilarly from the widely studied case of a sequential Z' boson, where the new boson decays dominantly to dijets, the elusive Z' couples more strongly to leptons than to quarks (due to the B-L number). Furthermore, it has large couplings to the SM singlets, specially χ and ω which carry large B-L charges. Thus, typical branching ratios are ∼70% invisible (i.e. into SM neutrinos and ω), ∼12% to quarks and ∼18% to charged leptons.[If the decaychannels to the other SM singlets are kinematically accessible,specially into χ and into theN_R, N'_R pseudo-Dirac pairs, the invisible branching ratio can go up to ∼ 87%, making the Z' even more elusive and rendering these collider constraints irrelevant with respect to direct DM searches.]LHCZ'→ e^+e^-,μ^+μ^- resonant searches <cit.> can be easily recast into constraints on the elusive Z'. The production cross section times branching ratio to dileptonsis given by σ(pp→ Z'→ℓℓ̅)=∑_qC_qq/s M_Z'Γ(Z'→ q q̅) BR(Z'→ℓℓ̅), where s is the center of mass energy, Γ(Z'→ q q̅) is the partial width to q q̅ pair given by Eq. (<ref>),and C_qq is the q q̅ luminosity function obtained here using the parton distribution function MSTW2008NLO <cit.>. To have some insight on what to expect, we compare our Z' with the usual sequential standard model (SSM) Z', in which all couplings to fermions are equal to the Z couplings. The dominant production mode is again q q̅→ Z' though the coupling in our case is mostly vectorial.The main dissimilarity arrives from the branching ratio to dileptons, as there are many additional fermions charged under the new gauge group.In summary, only 𝒪(1) differences in the gauge coupling bounds are expected, between the SSM Z' and our elusive Z'.§ RESULTS We now combine in fig. <ref> the constraints coming from DM relic abundance, DM direct detection experiments and collider searches. We can clearly see the synergy between these different observables. Since the DM candidate in our model is a thermal WIMP, the relic abundance constraint puts a lower bound on the gauge coupling, excluding the blue shaded region in the panels of fig. <ref>. On the other hand, LHC resonant searches essentially put a lower bound on the mass of the Z' (red shaded region), whilethe LUX direct detection experiment constrains the product g_ BL· M_Z' from above (orange shaded region). For reference, we also show the prospects for future direct detection experiments,namely, XENON1T (orange short-dashed line, projected sensitivity assuming 2t · y) and LZ (orange long-dashed line, projected sensitivity for 1000d of data taking). Finally, if the gauge coupling is too large, perturbativity will be lost.To estimate this region we adopt the constraint g_ BL· q_ max≤√(2π) and being thelargest B-L charge q_ max=5, we obtain g_ BL>0.5 for the non-perturbative region. The white region in these panels represents the allowed region. We present four different DM masses so as to exemplify the dependence on m_χ. First, we see that for DM masses at 1 TeV (upper left panel), there is only a tiny allowed region in which the relic abundance is set via resonant χχ̅→ Z'→ f f̅ annihilation. For larger masses, the allowed region grows but some amount of enhancement is in any case needed so that the Z' mass needs to be around twice the DM mass in order to obtain the correct relic abundance. For m_χ above 20 TeV (lower right panel), the allowed parameter space cannot be fully probed even with generation-2 DM direct detection experiments.On top of the DM and collider phenomenology discussed here, this model allows for a rich phenomenology in other sectors.In full analogy to the standard ISS model, the dynamical ISS mechanism here considered is also capable of generating a large CP asymmetry in the lepton sector at the TeV scale, thus allowing for a possibleexplanation of the baryon asymmetry of the Universe via leptogenesis <cit.>.Moreover, the heavy sterile states typically introduced in ISS scenarios, namely the three pseudo-Dirac pairs from the statesN_R and N_R^' can lead to new contributions to a wide array of observables <cit.> such as weak universality, lepton flavour violating or precision electroweak observables, which allow to constrain the mixing of the SM neutrinos with the extra heavy pseudo-Dirac pairs to the level of 10^-2 or even better for some elements <cit.>.§ CONCLUSIONSThe simplest extension to the SM particle content so as to accommodate the experimental evidence for neutrino masses and mixings is the addition of right-handed neutrinos,making the neutrino sector more symmetric to its charged lepton and quark counterparts. In this context, the popular Seesaw mechanism also gives a rationalefor the extreme smallness of these neutrino masses as compared to the rest of the SM fermionsthrough a hierarchy between two different energy scales:the electroweak scale – at which Dirac neutrino masses are induced – and a much larger energy scale tantalizingly close to the Grand Unification scaleat which Lepton Number is explicitly broken by the Majorana mass of the right-handed neutrinos.On the other hand, this very natural option to explain the smallness of neutrino masses automatically makes the mass of the Higgs extremely unnatural, given the hierarchy problem that is hence introduced between the electroweakscale and the heavy Seesaw scale. The ISS mechanism provides an elegant solution to this tension by lowering the Seesaw scale close to the electroweakscale, thus avoiding the Higgs hierarchy problem altogether. In the ISS the smallness of neutrino masses is thus not explained by a strong hierarchy between these scales but rather by a symmetry argument. Since neutrino masses are protected by the Lepton Number symmetry, or rather B-L in its non-anomalous version, if this symmetry is only mildly broken, neutrino masses will be naturally suppressed by the small parameters breaking this symmetry. In this work, the possibility of breaking this gauged symmetry dynamically has been explored. Since the ISS mechanism requires a chiral structure of the extra right-handed neutrinos under the B-L symmetry, some extra states are predicted for this symmetry to be gauged due to anomaly cancellation. The minimal such extension requires the addition of three new fields with large non-trivial B-L charges. Upon the spontaneous breaking of the B-L symmetry, two of these extra fields become a massive heavy fermion around the TeV scale while the third remains massless. Given their large charges, the Z' gauge boson mediating the B-L symmetry couples preferentially to this new dark sector and much more weakly to the SM leptons and particularly to quarks, making it rather elusive. The phenomenology of this new dark sector and the elusive Z' has been investigated. We find that the heavy Dirac fermion is a viable DM candidate in some regions of the parameter space. While the elusive nature of the heavy Z' makes its search rather challenging at the LHC, it would also mediate spin-independent direct detection cross sections for the DM candidate, which place very stringent constraints in the scenario. Given its preference to couple to the dark sector and its suppressed couplings to quarks, the strong tension between direct detection searches and the correct relic abundance for Z' mediated DM is mildly alleviated and some parts of the parameter space, not far from the resonance, survive present constraints. Future DM searches by XENON1T and LZ will be able to constrain this possibility even further. Finally, the massless dark fermion will contribute to the amount of relativistic degrees of freedom in the early Universe. While its contribution to the effective number of neutrinos is too small to be constrained with present data, future EUCLID-like surveys could reach a sensitivity close to their expected contribution, making this alternative probe a promising complementary way to test this scenario. § ACKNOWLEDGEMENTSVDR would like to thank A. Vicente for valuable assistance on SARAH and SPheno. JG would like to thank Fermilab for kind hospitality during the final stages of this project. This work is supported in part by the EU grants H2020-MSCA-ITN-2015/674896-Elusives and H2020-MSCA-2015-690575-InvisiblesPlus.VDR acknowledges support by the Spanish grant SEV-2014-0398 (MINECO) and partial support by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 and PROMETEOII/2014/084 (Generalitat Valenciana). EFM acknowledges support from the EU FP7 Marie Curie Actions CIG NeuProbes (PCIG11-GA-2012-321582), "Spanish Agencia Estatal de Investigación" (AEI)and the EU "Fondo Europeo de Desarrollo Regional" (FEDER) through the project FPA2016-78645-P and the Spanish MINECO through the “Ramón y Cajal” programme (RYC2011-07710)and through the Centro de Excelencia Severo Ochoa Program under grant SEV-2012-0249 and the HPC-Hydra cluster at IFT.The work of VN was supported by the SFB-Transregio TR33 “The Dark Universe". This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.JHEP | http://arxiv.org/abs/1707.08606v2 | {
"authors": [
"Valentina De Romeri",
"Enrique Fernandez-Martinez",
"Julia Gehrlein",
"Pedro A. N. Machado",
"Viviana Niro"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170726185542",
"title": "Dark Matter and the elusive $\\mathbf{Z'}$ in a dynamical Inverse Seesaw scenario"
} |
GBgbsn [c]Chinese Physics C Vol. xx, No. x (201x) xxxxxx [C]010201-[0]Received 31 June 2015 Scattering of massless fermions by Schwarzschild and Reissner-Nordström black holesCiprian A. Sporea^1;1)[email protected] 29, 2018 =================================================================================== ^1 West University of Timişoara, V.Pârvan Ave.4, RO-300223 Timişoara, Romania In this paper we are studying the scattering of massless Dirac fermions by Schwarzschild and Reissner-Nordström black holes. This is done by applying the partial wave analysis to the scattering modes obtained after solving the massless Dirac equation in the asymptotic regions of the two black hole geometries. We succeed to obtain analytic phase shifts with the help of which the scattering cross section is computed. The glory and spiral scattering phenomena are showed to be present like in the case of massive fermion scattering by black holes. black holes, scattering, massless fermions.04.70.-s, 03.65.Nk, 04.62.+v. [0]0.3ex2013 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd2§ INTRODUCTION The problem of black hole scattering is still an ongoing on, despite the numerous papers that have dealt with it so far. Most of these studies have been dedicated to investigating different aspects of (massless) scalar wave scattering by black holes <cit.>. However, there are also papers that studi the scattering of massless electromagnetic waves <cit.> and gravitational waves <cit.>. The problem of scattering of massive spinor 1/2 waves by black holes was discussed in <cit.>.The scattering of massless fermions was studied in <cit.>, where the scattering of massless fermions by a black hole with a cosmic string, respectively a dilatonic black hole was investigated and partially in <cit.> for a Schwarzschild black hole. The authors of <cit.> have used numerical methods to solve the Dirac equation in Schwarzschild black hole geometries in order to find numerical phase shifts using a partial wave analysis. In <cit.> we succeeded to obtain for the first time analytic expressions for the Schwarzschild phase shifts. Furthermore, in <cit.> we extended our study to include also the case of fermion scattering by charged Reissner-Nordström black holes, where we found again analytic phase shifts. Moreover, our study <cit.> is up to our knowledge the first one in the literature in which the problem of massive fermion scattering by Reissner-Nordström black holes was investigated.In this paper we are studying the scattering of massless fermions by spherical symmetric black holes, with a focus on Schwarzschild and charged Reissner-Nordström black holes. We derive analytic phase shifts that will allow us to write down analytic expressions for the scattering cross sections.Although it is generally assumed that astrophysical black holes are electrically neutral <cit.>, or at least have negligible charges, it has been recently showed in Ref. <cit.> that charged astrophysical black holes can in fact exist in the context of minicharged dark matter models <cit.>. These dark matter models predict new fermions that can have fractional electric charge or fermions that are charged under a U(1) hidden symmetry. Because these new charges have just a small fraction of the electron's charge, their coupling with the Standard Model electromagnetic sector is suppressed. This means that even in the case of massive fermions there could be no direct interaction between the Dirac field and the electromagnetic field of the black hole. This could open new possibilities for dark matter detection through neutrino-wave scattering by black holes, besides the gravitational-wave signatures discussed in <cit.>.In the original Standard Model of particle physics neutrinos are assumed to be massless. Thus our results will contain also the case of massless neutrino scattering by black holes (see also Appendix A). However, it has been observed experimentally <cit.> that neutrinos have a nonzero mass that it is currently bound to ∑ m_ν < 0.183 eV <cit.>. The electron neutrino mass could be as small as m_ν_e∼0.01 eV or smaller. Having this in mind one can easily assume that our results (presented in the following sections) will give also a good approximation for the cross sections in the case of scattering of astrophysical neutrinos by black holes.The paper is organised as follow: in Sec. <ref> we give a brief introduction of the general form of the massless Dirac equation in a curved spacetime with spherical symmetry. The Cartesian gauge is introduced and the separation of spherical variables is made. In the next Sec. <ref> we solve the massless Dirac equation in the case of Schwarzschild and Reissner-Nordström black hole geometries. We focus only on scattering mode solutions. The following Sec. <ref> deals with the partial wave analysis, where we give analytical expressions for the phase shifts that enter into the definitions of the scattering amplitudes and cross sections. Sec. <ref> is dedicated to the presentations of the main results obtained. The paper ends with Sec. <ref> where the final conclusions and some remarks are presented.In this paper we set G=c=ħ=1; the metric signature used is (+,-,-,-); the natural indices are labeled with Greek letters μ, ν, α, ..., while the local indices are labeled with a, b, c, ...; both can take values in the range (0,1,2,3). § PRELIMINARIES Starting from the gauge invariant action𝒮 =∫d^4x √(-g){i/2ψγ^a D_aψ - i/2( D_aψ )γ^aψ}we can immediately derive the following Dirac equation for a massless spinor field in a curved spacetimeiγ^a e^μ_a∂_μψ + i/21/√(-g)∂_μ(√(-g) e^μ_a)γ^aψ -1/4{γ^a , S^ b_ c}ω^ c_a bψ = 0where e^μ_a are the tetrad fields such that g^μν=η^a be_a^μ e_b^ν; γ^a are the point-independent Dirac matrices satisfying {γ^a,γ^b}=2η^a b; S^ b_c are the generators of the spinor representation of SL(2,ℂ) <cit.> such that S^a b=i/4[γ^a,γ^b];The covariant derivative D_a and the spin-connection readD_a = ∂_a + i/2S^ b_c ω^ c_a bω^ c_ab = e^μ_a e^ν_b ( ê^c_λΓ^λ_μν - ê^c_ν,μ)with ∂_a=e^μ_a∂_μ and Γ^λ_μν stand for the GR Christoffel symbols.The spacetime geometry of a spherically symmetric black hole is given by the following line elementds^2=h(r) dt^2-dr^2/h(r)-r^2( dθ^2+sin^2θ dϕ^2 ) In the Cartesian gauge <cit.>, the above line element can be obtained from ds^2=η_abê^aê^b with the following choice of the tetrad field ê^a(x)=ê^a_μ dx^μ (i.e. the 1-forms)ê^0 = h(r)dt ê^1 = 1/h(r)sinθcosϕ dr + rcosθcosϕ dθ - rsinθsinϕ dϕê^2 = 1/h(r)sinθsinϕ dr + rcosθsinϕ dθ + rsinθcosϕ dϕê^3 =1/h(r)cosθ dr - rsinθ dθThe main advantage of the above Cartesian gauge consists in the fact that in this gauge the Dirac equation (free or in central scalar potentials) is manifestly covariant under rotations. This means that the angular part of the equation can be solved in terms of the usual 4-component angular spinors form special relativity Φ^±_m,κ(θ,ϕ) <cit.>. Using the Cartesian gauge new (exact or approximative) analytical solutions of the Dirac equation in curved backgrounds were found <cit.>.Inserting the metric (<ref>) in eq. (<ref>) and after some calculations, one can put the Dirac equation into a Hamiltonian form Hψ(x)=i ∂_t ψ(x), withH_D=-iγ^0(γ⃗·e⃗_r)(h∂_r +h/r -√(h)/rK )and where K=2S⃗·L⃗ +1 = J^2 - L^2 + 1/4 is the spin-orbit operator <cit.>, whose eigenvalues κ are related to the ones of the total angular momentum operator (J) and of the orbital angular momentum (L) byκ={[ -(j+1/2)=-(l+1) for j=l+1/2;+(j+1/2)=l for j=l-1/2 ]. The radial part of the Dirac equation can be derived by searching for (particle-like) positive frequency solutions of energy E of the typeψ (x)= ψ_E,κ,m_j(t,r,θ,ϕ)==e^-iEt/r{f^+_E,κ(r)Φ^+_m_j,κ(θ,ϕ) +f^-_E,κ(r)Φ^-_m_j,κ(θ,ϕ) }One can show that the final form of the equations satisfied by the radial wave-functions f^±(r) (where for simplicity we have dropped the indices E and κ) read ([ 0 -h(r) d/ dr+κ/ r√(h(r));;h(r) d/ dr+κ/ r√(h(r)) 0 ]) ([ f^+(r); ; f^-(r) ])= E ([ f^+(r); ; f^-(r) ])2§ THE MASSLESS DIRAC EQUATION IN SCHWARZSCHILD AND REISSNER-NORDSTRÖM GEOMETRYAs already mentioned in the Introduction we are studying here only the scattering of massless fermions by black holes. For that we need first to find the scattering modes of the Dirac equation (<ref>) on which to apply the partial wave analysis (PWA) method. That will allow us to find the phase shifts and then to calculate all the physical quantities that are characteristic to the scattering phenomena. For PWA one only needs to know the asymptotic behaviour of the scattering modes. As showed bellow in the asymptotic region of both the Schwarzschild and RN back hole the Dirac equation (<ref>) can be brought to a simpler form that will allow us to solve it analytically. Furthermore, having analytical solution at our disposal will allow us to find analytical phase shifts.In the case of a Reissner-Nordström black hole the function h(r) entering the line element (<ref>) is defined byh(r)=1-2M/r+Q^2/r^2=(1-r_+/r)(1-r_-/r)where M is the mass of the black hole and Q the electric charge. The Cauchy (r_-) and black hole horizon (r_+) radii are easily found to ber_±=M±√(M^2-Q^2) (provided Q<M). If we make Q=0 in (<ref>) we obtain the Schwarzschild line element with r_-=r_+=r_0=2M.It proves useful to introduce a convenient Novikov-like dimensionless coordinate <cit.>x=√(r/r_+-1) ∈ (0,∞) Then the Dirac equation (<ref>) in the asymptotic region of the black hole becomes([ 1/ 2 d/ dx +κ/ x -ε(x+1/ x); ;ε(x+1/ x) 1/ 2 d/ dx -κ/ x ]) ([ f^+(x); ; f^-(x) ])=0where we denoted ε=r_+E. In obtaining (<ref>) we have used a Taylor expansion with respect to 1/x from which we neglected the O(1/x^2) terms and higher.After putting the terms proportional with x into diagonal form, using the transformation matrixM=√(ε)([ -ii;11;])that transforms (f^+, f^-)^T→ (f̂^+, f̂^-)^T=M^-1(f^+, f^-)^T, the final system of radial equations is obtained1/2df̂^+/dx- i ε(x+1/x)f̂^+ =κ/xf̂^-1/2df̂^-/dx+ i ε(x+1/x)f̂^- =κ/xf̂^+ The analytical solutions of the above equations can be found in terms of Whittaker M and W functions <cit.>f̂^+(x)=C_11/xM_ρ_+,s(2iε x^2) +C_21/xW_ρ_+,s(2iε x^2) f̂^-(x)=C_1s-i ε/κ1/xM_ρ_-,s(2iε x^2) - C_21/κ1/xW_ρ_-,s(2iε x^2)where the parameters s, ρ_± are related to κ and ε by the following relationss=√(κ^2-ε^2),ρ_±=∓1/2-iε These solution are the starting point for studying the scattering phenomena by Schwarzschild and Reissner-Nordström black hole with the help of partial wave analysis. The Whittaker functions M_ρ_±,s(2iε x^2)=(2iε x^2)^s+1/2[1+O(x^2)] are regular in x=0 (i.e. in r=r_+), while the Whittaker W_ρ_±,s(2iε x^2) are divergent as x^1-2s if s>1/2 <cit.>. As showed in the Appendices of ref. <cit.> and <cit.> one must impose the asymptotic condition C_2=0 in order to have elastic collisions with a correct Newtonian limit for large angular momentum. § SCATTERING CROSS SECTION AND PHASE SHIFTS The phase shifts that result after applying the partial wave analysis <cit.> on the scattering modes (<ref>) are defined byS_κ=e^2iδ_κ=(κ/s-iε) Γ(1+s-iε)/Γ(1+s+iε) e^iπ(l-s)The scattering amplitudes are defined by <cit.>f(θ)=∑_l=0^∞a_l P_l(cosθ) , g(θ)=∑_l=1^∞b_l P_l^1(cosθ)where a_l and b_l are the partial amplitudesa_l = 1/2ip[(l+1)(S_-l-1-1)+l(S_l-1)] b_l = 1/2ip(S_-l-1-S_l)Putting all together one gets the differential scattering cross sectiondσ/dΩ=|f(θ)|^2+|g(θ)|^2 In the next section we will give a selection of our key results for the scattering of massless fermions by Schwarzschild and Reissner-Nordström black holes. In the derivation of the plots we have used a method first proposed in <cit.> and further developed in <cit.> for improving the convergence of the partial wave series (<ref>). § RESULTS In Fig. <ref> we compare the scattering of massless (v=1) and massive (v≠1) fermions by a Schwarzschild black hole (q=0) for a fixed value of the (frequency) parameter ME. The case of massive fermion scattering by Schwarzschild and Reissner-Nordström black holes was studied in more detail in our previous papers <cit.>. Analyzing the differential cross section in the backward direction (near θ≈π) one can observe the presence of a minima in the scattering intensity. If the fermion is massive then the scattering intensity in the backward direction is higher compared with the massless case. Moreover, decreasing the fermion speed the minima will become eventually a maxima in the backward direction (see <cit.> for more details). < g r a p h i c s >(color online). Comparison between the scattering of massless (v=1) and massive (v≠1) fermions by a Schwarzschild black hole at ME=2.5. The presence of a minima in the backward direction can be observed. In optics the presence of a minima or maxima in the scattering intensity in the backward direction occurs when the deflection angle of a ray is a multiple of π. This is observed by the presence of a bright spot or hallo in the antipodal direction. If the orbit passes the scattering center multiple times, then spiral (or orbiting) scattering can occur. This can be seen by the presence of oscillations in the scattering intensity. As can be seen from Fig. <ref>-<ref> the phenomena of glory and spiral scattering also occurs in the case of massless fermion scattering by Schwarzschild and charged Reissner-Nordström black holes.In Fig. <ref> we plot the scattering intensity for a massless spinor wave of fixed frequency for a Schwarzschild black hole (q=0), a typical Reissner-Nordström black hole (with q=0.5,q=0.6,q=0.9) and respectively, an extremal Reissner-Nordström black hole (q=1). One can observe (very clearly for ME=3) that at a fixed frequency the glory width gets larger as the value of the charge-to-mass ratio q is increased. The same behaviour was also reported in <cit.> in the case of scattering of massless scalar waves by Reissner-Nordström black holes. In Ref. <cit.> the authors found that the linear mass density of the cosmic string produces a similar effect. Furthermore, the oscillations present in the scattering intensity become less frequent as we approach the extremal case q=1. This means that the spiral scattering becomes less important as the black hole gets more and more charge on it. < g r a p h i c s > < g r a p h i c s >(color online). Comparison between the massless fermion scattering cross section at fixed frequency ME=1.5 for q=0,q= 0.6, q=1 in one case and at ME=3 for q=0.5,q= 0.9, q=1 in the other case. Fixing the frequency the glory width gets larger as the value of the charge-to-mass ratio q is increased. In Fig. <ref> the differential scattering cross section for the massless fermion field is plotted, using a logarithmic scale, for different values of the incoming fermion frequency ME=2.5,3,3.5 at a chosen fixed value of the charge-to-mass ratio q=0 (Schwarzschild case), q= 0.5 (typical Reissner-Nordström) and q=1 (extremal Reissner-Nordström case). The first thing to observe is the fact that the width of the glory becomes narrower as the frequency increases. On the contrary the oscillations (indicating spiral scattering) present in the scattering intensity become more frequent as the value of ME is increased. This can be best seen for the extremal case q=1. < g r a p h i c s > < g r a p h i c s > < g r a p h i c s >(color online). Reissner-Nordström scattering intensity for ME=2.5, 3, 3.5 in the case of a black hole with no charge q=0, with charge q=0.5 and the extremal case with charge q=1. Increasing the frequency has as an effect the narrowing of the glory width. At the same time the oscillations in the scattering intensity become more frequent. Fig. <ref> shows the behaviour of the massless fermion differential scattering cross section at low frequency for a typical Reissner-Nordström black hole (q=0.5). In Fig. <ref> the extremal Reissner-Nordström case (q=1) is studied for a large variance of ME. We notice the absence of oscillations at very low frequency (ME=0.1) in the differential scattering cross section. However, as the value of ME is increased the spiral scattering and eventually glory start to occur. < g r a p h i c s >(color online). Differential scattering cross section for massless fermions at low frequencies (ME=0.4,0.6,0.8) for a typical Reissner-Nordström black hole with charge q=0.5.< g r a p h i c s >(color online). Scattering cross section for extremal Reissner-Nordström black hole (q=1) for ME=0.1,1,3. Increasing the value of ME, spiral scattering and glory start to occur in the scattering intensity.Exploring the parameter space (q,ME) we have found situations when two different sets of parameters present similar scattering patterns (see Fig. <ref>). In some cases (like in Fig. <ref>B) the scattering patterns are almost the same. For example the difference between the values in the scattering intensity of (0.6,1.1) and (1.0,0.2) curves is less than 17%. As a consequence one will need higher accuracy in the observed data in order to distinguish between the two data sets. < g r a p h i c s > < g r a p h i c s >(color online). The dependence of the scattering pattern with the black hole charge and with the frequency. One can observe the similarities between the scattering patterns for certain values in the parameter space (q,ME).§ CONCLUSIONS AND FINAL REMARKS In this paper we have studied the scattering of massless fermions by Schwarzschild and charged Reissner-Nordström black holes. We have showed that glory and spiral scattering phenomena could occur for both types of black holes analysed, similar to what happens in the case of massive fermion scattering by black holes <cit.>. As can be seen from Fig. <ref>-<ref> the scattering of massless fermions has always a minima in the backward direction (opposed to the massless scalar case <cit.>). However, if the fermion becomes massive (see Fig. <ref>), then this minima starts to increase and will become eventually a maxima <cit.>.The dependence of the scattering on the charge-to-mass ratio q=Q/M was analysed for typical values including the extreme case q=1. As showed in Fig. <ref> for a fixed value of ME the glory width gets larger as the value of q is increased. One the other hand, keeping fixed the value of q and varying the frequency ME we observe an increase in the number of oscillations present in the scattering intensity, as showed in Fig. <ref>.As already mentioned, at a fixed frequency, the glory pick is wider in the case of Reissner-Nordström black hole (q≠0) compared with a Schwarzschild black hole (q=0). As a consequence the glory phenomena will be more easily to observe it astronomically for a Schwarzschild black hole than for a charged Reissner-Nordström one. Moreover the glory for extremal Reissner-Nordström case is the hardest one for astronomy observation.We have used the parameter ME to label our figures. Restoring the units we can make the following dimensionless quantityϵ=GME/ħ c^3=π r_S/vλ_Cwhere r_S=2MG is the Schwarzschild gravitational radius, λ_C=h/p is the associated Compton wavelength of the particle and v=p/E it's speed (v=1 for massless fermions). One can interpret ϵ as a measure of the gravitational coupling. The results obtained in the previous sections show that glory and spiral scattering of massless fermion by black holes are significant when the gravitational coupling is of order of π. This implies that we must have r_S∼λ_C. Thus we can conclude that diffraction patterns of massless fermions (like the glory and spiral scattering) by black holes are significant if the condition r_S∼λ_C is fulfilled.Neutrinos have the smallest mass among the fermions known experimentally today. The current upper bound limit on the sum of the three known neutrinos is of ∑ m_ν < 0.183 eV <cit.>. If we assume the mass of the electron neutrino to be of m_ν_e∼0.01 eV, then the condition r_S∼λ_C implies a black hole mass of M∼10^22 kg, which is much smaller than the mass of an astrophysical stelar black hole M_BH∼ 10^31 kg. This means that neutrino glory and spiral scattering can be observed only for scattering by small black holes. Such types of primordial black holes could have been created in the very early universe. Another possible scenario for the existence of such small black holes is in the context of theories with large extra-dimensions <cit.>. In these circumstances the possibilities of observing and detecting diffraction patterns for massive fermion scattering by black holes are currently unavailable. However, in the case of existence of truly massless fermions (yet to be detected) we are no longer bound by the mass of the fermion (that as we saw constrains also the possible mass of the black hole), which means that glory and spiral scattering can in principle be observed for scattering of massless fermions (having appropriate energies) by real astrophysical black holes. § ACKNOWLEDGEMENTS I would like to thank Prof. I.I. Cotăescu for very useful discussions related to this subject, that helped to improve the manuscript and for suggesting to include Appendix A. I am also grateful to C. Crucean for discussions and for reading the manuscript. Last but not least, I would like to express my gratitude to the anonymous Referee for his/her comments and for suggesting to search for situations with different charge and frequency that present similar scattering patterns.This work was supported by a grant of the Ministry of National Education and Scientific Research, RDI Programme for Space Technology and Advanced Research - STAR, project number 181/20.07.2017.§.§ Appendix A Neutrino limit to the Dirac fieldThe aim of this Appendix is to show that the scattering of the Standard Model neutrino (which is a left-handed massless Dirac fermion) by black holes is contained in our results presented here regarding the scattering of massless Dirac fermions.The following combination of the (f^+_κ, f^-_κ) radial wave functionsf^L_κ=1/√(2)( f^+_κ - if^-_-κ)f^R_κ=1/√(2)( f^+_κ + if^-_-κ)correspond to the radial wave functions for a left-handed fermion, respectively a right-handed one.From eqs. (<ref>), (<ref>) and (f^+, f^-)^T=M(f̂^+, f̂^-) one getsf^+_κ =i√(ε)(f̂^-_κ - f̂^+_κ) =i√(ε)1/x[s-iε/κC_1^(κ)M_ρ_+,s(2iε x^2)- C_1^(κ)M_ρ_-,s(2iε x^2) ] f^-_-κ =√(ε)(f̂^-_-κ + f̂^+_-κ) =√(ε)1/x[s-iε/-κC_1^(-κ)M_ρ_+,s(2iε x^2)+ C_1^(-κ)M_ρ_-,s(2iε x^2) ]where we used the condition C_2=0 as already mentioned in the main text for obtaining the phase shifts (<ref>).Now by imposing the condition f^R_κ=0 and having in mind eq. (<ref>) together with the fact that M_ρ_+,s and M_ρ_-,s are linearly independent, we obtain in the end the following relationC_1^(κ)=C_1^(-κ) Calculating now f^L_κ using eq. (<ref>) we obtain in the endf^L_κ=√(2)f^+_κ The last equation tells us that by applying the partial wave analysis to the left-handed neutrino f^L_κ we will get the same phase shifts (<ref>) as obtained by applying the partial wave analysis to the fermion spinor f^+_κ.80mm0.1pt290 Ford K.W. Ford and J. A. Wheeler, Ann. Phys. (Leipzig) 7, 259 (1959).Darwin C. Darwin, Proc. R. Soc. 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"authors": [
"Ciprian A. Sporea"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170726110619",
"title": "Scattering of massless fermions by Schwarzschild and Reissner-Nordström black holes"
} |
[ Jorge Drumond Silva December 30, 2023 ======================= P-splines are penalized B-splines, in which finite order differences in coefficients are typically penalized with an ℓ_2 norm.P-splines can be used for semiparametric regression and can include random effects to account for within-subject variability. In addition to ℓ_2 penalties, ℓ_1-type penalties have been used in nonparametric and semiparametric regression to achieve greater flexibility, such as in locally adaptive regression splines, ℓ_1 trend filtering, and the fused lasso additive model. However, there has been less focus on using ℓ_1 penalties in P-splines, particularly for estimating conditional means.In this paper, we demonstrate the potential benefits of using an ℓ_1 penalty in P-splines with an emphasis on fitting non-smooth functions. We propose an estimation procedure using the alternating direction method of multipliers and cross validation, and provide degrees of freedom and approximate confidence bands based on a ridge approximation to the ℓ_1 penalized fit. We also demonstrate potential uses through simulations and an application to electrodermal activity data collected as part of a stress study. § INTRODUCTION Many nonparametric regression methods, including smoothing splines and regression splines, obtain point estimates by minimizing a penalized negative log-likelihood function of the form l_pen = -l(β) + λ P(β), where l is a log-likelihood, P is a penalty term, λ > 0 is a smoothing parameter, and β are the coefficients to be estimated. Typically, quadratic (ℓ_2 norm) penalties are used, which lead to straightforward computation and inference. In particular, ℓ_2 penalties typically lead to ridge estimators, which have both closed form solutions and are linear smoothers. The ℓ_2 penalty also has connections to mixed models, which allows the smoothing parameters to be estimated as variance components <cit.>.However, nonparametric regression methods that use an ℓ_1-type penalty, such as ℓ_1 trend filtering <cit.> and locally adaptive regression splines <cit.>, are better able to adapt to local differences in smoothness and achieve the minimax rate of convergence for weakly differentiable functions of bounded variation <cit.>, whereas ℓ_2 penalized methods do not <cit.>. The trade-off is that ℓ_1 penalties generally lead to more difficult computation and inference because the objective function is convex but non-differentiable, and the fit is no longer a linear smoother.In this article, we propose P-splines with an ℓ_1 penalty as a framework for generalizing ℓ_1 trend filtering within the context of repeated measures data and semiparametric (additive) models <cit.>. In Section <ref>, we discuss the connection between P-splines and ℓ_1 trend filtering which motivates the methodological development. In Section <ref>, we present our proposed model, and in Section <ref>, we discuss related work. In Section <ref>, we propose an estimation procedure using the alternating direction method of multipliers (ADMM) <cit.> and cross validation (CV). In Section <ref>, we derive the degrees of freedom and propose computationally stable and fast approximations, and in Section <ref>, we develop approximate confidence bands based on a ridge approximation to the ℓ_1 fit. In Section <ref>, we study our method through simulations and evaluate its performance in fitting non-smooth functions. In section <ref>, we demonstrate our method in an application to electrodermal activity data collected as part of a stress study. We close with a discussion in Section <ref>.§ P-SPLINES AND ℓ_1 TREND FILTERINGIn this section, we give brief background on P-splines and ℓ_1 trend filtering, and show the relation between them when the data are independent and identically distributed (i.i.d.) normal.P-splines <cit.> are penalized B-splines <cit.>. B-splines are flexible bases that are notable in part because they have compact support, which leads to banded design matrices and faster computation. This compact support can be seen in Figure <ref>, which shows eight evenly spaced first degree and third degree B-spline bases on [0,1]. We can define an order M (degree M-1) B-spline basis with j=1,…,p basis functions recursively as <cit.>ϕ_j^m(x)= x - t_j/t_j+m-1 - t_jϕ_j^m-1(x) + t_j+m - x/t_j+m - t_j+1ϕ_j+1^m-1(x),j = 1,…, 2M + c -m,1 < m ≤ Mϕ_j^1(x)= 1 t_j ≤ x < t_j+10otherwise,j = 1,…,2M +c -1where t_j are the knots, division by zero is taken to be zero, and c is the number of internal knots. For order M B-splines defined on the interval [x_min, x_max], in order to obtain j=1,…,p basis functions, we set 2M boundary knots (M knots on each side) and c=p-M interior knots. In general, one can set t_1 ≤ t_2 ≤⋯≤ t_M = x_min < t_M+1 < ⋯ < t_M+c < x_max = t_M+c+1≤ t_M+c+2≤⋯≤ t_2M + c. In order to ensure continuity at the boundaries, we set t_1 < t_2 < ⋯ < t_M-1 < t_M = x_min and x_max = t_M+c+1 < t_M+c+2 < ⋯ < t_2M + c. We also use equally spaced interior knots, which is important for the P-spline penalty, and drop the superscript on ϕ designating order when the order does not matter or is stated in the text.B-spline bases can be used to fit nonparametric models of the form y(x) = f(x) + ϵ(x), where y(x) is the outcome y at point x, f(x) is the mean response function at x, and ϵ(x) is the error at x. To that end, let y=(y_1,…,y_n)^T be an n × 1 vector of outcomes and x = (x_1 …, x_n)^T be a corresponding n × 1 vector of covariates. Also, let ϕ_1,…,ϕ_p be B-spline basis functions and let F be an n × p design matrix such that F_ij = ϕ_j(x_i), i.e., the j^th column of F is the j^th basis function evaluated at x_1,…,x_n. Equivalently, the i^th row of F is the i^th data point evaluated by ϕ_1,…,ϕ_p. For i.i.d. normal y, a simple linear P-spline model with the standard ℓ_2 penalty can be written asβ̂_0, β̂ = _β_0 ∈ℝ, β∈ℝ^p1/2y - β_0 1 - Fβ_2^2 + λ/2D^(k+1)β_2^2,where β_0 is the intercept, β is a p × 1 vector of parameter estimates, 1 is an n × 1 vector with each element equal to 1, λ>0 is a smoothing parameter, and D^(k + 1)∈ℝ^(p-k-1) × p is the k+1 order finite difference matrix. For example, for k=1D^(2)= [1 -21 ; 1 -21;⋱⋱⋱; 1 -21;1 -21 ]∈ℝ^(p-2) × pIn general, as described by <cit.>, D^(k+1) = D^(1) D^(k) where D^(1) is the (p-k-1) × (p-k) upper left matrix of:D^(1) =[ -11 ;-11 ;⋱⋱;-11 ]∈ℝ^(p-1) × p.Our proposed model builds on one in which the ℓ_2 penalty in (<ref>) is replaced with an ℓ_1 penalty:β̂_0, β̂ = _β_0 ∈ℝ, β∈ℝ^p1/2y - β_0 1 - Fβ_2^2 + λD^(k+1)β_1. Letting f(x) = ∑_j=1^p β_j ϕ_j^M(x), for order M=4 B-splines, <cit.> show that ∫_x_min^x_max( d^2/dx^2 f(x) )^2 dx = c_1 D^(2)β_2^2 + c_2 ∑_j=4^p ∇^2 β_j ∇^2 β_j-1where ∇^2 is the second-order backwards difference and c_1 and c_2 are constants. As shown in Appendix <ref>, a similar result holds for P-splines with an ℓ_1 penalty. In particular, for 0 ≤ k < M-1,∫_x_min^x_max| d^k+1/dx^k+1 f(x) | dx ≤ C_M,k+1D^(k+1)β_1where C_M, k+1 is a constant given in Appendix <ref> that depends on the order M of the B-splines and order k+1 of the finite difference. In other words, controlling the ℓ_1 norm of the (k+1)^th order finite differences in coefficients also controls the total variation of the k^th derivative of the function.ℓ_1 trend filtering is similar to (<ref>). In the case where x_1 < x_2 < ⋯ <x_n are unique and equally spaced, ℓ_1 trend filtering solves the following problem (the intercept is handled implicitly):β̂ = _β∈ℝ^n1/2y - β_2^2 + λD^(k+1)β_1.Problem (<ref>) differs from (<ref>) in that (<ref>) has one parameter per data point, and the design matrix is the identity matrix. D^(k+1) is also resized appropriately by replacing p with n in the dimensions of (<ref>) and (<ref>). However, under certain conditions noted in Observation <ref>, (<ref>) and (<ref>) are identical.[Continuous representation] For second order (first degree) B-splines with n basis functions, equally spaced data x_1 < x_2 < ⋯ < x_n with knots at t_1 < x_1, t_2 = x_1, t_3=x_2,…,t_n = x_n-1, t_n+1 = x_n, t_n+2 > x_n, and centered outcomes such that y(0) = 0, P-splines with an ℓ_1 penalty are a continuous analogue to ℓ_1 trend filtering. Under these conditions, for i=1,…,nϕ^2_j(x_i) = 1 i = j 0otherwise.To see this, note thatϕ_j^2(x_i)= x_i - t_j/t_j+1 - t_jϕ_j^1(x_i) + t_j+2 - x_i/t_j+2 - t_j+1ϕ_j+1^1(x_i) = t_i+1 - t_j/t_j+1 - t_jϕ_j^1(t_i+1) + t_j+2 - t_i+1/t_j+2 - t_j+1ϕ_j+1^1(t_i+1).Now, ϕ_j^1(t_i+1) = 1 t_j ≤ t_i+1 < t_j+10otherwiseandϕ_j+1^1(t_i+1) = 1 t_j+1≤ t_i+1 < t_j+20otherwise.We have ϕ_j^1(t_i+1) = 1 for i=j-1 and 0 otherwise, but for i=j-1, we have t_i+1 - t_j = t_j - t_j = 0. We also have ϕ_j+1^1(t_i+1)=1 for i=j and 0 otherwise, and for i=j, we have t_j+2 - t_i+1 = t_j+2 - t_j+1 > 0. It follows that for i=1…,n, (<ref>) evaluates to 1 if i=j and 0 otherwise.Let F be the design matrix in (<ref>), where F_ij = ϕ^2_j(x_i). Then from the previous result, we have F=I_n, where I_n is the n × n identity matrix. This, together with the assumption that β_0 = y(0) = 0, implies that the objective functions (<ref>) and (<ref>) are identical, which proves Observation <ref>. We note that <cit.> shows that ℓ_1 trend filtering has a continuous representation when expressed in the standard lasso form, and Observation <ref> gives a continuous representation of ℓ_1 trend filtering when expressed in generalized lasso form.ℓ_1 trend filtering can be applied to irregularly spaced data, such as with the algorithm developed by <cit.>. It might also be possible to extend ℓ_1 trend filtering to repeated measures data to account for within-subject correlations. However, due to Observation <ref>, we think it is beneficial to view ℓ_1 trend filtering as a special case of P-splines with an ℓ_1 penalty. We think this approach has the potential to be a general framework, because higher order B-splines could be used in combination with different order difference matrices just as can be done with P-splines that use the standard ℓ_2 penalty. Furthermore, expressing ℓ_1 trend filtering as P-splines with an ℓ_1 penalty may facilitate the development of confidence bands (see Section <ref>), which could help to fill a gap in the ℓ_1 penalized regression literature.In addition, there are connections between P-splines with an ℓ_1 penalty and locally adaptive regression splines. In particular, as <cit.> shows, the continuous analogue of ℓ_1 trend filtering is identical to locally adaptive regression splines <cit.> for k=0,1, and asymptotically equivalent for k ≥ 2.§ PROPOSED MODEL: ADDITIVE MIXED MODEL USING P-SPLINES WITH AN ℓ_1 PENALTYTo introduce our model, let y_i = (y_i1, …, y_i n_i)^T be an n_i × 1 vector of responses for subject i=1,…, N, and let y = (y_1^T, …, y_N^T)^T be the stacked n × 1 vector of responses for all N subjects, where n= ∑_i=1^N n_i. Let x_i = (x_i1, …, x_i n_i)^T be a corresponding n_i × 1 vector of covariates for subject i, and x = (x_1^T,…,x_N^T)^T be the n × 1 stacked vector of all covariate values. In many contexts, x is time. To account for the within-subject correlations of y_i, we can incorporate random effects into the P-spline model. To that end, let Z_i be an n_i × q_i design matrix for the random effects for subject i (possibly including a B-spline basis), and let b_i = (b_i1, …, b_iq_i)^T be the corresponding q_i × 1 vector of random effect coefficients for subject i. Also, letZ = [ Z_1; ⋱; Z_N ]be the n × q block diagonal random effects design matrix for all subjects, where q=∑_i=1^N q_i, and let b = (b_1^T,…,b_N^T)^T be the q × 1 stacked vector of random effects for all subjects. We propose an additive mixed model with j=1,…,J smooths (tildes denote quantities that will be subject to additional constraints, as described below):β_0 ∈ℝ, b∈ℝ^q, β̃_j ∈ℝ^p_j, j = 1,…,Jminimize 1/2y - β_0 1 - ∑_j=1^J F̃_j β̃_j- Z b_2^2 + ∑_j=1^J λ_j D̃_j^(k_j+1)β̃_j _1 + τ1/2b^T S bwhere F̃_j is a n × p_j design matrix of B-spline bases for smooth j, D̃_j^(k_j+1) is the k_j + 1 finite difference matrix, and σ^2_b S is the covariance matrix of the random effects b. For example, if b are random intercepts, then S=I_N and Z would be an n × N matrix such that Z_il = 1 if observation i belonged to subject l and zero otherwise. Alternatively, to obtain random curves using smoothing splines and a B-spline basis, we could setS = [ S_1; ⋱; S_N ]where S_j, il = ∫ϕ”_ji(t) ϕ”_jl(t) dt, and ϕ”_j1,…,ϕ”_jp_j are the second derivatives of the B-spline basis functions for the j^th smooth. We would then set Z to be the corresponding B-splines evaluated at the input points.We note that (<ref>) includes varying-coefficient models <cit.>. For example, as pointed out by <cit.>, if F̃_1 are B-splines evaluated at x, we could have F̃_2 = diag(x') F̃_1, where x'x is another covariate vector and diag(x') is a diagonal matrix with x'_i at the i^th leading diagonal position.As written, (<ref>) is not generally identifiable. To see this, suppose ŷ(x) = β̂_0 + f̂_1(x) + f̂_2(x), where neither f_1 nor f_2 are varying-coefficient terms. Then letting f̂'_1(x) = f̂_1(x) + δ and f̂'_2(x) = f̂_2(x) - δ for δ∈R, we also have ŷ(x) = β̂_0 + f̂'_1(x) + f̂'_2(x). To make (<ref>) identifiable, we follow <cit.> and introduce a centering constraint on each non-varying coefficient smooth, i.e. ∫f̂_j(x) dx = 0 for all smooths j=1,…, J such that F̃_j diag(x')F̃_l for some x' and lj. To this end, let ℰ = {j ∈{1,…, J} : F̃_j diag(x')F̃_lfor some x', lj} be the indices of the non-varying coefficient smooths, and let ℰ̅ = {j ∈{1,…, J} : j ∉ℰ} be its complement. We constrain 1^T F̃_j β̃_j = 0 for j ∈ℰ. We accomplish this by defining new p_j × (p_j - 1) orthonormal matrices Q_j, j=1, …, J, such that 1^T F̃_j Q_j = 0. If desired, one can also define a q × (q-1) matrix Q_J+1 such that 1^T Z Q_J+1 = 0. However, this last centering constraint is not necessary, because the penalty on the random effect terms pulls the coefficients themselves towards zero, as opposed to the finite order differences in coefficients.As <cit.> shows, Q can be obtained by taking the QR decomposition of F̃^T_j 1 and retaining the last p_j -1 columns of the left orthonormal matrix.[The matrices 1^T F̃_j, j=1,…, J are of rank 1, so the remaining p_j-2 columns are arbitrary orthonormal vectors. In<cit.>, when taking the QR decomposition of F̃^T 1, an appropriate matrix Q can be obtained as .] We can then re-parameterize the p_j constrained parameters β̃_j in terms of the p_j-1 unconstrained parameters β_j, such that β̃_j = Q_j β_j. For j ∈ℰ, let F_j = F̃_j Q_j and D_j = D̃_j^(k_j+1)Q_j. For j ∈ℰ̅, let F_j = F̃_j and D_j = D̃^(k_j+1)_j. If centering the random effects, then we redefine S := Q_J+1^T S Q_J+1 and Z := Z Q_J+1. Then we can re-write (<ref>) in the identifiable formβ_0 ∈ℝ, b∈ℝ^q, β_j ∈ℝ^p'_j, j = 1,…,Jminimize 1/2y - β_0 1 - ∑_j=1^J F_j β_j- Z b_2^2 + ∑_j=1^J λ_j D_j β_j _1 + τ1/2b^T S bwhere p_j' = p_j - 1 for j ∈ℰ and p_j' = p_j for j ∈ℰ̅.We note that the penalty matrix S given above for random subject-specific splines defines non-zero correlation between nearby within-subject random effect coefficients. This is in contrast to the approach of <cit.> for estimating subject-specific random curves, which focuses on the case in which nearby within-subject coefficients are not correlated. To see this, let d̂_i(x) = ∑_j=1^q_ib̂_ijϕ_ij(x) be the estimated difference between the i^th subject-specific curve and the marginal mean at point x. The smoothing spline approach above constrains ∫ (d̂”)^2(x)dx = b_i^T S_i b_i <C for some constant C>0, whereas the approach of <cit.> constrains b_i^T I_q_ib_i = ∑_j=1^q_ib̂_j^2 <C. Whereas the non-diagonal penalty matrix S implies correlations between nearby coefficients, the identity matrix in the approach of <cit.> implies zero correlation.Similar to the equivalence between Bayesian models and ℓ_2 penalized smoothing splines <cit.>, there is an equivalence between Bayesian models and ℓ_1 penalized splines. In particular, (<ref>) is equivalent to the following distributional assumptions, which we can use to obtain Bayesian estimates:y|b = β_0 1 + ∑_j=1^J F_j β_j + Z b + ϵ ϵ ∼ N ( 0, σ^2_ϵ I_n )b ∼ N(0, σ^2_b S^-1)for σ^2_b = σ^2_ϵ / τ ϵ ⊥b ( D_j β_j )_l∼Laplace(0, a_j)fora_j = σ^2_ϵ / (2 λ_j), l=1,…, p_j - k_j - 1, j=1,…,JThe last distributional assumption is an element-wise Laplace prior on the k_j+1 order differences in coefficients.In some cases, the random effects penalty matrix S may be positive semidefinite but not invertible. For example, the smoothing spline random curves outlined above lead to a penalty matrix S that is not strictly positive definite, but that is still positive semidefinite. This does not cause problems for the ADMM algorithm, but some changes are required for other algorithms as well as for Bayesian estimation. Following <cit.>, let S=U Λ U^T be the eigendecomposition of a positive semidefinite matrix S, where U U^T = I_q and Λ is a diagonal matrix with eigenvalues in descending order in the diagonal positions. Let b̆ = U^T b and Z̆ = Z U, so that b^T S b = b̆^T Λb̆ and Z̆b̆ = Z b. Let q_r be the number of strictly positive eigenvalues of S, where 0 < q_r < q, and let Λ_r be the q_r × q_r upper left portion of Λ. We can partition b̆ as b̆ = (b̆_r^T, b̆_f^T)^T, where b̆_r^T is a q_r × 1 vector of penalized coefficients and b̆_f^T is a q_f × 1 vector of unpenalized coefficients, where q_r + q_f = q. Then b̆^T Λb̆ = b̆^T_r Λ_r b̆_r, and it follows that b̆_r ∼ N(0, σ^2_bΛ_r^-1) and b̆_f ∝1.However, allowing for unconstrained random effect parameters leads to identifiability issues. Therefore, in practice if q_f > 0, we recommend using a normal or Cauchy prior on b̆_f. In particular, b̆_f,l∼ N(0, σ_f) or b̆_f,l∼Cauchy(0, σ_f), l=1,…, q_f with either a diffuse prior on σ_f and constraints to ensure σ_f > 0, or a diffuse prior on log(σ_f) without constraints. The Cauchy prior may be a preferable first choice, as it provides a weaker penalty and is similar to the recommendations of <cit.> for logistic regression. However, in some cases, such as in Section <ref>, it is necessary to use a normal prior.To further improve the computational efficiency of Monte Carlo sampling methods, we can partition Z̆ into Z̆ = [Z̆_r, Z̆_f] where Z̆_r contains the first q_r columns of Z̆ and Z̆_f contains the remaining q_f columns. We then set b̌_r = Λ_r^-1/2b̆_r and Ž_r = Z̆_r Λ_r^1/2, so that Ž_r b̌_r = Z̆_r b̆_r and b̌_r ∼ N (0, σ^2_b I), which allows for more efficient sampling <cit.>.§ RELATED WORKThere are many nonparametric and semiparametric methods for analyzing repeated measures data. For an overview, please see <cit.>. However, most existing methods use an ℓ_2 penalty <cit.>. Focusing on the optimization problem, our method puts a generalized lasso penalty <cit.> on the fixed effects and a quadratic penalty on the random effects. Unlike the elastic net <cit.>, we do not mix the ℓ_1 and ℓ_2 penalties on the same parameters, though this could be done in the future.The additive model with trend filtering developed by <cit.> is similar to our approach. <cit.> optimize θ_1,…, θ_J ∈ℝ^nminimize1/2y - y̅1 - ∑_j=1^J θ_j _2^2 + λ∑_j=1^J D^(k+1)θ_j _1 subject to1^T θ_j = 0, j=1,…,J.In contrast to (<ref>), (<ref>) has one smoothing parameter and constrains all smooths to be zero-centered. From Observation <ref>, we see that (<ref>) is equivalent to (<ref>) when there are is J=1 smooth and no random effects, in which case there would be only one smoothing parameter λ and no varying-coefficient smooths.<cit.> develop the theoretical and computational aspects of additive models with trend filtering, including the extension of the falling factorial basis to additive models. Similar to the B-spline basis, the falling factorial basis allows for linear time multiplication and inversion, which leads to fast computation <cit.>.When smooths j=1,…,J are expected to have similar degrees of freedom and n is not large enough to require dimension reduction, then (<ref>) with the addition of random effects and the relaxation of the zero-constraints for varying-coefficient smooths may be a viable alternative to (<ref>) that could potentially adapt better to local differences in smoothness because it would have one knot per data point.While not developed for analyzing repeated measures, the fused lasso additive model (FLAM) <cit.> is also similar to (<ref>). FLAM optimizes the following problem:θ_0 ∈ℝ, θ_j ∈ℝ^n, 1 ≤ j ≤ Jminimize1/2y - θ_0 1 - ∑_j=1^J θ_j _2^2 + αλ∑_j=1^J D^(1)θ_j _1 + (1-α) λ∑_j=1^J θ_j _2where 0 ≤α≤ 1 specifies the balance between fitting piecewise constant functions (α = 1) and inducing sparsity on the selected smooths (α = 0). From Observation <ref>, we see that (<ref>) is equivalent to our model (<ref>) when: α = 1, there is J=1 smooth, our design matrix has p=n columns, our B-spline bases have appropriately chosen knots, and our model has no random effects. As <cit.> show, FLAM can be a very useful method for modeling additive phenomenon, and as with the fused lasso <cit.>, jumps in the piecewise linear fits have the advantage of being interpretable.We also mention the sparse additive model (SpAM) <cit.> and sparse partially linear additive model (SPLAM) <cit.>. SpAM fits an additive model and uses a group lasso penalty <cit.> to induce sparsity on the number of active smooths. SPLAM fits a partially linear additive model and uses a hierarchical group lasso penalty <cit.> to induce sparsity in the selected predictors and to control the number of nonlinear features.One notable difference between our model and that of<cit.>, as well as FLAM, SpAM, and SPLAM, is that we allow for multiple smoothing parameters. In our applied experience with additive models and standard ℓ_2 penalties, we have found that in practice it can be important to allow for multiple smoothing parameters, particularly when the quantities of interest are the individual smooths as opposed to the overall prediction. This is equivalent to allowing each smooth to have different variance. However, this flexibility comes at a cost: estimating multiple smoothing parameters is currently the greatest challenge in fitting our proposed model. Perhaps due in part to these computational difficulties, several other authors also assume a single smoothing parameter in high-dimensional additive models <cit.>.There are fast and stable methods for fitting multiple smoothing parameters for ℓ_2 penalties paired with exponential family and quasilikelihood loss functions, notably the work of <cit.> using generalized cross validation (GCV) and <cit.> using restricted maximum likelihood. Furthermore, <cit.> extend these methods to larger datasets, and <cit.> extend these methods to likelihoods outside the exponential family and quasilikelihood form. However, similarly computationally efficient methods do not yet exist for fitting multiple smoothing parameters for ℓ_1 penalties.In addition to allowing for multiple smoothing parameters, we also propose approximate inferential methods, which is not typically provided for ℓ_1 penalized models. <cit.>, <cit.>, <cit.>, and <cit.> focus on prediction and provide bounds on the prediction risk and related quantities. These are important results, and we think that distributional results for individual parameters and smooths will also be useful to practitioners.We also note that <cit.> and <cit.> discuss a variant of P-splines for quantile regression, in which the ℓ_1 norm is used in both the loss and penalty function. However, we are not aware of existing P-spline methods that combine an ℓ_1 penalty with an ℓ_2 loss function.§ POINT ESTIMATION §.§ Regression parameters and random effectsTo fit (<ref>), we use the alternating direction method of multipliers (ADMM) <cit.>. ADMM has the advantage of being scalable to large datasets. To formulate (<ref>) for ADMM, we introduce constraint terms w_j and re-write the optimization problem asminimize1/2y - β_0 1 - ∑_j=1^J F_j β_j -Zb_2^2 + ∑_j=1^J λ_j w_j _1 + τ/2b^T S b subject to D_j β_j - w_j = 0, j = 1,…, J The augmented Lagrangian in scaled form (using u to denote the scaled dual variable) isL_ρ(β, b, w, u)∝1/2y - β_0 1 - ∑_j F_j β_j - Z b_2^2 + ∑_j λ_j w_j _1+ ρ/2∑_jD_j β_j - w_j + u_j _2^2 + τ/2b^T S bwhere ρ > 0 is the penalty parameter. The dimensions arey∈ℝ^n × 1, β_0 ∈ℝ, F_j ∈ℝ^n × p'_j, β_j ∈ℝ^p_j' × 1, Z ∈ℝ^n × q, b∈ℝ^q × 1, D_j ∈ℝ^(p_j-k_j-1) × p'_j, w_j ∈ℝ^(p_j-k_j-1) × 1, u_j ∈ℝ^(p_j-k_j-1) × 1, andS ∈ℝ^q × q, where p'_j = p_j - 1 if j ∈ℰ (non-varying coefficient smooths) and p'_j = p_j if j ∈ℰ̅ (varying coefficient smooths).ADMM is an iterative algorithm, and we re-estimate the parameters for updates m=1,2,… until convergence.[We use m to denote the iteration of the ADMM algorithm. This is unrelated to our use of m in Section <ref> to denote the order of the B-spline basis.] It is straightforward to derive the m+1 updates <cit.>:β_0^m+1 = 1/n1^T ( y - ∑_j F_j β_j^m - Z b^m )β^m+1_j:= _β_j L_ρ(β_0^m+1, β_j, β^m+1_l<j, β^m_l>j, b^m, w^m, u^m) = (F_j^T F_j + ρ D_j^T D_j )^-1(F_j^T y^(j,m) + ρ D_j^T(w^m_j - u^m_j) )b^m+1 := _b L_ρ(β_j=1,…, J^m+1, b, w^m, u^m) = (Z^T Z + τ S)^-1Z^T(y - β_0^m+11 - ∑_j F_j β^m+1_j)w_j^m+1 := _w_j L_ρ(β^m+1_j=1,…,J, b^m+1, w_j, u^m) = ψ_λ_j/ρ(D_j β_j^m+1 + u_j^m)u_j^m+1 := u_j^m + D_j β_j^m+1 - w_j^m+1where y^(j,m) = y - β_0^m+11 - ∑_l < j F_l β^m+1_l - ∑_l > j F_j β^m_l - Zb^m and ψ_λ/ρ is the element-wise soft thresholding operator, where for a single scalar element xψ_λ/ρ(x) = x - λ/ρx > λ/ρ0 |x| ≤λ / ρx + λ / ρx < -λ / ρTo initialize the algorithm, we set β_0 := y̅, b := 0, and β_j := 0, w_j := 0, and u_j := 0, for j = 1,…, J.As an alternative to the closed-form update (<ref>) for the random effects, it is also possible to update the random effects via a linear mixed effects (LME) model that is embedded into the ADMM algorithm. In particular, an LME model is fit to the residuals y - β_0^m+11 - ∑_j F_j β^m+1_j, and b^m+1 are updated as the best linear unbiased predictors (BLUPs). This update occurs at each step of the ADMM algorithm and replaces the update given by (<ref>). The LME update has the additional benefit of simultaneously estimating the variance of the random effects σ^2_b. In simulations, we have found that using an LME update leads to more accurate estimates of σ^2_b, which is important for subsequent estimates of degrees of freedom and confidence intervals.For stopping criteria, we use the primal and dual residuals (r^m and s^m, respectively):r^m= [ D_1 β_1^m - w_1^m; ⋮; D_J β_J^m - w_J^m ]∈ℝ^(p-k - J ) × 1s^m= -ρ[ D_1^T (w_1^m - w_1^m-1); ⋮; D_J^T (w_J^m - w_J^m-1) ]∈ℝ^p × 1where k=∑_j=1^J k_j, p = ∑_j=1^J p_j - |ℰ|, and |ℰ| is the cardinality of ℰ.Following the guidance of <cit.>, we stop when r^m_2 ≤ϵ^pri and s^m_2 ≤ϵ^dual, whereϵ^pri = ϵ^abs√(p - k - J) + ϵ^relmax{ D_1 β_1^m⋮D_J β_J^m _2, w^m_1⋮ w^m_J _2 } ϵ^dual = ϵ^abs√(p) + ϵ^relρ D_1^T u_1^m⋮D_J^T u_J^m _2.By default, we set ϵ^rel = ϵ^abs = 10^-4 and the maximum number of iterations at 1,000. §.§ Smoothing parametersTo estimate λ_1,…,λ_J we compute cross validation (CV) error for a path of values one smoothing parameter at a time. In the CV, we split the sample at the subject level, as opposed to individual observations, and ensure that there are at least two subjects in each fold per unique combination of factor covariates. First, we estimate a path for τ with λ_1,…, λ_J set to 0. Then we fix τ at the value that minimizes CV error and compute a path for λ_1, setting it to the value that minimizes CV error, and so on.We fit a path for each λ_j from λ_j^max to 10^-5λ_j^max evenly spaced on the log scale, where λ_j^max is the smallest value at which D_j β_j = 0. As shown in Appendix <ref>,λ_j^max = (D_j D_j^T)^-1 D_j (F_j^T F_j)^-1 F_j^T r_j_∞, where r_j = y - β_0 1 - ∑_ℓ j F_ℓβ_ℓ- Z b are the j^th partial residuals and for a vector a, a_∞ = max_j |a_j|. We also use warm starts, passing starting values separately for each fold, though warm starts appear to be minimally beneficial with ADMM. We set ρ = min( max ( λ_1, …, λ_J), c) at each iteration for some constant c>0 (e.g. c=5). When the number of smooths J is small (e.g. J ≤ 2) a grid search is also feasible.To estimate τ, we can either use CV and the close-form update given by (<ref>), or an LME update that is embedded in the ADMM algorithm, as described in Section <ref>. In simulations, we have found that the overall computation time to estimate the smoothing parameters is greater when using the LME update, and that the estimates of λ_1, …, λ_J do not appear sensitive to updates for b. However, the final estimates of σ^2_b, and consequently the width of confidence intervals can be improved by using the LME update. Consequently, we recommend using cross validation to estimate τ for the purposes of then estimating λ_1,…,λ_J, but using an LME update when estimating the final model.With both the closed-form and LME update, we cannot use the training sample to estimate the random effect parameters b for the test sample, because these parameters are subject-specific and the test subjects are not included in the training sample. Instead, we use the training sample to obtain estimates for the fixed effect parameters β_0, β_j, j=1,…,J and then use the test sample to estimate the random effects.To make our approach clear, we first fix notation. Let 𝒯^r ⊆{1,…,n} be the row indices for the observations in the test sample for both the fixed and random effect design matrices F_j, j=1,…,J, and Z. Also, let 𝒯^c ⊆{1,…,q} be the column indices of Z for observations in the test sample, and let 𝒯 = (𝒯^r, 𝒯^c) be the tuple of row and column indices designating the test sample. Let matrices F_j,𝒯 and F_j,-𝒯 be matrix F_j with only rows indexed by 𝒯^r retained and removed, respectively. Similarly, let matrices Z_𝒯 and Z_-𝒯 be matrix Z with only rows and columns indexed by 𝒯^r and 𝒯^c, respectively, retained and removed, respectively. Let matrices S_𝒯 and S_-𝒯 be matrix S with only rows and columns indexed by 𝒯^c retained and removed, respectively. Also, let y_𝒯 and y_-𝒯 be vector y with elements indexed by 𝒯^r retained and removed, respectively.We obtain out-of-sample marginal estimates as μ̂_𝒯 = β̂_0 1 + ∑_j=1^J F_j,𝒯β̂_j, where β̂_0 and β̂_j, j=1,…,J are estimated with y_-𝒯, F_j, -𝒯, and Z_-𝒯. If using the closed-form update (<ref>), we estimate subject-specific random effects as b̂_𝒯 = ( Z_𝒯^T Z_𝒯^T + τ S_𝒯)^-1 Z_𝒯^T (y_𝒯 - μ̂_𝒯) and obtain the out-of-sample prediction residuals as r_𝒯 = y_𝒯 - μ̂_𝒯 - Z_𝒯b̂_𝒯. Letting 𝒯_k be the tuple of indices for test sample (fold) k=1,…,K, we obtain the CV error as ∑_k=1^K r_𝒯_k_2^2.§ DEGREES OF FREEDOMIn this section, we obtain the degrees of freedom, with the primary goal of estimating variance (see Section <ref>). However, we note that degrees of freedom does not always align with a model's complexity in terms of its tendency to overfit the data <cit.>.In each of the approaches described in this section, the degrees of freedom (df) is a function of the smoothing parameters λ_1,…λ_J and τ. We always obtain the fixed effects smoothing parameters λ_1,…,λ_J from CV, but when using an LME update for the random effects b as described in Sections <ref> and <ref>, we do not directly obtain τ. Consequently, we cannot directly apply the results in this section to estimate df. However, from (<ref>), we have that τ = σ^2_b / σ^2_ϵ. Writing df = df(τ), and letting r = y - ∑_j=1^J F_j β̂_j - Z b̂ be an n × 1 vector of residuals and σ̂^2_ϵ = r_2^2 / (n - df(τ)) be an estimate of variance, we have thatτ̂ = σ̂^2_b/σ̂^2_ϵ = σ̂^2_b/r_2^2(n - df(τ̂) ).Therefore, lettingψ(τ) = τ - σ̂^2_b/r_2^2(n - df(τ) ),we numerically solve for τ̂ such that ψ(τ̂) = 0 and set df = df(τ̂). §.§ Stein's method Let g(y) = ŷ, where g : ℝ^n →ℝ^n is the model fitting procedure. For y∼ N(μ, σ^2I), the degrees of freedom is defined as <cit.>df = 1/σ^2∑_i=1^n(g_i(y), y_i).As <cit.> notes, (<ref>) is motivated by the fact that the risk Risk(g) = 𝔼g(y) - μ_2^2 can be decomposed asRisk(g) = 𝔼 g(y) - y_2^2 - n σ^2 + 2 ∑_i=1^n (g_i(y), y_i).Therefore, the degrees of freedom (<ref>) corresponds to the difference between risk and expected training error. Furthermore, if g is continuous and weakly differentiable, then df = 𝔼[∇· g(y)] <cit.> where ∇· g = ∑_i=1^n ∂ g_i / ∂ y_i is the divergence of g. Therefore, an unbiased estimate of df (also used in Stein's unbiased risk estimate <cit.>) is d̂f̂ = ∑_i=1^n ∂ g_i / ∂ y_i. To obtain an estimate of degrees of freedom, we transform the generalized lasso component of our model to standard form, similar to the approach of <cit.>. To do so, we use the following matrices described by <cit.>. LetD̃^*_j = [ D̃^(0)_j,1;⋮; D̃^(k_j)_j,1; D̃^(k_j+1)_j ]∈ℝ^p_j × p_jbe an augmented finite difference matrix, where D̃^(i)_j,1 is the first row of the finite difference matrix D̃^(i)_j, and D̃^(0)_j = I_p_j is the identity matrix.As shown by <cit.>, the inverse of D̃^*_j is given by M_j = M_j^(0) M_j^(1)⋯ M_j^(k) where[We denote the inverse matrix as M_j. This is unrelated to our use of M in Section <ref> to denote the order of the B-spline basis.]M_j^(i) =[ I_i; L_(p_j - i) × (p_j - i) ]∈ℝ^p_j × p_j,where L_(p_j - i) × (p_j - i) is the (p_j - i) × (p_j - i) lower diagonal matrix of 1s.Assuming our outcome y is centered, so that β_0 = y(0) = 0, and letting V_j = F̃_j M_j, D_j^* = D̃_j^* Q_j for j ∈ℰ and D_j^* = D̃_j^* for j ∈ℰ̅, and α_j = D^*_j β_j, we can write the penalized log likelihood (<ref>) as l_pen = 1/2y - ∑_j V_j α_j - Z b_2^2 + ∑_j=1^J λ_j ∑_l=k_j+2^p_j |α_jl | + 1/2τb^T S b. To avoid difficulties later differentiating with respect to the ℓ_1 norm, we remove the non-active ℓ_1 penalized coefficients from (<ref>). We also form the concatenated design matrix V = [V_1,…,V_J] and will need to index the active set of V. To these ends, let 𝒜_j = {l ∈{k_j+2, …, p'_j} : α̂_j,l 0 } be the active set of the penalized coefficients for smooth j, and let 𝒜_j^* = {1,…,k_j + 1 }∪𝒜_j be the active set for smooth j augmented with the unpenalized coefficients. Also, for a set 𝒜_j and constant c ∈ℝ, let 𝒜_j + c = {i + c : i ∈𝒜_j } be the set of elements in 𝒜_j shifted by c. Now let 𝒜^* = ⋃_j=1^J (𝒜^*_j + ∑_l =0^j-1 p'_l) be the augmented active set of V, where p'_0 = 0 and p'_j, j=1,…,J are the number of columns in V_j (equivalently F_j). Finally, let V_𝒜^* be matrix V subset to retain only those columns indexed by 𝒜^*. Similarly, let α̂ = (α̂_1^T, …, α̂_J^T)^T be the concatenated vector of estimated coefficients, and let α̂_𝒜^* be vector α̂ subset to retain only elements indexed by 𝒜^*. Then we can write the estimated penalized loss (<ref>) asl̂_pen = 1/2y - [V_𝒜^*, Z] [ α̂_𝒜^*; b̂ ]_2^2 + ∑_j=1^J λ_j ∑_l=k_j+2^p_j |α̂_jl | + 1/2τb̂^T S b̂ Taking the derivative of (<ref>) and keeping in mind that the first k_j + 1 elements of each α̂_j are unpenalized and |α̂_jl| > 0 for all l ∈𝒜_j, we have0_(|𝒜^*| + q) × 1 = ∂ l_pen/∂ (α̂_𝒜^*^T, b̂^T)^T = [ V_𝒜^*^T; Z^T ]([V_𝒜^*, Z] [ α̂_𝒜^*; b̂ ] - y) +[η; τ S b̂ ]where η =[ 0_k_1+1; λ_1sign(α̂_𝒜_1); 0_k_2+1; λ_2sign(α̂_𝒜_2 + p_1); ⋮; 0_k_J+1; λ_Jsign(α̂_𝒜_J + ∑_j=1^J-1 p_j) ],0_k_j + 1 is a (k_j + 1) × 1 vector of zeros, and the sign operator is taken element-wise. From <cit.>, we know that within a small neighborhood of y, the active set 𝒜 and the sign of the fitted terms α̂_𝒜 are constant with respect to y except for y in a set of measure zero. Therefore, ∂η / ∂y = 0_|𝒜^*| × n, where 0_|𝒜^*| × n is an |𝒜^*| × n matrix of zeros and |𝒜^*| is the cardinality of 𝒜^*. Then taking the derivative of (<ref>) with respect to y, we have 0_(|𝒜^*| + q) × n = ∂^2 l_pen/∂ (α̂_𝒜^*^T, b̂^T)^T ∂y =[ V_𝒜^*^T; Z^T ][V_𝒜^*, Z][ ∂α̂_𝒜^* / ∂y; ∂b̂ / ∂y ] -[ V_𝒜^*^T; Z^T ]+[0_|𝒜^*| × n; τ S (∂b̂ / ∂y) ].Solving for the derivatives of the estimated coefficients, we have[ ∂α̂_𝒜^* / ∂y; ∂b̂ / ∂y ] = ( [ V_𝒜^*^T; Z^T ][V_𝒜^*, Z]+ [ 0_|𝒜^*| × |𝒜^*| 0_|𝒜^*| × q; 0_q × |𝒜^*| τ S ])^-1[ V_𝒜^*^T; Z^T ].Now let A = [V_𝒜^*, Z] and Ω = [ 0_|𝒜^*| × |𝒜^*| 0_|𝒜^*| × q; 0_q × |𝒜^*| τ S ].Then since ŷ = A (α̂_𝒜^*^T, b̂^T)^T we have∂ŷ/∂y =∂ŷ/∂ (α̂_𝒜^*^T, b̂^T)^T∂ (α̂_𝒜^*^T, b̂^T)^T/∂y= A ( A^T A + Ω)^-1 A^T. From <cit.>, we know that g(y) = ŷ is continuous and weakly differentiable. Also, ∇ g = (∂ŷ / ∂y). Therefore, we can use Stein's formula (<ref>) to estimate the degrees of freedom asd̂f̂ = 1 + (A ( A^T A + Ω)^-1 A^T) = 1 + ( (A^T A + Ω)^-1 A^T A ),where we add 1 for the intercept. We note that this result is similar to the degrees of freedom for the elastic net <cit.> as well as for FLAM <cit.>.To obtain degrees of freedom for individual smooths j=1,…, J, let E_j be an (|𝒜^*|+q) × (|𝒜^*|+q) matrix with 1s on the diagonal positions indexed by 𝒜^*_j + ∑_l=0^j-1 |𝒜^*_l| and zero elsewhere, where |𝒜^*_j| is the cardinality of 𝒜^*_j and 𝒜^*_0 = ∅. Also, let f̂_j = V_j α̂_j be the estimate of the j^th smooth. Then as <cit.> note, f̂_j = A E_j(A^TA + Ω)^-1 A^T y. Therefore, d̂f̂_j = (A E_j(A^T A + Ω)^-1 A^T ) = (E_j(A^T A + Ω)^-1 A^T A ).In other words, the degrees of freedom for smooth j is the sum of the diagonal elements of (A^T A + Ω)^-1 A^T A indexed by 𝒜_j^* + ∑_l=0^j-1 |𝒜^*_l|.We note that when using the ADMM algorithm, or most likely any proximal algorithm, the fitted D_j β̂_j, or equivalently α̂_j, will typically have several very small non-zero values, but will not typically be sparse. However, the vector ŵ_j is sparse, where in the ADMM algorithm we constrain w_j = D_j β_j. Therefore, in practice we use w_j to obtain the active set 𝒜_j. §.§ Stable and fast approximationsIn some cases, such as the application in Section <ref>, the estimates based on Stein's method (<ref>) and (<ref>) cannot be computed due to numerical instability. In this section, we propose alternatives that are more numerically stable and which are also more computationally efficient.§.§.§ Based on restricted derivatives In this approach, we take derivatives of the fitted values restricted to individual smooths. In particular, from Section <ref>, we see that∂ŷ/∂α̂_𝒜^*_j∂α̂_𝒜^*_j/∂y = V_𝒜_j^* (V_𝒜_j^*^T V_𝒜_j^*)^-1 V_𝒜_j^*^T∂ŷ/∂b̂∂b̂/∂y = Z (Z^T Z + τ S)^-1 Z^T.We can then approximate the degrees of freedom for each individual smooth and the random effects byd̃f̃_j =((V_𝒜_j^*^T V_𝒜_j^*)^-1 V_𝒜_j^*^T V_𝒜_j^*) j=1,…,J((Z^T Z + τ S)^-1 Z^T Z ) j=J+1We estimate the overall degrees of freedom asd̃f̃ = 1 + ∑_j=1^J+1d̃f̃_jwhere we add 1 for the intercept.This approach is similar to one described by <cit.>, though in a different context and for a different purpose. In particular, whereas we use this approach to approximate the degrees of freedom after fitting the model, <cit.> use it to set the degrees of freedom before fitting the model in the context of ℓ_2 penalized loss functions.§.§.§ Based on ADMM constraint parameters In this approach, we propose estimates of degrees of freedom specific to the ADMM algorithm. As in the previous section, this approach is based on estimates for the individual smooths. Consider the model with J=1 smooth, no random effects, and centered y:y - F β_2^2 + λ D β_1.Suppose we make the centering constraints described Section <ref>, i.e. we set F = F̃ Q and D = D̃^(k+1) Q for an n × p design matrix F̃, a k+1 order finite difference matrix D^(k+1), and an orthonormal p × (p-1) matrix Q. Let 𝒜 = {l ∈{1,…,p - k-1 }: (D β̂)_l0} be the active set, and let |𝒜| be its cardinality. In our context, we expect the design matrices F to be full rank, in which case Theorem 3 of <cit.> (see the first Remark) states that the degrees of freedom is given by df = 𝔼[nullity(D_-𝒜)]. Here, nullity(D) is the dimension of the null space of matrix D, and D_-𝒜 is matrix D with rows indexed by 𝒜 removed. Now, D has dimensions (p-k-1) × (p-1), and we can see by inspection that for all k<p-1 the columns of D are linearly independent. Therefore, the rank of D_-𝒜 is equal to the number of rows p - k - 1 - |𝒜|, and the nullity is equal to the number of columns p-1 minus the number of rows. This gives d̂f̂ = nullity(D_-𝒜) = k + |𝒜| for centered smooths, i.e. the number of non-zero elements of D β̂ plus one less than the order of the difference penalty. This is similar to the result for ℓ_1 trend filtering, but we have lost one degree of freedom due to the constraint that 1^T F̃β̃ = 0. For uncentered smooths, D has dimensions (p-k-1) × p, which gives d̂f̂ = nullity(D_-𝒜)) = k + 1 + |𝒜|.As before, we note that in the ADMM algorithm, D β̂ will not generally be sparse, as ADMM is a proximal algorithm. However, the corresponding w is sparse, where in the optimization problem we constrain D β = w. Now considering a model with smooths j=1,…,J, a numerically stable and fast alternative to (<ref>) is given byd̃f̃_j^ADMM = 1[j ∈ℰ̅] + k_j + ∑_l=1^p - k - 11[w_jl 0].where ℰ̅ indexes the un-centered smooths and 1 is an indicator variable. We then combine (<ref>) with the restricted derivative approximation for the degrees of freedom of the random effects given above to obtain the overall degrees of freedomd̃f̃^ADMM = 1 + ∑_j=1^J d̃f̃^ADMM_j + ((Z^T Z + τ S)^-1 Z^T Z ),where we add 1 for the intercept. §.§ Ridge approximation Let U = [F_1, …, F_J, Z] be the concatenated design matrix of fixed and random effects andΩ^ridge = [ λ_1 D_1^T D_1; ⋱; λ_J D_J^T D_J; τ S ]be the penalty matrix. Then the hat matrix from the linear smoother approximation (see Section <ref>) is given by H = U (U^T U + Ω^ridge)^-1 U^T. Similar to before, we can get the overall degrees of freedom asd̂f̂^ridge = 1 + ((U^T U + Ω^ridge)^-1 U^T U ),where we add 1 for the intercept. To obtain degrees of freedom for individual smooths j=1,…, J, let E_j be a (p+q) × (p+q) matrix with 1s on the diagonal positions indexed by the columns of F_j and zero elsewhere. Also, let f̂_j = F_j β̂_j be the estimate of the j^th smooth. Then the ridge approximation for smooth j is given by f̂_j ≈ U E_j(U^T U + Ω^ridge)^-1 U^T y. Therefore, d̂f̂^ridge_j = ( E_j(U^T U + Ω^ridge)^-1 U^T U ) Similar to before, we also propose stable and fast approximations to the ridge estimate of degrees of freedom based on restricted derivatives. In particular, letd̃f̃_j^ridge =((F_j^T F_j + λ_j D_j^T D_j)^-1 F_j^T F ) j=1,…,J((Z^T Z + τ S)^-1 Z^T Z ) j=J+1Then we can estimate the overall degrees of freedom asd̃f̃^ridge = 1 + ∑_j=1^J+1d̃f̃_j^ridgewhere we add 1 for the intercept.As noted above, this approach is similar to one described by <cit.>, though for a different purpose. Whereas we use this approach to obtain the degrees of freedom after fitting the model, <cit.> use it to set the degrees of freedom before fitting the model.§ APPROXIMATE INFERENCEIn this section, we discuss approximate inferential methods based on ridge approximations to the ℓ_1 penalized fit and conditional on the smoothing parameters λ_j, j=1,…,J and τ. We use the ADMM algorithm to analyze the approximation. In particular, we note that we can write the ADMM update for β_j asβ_j^m+1 = (F_j^T F_j + ρ D_j^T D_j )^-1 F_j^T y^(j,m) + δ_j^mwhere δ_j^m = ρ(F_j^T F_j + ρ D_j^T D_j)^-1F^T_j D^T_j(w^m_j - u^m_j) and y^(j,m) = y - β_0^m+1 - ∑_l<j F_l β_l^m+1 - ∑_l>j F_l β_l^m - Z b^m. As we note in Observation <ref>, δ_j loosely represents the difference in the estimate of β_j obtained with the ℓ_1 and ℓ_2 penalties. With the ℓ_1 penalty, i.e. D_j β_j_1, in general δ_j^m0. However, with the ℓ_2 penalty, i.e. D_j β_j_2^2, and λ_j = ρ, we have δ_j^m = 0. Similar to the ridge update for b, if we changed λ_j D_j β_j_1 to (λ_j / 2) D_j β_j_2^2 in (<ref>) we could remove the w_j term and the constraint that D_j β_j^m = w_j from (<ref>) to obtain the ridge update β_j^m+1 = (F_j^T F_j + λ_j D_j^T D_j )^-1 F_j^T y^(j,m). Then since we assumed λ_j = ρ, we have β_j^m+1 = (F_j^T F_j + ρ D_j^T D_j )^-1 F_j^T y^(j,m). By comparison with (<ref>), we see that δ_j^m = 0. Observation <ref> motivates our approximate inferential strategy. Letting f̂_j be the j^th fitted smooth, and letting y^(j) = y - β̂_0 - ∑_lj F_l β̂_l - Z b̂, we havef̂_j = F_j β̂_j= F_j(F_j^T F_j + ρ D_j^T D_j)^-1 F_j^T y^(j) + F_j δ̂_j ≈ F_j(F_j^T F_j + ρ D_j^T D_j)^-1 F_j^T y^(j) (assumingF_j δ̂_j ≈0) ≈ F_j(F_j^T F_j + λ_j D_j^T D_j)^-1 F_j^T y^(j) (assuming λ_j ≈ρ) =H_j y^(j)where H_j = F_j(F_j^T F_j + λ_j D_j^T D_j)^-1 F_j^T. We obtain confidence intervals for the linear smoother (<ref>) centered around the estimated fit (<ref>), ignore F_j δ_j when estimating variance, and assume λ_j ≈ρ. We also condition on the smoothing parameters λ_1,…,λ_J and τ.Figure <ref> gives a visual demonstration of the approximation for the simulation presented in Section <ref> and the application shown in Section <ref>. As seen in Figure <ref>, in these examples the ℓ_1 fit and ridge approximation are very similar. If this holds in general, then this would suggest that 1) the approximate inferential procedures we propose might have reliable coverage probabilities, and 2) there may be minimal practical advantage to using an ℓ_1 penalty instead of the standard ℓ_2 penalty. However, as shown in Section <ref>, the ℓ_1 penalty appears to perform noticeably better in certain situations, including the detection of change points.Before presenting the confidence bands in greater detail, we discuss our approach for estimating variance in Section <ref>, which we then use to form confidence bands in Section <ref>. §.§ VarianceLet r = y - ∑_j=1^J F_j β̂_j - Z b̂ be an n × 1 vector of residuals. We estimate the overall variance as σ̂^2_ϵ = r^2_2 / d̂f̂_resid, where d̂f̂_resid is the residual degrees of freedom. When possible, we use the estimate based on Stein's method (<ref>) and set d̂f̂_resid = n - d̂f̂. If Stein's method is not numerically stable, then we use the restricted derivatives approximation (<ref>) and set d̂f̂_resid = n - d̃f̃. As another alternative, we could also use the ADMM approximation and set d̂f̂_resid = n - d̃f̃^ADMM. §.§ Confidence bandsIn this section, we obtain confidence bands for typical subjects, i.e. for subjects for whom b_i = 0. Since we assume a normal outcome, this is equivalent to the marginal population level response.§.§.§ Frequentist confidence bands Ignoring the distribution on D_j β_j and treating β_l, lj as fixed, y^(j) is normal with variance Var(y^(j)) = σ^2_ϵ I_n + σ^2_b Z S^+ Z^T, where S^+ is the Moore-Penrose generalized inverse of matrix S (as noted in Section <ref>, S may not be positive definite). Therefore, Var(f̂_j) ≈ H_j Var(y^(j)) H_j^T where Var(y^(j)) is an n × n estimate of Var(y^(j)) with σ̂^2_ϵ and σ̂^2_b plugged in for σ^2_ϵ and σ^2_b respectively, and f̂_j ·∼ N(f̂_j, H_j Var(y^(j)) H_j^T). The estimated variance of the fit at a single point x, which we denote as Var(f̂_j(x)), is the corresponding diagonal element of H_j Var(y^(j)) H_j^T. Therefore, asymptotic pointwise 1-α confidence bands take the form f̂_j(x) ± z_1-α/2√(Var(f̂_j(x))) where Φ(z_a) = a and Φ is the standard normal CDF, e.g. z_1-α/2 = 1.96 for α = 0.05.For the purposes of interpretation, we include the intercept term in the confidence band for the j=1 smooth, but not for the remaining smooths.§.§.§ Bayesian credible bandsMany authors, including <cit.>, recommend using Bayesian confidence bands for nonparametric and semiparametric models, because the point estimates are themselves biased. While Bayesian credible bands do not remedy the bias, they are self consistent.To this end, we replace the element-wise Laplace prior with the (generally improper) joint normal prior that is equivalent to the standard ℓ_2 penalty: β_j ∼ N (0, ( λ_j D_j^T D_j)^-1). This leads to the posteriorβ_j | y·∼ N (β̂_j, (W_jF^T_j Var(y^(j))^-1 F_j + λ_j D^T_j D_j)^-1).We can then form simultaneous Bayesian credible bands for f_j | y by simulating from the posterior (<ref>) and taking quantiles fromF_j β_j^b, b = 1,…,B. Alternatively, for a faster approximation we use frequentist confidence bands with F_j W_j^-1 F^T_j in place of H_j Var(y^(j)) H_j^T. In practice, we have found the simultaneous credible bands and the faster approximation to be nearly indistinguishable.[It appears that the latter (faster) method is the default in thepackage <cit.>. As in , we only need to compute the diagonal elements of F_j W_j^-1 F^T_j as ((F_j W_j^-1) ∘ F_j), where ∘ is the Hadamard (element-wise) product.]As before, for the purposes of interpretation, we include the intercept term in the credible band for the j=1 smooth, but not for the remaining smooths.§ SIMULATIONWe simulated data from a piecewise linear mean curve as shown in Figure <ref>. Each subject had a random intercept and is observed over only a portion of the domain. There are 50 subjects, each with between 4 and 14 measurements (450 total observations). The random intercepts were normally distributed with variance σ_b^2 = 1, and the overall noise was normally distributed with variance σ_ϵ^2 = 0.01.In all models, we used order 2 (degree 1) B-splines with p=21 basis functions. §.§ Frequentist estimation We fit models with J=1 smooth term and random intercepts. To obtain estimates for the ℓ_1 penalized model, we used ADMM and 5-fold CV to minimizeβ_0 ∈ℝ, β∈ℝ^p-1, b∈ℝ^Nminimize1/2y - β_0 1 - F β - Z b_2^2 + λD^(2)β_1 + τb^T b.where Z_il = 1 if observation i belongs to subject l and zero otherwise. As noted above, we used order 2 (degree 1) B-splines with p=21 basis functions, i.e. F ∈ℝ^n × (p-1) where n=450 and p=21. After estimating λ and τ via CV, we used LME updates to estimate σ^2_b and b in the final model. We also fit an equivalent model with an ℓ_2 penalty using thepackage <cit.>, i.e. with (λ / 2) D^(2)β_2^2 in place of λD^(2)β_1 in (<ref>). Figure <ref> shows the marginal mean with 95% credible intervals, and Figure <ref> shows the subject-specific predicted curves.As seen in Figures <ref> and <ref>, the results from the ℓ_1 and ℓ_2 penalized models are very similar. However, the ℓ_1 penalized model does slightly better at identifying the change points and the line segments. We explore this further in Section <ref>.Table <ref> compares the degrees of freedom and variance estimates from the ℓ_1 penalized fit against those from the ℓ_2 penalized fit. From Table <ref>, we see that the ridge degrees of freedom d̂f̂^ridge appears reasonable, as it is near the estimate for the ℓ_2 penalized model. The true degrees of freedom d̂f̂ also seems reasonable. Ideally, the degrees of freedom for the ℓ_1 penalized fit should equal six, as there are four change points and we are using a second order difference penalty (see Section <ref>).Table <ref> compares the different estimates of degrees of freedom. In this simulation, the degrees of freedom based on the ridge approximation is larger than that from Stein's formula, and the approximations based on restricted derivatives are equal or near the estimate with Stein's formula. §.§ Bayesian estimation We modeled the data as y | b = β_0 1 + F β + b + ϵ whereϵ ∼ N(0, σ^2_ϵ I)b ∼ N(0, σ^2_b I) D^(2)β ∼Laplace(0, σ^2_λ I) p(σ_ϵ)∝ 1 p(σ_b)∝ 1 p(log(σ_λ))∝ 1.We also fit models with normal and diffuse priors for D^(2)β.We fit all models with<cit.>, each with four chains of 2,000 iterations with the first 1,000 iterations of each chain used as warmup. The MCMC chains, not shown, appeared to be reasonably well mixing and stationary, and had R̂ values under 1.1 <cit.>.[As described by <cit.>, for each scalar parameter, R̂ is the square root of the ratio of the marginal posterior variance (a weighed average of between- and within-chain variances) to the mean within-chain variance. As the number of iterations in the MCMC chains goes to infinity, R̂ converges to 1 from above. Consequently, R̂ can be interpreted as a scale reduction factor, and <cit.> recommend ensuring that R̂<1 for all parameters.] Figure <ref> shows the marginal mean with 95% credible intervals, and Figure <ref> shows point estimates.As seen in Figures <ref> and <ref>, all models performed well and gave similar fits as above. Similar to before, the Laplace prior appears to better enforce a piece-wise linear fit, particularly around x=0.2. §.§ Change point detectionWe simulated 1,000 datasets with the same generating mechanism used to produce the data shown in Figure <ref> and measured the performance of the ℓ_1 and ℓ_2 penalized models on two criteria: 1) the number of inflection points found, and 2) the distance between the estimated inflection points and the closest true inflection point. To that end, let 𝒯={τ_1,…,τ_4} be the set of true inflection points, and M = max_x ∈𝒳|f̂”(x)| be the maximum absolute second derivative of the estimated function, where 𝒳={x_1,x_2,…} is the ordered set of unique simulated x values. We approximate f̂” byf̂”(x_i) ≈(f̂(x_i+1) -f̂(x_i))/(x_i+1 - x_i) - (f̂(x_i) -f̂(x_i-1))/(x_i - x_i-1)/x_i+1 - x_i.Then let ℐ = {x ∈𝒳 : |f̂”(x)| ≥ c M } be the set of estimated inflection points, where c ∈ (0,1) is a cutoff value defining how large the second derivative must be to be counted as an inflection point. Also, let n_ℐ = |ℐ| be the number of estimated inflection points, and d̅ = n^-1_ℐ∑_x ∈ℐmin_τ∈𝒯 |x - τ| be the mean absolute deviance of the estimated inflection points.Figure <ref> shows the results from 1,000 simulated datasets. The ℓ_1 penalized model was better able to 1) find the correct number of inflection points, and 2) determine the location of the inflection points. §.§ Coverage probability We simulated 1,000 datasets with the same generating mechanism used to produce the data shown in Figure <ref> and measured the coverage probability of the approximate Bayesian credible bands described in Section <ref> for the ℓ_1 penalized model, and simultaneous Bayesian credible bands for the ℓ_2 penalized model <cit.>. Figure <ref> shows the coverage probabilities for both approaches. As seen in Figure <ref>, the confidence bands perform similarly and are near the nominal rate over most of the x domain. Both approaches have difficulty maintaining nominal coverage at the edges of the x domain.§ APPLICATION §.§ Data description and preparation In this section, we analyze electrodermal activity (EDA) data collected as part of a stress study. In brief, all subjects completed a written questionnaire prior to the study, which categorized the subjects as having either low vigilance or high vigilance personality types. During the study, all participants wore wristbands that measured EDA while undergoing stress-inducing activities, including giving a public speech and performing mental arithmetic in front of an audience. The scientific questions were: 1) Is EDA higher among high vigilance subjects, and 2) when did trends in stress levels change? In this section, we demonstrate how P-splines with an ℓ_1 penalty can address both questions.The raw EDA data are shown in Figure <ref>. After excluding subjects who had EDA measurements of essentially zero throughout the entire study, we were left with ten high vigilance subjects and seven low vigilance subjects.To remove the extreme second-by-second fluctuations in EDA, which we believe are artifacts of the measurement device as opposed to real biological signals, we smoothed each curve separately with a Nadaraya–Watson kernel estimator using thefunction in . We then thinned the data to reduce computational burden, taking 100 evenly spaced measurements from each subject. Figure <ref> shows the results of this process for a single subject, and Figure <ref> shows the prepared data for all subjects. Because of the limited number of subjects, as well as issues of misalignment in the time series across individuals, the results presented here should be considered as illustrative rather than of full scientific validity.§.§ ModelsIn all models, we fit the structurey_i(x) = β_0 + β_1(x) + 1_high[i] β_2(x) + b_i(x) + ϵ_i(x)where x represents time in minutes, 1_high[i]=1 if subject i has high vigilance and 1_high[i]=0 if subject i has low vigilance, b_i(x) are random subject-specific curves, and ϵ_i(x) ∼ N(0, σ^2_ϵ). For β_1(x), β_2(x), and b_i(x), we used a fourth order B-spline basis with 31 basis functions each and a second order difference penalty (k=1).Written in matrix notation, the ℓ_1 penalized model ismin1/2y - β_0 1 - ∑_j=1^2 F_j β_j - Z b_2^2 + ∑_j=1^2 λ_j D^(2)β_j _1 + b^T S bwhere y is a stacked vector for subjects i=1,…,17, F_1 is an n × p design matrix where n=1,700 and p=31, and F_2 = diag( 1_high[i]) F_1 where i is an n × 1 vector of subject IDs. In other words, F_2 is equal to F_1, but with rows corresponding to low vigilance subjects zeroed out. We setZ = [Z_1; ⋱;Z_17 ]where each Z_i is an n_i × 31 random effects design matrix of order 4 B-splines evaluated at the input points for subject i, and S = [S_1; ⋱;S_17 ]where S_i,jl = ∫ϕ”_ij(t) ϕ”_il(t) dt are smoothing spline penalty matrices. We also mean-centered F_1 as described in Section <ref>, with the corresponding changes in dimensions.To fit a comparable ℓ_2 penalized model, in which λ_j D^(2)β_j _1 in (<ref>) is replaced with (λ_j/2) D^(2)β_j _2^2, we rotated the random effect design and penalty matrices Z and S as described in Section <ref>. To facilitate the use of existing software, we used a normal prior for the “unpenalized" random effect coefficients, i.e. b̆_f ∼ N(0, σ^2_f I).We also fit a Bayesian model using the same rotations and equivalent penalties as above. In particular, we modeled the data as y|b = β_0 1 + ∑_j=1^J F_j β_j + Ž_r b̌_r + Z̆_f b̆_f + ϵ whereb̌_r∼ N(0, σ^2_r I)b̆_f∼ N(0, σ^2_f I)( D_j β_j )_l∼Laplace(0, a_j)fora_j = σ^2_ϵ / (2 λ_j), l=1,…, p_j - k_j - 1, j=1,…,Jϵ ∼ N ( 0, σ^2_ϵ I_n ).§.§ Results§.§.§ Frequentist estimation We tried to use CV to estimate the smoothing parameters for the ℓ_1 penalized model. However, with only 17 subjects split between two groups, we only did 3-fold CV. CV did not find a visually reasonable fit so we set the tuning parameters by hand.Figure <ref> shows the estimated marginal mean and 95% credible bands for the ℓ_1 penalized model, and Figure <ref> shows the subject-specific predicted curves for the ℓ_1 penalized model. As seen in Figure <ref>, our model identified a few inflection points, particularly near minutes 40, 50, and 60. From Figure <ref> it appears that the difference in EDA between the low and high vigilance subjects was not statistically significant. Also, as seen in Figure <ref>, the subject-specific predicted curves are shrunk towards the mean, which is expected, because the predicted curves are analogous to BLUPs, although they are not linear smoothers.Figure <ref> shows the estimated marginal mean and 95% credible bands for the ℓ_2 penalized model, and Figure <ref> shows the subject-specific predicted curves for the ℓ_2 penalized model. The estimate shown in Figure <ref> is similar to that shown in Figure <ref>, though the inflection points are slightly less pronounced in Figure <ref>. The results in Figure <ref> are for the most part substantively the same as those in Figure <ref>; the ℓ_2 penalized model does not show a statistically significant difference between the low and high vigilance subjects, with the possible exception of minutes 45 to 66. As seen in Figure <ref>, the predicted subject-specific curves from the ℓ_2 penalized model are also shrunk towards the mean.Table <ref> shows the estimated degrees of freedom for the ℓ_1 penalized model. Stein's method d̂f̂ ((<ref>) and (<ref>)) and the ridge approximation d̂f̂^ridge ((<ref>) and (<ref>)) were numerically instable (A^T A + Ω and U^T U + Ω^ridge were computationally singular). Therefore we used the restricted derivative approximation d̃f̃ to estimate the variance, as described in Section <ref>. In the ℓ_2 penalized model, smooth F_1 had 14.2 degrees of freedom, and smooth F_2 had 6.96 degrees of freedom.§.§.§ Bayesian estimation We fit the model described in Section <ref> with an element-wise Laplace prior on D β given by (<ref>). To fit the model, we used<cit.> with four chains of 5,000 iterations each, with the first 2,500 iterations of each chain used as warmup. The MCMC chains, not shown, appeared to be reasonably well mixing and stationary with R̂ values under 1.1 <cit.>. Figure <ref> shows the marginal means with 95% credible bands, and Figure <ref> shows the subject-specific curves. Similar to the ℓ_2 penalized model, the Bayesian model found a slightly statistically significant difference between low and high vigilance between minutes 42 and 65. §.§ Alternative correlation structureFor comparison, we also fit ℓ_1 and ℓ_2 penalized models with alternative correlation structures similar to that recommended by <cit.>.For the ℓ_1 penalized model, in place of the correlation structure implied by the penalty matrix S described above, we set the penalty matrix to S:= I_q. While this is a simplification of the correlation structure recommended by <cit.>, we think it offers a similar amount of flexibility.Figure <ref> shows the estimated marginal mean and 95% credible bands, and Figure <ref> shows the subject-specific predicted curves. The point estimates shown in Figure <ref> are similar to that shown in Figure <ref>. However, the confidence intervals in Figure <ref> appear more reasonable. The subject-specific predicted curves shown in <ref> are not shrunk towards the mean as much as in Figure <ref>.For the ℓ_2 Penalized model, in place of the correlation structure implied by the penalty matrix S described above, we augmented each Z_i matrix on the left with the columns [1, x_i], where x_i is an n_i × 1 vector of measurement times for subject i. We then replaced Z_i b_i with [1, x_i, Z_i] (u_i^T, b_i^T)^T, and assumed (u_i^T, b_i^T)^T ∼ N(0, Σ_i) where Σ_i = [Σ'; σ_b^2 I_q_i ]and Σ' is a common 2 × 2 unstructured positive definite matrix. To model the within-subject correlations, we used a continuous autoregressive process of order 1. In particular, Cor(y_i(x_ij), y_i(x_ij')) = ζ^|x_ij - x_ij'| for a common parameter ζ>0.Figure <ref> shows the estimated marginal mean and 95% credible bands, and Figure <ref> shows the subject-specific predicted curves. The estimates shown in Figure <ref> are similar to that shown in Figure <ref>. While estimates of the difference between low and high vigilance subjects differs between this model and the ℓ_2 penalized model in Section <ref>, the more notable difference is in the subject-specific predicted curves. As seen in Figure <ref>, the predicted subject-specific curves are not shrunk towards the mean as much as in Figure <ref>.Table <ref> shows the mean squared error (MSE) and computing time for the ℓ_1 penalized and ℓ_2 penalized models. In Table <ref>, computing time for the ℓ_1 penalized model does not include cross-validation, because the parameters were hand-tuned (with only 17 subjects and a complex random effects structure, cross-validation did not find reasonable parameter values). As can be seen in Table <ref>, the alternative correlation structure led to smaller MSE for both the ℓ_1 and ℓ_2 penalized models, and less computing time for the ℓ_2 penalized model.§ DISCUSSION AND POTENTIAL EXTENSIONSAs demonstrated in this article, P-splines with an ℓ_1 penalty can be useful for analyzing repeated measures data. Compared to related work with ℓ_1 penalties, our model is ambitious in that we allow for multiple smoothing parameters and propose approximate inferential procedures that do not require Bayesian estimation. However, these are also the two aspects of our proposed approach that require additional future work. For P-splines with an ℓ_2 penalty, in most cases the knot placement is not critical so long as the number of knots is large enough <cit.>. We believe this also holds for P-splines with an ℓ_1 penalty, though further experimentation is needed to support this assumption. In practice, we recommend fitting models with a few different knot placements and widths to determine whether the model is sensitive to those choices for the data at hand.Regarding estimation, our current approach of using ADMM and CV appears to work reasonably well for random intercepts, but is not yet reliable for random curves. In the future, we plan to develop more robust estimation techniques, particularly for smoothing parameters. As one possibility, we have done preliminary work to minimize quantities similar to GCV and AIC instead of the more computationally intensive CV, though these approaches do not seem as promising as their ℓ_2 counterparts. It may also be helpful to set the degrees of freedom prior to fitting the model. When possible, Bayesian estimation may be the most reliable way to currently fit these models. Bayesian estimation also opens the possibility of using other sparsity inducing priors, such as spike and slab models <cit.>.Regarding inference, in future work it may be possible to use the δ quantity to bound difference between ℓ_1 and ℓ_2 penalized fits under certain assumptions on the data. It may also be helpful to investigate the use of post-selection inference methods to develop confidence bands for linear combinations of the active set, and to further investigate through simulations the performance of our proposed approximations of degrees of freedom. However, we note that our primary use of the degrees of freedom estimate d̂f̂ is to obtain the residual degrees of freedom d̂f̂_resid = n - d̂f̂, which we then use to estimate the variances σ̂^2_ϵ = r_2^2 / d̂f̂_resid. Therefore, when n ≫d̂f̂, σ̂^2_ϵ is not very sensitive to d̂f̂, in which case it is not critical for our purposes to obtain an exact estimate of degrees of freedom.As for P-splines with an ℓ_2 penalty, users must select both the order M of the B-splines and the order k+1 of the finite differences. These choices will depend on the scientific problem and analytical goals. Using k=1 (2^nd order differences) is likely an appropriate starting point for most applications, and larger k could be used to increase the amount of smoothness.For P-splines with an ℓ_2 penalty, in most cases the knot placement is not critical so long as the number of knots is large enough <cit.>. We believe this also holds for P-splines with an ℓ_1 penalty, though further experimentation is needed to support this assumption. In practice, we recommend fitting models with a few different knot placements and widths to determine whether the model is sensitive to those choices for the data at hand.Regarding the rate of convergence, from Observation <ref> and the work of <cit.>, we know that for equally spaced data and F = I_n, P-splines with an ℓ_1 penalty achieve the minimax rate of convergence for the class of weakly differentiable functions of bounded variation. When there are less knots than data points, we do not think it is possible to achieve the minimax rate of convergence. However, if the knots are selected well, it may be possible to achieve the same performance in practice.It could also be useful to extend these results to generalized additive models to allow for non-normal responses, and to extend the approach of <cit.> to include random effects and multiple smoothing parameters.§ SUPPLEMENTARY MATERIAL We have implemented our method in the R packageavailable at <https://github.com/bdsegal/psplinesl1>. All code for the simulations and analyses in this paper are available at <https://github.com/bdsegal/code-for-psplinesl1-paper>.§ ACKNOWLEGEMENTS We thank Margaret Hicken for sharing the data from the stress study.§ SIMULATED DEMONSTRATION WITH TWO SMOOTHS In this appendix, we simulated data similar to that in Section <ref>, but with an additional varying-coefficient smooth. In particular, we simulated data for two groups with 50 subjects in each group and between 4 and 14 measurements per subject (900 total observations). The data for subject i at time t was generated as y_it = β_0 + f_1(x_it) + f_2(x_it) 1[subjectiin Group 2] + b_i + ϵ_it where b_i ∼ N(0, σ_b^2) and ϵ_it∼ N(0, σ^2_ϵ) for σ_b^2 = 1 and σ_ϵ^2 = 0.01. The true group means f_1(x) and f_2(x) are shown in Figure <ref> and the simulated data are shown in Figure <ref>.We fit a varying-coefficient model with J=2 smooths to the data. In particular, we used ADMM and 5-fold CV to minimizeβ_0 ∈ℝ, β_1 ∈ℝ^p-1, β_2 ∈ℝ^p, b∈ℝ^Nminimize 1/2y - β_0 1 - F_1 β_1 - F_2 β_2 - Z b_2^2+ λ_1 D^(2)β_1 _1 + λ_2 D^(2)β_2 _1 + τb^T b.where F_1, F_2 ∈ℝ^n × p were formed with second order (first degree) B-splines and p=21 basis functions, F_2 = diag(u) F_1 where u_i = 1[subjectiin Group 2], and Z_il = 1 if observation i belongs to subject l and zero otherwise. We also fit an equivalent model with an ℓ_2 penalty using thepackage <cit.>, i.e. with (λ_j / 2) D^(2)β_j _2^2 in place of λ_j D^(2)β_j _1 in (<ref>), j = 1, 2.The estimated curves are shown in Figure <ref> for the ℓ_1 penalized model and in Figure <ref> for the ℓ_2 penalized model. We used 5-fold CV to estimate the smoothing parameters λ_1, λ_2 and τ in the ℓ_1 penalized model, and LME updates to estimate σ^2_b and b in the final model. As seen in Figures <ref> and <ref>, the fits are similar, but the results with the ℓ_1 penalized model are slightly closer to the truth.Table <ref> shows the degrees of freedom and variance estimates with the ℓ_1 penalized and ℓ_2 penalized models. As seen in Table <ref>, variance estimates from both the ℓ_1 and ℓ_2 penalized models are very near the true values.§ DETAILS FOR Λ_J^MAXLetting r_j = y - β_0 1 - ∑_ℓ j F_ℓβ_ℓ- Z b be the j^th partial residuals, we can write the terms in (<ref>) that involve β_j as (1/2) r_j - F_j β_j_2^2 + λ_j D_j β_j _1. Then taking the sub-differential of (<ref>) with respect to β_j, we have0 = -F_j^T(r_j - F_j β̂_j) + D_j^T λ_j s_jfor some s_j = (s_j, 1,…,s_j, p_j - k_j - 1)^T wheres_j,ℓ∈{1}if(D β̂_j)_ℓ > 1{-1}if(D β̂_j)_ℓ < 1 [-1, 1] if(D β̂_j)_ℓ = 0.Solving (<ref>) for β̂_j, we have β̂_j = (F_j^T F_j)^-1 F_j^T r_j - D_j^T λ_j s_j. Multiplying through by D_j and noting that D_j D_j^T is full rank and thus invertible, we have(D_j D_j^T)^-1 D_j β̂_j = (D_j D_j^T)^-1 D_j (F_j^T F_j)^-1 F_j^T r_j - λ_j s_j.Setting D_j β̂_j = 0 in (<ref>), we get that (D_j D_j^T)^-1 D_j (F_j^T F_j)^-1 F_j^T r_j = λ_j s_j where s_j, ℓ∈ [-1, 1] for all ℓ. This can only hold if λ_j = (D_j D_j^T)^-1 D_j (F_j^T F_j)^-1 F_j^T r_j_∞, which gives us λ_j^max.§ CONTROLLING TOTAL VARIATION WITH THE ℓ_1 PENALTYLet f(x) = ∑_j=1^p β_j ϕ_j^M(x). Suppose the knots are equally spaced, and let h_M-k-1 = (M-k-1) / (t_j+M-k-1 - t_j) for all j and 0 ≤ k < M - 1. Then on the interval [t_M = x_min, t_p+1 = x_max], from <cit.> we haved^k+1/dx^k+1f(x)= h_M-1⋯ h_M-k-1∑_j=k+2^p∇^k+1β_j ϕ_j^M-k-1(x)where ∇^k+1 is the (k+1)^th order backwards difference.Let a^M_k+1 = max_j ∈{k + 2,… p }∫_x_min^x_maxϕ_j^M-k-1(x) dx. We note that a^M_k+1 is finite and positive for all 0 ≤ k < M-1. Then from (<ref>), we have1/h_M - 1⋯ h_M-k-1 ∫_x_min^x_max| d^k+1/dx^k+1 f(x) | dx =∫_x_min^x_max| ∑_j=k+2^p ∇^k+1β_j ϕ_j^M-k-1(x) |dx = ∫_x_min^x_max| ∑_j=k+2^p (D^(k+1)β)_j-k-1ϕ_j^M-k-1(x) | dx ≤∫_x_min^x_max∑_j=k+2^p | (D^(k+1)β)_j-k-1ϕ_j^M-k-1(x) | dx = ∑_j=k+2^p ∫_x_min^x_max| (D^(k+1)β)_j-k-1ϕ_j^M-k-1(x) | dx = ∑_j=k+2^p | (D^(k+1)β)_j-k-1| ∫_x_min^x_maxϕ_j^M-k-1(x) dx ≤ a^M_k+1∑_j=k+2^p | (D^(k+1)β)_j-k-1| = a^M_k+1 D^(k+1)β_1where (<ref>) follows because ϕ_j^M-k-1(x) ≥ 0∀ x ∈ℝ.Rewriting (<ref>), for 0 ≤ k < M-1 we have∫_x_min^x_max| d^k+1/dx^k+1 f(x) | dx ≤ C_M,k+1D^(k+1)β_1where C_M, k+1 = a^M_k+1 h_M-1⋯ h_M-k-1 is a constant. This shows that controlling the ℓ_1 norm of the (k+1)^th order finite differences in coefficients also controls the total variation of the k^th derivative of the function. apalike | http://arxiv.org/abs/1707.08933v2 | {
"authors": [
"Brian D. Segal",
"Michael R. Elliott",
"Thomas Braun",
"Hui Jiang"
],
"categories": [
"stat.ME",
"62G08 (Primary), 62P10 (Secondary)"
],
"primary_category": "stat.ME",
"published": "20170727165522",
"title": "P-splines with an l1 penalty for repeated measures"
} |
firstpage–lastpage Novel Electronic State and Superconductivity in the Electron-Doped High-T_ c T'-Superconductors Y. Koike May 29, 2018 ===============================================================================================The Magellanic Bridge (MB) is a gaseous stream that links the Large (LMC) and Small (SMC) Magellanic Clouds. Current simulations suggest that the MB forms from a recent interaction between the Clouds. In this scenario, the MB should also have an associated stellar bridge formed by stars tidally stripped from the SMC by the LMC. There are several observational evidences for these stripped stars, from the presence of intermediate age populations in the MB and carbon stars, to the recent observation of an over-density of RR Lyrae stars offset from the MB. However, spectroscopic confirmation of stripped stars in the MB remains lacking. In this paper, we use medium resolution spectra to derive the radial velocities and metallicities of stars in two fields along the MB. We show from both their chemistry and kinematics that the bulk of these stars must have been tidally stripped from the SMC. This is the first spectroscopic evidence for a dwarf galaxy being tidally stripped by a larger dwarf.galaxies: interactions – Local Group – Magellanic Clouds.§ INTRODUCTIONThe Magellanic Bridge (MB) was discovered by <cit.> as angaseous structure connecting the Small (SMC) and the Large (LMC) Magellanic Clouds. Two further streams of gas – the Leading Arm and the Magellanic Stream – were later discovered by <cit.> and <cit.>, respectively. Initial attempts to reproduce the Magellanic System from dynamical simulations assumed multiple strong pericentric passages with the Milky Way. These were able to reproduce several features of the Magellanic System through the action of tidal stripping, ram pressure stripping, or a combination of the two <cit.>.However, recent measurements of the Magellanic Cloud's proper motions <cit.> have dramatically changed our view of the Magellanic system. Their large orbital velocities make the classical assumption of multiple encounters with the Milky Way unlikely. In this context, new hypotheses have emerged. <cit.> introduced the `first infall' scenario. In this model, the Magellanic Clouds were bound to one another for a long period of time and are just now on their first infall to the Milky Way. In this picture, the MB and tail form as a result of a recent tidal interaction between the Clouds, prior to their recent accretion onto the Galaxy <cit.>. An alternative scenario was proposed by <cit.>. In this model, the LMC and the SMC have had multiple passages around the Galaxy, becoming a close binary pair only ∼2 Gyr ago. The Magellanic Stream and Leading Arm form from the first interaction between the Clouds, while the latest interaction ∼ 250 Myr ago formed the MB. The star formation histories of the LMC and SMC exhibit two correlated bursts of star formation at ∼2 Gyr ago and ∼500 Myr ago that are consistent with such close interactions between the Clouds<cit.>.Independently of the initial assumptions about the Magellanic Cloud's orbits, all of the latest models favour tidal stripping as the primary mechanism for forming the Leading and trailing arms and the MB <cit.>. Such models predict that, alongside the already observed gaseous structures, there must be companion stellar tidal debris comprised of intermediate-age and old, ⪆ 1 Gyr, stars. What differentiates the different models are the number of stars stripped and their origin. For example, Model 2 from <cit.> predicts a factor to ∼5 more stars in the MB as compared with their Model 1. In both cases, these stars originate in the SMC. By contrast, <cit.> predict that the MB should contain stars stripped from both the SMC and LMC.The search for a stellar counterpart to the gaseous MB remains inconclusive. It is well known that there are young stars (≲100 Myr) in this area, most likely formed in-situ <cit.>. In fact, <cit.> have shown that these young stars form a continues stream linking both galaxies. However, intermediate-age and old stars have proven more elusive. <cit.> and <cit.> were unable to find any in the MB region. However, more recent studies with deeper data find an excess of stars in the regions of the color-magnitude diagram (CMD) that are expected to be populated by these stars, e.g. the red clump, at the distance of the MB. <cit.> found a distinct population in the LMC with amedian metallicity of [M/H]=-1.23 dex, substantially different from the bulk of the LMC disk population. This discrepancy in the metallicity lead Olsen et al. to claim that the kinematically distinct population found in the LMC was accretedfrom the SMC. <cit.> reported a large azimuthally-symmetric metal-poor stellar population up to ∼11 kpc from the SMC centre that was well-fitted by an exponential profile. However, they were not able to determine whether these stars constituted a bound stellar halo or extra-tidal stars. <cit.> found a bimodal velocity distribution of the red giant branch (RGB) stars to the east and south of the SMC centre. The second peak at larger radial velocities was interpreted as SMC stars tidally striped from the LMC. However, this result has not been confirmed by recent studies <cit.>. A stellar structure in front of the SMC main body in the eastern region at a distance of 4 was reported by <cit.>. They interpreted this as the tidally stripped stellar counterpart of thegaseous bridge. As part of the MAGellanic Inter-Cloud project <cit.> we found a population with ages between 1 and 10 Gyr with a spatial distribution more spread out than the younger stars. This older population has very similar properties to the stars located at ∼2 from the SMC centre suggesting than they were stripped from this region. <cit.> found evidence for an older population in the MB that they suggest comprised tidal debris in the region 1^ h20^ m≤α≤4^ h40^ m and -69≥δ≥-77. The radial density profiles of red clump stars in both galaxies, derived from the fourth phase of the Optical Gravitational Lensing Experiment (OGLE) by <cit.>, shows a strong deviation. They suggest that this owes to the overlapping stellar halos of the Magellanic Clouds. Very recently, <cit.> reported the existence of stellar tidal tail mapped with RR Lyrae in the Gaia DR1 database, not aligned with the gaseous MB and shifted by some ∼5 from the young main sequence bridge (see Fig. <ref>). Finally, a foreground population in the form of a distance bi-modality in the red clump distribution has been identified in the eastern SMC by <cit.> with the VISTA Magellanic Clouds (VMC) survey. The authors claim that the most likely explanation for this foreground population is tidal stripping from the SMC during its most recent encounter with the LMC. Thus, most of these studies support tidal stripping of stars from the SMC. However, the nature and origin of this stellar population in the inter-Cloud region is still not fully understood.In this paper, we set out to unequivocally test the tidal origin of the MB stellar populations older than ∼1 Gyr. To achieve this, we present the first spectroscopic analysis of the old stellar population in the MB region. If stars in the MB are indeed tidal debris, then they should have velocities and metallicities similar to those of the SMC and/or LMC stars. By determining whether the MB stars are more SMC-like or LMC-like, we will determine whether they were stripped primarily from the SMC, the LMC, or both.This paper is organised as follows. In Section 2, we describe the target selection, observations, and data reduction. Radial velocities and stellar metallicity determination procedures are presented in Section 3. Finally, in Section 4 we discuss the implications of our results in the context of the Magellanic Clouds' dynamical evolution.§ OBSERVATIONAL DATA We have selected two fields studied in previous MAGIC papers centred at [α = +024, δ = -720 (0224-7200)] andrespectively. The former is located at 69 and 141 from the SMC and LMC centres, respectively; the later at 65 and 140 from the SMC and LMC centres, respectively. The potential spectroscopic targets, selected from the expected position of the upper RGB in the CMDs, are shown in Fig. <ref> (large filled circles). This sample has been extended by selecting additional potential targets from the 2MASS <cit.>. To do that, we defined a region around the location of the objects previously selected from optical photometry (see Fig. <ref>). The observations were secured on the nights of the 12th and 13th November 2014 and 14th November 2016 with the AAOmega spectrograph <cit.> fed by the Two Degree Field (2dF) multi-object systems installed at the prime focus of the Anglo-Australian Telescope located at Siding Spring Observatory (Australia)[Program I.D.: ATAC/2014B/104, and S/2016B/04]. 2dF+AAOmega is a dual-beam spectrograph that allows to allocate up to 400 2"-size optical fibres within a 2 field of view. The blue arm was configured to observe blue targets located in the upper young main sequence, the analysis of which will be presented in a forthcoming paper. In the red arm, we used the grating 1700D centred on ∼8500 Å, providing a spectral resolution of R∼ 8500. Three different configurations were observed, two in a field at +024 and -720 (0224-7400), for which we acquired three exposures of 3600 s for each of them, and one in a field at +024 and -740 (0224-7400), for which we acquired a total of 6 exposures of 3600 s. The area covered by these fields is marked as dashed yellow circles in Fig. <ref>. In total, we obtained spectra with a signal-to-noise ratio larger than 5 for 514 stars listed in Table <ref>. The initial steps of the data reduction, including bias subtraction, flat-field normalisation, fibre tracing and extraction, and wavelength calibration, were performed with the dedicated 2dF data reduction pipeline <cit.>. Our own software was used to subtract the emission sky lines following the procedure described in detail by <cit.>. Briefly, the scale factor that minimises the sky line residuals is searched for by comparing the spectrum observed in each fibre with a master sky spectrum obtained by averaging the spectra of the nearest ten fibres placed on sky positions. After sky subtraction, the spectra are then normalised by fitting a low order polynomial. § RADIAL VELOCITIES AND METALLICITIES The radial velocities of the observed stars were calculated by comparing the observed spectra with a grid of synthetic spectra using the classical cross-correlation method. Details about the procedure and the grid of synthetic spectra used can be found in <cit.>. In brief, the velocities are determined in three steps. (1) Each object spectrum is cross-correlated with a reference synthetic spectrum to obtain an initial shift for all of them. In this case, we chose: [M/H]=-0.5 dex; [α/H]=+0.0 dex; ξ=1.5 km s^-1; T_ eff=4,500 k; and log g=2.0 dex. (2) After applying this initial shift, the observed spectrum is compared with the whole grid in order to identify the model parameters that best reproduce it through a χ^2 minimisation using FERRE <cit.>. (3) The best-fit synthetic spectrum is cross-correlated again with the observed spectrum in order to refine the shift between both. The heliocentric radial velocities derived for each star are listed in Table <ref>.The velocity distribution of observed stars is shown in top panels of Fig. <ref>. There is a peak centred between 20 and 30 km s^-1 with a clear tail that extends from 100 to 200 km s^-1. To better understand the obtained distribution we have plotted the prediction of the Besançon Galaxy Model[Available at <http://model.obs-besancon.fr/>.] <cit.> at the same position of the observed stars. This has been computed assuming the typical uncertainties for our radial velocities of 3 km s^-1. We restricted our comparison to those stars located in the same region of the CMD as that of our target star locations, scaled to reproduce the height of the observed velocity distribution.The BGM reproduces quite well the shape of the distribution between ∼ -10 to ∼90 km s^-1. However, the model does not predict any star above 100 km s^-1. The velocity distribution of SMC stars, bottom panels of Fig. <ref>, is centred at 147.8±0.5 km s^-1 with a dispersion of 26.4±0.4 km s^-1 according to <cit.>. This agrees with the tail at velocities larger than 100 km s^-1 of the observed distribution. In the case of the LMC, the radial velocities in the outskirts of the side that faces the SMC are typically larger than 200 km s^-1 <cit.>. This could explain the observed objects with the radial velocities around ∼200 km s^-1 and one star with a velocity of ∼270 km s^-1, but not the majority of stars between 100 and 200 km s^-1.We have complemented our analysis with the metallicities, [M/H], of observed stars obtained from an empirical relation between the strength of thetriplet lines and a luminosity indicator, e.g. the absolute magnitude. We have used the relation obtained by <cit.> using M_K_s as a luminosity indicator. The absolute magnitude for each observed star was derived assuming a distance modulus of (m-M)_0 = 18.6 <cit.>, and using a reddening E_B-V derived from the <cit.> extinction maps. Finally, the strength of each line was determined by fitting its profile with a Gaussian plus a Lorentzian within a given bandpass following the procedure described by <cit.>. The obtained metallicity distribution shown in Fig. <ref> has 14 stars: those with V_r≥100 km s^-1, signal-to-noise ratio larger than 10 and excluding the star with radial velocity ∼270 km s^-1. Our derived metallicities range from -0.9 to -2.4 dex with a median of -1.68 dex and a standard deviation of 0.47 dex. These values are similar to those obtained by <cit.> from RR Lyrae in the bridge, with an average of [M/H] = -1.79 dex.For comparison, we have over-plotted in Fig. <ref> the metallicity distribution of the innermost, smc0057 (green), and the outermost, smc0053 (red), fields studied by <cit.> in the SMC (located on the right of Fig. <ref>). Clearly, the innermost field, located at a distance of ∼ 1from the SMC centre, is more metal rich than the bridge with a peak at ∼ -1 dex. By contrast, the outermost field, at a distance of ∼ 4, has a distribution more similar to the bridge field, peaking at -1.64 dex. We have also over-plotted the metallicity distribution of the field LMC0356-7100 (blue) studied by <cit.>. This field is located at a distance of about ∼ 8and ∼ 14from the LMC and SMC centres, respectively. The metallicity distribution of this field is more metal rich than the outermost field studied in the SMC and in the MB, with a peak at -0.87 dex and reaching up to metallicities of -0.2 dex. We conclude that the stellar populations in the bridge are similar to the external parts of the SMC but more metal-poor than both the SMC centre and the outskirts of the LMC.§ DISCUSSION§.§ The SMC outskirts We have shown that stars older than ∼ 1 Gyr in the MB region, with radial velocities larger than 100 km s^-1, are unequivocally linked with the SMC. Except for one star with a velocity of 270 km s^-1, we do not find objects with kinematics compatible with that of the LMC. This is in agreement with the results obtained by <cit.> in the inner SMC, but contrasts with claims in <cit.> and dynamical simulations from <cit.>. Note, however, that the stars analysed here are located almost 2 further from the SMC centre than those studied by <cit.>.<cit.> found a negative metallicity gradient in the inner SMC regions at ≤4 from its centre, and an inversion in the outer regions between 4 and 5 towards the Bridge. This can be interpreted as the presence of metal-rich stars in the outer regions stripped from the inner regions of the SMC. These results are in good agreement with the findings of <cit.> who used red clump stars. We do not find stars more metal-rich than -0.9 dex and so we disfavour them coming from the inner ∼2. From the synthetic CMD fitting technique, we conclude that the star formation histories of the fields studied here are similar to those obtained by <cit.> at a distance of ∼2.5 and by<cit.> at ∼6 from the centre, both in the southern direction. However, the metallicity distributions obtained in the present work are more similar to the fields located at ∼4 in the same direction. This is in agreement with the stellar population with a shallow density profile reported by <cit.> at a radius between 3 and 75. The shallow extended component of the SMC described above can be explained both by a bound stellar halo or by extra-tidal stars. To investigate further these two scenarios, we have computed the velocity dispersion of the stars with radial velocities larger than 100 km s^-1 (excluding the star with a velocity of 270 km s^-1). There is no significant difference between the two fields studied (Table <ref>). All together, these stars have an average velocity of 151.7 km s^-1 and a dispersion of 33.3 km s^-1. The velocity dispersion is in good agreement with the value obtained for the SMC σ=26.4 km s^-1 by <cit.> but the systemic velocity, ⟨ V_r⟩=147.8 km s^-1, is slightly lower than we find here for the inter-Cloud population. To further explore the above, we obtained the average velocity and dispersion in three metallicity bins:-1.85 <[M/H]< -1.2; and [M/H] > -1.2, as listed in Table <ref>. Since the SMC has a well defined age-metallicity relationship <cit.>, these three metallicity bins can also be thought of as a proxy for three age bins. We find that the most metal-poor stars have a mean radial velocity very different from the other stars, and a small dispersion of just 9.6 km s^-1 (see Table <ref>). This suggests that these owe to foreground contamination from Milky Way halo stars. For the other two more metal-rich bins, we findvelocity dispersions in remarkable agreement with that of the SMC: 26.2 km s^-1 and 25.4 km s^-1, respectively (see Table <ref>). The excellent agreement between the velocity dispersion of the more metal-rich inter-Cloud stars and that observed for stars in the SMC reinforces our hypothesis that these inter-Cloud stars originated in the SMC. The SMC has a stellar mass of M_* ∼ 4.6 × 10^8 M_⊙ <cit.>. Thus, we expect from abundance matching that it would have inhabited a dark matter halo of mass M_200∼ 7-9 × 10^10 M_⊙ before infall <cit.>, consistent with dynamical models of the SMC <cit.>. Similarly, the latest abundance matching and dynamical mass estimates for the LMC place it in a M_200∼ 2 × 10^11 M_⊙ halo before infall <cit.>. Treating the Clouds as point masses moving on a pure radial orbit, and setting the tidal radius of the SMC to r_t = 2.1 kpc (based on the similarity of the stars in the inter-Cloud region with stars ∼2 from the SMC centre), we can derive the pericentric radius of the recent SMC-LMC encounter that formed the MB as: r_p ∼ r_t(M_ LMC/4 M_ SMC)^1/3∼ 1.8 kpc <cit.>. This is in excellent agreement with recent models of tidal interactions between Clouds, where such a close encounter has been proposed to explain the LMC's off-centre stellar bar <cit.>. This lends further support to a tidal origin for the inter-Cloud stars.§.§ Comparison with simulations The kinematics, photometry and chemistry of stars in the MB all point to them having been tidally stripped from the SMC. In this section we perform a more direct comparison with the dynamical simulations from <cit.>. In particular, we compare the location of tidally stripped stars in the MB region of the simulations with those in the locations of the fields sampled in this paper.Although the <cit.> simulations are based on the “bound scenario", in their favoured model the LMC and SMC suffer strong tidal interactions only recently, with two close passages ∼2 Gyr and ∼250 Myr ago. Portions of the gaseous disk of the SMC are stripped away during each of these strong encounters. In particular, the most recent one forms the gaseous bridge.Since the gravitational field of the LMC acts in the same manner on gas and stars, a tail of stars pulled out from the SMC into the MB is expected. <cit.> assume a multi-component SMC, consisting of a spherical dark matter halo, central disc and a more extended spheroidal component. They generated three different models where the size of the spheroid was varied: “model 1" with a Plummer scale radius of 7.5 kpc; “model 2" with 5.0 kpc; and “model 3" with 2.5 kpc, respectively. This is because it is still unclear how stellar populations of different ages and metallicities are distributed within the tidal radius of the SMC. The top panel of Fig <ref> shows the predicted distribution of stars for each model. It is clear that stripped stars in all three cases are found in the MB region. However, the number of stars stripped in model 3 is much lower than for the other two models, owing to its more concentrated spheroidal population. Another interesting result is that in all cases, stripped SMC stars are captured by the LMC. This is in agreement with the peculiar stellar populations reported in the LMC by <cit.>. In all three models, the SMC maintains an approximately spheroidal outer population after the interaction with the LMC. However, clear tidal distortions are observed towards the LMC in models 1 and 2. The position of these tails agree relatively well with location of the observed gaseous bridge (black connected dots). On contrast, the models do not predict a significant number of tidally stripped object in the position of the recently discovered RR Lyrae bridge (gray connected dots). Although model 3 remains dense and compact, stripped stars in the MB are still present. In model 1, the edge of the spheroid almost reaches the position of our fields (open black circles). In this case, the populations there would be a mix of both bound and unbound stars. By contrast, models 2 and 3 predict mainly tidally stripped stars from the SMC at the positions of our observed fields. This is not a particularity of the <cit.> simulations. All recent models based on the latest knowledge of the Magellanic System also predict the existence of tidally stripped stars in the inter-Cloud region to a greater or lesser extent <cit.>.These simulations also provide valuable information about the expected radial velocities of the model stars at the positions of our fields. The velocity distributions predicted for each model at the locations of fields 0224-7200 and 0224-7400 are shown in Fig.<ref>. In the three models, the bulk of stars have radial velocities between ∼90 and ∼200 km s^-1. However, model 1 predicts a continuum tail towards ∼300 km s^-1 that is not observed in our fields. By contrast, model 2 only predicts a few stars between 200 and 300 km s^-1 while model 3 does not predict any. Our kinematic results favour models 2 and 3 over model 1. However, the spatial distributions observed in the SMC periphery favour models 1 and 2 over model 3 <cit.>. Thus, taking the spatial and velocity data together, model 2 is closest to our observations. §.§ Dwarf/dwarf stripping The tidal stripping of satellite galaxies is expected to be a common process <cit.>. Indeed, we see direct evidence for such stripping events in the Milky Way <cit.>, M31 <cit.>, and other spiral galaxies in the Local Volume <cit.>. Being the most numerous type of galaxies, interactions between dwarf systems have been widely reported <cit.>. In fact, stellar tidal tails have been associated with the interaction of the Magellanic analog system formed by NGC 4485 and NGC 4490 <cit.>. The findings shown in this work confirm for the first time the existence of a stellar population older than 1 Gyr in the Magellanic inter-Cloud area, unequivocally related to the SMC. Moreover, these stars have a metallicity distribution similar to that of the SMC outskirts, as shown in Fig. <ref>. This represents the first spectroscopic evidence for dwarf-dwarf stripping in the Universe.§ CONCLUSIONS We have used medium resolution spectra for 514 red giant stars in the Magellanic Bridge (MB). Our key findings are as follows:* The chemistry and kinematics for 39 of the target stars are consistent with those located ∼ 2 deg from the centre of the SMC, but inconsistent with LMC stars. We conclude, therefore, that these stars were tidally stripped from the SMC. * We used the above to estimate the tidal radius of the SMC, finding r_t ∼ 2.1 kpc. Using the latest estimates of the pre-infall masses of the LMC and SMC, we then estimated the closest passage between the Clouds to be r_p ∼ 1.8 kpc. Such a close encounter has been invoked to explain the LMC's off-centre stellar bar <cit.>. * We compared the spatial location and kinematics of stars in the MB region to the simulations from <cit.>. We found that their “model 2” provided the best qualitative match to our data. In this model, in addition to a stellar disc and dark matter halo, the SMC has an outer spheroidal population with a Plummer scale length of 5 kpc. Models with a more concentrated spheroid produce too little tidal debris in the MB, while those with a more extended spheroid produce a tail to large radial velocities that is not observed. * Our results represent the first spectroscopic evidence for a dwarf galaxy being tidally stripped by a larger dwarf. § ACKNOWLEDGEMENTS We would like to thank the anonymous referee for their helpful comments that have improved this paper. We would also like to thank J. Diaz and K. Bekki for providing us with their N-body models. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2013-2016) under grant agreement number 312430 (OPTICON). BCC acknowledges the support of the Australian Research Council through Discovery project DP150100862. 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"authors": [
"Ricardo Carrera",
"Blair C. Conn",
"Noelia E. D. Noël",
"Justin I. Read",
"Ángel R López Sánchez"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170726115913",
"title": "The Magellanic Inter-Cloud Project (MAGIC) III: First spectroscopic evidence of a dwarf stripping a dwarf"
} |
Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana,SI-1000 Ljubljana, Slovenia Department of Physics and BME-MTA ExoticQuantumPhases Research Group, Budapest University of Technology and Economics, 1521 Budapest, HungaryDepartment of Physics, Faculty of Mathematics and Physics, University of Ljubljana,SI-1000 Ljubljana, SloveniaDepartment of Physics, Faculty of Mathematics and Physics, University of Ljubljana,SI-1000 Ljubljana, SloveniaDepartment of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USAWe developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systemswhere the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the un-driven (non-perturbed) Hamiltonian,while considering terms up to a finite order in the kick strength. As an application of the replica expansion, weanalyze an Ising spin 1/2 chain periodically kicked with magnetic field of strength h, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, the heating rate is nonperturbative in the kick strength bounded from above by a stretched exponential: e^- const h^-1/2. This guarantees existence of a very long pre-thermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion.Replica resummation of the Baker-Campbell-Hausdorff series Anatoli Polkovnikov December 30, 2023 ==========================================================Introduction.— Time-periodic modulation of interactions is a powerful tool to engineer properties of materials in both artificial and condensed matter systems <cit.>. In particular, high frequency driving is the cornerstone of various experiments and proposals inducing interactions such as spin-orbit coupling <cit.>, artificial gauge fields for uncharged particles <cit.>; it has been applied to dynamically tune or suppress hopping amplitude in optical lattices <cit.>, and also to change topological properties of materials <cit.>.Given a periodic driving protocol, however, determining the Floquet Hamiltonian that governs the stroboscopic evolution is usually a highly non-trivial task. Except for some special integrable cases <cit.>, one is compelled to apply approximate methods, e.g. variants of high frequency expansion (Magnus <cit.>, van Vleck <cit.> or Brillouin-Wigner <cit.> expansions). These provide a local effective Hamiltonian in each order of the expansion, however, until recently, not much had been known about the convergence properties of these series. A conjecture based on the generalization of the eigenstate thermalization hypothesis suggests that generic closed periodically driven systems heat up in the thermodynamic limit, i.e. they approach a completely structureless, infinite temperature steady state <cit.>. The convergence of the expansions of the effective Hamiltonian is intimately related to heating. Recently upper bounds on heating had been reported in the linear response regime <cit.> and for the Magnus expansion <cit.>, with the central result that the heating is at least exponentially suppressed in the driving frequency for periodically driven models characterized by local Hamiltonians with bounded energy spectrum. This theoretical finding implies that one can engineer nontrivial phases of matter, which remain stable for the experimentally relevant timescales.In some situations heating seems to be either absent completely or remain well below exponential bounds <cit.>. Another recent theoretical work showed that nontrivial non-equilibrium Floquet phases can be stabilized by weak coupling to environment <cit.>.One of the most studied driving protocols is a time-periodic sequence of sudden quenches between different Hamiltonians <cit.>, which is interpreted as kicked dynamics if one of the time intervals on which the Hamiltonians act is much shorter than the other, or equivalently, the strength of one of the Hamiltonians is smaller than the other. Such protocols naturally appear e.g. in the context of digital quantum simulation in trapped ions <cit.> and are frequently realized in other experimental platforms (see e.g. Refs. <cit.>).In the present work a novel expansion for the effective Floquet Hamiltonian is introduced for periodically kicked systems, which clearly outperforms the traditional high frequency expansions in a wide parameter range, as illustrated using two examples shown in Figure <ref>. Our approach uses the replica trick to calculate the logarithm of the time evolution operator describing a single period. The small parameter is the kick strength and we do not assume high frequency driving. The Magnus expansion is equivalent to the Baker-Campbell-Hausdorff (BCH) series in periodically kicked systems, and the replica expansion can be thought of as an infinite resummation of the BCH formula.As such, the possible applications of the replica expansion can reach far beyond periodically driven systems, including the theory of differential equations <cit.>, Lie group theory <cit.>, analysis of NMR experiments <cit.> or estimation of Trotterization errors in various numerical integration schemes. As an application of the replica expansion, we establish a conjecture for a non-perturbative – stretched exponential exp(- const h^-1/2) in the kick strength h – upper bound for the heating rate in the kicked tilted field Ising model. Our conjecture supplements the bounds introduced in Refs. <cit.>, which address only the regime of high frequency driving. Kicked dynamics.— The effective Floquet Hamiltonian evolving the kicked system is given by the logarithm of the stroboscopic time evolution operator over one period U=U_0 U_1=e^-i J H_0e^-i h H_1 as H_F=ilog(e^-i J H_0e^-i h H_1),where we incorporated the time intervals of the Hamiltonians H_0,1 to the coupling constants J,h. The BCH formula provides a series expansion for H_F assuming both J and h are small as H_F= J H_0+h H_1-i J h 1/2[H_0,H_1]-Jh1/12(J[H_0,[H_0,H_1]]+h[H_1,[H_1,H_0]]) +…In the usual setup of kicked systems, where J≫ h, that is, at intermediate frequencies and weak kick strengths, the terms with low order in H_1 but high order in H_0 are not negligible. A series expansion in the small parameter of h could be obtained formally by a resummation of the BCH series in H_0. The first order resummation is well known <cit.>,H_F=J H_0+-i J _H_0e^-i J_H_0/e^-i J_H_0-1h H_1+(h^2)where _X(Y)=[X,Y] is the Lie derivative. However, to the best knowledge of the authors, closed form expressions for the infinite resummation in higher orders of H_1 have not yet been reported in the literature.Replica expansion.— We tackle this problem by constructing a series expansion in h in Eq. (<ref>). Because of the noncommutativity of H_0 and H_1, the higher order derivatives of the logarithm of the time evolution operator cannot be obtained easily.To circumvent this obstacle, we apply the replica trick to express the logarithm, log U=lim_ρ→ 01/ρ(U^ρ-1). This idea has been proven to be uniquely useful in various fields of science, such as in the statistical physics of spin glasses <cit.>, machine learning <cit.>, and also in calculation of the entanglement entropy <cit.>.Assuming that the replica limit (∙)≡lim_ρ→ 01/ρ(∙) commutes with the differentiation, the series expansion of the Floquet Hamiltonian in Eq. (<ref>) reads H_F^(n)=∑_r=0^n h^r Γ_r ,withΓ_0=J H_0 andΓ_r= 1/r!∂^r_h U^ρ. The derivatives of the powers of the time evolution are easy to calculate at integer values of the replica index ρ, and the replica limit is taken following an analytical continuation to arbitrary real values. After evaluating the derivatives and neglecting the prefactor U_0^ρ, which approaches the identity operator in the replica limit, the r^th order correction reads Γ_r=(-i)^r-1/r!∑_0≤ m_1≤… m_r<ρm_rm_r-1…m_1 c_m_1 … m_rwhere m=U_0^-m H_1 U_0^m and c_m_1 … m_r=r!/n_0!n_1!… is the multinomial coefficient with n_s being the number of indices taking value s. Our method provides a remarkably simple derivation of the known first order term in Eq. (<ref>) giving a certain degree of confidence in the replica expansion (see <cit.>).It is not clear directly from Eq. (<ref>) if higher order terms can be expressed as nested commutators (multiple Lie derivatives), which would be expected from the resummation of the BCH series. One can show that the corrections to the Floquet Hamiltonian can be represented as sums of terms containing the commutators and extra terms, which vanish in the zero replica limit. For example at second order Γ_2=-i/2{∑_0 ≤ m_1 ≤m_2<ρ [m_2,m_1]+ (∑_0 ≤ m_1 <ρm_1)^2} ,where the second term is proportional to ρ^2, assuming that the replica trick works at first order, that is, ∑m∼ρ. Similar transformations that produce the nested commutator expression are given in the supplemental material up to the 5^th order.The resulting expansion of the replica Floquet Hamiltonian expressed in terms of nested commutators reads Γ_0 =J H_0 Γ_1 =∑_0≤ m < ρm Γ_r =(-i)^r-1/r!∑_0≤ m_1≤… m_r<ρ [m_r,… [m_2,m_1]]] c_m_2 … m_rwhere c_m_2 … m_r=(r-1)!/n_0!n_1!…. The expansion can be constructed similarly for different initial phases of the driving, U'=e^-i J H_0 (1-φ)e^-i h H_1 e^-i J H_0 φ=e^-i H_F', which results in the same equations as Eq. (<ref>) except for a simple substitution m_i→m_i+φ. Having established the first main result of this Letter, we now demonstrate its performance in the example of the kicked Ising model in a tilted field <cit.>.Kicked Ising model.—The time evolution is characterized by time-periodic quenches between the Hamiltonians H_0,1, H_0 =∑_i σ_i^z σ_i+1^z H_1 =∑_i σ_i^x+ σ_i^z ,whereandare shorthand notations for cosθ and sinθ. The purpose of introducing the tilt angle is to break the integrability of the modelatθ= 0 and π/2 <cit.>. Figure <ref> shows the performance of the replica expansion for the kicked Ising model in two different limits: when the kick parameter is the magnetic field h, or the Ising interaction J.The spectral norm of the difference between the approximate and exact time evolution operators, Δ_n=U-U^(n) shown in Figure <ref>, bounds the accuracy of the expansion for the dynamics of any observable A, as |A(t)-A_n(t)|≤ 2 t AΔ_n +(Δ_n^2), where A(t) (A_n(t)) is the expectation value of the observable following t periods with respect to the exact (approximate) time evolution, starting from an arbitrary initial state [Although Δ_n scales linearly with the system size L, this accounts for expectation values of nonlocal operators. The accuracy of the dynamics of local observables is independent of system size <cit.>.].As the two cases, kicking with H_0 or H_1, show qualitatively similar behavior, we discuss here only the replica expansion for kicking magnetic field. The time evolution of kicking Hamiltonian H_1 with respect to unperturbed dynamics H_0 for m periods reads explicitlym= ∑_iσ_i^z-(1/2sin(4mJ)(σ_i-1^zσ_i^y+σ_i^yσ_i+1^z)+ sin^2(2mJ)σ_i-1^zσ_i^xσ_i+1^z -cos^2(2mJ)σ_i^x),and is the main building block of the replica expansion. The computation of the nested commutators of these objects is trivial, following which one has to deal with the multiple sums and the replica limit. It is convenient to separate the operator part of the expansion from the replica coefficients by expressing H̃_m in Fourier harmonics as m=_0+e^i 4 J m_1+e^-i 4 J m_-1, where _0 =∑_i σ_i^z+/2(σ_i^x-σ_i^z σ_i+1^xσ_i+2^z) _± 1 =∑_i /4[σ_i^x+σ_i^z σ_i+1^xσ_i+2^z± i (σ_i^zσ_i+1^y+σ_i^yσ_i+1^z)], which brings us to the simplest formulation of the replica expansion,Γ_1 =∑_x_1_x_1_x_1 Γ_r =(-i)^r-1/r!∑_x_1,x_2,…,x_r_x_1 x_2 … x_r [_x_r,… [_x_2,_x_1]]where x_i∈{0,± 1} and we introduced the replica sum_x_1 x_2 … x_r =∑_0≤ m_1 ≤… m_r < ρe^i 4J m_1 x_1e^i 4J m_2 x_2… e^i 4J m_r x_r c_m_2 … m_r .These sums are evaluated gradually (with attention to the combinatorial factors) as ∑_m_j=0^m_j+1-1 m_j^y e^im_j =(∂/i ∂_)^ye^im_j+1-1/e^i -1 ,with m_r+1=ρ, and J̃ is an integer multiple of 4J. The prefactor m^y, 0≤ y∈ℕ may arise from the previous sum with respect to m_j-1, e.g. from the sum of constant terms. This way of evaluating the sums already defines the analytical continuation to arbitrary real values of ρ, allowing one to take the replica limit .This analytical continuation leads to e^i ρ-1=log e^i =i, which tries to enforce a Floquet Hamiltonian continuous in J at a price of breaking the periodicity H_F(J)=H_F(J+2π). Alternatively, one can choose a different branch of the logarithm, e.g. which folds J into the interval (-π,π] by applying a different analytical continuation <cit.>. This ambiguity in choosing the branch of the logarithm can be potentially used to further improve the expansion, restore the periodicity in J and eliminate divergences which are discussed below.The sum is especially simple in the first order correction: _0=0, _± 1=2J( 2J ∓ i), yieldingΓ_1=∑_i a_+σ_i^x+ a_-σ_i-1^zσ_i^x σ_i+1^z+σ_i^z+ J (σ_i-1^zσ_i^y+σ_i^yσ_i+1^z)with a_±=(J 2J ± 1/2). The second order correction is written in a compact form by noticing that_± 1^†=_∓ 1 and _x_1,x_2^*=_-x_1,-x_2,Γ_2=-i/2{(_10-_01)[_0,_1]+_1-1[_-1,_1]}+h.c.The replica coefficients are evaluated as_10-_01 =(1-2J2J)(1+i2J) _1-1 =1/2-isin 4J-4J/4sin^2 2J ,which finally yieldsΓ_2= ∑_ia_- (σ_i^y σ_i+1^x σ_i+2^z+σ_i^z σ_i+1^x σ_i+2^y) -a_- [σ_i^y+σ_i^z σ_i+1^y σ_i+2^z+2J(σ_i^x σ_i+1^z +σ_i^z σ_i+1^x )]+ (b+c)σ_i^z σ_i+1^xσ_i+2^x σ_i+1^z -b σ_i^y σ_i+1^y-c σ_i^z σ_i+1^z .The coefficients are b=^2/84Jcos 4J-sin 4J/sin^2 4J and c=^2/44J-sin 4J/sin^2 4J. The higher order corrections can becalculated similarly <cit.>. Notice that the first order correction diverges near J_k,1=kπ/2, which was identified as a signal of a heating (or nonergodicity–ergodicity) transition in a different spin model <cit.>, similar to the divergence of high-temperature expansion signaling phase transition in statistical physics.The source of this divergence is easily identified as the zero of the denominator in Eq. (<ref>). Similar to the high frequency expansion, the higher order corrections become less and less local due to the increasing number of commutators. The degree of divergence at J_k,1 also increases with the order, as the denominators from the consecutive sums become multiplied, and it can also increase because of the derivative in Eq. (<ref>), leading to a divergence ∼ |J-k π/2|^-r at the r^th order of expansion. These divergences restrict the convergence radius of the expansion.A finite convergence radius would imply no heating in the domain of convergence and would suggest the existence of a heating transition in the parameter J. However, at order r, in addition to the r^th order divergence at J=kπ/2, additional lower order divergencies may appear at J_k,m=kπ/2m, m=1… r. Consider e.g. the replica sum _11=2J ( 4J-i) appearing in the second order expansion, which diverges at J_k,2=k π/4. Many of these possible divergences do not enter the expansion because of the vanishing commutators in the operator part or due to cancellations, e.g. [_1,_1]=0, [_1,[_1,_0]]=0, etc. For instance, the divergence at k π/4 only appears at the 5^th order of the expansion, see Figure <ref>. In spite of the cancellations we conjecture that new divergences keep appearing in increasing orders similar to the dual case with interaction kicks (Figure <ref>(b)), and the expansion blows up near every rational fraction of π/2 (similar to KAM series).In spite of the divergences at high orders of the expansion, it can provide a very accurate estimate of the Floquet Hamiltonian at low orders. Moreover, the divergences determine the order at which to stop the expansion. That is, given a fixed J, similar to the method introduced in Ref. <cit.>, one can introduce an optimal order of expansion n^*, up to which the corrections increase the accuracy of the approximation of the Floquet Hamiltonian. It is natural to assume that the width of the resonances is proportional to the small parameter h, which is further supported by the analysis of the magnitude of the corrections Γ_r [We give a more rigorous discussion and estimate about the magnitudes of Γ_r in a subsequent publication.]. As an illustration, we give the scaling of the Hilbert-Schmidt norm Γ_r_ HS = √( Γ^†_r Γ_r)∼ f^r_J(J-π/4) near the resonance π/4,f^r_π/4(δ J) =c_π/4(r)/δ J^r-4+𝒪(δ J^-(r-5)) .The r dependence of the prefactor is illustrated in <cit.>. Up to the highest order we had access to, we found c_J(r) to decrease with r. For our purposes it is enough to assume that it grows at most exponentially ≲α^r, and we expect that at high orders this exponential growth indeed appears as the asymptote of c_J(r). Then the series ∑_r Γ_rh^r diverges for δ J< α h, which gives the width of the resonances. The optimal order of the expansion is hence estimated by the maximal order at which the closest resonance is located further than ∼α h. As the resonances appear at the rational fractions of π/2, J=kπ/2m, where m=1,…, n at the n^th order of the expansion, the question is how far one can get in the expansion without having a resonance approaching a fixed J.Rational approximation of irrational numbers has been thoroughly studied in the mathematical literature <cit.>, and is the cornerstone of the KAM theorem in classical dynamical systems, where the stability of the(quasi)periodic motion to integrability-breaking perturbations depends on the irrationality of the corresponding frequencies. The irrationality of a number is defined by how difficult it is to approximate by rational numbers. The number x is of type (K,ν) if it satisfies |x-p/q|>K q^-ν for all integer pairs (p,q) <cit.>.For example, the most irrational number in this sense is the golden ratio, which is of type (1/√(5),2). Such badly approximable numbers are generic in the sense that for any ν>2, almost all irrational numbers x are of type (K,ν) for some K <cit.>. In the following we choose a J for which 2J/π is of type (K,ν), such that|J-kπ/2 m|=π/2|2J/π-k/m|>π/2K/m^ν .Hence J is not affected by any resonances as long as n<n^*n^*=(π K/2α h)^1/ν ,which we set as the optimal order of expansion. By the construction of the expansion,U-e^-i H_F^(n)∼ h^n+1 ,which givesU-e^-i H_F^(n^*)∼ h^n^*+1∼ h^C/h^1/ν≲ e^-C'/h^1/2-ϵat the optimal order with some constants C, C' and arbitrary ϵ>0, by choosing ν close enough to 2. Consequently, the Floquet Hamiltonian in the optimal order is conserved for stretched exponentially long times in the inverse kick strength, and, if the steady state is the infinite temperature ensemble, it is approached at least stretched exponentially slowly. In the above analysis we gave an estimate for the accuracy of the replica expansion. We leave a more rigorous mathematical analysis, similar to the ones in Refs. <cit.>, to future work. Conclusion.— We have developed a novel expansion applicable to periodically driven systems where the driving consists of sudden quenches between different Hamiltonians. The expansion takes into account all orders in one of the Hamiltonians and is perturbative in the other. As such, it is an infinite resummation of the BCH formula, whose coefficients can be reproduced by taking the derivatives of the terms in the replica expansion <cit.>. We demonstrated that, similar to the high frequency expansions, the replica expansion isasymptotic for systems with unbounded Hamilton operators, that is, it may not converge, but performs very well when evaluated at an optimal order. The expansion suffers from resonances near rational frequencies, whose avoidance determines the optimal order of expansion. It is an interesting question whether these resonances have a physical meaning orare just an artifact of the expansion, and whether one could remove the resonances by a proper choice of analytical continuation in the replica trick. This research has been supported by the grants P1-0044 and N1-0025 of Slovenian Research Agency (ARRS), Hungarian-Slovenian (OTKA/ARRS) bilateralgrant N1-0055, ERC AdG grant OMNES, and grants by the Hungarian National Research, Development and Innovation Office - NKFIH K119442, SNN118028. A.P. was supported by NSF DMR-1506340, ARO W911NF1410540 and AFOSR FA9550-16-1-0334. apsrev § SUPPLEMENTARY MATERIAL FOR "REPLICA RESUMMATION OF THE BAKER-CAMPBELL-HAUSDORFF SERIES" § EVALUATION OF THE REPLICA LIMIT IN THE FIRST ORDER CORRECTIONThe first order correction in the replica trick from Eq. (<ref>) is simply written asΓ_1=∑_0≤ m <ρm=∑_0≤ m <ρ (U_0^ad)^m H_1=lim_ρ→ 01/ρ(U_0^ad)^ρ-1/U_0^ad-1=log(U_0^ad)/U_0^ad-1where we expressed m by the adjoint action of U_0 as m=U_0^-mH_1 U_0^m=(U_0^ad)^m H_1 and the substitution of U_0^ad=e^i Jad_H_0 brings us to Eq. (<ref>) of the main text.§ NESTED COMMUTATOR EXPRESSION OF REPLICA EXPANSIONA direct evaluation of the series expansion of the integer powers of the time evolution operator leads to Eq. (<ref>) of the main text. However, similar to the BCH expansion, it is expected that the corrections can be written in terms of nested commutators, which produce local operators if the original Hamiltonians are local. We denotethe sums in Eq. (<ref>) by η_r before evaluating the replica limit, that is, Γ_r=(-i)^r-1/r!η_r, andη_r=∑_0≤ m_1≤… m_r<ρm_rm_r-1…m_1 c_m_1 … m_r .The corresponding nested commutator expressions from Eq. (<ref>) are denoted by η̃_r,η̃_r=∑_0≤ m_1≤… m_r<ρ [m_r,… [m_2,m_1]]] c_m_2 … m_r .We show that η_r can be expressed as a sum of η̃_̃r̃ and terms which vanish in the replica limit, that is, η_r=η̃_r.A necessary condition for the replica trick to work up to the r^th order is to have an analytical continuation for which η_r=(ρ). This means that polynomials of at least second order of {η_1,…,η_r-1} vanish in the replica limit. The second order correction was discussed in the main text, in this notation η̃_2 =∑_0 ≤ m_1 ≤m_2<ρ [m_2,m_1] η_1^2=∑_0 ≤ m_1 ≤m_2<ρm_2m_1+∑_0 ≤ m_1 <m_2<ρm_1m_2 η_2=∑_0 ≤ m_1 ≤m_2<ρm_2m_1(2-δ_m_1,m_2)= η̃_2 +η_1^2Below we list the combinations of the operators η_s which produce the nested commutators in Eq. (<ref>) of the main text, up to the 5^th order (note the non-commutativity of the different η_ss). η̃_1= η_1 η̃_2= η_2-η_1^2 η̃_3= η_3-2 η_2 η_1-η_1 η_2+2η_1^3 η̃_4= η_4-3η_3 η_1-η_1 η_3-3 η_2^2+6η_2 η_1^2+3 η_1 η_2 η_1+3 η_1^2η_2-6η_1^4 η̃_5= η_5-4η_4 η_1-η_1 η_4-6η_3 η_2-4η_2 η_3+12η_3 η_1^2 +4η_1 η_3 η_1+4η_1^2 η_3+12η_2^2 η_1+12η_2 η_1 η_2+6 η_1 η_2^2 -24η_2 η_1^3-12η_1 η_2 η_1^2 -12 η_1^2 η_2 η_1-12η_1^3 η_2+24η_1^5We note that if the operators η_s commute then these equations reduce to the cumulant expansion.§ EXTRACTION OF THE BCH COEFFICIENTS FROM THE REPLICA EXPANSIONIt is easy to check that series expansion, or equivalently, the partial derivatives of the corrections Γ_r at J=0 reproduce the commutators from the BCH formula containing r instances of H_1, e.g. ∂_J Γ_1|_J=0 = ∑_j σ_j^yσ_j+1^z+σ_j^zσ_j+1^y=-1/2i [H_0,H_1] 1/2∂_J^2 Γ_1|_J=0 = -2/3∑_j σ_j^x+σ_j^zσ_j+1^xσ_j+2^z=-1/12[H_0,[H_0,H_1]] ∂_J Γ_2|_J=0 = 2/3∑_j(σ_j^x σ_j+1^z+σ_j^z σ_j+1^x)+ 2 (σ_j^y σ_j+1^y-σ_j^z σ_j+1^z) = -1/12[H_1,[H_1,H_0]] 1/2∂_J^2 Γ_2|_J=0 = 2/3∑_j(σ_j^y +σ_j^zσ_j+1^y σ_j+2^z) - (σ_j^z σ_j+1^x σ_j+2^y+σ_j^y σ_j+1^x σ_j+2^z) = -i1/24[H_1,[H_0,[H_0,H_1]]] § EXPECTATION VALUE OF LOCAL OPERATORSAn important measure of the accuracy of the replica expansion is how well it reproduces the dynamics of local observables. This can be quantified by the operator norm of the difference between the time evolved observable with respect to the exact and approximate time evolution: Δ̃_n(A)=U^adA-U^ad_(n)A, for a local operator A and notation U^adA ≡ U^-1A U. This measure is independent of the system size up to an exponentially small correction due to Lieb-Robinson bounds. In principle it could be possible that the resonances seen in Figure <ref> correspond to some non-local observables and they do not affect the dynamics of local operators. In contrast, we find that the accuracy measured by the time evolution of single spin operators σ_i^μ, μ∈{x,y,z} follows closely the estimate given by Δ_n, as shown in Figure <ref> for σ_i^x. The other spin components behave in qualitatively the same way. As discussed in the main text, the spectral norm of the difference of the exact and approximate time evolution operators Δ_n=U-U^(n) bounds the accuracy of the expectation value of any dynamical observable A, including non-local ones, as Δ̃_n(A)≤ (2 Δ_n +Δ_n^2)A. However, to account for the possibly non-local observables, Δ_n scales linearly with system size, and underestimates the accuracy of the replica expansion for the time evolution of local operators.§ HILBERT-SCHMIDT NORM OF THE CORRECTIONSThe Hilbert-Schmidt norm of corrections Γ_r are illustrated in Figure <ref> for both magnetic field and interaction kicks. Note that in the latter case the expansion parameter is J rather than h.The asymptotic behavior of the Hilbert-Schmidt norm near divergencies can be further analyzed, as in Eq. (<ref>) of the main text,f^r_J^*(δ J) =c_J^*(r)/δ J^r-r_J^*+𝒪(δ J^-(r-r_J^*)+1)f^r_h^*(δ h) =c_h^*(r)/δ h^r-r_h^*+𝒪(δ h^-(r-r_h^*)+1) ,where r_J^*+1 (r_h^*+1) determines the order of expansion at which the divergence at coupling J=J^* (h=h^*) first appears. The coefficient c_π/4(r) is plotted in Figure <ref> (a) as a function offor kicking magnetic field. It is more informative to analyze the model with interaction kicks, because we have much more resonances at hand in this situation. The coefficients c_h^*(r) are shown for that model in Figure <ref> (b). The new resonances appearing at higher orders are characterized by smaller coefficients than the previous ones, at least up to the 6^th order.§ PERIODICITY PRESERVING ANALYTICAL CONTINUATIONIn the manuscript we used the naive analytical continuation, that is, we evaluated the replica sums in Eq. (<ref>) with the prescription Eq. (<ref>), then we continued the resulting expression to arbitrary real values of the replica index ρ. This results in replica coefficients continuous in J except for the resonances. The denominators in Eq. (<ref>) do not break the 2π periodicity of the time evolution operator U in variable J. The linear in J factors originate in the replica limit of the expressions e^i 4 J z ρ-1,e^i 4 J z ρ-1= lim_ρ→ 0e^i 4 J z ρ-1/ρ=log e^i 4 J z ρ=i m_z(4 J z)where z is an integer. In the naive analytical continuation, for all values of z, we chose m_z as the identity function. However, we can choose a different branch cut of the logarithm, which is equivalent to dividing our analytical continued function by e^i 2πζρ, which is unity at integer values of ρ, and ζ is an integer defining the new analytical continuation. In principle we are allowed to define a different ζ for every single term in the expansion, here we consider it as a function of z. The periodicity H_F(J)=H_F(J+2π) can be enforced by substituting J by Jmod2π in the replica coefficients. This corresponds to the analytical continuation defined by m_z(4 J z)=4 z (Jmod2π).From the point of view of observables, the periodicity in J is shorter, since U only obtains a real phase factor (± 1) following a shift of π:U(J+π)=(-1)^b U(J), where b is the number of nearest neighbor bonds in the system. In open boundary conditions b=L-1 and in periodic boundary conditions (PBC) b=L. The period is further decreased in PBC, because then U(J+π/2)=i^L U(J) sincee^i π/2 σ_i^zσ_i+1^z=i σ_i^zσ_i+1^zand ∏σ_i^zσ_i+1^z is the identity operator if PBC is applied. Thus, at appropriate system sizes, the period of U in J is halved or even quartered, which can be similarly established by a corresponding analytical continuation, that is, by the proper choice of the function m_z. | http://arxiv.org/abs/1707.08987v1 | {
"authors": [
"Szabolcs Vajna",
"Katja Klobas",
"Tomaz Prosen",
"Anatoli Polkovnikov"
],
"categories": [
"cond-mat.str-el",
"cond-mat.quant-gas",
"cond-mat.stat-mech",
"quant-ph"
],
"primary_category": "cond-mat.str-el",
"published": "20170727182053",
"title": "Replica resummation of the Baker-Campbell-Hausdorff series"
} |
Department of Mathematics, Salt Lake City, UT 84112 [Gordan Savin][email protected] of Mathematics, Salt Lake City, UT 84112 [Michael Zhao][email protected] Binary Hermitian Forms and Optimal Embeddings Michael Zhao December 30, 2023 ============================================= Fix a quadratic order over the ring of integers.An embedding of the quadratic order intoa quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, thiscorrespondence isa discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.§ INTRODUCTION Let K be a field of any characteristic and fix L, a separable quadratic extension of K. Let n_L : L → K, n_L(l)=ll̅,be the norm map. In this paper we study relationship between embeddings of L into a quaternion algebra and vector spaces over L with a hermitian form,as well as an integral version of this problem.Let V be a vector space over L.A hermitian form on V is aquadratic form h: V → K such that h(lv) = n_L(l) h(v) for all l∈ L and v∈ V. This definition is different than what is customary inthe literature, where a hermitian symmetric form is a function s : V × V → L, linear in the first variable,and s(y,x)= s(x,y) for all x,y∈ V. However, given h, we show that there exists a unique s such that h(v)=s(v,v) for all v∈ V, even if the characteristic of K is 2. Let i: L → H be an embedding of L into a quaternion algebra H. Let n: H → K be the (reduced) norm. Then H is naturally a vector space over L of dimension 2,with a hermitian form h: = nrepresenting 1, sincen(1)=1. Conversely, let (V,v,h) be a triple consisting of a 2-dimensional vector space V over L, a non-degenerate hermitian form h on V, and a point v∈ V such thath(v)=1. Then one can define a quaternion algebra H_V such that H_V=V, as vector spaces over K,v is the identity element of H_V,and l ↦ l v is an embedding of L into H_V, wherel v is the scalar multiplication inherited from V. The reduced norm is, of course, equal to the hermitian form h. This discussion can be placed in a categorical context;the following two categories are isomorphic: one whose objects are pairs (i,L) where i is an embedding of L into a quaternion algebra, the other whose objects are triples (V,v,h) ofpointed, non-degenerate, binary hermitian spaces.We now observe that different choices of the point v representing 1 give isomorphic objects. Indeed, if u∈ V is another element in V representing 1, then u is a unit element in the quaternion algebraH_V arising from the triple (V, v, h). Right multiplication by u gives an isomorphismof the triples (V, v,h) and (V, u, h).As a consequence, there is a bijection between isomorphismclasses of embeddings of L into quaternion algebras and non-degenerate binary hermitian spaces (V,h) representing 1. As the next result, we establish an integral version of this bijection. More precisely, assume that K is the field of fractions of a Dedekind domain A, and Bthe integralclosure of A in L.An order O in H is an A-lattice, containing the identity element and closed under the multiplication in H. An embedding i: L → H iscalled optimal if i(L)∩ O = i(B). If this is the case, then O is naturally a projective B-of rank 2, such that the norm n takes values in A and represents 1.Conversely, let Λ⊂ V be a lattice, i.e. a projective B-module of rank 2, such that h is integral on Λ, i.e.h(Λ) ⊆ A. Such a pair (Λ, h) is called an integral binary hermitian form.Assume that there exists v∈Λ such that h(v)=1. We show that Λ is an order (denoted by O_Λ) in H_V, and thus we haveconstructed a quaternionic order and an embedding of B into it, given by b↦ bv. Just as in the field case, different choices ofv give isomorphic objects and, as a consequence,there is a bijection between isomorphismclasses of embeddings of B into quaternionic orders and non-degenerate integral binary hermitian forms(Λ,h) representing 1. We now turn to the question when a hermitian form represents 1, which is especially interesting in the integral case. To that end we establish a relationshipbetween invariantsof the objects studied. The determinant d(Λ, h) of an integral binary hermitian form (Λ, h) is a fractional ideal in K generated by(s(v_i, v_j)) for all pairs (v_1,v_2) of elements of Λ. We note that d(Λ, h) is not necessarily an integral ideal. Integrality of h implies only that s takes values in the different ideal of B over A in L. Since the discriminant D_B/A is the norm of the inverse different, the product D_B/A· d(Λ, h) is an integral ideal. This was previously observed in <cit.> where the discriminant of an integral binary hermitian form is defined as this product. Thus we follow the same convention and define the discriminant of(Λ,h)to be the integral idealΔ(Λ, h) =D_B/A· d(Λ, h). If h represents 1, so Λ is the order O_Λ, then we have an identity Δ( O_Λ)= Δ(Λ, h) of integral ideals where Δ( O_Λ) is the discriminant of the order O_Λ.If K=ℚ and L an imaginary quadratic extension, then these invariants can be refined to be integers, and this is an equality of two integers.The following is the most interesting result in this paper:Assume L is a tamely ramified, imaginary quadratic extension of ℚ. Let B be the maximal order in L.Let Δ<0 be a square free integer.There is a bijection between the isomorphism classes of embeddings of B into quaternionic orders of discriminant Δ and integral binaryB-hermitian forms of discriminant Δ.The key here is to show that an integral binaryhermitian form (Λ, h) of discriminant Δ represents 1.The condition Δ is square free assures that h represents 1 onΛ⊗ℤ_p for all p. The condition Δ <0 implies that h is indefinite on Λ⊗ℝ.The theorem follows from the integral version of the Hasse-Minkowski theorem for indefinitequadratic forms in 4 variables.§ HERMITIAN FORMSLet K be a commutative field of any characteristic. Let L be a separable quadratic K-algebra. In particular, there exists aninvolution l ↦l̅ on L such that the set of fixed points is K. A vector space over L is a free L-module.We go over some basic results about hermitian forms on vector spaces. Let V be a vector space overL.A hermitian symmetric form is a function s: V × V → L which is L-linear in the first argument and s(x,y) = s(y,x).Let h: V → K be the function defined by h(x)=s(x,x) for all x∈ V. This is a quadratic form, if we consider V a vector space over K. in particular,the function b(x,y) = h(x + y) - h(x) - h(y) is K-bilinear.If we let n_L(l) = l l, for l∈ L, thenh(lx) = n_L(l) h(x).These two facts are a sufficient characterization of hermitian forms for our purposes, due to theLet V be a vector space over L.Let h: V → K be a quadratic form, where V is considered a vector space over K, satisfying h(lx) = n_L(l) h(x) for all l ∈ K. Then there is a unique hermitian symmetric form s(x,y) such that h(x) = s(x, x).Assume that s exists. Then, for all x, y ∈ V and l ∈ L,s(x,y) + s(y,x) = b(x,y) l s(x,y) + ls(y,x) = b(lx, y).We view this as a 2× 2 system in unknowns s(x,y) and s(y,x). If l≠l̅ then this system has a unique solution given by s_l(x,y) = lb(x,y) - b(lx, y)l - l where the subscript l indicates that the solution, so far, depends on the choice of l.(Note that l - l is invertible, even when L≅ K^2.) We claim that s_l(x,y) is independent of l. Indeed, let l = c + d π, where L=K[π], and c,d∈ K. Then s_l(x, y) = l b(x, y) - b(lx, y)l - l = c b(x, y) + d π b(x, y) - b(cx + d π x, y)c + d π - c - dπ = d π b(x, y) - d b(π x, y)dπ - dπ = s_π(x, y).Hence s_l = s_π and we can write s for any of the s_l. It is then easy to check that s(x, x) = s_l(x, x) = h(x).Now we check that s(x,y) is linear in the first variable. It is clear thats(x_1 + x_2, y) = s_l(x_1 + x_2, y). Let l∈ L ∖ K. Then s(l x, y) = s_l(lx, y) = l b(l x,y) - b(n_L(l) x, y)l - l = l b(l x,y) - n_L(l) b(x, y)l- l = l l b(x, y) - b(lx,y)l -l = l s_l(x,y) = l s(x,y),which shows s is linear in the first argument.It remains to check that s(x, y) = s(y, x).Note that b(l x, l y) = n_L(l) b(x, y). Replacing x with lx, we find that b(n_L(l) x, l y) = n_L(l) b(lx, y), hence b(lx, y) = b(x, l y). Thens(y, x) = s_l(y, x) = l b(y, x) - b(l y, x)l -l = l b(x, y) - b(lx, y)l - l = s_l(x, y) = s(x, y),or s(x, y) = s(y, x), as desired. Uniqueness is clear.Thus, we can make the following definition. Let V be a vector space over L.A hermitian form is a quadratic form h: V → K, where V is a considered a vector space over K, such that h(lv) = n_L(l) h(v) for all l ∈ L, and for all v ∈ V, § QUATERNION ALGEBRASLet H be a quaternion algebra over K, i.e. a central simple algebra over K of dimension 4.We know that by general theory (e.g. <cit.>), every element x∈ H satisfies a reduced characteristic polynomial of degree two, associated to its action on the algebra via left-multiplication. The reduced trace (x) is the sum of the roots of the reduced characteristic polynomial of x. Let the reduced norm n(x) be the product of the roots.The trace furnishes H with an involution x := (x) - x such that n(x)=xx.Let L be a separable quadratic field extension of K and i: L → H an embedding. Since L is separable, the restriction of the involution of H to L is non-trivial.The norm n is multiplicative, i.e. n(xy)=n(x)n(y) for all x,y∈ H. Hencethe norm is a hermitian form on H. This hints at a correspondence between embeddings of L into quaternion algebrasand hermitian forms. The next section will formalize this idea. We finish this section with the following structural result. Let L be a separable quadratic field extension of K. If i: L → H is an embedding, then there exists u ∈ H such that u̅ =-u and θ∈ K^× satisfying H = L + Lu, u^2 = θ, and um = mu for all m ∈ L (cf. <cit.>). For every v=x+yu ∈L + Lu, n(v)= n_L(x) - n_L(y)θ, where n_L: L → K is the norm.By Proposition <ref>, there exists a unique, L-hermitian bilinear forms(v,w) : V × V → L such that s(v,v)=n(v) for all v∈ H. Take u≠ 0 with s(u, 1) = 0. Then H=L + Lu.For any l ∈ L ∖ K,0 = s(u, 1) = s_l(u, 1) = lb(u, 1) - b(lu, 1)l - l.Since b(u, 1) = n(u + 1) - n(u) - n(1) = (u) and b(lu, 1) = (lu), hence l(u) = (lu). If (u) ≠ 0, then l∈ K, a contradiction. Hence (u) = 0 and (lu) = 0. The first of these implies that u = -u, and so 0 = (lu) = lu + ul = lu - ul, i.e. that lu = u l, or u l= l u. Finally, since (u) = 0, u^2 = -n(u) ∈ K. Simplicity of H forces θ to be non-zero. The formula for n(v) follows from orthogonality of 1 and u.Note that the hermitian symmetric form s, attached to n, in the basis 1,u is given by a diagonal matrix with 1 and -θ on the diagonal. In particular,invertibility of θ implies that the hermitian form is non-degenerate.§ CORRESPONDENCE OVER A FIELD Here L continues to be a separable quadratic extension of K. The correspondence alluded to beforeProposition <ref> can be formalized as an isomorphism of the categories L of non-degenerate pointed hermitian spaces and L of embeddings of L into quaternion algebras over K.We will define these terms more precisely, and give a proof of the isomorphism.Let L be the category whose objects are pairs (i, H) where H is a quaternion algebra over K, and i : L → H is an embedding of L into H.The morphisms f : (i, H) → (i', H') are morphisms f : H → H' of K-algebras satisfying f ∘ i = i'. Note that any f must be an isomorphism of H with H'. Let L be the category whose objects are triples (V, v, h), called pointed hermitian spaces, where V is a two-dimensional L-vector space, h: V → K a non-degenerate hermitian form, and v ∈ V such that h(v) = 1. The morphisms from (V, v, h) to (V', v', h') are isomorphisms f: V → V' with f(v) = v' and h(x) = h'(f(x)).We can now define a functor F: L → L, due to the following easy lemma: Let F: L → L be given by F(i, H) = (H, 1, n), where n is the norm form of H, and 1 is the identity element of H. Given a morphism f: (i, H) → (i', H'), define F(f) := f. Then F is a functor.Now we construct a functor G: L → L by defining a multiplication · on a given (V, v, h), motivated by Proposition <ref>. Recall that there is a unique hermitian symmetric form s (v,w) on V with h(x) = s(x, x) for all x∈ V.Embed L into V via l ↦ lv where lv denotes the scalar multiplication.* v is the identity element, * for x ∈ V,and l∈ L, l · x := lx, the scalar multiplication, * for y ∈ L^⊥ = { w|s(v, w) = 0}, y · (lv) := ly, for l ∈ L.* for y ∈ L^⊥, y · y := -h(y)v.* conjugation on V is lv+y=lv -y,for l∈ L and y ∈ L^⊥, This gives V = L ⊕ L^⊥ a structure of quaternion algebra, as in Proposition <ref>, since we can choose u to be any element of L^⊥, and θ = -h(u). Non-degeneracy of s implies that θ≠ 0. The norm form is h.Let H_V denote this algebra. Then we have Define G(V, v, h) := (i_v, H_V), where i_v(l)=lv for l∈ L, and for a morphism f: (V, v, h) → (V', v', h'), define G(f) := f. Then G is a functor fromL to L. It is clear that the functors F and G are inverses of each other. Thus, we have The categories L andL are isomorphic.Let (V,v, h) be a pointed binary hermitian space and H_V the quaternion algebra with the identity element v arising from it.Let u∈ V be another element inV such that h(u)=1. Then v,u, understood as elements in H_V,satisfy v· u=u. Thus, the right multiplication by u gives an isomorphism of the pointed symmetric spaces (V,v,h) and (V, u, h).This shows that the isomorphism classes of objects in the category Ldo not dependon the choice of the point representing 1.Thus we have the following: The functors F and G give a bijection between the isomorphism classes of embeddings of L into quaternion algebras and the isomorphism classesof non-degenerate binary hermitian spaces (V,h) such that h represents 1. § CORRESPONDENCE OVER A DEDEKIND DOMAIN Assume that K is a field of fractions of a Dedekind domain A. Let B be the integral closure of A in L. It turns out anintegral version of the above isomorphism can be established, replacing the field with a Dedekind domain, quaternion algebras with orders, and binary hermitian forms on a vector space with integral binary hermitian forms on a B-projective module of rank 2. Let i: L → H, be an embedding of L into H.Recall that a lattice Λ in the quaternion algebra H is a finitely generated A-submodule containinga K-basis of H.If the lattice Λ is a B-module, then it is also projective of rank 2.An order in His a lattice containing the identity element and closed under the multiplication.Let B be the category whose objects are pairs (i, 𝒪) where i : B → is an embedding into an orderof quaternion algebra. The morphisms from(i, 𝒪) to(i', 𝒪')are isomorphisms f : 𝒪→𝒪' of A-algebras satisfying f ∘ i = i'. A binary B-hermitian form is a pair (Λ, h) where Λ is aprojective B-module of rank two, and h: Λ⊗_A K → K is an L-hermitian form. The form is integral if h(Λ) ⊂ A. In the above definition we have identified Λ with Λ⊗ 1 ⊂ V = Λ⊗_A K.Let B be the category whose objects are triples (Λ,v, h),called pointed integral hermitian spaces, where (Λ, h) is an integral, binary B-hermitian form, with h non-degenerate, and v ∈Λ with h(v) = 1. The morphisms from (Λ, v, h) to (Λ', v', h') are B-module isomorphisms f: Λ→Λ' preserving the pointed hermitian structure, i.e. for x ∈Λ, h(x) = h'(f(x)) and f(v) = v'. We will prove that the two categories are isomorphic. In one direction we havea functor given by the following:Let P: B → B be defined by P(i, 𝒪) := (𝒪, 1, n), where n is the norm of the quaternion algebra ⊗_A K, and for a morphism f: (i, 𝒪) → (i', 𝒪'), P(f) := f. Then P is a functor between these two categories.In order to define a functor Q: B → B, we need to define a multiplication on a pointed B-hermitian module (Λ, v, h). On V = Λ⊗_A K, as in the previous section, we have aquaternion algebra structureH_V arising from h and theassociated L-hermitian bilinear form s: V × V → L, such that v is the identity.We need the following:The B-module Λ, regarded as a subset of H_V (recall V = H_V as sets), is closed under the multiplication on H_V i.e. it is an order in H_V denoted by 𝒪_ΛWe first reduce to the case of A and B principal ideal domains.Letbe any prime of A, S_ = A -, and A_ = S_^-1 A, B_ = S_^-1 B the localizations of A and B respectively. Then B_ is a semi-local Dedekind domain, hence it is a principal ideal domain.Let Λ_ = S_^-1Λ. If we show, for x, y ∈Λ , that x · y ∈Λ_, for all , then using Λ = ⋂_Λ_, weconclude that x · y ∈Λ. Hence we can assume that B is a principal ideal domain. Then Λ is a free B-module of rank 2. Note that v is a primitive element of the lattice Λ since h(v) = 1. Hence vismember of a B-basis (v,w) of Λ <cit.>.Letv^⊥ = w - s(w, v) v, then s(v, v^⊥)= 0. Let γ = s(w, v) and θ = v^⊥· v^⊥ = -h(v^⊥). Then to check x · y ∈Λ, we just need to check thatw · w = (v^⊥ + γ v)· (v^⊥ + γ v)is an element of Λ_. But we can computew · w = (v^⊥ + γ· v) · (v^⊥ + γ· v)= v^⊥· v^⊥ + v^⊥·γ· v + γ· v · v^⊥ + γ· v ·γ· v= θ· v + γ· v^⊥ + γ· v^⊥ + γ^2 · v= (θ + γ^2) v + (γ + γ) v^⊥ = (θ + γ^2 - (γ)γ) v + (γ) w= (θ - n(γ)) v + (γ) wNow note that n(γ) = γγ= h(γ· v). Hence,-θ = s(v^⊥, v^⊥) = s(w, w) + s(-γ v, w) + s(w, -γ v) +s(-γ v, -γ v)= h(w) + h(γ v) - γ s(v, w) - γ s(w, v)= h(w) + n(γ) - 2 n(γ) = h(w) - n(γ),since s(v,w) = γ. Hence θ - n(γ) = -h(w), which is in A. Finally,(γ) = γ + γ= s(w, v) + s(v, w)= s(v + w, v + w) - s(v, v) - s(w, w)= h(v + w) - h(v) - h(w),which is also in A. Thus, the product from V preserves Λ. Then we have the lemmaLet Q: B → B be defined by Q(Λ, v, h) = (i_v, Λ),i_v(b)=b v forb∈ B, andfor a morphism f : (Λ, v, h) → (Λ', v',h'), define Q(f) := f. Then Q is a functor. Again, it is easy to check that the functors P and Q are inverses of each other. In particular: The categoriesB andB are isomorphic.Arguing in the same way as in the case of fields, Corollary <ref>, two pointed binary integral B-hermitian modules (Λ, v, h) and (Λ, u, h) are isomorphic. Thus the isomorphism classesin B are the same as the isomorphism classes of binary integral hermitian spaces (Λ, h) such that h represents 1.The functors P and Q give a bijection between the isomorphism classes of embeddings of B into quaternionic orders of and the isomorphism classesof binary integral B-hermitian spaces (Λ,h) such that h represents 1. All that remains is to examine in which situations there isexistence of a point with h(v) = 1. This can be done, in the field and domain cases, by using local-global principles.§ DISCRIMINANT RELATION We are in the setting of the previous section. Let Λ be a lattice in H. The fractional ideal d(Λ) in K generated by((u_i u_j)), for all quadruples(u_1, … , u_4) of elements in Λ is asquare. Assume firstly that A is a principal ideal domain. Then d(Λ) is a principal ideal generated by ((u_i u_j)) where (u_1, … , u_4) is any A-basisof Λ. Let Λ' be another lattice and (u'_1, … , u'_4) its basis. Let T: H → H be the linear transformation such that T(u_i)= u_i' for all i. Thend(Λ')= (T)^2 d(Λ).Hence, in order to show that d(Λ) is a square, it suffices to do so for one lattice. Write H=L + L u, as in Proposition <ref>.Let Λ= B + Bu. Assume that B= A[π] where π is a root of the polynomial x^2 + ax +b. Pick 1, π, u, π u as a basis of Λ. Then the matrix of traces is( 2 -a 0 0 -a a^2 - 2b 0 0 0 0 -2θaθ0 0 aθ-2 b θ ),and its determinant is (a^2 - 4b)(4b - a^2) θ^2 = -(a^2-4b)^2 θ^2.Then d(Λ)= (D_B/Aθ)^2 (as ideals) where D_B/A is the discriminant of B over A. The general case is now treated using localization. Letbe a maximal ideal in A. Then, one easily checks, d(Λ)_𝔭= d(Λ_𝔭). Since A_𝔭 is a local Dedekind ring, hence a principal domain, d(Λ_𝔭) must be an even power of 𝔭. The discriminant of an A-lattice Λ in H is the fractional ideal Δ(Λ) in K such that Δ(Λ)^2=d(Λ). If K=ℚ then we can refine the notion of Δ(Λ) to be a rational number, the generator of this ideal.We pick Δ(Λ) to be the positive generator ifH⊗_ℚℝ is the matrix algebra, and negative otherwise. Let Λ be a projective B-module of rank 2 and h a hermitian form on Λ, not necessarily integral. Let s be the bilinear hermitian form on Λ such thats(x,x)=h(x), for all x∈Λ. Let d(Λ, h) be the fractional ideal in K generated by (s(v_i,v_j)) for all pairs(v_1, v_2)of elements in Λ. If K=ℚ and L a complex quadratic extension we refine d(Λ,h) to be the positive generator of this ideal if h is a positivedefinitehermitian form on Λ⊗_ℤℝ and negative otherwise. In other words this sign is the sign of (s(v_i,v_j)), forany basis (v_1, v_2)of Λ⊗_ℤℝ. Let Λ be a lattice in H that is also a B-module. Then we have the following identity of fractional ideals in K:Δ(Λ)= D_B/A· d(Λ, n).If K=ℚ and L a complex quadratic extension, this is an identity of rational numbers.This is an identity of two fractional ideals, so it can be checked by localization at every prime ideal . The localizations A_𝔭 and B_𝔭 are a local and a semi-local, respectively, Dedekind domains.Thus they are principal ideal domains <cit.>.Hence, by localization, the proof can be reduced to the case of principal ideal domains. So we shall assume thatA and B are both principal ideal domains. In that case d(Λ, n) is a principal ideal generated by (s(v_i,v_j)) where v_1, v_2 is a B-basisof Λ.Let Λ' be another lattice in H that is also a B-module. Let v_1', v_2' be a B-basis of Λ'.Let T: H → H be the L-linear map defined by T(v_i)=v'_i. Thend(Λ',n)= n_L(_L(T))d (Λ, n). Now note that if we view T as a K-linear map, then _K(T)= n_L(_L(T)). Since Δ(Λ')= _K(T)Δ(Λ), it follows that both sides ofthe proposed identity change in the same way, if we change the lattice. Hence it suffices to check the identity for one lattice. Take Λ = B + B u, then both sidesare ideals generated by D_B/Aθ,where -θ= u^2, so the identity of ideals holds.The last statement is an easy check of signs, since the discriminant of complex quadratic fields is negative. We finish this section with the following proposition which defines the discriminant of an integral binary hermitian form, and establishes its integrality.Let Λ be a projective B-module of rank 2 and h an integral hermitian form on Λ. ThenΔ (Λ, h)= D_B/A· d(Λ , h) is an integral ideal, called the discriminant of (Λ, h). By localization, we can assume that A is local. Let v_1,v_2∈Λ and consider the entries of the matrix (s(v_i,v_i)). Sinces(v,v)=h(v) the diagonal entries are integral. However, the off-diagonal entries need not be integral. Indeed, for every l∈ B, such that l≠ l,we have the formula s(v_1, v_2) = lb(v_1,v_2) - b(lv_1, v_2)l - l. Ifsplits or is inert in B, then there exists l such that l - l is a unit. Hence (s(v_i,v_j)) is in A. Iframifies, then we can takel to be the uniformizing element in B. In this caseB=A[l ] and the discriminant D_B/A is precisely the norm of l -l. It follows thatD_B/A·(s(v_i,v_j)) is integral. § REPRESENTING ONE Here we address the question when a binary hermitian form represents 1, integral or otherwise. We first address the field case. The case of integral representations is significantly more complicated; a local-global principle is provided by <cit.>, which will reduce the problem of an integral representation of 1 to the problem of integral representations over p-adic rings. §.§ Over a FieldAssume that K is a number field. Any binary L-hermitian form h is a quaternary quadratic form over K. Recall that h is positive semi-definite at a realplace of K (i.e. an embedding of K into ℝ) if h is always greater than or equal to 0 on V⊗_Kℝ. We say that h ispositive semi-definite or indefinite if it is positive semi-definite or indefinite at all real places of K. Then we have the following The quadratic form h is positive semi-definite or indefinite if and only if h represents 1. Consider the 5-dimensional spaceV ⊕ K with the quadratic form f(v,x)=h(v)-x^2. Since V⊕ K is 5-dimensional, by the Hasse-Minkowski theorem, it represents 0 if and only if it represents 0 at allreal places. If h is positive semi-definite or indefinite, the form f issemi-define at all real places, so it represents 0 at all real places. Hence frepresents 0, and if x ≠ 0 in the representation, then h represents 1 by dividing by x. Otherwise, h represents 0, and so it represents any element of K. Conversely, if it represents 1, then it does so at every real place. Hence it cannot be negative semi-definite, and thus, it must be positive semi-definite or indefinite.§.§ Over integers We assume that K=ℚ and L a quadratic imaginary field. Thus, A=ℤ and B is the ring of integers in L.Let D=D_B/A be the discriminant ofL. We shall assume that L is unramified at 2, in order to avoid discussing the difficult 2-adic theory of integral hermitian and quadratic forms. Let Δ<0 be a square-free integer. Let Λ be a projective B-module of rank 2 andh an integral hermitian form on Λ of the discriminant Δ. Then there exists v∈Λ such that h(v)=1 Let ℤ_p be the p-adic completion of ℤ. Let B_p=B⊗ℤ_p and Λ_p=Λ⊗ℤ_p.Note that the subscript p denotes the completion and not localization in this proof.Since h is assumed to be indefinite,by <cit.>, it suffices toprove that for every p there exists v∈Λ_p such that h(v)=1. Assume that p does not divide D i.e. p does not ramify in L. Recallthat_p(Δ) =0 or 1, by the assumption on Δ.We claim that there exists u∈Λ such that h(u)=x where _p(x) =0. Indeed, if not,then v↦ h(v)/p is an integral form.But the discriminant of h/p is Δ/p^2, not an integer, contradicting Proposition <ref>.Furthermore, since p does not ramify in L,the norm map n_L: B_p^×→ℤ_p^× is surjective. Hence we can rescale u by an element in B_p^×to get v such that h(v)=1. Now assume p divides D, so p is tamely ramified in L. In particular, p is odd, and we can write B_p=ℤ[π] where π is a uniformizer of B_p. We can assume that π =-π and, for the price of changing the uniformizer in ℤ_p, that ππ=p.Pick a B_p-basis (v_1, v_2) of Λ_p and write any v∈Λ_p as a sum v=xv_1+ yv_2 where x,y∈ B_p. Thenh(v) = (x y )( α γ γ β)( x y) where α,β∈ℤ_p.By the assumption, we have_p(Δ) =0 or 1.If _p(Δ)=1 then _π (γ)≥ 0, and it is easy to see that the matrix can be diagonalized, bya change of B_p-basis in Λ_p. Hence, in this case,the quadratic form h on Λ_p is equivalent to a diagonal forma_1x_1^2+ a_2 x_2^2 + a_3 px_3^2 + a_4 px_4^2for somea_iin ℤ_p^×. Assume now that _p(Δ)=0. Then γ = (a+bπ)/π, for some a∈ℤ_p^× and b∈ℤ_p.Writing x = x_1 + x_2π and y = y_1 + y_2 π,the 4× 4 matrix representing the quadratic form h in the variables (x_1, x_2, y_1, y_2) is(α0 b/2 a/2 0α p a/2 p b/2 b/2 a/2β0 a/2 p b/2 0β p ).The determinant of this matrix modulo p is a^4/16 ∈ℤ_p^×.Recall that, since p is odd, every ℤ_p-integral form can be diagonalized. Sincethe determinant is p-adic unit, the form is integrally equivalent to a diagonal form a_1x_1^2+ a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2 for somea_iin ℤ_p^×. But in each of the two cases the diagonal quartic, quadratic form represents 1 by <cit.>. This completes the proof of lemma. The following is a combination of the previous lemma, Corollary <ref> and Theorem <ref>.We remind the reader that an integral binary B-hermitian form is a B-hermitianform h on a projective B-module Λ of rank 2, such that h(Λ) ⊆ℤ. Let L be a quadratic imaginary extension of ℚ of odd discriminant D <0. Let B be its maximal order. Let Δ<0 bea square free integer. Then there is a natural bijection between isomorphism classes of embeddings of B into quaternionic ordersof the discriminant Δ and isomorphism classes of integral binary B-hermitian forms of the discriminant Δ.§ ACKNOWLEDGEMENTS We would like to thank Tonghai Yang for suggesting the relationship between optimal embeddings and integral binary hermitian forms, and to Dick Gross for a conversation on this subject.The first author has been supported by an NSF grant DMS-1359774. The second author was supported by REU and UROP funding at the University of Utah.amsplain | http://arxiv.org/abs/1707.09001v1 | {
"authors": [
"Gordan Savin",
"Michael Zhao"
],
"categories": [
"math.NT"
],
"primary_category": "math.NT",
"published": "20170727190105",
"title": "Binary hermitian forms and optimal embeddings"
} |
#1sec. (<ref>) | http://arxiv.org/abs/1707.08573v1 | {
"authors": [
"Pilar Coloma",
"Pedro A. N. Machado",
"Ivan Martinez-Soler",
"Ian M. Shoemaker"
],
"categories": [
"hep-ph",
"astro-ph.HE",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170726180001",
"title": "Double Bangs from New Physics in IceCube"
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Wisture: RNN-based Learning of Wireless Signals for Gesture Recognition in Unmodified SmartphonesMohamed Abudulaziz Ali Haseeb, Ramviyas ParasuramanM. Abdulaziz is affiliated with the Robotics, Perception and Learning (RPL) Lab, KTH Royal Institute of Technology, Stockholm 100 44, and with Watty AB, Sweden.R. Parasuraman is with Purdue University, West Lafayette, 47906, USA. email: [email protected]; [email protected]=====================================================================================================================================================================================================================================================================================================================================================================This paper introduces Wisture, a new online machine learning solution for recognizing touch-less dynamic hand gestures on a smartphone. Wisture relies on the standard Wi-Fi Received Signal Strength (RSS) using a Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN), thresholding filters and traffic induction. Unlike other Wi-Fi based gesture recognition methods, the proposed method does not require a modification of the smartphone hardware or the operating system, and performs the gesture recognition without interfering with the normal operation of other smartphone applications. We discuss the characteristics of Wisture, and conduct extensive experiments to compare its performance againststate-of-the-art machine learning solutions in terms of both accuracy and time efficiency. The experiments include a set of different scenarios in terms of both spatial setup and traffic between the smartphone and Wi-Fi access points (AP). The results show that Wisture achieves an online recognition accuracy of up to 94% (average 78%) in detecting and classifying three hand gestures. Radio Signal, Gesture recognition, Wi-Fi, Smartphones, LSTM RNN, Traffic Induction. § INTRODUCTIONsmartphones have become the first high performance computing and sensing devices that are carried by a majority of the population in many countries.Today, smartphones are responsible for the majority of internet search traffic <cit.> as well as online media consumption <cit.>. Consumer reports reveal that users spend more time on smartphones than they do on desktop computers <cit.>.However, their inherent limitations in terms of physical size, screen size, computing capacity and battery power, calls for interaction modalities beyond the smartphone touch screens. Therefore, researchers and engineers are continually looking for novel methods to enrich and simplify how humans interact with their smartphones, such as natural language processing <cit.>, and touch-less gestures <cit.>.Modern gesture recognition methods mainly use two sensing modalities: intertial measurements <cit.> and images from the camera <cit.>. The advantages of these modalities include availability and accuracy. However, intertia-based systems require the smartphone to be held by the user, and image based techniquessuffer from drawbacks such as limited sensing range, sensitivity to lighting conditions, and more importantly high-power consumption.Therefore, in recent years, Radio Frequency signals and Received Signal Strength (RSS) have been exploitedfor sensing human activities and gestures <cit.>, mainly due to their advantages in terms of low power consumption, and being able to handlenon line-of-sight conditions. For instance, Google's project Soli <cit.> aims at creatinga rich gesture recognition interface using a specialized radar device that can be embedded into wearables, phones, computers, and Internet of Things (IoT) devices. The ubiquitous nature of Wi-Fi technology makes it attractivefor gesture recognition in smartphones. Thus, a number of novel methods have been proposed for using Wi-Fi RSS (e.g. <cit.>). However, they require either special hardware or special software modifications such as root privileges, blocked data traffic to other applications, etc. to operate. In this paper, we propose a novel machine learning method based on LSTM RNN for recognizing gestures in unmodified smartphones, based on artificially induced data traffic between the smartphone and the Wi-Fi Access Point (AP). The main contributions of this paper are: * We demonstrate that the Wi-Fi RSS can be used to recognize hand gestures near smartphones (see Fig. <ref>) using a fusion of machine learning techniques and custom signal processing algorithms.* We conduct several experiments under various conditions to validate the gesture classification performance of the proposed method, and compare it against state-of-the-art machine learning methods.* We release the experiment dataset (an extensive collection of labeled Wi-Fi RSS measurements corresponding to multiple hand gestures made near a smartphone under different spatial and data traffic scenarios.) to enable comparison with future approaches[The dataset is available at <https://goo.gl/2AQKdT>, and will be published to the public CRAWDAD repository (<http://crawdad.org>).].* We openly share the source codes of part of the Wisture[Accessible at <https://github.com/mohaseeb/wisture>. We also release the Android apps "Wisture" and "Winiff", available in Google Play Store.], specifically to record high frequency Wi-Fi RSS measurements using the induced traffic approach. § RELATED WORKS Radio signals, notably Wi-Fi signals, have been recently exploited for sensing and recognizing human activities <cit.>. For instance, customized hardware-based active sensing solution is introduced in <cit.> using transmit and receive arrays/antennas along with Fourier/Doppler analysis of the RSS data. They achieved a recognition accuracy of 94% to classify 9 whole-body gestures in a home environment. Similar custom hardware-based solutions (e.g. <cit.>) are limited to the application environments and the resources. Nevertheless, antenna-array based methods find sophisticated novel solutions like seeing through walls using radio signals <cit.>. In <cit.>, the authors used both the Wi-Fi RSS and the Channel State Information (CSI) to recognize hand gestures using a signal conditioning and thresholding based gesture recognition algorithm, and achieved a classification accuracy of 91% on a laptop.Note, the CSI provides detailed channel features including the sub-carrier level phase information, but is supported only by a very limited set of Wi-Fi devices[Currently, the CSI data is available only from Intel's Wi-Fi Link 5300 drivers, and not available on smartphones Wi-Fi devices/drivers.]. Several other works such as <cit.> also proposed CSI-based solutions for gesture or activity recognition, however they are subjected to the same limitations. K-Nearest Neighbors (K-NN) based classifiers have been widely used to recognize gestures from the RSS data. In <cit.> and <cit.>, the authors used window-based statistical features (e.g. mean, variance, maximum, number of peaks, etc.) applied to a K-NN classifier for recognizing hand gestures on a smartphone. They achieved accuracies of 50% (with K=20, 11 hand gestures) and 90% (with K=5, four hand gestures) respectively.However, their solutions require modified device firmware, root access to the OS, and dedicated applications that limits smartphone's Wi-Fi traffic.A Discrete Wavelet Transform (DWT) based approach is presented in <cit.> to transform the RSS data intro three primitive signals: rising edges, falling edges and pauses. It achieved high accuracy (90%) by using a classifier that compares the series of primitive signals to a set of pre-defined rules. However, such a solution requires high-frequency sampling of RSS and extensive computation abilities.We depart from the literature works in four different ways: (1) we do not make any modifications to the existing hardware or software applications of the phone; (2) we introduce a new traffic induction approach to enable high-frequency RSS measurements; (3) we use custom but simple signal processing techniques and an efficient LSTM-RNN machine learning method to classify over-the-air hand gestures; (4) we share the experiment datasets and the partial source codes of our solution for the community to build this research further.To the best of our knowledge, very few works uses deep learning or neural network based methods to radio signal based activity/gesture classification. For example, the authors in <cit.> used a Convolutional Neural Network (CNN) for classifying user driving behaviors based on narrow-band radio signals and achieved 88% accuracy. On a higher level, the CNN is ideal for image processing due to its nature of recognizing patterns within the data (across space), whereas the RNN is ideal for speech/time-series processing due to its between-data (across sequence) recognition capabilities. Thus we choose RNN as the core machine learning algorithm in our Wisture solution. § BACKGROUND §.§ Radio signal propagationAn RF signal propagating through a medium is subject to several environmental factors that impact its characteristics. In the absence of nearby obstacles, the signal strength will be reduced by the free-space path loss (FSPL) caused by the spreading out of the signal energy in space.The RSS is usually modeled using a log-distance path loss model (an extension of Friss transmission equation) <cit.>:P_r(dBm) = L_0 - 10n log_10(R)_Path Loss + χ_σ_Shadowing + ε_multipath,where L_0 is the reference RSS at 1 meter from the transmitter (which depends on transmitter and receiver antenna properties as well as the signal frequency), n is the decay exponent depending on the environment, R is the distance between the transmitter and receiver, χ_sigma = 𝒩(0,σ) is a zero-mean Gaussian variable with variance σ^2 used to represent the shadow fading caused by surrounding objects due to reflection, absorption, and diffractions <cit.>. ε is a random variable following the Nakagami distributionused to represent multi-path fading caused by the fact that the signals take multiple paths to the destination, and to represent interferences and spatio-temporal noise <cit.>.The combination of χ_σ and ε models the different ways by which human body presence and movements impacts the strength of the radio signal received by a nearby wireless device, such as a smartphone.Friis transmission equation (Eq. <ref>) describes that the received power is inversely proportional to the square of the distance between the transmitter and the receiver <cit.>.P_r(W) = P_tG_rG_t(λ/4π R)^2In the above equation, P_t and G_t are the transmitter output power and its antenna gain, P_r and G_r are the receiver input power and its antenna gain, R is the distance between the antennas and λ is the signal wavelength.§.§ Wi-Fi RSS measurementsWi-Fi technology is based on the IEEE 802.11 standards and typically use 2.4 or 5 GHz ISM frequency spectrum. The data is communicated as frames. For every received frame, the device measures the RSS and reports the Received Signal Strength Indicator (RSSI), usually in dBm.Figure <ref> shows an illustration of a sequence of RSSI measurements generated by a Wi-Fi receiver. The RSSI together with additional measurements (e.g Link Quality, Noise level, etc.) at the interface provide information on the transmission channel.The frequency by which the RSSI measurements are taken is not deterministic, but depends on the traffic sent to the Wi-Fi receiver. This burstiness nature of the RSSI measurements makes it hard to use for gesture detection. In an idle Wi-Fi network, the majority of the traffic is formed of so-called beacon frames. Beacon frames are sent periodically by the AP to signal the presence of the Wi-Fi network. The time between two beacon frame transmissions is configurable, but typically set to 102 milliseconds <cit.>. This means that a smartphone connected to a Wi-Fi network, and not actively receiving data, will have around nine RSSI measurements every second corresponding to the beacon frames received during that second.Note the RSSI measurements made on a laptop or a computer with commercial Wi-Fi adapters, can provide much stronger and high-frequency data compared to measurements on a smartphone. This is likely due to the antenna properties, device drivers, and limited resources on a smartphone, which alludes to the difficulty of using Wi-Fi RSSI for recognizing gestures in smartphones compared to computers. Figure <ref> shows how the RSSI stream recorded in a computer exhibits more and stronger RSSI variations compared to that recorded in a smartphone. These differences are likely attributed to the limited smartphone resources, which alludes to the difficulty of using Wi-Fi RSSI for recognizing gestures in smartphones compared to computers. §.§ Hand gesturesWe first show in Fig. <ref> the RSSI data recorded in a smartphone while a person is performing an activity such as walking past the phone, typing on a keyboard, and swiping his/her hand over the phone. Note the unique pattern in the RSSI stream created by the hand gesture.In this paper, we consider three hand gestures, see Fig. <ref>: swipe, push, and pull.* Swipe gesture:moving the hand about five centimeters above the smartphone from one side to the other and back to the starting point.* Push gesture: moving the hand downward towards the smartphone and holding it steadily about five centimeters above it for around two seconds.* Pull gesture: placing the hand about five centimeters above the smartphone, holding itthere for about two seconds before moving it upward.If more gestures are needed one could either introduce new gestures, or make combinations of these three primitive gestures as in <cit.>. The question of which option is best is however beyond the scope of this paper, as we focus on the best machine learning solution.§ PROBLEM FORMULATIONThe problem of recognizing hand gestures from RSS values can be viewed as a classification problem where the objective is to learn a mapping from the RSS values to the probability distribution over the possible hand gestures P(y|x).x → P(y|x;θ)Where x ∈ℝ^τ is a sequence of input RSS values with length τ, y∈{0,1,…,K} is the list of gestures, θ is a parameterization of the mapping.Given a dataset of m sample gestures that is formed from the inputs X=[x_1 x_2 … x_m] and their corresponding outputs Y=[y_1 y_2 … y_m]. The maximum likelihood (ML) method can then be used to find a good estimate of θ as below:θ_ML=*arg max_θ P(Y|X;θ) Assuming the dataset samples are independent and identically distributed (i.i.d.). Equation <ref> can be rewritten as follows:θ_ML=*arg max_θ∏_i=1^m P(y_i|x_i;θ)The above probability product can become very small and hence render the problem computationally unstable. This can be solved by taking the logarithm of the likelihood, which transforms the product of probabilities into a sum. θ_ML=*arg max_θ∑_i=1^m log P(y_i|x_i;θ)The estimate of θ can be expressed as minimizing a loss function L (also referred to as cost) defined as below L = ∑_i=1^m -log P(y_i|x_i;θ)This loss is known as negative log-likelihood (NLL).As will be described below, an RNN is used to model the conditional probability P(y|x;θ). An ML estimate of θ is foundusing the Stochastic Gradient Descent (SGD) algorithm, to minimize the NLL in a collected hand gesture dataset. § PROPOSED SOLUTION: WISTUREWe first describe the Wisture solution in general and then present how to train the system with the training data.§.§ Signal processing and machine learning methodFigure <ref> shows an overview of the proposed gesture recognition solution. The different submodules are described below. §.§.§ Traffic inductionAs discussed earlier, the smartphone Wi-Fi interface makes new RSSI measurement only when a new Wi-Fi frame is received. To guarantee that the wireless device makes enough updated RSSI measurements, we induce artificial traffic between the AP and the smartphone by sending a continuous stream of Internet Control Message Protocol (ICMP) echo requests to the AP. For every ICMP echo request, the AP will send an ICMP echo reply back to the smartphone which will make an updated RSSI measurement.This enables us to have enough RSSI measurements while avoiding the need for a custom firmware or putting the Wi-Fi interface in monitor mode as in <cit.>.§.§.§ RSSI collectionThis module extracts a stream of up to ∼200 RSSI values/second from the Wi-Fi interface. In our Android implementation, the RSSI measurements are collected using the wireless extension for Linux user interface <cit.>, which is exposed as a pseudo file named /proc/net/wireless.§.§.§ WindowingThe incoming RSSI stream is split into overlapping windows of T seconds length, and d seconds gaps between window starts. In all the experiments the gap d was set to 1 second. Different values of T are investigated and reported in the experiments section. Since the incoming RSSI stream rate is varying around 200 values per second, the output windows will have a variable length.§.§.§ Noise detectionWe note that only windows with high activity, identified by a theshold on the window variance, are likely to be caused by hand gestures. Thus only the windows that pass this criteria will be forwarded to the subsequent steps. All windows that have a variance less than the threshold will be predicted as no gesture or Noise. The process of estimating the variance threshold is described in Sec. <ref>.§.§.§ PreprocessingThis submodule transforms the incoming windows into windows of equal number of feature values (τ). Each incoming window is processed as below: * Mean subtraction: The window values are centered around zero, by subtracting the window mean from all RSSI values in the window. This increases the system robustness against changes in the RSSI values due to, for example, RSSI increases or decreases when the smartphone is moved close or away from the AP respectively.* Sampling: This steps samples τ feature values with a time difference between consecutive samples equal to T/τ on average. Different values of τ have been investigated and reported in the experiments section.* Standardizing: Each one of the τ feature values is reduced by the training data mean of that feature value. * Normalizing: Each one of the τ feature values (standardized in the previous step) is divided by the training data standard deviation of that feature value.A detailed description of the standardizing and normalizing steps is provided in Sec. <ref>.§.§.§ InferenceThe LSTM RNN model takes an input of τ feature values, and outputs three values proportional to the conditional probability of each possible gesture on the inputs. These values are referred to as logits, because they are the inputs of the softmax layer used in calculating the model loss during training. The Softmax function is a generalization of the logistic function, and its inputs are referred to as logits.§.§.§ Logits thresholdingThis step discards LSTM RNN model predictions that are below a specific threshold and predicts Noise for those inputs.§.§.§ Prediction decision rulesThis submodule keeps a short history of the previous predictions, and applies a set of rules to accept or reject the current prediction made by the preceding steps. These rules are: * Allow Pull gestures only after Push gestures. The Pull gesture RSSI signature appears as a pause followed by an increase. This signature is similar to those caused by some background activities, e.g. when an AP increases its output signal power. This rule reduces the number of false positive Pull predictions caused by such interfering background activities. Note: However, the solution is prone to confusing decreases in RSSI values caused by interfering background activities (e.g. the AP reducing its output signal power) as Push gesture, and no solution was proposed in this work to harden the system against such interference.* A prediction that is different from its immediate predecessor is ignored (and Noise is predicted instead). Exempted from this rule are:* Swipe or Push following a Noise prediction.* Pull prediction that follows a Push.* Noise predictions.The rationale for this rule is that each prediction window overlaps with the previous window (three seconds overlap in most experiments). In many cases, if the preceding window contained a gesture, the succeeding window RSSI stream signature might look similar to another gesture than the performed one. For example, the end of Swipe gestures look similar to Pull gestures (see Fig. <ref>).§.§ System trainingTraining the LSTM RNN model and selecting the different hyperparameters and thresholds are performed in an offline setting. The training procedure is detailed below.§.§.§ Data preprocessing The online machine learning solution preprocesses the incoming RSSI windows in the way described below. * The RSSI values are read from the collected data files, and then split into D windows (corresponding to the gestures), each being T seconds long.* For each window, the mean is calculated and then subtracted from the individual window values.* τ values that are equally spaced in time are then sampled from each window. The result is a dataset of shape D windows each having τ features.* The dataset is then randomly split into training (D_train=0.75D) and testing (D_test=0.25D) sets. Furthermore, when a model hyper parameter selection is done, 0.8D_train of the training set is used to train the model, and the remaining D_val=0.20D_train is used to select the hyper parameters (validation set).* Using the training set (D_train×τ), the mean and standard deviation of each one of the τ features is calculated as below.x^(i)_train_mean=1/M∑_j=1^M x^(i)_train, j x^(i)_train_std=√(1/M∑_j=1^M (x^(i)_train, j - x^(i)_train_mean)^2)Where M is the training set D_train size and x^(i)_train, j is feature i value of sample j from the training set.* All training and testing set windows were standardized and normalized using the training mean and standard deviation. Let x=[x^(1) x^(2)… x^(i)… x^(τ)] be some input window (from training or testing), the output of the standardization and normalization steps x_o=[x^(1)_o x^(2)_o … x^(i)_o … x^(τ)_o] can be described as below:x^(i)_o= x^(i)- x^(i)_train_mean/x^(i)_train_std§.§.§ LSTM RNN model trainingFigure <ref> shows an illustration of the LSTM RNN model used in this work (inspired by <cit.>). The model was trained to minimize the NLL loss, using a variant of SGD known as Adaptive Moment Estimation, or shortly ADAM.Most of the model hyper parameters (τ, N, number of layers and others) were selected by performing a grid search in the parameters space. Each parameter setting is evaluated using a four folds cross validation.§.§.§ Thresholds selectionThe variance threshold used by the Noise detection step, is initially estimated as the training data minimumwindow variance. This value is then manually optimized to maximize the online prediction accuracy. The same approach is used to select the logits thresholds. § EXPERIMENTS AND RESULTS §.§ Experiment DatasetThis section contains the details of the dataset used to train and evaluate the recognition system.§.§.§ Spacial setupFigure <ref> presents the experimental setup. The dataset was collected under two different spatial configurations of the Wi-Fi AP and the smartphone: * (Room A): The AP and the smartphone were placed two meters apart in room A with a line of sight (LoS) between them. The AP was placed on a table slightly lower than the table where the smartphone was placed.* (Room A & B): The AP was placed in room B and the smartphone in room A, see Fig. <ref>, and the distance between them was ∼4.5 meters. The two rooms were separated by a wall made mainly of wood and gypsum, thus there is no line of sight (nLoS). Both the AP and the smartphone were placed on tables of similar height.§.§.§ Traffic scenariosThree different WiFi traffic scenarios were considered when collecting the data: * (Internet access + traffic induction): in this scenario the AP was connected to the Internet andthe smartphone was connected to the AP. The smartphone was continuously sending ICMP requests to the AP (pinging) at a rate of ∼700 times/second.* (No Internet access + traffic induction): neither the AP nor the smartphone had Internet access, but the smartphone was continuously pinging the AP at a rate of ∼700 times/second.* (No Internet access + no traffic induction): neither the AP nor the smartphone had Internet access, and there was no traffic induction.§.§.§ Data collection procedureAn Android mobile application was developed specifically for recording the Wi-Fi RSSI data using the induced traffic approach (called "Winiff"). It records the RSSI measurements made by the smartphone at a frequency of ∼200 samples/second. A typical collection session is described below: * The AP and the smartphone are placed as per one of the spatial setups described earlier.* The subject performing the experiment sits in a chair facing the smartphone.* The smartphone is connected to the AP.* The RSSI collection application is started.* At a specific point in time (start time), the subject starts performing the gestures, leaving a gap of ten seconds between consecutive gestures (gap time). Both the start and gap times are noted down and used later to extract the gesture windows.* The collected RSSI stream is stored in a text file. The collected dataset details are summarized in Table <ref>.§.§.§ Evaluation metricThe system was evaluated using the accuracy measure, defined as the percentage of correctly predicted gestures. For a set of test gestures X=[x_1 x_2… x_i … x_m] with corresponding true labels Y=[y_1 y_2 … y_i … y_m], the accuracy is defined as:accuracy = 100 ×1/m∑_i=1^m I_y_i(ŷ_̂î)% Where ŷ_̂î is the system prediction for input gesture x_i, and I_y_i(ŷ_̂î) is 1 if ŷ_̂î = y_i and 0 otherwise. §.§ Results and DiscussionsHere we list the conducted experiments and the obtained results. The RNN model was trained as described in Sec. <ref>. The model parameters used in the experiments are shown in Table <ref>. The reported mean accuracies inexperiments are calculated by evaluating the RNN model ten times on the specific configurations being tested, each using a different random split of the data into training and testing sets. §.§.§ RNN model accuracy on different datasetsTable <ref> lists the recognition model accuracy when evaluated on the different datasets.Due to the increased data size of the collective dataset (Dataset1 + Dataset2 + Dataset4), the number of training iterations for this configuration is 1000 instead of 600.If we exclude the Swipe gesture in Dataset3, the prediction accuracy jumps to 97% (±1.5). The gesture recognition accuracy is poor for Dataset3, which can be attributed to fact that only a small number of (low-frequency) RSSI measurements were available in the dataset because both the internet and the traffic induction were disabled. The significant increase in accuracy from 78% to 91% when the induction is enabled demonstrates the high dependency of the accuracy on the amount of data traffic at the Wi-Fi interface.Figure <ref> compares the RSSI values recorded for the Swipe and Push gestures performed while the induction is enabled and disabled. It is clear that with no traffic induction, the RSSI values corresponding to the Swipe gesture are not distinguishable from those of the Push gesture, and consequently the classification ability is severely impacted. §.§.§ Comparison with the state-of-the-art methodsA set of time series classification algorithms, including state-of-the-art ones, were evaluated on Dataset1, to compare them to the RNN model (Table <ref>). The Collective of Transformation Ensembles (COTE), Elastic Ensemble (EE), Shapelet Transform ensemble (STE) <cit.>, which are ensemble methods that employs multiple classifiers underneath, performed better than or equal to the RNN model. The Learning Time-series Shapelets (LTS)[We implemented a python version of the LTS method, available at <https://github.com/mohaseeb/shaplets-python>.] <cit.> performed better in terms of accuracy but poorer in terms of training/prediction time.. The K-Nearest Neighbor Dynamic Time Warp (K-NN DTW) algorithm and the Fast Shapelets (FS) <cit.> performed worse, with the k-NN DTW having a very slow prediction time of almost one second. §.§.§ Impact of dataset size on the accuracyFigure <ref> shows the model accuracies when trained with different dataset sizes. It shows that the model accuracy increases and becomes more stable (reduced variance) as it is trained with more data. §.§.§ Impact of model parameters on the accuracy Model complexity As can be seen in Fig. <ref>, the experiments show that increasing the number of layers initially increases the model accuracy, but then decreases it, most likely due to overfitting.Samples per window The results in Fig. <ref> shows that the model accuracy increases as the number of samples per window increases, but after a specific point the accuracy decreases. This decrease in accuracy can be explained by the increase in model complexity as the number of input samples increases (and hence increases the RNN time steps) which causes the model to overfit the training data.Window size In Fig. <ref>, the impact of the prediction window length on the RNN model accuracy with the Dataset1 is shown. The model achieves the highest accuracy for the two seconds window length. However, online experiments showed that a two seconds window length provides a short context that results in a higher rate of confusion between gestures, see Fig. <ref>, and false positive predictions. Thus, a four seconds window is used in the online experiments.§.§.§ Impact of spacial setup on the accuracyTable <ref> and Fig. <ref> summarize the various online experiments performed[Experiments made using the Wisture Android App. The RNN model had the same parameters as in Table <ref>, except no dropout was used due to a limitation in the Tensorflow Android library.] and the obtained results[A video demonstration of sample results is available at <https://www.youtube.com/watch?v=rMv_bKkDtbU>.]. Accordingly it can be concluded that the solution: (1) generalizes to AP-smartphone spatial configurations that are different from the training ones, and (2) performs better in settings where there is a line-of-sight (LoS) between the smatphone and the AP (81% accuracy), than where there is no LoS (74% accuracy). The exception to this is the no-LoS online experiment number three, where an accuracy of 93% was achieved, which is higher than all recorded LoS experiments accuracies. This might be due to the peculiarity of the Wi-Fi signal path between the smartphone and the AP in these cases. Refer to Sec. <ref> for more information on RF signal propagation.Figure <ref> shows that the system has approximately equal performance in recognizing the different gestures. Although Push has the highest average accuracy, the standard deviation (std) of the individual gestures show that, Push and Pull accuracies have a wider variation across the different test scenarios (0.29 std for Push and 0.3 std for Pull) compared to the Swipe gesture (0.15 std). We believe the low accuracy achieved in experiment six is because of the fact that the AP and the smartphone were far apart (9 m), and the line-of-sight was blocked by a wall, a dishwasher and a fridge.§.§.§ False positive predictionsTo estimate the robustness of the system against RSSI changes caused by interfering background activities, the recognition application was left running for a period of thirty minutes on a table inside a room, while a person was in the same room typing on a computer (placed on the same table as the smartphone), and occasionally moving in the room. Table <ref> summarizes the results, which show an average false positive rate of 8%. §.§.§ Resource consumptionTheWisture app was found to utilize around ∼13% of the total CPU time (9% system time + 4% user time) when traffic induction is disabled. The utilization increased to ∼25% (13% system time + 12% user time) when induction is enabled. Note, the screen usage of the app are also counted into the resources (the screen was always on display to observe the experiment results). No CPU or power usage figures were reported by previous works. §.§ SummaryUsing the proposed recognition solution, the experiments showed that it is possible to detect and classify three different hand gestures with an accuracy of 78%, across a variety of spatial and traffic scenarios without modifying the smartphone hardware, operating system or firmware (on the contrary to the works in <cit.>[In <cit.>, the wireless interface was in the "Monitor" mode, and hence captured all Wi-Fi traffic exchanged by all smartphones and APs in the smartphone vicinity. As a result, the available RSSI measurements increased and the solution achieved a high accuracy ( 90%). However, in addition to the need of a custom firmware, the "Monitor" mode comes at the cost of prohibiting all smartphone applications from sending or receiving traffic over Wi-Fi, and a likely increase in battery power consumption.], <cit.>,<cit.>, and <cit.>). To the best of our knowledge, the work presented here is the first to demonstrate this. Note, although the works in <cit.> demonstrated gesture recognition on unmodified devices, the solutions are tailored to a PC based implementation with higher resources than a smartphone[In <cit.>, CSI data (currently available on Intel's Wi-Fi Link 5300 device driver) is used to identify hand gestures. In <cit.>, the RSS collection procedure is not described in detail.].§ LIMITATIONS AND FUTURE WORK §.§ Limitations§.§.§ Background activitiesThe solution was sensitive to interference to background activities, but we did not analyze such impacts. They may affect the prediction performance in general, and false positives specifically. §.§.§ New gesturesTo support new gestures, a new dataset for the new gesture is needed. The developed recognition solution is trained with a dataset of 1000+ sample gestures (300+ samples per gesture). The requirement of huge number of training samples can be seen as a limitation. However, we can see in Fig. <ref> that the training data size can be reducedto a tenth of the original set (100+ samples), with a modest decrease in accuracy to 83% from 94%.§.§.§ Calibration to different Smartphones or devicesSince the solution is based on training data from one particular smartphone, using the trained RNN on a different smartphone may result in a lower accuracy. To address this issue, further work is necessary to calibrate the training data to new smartphones or other devices (laptop, tablet, or a wearable device). §.§ Lessons learned Below we list some of the design and implementation choices that were discarded since they had negative or no impact on the model accuracy. Wi-Fi Link Quality Indicator (LQI) The initial exploration of the collected RSSI and LQI data showed a high correlation between the two, and neither provided additional information that is not present in the other. Consequently, we did not use the LQI data as an extra input signal. Smoothing the RSS data Exponential and moving window based RSSI smoothing techniques affected the LSTM-RNN classification accuracy negatively. We believe that the smoothing process removed a few important characteristics of the RSS signal variations, however further investigation is needed to analyze the impact. Smartphone Wi-Fi interfaces Our experiments showed that the model ability to recognize more gestures can be improved by measuring the Wi-Fi RSS at higher frequencies. Designing smartphone Wi-Fi interfaces that support improved radio signals sensing has a great potential for smartphone vendors and users. We believe the smartphone device manufacturers provide such capabilities in the future. §.§ Future work Leaving the recognition application running for a long time while no gesture is performed, increases the probability of false positive predictions and consumes the smartphone battery. This can be addressed by introducing a new mode of operation in which the recognition application waits for a special preamble gesture that has two characteristics: (1) easy to separate from background noise; (2) requires small resource utilization to be detected. After a preamble detection mode, the application could detect further complex gestures. In our further works, we plan to use the preamble gesture concept, for instance with a sequence of Push and Pull gesture as a preamble.The results in Table <ref> show that Push and Pull gestures can be recognized with an accuracy of 97% even when no traffic is induced between the smartphone and the AP. Thus, during the preamble detection operation mode, traffic induction can be disabled, significantly reducing the power consumption and resource utilization of the application. Further, we will also investigate the influence of the distance and spatial characteristics between the smartphone and the hand making the gestures. §.§ Improved sensing of Wi-Fi RSSA high resolution RSSI stream will contain features that increase the models ability to detect and classify an increased number of gestures with high accuracy, see Sec. <ref> and Fig. <ref>.Designing the smartphone Wi-Fi system with the goal of improving the radio signal sensing by: 1) allowing RSSI measurements at high frequency, 2) native support for traffic induction and 3) providing simplified access to the RSSI values, has several benefits. It will enabledescent gesture recognition capabilities that are superior to what is demonstrated in this work, at a lower cost and with a shorter time-to-market compared to other approaches, like designing specialized sensing hardware for detecting gestures, such as the Google Soli project <cit.>. § CONCLUSION In this work, we have demonstratedthat it is possible to recognize and classify contact-less moving hand gestures near smartphones without modification to the smartphone hardware or existing software. The proposed solution used a custom signal processing techniques, an artificial traffic induction approach, and a LSTM RNN based machine learning model to detect and classify the performed hand gesture from the smartphone Wi-Fi RSSI measurements. The solution achieved an average on-phone recognition accuracy of 78% on average, and up to 94% on specific datasets, when tested under several configurations (scenarios) including the ones that were different from the training data scenarios. We believe this accuracy qualifies the solution for non mission-critical applications.In our future works, we aim to reduce the limitations of the solution such as vulnerability to interfering background activities (for instance by introducing a preamble detection mode), and calibrating the training data to new devices/gestures.§ ACKNOWLEDGMENTWe would like to thank Petter Ögren for his invaluable input and advice on this work.IEEEtran | http://arxiv.org/abs/1707.08569v2 | {
"authors": [
"Mohamed Abudulaziz Ali Haseeb",
"Ramviyas Parasuraman"
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"categories": [
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"published": "20170726171515",
"title": "Wisture: RNN-based Learning of Wireless Signals for Gesture Recognition in Unmodified Smartphones"
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A Comparative Study of the Clinical use of Motion Analysis from Kinect Skeleton Data Sean Maudsley-Barton1, Jamie McPhee2, Anthony Bukowski1, Daniel Leightley3 and Moi Hoon Yap1 1School of Computing, Mathematics and Digital Technology Manchester Metropolitan University, Manchester, M1 5GD, UK Email: [email protected]§ 2School of Heathcare Science Manchester Metropolitan University, Manchester, M1 5GD, UK 3King's Centre for Military Health Research King's College, London, WC2R 2LS, UK December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================== The analysis of human motion as a clinical tool can bring many benefits such as the early detection of disease and the monitoringof recovery, so in turn helping people to lead independent lives. However, it is currently under used. Developments in depth cameras, such as Kinect, have opened up the use of motion analysis in settings such as GP surgeries, care homes and private homes. To provide an insight into the use of Kinect in the healthcare domain, we present a review of the current state of the art. We then propose a method that can represent human motions from time-series data of arbitrary length, as a single vector.Finally, we demonstrate the utility of this method by extracting a set of clinically significant features and using them to detect the age related changes in the motions of a set of 54 individuals, with a high degree of certainty (F1-score between 0.9 - 1.0).Indicating its potential application in the detection of a range of age-related motion impairments. § INTRODUCTION The analysis of human motion from video has a long history, beginning in the 1970s with the experiments of Johansson <cit.>, which demonstrated people can recognise a wide range of human motions, even if only the joints of an actor are highlighted.This work led directly to the large multi camera, marker based, motion capture systems, still regarded by many as the gold standard for motion capture (mocap).The introduction of the Kinect depth camera, in 2011, heralded a new age of inexpensive markerless motion capture. Most importantly, thanks the work of Shotton et al. <cit.>, Kinect reduces the depth image to a stick skeleton, who's joints while not strictly anatomically correct have been demonstrated to have a high correlation with its marker-based kinsman <cit.>.Originally conceived to rival the Wii, Kinect has now found applications in areas as diverse as facial analysis <cit.>, surveillance, security and health.In the nearly 6 years since its introduction, much effort has been put into using the Kinect to analyse human motion.Often but not always allied to some form of automatic classification and/or quantification.Initially this work used classical machine learning techniques such as Random Forests and Support Vector Machine (SVM).Although these techniques have provided excellent results <cit.>,the newer deep learning techniques offer the possibility of detecting finer detail.This paper provides a comparison of traditional machine learning approaches and the newer deep learning techniques. their relative success so far and details our own investigations using the K3Da dataset.Our initial findings demonstrate that deep learning can outperform traditional methods. §.§ Scope of reviewHuman motion analysis is a broad topic, it encompasses areas of facial movement, hand movement and sign language recognition, rehabilitation and many others. This study limits itself to the use of skeleton data for the analysis of motions in a clinical setting.Motion analysis is used by health professionals in the assessment of a range of diseases and conditions which affects both young and old.Prior to the introduction of Kinect, the cost and availability of marker-based systems meant that only a small section of the population had access to this type of assessment.The majority of assessments were, and still are done subjectively, by direct observation. Or rely on patients filling in a questionnaire.Even after the introduction of Kinect, many studies <cit.>, <cit.> and applications still require an expert to interpret the results.This is where machine intelligence can play a role, by providing quick and objective analysis of human movement.§ RELATED WORKFrom the introduction of Kinect in 2011, the possibilities for use in healthcare were recognised.Some researchers have concentrated on rehabilitation while others work on the quantification of movement.Before Kinect, attempts had been made to use RGB cameras in rehabilitation schemes.These systems used colour segmentation to identify limbs.This made them very sensitive to light, shadows and distance from the camera.Kinect's technology is indifferent to these issues. Many studies have used Kinect in rehabilitation, sometimes referred to as the gamification of physiotherapy.It both reduces the strain on therapeutic resources and encourages individuals, to repeat their beneficial exercises <cit.>, <cit.>, <cit.>.Stone et al. <cit.> used Kinect in gait analysis.They used it to calculate gait velocity and length in the lab.Followed up, a year later, by a proof that Kinect cameras can be used to continuously monitor gait in the home <cit.>. Dolatabadi et al. <cit.> wrote a case study, detailing how they used features derived from Kinect to unobtrusively monitor recovery from hip replacements, in the home. Pu et al. <cit.> used Kinect to assess balance in a selection of 100 people.These studies calculated hand crafted features from the Kinect data and analysed them to demonstrate their significance. As the popularity of Kinect has grown a cottage industry in validating Kinect against the gold standard for mocap has sprung up.One of the most extensive comparisons was carried out bya large team, led by Otte <cit.>.They validated both V1 and V2 of Kinect and found there was high agreement between both versions, however there was some spread between different joints, the head joint being the most accurately tracked and feet joints being the least. Yang et al. <cit.> demonstrated a small, linear discrepancy between the Centre of Mass (CoM), calculated using Kinect and the ground truth.As this discrepancy is linear, it can easily be accommodated or just ignored for calculations which look at the change of a quantity over time, for example postural sway.The studies,highlighted above, rely on expert analysis by humans.There has been no attempt to automate the process or leverage machine intelligence.The next set of papers look at, do just that. §.§ Machine IntelligenceThe most basic application of machine intelligence is to apply a heuristic approach, asBigy et al. <cit.> did in their real-time system which can detect Freeze of Gait (FoG) a problematic symptom of Parkinson's disease as well as falls and tremors <cit.>.Their simple rules-based approach makes this easily applied in real time.Alexiadis et al. <cit.>, took a more sophisticated approach.in their work related to an automatic method for evaluating a dancer's performance.They developed three metrics and techniques to temporally align the movements of amateur dancers with ground truth provided by professionals. Inspired by this work, Suet al. <cit.> developed a system which uses Dynamic Time Warping (DTW), a method previously used in relation to hand writing and audio recognition, to compare the movements of patients in their everyday life to a standard set of movements.The aligned sequences are then classified using an Adaptive Neuro-Fuzzy Inference System (ANFIS) which uses a combination of artificial neural network and fuzzy logic.A drawback with DTW is that it does not perform well for periodic movements like waving.Wang et al. <cit.> proposed the pairwise encoding of the relative joints, producing a much more discriminating feature set.This process relies on the normalisation of skeleton graphs so that all joint offsets are relative to those in the first frame.This type of normalisation has become the de-facto pre-processing step when calculating features from Kinect skeletons.Gabel et al. <cit.> extracted a feature set which expanded on gait analysis to include arm kinematics and additional body features namely, Centre of Mass (COM), Direction of Progress (DoP) and Acceleration.These measures have become common features of many skeleton based studies.A combination of Multiple Additive Regression Trees (MART) and a state model were used to predict the gait cycle.However, machine learning was not used to produce a predictive model for gait quality. Greene et al. <cit.> had demonstrated the use of logistic regression to quantify falls risk, with data derived from kinematic sensors but the groups using Kinect had yet to venture in to prediction.Cary et al. <cit.> decomposed each skeleton generated by Kinect into a 17-value, feature vector of spherical coordinates.This was calculated, in relation to torso bias, a 3-value vector produced by applying Principal Component Analysis (PCA) to the torso joints. They used these features to train an Artificial Neural Network (ANN) to recognise a range of movements.Kargar et al. <cit.>, extracted both gait and angle-based anatomical features from skeletons in order to quantify a person's physical mobility.To achieve this they used a SVM.In 2016, Leightley et al. <cit.> detailed the first end-to-end pipeline for the automated analysis and quantification of human movement, using Kinect.The system uses a variety of motion analysis techniques to first extract clinically significant features and then quantify them using SVM. §.§ Deep LearningIn the last few years, there have been a flurry of papers that use deep learning to either extract features or analyse human movement.LeCun et al. <cit.> demonstrated the ability of Convolutional Neural Networks (CNN) to outperform SVM models for static images. The first 3D CNN was created by Ji et al. <cit.> to address the issue of automatic feature extraction and recognition from RGB airport footage.Ijjina et al. <cit.> demonstrated the use of 3D CNNs with motion capture, Leightley et al. <cit.> applied a similar network structure to clinically significant motions, captured by Kinect.They demonstrated significant improvement over the previous best method (SVM) in differentiating between good and unstable motions. In this paper we demonstrate a hybrid approach to deep learning.We use manually extracted clinically significant features and then use a deep, fully connected neural network to classify the motion.When comparing this approach to random forests and SVM.The deep neural network out performs the other classifiers. §.§ Datasets Datasets are a constant issue when training deep neural networks. The largest dataset currently available for RGB data is the Sports-1M dataset <cit.>, which contains one million sports videos culled from YouTube and separated into 487 classes <cit.>.RGB-D (depth) datasets are much smaller.The largest currently available being NTU RGB+D <cit.> with 56,880 recordings of 50 individuals, carrying out 60 classes of movement.Zhang et al. <cit.>, identified a total of 44 RGB-D datasets.7 of these concentrate on tracking multiple individuals and 10 use an array of multiple cameras.Of the remaining 27, only 2 are captured in laboratory conditions.However, the movements they capture are not clinically relevant.To address the issue of small datasets, the creation of synthetic data has become popular, indeed the random forest used to generate Kinect skeletons uses synthetic data in its training set.Validation issues have made researchers in clinical studies, shy away from this approach to bulk out small datasets.Zhang et al. <cit.> proposed a method for synthesising data, within the network that improved the recognition rate of motions.It is reported that this form of enhancement, overcomes the limitations of the human defined pre-processing approaches. In timethis type of approach may offer answers to those objections found in the clinical sphere.To this end we demonstrate an automatic method for the production of a family of vectors that can fully describe a motion of any length from a time-series of Kinect skeletons.Currently, the K3Da dataset, highlighted in <cit.> is the only dataset that contains RGB-D data of clinically important actions.It contains 576 recordings of 54 individuals completing the 13 movements from the Short Physical Performance Battery (SPPB) <cit.>.This is the dataset used in our investigations.§ METHODSSomeone with good postural control, when standing, will be able to keep their Centre Of Mass (CoM) over their Base of Support (BoS).Someone with poor postural control, their CoM will more often be outside their BoS and they will initiate more frequent and larger postural corrections, evident as higher postural sway, or poor balance.The corrective actions are initiated through motor control pathways achieved via joint moments applied around the ankle, knee and the hip <cit.>. The extent to which these different joints are active depends on the extent of the postural challenge.When the CoM during quiet standing irrecoverably deviates from the base of support, the person will take a step to rescue from falling.During two-legged quiet standing, the base of support is stable in the medio-lateral (ML) directions, so the ankle and hip strategies <cit.> mainly work to minimiseinstability in the anterior-posterior (AP) axis.In less stable foot positions, such as one-foot stand, instability around the ankle mainly occurs in the ML-axis. To measure deviations in the AP axis, we calculate the body lean angle, that is the Euler angle between the ground plane and the middle of the spine.Norris et al. <cit.> point to the loss of postural control also resulting in movements in the ML axis.To account for this, we record the position of the spine joints in the ML axis, ignoring movements in the AP axis to reduce noise.Together these features give us a measure of postural sway.In addition to measuring sway,we directly estimate the CoM position, Leightley et al. <cit.> found this to be a useful measure of steadiness.We complete our features by calculating Euclidean distance between the base of the spine and the head and the Euler angle between the base of the spine and the neck. Ejupi et al. <cit.> discusses the importance of visual and somatosensory systems, as well as the vestibular system in maintaining balance.To factor these items into our study, we chose tasks that would challenge all three systems.The movements used and the reason for their inclusion are detailed in table <ref>. In this work we used the skeleton data from the K3Da dataset <cit.>, which is the largest data set of it's kind.It consists of, 26 young and middle aged people (18-48 years, 17 male and 9 female) and 28 older age people (61-81 years, 14 male and 14 female) carrying out the SPPB.None of the participants have any non-age-related movement issues.Using the methods outlined in fig. <ref>, we automated the process of feature encoding and classification of an individual, based upon their motion alone.The data processing step was carried out using Matlab.Matlab was also used for classification by traditional machine learning.The Theano framework was used to build the neural network.The following section covers each step in detail. §.§ Pose NormalisationFrom the skeleton data, a series of matrices were constructed, one for each frame of the movement (30 fps).The skeletons were normalised by aligning all frames to the Spine Base joint of the first frame, using equation <ref>. p_n,i(x,y,z)^*= P_n,i(x,y,z) - P_spinebase,1(x,y,z)§.§ Feature EncodingAfter normalisation, a set of features, shown inTable <ref>, was calculated. Although the Kinect camera provides coordinates for 25 joints, we found that features derived from just the torso joints, as defined in <cit.>, plus a calculation of CoM were enough to discriminate the differences in motions of young and older people.Euclidean Distance was calculated between the spine base and head, using equation <ref>. distance= √((x_1-x_2 )^2 + (y_1-y_2 )^2 + (z_1-z_2 )^2) Euler Angle was calculated between the Spine base and Neck joints, using equation <ref>. Θ = arctan( S.Q/SQ) Body Lean Angle is defined as the Euler Angle between the ground plane and the Spine mid joint.This represents changes in the AP axis.Centre of Mass (CoM) of any body, is the mean point that the mass of that body acts.For a human body standing erect, the centre of mass is located around the navel. CoM was calculated using equation <ref>, where J1 = Spine mid, J2 = Hip left, J3 = Hip right, where CoM_x=J1_x+ J2_x+J3_x/3CoM_y=J1_y+ J2_y+J3_y/3CoM_y=J1_z+ J2_z+J3_z/3 ML Axis This represents the frame-by-frame position of the torso joints in the ML axis. §.§ Motion Representation K-mean Clustering is used to convert a time-series of differing lengths (depending on individual recordings), into a set of representative poses of a known length. The k was determined empirically for each type of motion, 5 for chair rise and 2 for all other movements.The centroid poses of each cluster was identified and extracted.Next, the centroids were concatenated together in time-order to produce a 1D vector that provides an example of the whole motion.A label is then added, 1 for young and 0 for older.Simply choosing the centroid would not provide enough examples to train the models.More examples were collected, using the following method:- * Each member of a cluster were ranked using Euclidean distancefrom the centroid. * The closest 50% were selected.To ensure that each vector represents the whole motion, only n number of motion were produced, from each time-series, where n is the number of members in the smallest cluster. This method produces a family of feature-sets which are representative of a person's motions.By providing a family of similar examples, we increase the number of examples retrieved from a time-series, many times, without theneed to resort to synthesising data.This approach makes the models more robust as they are trained on a diverse but representativefeature-set. §.§ EvaluationWe compared 3 methods for classification, SVM, Random Forests and Deep Neural Networks.SVM being widely regarded as the best choice in traditional machine learning for a binary classification, but requires extensive tuning of hyper parameters to achieve top results.Random Forests, on the other hand are much simpler to train and provide excellent results for both binary and multi-class scenarios.We chose these methods in our study to allow us to compare the best of traditional machine learning with deep learning.The neural network consisted of several fully connected layers which learned to separate young from older based on hand crafted features neural networks were not used to extract features.Feature extraction by neural networks, is an area which will be explored in future work. §.§ Validation10-fold cross validation was used to assess the effectiveness of each approach. In K-fold validation every data point gets put into the test set exactly once, and into training set k-1 times. This allows the results to be averaged over the whole dataset.§ RESULTSThe results, summarised in Table <ref>, are similar to those found in the literature for traditional machine learning.In addition we were able to demonstrate that deep learning is able to outperform traditional methods.Using our method, that automatically identifies the essence of a motion and then collects many examples of that motion from a time-series of arbitrary length, we were generate enough examples to allow both traditional and deep learning methods to discriminate between young and older people, with a high degree of certainty.This is demonstrated by high F1-score and Matthews Correlation Coefficient (MCC) scores for all movements, with the chair rise producing the best overall classification.These results are encouraging. However, we do accept that although, the K3Da is one of the larger depth datasets and the only one currently that contains clinically significant movements, it is still a small dataset when compared with those that contain RGB information.Consequently, our results may suffer from overfitting.To address this issue, our future work will involve building a large dataset of clinically significant depth data.§ DISCUSSIONThere isa pressing need to develop a portable system that can help in the assessment of physical impairmentand frailty. Currently the assessment of individuals requires a high degree of training and experience, which can lead to inconsistency from one location to the next.We have taken the first steps in developing a tool which could be used by clinicians in detecting the changes in motion that advance with age.We demonstrate its utility by separating a random sample of young and older individuals from the K3Da dataset.§ CONCLUSION AND FUTURE WORK Our current feature set is working well for a simple two class solution.However, our future work will concentrate on the prediction of mobility issues.The current method may lack the power needed to discern the small changes needed to predict future issues.Hence, we intend to consider the use of deep learning auto-encoders and convolutions, to automatically extract feature sets from motions encoded in depth data.Using this line of research we may be able to produce a complete pipeline exclusively using deep learning.We recognise that in order to build a robust end-to-end deep learning solution we need many more examples that exists in the K3Da dataset.To this end we have taken the first steps towards building a larger dataset of clinically significant motions.We hope that in turn, this dataset maybe a useful resource for other researchers.Finally, to fulfil our ambition of developing a tool, useful to clinicians, we musthave a system that works in real time, taking the data feed directly from the Kinect camera.IEEEtran [1]© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. | http://arxiv.org/abs/1707.08813v2 | {
"authors": [
"Sean Maudsley-Barton",
"Jamie McPheey",
"Anthony Bukowski",
"Daniel Leightley",
"Moi Hoon Yap"
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"published": "20170727105543",
"title": "A Comparative Study of the Clinical use of Motion Analysis from Kinect Skeleton Data"
} |
Department of Computer Science, University of Warwick [email protected] This work was supported by the Leverhulme Trust 2014 Philip Leverhulme Prize of Daniel Král'. We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real r∈(0,1) and an integer d≥ 2, let N(r,d) denote the minimum number of points inside the d-dimensional unit cube [0,1]^d such that they intersect every axis-aligned box inside [0,1]^d of volume greater than r. We prove an upper bound on N(r,d), matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on r. This fully determines the rate of growth of N(r,d) if r∈(0,1) is fixed. A note on minimal dispersion of point sets in the unit cube Jakub Sosnovec December 30, 2023 =========================================================== § INTRODUCTIONThe geometric discrepancy theory is the study of distributions of finite point sets and their irregularities <cit.>. In this note, we study a notion closely related to discrepancy, the dispersion of a point set.The problem of finding the area of the largest empty axis-parallel rectangle amidst a set of points in the unit square is a classical problem in computational geometry. The algorithmic version has been introduced by Naamad et al. <cit.> and several other algorithms have been proposed over the years, such as <cit.>. The problem naturally generalizes to multi-dimensional variant, where the task is to determine the volume of the largest empty box amidst a set of points in the d-dimensional unit cube. An active line of research concerns general bounds on the volume of largest empty box for any set of points, in terms of the dimension and the number of points. An upper bound thus amounts to exhibiting an example of a point set such that the volume of any empty box is small, while the lower bound asks for the minimal value such that every set of points of given cardinality allows an empty box of that volume. The first results in this direction were given by Rote and Tichy <cit.>. Dumitrescu and Jiang <cit.> first showed a non-trivial lower bound, which was later improved by Aistleitner et al. <cit.>. An upper bound by Larcher is also given in <cit.>. Rudolf <cit.> has found an upper bound with much better dependence on the dimension. The problem has recently received attention due to similar questions appearing in approximation theory <cit.>, discrepancy theory <cit.> and approximation of L_p-norms and Marcinkiewicz-type discretization <cit.>.The following reformulation is of interest in the applications in approximation theory: If we fix r∈(0,1) to be the “allowed volume”, how many points in ℝ^d are needed to force that any empty box has volume at most r, in terms of d? In other words, we ask for the minimum number of points needed to intersect every box of volume greater than r.In this note, we establish the optimal asymptotic growth of this quantity for r fixed. §.§ NotationFor a positive integer k, let [k] denote the set {1,2,…,k}. If x∈ℝ^d is a vector, (x)_i denotes the i-th coordinate of x. Let 1 denote the vector (1,…,1)∈ℝ^d (where d will always be clear from context). For d≥ 2, we use [0,1]^d to denote the d-dimensional unit cube. A box B=I_1×⋯× I_d⊆ [0,1]^d is open (closed) if all of I_1,…,I_d are open (closed) intervals. We define ℬ_d as the family of all open boxes inside [0,1]^d. For a set T of n points in [0,1]^d, the volume of the largest open axis-parallel box avoiding all points from T is called the dispersion of T and is defined as(T)=sup_B∈ℬ_d,B∩ T=∅(B),where (I_1×⋯× I_d)=|I_1|⋯|I_d|. Note that the supremum in (<ref>) is attained, since there are only finitely many inclusion-maximal boxes B∈ℬ_d avoiding T. We further define the minimal dispersion for any point set as^*(n,d)=inf_T⊂[0,1]^d, |T|=n(T).Again, observe that the infimum in (<ref>) is actually attained, since any sequence of n-element point sets inside [0,1]^d has a convergent subsequence.The quantity we mainly consider in this paper is the inverse function of the minimal dispersion,N(r,d)=min{n∈ℕ:^*(n,d)≤ r},where r∈(0,1). Determining N(r,d) thus corresponds to the question of how many points are needed to intersect every box of volume greater than r.We remark that the functions ^*(n,d) and N(r,d) are of course tightly connected and any bounds on them translate between each other.§.§ Previous workThe trivial lower bound on ^*(n,d) is 1/(n+1), since we can split the cube into n+1 parts and use the pigeonhole principle. This was improved in <cit.> to^*(n,d)≥log_2d/4(n+log_2d). The inequality (<ref>) can be reformulated to give a lower bound on N(r,d) for r∈(0,1/4),N(r,d)≥1-4r/4rlog_2 d.In order to show (<ref>), the same authors prove an auxiliary lemma, which is equivalent to thatN(1/4,d)≥log_2(d+1).Thus for r∈(0,1/4] fixed, we have N(r,d)=Ω(log d).On the other hand, Larcher (with proof also given in <cit.>) has found the following upper bound of the right order in n,^*(n,d)≤2^7d+1/n.For large enough n, this is the best bound. However, the dependence on d is exponential, which was recently greatly improved by Rudolf <cit.> as follows, ^*(n,d)≤4d/nlog(9n/d). The best upper bound on the inverse of the minimal dispersion for r∈(0,1/4] fixed, which is the setting of this paper, is the reformulation of (<ref>),N(r,d)≤ 8d qlog(33q),where q=1/r. Thus for r∈(0,1/4] fixed, we have N(r,d)=O(d). §.§ Our resultsWe are interested in determining the asymptotic growth of N(r,d) for r∈(0,1) fixed and d tending to infinity. As it turns out, the rate of growth is different for r∈(0,1/4] and r∈(1/4,1). First, we show that for r∈(1/4,1), the number N(r,d) is in fact bounded by a constant depending only on r. This is in sharp contrast with (<ref>) which implies that N(r,d)→∞ for d→∞ if r<1/4.For every r∈(1/4,1), there exists a constant c_r∈ℕ such that for every d≥ 2,N(r,d)≤ c_r.In particular, c_r can be set as⌊1/(r-1/4)⌋ + 1. Then we show an upper bound for r∈(0,1/4] which matches the lower bound (<ref>) up to a multiplicative constant depending only on r.For every r∈(0,1/4], there exists a constant c'_r∈ℝ such that for every d≥ 2,N(r,d)≤ c'_rlog d.In particular, c'_r can be set as q^q^2+2(4log q + 1), where q=⌈ 1/r⌉.This fully determines the asymptotic growth of N(r,d) in terms of d.Let r∈(0,1) be fixed and d tend to infinity. If r∈(0,1/4], then N(r,d)=Θ(log d), otherwise N(r,d)=O(1). § PROOFSWe proceed with the proof of Theorem <ref>.First, consider the case r∈[1/2,1). We set c_r=1 and claim that the single central point 1/2·1∈[0,1]^d does not allow an empty box of volume greater than 1/2. Let B=I_1×⋯× I_d∈ℬ_d be such that 1/2·1∉ X, then there exists a coordinate i∈[d] such that 1/2∉I_i. Hence |I_i|≤ 1/2 and the claim follows.Let us now assume r∈(1/4,1/2). We setδ=r-1/4>0 and k_0=⌊1/δ⌋ and define the set X={kδ·1:k∈[k_0]}∪{1/2·1}.Note that X is thus a set of points all lying on the diagonal of the unit cube and |X|≤ k_0 + 1=c_r. Let B=I_1×⋯× I_d∈ℬ_d be a box such that B∩ X=∅. Again, let i∈[d] be such that 1/2∉I_i. If |I_i|≤ 1/4, then (B)≤ 1/4≤ r, so we can assume |I_i|>1/4≥δ. Also, we have either I_i⊂[0,1/2] or I_i⊂[1/2,1], without loss of generality assume the former (the argument for the other case is symmetric). Let α∈(0,1/2] be the right endpoint of the interval I_i and choose k∈[k_0] maximal so that kδ<α (such k exists, as |I_i|>δ). Observe that kδ∈ I_i, but we assumed kδ·1∉B, therefore there exists an index j∈[d], distinct from i, such that kδ∉ I_j. It follows that |I_j|≤ 1-kδ. Finally, by the definition of k, we have α-kδ≤δ and hence |I_j|≤ 1-α+δ. We conclude(B)≤|I_i|· |I_j|≤α(1-α+δ)≤α(1-α)+δ≤ 1/4+δ= r.To prove Theorem <ref>, we use the probabilistic method to construct a set of points that does not allow any empty box of volume greater than r. Let q be defined as ⌈ 1/r⌉. In the following, we will assume that 1/r is an integer – this is without loss of generality, as otherwise we can use the construction for 1/⌈ 1/r⌉. Let X be the set of n=q^q^2+2(4log q + 1)log dpoints inside [0,1]^d chosen independently and uniformly at random from the grid {1/q,2/q,…,(q-1)/q}^d. Let ℬ_d^r⊂ℬ_d be the set of all boxes of volume greater than r. We now have to show that the probability that X intersects every box from ℬ_d^r is positive.For a box B=I_1×⋯× I_d∈ℬ_d^r, let the number of coordinates i∈[d] such that |I_i|≤ 1-r be equal to m_B. We haver<(B)≤ (1-r)^m_B,obtaining that m_B≤ M for M=⌊log r/log (1-r)⌋. The critical part of our proof is the following observation. There exists a finite family 𝒬_d^r of closed boxes inside [0,1]^d such that every open box B∈ℬ_d^r contains an element of 𝒬_d^r as a subset. Moreover, the cardinality of 𝒬_d^r will be at most (dq)^M.To see this, observe that for every box B=I_1×⋯× I_d∈ℬ_d^r, we can find (possibly non-unique) indices 1≤ j_1<j_2<⋯<j_M≤ d and values k_j_1,…,k_j_M∈[q-1] such that* |I_i|> (q-1)/q for all i∈ [d]∖{j_1,…,j_M} and* k_j_ℓ/q∈ I_j_ℓ for all ℓ∈[M]. We can obtain <ref> from the fact that (q-1)/q=1-r and the definition of m_B, and <ref> from |I_i|>r=1/q for all i∈[d]. For every such combination of j_1,…,j_M∈[d] and k_j_1,…,k_j_M∈[q-1], we will include in 𝒬_d^r the closed box Q=J_1×⋯× J_d⊂ [0,1]^d with J_i=[1/q,(q-1)/q]if i∈ [d]∖{j_1,…,j_M}, {k_i/q}otherwise. We claim that 𝒬_d^r satisfies the desired condition. For every B∈ℬ_d^r and corresponding indices j_1,…,j_M∈ [d] and values k_j_1,…,k_j_M∈[q-1], the closed box Q as defined above is contained within B by <ref> and <ref>. Moreover, we can bound the cardinality of 𝒬_d^r as follows, |𝒬_d^r|=dM(q-1)^M≤(dq)^M.By the property of 𝒬_d^r, if X intersects every box in 𝒬_d^r, then it also intersects every box in ℬ_d^r. We thus only need to show that with positive probability, X intersects every box in 𝒬_d^r. Let us fix indices j_1,…,j_M∈[d] and values k_j_1,…,k_j_M∈[q-1] and the corresponding closed box Q=J_1×⋯× J_d∈𝒬_d^r. Clearly, J_i=[1/q,(q-1)/q] implies that any choice of (x)_i will intersect J_i. Hence, the only coordinates restricting intersections with X are j_1,…,j_M. Moreover, for each of (x)_j_1,…,(x)_j_M, there is exactly one choice of k∈[q-1] such that k=k_j_ℓ.We obtain that for a point x∈{1/q,2/q,…,(q-1)/q}^d chosen uniformly at random,ℙ[x∈ Q]= (1/q-1)^M>r^M.Therefore, we getℙ[X∩ Q = ∅]<(1-r^M)^n≤exp(-nr^M),and, by the union bound,ℙ[X∩ Q' = ∅ for some Q'∈𝒬_d^r] <(dq)^Mexp(-nr^M). If the right hand side of (<ref>) is bounded by one, then there exists a choice of X such that it intersects every box in 𝒬_d^r. We bound the logarithm of the right hand side. Using the fact that M ≤ q^2 and the definition of n, we obtain Mlog d+Mlog q - nr^M≤ q^2log d + q^2log q - nq^-q^2 = q^2log d + q^2log q - q^2(4log q + 1)log d = (1 - 4log d)q^2log q ≤ 0,where the last inequality follows from d≥ 2. This concludes the proof of Theorem <ref>. § ACKNOWLEDGEMENTS The author would like to thank Dan Král' for introducing him to the problem and Jan Vybíral for some initial thoughts and a major simplification of the proof of Theorem <ref>. The author would also like to acknowledge the discussions held in Oberwolfach Workshop Perspectives in High-Dimensional Probability and Convexity, as shared to him by Jan Vybíral. abbrv | http://arxiv.org/abs/1707.08794v2 | {
"authors": [
"Jakub Sosnovec"
],
"categories": [
"cs.CG",
"math.CO"
],
"primary_category": "cs.CG",
"published": "20170727093531",
"title": "A note on minimal dispersion of point sets in the unit cube"
} |
Dipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, ItalyDipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, ItalyDipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, ItalyDipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, ItalySchool of Physics, University of Sydney, NSW 2006, AustraliaDipartimento di Fisica INFN, Sezione di Pavia, Università di Pavia, Via Bassi 6 I-27100 Pavia, ItalyDipartimento di Fisica INFN, Sezione di Pavia, Università di Pavia, Via Bassi 6 I-27100 Pavia, ItalyDipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, Italy The metrological ability of carefully designed probes can be spoilt by the presence of noisy processes occurring during their evolution. The noise is responsible for altering the evolution of the probes in such a way that bear little or no information on the parameter of interest, hence spoiling the signal-to-noise ratio of any possible measurement. Here we show an experiment in which the introduction of an ancilla improves the estimation of an optical phase in the high-noise regime. The advantage is realised by the coherent coupling of the probe and the ancilla at the initialisation and the measurement state, which generate and then select a subset of overall configurations less affected by the noise process.Experimental ancilla-assisted phase-estimation in a noisy channel Marco Barbieri December 30, 2023 =================================================================Optimal design of measurements for parameter estimation exploits the selection of the more informative state among the whole possible configuration space of the probe. Such a state is the one most affected by the perturbation imposed by the interaction with the sample. Strategies based on the use of quantum resources take benefit from this to deliver enhanced estimation <cit.>.In the absence of noise, the evolution characterised by the parameter establishes the number of resources to be considered: in the common example of phase estimation in the context of quantum optics implementation, we need two optical modes, one acting as a reference, and the other probing the phase. In quantum mechanical terms this means setting the dimensionality of the optimal states, but similar considerations can be drawn in the classical case. The presence of noise effectively enlarges the set of possible configurations, for instance by introducing loss modes <cit.>: the effect is generally detrimental to the precision of the estimation, and appropriate strategies need to be put in place.At the same time, it is well known that expanding the configuration space provides more flexibility in measurement design: the first demonstration concerned the implementation of unambiguous discrimination of two non-orthogonal quntum states <cit.>, and it has since been extended to the realisation of generalised measurements <cit.>. Furthermore, the approach of introducing ancillary systems can provide advantages in the characterisation of quantum processes <cit.>, and in the compact design of quantum circuits <cit.>.Even though the use of ancillae was motivated by the realisation of generalised measuments, only recently it has been considered under the metrology perspective: we now know that introducing an ancilla is an helpful tool for parameter estimation in the presence of noise <cit.>, see Fig. <ref>. Here we show experimentally how an ancilla-assisted strategy pushes the precision limit in phase estimation, and discuss the nature of origin of this advantage. We used single photons to probe a phase in an environment imposing amplitude damping. We could verify that coupling the polarisation of the photons with their path at both probe initialisation and readout stages resulted in improved precision in the high-noise regime. From the quantum metrology perspective, our findings support and extend previous results <cit.> that demonstrated how collective measurements offers better performance at handling noise. For classical optics, our results open a new perspective in metrological methods inspired by an analysis of the apparatus at the quantum level <cit.>. Our experimental scheme implements a qubit phase estimation in the presence of amplitude damping (AD) noise by measuring a birefringent optical phase in the presence of controlled noise . We used single photons generated by Spontaneous Parametric DownConversion (SPDC) in a Type-I non-linear crystal (BBO) with a 3 mm, length pumped with an 80 mW continuous wave 405 nm-wavelenght laser. One of the two correlated photons produced in the SPDC has been sent to the interferometer, the other has been used as a trigger.The standard procedure consists in preparing these photons in the diagonal | D⟩=1/√(2)(| H⟩+| V⟩), where H and V are the horizontal and vertical polarisation state of the photon respectively, sending them through the noisy channel and then performing the optimal measurement, identified by optimising the highest Quantum Fisher Information (QFI). Clearly, the precision will be affected by the fact that a number of probe photons will be transferred to a noisy mode that has no coherence with the original modes. This is captured by the QFI which, in the presence of a damping rate η, decreases from the noiseless value 1 to F_s=1-η <cit.>. Thus the damping rate η plays the role of noise level. In our experiment we considered a different approach: the initial two-level system was coupled to an ancillary one that in our implementation corresponded to a different degree of freedom of the same photon. It is necessary that such coupling is realised in a coherent fashion; we will comment later on the exact nature of this operation. For experimental simplicity the ancillary degree of freedom was encoded in the different paths in a Sagnac interferometer, where a Spatial Light Modulator (SLM) replaces one of the mirrors, as schematically depicted in Fig. <ref>. Such a scheme allows for the implementation of an AD channel <cit.>: the H and V polarisation components of the input state are coupled to different paths using a Polarising Beam Splitter (PBS), hence it is possible to simulate a controlled noisy channel by a path-dependent change of the polarisation. Two Half Wave Plates (HWP) and the SLM, which imparts tunable birefringence, implement the polarisation-path coupling. The greyscale image on the SLM is chosen in such a way that each path hits a region corresponding to a different tone. The birefringence is chosen in such a way that the overall transformation leaves the H polarisation unaltered on the clockwise loop; on the counterclockwise loop, a fraction η of the initial V component is rotated to H: | V⟩→(√(1-η))| V⟩ + √(η)| H⟩. The phase ϕ to be estimated has been obtained via a unitary transformation U_ϕ=| H⟩⟨ H|+e^iϕ| V⟩⟨ V| by means an additional HWP inside the Sagnac interferometer. Once these two loops are back together on the PBS a new polarisation-path coupling occur: the components that have been left unaffected by the channel arecoherently recombined with the phase ϕ on the output Branch 1. The H contribution that originated from the damping emerges separately on Branch 2 (see Fig. <ref>).The optimal measurement strategy around ϕ∼π consists in projecting the polarisation of the photons on Branch 1 on the right (R) and left (L) circular basis, while the other branch can be analysed in any basis since it carries no information on ϕ; it is a convenient choice to use the H/V basis.This choice seconds the symmetry of the output state and gives the following outcome probabilities: p_R^(1)=1/4(2-η + 2v√(1-η)sinϕ) and p_L^(1)=1/4(2-η + 2v√(1-η)sinϕ), where v is the visibility of our interferometer, for the Branch 1, and p_H^(2)=η/2 and p_V^(2)=0 for Branch 2. It is demonstrated that this measurement achieves the ultimate limit represented by the quantum Cramér-Rao bound (QCRB) for the employed input state; the expression for the corresponding QFI is F_a=2v^2(1-η)/(2-η), which is always above the single-probe QFI F_s <cit.>. For phase estimation, we fixed a value for η and collected 50 repetitions of four experimental outcome frequencies corresponding to the four channels in Fig.<ref>, in coincidence with the trigger photon. For each repetition, data was accumulated for measurement times of 0.1s, corresponding to a coincidence count rate of nearly 2000 events per acquisition; 50 values of ϕ have then been collected. The variance of this sample,multiplied by the average number of the events, gives the error (Δϕ)^2, which is expected to converge to the ultimate limit established by the QCRB in the limit of a large number of repetitions; numerical simulations of the ideal case have verified that, for the whole range of η, the number of events we collected were sufficient to ensure a behaviour close to the asymptotic.Fig. <ref> summarises our results: the measured values of Δϕ are close to the expected trend of the uncertainty as a function of η, demonstrating the advantage of adopting the ancilla-assisted strategy in the high-noise regime. At low noise, technical imperfections namely additional noise sources - including non-unit visibility of the interference - prevent to observe any advantage, and make the single-probe strategy more convenient. The transformations of our device can be interpreted as a series of quantum operations on a two-qubit system, one associated to the polarisation, the other to the interferometric path of the single photon. In our experiment the polarisation qubit acts as the probe, while the path qubit represents the ancilla. The action of the first passage on the PBS is then described in these terms by that of a controlled-Not gate, in which the polarisation acts as the control <cit.>. The second passage, instead, allows to implement a measurement projecting on entangled states with a second application of the gate. These considerations, which actually informed the proposal of our experiment <cit.>, clarify the working principle of the protocol at a general level: through the introduction of the ancilla, one engineers the state in such a way that the information on the phase is concentrated in a specific subset of all possible configurations. This subset can be then addressed by a (possibly collective) measurement. The higher signal-to-noise ratio can compensate for the discard of noisy events, and result in an improved QFI.In our experiment, we can certainly attribute the metrological advantage to the fact that, at the single-particle level, polarisation and path qubits are in a non-separable state. However, the same outcome statistics could be replicated with an attenuated laser, achieving, in principle, a similar improvement of the Fisher information (FI). Indeed, considering polarisation-sensitive loss as one of the possible extensions of amplitude damping to infinite dimensions, introducing additional modes in the interferometer aids the estimation even in situations where entanglement does not play a role. The use of the ancillary modes delivers an improved FI, since polarisation-path coupling remains a coherent operation in this regime as well. The absence of entanglement in this regime does not conflict with the previous observations, since coherent states are knownto be sub-optimal for lossy interferometry <cit.>. It remains as yet an open question whether the optimal states would take advantage of entangled ancillas. But, more importantly, one can take inspiration from this feature of coherent states to look for applications and futher developments in the classical regime as well.We have demonstrated the advantage of ancilla-assisted protocols in a phase estimation experiment run with single photons, and discussed general considerations on the applicability of this ideas beyond this regime,opening to the perspective for looking at this asan appealing method for phase estimation in noisy environments. Note: During preparation of this manuscript we became aware that similar work was being independently carried out by the group of P. Xue.We thank M.A. Ricci and F. Somma for discussion, P. 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A 92, 032114 (2015).altorio2016 M. Altorio, M. G. Genoni, F. Somma, and M. Barbieri, Phys. Rev. Lett. 166, 100802 (2016).thiel2017V. Thiel, J. Roslund, P. Jian, C. Fabre and N. Treps, Quantum Sci. Technol. 2, 034008 (2017) | http://arxiv.org/abs/1707.08792v1 | {
"authors": [
"Marco Sbroscia",
"Ilaria Gianani",
"Luca Mancino",
"Emanuele Roccia",
"Zixin Huang",
"Lorenzo Maccone",
"Chiara Macchiavello",
"Marco Barbieri"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727092954",
"title": "Experimental ancilla-assisted phase-estimation in a noisy channel"
} |
1 2]Jarod Y.L. Lee 1 3]Peter J. Green 1 2 4]Louise M. Ryan [1]School of Mathematical and Physical Sciences, University of Technology Sydney, Australia. [2]Australian Research Council Centre of Excellence for Mathematical & Statistical Frontiers, The University of Melbourne, Australia. [3]School of Mathematics, University of Bristol, U.K. [4]Department of Biostatistics, Harvard T.H. Chan School of Public Health, U.S.On the “Poisson Trick” and its Extensions for Fitting Multinomial Regression Models [ December 30, 2023 =================================================================================== This article is concerned with the fitting of multinomial regression models using the so-called “Poisson Trick”. The work is motivated by <cit.> and <cit.> which have been criticized for being computationally inefficient and sometimes producing nonsense results. We first discuss the case of independent data and offer a parsimonious fitting strategy when all covariates are categorical. We then propose a new approach for modelling correlated responses based on an extension of the Gamma-Poisson model, where the likelihood can be expressed in closed-form. The parameters are estimated via an Expectation/Conditional Maximization (ECM) algorithm, which can be implemented using functions for fitting generalized linear models readily available in standard statistical software packages. Compared to existing methods, our approach avoids the need to approximate the intractable integrals and thus the inference is exact with respect to the approximating Gamma-Poisson model. The proposed method is illustrated via a reanalysis of the yogurt data discussed by <cit.>. 2mm Keywords:Discrete choice model; Longitudinal data; Mixed logit model; Multinomial mixed model; Nominal polytomous data;Unobserved heterogeneity.§ INTRODUCTION Data with correlated categorical responses arise frequently in applications. This may arise from units grouped into clusters (clustered data) or multiple measurements taken on the same unit (longitudinal data). For instance, we might expect the unemployment outcomes (employed, unemployed, not in labour force) of residents living in the same region to be correlated, due to similar job opportunities and socioeconomic levels. Ignoring the correlation structure and assuming that all observations are independent by fitting an ordinary multinomial regression model may result in biased estimates and inaccurate predictions. Multinomial mixed models can account for correlation by using group level random effects <cit.>. For multinomial mixed models, it is a common practice to assume a multivariate normal distribution for the random effects. The multivariate normal distribution is easy to interpret and is convenient when we want to build more complicated correlation structures into our model. However, the resulting likelihood involves multidimensional integrals that cannot be solved analytically. The computational effort to evaluate the likelihood increases with the number of groups and categories, making it not suitable for large scale applications. In fact, <cit.> showed that closed-form likelihoods for multinomial mixed models do not exist regardless of the random effect distribution, except for the special case when there are no covariates. Various methods have been proposed to circumvent the computational obstacle for fitting multinomial mixed models. Among them are quadrature <cit.>, Monte Carlo EM algorithm, pseudo-likelihood approach <cit.> and Markov Chain Monte Carlo methods <cit.>. <cit.> proposed a random effects estimation approach using a discrete probability distribution approximation. Simulation based methods such as method of simulated moments <cit.> and method of simulated maximum likelihood <cit.> are widely used in theeconometrics literature. Recently, <cit.> proposed a fast moment-based estimation method that scales well for large samples andwhich arguably can be extended for fitting multinomial mixed models. <cit.> used the fact that the multinomial model is a multivariate binary model and exploited a procedure proposed by <cit.> for model fitting. Their approach has been criticized by <cit.> as they failed to realize that a multivariate link function is needed in the context of multinomial models. An alternative strategy using clustered bootstrap was subsequently proposed by <cit.>. Although some authors have considered the Dirichlet-multinomial model that results in a closed-form likelihood, it does not allow the incorporation of individual level covariates.<cit.> advocate using Poisson log-linear or non-linear mixed models, both with random effects, as surrogates to multinomial mixed models. Their method capitalizes on existing mixed models software packages for fitting generalized linear models with random effects. This allows multinomial mixed models to be fitted using an approximate likelihood from the Poisson surrogate models. Their results are based on extensions of the well known “Poisson Trick” <cit.> that relates multinomial models with Poisson models via a respecification of the model formulae. Although clever, their methods have been criticized for being computationally inefficient <cit.> and sometimes producing nonsense results <cit.>. This might be due to the intractable likelihoods of their models and the various approximation methods being used in different software packages. The considerable execution time is especially problematic, where it can take up to months to fit the model to a moderate-sizeddataset <cit.>! In this article, we propose a new approach based on an extension of the Gamma-Poisson model <cit.>, where the likelihood can be expressed in closed-form. Using the proposed estimation procedure, the parameters can be estimated via readily available packages for fitting generalized linear models.The remaining paper is organized as follows. Section <ref> reviews the “Poisson Trick” for multinomial regression models with independent responses, and suggests a parsimonious fitting strategy when all covariates are categorical. In Section <ref> we propose a new approach for approximating the likelihood of multinomial regression models with random effects. The empirical performance of the proposed model is demonstrated via a simulation study and a reanalysis of the yogurt brand choice dataset as discussed by <cit.> in Section <ref>. Finally, we conclude with a discussion and a summary of our findings in Section <ref>. § "POISSON TRICK" FOR INDEPENDENT MULTINOMIAL RESPONSESThis section describes the relationship between multinomial and Poisson regression models for independent responses, which we refer to as the “Poisson Trick”. The results are based on the well known fact that given the sum, Poisson counts are jointly multinomially distributed <cit.>.§.§ Derivation Let Y_j = (Y_jq)_q=1^Q be the Q × 1 response vector for observation j with the corresponding probability vector p_j = (p_jq)_q=1^Q, where q indexes the multinomial category. A common approach to satisfy the two characteristics of probability (i) 0 ≤ p_jq≤ 1 for all j and q; and (ii) ∑_q=1^Q p_jq = 1 is via p_jq = ζ_jq/ζ_j+, where ζ_jq is a positive user-specified function of covariates x and fixed effects γ, and ζ_j+ = ∑_q=1^Q ζ_jq. Depending on the type of variable under consideration, x and γ can be indexed by various combinations of j and q <cit.>. Conditional on the multinomial sums Y_j+ = ∑_q=1^Q Y_jq, Y_js are independently multinomially distributed for each j, i.e.Y_j|Y_j+∼ℳ(Y_j+,p_j) .In multinomial models, Y_j+ = y_j+ is treated as fixed. Suppose we instead treat Y_j+ as random and assume Y_j+∼𝒫(δ_j ζ_j+),independently for each j. This results in a multinomial-Poisson mixture with the following joint probability function for each j:P(Y_j = y_j ∩ Y_j+=y_j+)=P(Y_j+=y_j+) P(Y_j = y_j| Y_j+=y_j+) = e^-δ_jζ_j+(δ_jζ_j+)^y_j+/y_j+!×y_j+!/∏_q y_jq!∏_q (ζ_jq/ζ_j+)^y_jq= ∏_q {e^-δ_j ζ_jq(δ_jζ_jq)^y_jq/y_jq!}iff Y_j+ = y_j+.The marginal probability of Y_j can then be obtained by summing the joint probability over all possible values of Y_j+: P(Y_j=y_j)= ∑_Y_j+=0^∞∏_q {e^-δ_j ζ_jq (δ_j ζ_jq)^y_jq/y_jq!}= ∏_q {e^-δ_j ζ_jq (δ_jζ_jq)^y_jq/y_jq!}.Thus, allowing the multinomial sums to be random according to a Poisson distribution results inY_jq∼𝒫(δ_j ζ_jq),independently for each j and q. Summing over all observations, the log-likelihood is∑_j ℓ^𝒫(δ_jζ_j+;Y_j+) + ∑_j ℓ^ℳ(ζ_j;Y_j|Y_j+) = ∑_j ∑_q ℓ^𝒫(δ_jζ_jq; Y_jq),where ℓ^𝒫 and ℓ^ℳ denote the Poisson and multinomial log-likelihood functions respectively, and ζ_j = (ζ_jq)_q=1^Q. The second term on the left hand side is the model we would like to fit, and the term on the right hand side is the model we actually fit.To show that the Poisson surrogate model is an exact fit to the multinomial model, first note the log-likelihood corresponding to the multinomial model is∑_j log(y_j+!) - ∑_j∑_q log(y_jq!) + ∑_j∑_q y_jqlogζ_jq - ∑_j y_j+logζ_j+,and the log-likelihood of the Poisson surrogate model is- ∑_jδ_j ζ_j+ + ∑_j y_j+logδ_j + ∑_j∑_q y_jqlogζ_jq - ∑_j log(y_j+!).Differentiating Equation <ref> with respect to δ_j and setting it to 0, we obtain δ̂_j = y_j+/ζ_j+. Plugging in the maximizing value of δ_j into Equation <ref>, we have-∑_j y_j+ + ∑_j y_j+log y_j+ - ∑_j y_j+logζ_j+ + ∑_j∑_q y_jqlogζ_jq.Equation <ref> is identical to Equation <ref>, up to an additive constant. It follows that the maximum likelihood estimates, their asymptotic variances and tests for the fixed effects can be exactly recovered under the Poisson surrogate model <cit.>. That is, likelihood inference for ζ_jq is the same whether we regard Y_j+ as fixed (multinomial) or randomly sampled from independent Poissons. This result applies to the fixed effects model, with any parameterization of ζ_jq, including: * Exponential transformations of linear combinations of categorical variables and regression coefficients <cit.>, * Exponential transformations of linear combinations of continuous variables and regression coefficients, * Any monotonic transformations of linear combinations of covariates and regression coefficients, * Nonlinear functions of covariates and regression coefficients, * Nonparametric formulations. The Poisson surrogate model eliminates ζ_j+ from the denominator of the multinomial probabilities. This makes sense intuitively, as we do not expect the multinomial sums to provide any useful information in estimating the fixed effects. Given that δ̂_j can also be obtained by setting the fitted values of the multinomial sums Ŷ_j+=E(Y_j+) equal to the observed counts y_j+ in the Poisson surrogate model, δ_j has the effect of recovering the multinomial sums. The key idea is to include a separate constant δ_j for each unique combination of covariates in the Poisson surrogate models.§.§ Specifying Model Formulae for Poisson Surrogate Models For purposes of exposition, the model formulae in this section are written in terms of thelanguage <cit.>, although this article is not concerned with software packages per se. Multinomial models are fitted using the multinom() function within the nnet package <cit.>; Poisson models are fitted using the glm() function within the stats package. For concreteness, consider the non-parallel baseline category logit models. The “baseline category logit” assumption refers to the following: treating category 1 as the baseline category with ζ_j1 = 1∀ j without loss of generality, we model log(p_jq/p_j1) = log(ζ_jq) as a linear function of covariates x and regression coefficients γ. This assumption is not necessary, but chosen so that the model formulae can be illustrated using functions within the stats package. The “non-parallel” assumption refers to covariate effects that vary across categories <cit.>, i.e. all elements of the γ vector are indexed by q. That is, if the set of logits are plotted against the covariate on the same graph, a set of straight lines with slopes that are in general different will be obtained. Later we shall discuss cases where we relax this assumption.Consider a hypothetical dataset with two predictors X_1 and X_2 (these can be categorical or continuous) and a multinomial outcome vector Y with Q=3 categories. In short format, each row of data represents an observation with a 3-dimensional outcome vector (Y_1,Y_2,Y_3). Poisson models treat the outcomes of each observation as independent and glm() requires data to be presented in long format. This requires an additional factor C that denotes the category memberships. Each row now comprises a scalar outcome, resulting in 3 rows of data per observation. The first few rows of data in both short and long format are shown in Table <ref>.Table <ref> shows the equivalant relationship between non-parallel multinomial models and the corresponding Poisson models, where the parameters satisfy the usual constraints for identifiability. The Poisson surrogate models possess several important features: * The model includes an indicator variable I (that corresponds to logδ_j in Section <ref>) for each observation, although this can be simplified when all covariates are categorical. This is to ensure the exact recovery of the multinomial sums, as the fixed sums are treated as random in the Poisson models. As a result, we do not interpret the coefficients of I since they are just nuisance parameters. * The category membership indicator C enters as a covariate in the Poisson models, where the coefficients correspond to the intercepts in the multinomial modelsv so that the counts are allowed to vary by category. * The model includes interaction terms between X and C (denoted by * in the model formula), where the coefficients correspond to the slopes in the multinomial models. This is due to the non-parallel assumption where each category has a separate slope, and also the fact that multinomial models treat the response counts jointly for each observation, whereas Poisson models treat each response count as a separate observation. It is important that these interaction terms are included even if they are not significant. For multinomial models where some (partial models) or all (parallel models) of the covariate effects do not vary across categories, the equivalent Poisson models can be obtained by modifying the interaction structure between X and C accordingly. For instance, in parallel models where all categories share the same covariate effects, there is no need to include the interation terms between X and C, since the slopes do not vary across categories.When writing the model formula, it is important to specify I and C as factors due to their categorical nature. This can be achieved via the factor() function in . Special Case: Categorical Covariates When all covariates are categorical, the model formulae in Table <ref> offer a more parsimonious option for fitting the Poisson models without having to estimate a separate parameter for each observation.As stated above, the key to achieving the 1-1 correspondence between multinomial and Poisson models (with the same link function) is to include a separate constant for each unique combination of covariates. For categorical covariates, this can be achieved by including the full interaction among the predictors in the Poisson model. When all the covariates are categorical, the interaction term has the precise effect of pooling groups of observations with identical covariates. Fitting such models is equivalent to fitting the observation index as a factor (Table <ref>), but the pooling results in a smaller effective data frame, and therefore smaller storage requirements and faster fitting speed, with no loss of information. Of course, if there are many factors, there may not be much saving, because it will be comparatively rare for different observations to have all the same factor level combinations.§ EXTENDING THE “POISSON TRICK” FOR CORRELATED MULTINOMIAL RESPONSES §.§ DerivationConsider a set of observations which fall into a collection of I groups and let λ_i = (λ_iq)_q=1^Q be a vector-valued random effect for group i. Each observation belongs to only a single group. Extending the notation in Section <ref>, the Q × 1 response vector for observation j in group i is Y_ij = (Y_ijq)_q=1^Q, with the corresponding probability vector p_ij = (p_ijq)_q=1^Q, where p_ijq = λ_iqζ_ijq/∑_q=1^Q λ_iqζ_ijq. Conditional on the multinomial sums Y_ij+ = ∑_q=1^Q Y_ijq and the random effects λ_i,the counts are Multinomially distributed:Y_ij|Y_ij+,λ_i ∼ℳ(Y_ij+, p_ij). In analogy to the results in Section <ref>, given the random effects, we treat Y_ij+ as random and assumeY_ij+|λ_i ∼𝒫(δ_ij∑_q=1^Q λ_iqζ_ijq),independently for each i and j. This givesY_ijq|λ_iq∼𝒫(δ_ijλ_iqζ_ijq),independently for each j and q. The probability argument in Equation <ref> still holds, now conditional on the random effects:∑_i ∑_j ℓ^𝒫(δ_ij∑_q=1^Q λ_iqζ_ijq;Y_ij+|λ_i) + ∑_i ∑_j ℓ^ℳ(ζ_ij;Y_ij|Y_ij+,λ_i) = ∑_i ∑_j ∑_q ℓ^𝒫(δ_ijλ_iqζ_ijq; Y_ijq|λ_iq), whereζ_ij = (ζ_ijq)_q=1^Q. If the random effects are observed, the conditional probability statement above imply a 1-1 exact correspondence between the multinomial and the Poisson surrogate models. However, due to the unobserved nature of the random effects, interest lies in the marginal distribution, obtained by integrating out the random effects. This results in an approximate relationship between the two models. It turns out that the marginal likelihood of the approximating Poisson surrogate model (right hand side of Equation <ref>) can be expressed in closed-form if we assume an independent Gamma model for the random effects, with E(λ_iq) = α_q/β_q and Var(λ_iq) = α_q/β_q^2, i.e. λ_iq∼𝒢(α_q,β_q) <cit.>.With this assumption for the distribution of the random effects, the marginal likelihood of the multinomial model that we would like to fit (second term on the left hand side of Equation <ref>) is given byL^ℳ = ∏_i ∫⋯∫∏_j {y_ij+!/∏_q y_ijq!∏_q (λ_iqζ_ijq/∑_q=1^Q λ_iqζ_ijq)^y_ijq}×∏_qβ_q^α_qλ_iq^α_q-1e^-β_qλ_iq/Γ(α_q) dλ_i1…dλ_iQ. This does not generally exhibit a closed-form solution regardless of the random effect distribution, unless in the special cases of no covariate or with only group specific covariates <cit.>. Numerical or simulation methods can be used to approximate the likelihood, with computational efforts increasing with increasing number of groups and categories. On the other hand, the marginal likelihood of the Poisson surrogate model can be expressed in closed-form:L^P= ∏_i {∫⋯∫∏_j ∏_q e^-δ_ijλ_iqζ_ijq(δ_ijλ_iqζ_ijq)^y_ijq/y_ijq!∏_qβ_q^α_qλ_iq^α_q-1e^-β_qλ_iq/Γ(α_q) dλ_i1…dλ_iQ}= ∏_i {∏_qΓ(α_q+y_i+q)β_q^α_q/Γ(α_q)(β_q+∑_jδ_ijζ_ijq)^α_q+y_i+q×∏_j∏_q(δ_ijζ_ijq)^y_ijq/y_ijq!}. The Poisson surrogate model is an extension of the Gamma-Poisson model as proposed by <cit.> and <cit.> to allow the modelling of counts for multiple categories.As a consequence of Equation <ref>, we have E(Y_ijq) = α_q/β_qδ_ijζ_ijq. Refer to the appendix for details. This is the population-averaged expected value and is not suitable for prediction in general, as it does not take into account the cluster effect. However, it can be useful for out of sample prediction, when there are no samples present in a particular group.Special Case: Var(λ_iq) approaches 0 When Var(λ_iq) approaches 0 for all q, the model reduces to the special case of no random effects as outlined in Section <ref>, and the exact correspondence between the multinomial and the Poisson models can be regained.§.§ Identifiability There is some lack of identifiability with the model formulation given by Equation <ref>, characterized by non-uniqueness of the maximum likelihood estimates.There is an identifiability issue between λ_iq and δ_ij, and also between λ_iq and the category intercepts. To fix this, we impose the constraint of α_q = 1/β_q so thatE(λ_iq) = 1. As a consequence, λ_iq∼𝒢(1/β_q,β_q) and Var(λ_iq) = β_q. Also, we only require a random effect for each logit, and thus a constraint for the random effects associated with the baseline category q=1 is needed. Denote u_iq = logλ_iq. Several authors such as <cit.> (pp.514) and <cit.> considered a multivariate normal distribution for the random effects, i.e. (u_iq)_q=2^Q ∼𝒩(0,Σ), where Σ is a Q-1 by Q-1 variance-covariance matrix. This is equivalent to saying that u_i1=0 for all i, or σ_11 = 0. The equivalent statement in our proposed model is to fix λ_i1 = 1 for all i. This is tantamount to saying Var(λ_i1) = β_1 approaches 0, and thus α_1 approaches ∞.§.§ Prediction of Random Effects and Fitted ValuesWe focus on the best predictor (BP) for random effects prediction, i.e. the predictor that minimises the overall mean squared error of prediction.<cit.> shows that the BP is given by the posterior expectation of the random effect. Under the proposed Poisson surrogate model, the BP is given by BP(λ_iq) = λ̂_iq≡λ^⋆argmin E(λ_iq-λ^⋆)^2 := E(λ_iq|y), which can be calculated via λ̂_iq = ∫_-∞^∞λ_iq f(λ_iq) f(y|λ_iq)dλ_iq/∫_-∞^∞ f(λ_iq) f(y|λ_iq)dλ_iq . Solving for the integral, the BP isλ̂_iq= Y_i+q + 1/β_q∑_jδ_ijζ_ijq + β_q,where Y_i+q = ∑_j Y_ijq. λ̂_iq depends on the parameters δ_ij, γ and β_q, in which we replace by their estimators, leading to the empirical best predictor (EBP). The fitted values can then be defined as Ŷ_ijq = δ_ijλ̂_iqζ_ijq,where we replace δ_ij and ζ_ijq by their respective estimators δ̂_ij and ζ̂_ijq.§.§ Parameter Estimation Consider the parameterization ζ_ijq = exp(η_ijq) which is widely adopted in practice, where η_ijq = x_ijq^Tγ, where x_ijq and γ are both vectors. The chosen index structure for x and γ encompasses a variety of possible scenarios: (i) category-specific predictors with generic coefficients x_ijq^Tγ, (ii) category-specific predictors with category-specific coefficients x_ijq^Tγ_q, and (iii) observation-specific predictors with category-specific coefficients x_ijγ_q. This can be achieved by creating the appropriate interaction terms between the predictor and the category indicator variable, thus modifying the model matrix. Note that observation-specific predictors must be paired with choice-specific coefficients. Otherwise they will disappear in the differentiation when we consider the log-odds.Denote θ = (γ,(β_q)_q=2^Q), where γ includes the incidental parameters log(δ_ij) for all i and j. Algorithm <ref> presents an Expectation/Conditional Maximization (ECM) algorithm <cit.> for parameter estimation of the Poisson surrogate model. Refer to the appendix for a detailed derivation. § YOGURT BRAND CHOICE DATASET We consider the yogurt brand choice dataset previously analyzed by <cit.> and <cit.>. <cit.> approximated the likelihood of a multinomial logit model with Gaussian random effects using a discrete distribution. <cit.> approximated the multinomial logit model using the Poisson log-linear model and Poisson nonlinear model, both with Gaussian random effects.The dataset consists of purchases of yogurt by a panel of 100 households in Springfield, Missouri, and were originally provided by A. C. Nielsen. The data were collected by optical scanners for about two years and correspond to 2,412 purchases. Variables collected include brand, price and presence of newspaper feature advertisements for each purchase made by households in the panel. Price and feature advertisements are choice-specific variables. We assume a parallel baseline logit model by assigning generic coefficients γ to these variables, as we do not expect the effect of price and feature advertisements on the probability of purchase to vary according to brands. The four brands of yogurt: Yoplait, Dannon, Weight Watchers, and Hiland account for market shares of 34%, 40%, 23%, and 3% respectively. Following <cit.>, we put Hiland as the reference brand. Table <ref> presents the yogurt data in both long and short format. The letters `f' and `p' represent the feature and price variables respectively, with the letter that follows denoting the brand. For instance, `fy' stands for `feature of Yoplait' and `pd' stands for `price of Dannon'.We fit the models proposed in Sections <ref> and <ref>, and compare our results to that of <cit.>, fitted using the SAS macro GLIMMIX and the SAS procedure NLMIXED. The results are presented in Table <ref>. The preference ordering of the brands are the same for all models, i.e. Yoplait is the most preferred brand, followed by Dannon, Weight Watchers and Hiland. The slope parameters estimates have the expected signs for all models. An increase in price is associated with a decrease in the probability of purchase. Feature advertisement tends to increase the chance of purchase. The household-to-household variation in the probability of purchase for Weight Watchers is much larger than the other brands, although none are significant (p > 0.05).In comparing the estimates between models, we note that the fixed effects model is likely to produce biased estimates as it did not take into account of the correlation induced by multiple purchases from the same household. The parameter estimates of NLMIXED and Gamma-Poisson are uniformly larger than that of GLIMMIX, except for the intercept associated with Weight Watchers (for NLMIXED) and for the slope associated with price (for Gamma-Poisson). The estimates of the standard errors from NLMIXED and Gamma-Poisson are also uniformly larger than that of GLIMMIX. These differences can be attributed to the different distributional assumptions of the random effects, and also the different approximations used in GLIMMIX and NLMIXED to estimate the intractable likelihood. In this regard, our model exhibit a closed-form likelihood that allows exact inference to be performed with respect to the approximating model.We tried to fit a simplified version of GLIMMIX using the glmer() function within the lme4 package in , with just a random effect per household (ignoring the choice effect). However, the model failed to converge within a few months, even though <cit.> claimed that the GLIMMIX model coverged in SAS.§ CONCLUDING REMARKS In this article, we presented methods for fitting various multinomial regression models via the so-called “Poisson Trick” and its extensions. The “Poisson Trick” for fitting fixed effects multinomial regression models is handy when the direct fitting of multinomial models is not supported, for instance the INLA package <cit.> in . For multinomial regression models with random effects, there exist a variety of experience for using the existing extensions proposed by <cit.>, from taking months to fit a moderate sized dataset <cit.>, producing nonsense results <cit.> to non-convergence in our experience of fitting the yogurt brand choice dataset. We proposed an extension of the “Poisson Trick” using Gamma (multiplicative) random effects. In contrast to the models by <cit.>, our model exhibits a closed-form likelihood and can be maximized using existing functions for fitting generalized linear models that are stable and heavily optimized, without having to approximate the integrals.§ APPENDIX§.§ Derivation of the Population-Averaged Expected Values in Equation <ref>Equation <ref> is also equivalent to∏_i {∏_q[ Γ(α_q+y_i+q)/Γ(α_q) y_i+q!(∑_j δ_ijζ_ijq/β_q+∑_jδ_ijζ_ijq) ^y_i+q( β_q/β_q+∑_jδ_ijζ_ijq)^α_q]×∏_q[ y_i+q!/∏_j y_ijq!∏_j (δ_ijζ_ijq)^y_ijq/(∑_jδ_ijζ_ijq)^y_i+q] }. This results in two different interpretations for the extended Gamma-Poisson surrogate model: * For each i and q, Y_ijq is independent negative multinomial 𝒩ℳ(α_q, δ_ijζ_ijq/β_q+ ∑_j δ_ijζ_ijq) (Equation <ref>). * For each i and q, the category sumsY_i+q are independent Negative Binomial 𝒩ℬ(α_q,∑_jδ_ijζ_ijq/β_q+∑_jδ_ijζ_ijq), and conditional on the Y_i+q, Y_ijq is independent multinomial ℳ(Y_i+q,δ_ijζ_ijq/∑_jδ_ijζ_ijq) (Equation <ref>).Taking expectation of both Equations <ref> and <ref> with respect to Y_ijq gives rise to the population-averaged expected value given in Equation <ref>. The definitions of negative multinomial and negative binomial distributions are given in the following subsections.§.§.§ Negative Multinomial DistributionThis is the distribution on the n+1>2 non-negative integers outcomes { X_0,…,X_n }, with corresponding probability of occurence p = { p_0,…,p_n } and probability mass functionΓ( ∑_i=0^n x_i ) p_0^x_0/Γ(x_0)∏_i=1^n p_i^x_i/x_i!, for parameters x_0 > 0 and p=(p_i)_i=1^n, where p_i ∈ (0,1) for all i, ∑_i=0^n p_i = 1 and Γ(·) is the Gamma function. We write Y ∼𝒩ℳ(x_0,p). For positive integer x_0, the negative multinomial distribution can be recognized as the joint distribution of the n-tuple { X_1,…,X_n } when performing sampling until X_0 reaches the predetermined value x_0. The mean vector of negative multinomial distribution is given by x_0/p_0 p.§.§.§ Negative Binomial DistributionThis is the distribution on the non-negative integers outcome X, with corresponding probability of occurence p and probability mass functionΓ(r+x)/x!Γ(r) (1-p)^r p^x,for parameters r>0 and p∈(0,1). We write X∼NB(r,p). For positive integer r, the negative binomial distribution can be recognized as the distribution for the number of heads before the rth tail in biased coin-tossing, but it is a valid distribution for all r>0. In engineering, it is sometimes called the Pólya distribution in the case where r is not integer. §.§ Derivation of the Expectation/Conditional Maximixation (ECM) Algorithm in Section <ref>Treating λ = λ_iq for all i and q=2 to Q as missing data and y = y_ijq for all i, j and q as observed data, the complete data is (y_ijq,λ). Denote θ = (γ,(β_q)_q=2^Q), where γ includes the incidental parameters log(δ_ij) for all i and j. The complete data log-likeliood ℓ(θ|y,λ) is-∑_i∑_q≠1λ_iq( ∑_j e^x_ijq^Tγ) + ∑_i∑_q≠1 y_i+qlogλ_iq + ∑_i∑_j∑_q x_ijq^Tγ y_ijq + ∑_i∑_j∑_q log(y_ijq!) + ∑_i∑_q≠1 (1/β_q-1)logλ_iq - ∑_i∑_q≠1λ_iq/β_q - ∑_i∑_q≠1logβ_q/β_q-∑_i∑_q≠1logΓ(1/β_q). The (t+1)th E-step involves finding the conditional expectation of the complete data log-likelihood with respect to to the conditional distribution of λ given y and the current estimated parameter θ^(t). Straightforward algebra establishes thatλ_iq|y_ijq,θ^(t)∼𝒢( y_i+q+1/β_q^(t) , (∑_j e^x_ijqγ^(t)+1/β_q^(t))^-1),independently for each i and q, where the gamma distribution is parameterized in terms of scale parameter. It follows thatλ̂_iq^(t+1) = E(λ_iq|(y_ijq)_j,θ^(t)) = y_i+q+1/β_q^(t)/∑_j e^x_ijqγ^(t)+1/β_q^(t) χ̂_iq^(t+1) = E(log(λ_iq)|(y_ijq)_j,θ^(t)) = ψ(y_i+q+1/β_q^(t)) - log(∑_j e^x_ijqγ^(t)+1/β_q^(t)). Thus, in the (t+1)th E-step, we replace λ_iq and χ_iq = log(λ_iq) in Equation <ref> with λ̂_iq^(t+1) and χ̂_iq^(t+1), giving Q(θ|θ^(t)). The (t+1)th CM-step then finds θ^(t+1) to maximize Q(θ|θ^(t)) via a sequence of conditional maximization steps, each of which maximizes the Q function over a subset of θ, with the rest fixed at its previous value. In our application, it is natural to partition θ into γ and β_q for each q=2 to Q. Differentiating Equation <ref> with respect to γ, we obtain-∑_i∑_j∑_q λ_iq x_ijq e^x_ijq^Tγ + ∑_i∑_j∑_q x_ijq y_ijq, which is the score equation of the Poisson log-linear model <cit.> with an additional offset λ_iq. This allows us to leverage existing functions for fitting generalized linear models available in most statistical software packages for maximizing γ in the CM step. This is an important feature as γ often contains a huge amount of parameters in our applications, due to the inclusion the incidental parameter log(δ_ij) for every unique combination of covariates. Existing functions for fitting generalized linear models are typically stable and heavily optimized, even for a large number of parameters. Maximizing β_q in the CM step for each q is straightforward, as it only involves univariate optimization.§ ACKNOWLEDGEMENTSLee's research is partially supported by the Australian Bureau of Statistics. The authors are grateful to Zhen Chen for providing the yogurt data. agsm | http://arxiv.org/abs/1707.08538v1 | {
"authors": [
"Jarod Y. L. Lee",
"Peter J. Green",
"Louise M. Ryan"
],
"categories": [
"stat.ME",
"stat.AP",
"stat.CO"
],
"primary_category": "stat.ME",
"published": "20170726165548",
"title": "On the \"Poisson Trick\" and its Extensions for Fitting Multinomial Regression Models"
} |
^1 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan;[email protected]; [email protected]; [email protected]^2Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada^3Institute of Astronomy and Department of Physics, National Tsing Hua University,Hsinchu 30013, Taiwan.^4Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK^5School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China We search for the gamma-ray counterparts of stellar-mass black holes using long-term Fermi archive to investigate the electrostatic acceleration of electrons and positrons in the vicinity of the event horizon, by applying the pulsar outer-gap model to their magnetosphere.When a black hole transient (BHT) is in a low-hard or quiescent state, the radiatively inefficient accretion flow cannot emit enough MeV photons that are required to sustain the force-free magnetosphere in the polar funnel via two-photon collisions. In this charge-starved gap region, an electric field arises along the magnetic field lines to accelerate electrons and positronsinto ultra-relativistic energies. These relativistic leptons emit copious gamma-rays via the curvature and inverse-Compton (IC) processes. It is found that these gamma-ray emissions exhibit a flaring activity when the plasma accretion rate stays typically between 0.01 and 0.005 percent of the Eddington value for rapidly rotating, stellar-mass black holes. By analyzing the detection limit determined from archival Fermi/LAT data, we find that the 7-year averaged duty cycle of such flaring activities should be less than 5% and 10% for XTE J1118+480 and 1A 0620-00, respectively, and that the detection limit is comparable to the theoretical prediction for V404 Cyg. It is predicted that the gap emission can be discriminated fromthe jet emission, if we investigate the high-energy spectral behaviour orobserve nearby BHTs during deep quiescence simultaneouslyin infrared wavelength and very-high energies. § INTRODUCTIONIn the past several years there has been increasing interest in the γ-ray emissions from the direct vicinity of accreting black holes (BHs). Although accreting BHs can emit radiation in various wavelengths in general, only a few of them have the confirmed counterparts in the high-energy (HE) γ-ray band. For example, Cyg X-3 (i.e., V1521 Cyg or 4U 2030+40) has a confirmed transient γ-ray detection using the AGILE data above 100 MeV <cit.> and the Fermi/LAT data between 100 MeV and 100 GeV <cit.>. Cyg X-1 (i.e., V1357 Cyg or 4U 1956+350) also has a confirmed γ-ray counterpart detected by Fermi/LAT between the energy range of 60 MeV and 500 GeV <cit.>.On the other hand, several similar cases did not gain a positive detection in the γ-ray band <cit.>. The observed γ-rays were concluded to be associated with the radio flares that are consistent with the radio flux level of the relativistic jets and shock formation in the accretion process. Shocks propagating in a jet <cit.> provide a possible explanation of the (very) HE emission for Cyg X-3 <cit.> and for Cyg X-1 <cit.>. Recently, rapidly varying, sub-horizon scale TeV emissions were discovered from IC 310 <cit.> using MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov) telescopes. The observed radio jet power, the expected cloud crossing time and the proton-proton cooling time cannot support such models as jet-in-a-jet <cit.> and clouds/jet interactions <cit.> for the shock-in-jet scenario. A likely mechanism for such event-horizon-scale γ-radiation, is the particle acceleration taken place in the vacuum gap that arises in the polar funnel of a rotating BH magnetosphere <cit.>.In additional to the emission site, there are several key differences between the shock-in-jet models and the BH gap models. In the former, the spectrum is determined by the energy releases in the shock and the spectrum evolves as the shock propagating along the jet, with different phases (from Compton to synchrotron to adiabatic phases). A positive correlation between accretion rate and a shock emission is expected, provided that jet or outflow become more powerful at higher accretion rates <cit.>. In the latter, BH gap model, the rotational energy of a rotating BH is electrodynamically extracted via the Blandford-Znajek process <cit.>, and dissipated in the form of charged-particle acceleration, which leads to a resultant γ-radiation from the direct vicinity of the BH. Since the magnetic-field-aligned electric field is less efficiently screened by the created pairs when the soft photon field is weak, the γ-ray luminosity of a BH gap increases with decreasing mass accretion rate,Ṁ (). What is more, although Cyg X-3 and Cyg X-1 are persistent BHs in high-mass X-ray binaries (HMXBs)with high accretion rates,Ṁ∼ 3.64× 10^-8 M_⊙ yr^-1 andṀ∼ 3.88× 10^-9 M_⊙ yr^-1(; hereafter T16),and serve as examples of the shock-in-jet scenario, the BH-gap model provides alternative explanation of HE emissions at lower accretion rate. Furthermore, it is demonstrated in H16 that a BH gap can emit photons mostly in the HE (GeV) range via the curvature process for stellar-mass BHs, and mostly in the very-high-energy (VHE; TeV) range via the inverse-Compton (IC) process for super-massive BHs.Motivated by the striking differences between the shock-in-jet and gap-emission scenarios, we expect that only BH transients (BHTs) in quiescence are plausible HE (GeV) gap emitters.As the first trial for seeking possible BH gap emission from stellar mass black holes, we select and concentrate on nearby BHTs with low accretion rates in low-mass X-ray binaries (LMXBs) and search for their HE emission by analyzing their 7-year archival data of the Fermi/Large Area Telescope (LAT). In <ref>– <ref>, we describe how we select the sources, estimated their mass accretion rate, Ṁ, and analyzed their LAT archival data. Then in <ref>, we outline the BH gap model. In <ref>, we derive the observational upper limits on their HE fluxes and constrain their Ṁ. Finally in <ref>, we compare the expected spectral behaviour obtained by the BH-gap and the shock-in-jet scenarios, and discuss the next targets to be observed in HE and VHE. The distinct spectral features between the two models should help us discriminate the emission models with future HE observation. Nevertheless, we also emphasise that even for a positive result of HE emission for the sources that have low accretion rates satisfying the requirement of BH gap models, the shock-in-jet model is not ruled out.The emission nature should be further determined by the overall spectral profile in the HE/VHE region, or/and its dependence on Ṁ.§ SOURCE SELECTIONUsing the recent BH gap model (H16), we can infer which BHTs will exhibit strong HE and VHE fluxes at Earth. Specifically, we can introduce the following conditions to search for plausible gap emitters:(1) The BH mass is large.(2) The BH spin is large.(3) The dimensionless accretion rate, ṁ, liesbetween 10^-4.25≲ṁ≲10^-4 near the horizon, where ṁ≡Ṁ/Ṁ_ Edd, and Ṁ_ Edd≡ 2.18 × 10^-8 (M/M_⊙) (η/0.1)^-1 M_⊙^-1 denotes the Eddington accretion rate with accretion efficiencyη∼ 10%.(4) The distance is short. (5) The observer's viewing angle, ζ_ obs,is relatively small with respect to the rotation axis (i.e. the binary system is nearly face-on;see 5.1.2 of H16 for details). Conditions (1)–(3) represent intrinsic properties,while (4) and (5) show positional conditions relative to us. In this letter, we define that a BHT is in deep quiescence' if condition (3) is met.In general, when the accretion rate is very low,the corresponding radiatively inefficient accretion flow (RIAF) cannot supply enough MeV photons via free-free process to sustain the force-free magnetosphere (e.g., fig. 1 of H16). In this charge-starved magnetosphere,an electric field inevitably arises along the magnetic field lineto accelerate electrons and positrons into ultra-relativistic energies. The resultant gap emission becomes particularly strong when condition (3) is met.It is noteworthy that persistent X-ray sources will not showstrong gap emissions, because their ṁ is much higher than condition (3). However, if a BHT has a lower mass-transfer rate (e.g., Ṁ≲ 10^-9 M_⊙^-1; ),we can expect a stronger gap emission from such objects. What is more, we can approximately estimate the gap luminosityfrom the BZ power ( 2 of H16), without solving the set of Maxwell-Boltzmann equations ( 4 of H16), if we impose a condition for the polar funnel to be highly charge-starved. Utilizing this approximation, and using the BH masses anddistances reported by T16,we select top four BH X-ray binaries whose BZ flux(i.e. BZ power divided by distance squared) is the greatest at Earth. Among the four objects, we exclude the high-mass X-ray binary Cyg X-1, because its persistent X-ray emission indicates ṁ>10^-4 in the entire period. Coincidentally, the other three targets all have firm detections of jets, and some studies mentioned in 7 are interested in the connection between the jet detection and the HE emission.They are: (i) 1A 0620-00 (i.e. 3A 0620-003 or V616 Mon; ),(ii) XTE J1118+480 (i.e. KV UMa; ), and(iii) V404 Cyg (i.e. GS 2023+338; ).In this paper, we will examine these three objects as potential gap emitters.§ DETECTABILITY OF BH GAP EMISSIONTo infer the feasibility of strong gap emissions, we must estimate the ṁfor individual sources. However, such an ṁ is, in general, difficult to be constrained on real-time basis unless the Faraday rotation is measured like the case of supermassive BHs <cit.>. Nevertheless, their long-term averaged values can beestimated as described below.For 1A 0620-00, the BH mass and the distance are inferred to beM ≈ 6.60 M_⊙ andd ≈ 1.06 kpc,respectively, and our viewing angle is reported to be ζ_ obs = 51.0± 0.9^∘ <cit.>. From the luminosity of the 1975 outbursts and the intervalfrom the previous one in 1917, the mass accretion rate of 1A 0620-00can be estimated as Ṁ=3× 10^-11 M_⊙ yr^-1<cit.>,which corresponds to the dimensionless accretion rateṁ = 2.08 × 10^-4. During quiescence, its accretion rate can be alternatively estimatedby observing its bright spot, and can reach up toṀ=3.4× 10^-10 M_⊙ yr^-1 <cit.>,or equivalently ṁ= 2.36 × 10^-3, which is one-order higher than the previous measurement. If ṁ is close to the former value, 2 × 10^-4,which is slightly higher than condition (3), 1A 0620-00 is expected to spend a certain fraction of timein the narrow ṁ range defined by condition (3),and hence to show HE and VHE flares. However, if the actual ṁ is close to the latter value, 2 × 10^-3,we do not expect that this source undergoes flaring activities during a significant fraction of time.Its ζ_ obs indicates that a strong gap emission may beemitted marginally toward us (H16),if the polar funnel exists down to the lower latitudes (e.g., to the colatitudes 80^∘) within 6 r_ g <cit.>. For XTE J1118+480, we have M ≈ 7.30 M_⊙ (T16),d ≈ 1.72 kpc <cit.>,and 68^∘ < ζ_ obs < 79^∘ <cit.>. Its time-averaged mass accretion rate is estimated to beṀ=1.55× 10^-11 M_⊙ yr^-1(updated table for T16[http://142.244.87.173/static/MassTransferRate.txt?]), or ṁ= 9.92 × 10^-5. This small ṁ shows that the BH gap of XTE J1118+480 will exhibit HE and VHE flares in a significant fraction of time. However, its large viewing angle indicates that the BH-gap emission probably propagates away from our line of sight. For V404 Cyg, we have M ≈ 7.15 M_⊙ (T16),d ≈ 2.39 kpc <cit.>, and ζ_ obs= 67^∘^+3_-1 <cit.>. Its time-averaged accretion rate is between2.7× 10^-10 – 3.5× 10^-9 M_⊙ yr^-1among its evolution track <cit.>, orṁ= 1.73 × 10^-3 – 2.24 × 10^-2. But if we consider the updated measurement calculated between1996 January - 2016 September (updated table for T16^1),the time-averaged accretion rate is only2.30× 10^-12 M_⊙ yr^-1corresponding to a much smaller ṁ= 1.47 × 10^-5. If the actual accretion rate is close to the former value, it is very unlikely for V404 Cyg to exhibit a BH gap emission during a large fraction of time. However, if it is close to the latter value, its BH gap will exhibit a strong HE and VHE emission, which may be stationary or non-stationary, with a large duty cycle. However, its relatively large ζ_ obs may prevent us from detecting these γ-rays, which may propagate away from our line of sight.Although the ζ_ obs's of these target sources are large,we examine these three sources as the first step, because the actual ζ_ obs might be different from the represented values mentioned above. In addition, BHTs in quiescence are known to be variable X-ray sources <cit.>. For instance, V404 Cyg can vary by a factor of 20 during quiescence<cit.>. Thus, during a certain fraction of time, there is a possibilitythat their ṁ enters this relatively narrow range to activate the gap. Given that current observational evidence are not precise enough to determine the time interval for a deep quiescent stage[In comparison, HE emission is preferentially expected to be accompany with a prominent jet detection or at the outburst stage in other traditional models <cit.>.], as the first trial, we analyze the archival LAT data and examine their time-averaged HE fluxes during August 2008 - November 2015.§ OBSERVATIONS AND DATA ANALYSISLAT is a wide-band γ-ray detector (in 20 MeV – 300 GeV)onboard the Fermi observatory, which provides all-sky surveyevery two orbits since August 2008. Until the end of 2015, we already have 7-year accumulation time for a long-term investigation for those stellar-mass BHTs. For all the targets in our interests summarized in 1, we downloaded the Pass 8 (P8R2) archive from the LAT data server[http://fermi.gsfc.nasa.gov/cgi-bin/ssc/LAT/LATDataQuery.- cgi] within a circular FOV in 10^∘ radius and considered the time distributed among August 2008 to November 2015 for further investigations. For V404 Cyg, which experienced two outbursts in 2015, the photons detected during the first outburst (June/July; ) are removed from the analysis and our data do not cover the epoch of the second outburst <cit.>, so that we collect only the photons during low accretion rates.In order to reduce and analyze data, we used the Fermi Science tools v10r0p5 package and constrain photons in the class for the point source or galactic diffuse analysis (i.e. evclass = 128).We also collected all the events converting both in the front- and back-section of the tracker (i.e. evtype = 3), and the corresponding instrument response described for the selected event type has been defined in the “P8R2_SOURCE_V6” IRF (Instrument Response Function) used throughout this study.Events with zenith angles larger than 90^∘ were also excluded to avoid the contamination from Earth albedo γ-rays, and only the good data quality was counted in the time selection to exclude those data within time intervals affected by some spacecraft events (i.e. DATA_QUAL>0). We performed the binned likelihood analysis using the “NewMinunit” optimization algorithm by defining a 7^∘× 7^∘ square region of interest (ROI), together with the long time span and the energy range of 0.07-300 GeV. The choice for the lower boundary of the energy range was constrained by the usage of IRF to generate the exposure map, and we included the energy dispersion correction when the analysis includes photons <100 MeV. The source model determined for likelihood analysis is based on the LAT 4-year point source (3FGL; ) catalog, and the spectral parameters of each source in ROI is freed to gain the best fit to the archival observation.The standard templates of Galactic and isotropic background (gll_iem_v06.fits & iso_P8R2_SOURCE_V6.txt) are included in our analysis as well. The central positions determined for our targets are summarized in Table <ref>, and they can be referred to <cit.> and <cit.>.Since all of our targets are not known Fermi sources, a simple power law was adopted to characterize their spectra throughout the analysis. The test-statistic (TS) values yielded with the best fit to observations of the different energy range are also reported in Table <ref>.§ THE BLACK HOLE GAP MODELTo compare the LAT observational constraints with the theoretical prediction, we apply the method described in 4 of H16 to individual BHTs. Namely, we solve the set of the following three differential equations. First, we solve the Poisson equation for the non-corotational potential (eq. [19] in H16) near the event horizon.The magnetic-field-aligned electric field, E_∥, can be computed from the Poisson equation through eq. (23) of H16. Second, we solve the Boltzmann equations of the produced e^±'s in the gap, assuming that their Lorentz factors saturate at the curvature-limited terminal value at each position. This assumption is valid for stellar-mass BHs, because the curvature process dominates the IC one, and because the acceleration length is shorter than the gap width particularly during the HE flare. Third, we solved the radiative transfer equation of the emitted photons, assuming they have vanishing angular momenta. We adopt the analytic solution of <cit.> to describe the RIAF soft photon, and solve the gap in the 2-D poloidal plane ( 4.2.5 of H16). Both the outer and inner boundary positions were solved from the free boundary problem. To estimate the greatest gap flux, we adopt a_∗=0.9 andΩ_ F=0.5ω_ H, where a_∗≡ a/r_g is the dimensionless BH's spin parameter, and r_ g≡ GMc^-2 represents the gravitational radius; Ω_ F and ω_ H denote the angular frequency of rotating magnetic field lines and a rotating BH, respectively. We assume that the magnetic field takes the equipartition value with the plasma accretion,B=B_ eq≈ 4 × 10^8 ṁ^1/2 M_1^-1/2,at r= 2 r_ g, where M_1 denotes the BH mass in ten solar-mass unit.§ RESULTS In spite of the long-term accumulation of the Fermi data, we did not detect any counterparts at a significant level for the three sources we have concerned. This result is in consistency with sources resolved in 4-year Fermi catalogue <cit.> and the examination for V404 Cyg using 7-year LAT data <cit.>. We also tried the energy-resolved investigations as well; however, the corresponding γ-rays emitted from the direction of the source position are still too few to show any clear signature in the binned likelihood analysis. Assuming a simple power-law spectrum, we find that the spectral parametershave very large uncertainties in this case.On these grounds, we list their 2σ flux upper limits in Table <ref>, according to the best spectral fit obtained in the analysis. It is also noteworthy that the VHE flux may be detectable below 1 TeV during a HE flare, as indicated by the sensitivity curves of CTA in Figs. <ref>–<ref> (dashed and dotted curves labelled with CTA 50 hrs”)[https://www.cta-observatory.org/science/cta-performance/- #1472563157332-1ef9e83d-426c]. In all the figures, we put these upper limits on the predicted spectra of their BH-gap emissions.Fig. <ref> shows the case of 1A 0620-00. The cyan dash-dotted, blue dotted, green dash-dot-dot-dotted, black solid, and red dashed lines correspond to the dimensionlessaccretion rates, ṁ=1.00 × 10^-3 , 10^-3.5, 10^-4, 10^-4.125, and 10^-4.25, respectively. The latter three lines show that the flux flares up in 0.1–3 GeV via curvature process and in 0.03–1 TeV via IC process, when the dimensionless accretion rate is between 5 × 10^-5 and 10^-4. We then put the flux upper limits (obtained from the 7-year LAT archival data as described above) with the two red down arrows in 0.1–1 GeV and 1–10 GeV. It follows that the flaring HE fluxes, which are represented by the green, black, and red lines,are predicted to be 10–20 times greater than the observational upper limits. Thus, we find that the 7-year averaged duty cycle of such flaring activities was less than 10% from August 2008 to November 2015 for 1A 0620-00, provided that a_∗ > 0.9 and ζ_ obs < 40^∘(H16), and adopting a conservative flux estimate at ṁ=10^-4. For instance, if the BH gap flares during 10% of the entire period, its flux is predicted to appear at (or slightly above) the 10% level of its peak,which is comparable with the observational upper limit (red down arrow). However, if its BH is rotating as slowly as a_∗∼ 0.12 <cit.>, there is no chance for such a slow rotator to emit a detectable flux at Earth from its gap; that is, we will not be able to obtain any constraints on the duty cycle of HE flares, using the BH gap model. In the same way, we also apply the BH gap model to the two other targets and generate the predicted spectra (Fig. <ref> & Fig. <ref>).Comparing with the observational upper limits (red down arrows) obtained by the binned likelihood analysis with long-term Fermi archive, we find that the duty cycle of the BH gap is less than0.05 (i.e. 5 %) for XTE J1118+480, provided that a_∗≥ 0.9and ζ_ obs < 40^∘.For V404 Cyg, the predicted HE flux lies at the same level of the observational upper limits. Thus, we cannot constrain the duty cycle of its gap flares.Lastly, we would like to comment on the dependence between a_∗ and BZ power. The BZ power, and hence the gap luminosity is approximately proportional to a_∗^2. (It is noteworthy that the BH gap emission and the jet HE emission are independent each other.) Thus, the predicted flux is approximately halved if a_∗ reduces from 0.90 to 0.45, for instance. In the case of a smaller a_∗,the upper limit of the gap flaring duty cycle will be less constrained. On the other hand, if a_∗>0.95 is observationally confirmed, it is no longer valid to assumea constant radial magnetic field, B^r ∝ F_θφ/√(-g), on the horizon, where F_θφ denotes meridional derivative ofthe magnetic flux function, and √(-g) the volume element <cit.>. Such an extremely rotating case will be investigated in a separate paper. § DISCUSSIONThe non-detection of the HE fluxes from any of the three sources, may indicate either that their long-term accretion rates are above condition (3) (i.e. ṁ > 10^-4, see <ref>) so the duty cycle of the gap activity is weak, or that our line of sight misses their γ-ray flares. It is also possible that the BH spin is actually less than what we assumed. For example, even for 1A 0620-00 or XTE J1118+480, it is very unlikely to detect their BH-gap emission with LAT even if its flaring duty cycle is nearly 100 %, if a_∗ < 0.25. Nevertheless, in general, there might be a higher possibility for a BH binary to show detectable gap emission if its time-averaged accretion rate is moderately small, and if we view the binary almost face-on. We can expect a small accretion rate for the sources that stay in long quiescence and their recurrence time is long. For instance, if we simply assume a BH mass of 10 M_⊙, XTE J1818-245 is estimated to have relatively close accretion rate to condition (3) (T16), although it does not have confirmed M, or ζ_ obs yet. Such a source is located within several kpc (e.g., in the near side of our Galaxy), and its BH gap emission may be detectable during its deep quiescence if we luckily view them almost face-on (condition 5). Since the accretion rate is variable,it is desirable to observe the aforementioned BHTs frequently during their quiescence in near-IR and/or VHE,in order not to miss their gap flares. For example, once the near-IR flux decreases enough, we suggest to observe the source with ground-based,Imaging Atmospheric Cherenkov Telescopes (IACTs) to detect their BH-gap emission below a few TeV. In this case, X-ray flux is predicted to be very weak.On the contrary, if we detect a HE flare contemporaneously with an X-ray flare (i.e., during its high accretion rate phase), it suggests that these photons are emitted from the jet, rather than the BH gap. Thus, contemporaneous observations at X-ray and HE ranges will help us discriminate the emission processes in accreting BH systems. One similar example is the investigation of V404 Cyg during its 2015 X-ray outburst in June using the Swift/BAT, INTEGRAL/ISGRI and Fermi/LAT data <cit.>. Even with the consideration of local time bin (i.e., 6-hours), the detection significance yielded from the unbinned likelihood analysisfor the γ-ray data is still less than 4σ. <cit.> considered the AGILE data in the 50–400 MeV energy band and improves the significance of the source detection to ∼ 4.3σ. Inverse Compton scattering of photons applied to explain the transient γ-rays detected for Cyg X-3 <cit.> can provide a similar scenario to support for the simultaneous detection of radio outburst, pair annihilation and HE γ-rays for V404 Cyg.However, the observed strong cut-off in the HE emission of ∼400 MeV may give a constraint to classify such a detection.Whether there exists a similar spectral component for the hard X-ray and γ-ray can also serve as another basis to definitely confirm the emission mechanism to the traditional shock-in-jet approach.Due to the non-detection of the HE emission or the correlation between the accretion rate and the HE flux, we cannot specify whether the shock-in-jet or the BH gap model is responsible for these sources. To gain more insights into the differences between the gap model and the shock-in-jet model, we can compare the spectral features of both models. In the gap of a stellar-mass black hole, essentially all the electrons have the same Lorentz factor at each position, because their motion is saturated by the curvature radiation drags.On the contrary, in the shock-in-jet scenario, the electron is heated at the shock, resulting in a wider energy distribution than those electromagnetically accelerated in the gap. There exista characteristic slope for the shock-in-jet model of the optically thin flux density, ν S_ν∝ν^-s/2+1, during the so-called “Compton stage"<cit.>. During this stage, the energy density of the magnetic field is much larger than the energy density of photons when the shock propagates along the jet.With s≈2.1 <cit.>, a characteristic slope of ν S_ν∝ν^-0.05 is estimated. Therange of the such slope is related to the electron energy distribution, and the normalization is determined by the strength of the shock. It is, therefore, possible to distinguish the responsible emission models according to the high energy spectral profile. Namely, in the gap model, a bumpy spectrum is expected due to the nearly mono-energetic distribution of electrons. On the other hand, in the shock-in-jet model, a smoother, singe power-law is expected due to a much wider, power-law energy distribution of electrons. With current result of non detection, there is no observational preference for either model.Moreover, inclusion of lower energy observations are also useful in discriminating the emission models. It follows from Figs. <ref>–<ref> that the gap HE and VHE fluxes increases with decreasing ṁ. That is, we can predict an anti-correlation between the IR/optical and HE/VHE fluxes (H16). It forms a contrast to the standard shock-in-jet scenario, in which the IR/optical and the HE/VHE fluxes will correlate. If their time-varying multi-wavelength spectra show anti-correlation, it strongly suggests that the photons are emitted from the BH gap. With the observational upper limits of the individual sources and the upper limits predicted by the BH-gap model, we can constrain their mass accretion rate, Ṁ, which are potentially important to discuss their binary evolution through the long-term accretion rate. Therefore, we propose to simultaneously observe nearby low-mass X-ray binaries that have more massive BHs during deep quiescence in near-IR/optical, X-ray, HE, and VHE (with CTA) in the future.One of the authors (K. H.) is indebted to Drs. A. Okumura and T. Y. Saito for a valuable discussion on the CTA sensitivity. This work made use of data supplied by the LAT data server of Fermi Science Support Center (FSSC), and of the computational facilities in Theoretical Institute for Advanced Research in Astrophysics (TIARA) in ASIAA. This work is supported by the Ministry of Science and Technology of the Republic of China (Taiwan) through grants 103-2628-M-007-003-MY3 and 105-2112-M-007-033-MY2 for A. K. H. K., through grant 105-2112-M-007-002 - for H. K. C., through grant 103-2112-M-001-032-MY3 for S. M., and P. H. T. T. is supported by National Science Foundation of China (NSFC) through grants 11633007 and 11661161010.Facilities: Fermi (LAT). 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"authors": [
"Lupin Chun-Che Lin",
"Hung-Yi Pu",
"Kouichi Hirotani",
"Albert K. H Kong",
"Satoki Matsushita",
"Hsiang-Kuang Chang",
"Makoto Inoue",
"Pak-Hin T. Tam"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170727125145",
"title": "Searching for High Energy, Horizon-scale Emissions from Galactic Black Hole Transients during Quiescence"
} |
Irstea - LISC, 9 avenue Blaise Pascal 63178 Aubière, FranceThis paper proposes a new method which builds a simplex based approximation of a d-1-dimensional manifold M separating a d-dimensional compact set into two parts, and an efficient algorithm classifying points according to this approximation. In a first variant, the approximation is made of simplices that are defined in the cubes of a regular grid covering the compact set, from boundary points that approximate the intersection between M and the edges of the cubes. All the simplices defined in a cube share the barycentre of the boundary points located in the cube and include simplices similarly defined in cube facets, and so on recursively. In a second variant, the Kuhn triangulation is used to break the cubes into simplices and the approximation is defined in these simplices from the boundary points computed on their edges, with the same principle. Both the approximation in cubes and in simplices define a separating surface on the whole grid and classifying a point on one side or the other of this surface requires only a small number (at most d) of simple tests. Under some conditions on the definition of the boundary points and on the reach of the surface to approximate, for both variants the Hausdorff distance between M and its approximation decreases like 𝒪(d n_G^-2), where n_G is the number of points on each axis of the grid. The approximation in cubes requires computing less boundary points than the approximation in simplices but the latter is always a manifold and is more accurate for a given value of n_G. The paper reports tests of the method when varying n_G and the dimensionality of the space (up to 9). classification marching cubes simplex star Recursive simplex stars Guillaume Deffuant January 15, 2018 =======================bigskip§ INTRODUCTION. This paper addresses the following problems: * How to approximate a d-1-dimensional manifold M separating a d-dimensional compact set X into two parts (one labelled -1, the other +1), using as efficiently as possible an oracle which, to any point of X, provides the value stating on which part of X the point is located ? * How to define an efficient algorithm computing the classification of a point by the approximate separation ? The main motivation is to improve algorithms derived from Viability Theory <cit.>. This theory, which provides methods and tools for maintaining a dynamical system within a constraint set, is used in many fields such as sustainability management <cit.>, economics <cit.> or food processing <cit.>. The main algorithms derived from Viability Theory <cit.> iterate the computation of approximate classification functions using the vertices of a grid labelled into two classes. The approximate classification methods currently used are the nearest vertex of the grid <cit.> or machine learning techniques such as support vector machines <cit.> or k-d trees <cit.>. Our main purpose is to develop a more efficient method. Nevertheless, deriving efficient approximate classification can be useful in other contexts. For instance, when a classification requires a heavy or difficult process, it is often interesting to compute an approximate but lighter classification function, based on a limited set of well chosen classified points. This problem is a particular case of meta (or surrogate) modelling. The field of reliability in material sciences for instance develops specific techniques to build such meta-models <cit.>. A problem closely related to the approximation of a classification boundary is the approximation of an isosurface, defined as the set of points such that f(x)=0, f being a continuous function from the considered space into ℝ. In this problem, it is possible to define local linear approximations of f around the values f(v) of the vertices v of the grid, which is not possible in the approximation of a classification boundary because the values at the vertices are either -1 or +1.In 3 dimensions, the problem of approximating an isosurface is very common, for instance to visualise surfaces from scanners or magnetic resonance imaging measurements, and several techniques are available. In particular, the algorithms deriving from the marching cubes <cit.> (see <cit.> for a review) build simplex-based surfaces. They firstly compute the boundary points approximating the intersections between the isosurface and the edges of a regular grid, generally using a linear interpolation. Then the marching cubes generally use a table of rules specifying the connections between boundary points to define a simplex based separating surface in each cube configuration. Once solved the problems of consistency between cubes <cit.>, these techniques represent efficiently the surface of 3D objects. Some variants include an adaptive refinement of the grid in order to guarantee that the approximation is isotopic with the surface to approximate <cit.>. However, extending these methods to spaces of more than 3 dimensions faces serious difficulties as the number of cube configurations is in 2^2^d, which leads to a very high number of rules specifying the simplices by cube configuration. For instance in 6 dimensions, there are 2^64 cube configurations which is beyond any current computer storage capacities. Moreover, the number of simplices grows exponentially with the dimensionality and so does the necessary memory space to store them. Currently, as far as we know, the available methods of marching cubes in arbitrary dimensionality are: * Breaking cubes into simplices <cit.>. This addresses the problem of the fast growth of the table of rules mentioned earlier, because the number of configurations in a simplex is much lower (it varies as d+1/2) than in a cube. However, a d-dimensional cube breaks into between d! or 2^d-1d! simplices, depending on the decomposition used <cit.>. The cited papers show examples in at most 4 dimensions. * Defining the simplices in a cube from the convex hull of a set of points including the boundary points and some cube vertices <cit.>. The simplices are defined with a single rule but computing the convex hull and storing the corresponding simplices is computationally demanding when d increases. Again, <cit.> shows only examples up to 4 dimensions. A different approach builds on the principles of Delaunay triangulation and defines simplex based surfaces approximating a manifold without using a grid, from a sampling of points on this manifold. In addition to practical algorithms, the researchers studied the topological and geometric closeness between the approximation and the manifold <cit.>. Moreover, some variants are based on iterative sampling <cit.> adapting the density of the sampling to the local complexity of the shape. Recent variants of the approach <cit.> approximate smooth manifolds of any dimensionality. However, the time complexity of the algorithm is exponential in d'^2 where d' is the dimensionality of the manifold to approximate, which makes it difficult to apply practically even for moderate values of d' (say d' > 5). The memory size needed to store the set of simplices also grows significantly with the dimensionality.Moreover, none of these approaches considers the problem of using these approximations for a classification purpose. Indeed, when the number of simplices is very large, as expected when the dimensionality increases, computing the classification is also expected to become very demanding. Solving viability problems requires classifying large numbers of points, hence the efficiency of this procedure is crucial in this context. A specificity of this paper is precisely to propose an efficient classification algorithm adapted to specific structures of simplices.The method proposed in this paper uses a regular grid like the marching cubes and generalises the method of centroids <cit.> to an arbitrary dimensionality. Like the dual marching cubes <cit.> it adds new points in cubes and faces. A noticeable difference with the standard marching cubes is that the boundary points cannot be approximated linearly and successive dichotomies are used instead. In a first variant of the proposed method, the simplices are defined in cubes of the grid, from the boundary points located on their edges. All simplices share the barycentre of the cube boundary points as a common vertex (thus shaping a "simplex star") and include simplices of lower dimensionality similarly defined in cube facets. The recursion ends when the considered face is an edge of the cube containing a boundary point. This method can be related to the barycentric subdivision which divides an arbitrary convex polytope into simplices sharing the barycentre of the polytope's vertices and this operation can be recursively applied to the faces of the polytope. The main difference is that the simplices of the proposed method are defined in the faces of the cube in which the boundary points are located, not in the faces of the polytope that they define.In the proposed approach, there is only one rule deriving the simplices in a cube (or a face) whatever the dimensionality. The simplices can easily be enumerated, going through all the faces of a cube. However, when there are 2D faces including 4 boundary points, this method "glues" together surfaces that should remain separated. This creates a non-manifold approximation which should be avoided in many applications. The second variant of the method addresses this problem. It uses the Kuhn triangulation to break the cubes into simplices like in <cit.>. The boundary points are computed on the edges of these simplices and the approximation is defined as previously from these boundary points, using the faces of a simplex instead of the faces of a cube. This variant always defines a manifold.The paper underlines the following properties of both variants: * It is possible to compute the classification of a point with at most d relatively simple operations;* Under some conditions on the computation of the boundary points and on the smoothness of M, the Hausdorff distance between M and its approximation decreases like 𝒪(dn_G^-2), n_G being the number of points on each axis of the grid.The remaining of the paper is organised as follows: Section 2 presents the variant of the approximation defined in cubes of the grid and its classification algorithm, section 3 does the same for the variant defined in the Kuhn simplices of the grid, section 4 establishes the theorem about the approximation accuracy, section 5 reports the results of a series of tests of the method when varying the space dimensionality and the size of the grid and finally section 6 discusses these results and potential extensions. § MANIFOLD APPROXIMATION WITH RESISTARS IN CUBES.Let M be a d-1-dimensional manifold separating the compact set X = [0, 1]^d into two parts (one labelled -1 the other +1) and let ℳ: X →{-1, 0, +1}, the function which outputs 0 if x belongs to M and otherwise the label of the part of X in which x is located.We consider a regular grid G of n_G^d points covering X and its boundary. The distance between two adjacent points of the grid is ϵ = 1/n_G-1. The cubes of the grid are d-dimensional cubes of edge size ϵ whose vertices are points of the grid. The faces of the grid are the faces of these cubes (cubic polytopes of dimensionality lower than d, of edge size ϵ and whose vertices are points of the grid). It is supposed that none of the grid points belongs to M. The function ℳ is slightly modified if necessary in order to ensure this, as it is done in the marching cube approach.The following notations are frequently used: * The set of the i-dimensional faces of a grid cube or a face C is denoted ℱ_i(C);* For any polytope P, 𝒱(P) denotes the vertices of P; * For a set S of points of X, S̅ denotes the barycentre of S, [S] the convex hull of S and ∂ S the boundary of S; * For two sets A and B such that B ⊂ A, A - B is the complementary of B in A. The next subsection focuses on the approximation in a single cube of the grid and the following subsection is devoted to the approximation on the whole grid.§.§ c-resistar in a single cube.§.§.§ Boundary points.The method requires first to define the boundary points in the cube. Let C be a d-dimensional cube of the grid and n_e ≥ 1 be an integer. A boundary point b_M([v_-,v_+]) is defined on an edge [v_-,v_+] of C such that ℳ(v_-) = -1 and ℳ(v_+)=+1, as follows:b_M([v_-,v_+]) = v_- + i+0.5/n_e(v_+ - v_-),Where i ∈{0,..,n_e-1} is such that:ℳ(v_- + i/n_e(v_+ - v_-)) = -1, ℳ(v_- + i+1/n_e(v_+ - v_-)) = +1.In practice, the boundary points are determined by q successive dichotomies with algorithm <ref>, and n_e = 2^q. Because ℳ(v_-) = -1 and ℳ(v_+)=+1, M cuts the edge [v_-,v_+] at least once, therefore b_M([v_-,v_+]) exists and there exists b ∈ M ∩ [v_-,v_+] such that:b - b_M([v_-,v_+])≤v_+ - v_-/2 n_e = 2^-q -1ϵ. If the manifold M cuts the edge an odd number of times, this algorithm returns a boundary point which is close to one of the intersection points. Of course, if there is an even number of intersections, no boundary point is computed because the classification of the vertices by ℳ is the same. As shown in section 4, under some conditions, it is possible to guarantee the accuracy of the approximation, despite the possibility of these situations. For any face or cube F, we denote B_M(F) the set of boundary points defined on the edges of F.For a boundary point b ∈ [v_-, v_+], such that ℳ(v_+) = +1 and ℳ(v_-) = -1, v_+ is denoted v_+(b) (resp. v_- is denoted v_-(b)) and called the positive (resp. negative) vertex of b. §.§.§ Definition and main properties of c-resistars.Let C be a d-dimensional cube of the grid such that B = B_M(C) ≠∅. The c-resistar approximation of M in C, denoted [B]^⋆, is the following set of simplices: [B]^⋆ =⋃_{F_1,...,F_d-1}∈ℱ^⋆(B,C)[{B̅_M(F_1),...,B̅_M(F_d-1}∪{B̅} ],with:{F_1,...,F_d-1}∈ℱ^⋆(B,C) ⇔ F_1 ∈ℱ_1(C), B_M(F_1) ≠∅, F_i ∈ℱ_i(C), F_i-1⊂ F_i, 2 ≤ i ≤ d-1. ,where i is an integer.The word resistar stands for "recursive simplex star". Indeed, the barycentre B̅ of the boundary points is a vertex common to all the simplices which organises them as a star. This star is recursive because the common vertex B̅ is connected to simplices of lower dimensionality in the facets (faces of dimensionality d-1) of C that share the barycentre B̅_M(F_d-1) of the boundary points located in this facet, and so on recursively until reaching an edge F_1 of C which include a boundary point (see examples on Figures <ref> and <ref>). We use the denomination c-resistar, with the prefix "c" standing for cube, in order to distinguish these resistars from the ones which are defined in Kuhn simplices, presented in section 3. Propositions <ref> and <ref> establish that the c-resistar is a d-1-dimensional surface without boundary inside the cube C. For all { F_1, ..., F_d-1}∈ℱ^⋆(B,C), the set [{B̅_M(F_1),..., B̅_M(F_d-1) }∪{B̅}] is a d-1-dimensional simplex.Proof. Consider{ F_1, ..., F_d-1}∈ℱ^⋆(B,C). Setting F_d = C, we will show that, for 1 ≤ i ≤ d-1:∃ b_i+1∈ B_M(F_i+1), b_i+1∉ B_M(F_i). Let F'_i ∈ℱ_i(C) be the face opposite to F_i in F_i+1. F'_i is such that 𝒱(F_i) ∪𝒱(F'_i) = 𝒱(F_i+1) (𝒱(F) being the set of vertices of face F). Two cases occur: * There are edges [v,v'] of F'_i such that ℳ(v).ℳ(v') = -1, then, for each of these edges, there is a boundary point of F_i+1 which is not in F_i; * All the vertices of F'_i have the same classification by ℳ. By hypothesis, B_M(F_i) ≠∅, thus there are vertices classified differently by ℳ in F_i, hence there are vertices v such that ℳ(v) ≠ℳ(v'), forv' vertex of F'_i. There are such couples (v, v'), for which [v,v'] is an edge of F_i+1, thus there is a boundary point b_i+1∈ [v,v'] such that b_i+1∈ B_M(F_i+1) and b_i+1∉ F_i.Therefore, for 1 ≤ i ≤ d-1, B̅_M(F_i+1) ∉ F_i. Let S = {B̅_M(F_1),..., B̅_M(F_d-1) }∪{B̅}. The set S includes d affinely independent points, therefore [S] is a d-1-dimensional simplex. □The boundary of the c-resistar [B]^⋆ is included in the boundary of C.Proof. Let S = {B̅_M(F_1),..., B̅_M(F_d-1) }∪{B̅}, with { F_1, ..., F_d-1}∈ℱ^⋆(B,C). [S] is a simplex of [B]^⋆. Consider a facet [S_F] of this simplex, with S_F = S - { p }, p ∈ S. The following cases arise: * p = B̅. By definition of [B]^⋆, S_F is included in the facet F_d-1 of C, thus [S_F] is included in the boundary of C; * p = B̅_M(F_i), 2 ≤ i ≤ d-1. Assume that vectors {u_1,...,u_i-1} are a basis F_i-1, vectors {u_1,...,u_i-1, u_i} a basis of F_i and vectors {u_1,...,u_i-1, u_i, u_i+1} a basis of F_i+1, then let F'_i ∈ℱ(C) be such that vectors {u_1,...,u_i-1, u_i+1} are a basis of F'_i and F_i-1⊂ F'_i. We have: F'_i ≠ F_i and F'_i ⊂ F_i+1, thus the set S' = S_F ∪{B̅_M(F'_i) } is such that [S'] is a d-1-dimensional simplex of [B]^⋆ and [S_F] = [S] ∩ [S']; * p = B̅_M(F_1) = b ∈ B. There exists b' ∈ B such that b' ∈ F_2 and b' ≠ b (using the proof of proposition <ref>). Let F'_1 be the edge of C such that b' ∈ F'_1. We have {F'_1, F_2, ..., F_d-1}∈ℱ^⋆(B,C). Hence S' = {B̅_M(F'_1) }∪ S_F is such that [S'] is a d-1-dimensional simplex of [B]^⋆ and [S_F] = [S] ∩ [S'].Finally, each simplex of [B]^⋆ has one of its facets which is included in ∂ C the boundary of C and shares all its other facets with other simplices of [B]^⋆. Therefore the boundary of [B]^⋆ is included in ∂ C. □ In 2 dimensions (see examples on Figure <ref>), the simplices of the c-resistar are segments [b, B̅], therefore, the number of simplices equals the number of boundary points. In 3 dimensions (see an example on Figure <ref>), the simplices are triangles [b, B̅_M(F_2), B̅], where F_2 is a 2D face of the cube including boundary point b. For each boundary point, there are two such faces, therefore, the number of simplices is twice the number of boundary points.More generally, for 2 ≤ i ≤ d-1, there are d - i +1 faces of dimensionality i including a face of dimensionality i-1, therefore, the number of simplices in a d-dimensional cube is (d-1)! times its number of boundary points. The number of simplices increases thus very rapidly with the dimensionality. However, this is not a problem in our perspective because, as shown further, it is possible to compute efficiently the classification of a point by a c-resistar without considering the simplices explicitly.§.§.§ Classification function.Propositions <ref> and <ref> imply that the c-resistar is a d-1-dimensional set dividing the cube C into several connected sets that we call classification sets. Definition <ref> expresses these sets as the union of polytopes sharing vertices with faces of C and with the simplices of [B]^⋆. The following propositions establish their properties. Finally these sets are used to define the resistar classification function. Let C be a d-dimensional cube of the grid, such that B = B_M(C) ≠∅. Let ℱ(C) be the set of all the faces of cube C (including C itself) and ℋ(B, C) be the set of the faces of C without boundary points:ℋ(B,C) = { F ∈ℱ(C), B_M(F) = ∅}. Let 𝒟(B,C) be the set of connected components of ℋ(B,C).For all D ∈𝒟(B,C), the classification set [B]^𝒟⋆(D) associated to D by resistar [B]^⋆ in C, is defined as follows: [B]^𝒟⋆(D) =⋃_{ F_h,..,F_d-1}∈ℱ^𝒟⋆(D,C) [𝒱(F_h) ∪{B̅_M(F_h+1),...,B̅_M(F_d-1) }∪{B̅}], with: { F_h,..,F_d-1}∈ℱ^𝒟⋆(D,C) ⇔ F_h ∈ℱ_h(C), F_h ⊂ D, 0 ≤ h ≤ d-1, F_i ∈ℱ_i(C), F_i-1⊂ F_i, h+1 ≤ i ≤ d-1, B_M(F_h+1) ≠∅,where i and h are integers, 𝒱(F_h) is the set of vertices of face F_h.Figure <ref> shows examples of c-resistar classification sets in 2D. For all D ∈𝒟(B,C), D ⊂ [B]^𝒟⋆(D), [B]^𝒟⋆(D) is a d-dimensional connected set and [B]^𝒟⋆(D) ∩ ([B]^⋆ - ∂ [B]^⋆) is the boundary of [B]^𝒟⋆(D) in C - ∂ C. Proof. Obviously, D ⊂ [B]^𝒟⋆(D), by definition. Indeed, for any face F_h ∈ℱ_h(C), such that F_h ⊂ D, we have [𝒱(F_h)] = F_h ⊂ [B]^𝒟⋆(D) directly from the definition of [B]^𝒟⋆(D).Let P = 𝒱(F_h) ∪{B̅_M(F_h+1),...,B̅_M(F_d-1) }∪{B̅}, with { F_h, ...,F_d-1}∈ℱ^𝒟⋆(D, C). [P] is a polytope of dimensionality d because F_h is of dimensionality h and the set P - 𝒱(F_h) includes d-h affinely independent points (see proposition <ref>), which are not located in F_h. Let P_F ⊂ P such that [P_F] is a facet of [P]. There are two possibilities: * P_F = P - { p}, with p ∈{B̅_M(F_h+1),...,B̅_M(F_d-1) }∪{B̅}. The following cases occur: * p = B̅. Then [P_F] is included in F_d-1, therefore [P_F] ⊂∂ C; * p =B̅_M(F_i), with h+1 ≤ i ≤ d-1, then two cases occur again: * h+1 < i. As shown in the proof of proposition <ref>, there exists F'_i ∈ℱ_i(C) such that F'_i ≠ F_i, F_i-1⊂ F'_i, and F'_i ⊂ F_i+1. Moreover, because i > h+1, F_h+1⊂ F_i-1⊂ F'_i. Hence, [P'], with P' = P_F ∪{B̅_M(F'_i) }, is a polytope of [B]^𝒟⋆(D) and [P_F] = [P] ∩ [P']. * i = h+1. Let F'_h+1∈ℱ_h+1(C) be such that F_h ⊂ F'_h+1 and F'_h+1⊂ F_h+2. There are two possibilities: * B_M(F'_h+1) = ∅, then P' = 𝒱(F'_h+1) ∪{B̅_M(F_h+2),...,B̅_M(F_d-1) }∪{B̅} is such that [P'] is a polytope of [B]^𝒟⋆(D) because F'_h+1 is connected to F_h and [P_F] = [P - B̅_M(F_h+1)] = [P] ∩ [P']. * B_M(F'_h+1) ≠∅, then P' = P_F∪{B̅_M(F'_h+1) } is such that [P'] is a polytope of [B]^𝒟⋆(D) and [P_F] = [P] ∩ [P']. * The d-1-dimensional face is obtained by removing vertices in 𝒱(F_h). Thus: * If h = 0, 𝒱(F_h) is a single vertex v∈ D, and the edge F_1 = [v, v'] is such that B_M(F_1) = { b } . Then P_F = P - {v} = {B̅_M(F_1), ..., B̅_M(F_d-1) }∪{B̅} and [P_F] ⊂ [B]^⋆. Moreover,P' = { v'}∪ P_F is such that [P_F] = [P] ∩ [P'] and there exists D' ∈𝒟(B,C), D' ≠ D such that v' ∈ D' and [P'] ⊂ [B]^𝒟⋆(D'). * If h > 0, P_F = 𝒱(F_h-1) ∪{B̅_M(F_h+1),..., B̅_M(F_d-1) }∪{B̅}, with F_h-1∈ℱ_h-1(F_h). Let F'_h∈ℱ_h(C) be such that F_h-1⊂ F'_h, F'_h ⊂ F_h+1 and F'_h ≠ F_h. There are two possibilities: * B_M(F'_h) ≠∅, then P' = P_F ∪{B̅_M(F'_h) } is such that [P'] is a polytope of [B]^𝒟⋆(D) and[P_F] = [P] ∩ [P']; * B_M(F'_h) = ∅, then F'_h ⊂ D, because F_h and F'_h are connected by sharing face F_h-1 and P' = 𝒱(F'_h) ∪{B̅_M(F_h+1), ..., B̅_M(F_d-1) }∪{B̅} is such that [P'] is a polytope of [B]^𝒟⋆(D) and[P_F] = [P] ∩ [P'] (because F_h-1⊂ F'_h). Finally, all the polytopes [𝒱(F_h) ∪{B̅_M(F_h+1),...,B̅_M(F_d-1) }∪{B̅}] of [B]^𝒟⋆(D) such that h > 0 have all their facets shared with another polytope of [B]^𝒟⋆(D) except one facet which is included in ∂ C; the polytopes such that h = 0 have all their facets shared with another polytope of [B]^𝒟⋆(D) except one which is included in ∂ C and one which is included in [B]^⋆. Therefore, [B]^𝒟⋆(D) is a connected set and its boundary in C - ∂ C is included in [B]^⋆.Moreover, the set of facets included in [B]^⋆ from polytopes such that h = 0 is:E =⋃_{F_1,...,F_d-1}∈ℱ^⋆(B,C), F_1 ∩ D ≠∅[{B̅_M(F_1),...,B̅_M(F_d-1}∪{B̅} ].We have [B]^⋆∩ [B]^𝒟⋆(D) ⊂ E. Indeed, for any set {F_h,...,F_d-1}∈ℱ^𝒟⋆(D,C), such that h > 0, there exists {F_1,...,F_d-1}∈ℱ^⋆(B,C), such that F_1 ⊂ F_h+1, F_1 ∩ D ≠∅ and {F_h+1,...,F_d-1}⊂{F_1,...,F_d-1}. Therefore, for all polytopes [P] such that h > 0 the part of the boundary of [P] which is included in [B]^⋆ is also included in E. Moreover, E ⊂ [B]^⋆∩ [B]^𝒟⋆(D) by definition. Therefore, E = [B]^⋆∩ [B]^𝒟⋆(D).Moreover, E - (E ∩∂ C) is the boundary of[B]^𝒟⋆(D) in C - ∂ C, hence the boundary of [B]^𝒟⋆(D) in C - ∂ C is [B]^𝒟⋆(D) ∩ ([B]^⋆ - ∂[B]^⋆). □ The union of the classification sets defined by the c-resistar [B]^⋆ in cube C is cube C itself: C = ⋃_D ∈𝒟(B,C) [B]^𝒟⋆(D). Proof. The proof of proposition <ref> shows that for all polytopes [P] ∈ [B]^𝒟⋆(D), a facet of [P] which is not in ∂ C is either shared with another polytope of [B]^𝒟⋆(D) or with another polytope of [B]^𝒟⋆(D'), with D' ≠ D. Therefore, because the sets [B]^𝒟⋆(D) are of dimensionality d, this union is C itself. □For (D,D') ∈𝒟(B,C)^2, D' ≠ D, ([B]^𝒟⋆(D) ∩ [B]^𝒟⋆(D')) ⊂ [B]^⋆.Proof Consider (D,D') ∈𝒟(B,C)^2, D ≠ D'. Consider polytope [P] ∈ [B]^𝒟⋆(D) with P = 𝒱(F_h) ∪{B̅_M(F_h+1),..., B̅_M(F_d-1) }∪{B̅} and polytope [P'] ∈ [B]^𝒟⋆(D') with P' = 𝒱(F'_h) ∪{B̅_M(F'_h+1),..., B̅_M(F'_d-1) }∪{B̅}. We have: [P] ∩ [P'] = [P ∩ P'] and F_h ∩ F'_h' = ∅. Therefore the intersection between [P] and [P'] is a simplex of dimensionality at most d-1, of vertices the points B̅_M(F_i) such that there exists (b, b') ∈ B_M(F_i)^2 with (v_+(b) ∈ D or v_-(b) ∈ D) and (v_+(b') ∈ D' or v_-(b') ∈ D'). Therefore ([B]^𝒟⋆(D) ∩ [B]^𝒟⋆(D')) ⊂ [B]^⋆.□These properties of the classification sets [B]^𝒟⋆(D) guarantee that for any point x ∈ C - [B]^⋆, there exists a unique set D such that x ∈ [B]^𝒟⋆(D). This leads to the definition of the resistar classification.Let C be a cube of the grid such that B = B_M(C), The resistar classification function [B]^⋆(.), from C to {-1, 0, +1}, is defined for x ∈ C as follows: * If B = ∅, then [B]^⋆(x) = ℳ(v), v ∈𝒱(C); * Otherwise, let 𝒟(B,C) be the set of connected faces of C without boundary points and, for D ∈𝒟(B,C), let [B]^𝒟⋆(D) be the classification associated to D by the c-resistar [B]^⋆. * If x ∈ [B]^⋆ then [B]^⋆(x) = 0, * Otherwise there exists a single set D ∈𝒟(B,C) such that x ∈ [B]^𝒟⋆(D) and [B]^⋆(x) = ℳ(v), v ∈𝒱(D). This classification function is consistent with the classification of the vertices of C by ℳ because, for any vertex v ∈𝒱(D), D ∈𝒟(B,C), [B]^⋆(v) = ℳ(v), and any point x ∈ [B]^𝒟⋆(D) is on the same side of [B]^⋆ as v, since [B]^⋆∩ [B]^𝒟⋆(D) includes the boundary of [B]^𝒟⋆(D) in C - ∂ C.§.§.§ Classification algorithm.Algorithm <ref> takes as input a point x of cube C and, if B = B_M(C) is empty, it returns the classification of a vertex of C by ℳ. Otherwise, if x is not equal to B̅, it computes x_d-1 = r⃗(B̅,x) ∩∂ C, where r⃗(B̅,x) is the ray from B̅ in the direction of x and ∂ C is the boundary of C. x_d-1 is located in a facet F_d-1 of C and algorithm <ref> repeats the same operations for x_d-1∈ F_d-1. When it reaches a face F_i without boundary points, the algorithm returns the classification of a vertex of F_i, or whenx_i = B̅_M(F_i) it returns 0 (see examples on Figure <ref>). Algorithm <ref> always terminates, because the dimensionality of face F_i decreases of 1 at each step, and in the worst case the algorithm reaches a face of dimensionality 0, which, by hypothesis, cannot include any boundary point.Algorithm <ref> applied to x ∈ C returns [B]^⋆(x), as specified in definition <ref>.Proof. Consider x ∈ C. * If x ∈ [B]^⋆, let S = {B̅_M(F_1),...,B̅_M(F_d-1) }∪{B̅}, with {F_1,...,F_d-1}∈ℱ^⋆(B,C), such that x ∈ [S]. At each step of algorithm <ref>, either x_i = B̅_M(F_i) and then the algorithm returns 0, or x_i-1∈ [B̅_M(F_1),...,B̅_M(F_i-1) ]. Therefore, in the worst case, the algorithm reaches x_1 ∈ [B̅_M(F_1)] = b, b ∈ B, in which case the only possibility is x_1 = b, and the algorithm returns 0. Therefore if x ∈ [B]^⋆, algorithm <ref> returns 0. * If x ∈ C - [B]^⋆, let {x_d = x, .., x_h } be the values of x_i at the successive steps of algorithm <ref>, hbeing the dimensionality of the face F_h such that F_h ∩ B = ∅ at the last step of the algorithm. There exists a unique set of connected faces without boundary points D such that F_h ⊂ D thus x_h ∈ D and x_h ∈ [B]^𝒟⋆(D). For h+1 ≤ i ≤ d the rays r⃗(B̅_M(F_i), x_i)do not cross any simplex of [B]^⋆, thus the segments [x_i, x_i-1] for h+1 ≤ i ≤ d do not cross [B]^⋆ either. Therefore, because the boundary of [B]^𝒟⋆(D) in C- ∂ C is included in [B]^⋆, x = x_d ∈ [B]^𝒟⋆(D). Therefore, algorithm <ref> returns [B]^⋆(x). □ Actually, it can easily be shown that P = 𝒱(F_h) ∪{B̅_M(F_h+1),..., B̅_M(F_d-1)}∪{B̅}, with the faces F_i defined by algorithm <ref> is such that [P] a polytope of [B]^𝒟⋆(D) and x ∈ [P].It appears finally that, even if the set [B]^⋆ includes a large number of simplices, the classification algorithm requires at most d relatively light computations (selecting boundary points in a face, computing their barycentre, projecting a point on the boundary of the face). Section <ref> presents a modification of this algorithm with a better management of the memory space.§.§ c-resistar approximation on the grid.The regular grid G comprises n_G^d points covering X and its facets. The values of the coordinates of the grid points are taken in { 0, 1/n_G-1, 2/n_G-1..., 1}. ℱ_i(G) denotes all the i-dimensional cubes or cube faces of G and ℱ(G) denotes the set of all cubes or cube faces of G.§.§.§ Definition.The definition of the c-resistar approximation of M on grid G comes directly from the definition of the c-resistars in the cubes.The c-resistar approximation of M on grid G, denoted [B_M(G)]^⋆, B_M(G) being the set the boundary points from of all the edges of the cubes of the grid, is the union of the c-resistars approximating M in the cubes of G: [B_M(G)]^⋆ = ⋃_C ∈ℱ_d(G), B_M(C) ≠∅ [B_M(C)]^⋆. The c-resistar approximation of M on grid G is a set of d-1-dimensional simplices and its boundary is included in the boundary of X.Proof. [B_M(G)]^⋆ is a set of d-1-dimensional simplices as the union of sets of d-1-dimensional simplices (see proposition <ref>). Let C be a cube of G, with B = B_M(C) ≠∅, let S = {B̅_M(F_1),...,B̅_M(F_d-1) }∪{B̅}, such that [S] is a d-1-dimensional simplex in [B]^⋆. The facet [S_F] = [S - {B̅_M(C) }] of S is located in facet F_d-1 of C. Two cases occur: * There exists a cube C', with B_M(C') = B', sharing facet F_d-1 with C. Then the simplex [S'] = [S_F ∪{B̅'̅}] is such that [S'] ∈ [B_M(C')]^⋆ and [S_F] = [S] ∩ [S']; * Otherwise, [S_F] is included in the boundary of X.Therefore,taking proposition <ref> into account, all the simplices of [B_M(G)]^⋆ share all their facets with other simplices of [B_M(G)]^⋆, except the facets which are in the boundary of X. □ §.§.§ Classification by the c-resistar approximation on the grid. The classification sets defined by the c-resistar approximation on the grid are derived from the classification sets in the grid cubes. Let B_G = B_M(G), ℋ(B_G,G) = { F ∈ℱ(G), B_M(F) = ∅} and 𝒟(B_G,G) be the connected components of ℋ(B_G,G). For D ∈𝒟(B_G,G) the classification set [B_G]^𝒟⋆(D) defined by [B_G]^⋆, the c-resistar approximation of M on grid G, is the set:[B_G]^𝒟⋆(D) = ⋃_C ∈ℱ_d(G) [B_M(C)]^𝒟⋆(D ∩ C),where:[B_M(C)]^𝒟⋆(D ∩ C) = ∅,D ∩ C = ∅, C,D ∩ C = C,and follows definition <ref> otherwise.For all D ∈𝒟(B_G,G), D ⊂ [B_G]^𝒟⋆(D), [B_G]^𝒟⋆(D) is connected and its boundary in X - ∂ X is [B_G]^𝒟⋆(D) ∩ ([B_G]^⋆ - ∂ [B_G]^⋆).Proof. * D ⊂ [B_G]^𝒟⋆(D) by definition.* All the polytopes of [B_G]^𝒟⋆(D) defined from face F_h ∈ D are connected to each other because they share face F_h. Because D is a connected set by definition, the polytopes defined from all faces F_h ∈ D are connected to each other through the connections between faces F_h. Therefore [B_G]^𝒟⋆(D) is a connected set.* All the polytopes of [B_G]^𝒟⋆(D) are of dimensionality d, because they are either cubes without boundary points, or polytopes of a classification set in a cube, which are all of dimensionality d. * Consider D ∈𝒟(B_G,G) and a cube C such that D ∩ C ≠∅.* If C ⊂ D, [B_M(C)]^𝒟⋆(C ∩ D) = C and for any d-1-dimensional face F_d-1 of C which is shared with another cube C' ∈ℱ_d(G), there exists a polytope [P'] ⊂ [B_M(C')]^𝒟⋆(D ∩ C') such that F_d-1⊂ [P']. Indeed, if C' ⊂ D, then [B_M(C')]^𝒟⋆(D ∩ C') = C' and F_d-1⊂ C ∩ C' by hypothesis. Otherwise, we have { F_d-1}∈ℱ^𝒟⋆(D ∩ C', C') because B_M(F_d-1) = ∅;* If D ∩ C ≠ C, let P = 𝒱(F_h) ∪{B̅_M(F_h+1),..., B̅_M(F_d-1)}∪{B̅_M(C) },with { F_h,..., F_d-1}∈ℱ^𝒟⋆(D ∩ C, C). [P] is a polytope included in [B_G]^𝒟⋆(D). Let P_F = P - {B̅_M(C) }. [P_F] is a face of [P] which is included in facet F_d-1 of C. If there exists a cube C' ∈ℱ_d(G), C' ≠ C, such that F_d-1⊂ C', then: * if C' ⊂ D, [P_F] = F_d-1 and F_d-1 is shared by [P] and C';* if C' ∩ D ≠ C', then B_M(C') ≠∅. The set P' = P_F ∪{B̅_M(C') } is such that [P'] ⊂ [B_M(C')]^𝒟⋆(D ∩ C') and [P_F] = [P] ∩ [P']. As shown in the proof of proposition <ref>, the other facets [P_F] of [P] which are on the boundary of [B_M(C)]^𝒟⋆(C ∩ D) are such that [P_F] ⊂ [B_M(C)]^⋆⊂ [B_G]^⋆.Therefore, the boundary of [B_G]^𝒟⋆(D) is either in faces of cubes which are in ∂ X or included in [B_G]^⋆. Moreover, in each cube C such that D ∩ C ≠∅ and B_M(C) ≠∅, the boundary of [B_M(C)]^𝒟⋆(D) in C - ∂ C is [B_M(C)]^𝒟⋆(D) ∩ ([B_M(C)]^⋆ - ∂ [B_M(C)]^⋆) (proposition <ref>). Taking the union of these sets for all cubes of G, it can easily be seen that the boundary of [B_G]^𝒟⋆(D) in X - ∂ X is [B_G]^𝒟⋆(D) ∩ ([B_G]^⋆ - ∂ [B_G]^⋆). □ For all points x ∈ X - [B_G]^⋆, there exists a unique set D ∈𝒟(B_G, G) such that x ∈ [B_G]^𝒟⋆(D).Proof. Let C ∈ℱ_d(G) be such that x ∈ C. If B_M(C) = ∅, then there exist a unique set D ∈𝒟(B_G, G) such that C ⊂ D.Otherwise, there exists a unique set D_C ∈𝒟(B_M(C), C) such that x ∈ [B_M(C)]^𝒟⋆(D_C) (because of propositions <ref> and <ref>) and there exists a unique set D ∈𝒟(B_G, G) such that D_C ⊂ D. □The classification function by the resistar approximation is defined directly from proposition <ref>. The classification [B_G]^⋆(.) by the resistar approximation on the grid is a function from X to {-1,0,+1} defined for x ∈ X as follows: * If x ∈ [B_G]^⋆, [B_G]^⋆(x) = 0 * Otherwise, proposition <ref> ensures that there exists a unique classification set D ∈𝒟(B_G, G) such that x ∈[B_G]^𝒟⋆(D), and [B_G]^⋆(x) = ℳ(v), v ∈𝒱(D).This definition is consistent with the classification of the points of the grid by ℳ. Indeed, for all grid points v ∈𝒱(D), D ∈𝒟(B_G, G), [B_G]^⋆(v) = ℳ(v) and any point x ∈ [B_G]^𝒟⋆(D) is on the same side of [B_G]^⋆ as v, because the boundary of [B_G]^𝒟⋆(D) in X - ∂ X is [B_G]^𝒟⋆(D) ∩ ([B_G]^⋆ - ∂ [B_G]^⋆). For all x ∈ X, let C_x be a cube of ℱ_d(G) such that x ∈ C_x. We have: [B_G]^⋆(x) = [B_M(C_x)]^⋆(x). Proof. Consider x ∈ X and C_x ∈ℱ_d(G) such that x ∈ C_x.There exists D ∈𝒟(B_G, G) such that x ∈[B_G]^𝒟⋆(D) (proposition <ref>). By definition, [B_G]^𝒟⋆(D) ∩ C_x = [B_M(C_x)]^𝒟⋆(D ∩ C_x) thus x ∈ [B_M(C_x)]^𝒟⋆(D ∩ C_x). Therefore [B_G]^⋆(x) = [B_M(C_x)]^⋆(x). □Computing [B_M(C)]^⋆(x) can thus be performed by first computing a cube C_x such that x ∈ C_x, and then applying algorithm <ref> to x in C_x. However, this approach would require to store the classification of all the vertices of the grid. The next subsection proposes a method requiring less memory space.§.§ Algorithm of classification avoiding to store the classification of all grid vertices This subsection describes the algorithm of classification of c-resistar approximation when storing the boundary points and only the classification by ℳ of the vertices of the edge on which the boundary point is located, instead of the classification by ℳ of all the vertices of the grid. §.§.§ Classification algorithm in the cubeThe modified classification algorithm is based on proposition <ref>. Let C be a cube of G. For all x ∈ C - [B]^⋆ and let F_h be the face of C such that B_M(F_h) = ∅ at the last step of algorithm <ref> applied to x. The face F_h+1 from the previous step is such that B_M(F_h+1) ≠∅ and there exists b ∈ B_M(F_h+1) such that v_+(b) ∈ F_h or v_-(b) ∈ F_h.Proof The proof comes directly from the argument used in the proof of proposition <ref>. □ Algorithm <ref> modifies algorithm <ref> using proposition <ref>: it tests the vertices of the boundary points in B_h+1 and the classification by ℳ of the first of these vertices found in F_h gives the final classification to return. Note that this algorithm supposes that B_M(C) ≠∅ (which is not the case of algorithm <ref>).Moreover, computing the ray r⃗(B̅_M(F_i),x) requires to perform a division by B̅_M(F_i)- x which may lead to very large numbers and strong losses of precision when B̅_M(F_i)- x is very close to 0. Therefore, in practice, the classification returns 0 when B̅_M(F_i)- x is smaller than a given threshold (we take 0.00001). With this modification, some points at small distance of [B]^⋆ are classified 0. Algorithm <ref> also includes this modification. §.§.§ Classification algorithm on the whole grid.Algorithm <ref> performs the classification of a point x ∈ X by the c-resistar approximation on the grid keeping in memory only the set C_G of cubes containing boundary points. It requires defining point m ∈ X which the centre of an arbitrarily chosen cube C_m in C_G. It computes the cube C'_x ∈ C_G which is the closest to x and such that C'_x ∩ [m,x] ≠∅ and the point x' ∈ C'_x ∩ [m,x] which is the nearest to x in cube C'_x. Then, it computes the classification of x' in the identified cube C'_x (see illustration on Figure <ref>).In practice, a hash table stores the set of couples (C, B_M(C)), for the cubes C such that B_M(C) ≠∅. Computing x' and C'_x can be done by computing the intersections of [x, m] with the facets of cubes of the grid. The procedure then requires checking which cubes intersecting [x, m] are in the hash table. The number of cubes crossing [x, m] varies linearly with n_G hence the number of requests to the hash table is also linear with n_G. When given a point x ∈ X as input, algorithm <ref> yields [B_M(G)]^⋆(x) as output.Proof. Let x' = x'(C'_x) as defined in algorithm <ref>. It always exists because m is the centre of a cube C_m containing boundary points, and C'_x = C_m if no other cube containing boundary points cuts the segment [x,m]. By construction, the segment [x, x'] does not cross [B_M(G)]^⋆, therefore [B_M(G)]^⋆(x) = [B_M(G)]^⋆(x') = [B_M(C'_x)]^⋆(x'). □ § MANIFOLD APPROXIMATION WITH RESISTARS IN SIMPLICES FROM KUHN TRIANGULATION.Figure <ref>, panel (b) and Figure <ref>, give examples of c-resistars that are not manifolds. Indeed, in both of these cases, it is impossible to find a continuous and invertible mapping from the neighbourhood of point B̅ in the resistar into a hyperplane. However, for numerous applications, including viability kernel approximation, it is highly recommended to build manifold approximations. In order to address this problem, we now break the cubes into simplices, using the Kuhn triangulation and we define resistars in these simplices.The next subsection focuses on the resistar in a single simplex and the following subsection on the approximation on the whole grid.§.§ K-resistar in a single simplex from Kuhn triangulation. §.§.§ Definition and main properties. The Kuhn triangulation of a cube C is defined by the set of 𝒫(d) permutations of set {1, 2, .., d}. Let P be one of such permutations, the corresponding simplex S_P is defined in cube C as:S_P = { x ∈ C, min_P(1)≤ x_P(1)≤ x_P(2)≤ .. ≤ x_(P(d)≤max_P(d)}. Where x_P(i) is the P(i)^th coordinate of point x,min_P(1) and max_P(d) are respectively the minimum value of the P(1)^th coordinate and the maximum value of the P(d)^th coordinate for points in cube C. We denote these simplices with the prefix "K", in order to distinguish them from the simplices used to approximate M.Examples of K-simplices in cubes are represented (by their edges) on Figures <ref> and <ref>. The set of i-dimensional faces of a K-simplex S_P is denoted 𝒦_i(S_P). The set of K-simplices of cube C is denoted 𝒦_d(C). The union of all the K-simplices defined by all the permutations is the cube itself:C = ⋃_S_P ∈𝒦_d(C) S_P. On each edge[v_-,v_+] of a K-simplex such that ℳ(v_-) = -1 and ℳ(v_+) = +1, we compute a boundary point b_M([v_-,v_+]), approximating the intersection between the edge and M, by q successive dichotomies, as previously. The value of q may be adjusted in order to ensure a given precision of the approximation even on the longest edge of the K-simplex. We denote B_M(S) the set of boundary points of a K-simplex or of a K-simplex face S. The resistars in K-simplices are defined similarly to the c-resistars, except that faces of the K-simplices are considered instead of faces of cubes (see examples on Figures <ref> and <ref>). We denote K-resistars the resistars defined in K-simplices. Let S_P be a K-simplex such that B = B_M(S_P) ≠∅. The K-resistar [B]^⋆ is the following set: [B]^⋆ = ⋃_{ F_1,...,F_d-1}∈𝒦^⋆(B,S_P) [{B̅_M(F_1),...,B̅_M(F_d-1) }∪{B̅}] with: { F_1,...,F_d-1}∈𝒦^⋆(B,S_P) ⇔ F_1 ∈𝒦_1(S_P), B_M(F_1) ≠∅ F_i ∈𝒦_i(S_P), F_i-1⊂ F_i, 2 ≤ i ≤ d-1.where i is an integer. Using the same arguments as in the proofs of propositions <ref> and <ref>, it can easily be shown that K-resistar [B_M(S_P)]^⋆ is a set of d-1-dimensional simplices and that its boundary is included in the boundary of S_P.Let S_P be a K-simplex such that B = B_M(S_P) ≠∅. [B]^⋆ is a manifold. Proof. Consider a point x ∈ [B]^⋆. There exists {F_1,...,F_d-1}∈𝒦^⋆(B, S_P) such that x ∈ [{B̅_M(F_1),..., B̅_M(F_d-1)}∪{B̅}] and there exist positive numbers (λ_i), i ∈{1,...,d} such that (setting F_d = S_P): x = ∑_i ∈{1,...,d}λ_i B̅_M(F_i) ∑_i ∈{1,...,d}λ_i = 1. The Vapnik-Chervonenkis dimension of the d-1-dimensional linear separators being d+1 <cit.>, for any cut into two sets of d+1 affinely independent points, there exists a d-1-dimensional hyperplane making this cut. Let H be a hyperplane separating the set V_+ of vertices v of S_P such that ℳ(v) = +1 from the set V_- of vertices v of S_P such that ℳ(v) = -1. To each boundary point b ∈ B, located on edge [v_-, v_+] with v_- ∈ V_- and v_+ ∈ V_+, we associate point b_H = [v_-, v_+] ∩ H. Let B_H be the set of points b_H so defined. For any face F_i of S_P, we denote B_H(F_i) = F_i ∩ B_H. We can associate uniquely to x the point x_H ∈ H defined as follows: x_H = ∑_i ∈{1,...,d}λ_i B̅_H (F_i), where the faces F_i and λ_i are defined in equation <ref>. Therefore there exists a continuous and invertible function from [B]^⋆ to hyperplane H. □§.§.§ Classification function and algorithm. As stated in details in proposition <ref>, a K-simplex S_P such that B_M(S_P) ≠∅ is always separated by its K-resistar into two classification sets, each containing a single face of the K-simplex without boundary points (Figure <ref> panel (a) shows examples of these classification sets in 2 dimensions). Let S_P be a K-simplex such that B = B_M(S_P) ≠∅. let ℋ(B,S_P) be the set of faces of S_P without boundary points :ℋ(B,S_P) = { F ∈𝒦(S_P), B_M(F) = ∅},where 𝒦(S_P) is the set of all the faces of S_P. ℋ(B,S_P), includes 2 faces of S_P: [V_+] where V_+ = { v ∈𝒱(S_P), ℳ(v) = +1 } and [V_-] where V_- = { v ∈𝒱(S_P), ℳ(v) = -1 }. The resistar [B]^⋆ separates S_P into two classification sets [B]^𝒟⋆([V_+]) and [B]^𝒟⋆([V_-]), such that [V_±] ⊂ [B]^𝒟⋆([V_±]), which are :[B]^𝒟⋆([V_±]) = ⋃_{ F_h,..,F_d-1}∈𝒦^𝒟⋆([V_±],S_P) [𝒱(F_h) ∪{B̅(F_1),...,B̅_M(F_d-1) }∪{B̅}],with: { F_h,..,F_d-1}∈𝒦^𝒟⋆([V_±], S_P) ⇔ F_h∈𝒦_h([V_±]), 0 ≤ h ≤ dim([V_±]) F_i ∈𝒦_i(S_P), F_i-1⊂ F_i, h+1 ≤ i ≤ d-1, B_M(F_h+1) ≠∅,where dim([V_±]) is the dimensionality of [V_±]. Moreover, [B]^𝒟⋆([V_+]) and [B]^𝒟⋆([V_-]) are connected sets, [B]^𝒟⋆([V_-])∩ [B]^𝒟⋆([V_+]) = [B]^⋆ and [B]^𝒟⋆([V_-]) ∪ [B]^𝒟⋆([V_+]) = S_P.Proof. The sets V_+ and V_- are such that the sets [V_+] and [V_-] are faces of S_P, because S_P is a simplex and any set of its vertices defines one of its faces. The polytopes of [B]^𝒟⋆([V_±]) include at least one face F_h of [V_±], and [V_±] ⊂ [B]^𝒟⋆([V_±]) and is a connected set, therefore [B]^𝒟⋆([V_±]) is a connected set.Using the same reasoning as in the proof of proposition <ref>, it can easily be shown that [B]^𝒟⋆([V_-]) ∩ [B]^𝒟⋆([V_+]) = [B]^⋆ and [B]^𝒟⋆([V_-]) ∪ [B]^𝒟⋆([V_+]) =S_P. □ The following definition of the classification function is consistent because proposition <ref> ensures that, if x ∉ [B]^⋆, either x ∈ [B]^𝒟⋆([V_+]) or x ∈ [B]^𝒟⋆([V_-]). The classification [B]^⋆(.) by resistar [B]^⋆ in K-simplex S_P is the function from S_P to {-1, 0, +1} which, is defined as follows for x ∈ S_P (setting B = B_M(S_P)): * If B = ∅, then [B]^⋆(x) = ℳ(v), v ∈𝒱(S_P);* Otherwise: * If x ∈ [B]^⋆, [B]^⋆(x) = 0; * Otherwise, let ℋ(B,S_P) = {[V_-], [V_+] }: * Ifx ∈ [B]^𝒟⋆([V_+]), [B]^⋆(x) = +1; * If x ∈ [B]^𝒟⋆([V_-]), [B]^⋆(x) = -1. Algorithm <ref> is easily adapted to a K-simplex instead of a cube, by replacing the faces of the cube by the faces of the K-simplex (see Figure <ref> panel (b)). It is direct to show that this algorithm returns [B]^⋆(x).§.§ K-resistar approximation on the grid. §.§.§ Definition.The definition of the K-resistar approximation of M on the grid is similar to one of the c-resistar approximation. Let B_M^K(G) be the set of all the boundary points of all the edges of the K-simplices of the grid. We call K-resistar approximation of M on grid G, denoted [B_M^K(G)]^⋆, the following set of simplices: [B_M^K(G)]^⋆= ⋃_S_P ∈𝒦_d(G), B_M(S_P) ≠∅ [B_M(S_P)]^⋆,where 𝒦_d(G) is the set of all the K-simplices defined in the cubes of G.The K-resistar approximation of M on grid G is a set of d-1-dimensional simplices and its boundary is included in the boundary of X.Proof. The proof is the same as the one of proposition <ref>, considering faces of K-simplices instead of cube faces. □The K-resistar approximation of M on grid G is a d-1-dimensional manifold. Proof. K-resistars in K-simplices are d-1-dimensional manifolds as shown in proposition <ref>. We need to show that the union of K-resistars in K-simplices sharing a K-simplex face is also a manifold in the neighbourhood of their common points belonging to this face.Let F_i be a i-dimensional face of a K-simplex, such that B_M(F_i) ≠∅. [B_M(F_i)]^⋆ = [B_M^K(G)]^⋆∩ F_i is a i-1-dimensional manifold and there exists a i-1-dimensional hyperplane H_i which separates the vertices of F_i classified positively by ℳ from the ones classified negatively. Let n_i be the normal vector of H_i, such that the vertices classified positively by ℳ are on the positive side of H_i. For each K-simplex S in 𝒦_d(G) containing F_i, there exists a d-1-dimensional hyperplane H_S separating the vertices of S classified positively by ℳ from the ones classified negatively. Let n_S be the normal vector ofH_S, such that the vertices classified positively by ℳ are on the positive side of ℋ_𝒮. We have n_i.n_S ≥ 0 because the hyperplane H_S also separates the positive and negative vertices of F_i. Therefore, we can choose each H_S such that H_S ∩ F_i = H_i.With this choice of the hyperplanes H_S, the continuous and bijective mapping between each simplex of the K-resistar in S and H_S, defined in the proof of proposition <ref>, is the same for the points in [B_M(F_i)]^⋆ for all K-simplices S sharing face F_i. Let H'_i, be the d-1 dimensional hyperplane extending H_i to the set X by extending the i-dimensional normal vector n_i to the d-dimensional normal vector n'_i of d-1-dimensional hyperplane H'_i which coincides with H_i in F_i, by setting to 0all the coordinates of n'_i which are not defined in F_i. The composition of the mappings from the K-resistar in S to the hyperplane H_S with the orthogonal projection on H'_i for all S defines a continuous and bijective mapping from the neighbourhood of [B_M(F_i)]^⋆ in the K-resistar approximation to hyperplane H'_i. □§.§.§ Classification by the K-resistar approximation on the grid.The classification sets in X defined by the K-resistar approximation on the grid G are defined similarly to those of the c-rersistar approximation on the grid. 𝒦_i(G) denotes the set of all the faces of K-simplices defined in the cubes of the grid (𝒦_d(G) denoting the set of all the K-simplices defined in the grid cubes). Let B_G = B_M^K(G) be the set of boundary points defined in all the K-simplices of grid G. Let ℋ^K(B_G,G) = { F ∈𝒦(G), F ∩ B_G = ∅}, where 𝒦(G) is the set of the K-simplices of the cubes of the grid G and of all their faces. Let 𝒟^K(B_G,G) be the connected components of ℋ^K(B_G,G). For D ∈𝒟^K(B_G,G), the classification sets [B_G]^𝒟⋆(D) defined by the K-resistar approximation are:[B_G]^𝒟⋆(D) = ⋃_ S_P ∈𝒦_d(G)[B_M(S_P)]^𝒟⋆(D ∩ S_P),with:[B_M(S_P)]^𝒟⋆(D ∩ S_P) = ∅,D ∩ S_P = ∅, S_P,D ∩ S_P = S_P,and is specified in proposition <ref> in the other cases. For all D ∈𝒟^K(B_G,G), D ⊂ [B_G]^𝒟⋆(D), [B_G]^𝒟⋆(D) is connected and its boundary in X - ∂ X is a subset of [B_G]^⋆.Proof. The proof is similar to the one of proposition <ref>, when using faces of K-simplices instead of cube faces. □For all points x ∈ X, such that x ∉ [B_G]^⋆, there exists a unique set D ∈𝒟^K(B_G, G) such that x ∈ [B_G]^𝒟⋆(D).Proof. The proof is similar to the one of proposition <ref>. □Let B_G = B_M^K(G). The classification [B_G]^⋆(.) by the resistar approximation from the grid is the function from X to {-1,0,+1} defined as follows for x ∈ X: * If x ∈ [B_G]^⋆, [B_G]^⋆(x) = 0 * Otherwise, proposition <ref> ensures that there exists a unique set D ∈𝒟^K(B_G,G) such that x ∈ [B_G]^𝒟⋆(D), and [B_G]^⋆(x) = ℳ(v), v ∈𝒱(D). For x ∈ X, let S_P(x) be the K-simplex of 𝒦_d(G) such that x ∈ S_P(x). We have: [B_M^K(G)]^⋆(x) = [B_M(S_P(x))]^⋆(x).Proof. The proof is similar to the one of proposition <ref>. □§.§.§ Classification algorithm.In practice, when classifying point x ∈ X, we also use algorithm <ref>, which provides a cube C'_x such that B_M^K(C'_x) ≠∅ and a point x' to classify in this cube, ensuring [B_M^K(G)]^⋆(x) = [B_M^K(C'_x)]^⋆(x'). Then a K-simplex of C'_x containing x' is determined by ordering the coordinates of x', this order providing the permutation P defining S_P. Then the procedure computes B_M^K(C'_x) ∩ S_P, the boundary points belonging to S_P. At this point, the classification of x' in K-simplex S_P is performed with algorithm <ref> (in its version adapted to K-simplices). The procedure of classification using K-resistars is thus a bit more complicated than the one of c-resistars because it requires identifying the K-simplex containing the point to classify and determining its boundary points. § ACCURACY OF THE APPROXIMATION WHEN THE SIZE OF THE GRID INCREASES.Theorem <ref> bounds the Hausdorff distance between a resistar approximation and the manifold to approximate, when this manifold is smooth enough. The distance d(x, A) from a point x ∈ℝ^d to a set A ⊂ℝ^d is defined as (inf denoting the infimum):d(x, A) = inf_y ∈ Ax - y.The Hausdorff distance d_H(A,B) between set A and set B, both subsets of ℝ^d, is defined as(sup denoting the supremum):d_H(A, B) = max(sup_x ∈ A d(x, B), sup_y ∈ B d(y, A)).The smoothness of the manifold is characterised by its reach <cit.>, which is the supremum of ρ such that for any point x of ℝ^d for which d(x, M) = ρ, there is only one point y ∈ M such that x - y= ρ. Note that if the reach of M is strictly positive, then M is twice differentiable <cit.>.Let M be a d-1-dimensional manifold cutting the compact X = [0,1]^d into two parts, G be a regular grid of n_G^d points covering X and its boundary, ϵ = 1/n_G-1 be the size of an edge of the grid.If the reach r of M is such that r > √(2) d ϵ, if for all i-dimensional faces F of X, M ∩ F is a i-1-dimensional manifold of reach r_F > √(2) i ϵ, and if all the boundary points are determined with q ≥ -log_2(ϵ) dichotomies, then the Hausdorff distance between M and its resistar approximation (in cubes or in K-simplices) decreases like 𝒪(d ϵ^2).The proof of theorem <ref> uses two lemmas presented in paragraph <ref>. Then it uses an induction on the space dimensionality (set in paragraph <ref>) with two parts: bounding the distance between the resistar approximation and M (paragraph <ref>) and bounding the distance between M and the resistar approximation (paragraph <ref>). §.§ Lemmas. Let r >0 be the reach of M and let δ > 0 be such that r > δ. For any couple of points (y_1,y_2) of M such that y_1 - y_2 ≤δ:y_2 - P_y_1(y_2) ≤δ^2/2r + 𝒪( δ^3),where P_y_1(y_2) is the orthogonal projection of y_2 on the hyperplane tangent to M at y_1.Proof. Let (y_1,y_2) ∈M^2 be such that y_1 - y_2 ≤δ < r, and n be the normal vector of the hyperplane T tangent to M at y_1. Let M_- (resp. M_+) be the set of points x ∈ X such that ℳ(x) = -1 (resp.ℳ(x) = +1). There exists ℬ(c_+,r) (resp.ℬ(c_-,r)) a ball tangent to M at y_1 such that M_- ∩ℬ(c_+,r) = ∅ (resp. M_+ ∩ℬ(c_-,r) = ∅). y_2 is located between the balls ℬ(c_+,r) and ℬ(c_-,r), and we can suppose that it is closer to ℬ(c_+,r) (the reasoning would of course be the same if it was closer toℬ(c_-,r)). Let y'_2 be the projection of y_2 on ∂ℬ(c_+,r) parallel to n. We have:y_2 - P_y_1(y_2) ≤y'_2 - P_y_1(y_2) ,and:P_y_1(y_2) - y_1 ≤δ,because P_y_1(y_1) = y_1 and the projection is contracting.Moreover (see figure <ref>, panel (a)):y'_2 - P_y_1(y_2)= r (1 - cosα),with:cosα≥√(1 - (δ/r)^2),Developing equation <ref> at the second order, we get:y_2 - P_y_1(y_2) ≤δ^2/2r + 𝒪(δ^3).□ Let 𝒞 be a set of j^d adjacent cubes of the grid, covering a cubic part of X of edge size j ϵ (ϵ = 1/n_G -1), including a non-void set of boundary points B = B_M(𝒞), computed on cube edges, or B = B_M^K(𝒞), computed on edges of K-simplices, each determined with q dichotomies. If the reach r of M is such that:r > j√(d)ϵ,then for any point x ∈ [B]^⋆ and for any point y ∈ (M ∩𝒞):x - P_y(x) ≤2^-qϵ + d j^2/2rϵ^2 + 𝒪(ϵ^3),where P_y(x) is the orthogonal projection of x on the hyperplane tangent to M at y.Proof. By construction, for any boundary point b of B located on edge [v_-(b), v_+(b)] there exists a point b_M ∈ M ∩ [v_-(b), v_+(b)] such that b - b_M ≤ 2^-q-1ϵ. Therefore, for each boundary point b of B, we can write: b- P_y(b) ≤b - b_M+ b_M - P_y(b_M) + P_y(b_M) - P_y(b), with: * b - b_M≤ 2^-q-1ϵ (see equation <ref>);* P_y(b_M) - P_y(b)≤2^-q-1ϵ, because the orthogonal projection is contracting;* b_M - P_y(b_M) ≤d j^2 ϵ ^2/2r + 𝒪(ϵ^3), because b_M - y≤ j√(d)ϵ, since both b_M and y belong to 𝒞, and applying lemma <ref>. Overall, we get: b- P_y(b) ≤ 2^-qϵ + dj^2 /2rϵ^2 + 𝒪(ϵ^3). Moreover, by definition of [B]^⋆, for any point x in [B]^⋆ there exists a set of positive numbers (λ_b)_b ∈ B such that: x = ∑_b ∈ Bλ_b b, ∑_b ∈ Bλ_b = 1. Therefore:x - P_y(x)≤∑_b ∈ Bλ_bb - P_y(b) ≤ 2^-qϵ + d j^2 /2rϵ^2 + 𝒪(ϵ^3). □§.§ Starting induction. Assume d = 1 and r > √(2)ϵ. M is a set of discrete points such that, for b_M and b'_Mtwo distinct points of M, b_M - b'_M >√(2)ϵ. Therefore, in any edge [v,v'] of the grid, v and v' being two consecutive points such that v-v' = ϵ, there is at most one point b_M of M in [v,v'] and for each point b_M of M, there exists a single boundary point b ∈ [v,v'] with b-b'_M≤ 2^-q-1ϵ, q being the number of dichotomies performed to get b. Moreover, by construction, there is no boundary point in a segment [v,v'] such that [v,v'] ∩ M = ∅. Therefore, choosing q ≥ -log_2(ϵ), ensures that theorem <ref> is true for d = 1.Now, we assume (induction hypothesis) that the theorem is true in a compact of any dimensionality lower or equal to d-1 and we consider a manifold M splitting a X = [0,1]^d of dimensionality d and its resistar approximation, both satisfying the conditions of theorem <ref>. In the next subsection, we bound the distance from the resistar approximation to M and in the following subsection, we bound the distance from M to the resistar approximation. In both cases, the neighbourhood of the boundary of X is a specific case requiring the induction hypothesis.§.§ Bounding the distance from the resistar approximation to M.Let C be a cube of the grid and [B]^⋆ the resistar (c-resistar or K-resistar) approximation of M in this cube, supposed non-void. Since there are boundary points in C, M cuts some edges of C and C ∩ M ≠∅. Let y_0 ∈ M ∩ C and let ℬ(c_+,r) and ℬ(c_-,r) be the positive and negative balls tangent to y_0 as defined in the proof of lemma <ref>. Applying lemma <ref>, for all x in [B]^⋆, because the maximum distance between two points of C is √(d)ϵ, and r > √(2) d ϵ > √(d)ϵ we have:x - P_y_0(x)≤ 2^- qϵ + d /2rϵ^2 + 𝒪(ϵ^3), where P_y_0(x) is the orthogonal projection of x on T, the hyperplane tangent to M at y_0. Moreover, P_y_0(x) - y_0 ≤√(d)ϵ, because the orthogonal projection is contracting. Let z_- and z_+ be the projection of x parallel to n, the normal vector to T, on respectively ℬ(c_-,r) and ℬ(c_+,r). Because of lemma <ref>, we have (see figure <ref> panel b):P_y_0(x) - z_-≤d/2rϵ^2 + 𝒪(ϵ^3) P_y_0(x) - z_+≤d /2rϵ^2 + 𝒪(ϵ^3). There exists y ∈ M ∩ [z_-, z_+] because M is a d-1-dimensional manifold located between ℬ(c_+,r) and ℬ(c_-,r), and we have: x - y≤x - P_y_0(x) + P_y_0(x) - y≤ 2^- qϵ + d/rϵ^2 + 𝒪(ϵ^3), because P_y_0(x) = P_y_0(y) and P_y_0(x) - y≤ P_y_0(x) - z_+ orP_y_0(x) - y≤ P_y_0(x) - z_-. Two cases can take place: * y ∈ X, which is guaranteed when the cube C is not at the boundary of X (see figure <ref>, panel (a)). Then if q ≥ - log_2(ϵ), x - y= 𝒪(d ϵ^2).* y ∉ X, which can happen when C is at the boundary of X (see figure <ref>, panel (b)). Let z = [x,y] ∩∂ X and let F be the facet of X such that z ∈ F. We have: [x,z] ∩ M = ∅, hence ℳ(z) ≠ [B]^⋆(z). Because of the induction hypothesis, there exists a point y' ∈ (M ∩ F) such that y' - z = 𝒪((d-1) ϵ^2).Since: x - z≤x - y≤ 2^- qϵ + d /rϵ^2 + 𝒪(ϵ^3), if q ≥ - log_2(ϵ), x - y'≤x - z + z - y' = 𝒪(d ϵ^2).Therefore, in all cases, if q ≥ - log_2(ϵ), for all x ∈ [B]^⋆, there exists y ∈ M ∩ X such that x - y = 𝒪(d ϵ^2). §.§ Bounding the distance from M to its resistar approximation.Consider a point y_0 ∈ M ∩ X. Let C be a cube of the grid such that y_0 ∈ C, let T be the hyperplane tangent to M at y_0, let n be its normal vector and ℬ(c_+,r) and ℬ(c_-,r) be the positive and negative balls tangent to y_0. Let 𝒞 be the cube of centre the centre of C, and of edge size 3 ϵ. The segment [y_0,c_+] (resp. [y_0,c_-]) cuts the boundary of 𝒞 at z_+ (resp. z_-), because r > √(2) d ϵ > 2 √(d)ϵ for d ≥ 2.We first consider the case where there exists a cube C' of the grid, adjacent to C such that z_+ ∈ C', then there exists F, facet of C' such that z_+ ∈ F (see Figure <ref>).We first show that [B]^⋆(z_+) = +1 because the whole facet F is included in ℬ(c_+,r). Let c'_+ be the orthogonal projection of c_+ on the hyperplane H_F defined by F. The intersection of ∂ℬ(c_+,r) with H_F is a sphere ∂ℬ_F(c'_+, r') in H_F of centre c'_+. Let { w } = r⃗(c_+, z_+) ∩∂ℬ_F(c'_+,r'). w is the point of ∂ℬ(c_+,r) ∩ H_F which is the closest to z_+. We have (see Figure <ref>, panel (a)): z_+ - w = r sinβ/cosα. where α is the angle defined by (c'_+, c_+, y_0) and β the angle defined by (y_0, c_+, w). The angle β is minimum when the distance from y_0 to F is ϵ. Indeed, if y_0 gets closer to H_F, keeping the same angle between H_F and c_+ - y_0, the radius r' of the sphere ∂ℬ_F(c'_+, r') decreases as well as the distance w - z_+ and so does β. Moreover, cosα is maximum at 1. With these values, we have: sinβ = √(1 - (r - ϵ/r)^2). Therefore:z_+ - w≥√( r^2 - (r - ϵ)^2). Then, expressing within equation <ref> that z_+ - w is larger than √(d-1)ϵ the maximum distance between two points in the facet, requires:r ≥dϵ/2, Therefore, as we assumed r ≥√(2) dϵ this condition is satisfied, thus z_+ - w≥√(d-1)ϵ, implying that the whole facet F is included in ℬ(c_+,r) and [B]^⋆(z_+) = +1.Similarly, if there exists a cube C' of the grid, adjacent to C and such that z_- ∈ C', then[B]^⋆(z_-) = -1.Because of propositions <ref> and <ref>, the segment [z_+,z_-] crosses the resistar (c-resistar or K-resistar) approximation [B]^⋆ defined in 𝒞 (with B = B_M(𝒞) or B = B_M^K(𝒞)). Let z ∈ [z_+,z_-] ∩ [B]^⋆. Noticing that P_y_0(z) = y_0 and applying lemma <ref>, we get: y_0 - z = 𝒪(d ϵ^2).Now, we consider the case where the cube C is on the boundary of X and [y_0,c_+] crosses the boundary of X in d-1-dimensional facet F of C, before crossing the boundary of 𝒞. Let z_+ = [y_0,c_+] ∩ F. Two cases can take place: * [B]^⋆ (z_+) = +1, then the same reasoning as previously applies; * [B]^⋆ (z_+) = -1, is an error (because z_+ ∈ℬ(c_+,r)) and because of the induction hypothesis, there exists a point z in [B_M(F)]^⋆ or in [B_M^K(F)], such that z_+ - z= 𝒪((d-1)ϵ^2). We have: P_y_0(z) - P_y_0(z_+)= P_y_0(z) - y_0 = 𝒪((d-1) ϵ^2), because the orthogonal projection is contracting, and:P_y_0(z) - z = 𝒪(d ϵ^2) because z - y_0≤√(d)ϵ and applying lemma <ref>. Therefore (see figure <ref>, panel (b)):y_0 - z≤y_0 - P_y_0(z)+ P_y_0(z) - z = 𝒪(d ϵ^2).Of course, the same reasoning applies to the negative side [y,c_-]. This concludes the proof of theorem <ref>. □ § EXAMPLES AND TESTS.§.§ Visualisation of examples with spheres and radial based functions.When the dimensionality d is higher than 3, the surface cannot be directly represented, but it is possible to visualise its intersection with d - 3 hyperplanes. Algorithm <ref> sketches the method. It is based on building polytopes whose vertices are the intersections of edges of another polytope with a hyperplane. Starting with a simplex of the resistar, a polytope of one dimension less is computed in this way successively with each of the hyperplanes. If the intersection of the resistar simplex with the intersection of the d-3 hyperplanes is not empty, the result is a 2D polygon in the 3D intersection of the d - 3 hyperplanes. These polygons define together a 2D surface in this 3D space. Even though it is possible to focus on a small percentage of all the simplices of the resistar approximation that have chances to intersect with the intersection of all the hyperplanes, this small percentage may still correspond to a high number of simplices in some cases. This can occur easily when the dimensionality of the space is higher than 6, especially with K-resistars. Figure <ref> shows examples of approximations of a sphere in 5D with K-resistars (panel (b)) and in 6D with c-resistars (panel (a)), for n_G = 5 points on each axis of the grid. Note that the K-resistar includes a larger number of boundary points and even larger number of simplices, although it is in 5 dimensions while the c-resistar is in 6 dimensions. In 6 dimensions, the K-resistar approximating the same sphere for n_G = 5 involves 199,322 boundary points and 1.1 billion simplices. In order to test more systematically our approach, we derived classifications from a radial-based function defined on two sets of randomly chosen points in the space E_p = {p_1, p_2,.., p_l} and E_n = {n_1, n_2,.., n_m}, as follows:ℳ(x) = (∑_i = 1^l ϕ(p_i - x/σ) - ∑_i = 1^m ϕ(n_i - x/σ)),where σ is a positive number and function ϕ is defined by:ϕ(x) = 100/1 + x^2. When varying the number of points in E_p and E_n and the value of σ, the resulting surface is more or less complicated and smooth. In our tests, we use two settings that are illustrated by Figure <ref>.§.§ Testing the classification error when the grid size and the space dimensionality vary. Generally, the accuracy of classification methods is measured by the misclassification rate of points randomly drawn in [0,1]^d. However, this measure requires drawing a very large number of test points when the dimensionality d of the space increases. In order to limit the number of tests, we generated the test points only in the cubes which include boundary points. This gives more chances to get misclassified points. Taking 100 test points in each cube containing boundary points, we used 50 million test points for the resistar approximation with the highest number of cubes in the following tests (in dimensionality 4 with n_G = 48). Figure <ref> shows that the classification error decreases like n_G^-2 for both K-resistars and c-resistars (estimated slopes in the log-log graph: -2.06, for K-resistars, -2.01 for c-resistars, with R^2 = 0.99 for both) in accordance with theorem <ref>, whereas the classification error of nearest vertex decreases like n_G^-1 (estimated slope in the log-log graph: -1.02, with R^2 = 0.99). It appears also that the error of K-resistars is significantly lower than the one of c-resistars, which is expected because K-resistars are based on a larger number of boundary points located on the edges of the Kuhn simplices. Note that the values of errors do not change significantly when the dimensionality changes.This is confirmed by Figure <ref> which shows the classification error for nearest vertex, c-resistar and K-resistars approximating radial based classification functions with the same parameters (E_p and E_n equal to 10, σ = 0.4) and the same number of points by axis of the grid n_G = 4, in dimensionality varying from 3 to 9. For each dimensionality, the tests are repeated for 10 radial-based classification functions with points E_p and E_n drawn at random in [0,1]^d. The error percentage is computed by classifying 100 points uniformly drawn in each cube which includes boundary points. We observe that the error percentages do not vary significantly with the dimensionality whereas the bound on the Hausdorff distance should vary linearly with d.Figure <ref> shows the number of boundary points and simplices when the number of points of the grid or the dimensionality of the space vary. These graphs confirm the very rapid growth of the number of boundary points and of simplices, particularly in K-resistars. In 9 dimensions, for n_G =4, the number of boundary points of the K-resistar surface is around 10 million, and the number of simplices is around 10^14 (see panel (b)). There are about 100 times less boundary points and 10,000 times less simplices in the c-resistar surface.The estimation of the slope of the logarithm of the number of simplices as a function of the logarithm of n_G for the tests in dimensionality 3, 4 and 5 presented on Figure <ref> (see the case of dimensionality 4 on figure <ref> panel (a)) is reported in table <ref>. For c-resistars, the number of boundary points appears thus to grow like 𝒪(n_G^d-1). For K-resistars the slope is a bit higher than d-1 and the difference increases with the dimensionality. This is due to the number of edges in all the K-simplices in a cube which increases much more rapidly than the number of edges of the cube. § DISCUSSION - CONCLUSIONThis paper shows examples of simplex based approximation and classification in 9 dimensions, which has never been done with marching cube or Delaunay triangulation. Indeed, the resistar classification is achieved through a few projections on facets and faces of a cube while the other methods would require to test the position of the point to classify with respect to a large number of simplices. This advantage of resistars starts in low dimensionality and becomes decisive in higher dimensionality as the number of simplices increases exponentially. The classification methods used in the algorithms of viability kernel approximation, such as nearest vertex, SVM or k-d trees, are based only on the classification of the vertices of a regular grid, hence their error cannot decrease more than 𝒪(n_G^-1) which is the intrinsic error in the learning sample. Resistar approximation does better because it is based on the boundary points which can be at a precision of 𝒪(n_G^-2) with an adequate number of dichotomies. Some other methods, for instance decision trees <cit.> could possibly be modified to learn efficiently from boundary points, but such a modification does not seem immediate. This specificity of resistar approximations allows them to ensure, under arguably reasonable conditions, that their Hausdorff distance to the manifold to approximate decreases like 𝒪(n_G^-2). This is a very significant advantage over current methods. Indeed, the resistar classification from the a grid of n_G^d points has the same accuracy as a standard classification based on a grid of (n_G^2)^d points. Computing the boundary points of resistars requires first classifying by ℳ the n_G^d grid points. Then, for c-resistars which generate 𝒪(n_G^d-1) boundary points, there are in total 𝒪 (q. n_G^d-1) point classifications by ℳ, because of the q successive dichotomies necessary to compute each boundary point. If q = log_2 (n_G) as in Theorem <ref>, the number of point classifications by ℳ required by c-resistars is n_G^d + log(n_G). n_G^d-1, which is very significantly lower than then_G^2d grid vertex classifications required by standard methods to get the same accuracy. For K-resistars, the number of boundary points increases also approximately like 𝒪 (n_G^d-1) in low dimensionality, but this number may be closer to 𝒪(n_G^d) in dimensionality 8 or 9. Overall, the number of classifications by ℳ remains still very significantly lower than the one required by usual methods to get the same accuracy. Considering the requirements in memory space, the advantage of resistars is very strong over the nearest vertex classification which needs to store the whole grid of n_G^2d points (or with some optimisation 𝒪(n_G^2d-1) points) whereas the c-resistars need to store 𝒪(n_G^d-1) boundary points and K-resistars at worst 𝒪(n_G^d), to get the same accuracy.The two variants of resistars have different strengths and weaknesses. The K-resistars approximations have the major advantage to be manifolds and their error rate is lower than the one of the c-resistars for a given grid size. However, the c-resistars are significantly lighter, especially when the dimensionality increases. In some cases, it is possible to use them while the K-resistars are too heavy. For both of them, the accuracy in 𝒪(n_G^-2) requires the manifold to approximate to be smooth, which is not always the case in viability problems. This manifold is indeed often the boundary of the intersection of several smooth manifolds, hence with a reach equal to zero. A challenging future work is to define new types of resistars approximating these intersections with an accuracy of 𝒪(n_G^-2) and with an efficient classification algorithm. § ACKNOWLEDGMENTI am grateful to Sophie Martin and Isabelle Alvarez for their helpful comments and suggestions on earlier versions of the paper.§ REFERENCESelsarticle-num | http://arxiv.org/abs/1707.08373v3 | {
"authors": [
"Guillaume Deffuant"
],
"categories": [
"cs.CG"
],
"primary_category": "cs.CG",
"published": "20170726110315",
"title": "Recursive simplex stars"
} |
Regensburg]Peter C. Bruns GWU]Maxim Mai[Regensburg] Institut für Theoretische Physik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, Germany[GWU] Department of Physics, The George Washington University, 725 21^ st St. NW, Washington, DC 20052, USA We discuss the impact of chiral symmetry constraints on the quark-mass dependence of meson resonance pole positions, which are encoded in non-perturbative parametrizations of meson scattering amplitudes. Model-independent conditions on such parametrizations are derived, which are shown to guarantee the correct functional form of the leading quark-mass corrections to the resonance pole positions. Some model amplitudes for ππ scattering,widely used for the determination of ρ and σ resonance properties from results of lattice simulations, are tested explicitly with respect to these conditions. Chiral symmetries Lattice QCD Light mesons Resonances 11.30.Rd, 12.38.Gc, 14.40.Be § INTRODUCTION Chiral Perturbation Theory (ChPT) and ab-initio lattice QCD (LQCD) simulations are currently the state of the art approaches for the exploration of low-energy QCD. The specifically beneficial overlap between these approaches arises from the fact that LQCD simulations can be (and usually are) performed at unphysical quark masses. Thus, the results of those cover the full quark mass vs. energy plane. At the same time ChPT relies on the expansion of QCD Green's functions in small momenta and quark masses, and allows for interpolations and extrapolations of the measured results in the low-energy region of that plane.The simplest non-trivial hadronic system in this regime is the ππ system, which has been studied very extensively in the context of ChPT, see e.g. <cit.>. Recently, high precision LQCD data became available in both I=0 and I=1 channels, see e.g. <cit.>. Usually such discrete data are extrapolated in energy and quark masses to e.g. determine properties of the (isovector) ρ and (isoscalar) σ resonances in these channels at the physical point, see <cit.> for some recent examples. Obviously, such inter-, extrapolations require a well-founded theoretical control over the scattering amplitude in these channels. When the quark mass is fixed, the scattering amplitude is constrained by analyticity and unitarity requirements, and crossing symmetry. Many parametrizations, used in the literature, fulfill these requirements to some extent, such as the Inverse Amplitude Method, Bethe-Salpeter Equation, Breit-Wigner or Chew-Mandelstam parametrizations. The dependence on the quark masses goes beyond these requirements, and the dynamics of the underlying field theory (QCD)has to be specified in more detail, yielding additional constraints. These constraints are mainly given by chiral symmetry, and the particular way it is broken in the real world. Close to the two-flavor chiral limit (m_u=m_d=0) ChPT exactly implements all these constraints, order by order in a low-energy expansion, and fixes the functional form of the quark-mass corrections to the chiral limit quantities. It is the purpose of this letter to introduce model-independent conditions on the parametrizations of the ππ scattering amplitude, which assure that the leading quark-mass corrections in the chiral extrapolations of the resonance properties, such as mass and width, are consistent with the chiral behavior of QCD. We find that these conditions are violated in some currently used approaches. § CHIRAL SYMMETRY CONSTRAINTS The effective degrees of freedom of ChPT are pseudo-Goldstone bosons (pions) of spontaneously broken chiral SU(2)×SU(2) symmetry. Resonance fields can be included as explicit (massive) fields in the effective theory, see e.g. <cit.>. The corresponding Lagrangians contain bare quantities such as the bare resonance mass and couplings to pion fields, which are renormalized order-by-order in the usual sense of perturbation theory. The latter requires a proper power counting scheme, which is more subtle when massive fields are involved. The reason is that the mass and width of the meson resonances do not vanish in the chiral limit, thus introducing a new ("heavy") mass scale not small compared to the hadronic scale of ∼ 1 GeV. Fortunately, one can employ tailor-made subtraction schemes which make these extended versions of the effective field theory well-defined, so that quark-mass corrections to resonance properties can be computed unambiguously, see e.g. <cit.>. In this letter, we focus on the purely mesonic sector of the strong interaction (in particular, ππ scattering), and work in the isospin-symmetric limit where m_u=m_d=:m_ℓ.Due to pion loops, any amplitude involving the strong interaction will in general depend in a non-analytic fashion on the pion mass in an expansion around the SU(2)× SU(2) chiral-symmetric limit (m_ℓ→0), as first noted in <cit.>. Denoting M^2:=2Bm_ℓ, so that M_π^2=M^2+𝒪(M^4log M) <cit.>, the mass of a "heavy" degree of freedom in the sense explained above, denoted hereafter by H, depends on the light-quark mass asm_H = ∘m_H + c_1^HM^2 + c_2^HM^3+c_3^HM^4log M + 𝒪(M^4) ,where "∘" henceforth denotes the quantity (here the mass) in the chiral limit. Examples are the mass of the nucleon <cit.>, vector meson masses <cit.> and also the mass of the kaon in a two-flavor framework where the strange-quark mass is considered as heavy compared to m_ℓ <cit.>. An expression of the same form holds for the widths of heavy meson resonances, see e.g. <cit.>. We note that the non-analytic c^H_2-term ∼ m_ℓ^3/2 is somewhat exceptional: it exists only if there is a vertex for H→π H' in the effective theory, where H' is mass-degeneratewith H (possibly identical to H, or belonging to the same isospin multiplet), and is related to the threshold production of a pseudo-Goldstone boson. In the following, we will exclude this exceptional case (which is permissible in the meson sector), but we shall add some pertinent comments in the course of the investigation. Let us now sketch an argument showing that the quark-mass expansion of the on-shell ππ scattering amplitude at fixed Mandelstam variables s,t0 contains only quark-mass logarithms with a prefactor of order M^4 or higher, to all orders in the low-energy expansion. For this purpose, we use a modified version of the general argument presented in <cit.>. The π^a(q_a)π^b(q_b)→π^c(q_c)π^d(q_d) scattering amplitude can be read off from the residue of the quadruple pion pole in the Fourier transformG_P(q) of the correlator ⟨ 0|TP^a(x)P^b(y)P^c(z)P^d(w)|0⟩ with respect to the space-time arguments x,y,z,w <cit.>. Here q collectively denotes the pion four-momenta q_a,…, q_d. Moreover, let G̃_P^ef(q,p_e,p_f) denote the analogous Fourier transform of the matrix element ⟨π^f(p_f)|TP^a(x)P^b(y)P^c(z)P^d(w)|π^e(p_e)⟩. Following the argument of <cit.>, the leading quark-mass logarithm of G_P(q) is generated by soft Goldstone bosons circulating in the loop indicated by the dashed line in Fig. <ref>, and can therefore be inferred from the integralI_log(q):=1/2∫d^4p/(2π)^4iδ^efG̃^ef_P(q,p,p)/p^2-M^2 .Here G̃_P^ef(q,p,p) can be taken in the (p^μ→ 0, M→ 0) limit, since terms in G̃^ef_P linear in p^μ vanish in the integral, while terms of order p^2,M^2 will generate terms ∼ M^4log M^2. In this limit, this matrix element can be expressed through four-point functions of the type of G_P(q) in the chiral limit (and terms without a quadruple pion pole, which we can neglect here) by virtue of current algebra and PCAC techniques. Thus, the leading logarithm in the integral can be computed in terms of ∘G_P(q), employing dimensional regularization for definiteness, and scales as ∼ M^2log M^2. However, these logarithmic terms are exactly absorbed by the renormalization of the matrix elements ⟨ 0|P^a|π^b⟩ = δ^abG_π <cit.>, with G_π=2BF(1-M^2/32π^2F^2logM^2+…) ,at the four operator insertions, and so no term ∼ M^2log M^2 is left as a correction to the remaining part of the quadruple pole term in G_P(q), which is exactly the ππ scattering amplitude. We point out that this in general requires a complicated cancellation among the Feynman graphs in the scattering amplitude, and can not be assured by power-counting arguments for individual graphs. There are some exceptions to the simple argument just given, corresponding to special cases where some combination of energy-invariants of the ππ process also approach zero, so that the momentum of an internal pion in G̃^ef_P(q,p,p) is forced to be also “soft” (of order ∼ M_π) when p^μ→ 0. In the generic case, however, the light-quark-mass derivative of the ππ amplitudeexists in the chiral limit (the same is also true for the scattering of pions off heavy mesons[A consideration very similar to the one of the previous paragraph applies for matrix elements ⟨ H(p')|P^a(x)P^b(y)|H(p)⟩, where H is a heavy meson, under the qualification mentioned below Eq. (<ref>). In the general case, one has to carefully analyze the Born graphs of the process ππ H →ππ H, which is beyond the scope of this study. However, Eq. (<ref>) is generally valid for heavy resonances (denoted by R) due to chiral symmetry and a simple power-counting argument. The considerations outlined above just serve to make plausible how this chiral-symmetry constraint is realized in resonant amplitudes ππ→ππ or π H→π H, which do not contain the resonance degree of freedom R explicitly.]).We have explicitly verified this constraint for the available two-loop representation for the ππ scattering amplitude <cit.> (and also for the explicit one-loop expressions for pion-kaon-scattering given in <cit.> and <cit.>): Fixing generic non-zero energy variables s,t,u=4M_π^2-s-t away from the s,t-and u-channel thresholds, the expansion in the light-quark mass shows only logarithmic terms with at least a prefactor ∼ M^4, as a consequence of the general argument referred to above. Note that this quark-mass expansion is different from the chiral low-energy expansion, where one assumes s,t∼𝒪(M_π^2). This is why the mentioned result is not in conflict with the corresponding one from <cit.>, where small s,t,u are presumed. We will see examples of the quark-mass expansion at fixed energy in the next section (see Eqs. (<ref>), (<ref>)). Since the on-shell scattering amplitude shows no terms ∼ f(s,t)M^2log M in this expansion, except for t=0 or u=0, we expect that the quark-mass expansion of the partial-wave amplitudes for ππ scattering will also be free of terms ∼f̃(s)M^2log M, and we find that this is indeed the case. The complex-energy position s_H of a resonance H appearing in a partial wave of angular momentum l, t_l(s), given by solving (t_l(s_H))^-1!=0 on the second Riemann sheet in the Mandelstam variable s, is therefore expected to show a non-analytic quark mass dependence of ∼ M^4log M or higher. This is nicely consistent with the chiral prediction of Eq. (<ref>) in the common case where c^H_2=0. Should there exist an exactly mass-degenerate resonance H', with possible transitions H→π H' for M_π→ 0, the above argument must be modified, to take into account the additional π H' branch point, and the general expectation is spoilt in this case, which makes the c^H_2-term necessary. But even in this case, terms ∼ M^2log M, which are the main concern of this study, are never present in Eq. (<ref>) as a consequence of chiral symmetry and chiral power-counting applied to the resonance self-energy, compare <cit.>. The vanishing of f̃(s), motivated above, will guarantee that the quark-mass dependence of the resonance position, encoded in the partial-wave amplitude, is consistent with the absence of “forbidden logarithms” ∼ M^2log M in Eq. (<ref>). This constraint can be seen as a consistency condition between two different approaches to resonances in effective field theories (explicitinclusion of resonances, and dynamical generation of resonance poles). Thus, if a given model for the ππ scattering amplitude leads to such “forbidden logarithms”, one will have to conclude that the predicted quark-mass dependence of this model is in conflict with QCD. Following these considerations, we propose a simple test for the partial-wave amplitudes generated by a given model to fulfill the model-independent requirement, demanded by chiral symmetry – the vanishing of terms of the form f̃(s)M^2log M in the quark-mass expansion for fixed s≠0.Note that: 1) In the standard low-energy expansion of ChPT, this vanishing of the "forbidden logarithms" can only be verified up to a certain order s^n in f̃(s); 2) The coefficient functions, such as f̃(s), in this expansion are chiral-limit quantities without quark-mass dependence, but may contain energy logarithms ∼log s; 3) the sigma terms pertaining to the resonances, which were recently adressed as important clues to the nature of these states <cit.>, would diverge in the chiral limit if the forbidden logarithmic terms were present in the mass formula (<ref>). § CRITICAL EXAMINATION OF MODEL AMPLITUDESPractically all currently used model amplitudes for ππ scattering are of the general formt_l(s)^-1 = 16π(K_l^-1(s)+I(s)) ,where K_l(s) is a real-valued function for 0<s<16M_π^2, usually referred to as K-matrix in cases with more channels, or generalized potential, and I(s) is the two-pion loop function.Note that the requirement of elastic unitarity fixes Im(16π I(s))=-2q(s)/√(s)for real s>4M^2, where q(s)=√(s/4-M^2), such that the form of the loop function is fixed (requiring the appropriate analytic properties) up to a real constant, which can be absorbed in K_l(s). In dimensional regularization with MS subtraction (also employed in <cit.>) the loop function reads16π^2I(s)= log(M^2/μ^2)-1 -4q(s)/√(s)artanh(-√(s)/2q(s)) .In any channel the resonance-pole positions s_l^* on the second Riemann sheet are determined as the solutions of the equation K_l^-1(s_l^*)+I^II(s_l^*) = 0 for I^II(s) = I(s)-iq(s)/(4π√(s)). Expanding I^II(s) in powers of M for 0≤ 4M^2<|s|, one finds16π^2I^II(s)= - (2π i + 1 + log(-μ^2/s))+ 2M^2/s (2π i - 1 + log(-M^2/s))+𝒪(M^4/s^2) .The generalized potential K_l(s) parametrizes the interaction of two pions in the corresponding channel, and can be chosen in various ways. Frequently utilized examples are contact interactions from the next-to-leading chiral Lagrangian <cit.>, full (including u and t-channel loops) chiral amplitude of the next-to-leading order <cit.>, or phenomenological Chew-Mandelstam forms <cit.>. The most general form of the expansion of such a parametrization in powers of M, for fixed s≠0, reads K_l^-1(s) =ω_l^(0)(s)+ ω_l^(1)(s)M^2+ ω_l^(2)(s)M^2log(M^2/μ^2) + 𝒪(M^4log M) .The scale dependence may partly cancel with ω_l^1(s) and the first term of Eq. (<ref>). The following condition,ω_l^(2)(s)!=-(8π^2s)^-1 ,ensures the exact cancellation of the "forbidden logarithms" in Eq. (<ref>).An alternative approach, not limited to models of the form of Eq. (<ref>), is given as follows. Insert an ansatz s_H=(m_H-iΓ_H/2)^2, with m_H(M) =∘m_H + c_1m^HM^2 + d_m^HM^2log M^2 +𝒪(M^3) , Γ_H(M) =∘Γ_H + c_1Γ^HM^2 + d_Γ^HM^2log M^2 +𝒪(M^3)for the resonance pole position s_l^* in Eq. (<ref>). The resulting equation must hold separately in every order in M, since it is nothing than the resonance pole condition at any given quark mass. Now expand this resulting equation in M, truncate the expansion after the terms quadratic in M, and solve (e.g. numerically) the obtained set of equations (two for each order M^0,M^2,M^2log M^2) for the unknowns ∘m_H,∘Γ_H,c_1m^H,c_1Γ^H,d_m^H,d_Γ^H. This procedure was also employed in <cit.> for the propagator of the σ resonance. Should the solution return non-vanishing d_m^H,d_Γ^H (the coefficients of the “forbidden logarithms” in s_H), the assumed model for the partial-wave amplitudes is in conflict with the strictures of chiral symmetry encoded in Eq. (<ref>). In all cases examined below, we find that both versions of the test for "forbidden logarithms" are equivalent, i.e.d_m^H,d_Γ^H≠0 ⟺ω_l^(2)(s)≠-(8π^2s)^-1 .Thus, using one or another approach for the test might be a matter of technical advantages. However, when any of those fails, the model should not be used to predict the quark-mass variation of the resonance parameters, even if it describes the experimentally measured energy-dependence of the ππ scattering process reasonably well. This is the main point we want to make in this contribution.As a first explicit demonstration let us adopt a “unitarized Weinberg term”, which leads to K_0^I=0=2s-M_π^2/2F_π^2 , K_1^I=1=s-4M_π^2/6F_π^2 for the channels of different isospin I.Taking into account the known quark-mass dependencies of F_π and M_π <cit.>, one notes that only the isoscalar s-wave described by this model fulfills the condition (<ref>), while the other partial waves violate this condition, even though these amplitudes are in accord with chiral symmetry on tree level. The pertaining resonance poles can therefore not be expected to vary as prescribed by Eq. (<ref>) and its analogue for the width Γ_H.Another typical example, frequently used for chiral extrapolations, is the Bethe-Salpeter-like approach with driving term from a local chiral potential, see e.g. <cit.>. As it is used for the analysis of the isovector p-wave amplitude <cit.>, the expression for K_1(s) in Eq. (<ref>) readsK_1^ BSE(s)= 32π q(s)^2/48π(F^2-8M_π^2l̂_1+4sl̂_2)+2q(s)^2(a(μ)+1) ,where a(μ) is a real-valued subtraction constant, and l̂_i are some linear combinations of SU(3) low-energy constants (see e.g. App. B of <cit.>). Similar to the result for K_1^I=1(s) in the first model studied above, we find that Eq. (<ref>) is not obeyed here (whether one takes into account the running of F_π=F+𝒪(M^2log M) with the quark mass or not), and that the ρ mass and width in this model show “forbidden logarithms”. We conclude that this kind of models can be useful when they are applied at a fixed pion mass, with their free parameters fitted at each pion mass data point separately, since the energy-dependence in the low-energy region is expected to be described reasonably well by such models. However, the quark-mass dependence of the resonance position close to the chiral limit is incompatible with the one demanded by chiral symmetry as verified by the proposed test. Therefore, ambiguities will arise when such models are used for the purpose of chiral extrapolation and the corresponding uncertainty estimates.Finally, we consider the Inverse Amplitude Method (IAM) in the one channel version <cit.>, see also <cit.>. It can be re-written in the form of Eq. (<ref>), such thatK_l^ IAM(s)= (16π t_l^(2)(s))^2/(16π t_l^(2)(s))-(16π t̃_l^ (4))(s) ,where t̃_l^ (4)(s) := t_l^(4)(s) + 16π(t_l^(2)(s))^2I(s), and t_l^(n) is the partial-wave scattering amplitude of the n^ th chiral order. As we have already anticipated in the previous section that there are no “forbidden logarithms” in t_l^(4)(s), it is evident that condition (<ref>) is fulfilled here. Thus, the quark-mass variation of the resonance position agrees with Eq. (<ref>) with c_2^H=0. In that respect,the use of the IAM for the purpose of studying the quark-mass dependence of resonance properties in a non-perturbative framework ispreferable. Even though it disagrees with ChPT amplitudes above a certain chiral order, the fact that the IAM uses only well-behaved, complete chiral amplitudes of a fixed order as building blocks turns out as an advantage over other “unitarization procedures”.There are two additional remarks we wish to make. First, some resonances become stable at high pion masses. In this regime, the quark-mass variation of the resonance position could still be described satisfyingly, since the unitarity-loop effects dominate over the variation of the quark-mass logarithms there, see e.g. <cit.> for a discussion of such effects. Second, in the case of the ρ resonance, there is an additional difficulty due to the ρ→πω vertex. The ω mass is very close to the ρ mass (while the difference in the widths is 𝒪(M_π^phys)), and the quark mass expansion around the ρ pole in the chiral limit might have a radius of convergence smaller than the physical pion mass. If one takes the ρ and ω to be mass-degenerate in the chiral limit (as is e.g. done in <cit.>) to avoid this problem, one arrives at an exceptional case c_2^ρ≠0 in Eq. (<ref>). It is hard to see how such a behavior could be accounted for in a simple unitarized model for the I=1 ππ scattering amplitude. For the σ, however, there is no such nearly mass-degenerate state σ' with a σ→πσ' vertex, so in this case the use of such models is justified, given the model in question satisfies the constraint of Eq. (<ref>).Concluding, we propose a simple test for the scattering amplitude parametrizations used for chiral extrapolations of the isovector and isoscalar ππ resonances. It is formulated as a model-independent condition for amplitude parametrizations of a rather general form, consistent with elastic unitarity. This condition is an implication of the chiral symmetry breaking pattern of QCD, implemented in ChPT, and ensures the correct form of the leading quark-mass correction to the resonance pole positions.We have tested two models frequently used for chiral extrapolations of the σ and ρ resonance poles, and found that only one is consistent with this condition.To select those parametrizations which pass our proposed tests will clearly reduce the model-dependence afflicting the extraction of those resonance properties from results of lattice QCD simulations.§ ACKNOWLEDGMENTSMM thanks the German Research Foundation (DFG, MA 7156/1) for the financial support as well as M. Döring and The George Washington University for hospitality and inspiring environment during this fellowship. The work of PCB was supported by the Deutsche Forschungsgemeinschaft, SFB/Transregio 55.99 Colangelo:2001dfG. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B 603, 125 (2001) [hep-ph/0103088]. Lang:2011mn C. B. 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"authors": [
"Peter C. Bruns",
"Maxim Mai"
],
"categories": [
"hep-lat",
"nucl-th"
],
"primary_category": "hep-lat",
"published": "20170727181318",
"title": "Chiral symmetry constraints on resonant amplitudes"
} |
empty A&A Program in Structures [0.1in]William E. Boeing Department of Aeronautics and Astronautics [0.1in]University of Washington [0.1in]Seattle, Washington 98195, USA2mm0.5in A Study on the Fracturing Behavior of Thermoset Polymer Nanocomposites Yao Qiao, Cory Hage Mefford, Marco Salviato INTERNAL REPORT No. 17-07/02ESubmitted to Composites Science and Technology July 2017[cor1]Corresponding Author, [email protected] address]Yao Qiao address]Cory Hage Mefford address]Marco Salviatocor1[address]William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA 1This work proposes an investigation on the fracturing behavior of polymer nanocomposites. Towards this end, the study leverages on the analysis of a large bulk of fracture tests from the literature with the goal of critically investigating the effects of the nonlinear Fracture Process Zone (FPZ). It is shown that for most of the fracture tests, the effects of the nonlinear FPZ are not negligible, leading to significant deviations from Linear Elastic Fracture Mechanics (LEFM). As the data indicate, this aspect needs to be taken into serious consideration since the use of LEFM to estimate mode I fracture energy, which is common practice in the literature, can lead to an error as high as 157% depending on the specimen size and nanofiller content. B. Fracture Size effect A. Nano composites C. Crack C. Damage Mechanics B. Non-linear behavior § INTRODUCTION The outstanding advances in polymer nanocomposites in recent years have paved the way for their broad use in engineering. Potential applications of these materials include microelectronics <cit.>, energy storage <cit.> and harvesting <cit.>, soft robotics <cit.>, and bioengineering <cit.>. One of the reasons of this success is that, along with remarkable enhancements of physical properties such as e.g. electric and thermal conductivity <cit.>, nanomodification offers significant improvements of stiffness <cit.>, strength <cit.> and toughness <cit.>. These aspects make it an excellent technology to enhance the mechanical behavior of polymers <cit.> or to improve the weak matrix-dominated properties of fiber composites <cit.>.While a large bulk of data on the mechanical properties of polymer nanocomposites is available already, an aspect often overlooked is the effect on the fracturing behavior of the region close to the crack tip featuring most of energy dissipation, the Fracture Process Zone (FPZ). This is an important aspect since, due to the complex mesostructure characterizing nanocomposites, the size of the non-linear FPZ occurring in the presence of a large stress-free crack is usually not negligible <cit.> leading to a significant deviation from the typical brittle behavior of thermoset polymers. This phenomenon cannot be captured by classical Linear Elastic Fracture Mechanics (LEFM) which inherently assumes the size of the FPZ to be negligible compared to the structure size. To seize the effects of a finite, non-negligible FPZ, the introduction of a characteristic (finite) length scale related to the fracture energy and the strength of the material is necessary <cit.>.This work proposes an investigation on the fracturing behavior of thermoset polymer nanocomposites with the goal of critically investigating the effects of the nonlinear Fracture Process Zone (FPZ). By employing Size Effect Law (SEL), a formulation endowed with a characteristic length inherently related to the FPZ size, and assuming a linear cohesive behavior <cit.>, a large bulk of literature data is analyzed. It is shown that for most of the fracture tests, the nonlinear behavior of the FPZ is not negligible, leading to significant deviations from LEFM. As the data indicate, this aspect needs to be taken into serious consideration since the use of LEFM to estimate mode I fracture energy can lead to an error as high as 157% depending on the specimen size and nanofiller content.§ QUASI-BRITTLE FRACTURE OF NANOCOMPOSITESIn nanocomposites, the size of the non-linear Fracture Process Zone (FPZ) occurring in the presence of a large stress-free crack is generally not negligible. The stress field along the FPZ is nonuniform and decreases with crack opening, due to a number of damage mechanisms such as e.g. discontinuous cracking, micro-crack deflection, plastic yielding of nanovoids, shear banding and micro-crack pinning <cit.>. As a consequence, the fracturing behavior and, most importantly, the energetic size effect associated with the given structural geometry, cannot be described by means of classical Linear Elastic Fracture Mechanics (LEFM) which assumes the effects of the FPZ to be negligible. To capture the effects of a finite, non-negligible FPZ, the introduction of a characteristic (finite) length scale related to the fracture energy and the strength of the material is necessary <cit.>. This is done in the following sections. §.§ Size effect law for nanocomposites The fracture process in nanocomposites can be analyzed leveraging on an equivalent linear elastic fracture mechanics approach to account for the presence of a FPZ of finite size as shown in Fig. <ref>. To this end, an effective crack length a=a_0+c_f with a_0= initial crack length and c_f= effective FPZ length is considered. Following LEFM, the energy release rate can be written as follows:G(α)=σ_N^2D/E^*g(α)where α=a/D= normalized effective crack length, E^*= E for plane stress and E^*= E/(1-ν^2) for plane strain, g(α)= dimensionless energy release rate and, D is represented in Fig. <ref> for Single Edge Notch Bending (SENB) and Compact Tension (CT) specimens respectively. σ_N represents the nominal stress defined as e.g. σ_N=3PL/2tD^2 for SENB specimens or σ_N=P/tD for CT specimens where, following Fig. <ref>, P is the applied load, t is the thickness and L is the span between the two supports for a SENB specimen as defined in ASTM D5045-99 <cit.>. At incipient crack onset, the energy release rate ought to be equal to the fracture energy of the material. Accordingly, the failure condition can now be written as:G(α_0+c_f/D)=σ_Nc^2D/E^*g(α_0+c_f/D)=G_fwhere G_f is the mode I fracture energy of the material and c_f is the effective FPZ length, both assumed to be material properties. It should be remarked that this equation characterizes the peak load conditions if g'(α)>0, i.e. only if the structure has positive geometry <cit.>.By approximating g(α) with its Taylor series expansion at α_0 and retaining only up to the linear term of the expansion, one obtains:G_f=σ_Nc^2D/E^*[g(α_0)+c_f/Dg'(α_0)]which can be rearranged as follows <cit.>:σ_Nc=√(E^*G_f/Dg(α_0)+c_fg'(α_0))where g'(α_0)=g(α_0)/α.This equation, known as Bažant's Size Effect Law (SEL) <cit.>, relates the nominal strength to mode I fracture energy, a characteristic size of the structure, D, and to a characteristic length of the material, c_f, and it can be rewritten in the following form:σ_Nc=σ_0/√(1+D/D_0)with σ_0=√(E^*G_f/c_fg'(α_0)) and D_0=c_fg'(α_0)/g(α_0)= constant, depending on both FPZ size and specimen geometry. Contrary to classical LEFM, Eq. (<ref>) is endowed with a characteristic length scale D_0. This is key to describe the transition from ductile to brittle behavior with increasing structure size. §.§ Calculation of g(α) and g'(α)§.§.§ Single Edge Notch Bending (SENB) specimens The calculation of g(α) and g'(α) for SENB specimens can be done according to the procedure described in <cit.>. This leads to the following polynomial expressions:g(α)=1155.4α^5-1896.7α^4+1238.2α^3-383.04α^2+58.55α-3.0796 g'(α)=18909α^5-31733α^4+20788α^3-6461.5α^2+955.06α-50.88 §.§.§ Compact Tension (CT) specimens In the case of CT specimens, the values for g(α) and g'(α) can be determined leveraging on the equations provided by ASTM D5045-99 <cit.>. Following the standard, the mode I Stress Intensity Factor (SIF), K_I, can be written as:K_I=P/t√(D)f(α)where α=a/D and D is the distance between the center of hole to the end of the specimen as defined in ASTM D5045-99 <cit.> (see Fig. <ref>b). The nominal stress σ_N for CT specimens can be defined as:σ_N=P/tD The mode I Stress Intensity Factor can be rewritten as follows by combining Eq. (<ref>) and Eq. (<ref>):K_I=√(D)σ_Nf(α)By considering the relationship between energy release rate and stress intensity factor for a plane strain condition, the mode I energy release rate results into the following expression:G_I=Dσ^2_N/E^*g(α) where g(α)=f^2(α)(1-υ^2), and f(α) is a dimensionless function accounting for geometrical effects and the finiteness of the structure (see e.g. <cit.>). Once g(α) is derived, the expression of g'(α) can be obtained by differentiation leading to the following polynomial expressions for g(α) and g'(α) respectively:g(α)=33325α^5-52330α^4+32016α^3-9019.1α^2+1230.1α-51.944 g'(α)=555868α^5-895197α^4+554047α^3-159153α^2+21035α-917.3 § FRACTURE BEHAVIOR OF THERMOSET NANOCOMPOSITES: ANALYSIS AND DISCUSSIONIn the following sections, several data on the fracturing behavior of nanocomposites are critically analyzed employing the expressions derived in Section <ref>. First, some recent tests on geometrically-scaled SENB specimens made of a thermoset polymer reinforced by graphene are reviewed to investigate how the FPZ affects the failure behavior. Then, leveraging on SEL and assuming a linear cohesive behavior, a large bulk of data from the literature originally elaborated by LEFM is re-analyzed to include the effects of the FPZ. §.§ Fracture Scaling of Graphene NanocompositesTo investigate the effects of the non-linear FPZ, it is useful to review some recent evidences on the scaling of the fracturing behavior. To this end, the fracture tests on geometrically-scaled SENB specimens reported by Mefford et al. <cit.>, who studied a thermoset polymer reinforced by graphene nanoplatelets, are analyzed and discussed. Figures <ref>a-d show the experimental structural strength σ_Nc and the fitting by SEL plotted as a function of the structure size D in double logarithmic scale. In such a graph, the structural scaling predicted by LEFM is represented by a line of slope -1/2 whereas the case of no scaling, as predicted by stress-based failure criteria, is represented by a horizontal line. The intersection between the LEFM asymptote, typical of brittle behavior, and the plastic asymptote, typical of ductile behavior, corresponds to D=D_0, called the transitional size <cit.>. The figure reports the size effect tests for various graphene contents, from the case of a pristine polymer to wt%=1.6. As can be noted from Figure <ref>a, the experimental data related to the pure epoxy system all lie very close to the LEFM asymptote showing that, for the range of sizes investigated (or larger sizes), linear elastic fracture mechanics provides a very accurate description of fracture scaling. This shows that, for the pure epoxy and sufficiently large specimens, the FPZ size has a negligible effect and LEFM can be applied, as suggested by ASTM D5045-99 <cit.>. However, this is not the case for graphene nanocomposites which, as Figures <ref>b-d show, are characterized by a significant deviation from LEFM, the deviation being more pronounced for smaller sizes and higher graphene concentrations. In particular, the figures show a transition of the experimental data from stress-driven failure, characterized by the horizontal asymptote, to energy driven fracture characterized by the -1/2 asymptote. This phenomenon can be ascribed to the increased size of the FPZ compared to the structure size which makes the non-linear effects caused by micro-damage in front of the crack tip not negligible. For sufficiently small specimens, the FPZ affects the structural behavior and causes a significant deviation from the scaling predicted by LEFM with a much milder effect of the size on the structural strength. On the other hand, for increasing sizes, the effects of the FPZ become less and less significant thus leading to a stronger size effect closely captured by LEFM. Further, comparing the size effect plots of nanocomposites with different graphene concentrations, it can be noted a gradual shift towards the ductile region thus showing that not only the addition of graphene leads to a higher fracture toughness but also to a gradually more ductile structural behavior for a given size. As the experimental data show, LEFM does not always provide an accurate method to extrapolate the structural strength of larger structures from lab tests on small-scale specimens, especially if the size of the specimens belonged to the transitional zone. In fact, the use of LEFM in such cases may lead to a significant underestimation of structural strength, thus hindering the full exploitation of graphene nanocomposite fracture properties. This is a severe limitation in several engineering applications such as e.g. aerospace or aeronautics for which structural performance optimization is of utmost importance. On the other hand, LEFM always overestimates significantly the strength when used to predict the structural performance at smaller length-scales. This is a serious issue for the design of e.g. graphene-based MEMS and small electronic components or nanomodified carbon fiber composites in which the inter-fiber distance occupied by the resin is only a few micrometers and it is comparable to the FPZ size. In such cases, SEL or other material models characterized by a characteristic length scale ought to be used. §.§ Effects of a finite FPZ on the calculation of Mode I fracture energy Notwithstanding the importance of understanding the scaling of the fracturing behavior, the tests conducted by Mefford et al. <cit.> represent, to the best of the authors' knowledge, the only comprehensive investigation on the size effect in nanocomposites available to date. All the fracture tests reported in the literature were conducted on one size and analyzed by means of LEFM. Considering the remarkable effects of the nonlinear FPZ on the fracturing behavior documented in the foregoing section, it is interesting to critically re-analyze the fracture tests available in the literature by means of SEL. This formulation is endowed with a characteristic length related to the FPZ size and, different from LEFM, it has been shown to accurately capture the transition from brittle to quasi-ductile behavior of nanocomposites.§.§.§ Application of SEL to thermoset polymer nanocomposites To understand if the quasi-brittle behavior reported in previous tests <cit.> is a salient feature of graphene nanocomposites only or if it characterizes other nanocomposites, a large bulk of literature data were re-analyzed by SEL using Eq.(<ref>) in order to study the effects of the FPZ. In this analysis, in the absence of data on the effective FPZ length, c_f, in the literature, it is assumed that c_f=0.44 l_ch which, according to Cusatis et al. <cit.>, corresponds to the assumption of a linear cohesive law. In this expression, l_ch=E^*G_f/f_t^2 is Irwin's characteristic length which depends on Young's modulus E^*, the mode I fracture energy G_f and the ultimate strength of the material f_t. Substituting this expression into Eq. (<ref>) and rearranging, one gets the following expression which relates the fracture energy calculated according to SEL to the fracture energy calculated by LEFM: G_f,SEL=G_f,LEFM/1-0.44 E^*g'(α_0)G_f,LEFM/Df_t^2 g(α_0)In this equation,G_f,LEFM=σ_Nc^2 Dg(α_0)/E^* represents the fracture energy which can be estimated by analyzig the fracture tests by LEFM (note that this expression lacks of a characteristic length scale). It can be observed from Eq.(<ref>) that, once g(α) and g^'(α) are calculated, the fracture energy corrected for the effects of the FPZ can be calculated by knowing three key parameters: (1) the fracture energy estimated by LEFM, (2) the Young's modulus of the specimen, and (3) the ultimate strength of the specimens at different nanofiller concentrations. For cases in which those parameters were not provided by the authors, the ultimate strength, Young's modulus, and Poisson's ratio were reasonably assumed to be 50 MPa, 3000 MPa, and 0.35 respectively. §.§.§ Mode I fracture energy of thermoset polymer nanocompositesSeveral types of nanofillers were investigated in this re-analysis including carbon-based nano-fillers (such as carbon black, graphene oxide, graphene nanoplatelets, and multi-wall carbon nanotubes), rubber and silica nanoparticles, and nanoclay. The fracture energy estimated from LEFM compared to the calculation through SEL, Eq. (<ref>), for nanomodified SENB and CT specimens are plotted in Figures <ref>-<ref> along with the highest difference.Figure <ref> shows data elaborated from Carolan et al. <cit.> who conducted fracture tests on SENB specimens nano-modified by six different combinations of nanofillers. As can be noted, while for the pristine polymer the difference between LEFM and SEL is negligible, this is not the case for the nanomodified polymers, the difference increasing with increasing nanofiller content. The difference varies based on the type of nanofiller used, with the greatest value being 42.6% for the addition of 8 wt% core shell rubber mixed with 25% diluent and 8% silica. This remarkable discrepancy confirms that for the SENB specimens tested in <cit.> the nonlinear behavior of the FPZ was not negligible, leading to a more ductile behavior compared to the pristine polymer. Similar conclusions can be drawn based on Figures <ref>a-f which report the analysis of fracture tests conducted by Zamanian et al. <cit.> and Jiang et al. <cit.> on polymers reinforced by silica nanoparticles and silica nanoparticle+graphene oxide respectively. For the data in <cit.>, the greatest percent difference of fracture energy between LEFM and SEL decreased as the size of silica nanoparticle increased, with the greatest difference being 28% for the addition of 6 wt% 12 nm silica nanoparticles. For all the systems investigated, the maximum deviation from LEFM was for the largest amount of nanofiller, confirming that nanomodification lead to larger FPZ sizes and more pronounced ductility. On the other hand, the data by Jiang et al. <cit.> exhibit an even larger effect of the FPZ with the greatest difference in fracture energy between LEFM and SEL reaching up to 51.8% for silica nanoparticle attached to graphene oxide. A milder effect of the FPZ can be inferred from the data by Chandrasekaran et al. <cit.> who investigated three types of carbon-based nano-fillers (Figure <ref>): (1) thermally reduced graphene oxide; (2) graphene nanoplatelets; and (3) multi-wall carbon nanotubes. In these cases, the difference between SEL and LEFM ranges from 4.9% to 8.8%, the lowest difference among all the data analyzed in this study. For these systems, the specimen size compared to the size of the nonlinear FPZ was large enough to justify the use of LEFM which provided accurate and objective results. On the other hand, a more significant effect of the FPZ can be inferred from the data reported by Konnola et al. <cit.> who studied three different types of functionalized and nonfunctionalized nano-fillers. In this case, the greatest difference in fracture energy ranges between 15.2% to 20.3%.SENB specimens nano-modified by nanoclay and carbon black respectively were tested by Kim et al. <cit.>. As Figure <ref> shows, in this case, the specimen size was enough to justify the use of LEFM as confirmed by the low difference with SEL (11.2% for nanoclay and 7.3% for carbon black). Similar conclusions can be drawn on the silica nanoparticles investigated by Vaziri et al. <cit.>. However, for the three different sizes of silica nanoparticles investigated by Dittanet et al. <cit.>, a significant difference between LEFM and SEL was observed, confirming that these specimens tested belonged to the transition zone between ductile and brittle behavior where the effects of the nonlinear FPZ cannot be neglected.Figure <ref> shows a re-analysis of the data reported by Liu et al. <cit.> who tested CT specimens nano-modified by four different combinations of silica nanoparticle and rubber. As can be noted, in this case, the FPZ indeed affects the fracturing behavior significantly. Adopting LEFM, which assumes the size of the FPZ to be negligible, for the estimation of G_f from the fracture tests would lead to an underestimation of up to 156.8% for the case of polymer reinforced by 15 wt% rubber only. This tremendous difference, the largest found in the present study, gives a tangible idea on the importance of accounting for the nonlinear damage phenomena occurring in nanocomposites which can lead to a significant deviation from the typical brittle behavior of thermoset polymers.§ CONCLUSIONSLeveraging on a large bulk of literature data, this paper investigated the effects of the Fracture Process Zone (FPZ) on the fracturing behavior of thermoset polymer nanocomposites, an aspect of utmost importance for structural design but so far overlooked. Based on the results obtained in this study, the following conclusions can be elaborated:1. The fracture scaling of pure thermoset polymers is captured accurately by Linear Elastic Fracture Mechanics (LEFM). However, this is not the case for nanocomposites which exhibit a more complicated scaling. The double logarithmic plots of the nominal strength as a function of the characteristic size of geometrically-scaled SENB specimens <cit.> showed that the fracturing behavior evolves from ductile to brittle with increasing sizes. For sufficiently large specimens, the data tend to the classical -1/2 asymptote predicted by LEFM. However, for smaller sizes, a significant deviation from LEFM was reported with data exhibiting a milder scaling, a behavior associated to a more pronounced ductility. This trend was more and more pronounced for increasing nanofiller contents;2. Following Bažant <cit.>, an Equivalent Fracture Mechanics approach can be used to introduce a characteristic length, c_f, into the formulation. This length is related to the FPZ size and it is considered a material property as well as G_f. The resulting scaling equation, known as Bažant's Size Effect Law (SEL), depends not only on G_f but also on the FPZ size. An excellent agreement with experimental data is shown, with SEL capturing the transition from quasi-ductile to brittle behavior with increasing sizes. 3. By employing Size Effect Law and assuming a linear cohesive behavior <cit.>, a large bulk of literature data on the mode I fracture energy of thermoset nanocomposites was critically re-analyzed. It is shown that for most of the fracture tests in the literature, the effects of the nonlinear FPZ are not negligible, leading to significant deviations from LEFM. As the data indicate, this aspect needs to be taken into serious consideration since the use of LEFM to estimate mode I fracture energy can lead to an error as high as 156% depending on the specimen size and nanofiller content.4. The deviation from LEFM reported in the re-analyzed results is related to the size of the Fracture Process Zone (FPZ) for increasing contents of nanofiller. In the pristine polymer the damage/fracture zone close to the crack tip, characterized by significant non-linearity due to subcritical damaging, was generally very small compared to the specimen sizes investigated. This was in agreement with the inherent assumption of LEFM of negligible non-linear effects during the fracturing process. However, the addition of nano-fillers results in larger and larger FPZs. For sufficiently small specimens, the size of the highly non-linear FPZ was not negligible compared to the specimen characteristic size thus highly affecting the fracturing behavior, this resulting into a significant deviation from LEFM; 5. The foregoing evidences show that particular care should be devoted to the fracture characterization of nanocomposites. LEFM, which inherently assumes the FPZ to correspond to a mathematical point, can only be used to estimate mode I fracture energy when the specimen size is large enough compared to the FPZ. 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"authors": [
"Yao Qiao",
"Cory Hage Mefford",
"Marco Salviato"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170726223756",
"title": "A Study on the Fracturing Behavior of Thermoset Polymer Nanocomposites"
} |
[email protected] RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, JapanHigh Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, The NetherlandsDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), The University of Tokyo, Tokyo 113-8656, JapanDepartment of Applied Physics and Quantum-Phase Electronics Center (QPEC), The University of Tokyo, Tokyo 113-8656, JapanHigh Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, The NetherlandsHigh Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, The NetherlandsRIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Applied Physics and Quantum-Phase Electronics Center (QPEC), The University of Tokyo, Tokyo 113-8656, JapanInteractions between the constituents of a condensed matter system can drive it through a plethora of different phases due to many-body effects.A prominent platform for this type of behavior is a two-dimensional electron system in a magnetic field, which evolves intricately through various gaseous, liquid and solids phases governed by Coulomb interaction <cit.>. Here we report on the experimental observation of a phase transition between the Laughlin liquid of composite fermions and the adjacent insulating phase of a magnetic field-induced Wigner solid <cit.>.The experiments are performed in the lowest Landau level of a MgZnO/ZnO two-dimensional electron system with attributes of both a liquid and a solid <cit.>.An in-plane magnetic field component applied on top of the perpendicular magnetic field extends the Wigner phase further into the liquid phase region.Our observations indicate the direct competition between a Wigner solid and a Laughlin liquid both formed by composite particles rather than bare electrons.Phase transition from a composite fermion liquid to a Wigner solid in the lowest Landau level of ZnO M. Kawasaki December 30, 2023 ====================================================================================================A magnetic field B applied perpendicularly to a two-dimensionalsystem modifies its density of states by arranging the charge carriers in discrete Landau levels (LLs).Additionally, the Coulomb interaction acting on the magnetic length scale =√(ħ/e B) can be tuned by the magnetic field, causing the high mobility carriers to evolve through various correlated phases <cit.> , see Fig. <ref>.In the lowest LL the Wigner phase competes with the liquid phase and manifests as a large magnetoresistance peakaround or below ν=1/3 <cit.>. Thus the electron system, when moving from ν=1/2 to lower ν's, has to undergo a reorganization of the ground state between the picture ofcomposite fermions describing the magnetoresistance oscillations around ν=1/2 and the bare electrons forming the Wigner solid at smaller ν's. An alternative concept for such a regime is the realization of a Wigner solid formed by the composite fermions(Fig. <ref>, top right) - an idea that was put forward in a number of theoretical works <cit.>.Recent experiments focusing on GaAs-based 2DES have been gradually accumulating evidences pointing towards the realization of CF Wigner solid <cit.>. Here, we study the magnetotransport in a ZnO heterostructure (see Methods)in the magnetic field region between the CF liquid phase formed at ν=1/2 and the insulating phase appearing at higher field, associated with the Wigner phase. The mangetotransport features in this region exhibit a character of both CF liquid and Wigner solid. The presence of such a regionwith this interlaced character has a number of plausible explanations, one of which isthe presence of the Wigner phase formed by composite fermions.Figure <ref> shows a full scan of the magnetotransport from 0 T to 33 T applied perpendicular to the 2DES plane. Several fractional quantum Hall states are observed around ν = 3/2, consistent withprevious results <cit.> and, in addition, developing minima are observed at ν =9/5, 12/7, 9/7, and 6/5. Furthermore, up to six fractional quantum Hall states are observed on both sides of ν = 1/2. Close inspection of the transport around ν = 1/2 reveals a distinct asymmetry;maxima between fractional quantum Hall states for ν < 1/2 are much larger than those for ν > 1/2. This increase becomes increasingly dramatic between ν = 2/5, 1/3 and 2/7.Such a high resistance phase between ν=1/3 and 2/7 has also been observed in GaAs,and was interpreted as the electron Wigner solid pinned by disorder. Two mechanisms have been identified for the appearance of the Wigner solid around these filling factors: one is the Landau level mixing, which modifies the ground state energies of fractional quantum Hall states and Wigner solid <cit.>; the other is short range disorder <cit.>.Both mechanisms are distinctively more pronounced in ZnO-heterostructures than in GaAs <cit.>, and the ZnO system is, therefore, ideal to access the competition between liquid and solid phases in the fractional quantum Hall regime.In Fig. <ref> the distinct high resistance phases are colored and markedasIP1, IP2 and IP3. On the basis of Wigner solid studies in other materials system, the characteristics of IP1, IP2 and IP3 are typical of the Wigner solid. The temperature dependence of IP1, IP2 and IP3 resembles the melting of the Wigner solid (Supplementary Fig. 1a). The non-linear current-voltage characteristics are associated with the depinning of the Wigner solid from the disorder,when a certain threshold force is exceeded, and its subsequent sliding along the disorder landscape (Supplementary Fig.1b). Thus, the trace of a Wigner solid appearsalready in IP1 between ν=3/7 and ν=2/5,whereas a largerandI-V non-linearity at IP2 and IP3 indicate an even more pronounced Wigner solid.IP1 represents an interesting region. While it shows the features of an emerging Wigner solid, it is at the same time a part oftheoscillations caused by the composite fermions' orbital motion in , and therefore can also be attributed to the liquid phase. Therefore, we now analyze the CF massaround ν=1/2 from the temperature dependence ofthe oscillation amplitude by using the Lifshitz-Kosevitch formalism (Supplementary Information).Figure <ref>b displaysaround ν=1/2, which extends the linear dependence ofon B to higher field <cit.>.More noticeable is the excessive increase ofover the linear trend when the2DES approaches the insulating phase IP1. The mass increase can be interpreted as a signature of the underlying particles becoming more inert due tothe formation of a solid phase. This is then in agreement with observing the traces of the Wigner solid character at IP1, and more strongly pronounced Wigner phases IP2 and IP3 at higher field. Since we cannot assume the coexistence of electrons and composite fermions, as it would require a simultaneous existence of two gauge fields, the dual character of IP1 and IP2 has to be consolidated within the model frame based on either electrons or composite fermions. Because of the multiple experimental evidences for the validity of CF picture <cit.>, it would be more natural to treat the transport anomaly in ZnO around ν=1/2 with the composite fermion as an underlying particle for both liquid and solid phases. Another factor favoring the Wigner solid formation from composite fermions is the residual interaction among composite fermions. Indeed, the logarithmic temperature dependence ofat ν=1/2 shown in Fig. <ref>c points towards a residual CF interaction <cit.>.The transport properties discussed above change dramatically when the sample is rotated in the magnetic field, that is, when an additional field component is applied parallel to the 2DES.Since the electron spin susceptibility for this structure is about 2, the opposite spin orientation branch of the lowest LL lies energetically high and is not populated. Thus the spin effects are not anticipated to play a role for the discussion below.Figure <ref>a depictstraces at several sample orientations θ obtained atbase temperature and shows the asymmetrical impact of the in-plane field on the transport for ν<1/2 and ν>1/2 (θ is the tilt angle between the normal of the 2DES plane and the magnetic field direction). Firstly, one notices thatofIP1, IP2 and IP3 increases gradually with an increasing θ, whileminima at ν= 3/7, 2/5 and 1/3 do not change significantly. Thus the Wigner phase becomes more pronounced by applying an in-plane field. The temperature dependence of the insulating phases is depicted for three representative θs in Fig. <ref>b. Furthermore, around ν=1/2 gains a background, which becomes larger with the increasing θ. Since magnetotransport experiments in GaAs demonstrate the extension of the tail of the insulating phase into the ν=1/2 region with an increasing θ <cit.>, we may also suppose that the background forming around ν=1/2 has the same origin and is associated with the insulating phase shifting towards ν=1/2 and above.It is noteworthy that theoscillations are not damped but rather persist on top of the background. We analyze the temperature dependence of theoscillation amplitude and estimatearound ν=1/2 for several θs (Supplementary Information). The inset of Fig. <ref> depicts the result of this analysis.For ν>1/2,does not show any noticeable change, but it shows a pronounced field and tilt angle dependence for ν<1/2, that is, for a given perpendicular magnetic field is heavier at a larger tilt angle. The mass increase serves as a sign of the CFs becoming more strongly localized. This is consistent with the growing insulating character of IP1 and the shift of the Wigner phase towards higher ν's withincreasing θ.Finally, the enhanced CF interaction with an added in-plane field also becomes apparent at ν=1/2: Figure <ref> presents the temperature dependenceat several θ's and shows that the slope of the logarithmic temperature dependence increases with θ. The slope at each θ reflects not only the CF residual interaction but also the melting of Wigner phase penetrating to higher filling factors with increasing θ. In order to further address this in-plane field induced stabilization of the Wigner solid we now analyse how much the in-plane field squeezes the electron wave function, as it effectively enhances the Coulomb interaction and can affect the transport properties <cit.>. In zero in-plane field the wave function of the heterostructure is about 10 nm wide.At θ=50^∘ it is squeezed down to 2.6 nm at B_=12 T, representing the region ν>1/2, and down to 2.2 nm at B_=17 T, representing the region ν<1/2 (Supplementary Information).Since the wave function width reduces significantly with in-plane field on both sides of ν=1/2 compared with zero in-plane field, the Coulomb interactionshould also be equally enhanced around ν=1/2.Nonetheless,defined by the interaction effects remains almost unchangedfor ν>1/2 and no large effect of in-plane field on transport characteristics is seen in this region.Consequently, the increase offor ν<1/2 is not mainly caused by the reduced wave function width.Rather, it supports our hypothesis of a Wigner solid formation. Since the solid phase gains over the liquid phase upon the application of the in-plane field,increasein ν<1/2 region reflects an effective localization of the composite fermions. The origin for the asymmetrical response of liquid and solid phases to the in-plane field remains an open question, but our experimental result can likely be the precursor for the new insulating state proposed by Piot et al. <cit.>.Our experimental data show that the electron system enters an unconventional correlation regime, which reflects the character of both solid and liquid phases for ν<1/2.One interpretation for such regime can be the formation (melting) of the Wigner solid upon increasing (decreasing) the magnetic field, where a composite fermion would be the underlying particle in both phases. In such a regime, the particles can form a hexatic phase characterized by bond-oriented nearest-neighbor ordering, while the phase transition obeys the Kosterlitz-Thouless model <cit.>. In another scenario, a transition between the liquid and solid phase in a two-dimensional system can be accompanied by the appearance ofintermediate phases, such as microemulsion phases associated with liquid crystalline phases <cit.>. It appears unlikely that the magnetic field regime with dual character can be modeled by assuming the co-existence ofelectrons and composite fermions, as it would then require a simultaneous existence of two gauge fields. Our experimental results are interpreted within the composite fermions approach, which has recently attracted renewed attention fromtheory predicting that the composite fermions can be Dirac particles <cit.>. This also introduces an exciting perspective for ZnO studies. Our experimental results sheds the light on the composite fermion paradigm in a system distinct from conventional semiconductors and also on how the charge carrier system translates between liquid and solid phases. MethodsSampleThe sample under study is a MgZnO/ZnO heterostructure with a charge carrier density n=1.7×10^11 cm^-2 and a mobility μ=600,000 cm^2/Vs at the base temperature of our dilution refrigerator T = 60 mK. Tilted-field magnetotransport The sample is mounted on a rotating stage allowing in-situ sample rotation in the magnetic field. The tilt angle θ is determined acurately from the shift ofresistance minima of the well-known fractional quantum Hall states.Acknowledgment We acknowledge the support of the HFML-RU/FOM member of the European Magnetic Field Laboratory (EMFL). We would like to thank M. Kawamura, A. S. Mishchenko, M. Ueda, K. von Klitzingand N. Nagaosa for fruitful discussion. Author ConrinbutionsM.K. initiated and supervised the project. D.M. concieved and designed the experiment. J.F. and Y.K. fabricated the samples. D.M., A.M., J.B. and U.Z. performed the high-field magnetotransport experiment and analysed the data. 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D. et al. titleThe half-filled Landau level: The case for Dirac composite fermions. journalScience volume352, pages197 (year2016).CFModel2 authorWang, C. & authorSenthil, T. titleComposite fermi liquids in the lowest Landau level. journalPhys. Rev. B volume94, pages245107 (year2016).CFModel3 authorSon, D. T. titleIs the composite fermion a dirac particle? journalPhys. Rev. X volume5, pages031027 (year2015).CFModel4 authorWang, C. & authorSenthil, T. titleHalf-filled landau level, topological insulator surfaces, and three-dimensional quantum spin liquids. journalPhys. Rev. B volume93, pages085110 (year2016). | http://arxiv.org/abs/1707.08406v1 | {
"authors": [
"D. Maryenko",
"A. McCollam",
"J. Falson",
"Y. Kozuka",
"J. Bruin",
"U. Zeitler",
"M. Kawasaki"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726122008",
"title": "Phase transition from a composite fermion liquid to a Wigner solid in the lowest Landau level of ZnO"
} |
All-electric single electron spin initialization G. Skowron December 30, 2023 ================================================The Erdős multiplication table problem asks what is the number of distinct integers appearing in the N× N multiplication table. The order of magnitude of this quantity was determined by Ford <cit.>. In this paper we study the number of y-smooth entries of the N× N multiplication table that is to say entries with no prime factors greater than y. § INTRODUCTIONThe multiplication table problem involves estimating A(x):=#{ab: a,b≤√(x), and a,b ∈ℕ}.This interesting question, posed by Erdős, has been studied by many authors. Erdős in <cit.>, showed that for all ε>0, we havex/(log x)^δ+ϵ≤ A(x)≤x/(log x)^δ-ε (x→∞),whereδ=1-1+loglog 2/log 2= 0.0860… . In <cit.>, Hall and Tenenbaum improved the upper bound(<ref>) as followsA(x)≤c_1 x/(log x)^δ√(loglog x),for some absolute constant c_1>0, whereδ is defined in (<ref>).A key step in determining the order of magnitude of A(x) is to estimate the number of representations of integers as products of ab with a,b≤√(x). The best estimate of A(x) is a result due to Kevin Ford <cit.>. He proved the following estimate, that significantly improved the order of magnitude of A(x) as followsA(x)≍x/(log x)^δ(loglog x)^3/2.Notation: In this paper, we use the notationf(x) ≍ g(x) if both f(x)≪ g(x) and g(x)≪ f(x) hold, where f(x)≪ g(x) or f(x)=O(g(x)) interchangeably to mean that |f(x)| ≤ cg(x) holds with some constant c for all x in a range which will normally be clear from the context. Also, the notation f(x)∼ g(x) means that f(x)/g(x)→ 1 as x→∞, and f(x)=o(g(x)) means that f(x)/g(x)→ 0 as x→∞.Also, u is defined asu:=log x/log y x≥ y≥ 2,and we let log_k x denote the k-fold iterated logarithm, defined by log_1x:= log x and log_k x= loglog_k-1x, for k>1.The bounds for A(x) are intimately connected with bounds for the function H(x,y,2y), which is the number of integers n≤ x having have a divisor in the interval (y, 2y]. More formally H(x,y,2y):= #{n≤ x : ∃d|n , y<d ≤ 2y}.This function, which its study was initiated by Besicovitch, has an intimate connection to the multiplication table problem via this inequality:H(x/2,√(x)/4,√(x)/2)≤ A(x)≤∑_k≥ 0 H(x/2^k,√(x)/2^k+1,√(x)/2^k).Hence, studyingA(x) boils down to understanding H(x, y, 2y), which is slightly easier to study. Ford <cit.> proved H(x,y,2y)≍x/(log x)^δ(loglog x)^3/2,and he subsequently deduced (<ref>) from (<ref>). It is worth mentioning that Koukoulopoulos in his Ph.D thesis <cit.> extended the multiplication table problem to higher dimensional table. Motivated by this background, in this paper we investigate the multiplication table problem for smooth integers. The set of y-smooth numbers, is defined byS(x,y):= {n≤ x : P(n)≤ y},where P(n) denotes the largest prime factor of an integer n ≥ 2, with the convention P(1)=1. SetΨ(x,y):= |S(x,y) |.Our main aim in this work is to studyA(x,y):= #{ab: a,b∈ S(√(x),y) }.Hence computing A(x,y) is equivalent to estimating the size ofS(√(x),y)· S(√(x),y).If we define H(x,y;z,2z) as the number of all y-smooth integers having at least one divisor in the given interval (z,2z], then by a simple argument one can arrive at the following inequalities for y-smooth integers,H(x/2,y;√(x)/4,√(x)/2)≤ A(x,y)≤∑_k≥ 0 H(x/2^k,y; √(x)/2^k+1,√(x)/2^k),where x≥ y >2. Thus one would be tempted to estimate A(x,y) by achieving upper and lower bounds for H(x,y;z,2z), The motivation of this paper is to understand the behaviour of the function A(x,y) directly instead of considering H(x,y;z,2z). This is a good moment to digress slightly to explain a connection between estimating A(x,y) and sum-product problem in additive combinatorics. For any non-empty subset A of integers, the sumset and productset of A are defined asA· A={a_1a_2: a_i∈ A}, A+A={a_1+a_2: a_i∈ A}.A famous conjecture of Erdős and Szemerédi states that the sumset and productset of a finite set of integers cannot both be small, more formally max{|A· A|, |A+A|}≫_ε |A|^2-ε. A simple approximation of Ψ(x,y) proved by Canfield, Erdős and Pomerance <cit.> states that for a fixed ϵ>0, we haveΨ(x,y)= x u^-u(1+o(1))asu→∞,for u≤ y^1-ϵ, that is y≥ (log x)^1+ϵ.By estimate (<ref>), one can see that for u large (or y small), the value of Ψ(x,y) is small. It counts the integers having large number of prime factors. Since in this case every n has a lot of small prime factors, we can find a and b such that n=ab and a,b≤√(x). If u is small (which means that y is large), then by (<ref>), one can deduce that the value of Ψ(x,y) is large compared to x. In this case, S(x,y) contains integers with large prime factors and we expect the size of S(√(x),y)· S(√(x),y) to be small. It is good to mention that by a connection to sum-product problem, Banks and Covert <cit.> by invoking combinatorial tools,have considered the behaviour of A(x^2,y)= |S(x,y)· S(x,y)| in different ranges of y, particularly for the cases when y is relatively small or large.By using combinatorial techniques, they could show that |S(x,y)· S(x,y)|= Ψ(x,y)^1+o(1)as x→∞,when y=o(log x), and |S(x,y)· S(x,y)|= Ψ(x,y)^2+o(1)as x→∞.when y/(log x) →∞. By <cit.> they could prove that Ψ(x^2/y,y) ≤ |S(x,y)· S(x,y)| ≤Ψ(x^2,y), also they could show thatΨ(x^2,y)= Ψ(x,y)^1+o(1)as x→∞,when y≥ 2 and y=o(log x). Therefore |S(x,y) · S(x,y)| = Ψ(x,y)^1+o(1),If we replace x by √(x) in (<ref>), we getΨ(x/y,y)≤ A(x,y)≤Ψ(x,y).Here we first present a simple argument to prove that the y-smooth integers up to x/y can be represented as a product of two y-smooth integers less than √(x), and by a lemma we will show thatΨ(x/y,y) ∼Ψ(x,y)(x→∞),when y=o(log x). Combining these up, gives the following theoremHere we present a simple idea to prove that A(x,y) has a same size as Ψ(x,y) when y is small compared to log x. Letn≤x/y be a y-smooth number. If n ≤√(x) then trivially we have n∈ A(x,y). Thus, we assume that √(x)≤ n.Let p_1≤ p_2≤⋯≤ p_k be prime factors of n. Consider the following sequence obtained by prime factors of n:n_0=1, n_j= ∏ _i=1^j p_i,1≤ j ≤ k. Since n≥√(x) then there exists a unique integer s, with 0≤ s <k such that n_s< √(x)≤ n_s+1. Each prime factor of n is less than y, therefore n_s ≤√(x)≤ n_s+1≤ n_s y. Setd=n_s, then √(x)/y≤ d≤√(x). Since n≤ x/y, then we easily conclude thatn/d≤√(x). Therefore, Ψ(x/y,y) ≤ A(x,y) ≤Ψ(x,y),and by a simple argument one can deduce that as x,y →∞ then Ψ(x/y,y) ∼Ψ(x,y) when y=o(log x), (see Lemma <ref>). This argument leads us to state the following theorem. If y=o(log x) then we have A(x,y) ∼Ψ(x,y) as x,y→∞. The problem gets harder, and hence, more interesting when y takes larger values compared to log x. We shall prove the following theorem for small values of y compared to x. We haveA(x,y)∼Ψ(x,y) as x,y→∞,when u and y satisfy the rangeulog u/(log y log_2 y log_3 y)^2→∞ ,which implies,y≤exp{(log x)^1/3/(log_2 x)^1/3 +ϵ},for ϵ>0 arbitrarily small. Theorem <ref> is proved in Section 3. The proof relies on some probabilistic arguments and recent estimates for Ψ(x/p,y) where p is a prime factor of n.If y takes values very close to x, which implies u is small compared to loglog y, then we will show the following theorem. Let ϵ>0 is arbitrarily small, then we haveA(x,y)= o(Ψ(x,y)) as x,y→∞,where u and y satisfying the rangeu < (L-ϵ)log_2 y, which implies,y ≥exp{log x/(L-ϵ)log_2 x},where L:= 1-log 2/log 2.Theorem <ref> is proved in Section 4, by applying an Erdős' idea <cit.>, suitably modified for y-smooth integers.In what follows, we will give a heuristic argument that predicts the behaviour of A(x,y) in ranges (<ref>) and (<ref>).We define the function τ(n; A,B) to be the number of all divisors of n in the interval (A,B]. In other words.τ(n; A,B) :=#{d: d|n ⇒ A< d ≤ B}.Let n∈ S((1-η)x,y) be a square-free number with k prime factors, where η→ 0 as x→∞. Assume that the set D(n):={log d: d|n} is uniformly distributed in the interval [0, log n]. So P(d∈ (A,B)):=τ(n)log B- log A/log n,where the sample space is defined byS:= { n≤ x: ω(n)= k },and n being chosen uniformly at random. By this assumption, the expected value of the function τ(n,(1-η)√(x),√(x)) is as follows𝔼[τ(n,(1-η)√(x),√(x))]= 2^k log (1/(1-η))/log√(x)≍2^k/ulog y. Alladi and Hildebrand in <cit.> and <cit.> showed that the normal number of prime factors of y-smooth integers is very close to its expected value u+log_2 y in different ranges of y. Hence, from (<ref>), we deduce that 𝔼[τ(n,(1-η)√(x),√(x))] ≍2^u+log_2 y/log y.If 2^u+log_2 y/log y→∞, then we expect that n will have a divisor d in the interval ((1-η)√(x),√(x)].We know n≤ (1-η) x. Thus, n/d ≤√(x), and we can deduce that n∈ A(x,y), this means that Ψ((1-η)x,y) ≤ A(x,y).Trivially A(x,y)≤Ψ(x,y). So bythis argument, we obtain A(x,y)∼Ψ(x,y),when η→ 0 as x→∞.On the other hand, if2^u+log_2 y/log y → 0, then we expect that none of integers inS((1-η)x,y) have a divisor in ((1-η)√(x), √(x)] (except a set with density 0), this means that A(x,y)=o(Ψ(x,y)) as x,y→∞.This heuristic gives an evidence for the following conjecture:If L:= 1-log 2/log 2, then we have the following dichotomy * : If u-Llog_2 y→+∞, which implies y≤exp{log x/Llog_2 x},Then, we haveA(x,y)∼Ψ(x,y) as x,y→∞. * : If u- Llog_2 y →-∞, which implies that for small ϵ>0 y≥exp{log x/(L-ϵ)log_2 x}, Then, we have A(x,y)=o(Ψ(x,y)) as x,y→∞ .Theorem <ref> and Theorem <ref> are in the direction of the first case and the second case of Conjecture (<ref>) respectively, but the claimed ranges in the conjecture are stronger than the claimed ranges in Theorem <ref> and Theorem <ref>, and the reason stems from uniformity assumption about D(n). AcknowledgementI would like to thank Andrew Granville and Dimitris Koukoulopoulos for all their advice and encouragement as well as their valuable comments on the earlier version of the present paper. I am also grateful to Sary Drappeau, Farzad Aryan and Oleksiy Klurman for helpful conversations.§ PRELIMINARIES In this section, we review some results used in the proof of our main theorems. We first fix some notation. In this chapter ρ(u) is the Dickman-de Bruijn function, as we defined in the introduction. By <cit.> we have the following estimate for ρ(u) ρ(u) = (e+o(1)/ulog u)^u as u→∞.The estimate Ψ(x,y)=xρ(u)(1+O_ϵ(log (u+1)/log y))holds uniformly in the rangex≥ 3, 1≤ u ≤log x/(log_2 x)^5/3+ϵ, that is, y≥exp((log_2 x)^5/3+ϵ),where ϵ is any fixed positive number.Combining (<ref>) with the asymptotic formula (<ref>), one can arrive at the following simple corollaryWe haveΨ(x,y) = xu^-(u+o(u)),as y and u tend to infinity, uniformly in the range (<ref>), for any fixed ϵ>0.We will apply this estimate in the proof of Theorem <ref>. However this estimate of Ψ(x,y) is not very sharp for large values of u, for which the saddle point method is more effective.Let α:=α(x,y) be a real number satisfying∑_p≤ ylog p/p^α-1=log x. One can show that α is unique. This function will play an essential role in this work, so we briefly recall some fundamental facts of this function that are used frequently. By <cit.> we have the following estimates for α. α(x,y)= log(1+y/log x)/log y{1+O(log_2 y/log y)} x≥ y≥ 2.For any ϵ>0, we have the particular casesα(x,y)= 1-ξ(u)/log y +O(1/L_ϵ (y)+ 1/u(log y)^2) ify≥ (log x)^1+ϵ,where L_ϵ(y)=exp{(log y)^3/5-ϵ},and ξ(t) is the unique real non-zero root of the equatione^ξ(t)= 1+tξ(t).Also for small values of y, we haveα(x,y)= log (1+y/log x)/log y{1+O(1/log y)} if2≤ y≤ (log x)^2.We now turn to another ingredient related to the behaviour of Ψ(x,y). The following estimate is a special case of a general result of de La Breteche and Tenenbaum <cit.>.There exist constants b_1 and b_2 and a function b=b(x,y,d) satisfying b_1≤ b≤ b_2 such that the following equation holds uniformly for all 2≤ y ≤ x: Ψ(x/d,y)= {1+O(1/u_y+t/u)}(1-t^2/u^2+u̅^2)^bu̅Ψ(x,y)/d^α.If d≤ y, then uniformly for x≥ y ≥ 2 we haveΨ(x/d,y) ={1+ O(1/u+log y/y)}Ψ(x,y)/d^α.We can deduce the following lemma by Theorem <ref> which completes the proof of Theorem <ref> If y≥ 2 and y=o(log x), then we haveΨ(x/y, y)∼Ψ(x,y) as x→∞. (i): Let y≥ (log_2 x)^2 and y=o(log x). By applying (<ref>), if d=y, we obtainΨ(x/y,y)=Ψ(x,y)/ y^α{ 1+O(log y/y)}. By combination of the above estimate along with (<ref>), we getΨ(x/y,y) = Ψ(x,y)/(1+y/log x)^1+O(1/log y){1+O( log y/y)}.We remark again that y=o(log x), so we obtain 1/(1+y/log x)^1+O(1/log y)→ 1when x →∞.Also, we havelog y/y→ 0 when x→∞,since y≥ (log_2 x)^2. Thus, by (<ref>), we conclude Ψ(x/y,y)/Ψ(x,y)→ 1 when x→∞.(ii): Let 2≤y ≤ (log_2x)^2, then by recalling Ennola's theorem <ref>, we getΨ(x/y,y) = 1/π(y)!∏_p≤ ylog x/y/log p{ 1+O(y^2/log x log y) } =1/π(y)!∏_p≤ ylog x/log p∏_p≤ y (1-log y/log x){ 1+O(y^2/log x log y) } =Ψ(x,y) (1+ O(π(y) log y/log x))= Ψ(x,y)( 1+O(y/log x)),which gives thatΨ(x/y,y) ∼Ψ(x,y) as x→∞,and this completes the proof.Finally, we defineθ (x,y,z) := #{ n≤ x : p|n ⇒ z≤ p≤ y } .This function has been studied extensively in the literature. Namely Friedlander <cit.> and Saias <cit.> gave several estimates for θ(x,y,z) in different ranges. The following theorem is due to Saias <cit.> which is used in Section 4. There exists a constant c>0 such that for x≥ y≥ z≥ 2 we have θ(x,y,z) ≤ cΨ(x,y)/log z. § PROOF OF THEOREM <REF>We begin this section by setting some notation. Let η be defined byη := 1/log_3 y,and setN:=⌊log_2 y -logη/log 2+2⌋,which play an essential role in process of the proof.The idea of the proof of Theorem <ref> is a combination of some probabilistic and combinatorial techniques. Before going through the details, we give a sketch of proof here.The first step of proving Theorem <ref> is to study the number of all prime factors of n in the narrow intervals J_i:=[(1-κ)y^1-1/2^i,y^1-1/2^i],1≤ i≤ N,of multiplicative length (1-κ)^-1, where κ is defined asκ := η/2N.Also, we define the tail interval J_∞:= [(1-κ)y,y].Let ω_i(n) be the number of prime factors of n in J_i for each i∈{1,2,…,N,∞}, more formallyω_i(n):= #{ p|n : p∈ J_i}.We define μ_i(x,y) to be the expectation of ω_i(n), defined byμ_i(x,y):=1/Ψ(x,y)∑_n∈ S(x,y)ω_i(n),In Proposition <ref>, we will prove that for almost all y-smooth integers the value of ω_i(n) exceeds μ_i(x,y)/2. We establish this by applying the Chebyshev's inequality#{n∈ S(x,y): ω_i(n)≤μ_i(x,y)/2}/Ψ(x,y)≤4σ_i^2(x,y)/μ_i^2(x,y),where σ_i^2(x,y):=1/Ψ(x,y)∑_n∈ S(x,y)(ω_i(n)-μ_i(x,y))^2,is the variance of ω_i(n) and i∈{1,2,…,N,∞}. We will conclude that there is at least one prime factor p_i in each J_i for 1≤ i≤ N and N prime factors q_1,…, q_N in J_∞. Then by using the product of these prime factors in Corollary <ref>, we will find a divisor D_j of n such that (1-κ)^N y^N-j/2^N≤ D_j ≤ y^N-j/2^N, for an integer j in {0,1,… 2^N-1 }.Then, we fix an integer n in S((1-η)x,y), and by defining m:= n/∏_i=1^Np_i q_i, we will easily show that there is a divisor d_j of n, such that√(n)/y^Ny^ j/2^N <d_j< √(n)/y^N y^(j+1)/2^N.Multiplying D_j and d_j and using the definitions of η, κ and N, gives a new divisor d of n that helps us to write n as the product of two divisors less than √(x).Before stating technical lemmas we get an estimate for the expected value of ω_i(n) for all 1≤ i≤ N and i=∞. By changing the order of summation in (<ref>), we can easily see that μ_i(x,y)=∑_p∈ J_iΨ(x/p,y)/Ψ(x,y).By (<ref>), we have the following estimateμ_i(x,y) =∑_p∈J_i1/p^α(1+O(1/u+ log y/y)), for all 1≤ i ≤ N and x≥ y≥ 2.Also, we obtain the following estimate for μ_i(x/q,y), where q is a prime divisor of n.μ_i(x/q,y)=∑_p∈ J_i1/p^α_q{1+O(1/u_q+ log y/y)},where u_q:= u-log q/log y. By substitution we obtain x/q =y^u_q. Set the saddle point α_q:=α (x/q,y), defined as the unique real number satisfying in ∑_p≤ ylog p/p^α_q-1=log(x/q). We are ready to prove the following lemma that shows the difference between μ_i(x/q,y) and μ_i(x,y) is small.Let q be a prime divisor of n∈ S(x,y), then we have|μ_i(x/q,y)-μ_i(x,y)|≪μ_i(x,y)/u. We use the estimate0<-α'(u):= -dα(u)/du≍u̅/u^2 log y,established in <cit.>, where u̅:= min{u, y/log y}.By (<ref>), we deduce|α^'(u)| ≪1/ulog y.Then applying (<ref>), gives thatα -α_q ≤∫_u_q^u|α^'(v)| dv≪∫_u^u_qdv/vlog y= 1/log ylog(u/u_q) ≍log q/log y log x.By expanding μ_i(x/q,y)-μ_i(x,y) and using (<ref>) and (<ref>), we get|μ_i(x/q,y)-μ_i(x,y)|= |∑_p∈ J_i(Ψ(x/pq,y)/Ψ(x/q,y)- Ψ(x/p,y)/Ψ(x,y))| ≤∑_p∈ J_i1/p^α{| p^α -α_q-1 |+O(1/u+ log y/y)}.By the Taylor expansion of the exponential function and invoking (<ref>) we obtainexp{(α-α_q)log p}-1 ≪log p log q/log y log x.We recall that p,q ≤ y for 1≤ i≤ N and i=∞. From this we infer that | p^α-α_q-1|≪1/u,this finishes the proof. In the following lemma we shall find an upper bound for σ_i^2(x,y) (defined in (<ref>)) for each i∈{1,2,…, N,∞}.We have σ_i^2 (x,y) ≪μ_i(x,y)+ μ_i^2(x,y)/u,where i∈{1,2,… ,N,∞}. By the definition of σ_i^2 (x,y) in (<ref>), we haveσ_i^2(x,y) =1/Ψ(x,y)∑_n∈ S(x,y)[ ω_i^2(n)-2μ_i(x,y)ω_i(n)+μ_i^2(x,y)].Using the definition of ω_i(n) in (<ref>), gives∑_n∈ S(x,y)ω_i(n)=∑_n∈ S(x,y)∑_ p∈ J_i1_p|n = ∑_p∈ J_iΨ(x/p,y),where the indicator function 1_p|n is 1 or 0 according to the prime p divides n or not. By the definition of μ_i(x,y) in (<ref>), one can deduce that∑_n∈ S(x,y)ω_i(n)= Ψ(x,y) μ_i(x,y). By applying (<ref>) and the equation above, we obtainΨ(x,y)σ_i^2(x,y)=∑_n∈ S(x,y)[ ω_i^2(n)-2μ_i(x,y)ω_i(n)+μ_i^2(x,y)] =∑_n∈ S(x,y)ω_i^2(n)-2Ψ(x,y)μ_i^2(x,y)+ψ(x,y)μ_i^2(x,y) =(∑_p,q ∈ J_jp≠ qΨ(x/pq,y))- Ψ(x,y)μ_i^2(x,y) +∑ _p∈ J_iΨ(x/p,y):=S_1 +S_2,where S_1:=∑_p,q ∈ J_jp≠ qΨ(x/pq,y)- Ψ(x,y)μ_i^2(x,y) and S_2:= ∑ _p∈ J_iΨ(x/p,y).We next find an upper bound for each S_i. We first consider S_1, by using (<ref>) we can get∑_p,q ∈ J_ip≠ qΨ(x/pq,y)- Ψ(x,y)μ_i^2(x,y) ≤∑_p∈ J_iΨ(x/p,y)(μ_i(x/p,y)-μ_i(x,y)). By Lemma <ref> and using (<ref>), we obtain the following upper bound for S_1S_1≤ C Ψ(x,y)μ_i^2(x,y)/u,where C is a positive constant. It remains to estimate S_2, from (<ref>) we haveS_2= Ψ(x,y)μ_i(x,y).By substituting the upper bounds for S_1 and S_2,we getσ_i^2(x,y) = S_1+S_2/Ψ(x,y)≪( μ_i(x,y)+μ_i^2(x,y)/u),and the proof is complete. Now we give an order of magnitude for μ_i(x,y), where i∈{1,2,…, N, ∞}We haveμ_i(x,y) ≍κY^1-1/2^i/log y,wherei∈{1,2,…, N,∞}, andY:= y^1-α.By the definition of each J_i, we obtain the following simple inequalities 1/y^α(1-1/2^i)#{ p∈ J_i}≤∑_p∈ J_i1/p^α≤1/(1-κ)y^α(1-1/2^i)#{ p∈ J_i}.By applying the prime number theorem, we obtain#{p: p ∈ J_i} =π(y^1-1/2^i)-π((1-κ)y^1-1/2^i)=y^1-1/2^i/log(y^1-1/2^i)-(1-κ)y^1-1/2^i/log((1-κ)y^1-1/2^i)+O(y^1-1/2^i/log^2 y) = y^1-1/2^i/(1-1/2^i) log y-(1-κ)y^1-1/2^i/(1-1/2^i)log y(1+O( log (1-κ)/log y))= κ y^1-1/2^i/(1-1/2^i)log y(1+o(1)),The last equality is true, since the given values of κ and N in (<ref>) and (<ref>) implyκ≍ 1/(log_2 y log_3 y).By substituting (<ref>) in (<ref>) we haveμ_i(x,y)≍κ Y^1-1/2^i/log y, By the above lemmas, we are now ready for proving the following proposition.If u and y satisfy in range given in(<ref>), we have#{n∈ S(x,y): ω_i(n)> μ_i(x,y)/2∀ i∈{1,…,N,∞}}∼Ψ(x,y) as x,y→∞, By the Chebyshev's inequality in (<ref>) and using the upper bound for σ_i^2(x,y) in lemma (<ref>), we get #{n∈ S(x,y): ω_i(n)≤μ_i(x,y)/2}≪Ψ(x,y)(1/μ_i(x,y)+ 1/u). By the above inequality, we obtain an upper bound for the following set M:= #{n∈ S(x,y): ∃ i ∈{1,…,N,∞}such that ω_i(n)≤μ_i(x,y)/2}≪Ψ(x,y)[1/μ_∞(x,y)+N/u+∑_i=1^N1/μ_i(x,y)].Our main task that finishes the proof is to find a range such that M/Ψ(x,y) tends to 0.By using Lemma <ref> and substituting the order of magnitude of μ_i(x,y) in (<ref>), we getM≪Ψ(x,y) [log y/κ Y+ N/u+ log y/κ∑_i=1^N1/Y^1-1/2^i].In what follows, we find a lower bound for Y in two different ranges of y (𝐢): If y≤ (log x)^2, then by (<ref>) α≤ 1/2+ o(1) as y→∞. Therefore, Y≥ y^1/2-o(1) ≥ y^1/3.By substituting this lower bound in (<ref>) and using the precise value of N in (<ref>), we have M ≪Ψ(x,y) [log y/κ y^1/3+ N/u+log y/κ y^1/3∑_i=1^Ny^1/3(2^i)]≪Ψ(x,y)[log_2 y/u+ y^1/6log y/κ y^1/3(1+ O(Ny^-1/12))]≪Ψ(x,y)log y/κ y^1/6,By using the asymptotic value of κ in (<ref>), we obtainM ≪Ψ(x,y) log y log_2 y log_3 y/y^1/6,and clearly we have M=o(Ψ(x,y)) as x,y→∞,this finishes the proof for the case y ≤ (log x)^2. (𝐢𝐢): If y≥ (log x)^2, by applying (<ref>), we have1-α = ξ(u)/log y +O(1/L_ϵ (y)+ 1/u(log y)^2).Using <cit.>, we have the following estimate of ξ ξ(t) =log (tlog t)+ O(log_2 t/log t) if t>3.Therefore,1-α = log (ulog u)/log y+ O(log_2 u/log y log u),Thus, we getY= u log u [1+O(log_2 u/log u)]≍ u log u.We now find an estimate for Y, when y≥log x. By applying the estimate in (<ref>), we have Y= exp{log y -log (1+log y/log x) +O(log(1+ y/log x)loglog y/log y)} =y/1+y/log x{1+ O(log (y/log x)loglog y/log y)}= y/1+y/log x{1+O(loglog y)}≍y log_2 y/1+y/log x.When y≥log x we have the following bounds for Y log x loglog y ≪ Y ≪ y loglog y. By combining the above with the estimate in (<ref>), and using the value of N in (<ref>), we getM≪Ψ(x,y) [ log y/κ u log u+ N/u+ log y/κ u log u∑_i=1^N (u log u)^1/2^i]≪Ψ(x,y) [ N/u+ log y/κ u log u((ulog u)^1/2 + (ulog u )^1/2^2+...+(ulog u)^1/2^N)]≪Ψ(x,y)[N/u+ log y/κ(ulog u)^1/2(1+O( N(ulog u)^-1/4))]≪Ψ(x,y) [ log_2 y/u+ log y/κ(ulog u)^1/2],By using the order of κ in (<ref>), one can arrive at the following upper bound of MM≪Ψ(x,y)log y log_2 y log_3 y/(u log u)^1/2.So there exists a constant c such that for all i∈{1,…, N, ∞}, we have#{ n∈ S(x,y) : ω_i(n)> μ_i (x,y)/2 ∀ i }≥Ψ(x,y)(1-clog y log_2 y log_3 y/ (u log u)^1/2), and this finishes the proof by lettingulog u/(log y log_2 y log_3 y)^2 →∞. If x and y satisfy the range (<ref>), then almost all n in S(x,y) are divisible by at least one prime factor p_i in J_i, and N prime factors q_1,..., q_N in J_∞. Moreover, the product ∏_i=1^Np_i q_i has a divisor D_j in each of intervals [(1-κ)^N y^N-j/2^N,y^N-j/2^N], where j∈{0,1,...,2^N-1}. The first part of Corollary is a direct conclusion of Proposition <ref>. For the second part, let n be a y-smooth integer satisfying the first part of Corollary. We fix the following divisor of n D:=∏_i=1^N p_i q_i,where p_i ∈ J_i and q_1,...,q_N ∈ J_∞.Let j be an arbitrary integer in {0,1,...,2^N-1}. Moreover, we define a_0:= N-∑_i=1^Na_i,where a_i's get the values 0 or 1 such that ∑_i=1^Na_i/2^i = j/2^N. We now define the divisor of D_j of D with the following formD_j:= ∏_i=1^N p_i^a_i∏_i=1^a_0 q_i,By using the bounds ofp_is and q_is, one can get the following bounds for D_j.(1-κ)^N y^N-∑_i=1^Na_i/2^i≤ D_j ≤ y^N-∑_i=1^Na_i/2^i, By using (<ref>), we have(1-κ)^N y^N-j/2^N≤ D_j ≤y^N-j/2^N,and this finishes our proof.We are ready now to prove Theorem <ref>.Let n≤ (1-η)x be a y-smooth integer with at least one prime factor p_i in eachJ_i , where i=1,..,N, and N prime divisors q_1,q_2, ...,q_N in J_∞. Set m:= n/∏_i=1^N p_i q_i.By this definition, we getn/∏_i=1^N p_iq_i≥n/y^2N >√(n), when 4N≤ u. Thus, m>√(n). Let {r_v} be the increasing sequence of prime factors of m and set d_v=r_1...r_v.Clearly, m has at least one divisor bigger than √(n)/y^N. We supposethat l is the smallest integer such that d_l ≥√(n)/y^N, and evidently we have d_l-1≤√(n)/y^N, So, we arrive at the following bounds for d_l √(n)/y^N≤ d_l ≤ yd_l-1≤√(n)/y^N-1,We pick k ∈{0,1,2...,2^N-1} such that √(n)/y^N y^k/2^N≤ d_l ≤√(n)/y^Ny^(k+1)/2^N.By the second part of Corollary <ref>, for every k in {0,1,...,2^N-1} there exists a divisor D_k such that (1-κ)^Ny^N-k/2^N≤ D_k ≤ y^N- k/2^N, We define d:= d_l D_k, we have(1-κ)^N√(n)≤ d ≤ y^1/2^N√(n), By using the values of N in (<ref>) and κ in (<ref>), we havee^-η/2√(n)≤ d ≤ e^η/2√(n).Applying the Taylor expansion for exponential functions, gives (1-η +η^2/2+O(η^3))^1/2√(n)≤ d ≤(1+η +η^2/2+O(η^3))^1/2√(n).By using the assumption n≤ (1-η)x in the upper bound and lower bound above, we obtain d≤(1-η^2/2+O(η^3))^1/2√(x)≤√(x), and n/d≤(1+η+η^2/2+ O(η^3))^1/2√(n)≤(1-η^2/2+O(η^3))^1/2√(x)≤√(x).Thus, we can write n∈ S((1-η)x,y) as the product of two divisors less than √(x), and we can deduce that Ψ((1-η)x,y)≤ A(x,y) ≤Ψ(x,y),By using (<ref>), we have Ψ((1-η)x,y)/Ψ(x,y) = (1-η)^α{ 1+O(1/u +log y/y)}→ 1 as x,y→∞,this finishes the proof.§ PROOF OF THEOREM<REF>In this section, we shall study the behaviourof A(x,y) for large values of y. When y takes values very close to x, then the set of y-smooth integers contains integers having large prime factors. As we explained in the heuristic argument, one can expect that A(x,y)= o(Ψ(x,y)). To show this assertion, we recall the idea of Erdős used to prove the multiplication table problem for integers up to x.We start our argument by giving an upper bound for A^*(x), defined byA^*(x):= #{ ab: a,b≤√(x)and (a,b)=1}.We shall find an upper bound of A^*(x) by considering the number of prime factors of a and b. We first define π_k(x) := #{n≤ x: ω(n)= k}Therefore,A^*(x) ≤∑_kmin{π_k(x) , ∑_j=1^k-1π_j(√(x)) π_k-j (√(x))}≤∑_kmin{cx/log x(log_2x)^k-1/(k-1)!, ∑_j=1^k-1c√(x)/log√(x)(log_2 √(x))^j-1/(j-1)!c√(x)/log√(x)(log_2 √(x))^k-j-1/(k-j-1)!},where in the last inequality, we used the well-known result of Hardy and Ramanujan that states there are absolute constants C and c such thatπ_k(x) ≤cx/log x(log_2x+C)^k-1/(k-1)!for k=0,1,2,.. and x≥ 2.By simplifying the upper bound in (<ref>) and using Stirling's formulan!∼ n^n+1/2e^-nwe obtainA^*(x) ≤∑_kmin{cx/log x(log_2 x)^k-1/(k-1)! , 4c^2 x/(log x)^2∑_j=0^k-21/(k-2)!k-2j(log_2 √(x))^k-2}= ∑_kmin{cx/log x(log_2 x)^k-1/(k-1)! ,4c^2 x/(log x)^2(2log_2 √(x))^k-2/(k-2)!}= ∑_k≤log_2x/log 24c^2 x/(log x)^2(2log_2√(x))^k-2/(k-2)! +∑_k> log_2 x/log 2cx/log x(log_2 x)^k-1/(k-1)!≪x/(log x)^1-1+loglog 2/log 2(log_2x)^1/2→ 0 as x→∞.We shall get the same upper bound for A(x). Let n≤ x and there are a and b less than √(x) such that n=ab. If (a,b)= 1 then n is counted by A(x) , and if (a,b)=d>1 then we can write n as n= a^' b^'d^2 such that (a^',b^')=1. So, n/d^2≤x/d^2, and n/d^2 will be counted by A(x/d^2). Therefore,A(x) ≤∑_d ≤√(x)A^*(x/d^2) ≪ A^*(x)By (<ref>), we getA(x) ≪x/(log x)^1-1+loglog 2/log 2(log_2x)^1/2.Thus,A(x)=o(x) as x→∞.Motivated by Erdős' idea for the multiplication table of integers up to x, we apply a similar method to find an upper bound for A(x,y). As we mentioned in introduction, Alladi and Hildebrand proved that the number of prime divisors of y-smooth integers is normally about u+loglog y (the expectation value of Ω(n)). So if u is smaller than loglog y, then by Alladi and Hildebrand's results, the value of Ω(n) in this range will be normally about loglog y. If u ≤loglog y, then y takes large values bigger than x^1/(loglog x), and we expect a same upper as (<ref>) bound for the following functionΠ(x,y;k):= ∑ _n∈ S(x,y) Ω(n)=k1.The first step of proof is to study the following function which plays a crucial role in this section. LetN_k(x,y,z) :=#{n∈ S(x,y): Ω_z(n)=k},where Ω_z(n) is the truncated version of Ω(n), only counting divisibility by primes not exceeding z with their multiplicities. In other wordsΩ_z(n):= ∑ _p^v||n p≤ zv.In the following lemma, by using induction on k, we shall find an upper bound of type (<ref>) for N_k(x,y,z). The reason of applying truncation is to sieve out prime factors exceeding some power of y which are the cause of big error terms as k increases in each step of induction. The upper bound of N_k(x,y,z) leads us to generalize Erdős' idea for y-smooth integers in a certain range of y.Let u ≤ (C-ϵ)loglog y, where C is a positive constant and ϵ>0 is arbitrarily small. Set the parameter z such thatloglog z≪ u.Then, there are constants A and B such that the inequalityN_k(x,y,z)≤AΨ(x,y)/log z(loglog z+B)^k/k!holds for every integer k>0. When k=0, by (<ref>), evidently we haveN_0 (x,y,z) = θ(x,y,z) ≤ cΨ(x,y)/log z,where c>0 is a constant. When k=1, we can represent n as n=pm, where p≤ z and every prime factor q of m is between z and y, then using the definition of θ(x,y,z) we have N_1(x,y,z)=∑_p≤ z∑_m≤ x/pq|m ⇒ z≤ q≤ y1=∑_p≤ zθ(x/p,y,z).By applying the estimate (<ref>) and (<ref>), there is constant c such thatN_1(x,y,z) ≤∑_p≤ zcΨ(x/p,y)/log z = cΨ(x,y)/log z∑_p≤ z1/p^α{1+O(1/u)}.For the last summand we have ∑_p≤ z1/p^α = ∑_p≤ z1/p(p^1-α)= ∑_p≤ z1/p{1+O((1-α)log p)},since(1-α)log p ≤ (1-α)log z, and (1-α)log z is bounded in our range (see (<ref>)). Therefore,∑_p≤ z1/p^α= log_2 z + O((1-α) log z),By using the estimate of α in (<ref>) and the upper bound of z, we get (1-α)log z≪log u/log ylog z ≪log u/log_2 y≪log_3 y/log_2 y,and we obtain∑_p≤ z1/p^α= log_2 z +O(log _3 y/log _2 y).Thus,∑_p≤ z1/p^α{ 1+O(1/u)} = loglog z+ O(1),since we have loglogz ≪ u. Substituting (<ref>) in the upper bound of N_1(x,y,z), gives N_1(x,y,z) ≤cΨ(x,y)/log z(log_2z+O(1)). We will show the lemma with A=c and B=O(1).We argue by induction: we assume that the estimate in (<ref>) is true for any positive integer k, we now prove it for n∈ S(x,y) with Ω_z(n)=k+1. There are k+1 ways to write n as n=pm_1m_2 such that p≤ z and Ω_z(m_1)=k and every prime factor of m_2 is greater than z. Then we haveN_k+1(x,y,z) = 1/(k+1)∑_p≤ z∑_m_1∈ S(x/(p),y) Ω_z(m_1)=k m_2 ∈ S(x/(pm_1),y) q|m_2 ⇒ q>z 1≤1/(k+1)∑_p≤ z∑_m_1∈ S(x/(p),y) Ω_z(m_1)=k 1= 1/(k+1)∑_p≤ zN_k(x/p , y, z)By the assumption for Ω_z(n)=k and (<ref>), we get N_k+1(x,y,z) ≤A(log_2z+B)^k/log z(k+1)!∑_p≤ zΨ(x/p,y)=AΨ(x,y)/log z(log_2z+B)^k/(k+1)!∑_p≤ z1/p^α{1+O(1/u)}. By applying the estimate in (<ref>), we arrive at the following bound for N_k+1(x,y,z) N_k+1(x,y,z)≤AΨ(x,y)/log z(log_2z+B)^k+1/(k+1)!, so we derived our desired result.For a small ϵ>0, we set u < (λ/log 2-ϵ)log_2 y, where λ is a fixed real number in the open interval (1-2log 2 , 1-log 2). We now set z satisfyingloglog z= log 2/λ u, so the given ranges of u and z satisfy the conditions of Lemma <ref>.By the definition of A(x,y), we have the following evident bound of A(x,y)A(x,y) ≤∑_kmin{∑_n∈ S(x,y) Ω_z(n)=k1 , ∑_j=1^k-1∑_a∈ S(√(x),y) Ω_z(a)=j1∑ _b∈ S(√(x),y) Ω_z(b)=k-j1}. We setL=⌊ H log_2 z⌋,where H:= 1-λ/log 2.We have 1-2log 2 < λ< 1-log 2. Thus, 1<H<2. By using (<ref>), we write the following bound for A(x,y)A(x,y)≤#{ n∈ S(x,y): Ω_z(n)> L} +#{ ab: a,b ∈ S(√(x),y), Ω_z(a)+Ω_z(b)≤ L}= ∑_k>LN_k(x,y,z) + ∑_ k≤ L∑_j=0^k N_j(√(x),y,z)N_k-j(√(x),y,z).By applying Lemma <ref>, we have A(x,y)≪∑_k> LΨ(x,y)/log z(log_2z+c)^k/k!+ ∑_k≤ L∑_j=0^kΨ^2(√(x),y)/log ^2 z(log_2 z +c)^j/j!(log_2 z+c)^k-j/(k-j)! =∑_k>LΨ(x,y)/log z(log_2 z+c)^k/k!+ ∑_k≤ LΨ^2(√(x),y)/log ^2 z∑_j=0^k1/k!k j(log_2 z+ c)^k=∑_k>LΨ(x,y)/log z(log_2 z+c)^k/k!+∑_k≤ LΨ^2(√(x),y)/log ^2 z(2log_2 z+c)^k/k!.By applying the simple form of Ψ(x,y) in Corollary <ref>, and using the assumption (<ref>), we getΨ^2(√(x),y)/Ψ(x,y)≍ (log z)^λas u,y→∞. Thus,A(x,y)≪Ψ(x,y)/log z∑_k>L(log_2 z+c)^k/k!+ (log z)^λΨ(x,y)/log ^2 z∑_k≤ L(2log_2 z+c)^k/k!.The maximum values of functions in the above summands (with respect to k) are attained at k=⌊log_2 z⌋ and k=⌊2log_2 z⌋ respectively. We have loglog z < L < 2 loglog z, so the function in the first summation in (<ref>) in decreasing for k>L, and by using Stirling's formula k!∼ k^k+1/2e^-k, we have ∑_k> L(log_2z)^k/k! = ∑_Hlog_2 z<k≤ elog_2 z(log_2 z)^k/k!+∑_elog_2z < k ≤ 2elog_2z(log_2z)^k/k!+∑_k> 2elog_2 z(log_2 z)^k/k!≪ (log_2z)((e/H)^Hlog_2z +1)≪1/(log z)^Hlog H-H.The function in the second summation in (<ref>) is increasing for k≤ L, and we have∑_k≤ L(2log_2 z+c)^k/k!≪ (log_2 z)(2e/H)^Hlog_2z= 1/(log z)^Hlog H -H -Hlog 2 Substituting the upper bounds obtained in (<ref>) and (<ref>) in(<ref>), and using the definition of H, givesA(x,y) ≪Ψ(x,y)/(log z)^ G(H),where G(H) := 1+Hlog H -H.The function G(H) is an increasing function in the interval (1,2) with a zero at H=1. Thus, for any arbitrary 1-2log 2 < λ < 1-log 2, we have A(x,y)= o(Ψ(x,y)) as x,y→∞,so we obtained our desired result.plain | http://arxiv.org/abs/1707.09048v1 | {
"authors": [
"Marzieh Mehdizadeh"
],
"categories": [
"math.NT",
"11N25"
],
"primary_category": "math.NT",
"published": "20170727210808",
"title": "The Multiplication table for Smooth integers"
} |
Complexity of AdS_5 black holes with a rotating stringKoichi Nagasaki[[email protected]]Department of Physics and Center for High Energy Physics, Chung Yuan Christian UniversityAddress: 200 Chung Pei Road, Chung Li District, Taoyuan City, Taiwan, 32023We consider computational complexity of AdS_5 black holes.Our system contains a particle moving on the boundary of AdS. This corresponds to the insertion of a fundamental string in AdS_5 bulk spacetime. Our results give a constraint for complexity.This gives us a hint for defining complexity in quantum field theories. § INTRODUCTION §.§ Background Computational complexity is originally introduced in physics of quantum information <cit.> and its application to black hole physics is considered in many works <cit.>.It behaves in a similar way to the entropy as it is expected to satisfy the second law of thermodynamics <cit.>. The growth of complexity is related to problems of black holes, for example, information problem, the transparency of horizons or the existence of firewalls <cit.>. Complexity-Action (CA) conjecture <cit.> predicts that complexity of the black hole is equal to the the bulk action integrated in the region called the Wheeler-de Witt patch (WdW). This conjecture is checked by many works <cit.>. The time dependence is studied in <cit.>. This duality is studied also in a deformed case by adding probe branes <cit.>. Nevertheless, the strict and valid definition of complexity is still unclear. The most standard one so far is by the method of quantum operators which are called gates. Let us define a reference state as the simplest quantum state. Quantum gates operate a state and change its state. Computational complexity of a test state is defined as the minimum number of gates to prepare it from the reference state. But this definition has some unsatisfied points.One is that the definition of the reference state is unclear. Another is that the number of the gates depends on the choices of the primitive gates.§.§ Our approach To treat this problem, in this paper, we shed light on a new property of complexity and suggest a new constraint which complexity should possess.The above works so far treat stationary systems.A generalization to this conjecture to the system other than stationary states is an important work. As an example, we consider here a system moved by a drag force.Such a work is motivated by jet-quenching phenomena in heavy ion collisions. Energy loss of a charged particle in quark-gluon plasma (QGP) is calculated in several approaches <cit.>. Especially in <cit.> a particle moving in AdS_5-Schewarzchild spacetime was considered and its action was calculated. The particle is moved on the boundary of the AdS_5 space. Because of the effect of shear viscosity in QGP, this particle losses the energy.This system is a dissipative system. While his work considered the action in the outside of the horizon, that is usual spacetime, we apply here this method also to the inside of the horizon.According to CA relation, we will obtain complexity of a time dependent system. In this work we consider a Wilson line operator located in AdS_5 spacetime by inserting a fundamental string.This Wilson line moves on a great circle in S^3 part in AdS_5. Such a non-local operator describes a test particle moving on the boundary gauge theory. This shows how complexity is deformed when adding a time-dependent operator, especially also non-local operator. Since inserting the Wilson loop is described by adding a Nambu-Goto (NG) term, the action is expected to consist of theEinstein-Hilbert term, the Nambu-Goto term and the boundary term. We study on the effect of the Wilson line operator focusing on the Nambu-Goto term and show the black-hole mass and particle's velocity dependences of the action. This will show the growth of complexity for a dissipative system. This paper is constructed as follows. In section <ref>, we explain our setup which includes the AdS black hole with a kind of non-local operator, a Wilson line, and the action. In section <ref>, in order to find the effect of the test particle, we focus on the NG action which is a part of the action we introduced the former section. Section <ref> gives an interpretation of our calculation and some suggestions of the constraint for complexity. § SETUPWe consider a Wilson loop inserted on AdS_5 spacetime. The black hole geometry in AdS_5 spacetime is described by the metric:ds_AdS^2= -f(r)dt^2 + dr^2/f(r) + r^2dΩ_3^2,f(r)= 1 - 16π GM/5Ω_3r^2 + r^2/ℓ_AdS^2 = 1 - r_m^2/r^2 + r^2/ℓ_AdS^2, r_m^2 := 16π GM/5Ω_3,where dΩ_3^2 is the square of the line element on three sphere, while dΩ_3 is the volume form on the three sphere and Ω_3 is the volume obtained by integrating it.[ The volume of (n-1)-dim unit sphere S^n-1 is Ω_n-1 = 2π^n/2/Γ(n/2). ]We see in appendix <ref> that ℓ_AdS and other parameters are related to the gauge theory parameters asr_m^2/ℓ_AdS^2= 16π GM/5Ω_3ℓ_AdS^2 = 8GM/5πℓ_AdS^2 = 4M/5√(α'g_YM)/N^7/4.This geometry has some characteristic length; one is the Schwarzschild radius defined in eq.(<ref>) and the other is r_h which is the radius of the black-hole horizon which is determined by the equation f(r) = 0:r_h = ℓ_AdS(-1+√(1+4r_m^2/ℓ_AdS^2)/2)^1/2.We parametrize S^3 part of the AdS spacetime (<ref>) by x^1= cosθcosϕ, x^2 = cosθsinϕ,x^3= sinθcosφ, x^4 = sinθsinφ;θ∈[0,π/2), ϕ,φ∈[0,2π).Let us take parameters τ and σ on the world-sheet of the fundamental string:t = τ, r = σ,ϕ = ωτ + ξ(σ),where ω is the constant angular velocity and ξ(σ) is a function which determines the shape of the string.Thus, the induced metric on the world-sheet isds_ind^2= -f(σ)dτ^2 + dσ^2/f(σ),f(σ)= 1 - r_m^2/σ^2 + σ^2/ℓ_AdS^2. The action consists of the Einstein-Hilbert term and the Nambu-Goto (NG) term:S_EH = 1/16π G∫ d^6x√(|g|)(ℛ-2Λ),Λ = -6/ℓ_AdS^2,S_NG = -T_s∫ d^2σ√(- g_ind),where T_s is the tension of the fundamental string. We are afraid for the necessity of adding a boundary term like the York-Gibbos-Hawking (YGH) term.The development of the Einstein-Hilbert action is given in <cit.> asdS_EH/dt = -1/2π Gℓ_AdS^2∫_Σ r^3dr dΩ_3= -1/8π Gr_h^4Ω_3/ℓ_AdS^2.This is the leading contribution to complexity. In the next section we consider the effect of the addition of the Wilson loop. § EVALUATION OF THE NG ACTION The addition of the Wilson loop corresponds to inserting a fundamental string whose world-sheet has a boundary on the Wilson loop. We calculate here the NG action of this fundamental string. The time derivative of the NG action is obtained by integrating the square root of the determinant of the induced metric over the WdW (figure <ref>):dS_NG/dt = T_s∫_0^r_h dσ√(1 - σ^2ω^2/f(σ) + σ^2ξ'(σ)^2f(σ)). The Lagrangian isℒ = T_s√(1 - σ^2ω^2/f(σ) + σ^2ξ'(σ)^2f(σ)).Then the equation of motion for ξ(σ) is 0 = 1/T_sd/dσ∂ℒ/∂ξ'(σ) = d/dσ(σ^2 ξ'(σ)f(σ)/√(1- σ^2ω^2/f(σ) + σ^2 ξ'(σ)^2 f(σ))),c_ξ :=σ^2 ξ'(σ)f(σ)/√(1- σ^2ω^2/f(σ) + σ^2 ξ'(σ)^2 f(σ)) =σ^2 ξ'(σ)f(σ)/ℒ/T_s,where by the second line we defined a conserved constant c_ξ. Solving it for ξ(σ), this function is given by integrating the following:ξ'(σ)= c_ξ/σ f(σ)√(%s/%s)f(σ) - σ^2ω^2σ^2f(σ) - c_ξ^2.In order for this expression to give real values, the denominator in the square root must be negative when the numerator factor f(σ) - σ^2ω^2 becomes negative.Thus they become zero coincidentally. From this condition, the constant c_ξ is determined to be c_ξ = ωσ_H^2= ω-1+√(1+4r_m^2(1/ℓ_AdS^2-ω^2))/2(1/ℓ_AdS^2-ω^2), σ_H^2= -1+√(1+4r_m^2(1/ℓ_AdS^2-ω^2))/2(1/ℓ_AdS^2-ω^2),where σ = σ_H is the solution for numerator of the square root in eq.(<ref>). We assumed that c_ξ is positive. Since the numerator and the denominator have the coincident solution, the cancelation givesξ'(σ) = ωσ_H^2/(σ^2 - r_h^2)(1/ℓ_AdS^2(σ^2 + r_h^2)+ 1)√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2)+ 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1, ℒ = T_sσ^2ξ'(σ)f(σ)/c_ξ = T_s√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2)+ 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1.The integral of (<ref>) over the WdW is dS_NG/dt = ∫_WdWℒ = T_s∫_0^r_h dσ√(%s/%s)(1/ℓ_AdS^2-ω^2)(σ^2 + σ_H^2) + 11/ℓ_AdS^2(σ^2 + σ_H^2) + 1.Using the integral formula (<ref>) shown in appedix,dS_NG/dt= -iT_s√(σ_H^2(1-ℓ_AdS^2ω^2)+ℓ_AdS^2) E[arcsin(ir_h/√(σ_H^2 + ℓ_AdS^2)), √(%s/%s)(σ_H^2 + ℓ_AdS^2)(1-ℓ_AdS^2ω^2)σ_H^2(1-ℓ_AdS^2ω^2) + ℓ_AdS^2 ].Substituting eq.(<ref>) and the horizon radius r_h, (<ref>), we can express the growth of the action asdS_NG/dt = -iT_sℓ_AdS(1+√(1+4s^2 (1-v^2))/2)^1/2× E[arcsin(i( (-1+√(4s^2+1))(1-v^2)/(1 - 2v^2) +√(1+4s^2(1-v^2)))^1/2), ((1 - 2v^2) + √(1+4s^2(1-v^2))/1+√(1+4s^2(1-v^2)))^1/2 ], where for simplicity we defined the following dimensionless constants:s := r_m/ℓ_AdS,v := ℓ_AdSω.Limit behaviorThe behavior in the limits s→ 0 and s→∞ are, respectively, as follows:dS_NG/dt|_s→ 0 = -i4T_sℓ_AdS E[0, √(1-v^2)] = 0, dS_NG/dt|_s→∞ = 4T_sℓ_AdS√(s)(1-v^2)^1/2.The behavior in the limits v→ 0 and v→ 1 are dS_NG/dt|_v=0= √(2)T_sℓ_AdS(-1+√(4s^2+1)/2)^1/2∼√(s)(s≫ 1), dS_NG/dt|_v→ 1 = -iT_sℓ_AdSarcsin i(-1+√(4s^2+1)/2(1+s^2))^1/2 = T_sℓ_AdSarcsinh(-1+√(4s^2+1)/2(1+s^2))^1/2.The figures <ref> and <ref> show these behavior and also the curves of all parameter ranges. The function (<ref>) has an extremum at s=√(2).§ DISCUSSION Figure <ref> shows the relation between the growth of the action and the angular velocity (v = ℓ_AdSω).We can see that the effect of the drag force to the growth of the action is larger as the mass increases when the string moves relatively slower. As the string moves faster the effect become smaller. For any mass, the effect vanishes in the limit v→ 1. This behavior is similar to the one that of a rotating black holes — the complexity growth bound changes from 2M to 2√(M^2 - J^2/ℓ_AdS^2), where J is the angular momentum of the black holes. Figure <ref> shows mass dependence of the growth of the action.When the angular velocity is small, it is a monotonically increasing function of the mass. As the angular velocity becomes larger, it ceases to increase rapidly and the extremum appears at about v≳ 0.97. That is a notable phenomena — complexity has different velocity dependence in the relativistic region.According to CA duality these represent the growth of the black-hole complexity. For a large mass black hole, complexity changes rapidly when the particle moves slowly and this effect becomes small for a relativistic particle. As the black hole becomes smaller, the complexity does not change whether the particle moves quickly or not. Complexity is expected to increase as the entropy increases by the second law of thermodynamics <cit.>. So the results shown in figure <ref> and figure <ref> are consistent with this expectation.The velocity dependence of energy loss is changed by particle's mass. It is due to collisions between quarks and gluons, and gluon bremsstrahlung and which is dominant depends on its velocity <cit.>. The behavior of the action for large mass, (∼√(s)), is consistent with the expected behavior from CA relation <cit.>, that is, the growth of complexity is bounded by two times mass.These behavior characterize the growth of complexity and gives a hint to define complexity in black hole spacetime. In our analysis the boundary term, which was detaily discussed in <cit.>, is not taken into consideration. An example of the boundary term calculation is given in <cit.>. They considered AdS boundaries. It is different from our case since in our calculation we need boundary terms which come from the boundary of region 1 (see figure <ref>).There is a cancelation and the contributions only from r=0 (black-hole singularity) and r=r_h (horizon) survive. The detailed analysis will be considered as a future work. However, while the Einstein-Hilbert term includes the second derivative of the metric, the NG action only includes the first derivative of it. The value at the boundary is completely determined by the metric and the derivative of it is unnecessary.Then we can expect that the boundary term is not need in this situation. § ACKNOWLEDGMENTSI would like to thank Satoshi Yamaguchi, Hiroaki Nakajima and Sung Soo Kim for useful discussions. This work is supported in parts by the Chung Yuan Christian University, the Taiwan's Ministry of Science and Technology (grant No. 105-2811-M-033-011)§ GAUGE THEORY PARAMETER The Newton's gravitational constant we used is related to the gauge theory parameter. Since 5-dimensional gravitational constant and AdS radius are as follows <cit.>:G_5 = πℓ_AdS^3/2N^2,ℓ_AdS^2 = α'√(g_YM^2N)⇒ G_5= π/2N^2α'^3/2(g_YM^2N)^3/4 = π/2(α' g_YM)^3/2/N^5/4,we obtainG_5/ℓ_AdS^2= π/2N^2√(α')(g_YM^2N)^1/4 = π√(α'g_YM)/2N^7/4.Note that some people may use the different convention where ℓ_AdS^2 is different by the factor √(2), for example <cit.>, but we follow the above definition in this paper. § BLACK HOLE TEMPERATUREHere we derive the temperature of AdS_5 black hole. This result can be seen in <cit.>. We consider the Euclidean version of the metric (<ref>) and take the coordinates:r = r_h(1+ρ^2).Near the horizon the metric is approximated asds_AdS^2≈2ρ^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2) dt_E^2+ 2r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1dρ^2+ r_h^2dΩ_3^2= 2r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1(dρ^2+ ρ^2/r_h^2(r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^2 dt_E^2)+ r_h^2dΩ_3^2.Seeing the period of Euclidean time t_E, the black hole temperature (1/β) is β = 2π r_h (r_m^2/r_h^2 + r_h^2/ℓ_AdS^2)^-1. Expressed it by using mass, this leads T= 1/2π r_h(r_m^2/ℓ_AdS^2/r_h^2/ℓ_AdS^2+ r_h^2/ℓ_AdS^2)= 1/2πℓ_AdS(-1+√(1+4x)/2)^-1/2(2x/-1+√(1+4x)+ -1+√(1+4x)/2), x := 8GM/5πℓ_AdS^2,Tℓ_AdS = 1/2π(x(-1+√(1+4x)/2)^-3/2 + (-1+√(1+4x)/2)^1/2).where we used eqs. (<ref>) and (<ref>) and in the last line we used dimensionless quantities Tℓ_AdS and x. This function is plotted in figure <ref>. § ELLIPTIC INTEGRALFor calculating the action in the Wheeer-de Witt patch (WdW) we use the elliptic integral of second kind. 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Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv.Theor.Math.Phys. 2 (1998) 231–252, http://arxiv.org/abs/hep-th/9711200arXiv:hep-th/9711200 [hep-th]. johnson2006d C. Johnson, D-Branes. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2006. <https://books.google.com.tw/books?id=pnClL_tua_wC>.Witten:1998zw E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2 (1998) 505–532, http://arxiv.org/abs/hep-th/9803131arXiv:hep-th/9803131 [hep-th]. | http://arxiv.org/abs/1707.08376v2 | {
"authors": [
"Koichi Nagasaki"
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],
"primary_category": "hep-th",
"published": "20170726111856",
"title": "Complexity of AdS_5 black holes with a rotating string"
} |
M[1]>m#1 | http://arxiv.org/abs/1707.08931v2 | {
"authors": [
"Luca Lionni",
"Johannes Thürigen"
],
"categories": [
"hep-th",
"gr-qc",
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"primary_category": "hep-th",
"published": "20170727165224",
"title": "Multi-critical behaviour of 4-dimensional tensor models up to order 6"
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equationsection Fourier coefficients of small automorphic representations]Fourier coefficients attached to small automorphic representations of _n(𝔸) O. Ahlén]Olof Ahlén Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Am Mühlenberg 114476 Potsdam, Germany [email protected]. Gustafsson]Henrik P. A. Gustafsson Department of PhysicsChalmers University of Technology SE-412 96 Gothenburg, Sweden [email protected]. Kleinschmidt]Axel Kleinschmidt Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Am Mühlenberg 114476 Potsdam, Germany Solvay Institutes ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium [email protected] B. Liu]Baiying Liu Department of MathematicsPurdue University150 N. University StWest Lafayette, IN, 47907 [email protected]. Persson]Daniel Persson Department of Mathematical Sciences Chalmers University of Technology SE-412 96 Gothenburg, Sweden [email protected][2000]Primary 11F70, 22E55; Secondary 11F30We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of _n(𝔸) are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro–Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory. [ [ December 30, 2023 =====================§ INTRODUCTION§.§ Background and motivation Let F be a number field andbe the associated ring of adeles.Letbe a reductive algebraic group defined over F and π an (irreducible) automorphic representation of () as defined in <cit.>.Fix a Borel subgroup B and let P ⊂ be a standard parabolic subgroup with Levi decomposition P=LU, and let ψ : U(F)\ U(𝔸)→ℂ^× be a global unitary character.Given any automorphic form φ∈π one can consider the following function on ():ℱ_U(φ, ψ; g)=_U(F)\ U(𝔸)φ(ug)ψ^-1(u) du .This can be viewed as a Fourier coefficient of the automorphic form φ with respect to the unipotent subgroup U. Fourier coefficients of automorphic forms carry a wealth of arithmetic and representation-theoretic information. For example, in the case of classical modular forms on the upper half-plane, Fourier coefficients are well-known to encode information about the count of rational points on elliptic curves. On the other hand, for higher rank Lie groupstheir arithmetic content is not always transparent, but they always encode important representation-theoretic information. Langlands showed that the constant terms in the Fourier expansion of Eisenstein series provide a source for automorphic L-functions <cit.>, and Shahidi extended this method (now called the Langlands-Shahidi method) to include also the non-constant Fourier coefficients <cit.>.Theta correspondences provide realizations of Langlands functorial transfer between automorphic representations π and π' of two different groupsand '. In this context automorphic forms attached to minimal automorphic representations play a key role <cit.>. The wave front set of a minimal representation π_min of a groupis the closure of the smallest non-trivial nilpotent coadjoint orbit 𝒪_min of<cit.>. The automorphic realizations of minimal representations are characterized by having very few non-vanishing Fourier coefficients <cit.>. Conversely, the method of descent <cit.> can be viewed as an inverse to the functorial lifting, in which an automorphic representation of a general linear group _n is transferred to a representation of a smaller classical group . Also in this case do Fourier coefficients of small representations enter in a crucial way. In general it is a difficult problem to obtain explicit formulas for Fourier coefficients for higher rank groups, let alone settle the question of whether an automorphic form φ can be reconstructed from only a subset of its Fourier coefficients. For cusp forms on _n this is possible due to the Piatetski-Shapiro–Shalika formula <cit.> that allows to reconstruct φ from its Whittaker coefficients; i.e. the Fourier coefficients with respect to the unipotent radical N of the Borel subgroup B ⊂. These coefficients are sums of Eulerian Whittaker coefficients on subgroups of , and their non-archimedean parts can be obtained from the Casselman–Shalika formula <cit.> as described in <cit.>. However, even if this gives us complete control of the Fourier expansion with respect to N it does not automatically give us a way of calculating an arbitrary Fourier coefficient ℱ_U(φ, ψ; g) with respect to some other unipotent subgroup U. Such coefficients play an important role in the construction of L-functions, and also carry information about non-perturbative effects in string theory as described in section <ref>.Expanding upon the classic results of <cit.>, Miller and Sahi proved in <cit.> that for automorphic forms φ attached to a minimal representation π_min of E_6 and E_7, any Fourier coefficient ℱ_U(φ, ψ; g) is completely determined by maximally degenerate Whittaker coefficients of the form_N(F)\ N(𝔸)φ(ng)ψ_α (n)^-1 dn,where ψ_α is non-trivial only on the one-parameter subgroup of N corresponding to the simple root α. This result maybe viewed as a global version of the classic results of Moeglin-Waldspurger in the non-archimedean setting <cit.>, and Matumoto in the archimedean setting <cit.>.For the special cases of _3 and _4 the Miller–Sahi results were generalized in <cit.> (following related results in <cit.>) to automorphic forms attached to a next-to-minimal automorphic representation π_ntm. It was shown that any Fourier coefficient is completely determined by (<ref>) and coefficients of the following form∫_N(F)\ N(𝔸)φ(ng)ψ_α, β (n)^-1 dn ,where ψ_α, β is only supported on strongly orthogonal pairs of simple roots (α, β) which here reduces to that [E_α, E_β]=0 <cit.>. The main goal of the present paper is to use the techniques of <cit.>, in particular the notion of Whittaker pair, to extend the above results to all of _n.§.§ Summary of results We now summarize our main results. In the rest of this paper we will consider _n for n ≥ 5 where we have fixed a Borel subgroup with the unipotent radical N.Let also T be the diagonal elements of _n(F) and, for a character ψ_0 on N, let T_ψ_0 be the stabilizer of ψ_0 under the action [h.ψ_0](n) = ψ_0(h n h^-1) for h ∈ T. DefineΓ_i(ψ_0)(_n-i(F))_Ŷ_n-i(F) 1 ≤ i ≤ n-2 (T_ψ_0∩ T_ψ_α_n-1)T_ψ_0i = n-1,where (_n-i(F))_Ŷ is the stabilizer of Ŷ = ^t(1, 0, 0, …, 0) ∈_(n-i)× 1(F) and consists of elements[ 1 ξ; 0 h ], with h ∈_n-i-1(F) and ξ∈_1 × (n-i-1)(F). When ψ_0 = 1 we write Γ_i(1) as Γ_i.Similarly, let (_j(F))_X̂ be the stabilizer of X̂ = (0, …, 0, 1) ∈_1× j(F) with respect to multiplication on the right, ψ_0 a character on N, and defineΛ_j(ψ_0)(_j(F))_X̂_j(F) 2 < j ≤ n (T_ψ_0∩ T_ψ_α_1)T_ψ_0j = 2 ,where, again, we denote Λ_j(1) = Λ_j.Define also the embeddings ι, ι̂: _n-i→_n for any 0 ≤ i ≤ n-1 asι(γ) = [ I_i 0; 0 γ ]ι̂(γ) = [ γ 0; 0 I_i ],where we for brevity suppress their dependence on i. Note that for i = 0, they are just the identity maps for _n.The following theorem expands an automorphic form φ attached to a small automorphic representation of _n in terms of highly degenerate Whittaker coefficients similar to how cusp forms on _n can be expanded in terms of Whittaker coefficients with the Piatetski-Shapiro–Shalika formula <cit.>. Expansion of non-cuspidal automorphic forms on _n in terms of Whittaker coefficients were discussed in <cit.>.Let π be a minimal or next-to-minimal irreducible automorphic representation of _n(), and let φ∈π. *If π=π_min, then φ has the expansionφ(g) = _N(F)\ N(𝔸)φ(ng) dn + ∑_i=1^n-1∑_γ∈Γ_i _N(F)\ N(𝔸)φ(n ι(γ) g) ψ^-1_α_i(n) dn . *If π=π_ntm,then φ has the expansionφ(g) = _N(F)N()φ(vg) dv + ∑_i=1^n-1∑_γ∈Γ_i_N(F) N()φ(v ι(γ) g) ψ^-1_α_i(v) dv ++ ∑_j=1^n-3∑_i=j+2^n-1∑_γ_i ∈Γ_i(ψ_α_j)γ_j ∈Γ_j_N(F) N()φ(v ι(γ_i) ι(γ_j) g) ψ^-1_α_j, α_i (v) dv. Note that the Whittaker coefficients in the last sum of case <ref> have characters supported on two strongly orthogonal (or commuting) simple roots.As mentioned in section <ref> and further described in <cit.>, the Whittaker coefficients are sums of Eulerian Whittaker coefficients on smaller subgroups _n, whose non-archimedean parts can be computed by the Casselman–Shalika formula <cit.>.The more degenerate a Whittaker coefficient is the smaller the subgroup we need to consider (and on which character becomes generic). Thus, maximally degenerate Whittaker coefficients, and the ones with characters supported on two commuting simple roots become particularly simple and are, in principle, one, or a product of two, known _2 Whittaker coefficients respectively.Next, we consider Fourier coefficients on maximal parabolic subgroups.Let P_m the maximal parabolic subgroup of SL_n with respect to the simple root α_m and let U = U_m be the unipotent radical and L_m be the corresponding Levi subgroup which stabilizes U_m under conjugation.For an element l∈ L_m(F) and a character ψ_U on U_m we obtain another character ψ_U^l by conjugation asψ_U^l(u) = ψ_U(lul^-1) .Fourier coefficients ℱ_U with conjugated characters are related by l-translates of their argumentsℱ_U(φ, ψ_U^l; g)= _U(F)U()φ(ug) ψ_U^-1(lul^-1) du = _U(F)U()φ(l^-1u'lg) ψ_U^-1(u') du' = ℱ_U(φ, ψ_U; lg) ,where we have first made the variable substitution u' = lul^-1 and then used the automorphic invariance since l ∈ L_m(F).This means that we only need to compute the Fourier coefficients of one character per L_m(F)-orbit.We show in section <ref> that a character can be parametrized by an element y ∈ g by (<ref>) denoted by ψ_y, which, under conjugation, satisfies ψ_y^l = ψ_l^-1 y l according to (<ref>).In section <ref> and appendix <ref>, we describe these orbitsfollowing <cit.> and construct standard characters ψ_y(Y_r(d)) on U_m based on anti-diagonal (n-m) × m rank r matrices Y_r(d), where d ∈ F^×/(F^×)^2 for n = 2r = 2m and d = 1 otherwise (in which case we suppress the d), and y(Y_r(d)) is defined asy(Y_r(d)) =[0_m0; Y_r(d)0_n-m ] . Let π be a minimal or next-to-minimal automorphic representations, andr_π = 1if π is a minimal automorphic representation2if π is a next-to-minimal automorphic representation. We will show that only the characters with rank r ≤ r_π≤ 2 give non-vanishing Fourier coefficients. Let us briefly define the characters with rank r ≤ 2 which will be used in the next theorem, postponing a more general definition to section <ref>. The rank zero character is the trivial character ψ_y(Y_0) = 1 and the corresponding Fourier coefficient has been computed in <cit.> as reviewed in <cit.>. The rank one character is ψ_y(Y_1) = ψ_α_m and the rank two character can be defined as followsψ_y(Y_2)(u) = ψ(u_m,m+1 + u_m-1, m+2)u ∈ U_m() . The following theorem, together with the known constant term, then allows us to compute any Fourier coefficient with respect to the unipotent radical of a maximal parabolic subgroups for automorphic forms attached to minimal and next-to-minimal automorphic representations in terms of Whittaker coefficients.Let π be a minimal or next-to-minimal irreducible automorphic representation of _n(), and let r_π be 1 or 2 respectively (which denotes the maximal rank of the character matrix Y_r). Let also, φ∈π, P_m be the maximal parabolic subgroup described above with its associated subgroups U≡ U_m and L_m, and let ψ_U be a non-trivial character on U_m with Fourier coefficient ℱ_U(φ, ψ_U; g) = ∫_U_m(F) U_m()φ(ug) ψ_U^-1(u) du. Then, there exists an element l ∈ L_m(F) such that ℱ_U(φ, ψ_U; g) = ℱ_U(φ, ψ_y(Y_r(d)); lg) for some standard character ψ_y(Y_r(d)) described above and in the proof.Additionally, all ℱ_U(φ, ψ_y(Y_r(d)); lg) for r > r_π vanish identically. The remaining (non-constant) coefficients can be expressed in terms of Whittaker coefficients on N as follows. *If π = π_min:ℱ_U(φ, ψ_y(Y_1); g)= _N(F)N() φ(ng) ψ_α_m^-1(n) dn . *If π = π_ntm:ℱ_U(φ, ψ_y(Y_1); g)= [t] _[N] φ(ng) ψ_α_m^-1(n) dn ++ ∑_j=1^m-2 ∑_γ∈Λ_j(ψ_α_m)_[N] φ(nι̂(γ) g) ψ_α_j, α_m^-1(n) dn ++ ∑_i=m+2^n-1 ∑_γ∈Γ_i(ψ_α_m) _[N] φ(n ι(γ) g) ψ_α_m, α_i^-1(n) dn .*If π = π_ntm:ℱ_U(φ, ψ_y(Y_2); g)= _C()_N(F)N() φ(nωcg)ψ_α_1,α_3^-1(n)dn dc,where ω is the Weyl element mapping the torus elements (t_1, t_2, …, t_n) ↦ (t_m-1, t_m+2, t_m, t_m+1, t_1, t_2, …, t_m-2, t_m+3, t_m+4, …, t_n) ,and the subgroup C of U_m will be detailed in the proof in section <ref>.As described in detail in section <ref>, F-rational nilpotent orbits of _n are characterized by ( p, d) where p is a partition of n and d ∈ F^×/(F^×)^k with k = ( p).If k = 1 we will often suppress the extra d = 1 and only write out the partition.There we will also see that, for each orbit, there are natural choices of unipotent subgroups and characters related by conjugations with elements γ∈_n(F) and the corresponding Fourier coefficients (<ref>) are related by γ-translates of their arguments. The orbits may be partially ordered and the minimal and next-to-minimal orbits are described by the partitions [21^n-2] and [2^21^n-4], respectively.Besides the trivial partition, these are the only partitions whose associated Fourier coefficients are non-vanishing for φ in a minimal or next-to-minimal irreducible automorphic representation.In section <ref> we choose standard representatives for these orbits and specify the associated standard Fourier coefficients which we denote by ℱ^[211…] and ℱ^[221…]. For n≥ 5, we have that the trivial, minimal and next-to-minimal orbit all have k=1.The following theorems express these standard Fourier coefficients associated with the two partitions above in terms of Fourier coefficients on maximal parabolic subgroups that, in turn, were written in terms of Whittaker coefficients in theorem <ref>. Let π be an irreducible automorphic representation of _n(), φ∈π and Y = Π_i=3^n X_e_i - e_2.Then,ℱ^[211…](φ; g) = ∑_y ∈ Y(F) _U_1(F) U_1()φ(u y^-1 g) ψ_α_1^-1(u) du,where U_1 is the unipotent radical of P_1 consisting of the first row of N. The Fourier coefficient ℱ^[211…] is for a particular standard choice of orbit representative detailed in the proof; all other choices are related simply by _n(F) translation. Let π be an irreducible automorphic representation of _n(), φ∈π, Y' = ∏_i=5^n X_e_i-e_4∏_i=5^n X_e_i-e_3 and ω be the Weyl element mapping the torus elements (t_1, t_2, …, t_n) ↦ (t_1, t_3, t_4, t_2, t_5, t_6, …, t_n).Then,ℱ^[221…](φ; g) = ∑_y ∈ Y'(F) _U_2(F) U_2()φ(u y^-1ω g) ψ_y(Y_2)^-1(u) du ,where U_2 is the unipotent radical of P_2 consisting of the first two rows of N and ψ_y(Y_2) is defined in (<ref>) with m = 2. The Fourier coefficient ℱ^[2^21…] is for a particular standard choice of orbit representative detailed in the proof; all other choices are related simply by _n(F) translation.§.§ Applications in string theory String theory is a quantum theory of gravity describing maps X: Σ→ M, where Σ is a Riemann surface (the string worldsheet) and M is a ten-dimensional pseudo-Riemannian manifold (spacetime). Its low-energy limit is a supersymmetric extension of Einstein's theory of gravity in 10 dimensions coupled to additional matter in the form of scalar fields Φ : M→ℂ and differential forms on spacetime M. Our main focus here will be the scalar fields. The scalar fields parametrize the space of string theory vacua, i.e. the moduli space ℳ. To make contact with a lower-dimensional world, one choice is to decompose spacetime into M=ℝ^1,9-n× T^n , where ℝ^1,9-n is the flat Minkowski space in 10-n dimensions and T^nis an n-dimensional torus. In the limit when the size of the torus is small, the physics looks effectively (10-n)-dimensional and one says that the theory has been compactified. As the size of the torus isincreased the moduli space ℳ gets larger and larger due to an increased number of scalar fields Φ. The moduli space for this toroidal compactification is always of the form ℳ=G(ℤ)\G(ℝ)/K ,where G(ℝ) is a semi-simple Lie group in its split real form, Kits maximal compact subgroup and G(ℤ) an arithmetic subgroup. The group G(ℤ) is known as the U-duality group and is a symmetry of the full quantum string theory. The extreme case when n=0, i.e. for no compactification, the moduli space is given by ℳ=_2(ℤ)\_2(ℝ)/SO_2. Another extreme case is n=6, corresponding to four space-time dimensions, for which the moduli space is given by <cit.>ℳ= E_7(ℤ)\E_7(ℝ)/(SU_8/ℤ_2) .Here E_7(ℝ) is the split real form and E_7(ℤ) its Chevalley group of integer points. The sequence of groups in between are obtained by successively removing nodes from the E_7 Dynkin diagram; see table <ref> for the complete list.Constraints from U-duality and supersymmetry ensure that certain quantum corrections to Einstein's gravitational theory involve functions f : ℳ→ℂ that must be eigenfunctions of the ring of G(ℝ)-invariant differential operators. In particular they are eigenfunctions of the Laplacian on G(ℝ)/K withspecific eigenvalues. In addition, they must havewell-behaved growth properties in certain limits corresponding to `cusps' of ℳ. Such quantum corrections are therefore controlled by automorphic forms on ℳ.It turns out that the relevant automorphic forms are very special and are precisely those attached to a minimal and next-to-minimal automorphic representation of the groups G <cit.>. The Fourier coefficients of such automorphic forms therefore have a direct physical interpretation: the constant terms encode perturbative quantum corrections, while the non-constant terms correspond to non-perturbative, instanton, effects <cit.>. For a recent book on automorphic representations and the connection with string theory, see <cit.>.Fourier coefficients with respect to different choices of parabolic subgroups P⊂G correspond to different limits in string theory, and reveal different types of effects. The ones of main interest are certain maximal parabolic subgroups. Let P_α= L_α U_α denote the maximal parabolic whose Levi subgroup is L_α = M_α×_1, where M_α is obtained by removing the node in the Dynkin diagram of G corresponding to the simple root α. There are three types of maximal parabolics of main interestin string theory (the numbering of nodes are according to the Bourbaki convention of the exceptional Lie algebras):* P_α_1: this is the perturbative, or string theory, limit where the Levi is of orthogonal type M_α_1=D_n; * P_α_2: this is the M-theory limit where the Levi is of type M_α_2=A_n; * P_α_n+1: this is the decompactification limit where the Levi is of exceptional type M_α_n+1=E_n (for n<6 these are strictly speaking not exceptional, but given by table <ref>). Theorem <ref>, together with its counterpart in <cit.>, then provides explicit results for the Fourier coefficients of automorphic forms in all these parabolics for the cases n=2 or n=3 when the symmetry groups are _2×_3 or _5, respectively. The case of _5will be treated in detail in section <ref>. §.§ Acknowledgements We have greatly benefitted from many discussions with Dmitry Gourevitch, Joseph Hundley, Stephen D. Millerand Siddhartha Sahi. We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University during the program on “Automorphic forms, mock modular forms and string theory” in the fall of 2016 during which part of the research for this paper was performed. The fourth named author is partially supported by NSF grant DMS-1702218 and by a start-up fund from the Department of Mathematics at Purdue University. § NILPOTENT ORBITS AND FOURIER COEFFICIENTS In this section, first, we introduce Whittaker pairs and nilpotent orbits with their associated Fourier coefficients following <cit.>, which is slightly more general and easier to use than the one given in <cit.>.Then we recall the parametrization of F-rational nilpotent orbits of _n in terms of partitions of n from <cit.> and a lemma for exchanging roots in Fourier integrals from <cit.>.As before, let F be a number field,be the adele ring of F and fix a non-trivial additive character ψ on F.Let alsobe a reductive group defined over F, or a central extension of finite degree, and let g be the Lie algebra of (F). For a semi-simple element s ∈ g, let g^s_i be defined as the eigenspace of s in g with eigenvalue i under the adjoint action decomposing g to a direct sum of eigenspaces over different eigenvalues. For any r ∈, we further define g^s_≥ r = ⊕_r' ≥ r g^s_r' and similarly for other inequality relations. For an element X ∈ g, we will also denote the centralizer of X in g asg_X = { x ∈ g| [x,X]=0} . Furthermore, a semi-simple element s is called rational semi-simple if all of its eigenvalues under the adjoint action on g are in . For such a rational semi-simple element s and a non-trivial nilpotent element u ∈ g^s_-2 we call the ordered pair (s, u) a Whittaker pair. If, for such a pair, s is also called a neutral element for u or (s, u) a neutral pair if the map g^s_0 → g^s_-2 : X ↦ [X,u] is surjective or, equivalently <cit.>, s ∈(ad(u)).An sl_2-triple is an ordered triple (u, s, v) of elements in g that satisfy the standard commutation relations for sl_2, [s, v] = 2v[s, u] = - 2u[v, u] = s ,where u is called the nil-negative element, v is called the nil-positive element and s is a neutral element for u.We have, from <cit.>, that a Whittaker pair (s,u) comes from an sl_2-triple (u,s,v) if and only if s is a neutral element for u.By the Jacobson–Morozov theorem, there exists an sl_2 triple for any nilpotent element u ∈ g.Moreover, the -conjugacy classes of sl_2-triples are one-to-one with the nilpotent orbits 𝒪_X = {gXg^-1| g ∈(F)} in g <cit.>.We will now construct the Fourier coefficient that is associated to a Whittaker pair (s, u). The pair defines a unipotent subgroup N_s and a character ψ_u on N_s as follows. Following <cit.>, letn_s = g^s_>1⊕g^s_1 ∩g_u ,which is a nilpotent subalgebra of g, and define N_s = exp( n_s) as the corresponding unipotent subgroup of . Then ψ_u, defined byψ_u(n) = ψ(⟨ u, log(n)⟩)n ∈ N_s() ,is a character on N_s() where ⟨·, ·⟩ is the Killing form.Note that if the Whittaker pair (s,u) comes from an 𝔰𝔩_2-triple (u,s,v), then, by sl_2 representation theory, (s) has integer eigenvalues with a graded decomposition of the Lie algebra g = ⊕_i ∈g^s_i and g_u ⊂⊕_i ≤ 0g^s_i <cit.>, and thus, 𝔫_s=𝔤^s_≥ 2 (for neutrals). Let π be an automorphic representation of () and φ an automorphic form attached to π. The Fourier coefficient associated with a Whittaker pair (s, u) is_s,u(φ)(g) = _N_s(F)N_s()φ(ng)ψ^-1_u(n)dn,g ∈() ,and let _s,u(π)={_s,u(φ) |φ∈π}.For convenience, we introduce the following notation for a unipotent subgroup U[U] = U(F)U() . Consider the Fourier coefficient associated with a neutral Whittaker pair (s, u), and let (s', u') = (γ sγ^-1, γ u γ^-1) which is also neutral for any γ∈ G(F).Because of the invariance of the Killing form we have that ψ_u'(n') = ψ_u(γ^-1 n' γ) where n' ∈ [N_s'], and because of (<ref>) we have that N_s' = γ N_s γ^-1. Thus, with a variable substitution n' = γ n γ^-1,_s',u'(φ)(g)= _[γ N_s'γ^-1]φ(n'g) ψ^-1_u(γ^-1 n' γ) dn = _[N_s]φ(γ n γ^-1 g)ψ_u^-1(n) dn = _s,u(φ)(γ^-1g) ,using the automorphic invariance of φ.Note the resemblance with (<ref>) where we made a conjugation keeping N_s invariant. In particular, (<ref>) means that if _s,u vanishes identically then so do all Fourier coefficients associated to neutral Whittaker pairs (s',u') where u' ∈_u. For an F-rational nilpotent orbit , we say that the coefficients _s,u with neutral s and u ∈ are Fourier coefficients attached to the nilpotent orbit .We define the (global) wave-front set 𝒲ℱ(π) of an automorphic representation π of () as the set of nilpotent orbits 𝒪 such that ℱ_s,u(π) is non-zero, for some (and therefore all) neutral Whittaker pairs (s,u) with u ∈𝒪. Note that nilpotent orbits can be partially ordered with respect to the inclusion of Zariski closures ' ≤ if '⊆.We recall <cit.> as follows.Let π be an automorphic representation of (), let (s,u) be a Whittaker pair, and (h, u) a neutral Whittaker pair such that _h,u(π) is zero.Then, _s,u(π) is zero. This means that if u ∈ where 𝒲ℱ(π) then, for any Whittaker pair (s, u), not necessarily neutral, the associated Fourier coefficient ℱ_s, u(φ) vanishes identically for φ∈π.In this paper, we focus on the group _n where we parametrize a character on N_s by u ∈ g^s_-2 as ψ_u(n)=ψ( (ulog(n)))n ∈ N_s().Then, for any l in the normalizer of N_s() in ()ψ_y^l(x)= ψ_y(l x l^-1) = ψ(( y log(l x l^-1))) = ψ(( y l log(x) l^-1)) = ψ((l^-1 y l log(x))) = ψ_l^-1 y l(x) . The nilpotent orbits of _n can be described by partitions p of n. Let us characterize the F-rational orbits of _n following <cit.>. Let p=[p_1 p_2 ⋯ p_r] be an ordered partition of n, with p_1 ≥ p_2 ≥…≥ p_r and let m = (p)=(p_1, p_2, …, p_r).For d ∈ F^×, define D(d) = (1, 1, …, 1, d) and let also J_p be the standard (lower triangular) Jordan matrix corresponding to p: J_p = (J_[p_1], J_[p_2], …, J_[p_3]), where J_[p] is a p× p matrix with non-zero elements only on the subdiagonal which are one.* For each d ∈ F^×, the matrix D(d)J_p is a representative of an F-rational nilpotent orbit of _n parametrized by p, and conversely, every orbit parametrized by p has a representative of this form.We say that the F-rational orbit represented by D(d)J_p is parametrized by (p,d). * The _n(F)-orbits represented by D(d)J_p and D(d')J_p' coincide if and only if p=p' and d ≡ d' in F^×/(F^×)^m. The F-rational orbit ([322], 1) of _7 is represented byJ_[322]=(J_[3], J_[2], J_[2])=[ 0 0 0 0 0 0 0; 1 0 0 0 0 0 0; 0 1 0 0 0 0 0; 0 0 0 0 0 0 0; 0 0 0 1 0 0 0; 0 0 0 0 0 0 0; 0 0 0 0 0 1 0 ] . Over F the F-rational orbits for different d become the same, meaning that they are completely characterized by partitions of n. There is partial ordering for partitions that agrees with the partial ordering of the F-orbits, where [p_1p_2… p_r] ≤ [q_1q_2… q_r] (possibly padded by zeroes) if <cit.>∑_1≤ j ≤ n p_j ≤∑_1≤ j ≤ n q_j for1 ≤ n ≤ r .The Zarisky topology over F is induced from that of F which means that we can use this partial ordering of partitions for the F-rational orbits as well.Thus, when discussing the partial ordering of orbits or the closure of orbits we will sometimes not specify the F-rational orbit, but only the partition, that is, the _n(F)-orbit.An automorphic representation π of _n() is called minimal if 𝒲ℱ(π) is the set of orbits in the closure of the minimal (non-trivial) orbit which is represented by the partition [21^n-2], and it is called next-to-minimal if it is instead the set of orbits in the closure of the next-to-minimal orbit [2^21^n-4].We will now recall a general lemma for exchanging roots in Fourier coefficients from <cit.>.In <cit.>, the groups considered are quasi-split classical groups, but the lemma holds for any connected reductive group with exactly the same proof.Let G be a connected reductive group defined over F and let C be an F-subgroup of a maximal unipotent subgroup of G.Let also ψ_C be a non-trivial character on [C] = C(F)C(), and X, Y two unipotent F-subgroups satisfying the following conditions: * X and Y normalize C;* X ∩ C and Y ∩ C are normal in X and Y, respectively, (X ∩ C)X and (Y ∩ C)Y are abelian;* X() and Y() preserve ψ_C under conjugation;* ψ_C is trivial on (X ∩ C)() and (Y ∩ C)();* [X, Y] ⊂ C;* there is a non-degenerate pairing (X ∩ C)()× (Y ∩ C)() →^× (x,y)↦ψ_C([x,y])which is multiplicative in each coordinate, and identifies [4](Y ∩ C)(F)Y(F) with the dual of X(F)(X ∩ C)()X(), and (X ∩ C)(F)X(F) with the dual of Y(F)(Y ∩ C)()Y(). Let B =CY and D=CX, and extend ψ_C trivially to characters of [B]=B(F) B() and [D]=D(F) D(), which will be denoted by ψ_B and ψ_D respectively. Assume that (C, ψ_C, X, Y) satisfies all the above conditions. Let f be an automorphic form on G(). Then for any g ∈G(), ∫_[B] f(vg) ψ_B^-1(v) dv = ∫_(Y ∩ C) ()Y()∫_[D] f(vyg) ψ_D^-1(v)dv dy .For simplicity, we will use ψ_C to denote its extensions ψ_B and ψ_D when using the lemma. § PROOF OF THEOREM <REF> Before we prove Theorem <ref> in this section, let us first introduce a few definitions and useful lemmas. Let V_i be the unipotent radical of the parabolic subgroup of type (1^i,n-i), that is, the parabolic subgroup with Levi subgroup (_1)^i × GL_n-i together with a determinant one condition. Then, N= V_n = V_n-1 is the unipotent radical of the Borel subgroup and V_i can be seen as the first i rows of N. For 1 ≤ i ≤ n-1, let α_i = e_i-e_i+1 be the i-th simple root of _n, and let ψ_α_i be the character of N defined by ψ_α_i(n)= ψ(n_i,i+1), ∀ n ∈ N() .For a list of simple roots, we let ψ_α_i_1, …, α_i_m = ψ_α_i_1⋯ ψ_α_i_m and we also regard ψ_α_j for j ≤ i as a character of V_i via restriction. Also, let R_i+1 be the subgroup of V_i+1, consisting of the elements v with conditions that v_p,q=0, for all 1 ≤ p ≤ i and p < q ≤ n, that is R_i+1 consists of the row i+1 in V_i+1. It is clear that R_i+1 V_iV_i+1 is an abelian subgroup of V_i+1.For a character ψ_N on N, we say that ψ_N is trivial along a simple root α_i if the restriction of ψ_N to R_i is identically zero. For _5 we have that V_3 = {[ 1 * * * *; 1 * * *; 1 * *; 1; 1; ]} R_3 = {[ 1; 1; 1 * *; 1; 1 ]}.Thus, we have that [R_i] ≅ (F )^n-i and the dual of [R_i] is F^n-i, which can be identified with the nilpotent subalgebra ^tr_i(F) = log(^tR_i(F)), where ^tR_i(F) is the transpose of R_i(F). Given y ∈^t r_i(F), the corresponding character ψ_y on [R_i] is given by (<ref>) asψ_y(x) = ψ( (y log x)), ∀ x ∈ [R_i] .For SL_5 with R_3 above, let y = [ 0; 0; 0; y_1 0; y_2 0 ]∈^t r_3(F) x =[ 1; 1; 1 x_1 x_2; 1; 1 ]∈ [R_3] . Then, ψ_y(x) = ψ((y log x)) = ψ(y_1 x_1 + y_2 x_2). Define(·) = (·) -1/n((·))and let s = s_V_is_V_i = (2(i-1), 2(i-2), …, 0, -2, …, -2)for which g^s_1 = ∅ and n_s =g^s_≥ 2 with the corresponding N_s = V_i. In particular, we have s_N = s_V_n-1= (2(n-2), …, 0, -2)Let φ be an automorphic form on _n(). Then, for 1 ≤ i ≤ n-2,∑_y ∈^t r_i(F) y ≠ 0 _[R_i]φ(xg) ψ^-1_y(x) dx = ∑_γ∈Γ_i _[R_i]φ(x ι(γ) g) ψ^-1_α_i(x) dx ,where Γ_i is defined in (<ref>) and ι(γ) in (<ref>). We note that the left-hand side of the equation in this lemma equals φ(g) up to constant terms corresponding to y=0. With Y ∈_(n-i)× 1(F), we parametrize y ∈^t r_i(F) asy(Y) = [ 0_i-1 0 0; 0 0 0; 0 Y 0_n-i ].Let Ŷ = ^t(1, 0, …, 0) ∈_(n-i)× 1(F). Then the surjective map _n-i(F) →_(n-i) × 1(F)^× defined by γ↦γ^-1Ŷ gives that _(n-i) × 1(F)^× (_n-i(F))_Ŷ_n-i(F) = Γ_ifrom (<ref>). We then have that,∑_y ≠ 0 _[R_i]φ(xg) ψ^-1_y(x) dx = ∑_γ∈Γ_i_[R_i]φ(xg) ψ^-1_y(γ^-1Ŷ)(x) dx . We now rewrite the character using that for any Y ∈_n-i(F)y(γ^-1Y) = [0_i-100;000;0 γ^-1 Y0_n-i ] = [ I_i-1 0 0; 0 1 0; 0 0γ^-1 ][ 0_i-1 0 0; 0 0 0; 0 Y 0_n-i ][ I_i-1 0 0; 0 1 0; 0 0 γ ] =l^-1 y l ,where we have introduced l = ι(γ) and denoted y(Y) simply as y, which according to (<ref>) gives, for any x ∈ [R_i], thatψ_ y(γ^-1 Y)(x) = ψ_l^-1 y l(x) = ψ_y(l x l^-1) . The element l is in the Levi subgroup of the parabolic subgroup corresponding to V_i, meaning that it preserves V_i under conjugation. In particular, it also normalizes R_i since for x ∈ R_i parametrized by X ∈_1× n-1l x(X) l^-1 =[ I_i-1 0 0; 0 1 0; 0 0 γ ][ I_i-1 0 0; 0 1 X; 0 0 I_n-i ][ I_i-1 0 0; 0 1 0; 0 0γ^-1 ] = [I_i-100;01 X γ^-1;00I_n-i ] = x(X γ^-1). We can thus make the variable substitution lxl^-1→ x in (<ref>) to obtain∑_γ∈Γ_i_[R_i]φ(x l g) ψ^-1_y(Ŷ)(x) dx ,where we have used the fact that φ is left-invariant under l^-1. Noting that ψ_y(Ŷ) = ψ_α_i this proves the lemma. We will now state a similar lemma for the last row R_n-1, that needs to be treated separately.The freedom in choosing a character ψ_0 in this lemma will be of importance later.Let φ be an automorphic form on _n(). Then, for any character ψ_0 on N trivial onR_n-1 and along (at least) two adjacent simple roots not including α_n-1,∑_y ∈^t r_n-1(F) y ≠ 0 _[R_n-1]φ(xg) ψ^-1_y(x) dx = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(x ι(γ) g) ψ^-1_α_n-1(x) dx ,where Γ_n-1(ψ_0) is defined in (<ref>). With Y ∈ F, we parametrize y ∈^t r_n-1(F) asy(Y) = [ 0_n-2 0 0; 0 0 0; 0 Y 0 ]. We recall from page tpsi0 that T_ψ_0 is the subgroup of diagonal elements in _n(F) stabilizing ψ_0 under conjugation of its argument and that y ∈^tr_n-1(F)F.The map T_ψ_0→^tr_n-1(F)^× : h ↦ h^-1 y(1) h is surjective, which can be shown as follows. The character ψ_0 is, by assumption, trivial along at least two adjacent simple roots not including α_n-1.Pick such a pair α_j-1 and α_j where 2 ≤ j ≤ n-2 and for an arbitrary m ∈ F^× let h = (1, …, 1, m, 1, … 1, 1/m) where the first non-trivial element is at the jth position.Then h ∈ T_ψ_0 since y_0 ∈^tn corresponding to ψ_0 is zero at both rows and columns j and n and h ↦ y(m) Because of (<ref>) we have that the centralizer of y(1) in T is T_ψ_α_n-1, and thus,^tr_n-1(F)(T_ψ_0∩ T_ψ_α_n-1)T_ψ_0 = Γ_n-1(ψ_0) . We then have that ∑_y ∈^t r_n-1(F) y ≠ 0 _[R_n-1]φ(xg) ψ^-1_y(x) dx = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(xg) ψ^-1_γ^-1 y(1) γ(x) dx = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(xg) ψ^-1_y(1)(γ x γ^-1) = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(x γ g) ψ^-1_α_n-1(x) ,after making the variable change γ x γ^-1→ x, which concludes the proof. For n ≥ 5 any character ψ_0 on N that is non-trivial along at most a single simple root which is not α_n-1 satisfies the character condition in lemma <ref>. The following lemma will be used to iteratively expand in rows. The lemma, which is valid for any automorphic representation, will be followed by two corollaries that specialize to the minimal and next-to-minimal representations respectively.Let φ be an automorphic form on _n(), 1 ≤ i ≤ n-2, and ψ_0 be a character on N trivial on the complement of V_i in N. For i = n-2 we also require that ψ_0 is trivial along (at least) two adjacent simple roots not including α_n-1. Then, _[V_i]φ(vg) ψ^-1_0(v) dv = _[V_i+1]φ(vg) ψ^-1_0(v) dv ++ ∑_γ∈Γ_i+1(ψ_0) _[V_i+1]φ(v ι(γ) g) ψ^-1_0(v) ψ^-1_α_i+1(v) dv . For x ∈ R_i+1(F) and v ∈ V_i() we have that φ(xvg) = φ(vg) and can thus Fourier expand along the abelian unipotent R_i+1 asφ(vg) = ∑_y ∈^tr_i+1(F)_[R_i+1]φ(xvg) ψ^-1_y(x) dx .Then, using lemma <ref> (for i+1 ≤ n-2) or lemma <ref> (for i+1 = n-1)φ(vg) = _[R_i+1]φ(xvg) dx + ∑_γ∈Γ_i+1(ψ_0) _[R_i+1]φ(xι(γ)vg) ψ^-1_α_i+1(x) dx. Let v ∈ V_i be parametrized asv =[ A B; 0 I_n-i-1 ] ,where A ∈_(i+1) × (i+1) is upper unitriangular and B ∈_(i+1) × (n-i-1) with the elements in the last row being zero.Since B does not intersect the abelianization [N,N] N (that is, the Lie algebra of B does not contain any generator of a simple root), we have, by assumption, that ψ_0 only depends on A.Similarly, we parametrize x ∈ R_i+1 asx = [ I_i+1B'; 0 I_n-i-1 ] ,where B' ∈(i+1) × (n-i-1) with non-zero elements only in the last row.Then,xv = [ AB + B'; 0 I_n-i-1 ],which means that ψ_0(v) = ψ_0(xv), and since ψ_α_i+1 only depends on the first column in B' which is the same as for B + B', we also have that ψ_α_i+1(x) = ψ_α_i+1(xv). * For 1 ≤ i ≤ n-3 with γ∈Γ_i+1, l = ι(γ) is in the Levi subgroup corresponding to V_i and we will now show that ψ_0(l^-1 v l) = ψ_0(v) for v ∈ [V_i].We have thatl^-1vl =[ I_i+1 0; 0γ^-1 ][ A B; 0 I_n-i-1 ][ I_i+1 0; 0 γ ] =[ A B γ; 0 I_n-i-1 ]and ψ_0(v) only depends on A. * For i = n-2 with γ∈Γ_n-1(ψ_0), l = ι(γ) = γ is in the stabilizer T_ψ_0 which normalizes V_i and, by definition, means that ψ_0(v) = ψ_0(lvl^-1). Thus, for 1 ≤ i ≤ n-2,_[V_i]φ(vg) ψ^-1_0(v) dv = _[V_i]_[R_i+1]φ(xvg) ψ^-1_0(v) dxdv ++ ∑_γ∈Γ_i+1(ψ_0) _[V_i]_[R_i+1]φ(x v l g) ψ^-1_α_i+1(x) ψ^-1_0(v) dx dv ,where we have made the variable change l v l^-1→ v.Using that R_i+1 V_i = V_i+1 the above expressions simplifies to_[V_i+1]φ(vg) ψ^-1_0(v) dv + ∑_γ∈Γ_i+1(ψ_0) _[V_i+1]φ(v ι(γ) g) ψ^-1_0(v) ψ^-1_α_i+1(v) dv . Let π be an irreducible minimal automorphic representation of _n(), φ∈π, and ψ_0 be a character on N trivial on the complement of V_i in N, 1 ≤ i ≤ n-2.Then, ℱ_ψ_0∫_[V_i]φ(vg) ψ^-1_0 dv can be further expanded as follows.*If ψ_0 = 1, thenℱ_ψ_0= _[V_i+1] φ(vg) dv + ∑_γ∈Γ_i+1 _[V_i+1] φ(v ι(γ) g) ψ^-1_α_i+1(v)dv, where Γ_i+1(ψ_0) with Γ_i+1 = Γ_i+1(1)is defined in (<ref>).*If ψ_0 = ψ_α_j (1 ≤ j ≤ i), thenℱ_ψ_0= _[V_i+1] φ(vg) ψ^-1_0(v)dv. We will use lemma <ref> where all the considered ψ_0 satisfy the character condition for the last row according to remark <ref>.For ψ_0 = 1, the expression is already in the form of lemma <ref>.This proves case <ref>.For ψ_0 = ψ_α_j with 1 ≤ j ≤ i we have that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_i+1(v) = ψ_u(v) for some u ∈ g which is in the next-to-minimal orbit.Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes for φ in the minimal representation which leaves only the constant (or trivial) mode in lemma <ref>.This proves case <ref>. [2]Let π be an irreducible next-to-minimal automorphic representation of _n(), φ∈π, and ψ_0 be a character on N trivial on the complement of V_i in N, 1 ≤ i ≤ n-2. Then, ℱ_ψ_0∫_[V_i]φ(vg) ψ^-1_0 dv can be further expanded as follows.*If ψ_0 = 1, thenℱ_ψ_0= _[V_i+1] φ(vg) dv + ∑_γ∈Γ_i+1 _[V_i+1] φ(vι(γ) g) ψ^-1_α_i+1(v)dv . *If ψ_0 = ψ_α_j (1 ≤ j < i), thenℱ_ψ_0= _[V_i+1] φ(vg) ψ^-1_α_j(v) dv +∑_γ∈Γ_i+1(ψ_α_j) _[V_i+1] φ(vι(γ) g) ψ^-1_α_j, α_i+1(v)dv . *If ψ_0 = ψ_α_i, thenℱ_ψ_0= _[V_i+1] φ(vg) ψ^-1_α_i(v)dv . *If ψ_0 = ψ_α_j, α_k (1 < j+1 < k ≤ i), thenℱ_ψ_0= _[V_i+1] φ(vg) ψ^-1_α_j, α_k(v)dv . Where Γ_i+1(ψ_0) with Γ_i+1 = Γ_i+1(1) is defined in (<ref>). We will use lemma <ref> where the considered ψ_0 in cases <ref>–<ref> satisfy the character condition for the last row according to remark <ref>.* For ψ_0 = 1, the expression is already in the form of lemma <ref>. This proves case <ref>. * For ψ_0 = ψ_α_j with 1 ≤ j < i we get that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_i+1(v). This proves case <ref>. * For ψ_0 = ψ_α_i we get that ψ_0(v) ψ_α_i+1(v) = ψ_α_i, α_i+1(v) = ψ_u(v) for some u ∈ g belonging to an orbit higher than the next-to-minimal.Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes both for φ in the minimal and next-to-minimal representations which leaves only the constant mode in lemma <ref>.This proves case <ref>. * Lastly, for ψ_0 = ψ_α_j,α_k with 2 ≤ j+1 < k ≤ i we first consider i ≤ n-3 with lemma <ref>.We get that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_k, α_i+1(v) = ψ_u(v) for some u ∈ g belonging to an orbit higher than the next-to-minimal. Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes for φ in next-to-minimal representation which leaves only the first term in (<ref>). For i = n-2, we expand along the last row and obtain a sum over characters ψ_u = ψ_0 ψ_y on N for all y ∈^tr_n-1(F) where only y = 0 gives a u ∈ g belonging to an orbrit in the closure of the next-to-minimal orbit. Again, using theorem <ref> only the constant mode remains. This proves case <ref> and completes the proof. Since φ(x_1g) = φ(g) for x_1 ∈ V_1(F) we can make a Fourier expansion on V_1 and then use lemma <ref> to obtainφ(g) = _[V_1]φ(v g) dv + ∑_γ_1 ∈Γ_1_[V_1]φ(v ι(γ_1) g) ψ^-1_α_1(v) dv . We will now make an iteration in the rows of the nilpotent, starting with the row i = 1 and continue until we reach the last row i = n - 1.* For case <ref>, that is, with φ in the minimal representation, the first step, using corollary <ref>, isφ(g) = [V_2]φ(vg) dv + ∑_γ_2 ∈Γ_2 _[V_2]φ(v ι(γ_2) g)ψ^-1_α_2(v) dv ++ ∑_γ_1 ∈Γ_1 _[V_2]φ(v ι(γ_1) g) ψ^-1_α_1(v) dv ,where we note that the extra second term comes from the constant term on V_1. We will, after the iteration end up withφ(g) = _[N]φ(ng) dn + ∑_i=1^n-1∑_γ∈Γ_i _[N]φ(n ι(γ) g) ψ^-1_α_i(n) dn .This completes the proof for the minimal representation. * For case <ref>, where φ is in the next-to-minimal-representation, we start again from (<ref>) and expand using corollary <ref>. We get, for the first step, thatφ(g) = ( _[V_2]φ(v g) dv + ∑_γ_2∈Γ_2_[V_2]φ(v ι(γ_2) g) ψ^-1_α_2(vg) dv ) ++∑_γ_1 ∈Γ_1_[V_2]φ(v ι(γ_1) g) ψ^-1_α_1(v) dv ,where the parenthesis comes from the expansion of the constant term in (<ref>). Expanding in the next row as well, this becomes(_[V_3]φ(vg) dv + ∑_γ_3∈Γ_3 _[V_3]φ(v ι(γ_3) g) ψ^-1_α_3(v) dv + ∑_γ_2∈Γ_2 _[V_3]φ(v ι(γ_2) g) ψ^-1_α_2(v) dv )++ ∑_γ_1 ∈Γ_1( _[V_3]φ(v ι(γ_1) g) ψ^-1_α_1(v) dv + ∑_γ_3 ∈Γ_3(ψ_α_1) _[V_3]φ(v ι(γ_3) ι(γ_1) g) ψ^-1_α_1, α_3(v) dv ) . For each expansion adding a row i, the constant term gives an extra sum over Γ_i of a Fourier integral with character ψ_α_i, and from all terms with characters ψ_α_j with j < i - 1 we get an extra sum over Γ_i(ψ_α_j) together with a character ψ_α_j, α_i.Corollary <ref> <ref> implies that these terms with characters non-trivial along two simple roots do not receive any further contributions.Thus, after repeatedly using corollary <ref> to the last row, we get thatφ(g) = _[N]φ(ng) dn + ∑_i=1^n-1∑_γ∈Γ_i _[N]φ(n ι(γ) g) ψ^-1_α_i(n) dn ++ ∑_j=1^n-3∑_i=j+2^n-1∑_γ_i ∈Γ_i(ψ_α_j)γ_j ∈Γ_j _[N]φ(n ι(γ_i) ι(γ_j) g) ψ^-1_α_j, α_i (n) dn ,which completes the proof of Theorem <ref>.§ PROOF OF THEOREM <REF> In this section, we prove Theorem <ref> which relates Fourier coefficients on a maximal parabolic subgroup with Whittaker coefficients on the Borel subgroup. Recalling that the constant terms are known from <cit.>, we only focus on non-trivial characters, but first we need to introduce some notation and lemmas. For 1 ≤ m ≤ n-1, let U_m be the unipotent radical of the maximal parabolic subgroup P_m with Levi subgroup L_m isomorphic to the subgroup of _m ×_n-m defined by {(g,g')∈_m ×_n-m: (g) (g')=1}. U_m is abelian and is isomorphic to the set of all m × (n-m) matrices. Write U_m as U_m = {[ I_m X; 0 I_n-m ] : X ∈_m × (n-m)}. Let U_m = ^t U_m be the unipotent radical of the opposite parabolic P_m. Then the Lie algebra of U_m can be written as𝔲_m = ^tu_m ={y(Y) = [ 0_m 0; Y 0_n-m ] : Y ∈_(n-m) × m}.It is clear that the character group of U_m can be identified with ^tu_m.L_m acts on ^tu_m via conjugation and with (<ref>) this becomes a conjugation of the corresponding character's argument.Because of (<ref>), the Fourier coefficients for characters in the same L_m(F)-orbit are related by translates of their arguments, which means that we only need to compute one Fourier coefficient for each orbit. We will therefore now describe the L_m(F)-orbits of elements y(Y) ∈^tu_m but leave the details to be proven in appendix <ref>.Starting first with F the number of L_m( F)-orbits is min(m,n-m)+1 and the orbits are classified by the rank of the (n-m) × m matrix Y.A representative of an L_m( F)-orbit corresponding to rank r can be chosen as y(Y_r) where Y_r is an (n-m) × m matrix, zero everywhere except for the upper right r × r submatrix which is anti-diagonal with all anti-diagonal elements equal to one. For each rank r, 0 ≤ r ≤min(m,n-m), the corresponding G(F)-orbit is parametrized by the partition [2^r 1^n-2r]. As shown in appendix <ref>, the L_m(F)-orbits are characterized by the same data as the G(F)-orbits with ([2^r 1^n-2r], d), 0 ≤ r ≤min(m,n-m), d ∈ F^×/(F^×)^k and k ∈([2^r 1^n-2r]) with representatives y(Y_r(d)) where Y_r(d) is of the same form as Y_r above, but with the lower left element in the r × r matrix equal to dY_r(d) =[0 [ 1; ⋱; 1; d ];00 ]∈_(n-m) × m(F).We will continue to write Y_r(1) = Y_r.Note that for 0 ≤ r ≤ 2 and n ≥ 5, k is equal to 1. Each such L_m(F)-orbit is also part of the G(F)-orbit of the same data. From (<ref>) the corresponding character on U_m isψ_y(Y_r)(u)=ψ( (y(Y_r) log (u))),u ∈ U_m().Let s_m be the semisimple element (1,1,…, 1, -1,-1, …, -1) with m copies of 1's and (n-m) copies of -1's. Then, for any automorphic form φ on _n(), the following Fourier coefficient∫_[U_m]φ(ug) ψ_y(Y_r(d))^-1(u)duis exactly the degenerate Fourier coefficient _s_m,y(Y_r(d))(φ). Note that in this paper, we focus on minimal and next-to-minimal representations, hence we only need to consider the cases of 0 ≤ r ≤ 2. Indeed, for 3 ≤ r ≤min(m,n-m), by definition, the generalized Fourier coefficient attached to the partition [2^r1^n-2r] is identically zero for minimal and next-to-minimal representations. By Theorem <ref> and since y(Y_r(d)) is in the G(F)-orbit [2^r1^n-2r], all the Fourier coefficients _s_m,y(Y_r)(φ) are also identically zero.This leaves r ∈{1, 2} and with our assumption that n ≥ 5, we thus only need to consider the representatives y(Y_1) and y(Y_2) with d = 1 since ([2^r1^n-2r])=1. The above arguments proves the first part of theorem <ref>, that there exists an element l ∈ L_m(F) such that ℱ_U(φ, ψ_U; g) = ℱ_U(φ, ψ_y(Y_r); lg) (note the slight difference in notation ψ_y(Y_r) instead of ψ_Y_r), and that all ℱ_U(φ, ψ_y(Y_r); lg) for r > r_π vanish identically where r_π_min = 1 and r_π_ntm = 2.We will now determine the remaining Fourier coefficients ℱ_U(φ, ψ_y(Y_r); g) in terms of Whittaker coefficients. For 1 ≤ m ≤ n-1, 0 ≤ i ≤ m-1, let U_m^i be the unipotent radical of the parabolic of type (m-i,1^i,n-m). Note that U_m^0=U_m.Note that the character ψ_y(Y_1) can be extended to a character of any subgroup of N containing U_m, still denoted by ψ_y(Y_1).Let C_m-i be the subgroup of U_m^i+1 consisting of elements with u_p,q=0 except when q = m-i and the diagonal elements. Note that C_m-i is an abelian subgroup and its character group can be identified with ^t 𝔠_m-i, the Lie algebra of ^t C_m-i.Write C_m-i as C_m-i = { c(X) =[ I_m-i-1 X 0; 0 1 0; 0 0 I_n-m+i ]}and ^t 𝔠_m-i as^t 𝔠_m-i = { y(Y)= [ 0_m-i-1 0 0; Y 0 0; 0 0 0_n-m+i ]}. For each y ∈^t 𝔠_m-i, the corresponding character ψ_y of C_m-i is defined by ψ_y(c)=ψ((y log(c)).For any g ∈_m-i-1, let ι(g)= [ g 0 0; 0 I_n-m+i 0; 0 0(g)^-1 ]∈_n .For _5 we have thatU_3 = {[ 1 * *; 1 * *; 1 * *; 1; 1 ]} U_3^1 = {[ 1 * * *; 1 * * *; 1 * *; 1; 1 ]} C_3 = {[ 1 *; 1 *; 1; 1; 1 ]}.Note that U_m^m-1 = V_m and U_m^i+1 = C_m-i U_m^i. We will sometimes use j = m-i instead to denote column as follows U_m^m-j+1 = C_j U_m^m-j.We will now construct a semi-simple element s = s_U_m^i for which g^s_1 = ∅ and such that n_s =g^s_≥ 2 corresponds to N_s = U_m^i. These conditions are satisfied by(2i, …, 2i, 2(i-1) …, 2, 0, -2, …, -2)with m-i copies of 2i and n-m copies of -2.Note that any character ψ on N trivial on the complement of U_m^i in N is also a character on U_m^i by restriction and can be expressed as ψ_y with y ∈ g^s_-2 where s = s_U_m^i such that (s,y) forms a Whittaker pair.Indeed, we have that y ∈ g^s_N_-2 where s_N = (2(n-1), 2(n-2), …, 0,-2) from (<ref>) and the complement of U_m^i is described by s-s_N meaning that [y, s-s_N] = 0 for ψ to be trivial on the complement and thus [y, s] = [y, s_N] = -2y. Let φ be an automorphic form on _n() and 2 ≤ j ≤ n. Let also ψ_0 be a character on N which, if j = 2, should be trivial along α_1 and (at least) two adjacent other simple roots. Then, ∑_y ∈^tc_j(F) y ≠ 0_[C_j]φ(xg) ψ_y^-1(x) dx = ∑_γ∈Λ_j-1(ψ_0)_[C_j]φ(x ι̂(γ) g) ψ^-1_α_j-1(x) dx .where Λ_j(ψ_0) is defined in (<ref>) and only depends on ψ_0 for j = 2.The proof is similar to those of lemmas <ref> and <ref>. * For 2 < j ≤ n, we parametrize y ∈^tc_j(F) by row vectors Y ∈_1×(j-1)(F) with representative X̂ = (0, …, 0, 1) such that ψ_y(X̂) = ψ_α_j-1. The surjective map _j-1(F) →^tc_j(F)^× : γ↦X̂γ gives that ^tc_j(F)^× (_j-1(F))_X̂_j-1(F) = Λ_j-1. As in lemma <ref>, we can write the action as a conjugation y(X̂γ) = ι̂(γ)^-1 y(X̂) ι̂(γ) and, using (<ref>), ψ_y(X̂γ)(x) = ψ_y(X̂)(ι̂(γ) x ι̂(γ)^-1). Since ι̂(γ) normalizes C_j a variable change gives the wanted expression. * For j = 2, with y ∈^tc_2(F)F we instead consider the map T_ψ_0→^tc_2(F)^× : h ↦ h^-1 y(1) h which is surjective by similar arguments as in lemma <ref> and thus gives ^tc_2(F)^× (T_ψ_0∩ T_ψ_α_1)T_ψ_0 = Λ_1(ψ_0). Writing the conjugation of y as a conjugation of the character's argument and then substituing variables in the Fourier integral thus proves the lemma.Let φ be an automorphic form on _n(), 1 ≤ m ≤ n-1, 2 ≤ j ≤ m and ψ_0 a character on N trivial on the complement of U_m^m-j in N.For j = 2, ψ_0 should also be trivial along (at least) two adjacent simple roots other than α_1._[U_m^m-j]φ(ug) ψ_0^-1(u) du = = _[U_m^m-j+1]φ(ug) ψ_0^-1(u) du + ∑_γ∈Λ_j-1(ψ_0) _[U_m^m-j+1]φ(u ι̂(γ) g) ψ_0^-1(u) ψ_α_j-1^-1(u) du . For 2 ≤ j ≤ m we have that φ(xug) = φ(ug) for x ∈ C_i(F) and u ∈ U_m^i() and since C_j is abelianφ(ug) = ∑_y ∈^tc_j(F)_[C_j]φ(xug) ψ_y^-1(x) dx .Using lemma <ref>, we get thatφ(ug) = _[C_j]φ(xug) dx + ∑_γ∈Λ_j-1(ψ_0)_[C_j]φ(x ι̂(γ) ug) ψ_α_j-1^-1(x) dx .Let u ∈ U_m^m-j be parametrized asu = [ I_j-1 B; 0 A ]where A ∈_(n-j+1)×(n-j+1) is upper unitriangular (with several upper triangular elements being zero) and B ∈_(j-1)×(n-j+1) with elements in the first column being zero.Since B does not intersect the abelianization [N,N] N (that is, the Lie algebra of B does not contain any generator of a simple root), we have, by assumption, that ψ_0 only depends on A. We also have that x ∈ C_j can be parametrized asx = [ I_j-1B'; 0 I_n-j+1 ]where B' ∈_(j-1)×(n-j+1) with only the first column non-zero. Thus,xu = [I_j-1 B + B' A;0A ]which means that ψ_0(u) = ψ_0(xu). The first column of B is zero and A is upper unitriangular which means that the first column of B+B'A is the same as the first column of B' and since ψ_α_j-1 only depends on the first column of B' this implies that ψ_α_j-1(x) = ψ_α_j-1(xu).* For3 ≤ j ≤ m with γ∈Λ_j-1 and l = ι̂(γ),l u l^-1 =[ γ 0; 0 I_n-j+1 ][ I_j-1 B; 0 A ][γ^-1 0; 0 I_n-j+1 ] =[ I_j-1 γ B; 0 A ]and since ψ_0, by assumption, only depends on A we have that ψ_0(u) = ψ_0(l u l^-1).* For j = 2 with γ∈Λ_1 and l = ι̂(γ) = γ is in the stabilizer T_ψ_0 which, by definition, means that ψ_0(u) = ψ_0(lul^-1). Hence, for 2 ≤ j ≤ m, and after making a variable change lul^-1→ u, we get that_[U_m^m-j]_[C_j]φ(x l ug) ψ_0^-1(u) ψ_α_j-1^-1(x)dx du == _[U_m^m-j]_[C_j]φ(xulg) ψ_0^-1(u) ψ_α_j-1^-1(x)dx du = _[U_m^m-j]_[C_j]φ(xulg) ψ_0^-1(xu) ψ_α_j-1^-1(xu)dx du = _[U_m^m-j+1]φ(ulg) ψ_0^-1(u) ψ_α_j-1^-1(u)du . After similar manipulations for the constant term we obtain_[U_m^m-j]φ(ug) ψ_0^-1(u) du = = _[U_m^m-j+1]φ(ug) ψ_0^-1(u) du + ∑_γ∈Λ_j-1(ψ_0) _[U_m^m-j+1]φ(ulg) ψ_0^-1(u) ψ_α_j-1^-1(u)du .We note that if ψ_0 is trivial along α_1 but not along at least two adjacent other simple roots we cannot use lemma <ref>, but we could still make an expansion over C_2 and keep the sum over y ∈^tc_2(F)F in the proof above. Since the character ψ_y has the same support as ψ_α_1 on N we still have that ψ_y(x) = ψ_y(xu) for x ∈ C_j() and u ∈ U_m^m-j() and since ψ_0 is still a character on N trivial on the complement of U_m^m-j it is still true that ψ_0(u) = ψ_0(xu). Thus, using (<ref>)_[U_m^m-2]φ(ug) ψ_0^-1(u) du = ∑_y ∈^tc_2(F)_[U_m^m-2]_[C_2]φ(xug) ψ_0^-1(u) ψ_y^-1(x)dx du = ∑_y ∈^tc_2(F)_[U_m^m-2]_[C_2]φ(xug) ψ_0^-1(xu) ψ_y^-1(xu)dx du = ∑_y ∈^tc_2(F)_[V_m]_[C_2]φ(vg) ψ_0^-1(v) ψ_y^-1(v) dv . Assume that π is an irreducible minimal automorphic representation of _n(), φ∈π.For 1 ≤ m ≤ n-1, 0 ≤ i ≤ m-2, and g ∈_n(), ∫_[U_m^i]φ(ug) ψ^-1_y(Y_1)(u)du = ∫_[U_m^i+1]φ(ug) ψ^-1_y(Y_1)(u)du .Using lemma <ref> with ψ_0 = ψ_y(Y_1) = ψ_α_m we get that_[U_m^i]φ(ug) ψ_0^-1(u) du = _[U_m^i+1]φ(ug) ψ_0^-1(u) du + ∑_γ∈Λ_m-i-1(ψ_0)ℱ(φ; m,i,γ,g) ,where we have introducedℱ(φ; m,i,γ,g) = _[U_m^i+1]φ(u ι̂(γ) g) ψ_0^-1(u) ψ_α_m-i-1(u) du . Let s = s_U_m^i+1 from (<ref>), and let u ∈sl_n with two non-zero entries, both being 1, at positions (m-i, m-i-1) and (m+1, m). Then, ℱ(φ; m,i,γ,g) = ℱ_s,u(φ)(ι̂(γ)g) and since u is not in the closure of the minimal orbit, theorem <ref> gives that ℱ_s,u(φ) is identically zero leaving only the constant mode in (<ref>). [3]Proof of Theorem <ref>.* Minimal representation. Assume that π be an irreducible minimal automorphic representation of _n(), and φ∈π. Applying Lemma <ref> repeatedly,we get that for each 1 ≤ m ≤ n-1, ∫_[U_m]φ(ug)ψ_y(Y_1)^-1(u) du = ∫_[U_m^m-1]φ(ug)ψ_y(Y_1)^-1(u) du .Note that U_m^m-1=V_m and ψ_y(Y_1) = ψ_α_m.Applying corollary <ref> repeatedly,we get that for each 1 ≤ m ≤ n-1, ∫_[U_m^m-1]φ(ug)ψ_y(Y_1)^-1(u) du= ∫_[N]φ(ng)ψ_y(Y_1)^-1(n) dn ,which is exactly∫_[N]φ(ng)ψ_α_m^-1(n) dn .* Next-to-minimal representation - rank 1.Let π be an irreducible next-to-minimal automorphic representation of _n() and let φ∈π. Recalling that U_m = U_m^0 and applying lemma <ref> with ψ_0 = ψ_y(Y_1) = ψ_α_m we get_[U_m]φ(ug) ψ_y(Y_1)^-1(u) du = _[U_m^1]φ(ug) ψ_0^-1(u) dusince ψ_0 ψ_α_m-1 = ψ_α_m,α_m-1 = ψ_u for some u that is not in the closure of the next-to-minimal orbit and thus the non-constant modes in lemma <ref> can be expressed as Fourier coefficients ℱ_s,u with s = s_U_m^1 from (<ref>) which vanish according to theorem <ref>.Let us make an iteration in 1 ≤ i ≤ m-2. Using lemma <ref> we have that_[U_m^i]φ(ug) ψ_0^-1(u) du = = _[U_m^i+1]φ(ug) ψ_0^-1(u) du + ∑_γ∈Λ_m-i-1(ψ_α_m) _[U_m^i+1]φ(uι̂(γ) g) ψ_α_m-i-1, α_m^-1(u) du . Since ψ_α_m, α_m-i-1 is a character on N trivial on the complement of U_m^i+1 we can expand the second term further with lemma <ref> (or remark <ref> if m-i-1 = 2 and ψ_α_m, α_m-i-1 is not trivial along at least two adjacent roots other than α_1). This would lead to characters ψ_u = ψ_α_m, α_m-i-1, α_m-i-2 [3](or ψ_u = ψ_α_m, α_m-i-1ψ_y with y ∈^tc_2(F) respectively) where u is not in the closure of the next-to-minimal orbit.Then, ℱ_s,u with s = s_U_m^i+2 from (<ref>) vanishes according to theorem <ref> and the second term only receives the constant mode contribution. Repeating these arguments for the second term in (<ref>), it becomes∑_γ∈Λ_m-i-1(ψ_α_m) _[V_m]φ(uι̂(γ) g) ψ_α_m-i-1, α_m^-1(u) du . Iterating over i, starting from i = 1 above, we get that_[U_m]φ(ug) ψ_y(Y_1)^-1(u) du =_[V_m]φ(ug) ψ_α_m^-1(u) du + ∑_j=1^m-2∑_γ∈Λ_j(ψ_α_m) _[V_m]φ(uι̂(γ) g) ψ_α_j, α_m^-1(u) du . For m = 1, U_1 = V_1 and for m = 2 we only get the first term in (<ref>).We will now use the methods of section <ref> to expand along rows. Using corollary <ref> case <ref>, we see that the second term in (<ref>) does not get any further contributions when expanding to N. Starting with the first term in (<ref>) and using corollary <ref> first with case <ref> to V_m+1 and then repeatedly with cases <ref> and <ref> it becomes_[V_m+1]φ(ug) ψ_α_m^-1(u) du = = _[N]φ(ng) ψ_α_m^-1(n) dn + ∑_i=m+2^n-1∑_γ∈Γ_i(ψ_α_m) _[N]φ(n ι(γ) g) ψ_α_m, α_i^-1(n) dn . Lastly,_[U_m]φ(ug) ψ_y(Y_1)^-1(u) du = _[N]φ(ng) ψ_α_m^-1(n) dn + + ∑_j=1^m-2∑_γ∈Λ_j(ψ_α_m) _[N]φ(nι̂(γ) g) ψ_α_j, α_m^-1(n) dn + + ∑_i=m+2^n-1∑_γ∈Γ_i(ψ_α_m) _[N]φ(n ι(γ) g) ψ_α_m, α_i^-1(n) dn .* Next-to-minimal representation - rank 2. Let π be an irreducible next-to-minimal automorphic representation of _n() and let φ∈π.We start from the integral∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u)du . For each root α, let X_α be the corresponding one-dimensional root subgroup in _n. Let C_1 = X_e_m-e_m+2∏_i=1^m-2 X_e_i - e_m+2 ,and R_1 =X_e_m-1-e_m∏_i=1^m-2 X_e_m-1 - e_i .Then C_1 is a subgroup of U_m. Let U_m' be the subgroup of U_mwith C_1-part identically zero. Then one can see that the quadruple (U_m', C_1, R_1, ψ_y(Y_2))satisfies all the conditions of Lemma <ref>. By this lemma, ∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u) du= ∫_C_1()∫_[R_1U_m']φ (ucg) ψ_y(Y_2)^-1(u)du dc . Let C_2 = ∏_i=1^m-2 X_e_i - e_m+1 ,and R_2 =∏_i=1^m-2 X_e_m - e_i .Then C_2 is a subgroup of R_1U_m'. Let U_m” be the subgroup of R_1U_m'with C_2-part identically zero. Then one can see that the quadruple (U_m”, C_2, R_2, ψ_y(Y_2))satisfies all the conditions of Lemma <ref>. Applying this lemma and by changing of variables, ∫_C_1()∫_[R_1U_m']φ (ucg) ψ_y(Y_2)^-1(u)du dc= ∫_C_1()∫_C_2()∫_[R_2U_m”]φ (uc_2c_1g) ψ_y(Y_2)^-1(u)du dc_2 dc_1= ∫_(C_1C_2)()∫_[R_2U_m”]φ (ucg) ψ_y(Y_2)^-1(u)du dc . Let ω be the Weyl element sending torus elements (t_1, t_2, …, t_n)to torus elements (t_m-1, t_m+2, t_m, t_m+1, t_1, t_2, …, t_m-2, t_m+3, t_m+4, …, t_n) .Conjugating ω cross from left, the integral in (<ref>) becomes∫_C()∫_[U_m^ω]φ (uω cg) ψ_y(Y_2)^ω, -1(u)du dc ,where U_m^ω = ω R_2U_m”ω^-1, C=C_1C_2, for u ∈ U_m^ω,ψ_y(Y_2)^ω(u) = ψ_y(Y_2)(ω^-1 u ω).U_m^ω = U_m^ω,1V_1 ,where elements u ∈ U_m^ω,1 have the following form[ I_2 0; 0u' ] ,and U_m^ω,1 normalizes V_1.Recall thatV_i be unipotent radical of parabolic subgroup of type (1^i,n-i). Note that ψ_y(Y_2)^ω|_V_1=ψ_α_1,ψ_y(Y_2)^ω|_U_m^ω,1=ψ_α_3.Recall that α_1=e_1-e_2, α_3=e_3-e_4.Hence, the integral in (<ref>) becomes∫_C()∫_[U_m^ω,1]∫_[V_1]φ (vuω cg)ψ_α_1^-1(v) ψ_α_3^-1(u)dv du dc .Since π is an irreducible next-to-minimal automorphic representation of _n(),by corollary <ref>, case <ref>,the integral in (<ref>) becomes∫_C()∫_[U_m^ω,1]∫_[V_2]φ (vuω cg)ψ_α_1^-1(v) ψ_α_3^-1(u)dv du dc .U_m^ω,1 still normalizes V_2, and U_m^ω,1V_2 = U_m^ω,2V_3 ,where elements u ∈ U_m^ω,2 have the following form[ I_4 0; 0u” ] ,u” is in the radical of the parabolic subgroup of type (m-2,n-m-2) in _n-4,and U_m^ω,2 normalizes V_3.Note that ψ_y(Y_2)^ω|_V_3=ψ_α_1,α_3 andψ_y(Y_2)^ω|_U_m^ω,2 is the trivial character.By corollary <ref>, case <ref>,the integral in (<ref>) becomes∫_C()∫_[U_m^ω,2]∫_[V_4]φ (vuω cg)ψ_α_1,α_3^-1(v)dv du dc . Applying corollary <ref>, case <ref>, repeatedly, the integral in (<ref>) becomes∫_C()∫_[U_m^ω,2]∫_[N]φ (nuω cg)ψ_α_1,α_3^-1(n)dn du dc ,which becomes ∫_C()∫_[U_m^ω,2]∫_[N]φ (nω cg)ψ_α_1,α_3^-1(n)dn du dc ,by changing of variables.Since ∫_[U_m^ω,2]du=1, we have obtained that ∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u)du = ∫_C()∫_[N]φ (nω cg)ψ_α_1,α_3^-1(n)dn dc . This completes the proof of Theorem <ref>.§ PROOF OF THEOREMS <REF> AND <REF>Proof of Theorem <ref>. Let π be any irreducible automorphic representation of _n() and let φ∈π. The generalized Fourier coefficient of φ attached to the partition [21^n-2] has been defined in Section <ref>. We recall it as follows. Let s=(1, -1, 0, …, 0), and let u = J_[21^n-2] which is a matrix zero everywhere except the (2,1) entry being 1.Then the generalized Fourier coefficient of φ attached to the partition [21^n-2] is as follows:ℱ^[211…] (φ;g)= _s,u (φ;g)= ∫_[N_s]φ(ng)ψ_u^-1(n)dn ,where elements in the one-dimensional unipotent N_s have the form [ 1 * 0; 0 1 0; 0 0 I_n-2 ] . Let X=∏_i=3^n X_e_1-e_i and Y=∏_i=3^n X_e_i-e_2.Then one can see that Y(F) can be identified with the character space of [X] as follows: given y ∈ Y(F), ψ_y(x)=ψ_u([x,y]), for any x ∈ [X].Note that both X and Y normalize N_s. Taking the Fourier expansion of _s,u (φ)(g) along [X], we obtain that _s,u (φ;g)=∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xng)ψ_u^-1(n) ψ_y^-1(x) dn dx .Since y^-1∈ Y(F) and φ is automorphic,the above integral becomes∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xng)ψ_u^-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(y^-1xng)ψ_u^-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(y^-1xnyy^-1g)ψ_u^-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn' y^-1g)ψ_u^-1(n) ψ_y^-1(x) dn dx ,where n'=n+[x,y].By changing variables, we obtain that ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn' y^-1g)ψ_u^-1(n) ψ_y^-1(x)dndx= ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn y^-1g)ψ_u^-1(n)ψ_u^-1(-[x,y])ψ_y^-1(x) dn dx . Note that ψ_u^-1(-[x,y])ψ_y^-1(x)= ψ_u([x,y])ψ_u(-[x,y]) = 1 .Hence, we have that _s,u (φ;g) = ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn y^-1g)ψ_u^-1(n) dn dx .Note that XN_s=U_1 and ψ_u=ψ_α_1.Therefore, we have that _s,u (φ;g) = ∑_y∈ Y(F)∫_[U_1]φ(u y^-1g)ψ_α_1^-1(u) du . This completes the proof of Theorem <ref>.Proof of Theorem <ref>. Let π be any irreducible automorphic representation of _n() and let φ∈π. The generalized Fourier coefficient of φ attached to the partition [2^21^n-4] has also been defined in Section <ref>. We recall it as follows. Let s=(1, -1, 1, -1, 0, …, 0), and let u = J_[2^21^n-4] which is a matrix zero everywhere except the (2,1) and (4,3) entries being 1.Then the generalized Fourier coefficient of φ attached to the partition [2^21^n-4] is as follows:ℱ^[221…](φ;g) =_s,u (φ;g)= ∫_[N_s]φ(ng)ψ_u^-1(n)dn ,where elements in N_s have the form [ 1 * 0 * 0; 0 1 0 0 0; 0 * 1 * 0; 0 0 0 1 0; 0 0 0 0 I_n-4 ] .Let ω be the Weyl element sending the torus element(t_1, t_2, …, t_n)to the torus element(t_1, t_3, t_4, t_2, t_5, t_6, …, t_n) .Conjugating ω across from left, we obtain that _s,u (φ;g)= ∫_[N_s^ω]φ(nω g)ψ_u^ω,-1(n)dn ,where N_s^ω = ω N_s ω^-1, and for n ∈ N_s^ω, ψ_u^ω(n) =ψ_u(ω^-1 n ω).Elements in n ∈ N_s^ω have the following form n=n(z)=[ I_2 z 0; 0 I_2 0; 0 0 I_n-4 ] ,and ψ_u^ω(n)=ψ(z_1,2+z_2,1).Let X'=∏_i=5^n X_e_1-e_i∏_i=5^n X_e_2-e_i and Y'=∏_i=5^n X_e_i-e_4∏_i=5^n X_e_i-e_3 .Then one can see that Y'(F) can be identified with the character space of [X'] as follows: given y ∈ Y'(F), ψ_y(x)=ψ_u^ω([x,y]), for any x ∈ [X'].Note that both X' and Y' normalize N_s. Taking the Fourier expansion of _s,u (φ)(g) along [X'], we obtain that _s,u (φ;g)=∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx .Since y^-1∈ Y'(F) and φ is automorphic,the above integral becomes∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(y^-1xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(y^-1xnyy^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn' y^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx ,where n'=n+[x,y].By changing variables, we obtain that ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn' y^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx= ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn y^-1ω g)ψ_u^ω,-1(n)ψ_u^ω,-1(-[x,y])ψ_y^-1(x) dn dx . Note that ψ_u^ω,-1(-[x,y])ψ_y^-1(x)= ψ_u^ω([x,y])ψ_u^ω(-[x,y]) = 1 .Hence, we have that _s,u (φ;g) = ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn y^-1ω g)ψ_u^-1(n) dn dx .Note that X'N_s^ω=U_2 and ψ_u=ψ_y(Y_2), using the notation from section <ref>. Therefore, we have that _s,u (φ;g) = ∑_y∈ Y'(F)∫_[U_2]φ(u y^-1ω g)ψ_y(Y_2)^-1(u) du dx . This completes the proof of Theorem <ref>.§ APPLICATIONSAs is evident from table <ref>, the case _5 appears in the list of symmetry and duality groups in string theory. It is related to compactification of type II string theory on a three-torus T^3 from ten to seven spacetime dimensions. Fourier coefficients of automorphic forms on _5 are related to non-perturbative effects as discussed in the introduction. Therefore we analyse here in some detail the structure of Fourier coefficients for automorphic forms attached to a minimal or next-to-minimal automorphic representation of _5 that are relevant to the first two higher-derivative corrections in four-graviton scattering amplitudes.We will give a detailed description of how the formalism developed above can be used to calculate explicit expressions for Fourier coefficients on maximal parabolic subgroups for automorphic forms attached to a minimal or next-to-minimal automorphic representation. Following a general discussion, we will treat two explicit examples for n = 5. §.§ GeneralitiesWith applications to string theory in mind, throughout this section we are restricting to F = and let ≡_. The types of expressions that are of interest are of the form:ℱ^(φ, ψ; g) = _U()U()φ(ug) ψ^-1(u) du ,where U() is a parabolic subgroup of G(), ψ is some rank-1 or rank-2 character on U() and φ is an automorphic form in the minimal- or next-to-minimal automorphic representations of G(). Any such coefficient can be brought to a standard form using the action of the arithmetic Levi subgroup L(). For rank-1 this form is ψ = ψ_y(kY_1) for some integer k≠ 0 and for rank-2 one has ψ(k_1 u_m,m+1+k_2 u_m-1,m+2) for integers k_1 and k_2, cf. (<ref>). For simplicity, we will restrict ourselves to the case Y_y(Y_2) corresponding to k_1=k_2=1 and demonstrate how to apply theorem <ref>. The techniques demonstrated here allow for the calculation of all such Fourier coefficients for automorphic forms in the minimal and next-to-minimal representations on _n.In order to apply theorem <ref>, we first perform an adelic lift <cit.>ℱ^(φ, ψ; g_∞) = ℱ^(φ, ψ; (g_∞, I_n, I_n, ⋯)) = _[U]φ(u(g_∞, I_n, I_n, …)) ψ^-1(u) du.The theorem now gives ℱ^ in terms of adelic Whittaker functions. These Whittaker functions will then be evaluated using the adelic reduction formulaW_ψ(λ, a) = ∑_w_c w_0' ∈𝒞_ψa^(w_c w_0')^-1λ + ρ M(w_c^-1, λ) W_ψ^a'(w_c^-1λ, 1)of <cit.>. The power of this formula lies in that it expresses a degenerate Whittaker function evaluated on the Cartan torus of a group G() as a sum of generic Whittaker functions on a subgroup G'(). This subgroup G'() is determined by deleting all nodes in the Dynkin diagram of G() on which ψ is not supported. λ denotes the weight of the Eisenstein series, w_0' denotes the longest Weyl word on G', 𝒞_ψ denotes the set𝒞_ψ = { w ∈𝒲 |w Π' < 0 }where Π' is the set of simple roots of G' and w_c is hence the summation variable and corresponds to a specific representative of the quotient Weyl group 𝒲/𝒲' described in <cit.>. ρ denotes the Weyl vector, M denotes the intertwinerM(w, λ) = ∏_α > 0 wα < 0ξ(⟨λ | α⟩)/ξ(⟨λ | α⟩ + 1)as featured in the Langlands constant term formula, where ξ is the completed Riemann zeta function and ψ^a denotes the “twisted character” both defined in appendix <ref>.The evaluation of a real Fourier coefficient ℱ^ over a unipotent schematically looks likeℱ^(φ, ψ; g_∞) = ℱ^(φ, ψ; (g_∞, I_n, I_n, ⋯)) Adelic lift= ∑_ψ∑_l∈Λ orl∈Γ W_ψ(l(g_∞, I_n, I_n, ⋯)) Theorem <ref>= ∑_ψ∑_l∈Λ orl∈Γ W_ψ((n_∞ a_∞ k_∞, n_2 a_2 k_2, n_3 a_3 k_3, ⋯)) Iwasawa-decomposition= ∑_ψ( ∏_p≤∞ψ_p(n_p) ) ∑_l∈Λ orl∈Γ W_ψ((a_∞, a_2,a_3, ⋯))W_ψ(nak) = ψ(n) W_ψ(a) = ∑_ψψ_∞(n_∞) ∑_l∈Λ orl∈Γ∑_w a^… M(⋯) W_ψ^a'(⋯, 1) Reduction formula (<ref>) .The fourth line extracts the unipotent n_p-dependence at each of the local places p ≤∞. In the fifth line we have used that only the archimedean unipotent n_∞ contributes. The reason that the p-adic unipotent matrices n_p of the p-adic Iwasawa-decomposition of l ∈G(F) ⊂G(_p) above drop out is as follows. In using theorem <ref>, we will be faced with evaluating Whittaker functions such asW_α_j, α_m(ι̂(λ_j) g)for j ≤ m-2 whereλ_j∈Λ_jandW_α_m, α_i(ι(γ_i) g)for i ≥ m+2 whereγ_i ∈Γ_i.We have that γ_i and λ_j are embedded in _n as (cf. (<ref>))ι̂(λ_j) =( [ λ_j; I_n-j ]) andι(γ_i) =( [ I_i; γ_i ]).It is clear from their block-diagonal form that the unipotent n_p in the p-adic Iwasawa-decomposition of ι̂(λ_j) (and ι(γ_i)) will feature the same block-diagonal form. Since W_α_j, α_m (and W_α_m, α_i) is only sensitive to the unipotent on rows j and m ≥ j+2 > j (on rows i and m ≤ i-2 ≤ i), the block diagonal structure of n_p implies ψ_α_j, α_m; p(n_p) = 1 (and ψ_α_m, α_i; p(n_p) = 1).For a real matrix g ∈_n(), we will denote its Iwasawa-decompositiong = n_∞ a_∞ k_∞ =([1 x_12⋯⋯ x_1n; 1⋱⋱⋮;⋱⋱⋮; 1 x_n-1, n;1 ]) ([ y_1; y_2/y_1; ⋱; y_n-1/y_n-2; 1/y_n-1 ]) k_∞.Similarly, for a p-adic matrix g ∈_n(_p) we denote it asg = n_p a_p k_p =n_p ([η_1, p; η_2, p/η_1, p; ⋱; η_n-1, p/η_n-2, p;1/η_n-1, p ]) k_p.Appendix <ref> contains closed formulae for the x's and the y's, as well as a closed formula for the p-adic norm |η_i, p|_p of the η's.In what follows, we will make use of all formulae that are derived or stated in appendices <ref>, <ref> and <ref> along with the following notation * A prime on a variable, eg. x', generally denotes x' ≠ 0.* For sums we write ∑_x ≡∑_x ∈.* We write ∑_x' f(x) ≡∑_x ∈{0} f(x) and ∑_x' ∈ f(x) ≡∑_x ∈{0} f(x). Note that the prime is used to indicate whether or not zero is included in the sum but the prime is omitted in the summand.* For products we write ∏_p ≡∏_pprime. Writing ∏_p ≤∞ denotes the product over all primes p (the non-archimedean places) as well as the element p = ∞ (the archimedean place).* For x ∈ we denote x≡ e^2π i x. §.§ Example: Rank-1 coefficient of pimin on Palpha4 in SL5 Here, we will calculate the real rank-1 Fourier coefficient (<ref>) for the minimal Eisenstein series E(λ; g) with λ = 2sΛ_1 - ρ in the maximal parabolicP_α_4 = (4) ×(1) × U_α_4⊂(5) subject to ((4) ×(1)) = 1 ,associated with removing the “last” node in the Dynkin diagram of (5). The unipotent radical isU() = U_α_4() ={( [ 1 *; 1 *; 1 *; 1 *; 1 ]) }. Theorem <ref> gives for the unramified character ψ_y(Y_1) thatℱ^(E(2sΛ_1 - ρ), ψ_y(Y_1); g) =W_α_4(g) .The Whittaker function is found by the reduction formula with data given in table <ref>. In this case, there is no diagonally embedded rational matrix l, or equivalently l = I_5, in the general procedure and hence we have |η_1, p|_p = |η_2, p|_p = |η_3, 4|_p = |η_4, p|_p = 1. We getW_α_4(λ; (g_∞, I_5, I_5, ⋯)) = = x_45( y_4^5-2sξ( 2s-3 )/ξ( 2s )∏_p < ∞ |η_4, p|_p^5-2s) B_s-3/2( y_4^2/y_3, 1 ) ×∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-3/2)) 1-p^-2(s-3/2)+1|η_4, p^2/η_3, p|_p^2(s-3/2)-1/1-p^-2(s-3/2)+1= x_45 y_4^5-2s1/ξ( 2s ) 2 | y_4^2/y_3|_∞^s-2 K_s-2( 2π| y_4^2/y_3|_∞) =2 x_45 y_3^2-s y_4 1/ξ( 2s )K_s-2( 2π| y_4^2/y_3|_∞) = ℱ^(E(2sΛ_1 - ρ), ψ_y(Y_1); g_∞) .The x's and y's are the Iwasawa coordinates for the matrix g_∞ as in (<ref>). The function B_s that appears is a more compact way of writing the _2 Whittaker vector defined explicitly in (<ref>).Parameterizing g_∞ asg_∞ = ue =[ I_4 Q; 0 1 ][ r^-1/4e_4 0; 0 r ]where e_4 ∈_4() ,we get in particular thaty_3 = r^-3/4||N e_4|| and y_4 = r^-1 ,where N =[ 0 0 0 1 ] so that N e_4 is equal to the last row in e_4. This is obtained using the formula (<ref>). We get in particular thaty_3^2-s y_4 = r^2s-5( r^-5/4 ||N e_4 || )^s-2andy_4^2/y_3 = r^-5/4 ||N e_4 ||.The more general (real) ramified Fourier coefficient has the expression∫ E( 2sΛ_1 - ρ;[ 1 u_1; 1 u_2; 1 u_3; 1 u_4; 1 ] g_∞) m_1 u_1 + m_2 u_2 + m_3 u_3 + m'_4 u_4 d^4u = = x_45 r^2s-52/ξ(2s)σ_4-2s(k) ( r^-5/4 ||N e_4 || )^s-2 K_s-2( 2π r^-5/4||N e_4|| )= x_45 r^3s/4-5/22/ξ(2s)σ_2s-4(k)/|k|^s-2 ||Ñ e_4 ||^s-2 K_s-2( 2π|k|r^-5/4||Ñ e_4|| )for integer m's while for non-integer rational m's it vanishes. Here g_∞ has been parametrized as above, N =[m_1m_2m_3 m'_4 ]=k Ñ, k = (N) and m'_4 ≠ 0. This expression can also be found by starting from ψ_y(kY_1) for the standard Fourier coefficient instead. This corresponds to N =[ 0 0 0 k ]and its L() orbit gives the general expression (<ref>).Formula (<ref>) agrees with <cit.> where the Fourier coefficients were computed by Poisson resummation technique after a translation of conventions. §.§ Example: Rank-1 coefficient of pintm on Palpha4 in SL5 Here, we will calculate the real rank-1[There is no rank-2 character for this parabolic.] Fourier coefficient (<ref>) for the next-to-minimal Eisenstein series E(λ; g) with λ = 2sΛ_2 - ρ in the maximal parabolicP_α_4 = (4) ×(1) × U_α_4⊂(5) subject to ((4) ×(1)) = 1associated with removing the “last” node in the Dynkin diagram of (5). The unipotent radical isU() = U_β_4() ={( [ 1 *; 1 *; 1 *; 1 *; 1 ]) }. Theorem <ref> givesℱ^(E(2sΛ_2 - ρ), ψ_y(Y_1); g) = =W_α_4(g) + ∑_λ_1 ∈Λ_1 W_α_1, α_4(λ_1 g) + ∑_λ_2 ∈Λ_2 W_α_2, α_4(λ_2 g) =W_α_4(g) +∑_z' W_α_1, α_4( ( [ z; 1; 1; 1; 1/z ]) _l_z g ) +∑_x', y W_α_2, α_4(( [ x^-1;yx;I_3;]) _l_xy g ) +∑_x' W_α_2, α_4( ( [ 0 -x^-1; x 0; I_3; ]) _l_x g ) ,using the representatives derived in appendix <ref>.The first Whittaker function is found by the reduction formula with the data of table <ref>. In this case, there is no diagonally embedded rational matrix l, or equivalently l = I_5, and hence we have |η_1, p|_p = |η_2, p|_p = |η_3, 4|_p = |η_4, p|_p = 1. We get W_α_4(λ; (g_∞, I_n, I_n, ⋯)) = = x_45 B_s-1( y_4^2/y_3, 1 ) ( y_1^2s-1 y_4^4-2sξ( 2s-2 )/ξ( 2s )∏_p < ∞ |η_1, p|_p^2s-1|η_4, p|_p^4-2s. + . y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)^2/ξ(2s) ξ(2s-1)∏_p < ∞ |η_1, p|_p^3-2s |η_2, p|_p^2s-2 |η_4, p|_p^4-2s. + + . y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-3)ξ(2s-2)/ξ(2s)ξ(2s-1)∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s)∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1|η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1= x_45( y_1^2s-1 y_4^4-2s1/ξ( 2s ) + y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)/ξ(2s) ξ(2s-1). + . y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-3)/ξ(2s)ξ(2s-1)) 2 | y_4^2/y_3|_∞^s-3/2 K_s-3/2( 2π| y_4^2/y_3|_∞) =2 x_45( y_1^2s-1 y_3^3/2-s y_4 1/ξ( 2s ) + y_1^3-2s y_2^2s-2 y_3^3/2-s y_4 ξ(2s-2)/ξ(2s) ξ(2s-1). + . y_2^4-2s y_3^s-3/2 y_4 ξ(2s-3)/ξ(2s)ξ(2s-1)) K_s-3/2( 2π| y_4^2/y_3|_∞) .The x's and y's are the Iwasawa coordinates for the matrix g_∞ as in (<ref>). The second Whittaker function is found by the reduction formula with the data given in table <ref>. The p-adic Iwasawa-decomposition of l_z has|η_1, p|_p = |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = |z|_p.We get∑_z'W_α_1, α_4( λ; l_z (g_∞, I_n, I_n, ⋯) ) = = ∑_z' x_12 + x_45 y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)/ξ(2s) B_s-1/2( y_1^2/y_2, 1 ) B_s-1( y_4^2/y_3, 1 ) ∏_p < ∞ |η_1, p|_p^3-2s |η_2, p|_p^2s-2 |η_4, p|_p^4-2s∏_p < ∞γ_p( η_1, p^2/η_2, p) ( 1 - p^-2(s-1/2)) 1-p^-2(s-1/2)+1| η_1, p^2/η_2, p|_p^2(s-1/2)-1/1-p^-2(s-1/2)+1××∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1=∑_z' ∈ x_12 + x_45 y_1^3-2s y_2^2s-2y_4^4-2sξ(2s-1)/ξ(2s)∏_p < ∞ |z|_p^5-2s 4| y_1^2/y_2|_∞^s-3/2| y_4^2/y_3|_∞^s-2 K_s-3/2( 2π| y_1^2/y_2|_∞) K_s-2( 2π| y_4^2/y_3|_∞) σ_-2(s-1/2)+1(|z|_∞) σ_-2(s-1)+1(|z|_∞) =∑_z' ∈4 x_12 + x_45 y_2^s-1/2y_3^2-sξ(2s-1)/ξ(2s) |z|_∞^2s-5 K_s-3/2( 2π| y_1^2/y_2|_∞) K_s-2( 2π| y_4^2/y_3|_∞) σ_2-2s(|z|_∞) σ_3-2s(|z|_∞) .The x's and y's are the Iwasawa coordinates for the matrix l_z g_∞.The third and fourth Whittaker functions are found by the reduction formula with the data from table <ref>. The p-adic Iwasawa-decomposition of l_xy has|η_1, p|_p^-1 = max{|y|_p, |x|_p}and |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = 1. We get∑_x', yW_α_2, α_4( λ; l_xy (g_∞, I_n, I_n, ⋯) ) = = ∑_x', y x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-2)^2/ξ(2s)ξ(2s-1) B_s-1( y_2^2/y_1 y_3, 1 ) B_s-1( y_4^2/y_3, 1 )∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s∏_p < ∞γ_p( η_2, p^2/η_1, pη_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_2, p^2/η_1, pη_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1××∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1= ∑_x', y ∈ x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2s1/ξ(2s)ξ(2s-1) 4 | y_2^2/y_1 y_3|_∞^s-3/2| y_4^2/y_3|_∞^s-3/2 K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_-2(s-1)+1(k) = ∑_x', y ∈4 x_23 + x_45 y_1^3/2-s y_2^1 y_4^11/ξ(2s)ξ(2s-1) K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_3-2s(k) ,where k = (|y|, |x|). Here, the x's and y's are the Iwasawa coordinates for the matrix l_xy g_∞.The p-adic Iwasawa-decomposition of l_x has|η_1, p|_p^-1 = max{|0|_p, |x|_p} = |x|_p and |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = 1.We get∑_x'W_α_2, α_4( Λ; l_x (g_∞, I_n, I_n, ⋯) ) = = ∑_x' x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-2)^2/ξ(2s)ξ(2s-1) B_s-1( y_2^2/y_1 y_3, 1 ) B_s-1( y_4^2/y_3, 1 )∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s∏_p < ∞γ_p( η_2, p^2/η_1, pη_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_2, p^2/η_1, pη_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1= ∑_x' ∈ x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2s1/ξ(2s)ξ(2s-1) 4 | y_2^2/y_1 y_3|_∞^s-3/2| y_4^2/y_3|_∞^s-3/2 K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_-2(s-1)+1(|x|_∞) = ∑_x' ∈4 x_23 + x_45 y_1^3/2-s y_2^1 y_4^11/ξ(2s)ξ(2s-1) K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_3-2s(|x|_∞) .The x's and y's are the Iwasawa coordinates for the matrix l_x g_∞.The complete Fourier coefficient ℱ^(E(2sΛ_2 - ρ), ψ_y(Y_1); g_∞) is then given by the combination of (<ref>), (<ref>), (<ref>) and (<ref>). We note that the our final result differs formally from the one given in <cit.> where the result is given as a convoluted integral over two Bessel functions whereas we do not have any remaining integral. The two results need not be in actual disagreement as there are many non-trivial relations involving infinite sums or integrals of Bessel functions.The automorphic formlim_s→ 1/22ζ(3)ξ(2s-3)/ξ(2s) E(2sΛ_2-ρ;g) = 2ζ(3) E(3Λ_4-ρ)lies in a minimal automorphic representation and controls the first non-trivial corrections that string theory predicts to the four-graviton scattering amplitude beyond standard general relativity <cit.>. The Fourier coefficients that we computed above can then be used to to extract so-called 1/2 BPS instanton contributions in the string perturbation limit of the amplitude. More precisely, they represent non-perturbative corrections to the scattering amplitude that, albeit smooth, are not analytic in the string coupling constant around vanishing coupling. They are therefore not visible in standard perturbation theory for small coupling but represent important correction nonetheless. Their interpretation is in terms of specific Dp-branes (p≤ 2) that are extended (p+1)-dimensional objects that can extend on non-trivial cycles of the torus T^3 that is present when _5 is the duality group. The detailed structure of the Fourier coefficient, in particular the arithmetic divisor sums appearing, can shed some light on the combinatorics of these D-branes similar to what is happening in the _2 case <cit.>.For the next non-trivial correction to the four-graviton scattering amplitude one requires an automorphic form in the next-to-minimal automorphic representation <cit.>. This function is not a single Eisenstein series of the type we have analysed above but a very special combination of two formally divergent Eisenstein series with some Fourier coefficients computed using the Mellin transform of a theta lift in <cit.>. § EULER PRODUCTS AND TWISTED CHARACTERSThis appendix contains details and explanations for section <ref>, which is why we restrict to the field F = with the corresponding ring of adeles = _.An Euler product is a product over the primes. The p-adic norm is denoted |·|_p and is defined for the p-adic numbers _p. The absolute value norm or “infinity norm” is denoted |·|_∞ and is defined for real numbers = _∞. The p-adic numbers as well as the real numbers (being completions of the rational numbers) all contain the rational numbers: ⊂_p for all p prime. The norm of an adele x = (x_∞, x_2, x_3, x_5, …) ∈ is denoted |·| (without ornaments) and is the product of norms at the local places|x| = ∏_p ≤∞ |x_p|_p.The rational numbersare diagonally embedded into the adeles ⊂in the sense that (q, q, q, q, …) ∈for q ∈.Product of norms For a rational number x ∈ with a decomposition into primes as x = ±∏_p p^m^(p) ,we get a particularly simple result for the adelic norm of x, namely∏_p ≤∞ |x|_p = |x|_∞∏_p p^-m^(p) = |x|_∞ |x|_∞^-1 = 1.This is most often used asx ∈⇒∏_p |x|_p = |x|_∞^-1.Greatest common divisor For a set of natural numbers { x_i } where each x_i has a decomposition into primes asx_i = ∏_p p^m^(p)_i ,one can express the greatest common divisor k ask ≡({x_i }) = ∏_p p^min_i { m_i^(p)} .Together with| x_i |_p = p^-m_i^(p) ,we are led to the expressionk = ∏_p p^min_i { m_i^(p)} = ∏_p min_i{ p^ m_i^(p)} = ∏_p min_i{ |x_i|_p^-1} = ∏_p ( max_i {| x_i |_p})^-1 .We also have the formula| k |_p = max_i {| x_i |_p } . Note that(x_1, ⋯, x_n, 0) = (x_1, ⋯, x_n) ,since every (nonzero) integer divides 0. Additionally, we define(x) = x ∀ x ∈ ,including x = 0. Divisor sum We have the identity ∏_p 1-p^-s| m |_p^s/1 - p^-s = ∑_d|md^-s≡σ_-s(m) ,for s ∈ and m ∈. The completed Riemann zeta function The Riemann zeta functionζ(s) = ∑_n=1^∞ n^-s , (s) >1can be written as an Euler product asζ(s) = ∏_p 1/1 - p^-s, (s)>1and can be analytically continued to the whole complex plane except at s = 0 and s = 1 where it has simple poles. This is done by defining the completed Riemann zeta functionξ(s) ≡Γ( s/2)π^-s/2ζ(s)which obeys the functional relationξ(s) = ξ(1-s)as shown by Riemann. p-adic gaussian The p-adic gaussian γ_p: _p →{0, 1} is defined asγ_p(x) = 1, |x|_p ≤ 1 0, |x|_p > 1 =1, x ∈_p 0, x _p .For a rational number x we then get∏_p γ_p(x) = 1, x ∈_p ∀ p 0,else =1, x ∈0,else . Notice also that for rational numbers x_1,…, x_n∈ and picking an x ∈ such that for all primes p|x|_p = max{|x_1|_p, ⋯, |x_n|_p } ,we haveγ_p(x) = ∏_i = 1^n γ_p(x_i) .A consequence of this is that for an eulerian function depending only on the p-adic norms of its argumentf(x) = ∏_p f_p(|x|_p) ,then with x as in (<ref>), we havef(x) ∏_p γ_p(x) = ∏_p f_p(|x|_p) γ_p(x) = ∏_p f_p(|k|_p) γ_p(x) = f(k) ∏_p γ_p(x) ,wherek = (|x_1|_∞, ⋯, |x_n|_∞) .This equation makes sense as ∏_p γ_p(x) ensures that the left- and right hand sides are nonzero only when each x_i is integer for which k is well defined. We now see how a sum over rationals with x as in (<ref>) can collapse to a sum over integers due to the p-adic gaussian∑_x_1, …, x_n f(x) ∏_p γ_p(x) = ∑_x_1, …, x_n f(k) ∏_p γ_p(x) = ∑_x_1, …, x_n ∈ f(k). (2) Whittaker function The ramified (meaning m not necessarily unity) (2, ) Whittaker function evaluated atg = ( g_∞, , , …) =(( [ 1 x; 0 1 ]) ( [ y 0; 0 1/y ]) k, , , …)written as an Euler product readsW_α( 2sΛ - ρ, m; g ) = W_α( 2sΛ - ρ, m; (( [ 1 x; 1; ]) ( [ y; 1/y; ]) k, , , …) ) = = mx B_s(m, y) ∏_p γ_p(m) ( 1-p^-2s)1-p^-2s+1|m|_p^2s-1/1 - p^-2s+1 ,where α is the simple root and Λ is the fundamental weight. HereB_s (m, y) ≡2π^s/Γ(s) y^1/2 |m|^s-1/2 K_s-1/2( 2π |m| y )should be seen as the archimedean (2)-Whittaker function and each factor in the Euler product as the non-archimedean Whittaker functions. The product∏_p γ_p(m)restricts to m ∈ as explained above. The expression can then be written asW_α( 2sΛ - ρ, m; g ) = mx2/ξ(2s)y^1/2 |m|^s-1/2σ_1-2s(|m|) K_s-1/2( 2π |m| y ) .Notice how the factors of the Eulerian expression for the Riemann zeta function in the non-archimedean part combines with π^s/Γ(s) in the archimedean part to form a completed Riemann zeta function. Twisted character Let m ∈ and ψ_p,m be an additive character on _p defined asψ_∞, m(x)= e^2πi m x ;m, x∈for real numbersψ_p, m(x)= e^-2πi [ m x ]_p ; m, x∈_p for p-adic numbers . An unitary multiplicative character on the unipotent radical N() of the Borel subgroup of (n, ) can then be parametrized by m_1, …, m_n-1∈ as ψ( n ) = ψ( e^∑_α∈Δ_+ u_α E_α ) = ψ( e^∑_α∈Π u_α E_α ) = ∏_p ≤∞∏_i=1^n ψ_p, m_i( ( u_α_i)_p ) ,where Δ_+ is the set of positive roots and Π = {α_1, …, α_n-1}⊂Δ_+ is the set of simple roots. The second equality is due to the fact that the additive character is only sensitive to the abelianization of N(). In the final equality, ( x_α_i)_p denotes the p-adic (or real) component of the adelic coordinate x_α_i. For an element a ∈ A(), we would like to evaluate the twisted characterψ^a(n) ≡ψ( a n a^-1) . Let x_α(t) = exp(t E_α) where t ∈ and E_α is the positive Chevalley generator for the root α.For t ∈^×, define w_α(t) = x_α(t) x_-α(-t^-1) x_α(t) and h_α(t) = w_α(t) w_α(1)^-1.An element a ∈ A() is then parametrized by <cit.>a = ∏_i=1^n-1 h_α_i(y_i)y_i ∈^×,where the different generators h_α commute for all simple roots α∈Π and are multiplicative in t_α. For a simple root α and a root β, we have that <cit.>h_α(y) x_β(u) h_α(y)^-1 = x_β(y^β(H_α) u) . Since the character ψ on N is sensitive only to the x_β with β a simple root, it is enough to consider h_α_i(y) x_α_j(u) h_α_i(y)^-1 = x_α_j(y^A_ij u)for the simple roots α_i and α_j, where A_ij is the Cartan matrix.We then have thatψ(a n a^-1) = ψ(exp(∑_j = 1^n-1(∏_i = 1^n-1 y_i^A_ij) x_α_j E_α_j) ) = = ψ(exp(y_1^2/y_2 x_α_1 E_α_1 +∑_j=2^n-2y_j^2/y_j-1 y_j+1 x_α_j E_α_j + y_n-1^2/y_n-2 x_α_n-1 E_α_n-1 + …) ) . We can interpret the transformation ψ→ψ^a as that the parameters m_itransform according tom_i→ m_i' = ( y_i^2/y_i-1y_i+1) m_i,i = 1, …, n-1 ,where we have defined y_0 = y_n = 1. Note that starting with rational parameters m_i, the transformed parameters m_i' are no longer necessarily rational. § IWASAWA-DECOMPOSITION Proof of the following results can be found in <cit.>. For a real matrix g ∈_n() written in Iwasawa formg = n_∞ a_∞ k_∞ =([1 x_12⋯⋯ x_1n; 1⋱⋱⋮;⋱⋱⋮; 1 x_n-1, n;1 ]) ([ y_1; y_2/y_1; ⋱; y_n-1/y_n-2; 1/y_n-1 ]) k_∞ ,we havex_μν =y_ν-1^2 ϵ( V_μ, V_ν+1, …, V_n; V_ν, V_ν+1, …, V_n ), μ < ν, andy_μ^-2 = ϵ( V_μ+1, …, V_n; V_μ+1, …, V_n ) ,where V_μ is the μth row of g (regarded as an n-vector). Furthermore, ϵ denotes the totally antisymmetric productϵ( A_1, …, A_m; B_1, …, B_m ) = δ_a_1 a_m^i_1 i_m( A_1)^a_1…( A_m )^a_m( B_1)_i_1…( B_m )_i_m ,where the A's and B's are n-vectors and δ_a_1 a_m^i_1 i_m = m! δ_[ a_1.^i_1…δ_. a_m]^i_m = 1/(n-m)!ϵ_a_1a_m α_m+1α_nϵ^i_1i_m α_m+1α_ndenotes the generalized Kronecker delta. Put in words, ϵ takes two sets of vectors and returns the sum of every possible product of scalar products between the two sets weighted by the signs of the given permutations. For exampleϵ(A_1, A_2; B_1, B_2) = (A_1· B_1)(A_2· B_2) - (A_1· B_2)(A_2· B_1). For a p-adic matrix g ∈_n(_p), the Iwasawa-decompositiong = n_p a_p k_p =n_p ([η_1, p; η_2, p/η_1, p; ⋱; η_n-1, p/η_n-2, p;1/η_n-1, p ]) k_pis no longer unique. The p-adic norms of the η's however are constant across the family of decompositions and are given by| η_n-k|_p = ( max_σ∈Θ_k^n{| g( [ n-k+1 … n;σ(1) …σ(k); ]) |_p })^-1 , where k ∈{1, …, n-1} ,and Θ_k^n detones the set of all ordered subsets of {1, …, n} of order k. Here,g( [ r_1 … r_k; c_1 … c_k; ]) denotes a minor of order k, given as the determinant of the submatrix of g obtained by only picking the k rows {r_i} and k columns {c_i}. For example, the matrix[1/11/21/31/4;1/51/61/71/8;1/9 1/10 1/11 1/12; 1/13 1/14 1/15 1/16;].has|η_2|_3 = max{| | [1/9 1/10; 1/13 1/14;]| |_3,| | [1/9 1/11; 1/13 1/15;]| |_3,| | [1/9 1/12; 1/13 1/16;]| |_3,| | [ 1/10 1/11; 1/14 1/15;]| |_3,| | [ 1/10 1/12; 1/14 1/16;]| |_3,| | [ 1/11 1/12; 1/15 1/16;]| |_3 } = = max{| 1/4095|_3,| 8/19305|_3,| 1/1872|_3,| 1/5775|_3,| 1/3360|_3,| 1/7920|_3,}= max{ 3^2, 3^3, 3^2, 3^1, 3^1, 3^2 } = 3^3 = 9. § PARAMETRIZING GAMMA_I AND LAMBDA_J Recall the definitionsΓ_i(ψ_0)(_n-i(F))_Ŷ_n-i(F) 1 ≤ i ≤ n-2 (T_ψ_0∩ T_ψ_α_n-1)T_ψ_0i = n-1 ,where(_n-i(F))_Ŷ = {[ 1 ξ^T; 0 h ] : h ∈_n-i-1(F), ξ∈ F^n-i-1} ;andΛ_j(ψ_0)(_j(F))_X̂_j(F) 2 < j ≤ n (T_ψ_0∩ T_ψ_α_1)T_ψ_0j = 2 ,where(_j(F))_X̂ = {[ h ξ; 0 1 ] : h ∈_j-1(F), ξ∈ F^j-1}.In this appendix, we will find convenient representatives for these coset spaces. We begin with a lemma.Let S_k(F) denote the set of all k× k matrices m over the field F satisfying m = 1. The coset space _k(F)S_k(F) can then be parametrized as_k(F)S_k = ⋃_a = 0^k-1{([ 0 0 0; I_a 0 0; 0 v I_k-a-1 ]) : v ∈ F^k-a-1} .We will use induction to prove the lemma. Assume that the result holds up to and including matrices of size k× k and consider the coset space _k+1(F)S_k+1(F). Start with a matrix m_k+1∈ S_k+1(F). Left action of the group _k+1(F) ∋ h_k+1 taking m_k+1→ h_k+1m_k+1 is equivalent to performing Gauss elimination among the rows of m_k+1. Since we have m_k+1 = 1 we can bring m_k+1 to the formm_k+1→[ 0 0; v m ] ,where v ∈ F^k and m satisfies m_k ≤ 1. We cannot have m ≥ 2 as we could then perform additional row manipulations to produce two zero rows in m and hence another zero row in m_k+1 which violates m_k+1∈ S_k+1(F). §.§.§ Case 1: m = 0 Here m is invertible and we can bring m_k+1 to the form[ 0 0; v m_k ]→[ 0 0; v I_k ]having relabelled v. We get the contribution. {[ 0 0 0; I_a 0 0; 0 v I_k+1-a-1 ] : v ∈ F^k+1-a-1}|_a=0.§.§.§ Case 2: m = 1 We now have m = m_k for some m_k ∈ S_k and we can apply the induction assumption which leads us to consider matrices of the form[ 0 0 0 0; v^(1) 0 0 0; v^(2) I_a 0 0; v^(3) 0 u I_k-a-1 ] , where a ∈ [0, k-2] ∩.We see that we must have v^(1)≠ 0 and with further row manipulations we can thus bring this to the form[ 0 0 0 0; v^(1) 0 0 0; v^(2) I_a 0 0; v^(3) 0 u I_k-a-1 ]→[ 0 0 0 0; 1 0 0 0; 0 I_a 0 0; 0 0 u I_k-a-1 ].We get the contributions⋃_a = 0^k-2{[ 0 0 0 0; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-1 ] : v ∈ F^k-a-1} = ⋃_a = 1^k-1{([ 0 0 0; I_a 0 0; 0 v I_k+1-a-1 ]) : v ∈ F^k+1-a-1}.This combines with the contribution from case 1 to give the form stated in the lemma.That the base case k = 1 has the correct form is trivial. Peano's axiom of induction now establishes the lemma. The coset space(_n-i(F))_Ŷ_n-i(F)1 ≤ i ≤ n-2can be parametrized as(_n-i(F))_Ŷ_n-i(F) = = {[ x'^-1 0 0; yx' 0; v 0 I_n-i-2 ] : x' ∈ F^×, y ∈ F, v ∈ F^n-i-2} ∪ ⋃_a = 0^n-i-2{[00 (-1)^a+1 x'^-10; x'000;0I_a00;00vI_n-i-a-2 ] : x' ∈ F^×, v ∈ F^n-i-a-2} .Denote k ≡ n-i. Consider a matrixG = [ s T^T; B m ]∈_k(F) ,where s is a scalar, m is a (k-1)× (k-1)-matrix and T and B (for “top” and “bottom”) are (k-1)-column vectors. The action of an elementM = [ 1 ξ^T; 0 h ]∈ (_k(F))_Ŷon G isG → MG =[ s + ξ^T B T^T + ξ^T m; h B h m ].Parametrizing the coset space (_k(F))_Ŷ_k(F) amounts to choosing ξ∈ F^k-1 and h ∈_k-1(F) such that the product MG takes a particularly nice form, manifestly with at most k degrees of freedom which is the dimension of this coset space.Even though h ∈_k-1(F) we will proceed with h ∈_k-1(F) and restore the unit determinant of h at the end by left multiplication of the matrix [ 1 0 0; 0x' 0; 0 0 I_k-2 ] where 0 ≠ x' = ( h)^-1. By having h ∈_k-1(F) we are free to perform Gauss elimination among the bottom k-1 rows in G.We consider the two cases m = 0 and m = 1. Note that the cases m ≥ 2 do not arise as with row elimination it would then be possible to produce two zero-rows in m and hence a zero-row in G which violates G ∈_k(F).§.§.§ Case 1: m = 0We choose h = m^-1 and ξ^T = -T^T m^-1. Since h has full rank, we can redefine hB → B without loss of generality and redefine s + ξ^T B → s. This leads to the representativeG →[ s 0; B I_k-1 ].We now restore the unit determinant to hG →[ 1 0 0; 0x' 0; 0 0 I_k-2 ][ s 0; B I_k-1 ] = [ s 0 0; yx' 0; v 0 I_k-2 ] ,where we have split the (k-1)-vector into a scalar y and a (k-2)-vector v. The condition G = 1 now sets s = x'^-1 leading toG= [ x'^-1 0 0; yx' 0; v 0 I_k-2 ].This is a nice form of the representative G which manifestly has k degrees of freedom.§.§.§ Case 2: m = 1We can no longer choose h = m^-1. Having promoted h to be an element of _k-1(F), we can make use of lemma <ref> which leads us to consider representatives of the formG →[ sT^(1)T T^(2)T^(3)T; B^(1) 0 0 0; B^(2) I_a 0 0; B^(3) 0 v I_k-a-2 ]for a ∈ [0, k-2] ∩and v ∈ F^k-a-2.We see that B^(1)≠ 0 in order for G to remain non-singular. With further row elimination we can therefore bring this to the form[ s T^(1) T T^(2) T^(3) T; B^(1) 0 0 0; B^(2) I_a 0 0; B^(3) 0 v I_k-a-2 ]→[ s T^(1) T T^(2) T^(3) T; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ].Next, using the ξ-freedom we can bring this to the form[ s T^(1) T T^(2) T^(3) T; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ]→[ 1 ξ^(1) ξ^(2) T ξ^(3) T; 0 1 0 0; 0 0 I_a 0; 0 0 0 I_k-a-2 ][ s T^(1) T T^(2) T^(3) T; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ]= [ s + ξ^(1) T^(1) T + ξ^(2) T T^(2) + ξ^(3) T v T^(3) T + ξ^(3) T; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ]→[ 0 0 T^(2) 0; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ] ,with a suitable choice of ξ and having redefined T^(2). We now restore the unit determinant to h[ 0 0 T^(2) 0; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ]→[ 1 0 0; 0x' 0; 0 0 I_k-2 ][ 0 0 T^(2) 0; 1 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ]→= [ 0 0 T^(2) 0;x' 0 0 0; 0 I_a 0 0; 0 0 v I_k-a-2 ].The condition G = 1 now sets T^(2) = (-1)^a+1 x'^-1, leading to the representative[00 (-1)^a+1 x'^-10; x'000;0I_a00;00vI_k-a-2 ].The coset space(_j(F))_X̂_j(F)2 < j ≤ ncan be parametrized as(_j(F))_X̂_j(F) = = {[ I_j-2 0 0; 0x' 0; v^T y x'^-1 ] : x' ∈ F^×, y ∈ F, v ∈ F^j-2} ∪ ⋃_a = 0^j-2{[I_a000;00I_j-a-20;000 x';0 (-1)^j+a+1 x'^-1v^T0 ] : x' ∈ F^×, v ∈ F^j-a-2} .Denote k ≡ j. Consider a matrixG = [ m T; B^T s ]∈_k(F) ,where s is a scalar, m is a (k-1)× (k-1)-matrix and T and B (for “top” and “bottom”) are (k-1)-column vectors. The action of an elementM = [ h h ξ; 0 1 ]∈ (_k(F))_X̂on G isG → MG =[ h(m + ξ B^T) h(T + s ξ);B^Ts ].Parametrizing the coset space (_k(F))_X̂_k(F) amounts to choosing ξ∈ F^k-1 and h ∈_k-1(F) such that the product MG takes a particularly nice form, manifestly with at most k degrees of freedom which is the dimension of this coset space.Even though h ∈_k-1(F) we will proceed with h ∈_k-1(F) and restore the unit determinant of h at the end by left multiplication of the matrix [ I_k-2 0 0; 0x' 0; 0 0 1 ] , where 0 ≠ x' = ( h)^-1. By having h ∈_k-1(F) we are free to perform Gauss elimination among the top k-1 rows in G.We consider the two cases s ≠ 0 and s = 0.§.§.§ Case 1: s = s' ≠ 0We choose ξ = -1/s' T. This leads to the representativeG →[ m - 1/s'T B^T 0; B^Ts' ].From the condition1 =G = ( m - 1/s'T B^T) s'we get that( m - 1/s'T B^T) ≠ 0 ,and hence the matrix m - 1/s'T B^T can be inverted using our h-freedom which leads to the representative[ m - 1/s'T B^T 0; B^Ts' ]→[ I_k-1 0; B^Ts' ].We now restore the unit determinant to hG →[ I_k-2 0 0; 0x' 0; 0 0 1 ][ I_k-1 0; B^Ts' ] = [ I_k-2 0 0; 0x' 0; v^T ys' ] ,where we have split B into a scalar y and a (k-2)-vector v. The condition G = 1 now sets s' = x'^-1 leading toG= [ I_k-2 0 0; 0x' 0; v^T y x'^-1 ].This is a nice form of the representative G which manifestly has k degrees of freedom.§.§.§ Case 2: s = 0We can no longer eliminate T with our ξ-freedom. The group element G takes the formG= [m T'; B'^T0;] ,where the vectors T' and B' must be non-zero (as indicated by the primes) in order for G to be non-singular. A ξ-transformation takes the form[m T'; B'^T0;]→[ m + ξ B'^T T'; B'^T0;].We now consider the k-1 distinct cases labelled by a ∈ [0, k-2] ∩ defined by that B'^T takes the form B'^T =[ 0_1× a b'v ], where v is a k-a-2-vector and 0 ≠ b' ∈ F. The ξ-transformation then lets us eliminate the (a+1)th column of m. This works since the (a+1)th column of the matrix ξ B'^T is b' ξ where b' ≠ 0 by assumption. We are led to the representative[m T'; B'^T0;]→[ m_1 0 m_2T'; 0b' v^T 0; ] ,where m_1 is a (k-1)× a-matrix and m_2 is a (k-1)× (k-a-2)-matrix.The (k-1)× (k-1) matrix [ m_1 0 m_2 ] clearly has column-rank at most k-2. Since column-rank and row-rank for matrices are equal, we know that the row-rank is also at most k-2 and with row manipulations we can thus produce a zero row[ m_1 0 m_2T'; 0b' v^T 0; ]→[000 t'; m_210 m_22T^-;0 b'v^T0;] ,where 0 ≠ t' ∈ F in order for G to be non-singular. With further row manipulations we can then eliminate the vector T^- and bring this to the form[000 t'; m_210 m_22T^-;0 b'v^T0;]→[0001; m_210 m_220;0 b'v^T0;].The (k-2)× (k-2)-matrix [ m_21 m_22 ] must have full rank in order for G to be non-singular and can thus be inverted, leading to the representative[0001; m_210 m_220;0 b'v^T0;]→[ 0 0 0 1; I_a 0 0 0; 0 0 I_k-a-2 0; 0b' v^T 0; ].We permute the first k-1 rows to get[ 0 0 0 1; I_a 0 0 0; 0 0 I_k-a-2 0; 0b' v^T 0; ]→[ I_a 0 0 0; 0 0 I_k-a-2 0; 0 0 0 1; 0b' v^T 0; ].Lastly, we restore the unit determinant to h[ I_a 0 0 0; 0 0 I_k-a-2 0; 0 0 0 1; 0b' v^T 0; ]→[ I_k-2 0 0; 0x' 0; 0 0 1 ][ I_a 0 0 0; 0 0 I_k-a-2 0; 0 0 0 1; 0b' v^T 0; ]→= [ I_a 0 0 0; 0 0 I_k-a-2 0; 0 0 0x'; 0b' v^T 0; ].The condition G = 1 now sets b' = (-1)^k+a+1 x'^-1, leading to the representative[I_a000;00I_k-a-20;000 x';0 (-1)^k+a+1 x'^-1v^T0;]. Another way of parametrizing the coset (_k(F))_X̂_k(F) is to parametrize the coset _k(F) / (_k(F))_X̂ which works completely analogously to the how the coset (_k(F))_Ŷ_k(F) was parametrized in lemma <ref> and then invert the resulting matrices. § LEVI ORBITS Let P_m for 1 ≤ m ≤ n-1 be the maximal parabolic subgroup of _n associated to the simple root α_m with Levi decomposition L U_m (where we drop the subscript for L for convenience) and let y ∈^t u_m(F) be parametrised by a matrix Y ∈_(n-m) × m(F) as in (<ref>). We will now study the L(F)-orbits of elements y.We parametrise an element l ∈ L(F) by the two matrices A ∈_m× m(F) and B ∈_(n-m)×(n-m)(F) with (A) (B) = 1 as l = (A, B^-1). This element acts by conjugation on y as ly(Y)l^-1 = y(AYB).Using unit determinant matrices A and B we may perform standard row and column additions to put Y on a form which has zero elements everywhere except for an anti-diagonal r × r matrix with non-vanishing determinant in the upper right corner where 0 ≤ r ≤min(m, n-m) is the rank of Y.This can be seen as follows.If the upper right element is zero, pick any non-zero element whose row-column position we denote (i, j) and add multiples of row i to the first row, and column j to the last column to make the upper right element non-zero. Then, use the non-zero upper right element to cancel all remaining non-zero elements on the first row and last column by further row and column additions. Repeat the procedure for the matrix obtained by removing the first row and last column. The induction terminates when we run out of rows or columns, or when the remaining elements are all zero.We will now rescale the anti-diagonal elements by conjugating y with diagonal matrices l, meaning that the ith diagonal element in l rescales both row i and column i (inversely). Each rescaling of an element in the anti-diagonal of the r × r matrix then leaves two less diagonal elements in l for further rescalings. Since the non-zero elements of Y at this stage do not share any rows or columns (because of the anti-diagonal r × r submatrix), we may then perform any and all rescalings until we run out of free diagonal elements in l.The number of free diagonal elements in l is n-1 because of the determinant condition which means we can make n-1/2 rescalings. We have that 1 ≤ m ≤ n-1 and r ≤min(m, n-m) ≤n/2.Thus, it is possible to rescale all anti-diagonal elements unless n = 2r = 2m, for which there will be one remaining anti-diagonal element d. Using conjugations with l = (a, 1, …, 1, 1/a), a ∈ F^× this d can be shown to be in F^× / (F^×)^2. We have now shown that the L_m(F)-orbits of elements y(Y) ∈^t u_m(F) are characterized by matrices Y = Y_r(d) ∈_(n-m) × m(F), where Y_r(d) which is non-zero only for an anti-diagonal r× r matrix in its upper right corner whose elements are all one except the lower left which is dY_r(d) =[0 [ 1; ⋱; 1; d ];00 ].For n = 2r = 2m, d ∈ F^× / (F^×)^2 and otherwise d = 1. For convenience we will denote Y_r(1) as Y_r.Thus, the L_m(F)-orbits on ^t u_m(F) are characterized by the same data as the _n(F)-orbits ([2^r1^n-2r], d) with d ∈ F^×/(F^×)^k, k = ([2^r1^n-2r]) and 0 ≤ r ≤min(m, n-m).By conjugations with the Weyl element w ∈_n(F) mapping torus elements (t_1, t_2, …, t_n) ↦ (t_1, … t_m-r, t_m, t_m+1, t_m-1, t_m+2, …, t_m-r+1, t_m+r, t_m+r+1, … t_n), where we have underlined the changed elements, we see that y(Y_r(d)) is put on the form of the standard representative for the _n(F)-orbit ([2^r1^n-2r], d) shown in proposition <ref>. In this paper we will always be able to find a representative on the form Y_r(1), that is, to rescale all elements, since we consider n ≥ 5 and r ≤ 2 where the latter restriction comes from the fact that higher rank elements have vanishing associated Fourier coefficients in a next-to-minimal or minimal automorphic representation according to theorem <ref>.Lastly, we note that if we instead consider L(F)-orbits the last remaining rescaling in the maximal rank n=2r case would be possible by conjugation with l = (√(d), 1, …, 1, 1/√(d)). utphys-alpha | http://arxiv.org/abs/1707.08937v1 | {
"authors": [
"Olof Ahlén",
"Henrik P. A. Gustafsson",
"Axel Kleinschmidt",
"Baiying Liu",
"Daniel Persson"
],
"categories": [
"math.RT",
"hep-th",
"math.NT",
"11F70, 22E55, 11F30"
],
"primary_category": "math.RT",
"published": "20170727171824",
"title": "Fourier coefficients attached to small automorphic representations of ${\\mathrm{SL}}_n(\\mathbb{A})$"
} |
High-Dimensional Simplexes for Metric SearchConnor et al.Department of Computer and Information Sciences, University of Strathclyde, Glasgow, G1 1XH, United Kingdom ISTI - CNR, Via Moruzzi 1, 56124 Pisa, Italy High-Dimensional Simplexes for Supermetric Search Richard Connor1 Lucia Vadicamo 2 Fausto Rabitti2 May 29, 2018 ==================================================== In 1953, Blumenthal showed that every semi-metric space that is isometrically embeddable in a Hilbert space has the n-point property; we have previously called such spaces supermetric spaces.Although thisisa strictly stronger property than triangle inequality, it is nonetheless closely related andmany useful metric spaces possess it. Theseinclude Euclidean, Cosine and Jensen-Shannonspaces of any dimension.A simple corollary of the n-point property is that, for any (n+1) objects sampled from the space, there exists an n-dimensional simplex in Euclidean space whose edge lengths correspond to the distances among the objects.We show how the construction of such simplexes inhigher dimensions can be used to give arbitrarily tight lower and upper bounds ondistances within the original space. This allows the construction of an n-dimensional Euclidean space, from which lower and upper bounds of the original space can be calculated, and which is itself an indexable space with the n-point property. For similarity search, the engineering tradeoffs are good: we showsignificant reductions in data size and metric cost with little loss of accuracy, leading to a significantoverall improvement insearch performance.§ INTRODUCTION§.§.§ Context To set the context, we are interested in searching a (large) finiteset of objects S which is a subset of an infinite set U, where (U, d) is ametric space. The general requirement is to efficiently find members of S which are similar to an arbitrary member of U, where the distance function d givesthe only way by which any two objects may be compared. There are many important practical examples captured by this mathematical framework, see for example <cit.>. Such spaces are typically searched with reference to a query object q ∈ U.A threshold searchfor some threshold t, based on a query q ∈ U, has the solution set{s ∈ S such that d(q,s) ≤ t}.There are three main problems with achieving efficiency when the search space is very large.Most obviously, for very large collections we always require scalability. This is achieved within metric search domains by techniques which avoid searching parts of the collection, typically by using data structures which take advantage of mathematical properties of the distance metrics used.Secondly, distance metrics are often expensive. When the search space is large, semantic accuracy is important to avoid huge numbers of false positive results – in the terminology of information retrieval, precision becomes relatively more important that recall.In such cases higher specificity will normally result in a much more expensive metric, for example Jensen-Shannon or Quadratic Form distances, which are much more expensive to compute.Finally, the data objects themselves may be large. For example in the domain of near-duplicate image search, GIST representations give a better semantic comparison then MPEG-7, but occupy around 2KB per image <cit.>.Even although huge memory is nowadays available, alarge collection of large objects will still require to be paged. For examplea 32-bit architecture can typically address less than 2GB;a collection of only one million GIST descriptors cannot be accommodated.§.§.§ ApproachesIn high-dimensional Euclidean spaces, the last two problems can be addressed by various dimensionality reduction techniques. In outline, these techniques reduce an n-dimensional Euclidean space space into an m-dimensional one, where m < n. This reduces both the size required to store the data, and the cost of the Euclidean (ℓ_2) metric. However this may resultsin a loss of precision, which can defeat the purpose if there is an accompanying loss of semantic accuracy with respect to the original data. In non-Euclidean metric spaces, such techniques are not applicable. There are however various other techniques which use reduced-size object surrogates for an initial indexing or filtering phase. Such techniques maygive approximate results or, if they are guaranteed to return a superset of the solution set, exact search can be performed by re-checking their output against the original data.§.§.§ Outline of our Contribution Here, we present a new technique which can be used in either of these approaches. Using properties of finite isometric embedding, we show a mechanism which allowsspaces with certain properties to be translated into a second, smaller, space. For a metric space (U,d), we describe a family of functions ϕ_n which can be created by measuring the distances among n objects sampled from the original space, and which can then be used to create a surrogate space:ϕ_n : (U,d) → (ℝ^n,ℓ_2) with the propertyℓ_2(ϕ_n(u_1),ϕ_n(u_2)) ≤ d(u_1,u_2) ≤ g(ϕ_n(u_1),ϕ_n(u_2))for an associated function g. Further, the cost of evaluating g and ℓ_2 together is almost exactly the same as the cost ofℓ_2.This family of functions can be defined for any metric space which is isometrically embeddable in a Hilbert Space, or equivalently for any space that meets the n-point property <cit.>. The advantages of the proposed technique are that (a) the ℓ_2 metric is very much cheaper thansome Hilbert-embeddable metrics; (b) the size of elements of ℝ^n may be much smaller than elements of U, and (c) in many cases we can achieve both of these along with an increase in the scalability of the resulting search space.While not applicable to all purposes, we show that this mechanism may be used to great effect in a number of “real-world" search spaces. Among other results, we show a benchmark best-performance for SISAP colors <cit.> data set. § RELATED WORK§.§.§ Finite Isometric Embeddingsare excellently summarised by Blumenthal <cit.>.He uses the phrasefour-point property to mean a space that is4-embeddable in 3-dimensional Euclidean space (ℓ_2^3), i.e. if for any four points x_1,x_2, x_3, x_4 ∈ U exist a mapping function f:U →ℓ_2^2 such that ℓ_2(f(x_i), f(x_j))=d(x_i,x_j), for i,j=1,2,3,4.Wilson <cit.> shows various properties of such spaces, and Blumenthal points out that results given by Wilson, when combined with work by Menger <cit.>, generalise to show that some spaces with the four-point property also have the n-point property: any n points can be isometrically embedded in a (n-1)-dimensional Euclidean space (ℓ_2^n-1).In alater work, Blumenthal <cit.> shows that any space which is isometrically embeddable in a Hilbert space has the n-point property. This single result applies to many metrics, including Euclidean, Cosine, Jensen-Shannon and Triangular <cit.>, and is sufficient for our purposes here.§.§.§ Dimensionality Reductionaims to produce low-dimensional encodings of high-dimensional data,preserving the local structure of some input data. See <cit.> for comprehensive surveys on this topic.The Principal Component Analysis (PCA) <cit.> isthe most popular of the techniques for unsupervised dimensionality reduction. Theidea is to find a linear transformation of n-dimensional to k-dimensionalvectors (k≤ n) that best preserves the variance of the inputdata.Specifically, PCA projects the data along the direction of its first k principal components, which are the eigenvectors of the covariance matrix of the (centered) input data. According to the Johnson-Lindenstrauss Flattening Lemma (JL)(see e.g. <cit.>), a random projection can also be used to embed a finite set of n euclidean vectors into a k-dimensional euclidean space space (k<n) with a “small” distortion. Specifically the Lemma asserts that for any n-pointsof ℓ_2 and every 0<ϵ<1 there is a mapping into ℓ_2^k that preserves all the interpoint distances within factor 1+ϵ, where k=O(ϵ^-2log n). The low dimensional embedding given by the Johnson Lindenstrauss lemma is particularly simple to implement. §.§.§ General metric spacesdo not allow either PCA or JL as they require inspection of the coordinate space. Mao et al. <cit.> pointed out that multidimetional-methods can be indirectly applied to metric space by using the pivot space model. In that case each metric object is represented by its distance to a finite set of pivots.In the general metric space context, perhaps the best known technique is metric Multidimensional Scaling (MDS) <cit.>. MDS aims topreserveinter-point distancesusingspectral analysis. However, when the number m of data points islarge the classical MDS is too expensive in practice due to a requirement for O(m^2) distance computations and spectral decomposition of a m× m matrix. The Landmark MDS (LMDS) <cit.> is a fast approximation of MDS. LMDS uses a set of k landmark pointsto compute k× m distances of the data points from the pivots. It applies classical MSD to thesepoints anduses a distance-based triangulation procedure to project the remaining data points.§.§.§ LAESA <cit.> is a more tractablemechanism which has been used for metric filtering, rather than approximate search.n reference objects are selected. For each element of the data, the distances to these points are recorded in a table. At query time, the distances between the query and each reference point are calculated. The table can then be scanned row at a time, and each distance compared; if, for any reference object p_i and data object s_j the absolute difference|d(q,p_i) - d(s_j,p_i)| > t, then from triangle inequality it is impossible for s_j to be within distance t of the query, and the object need not be paged into the main memory.§ THE N-SIMPLEX APICAL SPACE In this section we give an informal outline of our new observations on supermetric spaces. They are based on the fact mentioned above that, for any (n+1) objects in the original space, there existsa simplex in ℓ_2^nwhose edge lengths correspond to the distances measured in the original space.In <cit.> we showed a less general result, that any semi-metric which is isometrically embeddable in a Hilbert Space has the four-point property: that is, given all of the distances measured among any four objects in the space, it is possible to construct a tetrahedron in 3D Euclidean Spacewith edge lengths corresponding to those distances.In <cit.> we showed an important lower-bound property based on this tetrahedral embedding; this is illustrated in Figure <ref>, extendedhere with a matching upper-bound. The case in point here is when four objects within the original space have been identified, but only five of the six possible distances have been measured. This corresponds to the situation of an indexing structure based on two reference objects, p_1 and p_2, which are chosen before a data set S is organisedaccording to relative distances from these objects. The third object s represents an arbitrary element ofSwhich has been stored, and the fourth and final object q represents a query over the data. For all possible s, we wish to identify those which may be within a threshold distance of q, based on some partition of the space constructed before q was available.Figure <ref> shows an ℓ_2^3 space into which these four objects have been projected, where for each element a the notation v_a is used to denote a corresponding point in the ℓ_2^3 space. The only distance which has not been measured is d(s,q); however the 4-point property means that the corresponding distance ℓ_2(v_s,v_q) must be able to form the final edge of a tetrahedron. From this Figure, the intuition of the upper and lower bounds on d(s,q) is clear, through rotation of the triangle v_p_1v_p_2v_q around the line v_p_1v_p_2 until it is coincident with the plane in which v_p_1v_p_2v_s lies. The two possible orientations give the upper and lower bounds, corresponding to the distances between v_s and the two apexes ap_q_- and ap_q_+of the two possible planar tetrahedra.We now understand that this same intuition generalises into many dimensions.In the general form, we consider a set p_i, i ∈{1 … n}, of n reference objects, whose inter-object distances are used to form abase simplex σ_0, with vertices v_p_1,…, v_p_n, in (n-1) dimensions. This corresponds to the line segment v_p_1v_p_2 in the figure, this representing a two-vertex simplex in ℓ_2^1. The simplex σ_0 is contained within a hyperplane of theℓ_2^n space, and the distances from object s to each p_i are used to calculate a new simplex σ_s, in ℓ_2^n, consisting of a new apex point v_s set above the base simplex σ_0. Note that there are two possible positions in ℓ_2^n for v_s, one on either side of the hyperplane containing σ_0; we denote these as v_s^+, and v_s^- respectively. Now, giventhe distances between object q and allp_i, we canagain construct two possible simplexes for σ_q with two possible positions for v_q, which we denote by v_q^+ and v_q^-.Finally, we note that the act of rotating the triangle around its base also generalises to the concept of rotating the apex point of any simplex around the hyperplane containing its base simplex. Furthermore, the n-point property guarantees the existence of a simplex σ_1 in ℓ_2^n+1 which preserves the distance d(s,q) as ℓ_2(v_s,v_q). Fromthese observations we immediately have the following inequalities:ℓ_2^n(v_s^+, v_q^+) ≤ d(s,q) ≤ℓ_2^n(v_s^+, v_q^-) To back up this intuition, we include proofs of these inequalities in theappendix. Meanwhile, we answer the more pragmatic questions which allow these lower and upper bound properties to be useful in the context of similarity search. § CONSTRUCTING SIMPLEXES FROM EDGE LENGTHSIn this section, we show an algorithm for determining Cartesian coordinates for the vertices of asimplex, given only the distances between points. The algorithm isinductive, at each stage allowing the apex of an n-dimensional simplex to be determined given the coordinates of an (n-1)-dimensional simplex, and the distances from the new apex to each vertex in the existing simplex. This is important because, given a fixed base simplex over which many new apexes are to be constructed, the time required to compute each one is linear with the number of dimensions. A simplex is a generalisation of a triangle or a tetrahedron in arbitrary dimensions.In one dimension, the simplex is a line segment. In two dimensions it is a convex hull of a triangle, while in three dimensions it is the convex hull of a tetrahedron. In general, the n-simplex of vertices p_1,…,p_n+1 equals the union of all the line segments joining p_n+1 to points of the (n-1)-simplex of vertices p_1,…,p_n.The structure of a simplex in n-dimensional space is given as an n+1 by n matrix representing the cartesian coordinates of each vertex.For example, the following matrix representsfour coordinates which are the vertices of a tetrahedron in 3D space:[ 0 0 0; v_2,1 0 0; v_3,1 v_3,2 0; v_4,1 v_4,2 v_4,3 ] For all such matrices Σ, the invariant that v_i,j = 0 whenever j ≥ i can bemaintained without loss of generality; for any simplex, this can be achieved by rotation and translation within the Euclidean space while maintaining the distances among all the vertices. Furthermore, if we restrict v_i,j≥ 0 whenever j = i-1 then in each row this component represents the altitude of the i^th point with respect to a base face represented by the matrix cut down from Σ by selecting elements above and to the left of that entry.§.§ Simplex Construction This section gives an inductive algorithm (Algorithm <ref>) to construct a simplex in n dimensions based only on the distances measured among n+1 points. For the base case of a one-dimensional simplex (i.e. two points with a single distance δ) the construction is simply Σ=[ 0; δ ].For an n-dimensional simplex, where n ≥ 2, the distances among n+1 points are given. In this case, an (n-1)-dimensional simplex is first constructed using the first n points. This simplex is used as a simplex base to which a new apex, the (n+1)^th point, is added by the following ApexAddition algorithm (Algorithm <ref>). For an arbitrary set of objects s_i ∈ S, the apex ϕ_n(s_i) can be pre-calculated. When a query is performed,only n distances in the metric space require to be calculated to discover the new apex ϕ_n(q)in ℓ_2^n.In essence, the ApexAddition algorithm is derived from exactly the same intuition as the lower-bound property explained earlier. Proofs of correctness for both the construction and the lower-bound property are included as an Appendix for the interested reader. §.§ BoundsBecause of the method we use to build simplexes, the final coordinate always represents the altitude of the apex above the hyperplane containing the base simplex. Given this, two apexes exist, according to whether a positive or negative real number is inserted at the final step of the algorithm.As a direct result of this observation, and those given in Section <ref>, we have the following bounds for any two objects s_1 and s_2 in the original space:Letϕ_n(s_1) = (x_1, x_2,…, x_n-1,x_n) ϕ_n(s_2) = (y_1, y_2,…, y_n-1,y_n)then√(∑_i=1^n(x_i - y_i)^2)≤ d(s_1,s_2)≤√(∑_i=1^n-1(x_i - y_i)^2 + (x_n + y_n)^2)From the structure of these calculations, it is apparent that they arelikely to converge rapidly around the true distance as the number of dimensions used becomes higher, as we will show in Section <ref>. It can also be seenthat the cost of calculating both of these values together, especially in higher dimensions, is essentially the same as a simple ℓ_2 calculation. Finally, we note that the lower-bound function is a proper metric, but the upper-bound function is not even a semi-metric: even although it is a Euclidean distance in the apex space, one of the domain points isconstructed by reflection across a hyperplane and thus the distance between a pair of identical points is in general non-zero. § MEASURING DISTORTION We define distortion for an approximation (U',d') of a space (U,d) mapped by a function f:U → U' as as the smallest D such that, for some scaling factor rr · d'(f(u_i),f(u_j)) ≤ d(u_i,u_j) ≤ D · r · d'(f(u_i),f(u_j))We have measured this for a number of different spaces, and presentresults over the SISAP colors benchmark set which are typical and easily reproducible. Summary results are shown in Figure <ref>.In each case, the X-axis represents the number of dimensions used for the representation, with the distortion plotted against this. For Euclidean distance, there are two entries for : one for randomly-selected reference points, and the other where the choice of reference points is guided by the use of PCA. In the latter case we select the first n principal components (eigenvectors of the covariance matrix) as pivots.It canbe seen that outperforms all other strategies except for PCA, which is not applicable to non-Euclidean spaces. LMDS is the only other mechanism applicable to general metric spaces [The authors note it works better for some metrics than others; in our understanding, it will work well only for spaces with the n-point property.] ; this is a little more expensive than to evaluate,and performs relatively badly. The comparison with JL is a slightly unfair, as the JLlemma applies only for very high dimensions in an evenly distributed space; we have tested such spaces, and JL is still out-performed at by , especially at lower dimensions.The distortion we show here is only for the lower-bound function of . We have measured the upper-bound function also, which gives similar results. Unlike the lower-bound, the upper-bound is not a proper metric; however for non-metric approximate search it should be noted that the mean of the lower- and upper-bound functions give around half the distortion plotted here.The implications of these results for exact searchshould be noted. For Euclidean search, it seems that only around 20 dimensions will be required to perform a very accurate search, i.e. one-fifth of the original space. For Jensen-Shannon, more dimensions will be required, but the cost of the ℓ_2 metric required to search the compressed space is around one-hundredth the cost of the original metric. In the next section we present experimental exact search results consistent with these observations.§ EXACT SEARCH: INDEXING WITH N-SIMPLEX In this section we examine the use ofin the context of exact search, using the lower and upper-bound properties.Any such mechanism can be viewed as similar to LAESA <cit.>, in that there exists an underlying data structure which is a table of numbers, n per original object, with the intention of using this table to exclude candidates which cannot be within a given search threshold. In both cases, n reference objects are chosen from the space. For LAESA, each row of the table isfilled, for one element of the data, with the distances from the candidate to each reference object. For , each row is filled for one element of the data with the Cartesian coordinates of the new apex formed in n dimensions by applying these distances to an (n-1)-dimensional simplex formed from the reference objects.The table having been established, a query notionally proceeds bymeasuring the distances from the query object to each reference point object.In the case of LAESA, the metric for comparison is Chebyshev: that is, if any pairwise difference is greater than the query threshold, the object from which that row was derived cannot be a solution to the query. For , the metric used is ℓ_2: that is, if the apex represented in a row is further than the query threshold from the apex generated from the query, again the object from which that apex was derived cannot be a solution to the query.In both cases, there are two ways of approaching the table search. It can be performed sequentially over the whole table, in which case either metric can be terminated within a row if the threshold is exceeded, without continuing to the end of the row. Alternatively the table can itself be re-indexed using a tree search structure: this can be implemented with only a few extra words per item by storing references into the table within the tree structure. Although this compromises the amount of space available for the table itself, it may avoid many of the individual row comparisons.In the context of re-indexing we also note that, in the case of , the Euclidean metric used over the table rows itself has the four-point property, and so the Hilbert Exclusion property as described in <cit.> may be used.In all cases the result is a filtered set of candidate objects which is guaranteed to contain the correct solution set. In general, this set must be re-checked against the original metric, in the original space. For however theupper-bound condition is checked first; if this is less than the query threshold, then the object is guaranteed to be an element of the result set and does not require to be re-checked within the original space. §.§ Experiment - SISAP colors We first apply these techniques to the SISAP colors <cit.> data set, using three different supermetrics: Euclidean, Cosine, and Jensen-Shannon[For precise definitions of the non-Euclidean metrics used, see <cit.>.]. We chose this data set because (a) it has only positive values and is therefore indexable by all of the metrics, and (b) it shows an interesting non-uniformity, in that its intrinsic dimensionality for all metrics is much less than its physical dimensionality (112). It should thus give an interesting “real world" context to assess the relative value of the different mechanisms. For Euclidean distance, we used thethree benchmark thresholds; for the other metrics, we chose thresholds that return around 0.01% of the data. In all cases the first 10% of the file is used to query the remaining 90%. Pivots are randomly-selected both for LAESA and n-simplex approach.For each metric, we tested different mechanisms with different allocations of space: 5 to 50 numbers per data element, thus thespace used per object is between 4.5% and 45%of the original. All results reported are for exact search, that is the initial filtering is followed by re-testing within the original space where required. Five different mechanism were tested, as follows:sequential LAESA (L_seq) each row of the table is scanned sequentially,each element of each row is tested against the query and that row is abandoned if the absolute difference is greater than the threshold.reindexed LAESA (L_rei) the data in the table is indexed using a monotone hyperplane tree, searched using the Chebyshev metric.sequential (N_seq)each row of the table is scanned sequentially,for each element of each row the square of the absolute difference is added to an accumulator, the row is abandoned if the accumulator exceeds the square of the threshold, and the upper-bound is applied if the end of the row is reached before re-checking in the original space.reindexed (N_rei) the data in the table is indexed using a monotone hyperplane tree using the Hilbert Exclusion property, and searched using the Euclidean metric; the upper-bound is applied for all results, before re-checking in the original space.normal indexing (Tree) the space is indexed using a monotone hyperplane tree with the Hilbert Exclusion property, without the use of reference points. The monotone hyperplane tree is used as, in previous work, this has been found to be the best-performing simple indexing mechanism for use with Hilbert Exclusion.§.§.§ Measurements Three different figures are measured for each mechanism: the elapsed time, the number of original-space distance calculations performed and, in the case of the re-indexing mechanisms, the number of re-indexed space calculations. All code is available online for independent testing [https://[email protected]/richardconnor/metric-space-framework.git].The tests were run on a 2.8 GHz Intel Core i7, running on an otherwise bare machine without network interference. The code is written in Java, and all data sets used fit easily into the Java heap without paging or garbage collection occurring. §.§.§ Results As can be seen in Table <ref>, N_rei consistently and significantly outperforms the normal index structure at between 15 and 25 dimensions, depending on the query threshold. It is also interesting to see that, as the query threshold increases, and therefore scalability decreases, N_seq takes over as the most efficient mechanism, again with a “sweet spot" at 15 dimensions.Table <ref> shows the same experiment performed with Cosine and Jensen-Shannon distances. In these cases, the extra relative cost saving from the more expensive metrics is very clear, with relative speedups of 4.5 and 8.5 times respectively. In the Jensen-Shannon tests, the relatively very high cost of the metric evaluation to some extent masks the difference betweenN_seq and N_rei, but wenote that the latter maintains scalability while the former does not.Finally, in the essentially intractable Euclidean space, with a relatively much smaller search threshold, N_seq takes over as the fastest mechanism. §.§.§ Scalability Table <ref> shows the actual number of distance measurements made, for Euclidean and Jensen-Shannon searches of the colors data. The number of calls required in both the original and re-indexed spaces are given. Note that original-space calls are the same for both table-checked and re-indexed mechanisms; the number of original-space calls include those to the reference points, from which the accuracy of the mechanism even in small dimensions can be appreciated. By 50 dimensions almost perfect accuracy is achieved for Euclidean search 50 original-space calculations are made,but in fact even at 10 dimensions almost every apex value can bedeterministically determined as either a member or otherwise of the solution set based on its upper and lower bounds. At 20 dimensions, only 10 elements of the 101414-element data set have bounds which straddle the query threshold. This indeed reflects the results presented in Figure <ref> where it is shown that for n≥20 the n-simplex lower bound is practically equivalent to the Euclidean distance to search colors data.Equally interesting is the number of re-indexed space calls. This gave us a considerable surprise, and is the subject of further investigation: for , these are generallyless than for the original space, including for tests made which are not presented here. This seems to hold for all dataother than perfectly evenly-distributed (generated sets), for which the scalability is the same. The implication is that the re-indexed metric has better scalability properties than the original, although we would have expected indexing over the lower-bound function to be less, rather than more, scalable. § CONCLUSIONS AND FURTHER WORKWe have used the technique to give best-recorded benchmark performance for exact search over the SISAP colors data set for some different metrics. It should however be noted that here we are only trying to demonstrate the potential value of the bounds mechanism in a simple and reproduceable context; as noted it is likely to be most effective in cases where the data set does not fit into memory, and where the metric used is very expensive. We emphasise that in all of our tests the whole data fits in main memory, and a recheck into the original space is relatively cheap.We believe the real power of this technique will emerge with huge data sets and more expensive metrics, and is yet to be experienced. §.§.§ AcknowledgementsThe work was partially funded by Smart News, “Social sensing for breaking news", co-funded by the Tuscany region under the FAR-FAS 2014 program, CUP CIPE D58C15000270008. plain§ APPENDIX Let Σ_Base∈n× n-1 representing a (n-1)-dimensional simplex of vertices Σ_Base[i]∈ℓ_2^n-1, with Σ_Base[i][j]=0 for all j≥ i and Σ_Base[n][n-1]≥0. Let v_i the corresponding vertices in ℓ_2^n (obtained from Σ_Base[i] by adding a zero to the end of the vector) and let δ_i the distance between an unknown apex point and the vertex v_i. Let o=[ o_1 … o_n ] the outputof the ApexAddition Algorithm. Then o is a feasible apex, i.e. it is a point in n satisfying ℓ_2(o,v_i)=δ_i for all 1≤ i≤ n. The last component o_n is non-negative and represents the altitude of o with respect to a base faceΣ_Base. It is sufficient to prove that the output o=[ o_1 … o_n ] of the Algorithm <ref> has distance δ_i from the vertex v_i, i.e. satisfies the following equations o_1^2+… +o_n^2=δ_1^2(<ref>.1)∑_j=1^i-1 (v_i,j- o_j)^2+ ∑_j=i^no_j^2=δ_i^2(<ref>.i)∑_j=1^n-1 (v_n,j-o_j)^2+o_n^2=δ_n^2 (<ref>.n) Note that the i-th component of the output o is updated only at the iteration i and i+1 of the ApexAddition Algorithm. So, if we denote with o^(i) the output at the end of iteration i we have: o^(1)=[ δ_1 0 … 0 ] o_i= o^(h)_i,o_n=o^(n)_n, o^(i)_h=0 1≤ i < h≤ n o_i-1=o_i-1^(i-1)-δ_i^2-∑_j=1^i-2(v_i,j-o_j)^2-(v_i,i-1-o_i-1^(i-1))^2/2v_i,i-1 2≤ i ≤n ( o_i-1)^2=( o_i-1^(i-1))^2- (o_i^(i))^21≤ i ≤n-1 By combiningEq. (<ref>) and (<ref>) we obtain ∑_j=i^n o_j^2= (o_i^(i))^2 for all 1≤ i ≤n-2, and so Eq. (<ref>.1) clearly holds (case i=1). Moreover, it follows that o satisfies Eq. (<ref>.i) for all i=2,…,n: ∑_j=1^i-1 (v_i ,j-o_j)^2+ ∑_j=i ^n o_j^2 =v_i ,i-1^2 -2v_i ,i-1 o_i-1+∑_j=1^i-2 (v_i ,i-1-o_j)^2+(o_i-1^(i-1))^2 (<ref>)=δ_i ^2 Let (U,d) a space (n+2)-embeddable in ℓ_2^n+1. Let p_1,…,p_n ∈ U and, for any m≤ n, let σ_m the (m-1)-dimensional simplex generated from p_1,…,p_m by using the nSimplexBuild Algorithm. For any x∈ U, let x^(m)∈ℓ_2^m the apex point with distance d(x,p_1), …, d(x,p_m) from the vertices of σ_m, computed using the ApexAddition Algorithm. Then for all q,s∈ U, * ℓ_2^m-1 (s^(m-1),q^(m-1)) ≤ℓ_2^m(s^(m),q^(m))2≤ m ≤ n * g (s^(m-1),q^(m-1)) ≥ g(s^(m),q^(m))2≤ m ≤ n * ℓ_2^n(s^(n),q^(n)) ≤ d(s,q) ≤ g(s^(n),q^(n)) where, for any k∈ℕ, g:ℓ_2^k→ℓ_2^k is defined as g(x,y)=√(∑_i=1^k-1 (x_i-y_i)^2+(x_k+y_k)^2). By construction, for any m≤ n we have x_i^(m)= x_i^(m-1)i=1,…, m-2x_i^(i)≥ 0 i=1,…, m (x_m-1^(m))^2+( x_m^(m))^2 =(x_m-1^(m-1))^2 Condition <ref> directly follows from Eq. (<ref>)-(<ref>): ℓ_2^m (s^(m),q^(m))^2 =ℓ_2^m-1 (s^(m-1),q^(m-1))^2- (s^(m-1)_m-1-q^(m-1)_m-1)^2+ ∑_i=m-1^m(s^(m)_i-q^(m)_i)^2=ℓ_2^m-1 (s^(m-1),q^(m-1))^2 + 2[-s^(m)_m-1q^(m)_m-1-s^(m)_mq^(m)_m+√((s_m-1^(m))^2+(s_m^(m))^2 )√((q_m-1^(m))^2+(q_m^(m))^2 )]≥ℓ_2^m-1 (s^(m-1),q^(m-1))^2 where the last passage follows from the Cauchy–Schwarz inequality [Cauchy–Schwarz inequality in two dimension is: (a_1b_1+a_2b_2)^2≤ (a_1^2+a_2^2)(b_1^2+b_2^2) ∀ a_1,b_1,a_2,b_2 ∈ℝ, which implies(a_1b_1+a_2b_2)≤√((a_1^2+a_2^2))√((b_1^2+b_2^2))∀ a_1,b_1,a_2,b_2 ∈ℝ ]. Similarly, Condition <ref> also holds: g (s^(m), q^(m))^2 =g(s^(m-1),q^(m-1))^2 + 2[-s^(m)_m-1q^(m)_m-1+s^(m)_mq^(m)_m-√((s_m-1^(m))^2+(s_m^(m))^2 )√((q_m-1^(m))^2+(q_m^(m))^2 )]≤ g(s^(m-1),q^(m-1))^2. Now we prove that ℓ_2^n(s^(n),q^(n)) and g(s^(n),q^(n)) are, respectively, a lower bound and an upper bound for the actual distance d(s,q). The main idea is using the simplexσ_n spanned by p_1,…, p_n as a base face tobuild the simplexσ_n+1 spanned by p_1,…, p_n, s and then use the latter as base face to build the simplex σ_n+2 spanned by p_1,…, p_n, s,q. In this way, we have an isometric embedding of p_1,…, p_n, s,q into ℓ_2^n+1 that is the function that mapsp_1,…, p_n, s,qinto the verticesof σ_n+2. So, given the base simplex σ_n (represented by the matrix Σ_n), and the apex s^(n), q^(n)∈ℓ_2^n we have that the simplexσ_n+2is represented by Σ_n+2 = [[3c|4*Σ_n 2c4*0; 3c|; 3c|; s^(n)_1 ⋯ s^(n)_n-1 s^(n)_n 0; q^(n)_1 ⋯ q^(n)_n-1 q^(n+1)_n q^(n+1)_n+1 ]]∈n+2× n+1 where, by construction, (q^(n+1)_n+1)^2=(q^(n)_n)^2-(q^(n+1)_n)^2, s^(n)_n,q^(n+1)_n+1≥ 0, and d(q,s) equals the euclidean distance between the two last rows of Σ_n+2. It follows that d(q,s)^2 = ∑_i=1^n-1(s^(n)_i-q^(n)_i)^2+(s^(n)_n)^2+(q^(n)_n)^2-2s^(n)_nq^(n+1)_n; and, since q^(n)_n≥ |q^(n+1)_n|, we have d(q,s)^2=ℓ_2^n(s^(n),q^(n))^2 +2s^(n)_n (q^(n)_n-q^(n+1)_n) ≥ℓ_2^n(s^(n),q^(n))^2, and d(q,s)^2=g(s^(n),q^(n))^2 -2s^(n)_n (q^(n)_n+q^(n+1)_n)≤ g(s^(n),q^(n))^2 | http://arxiv.org/abs/1707.08370v1 | {
"authors": [
"Richard Connor",
"Lucia Vadicamo",
"Fausto Rabitti"
],
"categories": [
"cs.IR",
"H.3.3"
],
"primary_category": "cs.IR",
"published": "20170726105228",
"title": "High-Dimensional Simplexes for Supermetric Search"
} |
firstpage–lastpage Novel Electronic State and Superconductivity in the Electron-Doped High-T_ c T'-Superconductors Y. Koike May 29, 2018 ===============================================================================================We investigated the connection between the mid-infrared (MIR) and optical spectral characteristicsin a sample of 82 Type 1 active galactic nuclei (AGNs), observed with Infrared Spectrometer on Spitzer (IRS)and Sloan Digital Sky Survey (SDSS, DR12). We found several interesting correlations between optical and MIRspectral properties: i) as starburst significators in MIR increase, the EWs ofoptical lines HβNLR and FeII, increase as well; ii) as MIR spectral index increases,EW([OIII]) decreases, while fractional contribution of AGN (RAGN) is not connected with EW([OIII]);iii) The log([OIII]5007/HβNLR) ratio isweakly related to the fractional contribution of polycyclic aromatic hydrocarbons (RPAHs). We compare the twodifferent MIR and optical diagnostics for starburst contribution to theoverall radiation (RPAH and BPT diagram, respectively). The significant differences between optical and MIRstarburst diagnostics were found. The starburst influence to observed correlations between optical and MIR parameters is discussed.galaxies: active – galaxies: emission lines § INTRODUCTION Understanding the nature of coexistence of active galactic nuclei (AGN) and surrounding starburst (SB) is one of the main problems of galactic evolution. Coexistence of AGNs and SBs is found in various samples such ashyperluminous infrared (IR) galaxies <cit.>, ultra(luminous) IR galaxies <cit.>,Seyfert 1 and Seyfert 2 galaxies <cit.>, studied in models <cit.> and often discussed in the frame of AGN spectral properties <cit.>. There are some indications that AGNs maysuppress star formation and gas cooling <cit.>, and that star formation is higher in AGNs with lower black hole (BH) mass <cit.>. Nevertheless, some studies show acorrelation between star formation rate and AGN luminosity, at high luminosity AGNs <cit.>. To find SB contribution to the AGN emission there are several methods in the optical and MIR spectrum. The empirical separation between the low-ionization nuclear emission-line regions (LINERs), HII regionsand AGNs at the optical wavelengths is the BPT diagram <cit.>, usually givenas the plot of the flux ratio of forbidden and allowed narrow lines log([NII]6563/Hα NLR) vs.log([OIII]5007/Hβ NLR). The main diagnostic assumptionis that HII regions are ionized by young massive stars, while AGNs are ionized by high energetic photons emitted from the accretion disc. In the case that the Hα spectral range is not presentin the AGN spectra, the R=log([OIII]5007/HβNLR) ratio may indicate the contribution of SB to the AGNemission, as e.g. <cit.> suggested that the dominant SBs have R<0.5,while dominant AGNs have R>0.5. Mid-infrared (MIR) based probes for the star formation suffer much less from the extinction than ultraviolet, optical and near-IR observations. The first comparison between Type 1 and Type 2 AGNs at 2-11 μm range is done by <cit.>. The star-forming galaxies are expected to have stronger polycyclic aromatic hydrocarbon (PAH) features <cit.>. <cit.> tested the relation between both 5-8.5μm PAH and 15μm continuum emission with Lyman continuum and found that MIR dust emission is a good tracer for the star formation rate. <cit.> used the Hα emission to calibrate PAH luminosity as a measure of a star formation rate. <cit.> and <cit.> discussed the discrimination of SBs from AGNs using theF_15 μ m/F_30 μ m continuum flux ratio. SBs have steeper MIR spectrum since AGNs produce a warm dust component in MIR <cit.>. The F_15 μ m/F_30 μ m ratio measures the strength of the warm dust component in galaxies and therefore reflects SB/AGN contribution.<cit.> have similar conclusion, that SB galaxies havelog(F_ 30 μ m/F_ 15 μ m) of 1.55, while AGNs have log(F_ 30 μ m/F_ 15 μ m) of 0.2. As a consequence, 6.2μmPAH equivalent widths (EWs) correlate with 20-30μm spectralindex <cit.>. A lower 25 to 60 μm flux ratio in SB galaxies than in Seyfert galaxies isexplained by cooler dust temperature in SB galaxies. The higher PAH EWs in SB galaxies than in Seyfertgalaxies is due to the PAH destruction from the high energetic radiation from the AGN accretion disc<cit.>. Another reason for PAH absence inluminous AGNs is a strong MIR continuum that can wash-out the PAH features, reducing their EWs <cit.>. MIR lines that could indicate an AGN presence are [NeV]14.32, [NeV]24.3, [SIV]10.51μm <cit.>, [OIV]25.9μm can originate both from starforming regions orAGNs <cit.>, while nearby [FeII]25.99μm is primarily due star formation<cit.>. [NeII]12.8μm is strong in SB galaxies, but weak in AGNs <cit.>.Therefore, <cit.> and <cit.> used various methodsto distinguish AGNs from SBs: [NeIII]15.6μm/[NeII]12.8μm, [NeV]14.3μm/[NeII]12.8μm,[NeV]24.3μm/[NeII]12.8μm, [OIV]25.9μm/[NeII]12.8μm, [OIV]25.9/[SIII]33.48 ratios, as wellas a fit to the MIR spectral-slope and strength of the PAH features. There is a number of correlations between the UV, optical and IR spectral properties that can be caused by physical characteristics of AGNs, but also by contribution of SB to the AGN emission. Some of these correlations can be an indicator of SB and AGN coevolution. Note here shortly the results of <cit.> (BG92 afterwards), who performed principal component analysis (PCA) on variousAGN optical, radio and X-ray characteristics. They found a set of correlations between different spectralparameters, projected to eigenvector 1 (EV1). Some of these are anticorrelations EW([OIII]) vs.EW(FeII) and full width at half maximum (FWHM) of Hβ_broad vs. EW(FeII)at optical wavelengths. These correlations in AGN spectral properties are very intriguing and theirphysical background is not understood. Other authors performed PCA in different AGN samples in orderto explain these correlations <cit.>. <cit.> and <cit.> suggested that theSB affects EV1, while <cit.> found that PAH characteristics are correlated with EV1. <cit.>performed PCA on Spitzer spectra of QSOs and compared these results with the ones from BG92.Many authors have compared the optical and MIR observations of AGNs<cit.>. <cit.> and <cit.> compared the BPT diagram classificationand MIR AGN/SB diagnostics and concluded that the congruence is high. On the other hand, <cit.>showed that the optical BPT AGN classification does not always match the one obtained from spectral energydistribution (SED) fitting from UV to far-infrared (FIR) wavelengths. <cit.> and <cit.>used Spitzer data andfound that there may exist AGNs in half of the luminous IR galaxies without any evidence of AGN atnear-infrared and optical wavelengths, undetected because of the extinction. At the MIR wavelength range one can see hot, AGN heated dust component, from the pc-size regionsurrounding the central BH, that could be the reservoir that feeds the central BH during theaccretion phase <cit.>. At the optical observations of Type 1 AGNs, the contributionsof accretion disc and broad line region are seen. However, the optical and MIR emission of an AGN should berelated, as e.g. if the emission of central continuum source is stronger, one can expect that the inner part of torus is larger <cit.>. Therefore, the correlations between MIR and optical emission in these objects were expected and found <cit.>.In this work we use optical (SDSS) and MIR (Spitzer) data to investigate the correlations between optical and MIR emission of Type 1 AGNs, and compare the influence of SB/AGN to the optical and MIRspectra. The paper is organized as follows: in Section <ref>, we describe our sampleof Type 1 AGNs, in Section <ref> we explain our data analysis for optical and MIR,in Section <ref> we present the results, in Section <ref> we discuss ourresults, and in Section <ref> we outline our conclusions.§ THE SAMPLE §.§ The sample of Type 1 AGNsIn this research, we used the sample of Type 1 AGNs, found in the cross-matchbetween optical Sloan Digital Sky Survey (SDSS) spectra and MIR Spitzer Space Telescope spectral data. The SDSS Data Release 12 (DR12) <cit.> contains all SDSS observationsuntil July 2014. These observations were done with 2.5 m telescope at the Apache Point Observatory withtwo optical spectrograph (SDSS-I and BOSS). All data from prior data releases are included in DR12and re-analyzed, so that it contains in total 477,161 QSO and 2,401,952 galactic optical spectra. The SDSS-Ispectra covering the wavelength range of 3800 Å to 9200 Å with spectral resolution 1850–2200,while BOSS spectra are observed in the range 3650-10400 Å, with spectral resolution of 1560–2650.IR data used in this work are reduced and calibrated 5-35 μm IRS[The Infra-redSpectrograph <cit.> on-board the Spitzer Space telescope.] spectra, available in the6^th version of The Cornell Atlas of Spitzer/IRS Sources(CASSIS[The spectra are taken from CASSIS (also known as the Combined Atlas of Sources with Spitzer IRSSpectra) web-page: <http://cassis.sirtf.com/atlas/>.]) database, inlow-resolution of R∼ 60-127 <cit.>. To find the sample for this investigation, we used Structural Query Language (SQL) to search forall galaxies and quasars in SDSS DR12 which satisfy following criteria:* S/N>15 in g–band (4686Å), in order to obtain the spectra of an adequate quality for fitting procedure,* z<0.7, with z_warning=0 matching the spectra which cover the optical range near the Hβ line,* the objects are classified as 'QSO' or 'galaxy' in SDSS spectral classification.The resulting search contained 135,633 objects. These objects were cross-matched in TOPCAT with the latest (from November 2015) Spitzer catalog – Infrared Database of Extragalactic Observables from Spitzer (IDEOS[<http://ideos.astro.cornell.edu/redshifts.html.>]) of3361 extragalactic sources <cit.>. This cross-match resulted with 585 objects.From that sample we selected Type 1 AGN by visualinspection of the optical spectra, removing all objects which do not have broad emission linesin the λλ4000-5500 Å range. That resulted with 98 Type 1 AGNs. Additionally, 16 objectswere removed from the sample because of high noise and/or poor fitting of IR spectra (see Section <ref>). Finally, the sample contains 82 AGNType 1, with optical and MIR spectra of satisfactory quality (see Table <ref>). The angular size distances are given in the column (7), while the projected linear diameters of the IRS and SDSSapertures are given in the columns (8) and (9). The distributions of the SDSS and IRS aperture projections for Type 1 sample are given in the Fig. <ref>, on the two panels – left. The SDSS aperture size is 3^'' <cit.>, while we took 4.7^'' for the IRS aperture <cit.>.§.§ The sample of Type 2 AGNs from the literatureBecause of the difficulties in decomposition of the narrow Hα, Hβ and [NII] lines inthe initial sample of Type 1 AGNs, to check some relations between narrow lines, we also considera sample of Type 2 AGNs, taken from the literature (more in Section <ref>). The sampleis taken from <cit.> and <cit.>,as these authors performed the fitting of CASSIS spectra of ∼150 objects – Seyfert 2galaxies, LINERs and HII regions, using deblendIRS routine (we use the same routine in this work). We took the fitting results that they obtained (spectral index, α and fractionalcontributions of AGN and PAH in the spectra – RAGN and RPAH, respectively) for all objects for which wefound the log([OIII]/Hβ) and log([NII]/Hα) measurements in the other literature. The sample has49 Type 2 AGNs and the data that we use are given in the Table <ref>. The angular size distances are given in the column (9), while the projected linear diameters of the IRS and SDSSapertures are given in the columns (10) and (11). The distribution of these aperture projections is given onthe Fig. <ref>, on the two panels – right.§ ANALYSIS §.§ AGN optical properties The SDSS spectra were corrected for the Galactic extinction by using the standard Galactic-typelaw <cit.> for optical-IR range and Galactic extinction coefficients given by <cit.>,available from the NASA/IPAC Infrared Science Archive(IRSA)[<http://irsa.ipac.caltech.edu/applications/DUST/>]. Afterwards, the spectra were corrected for cosmological redshift and host galaxy contribution(Section <ref>), and fitted using model of optical emissionin λλ4000-5500 Å and λλ6200-6950 Å ranges (Section <ref>).§.§.§ Host galaxy subtractionTo determine the host galaxy contribution in the optical spectra we applied thePCA <cit.>. PCA is a statistical methodwhich enables a large amount of data to be decomposed and compressed into independent components. In the case of the spectral PCA, theseindependent components are eigenspectra, whose linear combination can reproduce the observed spectrum.Spectral principal component analysis is commonly used for classification of galaxies and QSOs<cit.>. <cit.> used 170,000 galaxySDSS spectra to derive the set of the galaxy eigenspectra, while <cit.> used 16,707 QSO SDSS spectrato construct several sets of eigenspectra which describe the QSO sample in different redshift and luminositybins.<cit.> introduced the application of this method for spectral decomposition into pure-host and pure-QSO part of an AGN spectrum. This technique assumes that the composite, observed spectrum can be reproduced well by the linear combination of two independent sets of eigenspectra derived from the pure-galaxy and pure-quasar samples. They applied different sets of galaxy and QSO eignespectra derived in <cit.>, and found that the first fewgalaxy and quasar eigenspectra can reasonably recover the properties of the sample.Following the procedure described in <cit.>, we used the first 10 QSO eigenspectraderived from high-luminosity (C1), low-redshift range (ZBIN 1), defined by <cit.>, and thefirst 5 galaxy eigenspectra derived in <cit.>. The galaxy eigenspectra are downloaded fromSDSS Web site[<http://classic.sdss.org/dr2/products/value_added/>], while the QSO eigenspectraare obtained in the private communication <cit.>. First, we re-bined the observed spectra and 15 eigenspectra to have the same range and wavelength bins.Afterwards, we fitted our spectra with a linear combination of QSO and galaxy eigenspectra.The host part of the spectrum is derived as a linear combination of galaxy, while the AGN part as a linear combination of QSO eigenspectra. We masked allnarrow emission lines from the host galaxy part, and subtract only the host galaxy continuum and stellarabsorption lines from the observed spectra. Finally, we performed the fitting on that spectra. Our model isdescribed in the Section <ref>, while measuring of optical parameters is explained inSection <ref>. The example of spectral PCAdecomposition is shown in Fig. <ref>. The host fraction, F_H is measured in the λλ4160-4210Å range, and presented in Table <ref>.In some cases, the fitting results may give a non-physical solutions, such as negative host or AGN part.<cit.> also had non-physical solutions in their Table 2. In our sample we had several objects where fit gives the negative host contribution. In these cases, we assumed that the host part isequal to zero, therefore we excluded galaxy eigenvectors from ourfitting and fitted only with the QSO eigenvectors, since we find that these fits are usually good inthe part of the spectra we were interested in (see Fig. <ref>).§.§.§ Model of the line spectra in the optical rangeAfter the host galaxy contribution is subtracted from the observed spectra, the power-lawSED, typical for the quasars, with the broad and narrow emission lines is obtained. The QSOcontinuum is estimated using the continuum windows given in <cit.>. The points of the continuumlevel are interpolated and the continuum is subtracted <cit.>. The optical emission lines were fitted in two ranges: λλ4000-5500 Å in order to cover Balmer lines (Hβ, Hγ, Hδ), optical FeIIand [OIII] λλ4959, 5007Å lines, and λλ6200-6950 Å,where Hα and [NII] λλ6548, 6583Å lines are present. For fittingthe emission lines we used a model of multi-Gaussian functions <cit.>, where each Gaussian is assumedto represent an emission from one emissionregion <cit.>In this model, the number of the free parameters is reducedassuming that lines or line components, whichoriginate from the same emission region, have the same widths and shifts.Therefore, all narrow lines in considered ranges ([OIII] λλ4959, 5007Å, narrowBalmer lines, [NII] λλ6548, 6583Å, etc.) have the same parameters of the widthsand shifts, since we assumed that they are originating in the Narrow Line Region (NLR). The [OIII]λλ4959, 5007Å lines are fitted with an additional component which describes the asymmetryin the wings of these lines, while the flux ratio of the λ4959:λ5007Å is taken as1:2.99 <cit.>. The ratio ≈3 is taken for [NII] doubletcomponents. The Balmer lines are fitted with three components, one narrow – Narrow Line Region (NLR), and two broad– Intermediate and Very Broad Line Region (ILR and VBLR), see <cit.>. All ILR components of the Balmer lines have the same widths andshifts. The same is for the VBLR components. Numerous optical FeIIlines in the λλ4000-5500Å range are fitted with the FeIItemplate[<http://servo.aob.rs/FeII_AGN/>] presented in <cit.> and <cit.>.In this FeII model, all FeII lines have the same widths and shifts, while relative intensitiesare calculated within the different FeII line groups, which have the same lower level of the transition<cit.>.The detailed description of this multi-Gaussian model and the fitting procedure is given in<cit.> and <cit.>. The examples of the best fit in the λλ4000-5500Å and λλ6200-6950Å ranges are given in Figs. <ref> and <ref>.§.§.§ Measuring the optical spectral parameters After performing the decomposition, we measured differentspectral parameters of all considered optical emission lines and their components. Kinematic parameters, Doppler widths and velocity shifts of emission lines are directly obtained as a product of fitting procedure. Additionally, we measured the FWHM of broad Hβ (ILR+VBLR component), FWHM(Hβ). The EWs of emission lines have been measured with respect to pure QSO continuum (after subtraction of the host contribution) below the lines <cit.>. The flux of the pure QSO continuum is measured at λ5100Å, and continuum luminosity is calculated using the formula given in<cit.>, with adopted cosmological parameters: Ω_M=0.3, Ω_Λ=0.7 and Ω_k=0, and Hubble constant H_0=70 km s^-1 Mpc^-1. The mass of the black hole (M_BH) is calculated using the improved formula from <cit.>, for the host light corrected L_5100, given in <cit.>. All measured properties that we used in this work are given in Table <ref>. The distributionsof measured parameters from optical spectra are given in Fig. <ref>.§.§ MIR properties of the AGNsTo study the AGN MIR properties of the sample, we need to disentangle the AGN emission from interstellarPAH, and stellar (STR) components, using some of the existing tools for spectraldecomposition of IRS data. We used deblendIRS[<http://denebola.org/ahc/deblendIRS>]routine, written in IDL <cit.>. Having a collection of real spectral templates, thissoftware chooses the best linear combination of one stellar template, one PAH template, and one AGN template,to model an IRS spectra, only in the spectral range 5.3–15.8μm. An example of fitting of an IRS spectrum is present on the Fig. <ref>. For the stellar templates, theroutine uses 19 local elliptical and S0 galaxies, the 56 PAH templates are IRS spectra of normalstarfoming and SB galaxies at z≤ 0.14, while the AGN templates are 181 IRS spectra of sources classified in the optical as quasars, Seyfert galaxies, LINERs, blazars, optically obscured AGNs and radio galaxies.The main fitting results are fractional contributionsof AGN, PAH and stellar components to the integrated 5–15μm luminosity, named RAGN, RPAH and RSTR,respectively (RAGN+RPAH+RSTR=1), spectral index of the AGN component(assuming a power law continuum f_ν∝ν^α, between 8.1 and 12.5μm), α andthe silicate strength of the best fitting AGN template, S_SIL <cit.>.They define S_SIL as a ln(F(λ_p)/F_C(λ_p)), where F(λ_p) andF_C(λ_p) are the flux densities of the spectrum and the underlying continuum, at the wavelength of the peak of the silicate feature. Other resultsof the fit are flux densities of AGN, PAH and STR components (spectra), names of the used spectral galactic templates, χ^2, thecoefficient of variation of the rms error (CV_RMSE), monochromatic luminosities of the source andfractional contribution to the restframe of the AGN component, at 6 and 12μm. The resulting parametersthat we use in this work are given in Table <ref>, while the distributions of the main parameters isgiven on the Fig. <ref>. In these histograms one can see that the AGN contribution is very dominantin the sample, comparing to PAH and stellar emission. Silicate featureis usually in the emission (positive S_SIL). From the fitting results we chose 82 successful fits, based on low reduced χ^2, CV_RMSE <cit.>, and the visual inspection.The 16 sources were rejected due to poor fitting, from which 6 are extended sources. All these 82 objectsare point sources except one, 0615-52345-0041. Independently of deblendIRS code, we calculated EWs of PAH features at 7.7 and 11.2 μm (given inTable <ref>), usingSTARLINK software <cit.> on redshift corrected IRS spectra.§.§ Broad-line Balmer decrement and nuclear extinction As can be seen on Figs. <ref> and <ref>, we decompose the Hα and Hβ into three components: NLR, ILR and VBLR. The narrow-line Balmer decrement, Hα^n/Hβ^n, is calculated as a flux ratio of HαNLR and HβNLR, while the broad-line Balmer decrement is found as Hα^b/Hβ^b, where Hα^b and Hβ^b are sums of the ILR and VBLR flux components. These fluxes and ratios are given in the Table <ref>. The broad-line Balmer decrement is often used for the estimation of the dust extinction in the BLR <cit.>. The distribution of broad-line Balmer decrement is given in the histogram on the Fig. <ref>. They are typically lower than the ones for the submm galaxies from <cit.> that have values from 5-20. Obtained Hα^b/Hβ^b ratios are comparable to the values from the large sample of Seyfert 1 galaxies from <cit.>, that are well described with a log-Gaussian, with a peak at 3.05. The median value of our decrements is 3.76. However, we should note that the other effects can affect the Hα^b/Hβ^b ratio, such as photoionization, recombination, collisions, self-absorption, dust obscuration, etc. <cit.>.If we use the equation (5) from <cit.>, we could estimate the minimum color excess, E(B-V), from the sample to be -0.263. For that value of E(B-V), one can use the reddening curve from <cit.> to calculate the extinction, A_λ at any specific wavelength.§ RESULTS We found that in the Type 1 AGN sample there is relatively small contribution of PAH, of RPAH<20%, forthe majority of objects (see histogram in Fig. <ref>). On the histogram on the Fig. <ref>, in Type 1 AGN sample we see more silicateemission (S_SIL>0), than absorption (S_SIL<0), as mentioned in <cit.> and<cit.>. <cit.> suggested that the QSOs are characterized by silicate emission, whileSy1s have equally distributed emission or weak absorption. That is in the agreement with our findings. The silicate feature in the absorption means there is a coolerdust between the observer and the hotter dust responsible for the MIR continuum.We created a linear correlation matrix for all MIR and optical parameters together. Pearson correlation coefficients and the P-values are given in the Table <ref>, higher and lower, respectively.Significant correlations are marked with stars. §.§ Expected and confirmed relationsOne of the most expected results is the correlation between the AGN continuum luminosity (L5100) atoptical wavelengths and total luminosities at 6 and 12μm, with ρ=0.67 and 0.66, whileP<0.00001 (Fig. <ref>). Also, there is the dependence of RAGN and RPAH with L5100, ρ=0.55 and -0.54, respectively,with P<0.00001, which is shown in numerous works <cit.>. Another expected relation that we found is that RAGNand RPAH are in trend with redshift; Pearson' coefficients are ρ=0.31;P=0.0038 and ρ=-0.24; P=0.03, respectively. Finally, RAGN and RPAH are in trend with theluminosities at 6μm (ρ=0.39, P=0.0003 and ρ=-0.27; P=0.014) and at 12μm (ρ=0.41;P=0.0001 and ρ=-0.27; P=0.014). S_SIL is only weakly correlated with α, with ρ=0.228,P=0.039, and that is already shown in the literature <cit.>.Among known anticorrelations that we expected to obtain is between RPAH and black hole mass,M_BH <cit.>, see Fig. <ref>. Here we obtained a slight trend with Pearson' coefficient ρ=-0.44, with P=0.00004; PAH may be more dominant in AGNs with lower M_BH.We obtained expected correlations between EW([OIII]) and EW(FeII) (ρ=-0.34; P=0.002), as well asbetween EW(FeII) and FWHM(Hβ) (ρ=-0.32; P=0.003, see Fig. <ref>), whichare a part of the EV1 from BG92.§.§ Starbursts at MIR wavelengths At MIR wavelengths, we may estimate the SB contribution to the total radiation based on the RPAH result.Here, we additionally, compare the RPAH with other two usual methods by which SB is estimated. The first is the ratio of the fluxes at 15 and 30 μm, the most accurate method, as suggested by <cit.>, see Fig. <ref>. There is a significant correlation between RPAH and this ratio. On this plot, x-axis is divided to the binsof the width 0.25 and binned data are shown with the triangles.The second criterion is the the strength of some PAH feature; here we present the EWs of the 7.7 and 11.2μm PAH features, on Fig. <ref>. There is asignificant dependence between EWs and RPAH. As suggestedby <cit.>, the objects with EW7.7 μm >1 are SB dominated, while the rest are AGN dominated. On this plot, EW7.7 μm >1 is present onlyfor two objects. Interestingly, only these two objects have RPAH > 50%. §.§ Comparison of the optical and MIR parametersIn the correlation matrix (Table <ref>) there are several trends between the optical and MIRparameters. As SB significators in the MIR (RPAH, EWPAH7.7, EWPAH11.2) increase, EWs of the optical linesFeII and HβNLR increase, as well. On the other hand, as the MIR spectral index, α increases,a well known AGN indicator, EW([OIII]) decreases (ρ=-0.37, P=6.6E^-4), while the EW([OIII]) is not related to the AGN or PAH fraction. We do not notice any trend of PAH EW or RPAH with the FWHM(Hβ) line, as suggested by <cit.>,who found that narrower broad Hβ lines have a stronger PAH emission in the Type 1 AGNs. However, ourlater analysis (Section <ref>, <ref>), will show that there is a connectionbetween RPAH and FWHM(Hβ).Considering this comparison it should be emphasized that the BPT and MIR SB/AGN diagnostics do not necessarily trace the contribution of an AGN to the total power of the galaxy. Therefore, there may exist some other effects which can affect one or both diagnostics.§.§ Comparison between starburst fraction at optical (BPT diagram) and at MIR wavelengths (RPAH) Traditionally, BPT diagram have been used for optical diagnostics between AGN, composites and SB<cit.>. In Fig. <ref> we show the BPT diagram of the Type 1 data sub-sample of69 objects with available range of 6200-6550Å which covers Hα and [NII] lines. To find the realratio between the AGN and PAH contribution in the MIR spectra and present it on BPT diagram, we excludedstellar contribution, by using the formula RPAH=RPAH/(RPAH+RAGN) from now on. On the diagram, the RPAH isquantified by the three different symbol sizes. Clearly, there are a couple of objects with a low RPAH, that lie below the solid separation curve from <cit.>; they should be above that curve, by the optical diagnostics. Similarly, a few SB dominated objects,according to the MIR fitting, lie on the AGN part of BPT diagram. These results suggest that there mightbe a significant difference between optical and MIR SB quantification.To confirm these doubts, taking into account that it is complicated to decompose the narrow lines in theType 1 AGNs <cit.>, we chose a sample of Type 2 AGNs (seeTable <ref> and Section <ref>). We made another BPT diagram, composed from the various LINER,HII regions and Seyfert 2 galaxies, from the samples of <cit.> and <cit.>.That BPT diagram is shown in the Fig. <ref>. Again, the symbol size represents RPAH value.Similarly as above, we notice that often these optical and MIR results give different information about SB and AGN ratio.<cit.> suggested that one axis of the BPT diagram, the ratioR=log([OIII]5007/ H βNLR), may be the significantindicator of the SB activity, where the objects with R<0.5 are SB dominated and the rest are AGN dominated. Furthermore, these authors showed that these two groups(R<0.5 and R>0.5) belong to the different populations of objects, since they show certain differentoptical characteristics, and that is the essence of the BPT diagram. Therefore, wecompare the RPAH (from MIR data) and this ratio R, on the Fig. <ref> for both Type 1 and Type 2 samples. Here we found quite weak trend of ρ=-0.26 and P=0.018 for Type 1 and stronger ρ=-0.6 and P=4.77× 10^-6 correlation for the Type 2 sample. Thisgraph shows that the objects with R<0.5 are not always SB dominated (based to MIR data), hence theremay be some other reason why they are different at optical wavelengths, or the origin of the optical andMIR radiation may be different. It can be seen in Fig. <ref> that, in general, the ratio log([OIII]5007/ H βNLR) is decreasing as RPAH is increasing.One should be aware that this comparison between MIR and optical diagnostics is limited by the factors such as the used data, radiation mechanisms, observed wavelength ranges and power distribution, and therefore these conclusions should not be taken literally. §.§ Principal Component analysis of the spectral parameters The dependence between all calculated optical and MIR parameters is complex and therefore we used the PCanalysis to understand the most important connections. Having our correlation matrix (Table <ref>),we chose several parameters which should contain a potentially unique information. The optical parametersare: EW([OIII]), EW(FeII), EW(HβNLR), EW(Hβ_broad) (ILR+VBLR),log([OIII]/HβNLR), FWHM(Hβ) and log(L5100), which are chosen to becompared with correlations found in the EV1 of BG92. The MIRparameters taken for analysis are: RPAH, α and S_SIL (since it is a possible indicator of theAGN geometry or inclination). The aim of this analysis is:* to check possible connections between BG92 EV1 correlations (between EW(FeII) vs. EW([OIII]) and EW(FeII) vs. FWHM(Hβ)) and some MIR spectral properties, which could give us some insight in physical cause of these correlations.* to compare the RPAH with log([OIII]5007/HβNLR), in order to clarify the previous results (see Section <ref>) about their inconsistency.For the PCA we used the task princomp with the option cor=true, in R. The resultsof the PCA are shown in the Table <ref>. The first four components together account for 71% ofthe variance. The first two components are the most important underlying parameters that governthe observed properties of the AGN sample. Both, the first and the second eigenvector account for≈20% variance, while each of the next two eigenvectors account for ≈10% variance.The PCA indicates that the first principal component is dominated by RPAH and EW(HβNLR). While RPAH, EW(HβNLR) and EW(FeII) have positive projections on the first eigenvector,log([OIII]/HβNLR), FWHM(Hβ) and log(L5100) have negative projections.Therefore, this eigenvector indicates that as the SB contribution, measured in IR, is stronger,EW(HβNLR), EW(FeII) increase, but the optical luminosity, FWHM(Hβ) and log([OIII]/HβNLR)ratio decrease. It seems that the strongest indicator of SB presence in optical is EW(HβNLR) (see Fig. <ref>), and consequently, the ratio of log([OIII]/HβNLR) isrelated to SBs as well. Note that the EW([OIII]) lines are not dependent on a SB presence (RPAH). Thiseigenvector shows that stronger SBs are present in AGNs with lower FWHM(Hβ) and stronger EW(FeII).This is one of the anticorrelationsfrom BG92 EV1 (EW(FeII) vs. FWHM(Hβ); see Fig. <ref>). This confirms the resultof <cit.>, who foundthat SB presence is stronger in AGNs with lower width of the broad Hβ lines. The second eigenvector is dominated with MIR spectral index, α and EW([OIII]). α andEW(FeII) have positive projections, as well as S_SIL and log(L5100), but weaker. On theother hand, EW([OIII]), and consequently log([OIII]/HβNLR), have the negative projections. EW(HβNLR)is not projected on this eigenvector. This implies that as the α grows, theEW([OIII]) and log([OIII]/HβNLR) decreases, but EW(FeII) increases. In this eigenvectoris projected the anticorrelation between EW([OIII]) and EW(FeII) from BG92 EV1, which seems to berelated with α.The third eigenvector is dominated by EW(Hβ_broad), which correlateswith the FWHM(Hβ) broad line and anticorrelates with EWs of the narrow lines. The fourth eigenvector is strongly dominated with the strength of the silicate feature (S_SIL)which is correlated with FWHM(Hβ). Thisimplies the connection between the widths of the broad lines and inclination angle or geometry of torus.§ DISCUSSION §.§ BPT diagram at MIR wavelengths In the Section <ref>, we found a certain disagreement between optical andMIR quantifying of the SB contribution to the AGN spectra, for both the Type 1 and Type 2 AGNs. Thepresence of the Type 1 AGNs on the SB part of the BPT diagram that we noticed here is earlierobserved <cit.>. We do not completely understand the reason of themisplacement of these objects on the BPT diagram. The extinction at the optical wavelengthsis one of the most often explanations, although some authors believe that the radiation may come from thedifferent regions and/or that the slit difference between SDSS and IRS might contribute to thisdisagreement <cit.>. We can not exclude the possibility of the imperfection of some of the methods, such as the decomposition or fitting. These results remind to the results of<cit.> and <cit.>, who found that there may exist AGNs in half of the luminousIR galaxies without any evidence of AGN at the near-infrared and optical wavelengths. It seems that the Type 1 AGNs have a lower disagreement than the Type 2. The possible reasons for that are: 1) Maybe since the AGN signature is more prominent in the Type 1 sample; 2) The column density should be lowerfor the Type 1 than for Type 2 AGNs, thus the probability to observe the different regions of the AGNin optical and MIR is lower; 3) Because the Type 1 sample is more homogeneous and may be more accurate. Another cause of this disagreement between optical and MIR SB/AGN dominance could be the difference in the emission from within the optical or MIR wavebands sampled by the data. The only way to properly estimate which power source dominates the emission in galaxy is to sample the full SED or to apply bolometric corrections to the data. Methods of this type have been done in analyses of ULIRGs from various groups <cit.>. §.§ Comparison between the optical and MIR SB/AGN diagnosticsAs we mentioned, the objects with R<0.5 have somewhat different certain opticalcharacteristics <cit.>; which is believed to be caused by SB presence. However, aswe obtained, on Fig. <ref>, objects with R<0.5 do not always have a high RPAH contribution, therefore there may exist some other reason why these objectshave special optical characteristics. PCA of the optical and MIR parameters confirms the results of BG92 and the other authors. Itshows that R is probablyinfluenced by more different physical properties of AGNs. Namely, EW(HβNLR) is correlated withSB strength in MIR, while [OIII] lines arecorrelated with α. This means that R is indeed influenced by the SB presence, but also influenced by some other physical property which affects α, which may bethe cause of disagreement between MIR and optical diagnostics of the SB presence.§.§ Connection between BG92 EV1 and MIR propertiesSince BG92 established the set of correlations between AGN Type 1 spectral properties (EV1 in their PCA),it has been many attempts to explain their physical origin. The most frequently proposed governing mechanismsare: 1) Eddington ratio, L/L_edd <cit.> 2) AGN orientation <cit.>, 3)and combination of these two properties <cit.>.It was proposed that BG92 EV1 correlations can be considered as a surrogate "H–R Diagram" for Type 1 AGNs,with a main sequence driven by Eddington ratio convolved with line-of-sight orientation<cit.>, for a review see <cit.>. Also, the BG92 EV1 is considered as anindicator of the AGN evolution <cit.>. The evolution of AGNs is probablyrelated with SB regions, assuming that there is stronger presence of the SB nearby thecentral engine of AGN in an earlier phase of AGN evolution, while in the later phases, the SBcontribution probably becomes weaker and/or negligible <cit.>.To understand the physical background of BG92 EV1 correlations, in Section <ref>, we performed PCAusing several optical and MIR spectral parameters. When interpreting the PCA results, it is important totake into account that an eigenvector is always specific to a certain sample, depending which observedparameters have been used and on the range of the parameters <cit.>. Therefore, each individualsample has its own eigenvectors, e.g. when using a set of different spectral parameters for PCA, BG92 EV1 can be projected on some other eigenvector, or divided into two or more eigenvectors. In this analysis, RPAH is chosen asa SB indicator <cit.>, α as a MIR spectralindex of the pure AGN continuum, and S_SIL should be an indicator of thegeometry and inclination <cit.>. The results of the PCA are summarized in Table <ref>, wherecorrelations between the optical and MIR parameters are denoted with uprising arrows, anti-correlations withdecreasing arrows, and lack of any connection (projections to eigenvectors<0.25) with zero. In cases ofweak connection (0.25<projections to eigenvector<0.30) we note "weak" in Table <ref>. Theseresults imply that the two the most interesting BG92 EV1 anti-correlations (EW(FeII) vs. FWHM(Hβ) andEW(FeII) vs. EW([OIII])) are relatedwith different MIR parameters, since they are projected into two different eigenvectors in our analysis. The first dominated with RPAH and the second with α. The EW(FeII)vs. FWHM(Hβ) anti-correlation is connected with SB presence (RPAH), while EW(FeII) vs. EW([OIII]) seems tobe connected with some physical property, reflected in α. The α is a complex parameter whichreflects MIR SED, and therefore depends on several physical properties as:i) accretion disc radiation (which depends on Eddington ratio, L/L_edd, <cit.>,ii) inclination, and iii) torus physical properties as geometry, dust distribution, optical depth, etc.<cit.>.Although it is supposed that they are dominantly driven by different physical properties, the relationbetween RPAH, α and S_SIL seems to be complex. <cit.> tested the validity of theMIR spectral decomposition of deblendIRS code (see their Section 3.2), and found thatα is in significant correlation with nuclear spectral index derived from ground-based observations.Althoughα should be a pure AGN property, it is in a weak anticorrelation with the RPAH, EWPAH7.7 andEWPAH11.2 in our sample (see Table <ref>). The trend between α and RPAH may be causedby the reverse dependence between RPAH and RAGN (see Fig. <ref>). The trend between the αand EW of PAHs is probably present because EWs aremeasured relative to the total continuum flux, with AGN flux included. We found a weak trend between αand S_SIL in our correlation matrix (Table <ref>) and Section <ref><cit.>. <cit.> used the MIR color α(60,25) as an indicatorof the SB presence, and found its correlation with BG92 EV1 correlations. Note, that the MIR colorα(60,25) is measured using the total (AGN+SB) flux, and therefore it contains information about theSB presence and AGN continuum slope, which are in this work separated in two parameters, RPAH and α.Our results are consistent with results of <cit.>, but we give more detailedinsight in origin of BG92 EV1 correlations.§ CONCLUSIONSHere we investigate the optical and MIR spectral properties of a sample of 82 Type 1 AGNs. Additionally, to check the results based on the narrow lines (which are superposed with broad lines in the Type 1 AGNs), we considered a sample of 49 Type 2 AGNs. We carefully fit the optical spectra using methods described in <cit.>, <cit.> and <cit.>. For fitting of MIR data we useddeblendIRS code described in <cit.>. Concerning our investigation, we canoutline following conclusions: * In sample of Type 1, we see more silicate emission, than absorption, which is expected, according to <cit.> and <cit.>.* We did not find any linear trend in the correlation matrix between EW(PAH) or RPAH with FWHM(Hβ) (see Table <ref>), but PCA shows anticorrelation between these properties, which is in an agreement with the result of<cit.>, who found that the narrower broad FWHM(Hβ) have stronger a PAH emission.* The separation between AGN and SB based on the BPT diagram does not give the same result as theone from MIR spectra in both Type 1 and Type 2 AGN samples. Some of the possible reasonsare extinction at optical wavelengths <cit.>, different sizes of slits, or the radiation may come from thedifferent regions <cit.>.* The weak correlation between the main optical (log([OIII]5007/HβNLR)) and MIR (RPAH)starburst estimators implies that the difference in the optical characteristics in the objects with log([OIII]5007/HβNLR)<0.5and >0.5 <cit.> may have a different reason than the star formationpresence.* PCA shows that the anticorrelations between EW(FeII) vs. FWHM(Hβ), as wellas EW(FeII) vs. EW([OIII]), from BG92 EV1, probably have a different governing mechanism: the formeris connected with SB presence, while the latter is more connected with the MIR AGN spectral index.* PCA implies that the ratio log([OIII]5007/HβNLR) is indeed influenced by the starburst presence, but also influenced by some other physical property (MIR AGN spectral index), which may bethe cause of disagreement between MIR and optical diagnostics of the starburst presence in AGN spectra.* A well known AGN indicator, EW([OIII]) is related to the MIR spectral index α, but not relatedto the AGN or PAH fraction.* Since the BPT and MIR SB/AGN diagnostics do not necessarily trace the contribution of an AGN to the total power of the galaxy, the dissagrement between the two methods is not overly unexpected.Finally, here we confirm some correlations between the optical and IR spectral properties that have been governed by presence of the SB contribution. EW(FeII) and EW(HβNLR) are correlated with RPAH. Anticorrelation EW(FeII) vs. FWHM(Hβ) may be also connected with the RPAH, since they areprojected on the same eigenvector as RPAH.§ ACKNOWLEDGMENTS This work is part of the project (146001) "Astrophysical Spectroscopy of Extragalactic Objects" supported by the Ministry of Science of Serbia.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 SDSS-III web siteis http://www.sdss3.org/.SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of theSDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, BrookhavenNational Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group,the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the MichiganState/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory,Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico StateUniversity, New York University, Ohio State University, Pennsylvania State University, University ofPortsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University ofUtah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.The Cornell Atlas of Spitzer/IRS Sources (CASSIS) is a product of the Infrared Science Center at CornellUniversity, supported by NASA and JPL.Much of the analysis presented in this workwas done with TOPCAT(<http://www.star.bris.ac.uk/m̃bt/topcat/>), developed by M. Taylor.We thank dr Ching Wa Yip, dr Nataša Bon, dr Marko Stalevski, dr Predrag Jovanović and dr Giovanni Lamura for help with important issues in this work. We also thank the referee for helpful and constructive suggestions.§ ESTIMATION OF THE NARROW EMISSION LINES FOR THE BPT DIAGRAMThe axes in the BPT diagram are the ratios of the particular narrow lines ([OIII]/HβNLR and[NII]/HαNLR) and therefore the accurate measurements of the fluxes of these lines is very importantfor correct AGN/SB diagnostics.In the Type 1 AGNs, these narrow lines overlap with the broad lines, and in some cases it is very difficultto distinguish them. We use the same fitting procedure as in <cit.>, where the confidence ofthe narrow Hβ component estimation and non-uniqueness in the solutions is tested and discussed<cit.>.As it is explained in Section <ref>, we use the same parameters for widths and shifts for allconsidered narrow lines ([OIII], HβNLR, [NII] and HαNLR). In this way, we are getting lessdegree of freedom in the fitting procedure and more confident fits in the case when one of these lines can notbe distinguished well from the broad lines.In our sub-sample of 69 objects with available optical range of λλ6200-6950Å (which covers[NII] and Hα), in 26 objects [NII] lines are are very weak and barely seen in the broad Hαprofiles. These objects are assigned in the BPT diagram as less confident (empty squares, seeFig. <ref>). The example of this kind of objects is shown in Fig. <ref>. On the other hand,in 43 objects from the sub-sample, [NII] lines can be well resolved from the broad Hα profile (see theexample in Fig. <ref>), and these object are assigned in the BPTdiagram as more confident (full squares in Fig. <ref>).[Alam et al.(2015)]2015ApJS...219..12A Alam, S., Albareti, F. 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"authors": [
"Maša Lakićević",
"Jelena Kovačević-Dojčinović",
"Luka Č. Popović"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170726115408",
"title": "The optical vs. mid-infrared spectral properties of 82 Type 1 AGNs: coevolution of AGN and starburst"
} |
mko]M.K. Olsen [email protected] of Mathematics and Physics, University of Queensland, Brisbane,Queensland 4072, Australia. We analyse the output quantum tripartite correlations from an intracavity nonlinear optical system which uses cascaded nonlinearities to produce both second and fourth harmonic outputs from an input field at the fundamental frequency. Using fully quantum equations of motion, we investigate two parameter regimes and show thatthe system producestripartite inseparability, entanglement and EPR steering, with the detection of these depending on the correlations being considered.Cascaded systems, entanglement, steering.§ INTRODUCTIONFourth harmonic generation has not received a huge amount of attention in the scientific literature, possibly because materials with the nonlinearity needed for a five wave mixing process are difficult to find. Despite this inherent problem, and the difficulty of finding materials that are transparent over two octaves, Komatsu have successfully produced fourth harmonic from Li_2B_4O_7 crystal, with a conversion efficiency of 20% <cit.>. The advent of quasi-periodic superlattices meant that higher than second order processes were now available, with Zhu producing third harmonic by coupling second harmonic (SHG) and sum-frequency generation in 1997 <cit.>. Using CsLiB_6O_10, Kojima were able to produce fourth harmonic at a 10 kHz repetition rate by 2000 <cit.>. Broderick have produced fourth harmonic from a cascaded SHG process using aHeXLN crystal <cit.> tuneable for both processes at the same temperature. Südmeyer produced fields at both second and fourth harmonics using an intracavity cascaded process with LBO and BBO crystals, with greater than 50% efficiency in 2007 <cit.>. More recently, Ji have generated light at 263 nm from a 1053 nm input, using KD^*P and NH_4H_2PO_4 crystals with non-critical phase matching <cit.>.The theoretical examination of the quantum statistical properties of fourth harmonic generation began with Kheruntsyan , who analysed an intracavity cascaded frequency doubler process <cit.>. The authors adiabatically eliminated the highest frequency mode to calculate squeezing in the lower modes, also finding self-pulsing in the intensities. Yu and Wang <cit.> performed an analysis of the system without any elimination, starting with the full positive-P representation <cit.> equations of motion. Linearising around the steady-state solutions of the semi-classical equations, they performed a stability analysis and examined the entanglement properties using the method of symplectic eigenvalues <cit.>. More recently, Olsen has examined the quantum statistical properties of the system <cit.>, finding that quadrature squeezing and bipartite entanglement and asymmetric Einstein-Podolsky-Rosen (EPR) steering <cit.> are available for some of the possible bipartitions.In this work we extend previous analyses by examining various correlations often used in continuous variable systems to detect tripartite inseparability, entanglement, and steering. We begin with the two types of van Loock-Furusawa (vLF) inequalities <cit.> and their refinements by Teh and Reid <cit.> for mixed states. Following from that, we will use three mode EPR <cit.> inequalities developed by Olsen, Bradley and Reid (OBR) <cit.>, to investigate whether two members of a possible tripartition can combine to steer the third and whether our results can indicate genuine multipartite steering <cit.>. We investigate two different regimes with changes in the pumping rate, the loss rates, and the ratio of the two χ^(2) nonlinearities. § HAMILTONIAN AND EQUATIONS OF MOTIONThe system consists of three optical fields interacting in nonlinear media, which could either be a periodically poled dielectric or two separate nonlinear crystals held in the same optical cavity. The equations of motion are the same for both. The fundamental field at ω_1, which will be externally pumped, is represented by â_1. The second harmonic, at ω_2=2ω_1, is represented by â_2, and the fourth harmonic, at ω_3=4ω_1, is represented by â_3. The nonlinearity κ_1 couples the fields at ω_1 and ω_2, while κ_2 couples those at ω_2 and ω_3. The unitary interaction Hamiltonian in a rotating frame is then written as H_int = iħ/2[ κ_1(â_1^2â_2^†-â_1^† 2â_2)+κ_2(â_2^2â_3^†-â_2^† 2â_3) ].The cavity pumping Hamiltonian isH_pump = iħ(ϵâ_1^†-ϵ^∗â_1),where ϵ represents an external pumping field which is usually taken as coherent, although this is not necessary <cit.>. The damping of the cavity into a zero temperature Markovian reservoir is described by the Lindblad superoperator Lρ = ∑_i=1^3γ_i(2â_iρâ_i^†-â_i^†â_iρ-ρâ_i^†â_i),where ρ is the system density matrix and γ_i is the cavity loss rate at ω_i. We will treat all three fields as being at resonance with the optical cavity, which means that we do not need to examine quadrature correlations at all angles, but that the canonical X̂ and Ŷ quadratures are sufficient.Following the usual procedures <cit.>, we proceed via the von Neumann and Fokker-Planck equations to derive equations of motion in the positive-P representation <cit.>,dα_1/dt = ϵ-γ_1α_1+κ_1α_1^+α_2+√(κ_1α_2) η_1, dα_1^+/dt = ϵ^∗-γ_1α_1^++κ_1α_1α_2^++√(κ_1α_2^+) η_2, dα_2/dt =-γ_2α_2+κ_2α_2^+α_3-κ_1/2α_1^2+√(κ_2α_3) η_3, dα_2^+/dt =-γ_2α_2^++κ_2α_2α_3^+-κ_1/2α_1^+ 2+√(κ_2α_3^+) η_4, dα_3/dt =-γ_3α_3-κ_2/2α_2^2, dα_3^+/dt =-γ_3α_3^+-κ_2/2α_2^+ 2,noting that these have the same form in either Itô or Stratonovich calculus <cit.>. The complex variable pairs (α_i,α_j^+) correspond to the operator pairs (â_i,â_j^†) in the sense that stochastic averages of products converge to normally-ordered operator expectation values, e.g. α_i^+ mα_j^n→⟨â_i^† mâ_j^n⟩. The η_j are Gaussian noise terms with the properties η_i=0 and η_j(t)η_k(t')=δ_jkδ(t-t'). § QUANTUM CORRELATIONSBefore defining the inequalities we will use, we define the amplitude quadratures of the three interacting fields asX̂_i = â_i+â_i^†, Ŷ_i =-i(â_i-â_i^†),with the Heisenberg uncertainty principal demanding that the product of the variances, V(X̂_i)V(Ŷ_i)≥ 1.For three mode inseparability and entanglement, we use the van-Loock Furusawa inequalities <cit.>, which have proven useful for other cascaded optical systems <cit.>. The first of these is V_ij = V(X̂_i-X̂_j)+V(Ŷ_i+Ŷ_j+g_kŶ_k) ≥ 4,for which the violation of any two demonstrates tripartite inseparability. The g_j, which are arbitrary and real, can be optimised <cit.>, using the variances and covariances, asg_i = -V(Ŷ_i,Ŷ_j)+V(Ŷ_i,Ŷ_k)/V(Ŷ_i).Teh and Reid <cit.> have shown that, for mixed states, tripartite entanglement is demonstrated if the sum of the three correlations is less than 8, with genuine tripartite EPR (Einstein-Podolsky-Rosen)-steering <cit.> requiring a sum of less than 4. The second set set of vLF inequalities,V_ijk = V(X̂_i-X̂_j+X̂_k/√(2))+V(Ŷ_i+Ŷ_j+Ŷ_k/√(2)) ≥ 4,requires the violation of only one to prove tripartite inseparability. Teh and Reid <cit.> also showed that for mixed states any one of these less than 2 demonstrates genuine tripartite entanglement, while one of them less than 1 demonstrates genuine tripartite EPR steering. Because our nonlinear system is held in a cavity which is open to the environment, we are working with mixed states here. For multipartite EPR-steering, Wang showed that the steering of a given quantum mode is allowed when not less than half of the total number of modes take part in the steering group <cit.>. In a tripartite system, this means that measurements on two of the modes are needed to steer the third. In order to quantify this, we will use the correlation functions developed by Olsen, Bradley, and Reid <cit.>. With tripartite inferred variances asV_inf^(t)(X̂_i)=V(X̂_i)-[V(X̂_i,X̂_j±X̂_k)]^2/V(X̂_j±X̂_k), V_inf^(t)(Ŷ_i)=V(Ŷ_i)-[V(Ŷ_i,Ŷ_j±Ŷ_k)]^2/V(Ŷ_j±Ŷ_k),we defineOBR_ijk = V_inf^(t)(X̂_i)V_inf^(t)(Ŷ_i),so that a value of less than one means that mode i can be steered by the combined forces of modes j and k. According to the work of He and Reid <cit.>, genuine tripartite steering is demonstrated wheneverOBR_ijk+OBR_jki+OBR_kij < 1.In this work we will use only the plus signs in Eq. <ref>, which will be denoted on the figure axes as OBR_ijk^+. We found that this gave greater violations of the inequalities in some cases, although the results were not qualitatively different. § STEADY-STATE SPECTRAL CORRELATIONSWe find that the semi-classical and quantum solutions for the intensities are identical until a certain pump power, after which the system enters a self-pulsing regime <cit.>. Below this pump power, the steady-state solutions for the field amplitudes found from the integration of the full positive-P equations and their semiclassical equivalentsare identical. The semiclassical equations are found by removing the noise terms from Eq. <ref>, and have been solved numerically here. The measured observables of an intracavity process are usually the output spectral correlations, which are accessible using homodyne measurement techniques <cit.>. These are readily calculated in the steady-state by treating the system as an Ornstein-Uhlenbeck process <cit.>. In order to do this, we begin by expanding the positive-P variables into their steady-state expectation values plus delta-correlated Gaussian fluctuation terms, e.g.α_ss→⟨â⟩_ss+δα.Given that we can calculate the ⟨â_i⟩_ss, we may then write the equations of motion for the fluctuation terms. The resulting equations are written for the vector of fluctuation terms asd/dtδα⃗ = -Aδα⃗+BdW⃗,where A is the drift matrix containing the steady-state solutions, B is found from the factorisation of the diffusion matrix of the original Fokker-Planck equation, D=BB^T, with the steady-state values substituted in, and dW⃗ is a vector of Wiener increments. As long as the matrix A has no eigenvalues with negative real parts and the steady-state solutions are stationary, this method may be used to calculate the intracavity spectra viaS(ω) = (A+iω)^-1D(A^-iω)^-1,from which the output spectra are calculated using the standard input-output relations <cit.>.In this caseA = [ γ_1 -κ_1α_2 -κ_1α_1^∗ 0 0 0; -κ_1α_2^∗ γ_1 0 -κ_1α_1 0 0;κ_1α_1 0 γ_2 -κ_2α_3 -κ_2α_2^∗ 0; 0κ_1α_1^∗ -κ_2α_3^∗ γ_2 0 -κ_2α_2; 0 0κ_2α_2 0 γ_3 0; 0 0 0κ_2α_2^∗ 0 γ_3 ],and D is a 6× 6 matrix with [κ_1α_2,κ_1α_2^∗,κ_2α_3,κ_2α_3^∗,0,0] on the diagonal. In the above, the α_j should be read as their steady-state values. Because we have set γ_1=1, the frequency ω is in units of γ_1. S(ω) then gives us products such as δα_iδα_j andδα_i^∗δα_j^∗, from which we obtain the output variances and covariances for modes i and j asS^out(X_i,X_j) = δ_ij+√(γ_iγ_j)(S_ij+S_ji). § RESULTSThis system has a very rich parameter regime, with κ_1, κ_2, ϵ, γ_2 and γ_3 all capable of changing independently within any physical constraints. We have performed extensive numerical experiments and present the results for two representative regimes. The first was found to maximise violations of bipartite correlations and give asymmetric steering <cit.>, while the second is interesting because of the different predictions of the various correlations. The first parameter set we present is that used previously for bipartite correlations <cit.>, with κ_1=5× 10^-3, κ_2=4κ_1, ϵ=105, γ_1=1 and γ_2=γ_3=γ_1/2, which was shown to give bipartite steering, both symmetric and asymmetric, in all bipartitions. The intention of this parameter set, with lower loss rates at the higher frequencies, is to give the two higher frequency fields more time to interact within the cavity. We find, as shown in Fig. <ref>, that all three possible pairs can steer the remaining mode, but according to the criteria of He and Reid, since the minimum of the sum of the three is 1.44, genuine tripartite steering is not present. We note that the vLF inequalities were not violated and that since steering is a strict subclass of entanglement, these have missed tripartite entanglement that is present. The better sensitivity of EPR type measures for detecting entanglement has previously been found with bipartite systems <cit.>, where the Reid EPR correlations <cit.> have detected entanglement missed by the Duan-Simon positive partial transpose measure <cit.>.In the second parameter regime, κ_1=10^-2, κ_2=0.5κ_1, ϵ=105, γ_1=1 and γ_2=2γ_1 and γ_3=γ_1/4. We see that only one of the V_ij correlations shown in Fig. <ref>, V_12, drops below 4. This result on its own could be taken to indicate that the system is not tripartite inseparable, but this is not the case. Two of the V_ijk shown in Fig. <ref> drop below a value of 4. Since only one of the possible three violating the inequality is sufficient to prove inseparability, this result is more than sufficient. It is not, however, sufficient to demonstrate genuine tripartite entanglement, since neither of the two V_ijk which violate the vLF inequality exhibit values of less than 2. For this parameter set we find that the vLF correlations are more efficient at finding separability than the OBR_ijk, which are shown in Fig. <ref>. We see that only OBR_123 drops below one, and then by an insignificant amount. This means that, while the participants receiving modes 2 and 3 can combine to steer mode 1 in a marginal fashion which would quite possibly be destroyed by experimental noise, the other two pairings cannot perform steering in any fashion at all by way of Gaussian measurements. Whether steering via non-Gaussian measurements is possible is outside the scope of this article. § CONCLUSIONSIn conclusion, we have analysed a system of cascaded intracavity harmonic generation in terms of tripartite correlations for the detection of inseparability, entanglement, and EPR steering. We have examined two different parameter regimes and found non-classical quantum correlations across two octaves of frequency difference in both of these. In the first regime, EPR like correlations were found to be the best indicator of inseparability and entanglement. In the second, one set of the vLF correlations indicated tripartite inseparability which was missed by the other set, while the EPR correlations were inconclusive, finding only marginal steering in one of the three partitions. Our system, which could have possible applications in multiplexing, is a good physical example of the difficulty of finding versatile measures for tripartite entanglement in mixed systems, with different correlation measures being efficient in different regimes. § ACKNOWLEDGMENTS I would like to thank Margaret Reid for the invitation to contribute to this special issue. 00 KumatsuR. Komatsu, T. Sugarawa, K. Sassa, N. Sarukura, Z. Liu, S. Izumida, Y. Segawa, S. Uda, T. Fukuda, and K. Yamanouchi, 70, (1997) 3492.ZhuS. Zhu, Y. Zhu, and N. Ming, Science 278, (1997) 843.KojimaT. Kojima, S. Konno, S. Fujikawa, K. Kasui, K. Yoshizawa, Y. Mori, T. Sasaki, M. Tanaka, and Y. Okada, 25, (2000) 58.BroderickN.G. R. Broderick, R.T. Bratfalean, T.M. Mumro, and D.J. 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Milburn, Quantum Optics (Springer-Verlag, Berlin, 1995).QNoiseC.W. Gardiner and P. Zoller, Quantum Noise, (Springer-Verlag, Heidelberg, 2000).SMCrispinC.W. Gardiner, Handbook of Stochastic Methods, (Springer, Berlin, 1985).AxMuzzJPBM.K. Olsen and A.S. Bradley, 39, (2005) 127.AxMuzzM.K. Olsen and A.S. Bradley, 74, (2006) 063809.ErwinE. Schrödinger, Proc. Cam. Philos. Soc. 31, (1935) 555.WisesteerH.M. Wiseman, S.J. Jones and A.C. Doherty, 98, 140402 (2007).halfplusM. Wang, Y. Xiang, Q. He, and Q. Gong, 91, (2015) 012112.SHGpulseK.J. McNeil, P.D. Drummond, and D.F. Walls, 27, (1978) 292.THGcascadeM.K. Olsen, arXiv:1706.05174.mjcC.W. Gardiner and M.J. Collett, 31, (1985) 3761.NGBHcav2M.K. Olsen, 95, (2017), 023623.MDREPRM.D. Reid, 40, (1989) 413.DuanL.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, 84, (2000) 2722.SimonR. Simon, 84, (2000) 2726. | http://arxiv.org/abs/1707.08960v1 | {
"authors": [
"M. K. Olsen"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727081229",
"title": "Tripartite correlations over two octaves from cascaded harmonic generation"
} |
§ INTRODUCTION Before reaching the site of their detection at Earth, Cosmic Rays (CRs) propagate across the Solar System, where they are subject to the interaction with the turbulent solar wind and with the Heliospheric Magnetic Field (HMF). The ensemble of effects that arise as the result of these interactions can significantly alter CR intensity, in particular at low energies. This mechanism goes typically under the name of solar modulation (we address the reader to <cit.> for a review). An accurate description of solar modulation represents a necessary ingredient in the study of CR properties, also in connection with the search for exotic CR sources, such as Dark Matter (DM). This is the reason why we have developed HelioProp, a tool to model the transport of CRs in the heliosphere by means of the realistic three-dimensional model based on the Stochastic Differential Equation (SDE) technique described in <cit.>. Results based on an earlier version of the code have been reported in <cit.>. The analyses described here have been conducted with a novel version of HelioProp which is currently under development and which will be released soon as a completely open-source package. In addition, HelioProp will be part of the Dragon2 project <cit.>. Together, the two codes aim at describing CR transport from the source to the observer under conditions that are as general and realistic as possible. This proceeding is organised as follows: in Section <ref> we briefly describe the most important ingredients that characterise the physics of solar modulation, while in Section <ref> we illustrate the main features of the numerical method that is adopted in HelioProp to model CR transport in the heliosphere. In Section <ref> we present some results obtained with the code, in particular for what concerns the modelling of CR proton and electron fluxes and the impact of solar modulation on DM searches. Lastly, in Section <ref> we report our conclusions.§ THE PHYSICS OF SOLAR MODULATION Following the model proposed by Parker <cit.>, the HMF is typically assumed to possess a spiral structure. If we consider a spherical coordinate system {r,θ,ϕ} with origin at the Sun location, the HMF can be written as:B⃗(r,θ,ϕ) = A_c B_0 ( r_0/r)^2 [ r̂ - ( Ω rsinθ/V_sw) ϕ̂], where r_0 = 1 AU and Ω is the differential rotation rate of the Sun, while V_sw represents the velocity of the solar wind, which is a radial flow continuously directed outward from the Sun. All the results that are shown in this work have been derived by assuming for the radial and latitudinal dependence of V_sw the expression given in <cit.>.Concerning the parameters that define the normalization of the HMF, B_0 can be related to the intensity of the HMF at Earth B_e, while A_c is the HMF polarity, defined as A_c = ± H (θ - θ') where the sign is known to shift with a period of roughly 11 years, while H is the Heaviside step function and θ' is the angular extent of the wavy Heliospheric Current Sheet (HCS). This quantity is defined as follows:θ' = π/2 + sin^-1[ sinα sin( ϕ - ϕ_0 + Ω r/V_sw)] where ϕ_0 is an (arbitrary) constant azimuthal phase and α∈ [0,π/2] is the HCS tilt angle <cit.>, which is a quantity that varies according to solar activity and reaches its lowest values (α≈ 10^∘) in solar minima periods. The transport of Cosmic-Rays (CRs) across the heliosphere is described in terms of a transport equation <cit.>:∂ f/∂ t = - V⃗_sw·∇ f+ ⟨v⃗_d⟩·∇ f + ∇· (K⃗_⃗s⃗·∇ f) + 1/3 (∇·V⃗_sw ) ∂ f/∂ln Rwhere R=pc/eZ is the rigidity of the CR particle under consideration while f is its distribution function, related to the CR differential intensity j by f = R^2j. The four terms on the r.h.s. are used to model, respectively, convection under the influence of the solar wind, drifts, spatial diffusion and adiabatic energy changes. Concerning the process of CR drifts, it is modelled through the drift velocity (averaged with respect to the pitch angle) ⟨v⃗_d⟩ that appears in Eq. (<ref>). Since drifts are caused both by the interaction of CRs with the gradient and curvature of the HMF and by the change in the field polarity at the crossing of the HCS, the drift velocity can be written as the sum of two components: ⟨v⃗_d⟩ = ⟨v⃗_d⟩_GC + ⟨v⃗_d⟩_HCS. These two components are implemented in HelioProp according to the prescriptions given in <cit.>, to which we address the reader for additional details. It is important to remark that, following <cit.>, we include in our model a progressive weakening of drifts towards small rigidities. This is done by multiplying the drift velocity by the reduction factor: f(R)_drifts = (R/R_d)^2/1+(R/R_d)^2 Concerning the process of CR spatial diffusion, it is related to the tensor K⃗_⃗s⃗, which represents the symmetric part of the diffusion tensor. More precisely, one can work in the reference frame of the average HMF (frame that is denoted here with the superscript B) and decompose the total diffusion tensor into a symmetric and an antisymmetric component as follows: K^B_ij = K^B,S_ij + K^B,A_ij = [ k_∥ 0 0; 0 k_⊥,r 0; 0 0 k_⊥,θ; ] + [000;00k_A;0 -k_A0;],where k_∥ is the diffusion coefficient along a direction parallel to the one of the HMF, while k_⊥,r and k_⊥,θ are the diffusion coefficients along the radial and polar perpendicular directions. The antisymmetric part of the diffusion tensor K^B,A_ij accounts for drifts and is already included in Eq. (<ref>) in terms of the drift velocity v⃗_ d = ∇×(K_A B⃗/|B|). On the other hand, as already said, the symmetric tensor K^B,S_ij, once translated into its counterpart in spherical coordinates K^S_ij, enters the transport equation to model spatial diffusion. As one can from Eq. (<ref>), diffusion is totally described once that the coefficients k_∥, k_⊥,r and k_⊥,θ are determined. As typical for the most realistic descriptions of solar modulation, HelioProp features a fully anisotropic diffusion, which means that it works under the assumption that k_∥ k_⊥,r k_⊥,θ. We express the three diffusion coefficients in terms of the Mean Free Paths (MFPs) in the associated direction k_i = v/3 λ_i with v being the CR velocity. We follow a common approach (see <cit.> for example) and define the parallel mean-free-path as a broken power-law with break at rigidity R_k and indices a and b below and above the break: λ_∥ = λ_0 (1 nT/B) ( R/1 GV)^a(( R_k/1 GV)^c + ( R_k/1 GV)^c /1 + ( R_k/1 GV)^c)^b-a/c.As for the MFPs along the perpendicular directions, HelioProp allow the user to implement any generic function. In this work, we assume: λ_⊥,r =f_⊥λ_∥ , λ_⊥,θ =f_⊥λ_∥ H(θ) , with the function H(θ) being defined as in <cit.>.§ NUMERICAL SOLUTION OF THE TRANSPORT EQUATIONHelioProp models solar modulation by numerically solving Eq. (<ref>). This is done by following the SDE technique described in <cit.>, which was originally proposed in the context of CR solar modulation in <cit.>. In brief, this method is realised by writing the transport equation as a backward Kolmogorov equation: -∂ j/∂ s = ∑_i ( A_i∂ j/∂ x_i) + 1/2∑_i,k( C_ik∂^2 j/∂ x_i ∂ x_k ) where j is the CR differential intensity, while the parameter s is a backward time, related to the standard time t by t = t_fin - s, where t_fin is the final time. The Itō's lemma <cit.> states that the above equation is equivalent to a set of stochastic processes: dx_i = A_i(x_i)ds + ∑_j B_ij (x_i) dW_iwhere we have used C = B^T B, while x_i = {r,θ,ϕ,E} and W_i is a Wiener process, i.e. a stochastic variable for which the increments Δ W_i = w_i - w_i-1 are independent one from the other and each of them has a gaussian probability distribution with zero mean and standard deviation equal to Δ s_i = t_i - s_s-1. We address the reader to <cit.> for the explicit form of the stochastic processes described by Eq. (<ref>).Under a practical point of view, when following this approach one models solar modulation by combining the backward trajectories in the phase space of N pseudo-particles. More precisely, these pseudo-particles are injected at Earth at s=0 with phase space coordinates x_Earth and then they are back-propagated through the heliosphere by following the stochastic processes described by Eq. (<ref>) until they reach the boundary of the heliopause (HP). The CR flux at Earth is then obtained by averaging the Local Interstellar (LIS) spectra evaluated at the coordinates x_HP that the pseudo-particles have at the HP: j_Earth (x_Earth) = 1/N∑_k=1^N j_LIS(x_HP)§ RESULTS§.§ Proton and electron fluxesThe first test that we perform consists in comparing the results of HelioProp with the proton and electron fluxes measured by PAMELA over short time intervals (one month for the protons, six months in the case of electrons) along the period of minimal solar activity and negative polarity of the HMF that goes from 2006 to 2009 <cit.>[Because of space limitations, here we restrict ourselves to 4 electron datasets out of the 7 released by the PAMELA Collaboration. ]. For each data-taking period that we consider, we fix the tilt angle α and the HMF at Earth B_e to the values reported in <cit.>. We are thus left with a set of free parameters that include the quantities λ_0, a, b, c and R_k that appear in Eq. (<ref>), the factor f_⊥ of Eq. (<ref>) and the rigidity R_d of Eq. (<ref>). For simplicity, we impose that electrons and protons share the same values for both R_d and f_⊥. In particular, we find that good fits to data can be found by assuming R_d = 0.32 GV, which is the same value used in <cit.>, and f_⊥ = 0.021, that is very close to the commonly used value 0.02 (see <cit.>).We assume the parallel MFP λ_∥ of protons to have a linear dependence on the CR rigidity as in the model of <cit.>. This means that, for protons, a = b = 1 (and therefore c and R_k do not play any role, as clear from Eq. (<ref>)). In the case of electrons, we find that we are able to reproduce the data behaviour by assuming a = 0, b = 1.55, c=3.5. A rigidity-dependence of the electron MFP in terms of abroken power-law of this kind (i.e. with a flattening at small rigidities) appears to be in a qualitative agreement with previous analyses of electron data, as the one reported in <cit.>. With this considered, the only parameters that are left free to vary from one dataset to the other are λ_0 (for both protons and electrons) and R_k (only for electrons). Concerning the proton and electron LIS fluxes, we adopt the parameterizations based on Voyager 1 and PAMELA data presented in <cit.>.A summary of the relevant parameters used in this analysis is reported in Table <ref>. Results are reported in Fig. <ref>, In the top row we show PAMELA data compared with the best-fit configurations obtained within the framework of our model, while in the bottom row the corresponding parallel MFPs at Earth are shown as a function of the CR rigidity. As it can be seen, the model that we are using can reproduce the observed behaviour of both protons and electrons. It is important to remark that the approach followed here is data-driven, since we fix the values of the the parameters characterising CR diffusion and drifts by fitting PAMELA data; as discussed in Section <ref>, HelioProp allows for a large freedom in the definition of the solar modulation setup and therefore one could instead follow a more theoretically motivated strategy and adopt for these parameters the results obtained in the context of turbulence transport models (see, e.g.,<cit.>).§.§ Antiprotons and consequences for DM searchesIn the context of indirect DM searches in charged CRs, it is customary to treat solar modulation in terms of the force field approximation <cit.>. Here we provide a simple example of what can be the impact of adopting a more realistic solar modulation setup, as the one implemented in HelioProp. We do this by comparing predictions of DM-generated antiproton fluxes obtained with HelioProp and with the force-field approach. Analyses along these lines have been already performed with the previous version of HelioProp and can be found in <cit.> (for the case of antiprotons) and in <cit.> (for the case of antideuterons). Since we are dealing with an exotic source of antiprotons contributing mostly at low energies, we cannot tune the solar modulation parameters by directly using antiproton observations and we must rely on proton data alone. Obviously, we have to consider a proton and an antiproton datasets derived from observations performed over the same time interval. This is the case of the PAMELA datasets presented in<cit.>. We find that we can fit the PAMELA proton dataset by using the same setup described in Section <ref> (with λ_∥ = 0.016 AU at 100 MV at Earth). In the framework of the force field approximation, a comparably good fit is provided by assuming a force-field potential of 450 MV. As for antiprotons, we adopt a LIS derived from a DRAGON run performed within the KRA propagation model (see <cit.> for details about this propagation setup)[It is important to remark that the DRAGON run has been performed by tuning the proton injection spectrum in order to match the LIS used for protons in this work.]. The antiproton fluxes at Earth obtained by modulating this LIS with HelioProp and with the FF approach are shown in Fig. <ref>: as one can see, the difference between the two modulation setups is minimal (below 10%) in the case of the antiproton flux produced by spallation processes. On the contrary, if one considers the antiproton flux produced by the annihilation of light DM, as we do here by assuming a DM particle with mass 10 or 20 GeV that annihilates into bb̅, the difference is larger and can reach the 35%. This is shown in the right panel of Fig. <ref>, where it is also shown that such a distance can be larger than the experimental uncertainty. This proves that when deriving constraints to DM properties from low-energy CR data, a realistic treatment of solar modulation is necessary.§ CONCLUSIONS We have presented here the new version of HelioProp the numerical code designed to model solar modulation in a three-dimensional and charge-dependent way by using the SDE approach described in <cit.>.HelioProp is designed to be part of the Dragon2 project <cit.>, but will also be released soon as an independent and fully open source package. We have briefly illustrated the main features of the code, under a physical and numerical point of view. In addition, we have presented a couple of basic tests aimed at investigating its performances and potentialities. In particular, we have shown that HelioProp is able to reproduce the observed behaviour of low-energy CR electrons and protons and we have illustrated how it can be an important tool for a more accurate treatment of DM indirect detection. | http://arxiv.org/abs/1707.09003v1 | {
"authors": [
"Andrea Vittino",
"Carmelo Evoli",
"Daniele Gaggero"
],
"categories": [
"astro-ph.HE",
"hep-ph",
"physics.space-ph"
],
"primary_category": "astro-ph.HE",
"published": "20170727190806",
"title": "Cosmic-ray transport in the heliosphere with HelioProp"
} |
=5mm∑ ∫ ⟨ ⟩ | http://arxiv.org/abs/1707.08602v1 | {
"authors": [
"B. G. Zakharov"
],
"categories": [
"nucl-th",
"hep-ph"
],
"primary_category": "nucl-th",
"published": "20170726183458",
"title": "Phenomenology of collinear photon emission from quark-gluon plasma in $AA$ collisions"
} |
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